The Dynamics of Natural Satellites of the Planets 0128227044, 9780128227046

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The Dynamics of Natural Satellites of the Planets
 0128227044, 9780128227046

Table of contents :
Contents
Author's preface
1 Objectives, current problems and general approach to the study of the dynamics of satellites
1.1 Introduction
1.2 Celestial mechanics—the basis for studying the dynamics of planetary satellites
1.3 Objectives of studying the dynamics of planetary satellites
1.4 Basic concepts of celestial mechanics and astrometry
1.5 General approach to studying the dynamics of planets and satellites based on observations
1.6 Special properties of necessary observations
2 Satellites of planets
2.1 Satellites of the planets—objects of the solar system
2.2 Classification and nomenclature of planetary satellites
2.3 Discovery of Neptune and its satellite, Triton
2.4 The history of the discovery of Pluto's satellite, Charon
2.5 Orbital and physical parameters of planets with satellites
2.6 Orbital parameters of planetary satellites
References
3 Equations of motion and analytic theories
3.1 Equations of motion and coordinate systems
3.2 Keplerian motion model
3.2.1 The basic formulas of the Keplerian motion
3.2.2 Calculation of coordinates in elliptical Keplerian motion
3.2.3 Calculation of velocity in elliptical Keplerian motion
3.2.4 Calculation of the partial derivatives of coordinates and velocity components with respect to elements of the Keplerian orbit in elliptical motion
3.2.5 Keplerian motion formulas with respect to nonsingular elements (Lagrange elements)
3.2.6 Examples of using Lagrange elements
3.3 Force function of attraction of a non-spherical planet
3.3.1 Force function expansion
3.3.2 Attraction in models and for real bodies
3.4 An approximate account of the influence of the main satellites on the motion of distant satellites of the planet
3.5 Various approaches and methods for constructing motion models of planetary satellites
3.6 A model of motion of a satellite of an oblate planet based on the solution of the generalized problem of two fixed centers
3.7 Constructing analytical theories of planetary satellite motion using perturbation theory methods
3.7.1 General scheme of perturbation theory
3.7.2 Circumstances in the motion of real celestial bodies, allowing the use of perturbation theory methods
Planetary problem
Satellite problem
The problem of satellite motion of a non-spherical planet
Other applications of perturbation theory
3.7.3 Equations for elements of the intermediate orbit
3.7.4 Solving equations for intermediate orbit elements. Small parameter method
3.7.5 Solving equations for intermediate orbit elements. Poisson Method
3.8 Expansion of the perturbing function with respect to the elements of the intermediate orbit of a planetary satellite
3.9 Determination of perturbations of elements of the planetary satellite intermediate orbit
Secular perturbation
Periodical perturbations
3.10 Constant perturbation of the semi-major axis of the satellite's orbit
3.11 Precessing ellipse model
3.12 Perturbed motion at small eccentricities of the orbits
3.12.1 Problem formulation
3.12.2 Constructing a model of circular perturbed motion
3.12.3 Transition to the elements of the Keplerian orbit
3.12.4 Osculating Keplerian elements of the satellite's orbit in perturbed motion with small eccentricities
3.13 Constructed analytical theories of planetary satellite motion
3.13.1 Analytical theory of the motion of Neptune's satellite Triton
3.13.2 Precessing Ellipse models for close Jupiter satellites
3.13.3 Special analytical theories of the main satellites of major planets, taking into account the mutual attraction of satellites
3.14 Influence of tides in viscoelastic bodies of planet and satellite on the satellite's orbital motion
3.14.1 Statement of the problem of influence of tides
3.14.2 Equations in rectangular coordinates
3.14.3 Solving the equations for rectangular coordinates
3.14.4 Transition to the differential equations in Keplerian elements
3.14.5 Some important conclusions about the influence of tidal deformations on satellite dynamics
References
4 Modeling the satellite motion. Numerical integration methods
4.1 The objective of solving the equations of motion of celestial bodies
4.2 General properties of methods for the numerical integration of equations of motion
4.3 Runge–Kutta integration method for ordinary differential equations
4.4 Algorithm for solving problems of motion of a celestial body by numerical integration methods
4.5 Instructions for the computational program for the numerical integration of ordinary differential equations by the Everhart method
4.6 Belikov program for numerical integration of ordinary differential equations
4.7 Testing and comparing some numerical integration procedures
4.8 Approximation of the rectangular coordinates of planets and satellites by truncated Chebyshev series
4.9 Overview of problems and methods of numerical integration. Book by Avdyushev
References
5 Observations of planetary satellites
5.1 General principles of observations
Important note
5.2 Determination of topocentric positions of planets and satellites
5.3 Planet observations
Comment
5.4 Observations of a planetary satellite
5.5 Observations of two satellites of the planet
5.6 Determination of angular measured values during observations of planetary satellites
5.7 Calculation of the angular distance between satellites and position angle
5.8 Determination of tangential coordinates of satellites
5.9 Determination of the coordinate difference between two satellites of the planet in the case of photometric observations of mutual eclipses of satellites
5.10 Conclusion regarding measured values during observations of planetary satellites
5.11 The moment of apparent approximation of planetary satellites as a measurable quantity during observations
5.12 Means and techniques of ground-based observations of planetary satellites
Speckle interferometric observations
5.13 Sources of observations from planetary satellites
5.14 Time scales and coordinate systems for observations of planetary satellites
Time scales
Coordinate systems
References
6 Construction of models for the motions of celestial bodies based on observations
6.1 Method of differential refinement of the motion parameters of celestial bodies based on observations. Application of least-squares method
6.2 Weak conditionality and ambiguity of solution
6.3 Overview of filtering algorithms
6.4 Calculation of measured values and partial derivatives of the measured values by refined parameters
6.4.1 General order of calculations
6.4.2 Differential equations for isochronous derivatives in the three-body problem. Refinement of the initial conditions of the equations of motion
6.4.3 Differential equations for isochronous derivatives in the three-body problem. Refinement of the mass of the perturbing body
6.4.4 Differential equations for isochronous derivatives in the satellite motion problem for a oblate planet
6.4.5 Construction of conditional equations for angular measurements of topocentric coordinates
6.5 Assigning weights to observations and conditional equations
6.6 Calculation of statistical characteristics of residuals
6.7 The problem of rejecting rough observations
References
7 Obtaining astrometric data from observations of mutual occultations and eclipses of planetary satellites
7.1 Description of phenomena
7.2 Method for obtaining astrometric data
7.3 A simplified model of mutual occultations and eclipses of planetary satellites
7.4 Photometric models of mutual occultations and eclipses of planetary satellites
7.4.1 General photometric characteristics
7.4.2 Photometric model of the mutual occultation of satellites
7.4.3 Photometric model of satellite eclipse
7.5 The laws of light scattering for planetary satellites
7.5.1 Lommel–Seeliger light scattering law
7.5.2 Hapke's light scattering law for a smooth surface
7.5.3 Hapke's light scattering law for a rough surface
7.5.4 Hapke's law parameters for the Galilean satellites of Jupiter
7.6 Disk-integrated photometric characteristics of the satellite
7.7 Photometric models of mutual occultations and eclipses of the main satellites of Saturn and Uranus
7.7.1 Photometric model of mutual occultations and eclipses of the main satellites of Saturn
7.7.2 Photometric model of mutual occultations and eclipses of the main satellites of Uranus
7.8 Relation for the accuracy of astrometric results of observations of various types
7.9 Worldwide campaigns on observations of satellites during their mutual occultations and eclipses
7.10 Obstacles to improving the accuracy of astrometric results
7.11 Periods of the phenomena in the future
References
8 Estimation of the accuracy of planetary satellite ephemeris
8.1 Factors determining ephemeris accuracy
8.2 Estimation of the ephemeris accuracy using observation-errors variance by the Monte Carlo techniques
8.3 Estimation of ephemeris accuracy by varying the composition of observations using ``bootstrap''-samples
8.4 Estimation of the accuracy of ephemeris by the method of motion parameter variation
8.5 The accuracy of the ephemeris of the distant satellites of major planets
References
9 The rotation of planetary satellites
9.1 General properties of the rotation of planetary satellites
9.2 Basic concepts of the rotation of planets and satellites
9.3 The rotation of Neptune and the orbit of Triton
9.4 Theory of rotation for Phobos
9.5 Rotation of the Galilean satellites of Jupiter, satellites of Saturn and Pluto
9.6 Chaotic rotation of planetary satellites. Rotation of Hyperion
References
10 The evolution of the orbits of the planetary satellites
10.1 The impact of various factors on the evolution of the orbits of planetary satellites
10.2 The evolution of the orbits of satellites subject to the predominant influence of planet oblateness
The case of small eccentricities
10.3 Evolution of the orbits of the planetary satellites under the action of the solar attraction
10.3.1 Averaging of the perturbing function
10.3.2 A special case—Hill problem
10.3.3 Analysis of families of possible changes in the eccentricity of e and the argument of the pericenter for a twice-averaged perturbing function in the Hill case
10.3.4 Orbit evolution in time for a double-averaged perturbing function in the Hill case
10.3.5 Applications of the constructed theory of orbit evolution in studying the dynamics of real planetary satellites
10.4 Refined models of the evolution of the orbits of planetary satellites. Numerical analytical method
10.5 The evolution of the orbits of planetary satellites under the combined influence of various factors
10.6 Classification of the orbits of the distant satellites of Jupiter, Saturn, Uranus, and Neptune according to the types and properties of the orbit evolution
10.7 The evolution of the orbits and rendez-vous occurrences of distant satellites of the planets
10.7.1 Modern knowledge about the evolution of the orbits of distant planetary moons
10.7.2 The problem of calculating and detecting the rendez-vous occurrences of distant planetary satellites
10.7.3 An analytical description of the evolution of satellite orbits
10.7.4 Determination of minimal distances between the orbits of distant planetary satellites
10.7.5 Proposed Internet source for the study of the evolution of the orbits and rendez-vous of distant planetary satellites
10.7.6 Examples of calculating the minimum distances between the orbits of satellites
10.7.7 Conclusion
10.8 Refinement of the Laplace–Lagrange secular perturbation theory
References
11 Physical parameters of planetary satellites
11.1 Introduction
11.2 Handbook of the physical parameters of planetary satellites
11.3 Detection of volcanoes on the satellite of Jupiter Io using ground photometry
11.4 Estimates of the physical parameters of distant planetary satellites
11.4.1 Features of distant planetary satellites
11.4.2 Overview of available photometric data for distant planetary satellites
11.4.3 Photometric model for distant planetary satellites
11.4.4 Determination of photometric parameters of satellites by photometry
11.4.5 Initial data and results of determining the photometric parameters of satellites
11.4.6 Comparison of results obtained by different authors
11.4.7 Conclusions on the estimates of the photometric parameters of distant planetary satellites
11.5 Determination of Himalia's mass from astrometric observations of other satellites
References
12 Recent models of planetary satellite motion. Information resources
12.1 Variants and change of version of motion theories and ephemeris of planetary satellites
12.2 Means of providing access to databases, motion models and ephemeris of planetary satellites
12.3 MULTI-SAT ephemeris server features
12.4 Theories and models in the MULTI-SAT ephemeris server
Satellites of Mars Phobos and Deimos
Galilean satellites of Jupiter
Close satellites of Jupiter
Distant satellites of Jupiter
New distant satellites of Jupiter
Main satellites of Saturn
Saturn's satellites, co-orbiting the main ones
Saturn's close co-orbiting satellites
Saturn's close satellites
The distant satellite of Saturn Phoebe
New distant satellites of Saturn
Main satellites of Uranus
Main satellites of Uranus
Close satellites of Uranus
New distant satellites of Uranus
Neptune's satellite Triton
Distant satellite of Neptune Nereid
Neptune's close satellites
Satellites of Pluto
12.5 Theories and models in the JPL ephemeris server
12.6 Planet satellites in virtual observatories
12.7 Standards of fundamental astronomy
References
A Nomenclature of planetary satellites
Nomenclature of the satellites of Mars
Announcement on the discovery of the satellites of Mars
Publication of the first observations of the satellites of Mars by Azaf Hall
Nomenclature of the satellites of Jupiter
References to the messages on discoveries of the satellites of Jupiter
Nomenclature of the satellites of Saturn
References to the messages on discoveries of the satellites of Saturn
The nomenclature of the satellites of Uranus
References to the messages on discoveries of the satellites of Uranus
The nomenclature of the satellites of Neptune
References to the messages on discoveries of the satellites of Neptune
The nomenclature of the satellites of Pluto
References to the messages on discoveries of the satellites of Pluto
B Orbital parameters of the natural planetary satellites
Orbital parameters of Mars's satellites
Orbital parameters of the satellites of Jupiter
Orbital parameters of Saturn's satellites
Orbital parameters of Uranus's satellites
Orbital parameters of Neptune's satellites
Orbital parameters of the satellites of Pluto
References
C Special functions in celestial mechanics
Inclination functions
Eccentricity functions
References
D Time scales
Creating time scales
Relationship UTC with TT scale
Time scales in publications of observations in past centuries
References
E Cholesky decomposition. Program in C-code
F Rotation parameters of planets and natural satellites
Definitions
Rotation parameters of planets having natural satellites
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Rotation parameters of natural satellites
Satellites of Mars
M1 Phobos
M2 Deimos
M1 Phobos
M2 Deimos
Satellites of Jupiter
J1 Io
J2 Europa
J3 Ganymede
J4 Callisto
J16 Metis
J15 Adrastea
J5 Amalthea
J14 Thebe
Satellites of Saturn
Satellites of Uranus
Satellites of Neptune
Pluto's satellite Charon
Satellites of Pluto P2 Nix, P3 Hydra, P4 Kerberos, and P5 Styx
References
G Physical parameters of the natural satellites of the planets
Satellites of Mars
Satellites of Jupiter
Satellites of Saturn
Satellites of Uranus
Satellites of Neptune
Satellites of Pluto
Bibliography of data sources
Index

Citation preview

THE DYNAMICS OF N AT U R A L S AT E L L I T E S OF THE PLANETS

THE DYNAMICS OF N AT U R A L S AT E L L I T E S OF THE PLANETS NIKOLAY EMELYANOV Sternberg State Astronomical Institute Lomonosov Moscow State University Moscow, Russia

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Inc. All rights reserved. The Russian version of this book was published in 2019: Emelyanov N.V., The Dynamics of Natural Satellites of the Planets Based on Observations. Vek-2, Fryazino. 576 pp. ISBN 978-5-85099-199-9. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-822704-6 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Candice Janco Acquisitions Editor: Peter Llewellyn Editorial Project Manager: Alice Grant Production Project Manager: Sruthi Satheesh Designer: Victoria Pearson Typeset by VTeX

Contents

v

Contents Author’s preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Celestial mechanics—the basis for studying the dynamics of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Objectives of studying the dynamics of planetary satellites . . . . . . . 3 1.4 Basic concepts of celestial mechanics and astrometry . . . . . . . . . . . . 4 1.5 General approach to studying the dynamics of planets and satellites based on observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Special properties of necessary observations . . . . . . . . . . . . . . . . . . . 12

Chapter 2 Satellites of planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Satellites of the planets—objects of the solar system . . . . . . . . . . . . 17 2.2 Classification and nomenclature of planetary satellites . . . . . . . . . . 18 2.3 Discovery of Neptune and its satellite, Triton . . . . . . . . . . . . . . . . . . . 22 2.4 The history of the discovery of Pluto’s satellite, Charon . . . . . . . . . . 23 2.5 Orbital and physical parameters of planets with satellites. . . . . . . . 31 2.6 Orbital parameters of planetary satellites. . . . . . . . . . . . . . . . . . . . . . . 34 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Chapter 3 Equations of motion and analytic theories . . . . . . . . . . . . . . . . 39 3.1 Equations of motion and coordinate systems . . . . . . . . . . . . . . . . . . . 39 3.2 Keplerian motion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.1 3.2.2

The basic formulas of the Keplerian motion. . . . . . . . . . . . . . 42 Calculation of coordinates in elliptical Keplerian motion . . 48

3.2.3 3.2.4

Calculation of velocity in elliptical Keplerian motion . . . . . . 48 Calculation of the partial derivatives of coordinates and velocity components with respect to elements of the Keplerian orbit in elliptical motion . . . . . . . . . . . . . . . . . . . . . . 49

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Contents

3.2.5

3.3

3.4 3.5 3.6 3.7

3.8 3.9 3.10 3.11 3.12

Keplerian motion formulas with respect to nonsingular elements (Lagrange elements) . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.6 Examples of using Lagrange elements . . . . . . . . . . . . . . . . . . 54 Force function of attraction of a non-spherical planet . . . . . . . . . . . . 55 3.3.1 Force function expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.2 Attraction in models and for real bodies . . . . . . . . . . . . . . . . . 56 An approximate account of the influence of the main satellites on the motion of distant satellites of the planet . . . . . . . . . . . . . . . . . 60 Various approaches and methods for constructing motion models of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A model of motion of a satellite of an oblate planet based on the solution of the generalized problem of two fixed centers . . . . . . . . 63 Constructing analytical theories of planetary satellite motion using perturbation theory methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.7.1 General scheme of perturbation theory . . . . . . . . . . . . . . . . . . 66 3.7.2 Circumstances in the motion of real celestial bodies, allowing the use of perturbation theory methods. . . . . . . . . 69 3.7.3 Equations for elements of the intermediate orbit . . . . . . . . . 74 3.7.4 Solving equations for intermediate orbit elements. Small parameter method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.7.5 Solving equations for intermediate orbit elements. Poisson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Expansion of the perturbing function with respect to the elements of the intermediate orbit of a planetary satellite. . . . . . . . 86 Determination of perturbations of elements of the planetary satellite intermediate orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Constant perturbation of the semi-major axis of the satellite’s orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Precessing ellipse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Perturbed motion at small eccentricities of the orbits . . . . . . . . . . . 103 3.12.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.12.2 Constructing a model of circular perturbed motion . . . . . . 104 3.12.3 Transition to the elements of the Keplerian orbit . . . . . . . . 106

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3.12.4 Osculating Keplerian elements of the satellite’s orbit in perturbed motion with small eccentricities. . . . . . . . . . . . . . 108 3.13 Constructed analytical theories of planetary satellite motion . . . . 114 3.13.1 Analytical theory of the motion of Neptune’s satellite Triton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.13.2 Precessing Ellipse models for close Jupiter satellites . . . . 120 3.13.3 Special analytical theories of the main satellites of major planets, taking into account the mutual attraction of satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.14 Influence of tides in viscoelastic bodies of planet and satellite on the satellite’s orbital motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.14.1 Statement of the problem of influence of tides . . . . . . . . . . 126 3.14.2 Equations in rectangular coordinates . . . . . . . . . . . . . . . . . . . 128 3.14.3 Solving the equations for rectangular coordinates. . . . . . . 131 3.14.4 Transition to the differential equations in Keplerian elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.14.5 Some important conclusions about the influence of tidal deformations on satellite dynamics . . . . . . . . . . . . . . . . . . . . 144 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Chapter 4 Modeling the satellite motion. Numerical integration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.1 The objective of solving the equations of motion of celestial bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 General properties of methods for the numerical integration of equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Runge–Kutta integration method for ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Algorithm for solving problems of motion of a celestial body by numerical integration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Instructions for the computational program for the numerical integration of ordinary differential equations by the Everhart method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Belikov program for numerical integration of ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 152 157 158

160 165

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4.7 Testing and comparing some numerical integration procedures. 167 4.8 Approximation of the rectangular coordinates of planets and satellites by truncated Chebyshev series . . . . . . . . . . . . . . . . . . . . . . 167 4.9 Overview of problems and methods of numerical integration. Book by Avdyushev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Chapter 5 Observations of planetary satellites . . . . . . . . . . . . . . . . . . . . . 173 5.1 General principles of observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Determination of topocentric positions of planets and satellites . 174 5.3 Planet observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.4 Observations of a planetary satellite . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.5 Observations of two satellites of the planet . . . . . . . . . . . . . . . . . . . . 178 5.6 Determination of angular measured values during observations of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.7 Calculation of the angular distance between satellites and position angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.8 Determination of tangential coordinates of satellites . . . . . . . . . . . 184 5.9 Determination of the coordinate difference between two satellites of the planet in the case of photometric observations of mutual eclipses of satellites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.10

Conclusion regarding measured values during observations of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.11

The moment of apparent approximation of planetary satellites as a measurable quantity during observations . . . . . . . . . . . . . . . . . 188

5.12

Means and techniques of ground-based observations of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.13

Sources of observations from planetary satellites . . . . . . . . . . . . . . 192

5.14

Time scales and coordinate systems for observations of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Chapter 6 Construction of models for the motions of celestial bodies based on observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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6.1 Method of differential refinement of the motion parameters of celestial bodies based on observations. Application of least-squares method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.2 Weak conditionality and ambiguity of solution . . . . . . . . . . . . . . . . . 212 6.3 Overview of filtering algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.4 Calculation of measured values and partial derivatives of the measured values by refined parameters. . . . . . . . . . . . . . . . . . . . . . . 217 6.4.1 General order of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.4.2 Differential equations for isochronous derivatives in the three-body problem. Refinement of the initial conditions of the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.4.3 Differential equations for isochronous derivatives in the three-body problem. Refinement of the mass of the perturbing body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.4.4 Differential equations for isochronous derivatives in the satellite motion problem for a oblate planet. . . . . . . . . . . . . 224 6.4.5 Construction of conditional equations for angular measurements of topocentric coordinates . . . . . . . . . . . . . . 227 6.5 Assigning weights to observations and conditional equations . . 231 6.6 Calculation of statistical characteristics of residuals . . . . . . . . . . . . 234 6.7 The problem of rejecting rough observations . . . . . . . . . . . . . . . . . . 236 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Chapter 7 Obtaining astrometric data from observations of mutual occultations and eclipses of planetary satellites . . . . . . . . . 239 7.1 Description of phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.2 Method for obtaining astrometric data . . . . . . . . . . . . . . . . . . . . . . . . 242 7.3 A simplified model of mutual occultations and eclipses of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.4 Photometric models of mutual occultations and eclipses of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.4.1 General photometric characteristics . . . . . . . . . . . . . . . . . . . . 248 7.4.2 Photometric model of the mutual occultation of satellites 249 7.4.3 Photometric model of satellite eclipse . . . . . . . . . . . . . . . . . . 251

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7.5 The laws of light scattering for planetary satellites . . . . . . . . . . . . . 253 7.5.1

Lommel–Seeliger light scattering law. . . . . . . . . . . . . . . . . . . 253

7.5.2

Hapke’s light scattering law for a smooth surface. . . . . . . . 254

7.5.3

Hapke’s light scattering law for a rough surface . . . . . . . . . 255

7.5.4

Hapke’s law parameters for the Galilean satellites of Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.6 Disk-integrated photometric characteristics of the satellite . . . . . . 258 7.7 Photometric models of mutual occultations and eclipses of the main satellites of Saturn and Uranus. . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.7.1

Photometric model of mutual occultations and eclipses of the main satellites of Saturn. . . . . . . . . . . . . . . . . . . . . . . . . 262

7.7.2

Photometric model of mutual occultations and eclipses of the main satellites of Uranus . . . . . . . . . . . . . . . . . . . . . . . . 264

7.8 Relation for the accuracy of astrometric results of observations of various types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.9 Worldwide campaigns on observations of satellites during their mutual occultations and eclipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.10

Obstacles to improving the accuracy of astrometric results . . . . . 267

7.11

Periods of the phenomena in the future . . . . . . . . . . . . . . . . . . . . . . . 273 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Chapter 8 Estimation of the accuracy of planetary satellite ephemeris. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.1 Factors determining ephemeris accuracy . . . . . . . . . . . . . . . . . . . . . . 277 8.2 Estimation of the ephemeris accuracy using observation-errors variance by the Monte Carlo techniques . . . . . . . . . . . . . . . . . . . . . . . 279 8.3 Estimation of ephemeris accuracy by varying the composition of observations using “bootstrap”-samples . . . . . . . . . . . . . . . . . . . . . . 281 8.4 Estimation of the accuracy of ephemeris by the method of motion parameter variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 8.5 The accuracy of the ephemeris of the distant satellites of major planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

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Chapter 9 The rotation of planetary satellites . . . . . . . . . . . . . . . . . . . . . . 293 9.1 9.2 9.3 9.4 9.5

General properties of the rotation of planetary satellites . . . . . . . . 293 Basic concepts of the rotation of planets and satellites . . . . . . . . . 295 The rotation of Neptune and the orbit of Triton . . . . . . . . . . . . . . . . 297 Theory of rotation for Phobos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Rotation of the Galilean satellites of Jupiter, satellites of Saturn and Pluto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.6 Chaotic rotation of planetary satellites. Rotation of Hyperion. . . . 306 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Chapter 10 The evolution of the orbits of the planetary satellites. . . . . . 315 10.1 10.2 10.3

10.4 10.5 10.6

10.7

The impact of various factors on the evolution of the orbits of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 The evolution of the orbits of satellites subject to the predominant influence of planet oblateness . . . . . . . . . . . . . . . . . . . 316 Evolution of the orbits of the planetary satellites under the action of the solar attraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 10.3.1 Averaging of the perturbing function . . . . . . . . . . . . . . . . . . . 319 10.3.2 A special case—Hill problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 324 10.3.3 Analysis of families of possible changes in the eccentricity of e and the argument of the pericenter ω for a twice-averaged perturbing function in the Hill case . . . . 325 10.3.4 Orbit evolution in time for a double-averaged perturbing function in the Hill case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 10.3.5 Applications of the constructed theory of orbit evolution in studying the dynamics of real planetary satellites . . . . . 341 Refined models of the evolution of the orbits of planetary satellites. Numerical analytical method. . . . . . . . . . . . . . . . . . . . . . . . 342 The evolution of the orbits of planetary satellites under the combined influence of various factors . . . . . . . . . . . . . . . . . . . . . . . . 346 Classification of the orbits of the distant satellites of Jupiter, Saturn, Uranus, and Neptune according to the types and properties of the orbit evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 The evolution of the orbits and rendez-vous occurrences of distant satellites of the planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

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10.7.1 Modern knowledge about the evolution of the orbits of distant planetary moons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 10.7.2 The problem of calculating and detecting the rendez-vous occurrences of distant planetary satellites . . 355 10.7.3 An analytical description of the evolution of satellite orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 10.7.4 Determination of minimal distances between the orbits of distant planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . 357 10.7.5 Proposed Internet source for the study of the evolution of the orbits and rendez-vous of distant planetary satellites. 358 10.7.6 Examples of calculating the minimum distances between the orbits of satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 10.7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Refinement of the Laplace–Lagrange secular perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Chapter 11 Physical parameters of planetary satellites . . . . . . . . . . . . . . 369 11.1 11.2 11.3 11.4

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Handbook of the physical parameters of planetary satellites . . . . 370 Detection of volcanoes on the satellite of Jupiter Io using ground photometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Estimates of the physical parameters of distant planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 11.4.1 Features of distant planetary satellites. . . . . . . . . . . . . . . . . . 372 11.4.2 Overview of available photometric data for distant planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 11.4.3 Photometric model for distant planetary satellites . . . . . . . 377 11.4.4 Determination of photometric parameters of satellites by photometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 11.4.5 Initial data and results of determining the photometric parameters of satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 11.4.6 Comparison of results obtained by different authors. . . . . 385 11.4.7 Conclusions on the estimates of the photometric parameters of distant planetary satellites . . . . . . . . . . . . . . . 388

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11.5 Determination of Himalia’s mass from astrometric observations of other satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Chapter 12 Recent models of planetary satellite motion. Information resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 12.1 Variants and change of version of motion theories and ephemeris of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 12.2 Means of providing access to databases, motion models and ephemeris of planetary satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 12.3 MULTI-SAT ephemeris server features . . . . . . . . . . . . . . . . . . . . . . . . 405 12.4 Theories and models in the MULTI-SAT ephemeris server . . . . . . 411 12.5 Theories and models in the JPL ephemeris server . . . . . . . . . . . . . 416 12.6 Planet satellites in virtual observatories . . . . . . . . . . . . . . . . . . . . . . . 417 12.7 Standards of fundamental astronomy . . . . . . . . . . . . . . . . . . . . . . . . . 417 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Appendix A Nomenclature of planetary satellites . . . . . . . . . . . . . . . . . . . 423 Appendix B Orbital parameters of the natural planetary satellites . . . . 437 Appendix C Special functions in celestial mechanics . . . . . . . . . . . . . . . 451 Appendix D Time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Appendix E Cholesky decomposition. Program in C-code . . . . . . . . . . . . 463 Appendix F Rotation parameters of planets and natural satellites . . . . 465 Appendix G Physical parameters of the natural satellites of the planets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Author’s preface

Author’s preface

A theory only makes sense when it is not only abstract constructions in the imagination of a fascinated theoretician, but wellestablished procedures that properly serve the goals of practical knowledge of nature. The dynamics of planetary satellites is a very interesting area of celestial mechanics. At first glance, the dynamics of satellites can be studied without observation. The researcher can come up with a satellite model that is close to reality or in general abstract one, try new methods on it, demonstrating his highest skill. There is another seductive line of activity: the explanation of why celestial bodies move exactly as they move. A new explanation of facts known in nature or an explanation of previously unexplained phenomena seems to be a significant scientific achievement. Of course, we thus hone our skills. However, in these matters, the researcher should at some point stop and ask himself the question: do we get new information about the nature around us? Of course, a scientific generalization of facts at some stage can create a qualitative leap in our ideas about nature. However, this is preceded by a time-consuming and sometimes exhausting process of accumulating factual knowledge. In the dynamics of planetary satellites, this way inevitably runs through the technical processing of information from observations, through the compilation of prohibitively cumbersome computing programs and the implementation of boring calculations. Which researcher will go there? Either a researcher who understands the harsh inevitability of this process, or one who has his own special scientific and technological predilections. To help just such brave researchers, this book was written. As in many other branches of astronomy, in the dynamics of planetary satellites, the criterion of truth is compliance with observations. Theorists know that the more observations, the worse it can be for a theory. We can congratulate such theorists—their theory has been replaced by a new one. It is precisely for such events this book is aimed. Both in ordinary life and in scientific research, we are often in search of a “help desk”. Nowadays, such a bureau is “the global virtual mind”—Internet. In studies of the dynamics of planetary satellites, as in many other scientific studies, only such data are

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required which are provided with information about who received this data and how, and for which the accuracy and reliability are clear. This book provides reference information on planetary satellites, with links to reliable sources. Scientific work is often successful when it is limited to a certain framework both in terms of objects and research methods. In such a harmonious process, annoying situations sometimes occur when it is necessary to go beyond familiar methods or information. In such situations, the proposed book may help. Finally, this book helps counteract the thought that the words “celestial mechanics” sound old-fashioned and that this is not a modern field of astronomy. In fact, celestial mechanics is not limited to the three-body problem and the determination of the orbit from three observations. Nowadays, this is the most practical and modern field of astronomy. It not only satisfies our natural curiosity, but also serves to solve two perennial problems of humanity: expanding our habitat and protecting against the dangerous forces of nature. Most of the book is based on the scientific results and publications of the author. For those subjects that the author himself was not directly involved in, the book provides brief reviews of publications of other specialists. An extensive bibliography is given for all sections of the book. This is necessary for a more detailed study of methods and scientific results. The bibliography itself is a reference material that is in demand for widespread use. A separate list is given for each chapter. Some links may be repeated in different chapters. The scientific work on the topic of this book was carried out by the author partially in collaboration with colleagues from the Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Paris, France. This, on the one hand, accompanied the work with good expertise, and on the other hand, ensured the relevance of the results obtained by the author. The possibility of writing the book was primarily provided by a high level of education, which was given to the author by M.V. Lomonosov Moscow State University. The author spent his entire scientific life within the Sternberg Astronomical Institute of Moscow State University, where this book was written. The author thanks the associate professor G.I. Shirmin for the final editing and proofreading of the book.

1 Objectives, current problems and general approach to the study of the dynamics of satellites 1.1 Introduction The understanding that the vast Universe extends in all directions has always worried mankind. This causes a double desire. Firstly, it would be nice to understand our place in the boundless space and the infinite diversity of the world. People often experience a slight discomfort from the lack of an answer to such a question. At the same time, a desire arises to extract benefits from the Cosmos to satisfy ever-increasing needs. People are even more worried when they discover a threat to their lives by the forces of nature. Nothing scares us so much as an incomprehensible phenomenon. It is surprisingly easy to reassure people by explaining terrible phenomena, even with not quite familiar words. The information that at least someone understands the processes of nature returns us to the usual comfort of everyday life. That is why we should be grateful to the few people who work to save us from painful questions about space and fate. Since ancient times, people have thought about the influence of celestial bodies on terrestrial life. Attempts to compare celestial phenomena with the fate of man were made by both scientists and investigative individuals not being scientists. However, at all times, a very unreliable result was obtained time and again. As for the fate of the celestial bodies themselves, astronomers and mathematicians have long calculated the surprisingly stable nature of their movement. The sizes and shapes of the orbits of the planets, or the slopes of the axes of their rotation, have not changed much even at cosmogonic time intervals. Natural scientists and philosophers have come to the conclusion that the main reason for the existence of a Cosmic Mind in the Universe is the function of cognition. Led by reason life is characterized by a desire to understand and explain what is happening as regards the phenomenon. At any stage of cognition of the Universe, we already have a more or less adequate model for it. New, more accurate obserThe Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00006-6 Copyright © 2021 Elsevier Inc. All rights reserved.

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vations may lead to a model mismatch with reality. At most, the required adjustment of the model is restored by clarifying the known parameters of the motion or the state of celestial bodies. Sometimes it is necessary to significantly improve theories, the model-constructing technique, or the calculation methods. This process is unconsciously aimed at discovering new, unexplained phenomena. At some stage, it is possible to get this much-needed “food” for the Mind, but this is always preceded by the colossal work of scientists—observers, theorists, and calculators. The motion models of celestial bodies are also valuable in that they allow us to predict their location at any time in the past or future. A theory only makes sense when it is not only by abstract constructions in the imagination of a fascinated theoretician, but also one needs well-established procedures that regularly serve the purposes of practical knowledge of nature. One of the main tools in this respect is practical celestial mechanics. It is practical celestial mechanics that gives us the most complete and accurate knowledge of the dynamics of planetary satellites.

1.2 Celestial mechanics—the basis for studying the dynamics of planetary satellites Celestial mechanics is the branch of science that studies the movements of celestial bodies under the action of natural forces. The subject of celestial mechanics is the mechanical form of the motion of matter. The objects of research are all kinds of material formations, from the smallest particles of cosmic dust to colossal systems such as star clusters, galaxies and clusters of galaxies. The purpose of celestial mechanics is to study the laws of nature that govern the mechanical movements of celestial bodies. For all natural sciences, celestial mechanics plays the role of a foundation, without which the study of the Universe and the exploration of the Cosmos are unthinkable. The significance of celestial mechanics for life on Earth is to gain knowledge about the motion of celestial bodies and the near Cosmos to better meet the needs of mankind and to result in protection from the forces of nature. The theory of motion of artificial satellites of Earth allows for the use of spacecraft for communication and research of terrestrial resources. The theory of motion of asteroids, comets and meteors gives an assessment of the danger of these bodies entering the atmosphere and falling to the Earth’s surface. Studies of the motions of the bodies of the solar system made it possible to cre-

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

ate a fundamental reference frame—a model of the inertial system implemented by celestial mechanics and astrometry in the form of astronomical yearbooks and the fundamental star catalogues. In the development of celestial mechanics many of the most effective methods of mathematical physics and computational mathematics arose, took shape and were furthered. As an example (and by no means the only one!), we can indicate methods for the numerical integration of differential equations describing various natural phenomena and man-made processes. Having arisen in celestial mechanics, these and other numerical methods are widely used in science and technology. In the 17–18th centuries, with the solution of astronomical problems by the methods of celestial mechanics, essentially all theoretical physics began. Not only the theory of systems of ordinary differential equations, as it occurred in the last century, is predominant, but, in fact, the entire set of modern tools of applied mathematics is used by modern celestial mechanics to model the movements of space objects.

1.3 Objectives of studying the dynamics of planetary satellites The primary objective of research into the dynamics of Solar System bodies is the determination of parameters of motion of planets and their satellites. This objective is relevant to the perennial challenge of mankind: expanding and exploring our habitat. Satellites of major planets are the most suitable targets for unmanned and manned landing missions. Research of the structure and dynamics of Solar System bodies is an integral part of dynamical astronomy. The methods of celestial mechanics and astrometric observations are used in this research. Interplanetary navigation, which attracted the interest of scientists in the second half of the 20th century, is a new problem of the dynamics of Solar System bodies. The general approach to studying the dynamics of celestial bodies consists in developing models of motion and ephemerides of planets, asteroids, and planetary satellites. Such models are built based on the general laws of nature, the physical parameters of celestial bodies, and, most importantly, observations. Advanced mathematical and computational techniques are used in the process. Ephemerides are the end result of this research and incorporate the entire body of knowledge on the dynamics of Solar System bodies.

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Ephemerides are used to determine the physical properties of celestial bodies and to study the origins and evolution of the Solar System. They are also needed to prepare and launch space missions to other planets and help discover new celestial bodies. In the middle of the 19th century, Urbain Le Verrier had used ephemerides to predict the existence of the then unknown planet Neptune, and new planets and satellites are still being discovered this way. Therefore, one may conclude that ephemerides also serve as a research tool, since they incorporate all the available data on the motion of planets and satellites. The results and conclusions of celestial mechanics are visibly and invisibly present in many other areas of science and human practice.

1.4 Basic concepts of celestial mechanics and astrometry We establish some basic concepts of practical celestial mechanics and astrometry, with which we will operate in the following presentation. The objects of our research are the planets and satellites of the Solar System. Thus, we operate with models of celestial bodies, which in nature do not exist, but which to a certain extent differ little from the behaviors of real celestial bodies. Examples of such objects are a material point and an absolutely solid homogeneous body bounded by the surface of a triaxial ellipsoid. Laws of motion. The real manifestation of the motion of celestial bodies is a change in their relative position, which is determined by the mutual distances. To set the motion of a system of celestial bodies, one should set the law of change in their mutual distances in time. The mathematical description of the laws of motion are these or other functions of time. For a convenient representation of the motion of celestial bodies, we use the concepts of a reference frame, coordinate system and time scale. The abstract concept of a coordinate system is somehow connected with real celestial bodies. Examples include the Greenwich meridian on Earth or extragalactic radio sources. The abstract concept of a time scale is associated with real physical processes. Examples include Earth’s rotation or electromagnetic radiation from an atom. Laws of interaction. The basis for studying the motion of celestial bodies is the laws of physics that are strictly established from observations, which describe the interactions of bodies or the effects on them of the environment in which they move. The

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

mathematical form of the laws of interaction of celestial bodies are ordinary differential equations, while the mutual distances between celestial bodies or their coordinates satisfy these equations. Mechanical model. In celestial mechanics, the concept of a mechanical model is used. The model is described by the composition of moving objects and their properties, by specifying the forces acting on the individual components of the model. Mechanical models are used either for an approximate description of the motions of celestial bodies or as a basis for the development of more accurate methods for describing their motions. The task of practical celestial mechanics is the creation and study of various mechanical models, as well as the study and description of the motion of real celestial bodies. A mechanical model, being, as a rule, an approximate description of the motions of a system of real celestial bodies, can fundamentally differ from them. In particular, the properties of bodies in the model may not correspond to reality, and the laws of the acting forces can be specified in a special way. Examples include the motion of a system of material points in which celestial bodies are dimensionless, or a restricted three-body problem that does not satisfy Newton’s third law. Observations. Measured values. The source of our knowledge of celestial bodies is observation. In observing, we cannot be content with stating the fact of the presence of a celestial body in the sky. During astronomical observations, measurements of various quantities are carried out using a variety of instruments. Unlike abstract coordinates, the measured value is always the real one. It is formed in the measuring device. Astronomers deal with a wide variety of instruments and measured values. Examples are the angles of rotation of the telescope axis relative to the vertical line and the meridian plane, the distance between images of celestial bodies on photographic plates, the time interval between the flash of the laser rangefinder and the fixation of the light pulse reflected from the celestial body, the background intensity from a single pixel of a semiconductor light detector, and the difference in recordings of the signal from a space radio source at two radio telescopes. Accuracy of observations. Instruments usually have measurement errors. Note that the mysteries of the processes occurring in measuring instruments leave us only with the opportunity to build hypotheses regarding measurement errors. The magnitude of the error of an individual measurement is never known. Often we assume that the errors are purely random, and we consider various statistical characteristics of the errors. Mostly, we use the con-

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Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

cept of the most probable root-mean-square error. The structural properties of measuring instruments sometimes make it possible to approximately establish the accuracy of measurements. In the general case, we are talking about the accuracy of observations. Time. Variation of the measured value in time is due to the motion of celestial bodies. Measurement is performed at some point in time. This time point is counted by the clock of the observatory. In practical celestial mechanics, a specific time of measurement is always ascribed to a measurable quantity. Time is an abstract concept and some instruments are needed to measure it. However, any device has its own measurement error. First, time was measured by the angle of rotation of the Earth. Such a time was called universal was and designated as UT (Universal Time). When discrepancies between the theory of the motion of the Moon and observations were discovered, it became clear that the Earth rotates unevenly, and time has become the standard, as an independent variable in the theory of motion of the Moon. Time, measured by observations of the Moon, was called ephemeris time and was denoted ET. However, the accuracy of the observations of the Moon is still limited. The search for a more accurate time meter led to an atomic clock. This time sensor is now the most accurate. Time, averaged over several of the most accurate atomic clocks in the world, is called international atomic time and is designated as IAT (International Atomic Time). In the future, we will talk about observations of celestial bodies, always assuming that one or another measured value is received at a certain point in time: measurement time. The accuracy of astronomical measurements has already reached such a level that the inadequacy of classical Newtonian mechanics for describing the observed motion of celestial bodies has become noticeable. In a more accurate theory of general relativity, time passes differently at any two points in space. To connect different time scales, it is necessary to take into account the motion of bodies and their masses. Motion parameters. When we study planets and satellites, stars and galaxies, we boldly assume that some parameters inherent in celestial bodies and their motion remain constant all the time. These include the mass, size and shape of bodies, orbit parameters and many other quantities. These parameters cannot be directly measured using existing instruments. However, their meanings really manifest themselves in the observed motion of celestial bodies. In the future we will call such quantities motion parameters of celestial bodies. Coordinate systems. Measured quantities do not give visual representations of the configuration of the system of celestial bod-

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

ies and are even less suitable for expressing general laws of motion. A convenient means of describing the spatial arrangement of bodies and directions of celestial bodies is the use of coordinate systems. When we talk about the position of the star or about the orientation of the body in a certain coordinate system, we mean the abstract coordinate axes in space and imaginary lines in the sky. Coordinate systems are chosen so as to give a clear idea of the laws and properties of the motion of celestial bodies. The choice of a coordinate system is due to the convenience of describing and studying the motion of a particular celestial body. The origin and coordinate axes are associated either with the details of the object, for example, the Earth’s Greenwich meridian, or with its dynamic properties, for example, with the principal axes of inertia of the body, or with the properties of motion, for example, with the rotation axis of the body, or with the position of the body at some time point, or we may choose a coordinate system in another particular way. Mostly, a system of rectangular or Cartesian coordinates is used, its origin is denoted by the letter O, and the axes by the letters x, y, and z. The system of spherical coordinates is often used with the designation of the central distance by the letter r, the latitude by the letter ϕ and the longitude by the letter λ. We refer to any coordinate systems with an origin located at the observation point as topocentric coordinate systems. In addition, we associate the axes of the topocentric system with the vertical line and the local meridian. When the origin of the coordinate system is placed at the mass center of the Earth, we are talking about geocentric coordinate systems. The laws of motion of celestial bodies are the dependences of the coordinates of bodies on time and motion parameters. Dependencies can take many forms. At most, analytical functions are used that describe the explicit dependence of the coordinates on time. In some cases, the dependence is given in implicit form, then the coordinates are obtained by calculations with formulas by way of successive approximations. The law of motion can take the form of numerical tables in which the coordinates of celestial bodies are given for a number of fixed points in time, usually defined with some constant step. With such a numerical specification of the law of motion, the dependence of the coordinates on the motion parameters of the celestial body is lost. In this case, it is difficult to analyze the properties of motion, and we are limited to the time interval for which the coordinates were calculated. The coordinates of celestial bodies are abstract concepts. They cannot be measured by any instruments. Coordinate systems are

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modeled using formulas and algorithms and form a constituent part of the motion model of celestial bodies. A model of motion of a celestial body. We do not know exactly how the celestial bodies are arranged and by what exact laws they move. Therefore, we have to be content with the study of motion models, putting forward the bold hypothesis that our models differ little from reality. In the general case, by a model of motion of a celestial body we will mean a certain construction that allows us to determine the values of the measured quantity at any given time instants for known values of the parameters of motion. Implementations of the model of motion of a celestial body can have very different forms. These can be mathematical formulas, written manually on paper or published as printed material. These can be printed numeric tables of coordinate values. Currently, both formulas and tables are displayed in computer memory units. In this case, the formulas are converted into calculation algorithms, and the tables are available to computational programs that solve certain problems. Even in the era of powerful computing technology, the coordinates of the principal celestial bodies calculated for several years in advance are created and printed in the form of astronomical yearbooks in several world research centers. Where do our ideas of the laws of motion of celestial bodies come from? In ancient times, they were established almost empirically from simple observations. Now, of course, the laws of motion are found in the process of solving differential equations of motion relative to the coordinates of celestial bodies. These equations are compiled on the basis of strictly established laws of physics, which describe the interactions of bodies or the effects on them of the environment in which they move. This is done as part of a mechanical model. All factors affecting the movement of each body of the system and included in the model under consideration are clearly fixed. The set of constructs of the laws of motion of celestial bodies, as well as its result, the laws of motion themselves, are called the theory of motion. This is what celestial mechanics addresses. In the vast majority of problems of celestial mechanics, it is impossible to obtain an exact solution of the equations of motion. One has to be content with either an approximate solution of the exact equations, or an exact solution of the approximate equations. Both analytical and numerical methods for solving differential equations are used. In both cases, the solution has an error. This error can be more or less reliably estimated using the theory itself.

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

The accuracy of the motion model of a celestial body. The initial data for the model of motion of a celestial body are motion parameters, which, in turn, are known with some error. This error will also affect the accuracy of the pre-calculation of the coordinates of the celestial body and the accuracy of the pre-calculation of the measured value. Furthermore, we will talk about the model accuracy, implying an error in the calculation of the measured value. In this case, we separate two sources of this error: the proximity of the obtained solution of the motion equations and the inaccuracy of the motion parameters. The error of the solution of the motion equations will also be called the error of the calculations or the error of the method. When we talk about the accuracy of the theory of motion of a celestial body, it is always necessary to clarify whether the inaccuracy of the motion parameters is included in the error of the theory or is in the accuracy of the theory under the assumption of absolutely accurate parameters. Research methods. From other astronomical disciplines celestial mechanics differs only in research methods, among which are analytical, numerical and qualitative approaches. Analytical methods make it possible to obtain a set of analytical relationships that allow us to calculate the approximate positions and velocities of celestial bodies at given time points, omitting its values at any intermediate time points. A feature of analytical methods is the great complexity and growing bulkiness of the calculations. In addition, analytical methods make it impossible to assess the properties of the studied motions at very large time intervals. Another drawback is that analytical methods are not applicable to all objects. The limitations inherent in analytical methods do not apply to numerical methods, which are suitable for calculating the motions of any celestial bodies and their systems with a predetermined accuracy. With the use of powerful computers in scientific research, the previously considered excessive laboriousness of numerical methods has ceased to be an obstacle to their application. But they have their own “Achilles’ heel”—this is the steady accumulation of error with increase in the integration interval, while rigorous estimates of the growth of this error are impossible. Another drawback of these methods is the numerical form of presenting the results and the inevitability of calculating the intermediate stages, although often the goal of the study is the final configuration after integration. Qualitative methods of celestial mechanics make it possible to judge the properties of the movements of celestial bodies without full integration of (analytical or numerical) differential equations.

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Analytical, numerical and qualitative methods continue to be applied in modern practical celestial mechanics, and the beauty and high efficiency of analytical methods are successfully combined with the simplicity and universality of numerical methods, and all this is complemented by the cosmogonic importance of the conclusions obtained by qualitative research methods.

1.5 General approach to studying the dynamics of planets and satellites based on observations A general approach to studying the dynamics of planets and satellites is the construction of a model of motion based on observations. It is the model of motion that is needed for the practical knowledge of nature. Fig. 1.1 shows a scheme for studying the dynamics of Solar System bodies based on observations. At any stage of research, we fix the composition of the studied system of celestial bodies. The laws of the interaction of bodies (gravitational attraction, resistance of the medium), currently established, allow for writing down the differential equations of motion. Using analytical methods, one can find a general solution of the equations of motion. After substituting the values of arbitrary constants (motion parameters) into this general solution, we obtain the required model of motion of the system of celestial bodies. When we solve equations of motion by methods of numerical integration under known initial conditions (motion parameters), we also obtain a model of motion of a system of celestial bodies. Some preliminary values of motion parameters are usually known from previous studies. To construct a model of motion, the values of the physical parameters entering the equations of motion through the laws of interaction (for example, the mass of bodies) will also be required. The main procedure for studying the dynamics of celestial bodies is to refine the model based on observations. Observations give us the values of the measured quantities. Call them measured values. On the other hand, we have a motion model that serves to pre-calculate measured values. We can calculate the measured values precisely at the times of observation. Results are called calculated measured values. Values different in origin of the same entity will differ from each other. We denote this difference of values in Fig. 1.1 symbolically by “O-C” (O for observatum, C for calculatum). The difference is a natural result, since it contains an error of observation and an error in the model of motion of a celestial body. However, in some cases, the differences “O-C” will

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

Figure 1.1. Scheme of studying the dynamics of celestial bodies.

exceed the model error and the observation error. New, more accurate observations reveal a model mismatch with reality. In these cases, the mismatch is attributed to the simplest and most probable cause—the inaccuracy of the accepted values of the motion parameters of the celestial body. A process called refinement of motion parameters from observations is included in this case (see “Parameter refinement methods” in Fig. 1.1). Mostly, the required agreement between the theory and observations is achieved by refining the parameters, and the differences “O-C” again fall within the errors of the model and observations. In some rare cases, the theory cannot be reconciled with observations—the differences “O-C” remain significant. Then we have to improve methods for solving the equations of motion and calculation methods. This is the most laborious part of celestial mechanics. The factors affecting the motion of each celestial body are being reconsidered. New, more accurate formulas of the theory are derived. As a result, the formulas become longer and more complex. In addition, more accurate calculation methods are being developed and applied. As a result, the required computing time is significantly increased. In even rarer cases, the mismatch of the theory with observations remains significant, no matter how hard the researchers try to refine the motion parameters and improve the motion model. As a result of the generalization of facts, testing of new hypotheses and higher tension of intelligence, a discovery is made. Previously unknown celestial bodies or new laws of the interaction of the known bodies may be discovered. In such a situation, our general ideas about the world around us expand significantly. A generalization of the basic laws of nature is made. The scheme presented here, like any scheme, is meagre and limited, it only in general terms reflects the mixture of scientific research and the accumulation of facts, fantasies and errors.

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Note that the described process also has a purely practical focus. The model of motion of celestial bodies is the basis for tracking possible dangers threatening from outer cosmic space. The model of the motion of celestial bodies is also directly used for the design and support of flights of automatic and manned nearEarth and interplanetary vehicles—artificial celestial bodies.

1.6 Special properties of necessary observations The motion of most real and imaginary celestial bodies is of the form of the circulations of some bodies around others. The proper rotations of celestial bodies are also being studied. The revolution or rotation of the body is described by an angle whose magnitude monotonically increases in time. Let us consider in more detail how these processes are determined from observations. The orbital revolution angle or the rotation angle of a celestial body is conventionally called the longitude and denoted here by λ. In most mechanical models, the rate of change in the longitude of λ˙ is approximately constant. Advancement can be achieved by increasing the accuracy of the observations. In this way, the discovery either of a new property of a known celestial body, or of a new planet or satellites, can occur. Let us illustrate this with an example. Let us assume that we have built a good model of motion and with its help calculated the so-called “O-C”, the differences between the observed and the theoretical orbital longitude values. If measurements are inaccurate, i.e. in the presence of observation errors, a plot of these differences may look like the one shown in Fig. 1.2a, where “noise” is the only apparent component. Let us assume that progress in observational techniques provided an opportunity to improve the accuracy of observations and suppress noise. A certain pattern then emerges (Fig. 1.2b), and sinusoidal variation of the “O-C” differences becomes clearly visible when observations get even more accurate (Fig. 1.2c). This “signal” helps one to determine the factors that were left unaccounted for by the theory. The orbital motion of celestial bodies is distinctive in that the orbital longitude increases monotonically with time. If one removes the function of the theoretical variation of orbital longitude from its observed values, a plot similar to the one in Fig. 1.3a may be obtained. Again, there is nothing interesting in it. If past and new observations of the celestial body under study are added to the data presented in Fig. 1.3a, the plot in Fig. 1.3b is obtained. It

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

Figure 1.2. Examples of O-C residuals of the orbital longitude of a celestial body at different levels of accuracy of observations.

Figure 1.3. Examples of O-C residuals of the orbital longitude of a celestial body at different time intervals.

can be seen that the longitude varies almost quadratically in time. This effect may be induced by the unaccounted dissipation of the mechanical energy of a celestial body, which, in its turn, may be attributed, for example, to tidal forces. It is now clear that the observation time interval should also be expanded in order to make progress in this field. At some stage, this may lead to the discovery of new phenomena. What is the relation between the observation time interval and the accuracy of ephemerides? Let us take a look at Fig. 1.4a. Here, the values of orbital longitude of a celestial body, which were derived from observations performed in time interval (t1 , t2 ), are shown. “Noise” and linear variation are apparent. Using theory and observational data, we may calculate the probable orbital longitude at time point tf of interest to us (vertical lines in this figure). If observations (with the same accuracy) are extended to time point t3 , the ephemeris becomes more accurate, which is seen in Fig. 1.4b.

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Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

Figure 1.4. Illustration of the dependence of accuracy of ephemerides of a celestial body on the observation time interval.

Naturally, it is not possible to measure the orbital longitude directly in the process of observations. One observes just the projection of orbital motion onto the image plane. However, an approximate formula can be derived regarding the accuracy of determining the orbital longitude. Similar conclusions can be drawn with respect to other characteristics of orbital motion. The theory of the process considered here is to describe a linear change in the orbital longitude of a celestial body in time. The parameter of the theory of the motion, determined from obser˙ The measured value is the longitude λ itself. Such vations, is λ. circumstances are characteristic of all natural satellites of the planets. When performing the same type of observations, the error in measuring the longitude will be the same at any time. Denote this error by σλ . Let the longitude measurement be performed at two points in time t1 and t2 . Then the error in determining the motion parameter σλ˙ is obtained from the relation σλ˙ =

σλ , t2 − t1

whence it can be seen that the accuracy of determining the motion parameter improves with increasing measurement interval t2 − t1 . Let us now see what the role is of the accuracy of the motion parameter in realizing the main objective of the theory—calculating the longitude at given time points. Suppose that we need to calculate the longitude at a time point tf in the future, that is, tf > t2 . The error of such a calculation is determined by the formula   σλ t1 + t2 tf − . σλ(tf ) = t2 − t1 2 This error also decreases with increasing measurement interval. Suppose that the observations continued after the time point t2 . Let the last new observation be performed at some time point

Chapter 1 Objectives, current problems and general approach to the study of the dynamics of satellites

t3 (t3 > t2 ). Now, the error in calculating the longitude at the time point tf has become equal to σλ σλ(tf ) = t3 − t1



t1 + t3 tf − 2



σλ < t2 − t1



 t1 + t2 . tf − 2

Thus, the accuracy of calculating longitude has improved. Let us see what the use of more accurate observations, but performed over a short time interval, can yield. Suppose, for example, in the middle of the measurement interval t1 –t2 at time points t1∗ , t2∗ , two measurements of longitude were made with an error of σλ∗ . Let us assume that σλ∗ = 0.5 σλ , t2∗ − t1∗ = 0.1 (t2 − t1 ),

t1∗ + t2∗ t1 + t2 = , 2 2

that is, new observations are twice more accurate than the previous ones, and the measurement interval is ten times less. The accuracy of pre-calculating the longitude based on more accurate observations is found from the relationships σ∗ σλ(tf ) = ∗ λ ∗ t 2 − t1

    t1∗ + t2∗ σλ t1 + t2 tf − tf − =5 , 2 t2 − t1 2

whence it is clear that this accuracy turns out to be five times worse than on the basis of previous, less accurate observations. The analysis of the general properties of constructing a model based on observations considered here leads to the following conclusions. 1. To build a motion model of any celestial body, we always try to use the set of all the observations that exist in the world, starting from the discovery of this celestial body. 2. Continuing observations of celestial bodies even with the same accuracy is useful. 3. The use of observations made with better accuracy does not always lead to a refinement of the model. The advantages of some observations over others are determined not only by their accuracy, but also by the time interval at which they are performed. 4. Any new observations, even more accurate ones, are almost always used only as an addition to an existing database. These conclusions constitute a feature of practical celestial mechanics in comparison with many other studies of celestial bodies, when new valuable scientific results are obtained on the basis of only the latest observations, which accurately overlap past ones.

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In practical celestial mechanics, a more accurate and more adequate model of the motion of a celestial body is built on the basis of a more complete database of observations. Naturally, novel observational techniques, which may provide new data on the motion of celestial bodies, should also be developed.

2 Satellites of planets 2.1 Satellites of the planets—objects of the solar system The naming of the studied objects “satellites of planets” (sometimes also called “planetary satellites”) refers to their property of accompanying planets in their motion around the Sun. Satellites are available on major and minor planets. There are planets without satellites. The Earth also has a satellite, the Moon. In our consideration, we confine ourselves to the satellites of Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. We are not considering the Moon here. It deserves special attention for a number of reasons. The main reason is that the Moon is the closest natural extraterrestrial object to us. Therefore, its observations are particularly high accuracy. The dynamics of the Moon is closely related to the shape and rotation of the Earth. To describe the motion of the Moon, special approaches and methods are required. This is the subject of an extensive and special science—the dynamics of the Moon. We will not consider here the satellites of dwarf planets and asteroids, except for the satellites of Pluto. The specificity of the satellites of dwarf planets and asteroids is due to the fact that the apparent distances of these satellites from their minor planets are so small that the accuracy of determining the orbits is very low. For most such satellites, only the presence of the satellites is known. Satellites of major planets and satellites of the dwarf planet Pluto are special objects of the Solar System, and they are of particular interest. The main satellites of Jupiter and Saturn have been discovered a relatively long time ago. Their dynamics are well studied. There are good conditions for observing most of these satellites from the Earth. Satellites of planets are of practical interest as the most suitable objects for expanding the human environment. A spacecraft is easier to land on many satellites than on a planet. The properties of satellite surfaces and the gravitational forces on them are most suitable for exploration. Another remarkable property of satellites is that their motion is subject to the gravitational attraction of the planet. Therefore, the motion of satellites can determine the mass and dynamical parameters of the planet. Moreover, the motion of the satellites is sensitive to the viscosity of the matter of the planet’s body. The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00007-8 Copyright © 2021 Elsevier Inc. All rights reserved.

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The diversity and complexity of the motion of planetary satellites has led to the improvement and rigorous testing of the methods of celestial mechanics. Studying the dynamics of satellites requires the most complex and sophisticated forms of mathematical technique. Being bodies formed together with the planets, the satellites left evidence of a complex pattern of the formation and evolution of the Solar System. By studying the dynamics of satellites, one can come closer to understanding this pattern. The presence of people as manifest outside the Earth has long been represented by its automatic messengers—spacecraft. These artificial objects shuttle around in the Solar System. Natural satellites of the planets fall into the field of view of spacecraft cameras and radars. In addition to ground-based observations, this gives us extensive data to study our extended habitat. As a result, we can conclude that the properties of satellites give us reason to single out the dynamics of planetary satellites in a special field of science worthy of special and close attention. The development of this science is of undoubted practical interest.

2.2 Classification and nomenclature of planetary satellites Classification of celestial bodies, as always, is a very thankless task. There will always be objects that go beyond any classification. Nevertheless, the satellites of the planets can be divided into three groups. The first is a group of the main satellites of the planet. The satellites of this group are characterized by the fact that they have the most significant size and almost circular orbits near the plane of the planetary equator. Since the equators of major planets are associated with the axial symmetry of their compressed bodies, the non-sphericity of the planet has a noticeable effect on the motion of the main satellites. For Jupiter, Saturn and Uranus, the orbits of the main satellites are not far from each other. Therefore, the mutual attraction of satellites also significantly affects their motion. Another characteristic feature of the main satellites is the presence of a large number of relatively accurate observations accumulated over long time intervals. This is because the observation conditions for these satellites are most favorable. The range of magnitudes of the main satellites is from 4m to 14m . Planets with their bright bodies do not really interfere with satellite observations. As a result, among all satellites, the dynamics of the main satellites is studied most. Until recently, analytical theories

Chapter 2 Satellites of planets

were used to construct the motion models of the main satellites. Currently, the methods for numerical integrating the equations of motion of satellites are most suitable for constructing and using motion models. The second group is the close (or inner) satellites of the planets. They are significantly smaller than the main satellites, and move inside the orbits of the main satellites in almost circular orbits in the plane of the equator of the planet. Ground-based observations of close satellites are rather difficult by their apparent proximity to the planets themselves. The bright body of the planet, which does not look very sharp in the images obtained from the Earth, hides faint spots of satellites with its halo. A number of close satellites were discovered using spacecraft. Later, some of them were also seen from the Earth. For these reasons, the known parameters of the orbits of close satellites are not very accurate. The dynamics of close satellites are most affected by the oblateness of the planet and features of its gravity field. The motion of the main satellites also significantly affects their movement. The motion models of close satellites in most cases can be represented by precessing ellipses. We consider such models in the corresponding section of the book. The third group is distant (or outer) satellites. These are also small bodies of the Solar System, they are significantly smaller than the main satellites. The orbits of distant satellites extend beyond the orbits of the main ones. The inclination and eccentricity of the orbits of distant planetary satellites vary widely. Their eccentricity reaches a value of 0.75. The inclination of the orbit of a distant satellite relative to the plane of the planet’s orbit can be even more than 90 degrees, these are the so-called retrograde satellites with reverse orbital motion relative to the orbital motion of the planet. The most significant perturbing factor for these satellites is the attraction of the Sun. The attraction of the main satellites and the oblateness of the planet are weaker. At a time when there were no powerful computers, attempts only were made to develop analytical theories of motion of distant planetary satellites. The difficulty here lies in the fact that the main perturbing factor, the attraction of the Sun, is significant. The corresponding small parameter in perturbation theory is not small enough, and its significant value limits the possibility of taking into account low-order perturbations. The necessary accuracy of the theory required the determination of high-order perturbations, which led to extremely cumbersome formulas. With the advent of the ability to quickly obtain the solution of differential equations of motion by methods of numerical integration, analytical methods have remained in demand only for studying the

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evolution of the orbits of distant satellites over large time intervals. To date, 71 distant satellites of Jupiter, 58 distant satellites of Saturn, nine distant satellites of Uranus and six distant satellites of Neptune have been discovered. The distant satellites of Jupiter can be divided into two subgroups. One of them is represented by satellites mainly with semimajor axes of orbits within 9–13 million km in direct orbital motion. The second subgroup is in general satellites with semi-major axes of orbits from 16 to 25 million km, performing retrograde orbital motion. Note a feature in the classification of the satellites of Pluto. The fact is that due to the discovery of many minor planets beyond the orbit of Neptune, according to their properties being comparable to Pluto’s, it was decided to classify this planet as a class of dwarf planets. The decision was made at the 26th General Assembly of the International Astronomical Union in 2006. Now this planet is designated as 134340 Pluto. According to the rules for minor planets, the satellites of Pluto should be designated, for example, S/2011 (134340) 1 for Kerberos, discovered in 2011. The need to give the year of discovery in the designation is not entirely convenient. Therefore, we will use the previously accepted notation P1, P2, ... and the names assigned to the satellites. When considering the entire complex of planetary satellites, we are interested in the simplest characteristic—the number of satellites. Over time, this number changes as more and more new satellites are discovered. Fig. 2.1 presents the variation of the number N of known planetary satellites with time since the first discovery of Jupiter’s satellites (so-called Galilean moons of Jupiter) by Galileo Galilei in 1610. To date, 209 natural satellites of Mars, Jupiter, Saturn, Uranus, Neptune, Pluto have become known. The distribution of the number of satellites in three groups for each of the planets is given in Table 2.1. The composition of all planetary satellites currently known is represented by a table called the Nomenclature of satellites. Each satellite is assigned a number and a name is given. Numbers and names of satellites are approved by the decision of a special commission of the International Astronomical Union. Names are given in Latin letters and are usually taken from mythology. Some recently discovered satellites do not yet have names. They are assigned a temporary designation. Russian equivalents of satellite names should be chosen using information from mythology recorded in Russian by professional philologists.

Chapter 2 Satellites of planets

Table 2.1 The number of satellites of each of the three types for each planet. Planet

Close

Main

Distant

In total

Mars



2



2

Jupiter

4

4

71

79

Saturn

16

8

58

82

Uranus

13

5

9

27

Neptune

7

1

6

14

Pluto



1

4

5

Figure 2.1. Growth of number N of known planetary satellites with time.

To identify the satellite, the nomenclature also contains an approximate value of the semi-major axis of the orbit. This immediately makes it possible to determine which group the satellites belongs to. The nomenclature of the natural satellites of the planets is available in Appendix A of this book. There are also given the circumstances of the discovery of each satellite: the year of discovery and a link to the publication of a message as regards the discovery.

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The list of publications is compiled separately for the satellites of each planet. Russian names of planetary satellites are selected on the basis of Russian translations of mythology. This work was done by Valentina Semenovna Uralskaya (Sternberg Astronomical Institute of Moscow State University).

2.3 Discovery of Neptune and its satellite, Triton The calculation of the planet ephemeris in some countries was a truly state matter. In past centuries, these calculations were most intensively carried out in France, where an astronomical yearbook called Connaissance des Temps was published since 1679. In 1795, the Bureau des Longitudes Institute was created in Paris, whose task was to calculate the ephemeris. A brief history of this institute can be found in the paper (Emelyanov, 1997). The ephemerides of the planets were created successively by a number of French scientists. The history and description of French ephemeris created before 1997 can be found in the book (Simon et al., 1997). The ephemerides of the planets were then called tables. In the mid19th century, Alexis Bouvard was doing this. For Jupiter, Saturn and Uranus, these were the Bouvard tables. By 1845, discrepancies between the positions of Uranus, which were given by the tables of Bouvard, and the results of observations, reached 2 . At the request of Dominique François Jean Arago, Urbain Le Verrier took up this problem. He calculated in the motion of Uranus second-order perturbations about the masses of the planets caused by the attraction of Jupiter and Saturn, and introduced corrections to the longitude of Uranus in the Bouvard tables on the order of 40 . He then compared his theory with observations and improved integration constants. Thus, he brought the discrepancies between theory and meridian observations of Uranus to 20 , which was still too much. Then Le Verrier began to look for the perturbations that one has on Uranus of a planet located approximately in the plane of the ecliptic and in the first approximation at the mean distance from the Sun, twice as large as Uranus (according to the Titius–Bode empirical law). He was able to reduce the longitude mismatch between theory and observations to at least 5.4 for meridian observations and to 20 for past observations, taking into account the perturbing planet with a mass of 0.000107 solar masses and orbit with a semi-major axis of 36.1539 AU (AU is for astronomical unit). He established that on January 1, 1847, the heliocentric longitude of the perturbing planet should be equal to

Chapter 2 Satellites of planets

326◦ 32 , and the distance should be 33.06 AU. Le Verrier reported the position of the planet on September 18, 1846 to the German astronomer Johann Gottfried Galle, who on September 23, 1846 discovered it at a distance of 52 from the position predicted by Le Verrier. It is well known that the English astronomer John Couch Adams solved this problem simultaneously with Le Verrier, but his results were published after the discovery of this planet. The position calculated by Adams differed from Galle’s observations by 2◦ 27 . Based on the first observations of Neptune and the old observations of Jérôme Lalande, who in 1795 took this planet to be a star, it was possible to determine the elements of the elliptical orbit. Then it was possible to calculate the mass of the planet from observations of the satellite of Neptune, Triton, discovered by William Lassell in 1846, shortly after the discovery of the planet. This result was significantly different from the mass predicted by Le Verrier. Félix Tisserand noted that the period of meridian observations corresponds to the time interval when the mutual perturbations of Uranus and Neptune are significant. He showed that the directions of the perturbing forces calculated by Le Verrier turned out to be close to real, but the accepted values of the eccentricity and the semi-major axis of the orbit gave too weak perturbations, which was partially offset by the large mass of Neptune. This allowed Le Verrier and Adams to correctly represent the heliocentric position of Neptune.

2.4 The history of the discovery of Pluto’s satellite, Charon Planet Pluto until recently was the least studied of all the bodies of the Solar System. The distance to it from the Earth is on average around 40 AU, i.e. 6 billion km. From the Earth, Pluto looks like a star of about 14th magnitude. Pluto’s satellite, named Charon, was discovered in 1978 in a photograph taken at the United States Naval Observatory (USNO) (Smith et al., 1978). This satellite was barely distinguishable in the form of a hump in the image of the planet. The maximum angular distance between Pluto and Charon is 0.9 arcseconds. The orbit of Charon was first determined in Christy and Harrington (1978) and then refined by additional observations (Christy and Harrington, 1980; Harrington and Christy, 1980). Charon was found on old 1965 photographic plates, on which the satellite’s

23

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Chapter 2 Satellites of planets

positional angles were measured. For the Charon orbit, the semimajor axis and the orbital period were independently determined, which made it possible to approximately estimate the mass of the planet–satellite system. However, the accuracy of determining the semi-major axis from photographic observations remained not high. By this time, progress in astronomical technique led to the appearance of speckle-interferometric observations. Their methodology is described in Chapter 5. Since 1980, the results of speckleinterferometric observations of the Pluto–Charon system began to appear (Bonneau and Foy, 1980). As an example of processing speckle-interferometric observations of this system, we can mention the paper (Baier and Weigelt, 1987). To obtain the seven positions of Pluto and Charon, 87,000 interferograms were processed. By 1985, 19 speckle-interferometric observations had accumulated. The most significant results in determining the Charon orbit from these observations were obtained in Tholen (1985). The author of this work determined the Charon orbit by the nonlinear least squares method taking into account the weights of the observations. Keplerian unperturbed orbit was taken as a model of Charon’s motion. An attempt to determine the eccentricity of the orbit based on observations led to a value of 0.008 ± 0.032. The error is four times the value itself, so the orbit was adopted circular one. The orbital radius turned out to be equal to (19360 ± 300) km, and the orbital period is (6.38764 ± 0.00018) days. An independent determination of the orbital period and the orbital radius made it possible to determine the total mass of the Pluto–Charon system; it turned out to be equal to (6.8 ± 0.5) · 10−9 solar masses. It turned out that the orbit of Charon is almost perpendicular to the plane of the Earth’s equator. The slope to the equator of the 1950.0 epoch was 93.9 degrees. Since Charon’s orbital period exactly coincides with the period of Pluto’s proper rotation and the period of Charon’s proper rotation, the Pluto–Charon system can be considered to have completed its tidal evolution (Farinella et al., 1979; Reinsch and Pakull, 1987). This, in turn, serves as an explanation for the fact that the observations produce an almost circular orbit of Charon (Tholen, 1985). Speckle-interferometric observations cannot reliably determine the size of bodies, which can be inferred from the description of the technique for such observations. Therefore, the question of the density of the Pluto–Charon system remained open for some time.

Chapter 2 Satellites of planets

An exceptional astronomical event of that era was the passage of the Charon orbit plane near the line connecting the Sun and Pluto. This phenomenon repeats itself with a period of 124 years (half the period of Pluto’s rotation around the Sun). According to orbital parameters determined by speckle-interferometric observations, since 1985 Pluto and Charon, as they are visible from the Earth, should periodically cover each other. In addition, the shadow cast by one body on another will be visible from the Earth. Accordingly, their total brightness will decrease. As it turned out later, this decrease can reach 0.7 magnitude. Such events are called in the literature mutual occultations and eclipses. Changes in the brightness of Pluto due to occultations and eclipses were indeed discovered in 1985 for the first time from observations made at the Mount Palomar and Mac Donald Observatories (Binzel et al., 1985). High-precision photometric observations of the Pluto–Charon system provide fundamentally new information for determining the parameters of the system. A detailed description of the phenomena modeling technique and corresponding formulas are given in Dunbar and Tedesco (1986). It is assumed that Pluto and Charon have a spherical shape. The measured value, the total brightness, substantially depends on the physical properties of the surfaces of the bodies. Therefore, the albedo for each of the bodies is included in the number of determined parameters. In the paper (Dunbar, Tedesco, 1986) it was assumed that the albedo is the same on the whole surface of the body. Fig. 2.2 shows several examples of the relative positions of Pluto, Charon and the shadow of one of the bodies, as seen from the Earth. The dimensions of the bodies and the shadow shift in the plots correspond to the system parameters obtained in (Tholen et al., 1987a; 1987b) from the first photometric observations of mutual phenomena in the Pluto–Charon system. The visible border of the shadow is a curve described by a rather complex equation. Dunbar and Tedesco (1986) proposed determining the visible illuminated areas of bodies numerically by calculating definite integrals. For approximate calculations, a simplification can be introduced—consider the visible border of the shadow as a circle with a radius equal to the radius of the body that casts the shadow (Dunbar and Tedesco, 1986). In this case, the visible illuminated areas can be expressed analytically and calculated using simple formulas, which, for example, are given in Dunbar and Tedesco (1986). The error in the calculated brightness of the system does not exceed 0.005 magnitude. Despite the simplicity of the formulas, the algorithm for calculating brightness is quite

25

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Chapter 2 Satellites of planets

Figure 2.2. Different phases of mutual occultations and eclipses in the Pluto–Charon system.

complicated due to the large number of options for the relative location of bodies and shadows. In 1985 and 1986, the eclipses were incomplete, i.e. never the visible disk of one body was completely inside the visible disk of another body or its shadow. Under these conditions, a reliably determined parameter is the sum of the radii of the planet and the satellite. The radius of each of the bodies is determined much less accurately. However, to determine the density of bodies, it is necessary to know their radii separately, and not just the sum. To separately determine the radii of Pluto and Charon, observations of total eclipses that occurred in 1987 and 1988 should be used. The eclipses in 1989 and 1990 are again incomplete and in 1990 they ended completely. When using photometric observations during eclipses, it is in principle impossible to separately determine the semi-major axis of the Charon orbit and the radii of the bodies. If, for example, we simultaneously increase N times the semi-major axis of the orbit and the radii of the bodies, then the light curve of the system will not change. On the other hand, to determine the average density of the matter of the Pluto–Charon system, it is enough to know only the ratio of the radii of the bodies to the semi-major axis of the orbit. Indeed, it is easy to obtain a formula for the average density ρ of the system’s matter: 3π ρ= GT 2



RP a

3



RC + a

3 −1 ,

where G is the universal gravitational constant, T is the Charon orbital period, a is the semi-major axis of its orbit, and RP and RC are the radii of Pluto and Charon, respectively. The period T , more

Chapter 2 Satellites of planets

precisely than other parameters, is determined both from speckleinterferometric observations and from photometric observations during intervals of mutual occultations and eclipses. The total mass of the system can be determined assuming that in addition to the period T , the semi-major axis of the satellite’s orbit is known, which could be determined only by speckleinterferometric observations. Since 1985, photometric observations of mutual eclipses in the Pluto–Charon system have been systematically carried out at the Mauna Kea Observatory in the Hawaiian Islands and at the MacDonald Observatory in Texas (USA). The first most significant results of determining the parameters of the Pluto–Charon system from photometric observations of eclipses were obtained in Tholen et al. (1987a; 1987b). The observations of partial eclipses of 1985 and 1986 were used. The observational results were not published, the comparison results were presented only in the form of drawings of a theoretical light curve after refinement of the parameters with the observational results for six eclipses in 1986. The inclination of the orbit is determined with reduced accuracy, since the information for its determination from photometry is provided only by the deviation of the shadow, which in itself is very small. The average value of the residual deviations of the measured brightness values from the theoretical values after refinement of the parameters for each eclipse did not exceed 0.002 magnitude. The maximum decrease in the brightness of the Pluto–Charon system in the eclipses of 1985 and 1986 was about 0.25 magnitude. Tholen et al. (1987a; 1987b) made an attempt to determine the albedo variations by the visible discs of Pluto and Charon and concluded that the equatorial regions of Pluto are slightly darker than the polar ones. This result is consistent with the infrared measurements of Pluto’s radiation made on August 16, 1983 with the IRAS (Infrared Astronomical Satellite) (Sykes et al., 1987). Pluto was supposed to have bright polar caps made of solid methane, extending at least to latitudes of ±45 degrees. As for Charon, an attempt was made to identify ice on its surface (Marcialis et al., 1987). For all photometric observations of eclipses, including total eclipses up to June 26, 1987, the parameters of the Pluto–Charon system were again refined (Tholen et al., 1987a; 1987b). Similarly, further refinements of the parameters and a forecast of the circumstances of the eclipses in 1989 were made (Tholen and Buie, 1988). The maximum decrease in the luminosity of the Pluto– Charon system due to eclipses occurred in February 1987 and amounted to 0.67 magnitude. The last determined values of the

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Chapter 2 Satellites of planets

Table 2.2 The parameters of the Pluto–Charon system obtained from photometric observations during their total mutual eclipses. Parameter

Value

Root-mean-square error

Semi-major axis of Charon’s orbit, km

19640

320

0.00009

0.00038

98.3

1.3

Eccentricity Inclination to the Earth’s equator (Epoch 1950.0), deg Orbital period of Charon, d

6.387230

0.000021

Radius of Pluto, km

1142

9

Radius of Charon

596

17

0.43–0.60



0.375

0.018

2.065

0.047

Geometric albedo of Pluto Geometric albedo of Charon Average density of Pluto–Charon system, g/cm3

system parameters taken from Tholen and Buie (1988) are given in Table 2.2. The development of methods and algorithms for pre-calculating eclipses in the Pluto–Charon system was also carried out at the Sternberg astronomical institute of Moscow State University. Eclipse ephemerides for 1989 were calculated, as well as the conditions for photometric observations of eclipses in 1990 (Emelyanov, 1989; Emelyanov and Medvedev, 1989). Mulholland and Gustafson (1987) and Tholen and Hubbard (1988) examined the phenomenon of diffraction during the mutual eclipses of Pluto and Charon. However, the question of the influence of this effect on the observation results remained open. Interpretation of observations of the system based on the Pluto albedo model, which includes two spots on the surface, is presented in Marcialis (1988). As a result, we will give a final general idea of the Pluto–Charon system based on data obtained after the last epoch of mutual occultations and eclipses in this system. Pluto and Charon are two spherical celestial bodies with a total mass of (6.8 ± 0.5) · 10−9 solar masses moving in circular orbits around a common center of mass at a distance of 19130 km with a period of 6.387217 days. The planes of the orbits are almost per-

Chapter 2 Satellites of planets

pendicular to the plane of the Earth’s equator (the inclination to the plane of the equator of the 1950.0 epoch is about 98.3 degrees). The ratio of the mass of the satellite to the mass of the planet under the assumption of equal density of bodies is 0.1526. Each of the bodies rotates around its own axis with a period equal to the orbital period of Charon around Pluto. The rotation axes are perpendicular to the plane of the Charon orbit. Thus, Pluto and Charon are constantly oriented to each other by the same sides. Questions about the difference between their bodies and spherical bodies and about the difference in the eccentricity of the satellite’s orbit from zero remain open. Note that the perturbations of the Keplerian orbit of Charon have not yet been studied. The secular change in the longitude of ascending node of the orbit due to the gravitational attraction of the Sun can be simply estimated; it is 3 · 10−21 degrees per year. On the surface of Pluto there are bright polar caps of solid methane. Charon probably consists of ice and has a uniform illumination. There are several types of observations of the Pluto–Charon system. The first type is speckle-interferometric observations, which consist in measuring the angular distance between the planet and the satellite and the position angle. The second type of observation is photometric measurements of the total brightness of the system during mutual occultations and eclipses, which lasts about six years and is repeated once every 124 years. A recent period of this type is from 1985 to 1990. The Pluto–Charon system was also observed with the Hubble Space Telescope (HST). And in 1993, six images were obtained (Tholen and Buie, 1997). From these observations, new orbital parameters were determined. In contrast to the assumptions based on the tidal evolution model, an unexpectedly large value of the orbital eccentricity, equal to 0.0076 ± 0.0005, was obtained. The semi-major axis turned out to be equal to (19636 ± 8) km. The gravitational parameter of the system was inferred to be equal to (981.5 ± 1.1) km3 /s2 . Olkin et al. (2003) reported observations of the Pluto–Charon system with HST in 1998 continuously for 4.4 days. An additional 12 positions of Charon regarding Pluto were obtained with HST in 2002–2003 (Buie et al., 2006). In 2005, two new satellites of Pluto, Nikta and Hydra, were discovered, and in 2011 and 2012, two more satellites, Kerberos and Styx, were revealed. These are very small objects compared to Charon moving at distances of 40 000–60 000 km from the planet. Modern data on the orbital properties of satellites were obtained from ground-based observations and space observations

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Chapter 2 Satellites of planets

Table 2.3 Parameters of a system consisting of Pluto and its satellites. Gm is the gravitational parameter. The semi-major axis of Charon’s orbit refers to its rotation around Pluto. For the other four satellites, the semi-major axes of the orbits relate to the rotation around the barycenter of the binary system. Its orbital inclinations are given relative to the plane of Charon’s orbit. For Pluto, the period of axial rotation is given and, for satellites, periods of rotation around the planet are presented. Body

Semi-major axis

Eccentricity

km

Inclination deg

Period Gm d

km3 /s2

6.3872

869.60 ± 1.8

Charon

19596

0.00005

0.0

6.3872

105.88 ± 1.0

Styx

42413

0.00001

0.0

20.1617 0.0000 ± 0.0001

Nikta

48690

0.00000

0.0

24.8548 0.0030 ± 0.0027

Pluto

Cerberus 57750

0.00000

0.4

32.1679 0.0011 ± 0.0006

Hydra

0.00554

0.3

38.2021 0.0032 ± 0.0028

64721

with HST (Brozovic et al., 2015). These data are reproduced in Table 2.3, which also contains the rotation period and the gravitational parameter of Pluto. A new step in the study of Pluto’s satellite system was made in 2006 by launching the New Horizons automatic spacecraft towards Pluto. In mid-2015, the spacecraft reached the vicinity of Pluto. From distances less than 12 000 km, numerous images of the planet were taken, including images of Charon and other satellites. Maps of the temperatures of the planet and satellite were made, and a thin atmosphere was discovered at Pluto, consisting of nitrogen and methane molecules. A lot of data has been received from the New Horizons spacecraft. Their processing will take some time. Preliminary results are given in Stern et al. (2015). Here are the surface maps of Pluto and Charon. The physical properties of their surfaces are described. Particular attention is paid to the atmosphere of Pluto. Based on data from the New Horizons apparatus new values of the radii of Pluto and Charon were obtained: (1187 ± 4) km and (606 ± 3) km, respectively. The semi-axes of the triaxial body of Nikta were determined: 54, 41 and 36 km. The semi-axes of the ellipsoid representing the body of Hydra were obtained: 43 and 33 km.

Chapter 2 Satellites of planets

A detailed presentation of the first results can be found in Stern et al. (2018), where mainly attention is paid to the physical properties of Pluto and its satellites. The results are presented in the following sections: geology and composition, atmosphere, subsoil, size and shape of objects, morphology, composition and color of surfaces. Particular attention is paid to the origin of the Pluto system. New positional observational data, on the basis of which it would be possible to refine the satellite orbits, have not yet been published. For applications of the methods of celestial mechanics, the double planet Pluto–Charon is a unique case in the Solar System. The maximum ratio of the mass of the satellite and the mass of the planet is 0.1217, the maximum ratio of the radius of the satellite and its distance to the planet is 0.0309 and we have the double synchronism of rotation of the planet and the satellite with orbital motion.

2.5 Orbital and physical parameters of planets with satellites In the problems of studying the dynamics of the natural satellites of the planets, it is necessary to calculate the perturbations in the motion of the satellites due to the attraction of major planets. In some cases, it is also necessary to take into account the non-sphericity of the planets. To this end, it is convenient to have at hand the necessary parameters. Therefore, we give here some orbital and physical parameters of Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. Table 2.4 gives orbital parameters taken from Simon and Francou (2016). For Mars, the parameters of the VSOP2013 theory are taken, and for other planets, the parameters of the TOP2013 theory. The following tables list some physical parameters of major planets. The gravitational parameter is the product of the universal gravitational constant and body mass. It is the gravitational parameter that appears in the equations of motion of celestial bodies. It is determined from observations, more precisely than the separately considered universal gravitational constant. It should be noted that the given values of the gravitational parameters of the planets can include the masses of satellites. The values for the planet along with the satellites are called the gravitational parameters of the system. Separately, the gravitational parameter of the planet is set for the planet.

31

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Chapter 2 Satellites of planets

Table 2.4 Orbital parameters of some major planets. The inclination is measured relative to the ecliptic of the J2000 epoch. The period is set in Julian years of 365.25 days. Planet

Semi-major axis a.u.

Orbital eccentricity

Inclination to ecliptic deg

Orbital period year

Mars

1.5236793402

0.093400630

1.84972607

1.88084832678

Jupiter

5.2026032025

0.048497982

1.30328059

11.8619832216

Saturn

9.5549104300

0.055548261

2.48887405

29.4571606779

Uranus

19.2184382726

0.046384032

0.77318296

84.0204024633

Neptune

30.1104158724

0.009454315

1.76994618

164.770405189

39.54461714

0.24905026

17.1420813

247.997694064

Pluto

The parameter of the planet’s gravitational field is also necessary for constructing a model of the satellite motion. The gravitational field model is represented by the expansion coefficients of the force attraction function in a series of spherical functions. The form of the development is given in the corresponding section of Chapter 3. Major planets are almost axisymmetric, but also have an equatorial plane of symmetry. For these reasons, only terms with the coefficients J2 , J4 , J6 are taken in the expansion. These coefficients are determined from observations for a given equatorial radius of the planet Re , together with its gravitational parameter. Therefore, it is necessary to use mutually agreeing values of these parameters. Note that, for the expansion of the force function of the planet’s attraction, it is assumed that the coefficient J2 characterizing its dynamical oblateness is positive. The gravitational constants for planetary systems taken from Folkner et al. (2014) are given in Table 2.5. These are the values that were used to construct the planetary motion model DE431. The listed values include satellite masses. We note that when using constants in a particular theory, we must take the values that are consistent with each other. In particular, some other constants complete with gravitational constants of the planets should be taken from the same publication (Folkner et al., 2014): – the astronomical unit is 149597870.700 km, – the speed of light is 299792.458 km/s, – the radius of the Sun is 696000.0 km,

Chapter 2 Satellites of planets

33

Table 2.5 Physical parameters of some major planets. The mean Rm , equatorial Re , and polar Rp radii are taken from Archinal et al. (2018). The gravitational parameters of planetary systems with satellites Gm correspond to the planetary theory DE431 (Folkner et al., 2014). Planet

Rm

Re

Rp

Gm 3

km

km /s2

Mars

3389.50 3396.19 3376.20

42828.375214

Jupiter

69911.0 71492.0 66854.0

126712764.80

Saturn

58232.0 60268.0 54364.0

37940585.200

Uranus

25362.0 25559.0 24973.0

5794548.600

Neptune

24622.0 24764.0 24341.0 6836527.10058

km

Pluto



km





– the gravitational parameter of the Sun is 132712440041.939400 km3 /s2 , – the dynamical oblateness of the Sun is J2 = 2.1106088532726840· 10−7 . A large (redundant) number of significant digits in these values is sometimes necessary to adjust the initial data in the calculations. Table 2.5 gives the average, equatorial and polar values of the radii of the planets. These values are recommended by the IAU Working Group on Cartographic Coordinates and Rotational Elements—IAU WG CCRE. They are published in the next report of this working group (Archinal et al., 2018). The radius of Pluto is not given in the table; it is given above as a result of observations using the New Horizons spacecraft. When constructing a model of the motion of planetary satellites based on observations, some physical parameters of the planet are also obtained. Consider the data from several such works. In each of them, the parameter values are consistent with each other. Therefore, the values of the coefficients J2 , J4 , J6 and the accepted values of the radii of the planets Re should be used together. The obtained values of the physical parameters are given in Tables 2.6 and 2.7, where the parameter determination errors are also given. Bibliographic references of the sources of this data are as follows: for the satellites of Jupiter—(Jacobson, 2013), for the satellites of Saturn—(Jacobson et al., 2006), for the satellites of Uranus—(Jacobson, 2014), for the satellites of Neptune—(Jacobson, 2009).

977.00

34

Chapter 2 Satellites of planets

Table 2.6 Physical parameters of some major planets obtained from observations of satellites (in units of km3 /s2 ). Planet Gravitational parameter of the Gravitational parameter of the planet system Jupiter

126686536.1 ± 2.7

126712764.1 ± 2.7

Saturn

37931208.0 ± 1.0

37940585.0 ± 1.0

Uranus

5793951.3 ± 4.4

5794556.4 ± 4.3

Neptune

6835100.0 ± 10

6836527.0 ± 10

Table 2.7 Physical parameters of some major planets obtained from satellite observations. Parameters of gravitational fields. Planet

Radius Re , km

J2 ×106

J4 ×106

J6 ×106

Jupiter

71492

14695.62 ± 0.29

−591.31 ± 2.06

20.78 ± 4.87

Saturn

60330

16290.71 ± 0.27

−935.83 ± 2.77

86.14 ± 9.64

Uranus

25559

3510.68 ± 0.70

−34.17 ± 1.30



Neptune

25225

3408.43 ± 4.50

−33.40 ± 2.90



2.6 Orbital parameters of planetary satellites The motion of planetary satellites is influenced by various factors. The equations of motion are solved by numerical integration methods or by perturbation theory methods. A convenient and visual way of representing the movement is the Keplerian orbit. In the presence of perturbations, the real motion in comparison with the solution of the two-body problem is a motion in the Keplerian orbit, the parameters of which vary with time. These changes with small perturbations will be small. Therefore, the satellite motion can be characterized by some average parameters of the Keplerian orbit. The average values of the parameters will turn out to be different when averaging over different time intervals. To understand the satellite’s place in space, we are primarily interested in the average values of the semi-major axis, eccentricity and inclination of the orbit with respect to some fixed plane. Usually, an invariable plane is chosen so that the inclination of the orbit with respect to it changes minimally.

Chapter 2 Satellites of planets

For close satellites, the influence of the planet’s oblateness dominates. Under oblateness, the orbit plane slowly precesses with an almost constant inclination around the axis of symmetry of the planet. The orbit itself also slowly rotates in its plane. Very small short-period perturbations are superimposed on this motion, which slightly change the motion pattern. For distant planetary satellites, the main perturbations are due to the attraction of the Sun. In this case, the orbit precesses with an almost constant inclination with respect to the plane of the orbit of the Sun around the planet. When modeling the motion of distant satellites, the attraction of other nearby planets should also be taken into account. The mutual attraction of the satellites, as well as the combined influence of various factors having approximately the same values, significantly changes such a simplified pattern of motion along the precessing Keplerian orbit. For distant planetary satellites, perturbations from the attraction of the Sun can be very significant. In these cases, the eccentricity and inclination can vary over a fairly wide range, and their average values will significantly depend on the time interval over which the averaging of the elements is done. For these reasons, the constant values of the semi-major axis of the orbit, eccentricity and inclination can only approximately represent the satellite’s motion. Nevertheless, even approximate values of these parameters show a pattern of motion and the satellite’s position in the space around the planet. Therefore, in this book an attempt is made to show the parameters of the orbits. Since there are quite a few known planetary satellites and tables with the parameters are voluminous, we placed them in Appendix B. Below we give brief descriptions of the tables. For most satellites, motion models are constructed by numerical integrating differential equations of motion. The most advanced of them are placed in the ephemeris service. One such service is the MULTI-SAT ephemerides server (Emel’yanov and Arlot, 2008) for natural planetary satellites. It can be used via the Internet at http://www.sai.msu.ru/neb/nss/html/multisat/index. html. This server allows us to calculate the values of the elements of the Keplerian osculating orbit averaged over a given time interval. The tables with orbital parameter for some of the planetary satellites in Appendix B are formed using the MULTI-SAT server. The time intervals at which the average elements are calculated are indicated in the table headers. The ephemerides server uses by default the most advanced recent satellite motion models. It is such models that were used in

35

36

Chapter 2 Satellites of planets

the tables in the appendix. We will describe these models in a special section of Chapter 12 of this book. Note that the elements of the osculating satellite orbit are calculated from the values of the rectangular coordinates and velocity components. In calculations, the values of the gravitational parameters of the planet and satellite are used. When calculating the inclination of the orbit to the planet’s equatorial plane, the angles that determine the equatorial plane are also used. These parameters may have different versions of values. The MULTI-SAT server uses the most reliable versions. Since the calculated average elements are approximate, the data presented do not specify at what values of the indicated parameters the elements were calculated. For a number of satellites, the mean orbital parameters were taken from publications in which they were obtained on the basis of some motion models based on observations. This is indicated in the headings of the tables, where data sources are also given. In particular, for 59 distant satellites of Jupiter, the average elements were obtained in Brozovic and Jacobson (2017). This is done by determining the parameters of the precessing ellipse using the satellite motion model constructed by numerical integration of the equations of motion and refined by observation. In this paper, the mean elements are calculated over a time interval of 1000 years from 1600 to 2600.

References Archinal, B.A., Acton, C.H., A’hearn, M.F., Conrad, A., Consolmagno, G.J., Duxbury, T., Hestroffer, D., Hilton, J.L., Kirk, R.L., Klioner, S.A., McCarthy, D., Meech, K., Oberst, J., Ping, J., Seidelmann, P.K., Tholen, D.J., Thomas, P.C., Williams, I.P., 2018. Report of the IAU working group on cartographic coordinates and rotational elements: 2015. Celestial Mechanics & Dynamical Astronomy 130, 22. Baier, G., Weigelt, G., 1987. Speckle interferometric observations of Pluto and its moon Charon on seven different nights. Astronomy & Astrophysics 174 (1–2), 295–298. Binzel, R.P., Tholen, D.J., Tedesco, E.F., Buratti, B.J., Nelson, R.M., 1985. The detection of eclipses in the Pluto–Charon system. Science 228, 1193–1195. Bonneau, D., Foy, R., 1980. Speckle interferometry with the 3.60 M CFH telescope. I – Resolution of the Pluto–Charon system. Astronomy & Astrophysics 92 (1–2), L1–L4. Brozovic, M., Showalter, M.R., Jacobson, R.A, Buie, M.W., 2015. The orbits and masses of satellites of Pluto. Icarus 246, 317–329. Brozovic, M., Jacobson, R.A., 2017. The orbits of Jupiter’s irregular satellites. Astronomical Journal 153 (4), 147. 10 pp. Buie, M.W., Grundy, W.M., Young, E.F., Young, L.A., Stern, S.A., 2006. Orbits and photometry of Pluto’s satellites: Charon, S/2005 P1 and S/2005 P2. Astronomical Journal 132 (1), 290–298. Christy, J.W., Harrington, R.S., 1978. The satellite of Pluto. Astronomical Journal 83, 1005, 1007, 1008.

Chapter 2 Satellites of planets

Christy, J.W., Harrington, R.S., 1980. The discovery and orbit of Charon. Icarus 44, 38–40. Dunbar, R.S., Tedesco, E.F., 1986. Modeling Pluto–Charon mutual eclipse events. I – First-order models. Astronomical Journal 92, 1201–1209. Emelyanov, N.V., 1989. Eclipses ephemeris in the Pluto–Charon system in 1989. Astronomical Circular (1535), 27–28. In Russian. Emelyanov, N.V., Medvedev, V.G., 1989. Mutual eclipse pre-calculation in the Pluto–Charon system. Preprint GAISH 10, pp. 1–14. In Russian. Emelyanov, N.V., 1997. Two hundred years of bureau of longitudes. We and the Universe 3, 94–97. In Russian. Emel’yanov, N.V., Arlot, J.-E., 2008. The natural satellites ephemerides facility MULTI-SAT. Astronomy & Astrophysics 487, 759–765. Farinella, P., Paolicchi, P., Ferrini, F., 1979. Planet formation process as a phase transition. III – Mass distribution in the outer solar system. Moon and the Planets 21, 405–408. Folkner, W.M., Williams, J.G., Boggs, D.H., Park, R.S., Kuchynka, P., 2014. The Planetary and Lunar Ephemerides DE430 and DE431. The Interplanetary Network Progress Report, V. 42-196, Jet Propulsion Laboratory, California Institute of Technology, pp. 1–81. Harrington, R.S., Christy, J.W., 1980. The satellite of Pluto. II. Astronomical Journal 85, 168–170. Jacobson, R.A., Antreasian, P.G., Bordi, J.J., Criddle, K.E., Ionasescu, R., Jones, J.B., Mackenzie, R.A., Pelletier, F.J., Owen Jr., W.M., Roth, D.C., Stauch, J.R., 2006. The gravity field of the Saturnian system from satellite observations and spacecraft tracking data. Astronomical Journal 132 (6), 2520–2526. Jacobson, R.A., 2009. The orbits of the Neptunian satellites and the orientation of the pole of Neptune. Astronomical Journal 137 (5), 4322–4329. Jacobson, R.A., 2013. JUP310 Orbit Solution. Jet Propulsion Laboratory, California Institute of Technology. Jacobson, R.A., 2014. The orbits of the Uranian satellites and rings, the gravity field of the Uranian system, and the orientation of the pole of Uranus. Astronomical Journal 148, 76–88. Marcialis, R.L., Rieke, G.H., Lebofsky, L.A., 1987. The surface composition of Charon – tentative identification of water ice. Science 237, 1349–1351. Marcialis, R.L., 1988. A two-spot albedo model for the surface of Pluto. Astronomical Journal 95, 941–947. Mulholland, J.D., Gustafson, B.A.S., 1987. Pluto eclipses of and by Charon must be unequal. Astronomy & Astrophysics 171 (1–2), L5–L7. Olkin, C.B., Wasserman, L.H., Franz, O.G., 2003. The mass ratio of Charon to Pluto from Hubble Space Telescope astrometry with the fine guidance sensors. Icarus 164 (1), 254–259. Reinsch, K., Pakull, M.W., 1987. Physical parameters of the Pluto–Charon system. Astronomy & Astrophysics 177 (1–2), L43–L46. Simon, J.-L., Chapront-Touzé, M., Morando, B., Thuillot, W., 1997. Introduction aux éphémérides astronomiques. Supplément explicatif à la connaissance des temps. BDL, Institute of Technology, Paris. 450 pp. In French. Simon, J.L., Francou, G., 2016. Construction des théories planétaires analytiques de l’IMCCE. Notes scientifiques et techniques de l’Institut de mécanique céleste. S103. Smith, J.C., Christy, J.W., Graham, J.A., 1978. IAU Circ 3241, 1. Stern, S.A., Bagenal, F., Ennico, K., Gladstone, G.R., et al., 2015. The Pluto system: initial results from its exploration by new horizons. Science 350 (6258). id.aad1815.

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Stern, S.A., Grundy, W.M., McKinnon Wm, B., Weaver, H.A., Young, L.A., 2018. The Pluto system after new horizons. Annual Review of Astronomy and Astrophysics 56, 357–392. Sykes, M.V., Cutri, R.M., Lebofsky, L.A., Binzel, R.P., 1987. IRAS serendipitous survey observations of Pluto and Charon. Science 237, 1336–1340. Tholen, D.J., 1985. The orbit of Pluto’s satellite. Astronomical Journal 90, 2353–2359. Tholen, D.J., Buie, M.W., Binzel, R.P., Frueh, M.L., 1987a. Improved orbital and physical parameters for the Pluto–Charon system. Science 237, 512–514. Tholen, D.J., Buie, M.W., Swift, C.E., 1987b. Circumstances for Pluto–Charon mutual events in 1988. Astronomical Journal 94, 1681–1685. Tholen, D.J., Buie, M.W., 1988. Circumstances for Pluto–Charon mutual events in 1989. Astronomical Journal 96, 1977–1982. Tholen, D.J., Hubbard, W.B., 1988. No effect of diffraction on Pluto–Charon mutual events. Astronomy & Astrophysics 204 (1–2), L5–L7. Tholen, D.J., Buie, M.W., 1997. The orbit of Charon. Icarus 125 (2), 245–260.

3 Equations of motion and analytic theories 3.1 Equations of motion and coordinate systems The motion of satellites is described by differential equations. The equations of motion contain all the information about what forces and causes affect the movement. The form of equations is a very important point in the study of dynamics. On the one hand, impeccable accuracy is needed here, because with a careless statement of the problem, all subsequent work will be useless. On the other hand, the applicability and simplicity of the methods by which the equations will be solved depends on the form of the equations. The equations of motion are based on the laws of mechanics. At first, it is assumed that the mechanics used are the classical mechanics of Newton. It is possible that this will be enough. At some stage of research, the question arises of the need to use the laws of the general relativity theory, as more adequate to the real nature surrounding us. At this stage, we face conflicting requirements. Using the equations of the general relativity theory extremely complicates the task. On the other hand, the need for such use should be clearly justified. Do we need to use the general gravity theory? It is very important to determine what criteria to use. In practical modeling of motion in order to bring the model as close as possible to reality, the criterion for choosing the type of applied mechanics model should be the correspondence of the accuracy of the model to the accuracy of observations. A comparison of the accuracy of equations and the accuracy of observations is not good here. With the accuracy of observations, it is necessary to compare the accuracy of the solution of the differential equations of motion, and not the equations themselves. However, evaluating the accuracy of a solution is always very difficult. In addition, many other factors that are not contained in the equations of motion themselves are involved in modeling the quantities that are measured in the observation process. The problem of using the mechanics of general relativity can be solved by the technique that is often used in celestial mechanics in The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00008-X Copyright © 2021 Elsevier Inc. All rights reserved.

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such cases. Namely, instead of using the equations of general relativity, we provide equations of motion in the framework of Newtonian mechanics, but at the same time we introduce some fictitious perturbing forces acting on the celestial body, so that the resulting solution will be close enough to the solution of the equations of general relativity. The need for even such a simplified method should be carefully evaluated. At this stage, we restrict ourselves to the equations of Newtonian mechanics. Let us consider the planetary system with satellites, which we number with the index i = 1, 2, 3, ..., N . The motion of the system of N satellites occurs under the influence of the planet’s attraction, the mutual attraction of satellites, the attraction of other satellites not included in the first N ones, as well as under the influence of the attraction of the Sun and other planets. The Sun, other planets and the satellites will be called external bodies. We will assume that the body of the planet has finite dimensions, and its gravitational potential differs from the potential of the material point. The differential equations of motion of the planet’s satellites are initially compiled as the equations of motion of the problem for the (N + 1)th material points. It does not matter in which of them the origin of the coordinate system is located. The form of the equations does not depend on this. It can be assumed that satellites, the Sun and other planets move around the main planet, in the center of which the origin of the coordinate system is located. We will call such a system a planetocentric system. Further refinement of the equations of motion of the satellites consists in taking into account the fact that the planet, as well as other satellites, are different from material points and attract like extended bodies with complicated gravitational fields. Consider the equations of motion of N satellites with coordinates xi , yi , zi (i = 1, 2, 3, ..., N ) in a planetocentric non-rotating coordinate system. The equations of motion in vector form have the form  d 2 ri ri  = −G(m + m ) − Gmj 0 i dt 2 |ri |3 N

j =1







N  j =1

Gmj

 ij

r j +  3  3 ij | | |r j |



 rj  ij − + ij |3 |rj |3 |

 + F(t, ri ) ,

(3.1)

where G is the universal gravitational constant, mi is the mass of the satellite, m0 is the mass of the planet, and mj is the mass of the external body (Sun, planet, satellite). In the equations we introduced the notation: ri = {xi , yi , zi } is the radius vector of the

Chapter 3 Equations of motion and analytic theories

satellite with the number i, r j = {xj , yj , zj } is the radius vector of the external body with the number j ,  ij = ri − rj , and  ij = ri − r j . We have designated the acceleration vector due to the non-sphericity of the planet through F(t, ri ). This vector can clearly depend on time, because the planet with its gravitational field moves independently of the satellite. However, this term can take into account the non-sphericity of the satellite itself, if its rotation is specified. In the first sum, the prime means that the term is omitted for j = i. Other forces can be taken into account in the satellite’s motion model. These can be forces of a non-gravitational nature, for example, forces of resistance of the environment or forces of tidal friction in the body of a planet or in the body of a satellite. Then the corresponding terms on the right-hand side of the equations of motion can also depend on the components of the satellite velocity. In the equations of motion, the vectors ri (i = 1, 2, 3, ..., N ) appear as the required functions, and the vectors r j must be computable at any given point in time. The coordinates of the Sun and other planets are determined by the ephemeris of the planets. The coordinates of other satellites, in addition to the first N , are calculated on the basis of previously developed models of their motion. The main force acting on the satellite is the attraction of the planet. Therefore, it is the satellite of this planet. The main action of the planet is its attraction as a material point located in the center of mass. This main action is defined by the first term on the right side of the equations of motion. Neglecting all other influences on the satellite, except the main one, we obtain the equations of motion of the problem of two material points. In this case, the origin coincides with one of them, with the planet. For simplicity, this problem is called the two-body problem. An alternative to the equations of motion in a relative coordinate system are the equations of motion in a barycentric system. Leaving only the planet and the satellite in consideration, we again come to the equations of motion of the two-body problem. The solution of the differential equations of the two-body problem is well known. This is the law and model of the Kepler movement. We will consider such a model in the next section. For a detailed and in-depth study of the two-body problem, the education manual can be recommended (Kholshevnikov and Titov, 2007). The manner of presentation in this manual is abstract-theoretical. In particular, metric orbit spaces are constructed and their topological properties are described. Attention is paid to the expansion of coordinates in series and to the problems of determining orbits. At the end of the manual we provide a

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Chapter 3 Equations of motion and analytic theories

list of references containing links to handbooks, guides, manuals and problem books on celestial mechanics.

3.2 Keplerian motion model 3.2.1 The basic formulas of the Keplerian motion The Keplerian motion model appears as a solution to the twobody problem. Bodies are considered as material points isolated from other bodies and any other influences. They move only under the influence of mutual attraction according to Newton’s law of squares of reciprocal distances. Such a motion is also called unperturbed motion. The relative motion in the two-body problem is considered, when the origin of the reference system is placed at the center of mass of one of them, and the barycentric motion, at which the origin is placed at the center of mass of both bodies. The trajectory of motion in the two-body problem always lies in an unchanged plane. It is an ellipse, parabola or hyperbola. The central body in relative motion or the barycenter of two bodies in a barycentric motion is located in the focus of one of these curves. There are also rectilinear trajectories in the two-body problem. A point with a minimum distance to the origin of the reference system is called the pericenter, and a point with a maximum distance (in the case of elliptical motion) is called the apocenter. The apocenter may be located at infinity (hyperbolic and parabolic trajectories). The line connecting the pericenter of the orbit with its apocenter is called the line of apsides. Relative motion is described by a system of ordinary differential equations of the sixth order. Therefore, the general solution depends on six independent arbitrary constants. We consider the equations of motion of the two-body problem with masses m1 and m2 in an arbitrary non-rotating system of rectangular coordinates x, y, z. The origin of the coordinate system O is located either in the first body, or in the second, or in the barycenter of the system. For all three cases, the equations of motion will have the form d 2x μx d 2 y μy d 2 z μz = − , = − , =− 3 , 2 3 2 3 2 dt r dt r dt r  where r = x 2 + y 2 + x 2 and μ is a constant called the gravitational parameter. If the origin is placed in one of the bodies, then the equations describe the motion of the other of them, and the gravitational

Chapter 3 Equations of motion and analytic theories

parameter μ is given by the equation μ = G(m1 + m2 ), where G is the universal gravitational constant. If the origin is placed in the barycenter of the system, then the equations describe the motion of the first body with the gravitational parameter μ=G

m32 . (m1 + m2 )2

The equations of motion of the second body relative to the barycenter have the same form, but the gravitational parameter in this case is given by the equation μ=G

m31 . (m1 + m2 )2

There are seven first integrals of the equations of motion containing seven arbitrary integration constants, five of which are independent. The missing sixth independent arbitrary constant is a constant that determines the position of the body in orbit at a given time point. The trajectory of motion in the two-body problem is described by the parameters (elements) of the Keplerian orbit. Since we know that the motion of bodies occurs in a certain unchanged plane, we consider the coordinate system ξ , η, ζ with the origin at the point O, whose axis ξ is directed to the pericenter of the orbit, and the axis η is located in the plane of motion so that the radius vector of the body rotates from the axis ξ towards the axis η. We direct the third axis ζ so that the coordinate system is right. The coordinate system ξ , η, ζ is called the orbital system. The transition formulas from the Oξ ηζ orbital coordinate system to the Oxyz coordinate system are: x = Px · ξ + Qx · η + Rx · ζ, y = Py · ξ + Qy · η + Ry · ζ,

(3.2)

z = Pz · ξ + Qz · η + Rz · ζ, where Pi , Qi , Ri (i = 1, 2, 3) denotes the direction cosines of the line of apsides, the direction cosines of the perpendicular to the line of apsides in the orbit plane, and the direction cosines of the perpendicular to the plane of the orbit, respectively. The direction cosines are usually expressed through three angles of rotation of

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Chapter 3 Equations of motion and analytic theories

Figure 3.1. The geometry of coordinate systems and the orbital plane of Keplerian motion.

the orbital coordinate system Oξ ηζ with respect to the coordinate system Oxyz:  is the longitude of the ascending node of the orbit, ω is the angular distance of the pericenter from the ascending node of the orbit, i is the inclination of the orbit to the main plane. The expressions for the direction cosines are of the form Px Py Pz Qx Qy Qz Rx Ry Rz

= cos ω · cos  = cos ω · sin  = sin ω · sin i, = − sin ω · cos  = − sin ω · sin  = cos ω · sin i, = sin  · sin i, = − cos  · sin i, = cos i.

− sin ω · sin  · cos i, + sin ω · cos  · cos i, − cos ω · sin  · cos i, + cos ω · cos  · cos i,

The orientation of the orbit in space can be seen in Fig. 3.1, where the angles i, ω, and  are shown. The trajectory of motion in the two-body problem in the orbital coordinate system is described by the following relation: μ · r = c2 − f · ξ,

Chapter 3 Equations of motion and analytic theories

 where r = ξ 2 + η2 is the modulus of the radius vector of the body, c is the constant of the area integral, and f is the constant of the Laplace integral. The degenerate case of rectilinear motion in the two-body problem, when the constant of the area integral c is zero, we do not consider. Here we assume that c > 0, and consider the equation of the trajectory in polar coordinates. We introduce the polar coordinates of the object r and v in the orbit plane using the formulas ξ = r cos v,

η = r sin v,

where the angle v in the Keplerian motion is called true anomaly and is counted from the pericenter in the positive direction of the point. The focal equation of the orbit in polar coordinates has the form p r= , 1 + e cos v where p = c2 /μ, e = f/μ. Moreover, p is called the focal parameter of the orbit, and e is called its eccentricity. From the first integrals of the equations of motion in the twobody problem, it follows that the energy constant depends on the parameters introduced as follows: h=

μ2 2 (e − 1). c2

The change in the polar angle v in time is described by the differential equation  2 dv p = c. 1 + e cos v dt The solution to this equation is written in various forms depending on the type of movement. The type of motion is determined by the eccentricity value e. Therefore, the dependence of the coordinates on time t will be different. The following types of orbits exist. 1) C i r c u l a r orbit: e = 0, 2) E l l i p t i c a l orbit: 0 < e < 1, 3) P a r a b o l i c orbit: e = 1, 4) H y p e r b o l i c orbit: e > 1,

h < 0, r = const. h < 0. h = 0. h > 0.

Since we study the motion of planetary satellites on finite trajectories, we restrict ourselves to considering only circular and

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Chapter 3 Equations of motion and analytic theories

elliptical motions in the two-body problem. Then the eccentricity is e < 1. The circular motion will be at e = 0. In circular and elliptical movements, the semi-major axis of the orbit a is considered. In this case, the focal parameter and the energy integral are expressed through the semi-major axis as follows: μ p = a(1 − e2 ), h = − . a From the focal equation in polar coordinates it follows that the minimum distance of the body to the origin is a(1 − e), and the maximum is a(1 + e). To find r and v as functions of time, we introduce the eccentric anomaly E as an auxiliary variable:  r sin v = a 1 − e2 sin E, r cos v = a(cos E − e), r = a(1 − e cos E). The dependence of E on time t is described by the relations E − e sin E = M, M = n(t − t0 ) + M0 , n=

μ/a 3 ,

t0 is the initial moment of time (epoch) and M0 is an arbitrary integration constant. The value of M is called the mean anomaly, n is called the mean motion, and M0 is called the mean anomaly at epoch. The equation E − e sin E = M with respect to the eccentric anomaly E is called the Kepler equation. Its numerical solution is usually carried out by the method of successive approximations. The optimal construction of approximations is described below. Note that when M changes by 2π, the angles E and v also change by 2π. It is similar to the case of changing to π. The difference between them is that M varies uniformly in time, and E and v with the variable angular velocity. Only in the particular case of circular motion (e = 0) do all three angles coincide. In the practice of calculating coordinates in Kepler motion, the time function u = v + ω is sometimes used, which is called the latitude argument of the object. In fact, u is the central angle in the

Chapter 3 Equations of motion and analytic theories

plane of the orbit between the direction of the moving body and the direction of the ascending node of the orbit. We also use the value λ = v + ω + , called the orbital longitude, which is the sum of two angles u and  counted in different planes. We also consider the quantity λ = M + ω + , which is called the mean longitude. The orbital period T of the body is related to the mean motion n by the relation T=

2π . n

The initial elements of an elliptical orbit in the two-body problem, which determine the motion in the plane of the orbit, are three parameters n the mean motion, e the eccentricity, M0 the mean anomaly at epoch. There is an important relation between the mean motion n and the semi-major axis a n2 a 3 = μ, which corresponds to Kepler’s third law and has important practical value. If the mean motion n and the semi-major axis a are independently determined in the relative motion of two bodies from observations, then we immediately have the sum of the masses of two bodies μ m1 + m2 = . G Instead of n, we can set the semi-major axis a as the orbit parameter. However, it should be noted that the observed position of the body in orbit is determined by the mean anomaly M. Therefore, after observing several revolutions of the body in its orbit, the mean motion n is determined more precisely from the relation n=

M − M0 , t − t0

than the semi-major axis a. As parameters of the Keplerian orbit, we also consider the parameter = ω + , which is called the longitude of pericenter, as well as the parameter λ0 = M0 + ω + , which is called the mean longitude at the epoch. Instead of the mean anomaly in the epoch M0 , we sometimes set the dynamic time moment t  of passage by the body of the peri-

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Chapter 3 Equations of motion and analytic theories

center of the orbit. In other words, n(t  − t0 ) + M0 = 0. Consider the best way to solve the Kepler equation. In the classical method of successive approximations, iterations are performed according to the rules: 1. In the zeroth approximation, it is assumed that E0 = M. 2. Sequentially calculate En+1 = M + e sin En for n = 0, 1, . . . while |En+1 − En | > ε, where ε is the required accuracy of calculations. The method of successive approximations by Danby is more effective (Danby, 1995). Iterations are performed according to the scheme: 1. In the zeroth approximation, it is assumed that E0 = M + 0.85e. 2. Sequentially calculate En+1 = En − −

(M + e sin En − En )2 En − 2(M + e sin En ) + M + e sin(M +e sin En )

for n = 0, 1, . . . , while |En+1 − En | > ε.

3.2.2 Calculation of coordinates in elliptical Keplerian motion In some cases, instead of converting coordinates from the orbital system to a given system x, y, z, as described above (3.2), it is convenient to use the latitude argument u = v + ω and calculate the coordinates in a given system using the following formulas: x = r(cos u cos  − sin u sin  cos i), y = r(cos u sin  + sin u cos  cos i), z = r sin u sin i.

3.2.3 Calculation of velocity in elliptical Keplerian motion After the distance r and the true anomaly v are obtained, we can calculate the velocity modulus V , the radial velocity component Vr and the transverse velocity component Vn from the relations   2 1 − , V2 =μ r a

Chapter 3 Equations of motion and analytic theories

Vr = Vn =

μ e sin v, p

μ (1 + e cos v). p

Next we can find the rate of change of rectangular coordinates by the formulas x˙ =

x Vr + (− sin u cos  − cos u sin  cos i) Vn , r

y˙ =

y Vr + (− sin u sin  + cos u cos  cos i) Vn , r z˙ =

z Vr + cos u sin i Vn . r

3.2.4 Calculation of the partial derivatives of coordinates and velocity components with respect to elements of the Keplerian orbit in elliptical motion In practical celestial mechanics, the problems of refining the motion parameters of celestial bodies from observations are solved. As parameters, elements of the Keplerian orbit can be used. Orbit refinement is done by differential refinement using the least squares method. To construct the conditional equations, partial derivatives of the coordinates and velocity components with respect to the orbital elements are needed. We give here the formulas for these derivatives under the assumption that the coordinates and velocity are related to the parameters and time by the formulas of the elliptic Keplerian motion. The formulas follow in the order needed for the calculations. First, auxiliary values are calculated, c1 = sin i sin ,

c2 = − sin i cos ,

c3 = cos i,

l1 = cos ω cos  − sin ω sin  cos i. l2 = cos ω sin  + sin ω cos  cos i. l3 = sin ω sin i, r=



x 2 + y 2 + z2 ,

cos v =

l1 x + l2 y + l3 z , r

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Chapter 3 Equations of motion and analytic theories

sin v =

(c2 l3 − c3 l2 )x + (c3 l1 − c1 l3 )y + (c1 l2 − c2 l1 )z , r

a1 =

1 + e cos v , 1 − e2 r2 =

r1 = a1 cos v −

e , 1 − e2

n a1 sin v (1 + e cos v + e2 ) , (1 − e2 )3/2

p1 = −

μ , n r3

2 , 3n

q1 = 

p2 = −a1 cos v,

 1 q2 = a1 + sin v. 1 − e3

Then the required derivatives are found, ∂x x˙ = (t − t0 ) − q1 x, ∂n n

∂y y˙ = (t − t0 ) − q1 y, ∂n n

∂ x˙ 1 = p1 x(t − t0 ) + q1 x, ˙ ∂n 2

∂z z˙ = (t − t0 ) − q1 z, ∂n n

∂ y˙ 1 = p1 y(t − t0 ) + q1 y, ˙ ∂n 2

∂ z˙ 1 = p1 z(t − t0 ) + q1 z˙ , ∂n 2 ∂x = z sin , ∂i

∂y = −z cos , ∂i

∂z = y cos  − x sin , ∂i

∂ x˙ = z˙ sin , ∂i

∂ y˙ = −˙z cos , ∂i

∂ z˙ = y˙ cos  − x˙ sin , ∂i

x˙ ∂x = , ∂M n ∂ x˙ = p1 x, ∂M

∂y y˙ = , ∂M n

∂z z˙ = , ∂M n

∂ y˙ = p1 y, ∂M

∂ z˙ = p1 z, ∂M

∂x = c2 z − c3 y, ∂ω

∂y = c3 x − c1 z, ∂ω

∂z = c1 y − c2 x, ∂ω

∂ x˙ = c2 z˙ − c3 y, ˙ ∂ω

∂ y˙ = c3 x˙ − c1 z˙ , ∂ω

∂ z˙ = c1 y˙ − c2 x, ˙ ∂ω

∂x ∂x = q2 + p2 x, ∂e ∂ω

∂y ∂y = q2 + p2 y, ∂e ∂ω

∂z ∂z = q2 + p2 z, ∂e ∂ω

Chapter 3 Equations of motion and analytic theories

∂ x˙ ∂ x˙ ˙ = q2 + r2 x + r1 x, ∂e ∂ω

∂ y˙ ∂ y˙ ˙ = q2 + r2 y + r1 y, ∂e ∂ω

∂ z˙ ∂ z˙ = q2 + r2 z + r1 z˙ , ∂e ∂ω ∂x = −y, ∂

∂y = x, ∂

∂z = 0, ∂

∂ x˙ = −y, ˙ ∂

∂ y˙ = x, ˙ ∂

∂ z˙ = 0. ∂

In deriving these formulas, it was assumed that the independent parameters of the orbit are the six elements n, e, i, M0 , ω, and , the designations of which are explained in the previous section. Moreover, derivatives with respect to M0 are equal to derivatives with respect to M. In research practice, the problem of independent determination of the mean motion n and the semi-major axis a of the orbit from observations is often considered. This happens in cases where the mass of the attracting center is not known, and it is desirable to determine it from observations. Then partial derivatives of the coordinates or velocity components with respect to the elements n and a, considered as independent parameters, are needed. In these cases, in the above formulas for derivatives with respect to n, the second terms should be discarded, and derivatives with respect to a should be calculated according to the following equations: ∂x x ∂y y ∂z z ∂ x˙ x˙ ∂ y˙ y˙ ∂ z˙ z˙ = , = , = , =− , =− , =− . ∂a a ∂a a ∂a a ∂a 2a ∂a 2a ∂a 2a

3.2.5 Keplerian motion formulas with respect to nonsingular elements (Lagrange elements) Keplerian motion is the simplest law of motion of celestial bodies. The expressions for the coordinates and velocity components as functions of time follow from the general solution of the equations of motion of the two-body problem. The general solution also depends on six independent arbitrary constants. The choice of independent arbitrary constants is not unique. In the previous sections, elements of the Keplerian orbit, which have a clear geometric meaning, were considered as arbitrary constants. However, in a number of practical problems, the choice of Keplerian elements as the parameters of the orbit leads to a loss of accuracy in calculations with a limited number of significant digits in the values of the required variables. When applying the perturbation theory, problems arise associated with the unequal contribution of various terms in the expressions for perturbations

51

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Chapter 3 Equations of motion and analytic theories

of Keplerian elements. Ultimately, this also leads to a loss of accuracy of the theory. Such problems arise when the elliptical orbit is very close to circular one. A circular orbit is a degeneration of an elliptical orbit as the eccentricity tends to zero. A similar situation arises with very small inclinations of the satellite’s orbit. In these cases, the angular distance of the pericenter from the ascending node of the orbit and the longitude of the node are determined from observations with reduced accuracy with constant accuracy of the observations themselves. To overcome the indicated difficulties with small eccentricities and inclinations of the Keplerian orbit, the Lagrange elements selected as independent arbitrary integration constants in the general solution of the equations of the two-body problem allow. We consider below formulas that allow directly calculating the rectangular coordinates of the body directly from the Lagrange elements. The elements of the orbit are associated with an arbitrary nonrotating system of rectangular coordinates x, y, z, the origin of which is either placed in the center of mass of two bodies, or combined with one of the bodies. To introduce the Lagrange elements, we use the above notation for Keplerian elements of the orbit: n the mean motion, radian/time unit; e the eccentricity, dimensionless units; i the inclination (dihedral angle between the orbit plane and the main plane Oxy), radians; M0 the mean anomaly at the epoch (the value of the mean anomaly M at the initial moment of time—epoch), radians; ω the angular distance of the pericenter from the ascending node of the orbit, radians;  the longitude of the ascending node of the orbit (angle in the plane Oxy between the axis x and the line of nodes), radians; t0 the initial moment of time, the epoch of the elements; t the current moment of time at which the coordinates of the body are calculated. In some cases, instead of the mean motion of n, the semi-major axis of the orbit a associated with the n relation is considered as the initial parameter of the orbit a 3 n2 = μ, where μ the gravitational parameter of two bodies.

(3.3)

Chapter 3 Equations of motion and analytic theories

The mean anomaly M in any case is calculated by the formula M = n(t − t0 ) + M0 .

(3.4)

We assume that, for calculating the rectangular coordinates at a given time moment, five elements of the Kepler orbit a, e, i, ω,  and the mean anomaly M are known. The Lagrange elements are the quantities a, λ, k, h, q, p, five of which are determined by the relations λ = M + ω + , k = e cos(ω + ), h = e sin(ω + ), q = sin 2i cos , p = sin 2i sin .

(3.5)

As indicated above, the value λ is called the mean longitude and is a linear function of time. If the Lagrange elements a, λ, k, h, q, p are given, then the rectangular coordinates x, y, z and the velocity components x, ˙ y, ˙ z˙ can be calculated using the following sequence of formulas. First we calculate S = sin λ, C = cos λ, k  = k C + h S, h = k S − h C.

(3.6)

Next, we solve the equations by iterations, Cν = cos ν, Sν = sin ν, ν = h Cν + k  Sν ,

(3.7)

assuming that in the zeroth approximation ν = h . We also calculate the auxiliary values ν , √ 1 + 1 − k 2 − h2

(3.8)

Sν − ν  k  + h Cν − ν  h − k   , C = , 1 − k  Cν + h Sν 1 − k  Cν + h Sν

(3.9)

ν =

S =

Sλ = SC  + CS  , Cλ = CC  − SS  .

(3.10)

Now the central distance r and the rectangular coordinates of the body x, y, z can be obtained by the formulas r=

a (1 − k 2 − h2 ) , 1 + k Cλ + h S λ

(3.11)

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Chapter 3 Equations of motion and analytic theories

x = r Cλ (1 − 2p 2 ) + 2 r Sλ p q, y = r Sλ (1 − 2q 2 ) + 2 r Cλ p q,  z = 2 r 1 − p 2 − q 2 (q Sλ − p Cλ ).

(3.12)

To calculate the velocity components, it is necessary to find auxiliary quantities Vr = Vn =

μ (k Sλ − h Cλ ), a(1 − k 2 − h2 )

(3.13)

μ (1 + k Cλ + h Sλ ), a(1 − k 2 − h2 )

(3.14)

Rx = 2 Cλ p q − Sλ (1 − 2 p 2 ), Ry = Cλ (1 − 2 q 2 ) − 2 Sλ p q,  Rz = 2 1 − p 2 − q 2 (q Cλ + p Sλ ).

(3.15)

After that, the velocity components x, ˙ y, ˙ z˙ are calculated using the relations x˙ = xr Vr + Rx Vn , y˙ = yr Vr + Ry Vn ,

(3.16)

z˙ = rz Vr + Rz Vn .

3.2.6 Examples of using Lagrange elements We give two examples here, when the Lagrange elements found successful application. The first example is the theory of secular perturbations of the planets, constructed by Lagrange. In the perturbing function, only secular terms (independent of the longitudes of the planets) were saved. Considering the smallness of the eccentricities and mutual inclinations of the orbits of the major planets of the Solar System, the secular part of the perturbing function was expanded in a power series with respect to the eccentricities and mutual inclinations, and the terms were saved in the expansion to the second degree inclusively. In this case, the semi-major axes of the orbits of the planets were considered to be unchanged. Regarding the Lagrange elements, it was possible to use linear homogeneous differential equations with constant coefficients. The solution to these equations was obtained by Lagrange in the form of a sum of trigonometric functions with respect to time-linear arguments.

Chapter 3 Equations of motion and analytic theories

This made it possible to describe the evolution of planetary orbits over large time intervals and to discover interesting properties of the mutual perturbations of the planets. A good description of the Lagrange method can be found in the monograph by Subbotin (1968). The second example is actually the application of the Lagrange method to construct an analytical theory of the motion of the main satellites of Uranus (Laskar and Jacobson, 1987). Until recently, this theory was the only means of obtaining the most accurate ephemeris of the main satellites of Uranus.

3.3 Force function of attraction of a non-spherical planet 3.3.1 Force function expansion The influence of the planet non-sphericity is taken into account using the expansion of the force function of attraction of a non-spherical planet in a series of spherical functions. The expansion coefficients depend on the coordinate system relative to the planet’s body. Usually, a coordinate system is used, associated with the body of the planet and the main plane, which coincides with its equator. The equator of the planet is associated with the axis of its rotation. It is assumed that the axis of rotation is unchanged in the body of the planet. The bodies of major planets turn out to be close to axisymmetric bodies rotating around the axis of symmetry. Therefore, the main plane of the coordinate system in which the expansion of the force function of its attraction is carried out, is chosen so that the third axis (axis z) coincides with the axis of symmetry of some body, similar in shape to the planet. In the literature, there are various forms of expansion of the force function of the attraction of planets. Here is the formula recommended by Commission 7 of the International Astronomical Union. It has the form ∞   r0 n Gm0  Jn Pn (sin ϕ)+ U (r, ϕ, λ ) = 1− r r n=2

n  ∞   r0 n (k) + Pn (sin ϕ)(Cnk cos kλ + Snk sin kλ ) , (3.17) r n=2 k=1

where r, ϕ and λ are the central distance, latitude and longitude of a point, respectively, in the coordinate system associated with the (k) planet, Pn (sin ϕ) are the Legendre polynomials, and Pn (sin ϕ) are

55

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Chapter 3 Equations of motion and analytic theories

the associated functions Legendre whose argument is sin ϕ. The value of the constant r0 is fixed in advance equal to the radius of the circle closest to the equatorial section of the planet. Then the constant coefficients Jn , Cnk , Snk determine the planet’s gravitational field. Their values are determined from observations of various types. The expansion terms with coefficients Jn are called zonal harmonics. In much work, in the expansion of the force function of attraction of a non-spherical body, instead of the coefficients Jn , Cn0 = −Jn is used. Why it is customary to take the minus sign in the standard formula (3.17) before the zonal part of the expansion is explained below.

3.3.2 Attraction in models and for real bodies Important properties of expansion members in (3.17) should be noted. If the body is strictly axisymmetric and the main plane of the coordinate system is perpendicular to the axis of symmetry, then only the zonal part with coefficients Jn remains in the expansion. If, in addition, the body also has a plane of symmetry passing through the origin perpendicular to the axis of symmetry, then only zonal harmonics with even numbers remain in the expansion. It is known that major planets are oblate at the poles. For an oblate body J2 > 0. That is why in the standard formula (3.17), the minus sign should be written before the zonal part of the expansion. Then there will be no need to set the value of the coefficient J2 for major planets with a minus sign. Much work studies the surface shape of bodies using different models. The shape of the body indirectly affects the noncentrality of its gravitational field, however, the field substantially depends on the distribution of masses inside. In particular, a body oblate from the outside can create an elongated gravitational field. A common and convenient model is the model of a homogeneous triaxial ellipsoid. The expansion of the force function of a homogeneous triaxial ellipsoid contains an infinite number of terms, however, any coefficient with uneven n or uneven k in the expansion (3.17) is equal to zero, and all coefficients of Snk are also equal to zero. Formulas that relate the coefficients of some terms of the expansion of the force function with the dimensions of a uniform ellipsoid may be useful. Let the expansion be constructed so that the axes of the coordinate system x, y, z associated with the body are directed along the axes of the ellipsoid, and the half-axes are

Chapter 3 Equations of motion and analytic theories

a, b, c, respectively. Then the coefficients of the first few terms are expressed through the semi-axes of the ellipsoid as follows: −J2 = C20 =

−J4 = C40 = 3

C42 =

2c2 − (a 2 + b2 ) a 2 − b2 , C22 = , 2 10r0 20r02

3(a 4 + b4 ) + 8c4 + 2a 2 b2 − 8c2 (a 2 + b2 ) , 280r04

(a 2 − b2 )(2c2 − a 2 − b2 ) (a 2 − b2 )2 , C = . 44 280r04 2240r04

Potential theory is developed in fundamental work (Kondratyev, 2003; 2007). The expansions of gravitational potentials for various models of bodies are considered. Now consider the gravitational fields of major planets. Note that, for Jupiter, Saturn, Uranus and Neptune, only the values of the coefficients J2 , J4 , J6 are known with acceptable accuracy. When writing the equations of motion (3.1), the choice of the direction of the coordinate axes is not fixed at all. Usually an attempt is made to connect the directions of the axes with the plane of the Earth’s equator of the J2000 epoch, that is, with the International Celestial Reference Frame (ICRF). It should be noted that this relationship cannot be made by writing equations. The binding of the system in which the satellite coordinates are measured to the ICRF can only be done by refining the satellite’s motion parameters so that its model motion is closest to the observation results counted in the ICRF system. Only the proximity of the observation results to those in the ICRF system and the further refinement of satellite orbits based on observations can ensure the proximity of the coordinate system to the ICRF system. The values of the expansion coefficients of the force functions of attraction of some large planets are given in Sect. 2.5 of Chapter 2. In the equations of motion of satellites (3.1) the term F(ri ) appears, a function that defines the acceleration vector due to the non-sphericity of the planet. The argument to this function is the satellite coordinates in the coordinate system x, y, z. However, in the expansion of the force function (3.17), the coordinates appear in the system associated with the body, in particular, with the equator of the planet. The rectangular coordinates in this system are denoted by x, y, z. The relationship with the coordinates appearing in the expression (3.17) is obvious: x = r cos ϕ cos λ , y = r cos ϕ sin λ , z = r sin ϕ.

57

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Chapter 3 Equations of motion and analytic theories

To calculate the right-hand sides of the equations of motion (3.1), the acceleration components F = {Fx , Fy , Fz } in the coordinate system x, y, z, rigidly connected with the body. This is done by calculating the partial derivatives        ∂ Gm0 ∂ Gm0 ∂ Gm0 F= U− , U− , U− . ∂x r ∂y r ∂z r Then the components of the vector F are transferred to the coordinate system x, y, z using the transition matrix {Fx , Fy , Fz }T = R0 {Fx , Fy , Fz }T . Here, when recording coordinates, we omit for brevity the index denoting the satellite number. The matrix R0 is directly determined by the celestial coordinates of the planet’s pole α0 , δ0 and the angle of its rotation W , expressed in the coordinate system x, y, z. If ICRF is adopted as the main coordinate system, then the parameters α0 , δ0 , W can be taken from the publication of the report of the Working Group on Cartographic Coordinates and Rotation Elements of the International Astronomical Union (Archinal et al., 2018). These parameters are also given in Appendix F of this book. In the particular case when the planet is considered strictly axisymmetric, the x axis can be directed along the line of intersection of the planet’s equator with the plane (x, y). For definiteness, this should be done so that the axis x is directed to the ascending node of the plane x, y relative to the plane (x, y). In this particular case, the matrix R0 will have the form ⎞ ⎛ − sin α0 − cos α0 sin δ0 cos α0 cos δ0 R0 = ⎝ cos α0 (3.18) − sin α0 sin δ0 sin α0 cos δ0 ⎠ . 0 cos δ0 sin δ0 Since the planets Jupiter, Saturn, Uranus, Neptune and some other planets are almost axisymmetric, the question often arises of which of the poles of the planet is considered the north. To avoid confusion, the Working Group on Cartographic Coordinates and Rotation Elements of the International Astronomical Union decided that the north pole of the planet is the pole that lies on the north side with respect to the so-called unchanged plane of the Solar System (Archinal et al., 2018). The coordinates of the pole of the invariable plane of the Solar System (perpendicular to the plane) in the ISRF system of the J2000 epoch are equal: right ascension α0 = 273.85 degrees and declination δ0 = 66.99 degrees. Table 3.1 gives the celestial coordinates of the poles of Jupiter, Saturn, Uranus and Neptune. As can be seen in this table, the pole precession is known for Jupiter and Saturn.

Chapter 3 Equations of motion and analytic theories

59

Table 3.1 Pole coordinates in the ISRF system of the J2000 epoch for some major planets. The argument T is the time in the Julian centuries (36525 days each) from the J2000 epoch. Planet Right ascension α0 , deg

Declination δ0 , deg

Jupiter

268.056595 − 0.006499 T

64.495303 + 0.002413 T

Saturn

40.589 − 0.036 T

83.537 − 0.004 T

Uranus

257.311

−15.175

Neptune 299.36

43.46

Note that the north pole of Uranus is directed to the southern hemisphere of the celestial sphere, since the declination of the pole is negative. However, the planet rotates around this north pole in the opposite direction (from the y axis to the x axis), and the angular velocity vector is directed to the south pole. We present here the expressions for the components of the vector F in the planet equatorial coordinate system x, y, z taking into account only zonal harmonics with coefficients J2 and J4 :     r02 r04 z2 z4 z2 3 5 Fx = Gm0 J2 5 5 2 − 1 x + Gm0 J4 7 63 4 − 42 2 + 3 x, 2 8 r r r r r r2 3 Fy = Gm0 J2 05 2 r r2 3 Fz = Gm0 J2 05 2 r

   r04 z2 z4 z2 5 5 2 − 1 y + Gm0 J4 7 63 4 − 42 2 + 3 y, 8 r r r r





   r04 z2 z4 z2 5 5 2 − 3 z + Gm0 J4 7 63 4 − 70 2 + 15 z. 8 r r r r

When calculating the right-hand sides of the equations, the given coordinates are the satellite coordinates in the coordinate system x, y, z, and the coordinates x, y, z must be substituted into the previous formulas. These coordinates can be calculated using the transformation {x, y, z}T = R0 −1 {x, y, z}, where the matrix R0 is determined by the relationship (3.18). Here, instead of R0 −1 , we can use the transposed matrix R0 T , since R0 −1 = R0 T in force of orthogonality of these matrices.

60

Chapter 3 Equations of motion and analytic theories

3.4 An approximate account of the influence of the main satellites on the motion of distant satellites of the planet Let us consider here one technique that is used to take into account the influence of the attraction of the main satellites of Jupiter, Saturn, Uranus and Neptune in modeling the motion of distant satellites of these planets. The distances of distant satellites to the planet significantly exceed the size of the almost circular orbits of the main satellites, which move almost in the plane of the equator of the planet. It can be approximately assumed that the orbits of the main satellites are circular and are located in the plane of the equator of the planet. The periods of revolution of the main satellites around the planet are much shorter than the periods of revolution of distant satellites. We can average over time the movement of the main satellites, taking into account their influence on the distant satellites of the planet. Then we replace the real main satellites with infinitely thin massive circular rings located in the plane of the equator of the planet. The masses of the rings will be equal to the masses of the corresponding satellites, and the radii are equal to the radii of their circular orbits. In celestial mechanics, such rings are called Gaussian rings after K.F. Gauss, who first introduced them into consideration. Knowing the mass mc and the radius rc of the ring, one can easily find the force function Uc of its attraction in the form of expansion in spherical functions. We obtain

∞   r0 n (c) Gmc Jn Pn (sin ϕ) , Uc = 1− r r n=2

(c)

where the coefficients Jn are found from the relation Jn(c) = −

rcn Pn (0). r0n

Moreover, for uneven values of n it turns out that Pn (0) = 0, and for even n = 2k, where k is an integer, P2k (0) are calculated by the formula (2k!) P2k (0) = (−1)k 2k . 2 (k!)2 This technique of replacing the main satellites with attracting rings in the motion model of distant satellites was used in Emelyanov (2005); Emel’yanov and Kanter (2005), where it was

Chapter 3 Equations of motion and analytic theories

shown that, with the accuracy of modern ground-based observations, this replacement is quite acceptable.

3.5 Various approaches and methods for constructing motion models of planetary satellites The motion of the satellites of the planets is described by the law of motion, which is defined by functions representing the coordinates of the satellite as a function of time. We can obtain these functions in many ways. The methods for their presentation are also different. Usually, functions are understood as analytical expressions containing operations of addition, multiplication, division, elementary functions: logarithm, trigonometric functions, and also special functions. However, the law of motion can be set in a tabular description of functions: numerical values of coordinates for a number of time moments. Analytical methods immediately give a family of solutions defined by various parameter values that are literally included in the formulas. All models of family motion are described by the same formulas. The motion parameters are arbitrary constants of the analytical solution of differential equations. The choice of arbitrary integration constants can be done in various ways. One of them is the so-called initial conditions of motion, i.e., the coordinates and velocity components at a certain initial tine moment t0 . We can use the analytical solution as follows: we set the motion parameters once. Then we can define any set of time points. After substituting it into the analytical functions representing the solution of the equations of motion, we obtain the coordinates and velocity components. At each time point, the operation is the same. Therefore, the calculation time is proportional to the number of moments for which we need to know the coordinates, and does not depend on the interval between the initial and final moment. Note that obtaining analytical solution formulas is a very complex and time-consuming process. However, one thing is good, we have to do it only once. Since it is impossible to obtain an exact solution of the equations, except for a few simple cases, the solution always turns out to be approximate. The accuracy depends on how much we have developed an analytical solution. The more accurately a solution is needed, the more laborious is the process of obtaining it. The limit is imposed by the possibility of producing analytical calculations with huge formulas.

61

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In this matter, methods of computer algebra help well. However, these possibilities are not unlimited. An analytical solution to the differential equations of motion is obtained by perturbation theory methods. These methods are very diverse. Some of them and models of planetary satellites motion constructed in this way are considered below. A fundamentally different method is the numerical integration of equations. The main approach is as follows. We determine the coordinates and components of the satellite velocity at a certain initial time moment. Next, we select a certain time moment that is very close to the initial one, and by the approximate method we calculate the coordinates and velocity for this new moment. There are many ways to do this. Their main property is that they are all approximate, but the closer the given time moment is to the initial one, the more accurate is the result. After the coordinates and velocity are found at a new time point close to the initial one, this moment is taken as the initial one and the process repeats. So, step by step in time, we get to the time point at which we needed to know the position and velocity of the celestial body. We obtained the required result, however, for this we had to calculate the coordinates and velocity for many intermediate time moments. We need to find a result that will achieve the required time moment. So the computation time for the numerical integration of the differential equations of motion is proportional to the number of steps, and the number of steps is the greater, the better the accuracy we need. We can improve accuracy to some extent by decreasing the step size. However, a situation arises when the number of steps is so large that the accumulation of errors in calculations due to rounding errors in arithmetic operations begins to prevail over the influence of errors in obtaining a solution at every single step. As a result, we have the ultimate accuracy, which is obtained with some optimal step. This ultimate accuracy can be improved only by increasing the accuracy of the representation of numbers in the computer, thereby reducing rounding errors. Some insignificant improvement in accuracy can be achieved by improving computational methods in one step. The principal advantage of the methods of numerical integration of equations is that for their implementation we need to know only the expressions of the right-hand parts of differential equations. The fundamental drawback of these methods is the very time-consuming. Whenever we need to get a new solution, we have to start the process of numerical integration.

Chapter 3 Equations of motion and analytic theories

To date, many methods have been developed for the numerical integration of equations of motion. Their main properties of numerical integration are discussed below in Chapter 4. One of the drawbacks of numerical integration methods is that we need to somehow store all coordinate values at intermediate time points. These time points are fixed during numerical integration. There are usually a lot of them. If then we need to find the coordinates at some time point that does not coincide with any of those that were in the process of numerical integration, we will have to come up with something. The way in which the results of numerical integration are usually stored and how they are used is discussed below in a special section of Chapter 4. To date, extremely many methods have been developed for constructing models of motion of celestial bodies. An acceptable review of all of these methods would be extremely large. This book covers only a few of them. The choice of what is worth considering is justified only by the experience of the author.

3.6 A model of motion of a satellite of an oblate planet based on the solution of the generalized problem of two fixed centers Major planets are characterized by the oblateness of their figures. This is the most important non-sphericity factor. With the external oblate form of the planet, its gravitational field differs from the field of the material point. Therefore, the expansion of the force of attraction function (3.17) after the main term Gm r is dominated by the second zonal harmonic with coefficient J2 . The expansion was specially written in such a form that this coefficient was positive for the major planets of the Solar System. In celestial mechanics, various attempts have been made to take into account the second zonal harmonic in solving the differential equations of satellite motion. In a number of papers this is called the main problem. The model problem of the motion of a material point in the field of attraction of two fixed centers has long been considered. It is easy to see that the gravitational field of two stationary centers is close in structure to the field of an elongated body. However, all major planets are oblate. The force function of attraction of two fixed centers has the form W=

Gm1 Gm2 + , r1 r2

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where m1 and m2 are the masses of the centers, and r1 and r2 are the distances from the attracted mass to each of them. If in the rectangular coordinate system x, y, z the axis z is directed along the line connecting the centers of attraction, then the distances versus coordinates will have the form



r1 = x 2 + y 2 + (z − z1 )2 , r2 = x 2 + y 2 + (z − z2 )2 , where z1 , z2 are the coordinates of the centers on the axis z. For the first time, this problem for the case of plane motion was investigated and reduced to quadratures by Leonard Euler (1760; 1764). Therefore, the problem of two fixed centers is also called the Euler problem. At the beginning of the 20th century, Gaston Darboux (1901) pointed out the possibility of generalizing the Euler problem in the case of plane motion by introducing complex conjugate masses and the imaginary distance between them. The force function of attraction in this case always takes real values, and the solution of the problem also reduces to quadratures. Darboux obtained this generalization as some purely mathematical result and did not consider the possibilities for its practical application. The problem of the spatial motion of a material point in the field of two fixed centers with complex conjugate masses and imaginary distance between them is considered in Aksenov et al. (1961) and in a subsequent series of works by these authors. The authors called this model a generalized problem of two fixed centers. The problem is of primary interest due to the fact that the gravitational field of such a system of two fixed centers is close to the field of Newtonian attraction of an oblate planet. Next we will show it. Let us assume that √ √ m m m1 = (1 + −1σ ), m2 = (1 − −1σ ), 2 2 r1 = r2 =

 2 √ x 2 + y 2 + z − c(σ + −1) ,  2 √ x 2 + y 2 + z − c(σ − −1) ,

where c and σ are real parameters. Then the force function of attraction in this model will take the form √ √ G m  1 + −1 σ 1 − −1 σ  W = . (3.19) + 2 r1 r2

Chapter 3 Equations of motion and analytic theories

Obviously, being the sum of two complex conjugate quantities, the force function will be a real value. We expand the function W in a series in spherical functions. We write the expansion in the form

∞   r0 n  Gm W= Jn Pn (sin ϕ) , (3.20) 1− r r n=2

where r0 is an artificially introduced parameter, and the coefficients Jn are determined by the relations Jn = −

 √ √ √ √ 1  c n  (1 + −1 σ )(σ + −1)n + (1 − −1 σ )(σ − −1)n 2 r0 (n = 2, 3, ...).

In particular, we have J2 =

 c 2

J3 = 2

r0  c 3 r0

(1 + σ 2 ),

σ (1 + σ 2 ).

If we now represent the force function of the planet’s attraction U in the form U = W + R, then for R we obtain Gm   r0 n (Jn − Jn )Pn (sin ϕ). r r ∞

R=−

(3.21)

n=2

Now, the parameters c and σ , hitherto undefined, can be defined and it turns out that J2 = J2 , J3 = J3 . Then the sum in Eq. (3.21) will start with n = 4, i.e. will not contain second and third zonal harmonics. In addition, for Earth, Mars, Jupiter, Saturn, Uranus and Neptune, it turns out that |Jn | < |Jn | at least for n = 4, 5, 6. Now recall that the problem of two fixed centers is reduced by Euler to quadratures. This solution is also suitable for the force function (3.19). Aksenov took advantage of this fact (Aksenov, 1977). He performed the conversion of quadratures and built on this basis a new non-Keplerian intermediate orbit of a satellite of an oblate planet and an analytical theory of the motion of artificial Earth satellites. The theory has been successfully applied for

65

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several years, due to the fact that the perturbing function R turned out to be 1000 times smaller than when using the Keplerian intermediate orbit. The analytical theory of the motion of natural satellites of major planets can also be constructed on the basis of solving the generalized problem of two fixed centers. This turns out to be effective if it is sufficient to take into account the second and third zonal harmonics in the expansion of the force function of the planet’s attraction and first-order perturbations for other perturbing factors. To determine higher-order perturbations, this approach leads to rather cumbersome calculations and is ineffective. If it is necessary to use the satellite’s intermediate orbit, based on the solution of the generalized problem of two fixed centers, it is better to take formulas for calculating the coordinates and components of the satellite’s velocity for any time moment (Aksenov et al., 1988), where they are given in the form convenient for programming calculations. In addition, this work provides the listing of a computer program in Fortran.

3.7 Constructing analytical theories of planetary satellite motion using perturbation theory methods 3.7.1 General scheme of perturbation theory The simplest model of planetary satellite motion is Keplerian motion. The model is obtained by the exact solution of the differential equations of the two-body problem. The next stages of improving the model are taking into account the influence of perturbing factors. For this purpose, the corresponding additional terms are added to the equations of motion. At the same time, it is already impossible to find an exact analytical solution to the equations. Various methods of numerical integration of differential equations are always available. A special section of this book is devoted to these methods. Another way is to construct an approximate analytical solution by perturbation theory methods. Perturbation theory is used in many fields of science. The basic idea is the same everywhere. Only the forms of the methods and the form of the formulas differ. We consider here one of the methods of perturbation theory in the form most often used in celestial mechanics. For simplicity and clarity of presentation of the main idea, we confine ourselves to considering a mechanical model in which the

Chapter 3 Equations of motion and analytic theories

motion of a material point is described by differential equations d 2 x ∂U d 2 y ∂U d 2 z ∂U , , , = = = ∂x ∂y ∂z dt 2 dt 2 dt 2

(3.22)

where x, y, z are the coordinates of the material point in some system of rectangular coordinates, t is the time, and U is the force function. As a rule, in practical problems the force function has the form in which it is impossible to find the exact analytical solution of the equations of motion. The main idea of perturbation theory is as follows. We divide the force function into two terms, U = V + R, under the following two conditions: 1. After replacement in the equations of motion of the force function U to the function V their exact general analytical solution can be found. 2. At least in the region of the motion under consideration, the inequality |R|  |V | remains true. Of course, not in any problem such a partition is possible. At least the fulfillment of the first condition already allows us to formally construct the solution of the initial equations (3.22) by perturbation theory methods. However, of practical interest are cases where the second condition is also satisfied. The equations d 2 x ∂V = , ∂x dt 2

d 2 y ∂V = , ∂y dt 2

d 2 z ∂V = , ∂z dt 2

(3.23)

are called the equations of unperturbed motion, the original equations (3.22) are called the equations of perturbed motion, and R is a perturbing function. The original equations (3.22) can be written in the form of a system of six first-order equations, dx dt d x˙ dt

= x, ˙ =

dy dt

∂(V +R) ∂x ,

= y, ˙ d y˙ dt

dz dt

=

= z˙ ,

∂(V +R) ∂y ,

d z˙ dt

=

∂(V +R) ∂z ,

where the variables x, y, z, x, ˙ y, ˙ z˙ are the required functions.

(3.24)

67

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Chapter 3 Equations of motion and analytic theories

The general solution of the equations of unperturbed motion will have the form x = x(t, c1 , c2 , c3 , c4 , c5 , c6 ), y = y(t, c1 , c2 , c3 , c4 , c5 , c6 ), z = z(t, c1 , c2 , c3 , c4 , c5 , c6 ), x˙ = x(t, ˙ c1 , c2 , c3 , c4 , c5 , c6 ), y˙ = y(t, ˙ c1 , c2 , c3 , c4 , c5 , c6 ), z˙ = z˙ (t, c1 , c2 , c3 , c4 , c5 , c6 ),

(3.25)

where c1 , c2 , c3 , c4 , c5 , c6 are arbitrary integration constants. In the perturbation theory method, the last formulas are used as formulas for changing the variables x, y, z, x, ˙ y, ˙ z˙ for variables of the time function c1 , c2 , c3 , c4 , c5 , c6 in the equations of perturbed motion (3.24). We replace the dependent variables by the required functions. As a result of the transformations, differential equations of perturbed motion are obtained with respect to the new variables ci (t) (i = 1, 2, ..., 6) in the form dci = Ci (t, c1 , c2 , c3 , c4 , c5 , c6 ) (i = 1, 2, ..., 6). dt

(3.26)

An exact analytical solution to these equations, like Eqs. (3.22), cannot be found. However, they have one obvious advantage. If we put R = 0 in Eqs. (3.24), then they will turn into Eqs. (3.23), and in the corresponding solution (3.25) the arguments c1 , c2 , c3 , c4 , c5 , c6 will be constant values. Therefore, in the transformed Eqs. (3.26), the right-hand sides turn out to be equal to zero. For R, nonzero, and the second condition for applying the perturbation theory |R|  |V | the right-hand sides of Eqs. (3.26) will contain a factor of some small parameter. This allows one to find an approximate solution of the equations of perturbed motion by the small parameter method. The success of its application depends primarily on the value of the small parameter, that is, on the ratio |R|/|V |. Therefore, in the division of the force function U into the two terms V and R we have a natural idea to reduce the value |R| while maintaining the first condition for applying perturbation theory methods. In various problems of celestial mechanics, the form of the equations of motion when applying the methods of perturbation theory can be different, however, the general scheme of the approach presented here is the same.

Chapter 3 Equations of motion and analytic theories

3.7.2 Circumstances in the motion of real celestial bodies, allowing the use of perturbation theory methods In the general case, when considering the motion of an arbitrary number of celestial bodies, it is not at all obvious that the conditions for the application of perturbation theory methods are satisfied. However, there is a certain hierarchy in the ratio of the sizes of most real celestial bodies, the distances between them and the properties of their motion. The motion parameters of the planets of the Solar System and almost all of their satellites satisfy the conditions necessary for solving the equations of motion by perturbation theory methods. We consider several specific cases that lead to the fundamental problems of the theory of the motion of bodies of the Solar System. First, we simplify the consideration of the system of the Sun, planets and satellites, assuming that all these bodies are material points. Then a mechanical model of the motion of the n + 1 material points will be suitable for them. Among these points will be the Sun, planets and their satellites. We place the origin in one of them and describe the motion of the system n + 1 material points by the equations of relative motion ∂(Vi + Ri ) d 2 yi ∂(Vi + Ri ) d 2 zi ∂(Vi + Ri ) d 2 xi = , = , = , ∂xi ∂yi ∂zi dt 2 dt 2 dt 2 (3.27) where   n   xi xj + yi yj + zi zj 1 G(m0 + mi ) Vi = , Ri = G mj − , ri ij rj3 j =1

ij =



(xj − xi )2 + (yj − yi )2 + (zj − zi )2 , ri = rj =





xi2 + yi2 + zi2 ,

xj2 + yj2 + zj2 ,

G is the universal gravitational constant, xi , yi , zi , mi (i = 1, 2, ..., n) are the rectangular coordinates and masses of bodies, respectively, and m0 is the mass of the central body. A dash at the sum sign means that there is no term with j = i. Let us consider some practical problems. Planetary problem We will study the motion of n planets under the action of the attraction of the Sun and their mutual attraction. We neglect the

69

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small influence of other bodies. In the equations of relative motion, the central body will be the Sun. In this case, for the equations of unperturbed motion with Ri = 0 (i = 1, 2, ..., n) the general solution is known, since the system of equations decomposes into n independent systems of equations of motion of two bodies, for each of which a general solution is known. Thus, the first condition for the application of perturbation theory methods is satisfied. We now verify that the second condition is satisfied. Consider the relation Ri /Vi . From the equations of relative motion follows

n  ri (xi xj + yi yj + zi zj ) Ri  mj ri = − . (3.28) Vi m0 + mi ij rj3 j =1

Here i = 1, 2, ..., n. The orbital parameters of nine major planets are such that the planets experience neither close proximity to the Sun, nor close mutual proximity. Therefore, the quantities xi , yi , zi , ij , ri can be considered as approximately the same order of magnitude. On the other hand, the terms in Eq. (3.28) have factors mj , m0 + mi which are small parameters due to the small masses of the planets compared to the mass of the Sun. Thus, the second condition for the application of perturbation theory methods in the planetary problem is ensured by the small masses of the planets compared to the mass of the Sun. When solving the equations of perturbed motion (3.26) in a planetary problem, the small parameters are εj =

mj (j = 1, 2, ..., n) . m0

Satellite problem Consider the motion of a planetary satellite system under the influence of planetary attraction, the Sun and the mutual attraction of satellites. The attraction of other planets is neglected because of their remoteness. The attraction of the Sun, despite its remoteness, cannot be neglected, since it has a large mass. In the equations of relative motion, the central body will be the planet. Therefore, we will associate the origin of coordinates with this planet. The Sun will be considered the body number 1 (i = 1). The equations for i = 1 will not be considered, since they determine the relative motion of the planet and the Sun. In the satellite problem under consideration for the equations of unperturbed motion with Ri = 0 (i = 2, 3, ..., n) the general solution is also known, since the system of equations decomposes

Chapter 3 Equations of motion and analytic theories

into independent systems of equations of motion of two bodies. Thus, the first condition for the application of perturbation theory methods is satisfied. We now verify that the second condition is satisfied. Consider the expression n 

Ri  mj = Vi m0 + mi j =1



ri (xi xj + yi yj + zi zj ) ri − ij rj3

(3.29)

for i = 2, ..., n. The smallness of the terms at j = 2, ..., n in the same way as in the planetary problem is ensured by the small masses of the satellites mj in comparison with the mass of the planet m0 . The term with j = 1 (influence of the attraction of the Sun) requires special consideration. We denote this term in the value Ri by (Ri )1 , Ri i and in the value R Vi by ( Vi )1 . We use the relations xi x1 + yi y1 + zi z1 = ri r1 cos H1i , 2i1 = ri2 + r12 − 2 ri r1 cos H1i . Here ri is the planetocentric distance of the satellite, r1 is the heliocentric distance of the planet, and H1i is the angle between the planetocentric directions to the satellite and to the Sun. Obviously, while the satellite remains the satellite of the planet, the ratio ri /r1 will be small. We expand the value of 1i1 , and then (Ri )1 in a series in powers of the small parameter ri /r1 . Will have

  2  3 1 1 ri 2 cos H1i − + ... , (Ri )1 = Gm1 1+ r1 r1 2 2 where unwritten terms have a higher order of smallness than ( rr1i )2 . The perturbing function Ri enters into the equations of motion only under the sign of the partial derivatives with respect to xi , yi , zi . Derivatives of the first term will give zero. Therefore, it can be omitted. Leaving only the most significant term in the expansion, we obtain   ri2 3 1 2 (Ri )1 = Gm1 3 cos H1i − . 2 r1 2 Then, neglecting the mass of the satellite mi compared with the mass of the planet m0 , the term in the value RVii , due to the attraction of the Sun, can be written as 

Ri Vi



m1 ri3 = m0 r13 1



 3 1 2 cos H1i − . 2 2

71

72

Chapter 3 Equations of motion and analytic theories

Despite the fact that the mass of the Sun m1 is many times greater than the mass of the planet m0 , the ratio ( RVii )1 for planetary satellites turns out to be small due to the fact that the distances of satellites to the planet ri are small compared to the distance r1 of the planet to the Sun. Thus, the second condition for the application of perturbation theory methods in the satellite problem is ensured by the small masses of the satellites m2 , m3 , ..., mn compared to the mass of the planet m0 , as well as the small distances of the satellites to the planet compared to the distance of the planet to the Sun. When solving the equations of perturbed motion (3.26) in the satellite problem, the following small parameters are used: εj

3 m1 rj = (j = 2, 3, ..., n). m0 r13

Here the distances rj (j = 2, 3, ..., n) vary in time. In order for the parameters to be really constant, the changing distances are replaced by their approximate values equal to the semi-major axes of the Keplerian orbits, which are used as the zeroth approximation of the motion model. We can also use the mean Keplerian motion. Then the small parameters will be εj =

3 n21 m1 aj = (j = 2, 3, ..., n), m0 a13 n2j

where a1 , aj are the semi-major axes, and n1 , nj are the mean motions. The problem of satellite motion of a non-spherical planet Let us now consider an example of motion in a two-body problem, when one of the bodies (satellite) can be considered a material point, and the other (planet) creates a gravitational field different from the gravitational field of a material point or sphere with a spherical density distribution. The equations of motion of the satellite in this case can be written as d 2 x ∂(V + R) d 2 y ∂(V + R) d 2 z ∂(V + R) = = = , , , ∂x ∂y ∂z dt 2 dt 2 dt 2

(3.30)

where Gm Gm , R= J X(x, y, z), r r x, y, z are the planetocentric rectangular coordinates of the satellite, m is the mass of the planet, G is the universal gravitational V=

Chapter 3 Equations of motion and analytic theories

 constant, r is defined by the expression r = x 2 + y 2 + z2 , J is a constant parameter, and X(x, y, z) is some well-known function. The last two values can be ordered so that the function X(x, y, z) in the area of satellite motion took values slightly different from unity. The parameter J in this case will characterize the difference between the planet and the ball with a concentric density distribution. Known gravitational fields of the Earth, other planets and many natural satellites of the planets differ little from the field of attraction of the material point. Therefore, the conditions of applicability of perturbation theory methods are also satisfied in this case, and J is a characteristic small parameter. The shapes of major planets and their main satellites are close to the shape of an oblate axisymmetric body. Therefore, the coefficient J is taken as the coefficient for the second zonal harmonic of the expansion of the force function of the planet’s attraction in a series of spherical functions. Other applications of perturbation theory The configurations of celestial bodies discussed above are just examples of numerous applications of perturbation theory in celestial mechanics. We mention here only one group of problems when a particular solution of the equations of motion is considered as an unperturbed motion. The perturbed motion occurs near this particular solution. Small parameters in such problems characterize the difference in coordinates in perturbed and unperturbed motions, and an additional condition for the applicability of perturbation theory is to keep these differences small, at least in the time interval under study. When considering various small parameters in problems of celestial mechanics, one should single out parameters that characterize the smallness of the perturbing function. The perturbing function can be expanded in powers of other small parameters as well. This is often done to provide the ability to solve differential equations of perturbed motion in the form (3.26). We also note cases where the forces acting on a celestial body do not have a force function. An example of such forces are forces caused by tidal deformation of the viscoelastic bodies of a planet and a satellite. In such cases, the initial equations of motion in rectangular coordinates are written as d 2x d 2y d 2z = V + R , = V + R , = Vz + Rz , x x y y dt 2 dt 2 dt 2

(3.31)

where the terms Vx , Vy , Vz must be chosen so that it is possible to find a general solution of the equations of motion when discarding

73

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Chapter 3 Equations of motion and analytic theories

the terms Rx , Ry , Rz . These last terms are called the components of the perturbing acceleration. To be able to apply perturbation theory, it is necessary that Rx , Ry , Rz be small in comparison with the components of the main acceleration Vx , Vy , Vz .

3.7.3 Equations for elements of the intermediate orbit The equations of the perturbed motion of a celestial body in the perturbation theory in a general form have already been constructed above. In order for the presentation to be continuous, we will again proceed from the equations of motion in rectangular coordinates. For simplicity of understanding the general scheme of perturbation theory, we restrict ourselves to the case of the motion of one celestial body under the action of forces having a force function. The initial equations of motion in rectangular coordinates have the general form ∂R d 2 y ∂V ∂R d 2 z ∂V ∂R d 2 x ∂V + , + , + , = = = ∂x ∂x ∂y ∂y ∂z ∂z dt 2 dt 2 dt 2

(3.32)

where V and R are functions of the coordinates x, y, z and the time t. According to perturbation theory, the term V is chosen so that it is possible to find a general solution of the equations of unperturbed motion d 2 x ∂V d 2 y ∂V d 2 z ∂V = = = , , . 2 2 ∂x ∂y ∂z dt dt dt 2

(3.33)

Then R is called a perturbing function. For convenience of presentation, we reproduce here again some relations from the general scheme of perturbation theory. Assume that a solution to Eqs. (3.33) is found in the form x = x(t, c1 , c2 , c3 , c4 , c5 , c6 ), y = y(t, c1 , c2 , c3 , c4 , c5 , c6 ), z = z(t, c1 , c2 , c3 , c4 , c5 , c6 ), x˙ = x(t, ˙ c1 , c2 , c3 , c4 , c5 , c6 ), y˙ = y(t, ˙ c1 , c2 , c3 , c4 , c5 , c6 ), z˙ = z˙ (t, c1 , c2 , c3 , c4 , c5 , c6 ),

(3.34)

where c1 , c2 , c3 , c4 , c5 , c6 are arbitrary integration constants. This completes the auxiliary role of Eqs. (3.33). It is enough that they gave us the ratios (3.34). Further, these relations are considered as transformation formulas that relate the coordinates and

Chapter 3 Equations of motion and analytic theories

components of the velocity of the body in perturbed motion with some new unknown time functions c1 , c2 , c3 , c4 , c5 , c6 . Relations (3.34) are used as formulas for replacing variables in Eqs. (3.32) of the variables x, y, z, x, ˙ y, ˙ z˙ to the variables c1 , c2 , c3 , c4 , c5 , c6 . Thus, new equations of perturbed motion are obtained, dci = Ci (t, c1 , c2 , c3 , c4 , c5 , c6 ) (i = 1, 2, 3, 4, 5, 6), dt

(3.35)

whose advantage is that the right-hand sides of these equations vanish if R = 0 in the original equations (3.32). This allows us to solve them using the small parameter method. Eqs. (3.34) determine the law of motion, which is called an intermediate orbit if c1 , c2 , c3 , c4 , c5 , c6 are considered to be given constants. These formulas themselves are called intermediate orbit formulas. Eqs. (3.35) are called the equations for the elements of the intermediate orbit. The success of further actions essentially depends on how arbitrary constants c1 , c2 , c3 , c4 , c5 , c6 are chosen. We can select them in an infinite number of ways. In the practice of researchers over the past three centuries, many options have been considered and successfully applied. Consider here one of them, which had a large number of applications. As the force function of the unperturbed motion V , we choose the force function of the two-body problem, considered here as material points. This choice is due to the general hierarchy in the motion of the bodies of the Solar System. The movement of each planet occurs under the influence of the attraction of the Sun. The influence of other planets is relatively weak. A satellite is therefore a satellite of the planet, because it moves under the influence of its attraction. Other satellites and even the Sun only slightly distort the movement. Thus, in this consideration, we have V=

μ , r

(3.36)

where μ is a constant, and r is the distance between two bodies. Mostly, the origin is placed in one of the bodies, to which we assign the number 0. Then μ = G(m0 + m1 ), where G is the universal gravitational constant, and m0 + m1 is the sum of the masses of bodies. In the case when the origin is located in the barycenter of m3

0 two bodies, we have μ = G (m +m 2. 0 1) The solution of the two-body problem describes the motion, which is called Keplerian, because it occurs according to Kepler’s laws. Since we consider the motion of the bodies of the Solar System, which has a certain hierarchy, we restrict ourselves to the elliptical type of motion. In accordance with the Keplerian motion,

75

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Chapter 3 Equations of motion and analytic theories

arbitrary constants, which are called Keplerian orbital elements, are also chosen. We list here the Keplerian elements, as well as the time moments associated with them. n the mean motion, radian/time unit; e the eccentricity, dimensionless units; i the inclination (dihedral angle between the orbit plane and the main plane Oxy), radians; M0 the mean anomaly at the epoch (the value of the mean anomaly M at the initial moment of time—the epoch), radians; ω the angular distance of the pericenter from the ascending node of the orbit, radians;  the longitude of the ascending node of the orbit (angle in the plane Oxy between the axis x and the line of nodes), radians; t0 the initial moment of time, the epoch of the elements; t the current moment of time at which the coordinates of the body are calculated. Along with the mean motion n, we will also consider the semimajor axis of the orbit a associated with n with Kepler’s third law as an orbit parameter μ n= . a3 The change of variables in Eqs. (3.32) is done using the Keplerian motion formulas. These formulas are described in previous sections. The partial derivatives of the perturbing function R with respect to the coordinates are expressed in terms of the partial derivatives with respect to Keplerian elements. The procedure for such a replacement is described in detail in the literature (Duboshin, 1975). The procedure is called the “main operation”. The application of the perturbation theory turns out to be simpler if the quantities a, e, i, M, ω,  are taken as the new soughtafter functions. Instead of the mean anomaly at epoch M0 , the mean anomaly of M is taken, which in the Kepler motion is a known linear function of time M = M0 + n(t − t0 ). For brevity, writing equations for the elements of the intermediate orbit, we make the notation simpler: α1 = a, α2 = e, α3 = i, β1 = M, β2 = ω, β3 = .

(3.37)

Chapter 3 Equations of motion and analytic theories

As a result of the change of variables, the equations of the perturbed motion of a celestial body can be written in the following general form: dαi  ∂R = aij , dt ∂βj 3

j =1

 dβi ∂R = ni − aj i dt ∂αj 3

(3.38)

j =1

(i = 1, 2, 3). In these equations, ni , aij are functions that depend only on the elements α1 , α2 , α3 and the constant μ. In the case of the Keplerian intermediate orbit n1 , n2 , as well as some of the nine functions aij , are equal to zero. However, this fact does not simplify the solution of the equations. Writing Eqs. (3.38) in such a general form allows us to apply them also to some non-Keplerian intermediate orbits. Methods for solving these equations are discussed in the following sections. Note that in Eqs. (3.38) the perturbing function R is denoted by the same letter as in Eqs. (3.32). However, here it is a function of the elements α1 , α2 , α3 , β1 , β2 , β3 and the time t. We now consider the explicit form of the equations for the elements of the intermediate orbit in the case of elements (3.37) da 2 ∂R = , dt na ∂M de 1 − e2 ∂R = − dt ena 2 ∂M

√ 1 − e2 ∂R , ena 2 ∂ω

(3.39)

∂R ∂R cos i 1 di = − , √ √ 2 2 2 2 dt na 1 − e sin i ∂ω na 1 − e sin i ∂ dM 2 ∂R 1 − e2 ∂R =n− − , dt na ∂a ena 2 ∂e dω = dt

√ ∂R cos i 1 − e2 ∂R − , √ 2 ∂e ena na 2 1 − e2 sin i ∂i d ∂R 1 = . √ 2 2 dt na 1 − e sin i ∂i

(3.40)

77

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Chapter 3 Equations of motion and analytic theories

The factors in the right-hand sides of the equations can be replaced according to the relations 1 na 2 na 1 , = . = μ na μ na 2

(3.41)

In practical celestial mechanics, other versions of the elements of the Keplerian intermediate orbit are also used. Two of them deserve special attention. The orbits of many natural satellites of the planets are almost circular and lie near the plane of the equator of the planet. In such cases, the position of the satellite in orbit is primarily determined by the mean longitude λ, and the orientation of the orbit is determined by the longitude of the pericenter

. These quantities are related with M, ω,  using by the relations λ = M + ω + , = ω +  .

(3.42)

For reverse orbits, instead of the sum of ω and , their difference is used. The equations are compiled with respect to the functions λ,

and . The elements of the intermediate orbit are ,  and λ0 is the mean longitude at epoch t0 . In this case, instead of Eqs. (3.39), (3.40), we need to solve the equations 2 ∂R da = , dt na ∂λ √ √ de ∂R 1 − e2 ∂R e 1 − e2 =− − , √ dt ena 2 ∂

na 2 (1 + 1 − e2 ) ∂λ tan 2i di =− √ dt na 2 1 − e2



∂R ∂R + ∂

∂λ

 −

(3.43)

∂R 1 , √ na 2 1 − e2 sin i ∂

√ tan 2i dλ ∂R ∂R 1 − e2 2 ∂R =n− +e + , √ √ 2 2 2 2 dt na ∂a na (1 + 1 − e ) ∂e na 1 − e ∂i d

= dt

√ tan 2i ∂R 1 − e2 ∂R + , √ 2 ∂e ena na 2 1 − e2 ∂i

(3.44)

d ∂R 1 = . √ dt na 2 1 − e2 sin i ∂i The right sides of the equation for i and  have singularities at i = 0, and the right sides of the equations for e and ω have singularities at e = 0. These circumstances require special attention when solving equations.

Chapter 3 Equations of motion and analytic theories

Note that in many Russian publications and textbooks, instead of , the letter π is used to indicate the longitude of the pericenter. Now we consider a variant of the elements of the intermediate orbit, in which the equations have no singularities with the eccentricity and inclination of the orbit equal to zero. These elements were invented by Lagrange to study the secular perturbations of the planets. They are now called as Lagrange elements. The equations for a and λ have no singularities, and the remaining elements need to be replaced. Usually, Lagrange elements are denoted in the literature by h, k, p, q. With the elements considered above, they are related by the relations h = e sin , k = e cos ,

(3.45)

p = tan i sin , q = tan i cos .

(3.46)

After replacing the variables, the equations for the Lagrange elements will have the form √   k tan 2i dh 1 − e2 ∂R ∂R ∂R h = − + , √ √ 2 2 2 dt ∂k ∂λ na 2 1+ 1−e na 1 − e ∂i dk = dt



1 − e2 na 2

  k ∂R ∂R − − − √ ∂h 1 + 1 − e2 ∂λ −

h tan 2i ∂R , √ 2 2 na 1 − e ∂i

dp sec3 i p ∂R = − √ √ dt na 2 1 − e2 ∂q 2na 2 1 − e2 cos i cos2

 i 2

∂R ∂R + ∂

∂λ

dq sec3 i ∂R =− − √ 2 2 dt na 1 − e ∂p −

(3.47)



q

√ 2na 2 1 − e2 cos i cos2

i 2

 ,

(3.48)

 ∂R ∂R + . ∂

∂λ

For complete composition, the above equations should be added: da 2 ∂R = , dt na ∂λ √ tan 2i dλ ∂R 1 − e2 ∂R 2 ∂R =n− +e + . √ √ 2 2 2 2 dt na ∂a ∂e na (1 + 1 − e ) na 1 − e ∂i (3.49)

79

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The right-hand sides of Eqs. (3.47), (3.48), (3.49) contain partial derivatives of the perturbing function with respect to quantities that are not the required functions in these equations. It is good in this form, and should be left. The fact is that the perturbing function is obtained first, as a function of the variables a, e, i, λ, , . Therefore, it is better to differentiate it first with respect to these elements, and then move on to the variables h, k, p, q already in the derivatives obtained. Note that in the publications there are several options for introducing Lagrange elements into consideration, depending on the desired effect. Above in Section 3.2.5 we examined one of these options, in this section another.

3.7.4 Solving equations for intermediate orbit elements. Small parameter method Consider a method for solving Eqs. (3.38). These are the most commonly used equations. In cases where other variants of elements of the satellite’s intermediate orbit are used, the solution is constructed in the same way. The solution method is based on the smallness of the perturbing function R in comparison with the function V , therefore it is called the small parameter method. In the literature, this method is called the Poincaré method of small parameters. To state the method of a small parameter, one needs to introduce the concept of the order of smallness, which is ambiguous. The fact is that, in practice, the perturbing function R, expressed in terms of the elements of the intermediate orbit, contains many terms—the summands, which are different in magnitude. To express the perturbing function, series expansion in powers of various small parameters is used. Assignment to each term of one or another order of smallness is sometimes conditional. When constructing a solution, quantities of different orders of smallness are multiplied many times. Mostly, the maximum allowable order of smallness is assumed in advance. Each specific theory of motion has its own characteristics. The values of small parameters are determined both by the physical properties of celestial bodies, and by the relationships between their mutual distances. For a general presentation of the method, we assume that the smallness of the perturbing function is provided by some small parameter, which is contained in it as a common factor. We will consider this parameter of the first order of smallness. In practice, the perturbing function is expanded in powers of this and other

Chapter 3 Equations of motion and analytic theories

small parameters. Thus, its expansion will begin with a first-order term of smallness. For brevity’s sake, we will not explicitly write out the small parameters themselves. However, a certain order of smallness will be assigned to each value. This order will be written using the upper index of the quantity in parentheses. Now the expansion of the perturbing function will have the form R = R (1) + R (2) + R (3) + ... .

(3.50)

We introduce the notation Ai =

3 

aij

j =1

Bi = −

3  j =1

∂R , ∂βj (3.51)

∂R aj i ∂αj

(i = 1, 2, 3). Now we write the equations for the elements of the intermediate orbit in the form dαi (1) (2) = Ai + Ai + ..., dt dβi (0) (1) (1) (2) = ni + ni + ... + Bi + Bi + ... dt

(3.52)

(i = 1, 2, 3). Note that in the particular case of the Keplerian intermediate orbit (s) n2 = n3 = 0 and n1 = 0 for s > 1. In the last equations, the righthand sides depend on the required functions α1 , α2 , α3 , β1 , β2 , β3 and obviously depend on time. (j ) (j ) Each of the terms Ai , Bi (i = 1, 2, 3, j = 1, 2, ...) depends on (0) (1) all six elements of α1 , α2 , α3 , β1 , β2 , β3 . The terms ni + ni + ... depend only on α1 , α2 , α3 . We will seek a solution to Eqs. (3.52) in the form of series in powers of small parameters, that is, (0)

(1)

(2)

αi = αi + αi + αi + ..., (0)

βi = βi

(1)

+ βi

(2)

+ βi

+ ...,

(i = 1, 2, 3).

(3.53)

Here we consider the construction of a formal solution. Evidence for the existence of such a solution and the convergence of the

81

82

Chapter 3 Equations of motion and analytic theories

constructed series can be found in books (Duboshin, 1975; Subbotin, 1968). So, we substitute the series (3.53) into Eqs. (3.52). Then we equate the terms of the same order of smallness in the left- and right-hand sides of the equations. Each of the terms on the righthand sides of these equations will have to be expanded in a Taylor series in powers of a small parameter. The expansion is done according to the scheme     1 df 1 d 2f ε+ ε 2 + ..., f (a + ε) = f (x)|x=a + 1! dx x=a 2! dx 2 x=a where a is the value of the argument of the function f (x), relative to which a small increment of ε is counted. Thus, in the arguments (j ) (j ) (j ) of the functions Ai , Bi , ni (i = 1, 2, 3, j = 1, 2, ...) increments (1) (2) (1) (2) are the infinite sums αi + αi + ..., βi + βi + .... For terms of the zeroth order of smallness we get (0)

dαi = 0, dt (0) dβi (0) = (ni )0 . dt

(3.54)

Hereinafter, the symbols (...)0 denote the values of functions for (0) (0) the values of the elements αi = αi , βi = βi . Solving Eqs. (3.54) is trivial: (0)

αi (0)

(0)

(0)

= αi0 , βi

(0)

(0)

= (ni )0 (t − t0 ) + βi0 .

(3.55)

(0)

Here αi0 , βi0 (i = 1, 2, 3) are arbitrary integration constants. (0) (0) Moreover, the constants (ni )0 depend on αi0 . The solution (3.55) describes an intermediate unperturbed motion. Now we select and equate in the right- and left-hand sides of Eqs. (3.52) the terms of the first order of smallness, taking into account (3.53). We obtain (1)

dαi dt (1)

dβi dt

(1) = (Bi )0

(1) + (ni )0

(1)

= (Ai )0 ,

 (0)  3  ∂ni (1) + αj (i = 1, 2, 3). ∂αj j =1

(1)

(1)

(3.56)

0

Here (Ai )0 , (Bi )0 are known functions of the time t due to the (0) (0) substitutions αi = αi , βi = βi and Eqs. (3.55). Note that arbitrary

Chapter 3 Equations of motion and analytic theories

(0)

(0)

constants αi , βi0 enter here literally, their values have not yet been determined. The solution to the first three of six Eqs. (3.56) has the form (1) αi

 =

(1)

(1)

(Ai )0 dt + αi0 (i = 1, 2, 3),

(3.57)

(1)

where αi0 are the new arbitrary integration constants. We can leave these constants in the decision to attend literally and dispose of their meanings later. However, they are redundant, and they can immediately be set equal to zero. This is exactly what we will do for further constructions. To construct a solution, we need to implement indefinite integration on the right-hand side of Eq. (3.57). For this purpose, in specific cases, special methods are developed for the expansion of the perturbing function. Let us (1) assume that it succeeded. Then αi and the right-hand sides of the remaining three Eqs. (3.56) become known functions of time. (1) Now the solution for βi is expressed in terms of the indefinite integral (1)

βi

 =

⎤  (0)  3  ∂n (1) (1) i ⎣(B (1) )0 + (n(1) )0 + αj ⎦ dt +βi0 (i = 1, 2, 3). i i ∂αj ⎡

j =1

0

(3.58) (1) can be set equal to Here, the arbitrary integration constants βi0 zero. A solution is constructed similarly for terms of the second order of smallness. First we find ⎡  (1)   3  ∂Ai (2) (2) (1) αi = ⎣(Ai )0 + αj + ∂αj j =1 0 ⎤  (1)  3  ∂A (1) (2) i + βj ⎦ dt + αi0 (i = 1, 2, 3). (3.59) ∂βj j =1

0

(2)

The arbitrary constants αi0 are set equal to zero. The integrand in (3.59) turns out to be a well-known function of time t. Assume that this function was able to integrate. Then the solution for βi(2) is expressed as (2) βi

 =

⎡ ⎣(B (2) )0 i

 (1)  3  ∂Bi (1) + αj + ∂αj j =1

0

83

84

Chapter 3 Equations of motion and analytic theories

 (1)   (1)  3 3   ∂Bi ∂ni (1) (1) + βj + αj + ∂βj ∂αj j =1

0



1   ∂ 2 ni 2 ∂αj ∂αk 3

3

+

j =1

(0)

j =1 k=1

(3.60)

0





(1) (1) (2) αj αk ⎦ dt + βi0 (i = 1, 2, 3). 0

(2) The arbitrary constants βi0 are also assumed to be equal to zero. Thus, a solution is sequentially found for members of the following orders of smallness. Each time it is necessary to integrate a new known function of time, and to set new excessive arbitrary (0) (0) constants equal to zero. Since substitutions αi = αi , βi = βi were made in obtaining the integrands on the right-hand sides of the equations, taking into account (3.55) all terms series (3.53) (0) (0) turn out to be dependent on arbitrary constants αi0 , βi0 (i = 1, 2, 3). As a result, we obtain a solution of the equations of perturbed motion as a function of time t and six independent arbitrary constants, (0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

αi = αi (t, α10 , α20 , α30 , β10 , β20 , β30 ), (0)

(0)

(0)

(0)

βi = βi (t, α10 , α20 , α30 , β10 , β20 , β30 ) (i = 1, 2, 3).

(3.61)

3.7.5 Solving equations for intermediate orbit elements. Poisson Method The use of the small parameter method in the order described above leads to some fundamental complications that follow from the expansion properties of the perturbing function. These complications can be avoided by applying the technique proposed by the French mathematician Poisson. The problem arises at the stage of obtaining second-order perturbations by the method of the small Poincaré parameter. The equations for the elements of the intermediate orbit are considered in the form (3.52). We seek a solution to these equations in the form of series in powers of small parameters (3.53). Second-order perturbations in the elements αi are found by (2) Eqs. (3.59). The arbitrary constants αi0 can be set equal to zero, since they enter the solution of Eqs. (3.52) additively along with arbitrary constants appearing in terms of other orders of smallness. The integrand in (3.59) turns out to be a well-known function of time. If this function was integrated, then second-order perturbations in the elements βi can be found by Eq. (3.60).

Chapter 3 Equations of motion and analytic theories

Consider the terms in the integrands of the last two formulas,  (1)   (1)  ∂Ai ∂Bi (1) (1) βj , βj (i, j = 1, 2, 3). (3.62) ∂βj ∂βj 0

0

After substituting into the equations for the elements of the intermediate orbit a perturbing function in the form (3.65) (see below), it becomes clear that the left factors in Eqs. (3.62) can contain only periodic terms. On the other hand, we obtained earlier that (1) the first-order perturbations of βj can contain both periodic and secular, i.e. members linear in time. The product of periodic and secular terms will give mixed terms in the integrand, which, after integration, will lead to the appearance of mixed terms in the solution in second-order perturbations. By mixed terms we average products of the form t sin at, where t is time and a is constant. The appearance of such expressions in second-order perturbations is a consequence of the method used. This does not mean the inevitable unlimited increase in perturbations in time, since in this way we obtain only the initial terms of the expansion of perturbations in powers of small parameters. A complete series representing perturbations can turn out to be an expansion of a bounded function on an infinite time interval. It turns out that there is a method whose main idea was proposed by the French mathematician Poisson, which allows one to exclude mixed terms in the solution of at least the second, and possibly higher orders of smallness. Let us consider this method as applied to the equations of perturbed motion (3.52). Suppose that all perturbations of the elements αi do not contain secular and mixed terms and that all perturbations of the elements βi do not contain mixed terms, at least to some order of smallness. However, perturbations of the elements αi may contain periodic ones, while the perturbations of the elements βi may contain periodic and secular terms of any orders. We denote the (k) (k) sums of secular terms of a given order of smallness k by α i , β i (k) (k) and the sums of periodic terms of order k through α˜ i , β˜i . Now we present the expansion of the sought-for solution of the equations of perturbed motion in the form (0)

(1)

(2)

αi = α i + α˜ i + α˜ i + ..., βi = β i + β i + β i + ... + β˜i (0)

(1)

(2)

(1)

(1)

+ β˜i

(2)

+ ... (i = 1, 2, 3).

(3.63)

As was shown above, at least α i = 0 for i = 1, 2, 3 and the firstorder perturbations of all elements do not contain mixed terms.

85

86

Chapter 3 Equations of motion and analytic theories

Using the small parameter method, it is necessary to expand the right-hand sides of the equations in powers of the small parameter according to the scheme 1 f (a + ε) = f (x)|x=a + 1!



df dx



1 ε+ 2! x=a



d 2f dx 2

 ε 2 + ..., x=a

where a is the value of the argument of the function f (x), relative to which a small increment of ε is counted. According to the (0) Poisson method, the centers of the expansion of (a) are α i and (0)

(1)

(2)

infinite sums β i + β i + β i + ... of secular terms, and the incre(1) (2) ments of (ε) are considered to be infinite sums of α˜ i + α˜ i + ..., (1) (2) β˜i + β˜i + ... periodic summands. Under these assumptions, instead of Eqs. (3.62) in the formulas for perturbations of the second order, we will have  (1)   (1)  ∂Bi ∂Ai (1) (1) ˜ (3.64) βj , β˜j (i, j = 1, 2, 3). ∂βj ∂βj 0

0

These expressions are products of the sums of the periodic terms. Thus, mixed terms are excluded in second-order perturbations. It is easy to show that, if all perturbations of the elements αi do not contain secular and mixed terms and all perturbations of the elements βi do not contain mixed terms at least up to the order of smallness (k − 1) inclusive, then all perturbations of the order k will not contain mixed terms, but may be secular ones.

3.8 Expansion of the perturbing function with respect to the elements of the intermediate orbit of a planetary satellite In all previous sections, the perturbing function was considered as a function of the coordinates of the celestial body. Now, to apply the perturbation theory and the small parameter method, it is necessary to express it in terms of the elements of the intermediate orbit. Moreover, the expression should be such that the time integrals arising in the perturbation theory can be obtained in an analytical form. A general approach is to bring integration results to the form  1 cos(αt + β)dt = sin(αt + β), α where α and β are constants.

Chapter 3 Equations of motion and analytic theories

For each type of perturbing factors, this expansion has a special form. Let us consider here the expansions of the perturbing function in the theory of planetary satellite motion for the main factors, the attraction of other bodies and the non-sphericity of the planet. The derivation of such expansions are extremely complex and voluminous. We will refrain from demonstrating these findings here. Below are links to work where we can find relevant calculations and results. We give here the most general form of the expansion of the perturbing function,   (s) R=R+ Rk1 k2 k3 j1 j2 ...jn sin Dk1 k2 k3 j1 j2 ...jn + (c) +Rk1 k2 k3 j1 j2 ...jn cos Dk1 k2 k3 j1 j2 ...jn ,

(3.65)

where Dk1 k2 k3 j1 j2 ...jn = k1 M + k2 ω + k3  + j1 λ1 + j2 λ2 + ... + jn λn . Summation is carried out over the indices k1 , k2 , k3 , j1 , j2 , ... jn . (s) (c) Here the coefficients R, Rk1 k2 k3 j1 j2 ...jn , Rk1 k2 k3 j1 j2 ...jn depend on the elements a, e, i and do not depend on M, ω,  and time t. The quantities λ1 , λ2 , ..., λn are known linear functions of time. The term R is called the secular term of the expansion of the perturbing function. All these coefficients and quantities contain parameters characterizing perturbing factors. In particular, λ1 , λ2 , ... λn describe orbital motions or rotations of other bodies. It is easy to show that substituting the perturbing function in the form of the perturbation theory equations considered above, the resulting integrals over time will be made suitable to deal with in terms of elementary functions. In perturbations of the first order of smallness, only secular and periodic terms will be obtained, but in perturbations of higher orders, mixed terms of the form can also appear t cos(αt + β). The theory of the motion of a planet’s satellite takes into account the attraction of other satellites of the same planet, as well as the attraction of the Sun.The system of differential equations of relative motion has the same form, whether the Sun or the planet is taken as the dominant attracting body. Therefore, in the theory of satellite motion, it is assumed that the Sun moves around the planet, like other satellites. The attraction of other planets in analytical theory is much more difficult. We are not considering this very weak effect here.

87

88

Chapter 3 Equations of motion and analytic theories

We consider here the perturbing function due to the nonsphericity of the planet. We use the conclusion and results of the work (Brumberg, 1967). When deriving formulas, they proceed from the expression of the perturbing function in terms of the coordinates given above in (3.17). In this case, the system of rectangular coordinates is connected with the body of the planet, the main plane, with the equator of the planet, which in real cases of major planets is close to the axisymmetric body. In fact, the z axis is taken to coincide with the axis of dynamic symmetry. Eq. (3.17) contains the rotation angle of the planet S. In practice, the infinite limit in the first sum is replaced by a finite number N . Thus, the influence of higher-order harmonics is neglected. When expressing the function in terms of the elements of the Keplerian intermediate orbit, one has to make expansions in degrees of eccentricity e. Moreover, terms containing e to a higher degree than the given number K are neglected. Let us consider one modification of the expansion of the perturbing function due to the non-sphericity of the planet, which has advantages in practice and in the analysis of perturbations in the satellite’s motion. In the formula from Brumberg (1967), we change the summation order so that we select the terms with the same coefficients for M, ω, ( − S). We get 



q  j  N  R= (Aqj k cos Dqj k + Bqj k sin Dqj k ),

(3.66)

q=q  j =j  k=0

where Dqj k = qM + j ω + k( − S), 

Aqj k =

p 

Gm

p=p  

Bqj k =

p  p=p 

 Cnk

 =

 Snk =



Gm

r0n a n+1 r0n a n+1

−n−1,j

 (e)Cnk ,

(3.67)

−n−1,j

 (e)Snk ,

(3.68)

Fnkp (i)Xq

Fnkp (i)Xq

Cnk when n − k is even, −Snk when n − k is odd, Snk when n − k is even, Cnk when n − k is odd, n = j + 2p.

Chapter 3 Equations of motion and analytic theories

Hereinafter, in cases where the satellite mass is neglected compared to the planet mass, the planet mass is denoted by m. As above, G is the universal gravitational constant. The summation limits q  , q  , j  , j  , p  , p  are determined from the relations q  = −K − N, q  = K + N, j  = max{q − K, −N }, j  = min{q + K, N },     1 N −j  . p = −E − max{0, −2j, 2 − j, k − j } , p = E 2 2 

Here the function E(...) means the integer part of the number, that is, the nearest integer not exceeding the given one. By Snk , Cnk , as above, the coefficients of the expansion of the planetary gravity force function are denoted, but it is additionally accepted that Cn0 = −Jn , Sn0 = 0. Eqs. (3.67), (3.68) feature special functions of celestial mechan−n−1,j (e) are the ics: Fnkp (i) are the inclination functions and Xq eccentricity functions, which are also called the Hansen coefficients in this expansion. Methods for calculating these functions are described in books (Aksenov, 1986; Kaula, 1966) and in the article (Brumberg, 1967). Here we only note some of their properties. With the index value q = 0, the Hansen coefficients are expressed in the final form as a function of eccentricity. In addition, for all admissible index values, we can write the expansion Xq (e) = e|q−j | l,j

∞ 

l,j

Xq,s e2s ,

(3.69)

s=0 l,j

where Xq,s are some numbers, and the series converges for all e < 1. The inclination and eccentricity functions for some index values are given explicitly in Appendix C. Since the second zonal harmonic prevails in the expansion of the force function of attraction of major planets of the Solar System, it is customary to consider the coefficient J2 to be a small parameter of the first order of small quantity. The remaining harmonics are much smaller. Their contribution to the perturbations of the elements of the satellite’s intermediate orbit is considered to be a small second-order small quantity. We now consider the expansion of the perturbing function due to the attraction of external bodies. For distant planetary satellites, the main of these external bodies is the Sun. Expansion in a form

89

90

Chapter 3 Equations of motion and analytic theories

convenient for practical applications first appears in Kaula (1962). An elegant derivation of this expansion was also made in Brumberg (1967). The expansion is also given in Murray and Dermott (2000). We reproduce here the expansion of the perturbing function caused by the attraction of external bodies, using the notation used in this chapter for the Keplerian elements a, e, i, M, ω, , as well as other related quantities n, λ = M + ω + , = ω + . The same quantities, but related to the perturbing body, will be denoted by the same letters, but with a prime. The gravitational parameter of the perturbing body (the product of the universal gravitational constant by mass) is denoted by μ . In these notations, the expansion has the form R=

k k ∞ k ∞ ∞  a k+1 (k − m)! μ       × (2 − δm,0 )  a a (k + m)!  q=−∞  k=2 m=0 p=0 p =0

q =−∞

k,k−2p 

× Fkmp (i)Fkmp (i  )Xk−2p+q (e)Xk−2p +q  (e )× k,k−2p

(3.70)

× cos[(k − 2p + q)M − (k − 2p  + q  )M  + (k − 2p)ω− − (k − 2p  )ω + m( −  )] Here the argument under the cosine sign can be expressed in terms of the mean longitudes and longitudes of the pericenters as follows: (k − 2p + q)M − (k − 2p  + q  )M  + (k − 2p)ω− − (k − 2p  )ω + m( −  ) = = (k − 2p + q)λ − (k − 2p  + q  )λ − q

+ q   + (m − k + 2p) − (m − k + 2p  ) . The value δm,0 is equal to one at m = 0 and to zero otherwise. The inclination functions Fkmp (i) and the eccentricity funck,k−2p tions Xk−2p+q (e) included in the expansion are considered in Appendix C. The following three properties of these functions are important for constructing motion models and analyzing satellite perturbations. 1. When the inclination of the orbits are small or close to 180 degrees, the following property is valid (Brumberg, 1967):   i i Fkmp (i) = O (sin )|k−2p−m| , (cos )|k−2p−m| . 2 2

Chapter 3 Equations of motion and analytic theories

It follows that, if we choose the main plane of the coordinate system that coincides with the plane of the orbit of the external body, then only those terms remain for the expansion of the perturbing function for which |k − 2p  − m| = 0. 2. At zero eccentricity, only those eccentricity functions k,k−2p Xk−2p+q (e) are different from zero for which q = 0. This follows from Eq. (3.69). Therefore, for the satellite’s circular orbits, only terms with q = 0 remain in the expansion of the perturbing function, and only q  = 0 remain in circular orbits of the external body. k,k−2p 3. For k − 2p + q = 0, the functions Xk−2p+q (e) are expressed in the final form without the use of series expansion in eccentricity degrees. Appendix C gives explicit expressions for some inclination functions and eccentricity functions. We note that the above expansion of the perturbing function (3.70), taking into account the indicated properties of the tilt functions and eccentricity functions, is used to construct a model of the evolution of satellite orbits. A special chapter of the book is devoted to this. The expansion (3.70) was also used in constructing the analytical theory of the motion of a satellite of Neptune, Triton (Emelyanov, Samorodov, 2015). How this was done is described below in a special section.

3.9 Determination of perturbations of elements of the planetary satellite intermediate orbit Let us consider some aspects of constructing the analytical theory of satellite motion using perturbation theory methods. Naturally, secular perturbations of the elements of the satellite’s intermediate orbit are of greatest interest. At least it is from them that the conclusion of all perturbations in theory begins. It can be seen from the above formulas that secular perturbations caused by the non-sphericity of the planet are created only by even zonal harmonics of the expansion of the force function, i.e., when k = 0 and n is even. Periodic perturbations of elements can have periods equal to 2π , ˙ ˙ − S) q M˙ + j ω˙ + k(

(3.71)

where the letters with the upper dot indicate the rate of change of the corresponding quantities. Here, among the arguments in the

91

92

Chapter 3 Equations of motion and analytic theories

denominator, M has a maximum rate of change. Therefore, for q = 0, the perturbations have a minimal period and are called shortperiod ones. Secular perturbation Returning to secular perturbations, they can be obtained by substituting secular terms into the Lagrange equations with respect to the elements and first integrating them for first-order perturbations. In some problems, secular perturbations of the second order may also be present. For satellites of large planets of the solar system, in the expansion of the force function of the planet’s attraction among zonal harmonics, the second harmonic (n = 2) has a maximum value. The next most significant harmonic is the fourth zonal harmonic (n = 4). Secular perturbations are also generated by the attraction of external bodies—the Sun and other satellites. We give here formulas for the mentioned secular perturbations. When determining secular perturbations, we set M = M0 + n1 (t − t0 ), ω = ω0 + n2 (t − t0 ),  = 0 + n3 (t − t0 ) . The values a, e, i, M0 , ω0 , 0 are considered arbitrary integration constants. If only first-order perturbations are determined, then the coefficients n1 , n2 , n3 can be found as the values of the righthand sides of the Lagrange equations with respect to the elements M, ω, and , respectively, after substituting the values in them constant values a, e, and i. The secular perturbations of the second order will have a similar form. To distinguish between secular terms of different orders due to various perturbing factors, we represent n1 , n2 , n3 in the following form: n1 = n[1 + ν1 (J2 ) + ν1 (J22 ) + ν1 (J4 ) + ν1 (m )], n2 = n[ν2 (J2 ) + ν2 (J22 ) + ν2 (J4 ) + ν2 (m )], n3 = n[ν3 (J2 ) + ν3 (J22 ) + ν3 (J4 ) + ν3 (m )], where νj (J2 ) and νj (J22 ) (j = 1, 2, 3) are the terms of the first and second orders due to the second zonal harmonic in the expansion of the force of gravity of the planet, νj (J4 ) (j = 1, 2, 3) are the firstorder terms due to the fourth zonal harmonic, νj (m ) (j = 1, 2, 3) are terms due to the attraction of an external body (the Sun, another satellite). The first-order secular perturbations with respect to the coefficient J2 for the second zonal harmonic of the expansion of the

Chapter 3 Equations of motion and analytic theories

force of gravity of the planet are obtained as follows. In the perturbing function, we take only the secular part, substitute it in the equations for the elements of the intermediate orbit. We integrate the equations, considering a, e, i to be constant. Denote by r0 the mean equatorial radius of the planet and set s = sin i. Thus, we obtain 3  r0 2 2 − 3s 2 , (3.72) ν1 (J2 ) = J2 4 a (1 − e2 )3/2 3  r0 2 4 − 5s 2 ν2 (J2 ) = J2 , 4 a (1 − e2 )2

(3.73)

3  r0 2 cos i ν3 (J2 ) = − J2 . 2 a (1 − e2 )2

(3.74)

In some problems, we must immediately calculate the sum ν1 (J2 ) + ν2 (J2 ) + ν3 (J2 ) = √ 3  r0 2 4 − 5s 2 + 1 − e2 (2 − 3s 2 ) − 2 cos i = J2 . 4 a (1 − e2 )2

(3.75)

Similarly, we get expressions for ν1 (J4 ), ν2 (J4 ), ν2 (J4 ): ν1 (J4 ) = −

ν2 (J4 ) = −

45  r0 4 2 8 − 40s 2 + 35s 4 J4 e , 128 a (1 − e2 )7/2

15  r0 4 4(16 − 62s 2 + 49s 4 ) + 9e2 (8 − 28s 2 + 21s 4 ) J4 , 128 a (1 − e2 )4 ν3 (J4 ) =

15  r0 4 cos i(4 − 7s 2 )(2 + 3e2 ) J4 . 32 a (1 − e2 )4

The second-order secular perturbations are more complex actions. We use the results obtained by Brower (1959). In our notation, we have ν1 (J22 ) =

 1 3 2  r0 4 J2 {−15 + 16 1 − e2 + 25(1 − e2 )+ 128 a (1 − e2 )7/2  + [30 − 96 1 − e2 − 90(1 − e2 )] cos2 i+  + [105 + 144 1 − e2 + 25(1 − e2 )] cos4 i} ,

93

94

Chapter 3 Equations of motion and analytic theories

ν2 (J22 ) =

 1 3 2  r0 4 J2 {−35 + 24 1 − e2 + 25(1 − e2 )+ 2 4 128 a (1 − e )  + [90 − 192 1 − e2 − 126(1 − e2 )] cos2 i+  + [385 + 360 1 − e2 + 45(1 − e2 )] cos4 i} ,

ν3 (J22 ) =

 3 2  r0 4 cos i J2 {−5 + 12 1 − e2 + 9(1 − e2 )− 32 a (1 − e2 )4  − [35 + 36 1 − e2 + 5(1 − e2 )] cos2 i}.

For orbits with small inclinations and eccentricities, it is more convenient to use another form of these formulas, 1 3 2  r0 4 J ν1 (J22 ) = [3(40 − 80s 2 + 35s 4 )+ 128 2 a (1 − e2 )7/2  +16(4 − 12s 2 + 9s 4 ) 1 − e2 + 5(−8 + 8s 2 + 5s 4 )(1 − e2 )] , ν2 (J22 ) =

1 3 2  r0 4 J [5(88 − 172s 2 + 77s 4 )+ 128 2 a (1 − e2 )4

 +24(8 − 22s 2 + 15s 4 ) 1 − e2 + (−56 + 36s 2 + 45s 4 )(1 − e2 )] , ν3 (J22 ) =

3 2  r0 4 cos i J [5(−8 + 7s 2 )+ 32 2 a (1 − e2 )4

 +12(−2 + 3s 2 ) 1 − e2 + (4 + 5s 2 )(1 − e2 )] . Periodical perturbations Periodic perturbations of elements of the intermediate orbit, due to the non-sphericity of the planet, have small amplitudes. Therefore, the perturbed motion of a satellite of an oblate axisymmetric planet can be represented as motion near a certain ellipse, the plane of which precesses with a constant inclination around ˙ in the axis of symmetry of the planet with an angular velocity  the direction opposite to the orbital motion. In this case, the apse line precesses in the orbit plane with an angular velocity of ω. ˙ The direction of rotation of the apsid line is determined by the sign of the expression 4 − 5 sin2 i. The addition of M˙ to the angular velocity of orbital motion does not change the nature of the motion. We note a curious fact in the theory of the perturbed satellite motion of an oblate planet. Let us try to determine the periodic

Chapter 3 Equations of motion and analytic theories

perturbations of the elements generated by the expansion terms of the perturbing function for q = 0 and k = 0. Obviously, the precession rate of the line of apsides ω˙ has a first-order small value. The period corresponding to these perturbations by Eq. (3.71) is maximal among periodic perturbations. Therefore, these perturbations are called long-period. It is clear that the second zonal harmonic at n = 2 with the coefficient J2 will also generate longperiod perturbations in the elements of the satellite’s intermediate orbit. Since J2 is of the first order of smallness, we should expect to obtain a long-period perturbation of the first order. When integrating the Lagrange equations for the expansion terms of the perturbing function under consideration, the quantity ω, ˙ having the first order of smallness, enters the denominator of the expression. As a result, we obtain a perturbation of the zeroth order of smallness. At first glance, it turns out that the small parameter method refuses attempts to describe the perturbed satellite motion of an oblate planet. The situation is saved by a “gift of nature”. Namely, the required long-period perturbation for n = 2, q = 0 and k = 0 has eccentricity functions equal to zero, that is, X0−3,2 (e) = X0−3,−2 (e) = 0 for all e < 1. For this reason, long-period perturbations caused by the second zonal harmonic of the expansion of the planet’s gravitational force function are simply absent. Long-period perturbations caused by other harmonics of the expansion will be small, since the corresponding coefficients are of the second order of small quantity. We note an important property of perturbations of the elements of the intermediate satellite orbit of a non-spherical planet. Applying the small parameter method, as described above, we will obtain secular, periodic, and, possibly, mixed perturbations. The presence of secular perturbations in the elements a, e, and i could lead to catastrophic consequences, the planet’s loss of its satellites. Mixed perturbations would lead to the conclusion that the theory is untenable. However, as proved in Aksenov (1966), the perturbations of the elements of the satellite’s intermediate orbit due to the non-sphericity of the planet do not contain secular elements in a, e, i, and in all six elements a, e, i, M, ω,  there are no mixed perturbations of all orders of smallness. This can be proved by mathematical induction, applying it to a sequence of perturbations of each next order of smallness.

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The determination of perturbations caused by the attraction of other bodies, namely other satellites and the Sun, causes particular problems. We turn to the general form of the expansion of the perturbing function (3.65). As prescribed by the methods of perturbation theory, the secular perturbations of the elements of the intermediate orbit are first determined. The elements ω, , constant in the Keplerian motion, will now become linear functions of time with a coefficient at time proportional to the small parameter of the perturbing function. Then there are periodic members. In this case, terms containing only ω,  under the sign of trigonometric functions are encountered. When integrating the Lagrange equations over time, the small parameter will appear in the denominator and contract with the small parameter in the perturbing function. Here, when constructing the theory by the small parameter method, a contradiction arises: small perturbations are actually not small. Such perturbations turn out to be very long-period ones. If such a theory is applied over a short time interval, then such perturbations can be considered constant. They will actually be taken into account when determining motion parameters from observations. However, at large intervals of time, a theory like this cannot be constructed. Therefore, for distant satellites of major planets, acceptable analytical theories of motion have not yet been constructed. In some cases, when secular perturbations from the planet’s oblateness prevail, with the above integration of the Lagrange equations for the periodic terms of the expansion of the perturbing function, the denominator does not receive as small a value as a small parameter due to the attraction of another body. An example of such a favorable situation is discussed below. Thus, the analytical theory of the motion of the main satellite of Neptune, Triton, was successfully constructed (Emelyanov and Samorodov, 2015). Another problem of taking into account the attraction of other satellites is related to the need to expand the perturbing function in powers of the ratio of the semi-major axes of the orbits of the perturbed and perturbing bodies. This expansion is presented in the book (Murray and Dermott, 2000). In determining the perturbations due to the mutual attraction of the main satellites of Jupiter, Saturn and Uranus, the analytical theory cannot be constructed using the usual method of perturbation theory. Examples of special theories for these satellites are briefly discussed below in a special section.

Chapter 3 Equations of motion and analytic theories

3.10 Constant perturbation of the semi-major axis of the satellite’s orbit Secular perturbations of the satellite’s intermediate orbit elements are the main perturbations when considering motion over time intervals that significantly exceed the satellite’s orbital period. By definition, secular perturbations are obtained from the secular part of the expansion of the perturbing function. It is easy to see in the Lagrange equations that secular first-order perturbations in the elements a (semi-major axis), e (eccentricity) and i (inclination) are absent. As was stated above, the semi-major axis of the orbit a is related to the unperturbed value of the mean motion n by the relation n=

Gm , a3

(3.76)

where G is the universal gravitational constant and m is the mass of the planet. We can assume that the inclusion of secular perturbations in the elements M, ω, and  gives an acceptable model of satellite motion, since periodic perturbations proportional to the small parameter of the perturbing function are small. In fact, such a model of secular perturbations will not be the best when consistent with observations. It turns out that, for a given value of n, another value of the semi-major axis is better consistent with the observations. The fact is that the combination of short-period perturbations in the eccentricity e and the mean anomaly M gives a constant term in the perturbations of the satellite’s central distance r. However, in the theory of secular perturbations, short-period perturbations are discarded. The best model of secular perturbations is obtained if, instead of a, we take the value a, calculated by the formula a = a + δr,

(3.77)

where δr is the constant part of the perturbations of the central distance. The definition of δr will be made here for the case of perturbations caused by the second zonal harmonic of the expansion of the force of gravity of the planet. In addition, we will assume that the eccentricity of the satellite’s orbit e is so small that in expansions in eccentricity degrees one can restrict oneself to the lowest-order term. That is, we neglect the magnitude of the eccentricity compared to unity.

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With accepted accuracy, we can write δr = (1 − e cos M)δa − a cos Mδe + ea sin MδM,

(3.78)

where δa, δe, δM are the perturbations of the corresponding elements. We take the expansion of the perturbing function according to Eqs. (3.66), (3.67), and (3.68). We leave in the expansion only the terms corresponding to the second zonal harmonic of the expansion of the force of gravity of the planet. We obtain R = Gm

2 ∞   r02 −3,2−2p C F (i) Xq (e) cos[qM + (2 − 2p)ω] , 20 2,0,p a3 q=−∞ p=0

where C20 is the coefficient at the second zonal harmonic, and r0 is the mean equatorial radius of the planet, adopted when determining C20 . Moreover, C20 = −J2 . In the last sum, there will be only four short-period terms of the lowest order with respect to small eccentricity, which correspond to the following combinations of values of the summation indices p = 0, q = 1, p = 1, q = 1, p = 1, q = −1, p = 2, q = −1. Leaving only these terms in the sum, we find R = Gm

r02 C20 × a3

 F200 (i)X1−3,2 (e) cos(M + 2ω) + F201 (i)X1−3,0 (e) cos M+  −3,0 −3,−2 (e) cos(−M) + F202 (i)X−1 (e) cos(−M − 2ω) . F201 (i)X−1 The inclination and eccentricity functions included here with the accepted accuracy are of the form 3 3 1 F200 (i) = F202 (i) = − sin2 i, F201 (i) = (i) sin2 i − , 8 4 2 1 3 −3,−2 −3,0 X1−3,2 (e) = X−1 (e) = − e, X1−3,0 (e) = X−1 (e) = e. 2 2 Substituting them in the previous expression for the perturbing function, we have    r2 1 1 3 2 sin i − cos M + sin2 i cos(M + 2ω) . R = Gm 03 C20 3e 4 2 8 a

Chapter 3 Equations of motion and analytic theories

Now we write out the necessary equations of perturbed motion with respect to the Keplerian elements a, e and the function M: da 2 ∂R = , dt na ∂M de 1 − e2 ∂R = − dt ena 2 ∂M

√ 1 − e2 ∂R , ena 2 ∂ω

dM 2 ∂R 1 − e2 ∂R =n− − . dt na ∂a ena 2 ∂e After substituting the simplified expression derived above for the perturbing function, the equations take the form r2 da 6e = −Gm 03 C20 dt na a r2 de 3 = −Gm 03 C20 2 dt a na



  3 2 1 2 1 sin i − sin M + sin i sin(M + 2ω) , 4 2 8



  3 2 1 2 1 sin i − sin M − sin i sin(M + 2ω) , 4 2 8

r2 dM 3 = −Gm 03 C20 dt a ena 2



 3 2 1 sin i − cos M+ 4 2

 1 + sin2 i cos(M + 2ω) . 8 The integration of equations by the small parameter method gives the following first-order perturbations with respect to the coefficient C20 :    3 2 1 6e 1 sin i − cos M + sin2 i cos(M + 2ω) , δa = r02 C20 a 4 2 8 δe = r02 C20

δM

3 a2

= −r02 C20



3 ea 2

  3 2 1 1 sin i − cos M − sin2 i cos(M + 2ω) , 4 2 8



  3 2 1 2 1 sin i − sin M + sin i sin(M + 2ω) . 4 2 8

Substituting these expressions in Eq. (3.78), we find the main short-period perturbations of the central distance   1 1 3 2 sin i − (1 − e cos M) cos M+ δr = r02 C20 2 6ea 4 2 a

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 1 + sin2 i (1 − e cos M) cos(M + 2ω) − 8 1 −r02 C20 2 3a a



1 −r02 C20 2 3a a

  3 2 1 1 2 2 cos M − sin i cos(M + 2ω) cos M − sin i − 4 2 8



  3 2 1 1 2 2 sin i − sin M + sin i sin(M + 2ω) sin M . 4 2 8

Let us single out the constant part of the lowest-order perturbation with respect to the eccentricity δr =

3 r02 C20 a(2 − 3 sin2 i). 4 a2

As a result, it turns out that, for a given satellite mean motion n, the perturbed constant value of the semi-major axis (3.77) should be determined by the formula

3 r02 2 a =a 1+ C20 (2 − 3 sin i) , (3.79) 4 a2 where a is found from Eq. (3.76). Note that, from the observations, the perturbed value of the mean motion n1 is usually determined. Then the unperturbed value of n is found by iterations based on the relation n1 = n[1 + ν1 (J2 )], where ν1 (J2 ) is determined by Eq. (3.72), and the unperturbed value of the semi-major axis a is found from the relation n2 a 3 = Gm. For small inclinations and eccentricities, the perturbed mean motion in longitude n is best determined from observations. In this case, n is iterated from the relation n = n[1 + ν1 (J2 ) + ν2 (J2 ) + ν3 (J2 )],

(3.80)

where ν2 (J2 ) and ν3 (J2 ) are determined by Eqs. (3.73), (3.74). Let us specifically consider the case of small eccentricities and inclinations of the satellite orbit of an oblate planet. Neglecting the eccentricity and inclination in the relations (3.79) and (3.80), we obtain   r02 n = n 1 + 3J2 2 , (3.81) a

Chapter 3 Equations of motion and analytic theories



3 r2 a = a 1 − J2 02 2 a

 .

(3.82)

As a result, taking into account the main perturbations caused by the oblateness of the planet allows us to imagine a satellite circulating with a frequency of n in an orbit of radius a. Such a model would best fit observations. Kepler’s third law, modified due to perturbations, will take the form   3 r02 3 2 a n = Gm 1 + J2 2 . (3.83) 2 a The relationship Gm = n2 a 3 is used to determine the mass of the planet from satellite observations. As follows from Eq. (3.83) in the case of an oblate planet, the value of the right-hand side of this formula is actually obtained from observations. To take into account the oblatenes of the planet in this case, one must either accept some hypotheses regarding the magnitude of the dynamic oblateness, or try to determine the coefficient J2 from observations.

3.11 Precessing ellipse model To study the dynamics of the planet’s satellite, one or another model of motion is built on the basis of observations. In many cases, it is important to see the basic properties of motion, neglecting small perturbations. Sometimes approximate ephemeris is required. For some satellites, the accuracy of the observations is not high, and the simplest model of motion can give good agreement with the results of observations. In all these cases, some approximate model of motion, called the precessing ellipse, is suitable. In this model, the satellite moves along a certain ellipse, the plane of which precesses at a constant angular velocity with a constant inclination to some basic fixed plane. The line of the apses of the ellipse rotates in the plane of the orbit with a constant angular velocity. The satellite moves along a precessing ellipse according to the laws of the Kepler problem, however, the constant mean motion differs from that obtained from the semi-major axis of the orbit according to the known relation. From the previous sections it is clear that such a model can be created in the process of constructing an analytical theory of motion by taking into account secular perturbations caused by the main perturbing factors. The approximate solution of the

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exact differential equations of motion of the sixth order is obtained. Therefore, this solution contains six arbitrary integration constants. In practice, the determination of secular perturbations is difficult, inaccurate, or even impossible due to the fact that the parameters characterizing the perturbing factors are inaccurate or unknown. In such cases, proceed as follows. Changes to the M function and the ω and  elements are specified by linear time functions ˙ − t0 ), (3.84) ¯ − t0 ), ω = ω0 + ω(t ˙ − t0 ),  = 0 + (t M = M0 + n(t ˙ are constants. Elements of the Keplerian ¯ ω0 , ω, ˙ 0 ,  where M0 , n, satellite orbit e, i are also considered constant. The semi-major axis, which is directly included in the Keplerian motion formulas, is set independently of the parameter n. ¯ Since these parameters will no longer be connected by the laws of Keplerian motion, we denote the major axis by a. ¯ To determine the coordinates, Keplerian motion formulas are used. The mean anomaly M and the elements ω,  are calculated using Eqs. (3.84), the parameters a, ¯ e, i are independently substituted into the Keplerian motion formulas. ˙ ˙ 0 , , As a result, nine constant parameters a, ¯ n, ¯ e, i, M0 , ω0 , ω, Eqs. (3.84), and Keplerian motion formulas fully describe the motion model. The main plane relative to which the inclination i, the longitude of the ascending node , and the angular distance of the pericenter from the ascending node of the orbit are measured, are set so that in this coordinate system the changes in the inclination obtained from the observations are minimal. All nine parameters are refined from the observations so that the model is consistent with the observations. Such a model takes into account all secular perturbations caused by any possible reasons. The precessing ellipse model is also used to approximate a more accurate satellite motion model based on analytical theory, or constructed by numerically integrating differential equations of motion. An example of using the precessing ellipse model for four close moons of Jupiter is described in Emel’yanov (2015). The model parameters were determined from the satellite ephemeris calculated on the basis of two different models, each of which, in turn, was built on the basis of observations by numerically integrating the equations of motion.

Chapter 3 Equations of motion and analytic theories

3.12 Perturbed motion at small eccentricities of the orbits 3.12.1 Problem formulation In the classical approach to the theory of perturbed motion, when the Lagrange equations for the elements of the Keplerian orbit are solved by the method of the small Poincaré parameter, there are little studied cases that sometimes lead to unusual conclusions. Such cases include perturbed motion with small eccentricities of the intermediate orbit. When solving the Lagrange equations, it is first necessary to find secular perturbations, since they are the most significant in both the quantitative and the qualitative aspects of the description of motion. Secular perturbations are usually obtained by expanding the perturbing function and discarding the periodic terms on the right-hand sides of the Lagrange equations. So the model of a precessing ellipse was created and began to be widely used. For perturbations caused by the non-sphericity of the axisymmetric central body, for example, an oblate major planet, the orbit plane will precess around the axis of symmetry of the body with a constant inclination. The unperturbed Keplerian orbit can be reduced to a circular orbit, and the inclination in the unperturbed motion is determined only by the choice of a system of rectangular coordinates. The eccentricity is a natural parameter that describes the degeneracy of an orbit into a circular one. For a circular orbit, it is simply zero. One would expect that, for perturbed motion, when the eccentricity changes in time, its proximity or equality to zero should lead to the degeneracy of the orbit into a circular one. In fact, this is not so at all. The first mention of the existence of circular perturbed motion with nonzero eccentricity is found in Beletskii (1963). In this work, a particular solution of the equations for the elements of Kepler motion is indicated, in which the circular orbit in the osculating elements is described by an ellipse rotating with the angular velocity of the satellite around the central body, and the satellite is always located in the pericenter of this ellipse. However, in the above-mentioned paper by Beletskii it was noted that this example was due to Eneev. The widely used model of a precessing ellipse with an arbitrary constant and arbitrarily small eccentricity contradicts the example given in Beletskii (1963). It is interesting to find out how these two models relate to each other. It is important to have rigorous proof of the existence of a solution for circular perturbed motion

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with nonzero eccentricity. It would be useful to find a fairly simple solution that adequately describes the perturbed motion at small eccentricities. Such a study was performed in Emel’yanov (2015). In the next section, we give the main calculations and conclusions from this paper.

3.12.2 Constructing a model of circular perturbed motion Consider the motion of a material point in a central attraction field with a force function of the form μ U = [1 + f (r)], r where r is the central distance, μ is the gravitational parameter, and f (r) is some rather arbitrary dimensionless function. We expand the force function into two terms as follows: U = V + R, V =

μ μ , R = f (r), r r

where V is the force function of the Kepler problem, and R is the perturbing function. The motion at R = 0 is called unperturbed one. We show that, with some restriction on the function f (r), equations with the force function U will have a particular solution corresponding to circular motion. Let us consider in more detail the perturbed motions of a material point that are close to circular. We will try to construct approximate analytical models of the motion of real planetary satellites with small eccentricities of the orbits. For example, take four close moons of Jupiter. Since the force function depends only on the distance, the trajectory of motion lies in a certain unchanged plane in which we define a non-rotating coordinate system Oxy with the origin O in the attracting center. Then the differential equations of motion can be written as μx d 2y μy d 2x = − [1 + F (r)], = − 3 [1 + F (r)], 2 3 2 dt r dt r

(3.85)

where the dimensionless function F (r) is given by F (r) = f (r) − rf  (r), and the prime here means the first derivative with respect to r.  Here r = x 2 + y 2 .

Chapter 3 Equations of motion and analytic theories

In the region in which the condition F (r) > −1 is satisfied, there exist solutions of differential equations of motion corresponding to the circular motions of the point around the attracting center at any distance r with a linear velocity Vc depending on r. The centripetal acceleration of a point moving along a circle of radius r with a linear velocity Vc , regardless of the cause of the acceleration, is determined by the expression Vc2 . r The reason for the movement around the circumference is the gravitational force, which creates acceleration μ [1 + F (r)]. r2 Equating these two expressions for acceleration, we obtain Vc2 =

μ [1 + F (r)]. r

(3.86)

Without violating the generality of the problem, we will assume that at the initial instant t0 the point is on the axis x. Then the particular solution of the differential equations for circular motion in coordinates will have the form x = r cos nc (t − t0 ), y = r sin nc (t − t0 ),

(3.87)

where the frequency of revolution of the point nc is determined by the formula Vc nc = . r This will be a one-parameter family of particular solutions. As a parameter of this family, we can take r or nc . We pose the problem of finding the elements of the Keplerian osculating orbit as a function of time for circular motion, considering Eqs. (3.85) as the equations of the perturbed Keplerian motion. To do this, we need to replace the variables of the required functions x, y with elements of the Keplerian osculating orbit as a function of time and substitute the particular solution (3.87) in the variable replacement formulas instead of x, y. In addition, it is desirable to consider the equations of motion expressed in Keplerian elements, that is, the Lagrange equations, and to check whether the obtained expressions for Keplerian elements satisfy these equations.

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3.12.3 Transition to the elements of the Keplerian orbit Since we are considering plane motion here, it suffices to use four Keplerian elements to describe the motion: a is the semimajor axis, dimension length unit; e is the eccentricity, dimensionless units; M is the mean anomaly, radians; ω is the angular distance of the pericenter from the ascending node of the orbit, radians. Along with the semi-major axis a, we will also consider, as an orbit parameter, the parameter n, called the mean motion and connected with a using the law μ n= . a3 The dimension of the mean motion n is radians per time unit. The relationship of rectangular coordinates and velocity components with elements of the Keplerian orbit can be found in books (Duboshin, 1975; Subbotin, 1968). For Kepler motion, the velocity V and the distance r are related by the relation   2 1 V2 =μ − , (3.88) r a where a is a constant. In the case of circular Keplerian motion at e = 0, we have r = a and μ Vc2 = , r which is different from (3.86). Therefore, it turns out that a particular solution of the equations of perturbed motion (3.85) cannot be represented by elements of a Keplerian osculating orbit with a constant zero eccentricity. It is possible to represent perturbed circular motion in one of two ways depending on the sign of F (r). If F (r) > 0, then we put M = 0, r = a(1 − e), ω = nc (t − t0 ), and the elements a and e will be considered as constants. Comparing (3.86) and (3.88), we find e = F (r). Then we can find that the material point is always in the pericenter of the orbit, and the line of apsides rotates with the angular

Chapter 3 Equations of motion and analytic theories

velocity of nc . True v and eccentric E anomalies are constant and equal to zero. The element a can be found by the formula a=

r . 1−e

The frequency nc of the revolution of a point around the attracting center is expressed through the elements of the Keplerian osculating orbit by the formula √ μ 1+e . (3.89) nc = 3 a (1 − e)3/2 In the case −1 < F (r) < 0, we should put M = π, r = a(1 + e), where e = −F (r). In this case, it turns out that the point is always in the apocenter of the orbit, and the line of apsides rotates with the angular velocity of nc . The true and eccentric anomalies v and E are constant and equal to π. The semi-major axis of the orbit a and the velocity of rotation of the line of apsides can be found by the formulas √ μ 1−e r . a= , nc = 1+e a 3 (1 + e)3/2 Here we have found expressions for the elements of the osculating Keplerian orbit as functions of time for particular solutions of the equations of motion corresponding to the circular orbits of a material point around an attracting center. These formulas are obtained for an arbitrary function f (r) under the condition F (r) > −1. We consider a particular case of perturbed motion. Let the point move in the equatorial plane of an axisymmetric planet under its attraction. In the expansion of the force function of the planet’s attraction in a series of spherical functions, we take only the main term and the second zonal harmonic. For major planets, the dynamical oblateness of the planet in the motion of its satellite is thus taken into account. Then we have 1 1 1 1 3 1 R = μJ2 r02 3 , f (r) = J2 r02 2 , F (r) = J2 r02 2 . 2 2 2 r r r There is always F (r) > 0, and the circular motion at any given distance r is represented by a Keplerian osculating orbit with a rotating apsid line, and the point is constantly in the pericenter of

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the orbit. The eccentricity and semi-major axis of such an orbit are determined by the formulas 3 1 r , e = J2 r02 2 , a = 2 1−e r

(3.90)

and the velocity of rotation of the apsid line is found by Eq. (3.89). Let us try to compare the model of circular perturbed motion of the satellite constructed here with the model of circular perturbed motion, discussed in Sect. 3.10. In both cases, the perturbations are due to the second zonal harmonic in the expansion of the force function of gravity of the planet. In Sect. 3.10, the radius of the circular orbit was equal to the perturbed value of the semi-major axis a, and the satellite’s revolution frequency was equal to the perturbed value of the mean motion n. Therefore, in accordance with the circular motion model constructed here above, we must put a = r, n = nc . We now express the left side of Eq. (3.83) using Eq. (3.89) for nc and the above expression r = a(1 − e) for r. After simple transformations, we obtain   3 3 2 2 1 a n = μ 1 + J2 r0 2 , 2 a which exactly reproduces Eq. (3.83). Here, as in Sect. 3.10, we neglected small quantities of the order of J22 compared to unity. Thus, we have proved the complete correspondence of the two considered models of the circular perturbed motion of the satellite.

3.12.4 Osculating Keplerian elements of the satellite’s orbit in perturbed motion with small eccentricities In the often used model of precessing ellipse, for any values of the eccentricity of the orbit, including zero, the mean anomaly varies linearly in time. This contradicts the conclusion that perturbed motion in a circular orbit is possible only for some nonzero value of the eccentricity and the mean anomaly constant and equal to zero. It would be interesting to find out how the elements of the Keplerian osculating orbit change when the satellite moves under the attraction of an oblate planet. Since no exact analytical solution to this problem was found, analysis can be performed only on the results of numerical integration.

Chapter 3 Equations of motion and analytic theories

Consider again the motion of a satellite of an axisymmetric planet in the equatorial plane. Take the equations of motion (3.85) with a force function U=

r2 μ 1 + μJ2 03 . r 2 r

Here, from the expansion of the force function of the planet’s attraction, the main term and the second zonal harmonic are taken. The author performed a numerical integration of these Eqs. (3.85) and obtained the coordinates x, y and the satellite velocity components x, ˙ y˙ for a number of time instants with a constant step. Then, regardless of the possible approximate analytical solutions, in the exact solution obtained, we will replace the variables x, y, x, ˙ y˙ with the variables a(t), e(t), M(t), ω(t) according to the Keplerian motion formulas. Let us see how these variables change over time under various initial conditions. Such calculations were made. Below we present the obtained results. For our example, the gravitational parameter μ and the initial conditions were taken approximately corresponding to the close satellite of Jupiter, Adrastea. For Jupiter, μ = 126712763.92 km3 /c2 , J2 = 0.014736, r0 = 71398.0 km. We took the initial conditions so that at the initial instant of time, the coordinate y is equal to zero, and the vector of initial velocity is perpendicular to the axis x. The value of the initial distance x0 , approximately corresponding to the satellite of Jupiter, Adrastea, was taken equal to 127748.287979217545 km. The calculations are made for a number of values of the initial velocity V . In the first version, the velocity value was taken corresponding to the circular motion, V = Vc (3.86). In the following options, calculations were done for V > Vc . The integration time interval corresponded to two revolutions of the satellite around the planet. The results of the time variation of the osculating elements M, ω are presented in Fig. 3.2–3.5. The solid thin line shows the values of the mean anomaly M, the dashed line indicates the values of ω, and the thick line shows the values of the sum M + ω, that is, the mean orbital longitude. From Fig. 3.2 we can see that for V = Vc (circular motion) the mean anomaly M remains equal to zero, and the curve ω merges with the mean longitude curve M + ω and shows a linear rotation of the apsidal line with the angular velocity of the satellite around the planet. At the next higher value of the initial velocity in Fig. 3.3, it can be seen that strong oscillations are superimposed on the changes in M and ω in antiphase to each other, and the mean longitude is still linearly increasing in time. For the next value of the initial velocity in Fig. 3.4, the critical nature of the change in the elements occurs. The oscillations

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Figure 3.2. Change in the mean anomaly M, the angular distance of the pericenter from the orbit node ω and M + ω for the initial velocity of circular motion. The mean anomaly is zero. The line of apsides rotates with the angular velocity of circulation of the material point around the central body.

of the elements M and ω have an amplitude of up to half a revolution. At the same time, longitude still increases linearly in time. For the next value of the initial velocity, plots in Fig. 3.5 demonstrate that M and ω exchanged characters of change. Now, in the mean anomaly M, oscillations are superimposed on a linear increase in time, and ω has a periodic change with a small linear course. To distinguish between the two qualitative nature of the change in the mean anomaly, we will call the monotonous increase in the mean anomaly circulational change. Cases when the mean anomaly fluctuates with respect to the zero value will be called librational change. It can be seen from the considered examples that there exists some separatrix value of the initial velocity V = Vs that separates two different families of solutions. A change in eccentricity in the considered solution families is of particular interest. In Fig. 3.6 we showed with dots on concentric circles the curves for changes in the eccentricity e and the mean anomaly M for a number of the initial velocity. On the abscissa axis, we showed the values of e cos M, and on the ordinate axis we presented e sin M. Thus, the distance of the point from the origin of coordinates (0, 0) on the plots gives us the eccentricity value, and the angle between the central direction and the abscissa axis gives us the value of the mean anomaly M. The so-

Chapter 3 Equations of motion and analytic theories

Figure 3.3. Change in the mean anomaly M, the angular distance of the pericenter from the orbit node ω and M + ω for an initial velocity slightly exceeding the velocity of circular motion. The mean anomaly oscillates relative to zero with a period equal to the revolution period.

Figure 3.4. The critical nature of the change in the mean anomaly M, the angular distance of the pericenter from the orbit node ω, and the mean anomaly M + ω for some particular value of the initial velocity of motion. The mean anomaly and the angular distance of the pericenter from the node show antiphase oscillations with a large amplitude.

111

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Chapter 3 Equations of motion and analytic theories

Figure 3.5. Changes in the mean anomaly M, the angular distance of the pericenter from the orbit node ω and the mean longitude M + ω at an initial velocity exceeding the critical value. The mean anomaly increases monotonically over time.

lution corresponding to the circular motion (V = Vc ) is shown on the plot by the point at which M = 0, and the eccentricity of e has the value calculated by Eq. (3.90). Fig. 3.6 shows two solutions for V values greater than Vc , in which the mean anomaly M oscillates near zero, and the eccentricity varies within small limits. The “s” shows a particular solution in which the eccentricity changes from a certain maximum value to zero, and the mean anomaly ranges from (−π, π). The plots also show two solutions in which the mean anomaly M has a circulating character of motion, with the eccentricity changing in some small limits. The particular solution “s” separates two families of solutions with different properties of changes in eccentricity and mean anomaly. A solution with such initial conditions is called a separatrix solution. In Fig. 3.6 we have shown with thin lines another family of element changes. This family is obtained on the basis of an analytical solution to the generalized problem of two fixed centers, considered above in a special section. This figure demonstrates that this solution coincides with the results of numerical integration, at least with the accuracy of the image of the lines on the plots. From the obtained results, it follows that the model of a precessing ellipse with constant eccentricity as a variant of an approximate analytical solution does not reflect the qualitative picture of families of solutions.

Chapter 3 Equations of motion and analytic theories

Figure 3.6. Changes in the mean anomaly M and eccentricity e in the exact solution (points), in the refined model based on the solution of the generalized problem of two fixed centers (circles passing through the points) and in the model of precessing ellipse (concentric circles centered at the origin).

In Emel’yanov (2015), the first integrals of differential equations of motion in the problem under consideration are derived. They helped to find such initial conditions that separate two families of solutions with different properties of changes in eccentricity and mean anomaly, i.e. initial conditions for a separatrix solution. In the above paper, it is shown that in the separatrix motion, the distance varies from a certain minimum value at which the eccentricity is maximum, to a maximum value when the eccentricity is zero. Note that in the separatrix solution, at the moments when the true anomaly takes zero value at zero eccentricity, the mean anomaly also takes zero value. We studied the accuracy of describing the motion of a satellite of an oblate planet using the example of the motion of the Jupiter satellite discussed above (Emel’yanov, 2015) with the two models considered: the model of a precessing Kepler ellipse (PE) and a model based on the solution of the generalized problem of two fixed centers, and which we will designate as RM (the refined model). The family of solutions obtained by numerical integration of the equations of motion of an oblate planet over the time interval of two revolutions of the satellite around the planet was taken as reference solutions. The results of the study allow us to draw the following conclusions. The error in representing the exact solution by the model of solving the generalized problem of two fixed centers is 120 times less than the error in representing the precessing

113

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ellipse by the model for all variants of the initial conditions. For both models, the error decreases as the trajectory approaches the circular one. An attempt was also made in Emel’yanov (2015) to present, with the two analytical models considered (PE and RM), the ephemeris of four close moons of Jupiter, constructed by numerically integrating the equations of motion based on available observations. When constructing these ephemeris, perturbations from the attraction of the Galilean satellites of Jupiter were taken into account, the perturbing effect of which is significantly not taken into account in analytical models. Since the mismatches caused by neglect of these perturbing factors prevail, the mismatches turned out to be only slightly less when using the model based on the solution of the generalized problem of two fixed centers, in comparison with the precessing ellipse model. Both models represent with almost equal accuracy the motion of the moons of Jupiter mentioned above.

3.13 Constructed analytical theories of planetary satellite motion 3.13.1 Analytical theory of the motion of Neptune’s satellite Triton Let us consider in more detail the analytical theory of the motion of the satellite of Neptune, Triton, constructed in Emelyanov and Samorodov (2015). By the time this work was completed, several versions of the satellite’s motion model already existed, based on the numerical integration of the differential equations of motion. However, a very favorable ratio of perturbing factors and the suitable properties of the Triton orbit allowed the authors to demonstrate the advantages of the analytical theory in comparison with methods of numerical integration. The Triton–Neptune system is characterized by the following approximate values of the parameters (Jacobson et al., 1991). The equatorial radius of Neptune is 25225 km, the radius of Triton is 1350 km. The ratio of the mass of the satellite to the mass of the planet is 0.0002089. Triton revolves around the planet in an almost circular orbit with a period of 5.87685244989 days at a distance of 354700 km. The inclination of the orbit to the planet’s equator is approximately 156.66 degrees. The right ascension and declination of the North Pole of Neptune are approximately 299.5 and 43.6 degrees, respectively. The satellite’s motion is affected by the dy-

Chapter 3 Equations of motion and analytic theories

namic oblateness of Neptune and the gravitational attraction of the Sun. The attraction of the other Neptune moons is negligible. The criterion for the need to take into account certain perturbations is the accuracy of the observations. Therefore, an analysis of all possible perturbations in the elements of the Triton intermediate orbit was carried out. The terms of the expansion of the perturbing function, which contribute to the apparent coordinates of the satellite, not exceeding the accuracy of the observations, were discarded. From previous work by other authors, it was clear that the eccentricity of the orbit of Triton is negligible. Therefore, the orbit was previously assumed circular. The authors of the article (Emelyanov and Samorodov, 2015) estimated short-period perturbations in the elements of a circular Kepler orbit showed that they need not be taken into account. It remains necessary to take into account only secular and long-period perturbations of the first order of smallness. It follows from the Lagrange Eqs. (3.39) that there will be no long-period perturbations of the first order in the semi-major axis of the orbit. They can be generated only by terms of the perturbing function that do not contain the mean anomaly M, and such terms, when substituted to the right-hand side of the equation with respect to the semi-major axis, give zero. Therefore, in the cited paper (Emelyanov and Samorodov, 2015), the semi-major axis a was assumed to be constant. Triton by its attraction affects the rotational motion of Neptune. Therefore, the axis of rotation of the planet precesses synchronously with the movement of the plane of Triton’s orbit around the vector of the total angular momentum of the planet and satellite. The constant angle between these axes is approximately 0.506 degrees (Jacobson, 2009). The angles of inclination of the orbit relative to the axis of rotation of the planet and relative to the vector of the total angular momentum also remain constant. As shown in Jacobson (2009), such a model of planet and satellite motion is closest to reality. Small deviations from it can only be caused by perturbations from the attraction of the Sun. Consider the coordinate systems adopted in this theory. The basic coordinate system is associated with a fixed vector of the total angular momentum of the planet and satellite. The axis z of the system is directed along this vector, the axis x along the line of intersection of the plane xy with the plane of the plane of the Earth’s equator so that the axis y is inclined to the plane of the plane of the Earth’s equator at an acute angle. Below we will call such a system an orbital coordinate system.

115

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Chapter 3 Equations of motion and analytic theories

Figure 3.7. The location of the main planes in the theory of the motion of Triton.

The location of the orbits of the Triton and the Sun, as well as the plane of the plane of the Earth’s equator in the orbital coordinate system, is shown in Fig. 3.7. Denote by xg , yg , zg the axes of the planetocentric coordinate system connected with the plane of the Earth’s equator. The relationship of these coordinates with the coordinates of the orbital system is determined by the right ascension α0 and the declination δ0 of the z axis. The transition to the coordinates xg , yg , zg is carried out according to the formulas xg = − sin α0 x − cos α0 sin δ0 y + cos α0 cos δ0 z, yg = cos α0 x − sin α0 sin δ0 y + sin α0 cos δ0 z, zg = cos δ0 y + sin δ0 z.

(3.91)

The parameters α0 and δ0 are not known in advance. They can only be determined from observations of the satellite’s motion. The main factor in the non-sphericity of Neptune is the second zonal harmonic of the expansion of the gravitational attraction function. This factor cannot cause long-period perturbations of the first order. The axis of symmetry of Neptune remains stationary in the orbital coordinate system and relative to the plane of the orbit of Triton. As a result, in the elements of the Keplerian osculating orbit there can be no long-period perturbations caused by the non-sphericity of the planet. If we neglect the perturbations from the attraction of the Sun, then the satellite should move in a flat orbit that precesses with a constant angular velocity and constant inclination to the main plane. In the model under consideration, the orbit radius a, the inclination I , the latitude argument u, the longitude of the ascending

Chapter 3 Equations of motion and analytic theories

node  determine the position of the satellite relative to the center of the planet in the orbital coordinate system. Three of these quantities are functions of time I = I0 + δI (t), u = u0 + u(t ˙ − t0 ) + δu(t),

(3.92)

˙ − t0 ) + δ(t),  = 0 + (t ˙ are constants, t0 is some given initial era, where I0 , u0 , u, ˙ 0 ,  and δI (t), δu(t), δ(t) are long-period perturbations of the corresponding elements. By virtue of the accepted simplifications, the formulas of Kepler motion degenerate into the following simple relations: x = a (cos u cos  − sin u sin  cos I ), y = a (cos u sin  + sin u cos  cos I ), z = a sin u sin I.

(3.93)

To construct a model of Triton’s motion under the assumptions made, it remains to determine the long-period perturbations δI (t), δu(t), δ(t) due to the attraction of the Sun. The expansion of the perturbing function R  was taken from the book (Murray and Dermott, 2000). This expansion is a series in powers of the ratio a/a  , where a  is the semi-major axis of the Sun’s orbit around the planet. The series begins with a term containing (a/a  )2 . Additional valid simplifications are made. Since the semimajor axis of the Triton orbit is a = 354700 km, and the mean distance of Neptune from the Sun is 4504449760 km, then a/a  = 0.000078757455. In the theory of Triton’s motion, we neglect the higher powers of this ratio and leave only the terms with (a/a  )2 in the expansion. In addition, we assume that the Sun moves in a circular orbit in an unchanged plane. Denote by i  and  the inclination and longitude of the ascending node of the solar orbit in the orbital coordinate system. We assume that i  and  are known constant values, and the argument of the latitude of the Sun u is a wellknown linear time function u = u0 + u˙  (t − tS ), where tS is some given epoch. The values i  and  , u0 , u˙  can be determined using the mean elements of Neptune’s orbit, which can be taken, for example, from Simon et al. (1994). However, in Emelyanov and Samorodov (2015), the parameters of the solar motion model were determined by the least squares method using the heliocentric coordinates of the Sun calculated for a number of time instants in 10 day increments in the time interval from 1800 to 2200 using

117

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Table 3.2 The parameters of the motion model of the Sun along the mean elements from Simon et al. (1994) and determined using the INPOP10 ephemeris. Parameter

Based on mean elements

Based on INPOP 10 ephemerides

a  , km i  , deg  , deg u0 , deg u˙  , deg/d

4504449760 27.923658 200.788305 258.329018 0.00598182615

4499478064 27.923678 200.788181 258.727508 0.00598084154

the INPOP10 ephemeris (Fienga et al., 2011). The parameter values are given in Table 3.2. For the initial epoch tS of the parameter u0 , the moment JD = 2451545.0 (TT) was taken. As a result of the simplifications made, the perturbing function takes the form 2 2  Gm  a 2  (2 − k)!  R =  (2 − δ0,k ) F2k1 (I )F2kp (i  )× a a (2 + k)!  

k=0  

p =0



× cos[(2 − 2p )u + k( − )] , where m is the mass of the Sun, the Kronecker symbol is δ0,k = 1 for k = 0 and δ0,k = 0 for k = 0, and a prime at the sign of the second sum means that the term is omitted for k = 0 and p  = 1, corresponding to the secular term. Here we have designated the inclination functions using the term F2k1 (I ). As a result, the expression contains eight terms, two of which are equal to each other, since F200 (i  ) = F202 (i  ). The necessary inclination functions can be taken from the book (Kaula, 1966). The Lagrange equations for δI (t), δu(t), δ(t) were solved by the small parameter method. First-order perturbations were determined with respect to the small parameter (m /m)(a/a  )3 . In addition, we took into account the fact that secular perturbations from the second zonal harmonic with coefficient J2 in the elements u and  depend on the element I (t), which is subject to long-period perturbations. Here, second-order perturbations were actually determined in the method of the small Poincaré parameter by the Poisson method, as prescribed above. These perturbations are contained in the general formulas (3.60) and (3.64). Thus, combined perturbations proportional to (m /m)(a/a  )3 J2 were also taken into account.

Chapter 3 Equations of motion and analytic theories

119

Table 3.3 Coefficients in the formulas for long-period perturbations due to the attraction of the Sun. i

KI(i) , deg

1 0.0 2 0.00096486 3 0.00664661 4 0.00004687 5 0.00095976 6 −0.00037627 7 −0.00000225

Ku(i) , deg −0.00012327 −0.00279450 −0.04335622 −0.00017215 −0.00233686 0.00096231 0.00000730

(i) K , deg k1(i) k2(i)

0.00063339 −0.00178905 −0.01560091 −0.00009186 −0.00218071 −0.00037627 0.00000536

Long-period perturbations from the Sun were first determined in an analytical form. Then, the values of the above motion parameters of the Sun, as well as the values of the motion parameters of Triton found from observations, were substituted into the obtained formulas. As a result perturbations in the elements at a given point in time can be calculated using the following simple formulas: 7 

δI (t) =

i=1 7 

δu(t) =

i=1

δ(t) =

  (i) (i) (i) KI cos k1 u + k2 ( − ) ,   (i) (i) (i) Ku sin k1 u + k2 ( − ) ,

7  i=1

(3.94)

  (i) (i) (i) K sin k1 u + k2 ( − ) ,

where ˙ − t0 ). u = u0 + u˙  (t − tS ),  = 0 + (t

(3.95)

The coefficients included in these formulas are given in Table 3.3. The plots of changes in long-period perturbations due to the attraction of the Sun in the elements of the Triton orbit I , u, and  over the time interval from 1800 to 2200 are shown in Fig. 3.8. As a result, an analytical theory of the motion of the satellite of Neptune Triton was built. The arbitrary constants of the the˙ The parameters of ory were the parameters I0 , u0 , u, ˙ 0 , and . the theory were determined in Emelyanov and Samorodov (2015) from all available observations made over a time interval of 165 years from 1847 to 2012. At the same time, the geo-equatorial co-

2 2 0 −2 2 0 −2

0 1 1 1 2 2 2

120

Chapter 3 Equations of motion and analytic theories

Figure 3.8. Long-period perturbations from the attraction of the Sun in the elements of the orbit of Triton.

ordinates of the z axis of the orbital coordinate system α0 and δ0 were also included in the number of determined parameters. The Triton ephemeris, based on the considered analytical theory, differs insignificantly from the ephemeris of other authors. The differences are due to the different composition of the observations used. Thus, the advantage of the analytic theory in comparison with models of motion based on the numerical integration of differential equations of motion was shown. A more detailed description of the theory considered here can be found in publications (Emelyanov and Samorodov, 2015).

3.13.2 Precessing Ellipse models for close Jupiter satellites Four close satellites of Jupiter move around the planet almost in the equatorial plane and in almost circular orbits. In the order of increasing semi-major axes of the orbits, these are the satellites Metis, Adrastea, Amalthea and Thebe. In perturbations of the Keplerian orbits of satellites, the influence of oblateness of Jupiter prevails. Significantly less perturbations are exerted by the attraction of massive Galilean satellites. The effect of the attraction of the Sun is very small. Jacobson (1997) reported the construction of a motion theory of the close satellites of Jupiter. The models of precessing ellipses are applied taking into account secular per-

Chapter 3 Equations of motion and analytic theories

turbations from the non-sphericity of Jupiter, the attraction of the ˙ are Galilean satellites and the Sun. The precession rates ω˙ and  determined by perturbation theory. Researchers from Tomsk State University (TSU) Avdyushev and Ban’shikova (2008) construct a new motion model of four close satellites of Jupiter. The equations of motion were solved by the method of numerical integration. The motion parameters are refined based on all ground-based observations available up to 2008. It is shown in this paper that there are several solutions in the problem that represent observations approximately equally well. One of the solutions has been selected. On its basis, ephemeris were built on a time interval from 1954 to 2034. Ephemeris are presented in the form of expansions of rectangular planetocentric coordinates of satellites in series according to Chebyshev polynomials. The creation of the ephemeris of the Galilean and close satellites of Jupiter using the numerical integration of the equations of motion based on all available ground-based and space-based observations has been announced (Jacobson, 2013). These satellite ephemerides are accessible on the Internet at JPL’s Solar System data (Jet Propulsion Laboratory, NASA, USA). The service is described in an article (Giorgini et al., 1997). The author Emel’yanov (2015) presented the representation of the JPL ephemeris (Jacobson, 2013) and the ephemeris Avdyushev and Ban’shikova (2008) with two models: a precessing ellipse (PE) and a model based on the solution of the generalized problem of two fixed centers and taking into account periodic perturbations, which we previously designated as RM (refined model). Ephemeris (Avdyushev and Ban’shikova, 2008), presented in the form of expansions of the planetary rectangular coordinates of satellites in series according to Chebyshev polynomials, were transferred to the author of the work (Emel’yanov, 2015). Using each of the two options for ephemeris, tables of rectangular planetocentric coordinates of four close satellites of Jupiter were compiled in the time interval from August 1, 2014 to January 1, 2016 with a step of 0.1 days. JPL ephemeris was obtained via the Internet on April 3, 2014. Based on these data, the parameters of each of the two models were found by the method of differential refinement. The coordinates connected with the plane of the Earth’s equator α0 and δ0 of the pole of Jupiter were also included in the number of determined parameters. These coordinates actually determine the axis around which the orbital plane precesses with a constant slope to the equator of the planet. For each of the four satellites, α0 and δ0 were determined separately. At the same time, the Earth’s equator was considered as stationary and determined for the J2000 epoch.

121

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Table 3.4 Comparison between the representation of JPL ephemeris (Jacobson, 2013) and TSU ephemeris (Avdyushev and Ban’shikova, 2008) with two approximate models: PE is a model of precessing ellipses and RM is a refined model based on the solution of the generalized problem of two fixed centers. Satellite

JPL

a

TSU

km

σ¯ (PE) km

σ¯ (RM) km

σ¯ (PE) km

σ¯ (RM) km

Metis

127978.9

0.49

0.47

0.64

0.48

Adrastea

128979.9

0.49

0.48

1.87

0.58

Amalthea

181365.5

2.82

2.71

2.89

2.73

Thebe

221888.2

12.98

12.55

12.98

12.62

It should be noted that in the model of a precessing ellipse, only secular perturbations are actually taken into account. In the model based on the solution of the generalized problem of two fixed centers, short-period perturbations of the first order caused by the second zonal harmonic of the expansion of the planet’s gravitational force are also taken into account. The reference JPL and TSU models also take into account other perturbations from the non-sphericity of Jupiter, from the attraction of the Galilean satellites and the Sun. Therefore, the discrepancies between the analytical models and the results of numerical integration are due to differences in the perturbing factors considered. As a result, for four satellites based on each of the two sources of ephemeris, JPL and TSU, 11 parameters were determined using two motion models: nine parameters of the precessing ellipse ˙ as well as the parameters α0 ˙ 0 ,  model a, ¯ n, ¯ e, i, M0 , ω0 , ω, and δ0 . When refining the parameters, the root mean square value σ¯ of the deviations of the model coordinates from the reference coordinates calculated from the ephemeris was determined. The deviation for each time moment of the ephemeris was calculated as the distance between the model and ephemeris positions and measured in kilometers. The value σ¯ shows the accuracy of matching the analytical model with the reference model based on numerical integration. The obtained values for the satellites are given in Table 3.4. This table also shows the semi-major axis a obtained for the model of precessing ellipse from the JPL ephemeris. The results given in Table 3.4 show that the mismatch of analytical models with ephemeris is greater, the larger the semi-major

Chapter 3 Equations of motion and analytic theories

axis of the orbit. This can easily be explained by the fact that, with an increase in the semi-major axis of the orbit of a close satellite, the motion occurs closer to the orbits of the Galilean satellites, the perturbing effect of which is not significantly taken into account in analytical models. Since the mismatches caused by the differences in the perturbing factors taken into account prevail, the value σ¯ is only slightly less in the model based on the solution of the generalized problem of two fixed centers, compared with the model of precessing ellipse. Both models represent the motion of the close satellites of Jupiter with almost the same accuracy. To evaluate the found inconsistencies of the models on the scale of the geocentric angular coordinates of the satellites, it should be taken into account that 1 arcsecond corresponds to 3800 km at the mean distance of Jupiter from the Earth. A 12 km mismatch in the satellite’s orbit corresponds to a difference of geocentric angular coordinates of 0.003 arcseconds. Such a difference is at least 50 times less than the accuracy of available groundbased observations of the close satellites of Jupiter. The parameters of the precessing ellipses, which were determined using the JPL and TSU ephemeris, can be used to calculate the ephemeris using simple Keplerian motion formulas at any time. To do this, in Tables 3.5 and 3.6 we give the obtained values with the necessary number of significant digits for calculating the ephemeris. The parameters differ for different models and different reference ephemerides, however, each set of parameters is self-consistent. The coordinates of the Jupiter pole α0 and δ0 obtained from the JPL ephemeris turned out to be almost identical for four satellites and two models, namely α0 = 268.057 deg and δ0 = 64.497 deg. According to the TSU ephemeris, the coordinates of the pole of Jupiter are as follows: α0 = 268.049 deg and δ0 = 64.489 deg. Emel’yanov (2015) showed that, for the Metis and Adrastea satellites, the mean anomaly has a librational character of change in time. The change in the mean anomaly of the Thebe satellite is circulating in nature. For the satellite Amalthea the change in eccentricity and the mean anomaly almost coincides with the separatrix solution of the equations of motion.

3.13.3 Special analytical theories of the main satellites of major planets, taking into account the mutual attraction of satellites The mutual attraction of the main satellites in the well-known analytical theories of satellites was taken into account by special

123

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Table 3.5 Parameters of the precessing ellipse for the Jupiter satellites Metis and Adrastea obtained from JPL ephemeris (Jacobson, 2013) and TSU ephemeris (Avdyushev and Ban’shikova, 2008). The initial epoch of elements (TT scale) was 0 hours on August 1, 2014 (MJD = 56870.0). Parameter

JPL

TSU

Metis a, ¯ km

127978.860

127978.870

e

0.000504857

0.001274382

i, rad

0.000213446

0.000348744

M0 , rad

3.813296566

0.527952271

ω0 , rad

0.169346010

0.312420298

0 , rad

5.753821299

2.603426115

n, ¯ rad/d

21.164087429

21.164083095

ω, ˙ rad/d ˙ rad/d ,

0.300596369

0.300600575

-0.149768271 Adrastea

-0.149770172

a, ¯ km

128979.903

128979.840

e

0.000180935

0.005415531

i, rad

0.000225599

0.007701531

M0 , rad

2.545515933

1.724761655

ω0 , rad

3.034354065

2.395080996

0 , rad

5.712371588

0.919577232

n, ¯ rad/d

20.919404709

20.919415107

ω, ˙ rad/d ˙ rad/d ,

0.292385013

0.292382363

-0.145685219

−0.145690358

methods of perturbation theory. Without going into the details of these methods, we briefly describe what has been done. For the Galilean satellites of Jupiter, a certain heuristic method for constructing an analytical theory of satellite motion was proposed by the English researcher R. A. Sampson in 1921 (Sampson, 1921). The coordinate system was assumed to be cylindrical, the main plane being the equatorial plane of Jupiter. The solution of the differential equations of motion of the four Galilean satellites was found in the form of Poisson series with polynomial and trigonometric arguments. Expansions in series in powers of small parameters were applied, the smallness of which was ensured by

Chapter 3 Equations of motion and analytic theories

125

Table 3.6 Parameters of the precessing ellipses of the close satellites of Jupiter, Amalthea and Tebe, obtained from JPL ephemeris (Jacobson, 2013) and TSU ephemeris (Avdyushev and Ban’shikova, 2008). The initial epoch of elements (TT scale) was 0 hours on August 1, 2014 (MJD = 56870.0). Parameter

JPL

TSU

Amalthea a, ¯ km

181365.552

181365.561

e

0.003426003

0.004079207

i, rad

0.006565694

0.005659253

M0 , rad

3.839867712

4.038848183

ω0 , rad

4.598920930

4.476760700

0 , rad

4.630652745

4.556545020

n, ¯ rad/d

12.568437183

12.568436283

ω, ˙ rad/d ˙ rad/d ,

0.087582088

0.087583381

-0.043716407 Thebe

-0.043716439

a, ¯ km

221888.173

221888.157

e

0.017531954

0.016117934

i, rad

0.018706263

0.019443802

M0 , rad

1.526572934

1.603754238

ω0 , rad

4.294075517

4.238345375

0 , rad

4.125853541

4.115550534

n, ¯ rad/d

9.293210969

9.293215547

ω, ˙ rad/d ˙ rad/d ,

0.043193094

0.043188124

-0.021577028

-0.021574888

small eccentricities and inclinations of the orbits. Subsequently, this theory was developed in Lieske (1977) using computer algebra methods. The formulas of this theory contained 49 free parameters, which were refined from all the observations available then. In Arlot (1982), the parameters of the theory were revised based on all previous and then new observations. For the main satellites of Saturn, analytical theories were constructed using perturbation theory methods using expansions in powers of various small parameters (Harper and Taylor, 1993). The motion model of the main satellites of Saturn constructed in the

126

Chapter 3 Equations of motion and analytic theories

work (Vienne and Duriez, 1995) deserves special attention. In this work, the main frequencies of trigonometric series representing perturbations of the elements of the satellite’s intermediate orbit were found by the methods of analytical theory, and the amplitudes were refined by the motion model constructed by the method of numerical integration of the motion equations. The theory was called “synthetic.” For the main satellites of Uranus, the analytical theory was constructed using the secular Lagrange–Laplace perturbation method. This method is described in references (Duboshin, 1975; Subbotin, 1968), and the theory of satellite motion is given in Laskar and Jacobson (1987).

3.14 Influence of tides in viscoelastic bodies of planet and satellite on the satellite’s orbital motion 3.14.1 Statement of the problem of influence of tides In the last years, the problem of influence of tides raised on viscoelastic bodies of planets and satellites upon satellite orbital motion has become a topical issue. The accuracy of observations of the major satellites of Jupiter, Saturn, Uranus, and Neptune has greatly increased. In addition, in course of time, the intervals of observations have naturally become larger. These factors gave an impetus to attempts to determine from observations those physical parameters of planets and satellites that define the forces of tidal friction. The tidal bulge moves in body’s interior creating a torque acting upon satellite. Satellite motion depends on tidal deformation parameters. Therefore, the inverse problem can be solved: to determine these parameters from observations of satellite motion. The force is proportional to the ratio k2 /Q, where k2 is the Love number, which characterizes the deformability of a body and Q is the quality factor characterizing the viscosity of body’s interior. It follows from the equations of motion that observations do not allow us to obtain independent values of k2 and Q but only their ratio k2 /Q. Lainey et al. (2009a), taking all available at the time astrometric observations of the Galilean satellites of Jupiter, determined the ratio k2 /Q for both Jupiter and its satellite Io. Lainey et al. (2009a) obtained k2 /Q = (1.102 ± 0.203) · 10−5 for Jupiter and k2 /Q = 0.015 ± 0.003 for Io. In the same way, using astrometric observations, Lainey et al. (2012) obtained new values of tidal dissipation ratio of Saturn that

Chapter 3 Equations of motion and analytic theories

turned out to be 10 times greater than the value obtained from theoretical considerations. The value of the tidal dissipation ratio is k2 /Q = (2.3 ± 0.7) · 10−4 . Moreover, an unexpectedly high value of Mimas’ secular acceleration caused by the tides on the satellite’s body was obtained. When determining the parameters from observations, the usual practice is to carry out numerical integration of the equations of motion in rectangular coordinates. Hence, it is necessary to have expression for perturbing acceleration caused by tidal forces. Such equations have been deduced by earlier researchers (see the references below). The orbital evolution of a satellite caused by tidal forces is better to study by considering the changes in two key parameters: semi-major axis a and eccentricity e. It is the changes of these two parameters that determine the satellite’s fate, that is whether it falls down to planet or moves away from it. To this end, differential equations for these elements have been derived in a number of papers. Neglecting small short-period perturbations, two equations are usually obtained, which in general form look like de k2 da k2 = Aa (a, e), = Ae (a, e). dt Q dt Q Separate equations of this kind are composed for both the problem of taking into account the tides on the planet’s body and the problem of the tides on satellite. The two problems give different equations, but it is possible to take into account both effects in one system of equations. For the problem in consideration, differential equations in the Keplerian elements have been published in earlier papers. In particular, they can be found in Lainey et al. (2012) with reference to the source of formulas (Kaula, 1964) in the case of tides in the planet’s body and to the source (Peale and Cassen, 1978) in the case of tides in the satellite body. In the papers cited here, equations for orbital elements are derived separately from equations in rectangular coordinates using expressions for the right-hand sides of equations as functions of a and e. In order to improve methodology, to clarify the possibility of determining the parameters of tidal friction from observations as well as to study orbital evolution, it would be interesting to compare solutions of the equations in coordinates with solutions of the equations in Keplerian elements. This is the aim that was set in Emelyanov (2018). In this work, differential equations are newly derived that describe the changes in the semi-major axis and the eccentricity of the satellite’s orbit due to tides in the viscoelastic

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bodies of the planet and satellite. We consider here the conclusions and results obtained in Emelyanov (2018).

3.14.2 Equations in rectangular coordinates Let us consider differential equations of satellite motion in rectangular planetocentric coordinates published in earlier work. The equations necessary to solve the problem have been derived by Mignard (1979) who studied the influence of tides in the viscoelastic interior of the Earth upon motion of the Moon. We take Eq. (5) of this paper. Later the theory has been developed in Mignard (1980). Later, Lainey et al. (2007) obtained the solution of the problem of influence of tides on the body of Mars upon the motion of Phobos. The authors used the equations of satellite motion in rectangular coordinates taken from Mignard (1980). The corresponding formula in Lainey et al. (2007) has the reference (3). Lainey et al. (2009a) extended the formulas for perturbing acceleration caused by the tides in the planet’s body to the case of tides in viscoelastic satellite body influencing upon its orbital motion. However, there were no detailed derivations of the formulas. They were just declared and given without explanations in Supporting Information section of the paper [see the formulas (1) and (2) in Lainey et al. (2009b)]. The formulas have a generalized form for both tides raised on the planet and raised on the satellite. Later the same equations were also published in Lainey et al. (2012; see the formulas (1) and (2) therein). Let us use the generalized form of the equations of motion of planetary satellite as given in Lainey et al. (2009b) and Lainey et al. (2012). However, we shall make some simplifications leaving only the terms that are of fundamental importance for further analysis. First, we save only the main term corresponding to the planet’s attraction as a material point and the terms describing tidal effects. Secondly, we neglect satellite’s mass in comparison with that of the planet. This assumption is quite justifiable since the masses of the satellites are really small compared to the planetary masses. For the values in the formulas we shall use other notations than those in the papers mentioned earlier. As in these papers, the equations of motions we write in the satellite’s rectangular coordinates referred to the planetocentric reference frame. For convenience, we use equations for two separate problems: the problem of satellite motion influenced by the tides raised on planet’s viscoelastic body and that where satellite’s motion is perturbed by the tides raised on viscoelastic body of the satellite itself. We use the following notations:

Chapter 3 Equations of motion and analytic theories

R for the planet’s radius, Rs for the satellite’s radius, GM for the gravitational parameter of the planet, Gs for the gravitational parameter of the satellite, a for the semi-major axis of the satellite’s orbit, n for the mean motion of the satellite, k2 for the Love number of the planet (dimensionless), (s) k2 for the Love number of the satellite (dimensionless), tp for the time lag of tidal bulge in the planet’s body, ts for the time lag of tidal bulge in the satellite;s body, Qp for the quality factor of the planet, Qs for the quality factor of the satellite,  for the vector of the planet’s rotation rate,  s for the vector of the satellite’s rotation rate. Note that tp and ts are assumed to be positive. The satellite’s position and velocity are given by the vectors r and v. Referring to the general formulas (1) and (2) in Lainey et al. (2009b; 2012), under the assumptions made above and with adopted notations, we write the differential equations of satellite motion in the following form: – for the case of tides on the planet   2r(r v) d 2r GM 3k2 Gs R 5 =− 3 r− tp + [r  ] + v , dt 2 r r8 r2

(3.96)

– for the case of tides on the satellite   (s) 3k2 GM Rs5 GM d 2r 2r(r v) GM =− 3 r− ts + [r  s ] + v . (3.97) Gs dt 2 r r8 r2 Here [r  ] and [r  s ] are vector cross products, (r v) is vector dot product. To simplify further analysis, we introduce some new notations and slightly transform the equations. Let us introduce an arbitrary value a¯ whose magnitude is taken to be equal to averaged value of the satellite’s semi-major axis. We use the well-known relationship between the Keplerian elements n2 a 3 = GM . We introduce dimensionless constants Kp and Ks , which are defined as follows: Kp =

3R 5 Gs , a¯ 5 GM

(3.98)

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Chapter 3 Equations of motion and analytic theories

Ks =

3Rs5 GM . a¯ 5 Gs

(3.99)

The influence of viscoelastic tides on the satellite’s orbital motion is determined by the property of deformability of the body and the time lag of the tidal wave. For a planet’s body and a satellite’s body, the influence is characterized by the coefficients (p)

K2

(s)

= k2 tp , (s)

K2 = k2 ts ,

(3.100) (3.101)

respectively. Now, with the new notations the equations become as follows: – for the case of tides raised on the planet’s body   a¯ 5 a 3 2 2r(r v) GM d 2r (p) = − r − K K n + [r  ] + v , (3.102) p 2 dt 2 r3 r8 r2 – for the case of tides raised on the satellite’s body   d 2r a¯ 5 a 3 2 2r(r v) GM (s) = − 3 r − K2 Ks 8 n + [r  s ] + v . (3.103) dt 2 r r r2 The papers mentioned above explain the relationship between the time lags of tidal bulge and viscosity parameters. According to these explanations, we have tp =

1 1 , ts = . | − n) 2Qp (| Qs n

(3.104)

It is assumed that the planet is rotating faster than the satellite moves along its orbit. In this case, tp > 0. It was noted in Lainey et al. (2012) that the time lags tp and ts depend on the so-called tidal frequency, i.e. the velocity of the tidal wave run on the surface. Two variants are possible here. In the first variant, we assume that over an infinite time interval tp and ts remain constant. In the second variant, the coefficients (s) k2 /Qp and k2 /Qs are assumed constant. However, considering the orbital evolution at moderate time intervals, all these quantities can be considered constant. In our further analytical calculations, we do not fix the depen(p) (s) dence of the coefficients K2 and K2 on the velocity of the tidal wave. Note that, instead of the quality factor Q, the parameters arctan Q or arcsin Q are used in some papers. Such a change of parameters, however, is not substantial in this study.

Chapter 3 Equations of motion and analytic theories

3.14.3 Solving the equations for rectangular coordinates Solving the equations obtained above at sufficiently large time interval can give us the picture of evolution of the satellite’s orbital parameters caused by the tides raised on viscoelastic bodies of both planet and satellite. That is exactly what interests researchers in this problem. We shall try to obtain the sought properties of satellite motions. Since exact analytical solution is not possible in this case, we have to use the methods of numerical integration. We performed numerical integration of Eqs. (3.102) and (3.103) at a certain sufficiently large time interval and obtained planetocentric coordinates and velocities of satellite for a series of time instants with constant stepsize. For each time instant, osculating Keplerian elements were computed. The calculations were per(p) (s) formed for the special case when the coefficients K2 and K2 are considered constant. What is interesting for us here is the variation of the elements in time, particularly the changes in semimajor axis a and eccentricity e. The orbits of real main satellites of Jupiter, Saturn, Uranus, and Neptune have small inclinations to their planets’ equatorial planes. Let us consider a hypothetic case close to real one when a satellite moves in invariable plane, the axes of rotation of both planet and satellite being normal to this plane. Then the vectors [r  ] and [r  s ] lie in the plane of motion. Hence, all acting forces lie in the same plane and satellite motions occur in one plane too. That is why, in solving Eqs. (3.102) and (3.103), we can restrict ourselves to modeling 2D motions only. When carrying out the calculations, it is necessary to decide how to choose the value of the angular velocity of rotation of the satellite  s . There is no complete clarity on this issue. In this study, we do not study satellite rotation. We have to accept this or that hypothesis. In fact, two options are possible. In the first hypothesis, we assume that as a result of the planet’s influence on the tidal wave in the viscoelastic body of the satellite, it constantly maintains a synchronous rotation in which the angular velocity is equal to the mean orbital motion. As can be seen from the studies described below, just such a hypothesis is implicitly accepted in work on the problem under consideration. The second hypothesis is the assumption that the angular velocity of the satellite’s rotation does not remain constant during the evolution of orbital motion and, over time, may differ from the mean motion. As will be shown below, the result of the adoption of this second hypothesis contradicts the results of work on the problem under consideration. In our calculations, we accepted the first hypothesis of constant synchronous rotation of the satellite.

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Physical parameters were taken to be close to those of the Uranian satellites. The following constants were adopted as planetary parameters: GM = 5793939.3 km3 /s2 ,  = 501.1600928deg/day. The coefficients in the equations are taken to be as follows: Kp = 0.1 · 10−6 , Ks = 10.0 · 10−6 . These values do not correspond to real possible values of the viscosity parameters of Uranus and its satellites. However, these exaggerated values of the coefficients allow us to see the peculiarities of the solution. The constant a¯ was taken to be equal to the initial value of the satellite’s semi-major axis. Initial conditions for solution of the differential equations of motion were taken in two sets: 1. a = 190940.453 km, 2. a = 114820.064 km . The initial eccentricity in both cases is 0.002. The first set of orbital parameters is very close to those of the Uranian satellite Ariel. The second set is considered because it demonstrates some peculiarities in orbital evolution (see below). In performing the numerical integration, it was supposed that the satellite’s starting point is in the pericenter of its orbit. In order to see the character and magnitude of short-period changes in osculating elements, the values of semi-major axis and eccentricity were first computed at the time interval of 8 d with the stepsize 0.01 d. It is those changes computed for the second set of initial conditions in the problem of taking into account tidal dissipation in planetary body that are shown in Figs. 3.9 and 3.10. Because of strong secular perturbations, it is not possible to see short-period oscillations of semi-major axis in the plot. However, the eccentricity does manifest oscillations with the period equal to its orbital period. The plots demonstrate that short-period variations of the osculating elements a and e are extremely small and cannot characterize tidal evolution of the satellite’s orbit. In the same way, we obtained small amplitudes of short-period oscillations in the elements in all other cases that were considered. In studying the orbital evolution, integration was performed at the time interval of 80 200.0 d, i.e. about 220 yr. The results were output with the stepsize 100 d (for the data see below). Figs. 3.11–3.14 show the changes in semi-major axis and eccentricity of the satellite’s orbit caused by tidal friction in planetary interior for both sets of initial conditions. Note that, for the second set of initial conditions, the perturbing influence of tides raised on the planet’s body results in that

Chapter 3 Equations of motion and analytic theories

Figure 3.9. Changes in semi-major axis of satellite at 8-d time interval caused by tidal friction in the planet’s interior. The second set of initial conditions is used.

Figure 3.10. Changes in eccentricity of satellite at 8-d time interval caused by tidal friction in the planet’s interior. The second set of initial conditions is used.

the eccentricity is almost constant in the beginning of the time interval but increases with the growth of semi-major axis. Here the initial value of semi-major axis was especially chosen so that to

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Chapter 3 Equations of motion and analytic theories

Figure 3.11. Change in the satellite’s semi-major axis at 80 200-d (220-yr) time interval caused by tidal friction in the planet’s interior. The first set of initial conditions is used.

demonstrate the peculiarity of solution in this case. The way this value was obtained is explained below. Figs. 3.15–3.18 show the changes in semi-major axis and eccentricity of the satellite’s orbit caused by tidal friction in the satellite’s body for both sets of initial conditions. It is necessary to note that the changes in semi-major axes and eccentricities represented in the plots reflect exactly the real evolution of orbit due to the influence of tides in viscoelastic bodies of planet and satellite. The reliability of the results is based on the reliability of the equations of motion of satellite in rectangular coordinates that were taken from the work mentioned above.

3.14.4 Transition to the differential equations in Keplerian elements When studying planetary satellite motions at large time intervals, it is most interesting to look at the behavior of the semi-major axis a and the eccentricity e. It is these parameters that describe the satellite’s fate. Because of tidal dissipation of mechanical energy, a and e can change in such a way that the satellite can either fall to the planet or move away from it. That is why in much work devoted to the orbital evolution differential equations for semimajor axis and eccentricity are composed. We also made an attempt to compose and solve such equations.

Chapter 3 Equations of motion and analytic theories

Figure 3.12. Change in the satellite’s eccentricity at 80 200-d (220-yr) time interval caused by tidal friction in the planet’s interior. The first set of initial conditions is used.

Figure 3.13. Change in the satellite’s semi-major axis at 80 200-d (220-yr) time interval caused by tidal friction in the planet’s interior. The second set of initial conditions is used.

Since in this problem, without loss of generality, we can consider only planar motions, no inclinations or longitudes of ascending node are involved. It is also obvious that longitude of

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Chapter 3 Equations of motion and analytic theories

Figure 3.14. Change in the satellite’s eccentricity at 80 200-d (220-yr) time interval caused by tidal friction in the planet’s interior. The second set of initial conditions is used.

Figure 3.15. Change in the satellite’s semi-major axis at 80 200-d (220-yr) time interval caused by tidal friction in the satellite’s interior. The first set of initial conditions is used.

pericenter and mean anomaly at epoch do not determine the orbital evolution of satellite. It is for these reasons that we restricted ourselves to composing only equations for semi-major axis a and eccentricity e.

Chapter 3 Equations of motion and analytic theories

Figure 3.16. Change in the satellite’s eccentricity at 80 200-d (220-yr) time interval caused by tidal friction in the satellite’s interior. The first set of initial conditions is used.

Figure 3.17. Change in the satellite’s semi-major axis at 80 200-d (220-yr) time interval caused by tidal friction in the satellite’s interior. The second set of initial conditions is used.

To derive the sought equations, we use the equations in a and e taken from Subbotin (1968). They are as follows:   a(1 − e2 ) 2 da e sin f R + = √ T , (3.105) dt r n 1 − e2

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Chapter 3 Equations of motion and analytic theories

Figure 3.18. Change in the satellite’s eccentricity at 80 200-d (220-yr) time interval caused by tidal friction in the satellite’s interior. The second set of initial conditions is used.

de = dt



1 − e2 [sin f R + (cos f + cos E) T ], na

(3.106)

where f is the true anomaly, E the eccentric anomaly, R the radial component of perturbing acceleration, and T its tangential component. The perturbing acceleration is given by its components in the right-hand sides of Eqs. (3.102) and (3.103). The equations sought-after will be derived separately for tides raised on the planet’s body and for those raised on the satellite. Let us first consider the first problem. From Eqs. (3.102) we can obtain the following expressions for the components of perturbing acceleration: ¯ (p) a

R (p) = −Kp K2

¯ (p) a

T (p) = −Kp K2

5a3

r8 5a3

r8

 n2  n2

2r(r v) + [r  ] + v r2

(p)

2r(r v) + [r  ] + v r2

,

(3.107)

.

(3.108)

R

(p) T

The upper index (p) indicates that the expression is used in the problem of tides raised on the planet’s body. Here and below, the lower indices R and T denote corresponding components of vectors.

Chapter 3 Equations of motion and analytic theories

It is obvious that the first term in square brackets has only radial component, the second one has only tangential component, and the third one has both. We assume that the satellite’s orbital plane is normal to the vector of the planet’s angular rotation  . Hence the vector [r  ] lies in the orbital plane, it is normal to the vector r and points to the direction opposite to that of the satellite’s motion. To get the radial components, we used the fact that, for an arbitrary vector V, its radial component can be obtained from the expression (V r)/r. From the formulas of Keplerian motion we have r=

a(1 − e2 ) an an e sin f, vT = √ (1 + e cos f ), , vR = √ 2 1 + e cos f 1−e 1 − e2 ane (r v) = √ r sin f. 1 − e2

Taking into account these relationships, we obtain 



2r(r v) + [r  ] + v r2

(p) R

nae = 3√ sin f, 1 − e2

(3.109)

(p)

na a(1 − e2 ) |. =√ (1 + e cos f ) − | 1 + e cos f 1 − e2 T (3.110) Now, substituting Eqs. (3.109) and (3.110) into (3.107) and (3.108) and then substituting the results into (3.105) and (3.106), we have 2r(r v) + [r  ] + v r2

√ da ¯ 5 2 1 − e2 (p) a = Kp K2 5 na (1 + e cos f )8 × dt a (1 − e2 )8  | − × |

 n 2 2 2 (1 + 2e cos f + 3e − 2e cos f ) , (1 − e2 )3/2

(3.111)

√ ¯5 1 − e2 de (p) a (1 + e cos f )8 × = Kp K2 5 n dt a (1 − e2 )8  | × |

−√

n 1 − e2

1 − e2 (cos f + cos E)− 1 + e cos f

 [3e sin2 f + (cos f + cos E)(1 + e cos f )] .

(3.112)

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Chapter 3 Equations of motion and analytic theories

The derived equations exactly correspond to the initial equations (3.102) and (3.103) in rectangular coordinates. These equations are to be solved together with the equations for the argument of pericenter ω and mean anomaly M. Such a solution would correspond exactly to that of the equations in rectangular coordinates, since the orbital elements and the vectors of position and velocity remain interrelated by the formulas of Keplerian motion. As demonstrated above, when in solution of the equations in coordinates the transformation is made from coordinates and velocities to the Keplerian elements, the changes in semi-major axis and eccentricity look like monotone evolving functions with superimposed short-period oscillations. These oscillations are rather small so that, in analyzing the satellite’s orbital evolution, they can be neglected. We suppose that by averaging the right-hand sides of Eqs. (3.111) and (3.112) over time the solution of these equations will provide us with evolutionary changes in the elements free from shortperiod perturbations. It is possible to check this assumption by comparing the solutions of the strict equations (3.102) and (3.103) in coordinates with those of averaged equations in elements. Identity of averaged solution of the equations in coordinates with the solution of averaged equations in elements would allow us to study the long-term orbital evolution of satellites caused by tidal friction in planetary and satellite bodies by using only equations for a and e. To make such a check, it is first necessary to derive averaged equations for the elements a and e and then to solve them by numerical integration. We derived such equations and obtained their solution. Next, we were interested in comparing the two solutions under consideration. When averaging Eqs. (3.111) and (3.112), we had to make an expansion in powers of eccentricity neglecting the terms containing squared eccentricity. This simplification is acceptable because solution of the problem will supposedly be applied to the main satellites of major planets whose orbital eccentricities are really small. Our model orbits also had small eccentricities. Now let us proceed to carrying out the procedures described above. We denote averaged values by the bar above. In the process of averaging, the following relationships were used: e 1 1 cos f = −e, cos E = − , cos2 f = +O(e2 ), cos f cos E = +O(e2 ), 2 2 2

Chapter 3 Equations of motion and analytic theories

where O(e2 ) are terms of expansion in powers of e having the second order of smallness. In addition, we used expansions (1 + e cos f )k = 1 + ke cos f + O(e2 ), where k is an arbitrary integer. Only the first two terms of the expansion were used. At an intermediary stage of our actions we obtained the equations a¯ 5 da = 2Kp 5 na× dt a   | − n(1 + 2e cos f + 3e2 − 2e2 cos2 f ) , (3.113) × (1 + 8e cos f ) | a¯ 5 de |(1 + 7e cos f )(cos f + cos E)− = Kp 5 n {| dt a  n −√ [3e sin2 f (1 + 8e cos f ) + (cos f + cos E)(1 + 9e cos f )] . 1 − e2 (3.114) After averaging, we finally have ¯5 da (p) a | − n), = 2Kp K2 5 na(| dt a

(3.115)

¯5 de (p) 1 a | − 18n)ne. = Kp K2 · 5 · (11| dt 2 a

(3.116)

Now let us look at which equations are obtained in the case of perturbing action of tides raised on viscoelastic body of satellite. From Eqs. (3.103) we find that the radial and tangential components of acceleration have the form R

T

(s)

(s)

 ¯ 5 a 3 2 2r(r v) (s) a = −Ks K2 n r8 r2

+ [r  s ] + v

 ¯ 5 a 3 2 2r(r v) (s) a = −Ks K2 n r8 r2

+ [r s ] + v

(s) ,

(3.117)

.

(3.118)

R

(s) T

Here, the upper index (s) indicates that the expression is used in the problem of tides raised on satellite’s body. The lower indices R and T denote, as earlier, two components of the vectors.

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Chapter 3 Equations of motion and analytic theories

Supposing that the satellite’s angular rotation rate  s is normal to the orbital plane, similar to the previous case we have 



2r(r v) + [r  s ] + v r2

(s)

(s)

R

nae = 3√ sin f, 1 − e2

(3.119)

a(1 − e2 ) s |. | 1 + e cos f 1 − e2 T (3.120) Since we adopted the assumption that the satellite is in the s | = n. state of constant synchronous rotation, we assume that | Taking this into account and substituting (3.119) and (3.120) into (3.117) and (3.118) and then substituting the results into (3.105) and (3.106), we obtain 2r(r v) + [r  s ] + v r2

=√

na

(1 + e cos f ) −

da ¯ 5 a 3 2n2 a (s) a = −Ks K2 · × dt r 8 1 − e2   × 3e2 sin2 f + (1 + e cos f )2 − (1 − e2 )3/2 ,

(3.121)

de ¯ 5a3 2 (s) a = −Ks K2 n × dt r8    (1 − e2 )3/2 2 . (3.122) × 3e sin f + (cos f + cos E) 1 + e cos f − 1 + e cos f According to what was said earlier, we make simplifications in the right-hand sides of these equations, i.e. expand them in powers of eccentricity, saving only main terms of expansion, and average them over time. At intermediary stage of this process, after expansion in powers of eccentricity, the following equations are obtained:   da ¯5 9 2 (s) a = −Ks K2 5 2n2 a(1 + 8e cos f ) e − 2e2 cos2 f + 2e cos f , dt 2 a (3.123) de ¯5 (s) a = −Ks K2 5 n2 e[3 sin2 f + 2 cos f (cos f + cos E)]. dt a

(3.124)

Averaging over time gives the final result: da ¯5 (s) a = −19Ks K2 5 n2 ae2 , dt a

(3.125)

Chapter 3 Equations of motion and analytic theories

7 de ¯5 (s) a = − Ks K2 5 n2 e. dt 2 a

(3.126)

It is these equations that should describe the evolution of a and e caused by the dissipation of mechanical energy of the satellite’s orbital motion due to tidal friction in viscoelastic body of the satellite itself. The equations derived and cited above (3.115) and (3.116) describe the evolution of the elements a and e caused by tidal friction in the viscoelastic body of the planet. Note that these four equations were derived for an arbitrary (p) (s) dependence of K2 and K2 on the mean motion n, that is, for an arbitrary frequency dependence of the tidal time lag. Assuming Eqs. (3.104) to be valid, we reduce the equations to the form in which they appear in publications. As a result, we obtain k2 a¯ 5 da = Kp na, dt Qp a 5

(3.127)

| − 18n de 1 k2 a¯ 5 11| = Kp ne, · 5· | − n dt 4 Qp a |

(3.128)

(s)

k a¯ 5 da = −19Ks 2 5 nae2 , dt Qs a

(3.129)

(s)

de 7 k a¯ 5 = − Ks 2 5 ne. dt 2 Qs a

(3.130)

Now we should compare the solution of the equations in rectangular coordinates that was obtained earlier with that of Eqs. (3.115) and (3.116) of the first problem and with that of Eqs. (3.125) and (3.126) of the second problem. We carried out numerical integration of the latter equations with the same as(p) (s) sumption that the coefficients K2 and K2 were constant and with the same initial conditions that were set in solving the differential equations in rectangular coordinates. These solutions are shown in the same Figs. 3.11–3.14 and 3.15–3.18, corresponding to both problems. The lines of solutions coincide completely thus demonstrating exact identity (at least, within the limits of line’s thickness) of both solutions. More accurate numerical analysis proves that the solutions of the equations in orbital elements are exactly equal to the elements obtained from the solution of the equations in coordinates that were averaged to remove shortperiod oscillations.

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Chapter 3 Equations of motion and analytic theories

This result proves our assumption that the solution of Eqs. (3.115), (3.116), (3.125), and (3.126) in the elements reliably describes a satellite’s orbital evolution in both problems. Note that, for the second set of initial conditions, semi-major axis was chosen to satisfy the condition n=

11 |. | 18

In this case, at the initial moment, the right-hand side of Eq. (3.116) is equal to zero. Fig. 3.12 shows that, in the beginning of the interval, the mean value of eccentricity almost does not change. It is this behavior of this function that allows us to see short-period oscillations in Fig. 3.10. Our hypothesis about the constant synchronous rotation of the satellite can be called into question. Special research is needed on this issue. The assumption of a constant angular velocity of rotation of the satellite, independent of orbital motion, was also considered. In this case, the rate of change of the semi-major axis due to tidal friction in the satellite’s body will no longer be proportional to the squared eccentricity, as follows from Eq. (3.125). The numerical integration of the equations in the coordinates for the problem of taking into account viscoelastic tides in the satellite’s body was performed for the first version of the initial conditions under the assumption of a constant angular velocity of rotation of the satellite. As the result, the change in semi-major axis versus time is shown in Fig. 3.19. This figure shows that the change in the semi-major axis has a different character compared with the case of synchronous rotation of the satellite. The change is more significant. However, the change in eccentricity in this case does not differ from its change in the case of synchronous rotation of the satellite. Therefore, we do not give a corresponding plot.

3.14.5 Some important conclusions about the influence of tidal deformations on satellite dynamics As a result, we obtained the differential equations in the work (Emelyanov, 2018) for evolution of semi-major axis and eccentricity of the satellite’s orbit caused by tidal friction in both planet’s and satellite’s bodies. The departing point for these equations were differential equations in rectangular coordinates given in Lainey et al. (2012). The averaged solutions of the equations in coordinates are proved to be identical to the precise solution of averaged equations in elements derived in Emelyanov (2018).

Chapter 3 Equations of motion and analytic theories

Figure 3.19. Change in semi-major axis of the satellite’s orbit at 80 200-d (220-yr) time interval caused by tidal friction in the satellite’s body in the case of a constant angular velocity of the satellite’s rotation. The first set of initial conditions is used. The bold line shows the change in semi-major axis at a constant speed of the satellite’s rotation, the thin one shows the change in the semi-major axis during synchronous rotation. In the latter case, the apparent constancy of the semi-major axis is manifested due to the scale selected on the plot.

For the problem of tides raised on the planet’s body, the relationship was found between the satellite’s mean motion n and | when the rate of the eccentricity’s the planet’s rotation rate | change becomes equal to zero (Emelyanov, 2018). This happens  when n = 11 18 | |. It is of interest to compare the results obtained with those deduced by other authors. Let us consider several aspects. Tidal evolution in close binary systems was studied in the paper (Hut, 1981). The differential equations for particle motion in rectangular coordinates derived in this work, taking into account the tidal effect of the viscoelastic central body, coincide with the equations published in (Mignard, 1979; 1980; Lainey et al., 2009a; Lainey et al., 2012). The equations correspond to the case of perturbations of the satellite’s orbital motion under the influence of tides in the viscoelastic planet’s body. Hut (1981) transformed the equations for variables to the semi-major axis a and the eccentricity e. Our Eqs. (3.127) and (3.128) exactly coincide with the results of (Hut, 1981). Differential equations for the semi-major axis and the eccentricity of the satellite’s orbit, describing the influence of viscoelas-

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tic tides in the planet’s body and in the satellite’s body on the orbital motion published in Lainey et al. (2012). These equations have the same form as our Eqs. (3.127), (3.128), (3.129), and (3.130). However, only two of the four equations in Lainey et al. (2012) coincide with Eqs. (3.127) and (3.130). The other two are essentially different. Accordingly, they differ from the results of Hut (1981). A more general theory than that considered here to account for the tidal effect on the orbital motion of satellites was developed in (Efroimsky and Makarov, 2013; Makarov and Efroimsky, 2013; Makarov, 2015). Furthermore, the authors cite the previous work. It is shown that in the bodies of the planet and satellite, in addition to the main tidal deformation, which is considered above, other deformation waves arise, having different velocities of travel through the bodies and other time delays. The multiplicity of these waves is generated mainly by the difference between the satellite’s orbit and the circular one and its inclination to the plane of the planet’s equator. The parameters of the dissipation of the energy of the satellite’s orbital motion due to tidal deformations can be determined only from observations. However, the accuracy of observations is limited. Currently, accuracy is barely enough to determine these parameters for Jupiter, Saturn and their main satellites. As for the viscoelastic bodies of Uranus and Neptune, no attempts have been made to determine such parameters so far. The relationship between the accuracy of observations and the possible values of the tidal strain parameters of Uranus and Neptune was investigated in Lainey (2016). Such hypothetical values of the parameters were found as they could be determined from observations. These values correspond to a significant tidal deformation. We note here the results of determining the orbital acceleration of the satellite of Mars, Phobos. A feature of this satellite motion is that it rotates in orbit faster than Mars. The tides in the viscoelastic body of Mars, caused by the attraction of the satellite, are late from the direction to the satellite. As a result, the energy of orbital motion is lost, and the satellite “falls” onto Mars. The most developed theory of the orbital motion of Phobos and Deimos based on observations was published in Lainey et al. (2007). The orbital acceleration values taken from this work and from several earlier publications are given in Appendix B.

References Aksenov, E.P., Grebenikov, E.A., Demin, V.G., 1961. General solution to the problem of the motion of an artificial satellite in a normal field of gravity of the Earth. Artificial Satellites of the Earth Moscow 8, 64–71. In Russian.

Chapter 3 Equations of motion and analytic theories

Aksenov, E.P., 1966. Odin vid differentsialnykh uravnenii dvizheniia sputnika. Trudy Gosudarstvennogo Astronomicheskogo Instituta im. P.K. Sternberga, Moscow 35, 44–58. In Russian. Aksenov, E.P., 1977. Theory of Motion of Artificial Earth Satellites. Nauka, Moscow. 360 pp. In Russian. Aksenov, E.P., 1986. Special Functions in Celestial Mechanics. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. Nauka, Moscow. 320 pp. In Russian. Aksenov, E.P., Emel’yanov, N.V., Tamarov, V.A., 1988. Practical use of an intermediate satellite’s orbit. Formulas, programs, tests. Trudy Gosudarstvennogo Astronomicheskogo Instituta im. P.K. Sternberga, Moscow 59, 3–40. In Russian. Archinal, B.A., Acton, C.H., A’hearn, M.F., Conrad, A., Consolmagno, G.J., Duxbury, T., Hestroffer, D., Hilton, J.L., Kirk, R.L., Klioner, S.A., McCarthy, D., Meech, K., Oberst, J., Ping, J., Seidelmann, P.K., Tholen, D.J., Thomas, P.C., Williams, I.P., 2018. Report of the IAU working group on cartographic coordinates and rotational elements: 2015. Celestial Mechanics & Dynamical Astronomy 130, 22. Arlot, J.-E., 1982. New constants for Sampson–Lieske theory of the Galilean Satellites of Jupiter. Astronomy & Astrophysics 107 (2), 305–310. Avdyushev, V.A., Ban’shikova, M.A., 2008. Determination of the orbits of inner Jupiter satellites. Solar System Research 42 (4), 296–318. Beletskii, V.V., 1963. The orbit of an equatorial Earth satellite. Planetary and Space Science 11 (5), 553–560. Brower, D., 1959. Solution of the problem of artificial satellite theory without drag. Astronomical Journal 64, 378–397. Brumberg, V.A., 1967. Development of the perturbation function in satellite problems. Bulletin of the Institute of Theoretical Astronomy, Leningrad 11 (2), 73–83. In Russian. Danby, J.M.A., 1995. Fundamentals of Celestial Mechanics, second edition. Willmann-Bell, Inc., USA. Darboux, G., 1901. Sur un probleme de mechanique. Archives Neerlandaises Des Sciences Exactes et Naturelles. Ser. 2, Vol. 6, p. 371. Duboshin, G.N., 1975. Celestial Mechanics. Basic Problems and Methods. Izdatel’stvo Nauka, Moscow. 800 pp. In Russian. Efroimsky, M., Makarov, V.V., 2013. Tidal friction and tidal lagging. Applicability limitations of a popular formula for the tidal torque. Astrophysical Journal 764 (1), 26. 10 pp. Emelyanov, N.V., 2005. Ephemerides of the outer Jovian satellites. Astronomy & Astrophysics 435, 1173–1179. Emel’yanov, N.V., Kanter, A.A., 2005. Orbits of new outer planetary satellites based on observations. Solar System Research 39 (2), 112–123. Emel’yanov, N.V., 2015. Perturbed motion at small eccentricities. Solar System Research 49 (5), 346–359. Emelyanov, N.V., Samorodov, M.Yu, 2015. Analytical theory of motion and new ephemeris of Triton from observations. Monthly Notices of the Royal Astronomical Society 454, 2205–2215. Emelyanov, N.V., 2018. Influence of tides in viscoelastic bodies of planet and satellite on the satellite’s orbital motion. Monthly Notices of the Royal Astronomical Society 479 (1), 1278–1286. Euler, L., 1760. Un corps etant attire an raison reciproque quarree des distances vers deux points fixes donnes. Mem. Berlin, Vol. 228. Euler, L., 1764. De motu corporis ad duo centra virium fixa attracti. Novi commentarii Academiae Scientiarum Imperialis Petropolitanae 10, 207. 1765, Vol. 11, p. 152.

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Fienga, A., Laskar, J., Kuchynka, P., Manche, H., Desvignes, G., Gastineau, M., Cognard, I., Theureau, G., 2011. The INPOP10a planetary ephemeris and its applications in fundamental physics. Celestial Mechanics & Dynamical Astronomy 111 (3), 363–385. Giorgini, J.D., Yeomans, D.K., Chamberlin, A.B., Chodas, P.W., Jacobson, R.A., Keesey, M.S., Lieske, J.H., Ostro, S.J., Standish, E.M., Wimberly, R.N., 1997. JPL’s on-line solar system data service. In: Amer. Astron. Soc. DPS Meeting, N.28, N.25.04. Bulletin - American Astronomical Society 28, 1158 (1997). Harper, D., Taylor, D.B., 1993. The orbits of the major satellites of Saturn. Astronomy & Astrophysics 268, 326–349. Hut, P., 1981. Tidal evolution in close binary systems. Astronomy & Astrophysics 99, 126–140. Jacobson, R.A., Riedel, J.E., Taylor, A.H., 1991. The orbits of Triton and Nereid from spacecraft and earthbased observations. Astronomy & Astrophysics 247, 565–575. Jacobson, R.A., 1997. The orbits of the inner Jovian satellites. In: Amer. Astron. Soc. DDA Meeting, N.28 . Bulletin - American Astronomical Society 28, 1098 (1997). Jacobson, R.A., 2009. The orbits of the Neptunian satellites and the orientation of the pole of Neptune. Astronomical Journal 137, 4322–4329. Jacobson, R.A., 2013. The orbits of the regular Jovian satellites, their masses, and the gravity field of Jupiter. In: American Astronomical Society, DDA Meeting #44. Id. 402.04. Kaula, W.M., 1962. Development of the lunar and solar perturbing functions for a close satellite. Astronomical Journal 67, 300–303. Kaula, W.M., 1964. Tidal dissipation by solid friction and the resulting orbital evolution. Reviews of Geophysics 2, 661–684. Kaula, W.M., 1966. Theory of Satellite Geodesy. Applications of Satellites to Geodesy. Blaisdell, Waltham, Mass. Kholshevnikov, K.V., Titov, V.B., 2007. Two body problem. St.-Petersburg State University. In Russian. Kondratyev, B.P., 2003. Potential Theory and Equilibrium Figures. Edition of the Institute for Computer Research. Edition of the Institute for Computer Research. Izhevsk, Moscow. ISBN 5-93972-222-9. 624 pp. In Russian. Kondratyev, B.P., 2007. Potential Theory. New Methods and Tasks With Solutions. Mir, Moscow. ISBN 978-5-03-003798-1. 512 pp. In Russian. Lainey, V., Dehant, V., Patzold, M., 2007. First numerical ephemerides of the Martian moons. Astronomy & Astrophysics 465, 1075–1084. Lainey, V., Arlot, J.-E., Karatekin, O., van Hoolst, T., 2009a. Strong tidal dissipation in Io and Jupiter from astrometric observations. Nature 459 (7249), 957–959. Lainey, V., Arlot, J.-E., Karatekin, O., van Hoolst, T., 2009b. Strong tidal dissipation in Io and Jupiter from astrometric observations. Nature 459 (7249), 957–959. Supplementary information. Lainey, V., Karatekin, O., Desmars, J., Charnoz, S., Arlot, J.-E., Emelyanov, N., Le Poncin-Lafitte, Chr., Mathis, S., Remus, F., Tobie, G., Zahn, J.-P., 2012. Strong tidal dissipation in Saturn and constraints on Enceladus’ thermal state from astrometry. The Astrophysical Journal 752 (1), 14. Lainey, V., 2016. Quantification of tidal parameters from solar system data. Celestial Mechanics & Dynamical Astronomy 126 (1–3), 145–156. Laskar, J., Jacobson, R.A., 1987. GUST86 – an analytical ephemeris of the Uranian satellites. Astronomy & Astrophysics 188 (1), 212–224. Lieske, J.H., 1977. Theory of motion of Jupiter’s Galilean satellites. Astronomy & Astrophysics 56, 333–352. Makarov, V.V., Efroimsky, M., 2013. No pseudosynchronous rotation for terrestrial planets and moons. Astrophysical Journal 764 (1), 27. 12 pp.

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Makarov, V.V., 2015. Equilibrium rotation of semiliquid exoplanets and satellites. Astrophysical Journal 810 (1), 12. 9 pp. Mignard, F., 1979. The evolution of the lunar orbit revisited. I. Moon and the Planets 20, 301–315. Mignard, F., 1980. The evolution of the lunar orbit revisited. II. Moon and the Planets 23, 185–201. Murray, C.D., Dermott, S.F., 2000. Solar System Dynamics. Cambridge Univ. Press, Cambridge. 608 pp. Peale, S.J., Cassen, P., 1978. Contribution of tidal dissipation to lunar thermal history. Icarus 36, 245–269. Sampson, R.A., 1921. Theory of the four great satellites of Jupiter. Memoirs of the Royal Astronomical Society 63, 1. Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J., 1994. Numerical expressions for precession formulae and mean elements for the moon and the planets. Astronomy & Astrophysics 282, 663–683. Subbotin, M.F., 1968. Introduction to Theoretical Astronomy. Nauka, Moskva. 800 pp. In Russian. Vienne, A., Duriez, L., 1995. TASS1. 6: Ephemerides of the major Saturnian satellites. Astronomy & Astrophysics 297, 588–605.

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4 Modeling the satellite motion. Numerical integration methods 4.1 The objective of solving the equations of motion of celestial bodies Methods for the numerical integration of the equations of motion of celestial bodies are developed and applied to solve various problems of celestial mechanics. Together with analytical and qualitative methods, they are procedures that serve the purpose of practical knowledge of nature. The process of studying celestial bodies is to construct a model of their motion. The model is the core of all scientific research. It is constantly being updated based on ever new observations. To date, we already know quite a lot about the bodies of the Solar System, planets and satellites. Laws are established according to which planets and satellites interact. These laws are expressed in the form of differential equations for the coordinates of the mass centers of bodies and for the angles of their rotation. Using the model consists in predicting the coordinates and angular positions of bodies at any given time point. Such a point can be either the time moment of the next observation of a celestial body, or the time moment of a meeting of a spacecraft with a celestial body, or the time moment of observation from an artificial space object. The prediction of the orbital and rotational motions of celestial bodies can be done based on the analytical solution of the differential equations of motion, if any. However, almost all practically significant motion models are described by equations whose exact solution is unknown. As for the approximate analytical solutions, it very often happens that the error of the approximate solution is unacceptable, or with the required accuracy, the procedure for constructing the solution is excessively time-consuming even for the most powerful modern computers. The coordinates and angles of rotation of a body around its center of mass at a given time point can be calculated by the methods of numerical integration of ordinary differential equations. The advantages of numerical methods over analytical methods are The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00009-1 Copyright © 2021 Elsevier Inc. All rights reserved.

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the relative simplicity of their implementation in the form of algorithms and computer programs. In many cases, numerical methods have greater accuracy in predicting the coordinates of celestial bodies. These advantages are usually achieved with the high required accuracy of calculations. The disadvantages of the methods for the numerical solution of the equations of motion are the long computational time, the rapid growth of the error of the solution with the growth of the interval for calculating the motion, and the impossibility of a reliable estimate of the accuracy of the solution. For given equations of motion, initial values of the coordinates of the bodies, and a given moment of precalculation, the accuracy of the solution is limited, regardless of how perfect the applied numerical solution method is. Numerical methods are indispensable in problems for which it is impossible to obtain an analytical solution. Methods of numerical integration of systems of differential equations also serve to verify the accuracy and assess the accuracy of the obtained analytical solution.

4.2 General properties of methods for the numerical integration of equations of motion The formulation of the problem of solving equations of motion by numerical integration methods is divided into two independent stages. First, differential equations of motion are compiled. Secondly, a new one is being developed or one of the known methods for their numerical integration is selected. To relate these two stages, it is necessary to adopt some standard form of equations of motion. A single celestial body or system of bodies is considered with mutual gravitational interaction. The equations of motion can take into account the influence of many other factors. In particular, the consideration may include the attraction of other bodies with known laws of motion, the difference between gravitating bodies from material points, and the resistance of the medium in which the bodies move. The equations of motion are a system of ordinary differential equations of the first or second order relative to the coordinates of a celestial body. Usually a rectangular coordinate system is used. If a system of second-order equations is compiled, it can always be reduced to a system of first-order equations. This has to be done, in particular, if the acting forces directly depend on the speed of motion of the celestial body.

Chapter 4 Modeling the satellite motion. Numerical integration methods

So, in the problems of numerical integration of the equations of motion of celestial bodies, we accept the following most general form of differential equations: dxi = fi (x1 , x2 , ..., xn , t) (i = 1, 2, 3, ..., n), dt

(4.1)

where the xi (t) are the functions of time sought after. An essential condition is the specification of the functions fi (x1 , x2 , ..., xn , t) in such a way that they can be calculated for any given values of the arguments. These functions may also contain constant parameters whose values are specified. To date, many methods have been developed for the numerical integration of ordinary differential equations. They are different in their effectiveness and application area. A complete review of the methods of numerical integration of differential equations of motion used in celestial mechanics is given in Bordovitsyna (1984). There is even a theory of methods for the numerical solution of equations that generalizes many of these methods and provides a way to develop the most optimal integration schemes (Butcher, 1963; 1964a; 1964b; Hall and Watt, 1976). Various methods of numerical integration in their application have some common properties. These properties must be known when solving specific problems in order to achieve the best result and research efficiency. The main idea of the methods of numerical integration of ordinary differential equations can be easily explained on the simplest of them—the method of broken lines by Leonhard Euler (Leonhard Euler, 1707–1783). In this case, one can restrict oneself to considering a single differential equation of a general form. Assume that we have one differential equation, dx = f (x, t), dt in which t is an independent variable—time. One is required to find a function x(t) that satisfies this differential equation and the initial condition: x(t0 ) = x0 , where t0 is the initial moment of time, and x0 is the predetermined constant. To begin with, it is important to understand that no method of numerical integration will allow us to find the function x(t) itself, but only a series of its values for a finite number of time points. Consider a certain time point t1 , not far from the moment t0 . The

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difference t1 − t0 is denoted by h, that is, h = t1 − t0 . The exact value x(t1 ) is unknown to us, but if the value h is sufficiently small, and the function f (x, t) is continuous, then the approximate value x at time t1 can be determined by the formula x1 = x0 + f (x0 , t0 ) h. Obviously, the difference between x1 and the exact value x(t1 ) will be the less, the smaller h. The error is due to the possible nonlinearity of the function x(t) on the segment [t0 , t1 ]. Next, we can determine the approximate value of the function x(t) at time t2 = t1 + h by the formula x2 = x1 + f (x1 , t1 ) h. The error of the obtained value of the sought function at the time t2 will be the sum of the error x1 and the error caused by the nonlinearity of the function x(t) on the interval [t1 , t2 ]. Now the algorithm for determining the values of the sought function for consecutive time moments ti = ti−1 (i = 1, 2, ...) can be written as xi = xi−1 + f (xi−1 , ti−1 ) h.

(4.2)

In this algorithm, the value h is called a constant step of numerical integration. An error of the value xi will contain the sum of the errors committed at all previous integration steps. These errors can have different values and signs, but with a rather arbitrary function f (x, t), the total error, as a rule, increases. Obviously, an increase in the total error occurs with an increase in the number of completed integration steps. The more steps are used with a constant step value, the greater error appears in such a solution. Any problem of numerically integrating the equations of motion of a celestial body is usually posed in such a way that in the end it is required to find the coordinates of the body at a given finite time moment tk . If the integration step has already been selected, then the number of steps k needed to solve the problem can be calculated. It is obvious that   tk − t 0 + 1, k=E h where E(...) means the integer part. The method of broken lines belongs to the group of so-called one-step methods of numerical integration. In addition to them,

Chapter 4 Modeling the satellite motion. Numerical integration methods

extrapolation and multistep methods exist (Bordovitsyna, 1984). All these methods have some common properties. In particular, when applying them, general problems arise. One of the problems is the choice of the integration step. At first glance, it is clear that, if the integration step is reduced, then the error allowed at each step will decrease, and therefore the accuracy of solving the equations of motion of a celestial body will improve. For a given time interval for calculating the motion, that is, the integration interval tk − t0 , the number of steps taken will increase, which will increase the computational time consumption. In some cases, the maximum accuracy of the solution is determined only by the allowable computational time. At first glance, it seems that, if an increase in computational time consumptions is still acceptable, then improvements in accuracy can always be achieved by reducing the integration step. We can conduct such an experiment. Define specific equations of motion, the numerical solution of which can be verified. These can be equations that have an exact analytical solution, or equations that admit a known particular solution. Then we can set the integration interval [t0 , tk ], take some method of numerical integration and solve the problem many times, each time decreasing the integration step. Studying the dependence of the accuracy of the solution on the integration step, we will see that first, with a decrease in the step, the error in the numerical solution decreases. But for some very small values of the integration step, the accuracy will begin to deteriorate, no matter how much we reduce the step. What is going on? The fact is that all calculations are always done with a fixed accuracy of performing simple arithmetic operations. It can be improved using special methods for displaying numbers in the computer’s memory and using appropriate algorithms for arithmetic operations. This will lead to even greater computational consumptions. It is important that in any computer with the appropriate software, the maximum accuracy of the representation of numbers and the accuracy of arithmetic operations is fixed. Each operation commits some error. It is called the rounding error of numbers. When performing arithmetic operations sequentially, rounding errors accumulate. The smaller the integration step, the smaller the error of Eq. (4.2), the smaller the error made at each step, and the less its accumulation at a finite moment in time. On the other hand, the number of steps is increasing, and rounding errors are accumulating. Thus, there is some optimal integration step at which the error in calculating the value of the required function at a finite time is minimal. For the chosen integration method and the given

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specific task, accuracy that is better than with the optimal step is unattainable. By improving the methods of numerical integration, some improvement in accuracy can be achieved. However, modern methods are already so perfect that a further increase in accuracy almost does not occur. In practice, calculations with optimal steps, as a rule, are not productive—the computational consumptions are too high. The following problem is caused by the fact that the degree of nonlinearity of the required function can be different in different parts of the interval [t0 , tk ]. For a given constant step of integration, the total error is generated mainly in those areas where the nonlinearity is maximum. It makes no sense to choose the same small step in other areas and spend time in vain on calculations with excessive accuracy. It would be optimal to choose at each segment its integration step, depending on the degree of nonlinearity of the sought function. In perfect methods of numerical integration, the behavior of the solution is analyzed in the previous step in order to economically choose the next step. For this, of course, before starting the integration, a certain parameter is set that controls the accuracy of the calculations at each step. This parameter is called the given accuracy of integration, but this accuracy does not coincide with the accuracy of obtaining a solution at a finite time tk . These two accuracies are usually proportional with the same formulation of the problem, that is, with given equations of motion, initial conditions and a finite moment of time. To analyze the behavior of the solution, various techniques and formulas are used. One of them is that each step is done twice: first, one step of h, then two steps, of h/2. The difference between the two results is compared with a given constant—the accuracy of the calculations. If the difference is greater, then the step is halved. If this difference is ten or more times less than the specified accuracy, then the step is increased. The most advanced numerical integration algorithms use more complex techniques. Unfortunately, in practice, not always automatic step selection can be effective. In some cases, a constant integration step is still used. The biggest problem in the practice of numerical integration of the equations of motion of celestial bodies is the assessment of the accuracy of the solution. It turns out that there are no strict formulas or conditions allowing such estimates to be performed. In fact, some techniques are used, which nevertheless cannot be considered completely reliable. One of them is the “forward–backward” integration. That is, after obtaining the value of the function at a finite time moment sought-after, this value is taken as the initial one and integration is performed with a negative step to the initial moment. At the end, the result is compared with the initial condi-

Chapter 4 Modeling the satellite motion. Numerical integration methods

tion. The resulting difference in values is considered the accuracy of the integration. Sometimes such estimates are satisfactory, but mostly the real accuracy is worse. There are other methods for assessing the accuracy of numerical integration, but all of them are, in fact, unreliable.

4.3 Runge–Kutta integration method for ordinary differential equations The method of broken lines described in the previous section is almost not applied in practice. More accurate methods exist. One of them is the Runge–Kutta method (Bordovitsyna, 1984; Duboshin, 1976). This method is far from the most perfect, but at one time it was widely used in astronomical problems. The formulas of the Runge–Kutta method are quite simple (Duboshin, 1976). They can be easily programmed in any suitable programming language. In problems where it is not required to obtain extremely high accuracy of the numerical solution of the equations of motion, the Runge–Kutta method can be effectively applied. We describe this method and the formulas by which the calculations are performed. Consider the problem of numerical integration of a system of ordinary differential equations of the first or(0) (0) (0) der (4.1) with the initial conditions x1 = x1 , x2 = x2 , ..., xn = xn for t = t0 . The variables x1 , x2 , ..., xn will be called coordinates for convenience, and the independent variable t—time. The Runge–Kutta formulas given below make it possible to determine the coordinates at the time moment tk+1 = tk + h, if they are known at the time moment tk . The formulas are compiled on the basis of the method of interpolation by polynomials with respect to the integration step h, in which terms of a certain order of smallness are neglected with respect to the step size. In this case, all members are preserved up to fourth order inclusive. Runge– Kutta formulas have the form (k)

xi (k)

pi  (k) qi

(k)

ri

= fi

(k) x1

= xi (tk ),

  (k) (k) = fi x1 , x2 , ..., xn(k) , tk ,

 1 (k) (k) 1 (k) 1 (k) 1 (k) + hp1 , x2 + hp2 , ..., xn + hpn , tk + h , 2 2 2 2

  1 (k) (k) 1 (k) 1 1 (k) = fi x1 + hq1 , x2 + hq2 , ..., xn(k) + hqn(k) , tk + h , 2 2 2 2

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(k)

si

  (k) (k) (k) (k) = fi x1 + hr1 , x2 + hr2 , ..., xn(k) + hrn(k) , tk + h , (k)

xik+1 = xi

 1  (k) (k) (k) (k) + h pi + 2qi + 2ri + si 6 (i = 1, 2, 3, ..., n).

From the above formulas it is easy to see the scheme of the algorithm of one-step methods for the numerical integration of ordinary differential equations of the form (4.1). The algorithm consists of two relatively independent blocks. In the first block, calculations are performed using the Runge–Kutta formulas. The input data for this block are the values of the functions given in the right-hand sides of the equations at the time tk . The result of the block operation is the values of the sought functions at the time moment tk+1 . Of course, the initial parameters are the moments tk and tk+1 themselves, as well as the integration step h. The implementation of this block does not depend on the specific problem of celestial mechanics. The second block is the calculation of the right-hand sides of Eqs. (4.1) for any given arguments x1 , x2 , ..., xn , t. This block is completely determined by the formulation of the problem of the motion or rotation of celestial bodies and does not depend on the integration method.

4.4 Algorithm for solving problems of motion of a celestial body by numerical integration methods Consider the following class of problems on the motion of a system of celestial bodies. Let differential equations be given with respect to the coordinates of the bodies. The coordinates can be both the rectangular coordinates of the centers of mass of each body and the angles of rotation of each body relative to its center of mass in some given coordinate system. The equations must be expressed in the form (4.1), and their right parts are set so that the rules are known by which they can be calculated for any given values of the required coordinates and, possibly, time. In addition, coordinate values at some initial time moment t0 should be set. (0) (0) (0) Denote these initial values by x1 , x2 , ..., xn . One is required to determine the coordinate values for some other given time moment tk . The practical aims of solving the problem of the motion of bodies of a system dictate the need for sequential calculation of coordinates for other given moments t2 , t3 , ..., tk . Mostly, time

Chapter 4 Modeling the satellite motion. Numerical integration methods

moments are chosen equally spaced so that ti = ti−1 + H (i = 1, 2, ..., k),

(4.3)

where H is the given time step (but not the integration step). Thus, the result of solving the problem will be the values of the required (i) (i) (i) functions at the moments t1 , t2 , ... , tn , i.e. x1 , x2 , ..., xn (i = 1, 2, ..., k). The problem posed is typical not only in practical celestial mechanics, but also in various theoretical studies. According to the formulated statement of the problem, we can compose an algorithm for its solution. It will consist of the following blocks: 1. setting the initial moment of time, the initial values of the coordinates and the step of following the moments H ; 2. setting the next time moment; 3. performing numerical integration of equations and obtaining the values of the sought functions at the next time moment; 4. storing the obtained values in a file for later use; 5. verification of the attainment of the last predetermined time point and transition to item (2), if the last moment has not yet been reached; 6. termination of the algorithm. Item (1) is performed by entering numbers from a source data file or by setting values directly in the text of a computer program. Item (2) is satisfied by Eq. (4.3). The implementation of item (3) in a computing program should be independent of the problem, so that any specific problem can be solved by any method of numerical integration. Therefore, the integration method itself is usually composed as a separate procedure in the selected programming language. This procedure is usually followed by specialists in the methods of numerical integration of ordinary differential equations. To use the procedure in a specific problem, we only need to know the form of access to this procedure and the methods of exchanging data with it. The necessary information is usually contained in the instructions that accompany the procedure when it is transmitted or published. Item (3) of the algorithm is implemented according to the instructions for the procedure. An example of such an instruction is provided in the next section. At first glance, the algorithm described above completely lacks a block for calculating the right-hand sides of the equations being solved. In fact, this block is included in item (3), but it is implemented in the form of an independent procedure, which is composed by a specialist who poses and solves the problem. This procedure is used in the process of numerical integration, which is

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independent of the form of the right-hand sides of the equations. To organize the compilation of the algorithm and the program, some rules are established by which the procedure for calculating the right-hand sides of the equations of motion is referred to. These rules are also part of the instructions for the integration procedure. The algorithm for calculating the right-hand sides of the equations does not depend on the integration method, however, the heading of this procedure must be consistent with the instructions for the program. Items (4), (5), and (6) of the algorithm are simple enough for programming. We only note that item (4) can be implemented in a more sophisticated way. For example, it is possible to program the construction of graphic images on a computer screen depending on the obtained values of the sought functions. As a result, the successive change of pictures in the process of solving the problem will give a visual representation of the motion of the body system. We also note that when using some numerical integration programs, it is necessary to set certain operation parameters. These can be, for example, parameters that specify the accuracy of integration. At the same time, the result of the procedure can be various auxiliary data, which can also be used in the program for solving the problem.

4.5 Instructions for the computational program for the numerical integration of ordinary differential equations by the Everhart method In practical celestial mechanics, many different methods of numerically integrating the equations of motion are used. This science provides an incentive for the development and improvement of such methods. At the same time, the problems of celestial mechanics are a kind of proving ground for testing new developments in this area. One of the most recently applied methods of numerical integration in celestial mechanics is the Everhart method (Everhart, 1974). It is significant that its author is a specialist in the field of celestial mechanics. The method is described in more detail in the literature (Bordovitsyna, 1984; Avdyushev, 2015). Initially, the integration procedure was compiled in the Fortran programming language by Everhart himself, which is why it is called the Everhart procedure. This version of the program was then adapted for computers in whose memory floatingpoint numbers occupy 8 bytes of memory. The program is freely

Chapter 4 Modeling the satellite motion. Numerical integration methods

transferred between researchers of different scientific institutions. It is also available at the Sternberg Astronomical Institute of M.V. Lomonosov Moscow State University (SAI MSU), where the program was rewritten with the C programming language. At the same time, in the form of access to it, the properties of the parameters characteristic of the Fortran programming language were preserved. The main advantage of the Everhart method is the high achievable integration accuracy. The forfeit for accuracy is the high consumption of “machine” computing time. The integration procedure according to the Everhart method is designed so that with the help of two parameters the required accuracy of calculations can be set. Depending on the given accuracy, the necessary computational time changes. With the low required accuracy, the computational costs will be small, however, the best ratio between the accuracy of calculations and the time required is achieved with a high specified accuracy. The main parameter, ε, which determines the accuracy of calculations at each step, is expressed as ε = 10−l ,

(4.4)

where l is some given integer. The Everhart method uses special approximating polynomials in powers of the integration step. The degree of these Norder polynomials can be selected from the following list of integers: 7, 11, 15, 19, 23, and 27. In the procedure, we can take any of these values. However, with a low specified accuracy of integration, it is useless to set a high degree of polynomials. This can only lead to unjustified consumptions of computing time. Therefore, Norder is selected depending on the given accuracy of calculations within 3 l ≤ Norder ≤ 2l, 4

(4.5)

where l is the integer in Eq. (4.4). The Everhart procedure allows solving differential equations of motion both in the mode of automatic choice of the integration step and at a constant given step. Since in celestial mechanics most problems are described by ordinary second-order differential equations, for convenience of programming the problem, the Everhart procedure allows us to immediately solve second-order equations without decomposing them into twice the number of first-order equations of the form (4.1). Most of the integration time is spent on the calculations of the right-hand sides of the equations. Therefore, the effectiveness of the program is characterized by the number of calls made to the

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procedure for calculating the right-hand sides of equations in the process of computing in a specific problem. The number of completed integration steps in the case of automatic step selection also turns out to be interesting to evaluate the effectiveness of the procedure. Both quantitative characteristics are issued by the procedure through its output parameters. The Everhart method contains an iteration process for constructing polynomials in powers of the integration step in the first step. Usually it is sufficient to perform two iterations, but in some problems, increasing the number of iterations at this stage of the algorithm can lead to a slight increase in the efficiency of the procedure without significantly increasing the computational time. Therefore, the input parameter of the procedure is the specified number of iterations, with which we can control the effectiveness in some small limits. At the initial stage of solving a specific problem, little is known about the properties of equations in terms of the application of the Everhart procedure. Therefore, without a doubt, we can set the number of iterations to 2. We define a class of equations depending on what kind of equations should be solved by the Everhart method. The equations are divided into three classes. The equations of the first class have the form (1). The second class includes systems of equations of the second order of the form   d 2 xi dx1 dx2 dxn x = f , x , ..., x , , , ..., , t (i = 1, 2, 3, ..., n). i 1 2 n dt dt dt dt 2 (4.6) The equations of the third class differ from Eqs. (4.6) only in that the right-hand sides are independent of the velocity components. These are the most common equations in celestial mechanics. The class of such equations in a strange tradition is designated as −2. Class “−2” equations are of the form d 2 xi = fi (x1 , x2 , ..., xn , t) (i = 1, 2, 3, ..., n). dt 2

(4.7)

After describing all the features of the Everhart procedure, we will provide instructions for its use here. The instruction, of course, depends on the programming language used. Usually this is one of the languages of the procedural type, for example, Fortran, Pascal, or C. The procedure call forms in different languages are very similar. Suppose that the calculation program is compiled in the C programming language. The description of some parameters preserves the properties typical of older versions of

Chapter 4 Modeling the satellite motion. Numerical integration methods

the Fortran language. In particular, indexing variables in arrays starts with one. Here we first describe the parameters of the procedure, and then the procedure call itself. It is necessary to describe the variables used in the program and set their values. The corresponding fragments of the user program are of the form ... #define NV 3 ... double X[NV+1],V[NV+1]; double TI,TF,XL; int LL, NI, NF,NS, NCLASS, NOR; ... Rada27(X, V, TI, TF, XL, LL, NV, NI, &NF, &NS, NCLASS, NOR); We describe the type and meaning of each parameter. It is convenient to start with the input parameter NV. This is a parameter of int type. It is called by the procedure according to its value (“call by value”), that is, when accessing the procedure, any parameter of the int type can be in place of this parameter. The parameter NV sets the number of required functions n. In the above fragment of the user program, this parameter is specified by the symbolic constant NV. It is necessary to take into account the important fact that inside the text of the Rada27 procedure there are descriptions of internal arrays (indexed variables) of a given length. This length depends on the number of NV variables. We should check if there is enough space reserved for these internal arrays for a given NV. The NCLASS parameter specifies the class of solvable equations. It can have values 1, 2 or −2, respectively, as described above classification of equations. An NCLASS parameter of int type. It is called by value. Parameters X and V are one-dimensional arrays (indexed variables) in which indices are used, starting from one. These arrays are used to store and transfer the values of the required functions. The coordinates are specified in the X array, and the velocity components in the V array. Before accessing the procedure, these arrays must contain initial values at the time t0 . After return, the arrays X and V will contain the values of the required functions at the final moment tk . Array V is used only for equations of classes 2 and −2 to store and transmit the values of the first derivatives of the sought functions with respect to time.

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The parameters TI and TF are simple variables of the type double specify the start and end times t0 and tk , respectively. These parameters are transferred by value. An LL parameter of int type sets the accuracy of the calculation. It corresponds to the value of l in Eq. (4.4). This parameter is passed by value. Automatic selection of the integration step is made if the value of the LL parameter is set to positive. To enable the constant integration step mode, set LL to any negative number. In this mode, an XL parameter of double type is used to set the constant integration step. This input parameter is passed by value. When the integration step is automatically selected, the XL parameter is not used. To specify the degree of approximating polynomials Norder , use the NOR input parameter of int type. It is transferred to the procedure by value. To select the value of this parameter, use the inequalities (4.5). An NI parameter of int type sets the number of iterations when determining an approximating polynomial. This is an input parameter, it is transferred to the procedure by value. As already indicated above, the value of this parameter can be chosen equal to 2 if the most optimal value of this parameter is unknown. After the calculations, the output parameters NF and NS of int type transfer to the user program the number of calls made to the calculation of the right-hand sides of the equations and the number of integration steps performed, respectively. These parameters are transferred to the procedure by name (address). Now we will consider how the Everhart numerical integration procedure interacts with the procedure for calculating the righthand sides of the equations, which the user must make. In accordance with the form of accessing this procedure inside the Everhart procedure, its description (prototype) in the user program should be of the form void FORCE(double *Xc, *Vc, double TM, double *Fc); Arrays Xc and Vc at the entrance to the procedure contain the arguments of the functions that determine the right-hand sides of the equations being solved. In the Xc array, the coordinate values are stored, and in the Vc array, the values of the derivatives of the coordinates with respect to time. Arguments are placed in arrays according to their numbers, while the elements of the Xc[0] and Vc[0] arrays are never used. In addition, the parameter Vc is used only when class 2 equations are solved. An input parameter TM of double type sets the current time moment—the argument of the right-hand sides of the equations.

Chapter 4 Modeling the satellite motion. Numerical integration methods

It is transferred to the procedure by value. If the right-hand sides are time-independent, then the TM parameter is not used. The user must place the result of the calculations of the righthand sides of the equations in the elements of the output array Fc. For this, the elements of this array are sequentially used, starting from the first. The zero element of the Fc array is not used. When compiling a computer program for the numerical solution of the differential equations of motion of celestial bodies, some features associated with the applied programming language and other programming tools, for example, the compiler, may appear. In particular, we need to know that the Everhart procedure, written in the Pascal programming language, uses several working arrays for its work, which must be described outside the body of the procedure as global variables with fixed names. The length of these working arrays depends on the number of sought functions, that is, on the value of n in Eqs. (4.1), (4.6), and (4.7). The easiest way is to reserve some length of these arrays and remember the maximum allowable number of required functions, which cannot be exceeded. When programming the FORCE procedure, the calculation of the right-hand sides of the equations may require parameters that are entered or set in the main program. The transfer of these parameters can only be done through global variables declared in one of the program modules and connected via module interfaces.

4.6 Belikov program for numerical integration of ordinary differential equations Another highly efficient computational program for numerically integrating ordinary differential equations was developed by Belikov (1993). The integration method is based on the approximation of the solution at each integration step by segments of the series with respect to Chebyshev polynomials close to the best uniform polynomial approximation. The Rado–Hermite quadrature formula of the highest accuracy class is used in calculating the coefficients of the series based on the displaced Chebyshev polynomials. The implementation of the applied method in the form of a computer program was transferred in the 90s by the Institute of Theoretical Astronomy of the Russian Academy of Sciences (St. Petersburg) to the SAI MSU. Procedure originally compiled by Belikov M.V. with the programming language Fortran, was subsequently adapted by N.V. Emelyanov (SAI MSU) for the C pro-

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gramming language. In programming languages, the procedure is named DINCH. A significant advantage of the Belikov program is that it includes a procedure for approximating the results of numerical integration by segments of series in Chebyshev polynomials. After a single integration of the equations of motion in a special file, the coefficients of the expansion of the coordinates of the body in Chebyshev polynomials are accumulated. Using this file using a simple program, you can get the coordinates and speed of a celestial body at any given time. This technique is described in more detail below in a special section. Without going into details of the method, we consider some aspects of using this computing program. Parameter identifiers are those used in the text of the transferred procedure. The DINCH numerical integration procedure call form is similar to the Everhart procedure call (see above). The same terminology is used for the class of equations. The initial coordinate values and velocity components, as well as their resulting values, are placed in indexed variables as well as in the Everhart procedure. The (NORD) parameter is defined that determines the degree of approximation polynomials at the integration step. Integration accuracy is controlled by a special parameter, similar to the Everhart procedure. It is possible to integrate with a constant predetermined step size (XL). Additionally, some parameters that control the integration process are set. The meaning of these parameters can be clarified in the description of the method (Belikov, 1993). In particular, a certain parameter NNN is specified that controls the choice of the value of the integration step. When NNN = 0, a variable step is selected according to the absolute value of the last coefficient in the expansion of the right-hand sides of the equations. For NNN = 1, a relative criterion is used; the latter to the zero and the first. In the general case, when there is no information on how to choose this parameter, we can set NNN = 1. The parameters for choosing the number of iterations are set to determine the coefficients of the series from the displaced Chebyshev polynomials: ITS at the start step, ITC at the subsequent ones. If there is no additional information, it is recommended to select ITS = 6 and ITC = 2. The form for accessing the FORCE procedure for calculating the right-hand sides of the equations is exactly the same as for the Everhart procedure. It is important to take into account that the program text contains descriptions of arrays (indexed variables), the length of which depends on the number of required functions. These descriptions should be consistent with the user program.

Chapter 4 Modeling the satellite motion. Numerical integration methods

Some details of the Belikov procedure can be found necessary for transferring the program to the user. In conclusion, we note that the numerical integration program developed by Belikov was successfully used by the author of this book and showed high efficiency.

4.7 Testing and comparing some numerical integration procedures When we model the motion of celestial bodies by the methods of numerical integration of differential equations of motion, an urgent question arises about the choice of a particular integration procedure. Usually a search is made for the most effective method. The concept of effectiveness should be formulated strictly and unambiguously. Efficiency is characterized by two factors: the accuracy of the solution and the computational cost (computation time). A proper comparison of the two methods and clarification of the effectiveness is done as follows: the computational time consumptions are determined with equal accuracy, or the achieved accuracy is estimated at equal computational time consumptions. In some cases, it is difficult to choose the integration parameters so as to equate one of the indicators in the two methods. This can only be done approximately. As already noted above in a special section, the accuracy of solving differential equations of motion of a celestial body fundamentally depends on the following circumstances: – number of sought variables, – the form of the right-hand sides of the equations, – initial conditions of the solution, – time interval between start and end time points. Of course, a comparison of methods should be done under equal conditions. In this case, it is necessary to set such control parameters of the procedures that provide maximum accuracy. However, in some cases it is impossible to realize the maximum achievable accuracy of the solution due to unacceptably long computation times.

4.8 Approximation of the rectangular coordinates of planets and satellites by truncated Chebyshev series In the numerical integration of the differential equations of motion of a celestial body, rectangular coordinates are calculated

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at a number of time moments separated by a numerical integration step. There is a problem of storage and subsequent use of the results. Since the step of numerical integration can be very small, remembering the coordinates obtained at each step is difficult and impractical. In practical problems, an approximate representation of the results of numerical integration is applied using segments of series in the Chebyshev polynomials Tj (τ ). In principle, any orthogonal functions, for example, power functions, would be suitable for this. However, the advantage of expanding a function into Chebyshev polynomials is that with such a decomposition, the absolute calculation error is alternating and distributed more or less uniformly over the entire interval [−1,1] of the argument of the Chebyshev polynomials. Any time function on a finite interval can be represented by a segment of a series of Chebyshev polynomials with a length of N terms so that at N time moments at this interval such an approximate representation will exactly match the value of the function. Moreover, at intermediate time points, the value of the truncated series will differ from the function itself. The differences depend on the properties of the function and on the maximum degree of the polynomials N − 1. The entire time interval for which the equations of motion are numerically integrated is divided into equal subintervals of a certain length t = t2 − t1 , where t1 and t2 are the moments of the beginning and end of one such subinterval. At each such subinterval, a representation is constructed separately for each of the coordinates, for example, for the coordinate x. The new variable τ is set by the formula τ=

2t − t2 − t1 , t 2 − t1

(4.8)

which will be the argument of the Chebyshev polynomials. It is clear that for t = t1 we have τ = −1, and for t = t2 we will have τ = 1. Thus, at the time subinterval (t1 , t2 ), the argument τ varies within [−1, 1]. The function x(t) can be approximated by the formula x(t) ≈

N −1  j =0

1 Cj Tj (τ ) − C0 , 2

in which the dependence of τ on time is given by Eq. (4.8), and the coefficients Cj are calculated from the relation Cj =

N −1 2  x(τk )Tj (τk ) (j = 0, 1, 2, N − 1), N k=0

Chapter 4 Modeling the satellite motion. Numerical integration methods

in which the values of the argument τk are determined by the formula   π(k + 12 ) τk = cos (k = 0, 1, 2, N − 1). N Such a representation will exactly coincide with the function itself for the values of the argument τ equal to the roots of the polynomial TN (τ ), which are the numbers τk . From the above relationships it is clear that we need to know the coordinate values for a number of certain time points tk =

t2 + t1 + τk (t2 − t1 ) 2

inside the interval (t1 , t2 ). In practice, the calculation process is programmed as follows. An array of coefficients Cj (j = 0, 1, 2, N −1) is created in the memory. Before entering numerical integration in the interval (t1 , t2 ), all elements of this array are assigned zeros. Numerical integration is performed sequentially up to each moment tk . When the moment tk is reached, we have the values τk and x(τk ). Here the cycle is passed by the coefficient number Cj . Each coefficient Cj is incremented by (2/N )x(τk )Tj (τk ). In this cycle, the values of Tj (τk ) are calculated using the recurrence relation for Chebyshev polynomials Tj +1 (τk ) = 2τk Tj (τk ) − Tj −1 (τk ).

(4.9)

For the initial values of the index, we have T0 (τ ) = 1, T1 (τ ) = τ. Simultaneously, calculations are performed for the three coordinates x, y, z. Moreover, for each coordinate of the celestial body, its own array of coefficients Cj is calculated. After the numerical integration leaves the interval (t1 , t2 ), the coefficients obtained for each coordinate are stored in a file. At each subsequent subinterval, the same memory array is used for the coefficients Cj . As a result, if the entire integration interval was divided into K of equal subintervals, then the resulting file will contain 3KN numbers. If the user has a resulting file, then the calculation of the rectangular coordinates of the planet or satellite at a given time point t can be done by the following procedure. We find the time subinterval to which the given moment t belongs. We read from the file for each coordinate an array of coefficient values Cj (j = 0, 1, 2,

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N − 1). We calculate the coordinates by the formula x=

N −1 

Cj Tj (τ ).

j =0

Here the argument τ is found from Eq. (4.8), and the Chebyshev polynomials are calculated using the recurrence relations (4.9).

4.9 Overview of problems and methods of numerical integration. Book by Avdyushev Extensive information on the methods of numerical integration of the differential equations of motion of celestial bodies is contained in the monograph (Avdyushev, 2015). This book gives a lot of information necessary for modeling the motion of celestial bodies. Here is a brief summary of this monograph. The field under consideration is engaged in by many researchers in the world. For a good understanding between them, common terminology is needed. The monograph (Avdyushev, 2015) begins precisely with the clarification of terminological points. An important initial stage of motion simulation is the preparation of differential equations of motion for celestial bodies. Difficulties in obtaining a solution, in particular in cases of close approximation of bodies, led to the need to transform the equations. The monograph (Avdyushev, 2015) gives an extensive series of variants of differential equations of motion, which allow overcoming the problems that arise. The so-called stabilization methods of the system of equations of motion are considered. A comparative analysis of the effectiveness of various equations is given. Later in the monograph (Avdyushev, 2015), various methods of numerical integration are presented. In addition to one-step methods, extrapolation methods, multistep methods, and symplectic methods are considered. The result is a comparative analysis of the effectiveness of various methods. The third part of the monograph (Avdyushev, 2015) is devoted to the inverse problem of orbital dynamics, the determination of orbits from observations. In addition to refining the motion parameters using the least squares method, a number of other methods for solving the problem are considered. An important advantage of this material is that all methods are laid out as necessary for their practical application. The monograph (Avdyushev, 2015)

Chapter 4 Modeling the satellite motion. Numerical integration methods

provides a comparative analysis and evaluates the effectiveness of different methods. In fact, the monograph (Avdyushev, 2015) makes up for those important aspects that were omitted in Chapters 4 and 6 of this book due to space limitations.

References Avdyushev, V.A., 2015. Numerical Modelling the Celestial Body Motion. Izdatelsky Dom Tomsk State University, Tomsk. In Russian. Belikov, M.V., 1993. Methods of numerical integration with uniform and mean square approximation for solving problems of ephemeris astronomy and satellite geodesy. Manuscripta Geodaetica 18, 182–200. Bordovitsyna, T.V., 1984. Modern Numerical Methods in the Celestial Mechanics Problems. Nauka, Moscow. In Russian. Butcher, J.C., 1963. Coefficients for the study of Runge–Kutta integration processes. Journal of the Australian Mathematical Society 3, 185–201. Butcher, J.C., 1964a. On the Runge–Kutta processes of high order. Journal of the Australian Mathematical Society 4, 179–195. Butcher, J.C., 1964b. Implicit Runge–Kutta processes. Mathematics of computation. American Mathematical Society 18, 50–64. Duboshin, G.N., 1976. Spravochnoe rukovodstvo po nebesnoi mekhanike i astrodinamike. Nauka, Moscow. In Russian. Everhart, E., 1974. Implicit single-sequence methods for integrating orbits. Celestial Mechanics 10 (1), 35–55. Hall, G., Watt, J.M., 1976. Modern Numerical Methods for Ordinary Differential Equations. Clarendon Press. 336 pp.

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5 Observations of planetary satellites 5.1 General principles of observations We look at celestial bodies with “our eyes” through the devices with which we observe. Naturally, when we say “observations”, we mean “measurements”. What do we measure? Our instruments can measure the angles of directions to celestial bodies, the distances to them, as well as the rate of change of distance. In any case, we are dealing with a vector whose beginning is located at the point of observation, and the end is on the celestial body that we are observing. We will call this vector the observation vector. It is known that the propagation of light within the Solar System is described by the general theory of relativity. According to this theory, in particular, there is a curvature of light beams due to the mass of the Sun. In our consideration of observational processes, we will neglect the effects of the theory of general relativity. Each observation is ascribed with the moment of observation. This is the moment of registration in the photodetector of those photons that were emitted by the observed satellite or planet. We should keep in mind the fact that, due to the finiteness of the light velocity, the moment of the start of photons from an observed celestial body precedes the moment of observation. For practical actions, it is necessary to connect the observation vector on the one hand with the rectangular coordinates of celestial bodies, on the other hand, with those quantities that we measure with the help of instruments that we will call measurable quantities. Satellite motion parameters are determined based on observations using measured values. However, it turns out that it is impossible to derive a direct dependence of these parameters on the measured values. To determine the parameters, a special differential refinement method is used, which is discussed in the next chapter. According to this method, the inverse operation is necessary—the calculation of the measured value from the given motion parameters. Rectangular coordinates can be calculated using procedures developed in the construction of motion theories. For given moThe Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00010-8 Copyright © 2021 Elsevier Inc. All rights reserved.

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tion parameters, the procedure gives coordinates for any given time moment, which is called the input moment of the theory. By time we mean the time argument that appears in the differential equations of motion of a celestial body. In the following sections, we first derive the relationships between the observation vectors and the rectangular coordinates of celestial bodies. Then we obtain the relationship of the observation vectors with the measured values. At the end of this chapter, we will discuss some details of the technique for observing planetary satellites. Useful information on the issues discussed here can be found in Emel’yanov (2017a). Important note Attention should be paid to the widely used terminology associated with observations of planetary satellites. When we say “observations”, we mean not the process, but the measurement results themselves. Often, observations are data that are generated in the process of observing celestial bodies. Also in this book, in many cases, observations mean data: numbers, accompanied by explanations. Note that observations, representing the actual values of some actually measurable quantities, are inevitably connected with the concept of time and with such an abstract concept as a coordinate system. In fact, time is also measured using instruments, and coordinate systems are modeled by linking coordinates with some real objects. These circumstances are discussed in more detail below at the end of this chapter. Obviously, time has no beginning or end. However, it must be measured. For this, it is necessary to set a certain moment from which all other time moments will be counted. It is customary to consider this starting point to be 12 hours on January 1, 2000 in the same scale in which other moments are counted. This initial time moment is called and designated the J2000 epoch. The Julian day of the J2000 epoch is 2451545.0.

5.2 Determination of topocentric positions of planets and satellites Observations of celestial bodies are carried out from the Earth’s surface or from the orbit of an artificial Earth satellite. A special case of observations from the interplanetary spacecraft is not considered here.

Chapter 5 Observations of planetary satellites

The point on the Earth’s surface at which observations are made is usually called a topocenter. The position of the topocenter relative to the geocenter is determined by its position in the Earth’s coordinate system. In the problems considered here, this can be done approximately. Here we will assume that the topocenter coincides with the geocenter. Motion models of bodies in their orbital motion around the Sun are usually considered in the heliocentric coordinate system or in the barycentric coordinate system with the origin located in the barycenter of the Solar System. Theories and models of motion of planetary satellites are constructed in a coordinate system with a origin located either in the center of the planet or in the barycenter of the planet–satellite system. In the general case, we will call this point just a planet. Thus, in this section, the following points will be featured: the center of the Sun, the barycenter of the Solar System, the topocenter, the planet, and the satellite. The barycentric vectors of the Sun, topocenter, and planet will be denoted by H(t), T(t), and P(t), respectively. The vectors introduced here are considered as functions of time. For their calculation, specially developed motion models are used. The given input value of such computational procedures is the parameter t, which is an independent variable, conventionally called the time argument. The time argument in different theories can be the time counted in different scales, for example, TT and TDB. We will assume that before substituting into the procedure for calculating the position vector of a celestial body, time is expressed in the necessary scale.

5.3 Planet observations Let us first assume that we are observing a planet. Consider how the observation vector is determined in this case. Suppose that the observation is performed at time t0 . The observation vector has an origin in the topocenter, and its end is in the observed celestial body at its position that it had at the moment when it started those photons that hit the photodetector of the observer at the time t0 . We denote the planet’s observation vector by PT . At the start of photons we will assign different indices depending on the object of observation. If we observe a planet, then the moment of start of photons can be denoted by t1 . Fig. 5.1 shows the configuration of the planet and the topocenter at the corresponding time points. The letter B marks the barycenter, the letter T indicates the topocenter, and the letter P shows the planet. This figure shows that the observation vector

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is determined by the equation PT = P(t1 ) − T(t0 ).

(5.1)

The times t0 and t1 do not coincide. Their difference, called the light time, is given by the relation t0 − t1 =

|PT | , c

(5.2)

where c is the speed of light. Since the moment of observation t0 given, then Eqs. (5.1) and (5.2) can be solved by iterations, first setting t1 = t0 . The most accurate of the currently existing theories of planetary motion allow us to determine their rectangular coordinates in the coordinate system with the origin in the barycenter of the Solar System. Usually, the theory also gives the coordinates of the Sun in this system. We can always make the appropriate transformation and determine the heliocentric coordinates of the planets. Comment Here we make one important comment. In some work, one can meet with the assertion that the coordinate system is taken to be heliocentric, that is, the origin of the coordinate system is placed in the center of the Sun. The position of the topocenter and the position of the planet in this case are determined by heliocentric coordinates. It is required to determine the observation vector PT . The use of heliocentric coordinates here leads to an inaccuracy, which is found when comparing observations with theory. Let us explain the source of this inaccuracy. We show further that to calculate the observation vector, it is necessary to take the barycentric coordinates of the Sun and planets. We denote by Ph (t1 ) and Th (t0 ) the heliocentric vectors of the planet and topocenter at the corresponding time moments. By analogy with the previous arguments, the observation vector can be constructed as a difference, PT = Ph (t1 ) − Th (t0 ), where another notation is used here for the observation vector. In fact, such a construction is incorrect. For explanation, we introduce the barycentric vectors of the Sun at the corresponding time moments H(t0 ) and H(t1 ). The equations are obvious: Ph (t1 ) = P(t1 ) − H(t1 ), Th (t0 ) = T(t0 ) − H(t0 ).

Chapter 5 Observations of planetary satellites

Figure 5.1. The location of the Earth and planet during observations.

Figure 5.2. Erroneous construction of the observation vector based on the heliocentric vectors of the planet and the Earth.

After substituting these equations into the previous formula, we obtain PT = [P(t1 ) − H(t1 )] − [T(t0 ) − H(t0 )] = = P(t1 ) − T(t0 ) − [H(t1 ) − H(t0 )] = = PT − [H(t1 ) − H(t0 )]. The location of the vectors under consideration is illustrated in Fig. 5.2. To construct the vector PT we must combine the beginning of the vector Th (t0 ) with the origin of the vector Ph (t1 ). Now it is clear that PT differs from PT by the difference H(t1 ) − H(t0 ). This can be interpreted by stating that during the time t1 − t0 the Sun manages to shift by H(t1 ) − H(t0 ), which leads to the difference of the vectors under consideration. If the speed of light were infinite, it would turn out, according to Eq. (5.2), that t1 − t0 = 0, therefore H(t1 ) − H(t0 ) = 0 and PT = PT .

5.4 Observations of a planetary satellite Now we will consider a situation when we observe both a planet and a satellite of the planet. It is necessary to determine the observation vectors of these bodies.

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An analytical theory or a numerical model of the motion of a planetary satellite allows one to determine its rectangular planetocentric coordinates at any given time moment t. The satellite observation vector in this case has an origin in the topocenter, and the end is in the mass center of the satellite at that time t1 , into which the photons started, which then fell into the photodetector (1) at time t0 . We denote the satellite observation vector by ST , and (1) the planetocentric vector of the satellite at the time t1 by Sp . Now (1)

for the definition of ST we can write the relation (1)

(1)

ST = Sp (t1 ) + P(t1 ) − T(t0 ). To determine the moment t1 , one should now use the equation (1)

|S | t0 − t1 = T . c

(5.3)

Since the distance from the topocenter to the planet is different from the distance from the topocenter to the satellite, the moment the photons start from the planet, which we denote here by t2 , differs from t1 . The observation vector of the planet in this case is determined from the relations (see Fig. 5.3) PT = P(t2 ) − T(t0 ), t0 − t2 =

|PT | . c

(5.4) (5.5)

5.5 Observations of two satellites of the planet Consider the case when we observe simultaneously a planet and its two satellites. An analytical theory or a numerical model of the motion of satellites allows us to determine their rectangular planetocentric coordinates at any given time point t. The observation vector of each satellite has an origin in the topocenter, and the end is in the mass center of the satellite at the moment at which the photons started from it, which then hit the photodetector at (1) time t0 . We denote this vector of the first satellite by ST , and the (2) observation vector of the second satellite by ST . Photons that entered the photodetector at the time of observation t0 started from the first satellite at some time point t1 , and from the second one at the time t2 .

Chapter 5 Observations of planetary satellites

Figure 5.3. The location of the Earth, planet and satellite during observations. (1)

Let Sp (t1 ) denote the planetocentric vector of the first satellite at the time t1 . The barycentric vector of the planet at that moment was P(t1 ). The corresponding vectors for the second satellite will (2) be Sp (t2 ) and P(t2 ). All vectors considered here are shown in Fig. 5.4. (1) (2) Now, to determine the observation vectors ST and ST the following equations can be used: (1)

(1)

(2)

(2)

ST = Sp (t1 ) + P(t1 ) − T(t0 ), ST = Sp (t2 ) + P(t2 ) − T(t0 ), (1)

|S | t0 − t 1 = T , c (2)

t0 − t2 =

|ST | . c

These equations can be solved by successive approximations, first setting t1 = t0 , t2 = t0 .

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Figure 5.4. The location of the Earth, planet and satellites at time points corresponding to the moment of observation. Index 1 refers to satellite number 1 and index 2 refers to satellite number 2.

5.6 Determination of angular measured values during observations of planetary satellites An analytical theory or a numerical model of the motion of a planetary satellite makes it possible to determine its rectangular planetocentric coordinates at any given time point. When refining orbital parameters based on observations, it is necessary to calculate the angular measured values at the moments of observation. In the previous sections, methods for determining observation vectors from rectangular planetocentric coordinates of satellites and barycentric coordinates of planets are considered. Now it is necessary to connect the measured values with the vectors of observations of the planet and satellites. When refining the parameters of satellite motion, the angular topocentric coordinates are used as measured values. The angular coordinates are measured in the geoequatorial coordinate system associated with the equator and ecliptic of a certain epoch. In publications of past decades, coordinates are found in the system of the mean equator and equinox of various epochs. The sys-

Chapter 5 Observations of planetary satellites

tems of the true equator and equinox were also used. In this case, the precession and nutation of the Earth’s axis were simulated. In modern work, the equatorial coordinate system refers to the equator and ecliptic of the J2000 epoch. However, fixing the coordinate axes is no longer associated with the rotation of the Earth. The system relies on the positions of extragalactic sources of radio emission and is called the International Celestial Reference Frame (ICRF). In this section, we will assume that the axes of all considered rectangular coordinate systems are, respectively, parallel to the axes of some non-rotating geoequatorial system, for example, ICRF. The angular coordinates of the planet and satellites determined from the observation vectors discussed in the previous sections are called astrometric ones. Unlike astrometric coordinates, so-called apparent coordinates were also used. For apparent coordinates, the observation vectors are determined differently from discussed above. This is explained, for example, in reference (Simon et al., 1997). Here we consider only astrometric coordinates. In the geoequatorial system, along with rectangular, spherical coordinates are used. The latitude in the geoequatorial system is called the declination, and the longitude is called the right ascension. For any observation vector with components x, y, z, the right ascension α and the declination δ are determined from the relations z y . tan α = , tan δ =  2 x x + y2 Right ascension and declination of the celestial body, which are also called absolute coordinates, can be measured quantities. Along with absolute, relative coordinates are also used. This is the difference in the celestial coordinates of any two bodies: a satellite and a planet, two satellites. If the right ascensions of the first and second bodies are denoted by α1 and α2 , respectively, and the declination by δ1 and δ2 , then the differences, α = α2 − α1 , δ = δ2 − δ1 , called differential coordinates, are used as measured values. Measured values Xd = (α2 − α1 ) cos δ1 , Yd = δ2 − δ1 are more commonly used. Along with the differential angular coordinates Xd , Yd , the socalled tangential coordinates Xt , Yt are also considered, defined as follows. It is always the coordinates of one body relative to another.

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If the celestial body number 1 is displayed in the focal plane of the telescope on the optical axis, then Xt and Yt are the linear coordinates of body number 2 on the focal plane, expressed in fractions of the focal length. Moreover, it is believed that the Yt axis is parallel to the projection of the Earth’s rotation axis in the J2000 epoch and is directed to the Celestial North Pole. How to calculate the tangential coordinates is discussed below. To begin the process of refining the motion parameters of celestial bodies from observations, it is necessary to express the measured angular coordinates in terms of the components of the observation vectors. The expression of absolute coordinates through the components of the observation vector is given above. Let us consider here how to calculate the relative differential angular coordinates of two celestial bodies. Differential coordinates could be found as the differences in the absolute coordinates of two bodies. However, this method is not optimal. The fact is that the observed pairs of bodies are most often located in the sky very close to one another. Then the subtraction of the topocentric angular coordinates of the bodies consists in subtracting two close numbers, which leads to some loss of accuracy. The differences in the planetocentric coordinates of the two satellites are not related to the subtraction of two close numbers. It is these differences that should be used for calculations. For these reasons, we proceed as follows. Denote by X, Y , Z the components of the observation vector of the first satellite. Then we have (1)

{X, Y, Z} = ST . The components of the difference of the observation vectors of the second and first satellite are denoted by x , y , and z . Then we have (2)

(1)

(2)

(1)

{x , y , z } = ST − ST = Sp (t2 ) − Sp (t1 ) + P(t2 ) − P(t1 ). (2)

(1)

Here the vectors Sp (t2 ) and Sp (t1 ) by themselves are small, and the vectors Sp(2) (t2 ) and Sp(1) (t1 ) are calculated for very close time moments. Therefore, subtracting two close numbers does not lead to a loss of accuracy in calculating x , y , and z . Note that, if the first body is not a satellite but the planet itself, (1) then in the last relation we just need to put Sp (t1 ) = 0. To calculate the differential coordinates α and δ in various publications, simple approximate formulas are proposed based

Chapter 5 Observations of planetary satellites

on the smallness of the ratio (2)

(1)

|ST − ST | (1)

|ST | compared to unit. However, for the calculations, it is easy to program the exact formulas derived in Emelianov (1999). Calculations are performed according to the following chain of formulas: R2 = X2 + Y 2 , tg α =

−Y x +Xy , R 2 +Xx +Y y

A = 2R 2 Zz − 2Z 2 (Xx + Y y ) + R 2 2z − Z 2 (2x + 2y ),  B = R (X + x )2 + (Y + y )2 + Z(Z + z ),  C = R(Z + z ) + Z (X + x )2 + (Y + y )2 , tg δ =

(5.6)

A BC .

5.7 Calculation of the angular distance between satellites and position angle In specifying the orbits of the natural satellites of the planets, in addition to the differences in the angular coordinates of celestial bodies, the angular distance s between the satellites is also used as a measured quantity. The advantage of using such a measurable quantity is that it does not depend on the error in determining the orientation of the image with respect to the direction to the Celestial North Pole. We propose the following formula for calculating the angular distance between satellites s:  (Y z − Zy )2 + (Zx − Xz )2 + (Xy − Y x )2 . tg s = X 2 + Y 2 + Z 2 + Xx + Y y + Zz Here, the calculation of the differences does not increase the error in subtracting two close numbers. The position angle P can also be used as a measured quantity. This is the angle on the celestial sphere with the vertex at the point corresponding to the first satellite, between the large circle connecting the directions to the satellites, and the declination line (a large circle on the celestial sphere connecting the first satellite and the pole). To calculate the position angle, we must first calculate

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the absolute coordinates of the first satellite, which we will write here for brevity without indices as α, δ, then calculate the differential coordinates α, δ by the formulas which are given above. Then the position angle P can be found from the exact relation tan P =

cos(δ + δ) sin α sin δ + 2 cos(δ + δ) sin δ sin2

α 2

.

(5.7)

An approximate positional angle can be determined using the formula Xd . tan P = Yd

5.8 Determination of tangential coordinates of satellites Note that in the practice of processing astrometric observations of natural planetary satellites, in addition to differential angular coordinates, the so-called tangential coordinates of celestial bodies are also used. As indicated above, these are linear coordinates in the image plane. The origin coincides with the optical center of the field of view, one of the axes, Xt , is directed along the image of the sky parallel to the east, and the other, Yt , is perpendicular to the first axis to the north. The linear unit of measure is the focal length of the telescope. The approximate differences in the tangential coordinates of the two satellites coincide with their differential angular coordinates, however, the exact formulas for calculating the differences in tangential coordinates differ from the formulas for differential angular coordinates. Typically, the tangential coordinates of one celestial body relative to another are given. Then it is believed that the image of this other body is located exactly on the optical axis, and the image plane is perpendicular to this axis. Suppose that we need to calculate the tangential coordinates of satellite 1 relative to satellite 2. We use, as above, the topocentric radius vector of the second satellite (2)

ST = {X, Y, Z}, and the difference of satellite position vectors, which we denote by x , y , and z . Then the tangential coordinates can be determined by the following formulas: √ X 2 + Y 2 + Z 2 (−x Y + y X) Xt = √ , X 2 + Y 2 (X 2 + Y 2 + Z 2 + x X + y Y + z Z)

Chapter 5 Observations of planetary satellites

−x XZ − y Y Z + z (X 2 + Y 2 ) Yt = √ . X 2 + Y 2 (X 2 + Y 2 + Z 2 + x X + y Y + z Z) The tangential position angle Pt is found from the relation tan Pt =

Yt , Xt

(5.8)

and the tangential distance st by the formula  st = Xt2 + Yt2 .

5.9 Determination of the coordinate difference between two satellites of the planet in the case of photometric observations of mutual eclipses of satellites We consider here a special case of determining the observation vectors of satellites when determining the astrometric coordinates of planetary satellites based on the photometry of satellites during their mutual eclipses. Photometric observations of the mutual eclipses of two satellites of the planet make it possible to determine the relative positions of satellites with high accuracy. The phenomenon under consideration is that one of the satellites partially or completely falls into the shadow cast by another satellite. At the same time, its brightness decreases during observations from the Earth. The decrease in brightness of the eclipsed satellite can be measured using a photometer or a CCD. The brightness of the eclipsed satellite primarily depends on the heliocentric angular distance of the two satellites. It also depends on the angle between the directions from the eclipsed satellite to the Sun and to the Earth, that is, on the angle of the solar phase of the satellite. As for the heliocentric angular distance of two satellites, it is this quantity that determines the brightness changes of the eclipsed satellite in time, that is, the satellite brightness curve. When processing the photometric observations under consideration, it is necessary to accurately simulate the process of light propagation, taking into account the light velocity. The model of the phenomenon under consideration is illustrated in Fig. 5.5. The photons emitted by the Sun at the moment t3 , moving in a straight line, at the moment t2 reached satellite 2. We denote the barycentric vector of the Sun at the moment t3 by

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Figure 5.5. Light propagation during observations of mutual eclipses of planetary satellites. Sequence of points: t3 < t2 < t1 < t0 .

H(t3 ). Part of the emitted photons hit the surface of satellite 2. Other photons proceeded further. We denote the barycentric vector of satellite 2 at time t2 by S(2) (t2 ). Using planetary and satellite theories, it can be calculated by the formula (2)

S(2) (t2 ) = P(t2 ) + Sp (t2 ).

(5.9)

The considered group of photons reached satellite 1 at some point t1 , was scattered by its surface, and proceeded further towards the ground observer. At time t0 they reached the Earth and formed an image of an eclipsed satellite in the photodetector of the telescope. We denote the barycentric radius vector of satellite 1 at time t1 by S(1) (t1 ), which can be calculated by the formula (1)

S(1) (t1 ) = P(t1 ) + Sp (t1 ).

(5.10)

The moment differences t0 , t1 , t2 and t3 are given by the relations t2 − t3 =

t1 − t2 =

|S(2) (t2 ) − H(t3 )| , c

|S(1) (t1 ) − S(2) (t2 )| , c

t0 − t1 =

(5.11)

|T(t0 ) − S(1) (t1 )| . c

To determine the vectors H(t3 ), S(2) (t2 ) and S(1) (t1 ) we need to know the time moments t3 , t2 , and t1 . The required vectors and corresponding time instants can be found by solving the system of Eqs. (5.9), (5.10) and (5.11) by iterations. In the zeroth approximation, we set t3 = t2 = t1 = t0 .

Chapter 5 Observations of planetary satellites

Obviously, the degree of shadowing of satellite 1 depends on (2) (1) the angle between the vectors SH (t2 ) and SH (t1 ), defined by the relations (2)

(1)

SH (t2 ) = S(2) (t2 ) − H(t3 ) , SH (t1 ) = S(1) (t1 ) − H(t3 ) . We call this angle the effective heliocentric angular distance of two satellites and denote it by s ∗ . (2) (1) The components of the vectors SH (t2 ) and SH (t1 ) are denoted as follows: (1)

(2)

SH (t1 ) = {ξ, η, ζ }, SH (t2 ) = {ξ + ξ , η + η , ζ + ζ }. Then s ∗ is defined by the formula  (ηζ − ζ η )2 + (ζ ξ − ξ ζ )2 + (ξ η − ηξ )2 tg s ∗ = , ξ 2 + η2 + ζ 2 + ξ ξ + ηη + ζ ζ where small increments ξ , η , ζ should be determined by the formula (2)

(1)

(2)

(1)

{ξ , η , ζ } = SH (t2 ) − SH (t1 ) = P(t2 ) − P(t1 ) + Sp (t2 ) − Sp (t1 ). Note that when calculating s ∗ using the above formulas, there is no loss of accuracy due to the subtraction of two close numbers. If, in addition to the angular heliocentric distance, it is necessary to calculate the difference in the angular heliocentric coordinates of two satellites, we can use formulas similar to Eqs. (5.6).

5.10 Conclusion regarding measured values during observations of planetary satellites To determine the parameters of satellite motion from observations, it is necessary to apply the method of differential refinement, which is described in the next chapter. To apply this method, we have to calculate the values measured during observations from the given motion parameters. The calculations are carried out according to the following chain of operations: parameters ⇒ planetocentric rectangular coordinates ⇒ observation vectors ⇒ measured values. The first stage is based on satellite motion models. The above material is devoted to this. The second and third stages of operations are described above in this chapter. Now we have all ingredients necessary for applying the method of determination of satellite motion parameters from observations.

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Useful information on the issues discussed here can be found in Emel’yanov (2017a).

5.11 The moment of apparent approximation of planetary satellites as a measurable quantity during observations One of the types of measured values in observations of planetary satellites is the measurement of the mutual angular distances d of two satellites. Since this distance depends on the motion parameters of both satellites, it is possible to refine the orbits from the observational database with the inclusion of such measurements. An advantage is the independence of such measurements from the orientation of the image obtained during observations relative to the celestial pole. This eliminates the usual orientation error in such cases. Converting the measurement result on satellite images to the angular distance between satellites requires knowledge of the image scale, that is, the relationship between the linear distances in the image received in the camera and the angular distances in the celestial sphere. This scale is determined in various ways. However, in any case, the scale is accompanied by an error, which is unknown and different for different frames of images obtained overnight on the same telescope. This error will give the error of the obtained angular distance between the satellites, which is involved in determining the parameters, and ultimately the satellite’s ephemeris error. To avoid this error, namely, to eliminate it altogether, a special measurement method was proposed in Morgado et al. (2016). In this method, the angular distances between two satellites are first determined by a number of time points in such an interval during which some apparent approximation in the pair of satellites occurs. At this interval, there is a minimum of apparent distance. The minimum moment t0 does not depend on the distance itself. However, this moment depends on the motion of satellites and, therefore, depends on the parameters of movement. The idea of the method proposed in Morgado et al. (2016) is to use this moment of the minimum of the apparent distance t0 between satellites as a measured value when refining the parameters of satellite motion. The minimum moment is determined by approximating the dependence of the apparent distance on time by some suitable functions. We can use polynomials of a sufficiently high degree. Since the minimum moment does not depend on the distance itself, the image scale error is completely eliminated. However, the

Chapter 5 Observations of planetary satellites

measured distances contain random measurement errors in the images. Therefore, the found minimum moment will also contain some error. A series of such measurements was published in Morgado et al. (2016), namely, a number of moments of apparent approximation in the pairs of Galilean satellites of Jupiter. This approach required careful analysis. The fact is that the accuracy of the ephemeris depends not only on the accuracy of the observations, but also on the composition and type of the measured values. It was necessary to find out how the use of the moments of apparent satellite approximation as measured values will affect the accuracy of the ephemeris. It is the ephemeris that are the final result of creating a satellite motion model. For such an analysis, the author of this book had previously prepared the necessary tools. It is described in Chapter 8. A necessary analysis of the idea of the authors of the paper (Morgado et al., 2016) was performed in the work (Emelyanov, 2017b). For this purpose, an array of ephemeris of the Galilean satellites of Jupiter was formed. These ephemeris were used as observations for calculating, at the final stage, the ephemeris of satellites at time instants following the observation time interval. The moments of such artificial observations were selected at small intervals at which the mutual distances in the pairs of satellites had a minimum. Observations were modeled on a large number of such intervals. To analyze ephemeris errors, random errors with specified variances and the normal distribution of the probability density of errors were added to artificial measurements. The computer program used a random number sensor. Two variants of the composition of measurements by type were compared. In the first version (d), the apparent mutual distances were modeled as measurements. In the second version (t0 ), the moments of the apparent approximation of the satellites calculated from the mutual distances were used. Random errors in this experiment were set as follows. All simulated mutual distances were taken with random errors having the dispersion corresponding to the accuracy of modern real observations. In addition, at each time interval of approaching a pair of satellites, the same error was added to the model distances, simulating the scale error present in real observations. For different intervals of approximations, different values of the scale error were chosen. For the entire set of intervals, errors were randomly selected with a given variance of the error of the scale σ ds . In this way, the situation existing in real observations was modeled. From the simulated observations, the orbital parameters were first determined, and then the ephemeris were calculated for a number of time instants following the observation time interval.

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The process was repeated many times, each time with a new set of random observation errors. The values of the ephemeris coordinates turned out to be different in different sets of random observation errors. The accuracy of the ephemeris was estimated by the rms deviations of the ephemeris from some reference ephemeris obtained with zero observation errors. The experiment was carried out several times at different values of the error of the scale σ ds . Each time, the accuracy of the ephemeris was evaluated. Two sets of composition of observations, differing by types of measurements, were compared with each other. In the first set, the mutual distances d were used; while in the second set, the moments of apparent approximation t0 were used. As a result, we obtain the following. With a zero scale error (σ ds = 0), the ephemeris error in the second set (t0 ) turned out to be significantly larger than in the first (d). When increasing the scale error σ ds the ephemeris error in the first variant (d) increased, while in the second variant (t0 ) it remained unchanged. This is evident, since in the second variant the moments of mutual approach of satellites do not depend on scale error at all. After (eg) increasing σ ds for a certain value of σ ds = σ ds , the ephemeris errors in two cases were equal. After a further increase in σ ds , the ephemeris error in the first version (d) continued to increase. From the results of the experiment it follows that the new idea of the authors of the work (Morgado et al., 2016) gives an advantage only for sufficiently large scale errors. Unfortunately, the (eg) critical value of σ ds , which determines the choice of the type of observation, is very difficult to find in problems based on real observations. The work (Emelyanov, 2017b) only established the (eg) existence of such a critical value of σ ds . An experiment carried out in Emelyanov (2017b) shows that the use of new types of observations always requires a thorough analysis of the accuracy of the ephemeris obtained in this case.

5.12 Means and techniques of ground-based observations of planetary satellites The technique of astrometric observations of planetary satellites has been improved over the centuries. Until the 20th century, observers looked into the eyepiece of the telescope and saw, besides the images of stars, a cross of spider webs and another thread, the position of which could be changed by rotating the micrometer screw. The cross in the eyepiece was rotated by the observer so that one celestial body fell on the central crosshair of the threads, and the other on the crosshair with an additional

Chapter 5 Observations of planetary satellites

thread. With such observations, the angular distance between two celestial bodies s and the position angle P with a vertex in one luminary and rays directed to another luminary and to the North Pole were measured. Moreover, most often these two values were measured at different points in time. Such observations are called micrometric in the literature. In the 20th century, observers looked more through a microscope, examining images of celestial bodies imprinted on photographic plates. Such observations are called photographic. The relative coordinates were measured under a microscope, first in millimeters, then their values were converted into angular values. The coordinates of the planets and satellites under the microscope are counted relative to the stars. The necessary celestial equatorial coordinates of comparison stars were taken from star catalogs. Based on these data, the equatorial coordinates of the planets and satellites were deduced, which are called absolute in this case. Errors of the coordinates of stars from star catalogs directly fell into the absolute coordinates of the observed celestial bodies. To refine the satellite motion models, relative angular coordinates, that is, the difference of the coordinates of two satellites, can be used. According to such data, the parameters of the orbits of both bodies are refined simultaneously in one system of equations. Relative measurements are free from stellar catalog errors, however, the problem of the accuracy of these data is associated with the uncertainty in practice of knowing the scale and orientation of the image. At the end of the 20th century, photo-sensitive charge-coupled device (CCD) matrices were used instead of photographic plates. These photodetectors turned out to be much more sensitive and giving better image resolution. The problems of limiting the accuracy of observations remained the same as for photographic plates. However, the processing of results has been simplified with the use of powerful computers. Such observations are simply called CCD observations. Progress in the accuracy of stellar catalogs has led to a new unusual challenge. The fact is that photographic astrometric observations of planets and satellites have been made since the beginning of the 20th century. Based on the then available star catalogs, the coordinates of the observed celestial bodies were determined and published. Photographic plates of these epochs are still stored in the so-called “glass libraries”. Now it is possible to re-measure the coordinates of the planets and satellites on the surviving photographic plates and obtain new astrometric coordinates of these celestial bodies using modern stellar catalogs. This activity has been developing recently in various institutions of the

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world. The staff of the Paris Institute of Celestial Mechanics and Ephemeris Computing (IMCCE), France (Robert et al., 2016) supervises the work. Speckle interferometric observations Special consideration should be given to one particular method of astrometric observation. Advances in astronomical technology have led to speckle interferometric observations. The technique of such measurements is as follows. The diffraction pattern created at the focus of the telescope by light from the planet and satellite is recorded by a CCD camera with a very short exposure of about 10 ms (time of “freezing” of atmospheric turbulence). With this registration method, the image of the planet is obtained in the focal plane of the telescope in the form of a group of spots—speckles, randomly scattered inside a small area defined by the atmospheric blur circle of image. Each speckle corresponds to some fluctuation of light in the turbulent atmosphere of the Earth. The light from the satellite entering the telescope at a slight angle to the direction of the planet, provided that it passes through the same atmospheric fluctuations (isoplanatism), creates a group of speculi in the focal surface, similar to the group of speckles from the planet, but shifted according to the angular distance between the planet and the satellite and their positional angle. The accuracy of the location of speckles is limited by the size of the diffraction spot for a given telescope. Usually a long series of time-consistent frames is recorded. The measured linear coordinates of the speckles from all frames are entered into a computer and subjected to correlation analysis. Each speckle corresponding to the planet has a speckle corresponding to its satellite. Statistical processing of the results of such measurements gives the average values of the angular distance and their position angle. As a result, the accuracy of speckle interferometric observations is hundreds of times greater than the accuracy that ordinary photographic observations give.

5.13 Sources of observations from planetary satellites The ways of the results of observations from an observer to a researcher who uses this data to obtain new information or to create ephemeris are very different. In past centuries, researchers very often observed, and they themselves determined the orbits of satellites. However, quite quickly the usual method was developed: to publish observations in scientific journals. In this case, firstly, it was possible to save data for many years, and secondly,

Chapter 5 Observations of planetary satellites

the results of observations became available to anyone. This technique is still applied. For example, all observations of distant satellites of large planets are sent to the scientific publication Minor Planet Circular (MPC), the texts of which are now available on the Internet on the MPC website at https://minorplanetcenter.net/ iau/mpc.html. Until some time, the results of observations were placed directly on the pages of a scientific journal. Most observations over the centuries have been published in the magazines Monthly Notices of the Royal Astronomical Society (MNRAS), Astronomische Nachrichten (AN), Astronomy and Astrophysics (AA), Astronomy and Astrophysics Supplement Series (AAS), Astronomical Journal (AJ). Recently, some journals have created their electronic databases accessible via the Internet. At the same time, a detailed description of the observations is given in the text of the article, and the data themselves are placed electronically in the database. Often, the authors cite in their paper a small fragment of a table with observations placed in an electronic database in a paper. It happened at different times that observers published their observations in very inaccessible publications. Examples include the Proceedings of the Kazan City Astronomical Observatory and “Manuscripts deposited at VINITI (All-Russian Institute of Scientific and Technical Information, Russian Academy of Sciences)”. In such cases, the work of observers could be forgotten, and valuable scientific data lost. Fortunately, the creators of some common planetary satellite observation databases are engaged in “saving observations” by searching for data in inaccessible scientific journals and putting them in a database. In particular we mention the creators of the Natural Satellites Database (NSDB), the planetary satellites observational database, described in more detail in Chapter 12 of this book. Here are the NSDB addresses on the Internet: http://www.sai.msu.ru/neb/nss/html/obspos/ http://nsdb.imcce.fr/obspos/. Any observation consists, as a minimum, of two values: the time of observation and the value of the measured quantity. However, these two numbers are actually not enough. The fact is that time is always counted in one or another time scale, and the measured value is determined in one or another coordinate system. Without an indication of the time scale and the coordinate system, the data could be useless. Another necessary attribute of an observation is an indication of where the observation was made. In fact, the coordinates of the observatory are needed. In some epochs, all observers used the same time scale and the same coordinate system. However, these details were not indicated in the publications.

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Such circumstances sometimes hindered or completely precluded the use of such observations. Researchers had to use indirect information to recover the missing data. Sometimes in publications, the results of observations are provided with indications of the means of observation used: the diameter of the telescope, the type of photodetector, meteorological conditions. This information is useful for a preliminary assessment of the accuracy of observations or for the rejection of observational data. We note one of the special sources of planetary satellite observations. Data published in the journal Astronomy and Astrophysics is very often placed in a special electronic database VizieR On-line Data Catalog. This is part of the large astronomical database Center de Donnees astronomiques de Strasbourg – Strasbourg astronomical Data Center. We can find the results of observations in this database if we know the volume and first page numbers of the corresponding article published in the journal. The data address on the Internet will be http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/A+A/NV/ NP where NV is the volume number and NP is the page number. Recently, an article number has been used instead of a page number. An example of an address is the following line: http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/A+A/582/ A36. In addition to the observations made at ground-based observatories, astrometric observations of planetary satellites were also carried out using telescopes located on artificial Earth satellites. Many observations made using the Hubble Space Telescope are given in the NSDB database mentioned above. The databases also receive observations of distant planetary satellites carried out using a telescope on the WISE satellite. As a result, we can conclude that at present, to search for observations of natural planetary satellites, one should first of all turn to some electronic database. For more information on such databases, see Chapter 12 of this book.

5.14 Time scales and coordinate systems for observations of planetary satellites We note in advance that the development of time scales and coordinate systems is handled by the science of astrometry that serves this purpose. It is impossible to reproduce all the necessary information within the volume of this book. To solve all the astro-

Chapter 5 Observations of planetary satellites

metric problems that arise when studying the dynamics of the natural satellites of the planets, one should turn to the corresponding papers and books on astrometry. However, we will consider the most necessary information here. Time scales Time scales are determined by two factors. The first one is a type of physical time sensor, examples of which are atomic clocks and pulsars. The second one is associated with the effects of the theory of general relativity, because of which the clocks located in different places of the solar system do not go the same. The main and most universal time scale used in studying the motion of bodies in the Solar System is the TDB (Barycentric Dynamical Time) scale. This is the time of evenly running hours located in the barycenter of the solar system. Since in fact there are no watches there, the TDB scale is a model. Another universal scale is TT (Terrestrial Time). This is the time of evenly running hours located on a geoid (very close to the surface of the Earth). Unfortunately, in the lives of people on Earth, the TDB and TT timelines would be inconvenient. Our life is connected with the rising and setting of the Sun, which is determined by the rotation of the Earth. For us, the UT1 timeline associated with the rotation of the Earth could be convenient. Everything could be fine with the UT1 scale, but unfortunately, the Earth rotates not only unevenly, but also unpredictably unevenly. Earth may not be a suitable time sensor. It turns out that atomic clocks are an excellent sensor for uniform time. The corresponding timeline was called TAI (Time Atomic International). TT timeline linked to TAI scale by simple ratio T T = T AI + 32.184 s. For a convenient life on Earth, they came up with the UTC (Universal Time Coordinated) time scale, which never differs from UT1 by more than one second, but in some periods it constantly goes along the atomic time scale TAI, differing from it by a known constant value. However, due to the uneven course of UT1, sometimes we have to add a whole second to the UTC scale so that the difference of UTC-UT1 does not exceed 0.9 seconds. Such amendments have to be made about once a year, more often or less, depending on the quirks of the Earth’s rotation. As a result, for example, from July 1, 2015 to January 1, 2017, the difference in TT-UTC was constantly 68.184 seconds. At the same time, the difference between UTC-UT1 never once exceeded 0.9 seconds. The decision to introduce a leap second to UTC is made by the International Earth Rotation Service (IERS). A message about this

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decision a few months in advance appears on the IERS web page at https://hpiers.obspm.fr/iers/bul/bulc/bulletinc.dat. The table of differences TT-UTC and the corresponding schedule are located in Appendix D of this book. Currently, almost all observations of natural planetary satellites are conducted and published using the UTC time scale. However, this was not always the case. In the past, many different time scales have been used. If we try to use the data of observations made and published in past centuries, we need to know the relationship of the time scales used then with the UTC scale. Relationships of the various timelines that have been used in the past are given in Appendix D of this book. Now let us find out what to do with the time scale in which the satellite motion models are built. When solving the differential equations of motion, it is assumed that time flows uniformly. Therefore, only TDB or TT can be used as a time scale for the theory of satellite motion. Then we find the difference. There is a relation between these scales T DB = T T + P , where P is the sum of time-periodic terms. A term with a period of 1 year has a maximum amplitude of 0.001656 seconds. The remaining terms are much smaller. For a very accurate representation of P , we can take a series published in Fairhead and Bretagnon (1990). So, if we are satisfied with the time accuracy of 0.001656 seconds, it makes no difference which of these two scales to choose. Thus, from the previous statement it follows that the observation times published in the UTC scale must first be transferred to the TT scale using the table given in Appendix D, and then, if necessary, transferred to the TDB scale using the series published in Fairhead and Bretagnon (1990). In interpreting and using the results of astrometric observations published in past centuries, the following feature should be taken into account. When dating observations until 1925, the beginning of the day was considered noon, that is, 12 hours. At noon, every Julian day begins. The date of observation can be set in the form of a calendar date according to the Gregorian calendar or Julian day (JD). Astronomers also use Modified Julian Day (MJD) associated with JD using the ratio J D = MJ D + 2400000.5. Comments on timescales are also given in Appendix D.

Chapter 5 Observations of planetary satellites

Coordinate systems The results of astrometric observations of satellites are either angular geoequatorial coordinates, right ascension α and declination δ, or the difference of these coordinates for two objects: a satellite and a planet or two satellites. If α and δ are obtained directly, then the observations are called absolute; if the differences are coordinates, then the observations are considered relative. Absolute coordinates are obtained by measuring the linear coordinates of the object on a photographic plate or on a graphic image obtained using a CCD camera. Then the linear coordinates of the reference stars are measured. The coordinates of reference stars are taken from one or another catalog. Further, by modeling the relationship of linear coordinates in the image and angular coordinates in the sky, the right ascension and declination of the object are obtained. With such observations, the obtained satellite coordinates directly depend on the used star catalog. Relative coordinates are obtained by measuring the differences in the linear coordinates of two objects in the image. To translate these measured coordinate differences into the differences in celestial coordinates, we need to know the orientation and scale of the image. Various methods are used to determine these parameters. The scale is found by calibrating the camera from images of star clusters with known relative coordinates, or by measuring the focal length of the telescope using various technical methods. The orientation is sometimes determined by the clockwise movement of objects in the image frame obtained with the telescope guiding mechanism stopped. Other methods are also applied. Due to the limitations of our presentation, we do not consider other methods here. Usually the coordinates are counted in some non-rotating geoequatorial coordinate system of some fixed epoch of the equator and equinox. In publications of observations of natural planetary satellites, three such systems are found. One of them is ICRF (International Celestial Reference Frame), based on extragalactic radio sources. The second one is a dynamic coordinate system associated with the movement of the planets. The dynamic system is implemented by planetary ephemeris DE200/LE200, developed earlier in JPL (USA). The third one is the FK5 star catalog coordinate system. Often, observations are provided with a simple indication that the coordinate system of the J2000 epoch is being used. In this case, the name of the star catalog is given, which was used in the reduction of the observations. In this case, the coordinate system of this star catalog is implied. When sharing observations reduced using different stellar catalogs, it is necessary to convert the coordinates of the satellites

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from one system to another. To do this, we need to know the relationship of coordinate systems. The transition matrix between the FK5 star catalog system and the dynamic coordinate system is known. It is published in Standish (1982). The relationship of the coordinate system of the FK5 catalog and ICRF is given in Feissel and Mignard (1998). In the past, observations were processed using the coordinates of stars from the FK4 special star catalog. Its feature is a special simplified model of annual aberration. Due to this feature, the transition from coordinates in the FK4 system to coordinates in the FK5 system cannot be done using a simple rotation matrix. The transformation depends both on the coordinate reference time and on the coordinates of the object itself. How to make such a transformation is described in the literature (Aoki et al., 1983). Note that the coordinate system of the FK4 star catalog is associated with the geoecliptic and equator of the B1950 epoch. Therefore, in the publications of observations, the coordinate system of the B1950 epoch is often indicated. Actually, this means the coordinate system of the star catalog FK4. In fact, in astrometric observations of the 20th century there are various special cases of coordinate systems. When using observations, it is necessary to consider in detail the coordinate systems used in order to prevent errors. All the systems considered above are defined as astrographic coordinate systems. Since 1984, the astrometric coordinate system has become identical to the astrographic system. This is exactly the system that is adopted in Sects. 5.2–5.9. In addition to this system, during the observations of past years, other systems were also used. First of all, it is necessary to consider the coordinate system also astrometrically, but related to the moving geoequator and the equinox point. Consider the true equator and equinox at the same moment as the moment of observation. Then one talks about the true equator and equinox of the “date epoch”. Averaging the coordinates of the pole and the equinoxes during the nutation process leads to the mean equator and equinox of the “date epoch”. Geo-equatorial coordinates can be measured in the system of true or mean equator and equinox, respectively. When transforming from these systems to systems with a fixed equator and equinox, various models of precession and nutation can be used. One of the latest versions is the precession model adopted by the International Astronomical Union (IAU) in 1976, and the nutation model adopted by the International Earth Rotation Service (IERS) (Seidelmann, 1982). A reference to the publication of the MAC 1976 precession model can be found in Seidelmann (1982).

Chapter 5 Observations of planetary satellites

Despite progress in astrometry, the 1976 MAC model of precession is still in use. However, the nutation model is constantly being updated. In fact, the International Earth Rotation Service (IERS) abandoned such a simplified model. Instead, parameters are used that specify the orientation of the axis of rotation of the Earth. A constantly updated table is published on the IERS website, in which, with a step of 1 day for past epochs, the Earth Orientation Parameters (EOP) are given. The parameters that are needed to obtain corrections to the coordinates of the axis of rotation are given, which are determined by the nutation model MAC 1980 and parameters that directly determine the orientation of the axis. The Internet address of this table is as follows: https://datacenter.iers.org/data/latestVersion/223_EOP_C04_14. 62-NOW.IAU1980223.txt. In addition to astrometric coordinates, in the practice of observations in past years, other definitions of coordinate systems were also found. In particular, the concept of apparent coordinates was introduced. These are the coordinates that determine the direction of the vector from the point of position of the observer at the time of the start of the photons to the point of position of the object also at the time of the start of the photons from the observed object, those photons that reached the photodetector at the time of observation. The difference of moments is the light time from the observer to the object. When using apparent coordinates, additional assumptions are taken “by default”. In particular, the true position of the equator and the true position of the equinox are used. In each case, it is necessary to carefully study what assumptions were applied in processing the observations before they are published. The concept of apparent coordinates may include accounting for the curvature of the line of sight caused by the gravitational action of the Sun. This is the effect of the theory of general relativity. In particular, the effect of beam curvature is taken into account when calculating the apparent coordinates in the HORIZONS (JPL) ephemeris server, available on the Internet at https:// ssd.jpl.nasa.gov/horizons.cgi. A description of the coordinate systems considered here and the corresponding methods can be found in two books (Seidelmann, 1992; Simon et al., 1997).

References Aoki, S., Soma, M., Kinoshita, H., Inoue, K., 1983. Conversion matrix of epoch B 1950.0 FK 4-based positions of stars to epoch J 2000.0 positions in accordance with the new IAU resolutions. Astronomy & Astrophysics 128 (2), 263–267.

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Emelianov, N.V., 1999. Relationship between astrometric and theoretical coordinates of planetary satellites. Solar System Research 33, 133–137. Emel’yanov, N.V., 2017a. Current problems of dynamics of moons of planets and binary asteroids based on observations. Solar System Research 51 (1), 20–37. Emelyanov, N.V., 2017b. Precision of natural satellite ephemerides from observations of different types. Monthly Notices of the Royal Astronomical Society 469 (4), 4889–4898. Fairhead, L., Bretagnon, P., 1990. An analytical formula for the time transformation TB-TT. Astronomy & Astrophysics 229 (1), 240–247. Feissel, M., Mignard, F., 1998. The adoption of ICRS on 1 January 1998: meaning and consequences. Astronomy & Astrophysics 331, L33–L36. Morgado, B.E., Assafin, M., Dias-Oliveira, A., Gomas Jr., A., 2016. Astrometry of mutual approximations between natural satellites. Application to the Galilean moons. Monthly Notices of the Royal Astronomical Society 460, 4086–4097. Robert, V., Pascu, D., Lainey, V., Arlot, J.-E., De Cuyper, J.-P., Dehant, V., Thuillot, W., 2016. New astrometric measurement and reduction of USNO photographic observations of the main Saturnian satellites: 1974–1998. Astronomy & Astrophysics 596, A37. 10 pp. Seidelmann, P.K., 1982. 1980 IAU theory of nutation – the final report of the IAU working group on nutation. Celestial Mechanics 1982 (27), 79–106. Seidelmann, P.K. (Ed.), 1992. Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley, California. Simon, J.-L., Chapront-Touzé, M., Morando, B., Thuillot, W. (Eds.), 1997. Introduction aux éphémérides astronomiques. Supplément explicatif à la connaissance des temps. BDL, Institute of Technology, Paris. 450 c. Standish Jr., E.M., 1982. Orientation of the JPL Ephemerides, DE 200/LE 200, to the dynamical equinox of J2000. Astronomy & Astrophysics 114, 297–302.

6 Construction of models for the motions of celestial bodies based on observations 6.1 Method of differential refinement of the motion parameters of celestial bodies based on observations. Application of least-squares method In celestial mechanics, there are a number of problems that are different in the objects studied, but similar in the way they are solved. These problems can be formulated as “Refinement of the motion parameters of celestial bodies from observations”. A detailed explanation of what we mean by parameters and observations is given in the previous sections. Here we describe these concepts briefly. The motion parameters of celestial bodies are called the quantities on which the movement of bodies depends, and which are considered constant at least at some stage of the study or at a certain time interval. We consider three types of parameters. The parameters of the first type is the parameters that enter into the equations of motion. They exist even before solving any equations. The parameters of the second type appear in the process of solving differential equations of motion. These are either arbitrary constants in the general analytical solution of the equations, or the initial conditions for numerical integration, that is, the coordinates and velocity components of the bodies at the initial time moment. The parameters of the third type are included in the relations connecting the results of observations and the coordinates of the celestial body. They are not related to the motion of the object studied, but they depend on how we observe; they are called the parameters of the observation conditions. Examples of parameters of the first, second, and third types are the gravitational parameter of a celestial body, elements of its orbit, and geocentric coordinates of the observatory, respectively. In the process of observation, any quantities are measured depending on the position or velocity of the celestial body. They are The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00011-X Copyright © 2021 Elsevier Inc. All rights reserved.

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called measured quantities. Observations give us the values of the measured quantities at the moments of measurement. Several quantities can be measured at the same time. For simplicity and without loss of generality our presentation, we assume that all quantities are measured independently, each at one time. Moments of measurement of different quantities may coincide. Examples of measured quantities are the angular topocentric equatorial coordinates of a celestial body, the topocentric range of a celestial body, and the difference in angular coordinates of two celestial bodies. The measured quantity is always a real physical quantity obtained with the help of measuring instruments at a certain time point in a certain place. Measurement moments are counted by the clock located in the observation point. In this case, the relation between the time scale of observations and the time scale that appears in the motion model, that is, in the differential equations of motion, should be known. We formulate the problem of refining the motion model of a celestial body as follows: the results of observations are given, it is required to find the parameters of motion. Suppose that ξ is one of the measured quantities, and p1 , p2 , ..., pn are true but unknown values of the motion parameters of the celestial body. Measurement is taken at some time point t. Usually a lot of measurements are made, the number i is assigned to each measurement. As a result, we have a number of measured quantities of ξi for a number of time moments ti , i = 1, 2, ..., m, where m is the number of measurements. When constructing a model or theory of motion of a celestial body, one or another coordinate system is inevitably used. Usually these are the coordinates that appear in the differential equations of motion. Unlike measured quantities, which are always real, because they are obtained with real measuring instruments, coordinates are some abstract concepts. In most cases, they are impossible to measure. For example, we cannot directly measure the rectangular geocentric coordinates of an Earth satellite. The axes of any coordinate system are associated with any real object. For example, the axes of the terrestrial reference frame are associated with the positions of a number of reference observatories on the Earth’s surface. The axes of the celestial reference frame are associated with the positions of stars or extragalactic radio sources. In any case, some model of the coordinate system is constructed. This model can be changed and improved over time. The theory of motion of a celestial object gives us model values of some coordinates at a given time moment. It is often assumed that the axes of the coordinate system are non-rotating, that is, they are always mutually parallel to the axes of some inertial co-

Chapter 6 Construction of models for the motions of celestial bodies based on observations

ordinate system. As for time, it is assumed that time is uniform. However, these properties of coordinates and time are provided only by a model of link with some real celestial bodies or real periodic processes. The theory and model of motion give us coordinates at any given time moment t, which for definiteness we will consider to be Cartesian rectangular coordinates and denote them by x, y, z. The coordinates also depend on the motion parameters, which we denote here by p1 , p2 , ..., pj . Here, the number of parameters j for one celestial body can be equal to 6 or more, depending on the theory used. Thus, from the theory we have parametric functions of time x = x(t, p1 , p2 , ..., pj ), y = y(t, p1 , p2 , ..., pj ),

(6.1)

z = z(t, p1 , p2 , ..., pj ), called the law of motion. When performing observations, we must know how the measured value ξi is related to the coordinates of the celestial body. This relationship is defined by some measurement model. A model may include some parameters. Denote such parameters by pj +1 , pj +2 , ..., pn . The observation model gives us the function ξ = ξ(t, x, y, z, pj +1 , pj +2 , ..., pn ).

(6.2)

Now, after substituting the function (6.1) for the coordinates on the right side of Eq. (6.2), we obtain the model value of the measured quantity ξ c , as a well-known function of time t and motion parameters: ξ c = ξ(t, p1 , p2 , ..., pn ).

(6.3)

Any model contains errors. Denote the model error by δth . Then the true value of the measured quantity ξ is determined by eliminating the error δth : ξ = ξ(t, p1 , p2 , ..., pn ) − δth . In fact, the measured values are obtained from observations and therefore contain observation errors. Let ξ o be the observed value of the measured quantity, and δobs be its error. Subtracting the observation error, we again obtain the true value of the measured quantity, ξ = ξ o − δobs .

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After equating the right-hand sides of the last equalities, we obtain ξ o = ξ(t, p1 , p2 , ..., pn ) + δobs − δth . After performing measurements at time moments t1 , t2 , ..., tm , we obtain the system of equations (i)

(i)

ξio = ξ(ti , p1 , p2 , ..., pn ) + δobs − δth (i = 1, 2, 3, ..., m)

(6.4)

relative to the true values of the required parameters pk (k = 1, 2, ..., n). Quantities (i) (i) (i) δsum = δobs − δth

called the total errors of observation and theory. (i) When solving such a problem, the exact error values are δobs (i) and δth (i = 1, 2, 3, ..., m) remain unknown. They are usually considered as random variables with given probabilistic characteristics (distribution laws, moments, etc.). Thus, we have a system of m equations for m + n unknowns (i) δsum (i = 1, 2, ..., m), p1 , p2 , ..., pn ,

in which the number of unknowns is greater than the number of equations. In this situation, Eqs. (6.4) are replaced by the so-called system of conditional equations ξio = ξ(ti , p1 , p2 , ..., pn ) (i = 1, 2, 3, ..., m),

(6.5)

which is a system of m equations with n unknowns p1 , p2 , ..., pn . The system of conditional Eqs. (6.5) is incompatible, it has no solution, since it is obtained by subtracting from the right parts of (i) the exact Eqs. (6.4) random independent total errors δsum . We can try to find some approximate estimate of the required parameters. Moreover, the obtained values should, as far as possible, differ little from the true ones. An approximate estimation algorithm is called filtering algorithm. The main objective of this algorithm is the possible reduction (filtering) of the influence of theory errors and observation errors. The choice of a filtering algorithm is ambiguous, its structure depends on the available in(i) formation about the total error δsum . In practice, there is very little or no such information. Therefore, one has to be content with certain assumptions as regards the properties of the total error and a filtering algorithm based on these assumptions.

Chapter 6 Construction of models for the motions of celestial bodies based on observations

Relations (6.5) can be considered as equations for the required parameters p1 , p2 , ..., pn . It is not possible to solve these equations directly in practice for another reason. The fact is that ξ(ti , p1 , p2 , ..., pn ) is a purely non-linear function of its arguments. Most often it cannot even be written explicitly. Moreover, one cannot explicitly obtain the solution of Eqs. (6.5). We will carry out the solution of the problem according to the scheme that is already presented in Chapter 1. At almost any stage of research, some approximate values of the required parameters are known. We call these values prelimi(0) (0) (0) nary and denote them by p1 , p2 , ..., pn . Let the exact values of the parameters p1 , p2 , ..., pn differ from the preliminary approximate by corrections (0)

(0)

p1 = p1 − p1 , p2 = p2 − p2 , ... , pn = pn − pn(0) . Then (6.5) can be written as (0)

(0)

ξio = ξ(ti , p1 + p1 , p2 + p2 , ... , pn(0) + pn ).

(6.6)

For most celestial bodies, motion models are constantly developing. Therefore, at the next stage of refinement, the preliminary values of the parameters are already quite close to true. This allows us to read the corrections p1 , p2 , ... , pn , these being small, and expand the right-hand side of Eq. (6.6) in a Taylor series in powers of corrections: ξio

(0) (0) = ξ(ti , p1 , p2 , ... , pn(0) ) +

 n   ∂ξ pk + ... . ∂pk i

(6.7)

k=1

The derivatives in the right-hand sides are calculated at (0)

t = ti , p1 = p1 , ... , pn = pn(0) . We restrict ourselves to first-order smallness values with respect to the corrections pk and introduce the notation c(0)

ξi

(0)

(0)

= ξ(ti , p1 , p2 , ... , pn(0) ),  ak(i) =

∂ξ ∂pk

(6.8)

 ,

(6.9)

c(0)

(6.10)

i

ξi = ξio − ξi

.

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As a result, we obtain ξi =

n 

(i)

ak pk (i = 1, 2, ... , m).

(6.11)

k=1

The approximate relations (6.11) are called conditional equations for determining corrections to refined parameters. They are linear inhomogeneous algebraic equations for the sought corrections pk , (k = 1, 2, ..., n). Conditional equations are approximate for two reasons. Firstly, in the left parts the errors of observation and errors of the theory are discarded. Secondly, in the right-hand sides, all terms of the order of squares of corrections and higher are discarded. Eqs. (6.11) are sometimes called linearized with respect to the conditional Eqs. (6.5). If we make certain assumptions regarding theoretical errors and observation errors, we can choose one of the developed filtering algorithms and find an approximate solution of the conditional equations (6.11). Existing methods also allow us to estimate the error of the solution. After the corrections are found, we add them to the preliminary values of the parameters and get new and, we hope, more accurate their values. Such a method for determining the motion parameters of celestial bodies is called differential refinement of the parameters. Due to the inaccuracy of conditional equations and their approximate solution, the new parameter values will not be accurate enough. However, the refinement can be repeated several times. If the refinement process converges, that is, the corrections decrease from step to step, then the calculations can be stopped when the corrections become significantly less than their errors. In this case, we obtain the values of the motion parameters of the celestial body p 1 , p2 , ..., p n , corresponding to all the observations used in this case. This correspondence is uniquely determined by the given motion model (6.3) and the selected filtering algorithm. From a theoretical point of view, the convergence of differential refinement has not been studied. Examples can be given in which the process does not converge or converges to false values of the required parameters. When using the method of differential refinement of parameters, it should be taken into account that, from a certain step of refinement, corrections begin to fluctuate due to inevitable calculation errors. After the appearance of such fluctuations, further attempts to refine the parameters become useless. On the other hand, when the described process converges   well, ∂ξ , there is no need to accurately calculate the derivatives ∂p k i

Chapter 6 Construction of models for the motions of celestial bodies based on observations

since in the refinement process they are used to determine all decreasing corrections p1 , p2 , ..., pn . Under these conditions, reasonable errors in the calculation of these derivatives can only slightly increase the number of refinement steps, practically without affecting the accuracy of the final result. In the first and subsequent refinement steps after calculating the corrections p1 , p2 , ..., pn we can find the so-called residuals of conditional equations δi = ξi −

n 

(i)

ak pk

(i = 1, 2, ..., m).

(6.12)

k=1

After the refinement process is completed, and the corrections to the parameters become negligible, the residuals of the conditional equations will become the final discrepancies or residuals of the refined theory with observations δi = ξio − ξ(ti , p 1 , p2 , ..., p n ) (i = 1, 2, ..., m).

(6.13)

In literature and in practice, these discrepancies are symbolically denoted O–C (observatum minus calculatum). The set of residuals is often used to assess the quality of the resulting solution. However, to establish the proximity of the solution to the true solution, this is not enough. Among all the available filtering algorithms in practical celestial mechanics, the least-squares method (LSM) is most often used. This method has several advantages over other filtering algorithms. The main advantage is its simplicity. For brevity, we introduce the matrix notations ⎛ ⎜ ⎜ ⎜ p = ⎜ ⎜ ⎜ ⎝



p1 p2 . . . pn

(1)

a ⎜ 1 ⎜ a (2) 1 Ap = ⎜ ⎜ ... ⎝ (m) a1





⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ , ξ = ⎜ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

(1)

...

(2)

... ...

(m)

...

a2

a2 ... a2

ξ1 ξ2 . . . ξm ⎞





⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟, δ = ⎜ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠



δ1 δ2 . . . δm (1)

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

δsum

⎜ ⎜ δ (2) ⎜ sum ⎟ (2) ⎟ ⎜ . an ⎟ , δ ⎜ sum = ⎜ ⎟ ... ⎠ ⎜ . ⎜ . (m) ⎝ an (m) δsum (1)

an

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

207

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Chapter 6 Construction of models for the motions of celestial bodies based on observations

Then the system of conditional equations (6.11) and their residuals (6.12) can be written as ξ = Ap p ,

(6.14)

δ =  ξ − Ap  p .

(6.15)

Here, the elements of the matrix Ap are calculated by Eqs. (6.9). The least-square method is based on the conclusions of probability theory. It is justified under the following conditions (Eliasberg, 1976). 1. The motion model (6.3) is specified. 2. The error vector δsum is random. 3. The covariance error matrix is non-degenerate, i.e. the determinant of this matrix is not equal to zero. 4. The mathematical expectation of E(δsum ) of the total error is zero, i.e. E(δsum ) = 0. 5. The covariance matrix D(δsum ) is given up to some arbitrary factor, i.e. D(δsum ) = σ 2 K. The arbitrary factor σ 2 is refined by applying the least-squares method. Under the assumptions made, the least-squares filtering algorithm reduces to finding the vector  p from the condition of the absolute minimum of the quadratic form ξ − Ap  p]T K−1 [ ξ − Ap  p]. p) = δ T K−1 δ = [ S(

(6.16)

Note that in practice verification of compliance with these conditions is not possible. In particular, the covariance matrix K is almost always unknown. The randomness property of the error vector and the concept of its covariance matrix should be clarified for a better understanding of the subsequent presentation. In probability theory, the concept of “test” is considered. This is one of the random variable realizations. If, for example, we consider the result of throwing a coin under a random variable, then each such throw is a test, and the result is “heads” or “tails”—the realization of a random variable. In the least-square method, the set of observations made is one single “test” of random observation errors. And we do not and cannot have other tests of this random variable. Therefore, the covariance error matrix D(δsum ) is not available to us. However, considering the set of errors as a random variable, we imply

Chapter 6 Construction of models for the motions of celestial bodies based on observations

the existence of a covariance matrix and accept one or another hypothesis in relation to it. It is often accepted that the covariance matrix of errors is diagonal, which means that the errors are not correlated with each other, i.e. mutually independent. The widespread use and popularity of the least-square method often lead to an uncritical attitude to the results obtained using this method. In many cases, incorrect conclusions are drawn from the results. The reason for this is usually a mismatch between the real conditions in which this practical task is being solved and the conditions adopted when substantiating the method. However, LSM often lead to satisfactory results even if the specified conditions are not met. In practice, the covariance error matrix is unknown. In most problems and mechanical models, it can be accepted that observation errors are uncorrelated. Then the matrix K turns out to be diagonal. If there is some information about the accuracy of some observations with respect to the accuracy of others, then the matrix K can be made as an identity matrix. For this, each observation is assigned a certain weight wi (i = 1, 2, ..., m). Each conditional equation is multiplied term by term by the set weight. In this case, the errors also turn out to be multiplied by this weight. By choosing the weight of the observations accordingly, the observations can be reduced to equal ones. Then the matrix K becomes the identity matrix. How to select the weight of observations in practice is discussed below. In the case of the identity matrix K, the relation (6.16) takes the form p) = S(

m  i=1

δi2 =

m 

ξi −

i=1

n 

2 (i) ak pk

.

(6.17)

k=1

p) reduces to solving Finding the minimum of the function S( the system of equations p) ∂S( = 0 (k = 1, 2, ..., n). ∂pk

(6.18)

As can be seen from (6.17), Eqs. (6.18) contain only the zero and first degrees of the required corrections pk (k = 1, 2, ..., n). Therefore, the system (6.18) turns out to be a system of linear inhomogeneous equations. This system is called system of normal equations. After performing the differentiation in (6.18), the system of normal equations can be written as L  p = d,

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Chapter 6 Construction of models for the motions of celestial bodies based on observations

where L and d are the square matrix and the column matrix ⎛

l11 ⎜ l21 L=⎜ ⎝ ... ln1

l12 l22 ... ln2





⎜ l1n ⎜ ⎜ ⎟ l2n ⎟ ⎜ , d = ⎜ ... ⎠ ⎜ ⎝ lnn

... ... ... ...

d1 d2 . . . dn

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

(6.19)

whose elements are calculated by the formulas lkj =

m 

(i) (i)

ak aj

(k, j = 1, 2, ..., n),

(6.20)

(k = 1, 2, ..., n).

(6.21)

i=1

dk =

m 

(i)

ak ξi

i=1

For further calculations, we need another quantity: d0 =

m  (ξi )2 .

(6.22)

i=1

Let us consider the matrix of coefficients of the normal equations L. As can be seen from (6.20), the matrix L is symmetric one with positive diagonal elements. We find in one of the known ways the matrix L−1 , the inverse one of the matrix L. Then we obtain the solution of the system of normal equations from the matrix relation  p = L−1 d.

(6.23)

In order for a solution of a system of normal equations to exist and it can be found, the matrix L must satisfy certain conditions. In particular, its rank must be equal to n. In practice, it often turns out that the determinant of the matrix of normal equations is close to zero, and it is possible to calculate the inverse matrix L−1 with very limited accuracy. If corrections are nevertheless found, the root-mean-square errors of the required corrections can be determined as follows. First, we calculate the so-called mean-square error per unit weight σ0 by the formula σ02 =

 1  p) , d0 − (d m−n

(6.24)

Chapter 6 Construction of models for the motions of celestial bodies based on observations

where  p is the obtained correction vector. Now we multiply all the elements of the matrix L−1 by σ02 . The matrix thus obtained D = σ02 L−1 .

(6.25)

It is called the covariance matrix of correction errors, which is often called simply the covariance matrix of parameters. We write it in the form ⎛ ⎞ D11 D12 ... D1n ⎜ D21 D22 ... D2n ⎟ ⎟. D=⎜ ⎝ ... ... ... ... ⎠ Dn1 Dn2 ... Dnn According to the least-squares method, any diagonal element of the matrix D with number k is equal to the square of the meansquare error σk of the correction pk , that is σk =



Dkk .

(6.26)

From (6.24), (6.25), (6.26) it can be seen that the corrections errors of the refined parameters decrease with an increase in the number of observations m. They are approximately proportional to √1m , since in practice the number of observations is much larger than the number of specified parameters. A consequence of the least-squares method is the tendency to zero of the mean-square error of the solution with an increase in the number of observations. The conclusions drawn here are valid under certain conditions, imposed not only on the errors of theory and observations, but also on the accepted model of motion of a celestial body. These conditions are considered in more detail in the book (Eliasberg, 1976). The quantities rkj defined by the relation rkj =

Dkj σk σj

are called correlation coefficients between correction errors, and the matrix ⎞ ⎛ 1 r12 r13 ... r1n ⎜ r21 1 r23 ... r2n ⎟ ⎟ R=⎜ ⎝ ... ... ... ... ... ⎠ rn1 rn2 rn3 ... 1 is called the correlation matrix.

211

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Chapter 6 Construction of models for the motions of celestial bodies based on observations

After several refinement steps, when the corrections become sufficiently small, the corrections errors characterize the errors of the improved parameter values, caused by both theoretical errors and the observation errors. The quality of the agreement of the theory with observations after refining the motion parameters of the celestial body will be characterized by  d0 σ= , (6.27) m which, after a successful refinement of the parameters, is the rms value of the residuals δi (6.13). The least-squares method is described in more detail and together with the justification in the monograph (Eliasberg, 1976). It also investigates in detail the dependence of conditionality in the least squares on the composition of observations. The method of differential refinement of the parameters of motion of celestial bodies from observations with questions of its practical application is considered in the book (Emel’ianov, 1983). A simplified description of LSM is given in the textbook (Shchigolev, 1969). An essential part of the algorithm for refining the parameters of motion of celestial bodies from observations is the calculation of the values of the measured quantity at given times, as well as its derivatives by the improved parameters. For this purpose, one can apply both formulas of the analytical theory of motion of a celestial body, and methods of numerical integration of the equations of its motion. The number of computational operations may vary greatly. When using analytical theory, the calculation time will be proportional to the number of observations used. In this case, it does not depend on the time interval over which the observations were made. With numerical integration, the opposite is true: the time spent on calculations is proportional to the observation time interval and does not depend on the number of observations. The order of all such calculations is explained in the following sections.

6.2 Weak conditionality and ambiguity of solution The determination of motion parameters based on observations is obtained simply and accurately only in simple illustrative examples. As a rule, difficulties arise in practical problems using real observations, and the determination of parameters is inaccurate. Consider the most common problems here.

Chapter 6 Construction of models for the motions of celestial bodies based on observations

Using the LS method in practice often leads to unexpected problems. The fact is that in the conditions of research based on real observations of celestial bodies using theories of limited accuracy, those assumptions are not always strictly fulfilled in which the use of LSM is legitimate. If the errors of the theory prevail over the errors of observations, then the total error will not be a random variable. This will lead to the fact that with an increase in the number of observations, the accuracy of the result will not improve, and the solution, in turn, will depend on the composition of the measurements, that is, on what moments the measurements were taken. The presence of such a dependence makes the result not entirely reliable. In specific problems of refining the motion parameters of celestial bodies, it often turns out that the determinant of the matrix L is close to zero. In these cases, we are dealing with the so-called weakly conditioned systems of normal equations. When solving such systems, corrections to the parameters can be so crude that the refinement process will not converge. The reason for the weak conditioning is not associated with the least-squares method itself, but with the properties of the applied mechanical model. An example of a case with weak conditionality is the process of clarifying the longitude of the ascending node of the Keplerian orbit of a celestial body with a very small inclination. Another example is a joint refinement of the longitude of the pericenter of the orbit and the mean anomaly in the epoch with small eccentricities of the orbit. An indicator of weak conditioning may be the proximity to the unit value for modulus of one or more correlation coefficients. It is impossible to overcome weak conditioning, because its cause lies in the conditions of the problem. We can only replace the problem with another one. To reduce weak conditionality, one can exclude from the list of refined parameters that one that gives a strong correlation, fixing its preliminary value. This fixed value can be selected approximately. As a rule, the measured quantity weakly depends on the parameter giving a strong correlation. Therefore, the proximity of a fixed parameter may not affect the measured value. The success of refinement of motion parameters from observations substantially depends on the composition of the observations. In particular, if the observations cover only an insignificant part of the orbit of the celestial body, then a weak conditionality of the system of normal equations arises, and the determination of parameters may become impossible. Weak conditionality due to the composition of the observations can occur, for example, when determining the angular distance of the pericenter from the node ω for the Keplerian orbit of

213

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a celestial body. If all observations are concentrated in the pericenter and apocenter of the orbit, then uncertainty arises in determining ω. We can change ω, that is, rotate the orbit around the attracting center, with a constant value of the longitude of the celestial body, while the displacements of the points in orbit near the pericenter and near the apocenter will be insignificant. This will give a weak dependence of the measured values on the angle ω at constant longitude. If ω is fixed at a certain approximate value, this approximation will have little effect on the differences between the calculated and observed positions of the celestial body near the pericenter and near the apocenter of the orbit. The dependence of the accuracy of the found parameter values on the composition of observations is described in more detail in the book (Eliasberg, 1976). Let us consider another problem that arises in some observations. p) (6.17) is sometimes called the target or obThe function S( jective function. To obtain an estimate of the parameters, they search for its minimum. Above, the objective function was constructed for linearized conditional equations (6.11). However, it can also be constructed for Eqs. (6.5). In this more general case, the objective function is written as S(p1 , p2 , ..., pn ) =

m  2  o ξi − ξ(ti , p1 , p2 , ..., pn ) ,

(6.28)

i=1

and the problem is to find the minimum of this function on the set of parameter values p1 , p2 , ..., pn . In real problems, the function ξ(ti , p1 , p2 , ..., pn ) is non-linear by its arguments. Then the objective function S(p1 , p2 , ..., pn ) can have several minima. If we find all the minima, then we choose the smallest of them and hope that we get the most accurate estimate. However, such a formal choice may raise doubts. It may turn out that the observation errors were distributed in such a way that the exact estimate corresponds to not the smallest of the mentioned minima. On the other hand, solving linearized conditional equations (6.11), one can get to (0) (0) (0) such an initial approximation of the parameters p1 , p2 , ..., pn , which will lead to successive approximations to an incorrect result. In the paper (Avdyushev and Ban’shikova, 2008) it was shown that the problem of ambiguous determination of orbits can occur in problems of the dynamics of nearby planetary satellites with a small number of observations dispersed in several groups over a sufficiently large time interval. The problem of many solutions is discussed in detail in the book (Avdyushev, 2015). It is shown

Chapter 6 Construction of models for the motions of celestial bodies based on observations

that the graph of the function (6.28) can have a “ravine” structure, so that a whole family of values of the required parameters can produce equally small values of the function. In other cases, the function (6.28) may have several isolated minima. Then the problem of determining the parameters may be ambiguous. The problem of ambiguous determination of orbits can also arise in other cases, namely, when observations are grouped over a small time interval and cover a short orbital arc. Obviously, almost all new discovered satellites have such a composition of observations. We can reliably determine the orbits only for a part of distant satellites with a small number of observations. However, for some of these satellites problems nevertheless arise, and the reliability of the obtained orbits remains in doubt. These cases were considered in the work (Avdyushev and Ban’shikova, 2010). To solve this problem, obviously, additional observations are needed.

6.3 Overview of filtering algorithms Before talking about other filtering algorithms, we clarify some concepts. First of all, we recall that the problem is to search for the values of the required parameters based on the available measurements. Moreover, the obtained values should, as far as possible, differ little from the true ones. In practice, it turns out that the exact value of the required parameters cannot be found. Therefore, we are talking about an approximate parameter estimation. An approximate estimation algorithm is called filtering algorithm. The main objective of this algorithm is the possible reduction (filtering) of the influence of theory errors and observation errors. One of the characteristics of the estimate obtained by the filtering algorithm is consistency of the estimate, which is understood as the convergence of the estimate to true values when the number of observations tends to infinity. Extensive and detailed information on filtering algorithms is currently contained in the books (Eliasberg, 1976) and (Avdyushev, 2015). Observations in practical celestial mechanics mean measurements. The book (Eliasberg, 1976) is called Definition of motion from measurement results. The book discusses various traditional and non-traditional approaches to parameter estimation, in particular, a guaranteed approach when the limits of possible values of the required parameters are found for given sets of measurements. The value of the book (Eliasberg, 1976) lies in the combination of its focus on the practical solution of problems with the mathematical justification of the methods used. Examples of

215

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Chapter 6 Construction of models for the motions of celestial bodies based on observations

solving problems are considered. Conveniently, the first chapter contains the necessary information on probability theory. The following describes the characteristics of measurement errors. Most of the book is devoted to the mathematical justification of the LSM use. The conditions of its applicability are also given. The book by Eliasberg (1976) discusses a number of wellknown filtering algorithms: the maximum likelihood method, the least modulus method, the method of maximum posterior probability, and Kalman filters (discrete and continuous). In some practical situations, it is possible to choose the composition of the measurements or to choose some set of them from the available data. Here the problem arises of finding the optimal composition of measurements. This problem is given a lot of attention in the book by Eliasberg (1976). In particular, it is indicated that in real conditions an increase in the number of measurements may be useless, and in some cases even leads to a deterioration in the accuracy of the results. Another book, the monograph by Avdyushev (2015), is a good source of information about filtering algorithms. Two chapters of this book are devoted to the determination of orbits from observations and the estimation of the accuracy of the determined parameters. In addition to the LS method, the following methods are considered: the Newton method, the Gauss–Newton method, the Levenberg–Marquardt method, and ravine methods. A comparative analysis of the effectiveness of the methods is also given. The original approach to filtering algorithms adopted in Bakhshiyan et al. (1980). It examines the definition of motion and its control with the discrete nature of the measurement information and corrective actions. Particular attention is paid to assessing the accuracy of the results obtained and optimizing the strategy for solving the problems under consideration. In this case, an approach is used to guarantee the achievement of the required accuracy and reliability of the obtained solutions, provided that the error distribution functions of the source data are not exactly known, and only some sets are specified to which these functions can belong. This ensures the stability of the results. The mathematical programming apparatus used in solving the optimization problems under consideration is described. To get acquainted with some generalizations of the LS method, one can turn to the book (Gubanov, 1997). It contains a systematic presentation of the foundations of the theory of least squares and its generalizations, mean-square collocation and Kalman filtering. The main types of parametric and stochastic models of measurement data are considered. Examples are given of applying

Chapter 6 Construction of models for the motions of celestial bodies based on observations

the generalized theory of LSM to the processing of radio interferometric observations with extra-long bases, to the analysis of the Earth’s rotational motion, etc.

6.4 Calculation of measured values and partial derivatives of the measured values by refined parameters 6.4.1 General order of calculations In the process of differential refinement of the parameters of motion of celestial bodies from observations, it is necessary to calculate the values of the measured quantities and partial derivatives of the measured values by the specified parameters at the moments of observation. These calculations are made on the basis of the adopted law of motion of celestial bodies. The law of motion is described in any coordinate system. Often these are Cartesian rectangular coordinates x, y, z. The law of motion is defined by the functions x = x(t, p1 , p2 , ..., pj ), z = z(t, p1 , p2 , ..., pj ).

y = y(t, p1 , p2 , ..., pj ), (6.29)

In the previous section, the concept of measurable quantity was introduced. Observations give us the values of the measured quantities at the moments of measurement. Let ξ be one of the measured quantities. The measured value is related to the coordinates of the celestial body. This relationship is defined by some measurement model. The model may include some other parameters, denoted by pj +1 , pj +2 , ..., pn . The observation model gives us the function ξ = ξ(t, x, y, z, pj +1 , pj +2 , ..., pn ).

(6.30)

Thus, in the method of differential refinement of motion parameters, the measured value ξ , as a function of the refinement parameters p1 , p2 , ..., pn , is a complex function. Initially, it is defined as a function of the rectangular coordinates of a celestial body, which in turn, by virtue of the law of motion, are functions of time t and motion parameters. The dependence ξ(t, x, y, z, ...) is not related to the law of motion of a celestial body, however, it includes the time t and parameters, which can also be regarded as refined. In contrast to the motion parameters of a celestial body, they are called the parameters of the observation conditions. The function ξ(t, x, y, z, ...) is determined only by the choice of the measured

217

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Chapter 6 Construction of models for the motions of celestial bodies based on observations

quantity. In astronomical practice, a large number of quantities are used, measured in the process of observation. This section discusses some of them and gives explicit expressions for the function ξ(t, x, y, z, pj +1 , pj +2 , ..., pn ). Further, the dependences of the measured values on the parameters of the observation conditions and on the parameters of the movement are considered. As a result, these dependences are displayed as a function of time t and all specified parameters, ξ = ξ(t, p1 , p2 , ..., pn ).

(6.31)

It is this function that is used to compose conditional equations when refining parameters by the least-squares method. To apply the method of differential refinement of the parameters of motion of celestial bodies from observations, it is necessary, in addition to the measured quantities themselves, to also calculate the partial derivatives of the measured quantities by the specified parameters. The order of these calculations essentially depends on how the law of motion is obtained. The calculation of the coordinates of the celestial body x, y, z is done either according to the formulas of the previously constructed analytical theory of motion, or based on the numerical integration of differential equations of motion. The partial derivatives of the measured values by the specified parameters at the time of observation are calculated on the basis of the dependences considered above. Since the intermediate values are the coordinates of the celestial body x, y, z for the required derivatives, we can write the following relations: ⎛

∂ξ ∂p1





∂x ∂p1

⎜ ⎟ ⎜ ⎜ ∂x ⎜ ∂ξ ⎟ ⎜ ∂p ⎜ ∂p ⎟ ⎜ 2 ⎜ 2 ⎟ ⎟ = A · B, A = ⎜ ⎜ ⎜ ... ⎜ ... ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ ⎝ ∂ξ ∂pn

∂x ∂pn

∂y ∂p1 ∂y ∂p2

... ∂y ∂pn



⎛ ⎟ ⎟ ⎜ ∂z ⎟ ⎜ ∂p2 ⎟ ⎟, B = ⎜ ⎜ ⎟ ... ⎟ ⎝ ⎠ ∂z ∂p1

∂z ∂pn

∂ξ ∂x ∂ξ ∂y ∂ξ ∂z

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

(6.32) As we can see from these relations, the computational problem is divided into two independent parts. The first of them contains the values of the partial derivatives from rectangular coordinates by the motion parameters of a celestial body, i.e. matrix A. The second one is the derivatives of the measured quantity by the rectangular coordinates of the celestial body, the column matrix B. The vector B is calculated by the formulas obtained by differentiating the explicit expression for ξ(x, y, z). The calculation of the matrix of parameters A can be performed for each moment

Chapter 6 Construction of models for the motions of celestial bodies based on observations

of observation both according to the formulas following from the analytical theory of motion and during the numerical integration of differential equations specially constructed for partial derivatives of coordinates by motion parameters. In the latter case, the elements of the matrix A are called isochronous derivatives, since we need their values at the same time moments as the coordinates themselves. Differential equations for isochronous derivatives (they are sometimes called equations in variations) are integrated together with the equations of motion. Note that in the above calculations it is implicitly assumed that the measured value depends only on the coordinates of the celestial body. However, in practice, refinement of parameters from observations also uses measurable quantities depending on the components of the velocity vector x, ˙ y, ˙ z˙ . An example is the refinement of the motion parameters of an artificial Earth satellite based on ground-based radio-technical Doppler measurements. In this case, the radial velocity is measured, i.e. the rate of change of the topocentric distance of the satellite. Then the column vector B will acquire three more elements, ∂ξ ∂ξ ∂ξ , , , ∂ x˙ ∂ y˙ ∂ z˙ and the matrix A has three more corresponding additional columns. All further considerations and calculations will be similar to what is obtained if the measured value depends only on the coordinates. Since in the practice of ground-based observations of planetary satellites, Doppler observations are carried out only in exceptional cases, we restrict ourselves further to considering the dependence of the measured quantity ξ only on the Cartesian rectangular coordinates of the satellite. When studying the dynamics of the natural satellites of the planets, some studies construct and apply analytical theories of motion. The law of motion is found as a general solution of the differential equations of motion. Then the coordinates of the celestial body are represented by analytical functions of time and motion parameters. In analytic theories, motion parameters are often referred to as orbital elements, since they are related to the Keplerian motion model and Keplerian elements. The elements of the matrix A in such problems are obtained by analytic differentiation of the rectangular coordinates by the elements of the orbit. Formulas for these derivatives are usually given together with formulas for the coordinates and velocity components of a celestial body. This is exactly what is done in this book. The expressions for these derivatives in the case of an elliptic Keplerian motion are given in Sect. 3.2.4.

219

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Chapter 6 Construction of models for the motions of celestial bodies based on observations

In the following sections, formulas for calculating the elements of the matrix A in various concrete problems are first considered. Then formulas are given for calculating the measured quantities ξ and their partial derivatives by the rectangular coordinates and parameters of the observation conditions for various types of observations. Some recommendations are given regarding the preparation of conditional equations.

6.4.2 Differential equations for isochronous derivatives in the three-body problem. Refinement of the initial conditions of the equations of motion Let us consider the procedure of differential refinement of motion parameters in the case of a three-body problem. Since no exact analytical solution to the three-body problem has yet been found, the equations of motion in this problem are solved by numerical integration methods. In this case, the initial conditions are often considered as motion parameters, that is, the coordinates at a certain initial time moment t0 . The refinement parameters may also be constants, which are included in the differential equations of motion. In the problem considered, the motion parameters of the second body relative to the first one are determined, which is considered the main one, being the most massive. The movement occurs under the perturbing action of the third body, the movement of which is given by coordinates, as known functions of time. In the general case, the motion parameters of the second and third bodies in a single process of differential refinement can be determined from observations. Then the equations of motion, as well as the equations for changing the isochronous derivatives of the second and third bodies are integrated together. We will consider bodies as material points. The origin of the system of non-rotating rectangular coordinates is placed in the first of the bodies. The coordinates of the second body whose motion is studied, in contrast to the usual notation, are now denoted by x1 , x2 , x3 . The coordinates of the third perturbing body are denoted by x1 , x2 , x3 . The equations of motion of the second body in the accepted notation are written in the form    xi xi d 2 xi  xi − xi = Fi (i = 1, 2, 3), = −Gm − Gm + dt 2 r3 3 r 3

(6.33)

Chapter 6 Construction of models for the motions of celestial bodies based on observations

where G is the universal gravitational constant, m is the mass of the first body, and m is the mass of the perturbing body. In addition, we use the following notation: r=



=

x12



+ x22

+ x32 ,

 r = x1 2 + x2 2 + x3 2 , 

(x1 − x1 )2 + (x2 − x2 )2 + (x3 − x3 ).

The equations of motion of the third celestial body can be written similarly. In practice, instead of the masses of celestial bodies, their gravitational parameters μ = Gm, μ = Gm are considered. Note that the formulas given below will be suitable for the more general case, when the coordinates of the perturbing body are calculated on the basis of a more complex model that takes into account the influence of other bodies. The parameters of the studied motion of the second body will be the initial conditions, that is, the coordinates and velocity components (0)

(0)

(0)

(0)

(0)

(0)

x1 , x2 , x3 , x˙1 , x˙2 , x˙3 , given at the initial moment of time t0 . The sought-after partial derivatives necessary for the differential refinement of the parameters form a matrix, ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

∂x1 (0) ∂x1

∂x2 (0) ∂x1

∂x3 (0) ∂x1

∂x1 (0) ∂x2

∂x2 (0) ∂x2

∂x3 (0) ∂x2

∂x1 (0) ∂x3

∂x2 (0) ∂x3

∂x3 (0) ∂x3

∂x1 (0) ∂ x˙1

∂x2 (0) ∂ x˙1

∂x3 (0) ∂ x˙1

∂x1 (0) ∂ x˙2

∂x2 (0) ∂ x˙2

∂x3 (0) ∂ x˙2

∂x1 (0) ∂ x˙3

∂x2 (0) ∂ x˙3

∂x3 (0) ∂ x˙3

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(6.34)

For elements of this matrix, one can compose differential equations by differentiating the left- and right-hand sides of Eqs. (6.33) with respect to the parameter. Performing this operation sequentially for each of the parameters, we obtain the following system of

221

222

Chapter 6 Construction of models for the motions of celestial bodies based on observations

equations: d2 dt 2 d2 dt 2

∂xi

 =

(0)

∂xj

3  ∂Fi ∂xn , ∂xn ∂x (0)

(6.35)

3  ∂Fi ∂xn , ∂xn ∂ x˙ (0)

(6.36)

n=1

∂xi

 =

(0)

∂ x˙j

j

n=1

j

(i, j = 1, 2, 3), where ∂Fi 1 = Gm 3 ∂xn r



   3 3  1   xi xn − δin +Gm 3 (xi − xi )(xn − xn ) − δin , r2  2  δin =

1 with i = n, 0 with i = n.

Eqs. (6.35) and (6.36) can be written in matrix form. To do this, we introduce the matrix ⎞ ⎛ ∂F ∂F ∂F ⎜ ⎜ F=⎜ ⎜ ⎝

1

2

3

∂x1

∂x1

∂x1

∂F1 ∂x2

∂F2 ∂x2

∂F3 ∂x2

∂F1 ∂x3

∂F2 ∂x3

∂F3 ∂x3

⎟ ⎟ ⎟. ⎟ ⎠

(6.37)

Then Eqs. (6.35) and (6.36) can be written as d2 A = AF. dt 2

(6.38)

The numerical integration of Eqs. (6.35) and (6.36) should be performed together with the equations of motion (6.33). The initial conditions for Eqs. (6.35) and (6.36) are determined by the matrices ⎞ ⎛ 1 0 0 ⎟ ⎜ ⎜ 0 1 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 1 ⎟ ⎟ ⎜ (6.39) A0 = A|t=t0 = ⎜ ⎟, ⎜ 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 ⎟ ⎠ ⎝ 0 0 0

Chapter 6 Construction of models for the motions of celestial bodies based on observations



˙ t=t0 A˙ 0 = A|

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0

0

0

0

0

0

1

0

0

1

0

0

0



⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟. 0 ⎟ ⎟ ⎟ 0 ⎟ ⎠ 1

(6.40)

Here, the dot above the letters means time differentiation.

6.4.3 Differential equations for isochronous derivatives in the three-body problem. Refinement of the mass of the perturbing body Consider the movement of the second of three bodies under the influence of the attraction of the first body and the perturbing influence of the third gravitating body. From the observations of the motion of the second body, one can determine the parameters of its motion. In addition, together with the initial conditions of the second body, the mass of the perturbing body can be determined. Such a definition should be done necessarily together, since during the correction of the mass of the disturbing body, the motion parameters of the second body will be different. In such a task, the initial conditions, that is, the coordinates and velocity components, will be refined parameters (0)

(0)

(0)

(0)

(0)

(0)

x1 , x2 , x3 , x˙1 , x˙2 , x˙3 , given at the initial moment of time t0 for the second body and the gravitational parameter μ of the perturbing body. In this case, add one more row to the matrix (6.34),   ∂x3 . ∂x1 ∂x2 (6.41)    ∂μ ∂μ ∂μ For the elements of this line, the following differential equations can be composed: d2 dt 2



∂xi ∂μ





xi xi − xi + =− 3 r 3

 +

3  ∂Fi ∂xn . ∂xn ∂μ

(6.42)

n=1

Then we need to integrate the equations of motion together with (6.33), and Eqs. (6.35), (6.36) and (6.42). The initial conditions for

223

224

Chapter 6 Construction of models for the motions of celestial bodies based on observations

the variables (6.41) will be ∂x1 ∂μ |t=t0 d dt



∂x1 ∂μ



= 0,

|t=t0 = 0,

d dt

∂x2 ∂μ |t=t0



∂x2 ∂μ



= 0,

∂x3 ∂μ |t=t0

|t=t0 = 0,

d dt



= 0,

∂x3 ∂μ



(6.43) |t=t0 = 0.

6.4.4 Differential equations for isochronous derivatives in the satellite motion problem for a oblate planet Let us consider the procedure of differential refinement of the satellite motion parameters in the case of taking into account perturbations from the planet non-sphericity. In this problem, the perturbations of the elements of the satellite’s intermediate orbit can be determined by the methods of perturbation theory in an analytical form. However, the equations of motion of the satellite can also be solved by numerical integration. In this case, the initial conditions are considered as motion parameters, that is, the coordinate values at some initial time moment t0 . The force function of attraction of a nonspherical planet is used in the form of expansion in a series of spherical functions. This decomposition is discussed in detail in Sect. 3.3. Since the expansion of the force function in this case is written in the coordinate system associated with the axis of symmetry of the oblate body, the rectangular coordinates appear in the expansion in the system with the main plane coinciding with the equatorial plane of the planet. Denote these coordinates by x 1 , x 2 , and x 3 . However, in a number of problems it turns out that it is necessary to solve the equations of motion with respect to the coordinates x1 , x2 , x3 , not related to the equator of the planet. The relationship of coordinates in two systems is given by the relation {x1 , x2 , x3 }T = R0 {x 1 , x 2 , x 3 }T , where the matrix R0 is described in Section 3.3. First, for simplicity, we consider in the expansion of the force function only the main term describing the dynamical oblateness of the planet, namely, the second zonal harmonic. For other members of this expansion, equations for isochronous derivatives can be derived in a similar way. We write the equations of motion taking into account the second zonal harmonic of the expansion of the force function of the

Chapter 6 Construction of models for the motions of celestial bodies based on observations

planet’s attraction in the following form: d 2 xi = Fi (i = 1, 2, 3), dt 2

(6.44)

where {F1 , F2 , F3 }T = R0 {F 1 , F 2 , F 3 }T , moreover, in the equatorial coordinate system {F 1 , F 2 , F 3 } according to the formulas in Sect. 3.3 have the form

 r02 x 23 xi 3 F i = −μ 3 + μJ2 5 x i 5 2 − ei (i = 1, 2, 3) , (6.45) 2 r r r where μ is the planet’s gravitational parameter, J2 is the coefficient for the second zonal harmonic of the expansion of the planet’s gravitational force, and r0 is the mean equatorial radius of the planet. In addition, we use the following notation:  r = x 21 + x 22 + x 23 , e1 = 1, e2 = 1, e3 = 3. The parameters of the satellite motion under study will be the initial conditions, i.e., the coordinates and velocity components (0)

(0)

(0)

(0)

(0)

(0)

x1 , x2 , x3 , x˙1 , x˙2 , x˙3 , given at the initial moment of time t0 . The required partial derivatives necessary for the differential refinement of the parameters compose a matrix of the form (6.34). For the elements of this matrix, one can compose differential equations by differentiating the left- and right-hand sides of Eqs. (6.45) by the parameter. Performing this operation sequentially for each of the parameters, we obtain a system of equations, which we write here in matrix form:   d2 T , (6.46) A = AR F · R 0 0 dt 2 where the matrix F, similar to the matrix (6.37), has the general form ⎞ ⎛ ⎜ ⎜ ⎜ F=⎜ ⎜ ⎝

∂F 1 ∂x 1

∂F 2 ∂x 1

∂F 3 ∂x 1

∂F 1 ∂x 2

∂F 2 ∂x 2

∂F 3 ∂x 2

∂F 1 ∂x 3

∂F 2 ∂x 3

∂F 3 ∂x 3

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(6.47)

225

226

Chapter 6 Construction of models for the motions of celestial bodies based on observations

with elements defined by the formulas   ∂F i 1 3 =μ 3 x x − δ i n in + ∂x n r r2    x32 r02 x 23 3 x3xi xi xn + μJ2 5 5 2 − ei δin − 35 4 x i x n + 10 2 fn + 5 2 ei , 2 r r r r r (6.48) where we introduced the notation f1 = 0, f2 = 0, f3 = 1, and δin is the Kronecker symbol: δin = 1 for i = n and δin = 0 for i = n. The initial conditions for integrating Eqs. (6.46) should be taken by (6.39) and (6.40). Note that in cases where the main factor affecting the satellite’s motion is the oblateness of the planet, the main coordinate system x, y, z can be associated with the planet’s equator. Then the coordinates x 1 , x 2 , x 3 coincide with the coordinates x, y, z, acceleration components F 1 , F 2 , F 3 coincides with the components F1 , F2 , F3 , the matrix R0 turns out to be the identity matrix, and the matrix (6.47) matches the matrix (6.37). If in the solution we understand that the fourth zonal harmonic of the expansion of the force function of planet’s gravity must also be taken into account, then the corresponding terms should be added to the right-hand sides of Eqs. (6.45) and (6.48). In the notation adopted above, these additional terms have the form

 x 23 x i x 43 x i xi F i = ... + A ai 7 + bi 9 + c 11 , r r r where 5 A = μr04 J4 , 8 a1 = 3, a2 = 3, a3 = 15, b1 = −42, b2 = −42, b3 = −70, c = 63, and for Eqs. (6.48)   ∂F i (1) (2) (3) = ... + A ai Fin + bi Fin + cFin , ∂x n

Chapter 6 Construction of models for the motions of celestial bodies based on observations

where (1)

Fin = (2)

Fin = (3)

Fin =

δin 7x i x n − , r7 r9

x 23 δin 9x 23 x i x n 2x 3 x i − + fn 9 , 9 11 r r r 4x 33 x i x 43 δin 11x 43 x i x n − + f . n r 11 r 13 r 11

6.4.5 Construction of conditional equations for angular measurements of topocentric coordinates The construction of conditional equations for the differential refinement of the parameters of the orbits of celestial bodies based on observations is associated with the calculation of the partial derivatives of the measured quantity by the coordinates of the celestial body. In Sect. 6.4.1, these derivatives constitute the components of the column vector B. Of course, these derivatives depend on the type of observations and the type of measured quantity. In this section, we consider calculations for the cases of angular topocentric measurements. Chapter 5 introduces the concept of an observation vector. The beginning of this vector is located at the observation point, the topocenter, and the end at the center of the observed body. Denote the components of the observation vector by X, Y, Z. The geoequatorial angular coordinates are assigned to the observation vector: right ascension α and declination δ. The relationship of angular and rectangular coordinates is described by the formulas tan α =

Y Z , tan δ = √ . X X2 + Y 2

If the measured quantities are right ascension and declination, then the formulas for the partial derivatives of the measured quantities by the rectangular topocentric coordinates have the form ∂α − sin α ∂α cos α ∂α = , = , = 0, ∂X R cos δ ∂Y R cos δ ∂Z − cos α sin δ ∂δ − sin α sin δ ∂δ cos δ ∂δ = , = , = , ∂X R ∂Y R ∂Z R √ where R = X 2 + Y 2 + Z 2 .

227

228

Chapter 6 Construction of models for the motions of celestial bodies based on observations

The planetary motion model gives us the rectangular barycentric coordinates of the planet at given times. A planetary satellite motion model represents planetocentric rectangular coordinates. If the axes of all the systems under consideration are mutually parallel, then giving them the general designation x, y, z, we can present the expressions for the partial derivatives of the measured right ascension and declination in the coordinates of the planet or satellite in the following form: cos α ∂α ∂α − sin α ∂α = , = , = 0, ∂x R cos δ ∂y R cos δ ∂z ∂δ − cos α sin δ ∂δ − sin α sin δ ∂δ cos δ = , = , = . ∂x R ∂y R ∂z R In these formulas, for the case of observing a planet, x, y, z are the rectangular barycentric coordinates of the planet, and when observing a satellite, x, y, z are the rectangular planetocentric coordinates of the satellite. In any case, R is the topocentric distance of the observed object. If the differences of the right ascension and declination of a satellite and a planet, or two satellites α = α1 − α2 , δ = δ1 − δ2 , are measured, then the partial derivatives of the measured quantities by the coordinates of the bodies are expressed by the formulas cos α2 ∂α ∂α − sin α2 ∂α = , = , = 0, ∂x1 R cos δ2 ∂y1 R cos δ2 ∂z1 ∂δ − cos α2 sin δ2 ∂δ − sin α2 sin δ2 ∂δ cos δ2 , , , = = = ∂x1 R ∂y1 R ∂z1 R ∂α sin α2 ∂α − cos α2 ∂α = , = , = 0, ∂x2 R cos δ2 ∂y2 R cos δ2 ∂z2 cos δ2 ∂δ cos α2 sin δ2 ∂δ sin α2 sin δ2 ∂δ , , , = = =− ∂x2 R ∂y2 R ∂z2 R where the lower index means the number of the object and R is the topocentric distance of the corresponding object. If the second object is a planet, then the coordinates x2 , y2 , and z2 do not appear in the problem. Let us consider here one modification of the formulas for partial derivatives of the measured quantities, which is often applied

Chapter 6 Construction of models for the motions of celestial bodies based on observations

in practice. We write here again the conditional equations in the form as they are derived in Sect. 6.1:  n   ∂ξ pk (i = 1, 2, ... , m). ξi = ∂pk i

(6.49)

k=1

Here ξ is the general notation for the measured quantity, p1 , p2 , ...pn is for refined parameters, and m is the number of observations. Let us take into account the fact that the error in modeling the motion of planets and satellites in rectangular coordinates does not depend on how we observe them. However, at different topocentric distances, this error will manifest itself in different ways in the measured angular coordinates: right ascension and declination. To contribution the errors in the left-hand sides of the conditional equations became independent of the topocentric distance R, we need to multiply each conditional equation by R. Similar reasoning regarding observations at various declinations of the celestial body leads to the conclusion that the conditional equations in measurements of the right ascension α should be multiplied by cos δ. Now, taking into account the above circumstances and giving the measured value ξ alternately the specific meaning of ξ = α, ξ = δ, we write the conditional equations in the form R cos δαi

=

 n   ∂α 

pk (i = 1, 2, ..., m),

(6.50)

 n   ∂δ  pk (i = 1, 2, ..., m), ∂pk i

(6.51)

k=1

Rδi =

∂pk

i

k=1

where ⎛    ∂α ⎜ ∂p1 ⎜ ⎜   ⎜ ∂α ⎜ ∂p2 ⎜ ⎜ ⎜ ... ⎜ ⎜   ⎝ ∂α ∂pn

⎞ ⎟ ⎟ ⎛ ⎟ − sin α ⎟ ⎟ ⎜ ⎟ = A · B , B = ⎜ cos α α α ⎟ ⎝ ⎟ 0 ⎟ ⎟ ⎠

⎞ ⎟ ⎟, ⎠

(6.52)

229

230

Chapter 6 Construction of models for the motions of celestial bodies based on observations

⎛    ∂δ ⎜ ∂p1 ⎜ ⎜   ⎜ ∂δ ⎜ ∂p2 ⎜ ⎜ ⎜ ... ⎜ ⎜   ⎝ ∂δ

⎞ ⎟ ⎟ ⎛ ⎟ − cos α sin δ ⎟ ⎟ ⎜ ⎟ = A · B , B = ⎜ − sin α sin δ δ δ ⎟ ⎝ ⎟ cos δ ⎟ ⎟ ⎠

⎞ ⎟ ⎟. ⎠

(6.53)

∂pn

Here the matrix A is described above. On the left-hand sides of Eqs. (6.50) and (6.51) αi and δi are the differences between the measured and calculated values of the measured quantities, which can be both the right ascensions and the declinations of one celestial body α, δ, and their differences for two bodies, for example, a satellite and a planet or two satellites α = α1 − α2 , δ = δ1 − δ2 . In the latter case, if the conditional equation is constructed with respect to the parameters of the second celestial body, then the elements of the column vectors Bα , Bδ must be taken with the opposite sign. In practice, refinement of motion parameters from observations instead of the measured value α = α1 − α2 often uses the value Xd = α cos δ2 . For a uniform designation, Yd = δ is added to it. In this case, the conditional equations (6.50) and (6.51) are written as follows: (i)

R Xd =

 n   ∂α 

pk (i = 1, 2, ..., m),

(6.54)

 n   ∂δ  pk (i = 1, 2, ..., m), ∂pk i

(6.55)

k=1

(i)

R Yd =

∂pk

i

k=1

(i)

(i)

where Xd and Yd are the differences between the measured and calculated values of Xd and Yd , respectively, at the time of observation ti . Let us now consider how to construct conditional equations if the mutual angular distance between two bodies s and the corresponding position angle P are measured. In this case, the conditional equations have the form  (i)    n (i)   Xd ∂α  Yd ∂δ  R si = + pk (i = 1, 2, ... , m), si ∂pk i si ∂pk i k=1 (6.56)

Chapter 6 Construction of models for the motions of celestial bodies based on observations

 (i)    n (i)   Yd ∂α  Xd ∂δ  R si Pi = − pk (i = 1, 2, ..., m), si ∂pk i si ∂pk i k=1 (6.57) where si and Pi are the differences between the measured and calculated values of s and P , respectively, and the index i means that the values are taken at the time of observation ti . In Eqs. (6.54), (6.55), (6.56), and (6.57) the partial derivatives are calculated using Eqs. (6.52) and (6.53). For the tangential coordinates Xt , Yt , st , and Pt , the conditional equations are constructed similarly. When constructing the conditional equations in the problems under consideration, it should be taken into account that the coefficients under the corrections pk , (i = 1, 2, ..., m) can be calculated to some extent approximately, since refinement can be performed by successive approximations several times. However, the values of the measured values should be calculated with the greatest possible accuracy, since in successive approximations, the differences between the measured and calculated values of the measured quantity should tend to zero.

6.5 Assigning weights to observations and conditional equations For a justified application of the least-squares method, it is necessary that the hypothesis adopted with respect to the covariance matrix of the observation errors be as close as possible to reality. The off-diagonal elements of the covariance matrix in practice are most often unknown. Therefore, they are simply equated to zero. As for the diagonal elements, they characterize the accuracy of the observations, which may vary for different observations. However, diagonal elements can be made equal if weights are appropriately assigned to the observations. Let us take the conditional equations of the least-squares method, which were considered in Section 6.1 for the case of uniform observations: ξi =

n 

(i)

ak pk (i = 1, 2, ..., m).

(6.58)

k=1

Here ξi is the difference between the measured and calculated values of the measured quantity in the observation with number (i) i, p1 , p2 , ..., pn are the parameters to be specified, and ak are the numerical coefficients of the conditional equations. The set of

231

232

Chapter 6 Construction of models for the motions of celestial bodies based on observations

refined parameters can be represented by the vector  p = {p1 , p2 , ..., pn }T . To solve the problem according to the least-squares method, we need to compose a system of normal equations, which we write in the form L p = d , where L and d are the matrix and the vector ⎛

l12 l22 ... ln2

l11 ⎜ l21 L=⎜ ⎝ ... ln1

... ... ... ...





⎜ l1n ⎜ ⎜ ⎟ l2n ⎟ ⎜ , d = ⎜ ⎠ ... ⎜ ⎝ lnn

d1 d2 . . . dn

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

(6.59)

whose elements are calculated by the formulas lkj =

m 

(i) (i)

ak aj

(k, j = 1, 2, ... , n),

(6.60)

i=1

dk =

m 

(i)

ak ξi (k = 1, 2, ..., n).

(6.61)

i=1

The weights of the observations wi (i = 1, 2, ..., m) are introduced as follows. Multiplying the conditional equations (6.58) by √ wi , we obtain ξi



wi =

n  √ (i) wi ak pk (i = 1, 2, ... , m).

(6.62)

k=1

Instead of (6.60) and (6.61) we take the formulas lkj =

m 

(i) (i)

wi a k a j

(k, j = 1, 2, ... , n),

(6.63)

i=1

dk =

m 

(i)

wi ak ξi (k = 1, 2, ... , n).

(6.64)

i=1

Further, the solution is obtained in the same way as for uniform observations. The covariance matrix of the observation errors will have equal diagonal elements if the errors of the left parts of the conditional equations (6.62) are approximately the same for all i. This could be

Chapter 6 Construction of models for the motions of celestial bodies based on observations

done if the errors in the values of ξi were known. It is possible to achieve the indicated property of the covariance matrix if we put 1 √ wi = , σi where σi is the measurement error with number i. The problem is that usually we do not know the errors of σi . We have to accept some suitable hypothesis regarding measurement errors. Suppose that all observations can be divided into groups so that within each group the observations can be considered of the same precision. It can be assumed that a group of observations at one observatory is performed by one observer using the same tool for some time, when the conditions of the observations do not change. First, we refine the parameters, setting all wi = 1. After refinement for each observation, the value ξi will characterize the agreement of the theory with the observations. Assuming that observation errors dominate theory errors, we consider ξi as observation errors. Now, for each group, by the assumption of equal-precision observations, we calculate the root-mean-square value of all ξi belonging to the group. We denote it by σk , where k is the group number. Now for each observation, we can set the weight as follows: 1 √ wi = , σk where k is the number of the group to which observation number i belongs. By refining the parameters with the assigned weights, we can recalculate all observation errors and reassign the weights. Such an iteration can be done 2–3 times. As a result, we obtain the purpose of the weights in accordance with the accuracy of the observations, and the covariance matrix of the observation errors close to the real one. The least-squares method makes it possible to evaluate the accuracy of the obtained values of the specified parameters. Section 6.1 explains how to do this. In the case of weighted conditional equations, the sequence of actions is the same as without assigning weights. The matrix of normal equations and the vector of the right-hand sides are found by Eqs. (6.63) and (6.64). Additionally, the value is calculated of d0 =

m  i=1

wi ξi2 .

(6.65)

233

234

Chapter 6 Construction of models for the motions of celestial bodies based on observations

We calculate σ02 =

 1  p) , d0 − (d m−n

(6.66)

where  p is the obtained correction vector, and the components of the vector d are determined by Eq. (6.64). Now we multiply all the elements of the matrix L−1 by σ02 . The matrix thus obtained is D = σ02 L−1 ;

(6.67)

as its diagonal elements, it will have error squares σk of the corrections pk (k = 1, 2, ..., n), respectively.

6.6 Calculation of statistical characteristics of residuals In the description of the method of differential refinement of the parameters of motion of celestial bodies from observations, the quantities δi (i = 1, 2, ..., m) appear, called residuals. They are defined by Eqs. (6.12). Here m is the number of measurements. After a successful refinement of the parameters, when the corrections to them become negligible, the residuals turn out to be equal to the deviations ξi , which characterize the agreement between the theory and observations. In fact, these values contain observation errors and errors in the model of motion of a celestial body. In practice, they are always interested in the statistical characteristics of the residuals ξi . They provide useful information for further improving both the motion model and the observation methods. First, the arithmetic mean of the residuals is analyzed 1 ξi . m m

ξ =

i=1

A significant difference between this value and zero indicates the presence of a systematic error in the results of observations, which can be caused by imperfection of observational instruments or by an incorrect procedure for preliminary processing of observations. The proximity of ξ to zero does not yet indicate the overall quality of the observations. Next, we analyze the mean-square deviations of ξi . It is determined by the above formula (6.27)  σ=

d0 , m

(6.68)

Chapter 6 Construction of models for the motions of celestial bodies based on observations

in which the value d0 should be calculated according to Eq. (6.22). This characteristic includes both systematic and random observation errors. Both accurate and rough observations can be contained in the set of used observations. In this case, rough observations with large residuals will make the main contribution to the sum (6.22). If we apply observation weights, then the characteristic σ will not correspond to the way the data is used. Then we can analyze the so-called weighted average value of the residuals σ w , which is calculated by the same formula (6.68), but into which the value d0 , determined by Eq. (6.65), is substituted. In order to separate the systematic component of residuals from the random, the so-called unbiased dispersion of random residuals D(ξ ) is calculated, which is determined by the formula 1  (ξi − ξ )2 . m−1 m

D(ξ ) =

i=1

The square root of the variance, σ=



D(ξ ),

(6.69)

is called the standard deviation. This value characterizes the random component of the observation errors. In practical algorithms, it turns out to be irrational to store all the discrepancies ξi (i = 1, 2, ...m) in memory, then to calculate the statistical characteristics of ξ , σ , σ . It is especially difficult to do this with a very large number of observations. In the process of computing, observations are scanned sequentially one after another. The memory allocated for processing one observation is then used to process the next. In this process, recurrence relations can be used for the required characteristics. Suppose we computed ξ , σ , and σ for k observations. Denote the obtained values (k) by ξ , σ (k) , and σ (k) , respectively. When adding another observation with the number k + 1, that is, the discrepancies ξk+1 , new, (k+1) , σ (k+1) , σ (k+1) the following values of the characteristics ξ can be found from the recurrence relations ξ

(k+1)

= 

σ (k+1) =

1 (k) (k ξ + ξk+1 ), k+1 1 2 (k σ (k) + ξk+1 2 ), k+1

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σ

(k+1)

   2 1 1  2 (k) k σ (k) − k ξ + ξk+1 + ξk+1 2 . = k k+1

6.7 The problem of rejecting rough observations The presence of the observation errors creates difficulties in determining the parameters of the satellite motion based on observations. Moreover, the problem is not so much the presence of errors, but the uncertainty of their properties. Most often we have no information about this. It remains only to accept certain hypotheses. Among the specific errors of the observations, there may appear significant ones that are caused not by a real measurement error, but by an outstanding rare event that is unusual for the whole series of observations. Recognizing observations with such errors is almost impossible. Many researchers do the following. They specify the level of probability that the magnitude of the observation error is within certain limits. Then they find limits on the mean-square value of the residuals of the observations σ . After calculating σ for all observations, we set some factor κ. Further, all observations for which the residuals exceed κσ are discarded. Most often they take κ = 3. This ensures a certain level of probability that errors are enclosed within (−κσ, κσ ). This probability also depends on the number of specified parameters involved in the process. After such a rejection of observations, we can again calculate σ and repeat the rejection process. In this case, new observations may be discarded. When repeating the process several times, there is no guaranty that at some stage there will be no new discarded observations. If such a process stopped (new observations were not discarded), then there is no certainty that precisely those gross observations whose errors are exceptional atypical events were discarded. In practice, the factor κ is chosen between 3 and 6. The “Error Theory” has long been developed. This is a scientific discipline, which aims to determine the most reliable measurement results in experimental sciences. It can be considered an appropriate application of the statistical method. The history of this science can be found in the literature (Sheynin, 2007). However, this theory does not provide an unambiguous algorithm for rejecting rough observations in the absence of data on the properties of errors. The process of rejecting observations remains uncertain. The book (Sheynin, 2007) describes in detail how the classics of mathematics reasoned and acted. For example, a remark by Gauss

Chapter 6 Construction of models for the motions of celestial bodies based on observations

is quoted here: “... if too quick to reject observations, there is a danger of exaggerating their accuracy. It seems to me that this lesson is more like doing things in life where we rarely or never have mathematical rigor and where we have to act at the best possible discretion”. So the only recommendation that exists is to “act at the best possible discretion”. In practice, experienced researchers do just that. Having some informal information about the observations, either select the suitable factor κ in the above method, or set some limit σlim and discard all observations for which the discrepancy exceeds σlim . We have nothing more to recommend.

References Avdyushev, V.A., Ban’shikova, M.A., 2008. Determination of the orbits of inner Jupiter satellites. Solar System Research 42 (4), 296–318. Avdyushev, V.A., Ban’shikova, M.A., 2010. Alternative orbits of the new moons of Jupiter. Izvestiya Vuzov, Fizika 10, 27–30. In Russian. Avdyushev, V.A., 2015. Numerical Modelling the Celestial Body Motion. Izdatelsky Dom Tomsk State University, Tomsk. In Russian. Bakhshiyan, B.T., Nazirov, R.R., Elyasberg, P.E., 1980. Definition and Correction of Movement. Nauka, Moscow. 359 pp. In Russian. Gubanov, V., 1997. Generalized Least Squares Method. Nauka, St.-Petersbourg. ISBN 5-02-024860-6. 318 pp. In Russian. Eliasberg, P.E., 1976. Motion Determination Based on Measurement Results. Nauka, Moscow. ISBN 978-5-397-05719-6. 416 pp. New edition 2017. In Russian. Emel’ianov, N.V., 1983. Metody sostavleniia algoritmov i programm V zadachakh nebesnoi mekhaniki. Nauka, Glav. red. fiziko-matematicheskoi lit-ry, Moscow. 128 pp. In Russian. Shchigolev, B.M., 1969. Mathematical Processing of Observations. Nauka, Moscow. 344 pp. In Russian. Sheynin, O.B., 2007. Istoria Teorii Oshibok, Berlin. 141 pp. In Russian.

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7 Obtaining astrometric data from observations of mutual occultations and eclipses of planetary satellites 7.1 Description of phenomena The technique of observing celestial bodies is constantly being improved. Product excellence depends on the development of technology in the world. At the same time, astronomers, applying special knowledge, are looking for new, more advanced methods of observation. At some stage of the discovery and study of the natural satellites of the planets, it was discovered that the main satellites of Jupiter, Saturn, and Uranus move in orbits located almost in the same plane, coinciding with the equatorial plane of the planet, but inclined to the plane of its orbit. In this case, twice during the period of revolution of the planet around the Sun, a line connecting the planet with the Earth falls into this plane. Around the same epochs, the planet–Sun line passes through the plane of the satellite’s orbits. At some time points, the images of the disks of some pairs of satellites visible from the Earth are intersected. In the same epochs, the shadow from one satellite sometimes falls on another one, and this shadow is visible from the Earth. In both cases, the total brightness of the satellites temporarily decreases. This decrease in luminous flux can be measured even if we do not distinguish between satellite disk images. The so-called mutual occultations and eclipses of the planet’s satellites occur. The Sun–planet–satellites configuration in cases of such mutual phenomena is shown in Fig. 7.1. The duration of the drop in the brightness of satellites in most cases is from 4 to 15 minutes. Periods of this phenomena lasting 6–9 months are repeated after half a revolution of the planet around the Sun. The years in which the mutual occultations and eclipses of the main satellites occur are as follows. Satellites of Jupiter: ..., 1997, 2003, 2009, 2015, 2021, .... Saturn’s satellites: ..., 1995, 2009, 2025, ... Uranus’s satellites: ..., 1965, 2007, 2049, .... From 1 to 10 events per week occur. Each pheThe Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00012-1 Copyright © 2021 Elsevier Inc. All rights reserved.

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Figure 7.1. The configuration scheme of the heliocentric orbits of the Earth, the planet and the orbits of its satellites, in which mutual occultations and eclipses of satellites occur.

nomenon can be observed simultaneously only at 30% of groundbased observatories. To describe the phenomena, we introduce the concept of the plane of the phenomenon. This is a plane passing through an occulted or eclipsed satellite perpendicular to the beam directed to this satellite from the observer in the case of mutual occultation, or from the center of the Sun in the case of mutual eclipse. The position of the occulting or eclipsing satellite is projected onto this plane. The coordinates of this projection are determined in some coordinate system X, Y on the plane of the phenomenon. The origin is placed in an occulted or eclipsed satellite. The direction of the Y axis is chosen to the Celestial North Pole, and the X axis is chosen to the east. Obviously, the luminous flux coming from the satellites during the phenomenon depends on their apparent relative position, measured by the relative coordinates X, Y in the plane of the phenomenon. Figs. 7.2 and 7.3 show the decrease in the normalized luminous flux from satellites when the coordinates in the plane of the phenomenon change during mutual occultation or eclipse. In the process of photometric observation of satellites at a number of time points, the luminous flux from satellites is measured. The measured light curve is obtained. An example of the result of such an observation is shown in Fig. 7.4. Since the flux decay during the phenomenon depends on the coordinates of the satellites, it is possible to solve the inverse problem of obtaining astrometric data from the measured light curves of the satellites during their mutual occultations and eclipses. This method of observation has several significant advantages. First, the ratio of the speed of apparent motion to the photometric accuracy is such that the accuracy of the obtained astrometric results is

Chapter 7 Mutual occultations and eclipses

Figure 7.2. Three configurations of the mutual apparent arrangement of satellite disks during mutual occultation and the corresponding total light curve.

Figure 7.3. The dependence of the normalized luminous flux coming from a pair of satellites during the mutual occultation, on time.

several times higher than the accuracy of ordinary astrometric observations. Secondly, the necessary photometric observations can be carried out by modest means. The main satellites of the planets are bright enough, so very powerful telescopes are not needed. No star catalogs are needed. Image processing in order to obtain photometric data is performed by known procedures. For these reasons, photometric observations of the mutual occultations and eclipses of the main satellites of the planets can be performed by both professional observers and amateur astronomers. Each phenomenon occurs for a short time and is observed simultaneously only from a part of ground-based points. The participation of amateur observers in observation campaigns is very useful. The third advantage is that regular observations of various

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Figure 7.4. An example of the normalized flux of light from the satellite Io during its eclipse by another satellite and the corresponding model curve after specifying the model parameters.

types extend the time interval of satellite observations and thereby improve their ephemerides for the future.

7.2 Method for obtaining astrometric data A method for deriving astrometric data from photometry of mutual occultations and eclipses of planetary satellites was proposed as early as the 70s of the 20th century in Aksnes and Franklin (1976); Aksnes et al. (1984) and was subsequently developed in Vasundhara (1994); Noyelles et al. (2003). In this method, some simplifications of the phenomenon’s model have been adopted. Another, original method for processing photometric observations and obtaining astrometric results was developed in Emelianov (2003); Emelyanov and Gilbert (2006). Let us consider the basic principles of methods for obtaining astrometric data from photometric observations of satellites during their mutual occultations and eclipses. We will rely on the approach adopted in the work (Emelianov, 2003). The problem is formulated as follows. During the phenomenon under consideration, the flux of light from satellites is measured at a number of time moments. Using this data, we must determine the difference in the coordinates of two satellites at any one time moment during the phenomenon.

Chapter 7 Mutual occultations and eclipses

With satellite photometry, we obtain the measured flux of light E in a certain scale of values fixed for each given phenomenon. We do not know this scale in advance, but we do not need to know the absolute value of the flow. It can be measured accurate to any uncertain factor. We denote by S such a normalized flux of light from satellites that before its beginning and immediately after the end this value is equal to unity. We assume that these two values coincide. In the process of mutual occultation or eclipse, the flux of light decreases and becomes S < 1. Then we can establish the relation E = K S,

(7.1)

where K is some indefinite coefficient, which is assumed to be constant during the phenomenon. Obviously, S depends on the relative coordinates of X, Y in the system described above, and we define the function S(X, Y ). The coordinates X, Y can be calculated at any time t using the ephemerides of the planet and satellites. Denote these ephemeris values by Xth (t) and Yth (t). If we substitute them into the function S(X, Y ), and then substitute the function into Eq. (7.1), then we will not get the real value of the flux E due to the fact that the ephemerides have an error. Suppose that during the phenomenon the true values of the coordinates differ from the ephemerides by some constant Dx and Dy so that the real flux is determined from the relation   E = K S Xth (t) + Dx , Yth (t) + Dy . Let us assume that photometric observations have been carried out, i.e. at moments ti (i = 1, 2, ..., m) the measured values of Ei are obtained. Then we can write a system of conditional equations   Ei = K S Xth (ti ) + Dx , Yth (ti ) + Dy , i = 1, 2, ..., m, relatively unknown parameters K, Dx , and Dy . We linearize the function S with respect to its arguments and solve the system of linear conditional equations using the least squares method. After the solution is found, the astrometric result is expressed by the coordinates X(t ∗ ) = Xth (t ∗ ) + Dx and Y (t ∗ ) = Yth (t ∗ ) + Dy , where t ∗ is any moment during phenomenon. For definiteness, we choose as t ∗ the moment when the value of X 2 + Y 2 takes a minimum value, i.e. the apparent distance between satellites is minimal. The coordinates X(t ∗ ) and Y (t ∗ ) at the time point t ∗ no longer depend on the ephemerides used and are in the best way consistent with the results of all photometric measurements performed during the phenomenon.

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It is natural to assume that when there is no light from satellites, the measured value of E should be zero. To this end, we try to exclude, if possible, the background level of the sky or any instrumental light fluxes during the photometric processing of observations. However, in practice this is not entirely clear, and some level of the background P remains in the measurements. Then we must solve the conditional equations, composed as follows:   Ei = K S Xth (ti ) + Dx , Yth (ti ) + Dy + P (i = 1, 2, ..., m), and include the background level P among the defined parameters. However, the solution of such extended equations succeeds only in rare special cases of mutual apparent satellite motions. To implement the method, we need to be able to calculate S(X, Y ) as a function of its arguments. The flux of light comes from each point of the satellite and is summed in the photodetector. Each point on the surface of a satellite is characterized by its ability to scatter light. At all points, the direction of light incidence from the Sun and the direction of reflected light to the observer turn out to be different from the satellite surface. Of course, from the point obscured by the occulted satellite, the light does not come. Whether the point is occulted or not depends on the relative position of the satellites and the observer. During mutual eclipse of satellites, the light entering each point of the eclipsed satellite is formed by the sum of the fluxes from all points of that part of the solar disk that is not obscured by the eclipsing satellite. It should be borne in mind that there is a darkening of the disk of the Sun to the edge. If the telescope allowed us to distinguish between the satellite disks, then we would see a partially occulted or darkened satellite disk with uneven brightness and a dark edge due to the fact that the Sun illuminates the satellite a little from the side. In real observations, the occulted and occulting satellites give one common light in the photodetector. It is their total flux that is measured. Any used photodetector has a different sensitivity to the light of various wavelengths. Therefore, it is necessary to take into account the dependence of light scattering on the wavelength and the characteristics of the applied filter. In practical calculations, we divide the satellite hemisphere facing the Earth into finite elements, calculate the incoming flux from each element separately, and summarize all these fluxes. At this stage, one or another law of light scattering by a point on the surface of a satellite can be used. It is necessary to know a number of parameters that specify the light reflecting properties of the surface of a particular satellite. One of them is the satellite albedo distributed over the surface, taking into account the details

Chapter 7 Mutual occultations and eclipses

Figure 7.5. Mutual occultation of satellites. The uncovered portion of the satellite being occulted is located at the bottom right. The share of this part of the entire disk is characterized by the value k2 .

of the surface. Several parameters are considered here. In sum, all these circumstances make up the photometric model of the phenomenon. One of the most accurate photometric models of the mutual occultations and eclipses of Jupiter’s Galilean satellites is described in Emelianov (2003); Emelyanov and Gilbert (2006). Similar models for the main satellites of Saturn and Uranus are given in Arlot et al., (2012; 2013). Some simplified version of the model of mutual occultations and eclipses of satellites is considered below.

7.3 A simplified model of mutual occultations and eclipses of planetary satellites Consider the model of the mutual occultation of two satellites. In a simplified version, we will assume that there are homogeneous disks of two satellites. At the same time, the integral albedos on the disks are different. We denote them by p1 for the occulting satellite and p2 for the occulted satellite. The apparent radii of the satellites are denoted by r1 and r2 , respectively. Radii can be measured in arcseconds as disks are visible in the sky, or in kilometers in the plane of the phenomenon. We denote by d the distance between the centers of the disks, measured in the same units as the radii of the satellites. If d ≥ r1 +r2 , then no phenomenon is observed, and we get light from the full disks of both satellites. If disk occultation occurs, then we see only part of the disk of the satellite being occulted. Denote by k2 the visible fraction of the apparent disk of the satellite being occulted. Naturally, k2 depends on the distance between the centers of the disks d. This can be seen in Fig. 7.5. Thus, outside the phenomenon k2 = 1. If the distance between the centers of the disks is d ≤ |r1 − r2 |, then the disks completely

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overlap. Moreover, if r1 ≥ r2 , then the occulted satellite is not visible at all, and we have k2 = 0. In the case of r1 < r2 , the apparent fraction of the satellite’s disk is k2 = (r22 − r12 )/r22 . In the case of partial mutual occultation, when r1 + r2 > d > |r1 − r2 |, the value k2 can be calculated using the sequence of formulas    2  h = 2d 2 r12 + r22 − d 4 − r12 − r22 , a1 = d 2 − r22 + r12 , a2 = d 2 − r12 + r22 , tan ϕ1 =

h h , tan ϕ2 = , a1 a2

  1 r12 h k2 = 1 − ϕ1 + ϕ2 − 2 . π r22 2r2 Here, when calculating the angles ϕ1 and ϕ2 from the values of their tangents, it should be taken into account that the sign cos ϕ1 coincides with the sign a1 , and the sign cos ϕ2 coincides with the sign a2 . These formulas were obtained in Emel’yanov (1995). Another type of formulas for calculating the coefficient k2 was proposed in Assafin et al. (2009), namely cos α1 =

k2 = 1 −

r12 − r22 + d 2 r 2 − r12 + d 2 , cos α2 = 2 , 2r1 d 2r2 d

  1 r12 . (α − sin α cos α ) + α − sin α cos α 1 1 1 2 2 2 π r22

The angles α1 and α2 here should be calculated from the values of their tangents, given that the sines of these angles are positive. In the problems of processing photometric observations of mutual eclipces and eclipses of satellites in order to obtain astrometric data, one has to calculate the derivative of k2 with respect dk2 to d, i.e. d(d) . This can be done using the following chain of formulas:  2d  2 r1 + r22 − d 2 , h = h ϕ1 =

a1 h − 2hd a2 h − 2hd  , ϕ = , 2 a12 + h2 a22 + h2

1 dk2 =− d(d) π



r12

ϕ1 r22

+ ϕ2

 h − 2 . 2r2

Chapter 7 Mutual occultations and eclipses

Figure 7.6. Full occultation of one satellite by another over a period of time (t1 , t2 ) and the corresponding section on the curve of the total normalized flux from a pair of satellites.

Now consider how to calculate the normalized luminous flux. The function S with such a simplified model of phenomena depends on k2 , which in turn depends on d. The flux Rr12 p1 comes from the occulting satellite, and the flux Rr22 p2 k2 arises from the occulted one, where r1 and r2 are the radii of the disks, p1 and p2 are the satellite albedos, and R is the indefinite coefficient of proportionality. With mutual occultation, the flux from both satellites is always measured together. Under these conditions, the normalized flux S can be expressed by the formula

S(d) =

p1 r12 + p2 r22 k2 (d) p1 r12 + p2 r22

=

1+

p2 r22 k2 (d) p1 r12

1+

p2 r22 p1 r12

.

Outside the phenomenon, when there is no occultation, the entire occulted satellite is visible, and we have k2 = 1, S = 1, and E = K. Sometimes full occultations also happen. Fig. 7.6 shows the satellite configuration in this case. It is clear that in the time interval (t1 , t2 ) the occulted satellite is not visible at all, and k2 = 0. In this period we have S(d) =

1 1+

p2 r22 p1 r12

,

that is, the flux does not depend on the mutual distance of the satellites d. As can be seen from the simplified consideration of the photometric model, the normalized flux from satellites depends on

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the ratio of integral albedos over the surface. This dependence remains in more accurate photometric models. In processing the observations of the Galilean satellites of Jupiter (Emelyanov, 2009; Arlot et al., 2012), we average the apparent photometric properties of the satellite. At this stage, it is possible to take into account variations in the integral albedo value, which depend on the rotation angle of the satellite. More simply, the flux of light from a satellite depends on how sideways it faces us. We take data on variations in the integral albedo from publications (Morrison and Morrison, 1977; Prokof’eva-Mikhailovskaya et al., 2010; Abramenko et al., 2011). These data are not accurate enough. They can be a source of error in the photometric model of mutual occultations and eclipses of the Galilean satellites of Jupiter. The need for wide photometric observations of the Galilean satellites of Jupiter for different values of the angles of rotation of satellites from 0 to 360 degrees in different spectral bands is obvious.

7.4 Photometric models of mutual occultations and eclipses of planetary satellites 7.4.1 General photometric characteristics The simplified model of mutual occultations and eclipses of planetary satellites considered above does not provide the accuracy of astrometric data that can be achieved with available photometric observations. The most accurate model can be constructed in the following way. The Sun and satellites are extended objects. Each point on the surface has its own properties. Light emission and scattering depend on these properties. The signal that is measured in the photodetector is formed by the light that has entered it, but it also depends on the properties of the photodetector itself. Let us consider these processes in more detail. We will need general concepts and quantities that characterize the properties of light scattering at a given point on the surface, and other quantities needed to describe the integrated photometry of the satellite’s apparent disk. The law of light scattering is the dependence of the intensity of light scattered at a given point in the direction of the observer on parameters describing the circumstances of the phenomenon and the properties of the scattering surface. The magnitude of the considered intensity is usually normalized so that the integral from it over the entire apparent disk of the satellite divided by the disk

Chapter 7 Mutual occultations and eclipses

area is equal to the geometric albedo of the satellite, provided that the observer is in the direction of the light source. The amount of light scattered at a given point in the direction of the observer is called the scattering function. Denote this function by f . Light scattering is described by the following parameters: the angle of incidence of light i is the angle between the normal to the surface and the direction of light propagation from the source to the surface point, the angle of reflection e is the angle between the normal to the surface and direction from the observer to the scattering point, the phase angle α is the angle with the vertex at the point under consideration between the beams directed to the source and the observer, as well as the azimuthal angle  between the projections of these beams on the plane tangent to the surface at this point. Naturally, the intensity of the scattered light depends on the wavelength of light , since different areas of the satellite’s surface have different colors. The light scattering function has the general form f (ϕ, λ, i, e, , α, ), where the arguments ϕ and λ are introduced—the cartographic coordinates of a point on the satellite’s surface. To take into account the dependence on ϕ and λ, images from spacecraft can be used, provided that the images were taken at the same angles of the solar phase that would be observed from Earth. The geometric albedo p and the phase function (α), which is the ratio of the light coming from the satellite at a given phase angle to the incoming light at a phase angle equal to zero, are considered as integral characteristics of the reflective properties of the satellite. The phase function takes into account the integrated effect of uneven brightness of the satellite’s apparent disk and a decrease in the area of the illuminated part of the disk. The integral characteristics also depend on the wavelength of light . They are different for the various bands of the spectrum.

7.4.2 Photometric model of the mutual occultation of satellites Fig. 7.7 will help us here. It depicts a photometry scheme of the mutual occultation of two satellites. Light from the entire disk of the Sun is radiated towards the satellites. The intensity of the flux depends on the wavelength . The dependence is determined by the function F (), known from the physics of the Sun (Makarova et al., 1998). The light in each infinitesimal spectral band (,  + d) goes to the satellites and enters at some point of the occulted satellite. At this point, light is scattered in all directions, including towards the observer. The flux intensity depends on the properties of the surface point and the circum-

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Figure 7.7. Photometry of the mutual occultation of satellites.

stances of the incidence and reflection of the beam. In fact, the flux arriving at a point from the Sun needs to be multiplied by some function f (ϕ, λ, i, e, , α, ) to get the value of the flux going to the observer. Of course, the flux is proportional to the area of the infinitesimal disk element of the occulted satellite dS2 . Part of the function arguments are explained above. There are also cartographic coordinates of the point ϕ, λ on the satellite. The function f (ϕ, λ, i, e, , α, ), introduced above, is also called the scattering law. Defining this function is an extremely difficult problem. The surface can be glossy or porous, with minute pits and humps. Different scattering laws will be considered below. Now let us further trace the path of light. Light entering the photodetector causes its reaction: it sends a signal to the computer. The magnitude of this signal depends on the photosensitivity of the receiver, primarily on the wavelength. Here we must not forget that various filters can be used. The spectral sensitivity function of the photodetector is denoted by (). Then the magnitude of the signal from the photodetector will be obtained from the magnitude of the incoming flux by multiplying by (). We cannot measure the flux from each point on the surface of the satellite separately. For this, the image resolution is usually not enough. We can only measure the total flow from a pair of satellites: occulted and occulting ones. The total flux from the satellite being occulted is determined by the integral 2 () f (ϕ, λ, i, e, , α, ) F () dS2 d, G2 (X, Y ) = S2 1

where integration is carried out over the entire part of the satellite S2 uncovered and illuminated by the Sun, which depends on the

Chapter 7 Mutual occultations and eclipses

mutual coordinates of the satellites (X, Y ), and over the spectral band (1 , 2 ). The total flux from the occulting satellite is determined by a similar integral (0) G1

2 = () f (ϕ, λ, i, e, , α, ) F () dS1 d,

(7.2)

S1 1

where integration is carried out over the entire part of the satellite S1 illuminated by the Sun and visible from the Earth. The superscript 0 here indicates that this satellite is not occulted by anything. The integration limits 1 and 2 are taken as the limits of the spectral sensitivity of the photodetector. Of course, in practice, the detailed characteristics of the satellites are set in the form of tables, and instead of the integrals, the final sums are calculated for the small sections into which the satellite disks are divided. To determine the normalized luminous flux S, the total luminous flux from the occulted satellite G(0) 2 is necessary here under the assumption that it is not occulted by anything. This flow is determined in the same way as for G2 , but integration is carried out over the entire surface of the satellite being illuminated by the Sun and visible from the Earth. As a result, the value of the normalized flux S is determined from the relation S(X, Y ) =

G(0) 1 + G2 (X, Y ) (0)

(0)

G1 + G2

.

7.4.3 Photometric model of satellite eclipse During this phenomenon, the observed satellite is not illuminated by the full disk of the Sun. Part of the disk is obscured by an occulting satellite. The scheme of the phenomenon can be seen in Fig. 7.8. Some unscreened point on the solar disk radiates light toward the satellite. The intensity of this light depends both on the wavelength and on the location of the point on the disk, since the disk of the Sun has a darkening to the edge. This darkening is different in different parts of the spectrum. Therefore, the flux from the point of the solar disk will be determined by the value I (r, )dS0 d, where r is the apparent distance of the point from the center of the disk, dS0 is the infinitesimal element of the solar disk, and d is the infinitely small spectral bandwidth. The function I (r, ) in the

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form of tables can be found in publications on the physics of the Sun. We can use, for example, reference (Makarova et al., 1998). The light from the point of the solar disk, following to the eclipsed satellite, falls on its surface, in particular, at some point having reflective properties described by the scattering law f (ϕ, λ, i, e, α, ). The scattered light flux will be proportional to the area of the satellite disk element dS2 . As a result, we have the flux following the observatory f (ϕ, λ, i, e, , α, ) dS2 I (r, ) dS0 d. Reaching the photodetector, this light will cause a signal with a magnitude of () f (ϕ, λ, i, e, , α, ) dS2 I (r, ) dS0 d. We measure the flux from the eclipsed satellite, which is determined by the integral 2 G2 (X, Y ) =

() f (ϕ, λ, i, e, , α, ) I (r, ) dS0 dS2 d,

1 S2 S0

where integration is performed over the unshielded part of the solar disk S0 , the part of the satellite disk S2 illuminated by the Sun and visible from the Earth, and the spectral sensitivity band of the photodetector (1 , 2 ). The Sun and both satellites during a mutual eclipse are located almost on the same line. The Earth observer sees the eclipsed satellite a little from the side. What part of the solar disk is not obscured by the eclipse satellite, directly depends on the relative position of the satellites as they would be visible from the Sun. This relative position is determined by the relative heliocentric coordinates X, Y . Therefore, the total flux from the eclipsed satellite depends on X, Y . When observing planetary satellites during their mutual eclipses, an eclipsing satellite can also fall into the field of view along with the eclipse. The angular distance between the satellites may be so small that it may not be possible to measure the flux separately from the eclipsed satellite. In other cases, the flux from the eclipsed satellite can be measured separately. (0) The total flux from the occulting satellite G1 is determined by an integral similar to (7.2). However, in this case, the superscript 0 here indicates that the light comes to this satellite from the full disk of the Sun.

Chapter 7 Mutual occultations and eclipses

Figure 7.8. Photometry of the mutual eclipse of satellites.

To determine the normalized flux S, the flux from the eclipsed (0) satellite G2 is also necessary under the assumption that the Sun is not occulted by it. This flux is determined in the same way as for G2 , but integration is carried out through the full solar disk. Now the value of the normalized flux S is determined from the relation (0)

S(X, Y ) =

G1 + G2 (X, Y ) (0)

(0)

G1 + G2

,

if the flux of light from both satellites is measured together, or from the ratio G2 (X, Y ) S(X, Y ) = , (0) G2 if light comes from the eclipsed satellite only.

7.5 The laws of light scattering for planetary satellites 7.5.1 Lommel–Seeliger light scattering law In many studies of the characteristics and geometric structure of the surfaces of planets and satellites, the Lommel–Seeliger light scattering law is applied. Often it is used for rough surfaces with low or medium albedo. The Lommel–Seeliger law is described by the scattering function 2 cos i . f (i, e) = p cos i + cos e

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It depends only on the angles of incidence and reflection of light. At a zero phase angle, it is obvious that i = e, then f (i, e) = p, and we see a uniform satellite disk. At other phase angles, the highest brightness of the disk should be observed near the limb, and the darkening of the disk to the terminator will be visible. The phase function in the case of the Lommel–Seeliger law is determined by the expression (α) = 1 − sin

α α α tan ln cot . 2 2 2

(7.3)

The Lommel–Seeliger law can be applied only in very approximate models of phenomena. In more accurate models, only the general form of the dependence of scattering on the angles of incidence and reflection is used. Formulas are more complex.

7.5.2 Hapke’s light scattering law for a smooth surface Hapke’s laws, for various properties of the scattering surface, take into account the partial absorption of light and its multiple reflection by particles inside a certain layer of the satellite’s body. In the case of a perfectly smooth matte surface, Hapke’s law is described by the following function (Hapke, 1981): f (i, e, α) =

ω μ0 × 4 μ0 + μ



× 1 + B(B0 , α, h) P (α, g) + H (μ, ω)H (μ0 , ω) − 1 ,

where μ0 = cos i , μ = cos e , B(B0 , α, h) =

P (α, g) =

B0 , 1 + tan(α/2)/ h

1 − g2 , (1 + g 2 + 2g cos α)3/2

H (x, ω) =

1 + 2x . √ 1 + 2x 1 − ω

The parameters ω, B0 , h, g characterize the surface properties. These parameters in the literature have the following names: ω is the mean simple scattering albedo, B0 is the amplitude of the

Chapter 7 Mutual occultations and eclipses

opposition effect, h is the angular half-width of the opposition effect, and g is the asymmetry parameter. The parameter h is related to the porosity of the surface. The light scattering function in this law does not contain the wavelength  as its argument. However, the dependence on  is still taken into account, since the law parameters can be taken to be different for different spectral bands. The geometric albedo of the satellite in this case is obtained by the formula p= where

1 1 ω (1 + B0 )P (0, g) − 1 + r0 + r02 , 8 2 6 √ 1−ω r0 = . √ 1− 1+ω 1−

(7.4)

(7.5)

The phase function (α) is determined from the relation 

  1 ω  p (α) = 1 + B(B0 , α, h) P (α, g) − 1 + r0 (1 − r0 ) × 8 2   2 2r α α α + 0 sin α + (π − α) cos α . × 1 − sin tan ln cot 2 2 4 3π (7.6)

7.5.3 Hapke’s light scattering law for a rough surface In the case of a rough surface, Hapke’s law has the form (Hapke, 1984) f (i, e, α, ) =

ω μ0 × 4 μ0 + μ    

× 1 + B(B0 , α, h) P (α, g) + H μ , ω H μ0 , ω − 1 × × S(i, e, ) ,

where μ0 , μ , S(i, e, ) are functions of i, e,  and another parameter θ , called the mean slope of macroscopic surface roughness. These functions are calculated by different formulas in the cases i ≤ e and i > e. For i ≤ e we sequentially calculate μ (i, e, ) = C(θ )×   exp(− π1 cot2 θ cot2 e) − sin2 2 exp(− π1 cot2 θ cot2 i) , × μ + sin e tan θ 2 2 − exp(− π2 cot θ cot e) −  π exp(− π cot θ cot i)

255

256

Chapter 7 Mutual occultations and eclipses

μ0 (i, e, ) = C(θ ) [μ0 + + sin i tan θ

cos  exp(− π1 cot2 θ cot2 e) + sin2 2 − exp(− π2 cot θ cot e) −

μ0 μ (i, e, ) C(θ )×   μ (i, e, 0) μ0 (i, e, 0)  −1  μ0 × 1 − f () 1 − C(θ )  . μ0 (i, e, 0)

 π

 2

exp(− π1 cot2 θ cot2 i)

 ,

exp(− π2 cot θ cot i)

S(i, e, ) =

For i > e we have μ (i, e, ) = C(θ ) [μ+ + sin e tan θ

cos  exp(− π1 cot2 θ cot2 i) + sin2 2 − exp(− π2 cot θ cot i) −

 π

 2

exp(− π1 cot2 θ cot2 e)

exp(− π2 cot θ cot e)

μ0 (i, e, ) = C(θ ) [μ0 + + sin i tan θ

exp(− π1 cot2 θ cot2 i) − sin2 2 − exp(− π2 cot θ cot i) −

 2  π



exp(− π1 cot2 θ cot2 e) exp(− π2 cot θ cot e)

 ,

μ (i, e, ) μ0 C(θ )× μ (i, e, 0) μ0 (i, e, 0)  −1  μ × 1 − f () 1 − C(θ )  . μ (i, e, 0)

S(i, e, ) =

Two new functions are introduced in these formulas:   1  , f () = exp −2 tan . C(θ ) =  2 1 + π tan2 θ The phase function and geometric albedo, which in the case of a rough surface we denote by r (α) and pr , are calculated as follows. First, from Eqs. (7.4), (7.5), and (7.6) we determine (α) and p, then we find r (α) and pr by the formulas r (α) = K(α, θ )(α) ,   1 1 2 pr = p + C(ω, θ ) − 1 r0 + r0 . 2 6 Here the functions K(α, θ ) and C(ω, θ ) can be represented by empirical expressions, 

K(α, θ ) = exp −0.32 θ tan θ tan(α/2) − 0.52 θ tan θ tan(α/2) ,

,

Chapter 7 Mutual occultations and eclipses

Table 7.1 Parameters of Hapke’s photometric function for satellite Io. Filter

ω

h

B0

θ , degrees

g

V

0.94 0.028 ± 0.01 0.80 ± 0.2 −0.30 ± 0.06

30 ± 10

OR

0.96 0.028 ± 0.01 0.55 ± 0.4 −0.32 ± 0.04

30 ± 10

BL

0.90 0.028 ± 0.01 0.60 ± 0.4 −0.26 ± 0.04

30 ± 10

VI

0.67 0.028 ± 0.01 0.65 ± 0.4 −0.22 ± 0.04

30 ± 10

UV

0.33 0.028 ± 0.01 0.60 ± 0.4 −0.24 ± 0.04

30 ± 10

  2 2 C(ω, θ ) = 1 − 0.048 θ + 0.0041 θ r0 − 0.33 θ − 0.0049 θ r02 , where θ is expressed in radians.

7.5.4 Hapke’s law parameters for the Galilean satellites of Jupiter In the literature, we found only two papers in which complete sets of Hapke parameters for the Galilean satellites of Jupiter are published. The first of them (McEwen et al., 1988) gives the parameters of the Hapke model of a rough surface for the Io satellite in several versions obtained using ultraviolet (UV), violet (VI), blue (BL), orange (OR) filters and for V spectral band. The parameters were averaged over the entire surface of the satellite. These data are reproduced in Table 7.1. The second paper (Domingue and Verbiscer, 1997) is devoted to the improvement of the Hapke function for rough surfaces. It is proposed to replace the function P (α, g) with its more universal variants: the two-parameter function P2 (α, c2 , b2 ) or the threeparameter function P3 (α, c3 , b3 , d3 ). These functions have the form P2 (α, c2 , b2 ) = P3 (α, c3 , b3 , d3 ) =

(1 − c2 )(1 − b22 ) (1 + 2b2 cos α + b22 )3/2 (1 − c3 )(1 − b32 ) (1 + 2b3 cos α + b32 )3/2

+ +

c2 (1 − b22 ) (1 − 2b2 cos α + b22 )3/2 c3 (1 − d32 ) (1 + 2d3 cos α + d32 )3/2

, .

The parameters c2 , b2 and the parameters c3 , b3 , d3 together with other parameters of the Hapke function were determined for the Europe, Ganymede and Callisto satellites based on groundbased and space observations. The data were obtained separately for two wavelengths,  = 0.47 µm and  = 0.55 µm, for the two

257

258

Chapter 7 Mutual occultations and eclipses

Table 7.2 Hapke photometric parameters using the function P2 (α, c2 , b2 ). 

L/T

w

B0

h

θ ,◦

c2

b2

Europe 0.47 µm

L

0.922 0.50 0.0016 10.0

0.431

0.921

0.55 µm

L

0.964 0.50 0.0016 10.0

0.429

0.887

0.47 µm

T

0.897 0.45 0.0016 10.0

0.43

0.713

0.55 µm

T

0.940 0.50 0.0016 10.0 Ganymede

0.443

0.609

0.47 µm

L

0.830 0.62

0.003

28.0

0.282

0.960

0.55 µm

L

0.945 0.86

0.004

29.0

0.380

0.427

0.47 µm

T

0.870 1.00

0.074

35.0

0.039

0.989

0.55 µm

T

0.810 0.23 0.074 Callisto

35.0

0.307

0.962

0.47 µm

L

0.740 1.00

0.031

42.0

0.729

0.024

0.55 µm

L

0.540 1.00

0.031

42.0

0.132

0.949

0.47 µm

T

0.470 0.27 0.0277 42.0

0.432

0.542

0.55 µm

T

0.550 0.73 0.0277 42.0

0.206

0.958

hemispheres of the satellite: leading (L) and the trailing (T) ones. Values are given in Table 7.2 and Table 7.3. It was the Hapke photometric function with the above parameters that was used to obtain astrometric results on photometry of the Galilean satellites of Jupiter in (Emelyanov and Gilbert, 2006; Emelyanov, 2008; Emel’yanov and Vashkov’yak, 2009; Emelyanov, 2009; Arlot et al., 2014; Saquet et al., 2018).

7.6 Disk-integrated photometric characteristics of the satellite The reflective properties of different points on the surface of the satellite are different. The scattering function at each point also depends on the wavelength. In the scattering functions considered above, the surface characteristics were averaged either over the hemisphere or over the entire surface. The dependence of the scattering function on the wavelength is taken into account by substituting different parameters for different spectral bands.

Chapter 7 Mutual occultations and eclipses

Table 7.3 Hapke photometric parameters using the function P3 (α, c3 , b3 , d3 ). L/T



w

B0

h

θ,◦

c3

b3

d3

8.0

Europe L

0.934

0.49

0.0015

0.770

0.780

−0.459

0.55 µm

L

0.964

0.43

0.0016 10.0 0.726

0.945

−0.416

0.47 µm

T

0.897

0.51

0.0016 11.0

0.691

−0.417

0.55 µm

T

0.930 0.521 0.0016 11.0 0.083 Ganymede

0.784

−0.386

0.47 µm

L

0.830

0.63

0.003

28.0 0.090

0.960

−0.280

0.55 µm

L

0.930

0.91

0.003

28.0 0.200

0.450

−0.380

35.0

0.47 µm

0.0

0.47 µm

T

0.870

1.00

0.074

0.820

−0.046

0.55 µm

T

0.810

0.23

0.074 35.0 0.350 Callisto

0.970

−0.305

0.47 µm

L

0.510

0.82

0.031

42.0

0.033

−0.694

0.55 µm

L

0.605

1.00

0.031

42.0 0.005

0.015

−0.687

0.47 µm

T

0.530

0.50

0.0277 42.0 0.787

0.489

−0.400

0.55 µm

T

0.650

0.87

0.0277 42.0 0.988

0.618

−0.238

0.0

0.0

There are other ways to approximate the scattering function. One suitable method is as follows. We take some light scattering law, for example, Hapke’s law. The parameters included in this law are taken averages over the entire surface of the satellite and we relate them to a specific wavelength. The accepted function can take into account the dependence on the phase angle of the Sun. It is well known that the main satellites of the planets rotate almost synchronously with their orbital motion. The axis of rotation of each satellite with high accuracy is perpendicular to the plane of its orbit. In addition, these axes deviate little from the plane perpendicular to the line of sight of the Earth’s observer. Therefore, changes in the integral brightness of the satellite substantially depend on the rotation angle of the satellite with respect to the observer. Put more simply, the flux of light from a satellite depends on how sideways it faces us. However, this dependence may be different for different wavelengths and for different angles of the solar phase. Multiply the scattering function by some function A(θ, α, ),

259

260

Chapter 7 Mutual occultations and eclipses

Figure 7.9. Dependence of the relative magnitude of the flux of light coming from the Jovian satellite Io on the rotation angle θ. The line is plotted according to Morrison and Morrison (1977), and the dots show the results of Abramenko et al. (2011).

where θ is the rotation angle of the satellite relative to the observer, α is the angle of the solar phase, and  is the wavelength. The resulting function will approximately take into account both the local properties of light scattering and the disk-integrated properties. In work on this topic, the rotation angle is usually measured so that its value is zero in the superior conjunction, 90◦ in the eastern elongation, 180◦ in the inferior conjunction, and 270◦ in the western elongation. Such a determination of the rotation angle fully coincides with the determination of the rotation angle in the report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2015 (Archinal et al., 2018). The dependence of the satellite brightness on the rotation angle can be obtained from ground-based photometry of satellites. Such data for the Galilean satellites of Jupiter for the V-spectral band are published in Morrison and Morrison (1977). Only in 2011, repeated photometry of these satellites was performed in order to obtain the dependence of the brightness of the satellites on

Chapter 7 Mutual occultations and eclipses

Figure 7.10. Dependence of the relative magnitude of the flux of light coming from the Jovian satellite Europe on the rotation angle θ. The line is plotted according to Morrison and Morrison (1977), and the dots show the results of Abramenko et al. (2011).

the rotation angle (Prokof’eva-Mikhailovskaya et al., 2010; Abramenko et al., 2011). The data are based on observations in the V -spectral band and are reduced to a 6-degree angle of the solar phase. The measured dependences of the flux on the rotation angle for the Galilean satellites of Jupiter are shown in Figs. 7.9, 7.10, 7.11, and 7.12. These data are not accurate enough. The analysis by the author shows that the inaccuracy of knowledge of this dependence turns out to be one of the main sources of error in the photometric model of mutual occultations and eclipses of the Galilean satellites of Jupiter. The need for wide photometric observations of the Galilean satellites of Jupiter is obvious for different values of the rotation angles of satellites from 0 to 360 degrees in different spectral bands and at different angles of the solar phase in order to obtain refined tables for the function A(θ, α, ). The dependence of the light flux on the rotation angle taken from Morrison and Morrison (1977) with some modifications based on the work of (Abramenko et al. 2011; Prokof’evaMikhailovskaya et al., 2010) was used to obtain astrometric results

261

262

Chapter 7 Mutual occultations and eclipses

Figure 7.11. Dependence of the relative magnitude of the light flux coming from the Jovian satellite Ganymede on the rotation angle θ. The line is plotted according to Morrison and Morrison (1977), and the dots show the results of Prokof’eva-Mikhailovskaya et al. (2010).

from the photometry of the Galilean satellites of Jupiter in work of (Arlot et al., 2014; Saquet et al., 2018).

7.7 Photometric models of mutual occultations and eclipses of the main satellites of Saturn and Uranus 7.7.1 Photometric model of mutual occultations and eclipses of the main satellites of Saturn Once in 14 years, the eight main satellites of Saturn show mutual occultations and eclipses for an Earth observer. It occurred in 2009. The parameters of the Hapke law of light scattering for the main satellites of Saturn have not yet been obtained. The properties of Saturn’s satellites and the circumstances of mutual occultations and eclipses led to the need to construct a special photometric model. Such a model was developed (Arlot et al., 2012) based

Chapter 7 Mutual occultations and eclipses

Figure 7.12. Dependence of the relative magnitude of the light flux coming from Jovian satellite Callisto on the rotation angle θ. The line is plotted according to Morrison and Morrison (1977), and the dots show the results of Prokof’eva-Mikhailovskaya et al. (2010).

on photometric studies of the main satellites of Saturn published in Buratti (1984); Buratti and Veverka (1984); Buratti et al. (1998). The modified Lommel–Seeliger light scattering law was used. Local and integral surface characteristics were combined. Some parameters in this model were assumed to be common to the entire surface of the satellite. However, a coefficient was included in the model that takes into account the dependence of the satellite brightness on the rotation angle. As a result, the developed photometric model defines a light scattering function depending on the angle of incidence i, the reflection angle e, the angle of the solar phase α, and the satellite rotation angle θ . The function is described by the following chain of formulas:  pl −0.4 m μ0 10 f (α) + (1 − A)μ0 , f (i, e, α, θ ) = A μ + μ0 p 

where μ0 = cos i, μ = cos e,

263

264

Chapter 7 Mutual occultations and eclipses

f (α) =

(α)p − q , D

2 Af (0) p = (1 − A) + , 3 2 q=

2 (1 − A) sin α + (π − α) cos α , 3π

  Af (0) α α α D= 1 − sin tan ln cot , 2 2 2 4 (α) = 10−0.4βα , m = −

AR cos(θ − θ0 ). 2

In these formulas, A, f (0), β, pl , AR , and θ0 are the parameters related to the satellite. Their values are given in Table 7.4 for the five main satellites of Saturn. The parameter values are taken from Buratti (1984); Buratti and Veverka (1984); Buratti et al. (1998), with the exception of the parameters AR and θ0 for the satellites S2 Enceladus, S3 Tefiya, S4 Diona, and S5 Reya, which were taken from Kulyk (2008). For the other three main satellites, these parameters were not needed, since only the first five satellites participated in all the phenomena that were observed in the 2009 epoch. The photometric parameters of the satellites were obtained in Buratti (1984); Buratti and Veverka (1984); Buratti et al. (1998) for a wavelength of 0.9 µm. It was also shown that the light-reflecting properties of satellites slightly differ in the V , R, and I spectral bands. Since most of the observations were made in the R and I bands, the dependence of the scattering function on the wavelength was not taken into account. Using the developed photometric model of the mutual occultations and eclipses of main Saturnian satellites developed in Arlot et al. (2012), astrometric results were obtained based on the results of the 2009 worldwide observation campaign. 26 satellite light curves were processed.

7.7.2 Photometric model of mutual occultations and eclipses of the main satellites of Uranus In 2007, mutual occultations and eclipses of the main satellites of Uranus took place. These events occur once every 42 years. As a result of the worldwide observation campaign, 41 satellite light curves were obtained during the observed phenomena. Arlot et

Chapter 7 Mutual occultations and eclipses

265

Table 7.4 Parameters for the photometric model of mutual occultations and eclipses of Saturn’s satellites in 2009. Satellite

A

f (0)

S1 Mimas

0.70

1.10

0.021 0.720 0.100

270

S2 Enceladus 0.40

2.40

0.017 1.040 0.060

270

β

pl

AR

θ0 , deg

S3 Tefiya

0.70

1.45

0.016 0.830 0.070

90

S4 Dione

1.00

1.00

0.023 0.625 0.364

103

S5 Rhea

0.95

1.10

0.024 0.705 0.210

100

al. (2013) reports on the development of a special photometric model of these phenomena to obtain astrometric results. All observations of the worldwide campaign were processed using this model. New astrometric results were obtained. The law was based on the Lommel–Seeliger law to take into account the distribution of brightness over the disk. The factor A(α, ) was chosen as a factor for the light scattering function, taking into account the dependence of the integral brightness of the satellite on the angle of the solar phase α and on the wavelength . The type of function was taken from Karkoschka (2001): A(α, ) = A0 1 + γ ( − 0.55) × 10−0.4(βα+0.5α/(α0 +α)) , where  is measured in µm, and A0 , γ , β, and α0 are the photometric parameters of the satellite. The parameter values were again found by the least-squares method in Arlot et al. (2013) based on photometric data published in Karkoschka (2001). This satellite photometry was performed using the Hubble Space Telescope.

7.8 Relation for the accuracy of astrometric results of observations of various types Different methods of positional observations of planetary satellites give different accuracy of astrometric data. Observations have been carried out for a very long time. Published papers provide differing accuracy estimates. Based on numerous data, only a few expert opinions can be given. Table 7.5 gives approximate accuracy estimates of the astrometric data. The estimates are obtained from the analysis of random observation errors. They show

266

Chapter 7 Mutual occultations and eclipses

Table 7.5 The accuracy of astrometric data obtained from observations of various types. Internal estimates of errors caused by random observation errors are given. Accuracy in arcseconds refers to ground-based observations. Estimates of the accuracy of the position of the satellites are given taking into account the topocentric distance of the planet (in kilometers). Type of observations

Accuracy in arcseconds in kilometers

Galilean satellites of Jupiter Eclipses of satellite by Jupiter

0.150

450

Photographic plates

0.100

300

Meridian circle

0.060

180

CCD cameras

0.040

120

Mutual phenomena

0.015 The main satellites of Saturn

45

Photographic plates

0.100

600

Meridian circle

0.030

200

CCD cameras

0.030

200

0.005 The main satellites of Uranus

30

CCD cameras

0.040

400

Mutual phenomena

0.006

60

Mutual phenomena

the relationship between the accuracy of observations of various types.

7.9 Worldwide campaigns on observations of satellites during their mutual occultations and eclipses Mutual occultations and eclipses of the Galilean satellites of Jupiter occur every 6 years for about 9 months. Given that each event lasts only a few minutes, only 30 per cent of all such phenomena can be observed at a given observatory, implying the need for international campaigns of photometric observations. About 400 phenomena occur during each epoch of mutual occultations and eclipses of the satellites of Jupiter, Saturn, and Uranus

Chapter 7 Mutual occultations and eclipses

within 9–14 months. Each phenomenon lasts 5–15 minutes and is observed only on a small part of the Earth facing the planet. In order to observe as many phenomena as possible, it is necessary to organize worldwide campaigns for photometric observations of satellites. Institut de Mécanique céleste et de calcul des éphémérides (IMCCE, France) has coordinated an international campaigns since 1985. Within the framework of these campaigns, all results of the photometric observations after each campaign have been collected and stored in a common data base. The astrometric processing of all data is carried out after a time after the observations. The obtained relative coordinates of satellites form a database of astrometric results of the observation campaign. Papers describing the obtained light curves of satellites and the final astrometric results are published two or three years after the observations. All observers participating in the observations become co-authors of these publications. In some cases, astrometric results were published separately. Astrometric processing of the results of the observation campaign was carried out, as a rule, by one researcher using his own method. Table 7.6 gives the characteristics of the worldwide observation campaigns for the satellites of Jupiter, Saturn, and Uranus during their mutual occultations and eclipses. At the end of this chapter, a schedule of future seasons is given when new observation campaigns can be conducted.

7.10 Obstacles to improving the accuracy of astrometric results There are serious unresolved problems in processing photometric observations of the mutual phenomena of satellites in order to obtain the most accurate astrometric data. There are two sources of errors in the resulting astrometric coordinates of satellites: random photometry errors and the inaccuracy of the photometric model. The analysis shows that the errors caused by the inaccuracy of the model are three or four times greater than the errors due to random photometry errors. To make it easier to understand the reasons, we take a simplified photometric model of the mutual occultation of satellites, discussed above. In this model homogeneous disks of satellites appear. During mutual occultation, we always measure the light flux from both satellites together. Under these conditions, the normalized value of the light flux S can be expressed by the following

267

268

Chapter 7 Mutual occultations and eclipses

Table 7.6 Characteristics of the worldwide observation campaigns of the satellites of Jupiter, Saturn, and Uranus during their mutual occultations and eclipses. The first column shows a planet with a satellite system and years of observation. Designations: Nc is the number of obtained light curves of satellites and No is the number of participating observatories. System, epoch of observation

Nc No Authors of methods and references to publications

Jupiter, 1973

46

18 Aksnes K. (Aksnes et al., 1984)

Jupiter, 1979

19

11 Aksnes K. (Aksnes et al., 1984)

Saturn, 1979–1980

14

6 Aksnes K. (Aksnes et al., 1984)

Jupiter, 1985

166

28 Arlot J.-E. (Arlot et al., 1992) 56 Arlot J.-E. (Arlot et al., 1997)

Jupiter, 1991

374

Saturn, 1995

66

Jupiter, 1997

292

16 Noyelles B. (Noyelles et al., 2003) 42 Vasundhara R. (Vasundhara et al., 2003; Arlot et al., 2006) Emelyanov N.V., Vashkovyak S.N. (Emel’yanov and Vashkov’yak, 2009)

Jupiter, 2002–2003

377

42 Emelyanov N.V. (Emelyanov, (Arlot et al., 2009)

Uranus, 2007

41

19 Emelyanov N.V. (Arlot et al., 2013)

Jupiter, 2009

457

2009);

74 Emelyanov N.V., Varfolomeev M.I. (Arlot et al., 2014)

Jupiter, 2015

609

75 Emelyanov N.V. (Saquet et al., 2018)

formula: S(d) =

1+

p2 r22 k2 (d) p1 r12

1+

p2 r22 p1 r12

,

where r1 and r2 are the radii of the apparent disks of the satellites, p1 and p2 are the albedos of the occulting and occulted satellites, respectively. The coefficient k2 determines the visible portion of

Chapter 7 Mutual occultations and eclipses

Figure 7.13. Full occultation of one satellite by another over a period of time (t1 , t2 ) and the corresponding part on the light curve of the total normalized flux from a pair of satellites.

the apparent disk of the occulted satellite, which depends on the mutual distance of the satellites d. Beyond the phenomenon, when there is no occultation and the occulted satellite is visible, we have k2 = 1 and S = 1. In the case of full occultation, the satellite configuration is shown in Fig. 7.13. It is clear that in the time interval (t1 , t2 ) the occulted satellite is not seen at all, and k2 = 0. In this period we have S=

1 1+

p2 r22 p1 r12

,

that is, the light flux does not depend on the mutual distance of the satellites d. The problem arises because in many cases the observed value of the flux during full occultation is not equal to the calculated value, i.e. Eobserved = K

1 1+

p2 r22 p1 r12

,

where K is equal to the flux beyond the phenomenon. Figs. 7.14, 7.15, and 7.16 show examples of such situations. Here we present the values of Eobserved /K obtained from the measured values of the flux (points) and model changes of S. These figures demonstrate that the additional flux is present in the measured values during full occultation, and this flux is negative. There are two ways to correct the model. The first way is to set Eobserved = K

1 1+

p2 r22 p1 r12

+ P,

269

270

Chapter 7 Mutual occultations and eclipses

Figure 7.14. Example of the light curve for the measured normalized flux from the satellite Io during its full occultation by another satellite and the corresponding model curve after refining the model parameters. The observation performed on November 2, 2014. On the horizontal axis, we showed the time in minutes.

where P is the spurious light flux from an unrecorded background. The second way is to set Eobserved = K

1 1+m

p2 r22 p1 r12

,

where m is some additional factor that appears due to the fact that we inaccurately know the ratio of the satellites’ albedo, and corrects this inaccuracy. There is a dilemma which method we should choose among two methods. Equivalence of methods leads to equality K

1 1+

p2 r22 p1 r12

+P =K

1 1+m

p2 r22 p1 r12

.

The fact that in most such cases the spurious flux in the observations is negative suggests its real presence, rather than the influence of inaccurate knowledge of satellite albedo.

Chapter 7 Mutual occultations and eclipses

Figure 7.15. Example of the light curve for the measured normalized flux from the satellite Io during its full occultation by another satellite and the corresponding model curve after refining the model parameters. The observation performed on December 28, 2014. On the horizontal axis, we showed the time in minutes counted from December 29, 2014. A negative background level in the measured flux is clearly visible here.

When processing partial mutual occultations of satellites, we do not suspect the presence of a spurious background in the measurements and do not know about the inaccuracy of the accepted values of the satellites’ albedos. Therefore, we have to add the false correction to the mutual apparent distance between the satellites to match the model with the observations, which leads to systematic errors in astrometric results. This fact is illustrated by the following relation:

Eobserved = K

1+

p2 r22 k2 (d+ ) p1 r12

1+

p2 r22 p1 r12

.

Here, in our consideration, for a better understanding of the situation, we simplified the photometric model of the phenomenon. However, the problem is also reproduced in our processing of observations using the perfect model described in Emelianov (2003); Emelyanov and Gilbert (2006). The same prob-

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Figure 7.16. Example of the light curve for the measured normalized flux from the satellite Io during its full occultation by another satellite and the corresponding model curve after refining the model parameters. The observation performed on March 3, 2015. On the horizontal axis, we showed the time in minutes counted from March 3, 2015. A negative background level in the measured flux is clearly visible here.

lem arises when processing observations of mutual eclipses of satellites. We are forced to look for the sources of the above errors. Spurious light can get on the photodetector from a spurious sky background. It can be light scattered through a telescope and camera. A spurious background can be created by the photodetector itself. The calculation of the light flux from satellites from their images on CCD frames is performed by one of the special methods of photometric processing. The error of this method can give some level of background. Only this source can produce a negative background level. Therefore, such a source is the most likely cause of the situation. It is in the method of photometric processing that we should look for the source of spurious background. References to descriptions of existing photometric image processing methods would take up too much space in this book. We restrict ourselves to listing the names of the methods that appear in the explanations that accompany the data received in IMCCE: Source Extractor, DAOPHOT (IDL), Audela, Tangra, and LiMovie. Different ob-

Chapter 7 Mutual occultations and eclipses

servatories use their own special methods. Obviously, a revision of the methods used is necessary to determine the sources of the systematic errors. As we can see from the above simplified consideration of the photometric model, the normalized light flux from satellites depends on the ratio of the surface-integral albedos. This dependence remains in more accurate photometric models. In our processing of observations of the Galilean satellites of Jupiter (Emelyanov, 2009; Arlot et al., 2012), we average the apparent photometric properties of the satellite, but take into account variations in the integral albedo, depending on the rotation angle of the satellite. More simply, the light flux from a satellite depends on how sideways it faces us. We use data on variations in the integral albedo from publications (Morrison and Morrison, 1977; Prokof’eva-Mikhailovskaya et al., 2010; Abramenko et al., 2011). These data are not accurate enough. They can be a source of error in the photometric model of mutual occultations and eclipses of the Galilean satellites of Jupiter. The need for intensive photometric observations of the Galilean satellites of Jupiter for different values of the angles of rotation of the satellites from 0 to 360 degrees in different spectral bands is obvious. A special processing of such observations will give refined dependences of the light flux on the angle of rotation of the satellite.

7.11 Periods of the phenomena in the future The author of this book has calculated the ephemerides of mutual occultations and eclipses of planetary satellites for a period of time up to 2027 inclusive. Since the phenomena are repeated every six years for the satellites of Jupiter, in the future they should be expected only by 2021. The schedule of mutual occultations and eclipses of the Galilean satellites of Jupiter and the main satellites of Saturn is as follows: – mutual occultations and eclipses of the Galilean satellites of Jupiter in 2021, – eclipses of the inner satellites of Jupiter by the Galilean satellites in 2021, – mutual occultations and eclipses of the Galilean satellites of Jupiter in 2026–2027, – eclipses of the inner satellites of Jupiter by the Galilean satellites in 2026–2027, and – mutual occultations and eclipses of the main satellites of Saturn in 2024–2026. The ephemerides of all these phenomena with detailed descriptions of circumstances, the moments of the beginning and end of

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each event, and other characteristics can be found on the Internet at the sites of the Sternberg Astronomical Institute of Moscow State University and the Institut de Mécanique céleste et de calcul des éphémérides (IMCCE) at http://www.sai.msu.ru/neb/nss/html/multisat/nssephmr.htm http://nsdb.imcce.fr/multisat/nssephmr.htm. Ephemerides were calculated using the software tool “Search for mutual occultations and eclipses of satellites and eclipses by the planet”, available on the same sites. This tool can be used to calculate the ephemerides of phenomena after 2027. Probably, in future periods of mutual occultations and eclipses of planetary satellites, worldwide observation campaigns will be conducted. Previous experience has shown the high effectiveness of such campaigns to produce new highly accurate positional data. A review of past achievements reached in this way and projected plans for the future are given in Arlot and Emelyanov (2019).

References Abramenko, A.N., Baida, G.V., Zakrevskii, A.V., Karachkina, L.G., Prokof’eva-Mikhailovskaya, V.V., Sergeeva, E.A., 2011. Photometry of Io and Europa at the Crimean Astrophysical Observatory and Reasons for Differences between Ground-Based and Space Observations. Bulletin of the Crimean Astrophysical Observatory 107, 113–121. Aksnes, K., Franklin, F., 1976. Mutual phenomena of the Galilean satellites in 1973. III – Final results from 91 light curves. Astronomical Journal 81, 464–481. Aksnes, K., Franklin, F., Millis, R., Birch, P., Blanco, C., Catalano, S., Piironen, J., 1984. Mutual phenomena of the Galilean and Saturnian satellites in 1973 and 1979/1980. Astronomical Journal 89, 280–288. Archinal, B.A., Acton, C.H., A’hearn, M.F., Conrad, A., Consolmagno, G.J., Duxbury, T., Hestroffer, D., Hilton, J.L., Kirk, R.L., Klioner, S.A., McCarthy, D., Meech, K., Oberst, J., Ping, J., Seidelmann, P.K., Tholen, D.J., Thomas, P.C., Williams, I.P., 2018. Report of the IAU working group on cartographic coordinates and rotational elements: 2015. Celestial Mechanics and Dynamical Astronomy 130, 22. Arlot, J.E., Thuillot, W., Barroso Jr., J., Bergeal, L., Blanco, C., Boninsegna, R., Bouchet, P., Briot, D., Bulder, H., Bourgeois, J., 1992. A catalogue of the observations of the mutual phenomena of the Galilean satellites of Jupiter made in 1985 during the PHEMU85 campaign. Astronomy and Astrophysics Supplement Series 92, 151–205. Arlot, J.E., Ruatti, C., Thuillot, W., Arsenijevic, J., Baptista, R., Barroso Jr., J., Bauer, C., Berthier, J., Blanco, C., Bouchet, P., et al., 1997. A catalogue of the observations of the mutual phenomena of the Galilean satellites made in 1991 during the PHEMU91 campaign. Astronomy and Astrophysics Supplement Series 125, 399–405. Arlot, J.-E., Thuillot, W., Ruatti, C., et al., 2006. The PHEMU97 catalogue of observations of the mutual phenomena of the Galilean satellites of Jupiter. Astronomy and Astrophysics 451, 733–737.

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Arlot, J.-E., Thuillot, W., Ruatti, C., Ahmad, A., Amosse, A., Anbazhagan, P., Andreyev, M., Antov, A., Appakutty, M., Asher, D., et al., 2009. The PHEMU03 catalogue of observations of the mutual phenomena of the Galilean satellites of Jupiter. Astronomy and Astrophysics 493, 1171–1182. Arlot, J.-E., Emelyanov, N.V., Lainey, V., Andreev, M., Assafin, M., Braga-Ribas, F., Camargo, J.I.B., Casas, R., Christou, A., Colas, F., Da Silva Neto, D.N., Dechambre, O., Dias-Oliveira, A., Dourneau, G., Farmakopoulos, A., Gault, D., George, T., Gorshanov, D.L., Herald, D., Kozlov, V., Kurenya, A., Le Campion, J.F., Lecacheux, J., Loader, B., Massalle, A., Mc Brien, M., Murphy, A., Parakhin, N., Roman-Lopes, A., Schnabel, C., Sergeev, A., Tsamis, V., Valdes Sada, P., Vieira-Martins, R., Zhang, X., 2012. Astrometric results of observations of mutual occultations and eclipses of the Saturnian satellites in 2009. Astronomy and Astrophysics 544, A29. 7 pp. Arlot, J.-E., Emelyanov, N.V., Aslan, Z., Assafin, M., Bel, J., Bhatt, B.C., Braga-Ribas, F., Camargo, J.I.B., Casas, R., Colas, F., Coliac, J.F., Dumas, C., Ellington, C.K., Forne, E., Frappa, E., Khamitov, I., Miller, C., Modic, R.J., Sahu, Dk., Sicardy, B., Tanga, P., Valdes Sada, P., Vasundhara, R., Vieira-Martins, R., 2013. Astrometric results of observations of mutual occultations and eclipses of the Uranian satellites in 2007. Astronomy and Astrophysics 557, A4. 6 pp. Arlot, J.-E., Emelyanov, N., Varfolomeev, M.I., Amosse, A., Arena, C., Assafin, M., Barbieri, L., Bolzoni, S., Bragas-Ribas, F., Camargo, J.I.B., Casarramona, F., Casas, R., Christou, A., Colas, F., Collard, A., Combe, S., Constantinescu, M., et al., 2014. The PHEMU09 catalogue and astrometric results of the observations of the mutual occultations and eclipses of the Galilean satellites of Jupiter made in 2009. Astronomy and Astrophysics 572, A120. 9 pp. Arlot, J.-E., Emelyanov, N., 2019. Natural satellites mutual phenomena observations: achievements and future. Planetary and Space Science 169, 70–77. Assafin, M., Vieira-Martins, R., Braga-Ribas, F., Camargo, J.I.B., da Silva Andrei, A. H, Neto, D.N., 2009. Observations and analysis of mutual events between the Uranus main satellites. Astronomical Journal 137, 4046–4053. Buratti, B., 1984. Voyager disk resolved photometry of the Saturnian satellites. Icarus 59, 392–405. Buratti, B., Veverka, J., 1984. Voyager photometry of Rhea, Dione, Tethys, Enceladus and Mimas. Icarus 58, 254–264. Buratti, B.J., Mosher, J.A., Nicholson, P.D., McGhee, C.A., French, R.G. Near-Infrared, 1998. Photometry of the Saturnian satellites during ring plane crossing. Icarus 136, 223–231. Domingue, D., Verbiscer, A., 1997. Re-analysis of the solar phase curves of the icy Galilean satellites. Icarus 128, 49–74. Emel’yanov, N.V., 1995. Features of mutual occultations and eclipses in the system of Saturn’s satellites. Astronomy Reports 39 (4), 539–542. Emelianov, N.V., 2003. A method for reducing photometric observations of mutual occultations and eclipses of planetary satellites. Solar System Research 37 (4), 314–325. Emelyanov, N.V., Gilbert, R., 2006. Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2003. Astronomy and Astrophysics 453, 1141–1149. Emelyanov, N.V., 2008. Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002–2003. Planetary and Space Science 56, 1785–1790. Emel’yanov, N.V., Vashkov’yak, S.N., 2009. Mutual occultations and eclipses of the Galilean satellites of Jupiter in 1997: astrometric results of observations. Solar System Research 43 (3), 240–252.

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Emelyanov, N.V., 2009. Mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002-2003: final astrometric results. Monthly Notices of the Royal Astronomical Society 394, 1037–1044. Hapke, B., 1981. Bidirectional reflectance spectroscopy: 1 theory. Journal of Geophysical Research 86, 3039–3054. Hapke, B., 1984. Bidirectional reflectance spectroscopy: 3. correction for macroscopic roughness. Icarus 59, 41–59. Karkoschka, E., 2001. Comprehensive photometry of the rings and 16 satellites of Uranus with the Hubble space telescope. Icarus 151, 51–68. Kulyk, I., 2008. Saturnian icy satellites: disk-integrated observations of the brightness opposition surge at low phase angles. Planetary and Space Science 56, 386–397. Makarova, E.A., Kharitonov, A.V., Kazachevskaya, T.V., et al., 1998. Observable characteristics of solar radiation (revised tables). Baltic Astronomy 7, 467–494. McEwen, A.S., Johnson, T.V., Matson, D.L., 1988. The global distribution, abundance and stability of SO2 on Io. Icarus 75, 450–478. Morrison, D., Morrison, N.D., 1977. Photometry of the Galilean satellites. In: Planetary Satellites. University of Arizona Press, Tucson, pp. 363–378. Noyelles, B., Vienne, A., Descamps, P., 2003. Astrometric reduction of lightcurves observed during the PHESAT95 campaign of Saturnian satellites. Astronomy and Astrophysics 401, 1159–1175. Prokof’eva-Mikhailovskaya, V.V., Abramenko, A.N., Baida, G.V., Zakrevskii, A.V., Karachkina, L.G., Sergeeva, E.A., Zhuzhulina, E.A., 2010. On the cause of the discrepancy between groundbased and spaceborne LIghtcurves of Ganymede and Callisto in the V band. Bulletin of the Crimean Astrophysical Observatory 106, 68–81. Saquet, E., Emelyanov, N., Robert, V., Arlot, J.-E., Anbazhagan, P., Baillie, K., Bardecker, J., Berezhnoy, A.A., et al., 2018. The PHEMU15 catalogue and astrometric results of the Jupiter’s Galilean satellite mutual occultation and eclipse observations made in 2014–2015. Monthly Notices of the Royal Astronomical Society 474 (4), 4730–4739. Vasundhara, R., 1994. Mutual phenomena of the Galilean satellites: an analysis of the 1991 observations from VBO. Astronomy and Astrophysics 281, 565–575. Vasundhara, R., Arlot, J.E., Lainey, V., Thuillot, W., 2003. Astrometry from mutual events of Jovian satellites in 1997. Astronomy and Astrophysics 410, 337–341.

8 Estimation of the accuracy of planetary satellite ephemeris 8.1 Factors determining ephemeris accuracy First of all, we note that by ephemeris we mean the satellite coordinates calculated at a given time point. We also include a coordinate calculator in the concept of ephemeris. The basis of the ephemeris is always observations. The creation of ephemeris based on observations is done according to the scheme and methods described in previous chapters. The answer to the question of the ephemeris accuracy is not quite simple. Note that people using ephemeris do not always ask themselves such a question. Ephemerides published in astronomical yearbooks are not provided with accuracy data. It is often supposed that the annual data are completely accurate, and only the accuracy of what we compare with the ephemeris is questioned. In fact, the accuracy of the ephemeris is limited and in many cases is unknown. Let us consider the causes of errors in calculating the coordinates of satellites. Errors are supposed at different stages of ephemeris creation. First of all, we have observation errors arising from inaccurate measurements. Furthermore, the data that we consider as the results of observations are actually obtained with some processing of the measured values. At this stage, certain errors are introduced, caused by the inaccuracy of the processing method. Then a satellite motion model is constructed. If the differential equations of motion are numerically integrated, a computational integration error occurs. In constructing the analytical theory of motion, we take a limited number of terms in the expansion of the solution in powers of various small parameters, and discard the remaining terms. In this way, the error of the theory is introduced. To connect the measured values with the coordinates of the satellite, a certain model is created in which certain simplifications are accepted that generate additional errors. The dependence of the measured quantities on the motion parameters can be close to degenerate cases, when the same proximity of the theoretical positions of the satellite to the real observables provides a whole family of possible values of the parameters. In these sitThe Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00013-3 Copyright © 2021 Elsevier Inc. All rights reserved.

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uations, the reliability of assessing the accuracy of an ephemeris becomes low. The orbital motion of celestial bodies has one feature. The motion occurs near a certain plane. In the projection of the trajectory onto this plane, the radius vector rotates with an almost constant angular velocity. The angle between the radius vector and some fixed direction λ is called orbital longitude. This angle monotonously and almost uniformly increases with time, constantly “spinning up” turns. Suppose that over a certain time interval (t1 , t2 ), observations are made from which we can determine the values of λ at these extreme time points with some error σλ . From these values it is possible to determine the ephemeris value of orbital longitude at some point tf with some error λ . The properties of the orbital motion allow us to derive an approximate formula for the dependence of the accuracy of the ephemeris λ on the time intervals between the initial t1 and final t2 moments of observation and the moment tf for which the ephemeris is calculated. This formula has the form λ =

2 tf − t1 +t 2 σλ . t2 − t1

(8.1)

Obviously, we cannot measure orbital longitude directly from observations. We observe only the projection of the orbital motion on the sky plane. However, the approximate formula for longitude given here allows us to draw some conclusions. The accuracy of the ephemeris is proportional to the accuracy of the observations, but also inversely proportional to the time interval of the observations. From the above analysis, it is clear what exactly needs to be done to improve the accuracy of the ephemeris. It is necessary to increase the accuracy of observations. It is necessary to continue regular observations, even with the same accuracy. Eq. (8.1) gives only a general approximate idea of the main factors on which the accuracy of the ephemeris depends. In practice, we are interested in a more detailed description of the accuracy of the ephemeris precisely for those quantities that we measure or the coordinates of the satellites that we use. In this case, statistically valid estimates are needed. Only at the end of the last and the beginning of this century, special studies of the accuracy of the ephemeris of celestial bodies were conducted. It turned out that methods that give more or less reliable and accurate estimates are rather complicated. Below we list some significant work on this problem. Muinonen and Bowell (1993) proposed a special statistical approach to the problem of determining orbits from observations. Milani (1999) proposed a

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new algorithm to determine the region of possible location of lost asteroids. Virtanen et al. (2001) studied the posterior distribution of asteroid orbits and proposed a method for searching for lost asteroids in the main belt and asteroids approaching the Earth. In reference (Desmars et al., 2009), some results were obtained for assessing the accuracy of ephemeris for the two main satellites of Saturn based on an artificially generated array of observations. Around the same time, Emelyanov (2010) investigated the accuracy of the ephemeris of all the distant satellites of major planets. In this work, three different statistical methods for assessing the accuracy of ephemeris were used. All of them are based on variations of observational data. These methods can be considered basic in this problem. We will consider them in more detail in the following sections.

8.2 Estimation of the ephemeris accuracy using observation-errors variance by the Monte Carlo techniques The approach in the proposed method is quite simple. The general scheme is as follows. We form a large number of variants of the same observations that differ from those actually performed by the set of errors that are generated by the Monte Carlo method. For each option, we again determine the orbit by which we calculate the ephemeris at a given time point. Statistical estimates of ephemeris variations among these options provide an estimate of accuracy. Let us consider the method in more detail. Suppose we have a set of real satellite observations. One is required to evaluate the accuracy of the ephemeris constructed on the basis of these observations and calculated at a given time point. The algorithm consists of the following steps. 1. We refine the motion parameters based on available observations. 2. After refining the parameters, we obtain the mean square value σ of the deviations of the measured values from those calculated on the basis of the motion model. 3. Using the parameters found, we calculate the satellite ephemeris at a given time point, that is, the coordinates whose accuracy must be determined. We will call this ephemeris the reference ephemeris. 4. Based on the found parameter values, we calculate the values of the measured coordinates at the moments of observation. We will call this set the reference observations.

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5. We create a set of artificial observations by adding random errors distributed according to the normal probability law with a given rms value σ to the reference observations. We obtain random values in a computing program using a random-number generator. We will call this set observations with error variations. 6. We determine the satellite’s motion parameters by the differential refinement method based on a set of observations with error variations. 7. Based on the obtained parameters, we calculate the ephemeris at a given time point, that is, the coordinates whose accuracy we want to determine. We calculate and store the deviations x of these coordinates from the reference ephemeris. 8. We return to step 5 with a new set of random numbers. 9. After returning to step 5 many times, we exit this cycle and do a statistical analysis of the calculated deviations of x for all variations of the observation-error variances. 10. We accept the standard deviation x as the accuracy of the ephemeris. In this method, a set of reference observations was obtained for the moments of real observations from the motion parameters previously found from real observations. This set of reference observations is considered as a variant of accurate observations. Artificially created random errors are added to the exact values of the measured quantities. A scheme of the described algorithm is shown in Fig. 8.1. The calculation of reference ephemeris is performed simultaneously with the calculation of reference observations. The operations “adding error variations to reference observations”, “refinement of parameters”, and “calculation of ephemeris” are done in a cycle many times. The calculation of the deviation from the reference ephemeris and the standard deviation is carried out simultaneously with the calculation of the ephemeris. The standard deviation is calculated using the recurrence formula given in Sect. 6.6. Upon exiting the cycle, the accuracy of the ephemeris as the standard deviation is already calculated. Evaluation of the accuracy of ephemeris can be done simultaneously at a number of time points. Then we can plot the accuracy of the ephemeris versus time. By saving the deviations of the ephemeris from the reference one at a certain time point during the calculations in a separate file, we can construct the distribution of deviations by their values. The considered algorithm has two significant features. It is necessary to carry out estimates of the accuracy of observations in

Chapter 8 Estimation of the accuracy of planetary satellite ephemeris

Figure 8.1. Scheme of the algorithm for observation-error variance. Thick lines indicate data and data streams. Thin lines mark the actions of the algorithm and transitions between operations.

order to specify the error variations. These estimates may not be entirely reliable, especially if the observations are of unequal accuracy. The second feature consists in significant computational time, since for a good statistical estimation of the accuracy of the ephemeris, it is necessary to calculate the deviation of the ephemeris from the reference one many times.

8.3 Estimation of ephemeris accuracy by varying the composition of observations using “bootstrap”-samples The general scheme of this method is similar to the scheme observation-error variances using the Monte Carlo method. We form a large number of variants of random samples from a set of real observations. Samples are made using the “bootstrap”

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method by a random-number generator. For each option, we again determine the orbit by which we calculate the ephemeris at a given time point. Statistical estimates of ephemeris variations among these options provide an estimate of accuracy. Let us consider the method in more detail. Suppose we have a set of real observations of the satellite. It is required to evaluate the accuracy of the ephemeris built on the basis of these observations and calculated at a given time point. The algorithm for solving the problem consists of the following steps. 1. We refine the motion parameters based on available observations. 2. Using the parameters found, we calculate the satellite ephemeris at a given time point, that is, the coordinates whose accuracy must be determined. We will call this ephemeris the reference ephemeris. 3. We make a random sample of observations from a set of real observations using the “bootstrap” method. We call this set the “bootstrap”-sample of observations. 4. Using the differential refinement method, we determine the satellite’s motion parameters based on the “bootstrap”-sample of observations. 5. Using the obtained parameters, we calculate the ephemeris at a given time point, that is, the coordinates whose accuracy we want to determine. We calculate and save the deviations x of these coordinates from the reference ephemeris. 6. We return to point 3 with a new set of random numbers. 7. After returning to step 3 many times, we exit this cycle and do a statistical analysis of the calculated deviations of x for all variations of the observation-error variances. 8. The standard deviation x we accept as the accuracy of the ephemeris. In this method, a set of reference observations is considered as a sample variant. In the statistical analysis of deviations of x , we first calculate the average value and the standard deviation from the average value. We should not calculate the average value, but determine these two values in the course of calculations using the recurrence relations given in Sect. 6.6. A scheme of the described algorithm is shown in Fig. 8.2. The calculation of reference ephemeris is performed simultaneously with the calculation of reference observations. The operations “formation of bootstrap sample of observations”, “refinement of parameters”, and “calculation of ephemeris” are done in a cycle many times. The calculation of the deviation from the reference ephemeris and the standard deviation is carried out simultaneously with the calculation of the ephemeris. The standard devia-

Chapter 8 Estimation of the accuracy of planetary satellite ephemeris

Figure 8.2. Scheme of the algorithm for estimating the accuracy of ephemeris using “bootstrap”-samples of observations. Thick lines indicate data and data streams. Thin lines mark the actions of the algorithm and transitions between operations.

tion is calculated using the recurrence formula given in Sect. 6.6. Upon exiting the cycle, the accuracy of the ephemeris as the standard deviation is already calculated. We can make an estimate the accuracy of ephemeris simultaneously at a number of time points. Then we can plot the accuracy of the ephemeris versus time. By saving the deviations of the ephemeris from the reference one at a certain time point during the calculations in a separate file, we can construct the distribution of deviations by their values. The considered algorithm has one advantage over the variation of observation errors. Namely, no information on the accuracy of observations is needed to implement this algorithm. Let us explain here what is a “bootstrap” sample. Suppose we have a set of any numbered elements in the amount of N pieces. Consider a random variable that can take values from 1 to N with a constant probability density. We will set this random variable N times. In each case, we obtain a certain value n of a random value in the range from 1 to N , then we select from the initial set an element with the number n. We take a copy of this element for the resultant set and put them back in the initial set. As a result, we get again N elements. However, in the resultant set, some elements from the initial set may be selected several times, and some ones

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may never be selected. This method is also called the sampling with returning. In our case, each “bootstrap”-sample of N initial satellite’s observations gives us again a set of N observations, which is involved in the algorithm for estimating the accuracy of the ephemeris. When applying the method of “bootstrap” samples of observations described here, one should take into account its specific property; we do not vary the individual observation errors, but their complete composition. Since in some samples there may be no observations at the ends of the observation time interval, a shortened interval can lead to a deterioration in the accuracy of the parameters and the accuracy of the ephemeris. The correction of this drawback of the method can be done in the following way. First, it is necessary to calculate all the observation errors, that is, the deviations of the actual observations from the reference ones obtained by calculating the ephemeris at the moments of observation according to the specified parameters. Then, in the calculations, we can do “bootstrap”-samples of not the observations themselves, but only their errors. Thus, all moments of observations will be saved. However, the errors of some observations can be attributed to others. At the same time, some observation errors can be increased several times, and some ones will be set equal to zero. This technique was tested by the author of this book and led to positive results.

8.4 Estimation of the accuracy of ephemeris by the method of motion parameter variation The main idea of this method is as follows. Based on the initial set of observations, we determine the parameters of the motion of a celestial body by the differential refinement method. This yields a column vector of the obtained parameters P0 and the corresponding covariance error matrix of the parameters D, as described in Chapter 6. The vector of defined parameters contains n elements. The matrix D has the dimension (n × n). Using the parameters P0 , we calculate the reference ephemeris. In turn, using the random-number generator, we create a variant of the parameter values P, different from P0 by some vector of random variables. Thus, we form a large number of variants of random values of the motion parameters. For each option, we calculate the ephemeris at a given time point. Statistical estimates of ephemeris variations among these options provide an estimate of the accuracy.

Chapter 8 Estimation of the accuracy of planetary satellite ephemeris

This raises the important question of how to generate a random variation of the parameters. The method, justified by probability theory, is as follows. For the well-known covariance error matrix of the parameters D, we determine the Cholesky decomposition matrix L based on the relation LLT = D. Using the random-number generator, we form a random vector η, consisting of n components. Each component should, independently of the others, take random values with a probability density distributed according to the normal probability law, with zero mathematical expectation and dispersion equal to unity. A set of random parameter values is then calculated by the formula P = P0 + Lη. Note that the matrix L turns out to be a lower triangular matrix. For its calculation, there are special programs in the wellknown mathematical software packages MATLAB , Maple, Mathematica and others. We can use a program compiled in the Cprogramming language and given in Appendix E. Let us consider in more detail the algorithm for solving this problem. Suppose we have a set of real satellite’s observations. It is required to evaluate the accuracy of the ephemeris constructed on the basis of these observations and calculated at a given time point. 1. We refine the motion parameters based on available observations. We obtain the parameter vector P0 and the corresponding covariance matrix of parameter errors D. 2. Based on the obtained parameters, we calculate the satellite’s ephemeris at a given time point, that is, its coordinates, the accuracy of which must be determined. We will call this ephemeris the reference ephemeris. 3. We find a random version of the parameter vector P as described above using a random-number generator. 4. Based on the parameters found, we calculate the ephemeris at a given time point, that is, the coordinates whose accuracy we want to determine. We calculate and remember the deviations x of these coordinates from the reference ephemeris. 5. We return to step 3 with a new set of random numbers. 6. After going to step 3 many times, we exit this cycle and perform a statistical analysis of the calculated deviations of x for all variations of the observation-error variations. 7. The standard deviation x is taken as the accuracy of the ephemeris.

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In the statistical analysis of deviations of x , the reference ephemeris is considered as an exact ephemeris, since it was calculated on the basis of a set of real observations. The root-mean-square deviations of x can be determined in the course of calculations using the recurrence relation given in Sect. 6.6. A scheme of the described algorithm is shown in Fig. 8.3. The calculation of reference ephemeris is performed simultaneously with the calculation of reference observations. The operations “calculating a variant of parameters with variations” and “calculating the ephemeris” are done in a cycle many times. The calculation of the deviation from the reference ephemeris and the standard deviation is performed using the recurrence formula simultaneously with the calculation of the ephemeris. Upon exiting the cycle, the accuracy of the ephemeris as the standard deviation is already calculated. Evaluation of the accuracy of ephemeris can be done simultaneously at a number of time points. Then we can build a plot of the dependence of accuracy on time. By saving the deviations of the ephemeris from the reference one at a certain time point during the calculations in a separate file, we can construct the distribution of deviations by their values. The considered algorithm has an advantage in comparison with other methods considered above. The computation time is significantly reduced, since the refinement of parameters from observations needs to be done only once. Another advantage is that the statistical characteristics of observation errors are automatically taken into account by using the covariance matrix of parameter errors.

8.5 The accuracy of the ephemeris of the distant satellites of major planets The distant satellites of major planets are of great interest to the dynamics of the Solar System. Knowledge of the orbits of these satellites helps to establish a picture of the evolution of the system and provides information to confirm hypotheses about the origin of satellites. Distant satellites of the planets are subject to strong perturbations from the Sun. The determination of these perturbations by analytical methods is very difficult, although such attempts have been made in past centuries. Many new distant satellites of the planets were discovered at the end of the 20th century and the beginning of the 21st century. The total number of known distant satellites has increased recently from 10 to 148. The ephemerides of these satel-

Chapter 8 Estimation of the accuracy of planetary satellite ephemeris

Figure 8.3. Scheme of the algorithm for estimating the accuracy of ephemeris using variations of parameter errors. Thick arrows indicate data and data streams. Thin arrows mark the actions of the algorithm and transitions between operations.

lites are necessary to monitor the completed and new observations. For these purposes, one must know a priori the accuracy of the ephemeris. The distances of distant satellites from the planet range from 10 to 30 million km, and the orbital periods are from 2 to 4 years. Only two distant satellites of Neptune move at a distance of 46 million km from their planet. The ephemeris of distant planetary satellites are being developed at several world scientific centers. The results of Jacobson (2000); Brozovic and Jacobson (2009) were used to produce ephemeris available on the JPL ephemeris server (Giorgini et al., 1997). Motion models and ephemerides of distant satellites of major planets were built in Emelyanov (2005, 2007); Emel’yanov and Kanter (2005). The most accurate model of the motion and ephemeris of the satellite of Saturn, Phoebe, was constructed in Desmars et al. (2013). For the distant satellite of Neptune, Nereid, a motion model is constructed in Emelyanov and Arlot (2011). All of these ephemeris are available on the MULTI-SAT planetary satellite ephemeris server (Emel’yanov and Arlot, 2008).

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Figure 8.4. Evaluations of the accuracy of the ephemeris of Jupiter’s satellite J23, Kalike, by three methods: points indicate observation-error variances, circles mark “bootstrap”-samples, and crosses show variations of parameter errors. The ratio of the observation interval and the orbital period of the satellite is 1.52.

All new satellites discovered after 1997 have a low brightness. Their magnitudes are in the range of 21–24. These satellites can be observed only with the help of very powerful telescopes. The analysis shows that, for all distant satellites and for all ground-based observatories, the accuracy of observations is mainly 0.2–0.6 arcseconds (Emelyanov, 2005; 2007; Emel’yanov and Kanter, 2005). For the first time, the accuracy of the ephemeris of all distant satellites of major planets was determined based on observations of the work (Emelyanov, 2010). In this work, the methods are described in detail and the results of the estimates are given. For many of the distant satellites, the orbits were refined based on a longer series of observations compared to those used in previously published work. It turned out that, for distant satellites discovered at the beginning and middle of the 20th century, the accuracy of the ephemeris remains at the level of 0.06 arcseconds until 2020. For other satellites, the accuracy of the ephemeris is much worse and drops sharply with a decrease in the observation time interval. There are satellites for which the observation time interval is 30 days, which is 0.04–0.07 of the orbital period. For some of these

Chapter 8 Estimation of the accuracy of planetary satellite ephemeris

Figure 8.5. Evaluations of the accuracy of the ephemeris of Jupiter’s satellite J31, Etne, by three methods: points indicate observation-error variances, circles mark “bootstrap”-samples, and crosses show variations of parameter errors. The ratio of the observation interval and the orbital period of the satellite is 1.02.

satellites, the accuracy of the ephemeris is comparable to the apparent size of the orbit. This means that these satellites can be considered lost. They need to be searched and rediscovered. According to the results of Emelyanov (2010), there were 21 such lost satellites. The accuracy of the satellite ephemeris in Emelyanov (2010) was determined for a large number of time moments until 2020 and in most cases included the observation time interval. For all satellites, the three methods discussed above were used. A comparison is made of the accuracy estimates of the ephemeris with the three methods used. It turned out that, for satellites with a sufficiently long observation time interval, the three methods give completely identical results. However, in cases of short observation time intervals, the “bootstrap” method of observation samples gives completely unrealistic estimates. In this case, the methods of observation-error variances and the method of variation of parameters are mutually agreed. Several examples of ephemeris accuracy estimates are shown in Figs. 8.4, 8.5, and 8.6. In the captions to the Figures, the relations of the observation interval to the orbital period of the satellite are given. The plots in the figures confirm the conclusions made.

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Figure 8.6. Evaluations of the accuracy of the ephemeris of Jupiter’s satellite S/2003 J 3 by three methods: points indicate observation-error variances, circles mark “bootstrap”-samples, and crosses show variations of parameter errors. The ratio of the observation interval and the orbital period of the satellite is 0.04. On this plot, in most cases, the points merge with the crosses.

Jacobson et al. (2012), on the basis of all the observations available at that time, the orbits and ephemeris of all the distant satellites of Jupiter, Saturn, Uranus, and Neptune were specified. Estimates of the accuracy of the satellite ephemeris are made. The results obtained are similar to the results of Emelyanov (2010). For a number of satellites, the ephemeris turned out to be more accurate, since longer series of observations were used. It is important that Jacobson et al. (2012) performed new observations and searched for satellites previously lost due to inaccuracies in the ephemeris. Several satellites were again found. However, Jacobson et al. (2012) stated that 16 previously discovered distant satellites of Jupiter and Saturn remain lost. New work to refine the orbits of distant planetary satellites continued. Brozovic and Jacobson (2017) made a new definition of the orbits of the distant satellites of Jupiter using new observations. But even after this work, 11 among 71 Jupiter’s distant satellites remain lost.

References Brozovic, M., Jacobson, R.A., 2009. The orbits of the outer Uranian satellites. Astronomical Journal 137, 3834–3842.

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Brozovic, M., Jacobson, R.A., 2017. The orbits of Jupiter’s irregular satellites. Astronomical Journal 153 (4), 147. Desmars, J., Arlot, S., Arlot, J.-E., Lainey, V., Vienne, A., 2009. Estimating the accuracy of satellite ephemerides using the bootstrap method. Astronomy and Astrophysics 499, 321–330. Desmars, J., Li, S.N., Tajeddine, R., Peng, Q.Y., Tang, Z.H., 2013. Phoebe’s orbit from ground-based and space-based observations. Astronomy and Astrophysics 553, A36. 10 pp. Emelyanov, N.V., 2005. Ephemerides of the outer Jovian satellites. Astronomy and Astrophysics 435, 1173–1179. Emel’yanov, N.V., Kanter, A.A., 2005. Orbits of new outer planetary satellites based on observations. Solar System Research 39, 112–123. Emelyanov, N.V., 2007. Updated ephemeris of Phoebe, ninth satellite of Saturn. Astronomy and Astrophysics 473, 343–346. Emel’yanov, N.V., Arlot, J.-E., 2008. The natural satellites ephemerides facility MULTI-SAT. Astronomy and Astrophysics 487, 759–765. Emelyanov, N.V., 2010. Precision of the ephemerides of outer planetary satellites. Planetary and Space Science 58, 411–420. Emelyanov, N.V., Arlot, J.-E., 2011. The orbit of Nereid based on observations. Monthly Notices of the Royal Astronomical Society 417 (1), 458–463. Jacobson, R.A., 2000. The orbits of the outer Jovian satellites. Astronomical Journal 120, 2679–2686. Jacobson, R., Brozovic, M., Gladman, B., Alexandersen, M., Nicholson, P.D., Veillet, C., 2012. Irregular satellites of the outer planets: orbital uncertainties and astrometric recoveries in 2009–2011. Astronomical Journal 144, 132. 8 pp. Giorgini, J.D., Yeomans, D.K., Chamberlin, A.B., Chodas, P.W., Jacobson, R.A., Keesey, M.S., Lieske, J.H., Ostro, S.J., Standish, E.M., Wimberly, R.N., 1997. JPL’s on-line solar system data service. American Astronomical Society DPS meeting N.28, N.25.04. Bulletin - American Astronomical Society 28, 1158 (1997). Milani, A., 1999. The asteroid identification problem. I. Recovery of lost asteroids. Icarus 137, 269–292. Muinonen, K., Bowell, E., 1993. Asteroid orbit determination using Bayesian probabilities. Icarus 104, 255–279. Virtanen, J., Muinonen, K., Bowell, E., 2001. Statistical ranging of asteroid orbits. Icarus 154, 412–431.

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9 The rotation of planetary satellites 9.1 General properties of the rotation of planetary satellites Moving in orbits around their planets, satellites also rotate around their mass centers. The dynamics of rotation is very complex. Rotational and orbital motions are mutually connected. The rotation is affected mainly by conservative gravitational forces, but also by forces of a dissipative nature. By rotation, satellites can be divided into three groups. The first group of satellites has a rotation synchronous with orbital motion. This means that the satellite is always facing the planet the same side. Almost all of the major planetary satellites rotate in this way. The exception is Saturn’s satellite Hyperion, whose rotation is unusual. Satellites having synchronous rotation, as a rule, have an elongated shape, and the axis of elongation is always directed towards the planet. Rotating synchronously, the satellites are not always precisely oriented in the direction of the planet. They make small vibrations of their bodies relative to this direction, the so-called librations. This is due to the attraction of the planet and the mutual attraction of the satellites. The exact libration theories of these satellites are very complex. The second group is satellites with an unusual rotation, which is usually chaotic one. Hyperion is one of such satellites—one of the main satellites of Saturn. An analysis was presented in a theoretical paper (Wisdom et al., 1984), which revealed a large zone of possible states of chaotic rotation around the resonant spin–orbit state of Hyperion. Even earlier (Goguen et al., 1983) from the analysis of the light curves of this satellite, obtained from the photometric observations in 1983, its chaotic rotation was established. The random rotation of planetary satellites and the rotation of Hyperion are discussed below in a separate section. The third group includes satellites, the rotation of which is unknown. These are mainly distant satellites of major planets, very small objects. Knowledge of the rotation parameters of the natural satellites of the planets is extremely in demand in solving many practical The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00014-5 Copyright © 2021 Elsevier Inc. All rights reserved.

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problems of the dynamics of the Solar System. Therefore, it is important to have some specific data source that reflects modern knowledge, continually updated as new observation results become available. The IAU Working Group on Cartographic Coordinates and Rotational Elements (IAU WG CCRE) – is engaged in this problem. Until recently, this working group regularly published its report every three years, in which a summary of the latest information on the rotation of planets and satellites is being given. The last report, which we will consider later, is a publication of Archinal et al. (2018). However, after the publication of such a report, studies are again being carried out to refine previous models. The rotation parameters of natural satellites and planets, taken from the latest report of the IAU WG CCRE Working Group, are given in Appendix F. The new data that appeared after the publication of the report is also given and provided with special comments in this chapter. When describing the rotation of a planet or satellite, differences in the accepted definitions such as the North Pole and the initial meridian may arise. Therefore, in the next section we give a definition of the basic concepts adopted by the International Astronomical Union (IAU). Rotations of satellites are observed using spacecraft. Photometric indicators of rotation are also found in ground-based observations. Rotation is significantly dependent on the internal structure of the satellites. Based on observations of rotation, hypotheses about the internal structure can be constructed, and mass distribution inside the satellites can be modeled. Typically, the satellite’s gravitational field is modeled by expanding the force of the attraction function in a series of spherical functions. Based on these data, differential equations of satellite rotation around its mass center are composed. The right-hand sides of the equations take into account factors that influence rotation. This is mainly the attraction of the central planet and other satellites. The most accurate satellite orbital motion models are used. Satellite’s images are obtained using spacecraft. The shape of their bodies is determined from the images. These data are consistent with rotation models. To solve the differential equations of rotational motion, the initial conditions are necessary. They are determined from observations. The positions of the satellites are determined by the images obtained using the spacecraft. To study the rotation of satellites, ground-based photometric measurements of the satellite’s integrated brightness are also used. In this case, one or another model of the distribution of reflectivity over the surface of the satellite is used. The result is a satellite rotation model that allows us to deter-

Chapter 9 The rotation of planetary satellites

mine libration angles at any given time point. When modeling the rotation, models of an absolutely rigid body, a deformable elastic or viscoelastic body are considered. All of these models are the subject of numerous works published in scientific journals. Below is a brief overview of the works and results on the study of the rotation of the natural satellites of major planets based on observations. The sections correspond to individual satellites having certain rotation features.

9.2 Basic concepts of the rotation of planets and satellites To avoid potential confusion, the IAU WG CCRE has made recommendations in defining basic concepts about the rotation of planets and satellites. Coordinate systems on planets and natural satellites are defined relative to the mean axis of rotation. The concept of longitude depends on the type of celestial body. For most bodies with observable details on the surface, the longitude is determined relative to any detail. Other definitions of the longitude are also applied, which we will discuss below. Approximate expressions for rotation angles are given relative to the International Celestial Reference Frame (ICRF). Time is counted in the TDB scale. The reference epoch is J2000, i.e. JD = 2451545.0 (12 hours January 1, 2000) TDB. Time is measured in days (86,400 seconds of the SI system) or in centuries of 36525 days on the TDB scale. The North Pole of the rotation axis is the one of the two poles, which is located in the northern hemisphere relative to the invariable plane of rotation of the Solar System. Thus, the angle between the north direction of the rotation axis and the fixed momentum vector of rotation of the Solar System is always less than 90 degrees. The coordinates of the North Pole of rotation are set by the right ascension α0 and the declination δ0 in the ICRF system. With this definition of the North Pole, we have two points of intersection of the body’s equator and the main plane of the ICRF. We call the ascending node Q one of these points with right ascension α0 + 90◦ . The inclination of the equatorial plane of the body to the equator of the ICRF system is 90◦ − δ0 . It is assumed that the zero meridian of the body rotates monotonously and almost uniformly in time. Let us determine the position of the zero meridian at a given time moment by the angle W counted at the body’s equator eastward of the ascending node Q to the point B of the intersection of the zero meridian with

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Figure 9.1. The coordinate system used to determine the orientation of a planet or satellite.

the body’s equator. The concepts discussed here are illustrated in Fig. 9.1. If W increases with time, then we say that the planet or satellite has a direct (prograde) rotation. If W decreases, then we consider the rotation to be retrograde. The angle W determines the ephemeris position of the zero meridian. For planets or satellites that do not have pronounced details on the surface, we can specify the expression for W versus time as the prime meridian. In some papers, the coordinate axes, rigidly connected with the body, refer not to the rotation axis, but to the main central axes of inertia. Such cases are highlighted in publications. For some planetary satellites, the position of the zero meridian is associated with the mean direction to the center of the planet. With the above definition of the planet’s North Pole, it turns out that Uranus has a reverse rotation together with its main satellites. In some papers, the direction of the angular velocity of the planet’s rotation is used as the main direction of the axis. However, unless otherwise specified, the definitions recommended by the IAU WG CCRE are accepted. Note that the definition adopted above for the North Pole of the body is unsuitable for some minor planets and comets. The fact is that due to the rapid precession of the rotation axis, the pole can change to the opposite. By this reason, for small planets, dwarf planets and comets, the so-called “positive pole” is associated with the angular velocity vector of rotation according to the well-known “rule of the right hand”. Such a pole may not be connected in any way with the concept of a northern or southern position.

Chapter 9 The rotation of planetary satellites

In this regard, the position of the Pluto’s pole, as related to dwarf planets, was reversed already in the report of the 2011 IAU Working Group CCRE compared to the direction of the Pluto’s pole adopted in previous reports of the IAU Working Group CCRE. And now in the report (Archinal et al., 2018), the positions of the pole of Pluto and its satellite Charon are fixed according to the rule adopted for dwarf planets. We recall here again that the rotation parameters of natural satellites and planets, taken from the latest report of the IAU Working Group CCRE (Archinal et al., 2018), are given in Appendix F. The new data that appeared after the publication of this report are also given and accompanied by special comments. Of particular note is the new Mars satellite rotation model published in 2017 (Jacobson, 2017). The rotation parameters from this paper are given separately in Appendix F.

9.3 The rotation of Neptune and the orbit of Triton The Neptune’s satellite system has special properties. Neptune has one very massive satellite Triton, as well as several very small distant and close satellites. One of them is the distant companion, Nereid. It revolves in a very elongated orbit with an eccentricity of about 0.745. The influence of small satellites on the motion of the massive Triton is rather weak. In the latest published models of rotational motion, this influence is still neglected. Consider a mechanical system consisting of Neptune rotating around its axis and the satellite Triton, orbiting around it. The attraction of the Sun and other satellites can be neglected, and Triton can be considered as a point mass. The mass center of the system will be fixed, and the constant angular momentum of the system relative to the mass center will be equal to the sum of the momentum of the rotating planet and the momentum of Triton orbiting the mass center of the system. Due to the dynamic oblateness of Neptune, the orbit of Triton will precess. Neglecting short-period perturbations from this oblateness, we can conclude that the orbital precession will occur with a constant angular velocity and a constant inclination to an invariable plane perpendicular to the angular momentum vector of the system. The configuration of the Neptune’s rotation and the orbital motion of Triton is explained in Fig. 9.2. The origin is placed at the mass center of the system, the z axis is directed along the vector of the total angular momentum. Here we have shown the lines

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Figure 9.2. The configuration of the rotation of Neptune and the orbital motion of Triton. The vector of the rotational torque of Neptune is denoted by N. The vector T is the torque vector of the orbital motion of Triton. The z axis is directed along the total vector N + T.

of intersection of planes with a certain sphere of arbitrary radius. The rotational torque vector of Neptune is denoted by N, and the torque vector of the orbital motion of Triton is denoted by T. Both vectors and the z axis lie in the same plane. The angle  between the vector N and the axis z is 0.461627 degrees, and the inclination of the Triton orbit to the invariable plane I = 156.870 degrees (Jacobson, 2009). For illustration purposes, the angle  is intentionally increased. In Fig. 9.2, the direction to the ascending node of the Triton orbit is indicated by . The arrows mark the directions of the precession of the vectors and the plane of the orbit. The configuration of the Neptune’s system discussed here is similar to the configuration of the rotation of the Moon and its orbital motion. For the Moon, such properties were discovered by the French scientist Jean-Dominic Cassini (1625–1712). Now these laws are called the Cassini laws. Jacobson (1990) provides a detailed quantitative description of the considered dynamical system and gives the relationships between the quantities characterizing the motion. Based on such a model, Jacobson (2009) derives approximate formulas for the equatorial coordinates of the Neptune pole—the right ascension α0 and the declination δ0 , as functions of time. The constant parameters in these formulas are: angle , right ascension αr , and declination δr of the total vector of angular momentum of the system. Time enters into the formulas through the longitude of the ascending node of the orbit of Triton  with respect to the invari˙ − t0 ) is set by able plane. The time-linear change in  = 0 + (t ˙ and 0 . In Jacobson (2009), all these paramethe parameters 

Chapter 9 The rotation of planetary satellites

ters are determined directly from observations. All observations made up to 2008 were used. The equations of motion, which were solved by the method of numerical integration, also took into account perturbations from other celestial bodies. The formulas for the equatorial coordinates of the Neptune’s pole have the form (Jacobson, 2009) 1 α0 = αr +  sec δr sin T −  2 sec δr tan δr sin 2T , 2 1 δ0 = δr −  cos T −  2 tan δr (1 − cos 2T ), (9.1) 4 where T is the longitude of the ascending node of the equator of Neptune with respect to a fixed plane. From the condition that the rotational axis of the planet and the torque vector of the orbital motion of Triton are located on opposite sides relative to the axis z, the relation between T and  follows: T =  + 180◦ . In Eqs. (9.1), expansion terms containing degrees of  greater than 2 are discarded. We now substitute the values of the parameters obtained in Jacobson (2009) from the observations in Eqs. (9.1): αr = 299.460861, δr = 43.404811, and  = 0.461627. Here the ICRF coordinate system is adopted, and the angles are given in degrees. We obtain for the coordinates of the pole of Neptune the following relationships: α0 = 299.460861 + 0.635397 sin T − 0.002421 sin 2T , δ0 = 43.403932 − 0.461627 cos T + 0.000879 cos 2T .

(9.2)

The expression for T , taken from Jacobson (2009), is reduced to the initial epoch J2000. Thus we obtain T = 358.177292 + 52.383621844611 T , where the angles are measured in degrees, and T is the time interval between the current moment and the J2000 epoch, expressed in Julian centuries for 36525 days. Note that the angular values are taken from the paper in the form as they are presented there, with an excessive number of significant digits. In fact, in Jacobson (2009), the accuracy of determining the angle αr is 0.14, and the angle δr is 0.03 degrees. The coordinates of the pole in the relations (9.2), respectively, have the same accuracy.

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The Triton’s motion model was also independently constructed in Emelyanov and Samorodov (2015). An analytical theory of the motion of this satellite has been developed. Perturbations from the attraction of the Sun are taken into account by the analytical method. An advantage over the approach in Jacobson (2009) is that the time period of the observations used is extended to 2012. Emelyanov and Samorodov (2015), in addition to the parameters of the Triton’s orbit, specified the equatorial coordinates of the constant motion vector of the system αr and δr directly from observations (They are denoted in this paper by α0 and δ0 ). The angle  is not specified. A linear change in the longitude of the ascending ˙ T is also determined from observations. The panode  = 0 +  rameters were determined for two variants of the composition of the observations: without using the observations made using the Voyager-2 spacecraft, and with the inclusion of these observations in a common database. We can now take the parameter values obtained in Emelyanov and Samorodov (2015) and write down formulas for them for the coordinates of the Neptune’s pole. The angle value  can be taken from Jacobson (2009). Since observations from the Voyager2 spacecraft are used in this work, we will take from Emelyanov and Samorodov (2015) a variant of the results also obtained using these observations. According to these results, we have αr = 299.090 and δr = 43.019 (angles are measured in degrees here). For the coordinates of the Neptune’s pole in the ICRF system, we obtain the following relations: α0 = 299.090 + 0.631391 sin T − 0.002373 sin 2T , δ0 = 43.018132 − 0.461627 cos T + 0.000868 cos 2T .

(9.3)

The expression for T based on the results of Emelyanov and Samorodov (2015), reduced to the initial epoch J2000, has the form T = 359.460800221 + 53.05102845 T . The expressions for the coordinates of the pole of Neptune obtained from Jacobson (2009) and Emelyanov and Samorodov (2015) can be considered alternative to the formulas recommended in the report of the IAU Working Group CCRE (Archinal et al., 2018).

9.4 Theory of rotation for Phobos Among the two known satellites of Mars, the natural satellite Phobos is the most studied. A detailed study of its rotation is pub-

Chapter 9 The rotation of planetary satellites

lished in Rambaux et al. (2012). This study is motivated by the concepts of new space missions for high-precision measurements of Phobos rotation. The main task of these missions is to obtain information about the internal structure of the satellite. It is well known that variations of rotation strongly depend on the internal structure. Numerous high-resolution images of Phobos were obtained with the Mars EXpress (MEX) spacecraft. New data on satellite librations are derived from these images by Willner et al. (2010). Librations have an amplitude of 1.2 degrees and are determined with an accuracy of 0.15 degrees. The prospects for improving the accuracy of this analysis and the proposed new missions to Mars required the development of a new, more accurate theory of the rotation of Phobos. The task was to determine what information can be obtained from the planned libration measurements with an accuracy of 0.0001 degrees. Phobos in its rotation relative to the mass center is in synchronous resonance with orbital motion around Mars. Due to the non-zero eccentricity of the orbit, there are variations in the velocity of the orbital motion. As a result, the satellite shows 52% of its surface towards Mars. The planet creates by its attraction a moment of force, depending on time and affecting the rotation. The response of Phobos to this effect is physical libration, i.e. deviations from uniform rotation. The librational response depends on the distribution of masses inside the satellite’s body. As far back as the 90s, analytical theories of Phobos libration developed on the basis of data obtained using the spacecraft Viking and as a result of the project Phobos-2. Such theories are published in Borderies and Yoder (1990); Chapront-Touze (1990); Pesek (1991). In Rambaux et al. (2012), the Phobos rotation relative to the mass center is modeled by numerically integrating the Euler differential equations of satellite rotation. The Euler angles specify the position of a non-rotating coordinate system associated with the Mars equator of a fixed epoch J2000 in the coordinate system associated with the main central moments of inertia of the satellite. The values of the Phobos’s moments of inertia based on images from the MEX spacecraft were used in Willner et al. (2010). To simulate the rotation of Phobos, Rambaux et al. (2012) took the values of the moments of inertia A = 0.3615, B = 0.4265, and C = 0.5024 from Willner et al. (2010), where they were determined by the satellite topography. The mean radius of Phobos R0 was taken to be 11.27 km. The expansion of the Phobos gravitational potential in a series of spherical functions is applied up to the third order.

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Phobos is usually regarded as a homogeneous body. However, the librations measured in Willner et al. (2010) do not exclude inhomogeneities within Phobos. Knowing the density variations along the surface could provide a key to understanding the origin of the satellite. The model of the orbital motion and the Phobos ephemeris of Lainey et al. (2007) is used. The accuracy of these ephemeris is 1 km. In the theory of motion, perturbations from the Sun, Jupiter, Saturn, the Earth–Moon system, the noncentrality of the Mars gravitational field, and tidal effects in the Mars–Phobos system are taken into account. In order to take into account the influence of Mars attraction on the satellite’s rotation, it was necessary to give an analytical description of the orbital motion of Phobos in the coordinate system associated with the mean equator of Mars of the date epoch, since the main factor determining the orbital precession and satellite’s rotation is the planet’s dynamic oblateness. In Rambaux et al. (2012), for this description, Delaunay elements counted in the system of the mean equator of Mars from the epoch of date were used. The angles describing the motion of the pericenter of the Mars’s orbit and its rotation were also added as arguments to the trigonometric functions that are part of the theory. The numerical values of the frequencies corresponding to these angles are given in this work. The advantages of using numerical integration compared to the analytical solution lies in the possibility of taking into account the combined effects of various perturbing factors. This simplified the accounting of nonlinear changes due to the proximity of the frequencies of own librations and the frequencies of external forces. Rambaux et al. (2012) showed that the influence of third-order terms in the expansion of the Phobos’s gravitational potential and the effect of the dynamic oblatness of Mars have approximately the same order of smallness. The initial conditions for solving the equations of rotation of Phobos were taken from the condition that the eigenmodes of rotation of Phobos, also called free libration, are equal to zero. Thus, it was assumed that tidal dissipation is sufficient to quench free libration, and there are no mechanisms that can excite it. The classical method of such a search for the initial conditions turned out to be unsuitable here because of the proximity of the frequencies of free libration and forced oscillations. Therefore, artificial quenching of free libration was included in the numerical integration, and the decay time was selected by iterations.

Chapter 9 The rotation of planetary satellites

Phobos’s rotation angles were represented using sums of trigonometric functions of known arguments. The amplitudes and phases of the trigonometric terms were obtained with the frequency analysis of the results of numerical integration. As a result of the cited work, a description of the Phobos librations in longitude and the rotation of the Phobos body in space was obtained. The libration angle in longitude describes the difference between a real satellite rotation and a uniform one. The fundamental harmonic of libration in longitude has a period of 0.3190 days and an amplitude of 1.0998 degrees. The motion of the Phobos pole, associated with the main moment of inertia, relative to the mean equator of Mars of the epoch of date, occurs along a weakly elliptical trajectory with slight additional fluctuations. The radius of the circle close to the elliptical trajectory of the end of the axis of rotation at a distance of 11.3 km from the satellite’s center is approximately 220 meters. The period of rotation of the axis is equal to the draconic period of the orbital motion of the satellite (from node to node). Thus, the axis (the main moment of inertia) moves along a cone with an angular distance from the mean position of approximately 1.078 degrees. Ten trigonometric terms thus obtained are given in Rambaux et al. (2012) in the representation of libration in longitude and 12 terms in the motion of the Phobos pole. For each term in the expansion, frequency, phase, and amplitude are given. The amplitudes of the other terms in the expansion of the pole coordinates, except the main one, are one hundred times less than the amplitudes of the main term. These terms are mainly due to perturbations in the inclination of the Phobos orbit. A detailed analysis of the angles describing the rotation of the satellite as a function of time is given. In particular, the contribution of various perturbations to the librational motion is analyzed. The theory of Phobos rotation constructed in Rambaux et al. (2012) was used to study the possibility of refining the libration model from observations (LeMaistre et al., 2013). Two types of measurements were assumed. The first type of measurement (Direct-To-Earth (DTE) Doppler) is the ground-based measurement of the Doppler frequency change of a radio signal from a spacecraft placed on the Mars’s satellite. In the second method (Star Tracker), astrometric observations of stars from the spacecraft are carried out. Both methods give signals that directly depend on the rotation angles of Phobos. Since such observations were just planned, estimates of the accuracy of the rotation parameters determined in this way were preliminarily carried out on the basis of artificially created observation results, the so-called “simulations”. Assessments were also made of the possibility of

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building a model of the satellite’s internal structure in such an experiment. In LeMaistre et al. (2013), the effects of viscoelastic deformations of the Phobos body and the possibility of determining the corresponding parameters from observations were considered. An interesting material contained in LeMaistre et al. (2013) is the transformation of the parameters of the Phobos rotation found in Rambaux et al. (2012) to the form of representing the right ascension and declination of the pole of the Phobos rotation axis as functions of time. These functions are represented by the sum of linear in time and periodic terms. Note that the satellite pole in this work sets the direction of the figure’s axis of the body. The paper does not specify how this axis relates to the axis of rotation. It is only said that the polar motion of Phobos is negligible. From the context, we can conclude that these axes coincide with each other. If we accept according to the captions to Tables 1 and 2 of LeMaistre et al. (2013) that the tables show the motion of the Phobos rotation pole, comparable with the data in the report of the IAU WG CCRE Working Group (Archinal et al., 2018), then the data in the tables represent an alternative source for representing the rotation of Phobos. Phobos rotation parameters according to (LeMaistre et al., 2013) are given in Appendix F.

9.5 Rotation of the Galilean satellites of Jupiter, satellites of Saturn and Pluto In recent work on the rotation of the Galilean satellites of Jupiter, the so-called “librational response” of the internal structure of the satellites was studied. Attention was drawn to the effects of rotation caused by the orbital motions of the satellites. Various models of the internal structure of satellites were considered. It was assumed that on the satellites Europe, Ganymede, Callisto, under the surfaces there are layers of liquid (oceans). A spectrum of satellite libration frequencies was constructed in Rambaux et al. (2011). The harmonics associated with various external effects have been identified: orbital motion, the attraction of other satellites, and the attraction of the Sun. It is shown that the amplitudes of short-period terms in libration carry information about the internal structure. In Noyelles (2010a), a model of rotation of the Galilean satellites of Jupiter was constructed by analytical and numerical methods. The rotation angles, as a function of time, are not given, but the influence of the internal structure of the satellites on the prop-

Chapter 9 The rotation of planetary satellites

erties of rotation was investigated. It is shown that from observations of the rotation it is possible to draw conclusions about the internal structure of satellites. Over the past decade, much work has been published on the rotation of Saturn’s satellites. This is due to the implementation of new high-precision and highly informative observations using the spacecraft Cassini. In particular, new data were obtained on the gravitational fields of the satellites. The rotation of Titan was studied in Noyelles (2008); Noyelles et al. (2008); Van Hoolst et al. (2009); Richard and Rambaux (2014). A detailed analysis of the frequency spectra of librations and the dependences of libration motion on various external influences and the physical properties of the satellite, in particular, the viscosity of the substance, has been carried out. An analysis of the rotation of the satellite Mimas and the relationship of the properties of its rotation with the internal structure was performed in Noyelles et al. (2011); Tajeddine et al. (2014); Noyelles (2017). A description and analysis of the physical libration of the coorbiting satellites Janus and Epimetheus are given in Noyelles (2010b); Robutel et al. (2010, 2012). The mission New Horizons showed that the small satellites of Pluto P2 Nikta, P3 Hydra, P4 Kerberos, and P5 Styx did not reach the state of synchronous rotation in the tidal evolution of rotations and have significant inclination angles of the rotation axes relative to the normal to the orbit plane of the massive P1 Charon satellite (Weaver et al., 2016). In the satellite system P1 Charon, P5 Styx, P2 Nikta, resonant relationships between their orbital motions were detected. This was the reason for the analysis of the rotation of these satellites. Such an analysis was made in Correia et al. (2015); Quillen et al. (2017a,b). The method of constructing variants of artificial motions (“simulations”) was used in the many-body problem of finite dimensions with the effect of a “damped spring”. Models with tidal dissipation did not show significant inclination variations, and this is explained by large values of rotation velocity and small eccentricities of satellite orbits. However, the P5 Styx satellite in tidal evolution shows intermittent variations in inclination and rollover episodes. In the proposed process of migration of the P1 Charon satellite with increasing distance to Pluto, the satellites can be captured into resonances of mean motions with Charon, which can cause variations in the inclination of the rotation axes of the satellites P5 Styx and P2 Nikta. The reason may be the resonance between the Charon’s orbital motion and the precession of the axis of rotation of the small satellite.

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9.6 Chaotic rotation of planetary satellites. Rotation of Hyperion Modeling the rotation of planetary satellites is a particular problem. There are so many solutions and there is more published work on this topic. Research is conducted in two interrelated directions. The first one is the theoretical issues of rotation of non-spherical satellites, the development of methods for studying rotational dynamics, the development of criteria for evaluating the properties of rotation. The second direction is to make and interpret observations. Rotation is studied on the basis of groundbased photometric measurements of the brightness of satellites, as well as using images obtained with spacecraft flying near other planets. The rotation occurs under the force moment of the planet on a non-spherical satellite body in conditions of its uneven motion in an elliptical orbit. When comparing models with observations in some cases, the attraction of other satellites is also taken into account. The rotation of different planetary satellites has a different character. One of the rotation modes is synchronous resonance, when the mean angular velocity of rotation is exactly equal to the mean angular velocity of the satellite’s orbital motion around the planet. In the simplest version, the axis of rotation is orthogonal to the plane of the orbit and coincides with the axis of the maximum principal central moment of inertia of the satellite. In this case, the axis of maximum elongation of the satellite’s body is always located near the direction from the satellite to the planet. Then, in fact, the satellite is always facing the planet with the same side. During the long-term evolution of rotational motion, most large satellites are similar in shape to spherical, entered such a final mode. With synchronous resonance, the axis of elongation of the body is forced to deviate from the direction to the planet due to uneven motion of the satellite in an elliptical orbit. Thus, even if the satellite rotates uniformly in a non-rotating coordinate system, the satellite’s body sways relative to the direction to the planet, since the vector of this direction does not rotate uniformly. In real cases, the axis of rotation may deviate slightly from the perpendicular to the plane of the orbit, and the rotation angle of the satellite, measured in a non-rotating coordinate system, may deviate from the angle of uniform rotation. Changes in these deviations in time have very intricate properties; they are described by solving rather complex equations. These two indicated deviations define a phenomenon called physical libration of satellites.

Chapter 9 The rotation of planetary satellites

With regular rotation of the satellite, the angles of rotation are described by periodic functions. In this mode, for a single value of the angle, the angular velocity can have one of several fixed possible values. In other cases, the satellite’s rotation can be chaotic, when for one angle the rate of change can take random values in a certain range. For small satellites having a strongly pronounced non-spherical shape, rotation can be a three-dimensional somersault. Among more than 170 satellites discovered to date, the rotation state from observations was determined in 33. Almost all of them are in synchronous resonance. A huge number of works are devoted to the rotation of planetary satellites. There is no possibility to write a review that gives a complete picture of knowledge in this scientific field. Here we consider only some of the work on chaotic rotation. In this review, only characteristic problems and individual attempts to solve them will be mentioned. Attention will be paid mainly to satellites for which the randomness of their rotation is proved or they are suspected of the possibility of such chaos. The first reports of the IAU Working Group CCRE present rotation parameters of Saturn’s satellite, Hyperion, based on orbital data. However, an analysis was presented in a theoretical paper (Wisdom et al., 1984), which revealed a large zone of possible states of chaotic rotation around the resonance spin–orbit state of Hyperion. The chaotic rotation is caused by the strong nonsphericity of the satellite and the significant ellipticity of its orbit. Even earlier in Goguen et al. (1983) from the analysis of the light curves of this satellite, obtained from the photometric observations of 1983, its chaotic rotation was established. This resulted in instability of the period and brightness amplitude at eight revolutions of the satellite. The orientation and rotation velocity of Hyperion change randomly with time. The reason for the chaos lies in the “overlap” of spin–orbit resonances, which arises due to the strong nonsphericity of the Hyperion shape and the significant difference between its orbit and circular. The phenomenon and criteria for resonance overlap can be found in Chirikov (1979); Wisdom et al. (1984). In 1989, observational data were published (Klavetter, 1989) over a 64-day time interval. The amplitude of the brightness variations of Hyperion was 0.6 magnitude. No periodic modulations of the oscillations were found, and a conclusion was drawn about the randomness of rotation. For these reasons, since 1986, Hyperion has been excluded from the reporting tables of IAU Working Group CCRE.

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The numerical integration of the differential equations of rotation of Hyperion was performed in Black et al. (1995) for a wide range of initial conditions. It is concluded that Hyperion’s transition from chaotic “acrobatics” to quasi-regular rotation and back in a short period of time is more a rule than an exception. Photometric observations of Hyperion were made at the Main Astronomical Observatory of the Russian Academy of Sciences (Pulkovo) from September 1999 to March 2000 and in September– October 2000 (Devyatkin et al., 2002). The light curve of the satellite is analyzed and conclusions are drawn on the nature of its rotational dynamics. During the observation period, Hyperion was in a chaotic rotation mode. Attempts to find hidden periodicities showed that there are no periodic components in the light curve. Observations of Hyperion using the spacecraft Cassini gave new information about the rotation of the satellite. In 2005, the spacecraft passed three times near the satellite, which was observed at time intervals of 40 and 67 days. In all three passes, the satellite rotated with a frequency of 4.2 and 4.5 times greater than during rotation synchronous to orbital motion (Harbison et al., 2011). In this case, the axis of rotation moved both in the satellite’s body and in space. It was located approximately (within 30 degrees) equally in the satellite’s body in all three cases along the axis of maximum elongation. This contradicts the results of dynamics modeling performed in the same work. This means that at first the values of the moments of inertia of Hyperion were incorrectly selected. In Harbison et al. (2011), models of satellite rotation were constructed based on the moments of inertia following from the shape of the satellite and based on their corrected values. The inverse problem was also solved—the determination of the moments of inertia from observations of rotation and the shape of Hyperion. The result depends on the model of mass distribution within the satellite, which remains unknown. It remains to assume the uniformity of the density distribution. Several variants of the values of the moments of inertia were found, from which it can be concluded that the moments of inertia are not determined confidently. Harbison et al. (2011) concluded that the available data are insufficient for accurate definitions. Note that according to the data received from the spacecraft Cassini, Hyperion has an average density of (544 ± 50) kilogram per cubic meter (Thomas et al., 2007). It has many voids inside and looks like a sponge. In addition to Hyperion, are there any other satellites in the solar system randomly tumbling about their own center of mass? There is no answer to this question (Kouprianov and Shevchenko, 2006).

Chapter 9 The rotation of planetary satellites

Melnikov and Shevchenko (2010) investigated the stability of possible synchronous rotations of small planetary satellites, for which the rotation properties have not yet been obtained from observations. It is shown that most satellites with unknown rotation states cannot rotate synchronously with orbital motion, since for them there are no stable states of synchronous rotation. They rotate either faster than orbital motion, which is less likely, or experience chaotic rotation. Theoretical studies (Kouprianov and Shevchenko, 2005; Melnikov and Shevchenko, 2008) showed that satellites of Saturn Prometheus and Pandora are also in a state of chaotic rotation, apart from Hyperion. The nature of their rotation is currently unknown. The question is very important: is there a preferential orientation of the satellite during chaotic rotation, or are all its orientations in this case equally probable? Calculations show (Melnikov and Shevchenko, 2008) that, in the case of their chaotic rotation, Prometheus and Pandora have a predominant orientation of the largest axis of the satellite shape in the direction of Saturn. This makes it difficult to draw conclusions about the nature of the rotation of these satellites from observations, since the chaotic regime is somewhat similar to ordinary synchronous rotation. According to numerical experimental and analytical estimates (Shevchenko and Kouprianov, 2002; Kouprianov and Shevchenko, 2005), the chaotic rotation of small satellites of the planets of the Solar System should manifest itself at relatively small time intervals, and can be detected from observations. However, chaotic rotation has so far been observed only in the case of Hyperion. When constructing models of the actual rotation of satellites, attempts were made to study generally the properties of possible rotational motions. First, the so-called plane rotation was studied, when the axis of rotation is constant in space, and rotation is described by the change in time of only one angle. In the work on this problem, it was√found that the rotation modes depend on two parameters: ω0 = 3(BA)/C, where A, B, and C are the main central moments of inertia of the satellite (A < B < C) and the eccentricity of the orbit e. The rotation is described by the angle θ between the axis of the smallest principal central moment of inertia and the planetocentric radius vector of the satellite. Two synchronous rotation modes were detected. They differ in the ranges of values on the phase plane of “angle–rate of change”. The presence of these two different types of synchronous resonance was first noted by Wisdom et al. (1984), analyzing the results of their numerical experiments in the study of the rotational dynamics of Hyperion. An exact description of these types of rotation is contained in the papers of Mel’nikov and Shevchenko (1998, 2000). The ranges of the

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values of the parameters ω0 and e are revealed for which the rotation of each of the two types takes place. The results were applied to the satellite of Jupiter, Amalthea. The parameters for this satellite fall on the area of both types of rotation. In the same work, the stability of synchronous rotations with respect to the angle of inclination of the axis of rotation to the plane of the orbit was studied. It was found that for Amalthea one of the two types mentioned is an unstable rotation. Mel’nikov and Shevchenko (2000) showed that the probability of stability of synchronous resonance with respect to the inclination of the axis of rotation for the satellites of Mars, Phobos and Deimos is equal to one. An analysis of the possible dynamics of all planetary satellites known by 2007 was performed in Mel’nikov and Shevchenko (2007). Particular attention is paid to the satellite of Jupiter, Amalthea, and the satellite of Saturn, Prometheus, for which synchronous resonances of three different types are possible. In the study of chaotic motions, a parameter is used that characterizes the degree of randomness—this is the Maximal Lyapunov Characteristic Index (MLCI). This parameter is the coefficient of time in the exponent that describes the divergence of the solutions of the equations of motion or rotation, with small variations in the initial conditions. It is assumed that in the case of regular motions, the divergence of solutions occurs according to a linear law and the randomness index is zero. The reciprocal of the of the MLCI shows the predictability of the motion. The calculation of MLCI in the study of chaotic motions is a very laborious and time-consuming computational problem. A bibliography on this topic in the case of rotational motions can be found in Shevchenko (2002). The method of calculating Lyapunov characteristic exponents for studying the rotational motion of satellites is implemented as a complex of computational programs in Shevchenko and Kouprianov (2002); Kouprianov and Shevchenko (2005) The main result of Shevchenko (2002) is that a simple method for estimating the MLCI motion in a chaotic layer in the vicinity of the nonlinear resonance separatrix is proposed and justified. Using this method, for a number of natural planetary satellites, MLCI estimates of chaotic rotation in the main chaotic layer in the vicinity of the synchronous resonance separatrix are obtained. The value of this parameter substantially depends on the eccentricity of the satellite’s orbit and the parameter of its dynamic asymmetry. The formulas derived in Shevchenko (2002) for estimating the MLCI of the chaotic rotation of a non-spherical satellite in an el-

Chapter 9 The rotation of planetary satellites

liptical orbit make it possible to find the temporal scale of the predictability of rotational motion. According to Shevchenko (2002), the theoretical estimate of the predictability of the rotation time of the Saturn satellite Hyperion is approximately 30 days, which is only one and a half times the period of the satellite’s orbit. This estimate is consistent with direct calculations for the case of chaotic satellite rotation with Hyperion parameters in Wisdom et al. (1984). Calculation of the maximum Lyapunov exponent by the Shevchenko method (2002) when modeling the light curve of the distant satellite of Saturn, Phoebe, showed that the MLCI is zero (Kouprianov and Shevchenko, 2006). Consequently, the rotation of the Phoebe is regular. Mel’nikov and Shevchenko (2007) made conclusions about the theoretical possibility of three different types of synchronous resonance of rotation of the Saturn satellite Prometheus based on a significantly larger amount of data on the parameters of the satellite shapes. It is also shown that there is no stability with respect to the inclination of the rotation axis for one of these types of rotation. Note that the great importance of the eccentricity of the orbit of the satellite of Neptune, Nereida, served as the basis for the assumption of the randomness of its rotation. However, observations of this satellite (Grav et al., 2003) showed that rotation is regular.

References Archinal, B.A., Acton, C.H., A’hearn, M.F., Conrad, A., Consolmagno, G.J., Duxbury, T., Hestroffer, D., Hilton, J.L., Kirk, R.L., Klioner, S.A., McCarthy, D., Meech, K., Oberst, J., Ping, J., Seidelmann, P.K., Tholen, D.J., Thomas, P.C., Williams, I.P., 2018. Report of the IAU working group on cartographic coordinates and rotational elements: 2015. Celestial Mechanics and Dynamical Astronomy 130, 22. Black, G.J., Nicholson, P.D., Thomas, P.C., 1995. Hyperion: Rotational dynamics. Icarus 117 (1), 149–161. Borderies, N., Yoder, C.F., 1990. Phobos’ gravity field and its influence on its orbit and physical librations. Astronomy & Astrophysics 233, 235–251. Chapront-Touze, M., 1990. Phobos’ physical libration and complements to the ESAPHO theory for the orbital motion of PHOBOS. Astronomy & Astrophysics 235, 447–458. Chirikov, B.V., 1979. A universal instability of many-dimensional oscillator systems. Physics Reports 52 (5), 263–379. Correia, A.C.M., Leleu, A., Rambaux, N., Robutel, P., 2015. Spin-orbit coupling and chaotic rotation for circumbinary bodies. Application to the small satellites of the Pluto–Charon system. Astronomy & Astrophysics 580, L14.

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Devyatkin, A.V., Gorshanov, D.L., Gritsuk, A.N., Mel’nikov, A.V., Sidorov, M.Yu, Shevchenko, I.I., 2002. Observations and theoretical analysis of lightcurves of natural satellites of planets. Solar System Research 36 (3), 248–259. Emelyanov, N.V., Samorodov, M.Yu., 2015. Analytical theory of motion and new ephemeris of Triton from observations. Monthly Notices of the Royal Astronomical Society 454, 2205–2215. Jacobson, R.A., 1990. The orbits of the satellites of Neptune. Astronomy & Astrophysics 231 (1), 241–250. Jacobson, R.A., 2009. The orbits of the neptunian satellites and the orientation of the pole of Neptune. Astronomical Journal 137, 4322–4329. Jacobson, R.A., 2017. The Orientations of the Martian Satellites from a Fit to Ephemeris MAR097. Jet Propulsion Laboratory, Interoffice Memorandum 392R-17-004, April 20, 2017. Goguen, J., Cruikshank, D.P., Hammel, H., Hartmann, W.K., 1983. The rotational lightcurve of Hyperion during 1983. Bulletin - American Astronomical Society 15, 854. Grav, T., Holman, M.J., Klavetter, J.J., 2003. The short rotation period of Nereid. Astrophysical Journal 591, L71–L74. Harbison, R.A., Thomas, P.C., Nicholson, P.C., 2011. Rotational modeling of Hyperion. Celestial Mechanics and Dynamical Astronomy 110 (1), 1–16. Klavetter, J.J., 1989. Rotation of Hyperion. I. Observations. Astronomical Journal 97, 570–579. Kouprianov, V.V., Shevchenko, I.I., 2005. Rotational dynamics of planetary satellites: a survey of regular and chaotic behavior. Icarus 176, 224–234. Kouprianov, V.V., Shevchenko, I.I., 2006. The shapes and rotational dynamics of minor planetary satellites. Solar System Research 40 (5), 393–399. Lainey, V., Dehant, V., Patzold, M., 2007. First numerical ephemerides of the Martian moons. Astronomy & Astrophysics 465 (3), 1075–1084. LeMaistre, S., Rosenblatt, P., Rambaux, N., Castillo-Rogez, J.C., Dehant, V., Marty, J.-C., 2013. Phobos interior from librations determination using Doppler and star tracker measurements. Planetary and Space Science 85, 106–122. Mel’nikov, A.V., Shevchenko, I.I., 1998. The stability of the rotational motion of nonspherical natural satellites, with respect to tilting the axis of rotation. Solar System Research 32 (6), 480–490. Mel’nikov, A.V., Shevchenko, I.I., 2000. On the stability of the rotational motion of nonspherical natural satellites in a synchronous resonance. Solar System Research 34 (5), 434–442. Mel’nikov, A.V., Shevchenko, I.I., 2007. Unusual rotation modes of minor planetary satellites. Solar System Research 41 (6), 483–491. Melnikov, A.V., Shevchenko, I.I., 2008. On the rotational dynamics of Prometheus and Pandora. Celestial Mechanics and Dynamical Astronomy 101, 31–47. Melnikov, A.V., Shevchenko, I.I., 2010. The rotation states predominant among the planetary satellites. Icarus 209, 786–794. Noyelles, B., Lemaitre, A., Vienne, A., 2008. Titan’s rotation. A 3-dimensional theory. Astronomy & Astrophysics 478, 959–970. Noyelles, B., 2008. Titan’s rotational state. Celestial Mechanics and Dynamical Astronomy 101, 13–30. Noyelles, B., 2010a. Theory of the rotation of the Galilean satellites. In: Barbieri, C., Chakrabarti, S., Coradini, M., Lazzarin, M. (Eds.), Proceedings IAU Symposium No. 269. International Astronomical Union, pp. 240–244. Noyelles, B., 2010b. Theory of the rotation of Janus and Epimetheus. Icarus 207, 887–902. Noyelles, B., Karatekin, O., Rambaux, N., 2011. The rotation of Mimas. Astronomy & Astrophysics 536. A61.

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Noyelles, B., 2017. Interpreting the librations of a synchronous satellite – how their phase assesses Mimas’ global ocean. Icarus 282, 276–289. Pesek, I., 1991. Theory of rotation of PHOBOS. Bulletin of the Astronomical Institutes of Czechoslovakia 42 (5), 271–282. Quillen, A.C., Nichols-Fleming, F., Chen, Y.-Y., Noyelles, B., 2017a. Obliquity evolution of the minor satellites of Pluto and Charon. Icarus 293, 94–113. Quillen, A.C., Chen, Y.-Y., Noyelles, B., Loanel, S., 2017b. Tilting Styx and Nix but not Uranus with a Spin-Precession-Mean-motion resonance. arXiv:1707.03180. Rambaux, N., Van Hoolst, T., Karatekin, O., 2011. Librational response of Europa, Ganymede and Callisto with an ocean for a non-Keplerian orbit. Astronomy & Astrophysics 527, A118. Rambaux, N., Castillo-Rogez, J.C., LeMaistre, S., Rosenblatt, P., 2012. Rotational motion of Phobos. Astronomy & Astrophysics 548, A14. 11 pp. Richard, A., Rambaux, N., 2014. Complements to the longitudinal librations of an elastic 3-layer titan on a non-Keplerian orbit. Proceedings IAU Symposium 310, 21–24. Robutel, P., Rambaux, N., Castillo-Rogez, J., 2010. Analytical description of physical librations of saturnian coorbital satellites Janus and Epimetheus. Icarus 211, 758–769. Robutel, P., Rambaux, N., El Moutamid, M., 2012. Influence of the coorbital resonance on the rotation of the Trojan satellites of Saturn. Celestial Mechanics and Dynamical Astronomy 113, 1–22. Shevchenko, I.I., 2002. Maximum Lyapunov exponents for chaotic rotation of natural planetary satellites. Cosmic Research 40 (3), 296–304. Shevchenko, I.I., Kouprianov, V.V., 2002. On the chaotic rotation of planetary satellites: the Lyapunov spectra and the maximum Lyapunov exponents. Astronomy & Astrophysics 394, 663–674. Tajeddine, R., Rambaux, N., Lainey, V., Charnoz, S., Richard, A., Rivoldini, A., Noyelles, B., 2014. Constraints on Mimas’ interior from Cassini ISS libration measurements. Science 346, 322–324. Thomas, P.C., Armstrong, J.W., Asmar, S.W., Burns, J.A., Denk, T., Giese, B., Helfenstein, P., Iess, L., Johnson, T.V., McEwen, A., Nicolaisen, L., Porco, C.C., Rappaport, N.J., Richardson, J., Somenzi, L., Tortora, P., Turtle, E.P., Veverka, J., 2007. Hyperion’s sponge-like appearance. Nature 448, 50. Van Hoolst, T., Rambaux, N., Karatekin, O., Balanda, R.-M., 2009. The effect of gravitational and pressure torques on Titan’s length-of-day variations. Icarus 200, 256–264. Weaver, H.A., Buie, M.W., Buratti, B.J., Grundy, W.M., Lauer, T.R., Olkin, C.B., Parker, A.H., Porte, S.B., et al., 2016. The small satellites of Pluto as observed by new horizons. Science 351 (6279). Id.aae0030. Willner, K., Oberst, J., Hussmann, H., et al., 2010. Phobos control point network, rotation, and shape. Earth and Planetary Science Letters 294 (3–4), 541–546. Wisdom, J., Peale, S.J., Mignard, F., 1984. The chaotic rotation of Hyperion. Icarus 58, 137–152.

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10 The evolution of the orbits of the planetary satellites 10.1 The impact of various factors on the evolution of the orbits of planetary satellites We understand the evolution of the orbits of planetary satellites as slow changes in orbital elements that are considered over long time intervals. The study of evolution is directly related to the origin and fate of the Solar System. This topic is not central to this book. Therefore, we confine ourselves to a review of some of the achievements in this area. We note, however, that to model the evolution of the orbits of planetary satellites, reliable parameter values are needed that can only be found from observations. Therefore, the role of observations in the problems of studying the evolution of orbits is crucial. A description of the evolution of the motion of real satellites of the planets of the Solar System is carried out by constructing models of the evolving Keplerian satellite orbits. Speaking about the properties of evolution, we mean the behavior at large time intervals of those models of motion that we constructed based on observations. The specificity of this problem is such that small periodic deviations of the real motion from the model one are less interesting here. Mainly attention is paid to the adequacy of the behavior of the mean values of the parameters of the orbits to the real motion over long time intervals. Hypothetically, we extend the behavior of models to infinite time, but we recognize that no solution to the differential equations of motion is exact over an infinite time interval. All conclusions about the evolution of orbits remain approximate and limited in time. Satellites, under the influence of the planet’s attraction, revolve around it in almost Keplerian orbits. Their motion is influenced by other factors. The main ones are the non-sphericity of the planet, the attraction of the Sun and the mutual attraction of satellites. In most cases, for a particular satellite, one of the listed effects prevails, and the evolution of the orbit can be modeled taking into account only one, the most significant factor. However, the The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00015-7 Copyright © 2021 Elsevier Inc. All rights reserved.

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joint consideration of several factors can significantly change our knowledge of the evolution of the orbit. Let us examine in turn the influence of each factor on the evolution of the orbits of the natural satellites of giant planets and the processes of orbit evolution characteristic of different satellites.

10.2 The evolution of the orbits of satellites subject to the predominant influence of planet oblateness The non-sphericity of the planets consists mainly in their dynamic oblateness. The effect of this oblateness is described by the second zonal harmonic of the expansion of the force function of the attraction of the central planet in a series of spherical functions. To model the evolution of satellite orbits, only this decomposition term is usually taken. Note that the dynamic oblateness of the planet is a determinant factor in the evolution of orbits only for nearby planetary satellites. On the evolution of the orbits of the main satellites of the planet, its oblateness has a strong influence along with the influence of the mutual attraction of the satellites. How the orbit evolves, if only the oblateness of the planet is taken into account, is discussed above in Chapter 3. We reproduce here the formulas describing such evolution. We will use the notation adopted in Chapter 3 for elements of the Keplerian orbit: n is for the mean motion, radian/time unit; e is for the eccentricity, dimensionless units; i is for the inclination (dihedral angle between the orbit plane and the main plane Oxy), radians; M0 is for the mean anomaly at the epoch (the value of the mean anomaly M at the initial moment of time — epoch), radians; ω is for the angular distance of the pericenter from the ascending node of the orbit, radians;  is for the longitude of the ascending node of the orbit (angle in the plane Oxy between the axis x and the line of nodes), radians; t0 is for the initial moment of time, the epoch of the elements; t is for the current moment of time at which the coordinates of the body are calculated.

Chapter 10 The evolution of the orbits of the planetary satellites

Along with the mean motion of n, we also consider the semimajor axis of the orbit a associated with the third Kepler’s law  n=

μ , a3

where μ is the gravitational parameter of the planet. Considering the oblateness of the planet as the dominant factor in the evolution of the orbit, we locate the main coordinate plane perpendicular to the axis of dynamic symmetry of the planet. When studying the evolution of the orbits of planetary satellites, the averaging the perturbing function over rapidly changing arguments is performed. In this case, this argument is the mean anomaly of M. In perturbation theory, the expansion of the perturbing function in small parameters is applied. As a result, the expression for it takes the form of a multiple expansion of the mean anomaly M, the argument of the pericenter ω, and the longitude of the ascending node . The averaging over M actually consists in discarding short-period terms containing M under the sign of trigonometric functions. The dynamic structure of major planets is such that oblateness prevails in its non-sphericity, and approximately the planet can be considered an axisymmetric body. Then in the expansion of the perturbing function R we take only the second zonal harmonic with coefficient J2 and, after averaging over M, we have r02  −3,2−2p J2 F20p (i)X0 (e) cos[(2 − 2p)ω], a3 2

R = −μ

p=0

−3,2−2p

where F20p (i) and X0 (e) are the inclination and eccentricity functions, and r0 is the mean equatorial radius of the planet. Their nature turned out to be so arranged that the eccentricity functions X03,−2 (e) and X03,2 (e) are equal to zero. Therefore, the mean perturbing function due to the second zonal harmonic takes the form R = −μ

r02 J2 F201 (i)X0−3,0 (e). a3

After substituting explicit expressions for the inclination function and the eccentricity function from Appendix C, we obtain R = −μ

  r02 1 2 −3/2 3 2 sin . J (1 − e ) i − 2 4 2 a3

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Thus, only the secular term remains in the perturbing function. To determine the perturbations, it must be substituted into the Lagrange equations with respect to the satellite’s orbit. Since the secular term is independent of the elements M, ω, and , the right-hand sides of the Lagrange equation for the elements a, e, and i will be equal to zero, and these elements themselves will not contain secular perturbations. At the same time, one should not forget about the constant value that must be added to the semi-major axis, as explained in Sect. 3.10. When determining secular perturbations, we assume M = M0 + (n + n1 )(t − t0 ), ω = ω0 + n2 (t − t0 ),  = 0 + n3 (t − t0 ). If only first-order perturbations are determined in the perturbation theory, then the coefficients n1 , n2 , and n3 can be found as the values of the right-hand sides of the Lagrange equations with respect to the elements M, ω, and , respectively, after substituting constant values a, e, and i. In the case of perturbations caused by the second zonal harmonic, the expressions for n1 , n2 , and n3 have the following form: 3  r0 2 2 − 3 sin2 i , n1 = n J2 4 a (1 − e2 )3/2

(10.1)

3  r0 2 4 − 5 sin2 i n2 = n J2 , 4 a (1 − e2 )2

(10.2)

3  r0 2 cos i n3 = −n J2 . 2 a (1 − e2 )2

(10.3)

We remind the reader that we measure the elements i, ω, and  relative to the equator of the planet, which is perpendicular to the axis of its dynamic symmetry, as well as relative to the direction of the main axis Ox and the line of nodes. An essential property of secular changes in the three elements of the orbit is the smallness of the coefficients n1 , n2 , and n3 compared to the mean motion of n. From the above formulas it is seen how the orbit evolves only under the influence of the perturbing factor indicated here. If we neglect small deviations from a certain Keplerian orbit, it turns out that the plane of the satellite’s orbit slowly precesses with an almost constant inclination to the symmetry axis of the planet, the apsid line also slowly precesses, and the satellite moves along this orbit with a revolution frequency around the planet slightly changed in compared to Keplerian mean motion. The properties of the precessions are such that the orbital node always moves in

Chapter 10 The evolution of the orbits of the planetary satellites

the direction opposite to the satellite’s motion, and the apsid line, depending on the inclination, can rotate either one way or the other, remaining motionless in the plane of the osculating orbit with its inclination close to 63 degrees. The non-sphericity of the planet is not limited to its oblateness. Other terms of the expansion of the force function of planet’s gravity can also affect the evolution of the orbit. At this stage of the study, an important fact in perturbation theory is that, no matter how accurate we calculate perturbations from the non-sphericity of the planet, we will not get secular or even mixed perturbations in the elements a, e, and i. The proof can be done by mathematical induction. Instead, we simply refer to the work already cited above (Aksenov, 1966). The case of small eccentricities The evolutionary properties of the orbits of the satellites of a oblated planet described here remain valid only with significant eccentricities. If the eccentricity is small and has a value of the  2 order of J2 ra0 , then the properties of the evolution of the orbit change qualitatively. With decreasing eccentricity, short-period perturbations in the argument of the pericenter ω acquire an increasing amplitude, and as a result, its change becomes monotonic with the satellite’s velocity around the planet. At the same time, short-period perturbations of the mean anomaly M transform its change from monotonic to librational with respect to the value M = 0. This case of the motion of a satellite of a oblated planet was first studied in detail in Emel’yanov (2015). It is also considered in a special section of Chapter 3.

10.3 Evolution of the orbits of the planetary satellites under the action of the solar attraction 10.3.1 Averaging of the perturbing function It is very difficult to take into account the influence of the solar attraction on the motion of planetary satellites in the analytical theory, especially if this factor is dominant. The difficulty is due to the fact that the coefficients of secular perturbations of the elements M, ω, and  have the same order of smallness as the longperiod terms of the expansion of the perturbing function. The small parameter method in the usual perturbation theory is no longer effective. Moreover, so far it has not been possible to prove the absence of secular and mixed perturbations of high orders, as

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was done in the case of perturbations from the oblateness of the planet. However, an approximate model of the orbit evolution can be investigated. Some properties of evolution are revealed even without obtaining exact solutions of the equations of motion. Of course, when studying the evolution of orbits under the influence of the solar attraction, the motion of a planetary satellite is considered in the framework of the restricted three-body problem. The origin is located in the center of the planet. It is assumed that the Sun moves around the planet in a given unperturbed Keplerian orbit. The equations of satellite motion in rectangular coordinates can be seen in Chapter 3. The usual way to solve such problems is perturbation theory methods. The general course of the solution by perturbation theory methods is described in Chapter 3. The force function U in the equations of motion is divided into two terms U = V + W , where V is the force function of the Kepler problem. The function W is called the perturbing function. According to perturbation theory, we pass from the equations of motion in rectangular coordinates to the Lagrange equations with respect to the elements of the osculating Keplerian orbit. The equations contain the same perturbing function W , however, it must be expressed in terms of the required elements of the osculating orbit. This can only be done by expanding the perturbing function in series in powers of various small parameters. The general expansion of the perturbing function due to the attraction of the external body is discussed above in Chapter 3. To simplify the understanding of the theory of satellite orbit evolution, we reproduce here this expansion again using the accepted notation for Keplerian elements a, e, i, M, ω, and , as well as other related quantities n, λ = M + ω + , and  = ω + . The same quantities, but related to the perturbing body, we marked with the same letters, but primed. The gravitational parameter of the perturbing body (the product of the universal gravitational constant by its mass) is denoted by μ . The development under consideration has the form (Brumberg, 1967) W=

k k ∞ k ∞ ∞  a k+1 (k − m)! μ       × (2 − δm,0 )  a a (k + m)!  q=−∞  k=2 m=0 p=0 p =0

× Fkmp (i)Fkmp (i



q =−∞

k,k−2p k,k−2p  )Xk−2p+q (e)Xk−2p +q  (e )×   

(10.4)

× cos[(k − 2p + q)M − (k − 2p + q )M + + (k − 2p)ω − (k − 2p  )ω + m( −  )].

Here the argument under the cosine sign can be expressed in terms of the mean longitudes and longitudes of the pericenters

Chapter 10 The evolution of the orbits of the planetary satellites

as follows: (k − 2p + q)M − (k − 2p  + q  )M  + (k − 2p)ω− − (k − 2p  )ω + m( −  ) = = (k − 2p + q)λ − (k − 2p  + q  )λ − q + q    + + (m − k + 2p) − (m − k + 2p  ) . The expansion includes the inclination functions Fkmp (i) and the k,k−2p eccentricity functions Xk−2p+q (e). To construct a model of the evolution of satellite orbits, we use the following properties of these functions. 1. For small inclinations of the orbits, such a property of the inclination functions (Brumberg, 1967) is valid that   i |k−2p−m| i |k−2p−m| Fkmp (i) = O (sin ) , (cos ) . 2 2 It follows that, if we choose the main plane that coincides with the plane of the orbit of the external body, we get i  = 0, and only those terms for which |k − 2p  − m| = 0 remain. k,k−2p 2. At zero eccentricity, only those eccentricity functions Xk−2p+q (e) for which q = 0 are non-zero. Therefore, in the circular orbit of the external body, only terms with q  = 0 remain. k,k−2p 3. For k − 2p + q = 0, the functions Xk−2p+q (e) are expressed in the final form without the use of series expansion in eccentricity degrees. Since we are only interested in the evolution of the satellite’s orbit, short-period changes in the elements of the orbit can be neglected. Therefore, to simplify the problem, we discard all terms containing the mean longitudes of the satellite and the Sun—these are the shortest-period (fast-oscillating) terms from all terms contained in the expansion. In the literature, this operation is also called averaging over mean longitudes. It shows the so-called “twice-averaged value of the perturbing function”. Averaging over the mean longitudes λ, λ forces us to put k − 2p + q = 0, k − 2p  + q  = 0. Along with the indicated averaging of the perturbing function, we simplify the problem by considering the motion of the planet’s satellite in the framework of the circular restricted three-body problem. This means that the motion of the Sun is assumed to occur in a circular orbit in an unchanged plane. Since we consider the evolution of the satellite’s orbit only under the influence of the solar attraction, nothing prevents us from choosing the main coordinate plane that coincides with the plane of the Sun’s orbit.

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From these simplifications it follows that i  = 0 and e = 0. Then, from the above properties of the inclination functions and eccentricity functions, it follows, in turn, that only those terms for which |k − 2p  − m| = 0 and q  = 0 remain. Combining all these restrictions on the summation indices, we reduce the perturbing function to the following form: W (a, e, i, ω) =

∞  k μ    a k+1 × a a k=2 p=0

k,k−2p

× Fk0p (i)Fk,0,k/2 (0)X0

(e)X0k,0 (0) cos[(k − 2p)ω], (10.5)

where the prime at the sum sign means that the summation is performed only over the even values of the index k. The expression thus obtained for the perturbing function has the following properties: 1) it depends only on the elements a, e, i, ω, 2) is expressed in the final form with respect to e, i, ω, 3) remains in a series in even powers of the ratio of the semimajor axes of the satellite and the Sun’s orbits a/a  . Let us now consider what form the Lagrange equations will take. Given the first property of the perturbing function, we obtain √ da de 1 − e2 ∂W = 0, =− , dt dt na 2 e ∂ω di cos i 1 ∂W d ∂W = , = , √ √ dt dt na 2 sin i 1 − e2 ∂ω na 2 sin i 1 − e2 ∂i dω = dt

(10.6)

√ cos i ∂W 1 − e2 ∂W − . √ 2 2 na 2 e ∂e na sin i 1 − e ∂i

The fact that the perturbing function now depends only on a, e, i, ω allows us to derive the first three integrals of the Lagrange equations (10.6). One of them follows from the first equation: a = const. To derive the second integral, we perform the following operations: multiply the second equation term by term by e cos i −√ , 1 − e2

Chapter 10 The evolution of the orbits of the planetary satellites

√ multiply the third equation by − 1 − e2 sin i and add the results term by term. We get e cos i de

di −√ − 1 − e2 sin i = 0. dt 1 − e2 dt The left-hand side of this equality is represented as the total time derivative of the product of two functions that separately depend on e and i, and we have  d 

1 − e2 cos i = 0, dt where, integrating over time, we get

1 − e2 cos i = c1 ,

(10.7)

where c1 is an independent arbitrary integration constant. The physical meaning of the integral (10.7) consists in the constancy of √ the projection of the satellite’s momentum vector (normalized to a) on the normal to the plane of the solar orbit. This is a consequence of the axial symmetry of the doubly averaged problem (i.e., a consequence of the independence of the function W from ). The third integral is obtained by determining the total time derivative of the perturbing function W by the Lagrange differential equations (10.6). Since W depends on time only through its arguments a, e, i, ω, we have dW ∂W da ∂W de ∂W di ∂W dω = + + + . dt ∂a dt ∂e dt ∂i dt ∂ω dt Substituting now the expressions entering here for the derivatives of the elements with respect to time according to the Lagrange equations and summing up, we obtain dW = 0, dt whence, integrating over time, we have another first integral of the Lagrange equations W (a, e, i, ω) = cw ,

(10.8)

where cw is another independent arbitrary constant, and W is defined by Eq. (10.5). Eq. (10.6) is a doubly averaged modification of the well-known Jacobi integral of the circular restricted threebody problem.

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Note that the derivation of these first three integrals in this problem was first performed in Moiseev (1945), and double averaging itself was called the Gauss scheme. From the integral (10.7), one can express the inclination i as a function of the eccentricity of e and c1 . Substituting this inclination expression into the argument of the function W , we obtain the relation (e, ω, a, c1 , cw ) = 0, which defines for fixed constants a, c1 , cw the eccentricity of e as an implicit function of ω, or ω as a function of e. Determining the constants a, c1 , cw from the initial conditions, we obtain the properties of changes in the elements of the orbit of a particular satellite under the influence of the solar attraction, but without reference to time. A search for the dependence of orbital elements on time leads to quadratures whose inversion in known elementary or special functions for arbitrary values of the ratio a/a  has not yet been performed.

10.3.2 A special case—Hill problem Under the conditions of such an unsolved problem, attempts to simplify something are natural by replacing more accurate equations with less accurate ones. One of such paths is obviously due to the fact that for all planet satellites the semi-major axes of the orbits are much less than the distance of the planet from the Sun. By making a survey of the orbits of the satellites, we find that the largest value of the ratio a/a  = 0.036 is for the distant satellite of Jupiter S/2003 J2. For all other satellites currently open, this ratio is much smaller. Further, we restrict the expansion of the perturbing function in powers of the ratio of the semi-major axes so that we leave only the main term, which contains the third degree of this ratio. This version of the perturbing function is called the Hill case. Since in the expansion of the perturbing function (10.5) the summation is carried out only over the even values of the index k, the next term of the expansion after the main one contains the fifth power of the ratio of the semi-major axes of the orbits. Thus, we neglect the small value of (a/a  )2 compared to unity. For the Jovian satellite S/2003 J2, we have the value (a/a  )2 = 0.0013. As a result, the perturbing function in the Hill case has the form 2  μ  a 3 2,2−2p 2,0 F2,0,1 (0) X0 (0) F20p (i)X0 (e) cos[(2 − 2p)ω]. W2 = a a p=0

(10.9)

Chapter 10 The evolution of the orbits of the planetary satellites

Substituting here explicit expressions for the inclination functions and eccentricity functions, which can be taken from Appendix C, we obtain μ  a 3 3 4 2 2 2 2 2 2e . − sin i(2 + 3e ) + 5e sin i cos 2ω + a a  16 3 (10.10) The constant term 4/3 in square brackets on the right-hand side of Eq. (10.10) can be discarded, because the perturbing function enters the equations only under the sign of partial derivatives with respect to e, i, ω, and this term at differentiation gives zero. We list the admissions and assumptions made in deriving the perturbing function (10.10): – the function is averaged over the mean longitudes of the satellite and the Sun, – neglect the relation (a/a  )2 compared to unity, – we consider the plane of the planetocentric circular orbit of the Sun to be the main one. Under the assumptions made, the first integral of the Lagrange equations remains valid W2 =

W2 = c2 , where c2 is an independent arbitrary integration constant.

10.3.3 Analysis of families of possible changes in the eccentricity of e and the argument of the pericenter ω for a twice-averaged perturbing function in the Hill case The integrals a = const,



1 − e2 cos i = c1 , W2 = c2

were first and almost simultaneously subjected to detailed analysis in Lidov (1961, 1962, 1963); Kozai (1962) to describe the evolution of orbits whose elements satisfy the Lagrange equations (10.6). Moreover, in Lidov (1961, 1962, 1963), the evolution of the orbits of artificial Earth satellites was studied, and in Kozai (1962) the integrals were considered with respect to the equations describing the evolution of the asteroid orbit under the perturbing influence of Jupiter, taking into account in the function (10.5) terms with k = 2, 4, 6, 8.

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In Lidov (1961), by combining the second and third integrals, the relation was obtained:   2 2 2 2 − sin i sin ω = c2 , (10.11) e 5 where c2 is an independent arbitrary constant, and the integral (10.7) was written as (1 − e2 )(1 − sin2 i) = c1 ,

(10.12)

where c1 is an independent arbitrary constant. The first integrals of the Lagrange equations (10.11), (10.12) allow us to express two variables of three e, i, and ω in one. Substituting their expressions into the equations, we obtain a differential equation for one of these variables e of the form de = E(e), dt where E(e) is the well-known function of its argument. This equation is reduced to quadratures, and the inverse of quadrature, i.e. the expression of the function e(t) can be done in elliptic functions. By whom and how this was done, consider below. The jointly considered relations (10.11), (10.12), and the integral a = const made it possible to derive a number of interesting properties of the evolution of orbits even without solving the Lagrange equations. This was done for the first time in a detailed work (Lidov, 1961), devoted to the study of the evolution of artificial satellites orbits under the influence of lunar–solar perturbations. We reproduce here the main and most interesting results of this work. We also use the interpretations from the book of Shevchenko (2017). First of all, we determine the allowable values of the constants c1 and c2 . Obviously, 0 ≤ sin2 i sin2 ω ≤ 1, which implies that −3/5 ≤ c2 ≤ 2/5. It is also obvious that 0 ≤ c1 ≤ 1. However, Eqs. (10.11) and (10.12) impose narrower restrictions. The areas of possible values of c1 and c2 are shown in Fig. 10.1. These areas are bounded by the triangles 0AB and ED0. In this case, one of the sides of ED is given by a curve line, a function graph, 2

 3 5 3 (10.13) c1 = − − c2 . 3 5 5

Chapter 10 The evolution of the orbits of the planetary satellites

Figure 10.1. Areas of possible values of the constants c1 and c2 . The region L contains the values of the constants corresponding to the librational changes in the elements of the orbit e, ω, the region C corresponds to the circulation change in ω. The families of points S1 and S2 correspond to the families of orbits shown in the following figures.

The analysis shows that the argument of the pericenter ω can have one of two properties of a change in time: monotonous circulation change or oscillations with a limited amplitude relative to one of the values of 90 or 270 degrees. Such vibrations are called √ librational change. Their amplitude does not exceed arccos 2/5. For the first time, the existence of librational changes in the argument of the pericenter in this problem was discovered and studied in Lidov (1961) and Kozai (1962). Therefore, such changes are now called Lidov–Kozai resonances. Among the possible changes in the argument of the pericenter ω and the eccentricity e there are special cases, which we will consider below. The most interesting cases are those corresponding to the boundary lines of the regions of possible values 0 ≤ c1 ≤ 1, −3/5 ≤ c2 ≤ 2/5. The vertical line 0A of the plot in Fig. 10.1 separates the region of librational changes of ω at c2 < 0 and the region of circulation changes at c2 > 0. Libration orbits exist only for 0 ≤ c1 ≤ 3/5. Circulation motions are possible for the entire range of possible values of 0 ≤ c1 ≤ 1.

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The ED curve corresponds to orbits for which e and i are constants related by the equation e2 = 1 − 53 cos2 i. In this case, ω = ±90 degrees. These equilibrium points correspond to the centers of the so-called Lidov–Kozai resonance. Such orbits are constant ellipses that precess around an axis perpendicular to the main plane, the plane of the orbit of the external body, with a constant inclination to it. The segment AB corresponds to elliptical orbits with constant eccentricity. The orbits lie in the plane of motion of the external body. The inclination is zero or 180 degrees. The apse line rotates evenly with increasing ω. For these orbits, the relation 2c1 + 5c2 = 2 holds, whence, using expressions for the integrals (10.11) and (10.12), we derive (5e2 sin2 ω + 2 − 2e2 ) sin2 i = 0. Here, the first factor is equal to zero only at the point B, where e = 1 and sin ω = 0 are simultaneously. In addition to the point B, for the rest of the segment AB there should be sin i = 0. The segment BE corresponds to the orbits at c1 = 0. This condition applies either to polar orbits (cos i = 0) or to straight-line ones (e = 1). In the case of a polar orbit with c1 = 0, the relationship between the eccentricity e and the argument of the pericenter ω is described by the relation   2 2 2 − sin ω = c2 . (10.14) e 5 In the case of a rectilinear orbit with c1 = 0, the relationship between the inclination i and the argument of the pericenter ω is described by the relation 2 − sin2 i sin2 ω = c2 . 5

(10.15)

Plots for these dependencies are given in Figs. 10.4 and 10.5. The segment DA (c2 = 0, 3/5 < c1 < 1) corresponds to circular orbits (e = 0). The segment 0D (c2 = 0, c1 < 3/5) corresponds to the so-called separatrix solutions that separate the areas of libration and circulation changes in the pericenter argument ω. On these separatrices, the eccentricity asymptotically tends to zero, possibly passing through its maximum value if the initial value of ω is less than 90 degrees. For these orbits, the relation sin2 i sin2 ω = 2/5 is preserved. The points A, B, and E determine the following possible changes. The point A (c1 = 1, c2 = 0) corresponds to a circular

Chapter 10 The evolution of the orbits of the planetary satellites

equatorial orbit (e = 0, sin i = 0). The point B (c1 = 0, c2 = 2/5) corresponds to a rectilinear orbit (e = 1) with an arbitrary inclination and sin ω = 0. The point E (c1 = 0, c2 = −3/5) corresponds to polar orbits (cos i = 0) with arbitrary eccentricity and a value of ω satisfying the relation e2 (5 sin2 ω − 2) = 3, and in the case of e = 1 we have a rectilinear orbit with sin ω = ±1. The point D (c1 = 3/5, c2 = 0) corresponds to circular orbits (e = 0) with a critical inclination defined by the relation cos2 i = 3/5. Such an inclination value i is called critical due to the fact that the point D is a bifurcation point, so that when c1 decreases from the value 3/5 for c2 < 0, orbits with librational change ω appear. The point 0 (c1 = 0, c2 = 0) corresponds to the orbits of the following three types: 1) circular polar orbits (e = 0, cos i = 0) with arbitrary ω, 2) elliptical polar orbits (cos i = 0) with arbitrary e and with constant value ω (sin2 ω = 2/5), 3) rectilinear ones (e = 1) with an arbitrary inclination of the orbit under the condition sin2 i sin2 ω = 2/5. In addition to the features of the orbit evolution at the boundaries of the regions of possible values of the constants c1 and c2 , there are other special cases of evolution of orbits. Orbits with inclination greater than critical (cos2 i = 3/5) can experience large eccentricity variations, especially large if the orbit is close to the separatrix. The reason is that just at inclinations greater than critical, there is a Lidov–Kozai resonance. In the case where the inclination is 90 degrees, the eccentricity always tends to unity, whatever its initial value. Thus, the pericenter distance tends to zero, and such orbits exist only for a finite time. Note that all the special and critical cases of the properties of changes in the elements of the satellite’s orbit that correspond to the boundaries of the possible values of the constants c1 and c2 considered here are of purely theoretical interest. Among existing real satellites and possible states of their motion, such situations do not occur. Eqs. (10.11) and (10.12) allow us to construct lines on the coordinate plane (ω, e) describing the evolution of orbits. Each pair of fixed values of the constants c1 and c2 corresponds to a line on the chart. In one figure, we can construct a family of curves for one value of the constant c1 and a series of values of c2 . Let us consider two examples of such families. Fig. 10.2 shows a family of lines showing changes in ω and e. The lines correspond to the value c1 = 0.114570821 and to the series c2 in the entire range of possible values. The family of these constant values is also plotted in Fig. 10.1, where it is denoted by S1 . The leftmost point lying on the ED curve corresponds to an or-

329

330

Chapter 10 The evolution of the orbits of the planetary satellites

Figure 10.2. Family of orbits with c1 = 0.114570821 and a range of values −0.190195030 ≤ c2 ≤ 0.354171672. The horizontal axis represents the values of ω in degrees, the vertical axis represents the values of the eccentricity of e. In Fig. 10.1 this family corresponds to points on the line S1 .

bit with constant elements e = 0.750346703, i = 59.2 degrees, and c2 = −0.190195030. In Fig. 10.2 this orbit corresponds to a point— the center of the Lidov–Kozai resonance. Around this point, the concentric closed curves show the libration changes of the elements ω and e for other orbits of the family. These orbits correspond to points of the family S1 in Fig. 10.1, following from left to right. For a point with c2 = 0 in Fig. 10.2, the dashed line shows the separatrix of two types of orbits. The values e = 0.899471307 and i = 34.608220202 degrees correspond to the separatrix. The family continues with orbits with a circulation change of the elements ω and e. Finally, for a point in Fig. 10.1, lying on the boundary of AB, with c2 = 0.354171672, Fig. 10.2 shows a horizontal line of variation of ω with a constant value of e = 0.940972465. This is the orbit lying in the plane of motion of the external body, with i = 0. Note that Fig. 10.2 shows only half the range of variation of the angle ω. The second half repeats the first symmetrically with respect to the value of ω = 180 degrees.

Chapter 10 The evolution of the orbits of the planetary satellites

Figure 10.3. Family of orbits with c1 = 0.719185573 and a range of values −0 ≤ c2 ≤ 0.112325771. The horizontal axis represents the values of ω in degrees, the vertical axis represents the values of the eccentricity of e. In Fig. 10.1 this family corresponds to points on the line S2 .

Another family of orbits is shown in Fig. 10.3. These are exclusively only orbits with circulation change of the elements ω and e. This family corresponds to the family of points S2 in Fig. 10.1 with c1 = 0.719185573. The value c2 = 0 corresponds to a circular orbit with e = 0 and i = 32.0 degrees and the line in Fig. 10.3, which coincides with the horizontal axis. Further, for the points of the family following from Fig. 10.1 from left to right, we have the family of lines in Fig. 10.3 following from bottom to top. The family ends with a point with c2 = 0.112325771 on the line AB and a horizontal line in Fig. 10.3 with the value e = 0.529919264. The horizontal lines in Figs. 10.2 and 10.3 √ correspond to the boundary values of the eccentricity equal to 1 − c1 . Fig. 10.3 shows only half the range of the angle change ω. The second half repeats the first symmetrically with respect to the value of ω = 180 degrees. Note that it is possible to choose such initial conditions and the corresponding values of the constants c1 and c2 , for which during

331

332

Chapter 10 The evolution of the orbits of the planetary satellites

the evolution of the orbit the eccentricity e will change from very small values to close to unity during the evolution of the orbit. A special case is represented by orbits with initial conditions giving c1 = 0. This generates either polar orbits (sin i = 1) or rectilinear orbits (e = 1). The connection of e with ω in these special cases is given above by Eqs. (10.14) and (10.15). The plots of the possible values of e and ω for polar orbits for different values of c2 in this case are shown in Fig. 10.4. For each value of c2 we have three graphs: one in the area between the separatrices, the other from the values of 0 and 180 degrees to the sides of the separatrices. As can be seen in the plots, all such polar orbits in their evolution come to the value e = 1, which leads to the satellite falling onto the planet in a finite time, regardless of the initial values of e and ω. This property of polar (and near-polar) orbits in the literature is called the“Lidov–Kozai effect”. For rectilinear orbits with c1 = 0, the dependences of sin i on ω are shown in Fig. 10.5. The rectilinear orbits of the satellites have no practical meaning. Such satellites, if they were, would have long fallen to their planets. Here we have taken the case c1 = 0 only for completeness of the consideration of this theory. The dependences of e on ω found above do not give an idea of the direction in which the evolution of the orbits takes place. It remains to find out in which parts of the graphs ω is increasing, and in which it is decreasing. For the answer it is necessary to involve the last of the differential Eqs. (10.6). After substituting into it instead of the function W a simplified expression of this perturbing function (10.10), we obtain the explicit form of the equation for ω dω 15 μ  a 3 2 2 2 2 2 = (1 − e n (e − sin i) sin ω + ) , (10.16) dt 4 μ a 5 where μ = n2 a 3 . Given (10.11), this equation can be written   dω 15 μ  a 3 c2 2 2 2 = . n 2 + e sin ω − dt 4 μ a 5 e We now determine the value of the derivative dω dt for ω = π/2. We get    dω  15 μ  a 3 c2 3 2 = n + e . dt ω=π/2 4 μ a e2 5 It can be seen from the last equality and from the plots that in all cases of circulating change in ω (c2 > 0), including the separatrix (c2 = 0), ω increases. For all c2 < 0, excluding the case (10.13), the eccentricity of e can take two values: minimum and maximum,

Chapter 10 The evolution of the orbits of the planetary satellites

Figure 10.4. Family of polar orbits with c1 = 0 and different values of c2 . The horizontal axis represents the values of ω in degrees, the vertical axis represents the values of the eccentricity of e. The dashed line shows the separatrices of two families.

for which dω dt has different signs. At the minimum eccentricity, ω decreases, at the maximum, it increases. All directions of change ω obtained here are shown in Figs. 10.2 and 10.3. For the special case of polar orbits (c1 = 0) we have sin i = 1, and the equation for ω (10.16) takes the form   dω 15 μ  a 3 2 2 2 = − sin ω . n (1 − e ) dt 4 μ a 5 Substituting ω = 0 and ω = π/2 here sequentially, we obtain the directions of ω change shown in Fig. 10.4. The direction of change of ω for rectilinear orbits (c1 = 0) is similarly found. The equation for ω in this case (e = 1) takes the form dω 15 μ  a 3 = n (1 − sin2 i) sin2 ω > 0. dt 4 μ a The direction of change of ω in this case is shown in Fig. 10.5.

333

334

Chapter 10 The evolution of the orbits of the planetary satellites

Figure 10.5. Family of straight orbits with c1 = 0 and different values of c2 . The horizontal axis represents the values of ω in degrees, the vertical axis represents the values of sin i.

10.3.4 Orbit evolution in time for a double-averaged perturbing function in the Hill case Let us now consider how to obtain functions that describe the change in the elements of the orbits in time under the conditions of this problem. It turned out that solving equations leads to quadratures, the inversion of which can be performed only with the help of elliptic functions. Even in Kozai (1962), it was demonstrated that the solution of the problem in the Hill approximation (k = 2) can be expressed in terms of the elliptic Weierstrass functions. For a special case of initial values ω = 0, ±π/2, a solution for the elements e, i, and ω was obtained in Gordeeva (1968). A general solution for all four elements e, i, ω, and  was constructed in Vashkov’yak (1999) and almost simultaneously published in Kinoshita and Nakai (1999). Let us reproduce the results of Vashkov’yak (1999). To obtain the required functions, it is necessary to solve the Lagrange equations for the elements e, i, ω, and . We make simplifying transformations. Substituting the expression for the perturb-

Chapter 10 The evolution of the orbits of the planetary satellites

ing function (10.10) into Eqs. (10.6) and replacing the independent time variable t with the dimensionless variable τ by the formula τ=

3 μ  a 3 n (t − t0 ) , 16 μ a 

where μ = n2 a 3 , we derive the equations for the elements of the osculating orbit in the following form:

de = 10e 1 − e2 sin2 i sin 2ω, dτ

(10.17)

di = −10e2 1 − e2 sin i cos i sin 2ω, dτ

(10.18)

dω 2 =√ [5 cos2 i − 1 + e2 + 5(sin2 i − e2 ) cos 2ω], dτ 1 − e2 cos i d = 2√ (5e2 cos 2ω − 3e2 − 2). dτ 1 − e2

(10.19)

(10.20)

Eqs. (10.17), (10.18), and (10.19) have two integrals, (10.11) and (10.12). Excluding with the help of these integrals the variables i, ω from Eq. (10.17) and introducing a new variable z by the formula z = e2 , we obtain the quadrature for determining τ as a function of z sign(sin 2ω0 ) τ= √ 8 6

z √ z0

dζ , Q(ζ )

where Q(z) = (z1 − z)(z − z2 )(z − z3 ) ≥ 0, z0 is the initial value of e2 for τ = 0 (t = t0 ),  1 2 1 − c3 + (1 + c3 ) − (20/3)c1 , z1 = 2 z2 =

 1 1 − c3 − (1 + c3 )2 − (20/3)c1 , 2 5 z3 = c2 , 2

(10.21)

335

336

Chapter 10 The evolution of the orbits of the planetary satellites

5 c3 = (c1 + c2 ). 3 The general solution of Eqs. (10.17)–(10.20) will depend on four arbitrary constants e0 , i0 , ω0 , and 0 , the values of the corresponding functions in the initial point in time t0 (τ = 0). We will replace the three independent constants e0 , i0 , and ω0 with the other three independent constants z0 , c1 , and c2 related to the first relations: z0 = e02 ,

(10.22)

(1 − e02 )(1 − sin2 i0 ) = c1 ,

(10.23)

and

 e02

 2 2 2 − sin i0 sin ω0 = c2 . 5

(10.24)

The constants z0 , c1 , and c2 will be used later in the solution. The quadrature inversion (10.21) is performed in different ways depending on the value of c2 . Case for 0 < c2 < 2/5. This is a case of circulating change of ω. The roots of the polynomial Q(z) satisfy the inequalities z2 < 0 < z3 ≤ z(τ ) ≤ z1 , where z1 and z3 are the maximum and minimum values of e2 , respectively. Eccentricity is determined by the formula e(τ ) = where



 z(τ ) =

z3 − z2 k 2 sn2 u , 1 − k 2 sn2 u

(10.25)

u = 4 6(z1 − z2 )τ + u0 , k2 =

z1 − z3 < 1, z1 − z2

u0 = sign(sin 2ω0 )F (ϕ0 , k 2 ), sin2 ϕ0 =

(10.26)

(z1 − z2 )(z0 − z3 ) , (z1 − z3 )(z0 − z2 )

and sn u and F (ϕ, k 2 ) are, respectively, the Jacobi elliptic sine and the incomplete elliptic integral of the first kind with the module k.

Chapter 10 The evolution of the orbits of the planetary satellites

The oscillation period of the eccentricity e is given by the formula K(k 2 ) , Te = √ 2 6(z1 − z2 )

(10.27)

where K(k 2 ) is a complete elliptic integral of the first kind with the module k. The circulation period ω is twice the period of oscillations of the eccentricity e Tω = √

K(k 2 ) . 6(z1 − z2 )

(10.28)

By virtue of the integral (10.12), the period of oscillations of the inclination i is equal to the period of oscillations e. Case for −3/5 < c2 < 0. This case is characterized by the librational change of ω. Here z3 < 0 < z2 ≤ z(τ ) ≤ z1 , where z1 and z2 are the maximum and minimum values of e2 , respectively. The eccentricity is determined by the formula e(τ ) = where



 z(τ ) =

z2 − z3 κ 2 sn2 u , 1 − κ 2 sn2 u

(10.29)

u = 4 6(z1 − z3 )τ + u0 , κ2 =

z1 − z2 < 1, z1 − z3

u0 = sign(sin 2ω0 )F ( 0 , κ 2 ), sin2 0 =

(10.30)

(z1 − z3 )(z0 − z2 ) . (z1 − z2 )(z0 − z3 )

Here the module of elliptic functions is κ. The periods of oscillation of the eccentricity e, the inclination i and the argument of the pericenter ω coincide and are determined by the formula K(κ 2 ) T e = Ti = Tω = √ . 2 6(z1 − z3 )

(10.31)

337

338

Chapter 10 The evolution of the orbits of the planetary satellites

We mention here a fact important for programming computing. Eqs. (10.29)–(10.31) are obtained from Eqs. (10.25)–(10.27) on replacing z3 by z2 and z2 by z3 . A special case with c2 = 0 and c1 < 3/5. This is a borderline between the two solutions discussed above. The elements e and ω change along the separatrix according to the equation   2 c1 sin2 ω = 0. − 1− 5 1 − e2 Maximum eccentricity  emax =

5 1 − c1 3

is reached at angles ω that satisfy the relation sin2 ω = 1. When the eccentricity reaches zero, the argument of the pericenter has one of the values satisfying the relation 2 sin2 ω = (1 − c1 ). 5 The time dependence of the elements e, ω, and i for this special case was obtained in Vashkov’yak and Lidov (1991). We reproduce the result here. Without compromising the generality of the statement of the problem and for simplicity of formulas, we assume that at the initial moment t0 the elements have values e = emax and ω = 90 degrees and introduce the parameter: √ √ 3 6 μ 3 6 μ  a 3 β =− e = − emax . max 4 na  3 4 μ a Now the dependence of the eccentricity on time for this special case is determined by the formula e(t) =

emax . cosh[β(t − t0 )]

It is important to determine in which direction the argument of the pericenter ω changes at the moment  t0 , when the eccentricity

reaches the maximum value e = emax = 1 − 53 c1 . At this point, ω = 90 degrees. Substituting these values in Eq. (10.19), we obtain    dω 5 3 1 − c1 > 0. = 12 dt 5c1 3

Chapter 10 The evolution of the orbits of the planetary satellites

Therefore, ω increases at the moment when e = emax . Then, as can be seen in the separatrix graph in Fig. 10.2, ω increases at all time moments. A special case with c1 = 0. In this case, the orbit belongs to one of two types: polar (sin i = 1) or linear (e = 1). Possible values of elements with c1 = 0 and different values of c2 are shown above in the plots. The motion of satellites in all these orbits causes the satellite to fall on the planet. Such orbits are not of practical interest. Therefore, we do not consider the dependence of elements on time. Other special cases. 1. If c2 = 0 and c1 > 3/5, then the eccentricity is always zero, and the argument of the pericenter is not defined. 2. If the initial conditions are such that between the constants c1 and c2 the relation 2

 5 3 c1 , c2 = − 1− 5 3 and at the same time c1 < 3/5, then all three elements e, ω, and   i remain constant: ω = ±90 degrees and e = 1 − 53 c1 . 3. The case c1 = 1 corresponds to a circular orbit lying in the plane of motion of the external body. 4. In the case when c2 = 25 (1 − c1 ), we have an inclination i = √ 0, a constant eccentricity equal to em = 1 − c1 , and an argument of the pericenter, constantly increasing with angu√ lar velocity, which varies from 8 c1 at ω = k · 180 degrees to √ 20(1 − 3/5c1 )/ c1 at ω = 90 + k · 180 degrees, where k is any integer. The circulation period ω, as in the general case, is found by Eq. (10.28). At this stage of the presentation, we have expressions for e(τ ) as a function of time for all possible values of the constants c1 , c2 , and z0 . The required functions i(τ ) and ω(τ ) can now be found from the relations  c1 , i(τ ) = arccos 1 − e2 (τ )  2  − 2c2  5 e (τ ) ω(τ ) = arcsin . sin2 i(τ ) Now we just have to find the function (τ ). We continue to reproduce here the results of the work (Vashkov’yak, 1999). First, we perform the actions for the case 0 < c2 < 2/5.

339

340

Chapter 10 The evolution of the orbits of the planetary satellites

We use the formulas (10.26) to determine the relationship of the variable u with τ . Then we define a new variable u, setting u = mK(k 2 ) + u, where m is the integer closest to u/K(k 2 ) not exceeding this number. The number m will be remembered for further actions. Next, we calculate successively the constants l2 = k2

C=

1 − c1 − z2 , 1 − c1 − z3

(5c2 − 2z2 )k 2 , (1 − c1 − z3 )K(k 2 )

I(l 2 , k 2 ) =

(10.32)

 (l 2 , k 2 ) − K(k 2 ) , l2

where  (l 2 , k 2 ) is the complete elliptic integral of the third kind. Now we define some function J (x, l 2 , k 2 ) from its argument x as follows: J (x, l 2 , k 2 ) =

(ϕ, l 2 , k 2 ) − F (ϕ, k 2 ) , l2

(10.33)

where ϕ = arcsin(sn x), where (ϕ, l 2 , k 2 ) and F (ϕ, k 2 ) are incomplete elliptic integrals of the third and first kind, respectively. Next, we define the function Jm (u, l 2 , k 2 ), depending on the evenness of the number m, by the formula Jm (u, l 2 , k 2 ) =

⎧ ⎨ J (u, l 2 , k 2 ), if m − even,

⎩ I(l 2 , k 2 ) − J (K(k 2 ) − u, l 2 , k 2 ), if m − uneven. (10.34) Finally, the function (τ ) is calculated by the formula  √ (τ ) = 0 − 4 c1 sign[cos i(τ )] [1 + C I(l 2 , k 2 )]τ +  C + √ (u0 − u)I(l 2 , k 2 )+ 4 6(z1 − z2 )    Jm (u, l 2 , k 2 ) − J (u0 , l 2 , k 2 ) K(k 2 ) , (10.35)

where 0 is another independent arbitrary constant. It can be shown that in the case under consideration of the circulating change of ω k 2 < l 2 < 1.

Chapter 10 The evolution of the orbits of the planetary satellites

The period of circulation change  is determined by the approximate formula π . T = √ 2 c1 [1 + CI(l 2 , k 2 )] In the case of 3/5 < c2 < 0 (libration ω), the dependence (τ ) is defined by the formulas (10.32)–(10.35), in which z2 is necessary to replace by z1 , z3 by z2 , k 2 by κ 2 , ϕ0 by 0 , and use Eqs. (10.30) for the connection of u with τ . As a result, the found dependences e(τ ), i(τ ), ω(τ ), (τ ) for given at the initial time moment t = t0 (τ = 0) arbitrary constants e0 , i0 , ω0 , 0 describe the evolution of the satellite’s orbit. Let us recall here what assumptions were made in this statement of the problem: – only perturbations from the attraction of the Sun act on the satellite, – the perturbing function is averaged over the mean longitudes of the satellite and the Sun, – neglect the relation (a/a  )2 compared to unity, – we consider the plane of the planetocentric circular orbit of the Sun to be the main one. Based on the constructed theory of Lidov (1961), an important property of the evolution of satellite orbits was discovered: for c1 ≈ 0 and i ≈ π/2, the eccentricity approaches one, the pericentric distance becomes equal to the physical radius of the planet, and the satellite will inevitably fall to the surface. The existence of the main satellites of Uranus, which are also subject to this effect, is explained by Lidov (1963) by the stabilizing effect of the planet’s oblateness (Lidov, 1963).

10.3.5 Applications of the constructed theory of orbit evolution in studying the dynamics of real planetary satellites The constructed model of the evolution of satellite orbits under the solar attraction can be tested on real distant satellites of the planets, since it really is the dominant factors for these satellites. It should be borne in mind that the constructed model is approximate, since some assumptions are used. Therefore, the conclusions of the theory must be verified by different methods. Note that the solution to the problem of determining the satellite’s orbit as a function of time, performed in Vashkov’yak (1999), was used in the same paper to study the evolution of the orbits of two distant satellites of Uranus, discovered in 1997. These satellites were later named U16 Caliban and U17 Sycorax.

341

342

Chapter 10 The evolution of the orbits of the planetary satellites

Applications of the theory of orbit evolution discussed above were also made in Kinoshita and Nakai (1999) for the distant satellite of Uranus, U17 Sycorax, and the distant satellite of Neptune, Nereid. In both aforementioned works, the amplitudes and periods of oscillations of the eccentricities of the orbits of these two satellites were determined from the values of the elements of the orbits found from the observations. Table 10.1 gives the initial data and the results of such studies. The initial values of the orbital elements were taken from the IAU circular (Marsden, 1998). It turned out that both satellites have a circulation change in the argument of the pericenter ω. Vashkov’yak (1999) indicated that the results obtained using the analytical model of evolution “practically coincide” with the results of the control numerical integration of the evolution equations (10.17)–(10.20), to which the terms corresponding to the effect of the oblateness of Uranus were added. Similar results for the satellite U17 Sycorax were obtained in Kinoshita and Nakai (1999). These results, together with what is obtained by numerically integrating the equations, are also given in Table 10.1. In Kinoshita and Nakai (1999) for the distant satellite of Neptune, Nereid, the parameters of the orbit evolution were also obtained using the formulas of the theory considered above. The satellite Nereid has a very large eccentricity of the orbit (approximately 0.76). Its motion, in addition to perturbations from the solar attraction, is also affected by the attraction of the nearby massive satellite Triton. The obtained evolution parameters for Nereid were compared with the results of numerical integration and with the analytical theory of satellite’s motion constructed in Mignard (1981). These results are given in Table 10.2. Nereid’s orbit has a circulating character of the change in the argument of the pericenter ω. Later, after the work of Kinoshita and Nakai (1999), the orbit evolution of Nereid was investigated in Vashkov’yak and Teslenko (2010). The authors of this work built a more detailed model of the evolution of Nereid using more advanced methods.

10.4 Refined models of the evolution of the orbits of planetary satellites. Numerical analytical method In reference (Vashkov’Yak, 2005), a combined numerical analytical method was developed for modeling the evolution of the orbits of distant satellites of giant planets. This method was pro-

Chapter 10 The evolution of the orbits of the planetary satellites

343

Table 10.1 The initial parameters of the satellites and the results obtained on the evolution of the orbits. The notation “analytical theory” means that the results were obtained using the orbit evolution model described above. Parameter

U16 Caliban

U17 Sycorax

Epoch of elements

1998, July 6.0

1998, July 6.0

The semi-major axis, AU

0.047921

0.081643

Period of circulation, deg/day 0.047921

0.081643

Eccentricity

0.82347

0.509386

Inclination (geo-eclipt.), deg

139.6813

152.6686

ω, deg

339.4621

18.0055

, deg

174.9928 255.8085 From the work (Vashkov’yak, 1999) analytic theory

emin

0.07686

0.4964

emax

0.2861

0.5848

imin , deg

139.65

151.72

imax , deg

142.47

160.46

Tω , year

8272.1

1239.2

T , year 6577.4 1734.1 From the work (Kinoshita and Nakai, 1999) analytic theory emax



0.603

imax , deg



160.5

Tω , year



1220

T , year – 1780 From the work (Kinoshita and Nakai, 1999) numerical integration. emax



0.605

imax , deg



160.8

Tω , year



1350

T , year



1770

posed for a long time ago, and with its help many problems of satellite dynamics were solved. The idea of the method and related studies performed by Lidov together with students were reflected in the article (Lidov, 1978). The essence of the method is as follows. We take differential equations for the six elements of the Keplerian orbit with a perturbing function due to the solar attraction. In

344

Chapter 10 The evolution of the orbits of the planetary satellites

Table 10.2 The parameters of the evolution of the orbit of the satellite of Neptune, Nereid, obtained in Kinoshita and Nakai (1999) by numerical integration of Eqs. (Num. integr.), according to the analytical theory of motion of this satellite in Mignard (1981) and according to the formulas of the theory of evolution in Kinoshita and Nakai (1999). Parameter

Num. integr.

Mignard (1981)

K.,N., 1999

Tω , year

13600

13400

13670

T , year

17690

15000

17980

Amplitude e

0.00546

0.0060

0.00548

3.09

3.137

Amplitude i, deg 3.123

the expansion in powers of the ratio of the semi-major axes of the satellite and the Sun, we take only the main term, that is, the Hill approximation. The rectangular coordinates of the Sun are calculated by the elements of the Keplerian orbit of the planet. The equations for the elements of the satellite’s orbit were written in canonical form Vashkov’Yak (2005). The original equations contained all six variables. Next, the Zeipel method was used to exclude members from the equations with the period of revolution of the satellite and the period of revolution of the Sun. Since these short-period terms are not discarded in advance in the Zeipel method, members of different orders appear in the resulting equations with respect to the small parameter characterizing the perturbing factor. In the above work, we used the small parameter m determined by the relation   2 μ  a 3 n m = = .  μ a n 2

After canonical transformations, it turned out that in the new variables the semi-major axis of the satellite’s orbit remains constant, and four of the required functions remain. For the Sun, the mean anomaly disappears from consideration, and the semi-major axis of the orbit is assumed constant. We also use expansion in powers of small quantities e and i  . Elements of the solar orbit e , i  , ω , and  are considered to be some given functions of time. These time functions are found from the Lagrange theory of secular perturbations, where the Lagrange elements associated with the elements e , i  , ω , and  are used by the relations h = e sin(ω +  ), k  = e cos(ω +  ),

Chapter 10 The evolution of the orbits of the planetary satellites

p  = sin i  sin  , q  = sin i  cos  . For each of these elements, the sum of trigonometric functions of the form   N  sin (j ) (j ) (j ) (νk t + βk ), Ak (10.36) ej = cos k=1

where

ej

(j = 1, 2, 3, 4) is one of the Lagrange elements, N is the (j )

(j )

(j )

number of planets in theory, and Ak , νk , βk (k = 1, 2, ..., N ), some numbers, are constant of theories of planetary motion, which are found from observations. Eq. (10.36) takes the sine function for the elements h , p  and the cosine for k  , q  . The evolution equations obtained in this way (Vashkov’Yak, 2005) are not presented, but it is said that they are then solved by the method of numerical integration at time intervals of several thousand orbital periods of the aforementioned satellites. Due to the fact that the right-hand sides of the equations obtained for the mean elements do not contain rapidly oscillating functions, their numerical integration is performed with a large step, of the order of several orbital periods of the satellite’s revolution, i.e. much more effective than equations in rectangular coordinates or in osculating elements. Note that the equations obtained in this way contain expansions in powers of small parameters. The terms are proportional to the following combinations of small quantities: m2 , m3 , m4 , m2 e , and m2 i  . We also note that not all terms proportional to m4 are taken into account in the equations in this way, since only the main term of the Hill problem is taken in the original expansion of the perturbing function. In the above-mentioned work and subsequent papers that use this method in the numerical integration of evolutionary equations, the theory of secular perturbations of Brauer–Wurcom was used to represent the motion of the Sun, and the numerical values of the constants of the theory of planetary motion were taken from the work (Sharaf and Budnikova, 1967). The initial values of the satellite orbit elements in the work (Vashkov’Yak, 2005) and in the subsequent work of this author were taken from the publications MPECs (Minor Planets Electronic Circulars). On time intervals of the order of 105 –106 years, extreme values of the eccentricities and inclinations of a number of distant satellites of major planets, as well as periods of change in the arguments of the pericenters and longitudes of the ascending nodes of their orbits, have been obtained. At times of the order of the periods of circulation of longitudes of nodes (102 –103 years), a comparison is made with the results of

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numerical integration of the equations of motion written relative to the rectangular coordinates of the satellite and the Sun. In numerical integration, the rectangular coordinates of the Sun were calculated from the mean elements of Keplerian orbits, expressed as series in powers of time. The series themselves were taken from Bretagnon and Francou (1988). The results of such modeling of the orbits of distant planetary satellites are published in Vashkov’Yak and Teslenko (2005); Vashkov’yak and Teslenko (2008a,b). In two papers (Vashkov’yak and Teslenko, 2009; Vashkov’yak, 2010), the analytical method was improved by taking into account additional terms of the expansion of the perturbing function. In addition to the main term proportional to m2 , the terms proportional to m3 and m4 are also taken into account, respectively. Using the proposed solution, improved analytical time dependences were obtained for the elements of the evolving orbits of a number of distant satellites of giant planets in comparison with the solution of the twice-averaged Hill problem and thereby achieved their best agreement with the results of numerical integration of the equations of perturbed satellite motion in rectangular coordinates.

10.5 The evolution of the orbits of planetary satellites under the combined influence of various factors Attempts to construct a theory of the evolution of satellite orbits with the combined influence of various factors lead to a very difficult problem. We will further consider only those satellites that move in outer orbits with respect to the main satellites of the planet. In the problem of the evolution of the orbits of such planetary moons, the main factors are: – non-sphericity of the planet (mainly its dynamic oblateness), – the attraction of the Sun, – the attraction of the main satellites (the most massive satellites, the movement of which is considered given). The basis for obtaining the evolutionary system of equations is the secular part of the full perturbing function W , which is found by its independent averaging over all “fast” variables: the mean planetocentric longitudes of the Sun, main satellites, and the studied (real or hypothetical) satellite, i.e. excluding the short-period part. Thus, the function W depends only on five planetocentric Keplerian orbital elements:

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a, the semi-major axis, e, the eccentricity, i, the inclination, ω, the pericenter argument, and , the longitude of the ascending node. As follows from the Lagrange equations, in the elements, due to the independence of W from the mean longitude of the probe satellite, the semi-major axis of its orbit remains constant, and this function itself gives us the first integral of the evolution system W = const. In accordance with the perturbing factors accepted here for consideration, we divide the full perturbing function into three terms W = W0 + W1 + W2 , where W0 is due to the oblateness of the planet (second zonal harmonic), W1 is due to the solar attraction, and W2 is caused by the attraction of the main satellites. In the coordinate system with the main plane coinciding with the plane of the orbit of the Sun or with the plane of the equator of the planet, the terms W0 and W2 depend on the five elements of the satellite’s orbit a, e, i, ω, and . Therefore, the Lagrange equations will have only the first two integrals a = const and W = const. Their general analytical solution regarding orbital elements under arbitrary initial conditions was not found. In the work devoted to this problem, the term W0 is associated with the secular term, which depends on the inclination of the satellite’s orbit to the equator of the planet and on its eccentricity. In the function W1 , only the main term of the expansion in powers of the ratio of the semi-major axis of the satellite to the radius of the circular orbit of the Sun is taken (Hill approximation). As for the function W2 , the most advanced work on this topic (Vashkov’yak et al., 2015) uses the expansion in powers of the ratio of the semi-major axis of the orbit of the perturbing main satellite to the semi-major axis of the satellite whose motion is being studied. Due to the complexity of the problem and the vastness of work on this topic, we restrict ourselves here only to a review of the work carried out from 1961 to the present, and a description of the main results. The work of Lidov (1961) and Kozai (1962) revealed the main features of the evolution of satellite orbits under the influence of secular perturbations only from an external attracting point. This, in particular, is the effect of a strong increase in the eccentricity of the orbit with a constant semi-major axis with a simultaneous

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decrease in the distance of the pericenter, until the satellite falls to the surface of the planet. This effect, called the Lidov–Kozai mechanism, occurs when the satellite orbit is inclined to the plane of motion of the pertutbing point by an angle close to 90 degrees. Since the orbits of the overwhelming majority of known satellites are quite far from the orthogonal location relative to the plane of the heliocentric orbit of the planet, the fall effect for them cannot manifest itself. An exception are the main and close satellites of Uranus. The almost equatorial and almost circular orbits of these satellites are tilted to the plane of the planet’s orbit at angles that differ from the right angle only by about 8 degrees. In Lidov (1963), the problem of the evolution of satellite orbits under the combined influence of the solar attraction and the oblateness of the planet was considered. Estimates of the minimum and maximum values of the eccentricity in the presence of both pertubing factors are obtained. These values significantly depend on the relationship between the parameters characterizing the effect of planet oblateness and the attraction of the external body. In Lidov (1963), the most remote of the main satellites of Uranus, Oberon, was taken as an example. For this satellite, it turned out that, taking into account the combined effect of the solar attraction and the predominant effect of planet oblateness, the amplitude of the long-period oscillations of the eccentricity is approximately 10−5 . The oblateness of Uranus abundantly compensates for secular solar perturbations. In Lidov (1963), it was noted that “a significant change in the nature of evolution when taking into account the non-Keplerian main field is not typical of the planets of the Solar System”. An example of a hypothetical Earth satellite similar to the Moon, but with an inclination of the orbit to the ecliptic plane of 90 degrees, is considered. Estimates show that such a satellite could only make 52 turns, that is, it would exist in orbit for about 4 years. In the exact solution of this problem, obtained by numerical integration of the equations of motion, the minimum distance of the satellite’s orbit became less than the radius of the Earth after 55 revolutions. The study of the evolution of satellite orbits under the combined influence of the solar attraction and the oblateness of the planet continued. Following attempts were made to find particular cases of the problem when the averaged equations could be integrated. In Lidov and Yarskaya (1974), all special cases of integrability of the Lagrange equations in this problem are listed. This includes extreme cases of the mutual orientation of the orbit of the Sun, the equator of the planet and the plane of the satellite’s

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orbit (coplanarity, orthogonality). A special case of the satellite’s circular orbit was also considered. The search for solutions with special relationships between the parameters characterizing the effect of planet oblateness and the attraction of an external body continued. See the work of (Vashkov’yak 1996; 1998a; 1998b; Vashkov’yak and Teslenko, 2001) and the references given therein. They consider the special cases of the relations between: – the initial elements of the satellite’s orbit, – the angle between the planes of the orbit of the Sun and the equator of the planet, – the parameter γ=

μr02 J2 a  3 μ a  5

.

Particular stationary solutions for the elements e and ω are found. Families of periodic changes in these elements are found. The stability of these solutions is investigated. Some features of the satellite system of Uranus are investigated in the work of (Vashkov’yak, 2001a,c; Vashkov’yak and Teslenko, 2002). The combined effect of the attraction of the Sun and the oblateness of the planet is taken into account. The reasons for the lack of satellites in the equatorial plane in some areas around the planet are revealed. It is noted that the same features cannot explain the distribution of orbits over distances for the distant satellites of Jupiter and Saturn. In Vashkov’yak et al. (2015), evolutionary equations of the problem of secular perturbations of a satellite’s orbit are obtained under the combined influence of three perturbing factors (oblateness of the central planet, the attraction of its main satellites and the attraction of the Sun), integrable cases are described and possible ways of their research are outlined. In this new evolutionary restricted many-body problem, the most interesting is the study of the area of near-planet space in which the influence of the indicated perturbations on the satellite in pairs or in aggregate is comparable in magnitude. Based on the theoretical findings, the satellite system of Uranus is considered. For a wide range of semi-major axes of the orbits, using the obtained approximate analytical dependences and numerical estimates, the influence of the main satellites of Uranus on the evolution of the orbits of some of its real and hypothetical satellites is revealed.

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10.6 Classification of the orbits of the distant satellites of Jupiter, Saturn, Uranus, and Neptune according to the types and properties of the orbit evolution With the beginning of the discoveries in 1997 of the first distant satellites of Uranus, the new distant satellites of Neptune, Jupiter, and Saturn, work began on the classification of the types of evolution of their orbits. The properties of the solutions of evolution equations for the mean perturbing function in the Hill case were taken as the basis, i.e. the main term of its expansion in the relations of the semi-major axes of the orbits of the satellite and the Sun. This is the case that is discussed in detail in Sects. 10.3.2–10.3.5. We will give here a brief review of the work and the results obtained on this problem. After the first applications of the found solutions of the equations for the elements e, i, ω, and  in Vashkov’yak (1999), the author of this paper continued the analysis for newly discovered satellites. The orbital evolution of the new outer satellites of Uranus (Vashkov’yak, 2001a; 2001c), the new outer satellites of Saturn (Vashkov’yak, 2001b), and the new outer satellites of Jupiter (Vashkov’yak, 2001d) was analyzed. Vashkov’yak (2003) analyzed the evolution of the orbits of the new distant satellites of Neptune. A comparison of the libration type orbits for the distant satellites of Jupiter, Saturn, and Neptune is made. Interesting features of the evolution of these orbits are revealed. The features of the secular evolution of the orbits of the hypothetical satellites of Uranus are described in Vashkov’yak (2016). In addition to dividing the orbits into two types of evolution: with the librational change in the argument of the pericenter and with its circulation change, there are other parameters of the properties of evolution. A very detailed analysis of the types of orbit evolution was made in Cuk and Burns (2004). The applications of this analysis to the problem of the origin of distant satellites of large planets are also considered. A special case was the evolution of the orbit for the distant satellite of Jupiter J34 Euporie. According to the classification based on solving equations with a mean perturbation function in the Hill case and the circular orbit of the Sun (see Sects. 10.3.2–10.3.5), the change in the pericenter argument should have a circulating character, since the value of the constant c2 in the integral (10.11) calculated from the known elements of the orbit from the observations is positive. However, an analysis of the time

Chapter 10 The evolution of the orbits of the planetary satellites

variations of the osculating element ω obtained from the numerical integration of the equations of motion in rectangular coordinates shows that in the time interval of 1000 years this element experiences fluctuations with respect to the value of 90 degrees with an amplitude of about 20 degrees. In the work (Vashkov’yak and Teslenko, 2007) this case was investigated. A model of the evolution of the orbit of the satellite J34 Euporie is constructed with averaging the perturbing function only over the mean satellite longitude. The dependences of the orbit elements on the mean longitude of the Sun are obtained. It turned out that in the element ω, perturbations associated with the period of the revolution of the Sun around the planet dominate. It is these periodic perturbations that are found in the change in the osculating element of ω. In Brozovic and Jacobson (2017), the orbits of all the currently known distant satellites of Jupiter based on observations were refined. Motion modeling was carried out by numerical integration of the equations of motion. Integration was performed not only for the time interval of observations and the time of the necessary ephemeris, but also for the time interval of 1000 years to determine the types of evolution of satellite orbits. Changes in the longitudes of the pericenters are estimated. If the rate of change of the pericenter argument is less than the precession speed of the orbit node, then the satellite is called a reverse circulator (RC). In the opposite situation, the satellite is called a direct circulator (DC). The relationship between the rate of change of the longitude of the satellite pericenter and the planet’s longitude is considered. If these speeds are equal, they speak of the secular resonance of a satellite with a planet. In Cuk and Burns (2004); Brozovic and Jacobson (2017), a parameter is also calculated that characterizes the degree of proximity of the state of evolution of the orbit to secular resonance. A table is given containing for all the distant satellites of Jupiter the values of the elements of the osculating orbits: mean, minimum, maximum, as well as the periods of changes in the longitudes of the pericenters and longitudes of the nodes. A type of evolution is noted for each satellite. The enumeration of all distant satellites indicating the type of evolution would be too voluminous for this presentation. We do not provide this data. If necessary, they can be found in the work cited above. We only note that the satellites are grouped into some “clusters” with similar types of orbit evolution.

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10.7 The evolution of the orbits and rendez-vous occurrences of distant satellites of the planets 10.7.1 Modern knowledge about the evolution of the orbits of distant planetary moons Studying the orbital evolution and rendez-vous occurrences of distant satellites serves the purpose of establishing their origin. It is also important for understanding the spread of matter in the early stages of the evolution of the Solar System. The mutual attraction of satellites is very weak due to their small size and mass. However, at very large time intervals, the mutual rendez-vous can be close enough to significantly change the orbits of the satellites. The motion of distant satellites of planets is of great interest because of a number of features of their orbits. The orbital motion of distant satellites is subject to a strong influence of the solar attraction. The orbital planes of these satellites, continuously changing their position in space, are not associated with either the equator plane or the planet’s orbital plane. Most of these satellites move around the planet in the opposite direction with respect to its orbital motion. The eccentricities of the orbits are significant and, in addition, undergo significant changes. The listed properties suggest that distant satellites appeared as a result of their capture by the planet from heliocentric orbits. Within the framework of the three-body problem model (the Sun – planet – satellite), such a capture can only be temporary. Therefore, to confirm the capture hypothesis, it is necessary to find the reason for the transition of these bodies from the state of temporary capture to a stable state. In the work (Grav and Bauer, 2007) at least five processes are mentioned that could lead to such a transition. The first possible reason is the collision of a temporarily captured body with another, already existing satellite of the planet. The second one is capture due to an increase in the mass of the planet. The third reason is the inhibitory effect of interplanetary gas or a gas cloud around the planet. The fourth possible reason is the capture of two mutually gravitating bodies at the same time. The fifth reason is the capture during the passage of the resonance state of the satellite’s orbital motion with the planet’s orbital motion. A number of studies performed in recent years have investigated various specific capture mechanisms. These are including the capture mechanism in the framework of the four-body problem – the Sun, planet, double asteroid (Gaspar et al., 2011), capture

Chapter 10 The evolution of the orbits of the planetary satellites

in the three-body model with the additional influence of gas resistance (McGleam et al., 2007) or changes in the mass of the planet (de Oliveira et al., 2007), capture taking into account the tidal disruption of a smaller body (Philpott et al., 2010), capture as a result of mutual collisions of small bodies immediately after the formation of regular satellites (Ma et al., 2009), capture during the passage of a resonance (Cuk and Gladman, 2006), capture of objects from a planetesimal disk during close passages of the planets of the Jovian group (Nesvorny et al., 2007a,b), capture of comet type objects in the early stages of the formation of major planets of the Solar System (Jewitt and Haghighipour, 2007). Here it is beyond our intentions to provide an exhaustive and detailed review of the work related to the orbital dynamics of irregular satellites of giant planets. However, it is advisable to identify some areas of such research. A new model of secular evolution of the irregular satellites of Saturn was proposed in Cuk and Burns (2004), and the work (Burns et al., 2004) was devoted to the evolution of orbits close to the apsidal libration orbit of the satellite of Saturn S24 (Kiviuq). We point out a detailed work (Nesvorny et al., 2003), in which, mainly by numerical methods, the “orbital and collisional evolution” of 60 thousand fictitious satellites of giant planets, as well as 50 real distant satellites, were studied. Using calculations performed on a time interval of 108 years, in particular, the stability regions of the orbits of irregular satellites, as well as their individual groups, similar to families of asteroids, were identified. A number of studies have addressed the celestialmechanical aspects of the motion of irregular satellites, including orbital and secular resonances (Beauge and Nesvorny, 2007; Correa et al., 2010), as well as the presence of phase space regions with periodic, conditionally periodic, and chaotic motions (Hinse et al., 2010; Frouard et al., 2009; 2010; Tsirogiannis et al., 2009). Analytical methods for studying the orbit evolution of irregular satellites are the subject of studies that refine the well-known (Lidov, 1961; 1962; 1963; Kozai, 1962) model of the twice-averaged Hill problem taking into account third-order terms with respect to a small parameter, the ratio of the mean planetary and satellite motions (Kovalevsky, 1964; Orlov, 1965a,b; Beauge et al., 2006; Vashkov’yak and Teslenko, 2009). We point out that to study the evolution of the orbits of “ultradistant” satellites, such as, for example, the satellites of Neptune N10 (Psamathe) and N13 (Neso), a non-standard fourth-order constructive-analytical method was proposed (Vashkov’yak, 2010). We also note the studies of the orbital instability of satellites in the intermediate regions of the near-planet space between the orbit of the farthest of regular satellites and the orbit of the closest of

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irregular ones. In these areas, the lifetimes of hypothetical satellites, although they vary depending on the eccentricities and inclinations of their orbits, are, however, cosmogonically very small (Vashkov’yak, 2001c; Vashkov’yak and Teslenko, 2002; Haghighipour and Jewitt, 2006). In addition, we also mention a survey and bibliographic study of the dynamics of the natural satellites of the planets made by Ural’skaya (2003). Two among the five previously mentioned reasons for permanent capture are associated with the mutual gravitational attraction of satellites. Since the masses of these bodies are very small, a significant change in the properties of motion can occur only as a result of rather close mutual rendez-vous of satellites. We can consider the satellite motion as motion along the osculating Keplerian orbit. Due to the strong influence of the solar attraction, the eccentricity and orientation of the osculating orbit in space significantly change. This can lead to rapprochement and even to the intersection of the trajectories of different satellites. To detect them, it is necessary to simulate the motion of satellites over very long time intervals. Due to the limited accuracy of observations and the very short observation time intervals of most distant planetary satellites, it is impossible to calculate ephemeris at large time intervals with the accuracy necessary for reliable detection of satellite proximity. This is confirmed by the analysis of the accuracy of the ephemeris of distant satellites performed in Emelyanov (2010). However, the orientation of the elliptical osculating orbit in space, its eccentricity and the semi-major axis can be determined with acceptable accuracy over long time intervals. In this case, the position of the satellite in a planet-centric orbit can be known with an accuracy of 180 degrees. Therefore, only large sizes, shape and orientation of orbits can be simulated at large time intervals. Only the evolution of the orbits can be studied without asking the question of where the satellite itself is in orbit. Then, in the problem of the approach of satellites, it is possible to determine only the minimal distances between the orbits, taking into account that the approach of satellites takes place only in the “approaching orbits”. The rendez-vous occurrences of the satellites themselves can only be considered as a random process. The orbits of many distant satellites are determined at such short time intervals that the reliability of the configurations of their orbits modeled at large time intervals can be as low as the accuracy of the calculated positions of the satellites in orbit. In such cases, the approach of the orbits themselves can also be considered as a random process and only its probabilistic characteristics can be estimated. One can also count on increasing the reliability

Chapter 10 The evolution of the orbits of the planetary satellites

of the description of the orbital configurations when new observations of distant planetary satellites appear. At long time intervals, the facts of the rendez-vous occurrences of satellites certainly also affect the evolution of orbit configurations. The study of these processes is a special problem. The picture of the evolution of the orbits of planetary satellites must be constantly updated. We are still far from the final confirmation of the hypotheses.

10.7.2 The problem of calculating and detecting the rendez-vous occurrences of distant planetary satellites The above circumstances give rise to the urgent problem of studying the evolution of orbits and the possible rendez-vous occurrences of distant planetary satellites. In Emelyanov and Vashkov’yak (2012) an attempt was made to solve this problem. In particular, a method and computational programs have been developed that allow one to pre-calculate mutual paired “orbital rapprochements” of distant planetary satellites at long time intervals. The solution of the formulated problem consisted of the following steps. First, we must determine the satellite motion parameters from the observations or the initial conditions for solving the equations of motion. Then, according to the parameters found, it is necessary to carry out a numerical integration of the equations of satellite motion over a time interval acceptable for solving the problem. Next, we need to select suitable fairly simple analytical functions that will represent the results of numerical integration and make it possible to calculate the parameters of the orbits at any time in a long interval. And finally, we need to develop a method that allows us to calculate the minimum distance between the orbits at any time. These tools can be used to search for moments of “rapprochements” of the orbits of distant satellites of the planets and the probable rendez-vous occurrences of satellites. The first stage was performed earlier in the work of Emelyanov (2005); Emel’Yanov and Kanter (2005) for all 107 distant satellites of Jupiter, Saturn, Uranus, and Neptune. The found initial conditions for solving the equations of motion of satellites are available on the MULTI-SAT planet satellite ephemeris server (Emel’Yanov and Arlot, 2008) at http://www.sai.msu.ru/neb/nss/ html/multisat/index.html. To find this data, we need to successively follow the hyperlinks “instructions”, “sources”, “Original numerical models of motion of distant planetary satellites”, and

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“Initial conditions for integration ...”. Note that the parameters of the satellite orbits are regularly updated as new observations appear. The latest version of this data was taken at the time of writing (Emelyanov and Vashkov’yak, 2012). Changes in the orbits of the distant satellites of Jupiter, Saturn, Uranus, and Neptune over time intervals of several thousand years were found by numerical integration of the equations of motion in Vashkov’yak and Teslenko (2008a,b). In this work, the evolution of satellite orbits was also studied by analytical and semi-analytical methods. In view of the small discrepancies in the results obtained by three different methods, it seems to us appropriate to use the results of numerical integration. Since after the publication of the indicated papers, the initial conditions of motion of a number of satellites were recalculated, an independent numerical integration of the equations of motion according to the new values of the initial conditions was performed in Emelyanov and Vashkov’yak (2012). Based on the results of numerical integration, tables for five elements of the osculating orbits of all 107 distant satellites were compiled at intervals of 300 to 15,000 years (for satellites of different planets) with a constant time step. The time interval and step in the tables were chosen so that for each period of the satellite’s revolution at least several tabular moments fell, and the entire interval included one or more periods of motion of the node line and the apses of the orbit. The compiled tables contain the following elements of the Keplerian osculating orbit: semi-major axis, eccentricity, inclination to the ecliptic plane, longitude of the ascending node and the angular distance of the pericenter from the ascending node in the ecliptic coordinate system.

10.7.3 An analytical description of the evolution of satellite orbits According to the theory of motion of distant planetary satellites, changes in the considered osculating elements of the orbit in time can be represented by trigonometric series. The terms of these series are called “harmonics”. Harmonic frequencies are linear combinations with integer coefficients of a finite set of generating frequencies. These frequencies occur from secular and periodic changes in mean longitudes, longitudes of nodes and longitudes of the pericenters of the orbits of satellites, the planet and other bodies that perturb the motion of satellites. Among the harmonics of the expansion of the osculating elements of the satellite orbits, there are a small number of dominant harmonics and many other harmonics with smaller amplitudes, but of approxi-

Chapter 10 The evolution of the orbits of the planetary satellites

mately the same order. The time dependences of the longitudes of the ascending nodes and the arguments of the pericenters in the ecliptic coordinate system, as a rule, have secular terms. It is interesting to note that in a fairly large ensemble of distant satellite orbits of giant planets, only a few orbits with librational change in the arguments of the pericenters are known. The first identified omega librators are Saturn’s satellites S24 (Kiviuq) and S22 (Ijiraq). The nature of the changes in the osculating elements of the orbits of the distant planetary satellites does not allow us to practically represent all harmonics frequencies as combinations of generating frequencies. The widely used Fourier analysis method in this case does not give an acceptable result. Therefore, in Emelyanov and Vashkov’yak (2012), an attempt was made to develop a method for constructing functions that approximate changes in orbital elements in the form of trigonometric sums with different, unrelated frequencies. Such a method was developed, algorithmized and applied. In fact, a new original method of frequency analysis was proposed in this work. This method has a significant feature. In those cases when a number of harmonics with precisely expressed frequencies and amplitudes are present in the signal under investigation, these harmonics are recognized exactly. Their frequencies and amplitudes are calculated with the accuracy of the representation of numbers in a computer.

10.7.4 Determination of minimal distances between the orbits of distant planetary satellites The approximating functions of five elements of the osculating orbits for all 107 distant satellites of Jupiter, Saturn, Uranus, and Neptune were found using the frequency analysis described above. These functions can be used to study the evolution of satellite orbits over long time intervals. The question of the accuracy of the representation of elements of osculating orbits beyond the time interval of the initial sequence of values by which the approximation was made remains unexplored. We confine ourselves to the note that accuracy slowly deteriorates with time from the boundaries of this interval. One of the applications of the functions found can be the study of the circumstances of mutual rendez-vous occurrences of satellites. These rendez-vous occurrences can occur very rarely. However, at very long time intervals, many rendez-vous occurrences accumulate and they can determine the evolution of the orbits, in particular, turn the satellite’s temporary capture (from the heliocentric orbit to planetocentric) into permanent.

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If the osculating orbits of the two satellites at some point in time have areas that are close in space, then there is the possibility of close rendez-vous occurrences of the satellites themselves. Of interest is the problem of establishing all the facts of “reapprochment” or intersection of the orbits of pairs of satellites over a sufficiently long time interval. So we arrive at the problem of calculating the minimal distance in space between two confocal ellipses given by five parameters. In the literature, the term Minimum Orbital Intersection Distance (MOID) has been established to denote this value. An algorithm for approximating the required minimum distance can be very simple. We should evenly mark each orbit with points and calculate the distances between all pairs of points on two different ellipses. Then choose the minimum of the obtained values. It is clear that such a method will require considerable computational time and will give a result limited in accuracy. The problem of determining MOID was considered in Baluev and Kholshevnikov (2005). The authors of this work showed that the solution of the problem is reduced to the search for the real roots of the trigonometric polynomial of the eighth degree. For the practical solution of the problem, a computer program was kindly provided by the authors of the work (Baluev and Kholshevnikov, 2005). Elements of their Keplerian orbits are set as input data for the program for each of the two satellites: semi-major axis a, eccentricity e, inclination i, angular distance of the pericenter from the ascending node ω, and longitude of the ascending node . The result of the program is the values of the true anomalies for the points of each orbit, between which the minimum distance and the calculated value of this distance are reached. The procedure for determining the minimum distance between the orbits of pairs of satellites for a sequence of time instants with a given step at a given interval was compiled using the specified computing program. Elements of the orbits are calculated by the approximating functions found, as described above. The procedure also identifies cases where between two consecutive moments of time there were intersections of orbits, as two lines in space.

10.7.5 Proposed Internet source for the study of the evolution of the orbits and rendez-vous of distant planetary satellites All the methods and procedures described above are implemented in the form of a computer program, the use of which is

Chapter 10 The evolution of the orbits of the planetary satellites

organized via the Internet. The interface with the program is arranged on the pages of a special site on the Internet. After selecting a satellite system by the name of the planet, the user is offered the following actions. 1. Displaying on a separate page the parameters of trigonometric polynomials approximating the change of the five elements of the Keplerian osculating orbit of one distant satellite of the planet selected from the proposed list. In addition to the coefficients and frequencies, some related data are issued. 2. Display on the screen on a separate page of the table of values of all five elements of the Keplerian osculating orbit of the selected satellite with a given time step at a given interval. At the user’s request, instead of the elements e, i, ω, and , functions of the elements e sin ω, e cos ω, sin i sin , and sin i cos  can be given. 3. Display on the screen on a separate page of the table with the minimum distances between orbits of two selected satellites with a given time step at a given interval. The table also gives the values of the mutual inclination of the two orbits. After this table, a list of moments of all intersections of two orbits that occurred on a given time interval is displayed. These moments are calculated with the precision up to a time step in the table of minimum distances. The proposed Internet source is available in the “Planet Natural Satellites Service” at http://www.sai.msu.ru/neb/nss/html/multisat/index.html We need to follow the hyperlink “Outer Natural Satellites Orbits Ephemerides” at the end of the page.

10.7.6 Examples of calculating the minimum distances between the orbits of satellites Let us now consider an example of computing the time dependence of the minimum distance between the orbits of two satellites. For a pair of satellites J6 (Himalia) and J7 (Elara), this dependence is shown in Fig. 10.6, and for satellites J6 (Himalia) and J8 (Pasiphae) in Fig. 10.7. The points on the time axis show the moments of intersections of the orbits. Note that the satellites J6 (Himalia) and J8 (Pasiphae) belong to different families of orbits of the distant satellites of Jupiter. The semi-major axes of their orbits differ by about half. It turns out that the intersections of the orbits of these satellites happen as often as the intersections of the orbits of the two satellites J6 (Hi-

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Figure 10.6. Change of the minimum distance between orbits of satellites J6 (Himalia) and J7 (Elara). The dots show the moments of intersection of two orbits. The lines do not reach zero due to the discreteness of the reference points of the graph.

malia) and J7 (Elara), which belong to the same group and have approximately equal semi-major axes of the orbits. For 54 distant satellites of Jupiter, the mutual distances and moments of orbit intersection were calculated for all 1431 combinations of satellite pairs over a thousand-year time interval from 2010 to 3010. Only 155 out of 1431 combinations of satellite pairs did not have a single intersection of orbits in this time interval. For the remaining pairs, 2 to 330 intersections occurred. The maximum number of intersections of 330 was found at the orbits of the satellites J33 (Euanthe) and J40 (Mneme). The extreme values of the elements of the evolving orbits of these satellites, as well as the periods of circulation of the nodes and arguments of the pericenters, are very close to each other, as can be seen from the data presented, in particular, in Vashkov’yak and Teslenko (2008a). For all considered pairs of satellites, the total time intervals were also calculated during which the mutual distances between the orbits did not exceed 100 thousand km. Only in 214 pairs among 1431 satellites orbits did not come closer to each other closer than 100 thousand km in the considered interval of a thousand years.

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Figure 10.7. Change of the minimum distance between orbits of satellites J6 (Himalia) and J8 (Pasiphae). The dots show the moments of intersection of two orbits. The lines do not reach zero due to the discreteness of the reference points of the graph.

In the remaining pairs, the orbits were at a closer distance for a total time of 2 to 313 years. The maximum total time of approaching distances of less than 100 thousand km was 313 years for the pair J23 (Kalyke) and S/2003 J9. For this pair, the extreme values of the inclinations, as well as the periods of circulation of the nodes and the arguments of the pericenters, turn out to be rather close: 160◦ ≤ i ≤ 168◦ , 87.4 years, 80.4 years, and 158◦ ≤ i ≤ 167◦ , 86.8 years, and 76.6 years, respectively. Similar calculations on the same time interval of a thousand years were made for 38 distant satellites of Saturn. In 703 combinations of satellite pairs, only 46 did not have orbit intersections. We have the maximum number of intersections, 124, for the orbits of the satellites S19 (Ymir) and S/2007 S3. For this pair, only the extreme inclinations are close enough: 169◦ ≤ i ≤ 176◦ and 171◦ ≤ i ≤ 177◦ , respectively (Vashkov’yak and Teslenko, 2008b). In 41 pairs, the orbits did not approach at distances less than 100 thousand km. The maximum total time of orbit rendez-vous was 149 years for a pair of satellites S/2004 S12 and S/2004 S13.

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A similar analysis for nine distant satellites of Uranus should be done over a longer time interval, since the periods of revolution of the lines of nodes and apses of the orbits of these satellites are much larger than those of the orbits of the satellites of Jupiter and Saturn. We took a time interval of 4 thousand years, starting in 2010. Of the 36 combinations of satellite pairs for only 10 pairs, the orbits did not intersect and did not “come closer” to a distance of less than 100 thousand km. The orbits of the satellites U19 (Setebos) and U21 (Trinculo) had a maximum number of intersections (91 events). It is interesting to note that the extreme values of the semi-major axis of the orbit of satellite U19 are more than double the corresponding values for U21. Apparently, the maximum number of intersections of the orbits of this pair for the Uranus system is due to the possibility of a noticeable increase in the eccentricity of the U19 orbit to a value of approximately 0.7. The maximum total orbital convergence time of 485 years in the studied interval of 4 thousand years was found for a pair of satellites U16 (Caliban) and U21 (Trinculo). Orbits of six distant satellites of Neptune, including N2 (Nereid), experience much less intersections and rendez-vous. The studies were conducted on a time interval of 8000 years, since the periods of revolution of the lines of nodes and lines of apses of the orbits of distant satellites of Neptune are very large. With the orbit of Nereid, there were six intersections of only the orbit of satellite N9 (Halimede). The orbits of the other five distant satellites had mutual intersections and “rapprochements” at distances of less than 100 thousand km. The maximum number of intersections (73 events) in the time interval of 8000 years, had satellites N10 (Psamathe) and N13 (Neso). These are the most distant satellites from Neptune, for which solar perturbations lead to a very noticeable evolution of their orbits and, in particular, to an increase in eccentricities to values approximately equal to 0.87. Note that the orbit of satellite N13 has a librational change in the argument of the pericenter, under the conditions of the well-known Lidov– Kozai resonance. On the contrary, the argument of the pericenter of the satellite’s orbit N10 changes as circulation. Apparently, such a qualitative difference in the evolution of these orbits, together with the possibility of their strong “stretching”, contributes to the relatively frequent orbital intersections of this pair. The maximum time of “rapprochement” of 167 years, at a distance of less than 100 thousand km was at the orbits of satellites N9 (Halimede) and N12 (Laomedeia). It is interesting to note that unlike all the above pairs of satellites with reverse motions, one of the satellites of this pair N12 moves in orbit with an inclination less than 90◦ . This leads to the “oncoming” motion of the orbital nodes, which,

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apparently, also contributes to their frequent intersections during evolution. Thus, the developed method and computational programs make it possible to carry out various and extensive studies of mutual spatial displacements, as well as changes in the size and shape of the evolving orbits of the distant satellites of Jupiter, Saturn, Uranus, and Neptune. The above examples only demonstrate the capabilities of the proposed tools.

10.7.7 Conclusion The evolution of the orbits of the distant satellites of the planets is determined by significant perturbations in their motion from the Sun. These perturbations cause significant changes in the elements of the osculating Keplerian orbits. However, changes in the eccentricities and inclinations of the orbits are limited by known limits. The mutual attraction of satellites is very small due to their small size and mass. However, at very large time intervals, the mutual approach of satellites can be so close that their orbits can undergo large changes. Rendez-vous occurrences can affect the status of such a celestial body as a satellite of the planet. The detection of satellite rendez-vous is very difficult due to the impossibility of a sufficiently accurate calculation of their positions over very long time intervals. A promising way of research in this direction may be the pre-calculation of mutual configurations of the orbits of distant planetary satellites. The methods and means of calculation developed and proposed with this tool are designed to help advance on this path. At least now, we can conclude that the orbits of the ensemble of distant satellites at multiple time intervals of several thousand years have multiple intersections and “rendez-vouse” occurrences at which the satellites themselves can come together. The classification of satellites into separate groups is not directly related to the possible rendez-vous occurrences of satellites belonging to different groups.

10.8 Refinement of the Laplace–Lagrange secular perturbation theory In Nikonchuk (2012), a method was developed for calculating secular perturbations in the problem of the motion of a system of small bodies around a massive central body. Compared with the classical Laplace–Lagrange theory based on linear equations, third-degree terms with respect to eccentricities and orbital incli-

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nations are taken into account. The corresponding refinement of the solution turns out to be significant when studying the evolution of orbits over long time intervals. The proposed method was applied to study the motion of the main satellites of Uranus. It is shown that at time intervals of more than 100 years, the contribution of the new obtained secular perturbations for the Miranda satellite is of the order of the eccentricity of the orbit, which corresponds to several thousand kilometers. For other satellites, the effect of accounting for nonlinear terms is smaller. Obviously, when constructing the general analytical theory of motion of the main satellites of Uranus, it is necessary to take into account nonlinear terms in the equations for secular perturbations. The corresponding method was constructed, investigated, and proposed in Nikonchuk (2012).

References Aksenov, E.P., 1966. Odin vid differentsialnykh uravnenii dvizheniia sputnika. Trudy Gosudarstvennogo astronomicheskogo instituta im. P.K. Sternberga, Moscow 35, 44–58. In Russian. Baluev, R.V., Kholshevnikov, K.V., 2005. Distance between two arbitrary unperturbed orbits. Celestial Mechanics and Dynamical Astronomy 91, 287–300. Beauge, C., Nesvorny, D., Dones, L., 2006. A high-order analytical model for the secular dynamics of irregular satellites. Astronomical Journal 131 (4), 2299–2313. Beauge, C., Nesvorny, D., 2007. Proper elements and secular resonances for irregular satellites. Astronomical Journal 133 (6), 2537–2558. Bretagnon, P., Francou, G., 1988. Planetary theories in rectangular and spherical variables—VSOP 87 solutions. Astronomy and Astrophysics 202, 309–315. Brozovic, M., Jacobson, R.A., 2017. The orbits of Jupiter’s irregular satellites. Astronomical Journal 153, 147. Brumberg, V.A., 1967. Development of the perturbation function in satellite problems. Bulletin of the Institute of theoretical astronomy, Leningrad 11 (2), 73–83. In Russian. Burns, J.A., Carruba, V., Nesvorny, D., Cuk, M., Tsiganis, K., 2004. Chaos effects of planetary migration for the saturnian satellite Kiviuq. American Astronomical Society, DDA meeting # 35, # 07.06 Bulletin of the American Astronomical Society 36, 860. Correa, O.J., Leiva, A.M., Giuppone, C.A., Beauge, C., 2010. Mapping the νsolar secular resonance for retrograde irregular satellites. Monthly Notices of the Royal Astronomical Society 402 (3), 1959–1968. Cuk, M., Burns, J.A., 2004. On the secular behavior of irregular satellites. Astronomical Journal 128, 2518–2541. Cuk, M., Gladman, B.J., 2006. Irregular satellite capture during planetary resonance passage. Icarus 183, 362–372. de Oliveira, D.S., Winter, O.C., Neto, E.V., de Felipe, G., 2007. Irregular satellites of Jupiter: a study of the capture direction. Earth, Moon, and Planets 100 (3–4), 233–239. Emelyanov, N.V., 2005. Ephemerides of the outer Jovian satellites. Astronomy and Astrophysics 435, 1173–1179.

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Emel’Yanov, N.V., Kanter, A.A., 2005. Orbits of new outer planetary satellites based on observations. Solar System Research 39 (2), 112–123. Emel’Yanov, N.V., Arlot, J.-E., 2008. The natural satellites ephemerides facility MULTI-SAT. Astronomy and Astrophysics 487, 759–765. Emelyanov, N.V., 2010. Precision of the ephemerides of outer planetary satellites. Planetary and Space Science 58, 411–420. Emelyanov, N.V., Vashkov’yak, M.A., 2012. Evolution of orbits and encounters of distant planetary satellites. Study tools and examples. Solar System Research 46 (6), 423–435. Emel’yanov, N.V., 2015. Perturbed motion at small eccentricities. Solar System Research 49 (5), 346–359. Frouard, J., Fouchard, M., Vienne, A., 2009. Chaoticity of the jovian irregular satellites. In: American Astronomical Society, DPS Meeting # 41, # 38.08. Frouard, J., Fouchard, M., Vienne, A., 2010. The Long-term evolution of the Jovian irregular satellites. American Astronomical Society, DDA meeting # 41, # 9.11 Bulletin of the American Astronomical Society 41, 936. Gaspar, H.S., Winter, O.C., Vieira Neto, E., 2011. Irregular satellites of Jupiter: capture configurations of binary-asteroids. Monthly Notices of the Royal Astronomical Society 415, 1999–2008. Gordeeva, Yu.F., 1968. The time-dependence of orbital elements in long-period oscillations in the restricted three-body problem. Cosmic Research 6 (4), 450–453. Grav, T., Bauer, J., 2007. A deeper look at the colors of the saturnian irregular satellites. Icarus 191, 267–285. Haghighipour, N., Jewitt, D., 2006. Stability of Jovian irregular satellites between Callisto and Themisto. American Astronomical Society, DPS meeting # 38, # 64.09 Bulletin of the American Astronomical Society 38, 613. Hinse, T.C., Christou, A.A., Alvarellos, J.L.A., Gozdziewski, K., 2010. Application of the MEGNO technique to the dynamics of jovian irregular satellites. Monthly Notices of the Royal Astronomical Society 404, 837–857. Jewitt, D., Haghighipour, N., 2007. Irregular satellites of the planets: products of capture in the early solar system. Annual Review of Astronomy and Astrophysics 45 (1), 261–295. Kinoshita, H., Nakai, H., 1999. Analytical solution of the Kozai resonance and its application. Celestial Mechanics and Dynamical Astronomy 75 (2), 125–147. Kovalevsky, J., 1964. Sur la theorie du mouvement d’un satellite a fortes inclinaison et excentricite. The Theory of Orbits in the Solar System and in Stellar Systems. In: Kontopoulos, Georgios Ioannou (Ed.), Proceedings From Symposium No. 25, Held in Thessaloniki. August 17–22, 1964. In: International Astronomical Union. Symposium, vol. 25. Academic Press, London, p. 326. Kozai, Y., 1962. Secular perturbations of asteroids with the high inclination and eccentricities. Astronomical Journal 67, 591–598. Lidov, M.L., 1961. An approximate analysis of the evolution of artificial satellites. In: Problems of Motion of Artificial Celestial Bodies: Reports at the Conference on General and Applied Problems of Theoretical Astronomy. Moscow, November 20–25, 1961, pp. 119–141. In Russian. Lidov, M.L., 1962. The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Planetary and Space Science 9 (10), 719–759. Lidov, M.L., 1963. On the approximated analysis of the orbit evolution of artificial satellites. In: Roy, M. (Ed.), Dynamics of Satellites. Symposium Paris, May 28–30, 1962. Springer-Verlag, Berlin, pp. 168–179.

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Lidov, M.L., Yarskaya, M.V., 1974. Integrable cases in the problem of the evolution of a satellite orbit under the joint effect of an outside body and of the noncentrality of the planetary field. Cosmic Research 12 (2), 139–152. Lidov, M.L., 1978. Semi-analytical methods for computation of satellite motion. Tr. Inst. Teor. Astron. Leningrad. 17, 54–61. In Russian. Ma, Y., Zheng, J., Shen, X., 2009. On the origin of retrograde orbit satellites around Saturn and Jupiter. In: Icy Bodies of the Solar System, Proceedings of the Int. Astron. Union. IAU Symposium, vol. 263, pp. 157–160. Marsden, B.G., 1998. Central Bureau for Astronomical Telegrams, 6869, p. 6870. McGleam, C., Hamilton, D.P., Agnor, C.B., 2007. Three-body capture of irregular satellites. In: American Astronomical Society, DDA Meeting # 38, # 3.01. Mignard, F., 1981. The mean elements of Nereid. Astronomical Journal 86, 1728–1729. Moiseev, N.D., 1945. On nekotorykh osnovnykh uproshchennykh skhemakh nebesnoi mekhaniki, poluchaemykh pri pomoshchi osredneniia ogranichennoi krugovoi problemy trekh tochek. 2. Ob osrednennykh variantakh prostranstevennoi ogranichennoi krugovoi problemy trekh tochek. Trudy Gosudarstvennogo astronomicheskogo instituta im P.K. Sternberga 15 (1), 100–117. Publications of the Sternberg State Astronomical Institute, Izd. MGU, Moskva. In Russian. Nesvorny, D., Alvarellos, J.L.A., Dones, L., Levison, H.E., 2003. Orbital and collisional evolution of the irregular satellites. Astronomical Journal 126, 398–429. Nesvorny, D., Vokrouhlicky, D., Morbidelli, A., 2007a. Capture of irregular satellites during planetary encounters. American Astronomical Society, DPS meeting # 39, # 32.11 Bulletin of the American Astronomical Society 39, 475. Nesvorny, D., Vokrouhlicky, D., Morbidelli, A., 2007b. Capture of irregular satellites during planetary encounters. Astronomical Journal 133, 1962–1976. Nikonchuk, D.V., 2012. Nonlinear theory of secular perturbations of satellites of an oblate planet. Astronomy Letters 38 (12), 813–828. Orlov, A.A., 1965a. Luni-solar perturbations in the moving of the artificial Earth satellites. In: Proc. of 15th Intern. Congr. on Astronaut. Paris: Gautier-Villard, vol. 1. PWN-Polish Sci. Publ., Warstzawa, pp. 141–157. Orlov, A.A., 1965b. An approximate analytical representation of spatial motions in the Hill problem. Bulletin ITA (Leningrad) 10 (5), 360–378. In Russian. Philpott, C.M., Hamilton, D.P., Agnor, C.B., 2010. Three-body capture of irregular satellites: application to Jupiter. Icarus 208 (2), 824–836. Sharaf, S.G., Budnikova, N.A., 1967. On the secular changes in the elements of the Earth’s orbit, affecting the climates of the geological past. Bulletin ITA (Leningrad) 11 (4), 231. In Russian. Shevchenko, I.I., 2017. The Lidov–Kozai Effect—Applications in Exoplanet Research and Dynamical Astronomy. Astrophysics and Space Science Library, vol. 441. Springer International Publishing, Switzerland. ISBN 978-3-319-43520-6. Tsirogiannis, G.A., Perdios, E.A., Markellos, V.V., 2009. Improved grid search method: an efficient tool for global computation of periodic orbits. Application to Hill’s problem. Celestial Mechanics and Dynamical Astronomy 103 (1), 49–78. Ural’skaya, V.S., 2003. Dynamics of planetary satellites in the solar system. Solar System Research 37 (5), 337–365. Vashkov’yak, M.A., Lidov, M.L., 1991. Evolution of certain types of satellite orbits. Cosmic Research 28, 689–692.

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Vashkovyak, M.A., 1996. On the special particular solutions of a double-averaged Hill’s problem with allowance for flattening of the central planet. Astronomy Letters 22 (2), 207–216. Vashkov’yak, M.A., 1998a. On the families of periodically evolving orbits in Hill’s averaged problem with allowance for oblateness of the central planet. Astronomy Letters 24 (2), 185–192. Vashkov’yak, M.A., 1998b. On the stability of stationary solutions of the double-averaged hill’s problem with an oblate central planet. Astronomy Letters 24 (5), 682–691. Vashkov’yak, M.A., 1999. Evolution of the orbits of distant satellites of Uranus. Astronomy Letters 25 (7), 476–481. Vashkov’yak, M.A., Teslenko, N.M., 2001. Stability of periodic solutions for Hill’s averaged problem with allowance for planetary oblateness. Astronomy Letters 27 (3), 198–205. Vashkov’yak, M.A., 2001a. Orbital evolution of Uranus’s new outer satellites. Astronomy Letters 27 (6), 404–409. Vashkov’yak, M.A., 2001b. Orbital evolution of Saturn’s new outer satellites and their classification. Astronomy Letters 27 (7), 455–463. Vashkov’yak, M.A., 2001c. Celestial-mechanical peculiarities of Uranus’s satellite system. Astronomy Letters 27 (7), 464–469. Vashkov’yak, M.A., 2001d. Orbital evolution of Jupiter’s new outer satellites. Astronomy Letters 27 (10), 671–677. Vashkov’yak, M.A., Teslenko, N.M., 2002. Peculiarities of Uranus’s satellite system. Astronomy Letters 28 (9), 641–650. Vashkov’yak, M.A., 2003. Orbital evolution of new distant neptunian satellites and omega-librators in the satellite systems of Saturn and Jupiter. Astronomy Letters 29 (10), 695–703. Vashkov’Yak, M.A., 2005. A numerical–analytical method for studying the orbital evolution of distant planetary satellites. Astronomy Letters 31 (1), 64–72. Vashkov’Yak, M.A., Teslenko, N.M., 2005. Orbital evolution of the distant satellites of the giant planets. Astronomy Letters 31 (2), 140–146. Vashkov’yak, M.A., Teslenko, N.M., 2007. Peculiarities of the orbital evolution of the Jovian satellite J34 (Euporie). Astronomy Letters 33 (11), 780–787. Vashkov’yak, M.A., Teslenko, N.M., 2008a. Evolution characteristics of Jupiter’s outer satellites orbits. Solar System Research 42 (4), 281–295. Vashkov’yak, M.A., Teslenko, N.M., 2008b. Evolutionary characteristics of the orbits of outer Saturnian, Uranian, and Neptunian satellites. Solar System Research 42 (6), 488–504. Vashkov’yak, M.A., Teslenko, N.M., 2009. Refined model for the evolution of distant satellite orbits. Astronomy Letters 35 (12), 850–865. Vashkov’yak, M.A., Teslenko, N.M., 2010. On the evolution of the orbit of Nereid. Solar System Research 44 (1), 44–54. Vashkov’yak, M.A., 2010. Constructive analytical solution of the evolution Hill problem. Solar System Research 44 (6), 527–540. Vashkov’yak, M.A., Vashkov’yak, S.N., Emel’yanov, N.V., 2015. On the evolution of satellite orbits under the action of the planet’s oblateness and attraction by its massive satellites and the sun. Solar System Research 49 (4), 247–262. Vashkov’yak, M.A., 2016. Secular evolution of the orbits of hypothetical satellites of Uranus. Solar System Research 50 (6), 390–401.

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11 Physical parameters of planetary satellites 11.1 Introduction There are several sources of knowledge of the physical parameters of planetary satellites. Before the start of space missions to other planets, research was carried out using ground-based observations. Spacecraft observations provide a wealth of new information about the physical properties of satellites. For this, optical and radio-technical measuring instruments are installed on board. Ground-based positioning of the spacecraft is performed with very high accuracy. This allows us to determine the masses of satellites by their gravitational influence on the motion of the spacecraft. Each new space mission gives a new version of the physical parameters of the satellites. The disadvantage of such studies is the short duration of measurements using spacecraft. However, some data can only be obtained by long-term systematic measurements. In contrast to short-term measurements from the spacecraft, ground-based observations can be carried out continuously. There are two fundamentally different ways of obtaining data on the physical properties of satellites. The first one is the direct measurement of physical parameters. Thus one measures, for example, the brightness or magnitude of an object. Photometry using various filters and spectral measurements provide information on the properties of surface material. Another way of determining physical parameters is indirect, by the manifestations of the physical properties of satellites in their motion. Astrometric observations of one satellite make it possible to determine the mass of another satellite, affecting the motion by its gravitational attraction. In practice, we do this: when determining motion parameters from observations, the masses of satellites are included in the number of refined parameters. The equations necessary for this are given in Chapter 6. Note that, in fact, due to the very small masses of the satellites, the manifestation of their mutual attraction in the observation results is at the limit of the accuracy of observations. The only successful case of determining the mass of one satellite from astrometric observations of another satellite is described here below in a special section. The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00016-9 Copyright © 2021 Elsevier Inc. All rights reserved.

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Determining the motion of satellites by astrometric observations also allows us to obtain the physical parameters of the planet: its mass and the parameters of its gravitational field. This is also done by including the required parameters in the number of refined ones. Moreover, it is necessary to take into account all significant perturbing factors so that the influence of an unaccounted perturbation does not lead to false results in the determination of the required parameters. According to astrometric observations of satellites, it is possible to determine not only the parameters of the planet’s gravitational field, but also the parameters characterizing the visco-elastic properties of the planet’s bodies and the satellite. The necessary equations are given at the end of Chapter 3. The reflective properties of satellite surfaces also relate to physical parameters. Modeling these properties from photometric observations is a difficult task. Parameters of the reflective properties of some satellites are given in Chapter 7, which describes the process in which the reflective properties are used. The physical parameters of planetary satellites in this book are background information. This data should be constantly updated as new information becomes available. Appendix G gives a current version of the parameters. All values are provided with references to information sources. The determination of the photometric parameters of satellites is not directly related to their dynamics. However, since photometric data are usually associated with the results of astrometric measurements and published with them, the determination of the magnitudes of satellites is sometimes accompanied by the study of dynamics. An example of such an activity is described in this chapter in a separate section.

11.2 Handbook of the physical parameters of planetary satellites Appendix G gives the following physical parameters of almost all known planet satellites: the radii or semi-major axes of an ellipsoid approximating the shape, the gravitational parameter of the satellite, and the magnitude of the satellite visible from Earth in the planet’s mean opposition. Each given value is provided with a citation of the data source, and the corresponding bibliographic references are given at the end of the Appendix. The values given come from various determination methods. These are ground-based measurements of dimensions, measurements using spacecraft, as well as indirect determination of pa-

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rameters by the influence of satellites on the motion of a spacecraft or on another satellite. The measurement method can be found in the cited literature. Note that for many distant planetary satellites the sizes and masses are known very approximately. They are determined by magnitudes in some assumptions about the albedo and the density of the satellite material. For each group of satellites similar in properties, the data are given in separate tables.

11.3 Detection of volcanoes on the satellite of Jupiter Io using ground photometry The effect of reducing the brightness of satellites during their mutual occultations and eclipses is known and well studied. This is the subject of Chapter 7 of this book. As a result of photometric observations of Jupiter’s satellite, Io, one discovered an unusual effect of a sharp decrease in brightness with the subsequent restoration of the previous level. This was in the process of occultation of the satellite Io with another Jupiter satellite, Europe, on February 20, 1991, when observed at the European Southern Observatory (ESO). Measurement of the luminous flux was done with a filter that transmits infrared light with a wavelength of 3.8 µm. Fig. 11.1 reproduces the plot of the results of these measurements of Descamps et al. (1992). The total luminous flux from the satellites Io and Europe was measured. An interpretation of the observed phenomenon has been found. Previous observations from the Voyager spacecraft showed that there are active volcanoes on the satellite Io. The largest of them, Loki, in infrared light “shines” on the satellite, like a large flashlight. In moments of being occulted by another companion, the luminous flux decreases sharply and is also sharply then restored. The plot also shows changes in brightness during an occultation of another volcano on the satellite, Pele. We note that the manifestation of volcanoes on the satellite Io in photometric observations was noticed earlier and published in Goguen et al. (1988). The coordinates of the Loki volcano were refined using the constructed light curve of satellites (Descamps et al., 1992). More details about the studies described can be found in the work cited here.

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Figure 11.1. Photometric results of the Jupiter satellites during the occultation of the satellite Io by the satellite Europe. Figure taken from Descamps et al. (1992).

11.4 Estimates of the physical parameters of distant planetary satellites 11.4.1 Features of distant planetary satellites The distant satellites of the planets are very special objects of the Solar System. This is clear in a number of circumstances. First: among 148 distant satellites of Jupiter, Saturn, Uranus, and Neptune known to date, 138 ones (93%) were discovered after 1998. Second: for 11 distant planetary satellites, ephemeris are currently known with the precision of half the satellite’s revolution around the planet (Emelyanov, 2010; Brozovic and Jacobson, 2017). Therefore, they can be considered as lost ones. Third: with the exception of the satellite of Saturn, Phoebe, and the satellite of Jupiter, Himalia, spacecrafts were not approaching any of the distant satellites to see their size or determine the mass of the satellite by its gravitational influence on the motion of the spacecraft. Fourth: in contrast to regular satellites, which were formed from a protoplanetary gas–dust disk together with the planet and have almost circular orbits with a small inclination to the equatorial plane of the planet, the orbits of distant satellites have significant eccentricities and very diverse inclinations to the plane of the planet’s orbit. Therefore, distant satellites are often called irregular satel-

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lites. Most of them have a motion opposite to the orbital motion of the planet. Irregular satellites could not form from a protoplanetary cloud in modern orbits. Therefore, it is assumed that they are captured from orbits around the Sun in the early epoch of the formation of the Solar System. With the exception of the satellite of Jupiter, Himalia, and the satellite of Saturn, Phoebe, the dimensions and masses of the distant satellites were not directly measured. No observations give us such data. However, in some reference sources of information, for example, JPL Solar System Dynamics (https://ssd.jpl.nasa.gov/), we can find the values of the diameters and masses of all the distant satellites of large planets. How were these values found? Directly from observations, we only have measurements of their brightness (magnitude). However, if we accept the very bold hypothesis about the reflective properties of the satellite surface (albedo), then we can calculate their size. And if we accept an even bolder hypothesis about the density of the substance of which they are composed, then we can calculate the masses of satellites. As a result, the mass values are obtained with an accuracy no better than 100–200%. If we take the mass of a distant satellite from the directory, then we can make a mistake several times. The only case when the mass of a satellite was determined directly from its gravitational influence on another satellite whose motion was observed from the Earth is the determination of the mass of the satellite of Jupiter, Himalia, made in Emelyanov (2005a). The Himalia’s gravitational parameter turned out to be equal to (0.28 ± 0.04) km3 /s2 , which is significantly different from the value of 0.45 km3 /s2 given in JPL Solar System Dynamics (https://ssd.jpl.nasa.gov/). This work is described in more detail below in a special section. Emel’yanov et al. (2007) have shown that from ground-based observations it is impossible to obtain the mass of any other satellite in the same way. For a number of irregular satellites, special photometric observations were performed, as a result of which the reflective characteristics of the surfaces were obtained. These results are published in accessible scientific journals. However, these data concern only a small fraction (31%) of all distant planetary satellites discovered to date. For most satellites discovered after 1998, absolute magnitudes and other photometric parameters based on observations have not been published in scientific journals. In view of the insufficiency of published photometric results of distant planetary satellites in Emel’Yanov and Ural’Skaya (2011), an attempt was made to determine the photometric parameters of satellites from the photometric data that provide astrometric

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observational results published in Minor Planet Circulars (MPC). The value of this data is that it is available for all distant satellites. This chapter presents the methods and results of determining the photometric parameters of all the distant satellites of Jupiter, Saturn, Uranus, and Neptune according to observations published in the MPC. Grav and Bauer (2007) indicated that these results are “not very reliable for serious applications”, but, in the absence of other data sources for all distant planetary satellites, one can be content with what obtained in Emel’Yanov and Ural’Skaya (2011). The next section provides a brief overview of the publications of the photometric parameters of distant planetary satellites.

11.4.2 Overview of available photometric data for distant planetary satellites We reproduce here the review presented in the article Emel’Yanov and Ural’Skaya (2011). The physical characteristics of ten distant planetary satellites, known even before 1998, have been published in many references. Two irregular satellites J6 Himalia and S9 Phoebe were in the field of view of spacecraft. Consider briefly the found physical properties of these two satellites. The Cassini spacecraft received images of the Himalia, from which the Cassini Observation Processing Group, headed by K. Porco, determined its sizes along two axes of (150 ± 20) and (120 ± 20) km (Porco et al., 2003). The reflectance of Himalia, the albedo, was very low: 0.05 ± 0.01. This value is comparable with the albedo of large Jupiter asteroids defined in Fernandez et al. (2003), and in combination with the gray color of the satellite (Degewij et al., 1980a,b) suggests that the surface is rich in carbon. This is characteristic of C-type asteroids, which are distributed mainly in the outer part of the Main Asteroid Belt. Among the irregular satellites of the Saturn’s system, the satellite Phoebe has been well investigated. The passage of the Voyager and Cassini spacecraft made it possible to determine its gravitational parameter, which turned out to be (0.5517 ± 0.0007) km3 /s2 (Rappaport et al., 2005). The Phoebe’s surface is rather dark with a geometric mean albedo in the spectral V band equal to 0.081 ± 0.002 (Simonelli et al., 1999). However, the albedo varies approximately twice along the surface, showing a significant difference in the materials of which this satellite is composed. Combining these data with known sizes obtained using the Voyager spacecraft (mean radius 110 km), the Phoebe density was determined, which turned out to be (1630 ± 45) kg/m3 . This is much more than the

Chapter 11 Physical parameters of planetary satellites

density of Saturn’s regular satellites, which is about 1300 kg/m3 , but smaller than the density of Triton and Pluto 1900 kg/m3 . This average density is typical for a mixture of water ice and stone. Other studies show the presence of iron and silicon minerals, carbon dioxide, and organic compounds in Phoebe (Buratti et al., 2008). The photos of Phoebe show a highly cratered old surface, significantly different from the surface of asteroids, which indicates its origin outside the Saturn’s system. Apparently, the satellite’s capture occurred from the outer regions of the Solar System, possibly from the Kuiper belt. BVRI photometry of 13 distant satellites of Jupiter was carried out (Grav et al., 2003), of which only 8 are new, discovered after 1998. Using the spectra obtained from the observations, the authors of the cited work identified Pasiphae and Himalia as Ctype asteroids, Ananke and Themisto as P-type asteroids, while Carme, Sinope and Callirrhoe are similar to D-type asteroids. Asteroids with a low albedo and reddish color, classified as D- and P-types (D-bodies are redder), are the oldest, unmodified bodies of the Solar System. Such properties are silicates rich in carbon or organic substances. These types of asteroids are characteristic of the outer part of the Main Asteroid Belt and have low albedos of 0.04–0.06, as well as Jupiter’s Trojan asteroids and “dead” comets. The colors of Jupiter’s brightest external satellites show that a group of satellites with direct motion is redder and more compact in color than satellites with retrograde motion. Infrared photometry also showed that four members of the Himalia family (Himalia, Elara, Leda, and Lysithea) have uniform colors with average colors J − H = 0.34 ± 0.02, H − K = 0.27 ± 0.02 and J − K = 0.65 ± 0.02. For faint satellites with a magnitude of more than 24m , it is impossible to obtain accurate photometric data, since nothing is known except their magnitude. A unique series of photometric observations of 7 distant satellites of Saturn (S9, S19, S20, S21, S22, S26, and S29) was obtained in 2005 by Bauer et al. (2006) using the Palomar 200-inch telescope and the Cerro Tololo Inter-American Observatory’s 4-m Blanco telescope in two filters B and R. The results of photometric observations of 13 distant satellites of Saturn, including Phoebe, are given in Grav et al. (2003); Grav and Bauer (2007). These are satellites S9, S19, S20, S21, S22, S23, S24, S25, S26, S27, S28, S29, and S30. Studies of Saturn’s irregular satellites have shown significant color differences from the neutral color of S9 Phoebe, and S25 Mundilfari, to the moderately red color of S26 Albiorix, and S22 Ijirak. On average, they are redder than the irregular satellites of Jupiter, but do not show the presence on the surface of the very red material that is observed in the

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Kuiper Belt. Their colors are similar to active cometary nuclei and “dead” comets. To date, nine distant satellites of Uranus are known. Photometry was performed only for six of them (U16, U17, U18, U19, U20, U21). These irregular satellites of Uranus have a wide range of colors from the bluest to the reddest. In Maris et al. (2001), U16 Caliban, and U17 Sycorax, were investigated. Both satellites were on average red, with U17 Sycorax, slightly bluer than U16 Caliban. Romon et al. (2001) shown that the satellite U17 Sycorax, is more similar to trans-neptunian objects and Centaurs than to “Trojans” and irregular satellites of Jupiter. Photometric studies of distant satellites of Uranus were also continued in Maris et al. (2007). Grav et al. (2004) determined the absolute magnitude in the spectral V band and the color indices B–V and V–R for six distant satellites of Uranus, which turned out to be similar to the same magnitudes of the satellites of Jupiter and Saturn. Their colors are divided into two groups. One group of neutral colors contains U18 Prospero, and U19 Setebos. The other group is slightly red and contains two large satellites of Uranus—U17 Caliban, and U17 Sicorax. Among six irregular satellites of Neptune, only N2 Nereid, discovered in 1949, has reliable photometric data. The albedo of N2 Nereid, turned out to be 0.2 (Thomas et al., 1991). This is significantly higher than the albedo of Neptune’s internal satellites (0.06), Uranus’ small internal satellites (0.07), Himalia J6 (0.05), and Phoebe’s S9 (0.08). Schaefer and Schaefer (2000) determined the color of the satellite N2 Nereid based on 224 photometric measurements of N2 Nereid, in the VBRI bands. In turn, Brown et al. (1999) showed that N2 Nereid, has water ice using infrared spectra. Grav et al. (2004) determined the photometric parameters of N2 Nereid, and one of the five new irregular satellites of Neptune, N9 Halimede. The observed colors of N2 Nereid, and N9 Galimede, are approximately the same and suggest that their surface compositions are similar. A study of the possibility of collisions between the irregular satellites of Neptune shows that N2 Nereid, and N9 Galimede, had a high probability of collision about 4.5 billion years ago and, probably, N9 Galimede, is a fragment of N2 Nereid, (Grav and Holman, 2004). Both satellites have a neutral color, and the colors, albedo and spectra are similar to the same characteristics of U4 Oberon, and U2 Umbriel, and some objects of the Kuiper belt. As a result of this review, it is clear that photometric parameters are known only for 8 out of 46 (17%) new irregular satellites of Jupiter, for 12 among 37 (32%) new irregular satellites of Saturn,

Chapter 11 Physical parameters of planetary satellites

for six among nine irregular satellites of Uranus and one of five new irregular satellites of Neptune. For 97 distant planetary satellites, apart from the satellites Pluto, there are no publications of photometric parameters yet. Due to such limited data on the photometry of distant planetary satellites, Emel’Yanov and Ural’Skaya (2011) attempted to determine the photometric parameters of all satellites from photometric data published in Minor Planet Circulars (MPCs).

11.4.3 Photometric model for distant planetary satellites During astrometric observations of planetary satellites, satellite photometry can also be performed. The tables published in Minor Planet Circulars (MPCs) give the astrometric coordinates of the satellites. In addition, in a special column, observers report estimates of the apparent magnitude of the satellites. These estimates are given only for some time points for which the coordinates are measured. Magnitudes are also provided with a sign of the spectral band in which the observations were made. Mostly filters with the R band are used, but for some observations with the V band. Thus, from the observations we obtain the magnitude of the satellite m measured in a certain spectral band. This value depends on the distance of the observer to the satellite , the distance of the satellite to the Sun r and the phase angle α (the objectcentric angle between the directions to the observer and the light source). Therefore, the measured magnitude of the satellite should be written as a function of the three arguments m(r, , α). The absolute magnitude of the satellite m(1, 1, α), independent of distances, can be calculated by the formula m(1, 1, α) = m(r, , α) − 5 log(r ),

(11.1)

where the distances r and  are expressed in astronomical units. We recall here the relationship between the difference in magnitude m1 and m2 of any two objects and the ratio of the corresponding light fluxes E1 and E2 , m1 − m2 = −2.5 log

E1 , E2

(11.2)

which may be useful in photometric studies. We note one interesting property that follows from Eq. (11.2). If we increase the luminous flux E1 by 1 + ε times, then for sufficiently small ε the difference in magnitude m1 − m2 will decrease by ε.

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The dependence of the absolute magnitude of the satellite on the phase angle is determined by the fraction of the unilluminated part of the visible disk and by the reflective properties of the surface. The bodies of the Solar System have a variety of physical properties. For planets and satellites having an atmosphere, the intensity of the reflected light weakly depends on the phase angle. Rough surfaces are characterized by a sharp surge of satellite brightness at a zero phase angle. Concerning the origin of distant planetary satellites, the hypothesis of their capture from heliocentric orbits is mainly considered (Gladman et al., 2001; Grav and Bauer, 2007). If this hypothesis is true, then the physical properties of satellites can be considered to be similar to the properties of asteroids. Then, for satellites, we can apply the two-parameter photometric system adopted by Commission 20 of the International Astronomical Union in 1985 (Marsden, 1986) for asteroids. The function describing this model has the form (Bowell et al., 1989) m(1, 1, α) = H − 2.5 log[(1 − G)1 (α) + G2 (α)].

(11.3)

Two parameters H and G characterize a particular satellite. It is assumed that the reflective properties of the satellite are independent of the angle of rotation. If the surface of the satellite is heterogeneous, then we take a certain average value. The function 1 (α) corresponds to a sharp dependence of the satellite brightness on the phase angle (usually bodies with a low albedo value), and the function 2 (α) corresponds to a weak dependence (with a high albedo). These functions are calculated from the following relationships: i = W i + (1 − W )i (i = 1, 2), W = exp[−90.56 tan2 (α/2)], i = 1 −

Ci sin α 0.119 + 1.341 sin α − 0.754 sin2 α

,

  i = exp −Ai [tan(α/2)]Bi , A1 = 3.332, B1 = 0.631, C1 = 0.986, A2 = 1.862, B2 = 1.218, C2 = 0.238. The functions 1 and 2 are determined empirically.

Chapter 11 Physical parameters of planetary satellites

Figure 11.2. Accepted photometric model: dependence of the increment of the satellite’s absolute magnitude on the phase angle for various parameters G.

With these parameters H and G, the photometric model allows one to calculate the magnitude of the satellite at any phase angle. For a visual representation of the photometric model under consideration, we show in Fig. 11.2 the dependences of the magnitude increment as a function of the phase angle α, m = −2.5 log [(1 − G)1 (α) + G2 (α)] , for various values of the parameter G. If measurements of the magnitude m(r, , α) were made in the spectral bands R, V or B, then the parameters H and G found according to the photometric model (11.3) will be assigned the corresponding indices are HR , GR , HV , GV or HB , GB .

11.4.4 Determination of photometric parameters of satellites by photometry The parameters G and H can be determined from the photometric observations of satellites. Formally, two measurements of

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the magnitude at different phase angles are sufficient for this. In practice, photometric measurements are made with errors. Therefore, it is desirable to have as many measurements as possible with a uniform distribution over the phase angle values. To determine the parameters G and H from a set of observations, a conditional equation for the required parameters is drawn up for each measurement to find the parameter estimates using the least-square method. Regarding the parameters G and H , the conditional equations are nonlinear. One can compose linear conditional equations if one determines the parameters a1 and a2 related to G and H by the relations H = −2.5 log(a1 + a2 ), G =

a1 . a1 + a2

(11.4)

Regarding a1 and a2 , the conditional equations are written in the form 10−0.4m(1,1,α

(i) )

= a1 1 (α (i) ) + a2 2 (α (i) ) (i = 1, 2, ..., N ),

where m(1, 1, α (i) ) is the absolute magnitude of the satellite, obtained at the phase angle α (i) , and N is the number of measurements. The absolute magnitude is calculated by the result of each measurement using Eq. (11.1). Once the conditional equations are compiled, the parameter estimates by the least-square method are found. In this case, not only the parameter values are obtained, but also we determine the standard deviation of the measured absolute magnitude from its model value for all observations. This value, denoted by σ , can be considered as the accuracy of determining the satellite’s magnitude from photometric observations for a given satellite. After determining the parameters a1 an a2 using the leastsquare method, the photometric parameters of the satellite G and H can be calculated using Eqs. (11.4). Moreover, σ will characterize the accuracy of determining the parameter H , as well as the absolute magnitude values obtained from the model.

11.4.5 Initial data and results of determining the photometric parameters of satellites Astrometric observations of all distant planetary satellites are published in Minor Planet Circulars (MPC). These observations are also collected in the NSDC (Natural Satellites Data Center)

Chapter 11 Physical parameters of planetary satellites

database (Arlot and Emelyanov, 2009). The NSDC database is located on the Internet at the addresses: http://www.sai.msu.ru/neb/nss/html/obspos/index.html, http://nsdb.imcce.fr/obspos/. Some astrometric measurements are provided with the values of the observed magnitude of the satellite. From the observational data, estimates of the magnitudes of the satellites were selected. Each value is attached to the moment of observation. The estimates were made mainly in the spectral R band. There are a small number of measurements in the spectral V band. However, they are very few and therefore unsuitable for determining the required parameters. Therefore, all our results relate to the spectral R band. For each observation, the distances r, , and the phase angle α were determined from the ephemeris, the absolute magnitude of the satellite was calculated, and the conditional equation was compiled as described above. Then, using the least-square method, we obtained estimates of photometric parameters. For Jupiter’s satellites J6–J13, Saturn’s satellite S9 Phoebe and Neptune’s satellite N2 Nereid, the publications contain fairly reliable values of photometric parameters. Therefore, we did not make parameter estimates for these satellites. For some satellites, it was not possible to determine the parameters due to the insufficient number of photometric measurements for different values of the phase angle. In these cases, we determined only the mean absolute magnitude of the satellite from the available data. As a result, the parameters G and H are determined for 70 distant satellites of Jupiter, Saturn, and Uranus. For the remaining 27 distant satellites, the mean absolute magnitude was determined. Analysis of various observations of planetary satellites shows that the maximal number of observations is made at some mean values of the phase angle α ∗ . For Jupiter, this angle is approximately equal to 6 degrees, for Saturn – 3 degrees, for Uranus – 1 degree, and for Neptune – 0.8 degrees. Therefore, using the constructed photometric satellite models, the absolute magnitudes m(1, 1, α ∗ ) for the indicated mean phase angles, as well as the apparent magnitudes of the satellites reduced to planetary distances to the mean opposition m0 (α ∗ ). It is convenient to have the latter value for a simple estimate of the most probable apparent magnitude of the satellite. For satellites for which the mean absolute magnitude was determined instead of the parameters, we identified it with m(1, 1, α ∗ ), according to which the value m0 (α ∗ ) was also calculated—the value m(r, , α) in the mean opposition.

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Figure 11.3. The absolute magnitudes of the satellite J22 Harpalyke, derived from the MPC data (points) for different phase angles, and the refined photometric model of this satellite (line).

All obtained photometric parameters of 97 distant satellites of Jupiter, Saturn, Uranus, and Neptune are available on the NSDC website at the addresses: http://www.sai.msu.ru/neb/nss/html/multisat/index.html http://nsdb.imcce.fr/multisat/index.html in the section “Parameters and constants”. Note that the number of satellites specified here with certain parameters, as described above, corresponds to the publication of Emel’Yanov and Ural’Skaya (2011). In the future, as the data on the NSDC website is updated, the number of satellites with certain parameters may increase. Examples of comparing the model dependences of m(1, 1, α) with the results of observations from which they were obtained are shown in Figs. 11.3–11.6. If we have independently determined satellite albedos, then the photometric parameter H can be used to find estimates of their sizes. To estimate the radii of spherical bodies, Bowell et al.

Chapter 11 Physical parameters of planetary satellites

Figure 11.4. Absolute magnitudes of the satellite J28 Autonoe, derived from the MPC data (points) for different phase angles, and an updated photometric model of this satellite (line).

(1989) proposed the relationships log pB = 6.521 − 2 log(2Rs ) − 0.4HB , log pV = 6.259 − 2 log(2Rs ) − 0.4HV , where Rs is the satellite radius, pB and pV are the accepted values of the satellite geometric albedo in the spectral bands B and V, and HB and HR are the photometric parameters in the corresponding spectral bands. In these formulas, the geometric albedo defined for the zero phase angle should be specified. In cases where observations are performed in the spectral R band, a similar relation can be used, log pR = 6.114 − 2 log(2Rs ) − 0.4HR ,

(11.5)

which is easily derived from the corresponding formula proposed in Jewitt and Haghighipour (2007). From the obtained values of the radii of the satellites, one can determine their gravitational parameters or masses if we take

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Figure 11.5. The absolute magnitudes of the satellite S20, Paaliaq, derived from the MPC data (points) for different phase angles, and the refined photometric model of this satellite (line).

some hypothetical values of the density of the matter of the satellites ρ. The gravitational parameter of the satellite Gm can be found by the formula 4 Gm = GρπRs3 , 3

(11.6)

where G is the gravitational constant. Following the possibilities described here for estimating the size and gravitational parameters of satellites, we performed the corresponding calculations using Eqs. (11.5) and (11.6). Since the values of pR and ρ can only be hypothetical, we have accepted some values found in the literature. For the satellites of Jupiter, we took pR = 0.04, ρ = 2.6 g/cm3 , for Saturn’s satellites pR = 0.06, ρ = 2.3 g/cm3 , for the satellites of Uranus pR = 0.04, ρ = 1.5 g/cm3 , for Neptune’s satellites pR = 0.04, ρ = 1.5 g/cm3 . If it becomes necessary to obtain the dimensions and gravitational parameters of some satellites for other hypothetical values pR and ρ, they can be easily calculated using the above formulas.

Chapter 11 Physical parameters of planetary satellites

Figure 11.6. The absolute magnitudes of the satellite S22, Ijiraq, derived from the MPC data (points) for different phase angles, and the refined photometric model of this satellite (line).

All the results obtained are available in the NSDC planet natural satellite database (Arlot and Emelyanov, 2009), available via the Internet at the following addresses: http://www.sai.msu.ru/neb/nss/html/multisat/paramhr.htm http://nsdb.imcce.fr/multisat/paramhr.htm.

11.4.6 Comparison of results obtained by different authors From astrometric observations of distant satellites of major planets published in the MPC, the parameters G and H for 70 distant satellites of Jupiter, Saturn, Uranus, and Neptune were determined in Emel’Yanov and Ural’Skaya (2011). For 27 distant satellites from observations, their mean absolute magnitudes were found. Photometric parameters were obtained for 46 distant satellites of Jupiter. Only 8 of them have photometric data in Grav et al. (2003). In this work, the parameter H was determined for the spec-

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Table 11.1 The photometric parameters of the eight distant satellites of Jupiter, published in Grav et al. (2003), and the results of Emel’Yanov and Ural’Skaya (2011) for these satellites. To compare the results, the values of HV can be reduced to the spectral R band using the color indices V – R. Satellites N name J17 Callirrhoe J18 Themisto J19 Megaclite J20 Taygete J22 Harpalyke J23 Kalyke J24 Iocaste J27 Praxidike

Grav et al., 2003 HV 13.92 ± 0.02 12.94 ± 0.01 15.12 ± 0.05 15.63 ± 0.04 16.03 ± 0.12 15.28 ± 0.04 15.27 ± 0.03 15.24 ± 0.03

Emel’Yanov and Ural’Skaya, 2011 V −R HR GR

0.50 ± 0.02 14.00 ± 0.17 0.46 ± 0.01 14.30 ± 0.40∗ 0.41 ± 0.07 14.65 ± 0.18 0.52 ± 0.04 15.10 ± 0.25 0.62 ± 0.22 15.33 ± 0.29 0.70 ± 0.05 14.88 ± 0.21 0.36 ± 0.04 14.96 ± 0.21 0.34 ± 0.03 14.44 ± 0.45

0.40 − 0.17 0.21 −0.03 −0.08 0.03 −0.20

tral V band. To compare the results, Table 11.1 gives the values of HV and the color index V – R obtained in Grav et al. (2003) and the parameters HR and GR for eight distant satellites of Jupiter found in Emel’Yanov and Ural’Skaya (2011). The errors given in this and the following tables are the standard errors (1σ ) of the corresponding parameters. Among the 37 distant satellites of Saturn for which photometric parameters were determined, only 12 have similar data published in Grav et al. (2003); Bauer et al. (2006); Grav and Bauer (2007). The data of reference (Grav and Bauer, 2007) together with the results of reference (Emel’Yanov and Ural’Skaya, 2011) are given in Table 11.2. From the observations contained in the MPC, photometric parameters were determined for nine distant satellites of Uranus and five distant satellites of Neptune. Among these satellites, only six satellites of Uranus and one satellite of Neptune presented in Grav et al. (2004) give values of HV and color index V–R. These data along with the results of Emel’Yanov and Ural’Skaya (2011) are given in Table 11.3. For satellite N9, Halimede, two versions of the parameters were published in Grav et al. (2004). The first of them was obtained from observations made with the 6.5-meter Clay telescope at the Magellan Observatory. The second option (marked with ** in Table 11.3) was obtained using a 10-meter Keck II telescope.

Chapter 11 Physical parameters of planetary satellites

387

Table 11.2 Photometric parameters of the 12 distant satellites of Saturn, published in two references. To compare the results, the values of HV can be reduced to the spectral R band using the color indices V – R. Satellites N name S19 Ymir S20 Paaliaq S21 Tarvos S22 Ijiraq S23 Suttungr S24 Kiviuq S25 Mundilfari S26 Albiorix S27 Skathi S28 Erriapo S29 Siarnaq S30 Thrymr

Grav and Bauer, 2007 HR

GR

11.81 ± 0.02 11.27 ± 0.04 12.61 ± 0.07 12.85 ± 0.12 14.08 ± 0.08 12.43 ± 0.16 14.28 ± 0.08 10.87 ± 0.01 14.04 ± 0.11 13.27 ± 0.15 10.24 ± 0.02 13.73 ± 0.08

0.01 ± 0.06 −0.04 ± 0.12 0.19 ± 0.15 −0.14 ± 0.22 0.68 ± 0.46 0.27 ± 0.32 0.95 ± 0.52 0.42 ± 0.06 0.64 ± 0.52 0.57 ± 0.34 0.27 ± 0.04 −0.22 ± 0.30

Emel’Yanov and Ural’Skaya, 2011 HR GR 11.97 ± 0.14 11.62 ± 0.20 12.65 ± 0.22 13.03 ± 0.27 14.50 ± 0.22 12.32 ± 0.29 14.43 ± 0.29 11.15 ± 0.39 14.26 ± 0.14 13.28 ± 0.25 9.91 ± 0.10 14.01 ± 0.19

0.25 0.78 0.77 0.64 1.69 0.61 1.48 0.81 1.20 0.27 −0.30 0.48

Table 11.3 Photometric parameters of six distant satellites of Uranus and one satellite of Neptune, published in Grav et al. (2004) and Emel’Yanov and Ural’Skaya (2011). For comparison, the values of HV can be reduced to the spectral R band using the color indices V – R. Satellites N name U16 Caliban U17 Sycorax U18 Prospero U19 Setebos U20 Stephano U21 Trinculo N9 Halimede –

Grav et al., 2003 HV 9.16 ± 0.04 7.50 ± 0.04 10.56 ± 0.05 10.57 ± 0.05 11.69 ± 0.17 11.92 ± 0.18 9.01 ± 0.07 9.74 ± 0.08

Emel’Yanov and Ural’Skaya, 2011 V −R HR GR

0.57 ± 0.03 0.62 ± 0.01 0.39 ± 0.04 0.35 ± 0.03 0.67 ± 0.22 0.35 ± 0.19 0.29 ± 0.08 0.47 ± 0.12∗∗

8.86 ± 0.21 7.59 ± 0.19 10.28 ± 0.14 10.12 ± 0.21 11.13 ± 0.20 12.42 ± 0.22 9.44 ± 0.29∗ −

1.14 1.45 1.22 −0.28 −0.13 1.25 − −

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In Tables 11.1 and 11.3 we marked with an asterisk the cases in which the parameter G was not determined, and instead of HR the values m(1, 1, α ∗ ) are given. From a comparison of the data in Tables 11.1, 11.2, and 11.3, we can conclude that the results of determining the parameter HR for the satellites considered in the work of different authors mainly agree within the error 3σ . As for the parameter GR , this parameter in reference (Emel’Yanov and Ural’Skaya, 2011) served as the matching parameter of the model. Its values in most cases cannot be used to draw conclusions about the properties of satellite surfaces. From the plots showing examples of matching the model with observations, it is clear that the accuracy of the photometric measurements used is clearly not enough to reliably determine the parameter G.

11.4.7 Conclusions on the estimates of the photometric parameters of distant planetary satellites As a result of the work performed by Emel’Yanov and Ural’Skaya (2011), the parameters of the photometric model were obtained from the observations published in MPC, i.e. the absolute magnitude H at zero phase angle and the matching parameter G for all 97 distant satellites of Jupiter, Saturn, Uranus, and Neptune. The found parameters relate to the spectral R band. The accuracy of determining the absolute magnitudes for most satellites is 0.1–0.3. The accuracy of the observations used is not enough to reliably determine the parameter G. Comparison of the obtained values of H with the results of photometry published by other authors for 27 satellites shows differences not exceeding 3σ . Based on the obtained values of the parameter H , hypothetical values of the albedo and density of matter, the sizes of satellites and their gravitational parameters are determined. These parameters are given in our sites, http://www.sai.msu.ru/neb/nss/html/multisat/paramhe.htm http://nsdb.imcce.fr/multisat/paramhe.htm.

11.5 Determination of Himalia’s mass from astrometric observations of other satellites To solve the problems of the origin and evolution of the Solar System, we need to know the physical parameters of the plan-

Chapter 11 Physical parameters of planetary satellites

ets and satellites, as well as the properties of their orbital motions. Distant satellites of planets are very small celestial bodies. Only magnitudes can be determined for them based on observations from the Earth. The masses of distant satellites, their size and albedo remain unknown. However, if we accept any hypothesis about the reflective properties of the surface of satellites, then we can calculate their sizes. If we also accept the hypothesis about the density of the matter of which they consist, then we can calculate the masses of satellites. Knowledge of the physical properties of distant satellites becomes very important, especially when we consider the numerous discoveries of new such satellites in the late XX and early XXI centuries. The mass of only one distant satellite of Saturn, S9 Phoebe, was determined directly during the passage of the Cassini spacecraft near Saturn in June 2004 (Jacobson et al., 2004). The mass of the distant satellite of Jupiter Himalia was the subject of special attention in Christou (2005). The author of this work demonstrated the ability of Himalia to disperse a group of satellites with close orbits over a long time interval and showed that this dispersion is critically dependent on the mass of the Himalia. Researchers know an independent method for determining the masses of satellites whose motion is affected by the attraction of other satellites. Such a general approach was, in particular, applied to determine the masses of asteroids during their mutual rapprochement (Michalak, 2001). The possibility of determining masses in this way substantially depends on the relationship between the accuracy of observations and the magnitude of the mutual perturbations of celestial bodies. An attempt to determine the mass of Jupiter’s satellite Himalia directly by its gravitational influence on another satellite whose motion was observed from the Earth was made in Emelyanov (2005a). First, the possibility of determining the masses of the distant satellites of Jupiter in the considered way was fundamentally evaluated. Recently discovered satellites were excluded from consideration, since it was clear from estimates of their magnitude that they had very small masses. Two independent satellite groups were taken. The first group is satellites at distances of 10–13 million km from the planet. These are the satellites Himalia (J6), Elara (J7), Lysithea (J10), and Leda (J13). The second group consists of satellites at distances of 20–25 million km; these are Pasiphae (J8), Sinope (J9), Carme (J11), and Ananke (J12). The problem of determining the mass of the most massive satellite in each group from its perturbations exerted on the motion of the others was

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considered. In the first group, the satellite Himalia (J6) was such a satellite, and the satellite Pasiphae (J8) was such a satellite in the second group. Further studies were performed based on the results of artificially created observations, the so-called “simulations”. The motion model of the distant satellites of Jupiter, constructed on the basis of all available observations (Emelyanov, 2005b), was used. For the satellites of each of the two groups, based on the motion model, geocentric right ascensions and declinations of the satellites were calculated on the time interval from 1905 to 2003 in increments of 90 days. The results of such artificial observations were supplemented by random errors obtained by the random number generator, with zero mathematical expectation and a given dispersion. The mass of only the most massive satellite was assumed to be nonzero. For Himalia (J6), the value of the gravitational parameter was taken equal to 0.45 km3 /s2 , and for Pasiphae (J8) to 0.013 km3 /s2 . These are the values that were known at that date from publications. The method of differential refinement of the motion parameters from the observations, described in Chapter 6, was used. Among the refined parameters, we included the initial conditions for solving the differential equations of satellite motion and the gravitational parameter Gm of the most massive of them. At the same time, we jointly refined the parameters for four satellites in each group. The variance of observation errors varied within 0.2 arcseconds. Next, we calculated an estimate of the accuracy of the determination of parameters by the least-square method. We obtained the following results based on such “simulations”. The obtained masses of the satellites differ from their model values by less than their errors estimated by the least-square method. We determined the relationship between model observation errors and the error in determining the mass of the perturbed satellite. The observation error was defined as the angular distance between the exact and erroneous geocentric positions of the satellite. We also calculated the rms value σd of these errors. The rms value of errors σGm of the gravitational parameters of the perturbing satellites was determined by the least-square method. The following numerical relations were obtained: for Himalia (J6) σGm = 0.031 σd , for Pasiphae (J8) σGm = 0.32 σd .

Chapter 11 Physical parameters of planetary satellites

Here σd is expressed in arcseconds and σGm in km3 /s2 . From these relations it follows that, if, for example, the observation error is 0.2 arcseconds, then the error in determining the gravitational parameter of the Himalia (J6) will be 0.0062 km3 /s2 , which is 1.4 % of the model value. For Pasiphae (J8) in this case the error will be 0.064 km3 /s2 , which is five times the estimated value of Gm. Research performed based on “simulations” is an ideal case for solving the problem. In determining the parameters based on real observations, one should expect more rough results. Given that the accuracy of today’s ground-based observations of distant satellites is approximately 0.2 arcseconds, it becomes clear that the determination of the mass of Pasiphae (J8) in this way is impossible. However, for Himalia (J6), the hope remains to make a mass determination. Further, Emelyanov (2005a) investigated the sensitivity of observations to mass variations of the perturbing satellite. As a result, the sensitivity function (t) is determined depending on the time t,      ∂α 2 ∂δ 2 (t) = + . (11.7) ∂m ∂m Large values of the sensitivity function are showing good possibility of determining the required mass. It turned out that a sharp increase in the sensitivity function occurs after cases of approaching two satellites. The most significant increase in function occurred when the distance between Himalia (J6) and Elara (J7) drop to 65031 km. This happened on July 15, 1949. This is a significant rendez-vous, since the mean distance of both satellites to the planet is only about 12 million km. The satellite Lysithea (J10) after its discovery in 1938 experienced two sharp rendez-vous with the Himalia (J6): on July 9, 1954, to a distance of 454216 km, and November 19, 1954, to a distance of 168891 km. Artificial observation experiments have shown that the best composition of observations to determine the mass of a perturbing satellite is obtained when the observation time intervals before and after the rendez-vous are the same. The accuracy of the mass determination is improved with increasing size of these intervals. Obviously, at smaller distances of close rendez-vous, a more accurate mass determination result is obtained. Regarding the determination of the mass of Himalia (J6), the accuracy turned out to be unchanged when adding the observations of Lysithea (J10) to the observations of Elara (J7). An impor-

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Table 11.4 The results of determining the gravitational parameter Gm of Himalia (J6) for various compositions of observations. Designations: N J 7 is the total number of observations of Elara (J7); N  J 7 is the number of used observations of Elara (J7) made before 1949; and N J 10 is the number of used observations of Lysithea (J10). N

N

N

σd

σGm

Gm

arcsec

3

km3 /s2

km

/s2

J7

J7 J10

326

38

143

0.564

0.039

0.270

326

38



0.551

0.041

0.310

315

36

136

0.529

0.038

0.278

315

36



0.521

0.039

0.297

280

24

120

0.454

0.037

0.278

280

24



0.442

0.039

0.284

tant factor in this result was the exceptional rendez-vous between Himalia (J6) and Elara (J7) on July 15, 1949. The observational database for determining the mass of Himalia (J6) was the same as for determining the parameters of the satellite orbits in Emelyanov (2005b). A small number of observations were excluded being some evidently inaccurate data. The dependence of the obtained satellite mass on the composition of the observations used was investigated. Several options for data composition were considered. The results are shown in Table 11.4. Here we can conclude that the dependence of the obtained value on the composition of observations is rather weak compared to the error, which indicates the reliability of the result. Emelyanov (2005a) concluded that the most reliable value of the gravitational parameter of the distant satellite of Jupiter Himalia (J6), found from observations, is the following: (0.28 ± 0.04) km3 /s2 . The resulting value is significantly different from the value of 0.45 km3 /s2 given in JPL Solar System Dynamics (https:// ssd.jpl.nasa.gov/). In turn, Emel’yanov et al. (2007) have shown that from groundbased observations it is impossible to obtain the mass of any other distant satellite of Jupiter, Saturn, Uranus, and Neptune in the same way. Twelve years after the publication of reference (Emelyanov, 2005a), work (Brozovic and Jacobson, 2017) was done on deter-

Chapter 11 Physical parameters of planetary satellites

mining the mass of the distant satellite of Jupiter Himalia (J6). An extensive set of observations was used over a significantly wider time interval than was done in Emelyanov (2005a). One has taken 129 observations of Elara (J7), made before 1949, and 1876 observations in total over the time interval 1905–2016. Brozovic and Jacobson (2017) used 44 observations of Lysithea (J10) until its rendez-vous with the Himalia (J6) and 772 observations in total over the time interval between 1938 and 2016. The following formal value of the gravitational parameter of Himalia (J6) is obtained: Gm = (0.13 ± 0.02) km3 /s2 (accuracy estimate by the 1σ rule). However, Brozovic and Jacobson (2017) concluded that the gravitational parameter of the Himalia (J6) can be in the range of 0.13–0.28 km3 /s2 . In this work, the density of matter of the Jupiter satellite Himalia (J6) was also estimated based on the mean size of the satellite based on observations from the Cassini spacecraft (Porco et al., 2003). The radius of Himalia (J6) was taken to be equal to 67.5 km. It turned out that, if we take Gm = 0.13 km3 /s2 , then the density will be equal to 1.55 g/cm3 . With Gm = 0.28 km3 /s2 , the density is 2.26 g/cm3 .

References Arlot, J.-E., Emelyanov, N.V., 2009. The NSDB natural satellites astrometric database. Astronomy and Astrophysics 503, 631–638. Bauer, J.M., Grav, T., Buratti, B.J., Hicks, M.D., 2006. The phase curve survey of the irregular saturnian satellites: a possible method of physical classification. Icarus 184, 181–197. Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J., Harris, A.W., 1989. Application of photometric models to asteroids. In: Binzel, R.P. (Ed.), Asteroids II. Univ. Arizona Press, Tucson, pp. 524–556. Brown, R.H., Cruikshank, D.P., Pendleton, Y., Veeder, G.J., 1999. NOTE: water ice on Nereid. Icarus 139 (2), 374–378. Brozovic, M., Jacobson, R., 2017. The orbits of Jupiter’s irregular satellites. Astronomical Journal 153 (4), 147. Buratti, B.J., Soderlund, K., Bauer, J., et al., 2008. Infrared (0.83-5.1 µm) photometry of Phoebe from the Cassini visual infrared mapping spectrometer. Icarus 193 (2), 309–322. Christou, A.A., 2005. Gravitational scattering within the Himalia group of Jovian prograde irregular satellites. Icarus 174 (1), 215–229. Degewij, J., Zellner, B., Andersson, L.E., 1980a. Photometric properties of outer planetary satellites. Icarus 44, 520–540. Degewij, J., Cruikshank, D.P., Hartmann, W.K., 1980b. Near-infrared colorimetry of J6 Himalia and S9 Phoebe – a summary of 0.3- to 2.2-micron reflectances. Icarus 44, 541–547. Descamps, P., Arlot, J.E., Thuillot, W., Colas, F., Vu, D.T., Bouchet, P., Hainaut, O., 1992. Observations of the volcanoes of Io, Loki and Pele, made in 1991 at the ESO during an occultation by Europa. Icarus 100 (1), 235–244.

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Emelyanov, N.V., 2005a. The mass of Himalia from the perturbations on other satellites. Astronomy and Astrophysics 438, L33–L36. Emelyanov, N.V., 2005b. Ephemerides of the outer Jovian satellites. Astronomy and Astrophysics 435, 1173–1179. Emel’yanov, N.V., Vashkov’yak, S.N., Sheremet’ev, K.Yu, 2007. Determination of the masses of planetary satellites from their mutual gravitational perturbations. Solar System Research 41 (3), 203–210. Emelyanov, N., 2010. Precision of the ephemerides of outer planetary satellites. Planetary and Space Science 58 (3), 411–420. Emel’Yanov, N.V., Ural’Skaya, V.S., 2011. Estimates of the physical parameters of remote planetary satellites. Solar System Research 45 (5), 377–385. Fernandez, Y.R., Sheppard, S.S., Jewitt, D.C., 2003. The Albedo distribution of Jovian Trojan asteroids. Astronomical Journal 126 (3), 1563–1574. Gladman, B., Kavelaars, J.J., Holman, M., et al., 2001. Discovery of 12 satellites of Saturn exhibiting orbital clustering. Nature 412 (6843), 163–166. Goguen, J.D., Sinton, W.M., Matson, D.L., Howell, R.R., Dick, H.M., Johnson, T.V., Brown, R.H., Veeder, G.J., Lane, A.L., Nelson, R.M., Mclarren, R.A., 1988. Io hot spots: infrared photometry of satellite occultations. Icarus 76, 465–484. Grav, T., Holman, M.J., Gladman, B.J., Aksnes, K., 2003. Photometric survey of the irregular satellites. Icarus 166 (1), 33–45. Grav, T., Holman, M.J., 2004. Near-infrared photometry of the irregular satellites of Jupiter and Saturn. Astrophysical Journal 605 (2), L141–L144. Grav, T., Holman, M.J., Fraser, W.C., 2004. Photometry of irregular satellites of Uranus and Neptune. Astrophysical Journal 613 (1), L77–L80. Grav, T., Bauer, J., 2007. A deeper look at the colors of the saturnian irregular satellites. Icarus 191 (1), 267–285. Jacobson, R.A., Antreasian, P.G., Bordi, J.J., Criddle, K.E., Ionasescu, R., Jones, J.B., Meek, M.C., Owen Jr., W.M., Roth, D.C., Roundhill, I.M., Stauch, J.R., 2004. The orbits of the major Saturnian satellites and the gravity field of the Saturnian system. American Astronomical Society, DPS meeting N36, id. 15.02 Bulletin of the American Astronomical Society 36, 1097. Jewitt, D., Haghighipour, N., 2007. Irregular satellites of the planets: products of capture in the early solar system. Annual Review of Astronomy and Astrophysics 45 (1), 261–295. Maris, M., Carraro, G., Cremonese, G., Fulle, M., 2001. Multicolor photometry of the Uranus irregular satellites Sycorax and Caliban. Astronomical Journal 121, 2800–2803. Maris, M., Carraro, G., Parisi, M.G., 2007. Light curves and colours of the faint Uranian irregular satellites Sycorax, Prospero, Stephano, Setebos and Trinculo. Astronomy and Astrophysics 472 (1), 311–319. Marsden, B.G., 1986. Notes from the IAU General Assembly. Minor Planet Circulars. Nos. 10193 and 10194. Michalak, G., 2001. Determination of asteroid masses. II (6) Hebe, (10) Hygiea, (15) Eunomia, (52) Europa, (88) Thisbe, (444) Gyptis, (511) Davida and (704) Interamnia. Astronomy and Astrophysics 374, 703–711. Porco, C.C., West, R.A., McEven, A., et al., 2003. Cassini imaging of Jupiter’s atmosphere, satellites, and rings. Science 299 (5612), 1541–1547. Rappaport, N.J., Iess, L., Tortora, P., et al., 2005. Gravity science in the saturnian system: the masses of Phoebe, Iapetus, Dione, and Enceladus. Bulletin of the American Astronomical Society 37, 704. Romon, J., Bergh, C., Barucci, M.A., et al., 2001. Photometric and spectroscopic observations of Sycorax, satellite of Uranus. Astronomy and Astrophysics 376, 310–315. Schaefer, B.E., Schaefer, M.W., 2000. Nereid has complex large-amplitude photometric variability. Icarus 146 (2), 541–555.

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Simonelli, D.P., Kay, J., Adinolfi, D., Veverka, J., Thomas, P.C., Helfenstein, P., 1999. Phoebe: Albedo map and photometric properties. Icarus 138, 249–258. Thomas, P., Veverka, J., Helfenstein, P., 1991. Voyager observations of Nereid. Journal of Geophysical Research Supplement 96, 19253–19259.

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12 Recent models of planetary satellite motion. Information resources 12.1 Variants and change of version of motion theories and ephemeris of planetary satellites The history of the development of motion theories of satellites of major planets of the Solar System is very extensive. Attempts to model the motion of satellites began from the moment they were discovered. In past centuries, such theories relied on models of Keplerian motion. In the process of developing the theory, perturbations from various factors were added to the Keplerian motion model. The precessing ellipse model has been widely used and is still being applied. In attempts to construct the most accurate analytical theory of planetary satellite motion, researchers derived extremely voluminous formulas. Such analytical models simultaneously served both to calculate ephemerides and to describe the evolution of orbits. With the advent of high-speed computers, the general problem of modeling the motion of planetary satellites was divided into two methodically different problems in accordance with the goals. The first and immediate goal is to obtain ephemeris in the near future for new observations and for space missions. The time interval over which the satellite’s motion was modeled is relatively small one in this case. Therefore, the numerical integration of the differential equations of motion on high-speed computers turned out to be very effective. The simplicity of the implementation of the methods, when it is only necessary to program the calculation of the right-hand sides of the equations of motion, has led many researchers along this path. A somewhat more complicated problem arises when trying to refine the motion parameters from observations, when it is also necessary to calculate the partial derivatives of the measured quantities with respect to the specified parameters. Here the difficulty arises because in this problem it is necessary to model the The Dynamics of Natural Satellites of the Planets https://doi.org/10.1016/B978-0-12-822704-6.00017-0 Copyright © 2021 Elsevier Inc. All rights reserved.

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satellite’s motion over the entire observation time interval. For some satellites, this interval has already reached 300 years. But even these problems are mainly solved by numerical integration methods. Another goal of modeling the motion of planetary satellites is to describe the evolution of orbits over the maximum possible time interval. This is necessary to clarify the history of the formation of planets and satellites, as well as to clarify their fate in the future. In this setting, the solution by numerical integration methods is very difficult, and in some cases the problem is simply impossible. Here, the prohibitive cost of computation time, the impossibility of a reliable estimate of solution accuracy, and, in general, the limited accuracy, become obstacles. Under these circumstances, we are forced to resort to attempts to construct an analytical solution of the differential equations of motion. Except for a few simple mechanical models, it is impossible to obtain an exact analytical solution to the general problem. Moreover, no approximate analytical solution will be valid on an infinite time interval. Nevertheless, analytical methods make it possible to build models of the evolution of orbits over very long time intervals. The problems of evolution inhabited by planetary satellites are discussed above in Chapter 10. Here, the story will be devoted to theories and models of motion, which were built in order to calculate the most accurate ephemeris of planetary satellites. The path on which progress is being made here is as follows. At some current moment, a motion model already exists, based on all the observations available at that moment. Observers continue to observe, obtain new astrometric data and publish them in accessible scientific publications. There comes a time when it becomes possible to build a new version of the motion model and ephemeris of a particular group of satellites. Researchers who do this systematically, taking advantage of the situation, create a new model. Usually it is implemented in the form of available means of computing ephemeris and is at the disposal of everyone who wants to use it. Sometimes observers in their work cooperate with colleagues who are able to construct theories of motion, and jointly publish a new model of motion. In these cases, the “owners” of observations have an advantage over their equally capable colleagues. When changing versions, the time at which the motion model remains the most accurate and best in the world is very different. Some theories are replaced by new ones in a year or two, and some “live” for 20 years. Two circumstances are very important in this change of satellite motion model versions. Firstly, the normal circumstance is the simultaneous presence of two or three independently constructed

Chapter 12 Recent models of planetary satellite motion. Information resources

models. This allows us to check and evaluate the accuracy of the means of calculating the ephemeris. If there are three models, two of which give similar results, and the results of the third model differ significantly from the results of the first two, the merits of this third model become doubtful. Secondly, it is necessary to save the previous satellite motion models. Many new scientific results obtained on the basis of some theory of satellite motion sometimes need to be reproduced and verified. To make the calculations consistent, we need to use the tools that have already been used previously. Some implementers of motion models from time to time improve their work most often by expanding the observational base. Then new versions of the same model appear. At some time periods, two or more perfect satellite motion models constructed by different authors may exist. Then we are not talking about versions, but about different variants of the models. As for the set of perturbing factors taken into account and the accuracy of the methods used, it must be monitored so that the applied assumptions of the theory and the approximation calculation methods do not worsen the model by more than observation errors. Moreover, it is not enough to evaluate the accuracy of the determined parameters. For each discarded perturbing factor, it is necessary to determine its effect on the accuracy of the ephemeris. Usually, motion model implementers seek to make the theory more accurate than observations. It is now clear that, as soon as it becomes necessary to apply the theory of motion or the means of computing the ephemeris of a particular satellite for research or for practical purposes, it is necessary to find out which version is currently the most accurate, reliable and, in general, the most suitable. There are several selection criteria. Reliability is determined by the set of observations used and the observation time interval. It is necessary to find out how convenient the implementation of the model is for application, which model is best suited to solve the problem. We need to know how access to a particular model of satellite motion is organized. The next section of our book is devoted to this last problem.

12.2 Means of providing access to databases, motion models and ephemeris of planetary satellites In the modern epoch of information technology, the possibilities of using the fruits of labor of designers of motion models of planetary satellites are very diverse and great. Naturally, there is

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a technical merger of databases, means of solving problems and presenting the results. Consider how this is implemented in practice. First, we point out three basic methods for solving problems related to databases and ephemeris calculation tools. In the initial state, these tools consist of two elements: data files and computer programs that ensure the use of this data. The user’s work techniques are illustrated in Fig. 12.1. The first method is that the user copies all the necessary files to his computer and inserts those routines that provide access to the data into his problem solving program. The user program solves the problem and, as necessary, call the subroutines for data. Everything happens inside one problem solving program. A good example of such tools is the SPICE program and data system developed at the American Jet Propulsion Laboratory (Acton et al., 2015). It can be found on the Internet at https://naif.jpl.nasa.gov/ naif/toolkit.html. The second method is that the database is located on only one computer in the world. There is also a computer program that provides access to data via the Internet. This is usually a so-called web server. A user sitting at his computer launches one of the Web browsers and communicates with the corresponding Web server through the Internet pages. A Web browser program (Internet Explorer, Microsoft Edge, ...) allows the user to generate a request and receive the data of interest to him/her in the form of files. Then the user can run a program for solving his problem, which will use the files thus obtained. Several such tools are available and they are described below. The third method is effective when the user’s computer program in its work itself accesses the database located in another computer or even in another country via the Internet, takes the necessary data from the remote computer and continues the calculation. To do this, a program called a Web service must function on the remote computer. This technique is used mainly by users of high computer competence. Note that the three methods described are used both for sampling from databases and for calculating the ephemeris of planetary satellites. The source element is the database. In our case, these are mainly satellite observation databases. There are two approaches at the same time. The first of them is the desire to automate the process of using the results of observations. This inevitably leads to the need for unification of the presentation of results. Unification is associated with some preliminary data processing based on certain methods and models of coordinate systems and time

Chapter 12 Recent models of planetary satellite motion. Information resources

Figure 12.1. Three methods for working with databases and planetary satellite ephemeris services for the user. See explanations in the text.

scales. As a result, the data become model dependent. This leads to difficulties, and sometimes to misunderstandings. For example, when translating the source coordinates of satellites obtained using the old star catalog in the equatorial coordinate system of the B1950 epoch, various precession models can be used to go to the coordinate system of the J2000 epoch. If the coordinates are already corrected for the precession in the database, then it will not be possible to rework them with another precession model. Even worse are situations where it is not known what reductions were made before the results were put into the database. For these reasons, another approach is also used: to put in the database the results of observations of planetary satellites in the composition and in the form as given by the observers. The data should be accompanied by explanations taken from the same source. Currently, there are not many planetary satellite observational databases available via the Internet. Without going into details of the data composition, we list the observational databases available on the Internet. First of all, this is a database developed and is supported through the collaboration between the Sternberg Astronomical Institute of Moscow State University (SAI MSU) and the Insti-

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tut de Mécanique céleste et de calcul des éphémérides (IMCCE, Paris, France). It has the abbreviation NSDB for Natural Satellites Database. Addresses of these data are available through the Internet at the SAI MSU site (http://www.sai.msu.ru/neb/nss/indexr. htm) and at the site IMCCE (http://nsdb.imcce.fr/obspos/). Note that the pages of this database on the Internet are provided in three languages: Russian, English, and French. The announcement of the creation of the NSDB database and a brief description of it were published in Arlot and Emelyanov (2009). The observation database consists of two parts. The first one, tentatively referred to as NSDB1, has been compiled for a long time. The principle of placing observational data exactly in the composition and form as they were published is used. In fact, these are digital copies of data tables taken from publications. They are accompanied by descriptions necessary for use, taken also from publications. The second part (Standard astrometric data) contains the same data as NSDB2, but the data have a standardized form. This database is available at http://nsdb.imcce.fr/nsdb/home.html. The NSDB database contains all published observations of all known planetary satellites. The NSDB database is described in more detail below. Another observational database is available on the Minor planet center (MPC) website. As for natural satellites this database is limited to observations of only distant planetary satellites. The results published in MPC circulars were provided here. The format of observation data in this database is standard. It is about the same as for the publication of observations of minor planets. A description of the data format can be found on the MPC website. The page addresses on this site sometimes change, so here we will first give the shared address for the MPC site: http:// minorplanetcenter.net/iau/mpc.html. A file with observations of distant satellites of major planets can be copied from the page at http://www.minorplanetcenter.net/iau/ ECS/MPCAT-OBS/MPCAT-OBS.html To do this, follow the hyperlink Natural outer irregular satellites of the giant planets. Some observatories have their own database of observations of planetary satellites. An example of such a base is the Pulkovo Database of Observations of Planets and Their Satellites. This database addresses in the Internet is http://puldb.ru/db/sdb.php. A more general database of observations of the bodies of the solar system of the Pulkovo Observatory is located at http://puldb.ru/ db/index.php. Access pages to this database allow us to select the observations of a particular satellite for a given year of observation or for all years.

Chapter 12 Recent models of planetary satellite motion. Information resources

The United States Naval Observatory (USNO) Flagstaff has its own database of observations of planets and satellites. On the page at http://www.usno.navy.mil/USNO/astrometry/optical-IRprod/solsys/fastt-plansat we can access observations made with a telescope called the Flagstaff Astrometric Scanning Transit Telescope. Access should be through hyperlinks that indicate years. Astrometric observations of planets and satellites performed at Table Mountain Observatory (California, USA) are available on the Internet at https://ssd.jpl.nasa.gov/dat/planets/ccd1.tmo.html. In addition to planetary satellite observation databases, databases of the physical parameters of satellites can be useful. There are also few such databases available on the Internet. We list only some of them. Within the NSDB database (SAI MSU – IMCCE) there are pages “Parameters and constants”. To access it, we must first enter the NSDB at the above addresses, and then follow the hyperlinks: Databases—Parameters and constants. Then we can go to the pages: – Masses of satellites, – Sizes of satellites, – Photometric properties of satellites, – Satellite rotation parameters. Another source of data on the physical properties of planetary satellites is the reference and information system Dynamics of the Solar System at the Jet Propulsion Laboratory of California Institute of Technology (USA). It is available on the Internet at https:// ssd.jpl.nasa.gov/. There is the only specialized bibliographic database of natural planetary satellites created within the NSDB (SAI MSU – IMCCE). To get there via the Internet, we must first enter the NSDB at one of the addresses listed above, and then follow the hyperlinks: Databases – Bibliography. Of course, the entire bibliography on the research of natural planetary satellites is covered by the bibliographic database on astronomy: The SAO/NASA Astrophysics Data System Abstract Service. Access to bibliography searches is available at http:// adsabs.harvard.edu/abstract_service.html. Now consider the available means of calculating the ephemeris of natural planetary satellites, available via the Internet. Note that such funds are also called ephemeris servers or simply ephemeris. One of the most developed tools for calculating the ephemerides of planetary satellites is the MULTI-SAT server. It was created, maintained and accompanied jointly by the Institut de Mécanique céleste et de calcul des éphémérides (Paris, France), and the Department of Celestial Mechanics of SAI, Moscow State University (Moscow, Russia). The announcement of the creation of

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the MULTI-SAT service and a brief description of it are contained in the publication (Emel’yanov and Arlot, 2008). Addresses for the MULTI-SAT service are available through the Internet at the website IMCCE (http://nsdb.imcce.fr/multisat/) and at the SAI MSU website (http://www.sai.msu.ru/neb/nss/html/multisat/). This tool is described in more detail below. Another tool developed for calculating the planetary satellite ephemeris is the HORIZONS JPL ephemeris server, USA (Giorgini et al., 1996), which provides access to ephemeris via the Internet web page. The ephemeris request page address is http://ssd.jpl. nasa.gov/horizons.cgi. This means of calculating the ephemerides of not only the natural satellites of the planets, but also almost all the bodies of the Solar System. The Minor Plant Center (MPC) has a web service for calculating the ephemeris of distant planetary satellites. Access is provided at http://www.minorplanetcenter.net/iau/NatSats/NaturalSatellites. html. The Institute of Applied Astronomy of the Russian Academy of Sciences (IAA RAS, St. Petersburg, Russia) developed original models of motion of the main planetary satellites (Kosmodamianskii, 2009; Poroshina, 2013). A significant achievement in the modeling of planetary motion based on observations is a series of EPM (Ephemeris of Planets and Moon) works performed at the IAA RAS. EPM ephemeris include high-precision orbits of the planets of the Solar System, the Sun, the Moon, the three largest asteroids (Ceres, Pallas, and Vesta) and four trans-Neptune objects (Eris, Makemake, Haumea, and Sedna). In addition, the EPM includes the ephemeris of the physical libration of the Moon and the difference in dynamic and terrestrial time TT-TDB. EPM ephemeris cover a time interval of more than 400 years (1787–2214). A detailed description of the EPM ephemeris is available at http://iaaras.ru/en/dept/ ephemeris/epm/. An interactive site was created at the IAA RAS for calculating the ephemeris of planets and natural satellites. The website address is http://iaaras.ru/en/dept/ephemeris/online/. Different sets of values are supported for choosing calculated values: observation dates and calculated values in different formats, starting date and step, geocentric and topocentric coordinates for various points on Earth, etc. To calculate the ephemeris of the planets and natural satellites of Jupiter, Saturn, Uranus, and Neptune, models based on the numerical integration of differential equations of motion developed at the IAA RAS are used; in addition, theories of various other authors for the corresponding

Chapter 12 Recent models of planetary satellite motion. Information resources

satellites are also available, as well as versions of different variants of the ephemeris of the planets DE and INPOP (indicated in the “References” at the end of the interactive site). Note that the ephemeris request forms on the pages of the MULTI-SAT, HORIZONS, MPC, and IAA ephemeris servers are significantly different. An important difference is the number of significant digits that are given when calculating the ephemeris. The MPC server gives the topocentric coordinates of the satellites with an accuracy of units of arc seconds, the HORIZONS server with an accuracy of 0.001 arcseconds, and the MULTI-SAT and IAA servers give six significant digits after the decimal point (in arcseconds). These properties do not reflect the accuracy of the ephemeris issued by different services. These are only their technical capabilities and limitations.

12.3 MULTI-SAT ephemeris server features One of the most advanced tools providing planetary satellite ephemeris is the MULTI-SAT ephemeris server. It was created and developed, starting in 2002, on the basis of ephemeris means, which were developed at the Sternberg Astronomical Institute of M.V. Lomonosov Moscow State University. In collaboration with the Institut de Mécanique céleste et de calcul des éphémérides, this planetary satellite ephemeris service is constantly supported and developed. When new satellites are discovered, their ephemeris are added to the server. With the advent of new, more advanced motion models, they are added to the service tools. A report on the creation of the MULTI-SAT ephemeris server was published in Emel’yanov and Arlot (2008). An important property of the MULTI-SAT server is that it allows us to calculate the ephemeris of not only satellites, but also the ephemeris of all major planets from Mercury to Pluto, as well as the ephemeris of the Sun and Moon. There are various planetary satellite ephemeris services in the world. The MULTI-SAT ephemeris server has features and advantages over other analogs. We list here the main ones. 1. Ephemeris is calculated for all natural satellites of Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto, for which there are someone developed motion models based on observations. 2. Request for ephemeris is carried out from the Internet pages in a convenient form with a wide variety of choice of coordinate systems, time scales, types of output values. 3. It is possible to enter a table of satellite observation results and instantly receive differences “O – C”—the result of com-

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paring the observations with theory. The number of observations is not limited. 4. It is possible to graphically display the apparent location of satellites relative to the planet. 5. It is possible to obtain rectangular planetocentric coordinates of satellites, elements of osculating Keplerian orbits, and their mean values. 6. In addition to the coordinates of the satellites, astrometric coordinates of the planets can also be calculated. 7. We can choose one of ten modern planetary theories developed at IMCCE, JPL, and the Institute of Applied Astronomy of the Russian Academy of Sciences (Pitjeva, 2013; Pitjeva and Pavlov, 2017). 8. The ephemeris of the mutual eclipses of the satellites is calculated. In this case, the heliocentric relative coordinates of the satellites are issued with a special method for calculating light time. 9. We can get the ephemeris and circumstances of all the mutual occultations and eclipses of satellites, as well as eclipses of satellites by the planet. 10. Various values accompanying the satellite ephemeris are given out: the apparent magnitudes of the satellites, the planetocentric angular coordinates of the satellites, the Earth and the Sun, the solar phase, the apparent dimensions of the planet, the horizontal coordinates of the satellites at the observation site. Let us now consider in more detail the possibilities of choosing options for time scales and coordinate systems. The time can be set in the UTC scale or in the TT scale. The menu for choosing the equator and equinox of the geoequatorial celestial coordinate system offers the following sections: J2000—the coordinate system is determined by the choice of planetary theory. For theories developed in JPL (DE405/LE405, DE406/LE406, ...), IMCCE (INPOP) or IAA RAS (EPM-2015), this is the ICRF system. If the planetary theory VSOP87 or DE200 is selected, the coordinate system will be consistent with these theories. ICRF—all source coordinates are translated into this system. If the planetary theory DE200/LE200 or VSOP87 is chosen, then the coordinates of the satellites or planets are first transferred to the FK5 system as described in Standish (1982), and then to the ICRF system using the transition formulas described in Feissel and Mignard (1998).

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FK5—a conversion is made to this system as prescribed in the work specified in the previous paragraph. Mean of the date (IAU76)—first, the coordinates are converted to the FK5 system, and then to the equator and equinox systems of the epoch of the date according to the precession formulas prescribed by the IAU resolution in 1976. True of the date (IAU76, IAU80)— first, the coordinates are converted to the mean equator and the equinox of the date epoch, as described in the previous paragraph, and then to the true equator and equinox by taking into account nutation according to the algorithm prescribed by the IAU regulation 1980 (Seidelmann, 1980). Apparent (IAU76, IAU80)—this is the same transformation to the true equator and equinox, as described in the previous paragraph, but the coordinates will no longer be astrometric, but apparent. How apparent coordinates are obtained is described in reference books (Seidelmann, 1992; Simon et al., 1997). Note that in the MULTI-SAT server, when we enable nutation accounting according to the IAU80 standard, in fact, more accurate data on the Earth’s rotation are used. How this is done is described at the end of Chapter 5. B1950—coordinates are converted to a system based on the FK4 star catalog. For this, formulas and an algorithm recommended by the IAU resolution and described in Aoki et al. (1983) are used. This conversion takes into account the E-term in the expression for aberration and depends on the time of observation or ephemeris. B1900—similarly to the previous paragraph, only after the coordinate transformation in the FK4 system are they additionally transferred to the system of the middle equator and the equinox of the B1900 epoch according to the Newcomb precession formulas. There are also menu items that provide transformations similar to the previous paragraphs, but with the Newcomb precession model and nutation according to the IAU resolution of 1948. These are the following items: Mean of Date (Newcomb), Mean of Jan 1 of the year (Newcomb), True of Date (Newcomb, IAU1948), Apparent (Newcomb, IAU1948). It should be noted that in the MULTI-SAT server, all ephemeris in the apparent coordinate system are calculated without taking into account the curvature of the light beam caused by gravity of the Sun. Note that other coordinate systems, except ICRF, have not been used recently. However, the MULTI-SAT server makes it possible to convert ephemeris to systems that were used in past centuries.

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This is done so that it is possible to compare the results of old observations published in papers of past centuries with ephemeris. In the MULTI-SAT planetary satellite ephemeris server for given time points, either right ascension and declination of the selected satellite or planet, or the difference of the coordinates of two bodies (satellite and planet or two satellites), called relative coordinates, are issued. Moreover, the relative coordinates can be differential or tangential, as described in Chapter 5. In the special menu of the MULTI-SAT server, the following options for the types of output data are offered. Alpha, Delta (h, m, s.decimals)—at specified times, right ascensions and declinations are issued. Delta(Alpha), Delta(Delta) (arcsec)—at given points in time, differences of right ascensions and declinations of two bodies are issued. Diff. X, Y (arcsec)—relative coordinates X = α cos δ, Y = δ, where α, δ are the differences of right ascensions and the declination of two bodies, and δ is the declination of the body relative to which the relative coordinates are determined. Diff. S (arcsec), Position angle (degrees)—similar to the previous paragraph, however, instead of X, Y , a differential mutual angular distance, and a position angle are issued. Diff. Inclination (deg), A, B (arcsec)—differential relative coordinates are generated in the celestial relative coordinate system A, B, rotated by the position angle of the planet’s pole I , so that the coordinate relationship is given by the formulas X = A · cos I + B · sin I , Y = −A · sin I + B · cos I . The position angle of the planet’s pole I is given in front of the coordinates A, B. Tang. X, Y (arcsec), Tang. S(arcsec), Position angle (degrees), Tang. Inclination (deg), A, B (arcsec)—similar to the previous paragraphs, but the coordinates are tangential (see the explanation in Chapter 5). Pseudo-heliocentric X, Y Pseudo-heliocentric S, Position angle—similar to the previous paragraphs, but the coordinates are heliocentric with a special procedure for determining light time to model the mutual eclipses of two satellites. An explanation of such coordinates is given in Chapter 5. x, y, z, Vx, Vy, Vz Geo-equatorial—rectangular coordinates and velocity components of the selected satellite relative to the center of the planet or relative to another selected satellite are generated. The coordinate axes are mutually parallel to the axes of the geoequatorial coordinate system.

Chapter 12 Recent models of planetary satellite motion. Information resources

x, y, z, Vx, Vy, Vz Geo-ecliptic—similar to the previous paragraph, but the coordinate axes correspond to the geoecliptic. x, y, z, Vx, Vy, Vz Geo-planetocentric—similar to the previous paragraph, but the coordinate axes correspond to the plane of the equator of the planet. In this system, the x axis is directed to the ascending planet’s equator node at the geoequator. Kepler osculat. orbit(geo-equat.)—elements of the Keplerian osculating orbit with a moment of osculation corresponding to a given point in time are issued. Elements belong to the geoequatorial coordinate system. At the end of the table, the minimum, mean, and maximum values of the mean motion, semi-major axis, eccentricity, and inclination of the orbit are displayed. Kepler osculat. orbit(geo-eclipt.)—similar to the previous paragraph, however, the coordinate system is geoecliptic. Kepler osculat. orbit(planeto-equat.)—similar to the previous paragraph, however, with the planet-equatorial coordinate system. R, Alpha, Delta Geo-planetocentric—planetocentric distance (in km), right ascension and declination (in degrees) in the geoequatorial coordinate system. Plan-equat.lat.,long.,sideral of sat.—sidereal latitude and longitude of the satellite relative to the planetary equator. Longitude is measured from the ascending node of the equator of the planet on the plane of the Earth’s equator. Plan-equat.lat.Earth.,lat.long.synodic of sat.—planetocentric latitude of the Earth relative to the equator of the planet, synodic planetocentric latitude and longitude of the satellite in the planetequatorial coordinate system. The longitude of the satellite is measured relative to the projection of the radius vector of the Earth on the equator of the planet. Plan-equat.lat.Sun,lat.long.heliocentr.of sat.—planetocentric latitude of the Sun relative to the equator of the planet, planetocentric latitude and longitude of the satellite in the planet-equatorial coordinate system. The longitude of the satellite is measured relative to the projection of the radius vector of the Sun on the equator of the planet. Topoc.alt.,azim.of sat.,alt.,azim.of the Sun—topocentric angular coordinates of the satellite: altitude, azimuth, and hour angle. Topocentric angular elevation and azimuth of the Sun. Moon phase (0.0 for the new moon and 1.0 for the full moon). If the geocenter is selected as the observation point, then zero values are given for the coordinates. It should be noted that when translating coordinates into a horizontal system for a given ground point, only

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the precession of the Earth’s rotation axis is taken into account. Atmospheric refraction is not taken into account. The gravitational curvature of a light beam is also not taken into account. Dist.Sun.-Plan.,app.plan.R,Phase,magn.of sat.—geocentric angular distance between the planet and the Sun, the apparent angular radius of the planet, and the magnitude of the selected satellite. Let us make some important remarks regarding the types of various data that are issued by the MULTI-SAT planetary satellite ephemeris server. 1. The planetocentric planet-equatorial coordinates of the Earth and the Sun allow us to calculate the periods of possible mutual occultations and eclipses of the main satellites of major planets. Since the orbits of these satellites lie near the plane of the equator of the planet, mutual phenomena occur precisely in those time periods when the latitudes of the Earth and the Sun are close to zero. 2. The synodic planetocentric longitude of the satellite (measured relative to the direction to the Earth) in the case of synchronous rotation of the satellite allows us to immediately determine which side the satellite is facing toward the observer. In fact, the central apparent meridian on the satellite at any given time has a satelliteographic longitude equal to the synodic planetocentric longitude of the satellite plus 180 degrees. The satelliteographic longitude of a synchronous satellite is measured on the satellite’s body from the direction to the planet in the direction opposite to the satellite’s rotation. Simply put, if the synodic planetocentric longitude of a satellite is zero, then for an observer on Earth the satellite is located in front of the planet. If this longitude is 180 degrees, then the satellite is behind the planet. 3. If we want to determine the orientation of the rotation axis of the planet with respect to the Earth’s observer, this can be done by two issued angles. Before the coordinates A, B (see above), the positional angle of the planet’s pole I is displayed. The inclination of the rotation axis of the planet to the sky plane is equal to the planetocentric planet-equatorial latitude of the Earth (see above). If this angle is positive, it means that the north pole of the planet is deflected towards the observer. In particular, it is possible to determine the orientation of the rings of Saturn visible from the Earth. 4. For satellite observations, it is important to know the angle of the solar phase and the Moon phase and the magnitude of the satellite. These values are displayed when the menu item is selected (see above).

Chapter 12 Recent models of planetary satellite motion. Information resources

12.4 Theories and models in the MULTI-SAT ephemeris server The creators of theories and models of motion of natural planetary satellites are constantly working on the development of ever new versions. Progress in this matter is mainly ensured by the emergence of new observational results. Naturally, the desire of researchers is to use the latest versions. However, when comparing the results in the literature, it is sometimes necessary to use also previous versions of planetary satellite motion models. Therefore, in the MULTI-SAT ephemeris service (SAI MSU—IMCCE), the ability to calculate ephemeris using some previous versions is left. Modern and most advanced models of planetary satellite motion are used in the services of the ephemeris MULTI-SAT (SAI MSU—IMCCE) (Emel’yanov and Arlot, 2008) and HORIZONS JPL, USA (Giorgini et al., 1996). On the pages of the MULTI-SAT ephemeris service there is a table in which a list of all possible models of satellite motion is available. Of course, by default, the service includes the latest and most advanced versions for each satellite. The most significant characteristics of the motion models are as follows: – time period and types of observations used, – time period for the presentation of the ephemerides, – type of model. Here, the type of model can be one of the following: numerical integration of the equations of motion, analytical theory of motion, analytical representation of the results of numerical integration, a precessing ellipse, or a more complex combination of methods. Here is a description of only those models that are offered by default as the most advanced ones in the MULTI-SAT ephemeris service. In some cases, alternative models are also considered. Some models are built immediately for a number of satellites in some groups. The sequence will correspond to the planets (from Jupiter to Pluto) and the types of satellites: main, close, and distant. The designations of some models will indicate the choice in the menu in the MULTI-SAT ephemeris service. The authors are listed here below in the model descriptions. Next, calendar dates are given in the form YYYY/MM/DD (year, month, day). Satellites of Mars Phobos and Deimos The choice is from the menu as the item ’Lainey (2015)’. Model built by French researcher Valery Lainey at IMCCE (Paris, France). Differential equations of motion were solved by the method of numerical integration. All ground-based observations over the

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time interval 1877–2014 were used, as well as observations from spacecraft, including the latest data from the Mars Explorer (MEX) spacecraft. The time period for the presentation of ephemerides is from 1869/12/31 to 2123/07/03. A report on the creation of a model is published in an article (Arlot et al., 2017). This model differs from that published in Lainey et al. (2007) only in the composition of the observations used. Galilean satellites of Jupiter The choice is from the menu of J1–J4 as the item ’Lainey 2009, V2.0’. The model was built by French researcher Valery Lainey at IMCCE (Paris, France). Differential equations of motion were solved by the method of numerical integration. All ground-based photographic observations and observations on the meridian circle in the time interval 1891–2007 were used, astrometric results of photometric observations of satellites during their mutual occultations and eclipses of 1973–2003 (every 6 years). The time period for the presentation of ephemerides is from June 1, 1903 to February 13, 2043. A report on the creation of a model was published in Lainey et al. (2009). Close satellites of Jupiter The choice is from the menu of J5, J14–J16 as the item ’Adjusted on (Jacobson, 2009)’. The author constructed the model in Emel’yanov (2015) by refining the parameters of the precessing ellipse based on the ephemeris of these satellites, calculated according to the theory from the publication (Jacobson, 2013). The time interval of the satellite ephemeris is limited only by the time interval of planetary ephemeris. Distant satellites of Jupiter Eight distant satellites of Jupiter (J6–J13). The model is published in Emelyanov (2005). Differential equations of motion were solved by the method of numerical integration. After publication, the model was repeatedly revised based on wider series of observations. For the current version of the ephemeris, ground-based observations of 1905–2016 were used. The time interval for the presentation of the ephemerides is from 1905 to 2049. New distant satellites of Jupiter The model was published in Emel’yanov and Kanter (2005). Differential equations of motion were solved by the method of numerical integration. After publication, the model was repeatedly revised based on wider series of observations. We used ground-

Chapter 12 Recent models of planetary satellite motion. Information resources

based observations at different time intervals for different satellites (from 30 days to 12 years). The latest observations used were made in 2018. The time interval for the presentation of ephemerides for most of these satellites is from 1974 to 2049. For some satellites, the interval is smaller. Main satellites of Saturn The choice is from the menu of S1–S8 as the item ’Lainey et al. (2015)’. The model was built by French researcher Valery Lainey at IMCCE (Paris, France). Differential equations of motion were solved by the method of numerical integration. All groundbased observations were used over the time interval from 1885 to 2009, astrometric results of photometric observations of mutual phenomena in 1995 and 2009, and observations from Cassini spacecraft (2004–2012). A report on the creation of a model was published in Arlot et al. (2017). The time period for the presentation of ephemerides is from January 1, 1950, to January 1, 2048. Saturn’s satellites, co-orbiting the main ones Helene S12, Telesto S13, Calypso S14, and Polydeuce S34. The choice from the menu is as the item ’Lainey et al. (2015)’. Posted by Arlot et al. (2017). Differential equations of motion were solved by the method of numerical integration. Ground-based observations (1980–1996) and observations from the Cassini spacecraft (2004–2012) were used. The time period for the presentation of ephemerides is from January 1, 1950 to May 16, 2049. A report on the creation of a model was published in Arlot et al. (2017). Saturn’s close co-orbiting satellites The choice is from the menu of S10, S11 as the item ’Nicholson et al. (1992)’. Satellites S10 Janus, S11 Epimetheus. Analytic theory. The time interval of the satellite ephemeris is limited only by the time interval of planetary ephemeris. Published in Nicholson et al. (1992). Saturn’s close satellites The choice is from the menu of S15–S18 as the item ’Jacobson et al. (2008)’. Satellites Atlas S15, Prometheus S16, Pandora S17, Pan S18, Methone S32, Pallene S33, and Daphnis S35. Precessing ellipse model. Based on observations from the Cassini spacecraft (2004–2012). The time interval of the satellite ephemeris is limited only by the time interval of planetary ephemeris. Published in Jacobson et al. (2008).

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The distant satellite of Saturn Phoebe Differential equations of motion were solved by the method of numerical integration. We used ground-based observations (1898–2012). and observations from the Cassini spacecraft (2004– 2012). The time period for the presentation of ephemerides is from July 1, 1875 to June 30, 2022. A report on the creation of a model was published in Desmars et al. (2013). New distant satellites of Saturn The model was published in Emel’yanov and Kanter (2005). Differential equations of motion were solved by the method of numerical integration. We used ground-based observations at different time intervals for different satellites (from 30 days to 12 years). After the publication of the model, it was repeatedly revised on the basis of wider series of observations. The time period for the presentation of ephemerides for most of these satellites is from 1974 to 2049. For some satellites, the interval is smaller. Main satellites of Uranus The choice is from the menu as the item ’Lainey et al. (2015)’. Differential equations of motion were solved by the method of numerical integration. Ground-based observations (1874–2012), observations from the Voyager 2 spacecraft were used, and astrometric results of photometric observations of mutual phenomena (2007–2008). The time period for the presentation of ephemerides is from January 0, 1847 to January 2, 2145. The first version of the model was published in Lainey (2008). A revised model based on a wider range of observations was published in Arlot et al. (2017). Main satellites of Uranus Alternative model. The choice is from the menu as the item ’Emelyanov, Nokonchuk (2013)’. Model is published by Emelyanov and Nikonchuk (2013). Differential equations of motion were solved by the method of numerical integration. We used groundbased observations (1787–2008), observations from the Voyager 2 spacecraft, and astrometric results of photometric observations of the mutual phenomena (2007–2008). The time period for the presentation of ephemerides from February 12, 1787 to January 9, 2032. Close satellites of Uranus Precessing ellipse model. The parameters are published in Jacobson (1998); Pascu et al. (1998). The model is based on observations from the HST space telescope in 1994 and observations

Chapter 12 Recent models of planetary satellite motion. Information resources

of the Voyager-2 spacecraft in 1985–1986. The representation of ephemerides is limited in time only to planetary ephemeris. New distant satellites of Uranus The model was published in Emel’yanov and Kanter (2005). Differential equations of motion were solved by the method of numerical integration. We used ground-based observations for different satellites at different time intervals. After publication, the models could be revised based on wider series of observations. The last year of the observations is 2016. The time period for the presentation of the ephemerides for satellites U16 and U17 is from 1974 to 2049, for the rest from 1974 to 2026. Neptune’s satellite Triton The choice is from the menu as the item ’Triton by Emelyanov, Samorodov (2015)’. The analytical theory of the satellite is based on all ground-based observations made from 1847 to 2012 and observations from the Voyager-2 spacecraft. The representation of ephemerides is limited in time only to planetary ephemeris. Published in Emelyanov and Samorodov (2015). Distant satellite of Neptune Nereid The model was published in Emelyanov and Arlot (2011). Differential equations of motion were solved by the method of numerical integration. The model was updated after publication. Ground-based observations from 1949 to 2017 and observations from the Voyager-2 spacecraft were used. The time period for the presentation of the ephemerides is from 1920 to 2049. Neptune’s close satellites Precessing ellipse model. The parameters are published in Owen et al. (1991); Pascu et al. (2004); Jacobson (2009). The model is based on observations from the HST space telescope in 1997 and observations of the Voyager-2 spacecraft. The representation of ephemerides is limited in time only to planetary ephemerides. Satellites of Pluto The model is published in Beauvalet et al. (2013). Differential equations of motion were solved by the method of numerical integration. Observations from the ground-based telescope VLT-UT4 and the HST space telescope of the Charon satellite in the time interval 1992–2010, and for the Nikta and Hydra satellites in the interval 2002–2006 were used. The time period for the presentation of ephemerides is from January 1, 1950 to December 31, 2029.

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12.5 Theories and models in the JPL ephemeris server The modern and most advanced models of planetary satellite motion are used in the HORIZONS Solar System ephemeris server developed at JPL, USA (Giorgini et al., 1996). A list of all publications describing motion models is available on the Internet at https://ssd.jpl.nasa.gov/?sat_ephem. We will briefly review these models. Models are constructed on the basis of all available groundbased observations and observations using spacecraft. For most of the models created by Jacobson and Brozovic, reference to publications is not provided. Only model identifiers are given for identifying ephemeris. For the satellites of Mars, Phobos, and Deimos the theory described in Jacobson and Lainey (2014) is used and based on all available ground-based observations and observations with spacecrafts. The equations of motion are solved by numerical integration methods. For the Galilean and close satellites of Jupiter, the JUP310 model is used. A link to the publication is not provided. The motion model of the distant satellites of Jupiter is described in Brozovic and Jacobson (2017). The equations of motion are solved by numerical integration methods. For the main, close, and distant satellites of Saturn, the original models SAT389, SAT393, and SAT368 are used. References to publications are not provided. A model of the motion of the main satellites of Uranus is published in Jacobson (2014). The equations of motion are solved by numerical integration methods. For close satellites of Uranus, the URA091 model was built, for distant satellites, the URA112 model. References to publications are not provided. The motion model for satellites of Neptune Triton, Proteus, and Nereid was published in Jacobson (2009). For Neptune’s internal satellites, the original NEP088 model is used. A reference to the publication is not provided. A motion model of distant satellites of Neptune was published in Brozovic et al. (2011). Based on the data obtained from the New Horizons spacecraft, new motion models for the satellites of Pluto were constructed (Jacobson et al., 2015).

Chapter 12 Recent models of planetary satellite motion. Information resources

12.6 Planet satellites in virtual observatories The so-called Virtual Observatories (VOs) are being developed in the world. There is as yet no well-established understanding of this term. Most often, a virtual observatory is a set of tools for working with astronomical data resources. The data is created by the astronomers themselves; experts in the relevant fields. The virtual observatory accumulates information about these resources, describes them, classifies and develops methods for accessing them: formats, standards, etc. Attempts to find a virtual observatory containing planetary satellites give the only result—this is the Virtual Observatory of the Institut de Mécanique céleste et de calcul des éphémérides (Paris, France). Access to this virtual observatory is at http://vo.imcce.fr/. Most of the components of this VO are under development. Attempts to find tools related to planetary satellites here lead only to the possibility of calculating the planetary satellite ephemeris in a tabular form. The address of this tool is http://vo.imcce.fr/ webservices/skybot/?forms=resolver. The set of satellites in this virtual olbservatory is very limited. These are the satellites of Mars (2), the Galilean satellites of Jupiter (4), the main satellites of Saturn (8), the main satellites of Uranus (5), the satellites of Neptune Triton and Nereida, some close satellites of Uranus (3), and close satellites of Neptune (6). These 30 objects make up a small part of all the known 209 planet satellites.

12.7 Standards of fundamental astronomy Scientific activity in the field of the dynamics of the Solar System is developing and practicing in many large scientific centers and in individual laboratories of the world. The modern technology of scientific research consists of the following main practical actions: – production of observations, – accumulation and maintenance of databases, and – compilation of computer programs for data processing. Each of these actions is carried out in a huge network of large and small teams of researchers. Relationships are established with concomitant rivalry. There are two main motivations: financial support and satisfaction of ambitions. A prerequisite for the optimal practice on a global scale is the flexible coordination of scientific research. For this purpose, coordination organizations such as international Centers, Councils and Unions are needed. One of the functions of such centers is the development of standards.

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Standards are necessary for the coherence of research, comparison of results. Often standards are dictated by some of the most powerful and dominant research centers. Sometimes standards are set as a result of the creation of a large complex of dynamic models of celestial bodies, based on a large database of observations. We list the largest scientific enterprises in the field of the dynamics of the bodies of the Solar System. International Astronomical Union (IAU) approves the proposals of astronomical standards at its regular assemblies. Its internet address is https://www.iau.org/. Minor planet center (MPC) contains and maintains a database of all observations of small bodies: asteroids, dwarf planets, comets, and distant planetary satellites. MPC collects data, models the motion of small bodies, and also provides the world community with a number of research tools: ephemeris computing programs and data analysis tools. As a dominant scientific enterprise, MPC is the source and conductor of standards in its field. The address of MPC is available through the Internet at https:// minorplanetcenter.net/iau/mpc.html. In particular, the MPC establishes a standard format for the presentation of small body observations and a standard for assigning observatory codes. These standards can be found on the Internet at https://minorplanetcenter.net/iau/info/ObsFormat.html https://minorplanetcenter.net/iau/lists/ObsCodesF.html. Navigation and Ancillary Information Facility (NAIF) of the National Aeronautics and Space Administration (NASA) was created at the Jet Propulsion Laboratory (JPL), California Institute of Technology to guide the development and implementation of the SPICE Information System “(The SPICE Toolkit). The SPICE is used throughout the entire life cycle of NASA’s planetary scientific missions to help scientists and engineers design missions, plan scientific observations, analyze scientific data and perform various engineering functions related to flight projects. The NAIF team is involved in the production of highly accurate, clearly documented and easily used information needed by scientists and engineers. These are the main tasks of the NAIF team. The SPICE Toolkit is a system of databases and computing programs for solving problems of fundamental astronomy. Addresses of NAIF and SPICE are available through the Internet at https://naif.jpl.nasa.gov/naif/about.html and https://naif.jpl.nasa.gov/naif/aboutspice.html, respectively. It is natural that the NAIF-SPICE system sets completely clear and strict standards for databases and software. These standards en-

Chapter 12 Recent models of planetary satellite motion. Information resources

sure the functioning of all parts of the system. The programs provided in the SPICE system are recorded in several versions in accordance with the programming languages Fortran, C, and some others. Naturally, researchers around the world using these tools have to follow NAIF-SPICE standards. Standards of Fundamental Astronomy (SOFA) This is a service with the task of establishing and maintaining an accessible and authoritative set of algorithms and procedures that implement standard models used in fundamental astronomy. The service is managed by an international commission, the SOFA Council, appointed by Division A – Fundamental Astronomy of the International Astronomical Union (IAU). SOFA also works closely with the International Earth Rotation and Reference System (IERS). Algorithms and programs in the SOFA service are written in the Fortran and C programming languages (ANSI C). SOFA internet address is http://www.iausofa.org/. International Earth Rotation and Reference System Service (IERS). The main objectives of IERS are to service astronomical, geodetic, and geophysical communities by providing data and standards related to the Earth’s rotation and reference systems. The IERS internet address is https://www.iers.org/IERS/EN/ Home/home_node.html.

References Acton, C., Bachman, N., Folkner, W.M., Hilton, J., 2015. SPICE as an IAU recommendation for planetary ephemerides. In: IAU General Assembly Meeting #29, #2240327. Aoki, S., Soma, M., Kinoshita, H., Inoue, K., 1983. Conversion matrix of epoch B 1950.0 FK 4-based positions of stars to epoch J 2000.0 positions in accordance with the new IAU resolutions. Astronomy & Astrophysics 128, 263–267. Arlot, J.-E., Emelyanov, N.V., 2009. The NSDB natural satellites astrometric database. Astronomy & Astrophysics 503, 631–638. Arlot, J.E., Cooper, N., Emelyanov, N., Lainey, V., Meunier, L.E., Murray, C., Oberst, J., Pascu, D., Pasewaldt, A., Robert, V., Tajeddine, R., Willner, K., 2017. Natural satellites astrometric data from either space probes and ground-based observatories produced by the European consortium “ESPaCE”. Notes scientifiques et techniques de l’Institut de Mécanique céleste et de calcul des éphémérides S105. Beauvalet, L., Robert, V., Lainey, V., Arlot, J.-E., Colas, F., 2013. ODIN:a new model and ephemeris for the Pluto system. Astronomy & Astrophysics 553, A14. 22 pp. Brozovic, M., Jacobson, R.A., Sheppard, S.S., 2011. The orbits of Neptune’s outer satellites. Astronomical Journal 141 (4), 135. 9 pp. Brozovic, M., Jacobson, R.A., 2017. The orbits of Jupiter’s irregular satellites. Astronomical Journal 1531 (4), 147. 10 pp. Desmars, J., Li, S.N., Tajeddine, R., Peng, Q.Y., Tang, Z.H., 2013. Phoebe’s orbit from ground-based and space-based observations. Astronomy & Astrophysics 553, A36. 10 pp.

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Emelyanov, N.V., 2005. Ephemerides of the outer Jovian satellites. Astronomy & Astrophysics 435, 1173–1179. Emel’yanov, N.V., Kanter, A.A., 2005. Orbits of new outer planetary satellites based on observations. Solar System Research 39 (2), 112–123. Emel’yanov, N.V., Arlot, J.-E., 2008. The natural satellites ephemerides facility MULTI-SAT. Astronomy & Astrophysics 487, 759–765. Emelyanov, N.V., Arlot, J.-E., 2011. The orbit of Nereid based on observations. Monthly Notices of the Royal Astronomical Society 417 (1), 458–463. Emelyanov, N.V., Nikonchuk, D.V., 2013. Ephemerides of the main Uranian satellites. Monthly Notices of the Royal Astronomical Society 436, 3668–3679. Emel’yanov, N.V., 2015. Perturbed motion at small eccentricities. Solar System Research 49 (5), 346–359. Emelyanov, N.V., Samorodov, M.Yu., 2015. Analytical theory of motion and new ephemeris of Triton from observations. Monthly Notices of the Royal Astronomical Society 454, 2205–2215. Feissel, M., Mignard, F., 1998. The adoption of ICRS on 1 January 1998: meaning and consequences. Astronomy & Astrophysics 331, L33–L36. Giorgini, J.D., Yeomans, D.K., Chamberlin, A.B., Chodas, P.W., Jacobson, R.A., Keesey, M.S., Lieske, J.H., Ostro, S.J., Standish, E.M., Wimberly, R.N., 1996. JPL’s on-line solar system data service. Bulletin - American Astronomical Society 28, 1158. Jacobson, R.A., 1998. The orbits of the inner Uranian satellites from Hubble space telescope and voyager 2 observations. Astronomical Journal 115 (3), 1195–1199. Jacobson, R.A., Spitale, J., Porco, C.C., Beurle, K., Cooper, N.J., Evans, M.W., Murray, C.D., 2008. Revised orbits of Saturn’s small inner satellites. Astronomical Journal 135, 261–263. Jacobson, R.A., 2009. The orbits of the neptunian satellites and the orientation of the pole of Neptune. Astronomical Journal 137, 4322–4329. Jacobson, R.A., 2013. The orbits of the regular Jovian satellites, their masses, and the gravity field of Jupiter. In: American Astronomical Society, DDA Meeting, #44, #402.04. Jacobson, R.A., 2014. The orbits of the Uranian satellites and rings, the gravity field of the Uranian system, and the orientation of the pole of Uranus. Astronomical Journal 148 (5), 76. Jacobson, R.A., Lainey, V., 2014. Martian satellite orbits and ephemerides. Planetary and Space Science 102, 35–44. Jacobson, R.A., Brozovic, M., Buie, M., Porter, S., Showalter, M., Spencer, J., Stern, S.A., Weaver, H., Young, L., Ennico, K., Olkin, C., 2015. The orbits and masses of Pluto’s satellites after new horizons. In: American Astronomical Society. DPS Meeting #47. id. 102.08. Kosmodamianskii, G.A., 2009. Numerical theory of the motion of Jupiter’s Galilean satellites. Solar System Research 43 (6), 465–474. Lainey, V., Dehant, V., Patzold, M., 2007. First numerical ephemerides of the Martian moons. Astronomy & Astrophysics 465, 1075–1084. Lainey, V., 2008. A new dynamical model for the Uranian satellites. Planetary and Space Science 56, 1766–1772. Lainey, V., Arlot, J.-E., Karatekin, O., van Hoolst, T., 2009. Strong tidal dissipation in Io and Jupiter from astrometric observations. Nature 459 (7249), 957–959. Nicholson, P.D., Hamilton, D.P., Matthews, K., Yoder, C.F., 1992. New observations of Saturn’s coorbital satellites. Icarus 100, 464–484. Owen, W.M., Vaughan, R.M., Synnott, S.P., 1991. Orbits of the six new satellites of Neptune. Astronomical Journal 101, 1511–1515.

Chapter 12 Recent models of planetary satellite motion. Information resources

Pascu, D., Rohde, J.R., Seidelmann, P.K., Wells, E.N., Kowal, C.T., Zellner, B.H., Storrs, A.D., Currie, D.G., Dowling, D.M., 1998. Hubble space telescope astrometric observations and orbital mean motion corrections for the inner uranian satellites. Astronomical Journal 115 (3), 1190–1194. Pascu, D., Rohde, J.R., Seidelmann, P.K., Wells, E.N., Hershey John, L., Storrs, A.D., Zellner, B.H., Bosh, A.S., Hubble, Currie D.G., 2004. Space telescope astrometric observations and orbital mean motion corrections for the inner satellites of neptune. Astronomical Journal 127 (5), 2988–2996. Pitjeva, E.V., 2013. Updated IAA RAS planetary ephemerides-EPM2011 and their use in scientific research. Solar System Research 47 (5), 386–402. Pitjeva, E.V., Pavlov, D.A, 2017. EPM2017 and EPM2017H. http:// iaaras.ru/en/dept/ephemeris/epm/2017/. (Accessed 7 November 2017). Poroshina, A.L., 2013. Numerical theories of motion of Triton and Nereid. Astronomy Letters 39 (12), 876–881. Seidelmann, P.K., 1980. IAU theory of nutation—the final report of the IAU working group on nutation. Celestial Mechanics 1982 (27), 79–106. Seidelmann, P.K. (Ed.), 1992. Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley, California. Simon, J.-L., Chapront-Touzé, M., Morando, B., Thuillot, W. (Eds.), 1997. Introduction aux éphémérides Astronomiques. Supplément explicatif à la connaissance des temps. Bureau des Longitudes, Paris. Les éditions de physique, France. 450 pp. Standish Jr., E.M., 1982. Orientation of the JPL ephemerides, DE200/LE200, to the dynamical equinox of J2000. Astronomy & Astrophysics 114, 297–302.

421

A Nomenclature of planetary satellites The data provided here includes the names and numbers of natural planetary satellites adopted by the International Astronomical Union (IAU). The circumstances of the discovery of each satellite are also given: the year of discovery, a reference to the publication of the announcement of the discovery. To more easily identify the type of satellite, approximate values of the semi-major axis are given. In accordance with the rule adopted by the IAU for determining prograde and retrograde motion (see Chapter 9), satellites with retrograde motion are specially marked in Tables A.1–A.6.

Nomenclature of the satellites of Mars

Table A.1 Nomenclature of the satellites of Mars. Name

Latin name

Year of discovery

Semi-major axis, km

M1

Phobos

1877

9380.

M2

Deimos

1877

23460.

Announcement on the discovery of the satellites of Mars Rodgers John, Thompson R.W., Hall Asaph, 1877. Letter to the Hon. R.W. Thompson, Secretary of the Navy, Acing the discovery of satellites of Mars, by Rodgers, John;Thompson, R. W.; Hall, Asaph. Washington: U.S. Naval Observatory, [1877]. United States Naval Observatory.

423

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Appendix A Nomenclature of planetary satellites

Publication of the first observations of the satellites of Mars by Azaf Hall Hall A., 1877. Observations of the satellites of Mars, made with the 26-inch refractor of the U.S. Naval Observatory, Washington. Astronomische Nachrichten 91(2161), 11–14.

Nomenclature of the satellites of Jupiter Table A.2 The nomenclature of the satellites of Jupiter. For satellites without a name, temporary designations are given. Numbers of bibliographic references to the publication of reports of discoveries are given in square brackets. A bibliography is given after the table. In the column “Semi-major axis”, the r sign indicates satellites with retrograde motion. Name

Latin name

Year of discovery

Semi-major axis, km

J1

Io

1610 G. Galilei

421800 –

J2

Europa

1610 G. Galilei

671100 –

J3

Ganymede

1610 G. Galilei

1070400 –

J4

Callisto

1610 G. Galilei

1882700 –

J5

Amalthea

1892 [1]

181400 –

J6

Himalia

1904 [2]

11461000 –

J7

Elara

1904 [2]

11741000 –

J8

Pasiphae

1908 [3]

23624000 r

J9

Sinope

1914 [4]

23939000 r

J10

Lysithea

1938 [5]

11717000 –

J11

Carme

1938 [5]

23404000 r

J12

Ananke

1951 [6]

21276000 r

J13

Leda

1975 [7]

11165000 –

J14

Thebe

1979 [8]

221900 –

J15

Adrastea

1979 [9]

129000 –

J16

Metis

1979 [10]

128000 –

J17

Callirrhoe

1999 [11]

24103000 r

J18

Themisto

2000 [12]

7400000 –

J19

Megaclite

2000 [12]

23493000 r continued on next page

Appendix A Nomenclature of planetary satellites

Table A.2 (continued) Name

Latin name

Year of discovery

Semi-major axis, km

J20

Taygete

2000 [12]

23280000 r

J21

Chaldene

2000 [12]

23100000 r

J22

Harpalyke

2000 [12]

20858000 r

J23

Kalyke

2000 [12]

23566000 r

J24

Iocaste

2000 [12]

21061000 r

J25

Erinome

2000 [12]

23196000 r

J26

Isonoe

2000 [12]

23155000 r

J27

Praxidike

2000 [12]

20907000 r

J28

Autonoe

2001 [13]

24046000 r

J29

Thyone

2001 [13]

20939000 r

J30

Hermippe

2001 [13]

21131000 r

J31

Aitne

2001 [13]

23229000 r

J32

Eurydome

2001 [13]

22865000 r

J33

Euanthe

2001 [13]

20797000 r

J34

Euporie

2001 [13]

19304000 r

J35

Orthosie

2001 [13]

20720000 r

J36

Sponde

2001 [13]

23487000 r

J37

Kale

2001 [13]

23217000 r

J38

Pasithee

2001 [13]

23004000 r

J39

Hegemone

2003 [14]

23947000 r

J40

Mneme

2003 [15]

21069000 r

J41

Aoede

2003 [16]

23981000 r

J42

Thelxinoe

2003 [17]

21162000 r

J43

Arche

2002 [18]

22931000 r

J44

Kallichore

2003 [19]

24043000 r

J45

Helike

2003 [16]

21263000 r

J46

Carpo

2003 [20]

16989000 –

J47

Eukelade

2003 [16]

23661000 r

J48

Cyllene

2003 [21]

23951000 r

J49

Kore

2003 [21]

24011000 r

J50

Herse

2003 [21]

22992000 r continued on next page

425

426

Appendix A Nomenclature of planetary satellites

Table A.2 (continued) Name

Latin name

Year of discovery

Semi-major axis, km

J51

S/2010 J 1

2010 [22]

23189718 r

J52

S/2010 J 2

2010 [23]

20790825 r

J53

Dia

2000 [24]

12297000 –



S/2003 J 2

2003 [16]

29541000 r

J60

Eupheme

2003 [16]

20221000 r



S/2003 J 4

2003 [16]

23930000 r

J57

Eirene

2003 [16]

23495000 r



S/2003 J 9

2003 [19]

23384000 r



S/2003 J 10

2003 [19]

23041000 r



S/2003 J 12

2003 [19]

17582000 r

J58

Philophrosyne

2003 [21]

22627000 r



S/2003 J 16

2003 [21]

20957000 r

J55

S/2003 J 18

2003 [21]

20514000 r

J61

S/2003 J 19

2003 [20]

23533000 r



S/2003 J 23

2003 [25]

23563000 r

J72

S/2011 J 1

2012 [26]

20691211 r

J56

S/2011 J 2

2012 [26]

23233215 r

J54

S/2016 J 1

2017 [27]

20595483 r

J62

Valetudo

2018 [29]

18928094 –

J59

S/2017 J 1

2017 [28]

23483978 r

J63

S/2017 J 2

2018 [30]

23240956 r

J64

S/2017 J 3

2018 [31]

20639315 r

J65

Pandia

2018 [32]

11494801 –

J66

S/2017 J 5

2018 [33]

23169389 r

J67

S/2017 J 6

2018 [34]

22394681 r

J68

S/2017 J 7

2018 [35]

20571457 r

J69

S/2017 J 8

2018 [36]

23174445 r

J70

S/2017 J 9

2018 [37]

21429954 r

J71

Ersa

2018 [38]

11453003 r

Appendix A Nomenclature of planetary satellites

References to the messages on discoveries of the satellites of Jupiter [1] Barnard E.E., 1892. Discovery of a Fifth Satellite of Jupiter, September 8, 1892. Publications of the Astronomical Society of the Pacific 4 (25), 199–199. [2] Perrine C.D., 1904. Discovery, observations and approximate orbits of two new satellites of Jupiter. Lick Observatory Bulletin 3 (64) 52–52. [3] Melotte J., Perrine C.D. 1908. Recent Observations of the Moving Object Near Jupiter, Discovered at Greenwich. Publications of the Astronomical Society of the Pacific 20 (120), 184–184. [4] Nicholson S.B., 1914. Discovery of the Ninth Satellite of Jupiter. Publications of the Astronomical Society of the Pacific 26 (155), 197–197. [5] Nicholson S.B., 1938. Two New Satellites of Jupiter. Publications of the Astronomical Society of the Pacific 50 (297), 292–292. [6] Nicholson S.B., 1951. An Unidentified Object Near Jupiter, Probably a New Satellite. Publications of the Astronomical Society of the Pacific 63 (375) 297–297. [7] Kowal C., 1975. Probable New Satellite of Jupiter. IAUC 2845. [8] Synnott S.P., 1980. 1979J2—Discovery of a previously unknown Jovian satellite. Science 210, 786–788. [9] Jewitt D.C., Danielson G.E., Synnott S.P., 1979. Discovery of a new Jupiter satellite. Science 206, 951–951. [10] Synnott S.P., 1981. 1979J3—Discovery of a previously unknown satellite of Jupiter. Science 212, 1392–1392. [11] Scotti J.V., Spahr T.B., McMillan R.S., Larsen J.A., Montani J., Gleason A.E., Gehrels T., 1999. IAUC∗ 7460. [12] Sheppard S. S., Jewitt D.C., Fernandez Y., Magnier G., 2000. IAUC∗ 7525. [13] Sheppard S.S., Jewitt D.C., Kleyna J., 2002. MPEC2002-J54∗∗ . [14] Sheppard S.S., 2003. IAUC∗ 8088. [15] Sheppard S.S., Gladman B., 2003. IAUC∗ 8138. [16] Sheppard S.S., 2003. IAUC∗ 8087. [17] Sheppard S.S., Gladman B., 2003. IAUC∗ 8276. [18] Sheppard S.S., 2002. IAUC∗ 8035. [19] Sheppard S.S., 2003. IAUC∗ 8089. [20] Sheppard S.S., 2003. IAUC∗ 8125. [21] Sheppard S.S., 2003. IAUC∗ 8116. [22] Jacobson R., Brozovic M., Gladman B., Alexandersen M., 2011. CBAT IAU∗∗∗ 2734. [23] Veillet C., 2011. CBAT IAU∗∗∗ 2734.

427

428

Appendix A Nomenclature of planetary satellites

[24] Sheppard S.S., Jewitt D.C., Fernandez Y., Magnier G., 2000. IAUC∗ 7525. [25] Sheppard S.S., 2003. IAUC∗ 8281. [26] Sheppard S.S., 2012. CBAT IAU∗∗∗ 3002. [27] Sheppard S.S., 2017. MPEC∗∗ 2017-L08. [28] Sheppard S.S., 2017. MPEC∗∗ 2017-L47. [29] Sheppard S.S., 2018. MPEC∗∗ 2018-O09. [30] Sheppard S.S., 2018. MPEC∗∗ 2018-O10. [31] Sheppard S.S., 2018. MPEC∗∗ 2018-O11. [32] Sheppard S.S., 2018. MPEC∗∗ 2018-O12. [33] Sheppard S.S., 2018. MPEC∗∗ 2018-O13. [34] Sheppard S.S., 2018. MPEC∗∗ 2018-O14. [35] Sheppard S.S., 2018. MPEC∗∗ 2018-O15. [36] Sheppard S.S., 2018. MPEC∗∗ 2018-O16. [37] Sheppard S.S., 2018. MPEC∗∗ 2018-O17. [38] Sheppard S.S., 2018. MPEC∗∗ 2018-O18. ∗ IAUC—International Astronomical Union Circular. ∗∗ MPEC—Minor Planet Electronic Circulars – IAU Minor Planet Center. ∗∗∗ CBAT IAU—Central Bureau for Astronomical Telegrams IAU Circular.

Nomenclature of the satellites of Saturn Table A.3 The nomenclature of the satellites of Saturn. For satellites without a name, temporary designations are given. Numbers of bibliographic references to the publication of reports of discoveries are given in square brackets. A bibliography is given after the table. In the column “Semi-major axis”, the r sign indicates satellites with retrograde motion. The satellite S6 Titan was discovered by Christian Huygens van Zuilichem. The satellites S3 Tethys, S4 Dione, S5 Rhea and S8 Iapetus were discovered by Cassini Giovanni Domenico. The satellites S1 Mimas and S2 Enceladus were discovered by Herschel William. Name

Latin name

Year of discovery

Semi-major axis, km

S1

Mimas

1789

185600 –

S2

Enceladus

1789

238100 –

S3

Tethys

1684

294700 –

S4

Dione

1684

377400 –

S5

Rhea

1672

527100 – continued on next page

Appendix A Nomenclature of planetary satellites

Table A.3 (continued) Name

Latin name

Year of discovery

Semi-major axis, km

S6

Titan

1655

1221900 –

S7

Hyperion

1848 [1]

1464100 –

S8

Iapetus

1671

3560800 –

S9

Phoebe

1898 [2]

12944300 r

S10

Janus

1966 [3]

151500 –

S11

Epimetheus

1977 [4]

151400 –

S12

Helene

1980 [5], [6]

377400 –

S13

Telesto

1980 [7], [8]

294700 –

S14

Calypso

1980 [7], [8]

294700 –

S15

Atlas

1980 [9], [10]

137700 –

S16

Prometheus

1980 [9], [10]

139400 –

S17

Pandora

1980 [9], [10]

141700 –

S18

Pan

1990 [11]

133600 –

S19

Ymir

2000 [12], [13]

23130000 r

S20

Paaliaq

2000 [12], [13]

15198000 –

S21

Tarvos

2000 [14], [13]

18239000 –

S22

Ijiraq

2000 [15], [13]

11442000 –

S23

Suttungr

2000 [16], [13]

19465000 r

S24

Kiviuq

2000 [15], [13]

11365000 –

S25

Mundilfari

2000 [17], [13]

18722000 r

S26

Albiorix

2000 [18], [13]

16394000 –

S27

Skathi

2000 [17], [13]

15641000 r

S28

Erriapo

2000 [19], [13]

17604000 –

S29

Siarnaq

2000 [14], [13]

18195000 –

S30

Thrymr

2000 [17], [13]

20219000 r

S31

Narvi

2003 [20], [21]

18719000 r

S32

Methone

2004 [22], [21]

194000 –

S33

Pallene

2004 [22], [21]

211000 –

S34

Polydeuce

2004 [23], [21]

377400 –

S35

Daphnis

2005 [24]

136500 –

S36

Aegir

2005 [25], [26] 19460000 r continued on next page

429

430

Appendix A Nomenclature of planetary satellites

Table A.3 (continued) Name

Latin name

Year of discovery

Semi-major axis, km

S37

Bebhionn

2005 [25], [26]

17040000 –

S38

Bergelmir

2005 [25], [26]

19610000 r

S39

Bestla

2005 [25], [26]

19890000 r

S40

Farbauti

2005 [25], [26]

20580000 r

S41

Fenrir

2005 [25], [26]

23050000 r

S42

Fornjot

2005 [25], [26]

24030000 r

S43

Hati

2005 [25], [26]

20670000 r

S44

Hyrrokkin

2006 [27], [26]

18170000 r

S45

Kari

2006 [27], [26]

22320000 r

S46

Loge

2006 [27], [26]

22980000 r

S47

Skoll

2006 [27], [26]

17470000 r

S48

Surtur

2006 [27], [26]

22290000 r

S49

Anthe

2007 [28], [29]

197700 –

S50

Jarnsaxa

2006 [27], [29]

18560000 r

S51

Greip

2006 [27], [29]

18070000 r

S52

Tarqeq

2007 [30], [29]

17960000 –

S53

Aegaeon

2008 [31]

167500 –



S/2004 S 7

2005 [25]

20880000 r



S/2004 S 12

2005 [25]

20010000 r



S/2004 S 13

2005 [25]

18280000 r



S/2004 S 17

2005 [25]

19130000 r



S/2006 S 1

2006 [27]

18930000 r



S/2006 S 3

2006 [27]

21080000 r



S/2007 S 2

2007 [30]

16520000 r



S/2007 S 3

2007 [30]

19180000 r



S/2009 S 1

2009 [32]

117000 –



S/2004 S 20

2020 [33]

19211000 r



S/2004 S 21

2020 [33]

23810400 r



S/2004 S 22

2020 [33]

20379900 r



S/2004 S 23

2020 [33]

21427000 r



S/2004 S 24

2020 [33]

23231300 – continued on next page

Appendix A Nomenclature of planetary satellites

Table A.3 (continued) Name

Latin name

Year of discovery

Semi-major axis, km



S/2004 S 25

2020 [33]

20544500 r



S/2004 S 26

2020 [33]

26737800 r



S/2004 S 27

2020 [33]

19776700 r



S/2004 S 28

2020 [33]

21791300 r



S/2004 S 29

2020 [33]

17470700 –



S/2004 S 30

2020 [33]

20424000 r



S/2004 S 31

2020 [33]

17402800 –



S/2004 S 32

2020 [33]

21564200 r



S/2004 S 33

2020 [33]

26737800 r



S/2004 S 34

2020 [33]

24358900 r



S/2004 S 35

2020 [33]

21953200 r



S/2004 S 36

2020 [33]

23698700 r



S/2004 S 37

2020 [33]

16003300 r



S/2004 S 38

2020 [33]

23006200 r



S/2004 S 39

2020 [33]

22790400 r

References to the messages on discoveries of the satellites of Saturn [1] Bond W.C., 1848. Discovery of a new satellite of Saturn. Monthly Notices of the Royal Astronomical Society 9, 1–1. [2] Pickering E.C., 1899. Phoebe, ninth Satellite of Saturn discovered by W.W. Pickering. Harvard College Observatory Bulletin No. 49, pp. 1–1. [3] Dollfus A., 1967. Probable New Satellite of Saturn. IAUC 1987. [4] Fountain J.W., Larson S.M., 1977. A new satellite of Saturn. Science 197, 915–917. [5] Harris, A.W., Gibson J., Lecacheux J., Fort B. et al., 1980. IAUC 3463. [6] Lecacheux J., Laques P., Wierick G., Lelievre G. et al., 1980. IAUC 3483. [7] Smith B.A., Reitsema H.J., Larson S.M., 1980. 1980 S 2. IAUC 3456.

431

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Appendix A Nomenclature of planetary satellites

[8] Cruikshank D., Smith B.A., Reitsema H.J., Larson S.M. et al., 1980. Saturn. IAUC 3457. [9] Collins S.A., Cook A.F., Cuzzi J.N., Danielson G.E. et al., 1980. First Voyager view of the rings of Saturn. Nature 288, 439–442. [10] Smith B.A., Soderblom L., Beebe R.F., Boyce J.M. et al., 1981. Encounter with Saturn – Voyager 1 imaging science results. Science 212, 163–191. [11] Showalter M.R., Colas F., Lecacheux J., Laques P., Despiau R., 1990. Saturn. IAUC 5052. [12] Gladman B., 2000. S/2000 S 1 AND S/2000 S 2. IAUC 7512. [13] Satellites of Jupiter, Saturn, Uranus. IAUC 8177. [14] Gladman B., 2003. S/2000 S 3 AND S/2000 S 4. IAUC 7513. [15] Gladman B., 2000. S/2000 S 5 AND S/2000 S 6. IAUC 7521. [16] Gladman B., 2000. S/2000 S 12. IAUC 7548. 2000. [17] Gladman B., 2000. S/2000 S 7, S/2000 S 8, S/2000 S 9. IAUC 7538. [18] Gladman B., 2000. S/2000 S 11. IAUC 7545. [19] Gladman B., 2000. S/2000 S 10. IAUC 7539. [20] Sheppard S.S., 2003. S/2000 S 12. IAUC 8116. [21] Satellites of Saturn. IAUC 8471. 2005. [22] Porco C.C., 2004. S/2004 S 1 AND S/2004 S 2. IAUC 8389. [23] Porco C.C., 2004. S/2004 S 3, S/2004 S 4, AND R/2004 S 1. IAUC 8401. [24] Porco C.C., 2005. S/2005 S 1. IAUC 8524. [25] Jewitt D. Sheppard S, Kleyna J., 2005. New satellites of Saturn. IAUC 8523. [26] Satellites of Jupiter and Saturn. IAUC 8826. 2007. [27] Jewitt D., Sheppard S., Kleyna J., 2006. New satellites of Saturn. IAUC 8727. [28] Porco C.C., 2007. S/2007 S 4. IAUC 8857. [29] Satellites of Saturn. IAUC 8873. 2007. [30] Jewitt D., Sheppard S, Kleyna J., 2007. S/2007 S 1, S/2007 S 2, AND S/2007 S 3. IAUC 8836. [31] Porco C.C., 2009. S/2008 S 1. IAUC 9023. [33] Sheppard S.S., MPEC 2019-T126 – MPEC 2019-T161. ∗ IAUC—International Astronomical Union Circular. ∗∗ MPEC—Minor Planet Electronic Circulars—IAU Minor Planet Center.

Appendix A Nomenclature of planetary satellites

433

The nomenclature of the satellites of Uranus Table A.4 The nomenclature of the satellites of Uranus. For satellites without a name, temporary designations are given. Numbers of bibliographic references to the publication of reports of discoveries are given in square brackets. A bibliography is given after the table. In the column “Semi-major axis”, the r sign indicates satellites with retrograde motion. Name Latin name

Year of Semi-major discovery axis, km

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16 U17 U18 U19 U20 U21 U22 U23 U24 U25 U26 U27

1851 [1] 1851 [1] 1787 [2] 1787 [2] 1948 [3] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1986 [4] 1997 [5] 1997 [5] 1999 [6] 1999 [7] 1999 [7] 2001 [8] 2001 [9] 2003 [10] 2001 [11] 1999 [12] 2003 [13] 2003 [13]

Ariel Umbriel Titania Oberon Miranda Cordelia Ophelia Bianca Cressida Desdemona Juliet Portia Rosalind Belinda Puck Caliban Sycorax Prospero Setebos Stephano Trinculo Francisco Margaret Ferdinand Perdita Mab Cupid

190900 – 266000 – 436300 – 583500 – 129900 – 49800 – 53800 – 59200 – 61800 – 62700 – 64400 – 66100 – 69900 – 75300 – 86000 – 7231000 r 12179000 r 16256000 r 17418000 r 8004000 r 8504000 r 4276000 r 14345000 – 20901000 r 76416 – 97734 – 74800 –

434

Appendix A Nomenclature of planetary satellites

References to the messages on discoveries of the satellites of Uranus [1] Lassell W., 1851. Letter to the editor discovery of two satellites of Uranus. Astronomical Journal 2, 70–70. [2] Herschel W., 1787. An Account of the Discovery of Two Satellites Revolving Round the Georgian Planet. By William Herschel, LLD. F.R.S. Philosophical Transactions of the Royal Society of London 77, 125–129. [3] Kuiper G.P., 1949. The Fifth Satellite of Uranus. Publications of the Astronomical Society of the Pacific 61 (360), 129–129. [4] Smith B.A., Soderblom L.A., Beebe R., Bliss D. et al., 1986. Voyager 2 in the Uranian system—Imaging science results. Science 233, 43–64. [5] Gladman B.J., Nicholson P.D., Burns J.A., 1997. Satellites of Uranus. IAUC 6764. [6] Gladman B., 1999. Probable new satellites of Uranus. IAUC 7248. [7] Kavelaars J.J., Gladman B., Holman M., Petit J.-M., Scholl H., 1999. Probable new satellites of Uranus. IAUC 7230. [8] Holman M., Kavelaars J., Milisavljevic D., 2002. S/2001 U 1. IAUC 7980. [9] Holman M., Kavelaars J., Milisavljevic D., 2003. S/2001 U 3. IAUC 8216. [10] Sheppard S.S., Jewitt D.C., 2003. S/2003 U 3. IAUC 8217. [11] Holman M., Gladman B., 2003. S/2001 U 2 AND S/2002 N 4. IAUC 8213. [12] E. Karkoschka, 1999. S/1986 U 10. IAUC 7171. [13] Showalter M.R., Lissauer J.J., 2003. S/2003 U 1 AND S/2003 U 2. IAUC 8209. ∗ IAUC—International Astronomical Union Circular.

Appendix A Nomenclature of planetary satellites

435

The nomenclature of the satellites of Neptune Table A.5 The nomenclature of the satellites of Neptune. For satellites without a name, temporary designations are given. Numbers of bibliographic references to the publication of reports of discoveries are given in square brackets. A bibliography is given after the table. In the column “Semi-major axis”, the r sign indicates satellites with retrograde motion. Name Latin name Year of discovery Semi-major axis, km N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N13 N14

Triton Nereid Naiad Thalassa Despina Galatea Larissa Proteus Halimede Psamathe Sao Laomedeia Neso Hippocamp

1846 [1] 1949 [2] 1989 [3] 1989 [3] 1989 [3] 1989 [3] 1989 [3] 1989 [3] 2003 [4] 2003 [4] 2003 [4] 2003 [5] 2003 [6] 2004 [7]

354800 – 5513400 – 48200 – 50100 – 52500 – 62000 – 73500 – 117600 – 15686000 r 46738000 r 22452000 – 22580000 – 46570000 r 105250 –

References to the messages on discoveries of the satellites of Neptune [1] Lassell W., 1846. Discovery of supposed ring and satellite of Neptune. Monthly Notices of the Royal Astronomical Society 7, 157–157. [2] Kuiper G.P., 1949. Object near Neptune. IAUC 1212. [3] Smith B.A., Soderblom L.A., Banfield D., Barnet C., 1989. Voyager 2 at Neptune: Imaging Science Results. Science 246, 1422–1449. [4] Holman M., Kavelaars J., Grav T., Fraser W., Milisavljevic D., 2003. Satellites of Neptune. IAUC 8047. [5] Sheppard S.S., Jewitt D.C., Kleyna J., 2003. Satellites of Neptune. IAUC 8193. [6] Holman M., Gladman B., 2003. S/2001 U 2 AND S/2002 N 4. IAUC 8213. [7] Showalter M., 2004. Press release NASA13-215.

436

Appendix A Nomenclature of planetary satellites

The nomenclature of the satellites of Pluto Table A.6 The nomenclature of the satellites of Pluto. For satellites without a name, temporary designations are given. Numbers of bibliographic references to the publication of reports of discoveries are given in square brackets. A bibliography is given after the table. Name Latin name

Year of discovery

Semi-major axis, km

P1

Charon

1978 [1]

19500

P2

Nix

2005 [2]

49400

P3

Hydra

2005 [2]

64700

P4

Kerberos

2011 [3]

59000

P5

Styx

2012 [4]

42400

References to the messages on discoveries of the satellites of Pluto [1] Smith J.C., Christy J.W., Graham J.A., 1978. 1978 P 1. IAUC 3241. [2] Weaver H.A., Stern S.A., Mutchler M.J., Steffl A.J., Buie M.W., Merline W.J., Spencer J.R., Young E.F., Young L.A., 2005. S/2005 P 1 AND S/2005 P 2. IAUC 8625. [3] Showalter M.R., Hamilton D.P., Stern S.A., Weaver H.A., Steffl A.J., Young L.A., 2011. New Satellite of (134340) Pluto: S/2011 (134340) 1. IAUC 9221. [4] Showalter M.R., Weaver H.A., Stern S.A., Steffl A.J., Buie M.W., Merline W.J., Mutchler M.J., Soummer R., Throop H.B., 2012. New Satellite of (134340) Pluto: S/2012 (134340) 1. IAUC 9253.

B Orbital parameters of the natural planetary satellites Parameters are given in Tables B.1–B.16 distributed in the sections by satellite systems.

Orbital parameters of Mars’s satellites Table B.1 Mean orbital parameters of the satellites of Mars. The time interval is 2000–2040. The inclination of the orbit is measured from the equator of Mars. The orbital periods are calculated by the average value of the osculating mean motion. Data obtained using the MULTI-SAT ephemeris server. Satellite Semi-major axis Eccentricity Inclination km degrees

Period days

Phobos

9378.536

0.01511286

1.076095

0.319155947

Deimos

23458.954

0.00027719

2.041303 1.2625928067

Table B.2 Orbital acceleration of Phobos. The second column gives the value the mean motion of Phobos. Reference on source

Orbital acceleration, 10−5 deg /year2

Jacobson et al., 1989

124.9 ± 1.8

Emelyanov et al., 1993

129.0 ± 1.0

Bills et al., 2005

136.7 ± 0.6

Lainey et al., 2007

127.0 ± 1.5

1 dn 2 dt , where n is

437

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Appendix B Orbital parameters of the natural planetary satellites

Orbital parameters of the satellites of Jupiter Table B.3 Mean orbital parameters of the Galilean satellites of Jupiter. The time interval is 2000–2040. The inclination of the orbit is measured from the equator of Jupiter. The orbital periods are calculated by the average value of the osculating mean motion. Data obtained using the MULTI-SAT ephemeris server. Satellite J1 Io J2 Europa J3 Ganymede J4 Callisto

Semi-major axis Eccentricity Inclin. km degrees 421942.424 671122.114 1070399.250 1882744.546

0.00416355 0.00936389 0.00196240 0.00728706

Period days

0.036917 1.770620604 0.467883 3.551834882 0.147563 7.154143807 0.249848 16.689013764

Table B.4 Mean orbital parameters of the close satellites of Jupiter. The inclination of the orbit is measured from the equator of Jupiter. We used the parameters of the precessing ellipse refined in Emel’yanov (2015) applying the satellite motion model (Jacobson, 2013). Periods are given for mean longitude. Satellite J5 Amalthea J14 Thebe J15 Adrastea J16 Metis

Semi-major axis Eccentricity Inclin. km degrees 181365.552 221888.173 128979.903 127978.860

0.00342600 0.01753195 0.00018093 0.00050486

0.376187 1.071790 0.012926 0.012230

Period days 0.4981790697 0.6745359075 0.2982604262 0.2947788040

Table B.5 Mean orbital parameters of the outer satellites of Jupiter in the time interval from 1600 to 2600 (Brozovic and Jacobson, 2017). The inclination of the orbit is measured from the ecliptic of the J2000 epoch. Satellite Semi-major axis Eccentricity Inclin. Period km degrees days J6 Himalia J7 Elara

11460200 11740300

0.159 0.211

28.61 27.94

250.56 259.64 continued on next page

Appendix B Orbital parameters of the natural planetary satellites

Table B.5 (continued) Satellite J8 Pasiphae J9 Sinope J10 Lysithea J11 Carme J12 Ananke J13 Leda J17 Callirrhoe J18 Themisto J19 Megaclite J20 Taygete J21 Chaldene J22 Harpalyke J23 Kalyke J24 Iocaste J25 Erinome J26 Isonoe J27 Praxidike J28 Autonoe J29 Thyone J30 Hermippe J31 Aitne J32 Eurydome J33 Euanthe J34 Euporie J35 Orthosie J36 Sponde J37 Kale J38 Pasithee J39 Hegemone J40 Mneme

Semi-major axis Eccentricity Inclin. Period km degrees days 23629100 23942000 11717000 23400500 21253700 11164400 24098900 7503900 23813900 23362900 23180600 21106100 23564600 21272000 23285900 23231200 21147700 24037200 21197200 21297100 23316700 23146200 21039000 19336200 21158200 23790100 23305800 23091500 23574700 21033000

0.406 151.41 743.61 0.255 158.19 758.89 0.116 27.66 259.20 0.255 164.99 734.17 0.233 148.69 629.80 0.162 27.88 240.93 0.280 147.08 758.82 0.243 42.98 130.02 0.416 152.78 752.88 0.252 165.25 732.41 0.250 165.16 723.73 0.230 148.76 623.32 0.247 165.12 742.04 0.215 149.41 631.60 0.266 164.91 728.49 0.247 165.25 726.26 0.227 148.88 625.39 0.315 152.37 761.01 0.231 148.59 627.19 0.210 150.74 633.91 0.263 165.05 730.12 0.275 150.27 717.31 0.232 148.92 620.45 0.144 145.74 550.69 0.281 146.00 622.58 0.311 151.00 748.32 0.260 164.94 729.61 0.268 165.12 719.47 0.344 154.16 739.82 0.226 148.58 620.05 continued on next page

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440

Appendix B Orbital parameters of the natural planetary satellites

Table B.5 (continued) Satellite J41 Aoede J42 Thelxinoe J43 Arche J44 Kallichore J45 Helike J46 Carpo J47 Eukelade J48 Cyllene J49 Kore J50 Herse J51 2010 J1 J52 2010 J2 J53 Dia 2003 J2 J60 Eupheme 2003 J4 J57 Eirene 2003 J9 2003 J10 2003 J12 J58 Philophrosyne 2003 J16 J55 2003 J18 2003 J19 2003 J23 2011 J1 J56 2011 J2 J54 2016 J1 J59 2017 J1

Semi-major axis Eccentricity Inclin. Period km degrees days 23974100 21159700 23352000 23276300 21065500 17056600 23322700 23799600 24481800 23407900 23448500 21004200 12297500 28348600 20210000 23928700 23424100 23334700 22862300 17818600 22565200 21089700 20491400 23545900 23601700 23444400 23124300 20595500 23484000

0.432 0.220 0.249 0.251 0.150 0.432 0.262 0.415 0.331 0.254 0.249 0.227 0.232 0.410 0.197 0.362 0.251 0.266 0.475 0.491 0.191 0.228 0.090 0.256 0.276 0.253 0.349 0.140 0.397

158.27 151.39 165.01 165.10 154.84 51.62 165.26 150.33 145.17 164.96 165.10 148.67 28.63 157.29 147.63 149.59 165.24 165.03 168.79 151.08 146.90 148.74 146.20 165.13 146.51 165.34 153.60 139.84 149.20

761.40 628.03 731.90 728.24 626.33 456.28 730.33 751.98 776.84 734.52 736.50 618.85 278.21 980.59 583.84 755.25 735.40 730.93 719.55 489.67 689.79 622.89 598.14 741.03 734.64 736.33 718.40 603.83 735.21

Appendix B Orbital parameters of the natural planetary satellites

441

Orbital parameters of Saturn’s satellites

Table B.6 Mean orbital parameters of the main satellites of Saturn. The time interval is 2000–2040. The inclination of the orbit is measured from the equator of Saturn. The orbital periods are calculated by the average value of the osculating mean motion. Data obtained using the MULTI-SAT ephemeris server. Satellite

Semi-major axis Eccentricity Inclin. Period km degrees days

S1 Mimas S2 Enceladus S3 Tethys S4 Dione S5 Rhea S6 Titan S7 Hyperion S8 Iapetus

186021.35 238412.62 294976.58 377652.23 527235.48 1221952.96 1481540.33 3561752.42

0.01967966 0.00485416 0.00103366 0.00224433 0.00100497 0.02869581 0.10594669 0.02839588

1.572785 0.008796 1.091441 0.029095 0.334693 0.404763 1.010876 15.754312

0.94735329 1.37455344 1.89168521 2.74034631 4.52036866 15.94770268 21.29278135 79.37108889

Table B.7 Mean orbital parameters of the close coorbiting satellites of Saturn. The inclination of the orbit is measured from the equator of Saturn. For satellites S10 Janus and S11 Epimetheus, the parameters are taken from Nicholson et al. (1992). For the satellites S12 Helene, S13 Telesto, and S14 Calypso, the mean elements were obtained using the MULTI-SAT ephemeris server in the time interval 2000–2040. In this case, the periods were not determined, since the satellites are coorbital to the corresponding main satellites. Satellite S10 Janus S11 Epimetheus S12 Helene S13 Telesto S14 Calypso

Semi-major axis Eccentricity Inclin. km degrees 152026.525 152026.525 377557.343 294904.096 294903.047

0.00666424 0.00989707 0.00759738 0.00120295 0.00125402

Period days

0.148003 0.69991748 0.326212 0.69991748 0.212697 (S4 Dione) 1.180305 (S3 Tethys) 1.500476 (S3 Tethys)

442

Appendix B Orbital parameters of the natural planetary satellites

Table B.8 Mean orbital parameters of Saturn’s close satellites S15–S18, S32–S35, S49, and S53. The inclination of the orbit is measured from the equator of Saturn. Periods correspond to mean longitudes. The orbital parameters are taken from Jacobson et al. (2008), except for the S49 Anthe satellite, for which the parameters are taken from Cooper et al. (2008). Data for satellite S53 Aegaeon are taken from Porco (2009). Satellite S15 Atlas S16 Prometheus S17 Pandora S18 Pan S32 Methone S33 Pallene S34 Polydeuce S35 Daphnis S49 Anthe S53 Aegaeon

Semi-major axis Eccentricity Inclin. km degrees 137670.0 139380.0 141710.0 133584.0 194230.0 212280.0 377200.0 136505.5 197655.0 167500.0

0.0012 0.0022 0.0042 0.0000144 0.000 0.004 0.0192 0.0000331 0.0012 0.0002

0.0031 0.0075 0.0507 0.0001 0.0131 0.1813 0.1774 0.0036 0.0170 0.0010

Period days 0.60169240 0.61299003 0.62850414 0.57505072 1.00957630 1.15374576 2.73691945 0.59407983 1.03644246 0.80812000

Table B.9 Mean orbital parameters of the outer satellites of Saturn. The time interval is 2000–2022. The inclination of the orbit is measured from the ecliptic of the J2000 epoch. The orbital periods are calculated by the average value of the osculating mean motion. Data calculated using the MULTI-SAT ephemeris server. Satellite S9 Phoebe S19 Ymir S20 Paaliaq S21 Tarvos S22 Ijiraq S23 Suttungr S24 Kiviuq S25 Mundilfari

Semi-major axis Eccentricity Inclin. km degrees 12929039.9 22962695.0 14996742.1 18216574.2 11344466.9 19395933 11307200 18590897

0.1606324 0.3488370 0.4823732 0.5733807 0.3743473 0.1185987 0.1551990 0.2077999

Period days

173.09509 548.85427 172.55711 1298.51677 45.14496 685.63343 36.71391 917.70613 49.50231 451.11541 174.32725 1008.36326 48.85463 448.89544 169.41758 946.26206 continued on next page

Appendix B Orbital parameters of the natural planetary satellites

Table B.9 (continued) Satellite S26 Albiorix S27 Skathi S28 Erriapo S29 Siarnaq S30 Thrymr S31 Narvi S36 Aegir S37 Bebhionn S38 Bergelmir S39 Bestla S40 Farbauti S41 Fenrir S42 Fornjot S43 Hati S44 Hyrrokkin S45 Kari S46 Loge S47 Skoll S48 Surtur S50 Jarnsaxa S51 Greip S52 Tarqeq S/2004 S 7 S/2004 S 12 S/2004 S 13 S/2004 S 17 S/2006 S 1 S/2006 S 3 S/2007 S 2 S/2007 S 3 S/2004 S20 S/2004 S21 S/2004 S22 S/2004 S23

Semi-major axis Eccentricity Inclin. km degrees 16327637 15576511 17511661 17881078 20319707 19283228 20644613 17030786 19270288 20340498 20291660 22332629 24952935 19680664 18341509 22017148 22912847 17621701 22769483 19283446 18365735 17748123 20935811 19765284 18075271 19356909 18715128 21199569 16693361 19717834 19258627 23137217 20590994 21441207

Period days

0.5436476 35.84246 778.83551 0.2444519 148.86319 725.77430 0.5221874 38.16016 865.00410 0.4581336 45.78317 892.57443 0.4781182 174.49251 1081.10795 0.2933297 137.56415 999.63191 0.2591989 167.45641 1107.19916 0.3786360 41.56639 829.68887 0.1341646 158.02409 998.60492 0.6733796 145.68311 1082.82684 0.2102589 157.06470 1078.98982 0.1316001 163.36678 1245.61873 0.2031598 167.71320 1470.67610 0.3612598 162.60887 1030.59325 0.4051464 154.04512 927.29333 0.4047111 147.19929 1219.24861 0.1897239 166.31049 1294.40633 0.4332351 155.24493 873.23726 0.4327269 166.38362 1282.06719 0.2329932 163.74470 999.60686 0.3233132 172.73529 929.11893 0.1134880 50.37607 882.68959 0.5331217 165.59551 1130.54667 0.3460168 163.59877 1037.25182 0.2471338 167.57146 907.18131 0.1804826 167.78802 1005.32393 0.1252450 153.84562 955.78651 0.4172553 153.10901 1152.07421 0.1775148 176.60045 805.19585 0.1916931 177.17941 1033.54331 0.1909662 162.65375 997.68325 0.3394922 153.97072 1313.36999 0.2236645 177.39060 1102.88994 0.4235824 177.55744 1171.68389 continued on next page

443

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Appendix B Orbital parameters of the natural planetary satellites

Table B.9 (continued) Satellite

Semi-major axis Eccentricity Inclin. km degrees

S/2004 S24 S/2004 S25 S/2004 S26 S/2004 S27 S/2004 S28 S/2004 S29 S/2004 S30 S/2004 S31 S/2004 S32 S/2004 S33 S/2004 S34 S/2004 S35 S/2004 S36 S/2004 S37 S/2004 S38 S/2004 S39

23325074 20956892 26093204 19842231 21833247 17062210 20705485 17496508 21157909 23553327 24148963 21969729 23421217 15942530 22261654 23187204

0.0610325 0.5146853 0.1513586 0.1484213 0.1523616 0.4446027 0.1026236 0.2159875 0.2482777 0.4983495 0.2650205 0.2339878 0.6529387 0.4831924 0.4864136 0.1017577

35.52778 173.33898 171.22530 167.32280 170.34479 42.85640 157.13803 48.81282 158.09159 160.46126 165.36225 176.46758 152.68746 163.60092 152.58219 167.21208

Period days 1329.39162 1132.24963 1572.41328 1043.35146 1204.10729 831.96120 1112.14435 863.98398 1148.71754 1348.67135 1400.26138 1215.37118 1337.48336 751.48472 1239.52178 1317.73012

Orbital parameters of Uranus’s satellites Table B.10 Radii of orbits and periods of revolution of the main satellites of Uranus (Emelyanov and Nikonchuk, 2013). The values are obtained by refining the circular orbit in the plane of the equator of Uranus according to the results of numerical integration of the equations of motion in the interval from 1787 to 2032. The reference model was refined based on all available observations. Satellite U1 Ariel U2 Umbriel U3 Titania U4 Oberon U5 Miranda

Orbit radius, km

Period, day

190929.789 2.52037923705 265984.008 4.14417716318 436281.937 8.70586922413 583449.534 13.46323747494 129848.114 1.41347941664

Appendix B Orbital parameters of the natural planetary satellites

445

Table B.11 Mean values of the eccentricities and inclinations of the main satellites of Uranus in the time interval 2000–2040. The inclinations of the orbits are measured from the equator of Uranus towards the south pole. Data was obtained using the MULTI-SAT ephemeris server with a choice of the motion model of Arlot et al. (2017). Satellite U1 Ariel U2 Umbriel U3 Titania U4 Oberon U5 Miranda

Eccentricity Inclination, deg min. mean max. min. mean max. 0.0002 0.0030 0.0007 0.0008 0.0010

0.0007 0.0038 0.0019 0.0017 0.0013

0.0015 0.0045 0.0030 0.0030 0.0016

0.016 0.052 0.034 0.182 4.404

0.033 0.064 0.058 0.190 4.420

0.053 0.079 0.094 0.195 4.434

Table B.12 Parameters of the orbits of the close satellites of Uranus. The inclination of the orbit is measured from the equator of Uranus towards the south pole. Orbits are precessing ellipses. The periods of revolution correspond to the change in the mean longitudes of the satellites. For the satellites U6 Cordelia and U7 Ophelia, the parameters are taken from Jacobson (1998). For other satellites, the parameters are taken from Showalter and Lissauer (2006). Satellite U6 Cordelia U7 Ophelia U8 Bianca U9 Cressida U10 Desdemona U11 Juliet U12 Portia U13 Rosalind U14 Belinda U15 Puck U25 Perdita U26 Mab U27 Cupid

Semi-major axis Eccentricity Inclin. km degrees 49751.7220 53763.3900 59165.5621 61766.7199 62658.3825 64358.2307 66097.2873 69926.8179 75255.6102 86004.4919 97735.9095 74392.3409 76416.7306

0.000260 0.009920 0.000274 0.000203 0.000342 0.000052 0.000512 0.000579 0.000277 0.000389 0.002537 0.001335 0.003287

0.08479 0.10362 0.18110 0.03790 0.09820 0.04540 0.02550 0.09340 0.02820 0.32140 0.1335 0.0988 0.0676

Period days 0.335033842 0.376400393 0.434579025 0.463569377 0.473649687 0.493065462 0.513196030 0.558459595 0.623527138 0.761832904 0.922958342 0.612824737 0.638019137

446

Appendix B Orbital parameters of the natural planetary satellites

Table B.13 Parameters of the orbits of the outer satellites of Uranus. The inclination of the orbit is measured from the ecliptic. The orbital periods are calculated as averages over a long period of time and correspond to a change in the mean anomaly. The data were obtained using the tool described in Emelyanov and Vashkov’yak (2012). The remaining parameters were calculated over the time interval from 1960 to 2060 using the MULTI-SAT ephemeris server. Satellite

Semi-major axis Eccentricity Inclin. km degrees

Period year

U16 Caliban

min mean max

7160841 7166371 7170731

0.074576 0.078742 0.082372

139.388 139.726 1.586654 140.012

U17 Sycorax

min mean max

12147958 12193460 12253618

0.450552 0.492994 0.535105

151.620 153.325 3.521717 154.839

U18 Prospero

min mean max

16105679 16219543 16353362

0.316875 0.358308 0.400816

143.701 145.277 5.402682 146.930

U19 Setebos

min mean max

17313795 17520561 17734844

0.466791 0.533485 0.593424

145.866 148.114 6.066045 150.116

U20 Stephano

min mean max

7946520 7950882 7956250

0.137591 0.148893 0.160579

141.323 141.691 1.854024 142.039

U21 Trinculo

min mean max

8499736 8505890 8513782

0.201950 0.213832 0.224779

165.977 166.205 2.050397 166.440

U22 Francisco

min mean max

4274861 4275490 4276111

0.131887 0.138173 0.144428

147.362 147.500 0.731215 147.635

U23 Margaret

min mean max

14363104 14459011 14564006

0.722039 0.805199 0.869983

47.782 52.126 4.547646 57.380

min U24 Ferdinand mean max

19978168 20330388 20754127

0.321742 0.399019 0.474938

166.424 167.679 7.580276 168.758

Appendix B Orbital parameters of the natural planetary satellites

447

Orbital parameters of Neptune’s satellites Table B.14 Orbit parameters of the satellite Triton and close satellites. The inclinations of the orbits are measured from the equator of Neptune. Orbit parameters for Triton are taken from Emelyanov and Samorodov (2015), for satellites N3 Naiad and N4 Thalassa from Owen et al. (1991), for satellites N5 Despina, N6 Galatea, and N7 Larissa from Pascu et al. (2004), for satellite N8 Proteus from Jacobson (2009), and for satellite S/2004 N 1 from communication (Showalter, 2013). Satellite

Semi-major axis Eccentricity km

N1 Triton N3 Naiad N4 Thalassa N5 Despina N6 Galatea N7 Larissa N8 Proteus N14 Hippocamp

354696.8 48233.1 50069.2 52531.3 61945.1 73545.70 117646.0 105384.0

Inclin. degrees

Period days

0.0 157.268439 5.876714551 0.000328 4.7382 0.294395663 0.000156 0.2054 0.311484539 0.000139 0.0655 0.334655476 0.000120 0.0544 0.428744263 0.001386 0.2008 0.554653319 0.000510 0.0749 1.122314776 0.000000 0.0000 0.950000000

Table B.15 Parameters of the orbits of the outer satellites of Neptune. The inclination of the orbit is measured from the ecliptic. The orbital periods are calculated as averages over a long period of time and correspond to changes in the mean anomaly (Emelyanov and Vashkov’yak, 2012). The remaining parameters were calculated over the time interval from 1960 to 2060 using the MULTI-SAT ephemeris server. Satellite

Semi-major axis Eccentricity Inclin. km degrees

Period year

min mean max

5516655 5517274 5518238

0.744831 0.751105 0.755896

4.8560 4.9882 0.987018 5.1056

min N9 Halimede mean max

16565549 16588715 16607272

0.228959 0.250702 0.263546

111.8056 112.5882 5.144978 113.2500 continued on next page

N2 Nereid

448

Appendix B Orbital parameters of the natural planetary satellites

Table B.15 (continued) Satellite

Semi-major axis Eccentricity Inclin. km degrees

Period year

N10 Psamathe

min mean max

45652714 47371107 49449874

0.167440 0.279459 0.391994

117.1266 120.8545 24.82927 124.2182

N11 Sao

min mean max

22107254 22211127 22310554

0.113321 0.134501 0.153853

52.6235 53.6789 7.970776 54.4520

min N12 Laomedeia mean max

23345855 23476976 23673374

0.347880 0.387977 0.426845

37.6585 38.9636 8.663030 40.3566

min mean max

47889561 49735947 51483996

0.619595 0.739533 0.849778

134.5595 139.1570 26.68128 143.8458

N13 Neso

Orbital parameters of the satellites of Pluto Table B.16 Orbital parameters of the satellites of Pluto (Brozovic et al., 2015). The orbit of Charon is calculated relative to Pluto, the orbits of other satellites are calculated relative to the Pluto–Charon barycenter. The reference plane is set to Charon’s mean orbit that has the orbit normal at right ascension 133.03 deg and declination −6.23 deg in ICRF. Satellite P1 Charon P2 Nix P3 Hydra P4 Kerberos P5 Styx

Semi-major axis Eccentricity Inclin. Period km degrees year 19596 48690 64721 57750 42413

0.00005 0.0 0.00554 0.0 0.00001

0.0 0.0 0.3 0.4 0.0

6.3872 24.8548 38.2021 32.1679 20.1617

References Arlot, J.E., Cooper, N., Emelyanov, N., Lainey, V., Meunier, L.E., Murray, C., Oberst, J., Pascu, D., Pasewaldt, A., Robert, V., Tajeddine, R., Willner, K., 2017. Natural

Appendix B Orbital parameters of the natural planetary satellites

satellites astrometric data from either space probes and ground-based observatories produced by the European consortium «ESPaCE». Notes Scientifiques Et Techniques de L’Institut de Mécanique Céleste. S105.P.1–49. Bills, B.G., Neumann, G.A., Smith, D.E., Zuber, M.T., 2005. Improved estimate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos. Journal of Geophysical Research 110 (E7). CiteID E07004. Brozovic, M., Showalter, M.R., Jacobson, R.A., Buie, M.W., 2015. The orbits and masses of satellites of Pluto. Icarus 246, 317–329. Brozovic, M., Jacobson, R.A., 2017. The orbits of Jupiter’s irregular satellites. Astronomical Journal 153, 147. 10 pp. Cooper, N.J., Murray, C.D., Evans, M.W., Beurle, K., Jacobson, R.A., Porco, C.C., 2008. Astrometry and dynamics of Anthe (S/2007 S 4), a new satellite of Saturn. Icarus 195 (2), 765–777. Emelyanov, N.V., Vashkovyak, S.N., Nasonova, L.P., 1993. The dynamics of Martian satellites from observations. Astronomy and Astrophysics 267 (2), 634–642. Emelyanov, N.V., Vashkov’yak, M.A., 2012. Evolution of orbits and encounters of distant planetary satellites. Study tools and examples. Solar System Research 46 (6), 423–435. Emelyanov, N.V., Nikonchuk, D.V., 2013. Ephemerides of the main Uranian satellites. Monthly Notices of the Royal Astronomical Society 436 (4), 3668–3679. Emel’yanov, N.V., 2015. Perturbed motion at small eccentricities. Solar System Research 49 (5), 346–359. Emelyanov, N.V., Samorodov, M.Yu., 2015. Analytical theory of motion and new ephemeris of Triton from observations. Monthly Notices of the Royal Astronomical Society 454, 2205–2215. Jacobson, R.A., Synnott, S.P., Campbell, J.K., 1989. The orbits of the satellites of Mars from spacecraft and earthbased observations. Astronomy and Astrophysics 225, 548–554. Jacobson, R.A., 1998. The orbits of the inner Uranian satellites from Hubble space telescope and voyager 2 observations. Astronomical Journal 115 (3), 1195–1199. Jacobson, R.A., Spitale, J., Porco, C.C., Beurle, K., Cooper, N.J., Evans, M.W., Murray, C.D., 2008. Revised orbits of Saturn’s small inner satellites. Astronomical Journal 135, 261–263. Jacobson, R.A., 2009. The orbits of the Neptunian satellites and the orientation of the pole of Neptune. Astronomical Journal 137 (5), 4322–4329. Jacobson, R.A., 2013. The orbits of the regular Jovian satellites, their masses, and the gravity field of Jupiter. In: American Astronomical Society. DDA Meeting #44, #402.04. Lainey, V., Dehant, V., Patzold, M., 2007. First numerical ephemerides of the Martian moons. Astronomy and Astrophysics 465, 1075–1084. Nicholson, P.D., Hamilton, D.P., Matthews, K., Yoder, C.F., 1992. New observations of Saturn’s coorbital satellites. Icarus 100, 464–484. Owen, W.M., Vaughan, R.M., Synnott, S.P., 1991. Orbits of the six new satellites of Neptune. Astronomical Journal 101, 1511–1515. Pascu, D., Rohde, J.R., Seidelmann, P.K., Wells, E.N., Hershey, J.L., Storrs, A.D., Zellner, B.H., Bosh, A.S., Currie, D.G., 2004. Hubble space telescope astrometric observations and orbital mean motion corrections for the inner satellites of Neptune. Astronomical Journal 127 (5), 2988–2996. Porco, C.C., 2009. S/2008 S 1, IAUC 9023. P. 1.. Showalter, M.R., Lissauer, J.J., 2006. The second ring-moon system of Uranus: discovery and dynamics. Science 311 (5763), 973–977. Showalter, M.R., 2013. New satellite of Neptune: S/2004 N 1. CBET 3586.

449

C Special functions in celestial mechanics Inclination functions The inclination functions Fk,m,p (i) appear in the theory of perturbed satellite motion upon expansion of the perturbation function due to the non-sphericity of the planet, or from the attraction of an external body. The direction of the radius vector of the satellite is uniquely determined by its latitude ϕ and longitude, measured from the ascending node of the orbit, i.e. λ − . The direction of the satellite’s radius vector is also uniquely determined by the inclination of the orbit i and the latitude argument u (the angle between the radius vector and the direction to the ascending node). The relations between these pairs of angles are sin ϕ = sin i sin u, cos ϕ cos(λ − ) = cos u, cos ϕ sin(λ − ) = cos i sin u. Upon expansion of the perturbing function, a certain function arises, √ (m) Qkm = Pk (sin ϕ) exp −1m(λ − ), depending on ϕ and λ − . Using the last relations, this function can be expressed in terms of i and u as follows: k √ m−k+2E( k−m )  √ 2 −1 Fkmp (i) exp −1(k − 2p)u, Qkm =

(C.1)

p=0

where Fkmp (i) are special functions of celestial mechanics called as inclination functions. In this expression, the notation E(...) is used as the integer part of the number. The expression for Fkmp (i) for any index values in terms of sin i and cos i has the form  (2k − 2t)! Fkmp (i) = sink−m−2t i× 2k−2t t!(k − t)!(k − m − 2t)!2 t

451

452

Appendix C Special functions in celestial mechanics

 m   m coss i× s s=0   k − m − 2t + s   m − s  (−1)c−Ekm . p−t −c c c

Here Ekm is the integer part of (k − m)/2, t varies from 0 to p or Ekm (whichever is smaller), and the summation is performed for all values of c for which the binomial coefficients are nonzero. The expression for Fkmp (i) in terms of sin i/2 and cos i/2 can be found in Brumberg (1967). Convenient formulas for calculating the inclination functions and their derivatives, as well as the corresponding computational programs are given in Fominov and Filenko (1978). In view of the need to determine the inclination functions for large indices, one can use the method based on the special recurrence relations described in Emelyanov (1985). An effective method for calculating the inclination functions using recurrence relations was proposed in Emelianov and Kanter (1989). Explicit expressions for inclination functions with indices k = 2, 3, 4, m = 0, 1, ..., k, p = 0, 1, ..., k, are given by F200 (i) = − 38 sin2 i,

F201 (i) = 34 sin2 i − 12 ,

F202 (i) = − 38 sin2 i,

F210 (i) = 34 sin i(1 + cos i),

F211 (i) = − 32 sin i cos i,

F212 (i) = − 34 sin i(1 − cos i),

F220 (i) = 34 (1 + cos i)2 , F221 (i) = 32 sin2 i, F222 (i) = 34 (1 − cos i)2 , 5 F300 (i) = − 16 sin3 i,

F301 (i) =

15 16

sin3 i − 34 sin i,

3 3 F302 (i) = − 15 16 sin i + 4 sin i,

F303 (i) =

5 16

sin3 i,

2 F310 (i) = − 15 16 sin i(1 + cos i),

F311 (i) = F312 (i) =

15 16 15 16

sin2 i(1 + 3 cos i) − 34 (1 + cos i), sin2 i(1 − 3 cos i) − 34 (1 − cos i),

2 F313 (i) = − 15 16 sin i(1 − cos i),

F320 (i) = F321 (i) =

15 8 15 8

sin i(1 + cos i)2 , sin i(1 − 2 cos i − 3 cos2 i),

Appendix C Special functions in celestial mechanics

2 F322 (i) = − 15 8 sin i(1 + 2 cos i − 3 cos i), 2 F323 (i) = − 15 8 sin i(1 − cos i) ,

F330 (i) =

15 3 8 (1 + cos i) , 2 F331 (i) = 45 8 sin i(1 + cos i), 2 F332 (i) = 45 8 sin i(1 − cos i), 3 F333 (i) = 15 8 (1 − cos i) , 35 F400 (i) = 128 sin4 i, 15 4 2 F401 (i) = − 35 32 sin i + 16 sin i, 15 3 4 2 F402 (i) = 105 64 sin i − 8 sin i + 8 , 15 4 2 F403 (i) = − 35 32 sin i + 16 sin i, 35 F404 (i) = 128 sin4 i, 3 F410 (i) = − 35 32 sin i(1 + cos i), 15 3 F411 (i) = 35 16 sin i(1 + 2 cos i) − 8 sin i(1 + cos i), 105 3 F412 (i) = cos i( 15 4 sin i − 16 sin i), 15 3 F413 (i) = − 35 16 sin i(1 − 2 cos i) + 8 sin i(1 − cos i), 3 F414 (i) = 35 32 sin i(1 + cos i), 2 2 F420 (i) = − 105 32 sin i(1 + cos i) , 15 2 2 F421 (i) = 105 8 sin i cos i(1 + cos i) − 8 (1 + cos i) , 15 2 2 2 F422 (i) = 105 16 sin i(1 − 3 cos i) + 4 sin i, 15 2 2 F423 (i) = − 105 8 sin i cos i(1 − cos i) − 8 (1 − cos i) , 2 2 F424 (i) = − 105 32 sin i(1 − cos i) , 3 F430 (i) = 105 16 sin i(1 + cos i) , 2 3 F431 (i) = 105 8 sin i(1 − 3 cos i − 2 cos i), 3 F432 (i) = − 315 8 sin i cos i, 2 3 F433 (i) = − 105 8 sin i(1 − 3 cos i + 2 cos i), 3 F434 (i) = − 105 16 sin i(1 − cos i) , 4 F440 (i) = 105 16 (1 + cos i) , 2 2 F441 (i) = 105 4 sin i(1 + cos i) , 4 F442 (i) = 315 8 sin i,

453

454

Appendix C Special functions in celestial mechanics

F443 (i) = F444 (i) =

105 2 2 4 sin i(1 − cos i) , 105 4 16 (1 − cos i) .

Eccentricity functions The eccentricity functions appear in the theory of perturbed satellite motion during the expansion of the perturbing function due to the non-sphericity of the planet or the attraction of an external body. The following functions of the distance r and the true anomaly v have to be expanded in series in multiples of the mean anomaly M for small eccentricities e:  r n a

∞  √ √ n,j exp −1j v = Xq (e) exp −1qM,

(C.2)

q=−∞

n,j

where Xq (e) are special functions of celestial mechanics called as eccentricity functions. The conclusion of such a decomposition and a formula for calculating the eccentricity functions can be found in Brumberg (1967); Aksenov (1986). It turns out that the series recorded above in multiples of the mean anomaly converges for all eccentricity values smaller than unity. We note some properties of eccentricity functions. The number of calculations can be reduced using the ratio k,−j

k,j

X−q (e) = Xq (e). For all acceptable values of indices, we can write the expansion Xq (e) = e|q−j | k,j

∞ 

k,j

Xq,s e2s ,

s=0 k,j

where Xq,s are some numbers, and the series converges for all k,j e < 1. All the necessary coefficients Xq,s can be calculated using recurrence relations taken from Cherniack (1972) and reduced to a programming-friendly form in Fominov and Filenko (1978). Similar recurrence relations are given in Hughes (1981) and are presented in the book (Murray and Dermott, 2000). The eccentricity functions are addressed with much attention in the book (Aksenov, 1986). When the index q = 0, the eccentricity functions are expressed in the final form. For this special case, calculations can be performed using recurrence relations taken from Hughes (1981).

Appendix C Special functions in celestial mechanics

When calculating perturbations, the following property is important: X0−3,2 (e) = X0−3,−2 (e) = 0 for all e < 1. k,j Explicit expressions of the functions Xq (e) for k = −3, −4, −5 and some values j and q are given in reference (Kaula, 1966). k,j Note that in the literature the numbers Xq,s are also called Newcomb operators, and the eccentricity functions themselves are called Hansen coefficients. The expressions for some eccentricity functions at q = 0 are given by 3 1 5 X02,0 = 1 + e2 , X02,1 = −2e − e3 , X02,2 = e2 , 2 2 2 X0−3,0 = (1 − e2 )−3/2 , X0−4,1 = e(1 − e2 )−5/2 , 3 3 X0−5,2 = e2 (1 − e2 )−7/2 , X0−5,0 = (1 + e2 )(1 − e2 )−7/2 . 4 2

References Aksenov, E.P., 1986. Special Functions in Celestial Mechanics. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury, Nauka, Moscow. 320 pp. In Russian. Brumberg, V.A., 1967. Development of the perturbation function in satellite problems. Bulletin of the Institute of Theoretical Astronomy, Leningrad 11 (2), 73–83. In Russian. Cherniack, J.R., 1972. Computation of Hansen coefficients. SAO Special Report, N. 346. Emelyanov, N.V., 1985. Computing normalised inclination functions and their derivatives with big values of indexes. Tr. Gos. Astron. Inst. Sternberg 57, 83–91. In Russian. Emelianov, N.V., Kanter, A.A., 1989. A method to compute inclination functions and their derivatives. Manuscripta Geodaetica 14, 77–83. Fominov, A.M., Filenko, L.L., 1978. Computing Normalised Inclination Functions and Their Derivatives. Computing Hansen Coefficients and Their Derivatives. Algorithms of Celestial Mechanics, vol. 19. Institute of Theoretical Astronomy, Leningrad. In Russian. Hughes, S., 1981. The computation of tables of Hansen coefficients. Celestial Mechanics 25 (1), 101–107. Kaula, W.M., 1966. Theory of Satellite Geodesy. Applications of Satellites to Geodesy. Blaisdell, Waltham, Mass. Murray, C.D., Dermott, S.F., 2000. Solar System Dynamics. Cambridge Univ. Press, Cambridge. 608 pp.

455

D Time scales Creating time scales Time is a phenomenon that we dutifully accept as it is in nature. However, in astronomical practice, we need a source that tells us about time. The source is usually a physical process that is subject to observation and measurement. Astronomers always need the most uniform time, determined by one or another scale. Scientists are constantly searching for the most uniform time scale. In past centuries, the physical process that brought us time was the rotation of the Earth. The time associated with it was called universal time (UT). Astronomers set the universal time. One version of universal time is denoted by UT1. Since local solar time depends on the longitude of a place on the Earth, the concept of standard time was introduced. World time on the Greenwich meridian is designated as GMT— Greenwich Mean Time. After the uneven rotation of the Earth was discovered, the time scale began to be based on the process of the Earth’s motion around the Sun. Measurement of this process was done by astronomers. So ephemeris time (ET) is entered in astronomical practice. Ephemeris time was considered the most uniform one. The difference between ephemeris and universal time was measured and tabulated. Due to the unpredictability of the Earth’s variable speed of rotation, the difference UT1 – ET has always been known only for moments in the past after processing astronomical observations. The UT1 – ET difference tables were published in astronomical yearbooks. Since 1967, physicists began to engage in the definition of the concept of second. And since 1972, the concept of atomic time has been used in astronomical practice. The time sensor was physical processes in atoms that are accessible to observation and measurement. The atomic time scale is designated as IAT— International Atomic Time. This scale was considered as uniform as the ephemeris time scale. For unexplained historical reasons, the difference between ephemeris and atomic time (ET − I AT ) was set to 32.184 seconds. There was a contradiction between the desire to use the most uniform IAT time and the fact that our practical life on Earth is

457

458

Appendix D Time scales

associated with sunrise and sunset, and the Sun lives in the sky on a universal time scale. The contradiction was resolved in the following way. Astronomers came up with universal time coordinated (UTC) according to two simple rules: I AT − U T C is always equal to an integer number of seconds, the difference U T C − U T 1 never exceeds 0.9 seconds. Thus, the UTC, being uniform, tracks changes in the course of UT1 universal time. To implement these rules, it is necessary from time to time to abruptly change UTC for one second. This is done by the International Earth Rotation Service (IERS). Usually, if necessary, this is done either on January 1 or July 1. How often this is done can be found in the next section. The planned UTC time jump and the current I AT − U T C difference (leap second) can be found on the website https://hpiers.obspm.fr/iers/bul/bulc/bulletinc.dat. GMT (Greenwich Mean Time) is currently the same as UTC. By the end of the 20th century, the accuracy of astronomical observations reached such a level that we have to apply the model of the general theory of relativity instead of Newtonian mechanics. According to general relativity, time flows differently in different places of the Solar System. Therefore, the time scale began to be tied to the place in which the clock is stored. This is how the concepts of barycentric time (TDB—Time Dynamic Barycentric), in the barycenter of the Solar System, and time on the geoid (TT— Time Terestre) appeared. The implementation of TT was made possible due to the fact that the atomic clock is located on the geoid. It was assumed that TT – IAT = 32.184 seconds. Thus, it turned out that the TT time is a continuation of the ephemeris time ET, which is convenient for astronomical practice. Barycentric time is used to describe the dynamics of the bodies of the Solar System. Since there is virtually no clock, even atomic, in the barycenter of the Solar system, the TDB scale has to be modeled using knowledge of the distribution of masses, orbital and rotational motions of celestial bodies. Of course, the speed of the flow of time in the barycenter is different from the speed of its flow on the geoid. This difference has both a secular course and shows periodic changes. However, for convenience in astrometry, it was suggested that a clock is located in the barycenter, is running out on average at the same speed as the clock on the geoid. The periodically varying difference P = TDB – TT had to be modeled in huge trigonometric series, taking into account the periodic motions of many bodies of the Solar System. In fact, the value of P is represented by the sum of the periodic terms, each with its own period. One version of this series can be found in an article by Fairhead and Bretagnon (1990). The maximum amplitude of the periodic terms of the series is 0.001656 s.

Appendix D Time scales

Table D.1 The difference TT – UTC for each of the starting dates of its change. Julian Date, JD

Calendar Date

TT – UTC, seconds

2441317.5

1972 01 01

42.184

2441499.5

1972 07 01

43.184

2441683.5

1973 01 01

44.184

2442048.5

1974 01 01

45.184

2442413.5

1975 01 01

46.184

2442778.5

1976 01 01

47.184

2443144.5

1977 01 01

48.184

2443509.5

1978 01 01

49.184

2443874.5

1979 01 01

50.184

2444239.5

1980 01 01

51.184

2444786.5

1981 07 01

52.184

2445151.5

1982 07 01

53.184

2445516.5

1983 07 01

54.184

2446247.5

1985 07 01

55.184

2447161.5

1988 01 01

56.184

2447892.5

1990 01 01

57.184

2448257.5

1991 01 01

58.184

2448804.5

1992 07 01

59.184

2449169.5

1993 07 01

60.184

2449534.5

1994 07 01

61.184

2450083.5

1996 01 01

62.184

2450630.5

1997 07 01

63.184

2451179.5

1999 01 01

64.184

2453736.5

2006 01 01

65.184

2454832.5

2009 01 01

66.184

2456109.5

2012 07 01

67.184

2457204.5

2015 07 01

68.184

2457754.5

2017 01 01

69.184

In recent years, signals from extragalactic radio sources called pulsars have been used as a time sensor. Such pulsar time is considered the most uniform.

459

460

Appendix D Time scales

Figure D.1. Variation of the difference TT – UTC with time.

More detailed information on time scales and its measurement can be found in references (Simon et al., 1997; Works IAA RAS, 2004).

Relationship UTC with TT scale As follows from the previous section, the difference TT – UTC is 32.184 seconds plus some integer number of seconds. Table D.1 and Fig. D.1 give the difference TT – UTC for each of the starting dates of its change, starting in 1972. An abrupt change in the UTC scale for one second is set by the International Earth Rotation Service (IERS) at 0 hours of the starting date. For dates before 1972, the ET – UTC or ET – GMT difference was published in Astronomical Yearbooks.

Time scales in publications of observations in past centuries In publications of astrometric observations of planets, satellites, and asteroids in past centuries, scales of local mean solar time were used. The relationship of some of these scales with UTC or with the continuation of UTC into the past—GMT is given in Table D.2.

Appendix D Time scales

Table D.2 The relationship of the various timelines that have been used in the past. Time scale designation in publications

Difference: UTC – time scale (h is for hours, m for minutes, s for seconds)

Greenwich Mean Time

0h

Washington mean time

5 h 8 m 12.15 s

Central Standard Time

6h

Pacific Standard Time

8h

Pulkovo mean time

−(2 h 1 m 18.576 s)

Paris Mean Time

−(9 m 21 s)

Cambridge Mean Time

−(22.752 s)

Temps moyen de Nice

−(29 m 12.096 s)

Strasburg Mean Time

−(31 m 4.248 s)

Mount Hamilton Mean Time

8 h 6 m 34.92 s

90th Meridian Time

6h

Leander McCormick mean time

5 h 14 m 5.328 s

References Fairhead, L., Bretagnon, P., 1990. An analytical formula for the time transformation TB-TT. Astronomy and Astrophysics 229, 240–247. Simon, J.-L., Chapront-Touzé, M., Morando, B., Thuillot, W. (Eds.), 1997. Introduction aux éphémérides Astronomiques. Supplément explicatif à la connaissance des temps. Bureau des Longitudes, Paris. Les éditions de physique, France. 450 pp. In French. Works IAA RAS, 2004. Works of the Institut of the Applied Astronomy. Russian Academy of Science, No. 10. Ephemeris astronomy, St.-Petersbourg. In Russian.

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E Cholesky decomposition. Program in C-code To estimate the accuracy of the ephemeris by the method of variation of the motion parameters, it is necessary to calculate some matrix L, called the Cholesky decomposition matrix and satisfying the relation LLT = D, where the matrix D is the covariance matrix of parameter errors, which is obtained by refining the motion parameters of a celestial body from observations. The application of this matrix is described in Chapter 8. The matrix L turns out to be a lower triangular matrix. To calculate it, we can use the following program compiled in the C programming language: #include #include #include ... void decomp(double *A, int n, double *L) { int i,j,k; for(i=0;i