Physics and Evolution of Supernova Remnants [1st ed.] 9783030552299, 9783030552312

Table of contents :
Front Matter ....Pages i-xxiii
Introduction (Jacco Vink)....Pages 1-3
Supernovae (Jacco Vink)....Pages 5-31
Classification and Population (Jacco Vink)....Pages 33-53
Shocks and Post-shock Plasma Processes (Jacco Vink)....Pages 55-85
Supernova Remnant Evolution (Jacco Vink)....Pages 87-116
Neutron Stars, Pulsars, and Pulsar Wind Nebulae (Jacco Vink)....Pages 117-169
Dust Grains and Infrared Emission (Jacco Vink)....Pages 171-197
Optical Emission from Supernova Remnants (Jacco Vink)....Pages 199-219
Young Supernova Remnants: Probing the Ejecta and the Circumstellar Medium (Jacco Vink)....Pages 221-256
Middle-Aged and Old Supernova Remnants (Jacco Vink)....Pages 257-275
Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory (Jacco Vink)....Pages 277-321
Supernova Remnants and Cosmic Rays: Non-thermal Radiation (Jacco Vink)....Pages 323-377
Radiation Processes (Jacco Vink)....Pages 379-457
Summary and Prospects (Jacco Vink)....Pages 459-474
Back Matter ....Pages 475-521

Citation preview

Astronomy and Astrophysics Library

Jacco Vink

Physics and Evolution of Supernova Remnants

Astronomy and Astrophysics Library Series Editors Martin A. Barstow, Department of Physics and Astronomy, University of Leicester, Leicester, UK Andreas Burkert, University Observatory Munich, Munich, Germany Athena Coustenis, LESIA, Paris-Meudon Observatory, Meudon, France Roberto Gilmozzi, European Southern Observatory (ESO), Garching, Germany Georges Meynet, Geneva Observatory, Versoix, Switzerland Shin Mineshige, Department of Astronomy, Kyoto University, Kyoto, Japan Ian Robson, The UK Astronomy Technology Centre, Edinburgh, UK Peter Schneider, Argelander-Institut für Astronomie, Bonn, Germany Steven N. Shore, Dipartimento di Fisica “Enrico Fermi”, Università di Pisa, Pisa, Italy Virginia Trimble, Department of Physics & Astronomy, University of California, Irvine, CA, USA Derek Ward-Thompson, School of Physical Sciences and Computing, University of Central Lancashire, Preston, UK

More information about this series at http://www.springer.com/series/848

Jacco Vink

Physics and Evolution of Supernova Remnants

Jacco Vink Anton Pannekoek Institute/GRAPPA University of Amsterdam Amsterdam, The Netherlands

ISSN 0941-7834 ISSN 2196-9698 (electronic) Astronomy and Astrophysics Library ISBN 978-3-030-55229-9 ISBN 978-3-030-55231-2 (eBook) https://doi.org/10.1007/978-3-030-55231-2 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: The remnant of the historical supernova of 1604, also known as Kepler’s supernova remnant. The image is based on X-ray observations made by NASA’s Chandra X-ray Observatory. The colours red, green and blue correspond to X-ray emission lines from oxygen (0.5-0.7 keV), iron L-shell emission (0.7-1 keV) and silicon K-shell emission (1.7-1.9 keV), respectively. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

When I started working on my PhD project on X-ray spectroscopy of supernova remnants in 1995, one of the things that I missed was a good textbook covering the various aspects of supernova research. The library did contain two books on the subject, Iosif Shklovsky’s Supernovae (1968) and Tatjana Lozinskaya’s Supernovae and Stellar Wind in the Interstellar Medium (1991). The latter was not that old at the time, but it did not cover all aspects of supernova remnant research, and it was published before the advent of X-ray imaging spectroscopy and very highenergy gamma-ray astronomy. The book by Iosif Shklovsky was even then very dated, and it was as much about supernovae as their remnants. But it was (and still is) a fascinating read, coming from a person who contributed so much to our understanding of the physics of supernova remnants. Since I obtained my PhD in 1999, the situation concerning good background material on supernova remnants has not improved. There are no dedicated books on supernova remnants, although there are several books on supernovae, such as the recent Supernova Explosions (2017) by D. Branch and J. C. Wheeler. In addition, many aspects of supernova remnants are treated in various chapters in Handbook of Supernovae (2018), edited by A. Alsabti and P. Murdin. I contributed myself two chapters to this Handbook. Although the Handbook treats diverse aspects of supernova remnants, it does not provide a coherent treatment. Of course, several textbooks on the topics related to supernova remnants exist. I recommend in particular the Physics of the Interstellar and Intergalactic Medium (2011) by B. T. Draine, and in terms of physics of shocks and plasma processes, the monographs by Lyman Spitzer and the book Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena by Ya. Zel’dovich and Yu. Raizer provide continuous inspiration. Supernova remnants have been the subject of a number of extensive reviews. I was author among one of them, on X-ray spectroscopy of supernova remnant (2012) [1173]. Feedback on this review convinced me that a book on supernova remnants would be appreciated. The review itself focussed on the X-ray aspects—not treating multiwavelength aspects—although I smuggled in some multiwavelength perspectives, as well. v

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My aim with this book is to provide a complete overview of the relevant physical processes that shape the evolution and observational characteristics of supernova remnants. The book should be a good introduction to graduate students starting with a research project on supernova remnants, as well as provides the background knowledge for experienced researchers. Given the diversity of topics within supernova remnant research, it is difficult to have all knowledge at hand, even if supernova remnants is a researcher’s main field of expertise. It is this diversity that makes supernova remnants such fascinating objects to study. Supernova remnants bring together shock and plasma physics, cosmic-ray studies, aspects of the late stages of stellar evolution, the physics of pulsars, and dust formation and emission. However, supernova remnants are rarely a topic that is treated in a separate course, but at best a few classes will be dedicated to it. This is likely the reason that there are many books on galaxies, or active galactic nuclei, and no recent books on supernova remnants. However, I believe that some concepts detailed in this book could be given as background materials for students and researchers interested in shock physics, X-ray spectroscopy, or cosmic-ray acceleration. The current book treats all these aspects. Its format is a hybrid of a monograph on the basics of underlying physics and more review-type chapters on the current observational knowledge. The textbook aspects dominate the chapters on shock physics (Chap. 4), hydrodynamic evolution (Chap. 5), dust emission and formation (Chap. 7), optical emission (Chap. 8), and cosmic-ray acceleration (Chap. 11). The chapters on young (Chap. 9) and older supernova remnants (Chap. 10) are more of review type in nature. Finally, the chapters on the supernova remnant classification and population (Chap. 3) and pulsar wind nebulae and neutron stars (Chap. 6) are by themselves a hybrid of explaining basic concepts and providing review material. In my review of X-ray spectroscopy of supernova remnants, I provided an introduction to the relevant X-ray emission mechanisms. For this book, I extended the treatment of radiation mechanisms to include the various mechanisms for non-thermal radiation, including pion decay. I also included more material on forbidden-line formation, which is important for the optical properties of mature supernova remnants. As a large fraction of the material presented here is quite generic to many high-energy astrophysical sources, I placed the material on radiation physics in a separate chapter at the end of the book. Despite the more generic character, there are several radiative properties that are particularly of interest to supernova remnants, such as the appearance of cooling breaks in synchrotron spectra, non-equilibrium ionisation, and broad radiative recombination continua for X-ray spectra. Nevertheless, because of the generic aspects, I imagine that this chapter, as a standalone text, may be used for a course in radiative physics in astronomy, or a high-energy astrophysics course – I certainly will use it as such myself. Many of the figures in this book were made by myself. I opted for this as it allowed for more uniform style of the figures, and also because creating figures allowed me to come to grasp with the underlying physics and gain more handson knowledge. Moreover, I enjoyed this methodology of the book-writing process.

Preface

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This is in particular true for making multicolour images of supernova remnants. After all, one of the motivations for supernova remnant studies is that they make for such aesthetically pleasing objects. In writing the text, I did not shy away of sparingly putting in novel views. A text like this should be as much a compendium of accumulated knowledge, as well as provide a stimulant for further study and discussion. The reader should, therefore, be warned not to take the current text as a monolithic source of knowledge, but approach it critically, as a starting point for further study. But, I also encourage to consult the original source material. Although I write this preface in the first person, in the book I adopted the more neutral we. For one thing, most of what is written does not concern my own contribution, but the contributions of many researchers. So, the we stands for all the researchers who contributed to the current understanding of supernova remnants. And of course, the we stands for the author (me) and the reader together. Another textual aspect one encounters in writing an extensive book as this is the symbols for physical quantities. Many symbols are commonly associated with certain quantities, m for mass, v for velocity, p for momentum, etc. But conventions differ sometimes per field. For example, in cosmic-ray acceleration theory often u is used for velocity. Sometimes, the conventions clash. For example, the adiabatic index is usually denoted by γ , but so is the Lorentz factor. I used  for the Lorentz factor, but then  is also often used for the spectral index in γ -ray astronomy. Another example is E, which is used to denote energy, but also electric-field strength. I sometimes solved these clashes by using a different font style— I used E for the electric field and V for volume. But, it is impossible to invent a new symbol for all such cases, let alone coming up with symbols for the various tuning parameters α, β, η, etc. I tried to avoid confusion by clearly outlining the definition in case there are conflicts, or by using subscripts. The book contains an extended list of references. The astrophysical practice of using the author–date system for references can make texts sometimes look crammed. For that reason, I opted for the system of numbered references. An additional advantage is that the more concise format makes it easier to use these in tables. As already mentioned, I wrote an extensive review on X-ray spectroscopy of supernova remnants. My initial idea was to use that as the basis for this book. In the end, most of the text presented here is new, but some of the chapters contain text borrowed from the review. The review itself took quite some time to finish. So, I knew what I was up to when I expressed my ambition to write a book on the topic to Harry Blom, now vice-president Journals, Development, Policy and Strategy at Springer Nature, who was once a close colleague, when we were both PhD students at SRON, Utrecht. Our initial conversation led to a contract, and I started writing in earnest in 2015. Despite knowing beforehand that writing a book is a major timeconsuming endeavour, it still took me longer to finish the text than I anticipated. But, perhaps we need some misguided optimism to get started at all! The writing of the book was greatly helped by the involvement of the PhD students and postdocs in my group. They proofread some of the chapters, but it

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also helped that every now and then I provided them with the chapters as a source of information. This often led to sharpening the text, correcting mistakes, and it helped me to appreciate the value of having (already) a book at hand to refer to. Therefore, my special thanks to Maria Arias, Laura Driessen, Vladimir Domˇcek, Dimitris Kantzas, Sun Lei, Dmitry Prokhorov, Rachel Simoni, and Ping Zhou, past and current members of my research group during the writing process. I would also like to thank group members in a more distant past, who have helped to shape my knowledge of the topic: Sjors Broersen, Alexandros Chiotellis, Eveline Helder, Daria Kosenko, and Klara Schure. In this context, it is also worth mentioning my PhD supervisors Johan Bleeker and Jelle Kaastra, who introduced me to the field of high-energy emission from supernova remnants and X-ray spectroscopy. The final work on the book was carried out under peculiar circumstances, during the Covid-19 pandemic. On the one hand, it led to new distractions such as home schooling, and many online meetings. But on the other hand, it put to halt all the traveling that comes with being a scientist and instead allowed me to focus on completing the text. In this last phase, invaluable lessons on time management by Anne Baker were put to use, which helped me finishing the book. Finally, I would like to thank my family, Sonja, Tamar, and Guido, for indulging my frequent excuses for secluding myself in my office to finish this undertaking. Amsterdam, The Netherlands

Jacco Vink

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 The Optical Classification of Supernovae .. . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Spectroscopic Classification . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Supernova Light Curve Classification .. . . . . . . . . . . . . . . . . . . . 2.1.3 The Supernova Rates Per Type.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Core-Collapse Supernovae .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Pre-explosion Composition . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Neutron Star and Black Hole Formation . . . . . . . . . . . . . . . . . . 2.2.3 The Explosion Mechanism .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.4 Electron-Capture Supernovae of 8–10 M Stars . . . . . . . . . 2.2.5 Core-Collapse Supernova Ejecta Composition .. . . . . . . . . . . 2.3 Thermonuclear (Type Ia) Supernovae .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 The Single Degenerate Versus Double Generate Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Thermonuclear Explosions: Deflagration Versus Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 The Diversity Among Type Ia Supernova . . . . . . . . . . . . . . . . . 2.4 Detection of Radio-Active Elements from Supernovae . . . . . . . . . . . . 2.5 Light Echoes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5 8 8 10 12 13 13 14 15 16 17 18

Classification and Population . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Morphological Classification of Supernova Remnants .. . . . . . . . . . . . 3.2 The Galactic Supernova Remnant Population . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Finding and Naming Supernova Remnants.. . . . . . . . . . . . . . . 3.2.2 Measuring Distances to Supernova Remnants . . . . . . . . . . . . 3.2.3 The -D Relation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 The Spatial Distribution of Known Galactic Supernova Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 The Supernova-Remnant Population in the Magellanic Clouds . . . 3.5 Supernova Remnant Populations in Other Galaxies .. . . . . . . . . . . . . . .

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Shocks and Post-shock Plasma Processes . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 The Rankine-Hugoniot Jump Conditions . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Magnetohydrodynamical Shocks .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Collisionless Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Visocity and the Shock Transition Layer Thickness . . . . . . 4.3.2 The Collisional Mean Free Path . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 The Thermalisation Processes . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 The Expected Post-shock Electron-Ion Temperature Ratio . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.5 Post-shock Electron-Ion Temperature Equilibration .. . . . . 4.3.6 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Radiative Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Isothermal Shocks . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Magnetically Supported, Radiative Shocks . . . . . . . . . . . . . . . 4.5 Shock Waves Mediated by Magnetic Precursors . . . . . . . . . . . . . . . . . . . Supernova Remnant Evolution . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Supernova Remnant Evolution: Four Phases . . .. . . . . . . . . . . . . . . . . . . . 5.2 The Expansion Parameter .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 The Reverse Shock .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 The Reverse Shock Velocity in the Shock- and Observer-Frame.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.2 The Condition for Forming a Reverse Shock . . . . . . . . . . . . . 5.3.3 The Turning Around of the Reverse Shock.. . . . . . . . . . . . . . . 5.4 Self-similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 The Self-similar Sedov-Taylor Solution .. . . . . . . . . . . . . . . . . . 5.4.2 An Alternative Derivation of the Sedov-Taylor Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.3 The Sedov-Taylor solution for a stellar-wind profile . . . . . 5.4.4 The Expected Size Distribution of Supernova Remnants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 The Internal Structure of Self-similar Explosions . . . . . . . . . . . . . . . . . . 5.6 Self-similar Models for the Ejecta-Dominated Phase . . . . . . . . . . . . . . 5.6.1 The Chevalier Self-similar Model for Young Remnants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6.2 The Transition from Ejecta-Dominated to Adiabatic Phase . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 The Late Time Evolution of Supernova Remnants . . . . . . . . . . . . . . . . . 5.8 Supernova Remnant Evolution Inside Wind Bubbles . . . . . . . . . . . . . . 5.8.1 The Evolution of Main Sequence Wind Bubbles . . . . . . . . . 5.8.2 Supernova Remnant Evolution of Inside Wind Bubbles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.9 Rayleigh-Taylor Instabilities . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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Neutron Stars, Pulsars, and Pulsar Wind Nebulae .. . . . . . . . . . . . . . . . . . . . 6.1 The Internal Constitution of Neutron Stars . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 The Magnetic Dipole Model for Neutron Stars . . . . . . . . . . . 6.2.2 The Pulsar Braking Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.3 The Magnetosphere.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 The Inner Regions of Pulsar Wind Nebulae.. . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 The Pulsar Wind . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 The Kennel and Coroniti Model . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.3 Wisps, Jets, and Tori . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.4 The σ -Problem.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 The Evolution and Radiation of Pulsar Wind Nebulae . . . . . . . . . . . . . 6.4.1 A Self-similar Solution for the Expansion into the Ejecta .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 The Appearance and Dynamics of the Crab Nebula . . . . . . 6.4.3 Pulsar Wind Nebulae Interacting with the Reverse Shock .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.4 The Radiation from Pulsar Wind Nebulae.. . . . . . . . . . . . . . . . 6.4.5 The Electron/Positron Populations in Pulsar Wind Nebulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.6 The Frequency Dependent Sizes of Pulsar Wind Nebulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.7 The Large Extent of Some Pulsar Wind Nebulae in γ -Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.8 Pulsars Moving Through Hot Supernova Remnant Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Magnetars and Central Compact Objects . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5.1 Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5.2 Compact Central Objects (CCOs) . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Dust Grains and Infrared Emission . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Introduction: Interstellar Dust . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 The Supernova Connection . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Dust Heating and Radiation.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.1 Dust Emission . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.2 Collisional Dust Heating . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.3 Stochastic Dust Heating .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.4 Determining Dust Masses . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Dust Formation in Supernova Ejecta . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Dust Destruction in Supernova Remnants . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Infrared Observations of Supernova Remnants .. . . . . . . . . . . . . . . . . . . . 7.6.1 Infrared Emission from Young Supernova Remnants.. . . . 7.6.2 Observational Evidence for Dust Destruction .. . . . . . . . . . . .

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Optical Emission from Supernova Remnants . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Line Emission from Radiative Shocks Regions.. . . . . . . . . . . . . . . . . . . . 8.1.1 On the Prominence of Forbidden Line Emission . . . . . . . . . 8.1.2 Optical Emission from Young Supernova Remnants: Optical Emission from High-Density Clumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Balmer-Dominated Shocks . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 The Formation of the Narrow- and Broad-Line Components .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 The Broad- to Narrow-Line Ratio as a Diagnostic Tool .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.3 Measuring Distances to Balmer-Dominated Shocks .. . . . . 8.2.4 The Shock Structure in the Presence of Neutrals . . . . . . . . . 8.2.5 Complications: Pickup Ions and Non-thermal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.6 The Effects of Cosmic-Ray Acceleration.. . . . . . . . . . . . . . . . . Young Supernova Remnants: Probing the Ejecta and the Circumstellar Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Core-Collapse Versus Type Ia Supernova Remnants . . . . . . . . . . . . . . . 9.2 Type Ia Supernova Remnants .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.1 Hydrodynamical Plus Radiation Modelling .. . . . . . . . . . . . . . 9.2.2 X-ray Cr and Mn Line Diagnostics . . . .. . . . . . . . . . . . . . . . . . . . 9.2.3 The Ambient Medium of Type Ia Supernova Remnants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.4 The Case of the Missing Donor Stars. .. . . . . . . . . . . . . . . . . . . . 9.2.5 The Confusing Evidence Concerning Type Ia Progenitors .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Core-Collapse Supernova Remnants . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.1 Cas A and Other Oxygen-Rich Supernova Remnants . . . . 9.3.2 Asymmetric Ejecta: Donuts, Jets, Rings and Bubbles . . . . 9.3.3 SN 1987A: the making of a supernova remnant . . . . . . . . . .

10 Middle-Aged and Old Supernova Remnants . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 The Presence of Metal-Rich Ejecta in Middle-Aged Supernova Remnants .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Interaction with Molecular Clouds . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.1 Radiation from Molecules.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 The Interaction of Shocks with Molecular Clouds .. . . . . . . 10.2.3 Maser Emission from Supernova Remnants . . . . . . . . . . . . . . 10.3 Mixed-Morphology Supernova Remnants . . . . . .. . . . . . . . . . . . . . . . . . . .

199 200 201

206 207 209 212 215 215 215 217 221 221 224 227 228 230 236 237 238 239 243 249 257 260 262 263 264 267 268

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Introduction: Galactic Cosmic Rays. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.1 The Cosmic-Ray Spectrum.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.2 Cosmic-Ray Composition .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.3 Cosmic-Ray Transport in the Galaxy ... . . . . . . . . . . . . . . . . . . . 11.1.4 SNRs as the dominant sources for Galactic cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.5 Other Potential Sources of Galactic Cosmic Rays . . . . . . . . 11.2 The Theory of Diffusive Shock Acceleration .. .. . . . . . . . . . . . . . . . . . . . 11.2.1 Diffusive-Shock Acceleration .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.2 The Convection-Diffusion Equation .. .. . . . . . . . . . . . . . . . . . . . 11.2.3 The Acceleration Time Scale . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.4 The Maximum Size of the Cosmic-Ray Shock Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.5 The Effect of Adiabatic Losses on the Maximum Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.6 Particle Acceleration by Evolving Shocks . . . . . . . . . . . . . . . . 11.2.7 The Escape of Cosmic Rays . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.8 Radiative Losses: The Maximum Electron Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Non-linear Shock Acceleration .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Particle Acceleration and Magnetic Fields . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4.1 Resonant Particle-Wave Interaction . . .. . . . . . . . . . . . . . . . . . . . 11.4.2 Streaming Instabilities and Non-resonant Processes . . . . . . 11.4.3 The non-resonant Bell instability . . . . . .. . . . . . . . . . . . . . . . . . . . 12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation . . . . 12.1 Radio Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1.1 The Radio Spectral Index Distribution . . . . . . . . . . . . . . . . . . . . 12.1.2 The Minimum Energy Requirement and the Van der Laan Mechanism . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1.3 The Radio Evolution of Supernova Remnants . . . . . . . . . . . . 12.1.4 Radio Polarisation Measurements . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 X-ray Synchrotron Radiation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.1 The Implication of X-ray Synchrotron Radiation .. . . . . . . . 12.2.2 The Narrow Widths of the X-ray Synchrotron Regions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.3 The Case for Magnetic-Field Amplification.. . . . . . . . . . . . . . 12.2.4 Magnetic-Field Amplification Near the Reverse Shock .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.5 X-ray Synchrotron Flickering and Flux Decline . . . . . . . . . . 12.2.6 X-ray Synchrotron Peculiarities and (Possible) Consequences .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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277 277 278 280 284 289 290 294 294 297 300 302 303 303 305 305 308 313 314 317 318 323 323 324 327 330 335 338 340 341 345 349 350 353

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12.3 Gamma-Rays Observations: A Window on the Hadronic Cosmic-Ray Content of Supernova Remnants . .. . . . . . . . . . . . . . . . . . . . 12.3.1 A Brief Historical Overview of γ -Ray Astronomy . . . . . . . 12.3.2 Hadronic Versus Leptonic Emission . . .. . . . . . . . . . . . . . . . . . . . 12.3.3 A Few Words on Modelling Inverse Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3.4 Gamma-Ray Evidence for Escaping Cosmic Rays . . . . . . . 12.3.5 The Population of γ -Ray Emitting Supernova Remnants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3.6 Where Are the PeVatrons? . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13 Radiation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2 Radiation from Moving Charged Particles . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2.1 Thomson Scattering . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2.2 Inverse Compton Scattering .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3.1 The Synchrotron Power and Spectrum . . . . . . . . . . . . . . . . . . . . 13.3.2 Radiation from a Power-Law Electron Distribution . . . . . . 13.3.3 The Effects of Synchrotron Radiation Energy Losses . . . . 13.3.4 Cooling Breaks . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 Bremsstrahlung (Free-Free Emission).. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.1 Bremsstrahlung from a Single Electron-Ion Encounter .. . 13.4.2 The Collisional Cross-Section and Total Radiation Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.3 Relativistic Bremsstrahlung .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.4 Thermal Bremsstrahlung . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.5 Non-thermal Bremsstrahlung . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.6 Free-Free Absorption .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5 Line Emission, Ionisation and Recombination Processes . . . . . . . . . . 13.5.1 The Einstein Coefficients and Oscillator Strength . . . . . . . . 13.5.2 Some Basic Atomic Physics . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.3 The Atomic Shell Model and Electron Configurations . . . 13.5.4 Electron Transition Probabilities and the Einstein Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.5 Collisional Processes that Shape Emission Line Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.6 Radiative Recombination Continuum (Free-Bound Emission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.7 Non-equilibrium Ionisation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.8 X-ray Line Emission Diagnostics .. . . . .. . . . . . . . . . . . . . . . . . . . 13.5.9 Resonant Absorption and Line Scattering .. . . . . . . . . . . . . . . .

356 357 359 364 367 370 374 379 379 380 380 383 389 390 394 395 397 399 401 403 405 406 408 410 411 413 415 421 426 430 437 439 442 447

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13.6 Pion Production and Decay . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.6.1 Meson Production in Supernova Remnants . . . . . . . . . . . . . . . 13.6.2 The Energy Threshold for Pion Production . . . . . . . . . . . . . . . 13.6.3 The Formation of the γ -Ray Spectrum .. . . . . . . . . . . . . . . . . . .

449 451 452 453

14 Summary and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Knowledge Gained and Outstanding Questions . . . . . . . . . . . . . . . . . . . . 14.2 Future Observing Facilities . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3 The Emergence of Multimessenger Astronomy . . . . . . . . . . . . . . . . . . . . 14.4 The Extragalactic Transients Connection . . . . . . .. . . . . . . . . . . . . . . . . . . .

459 459 462 470 471

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 475 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 513 Astrophysical objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 519

List of Figures

Fig. 1.1 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 4.1 Fig. 4.2

The Veil Nebula as observed by the Hubble Space Telescope . . . . SN 1994D as observed by the Hubble Space Telescope.. . . . . . . . . . Title page of the book “De Stella Nova” . . . . . . .. . . . . . . . . . . . . . . . . . . . Supernova classification scheme.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Spectra of various types of supernovae.. . . . . . . .. . . . . . . . . . . . . . . . . . . . Light curves of various types of supernovae . . .. . . . . . . . . . . . . . . . . . . . The observed fractions of supernovae of different types . . . . . . . . . . Binding energy per nucleon .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Supernova yields of various abundant elements . . . . . . . . . . . . . . . . . . . The fractional occurrence of Type Ia subtypes.. . . . . . . . . . . . . . . . . . . . The light curves of SNe Ia, and the historical light curve of SN 1604 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Light echo images and chart . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Light echo spectrum of Cassiopeia A . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Light echo geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Supernova remnant classification: examples .. .. . . . . . . . . . . . . . . . . . . . The -D relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Histogram of the radio surface brightness.. . . . .. . . . . . . . . . . . . . . . . . . . The spatial distribution of known Galactic supernova remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The distribution of supernova remnants in the Galactic disk . . . . . . The positions of supernova remnants in the LMC and SMC . . . . . . Diameter distribution of Magellanic Cloud SNRs . . . . . . . . . . . . . . . . . Supernova remnant positions in M101 . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Spectral characteristics of SNRs and HII regions .. . . . . . . . . . . . . . . . . X-ray hardness for SNRs in M33; Venn diagrams of extragalactic SNRs . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The shock compression and Ms,2 as function of Mms,1 . . . . . . . . . . . Pressure and momentum changes in the shock, and shock thicknesses .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2 6 7 9 11 11 12 14 18 25 25 28 30 31 34 41 43 44 45 47 49 51 52 53 61 63

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Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. 6.11 Fig. 6.12 Fig. 6.13 Fig. 6.14

List of Figures

Shock transition thickness of a (solar system) collisionless shock .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Expected, and measured, electron/ion temperature ratios versus Mms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The temperature equilibration of different charged particles versus time .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The radiative cooling curve of an optically thin plasma .. . . . . . . . . . Cooling time and length scale as a function of shock velocity .. . . Density/temperature structure of a radiative shock . . . . . . . . . . . . . . . . Compression ratio of a radiative, isothermal, shock versus Mach number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Shocks with magnetic precursors .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Schematic view of the forward shock/reverse shock system . . . . . . The internal structure of an SNR according to the Sedov-Taylor solution .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Structure of young SNRs, according to the Chevalier model . . . . . Truelove and McKee models for the evolution both shocks.. . . . . . Optical image of Simeis 147 .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The wind energy of massive stars as a function of MS mass . . . . . . The structure and evolution of a wind bubble around an evolved star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Simulations of Rayleigh-Taylor instabilities in young SNRS . . . . . The P -P˙ diagram of known pulsars . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . X-ray image of N157B and the timing behaviour of its central pulsar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A sketch of the magnetosphere of a neutron star . . . . . . . . . . . . . . . . . . The composite supernova remnant G11.2-0.3 as observed by Chandra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Multiwavelength images of the Crab Nebula . .. . . . . . . . . . . . . . . . . . . . Kennel and Coroniti model for the velocities and magnetic fields in PWNe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Crab Nebula and Vela pulsar wind nebula in X-rays . . . . . . . . . X-ray snap shots of the Vela pulsar wind nebula and its wobbly jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Synchrotron emission model based on simulations of the Crab Nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The measured torus radii versus the pulsar spin-down energy E˙ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Schematic structure of a composite supernova remnant .. . . . . . . . . . The broad SED of the Crab Nebula, PWN N157B, and PWN N158A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The X-ray efficiency of PWNe and pulsars versus characteristic age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The extended PWN HESS J1303-631 as observed by H.E.S.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

66 70 72 76 77 78 80 83 90 98 100 103 105 108 111 115 123 125 127 129 130 133 134 135 136 137 139 146 148 152

List of Figures

Fig. 6.15 Fig. 6.16 Fig. 6.17 Fig. 6.18 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7 Fig. 7.8 Fig. 7.9 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 Fig. 8.10 Fig. 8.11 Fig. 8.12 Fig. 9.1

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The Mach number associated with a pulsar inside the SNR shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The supernova remnant IC 443 and its associated pulsar wind nebula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Two SNRs with central point sources: CTB 109 and PKS 1209-51/52 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . X-ray image and spectrum of Kes 73, containing a magnetar in its centre.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Dust depletion levels as a function of condensation temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The spectral energy distribution of Cassiopeia A from radio to X-ray .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Dust temperatures as expected for collisional grain heating . . . . . . Dust cooling time, maximum dust temperatures and stochastic heating .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Maximum grain-size as a function of time of grain formation onset .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . All sky infrared map based on IRAS observations . . . . . . . . . . . . . . . . . IR (70 μm) maps of SN1987A, SN 1604, SN 1572, and the Crab Nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The western part of the SNR IC 443 in IR by the WISE observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Three colour, infrared image of Cas A and its mid-infrared spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Two HST images of LMC SNRs; with radiative and non-radiative shocks.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Oxygen ionisation fractions behind a radiative shock . . . . . . . . . . . . . Optical spectrum of a filament in the Cygnus Loop . . . . . . . . . . . . . . . Level diagram for O III . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Illustration of the geometry of resonant line scattering .. . . . . . . . . . . Hubble Space Telescope image of a shock structure in SN 1006 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Spectral line shape for SNR 0509-67.5 in the LMC .. . . . . . . . . . . . . . Cross sections for hydrogen charge exchange, ionisation and excitation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The FWHM of broad-line Hα emission as function of shock velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Predicted Hα broad- to narrow-line ratio for Balmer-dominated shocks . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Density profile of cold, and charge-exchanged hydrogen, and hot protons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The effect of cosmic-ray acceleration on the broad-line Hα-width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . X-ray spectra of a typical core collapse and Type Ia SNR . . . . . . . .

155 156 157 165 172 175 179 181 186 190 192 193 194 200 201 202 204 205 208 210 210 213 214 216 218 222

xx

Fig. 9.2 Fig. 9.3 Fig. 9.4 Fig. 9.5 Fig. 9.6 Fig. 9.7 Fig. 9.8 Fig. 9.9 Fig. 9.10 Fig. 9.11 Fig. 9.12 Fig. 9.13 Fig. 9.14 Fig. 9.15 Fig. 9.16 Fig. 9.17 Fig. 9.18 Fig. 9.19 Fig. 10.1 Fig. 10.2 Fig. 10.3 Fig. 10.4 Fig. 10.5 Fig. 11.1 Fig. 11.2 Fig. 11.3 Fig. 11.4 Fig. 11.5 Fig. 11.6 Fig. 11.7 Fig. 11.8 Fig. 11.9 Fig. 12.1

List of Figures

The symmetry properties of Type Ia versus core-collapse SNRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Fe-K line luminosity of young SNRs versus Fe-K centroid energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Chandra images of supernova remnants in the LMC . . . . . . . . . . . . . . The radial stratification of oxygen, silicon and iron in SNR 0519-69.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The X-ray spectrum of Tycho’s SNR compared to a full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Cr and Mn K-shell emission in the spectrum of Tycho’s SNR . . . . The molecular shell around Tycho’s SNR. . . . . .. . . . . . . . . . . . . . . . . . . . Composite radio, infrared, X-ray image of W49B .. . . . . . . . . . . . . . . . Cas A and Cas N132D, as observed by the HST . . . . . . . . . . . . . . . . . . X-ray images of oxygen-rich supernova remnants.. . . . . . . . . . . . . . . . X-ray spectra of Fe-rich and Si-rich regions in Cas A .. . . . . . . . . . . . Doppler shifts maps of SNR 1E 0102.2-7219, obtained with the HETG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Images of Cas A’s jets and asymmetries . . . . . . .. . . . . . . . . . . . . . . . . . . . 3D morphology of the optical knots in Cas A .. . . . . . . . . . . . . . . . . . . . SN1987A in the 30 Doradus region and HST image of SN1987A in 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The evolution of SN1987A . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The expansion of SN1987A observed in X-rays and radio .. . . . . . . Combined Chandra HETGS count spectrum of SN 1987A .. . . . . . The Vela supernova remnant as observed by ROSAT and in Hα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . X-ray emission pattern of oxygen and iron in the Cygnus Loop.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . OH level diagram .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . OH maser emission from W28 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . G359.1-0.5 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The cosmic-ray spectrum . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Cosmic-ray composition . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Illustration of the leaky box model for cosmic-ray transport .. . . . . The boron-carbon ratio as a function of energy per nucleon . . . . . . Fermi γ -ray sky map .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Schematic illustration of the structure of a cosmic-ray modified shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The effects of non-linear cosmic-ray acceleration on shock properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Interaction between charged particles and the magnetic field (cartoon) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Bell instability: the evolution of magnetic=field strengths and structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Radio map and radio polarisation map of Cas A. . . . . . . . . . . . . . . . . . .

224 225 226 227 228 229 233 235 240 242 245 246 247 248 250 251 252 254 259 260 265 268 272 279 281 286 288 289 309 312 314 321 324

List of Figures

Fig. 12.2 Fig. 12.3 Fig. 12.4 Fig. 12.5 Fig. 12.6 Fig. 12.7 Fig. 12.8 Fig. 12.9 Fig. 12.10 Fig. 12.11 Fig. 12.12 Fig. 12.13 Fig. 12.14 Fig. 12.15 Fig. 12.16 Fig. 12.17 Fig. 12.18 Fig. 12.19 Fig. 12.20 Fig. 12.21 Fig. 12.22 Fig. 12.23 Fig. 12.24 Fig. 12.25 Fig. 12.26 Fig. 13.1 Fig. 13.2 Fig. 13.3 Fig. 13.4 Fig. 13.5 Fig. 13.6 Fig. 13.7 Fig. 13.8 Fig. 13.9 Fig. 13.10 Fig. 13.11

xxi

A histogram of the radio spectral index distribution .. . . . . . . . . . . . . . The radio spectrum of Cas A . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The flux decline of Cas A around 1420 MHz . .. . . . . . . . . . . . . . . . . . . . Models for the radio flux evolution .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Magnetic field orientation in a young and old supernova remnant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The radio polarisation fraction versus radio surface brightness .. . SN 1006 in X-rays .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . X-ray synchrotron filaments in Cas A, Kepler’s SNR, and Tycho’s SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . X-ray images of RX J1713.7-3946 and RCW 86 .. . . . . . . . . . . . . . . . . X-ray synchrotron profiles of supernova remnants . . . . . . . . . . . . . . . . The quantity B 2 /(8πρ) versus the shock velocity .. . . . . . . . . . . . . . . . X-ray synchrotron emission from reverse shock .. . . . . . . . . . . . . . . . . . Possible magnetic field configurations for SN 1006 . . . . . . . . . . . . . . . Tycho’s supernova remnant synchrotron “stripes” .. . . . . . . . . . . . . . . . The H.E.S.S. II imaging atmospheric telescope array in Namibia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . SED of the Vela Jr supernova remnant . . . . . . . . .. . . . . . . . . . . . . . . . . . . . H.E.S.S. excess map and Fermi-LAT spectrum of RX J1713.7-3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Gamma-ray SED of W44 and IC 443 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Galactic radiation fields . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B-field versus the electron cut-off energy in RX J1713.7-3946 .. . W28 in radio with H.E.S.S. significance contours . . . . . . . . . . . . . . . . . X-ray and VHE γ -ray surface brightness profiles of RX J1713.7-3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . γ -ray spectral indices of supernova remnants .. . . . . . . . . . . . . . . . . . . . The γ -ray SED of Cas A . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Cas A acceleration properties, past and future .. . . . . . . . . . . . . . . . . . . . Illustration of the principal of Thomson scattering . . . . . . . . . . . . . . . . The radiation pattern for Thomson scattering ... . . . . . . . . . . . . . . . . . . . Schematic view of (inverse) Compton scattering . . . . . . . . . . . . . . . . . . Number- and energy distribution of inverse Compton scattered photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Inverse Compton scattering in the Klein-Nishina regime . . . . . . . . . The number distribution of IC photons in the Klein-Nishina regime .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The spectral flux density distribution for synchrotron radiation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Electron spectra of a given age, affected by synchrotron radiation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Expected electron synchrotron spectra with cooling breaks.. . . . . . Illustration of the trajectory of an electron and its acceleration .. . The Lorentzian line profile .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

325 329 332 334 336 337 339 341 345 346 348 349 353 355 358 360 361 363 364 366 368 369 373 373 376 381 382 384 387 388 390 393 397 400 401 415

xxii

Fig. 13.12 Fig. 13.13 Fig. 13.14 Fig. 13.15 Fig. 13.16 Fig. 13.17 Fig. 13.18 Fig. 13.19 Fig. 13.20 Fig. 13.21 Fig. 13.22 Fig. 13.23 Fig. 13.24 Fig. 14.1 Fig. 14.2 Fig. 14.3 Fig. 14.4

List of Figures

Hydrogen electron orbitals .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ionisation energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Level diagram (Grotrian diagram) of O VII (He-like oxygen) .. . . Collisional processes .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fluorescence yield for K- and L-shell transitions.. . . . . . . . . . . . . . . . . Galactic interstellar extintion crosssection .. . . .. . . . . . . . . . . . . . . . . . . . Ion fractions in the CIE and NEI case as a function of temperature/net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . G-ratios and R-ratios for O VII and Si XIII . . . .. . . . . . . . . . . . . . . . . . . . Fe-K line energies as a function of ionisation state .. . . . . . . . . . . . . . . Possible evidence for resonant line scattering in DEM L71.. . . . . . Total and inelastic cross sections for proton-proton collisions .. . . Schematic illustration on the built-up of a pion-decay γ -ray spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The pion-decay spectrum expected from cosmic-ray proton collisions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Simulations of an XRISM image and spectrum of Cas A. . . . . . . . . . Monte-Carlo simulations of a XIPE observation of Cassiopeia A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Artist impression of CTA . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Sensitivity curve of CTA . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

418 423 429 431 431 433 441 445 446 449 453 456 456 466 467 468 469

List of Tables

Table 2.1 Table 2.2 Table 3.1 Table 5.1 Table 5.2 Table 6.1 Table 6.2 Table 7.1 Table 8.1 Table 9.1 Table 10.1 Table 11.1 Table 12.1 Table 12.2 Table 12.3 Table 12.4 Table 13.1 Table 13.2 Table 13.3 Table 13.4

Consecutive core-burning stages of a 15 M star . . . . . . . . . . . . . . . . . γ -ray signatures of radio-active elements in SNRs. . . . . . . . . . . . . . . . Least square solutions to the  − D relation ... . . . . . . . . . . . . . . . . . . . Reverse shock/forward shock properties of the Chevalier (1982) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Mass loss parameters for main-sequence stars . . . . . . . . . . . . . . . . . . . . List of known magnetars, based on the McGill Magnetar Catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . List of Central Compact Objects (CCOs) and their properties.. . . Common dust grain compositions . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Emission lines in spectra of the Cygnus Loop .. . . . . . . . . . . . . . . . . . . . Properties of the “oxygen-rich” supernova remnants . . . . . . . . . . . . . Mixed-morphology remnants.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Powerful Galactic source classes and their total mechanical power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B-fields and cosmic-ray energies based on the minimum-energy principle . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Radio polarisation measurements of young SNRs . . . . . . . . . . . . . . . . Magnetic-field strengths based on X-ray synchrotron filament widths .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Supernova remnants detected in VHE γ -rays, and their properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ionisation energies of atoms up to nickel . . . . . .. . . . . . . . . . . . . . . . . . . . Atomic transitions of helium-like ions . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Atomic transitions of hydrogen-like ions . . . . . .. . . . . . . . . . . . . . . . . . . . Properties of some mesons . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13 26 42 100 107 160 167 173 203 241 270 291 331 337 342 372 425 443 444 450

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Chapter 1

Introduction

Supernova remnants and pulsar wind nebulae are not among the most studied objects in the Universe, but their beautiful appearances provide the most attention grabbing images, capturing the attention of the general public. This holds true for both radio, optical and X-ray images of supernova remnants. Many popular articles about spacerelated news have been enlivened by an image of the Crab Nebula, or filaments inside the Cygnus Loop (Fig. 1.1), or the display of cosmic fireworks in Cassiopeia A. Even if the articles themselves were not always specifically about those objects, an image of a supernova remnant often conveyed the message that space objects are beautiful and mysterious. However, as professional astrophysicists we study supernova remnants not for their beauty, but for the fascinating physics behind their aesthetic appearances. And when it comes to physics supernova remnants and pulsar wind nebulae have much to offer. From the wider astrophysical perspective, supernova remnants link the last stages of stellar evolution to the study of galaxy evolution: when we observe an supernova remnant we see chemical evolution and supernova feedback in action. In particular in young supernova remnants we can measure the composition and distribution of fresh products of explosive nucleosynthesis. This provides a direct link to studying supernovae, but supernova remnants have the advantage that we can study many of them in the Milky Way and obtain a three-dimensional image of the distribution of the elements, rather than having to analyse the lightcurves and spectra of distant objects. Supernova remnants are part of the study of the interstellar medium. The supernova remnant shocks interacting with molecular clouds results in interesting astrochemical processes, and maser formation. In addition, supernova remnants play a central role in the debate on the origin of interstellar dust. We know dust is formed in the adiabatically cooling ejecta of core-collapse supernovae, which is often taken as evidence that the dust observed in ultraluminous infrared galaxies (ULIRGs) finds its origin in supernovae. However, this is still a contentious issue. In supernova remnants dust can be studied in the far infrared as the freshly made dust is © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_1

1

2

1 Introduction

Fig. 1.1 The “Veil Nebula” (a part of the Cygnus Loop supernova remnant) as observed with the WFC3/UVIS instrument on board the Hubble Space Telescope. Red colours are based on a combination of infrared and Hα + [NII] line emission, green corresponds to [SII] line emission and blue corresponds to a combination of V-band and [OIII] line emission (Credit: NASA/ESA, and the Hubble Heritage Team (STScI/AURA))

collisionally heated by the hot supernova remnant plasma. But collisional processes are also responsible for destroying dust. The study of supernova remnants may, therefore, help us to understand whether supernovae are on average dust factories or dust destroyers. Finally, an important aspect of stellar feedback on the interstellar medium is the production of cosmic rays, energetic atomic nuclei, which span an particle energy from a few hundred of MeV up to 1020 eV. The energy density of cosmic rays in the interstellar medium of the Milky Way is about 1 eV cm−3 comparable to the thermal energy density of interstellar gas, the magnetic-field energy density and the stellar radiation energy density. It is generally thought that the origin of cosmic rays with energies up to 1015–1016 eV have been accelerated by supernova remnant shock waves. There is no doubt that supernova remnants do indeed accelerate particles to high energies: radio to X-ray synchrotron radiation prove that electrons are accelerated from energies of a few GeV up to 100 TeV, and γ -ray observations have now shown that also energetic atomic nuclei (hadronic cosmic rays) are present. Nevertheless, it is still not proven that supernova remnant shocks accelerate particles to energies as high as 1015 eV and whether they convert ∼10% of the supernova explosion energy to cosmic rays, which would be both required to link the cosmic rays detected near (or on) Earth with the population of supernova remnants in the Milky Way.

1 Introduction

3

This book offers a multiwavelength perspective on the physics and evolution of supernova remnants, in which all the aforementioned aspects of supernova remnant will be described, as well as more general physical aspects. These general physical aspects may be of interest to other field as well: Collisionless shock heating is important for small and large scales shock encountered by solar coronal mass ejections on sub AU scales, and in cluster of galaxies on Mpc scales. X-ray emission from thin optical plasmas are also important for emission from stellar coronae, and the hot tenuous gas in clusters of galaxies, but with a twist: in supernova remnants plasmas tend to be out of ionisation and temperature equilibrium, i.e. the electron and ion temperature are not necessarily equal, and the ionisation fractions may not match the ionisation fractions in collisional ionisation equilibrium. And the acceleration of electrons/positrons in the inner regions of pulsar wind nebulae are the best local examples of relativistic particle acceleration, which may have implications for the physics of acceleration by active galactic nuclei. The layout of this book is such that all chapters give a general overview of a specific aspect of supernova remnant research. Some chapters, like Chaps. 9 and 10 provide reviews of young and old supernova remnants, whereas other chapters provide basic information about the physics of supernova remnants, like the chapters on collisionless shocks (4), and infrared and optical emission from supernova remnants. An important aspect of present-day supernova remnant research is the connection of Galactic cosmic rays with supernova remnants, which makes supernova remnants also an important source class for astroparticle physics research. Given that cosmicray physics is a special branch of research, two chapters are devoted to it; one devoted to the general aspects of cosmic rays, i.e. their spectrum and composition, as well as the theory of diffusive shock acceleration (Chap. 11), and another chapter providing a review of the observational aspects of cosmic-ray acceleration by supernova remnants (Chap. 12). Finally, background information on radiation theory of interest to supernova remnants was treated in a separate chapter. Most of the theory is not specific for just supernova remnants, and of generic interest to high-energy astrophysics and astroparticle physics. On the other hand, some aspects of radiation theory are specific for supernova remnants, such as synchrotron cooling breaks, and nonequilbrium ionisation. This is one reason why radiation processes are treated in this book, rather than referring to more general books on radiation aspects, such as the classic textbooks by Malcolm Longair [736] and Rybicki and Lightman [997]. The hope is that this book will be a good source of background information for those interested in the various aspect of supernova remnant and pulsar wind nebula research, but will also appeal to researchers working in related fields such as supernova, interstellar medium research and cosmic-ray research, whereas the general physical aspects may be of interest to all researchers with an affinity for high-energy astrophysics.

Chapter 2

Supernovae

For a proper understanding of supernova remnants we first need to discuss their origins: the stellar explosions known as supernovae—see the picture of SN1994D in Fig. 2.1 for a beautiful example. The supernova rate in the Milky Way is about 2–3 supernovae per century [1093]. Many of the past supernovae were probably too obscured for naked eyes detection due to interstellar extinction in the Milky Way disc, leaving us with only a handful of historical supernovae, all of which occurred more than four centuries ago. The historical supernovae have been reported by Asian, Middle Eastern and European astronomers, and were identified by them as “new stars” or “guest stars”. Some of the “guest stars” that can be reliably associated with supernovae (and supernova remnants) are the guest stars of AD 185, AD 1006, AD 1054, AD 1572, and AD 1604. The latter two occurred at the dawn of the scientific revolution, with two famous astronomers taking a keen interested in them, both writing monographs on the topic: Tycho Brahe published “De Nova Stella” (“On the New Star”) in 1573, and Johannes Kepler published “De Stella Nova” in 1606 (see Fig. 2.2). The supernova remnants that are associated with these supernovae are now often referred to as Tycho’s supernova remnant and Kepler’s supernova remnant. Since 1604 more new stars (“novae”) emerged and vanished again, but it was only by the 1920s–1930s that astronomers started to realise that some novae were much brighter than others. In particular, the “nova” of 1885 in the Andromeda Nebula was bright. But the absolute magnitude of the event depended on the distance to the Andromeda Nebula (M31), and at the time there was no general agreement that the M31 was a distant stellar system (an “Island Universe”), similar to the Milky Way, or whether the nebula was a cloud of gas located within the Milky Way. This debate was finally settled in 1925 when Edwin Hubble discovered Cepheid stars in M31 and M33 [554], which confirmed that the nebulae were distant stellar systems. The implication of this discovery was that the nova of 1885 was indeed much brighter

This chapter contains text previously published in [1173]. © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_2

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Fig. 2.1 The Type Ia supernova SN 1994D as imaged by the Hubble Space Telescope (Credit: NASA/ESA, The Hubble Key Project Team and The High-Z Supernova Search Team)

than “normal” novae, as pointed out in a series of papers in 1934 by Walter Baade and Fritz Zwicky [109, 110], in which they coined the term “super-nova” and also famously suggested that “the super-nova process represents the transition of an ordinary star into a neutron star”. Since then the word nova exclusively refers to the less powerful thermonuclear bursts that occur when runaway fusion starts in the hydrogen- or helium-layers that have accumulated on the surface of white dwarfs, in binary system. Although the word supernova suggest a single type of phenomenon, supernovae come in two broad classes that have two very different progenitors and whose physical explosion mechanisms also differ completely [552]. These two classes are (1) core-collapse supernovae (not very precisely, they are sometimes referred to as Type II supernovae), and (2) thermonuclear or Type Ia supernovae (SN Ia). Broadly speaking, core-collapse supernovae are the explosions of massive stars, and their explosion energy comes from the gravitational collapse of the stellar cores

2 Supernovae

7

Fig. 2.2 Left: Title page of the book “De Stella Nova” (Prague, 1606) by Johannes Kepler about the new star of 1604. Right: Finding chart of the “new star”—indicated by “N”—in the “foot” of Ophiuchus (“Serpent-bearer” or “Serpentarius”) (Credit: Images by Bryce Roberts, from an original print at the Milton S. Eisenhower Library at Johns Hopkins University)

into a neutron star or black hole, whereas thermonuclear supernovae involve the explosions of carbon-oxygen (CO) white dwarfs, and the source of energy for the explosion is nuclear fusion of carbon and oxygen into more massive elements, predominantly 56 Ni. Despite these very different origins and explosion mechanisms both types of explosions provide about 1051 erg of explosion energy. This canonical supernovae energy has become so familiar in astronomy that it merits its own unit: it is often referred to as 1 foe (for: fifty-one erg) and more recently as 1 Bethe, in honour of Hans Bethe (1906–2005), the Nobel Laureate who is best known for his seminal work on nuclear fusion in the Sun and other stars, but who was also keenly interested in supernovae, neutron stars and black holes. Supernovae have always been an important class of transients sky objects, studied by many of the giants of twentieth century astronomy like Walter Baade, Fritz Zwicky and Rudolph Minkowski. But over the last two decades the study of supernovae has greatly expanded. One reason is that thermonuclear supernovae have become “standardisable” light candles, which with the expansion of the Universe can be followed down to redshifts z  2. Their use has led to the discovery that the Universe is accelerating its expansion due to a repulsive energy (“dark energy”) [910, 980], which could be caused by a non-zero cosmological constant () or a an unknown scalar field, dubbed “quintessence”. The principal investigators of the two teams that made this discovery (Brian Schmidt, Adam Riess, and Saul Perlmutter) shared the 2011 Nobel Prize in Physics. These projects, and their follow-

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ups, rekindled the interest in supernovae, and led to many surveys targeted at finding supernovae, and other optical transients. This quest for optical transients continues unabated, with monitoring programs like the Palomar Transient Factor (PTF) [942], Pann-Stars, and in the future (2022) the Large Synoptic Survey Telescope (renamed Vera C. Rubin Observatory) [584]. To illustrate the success in finding new supernovae, one can look at the naming convention for supernovae. A new supernova is identified with the acronym SN followed by the year of discovered, followed by a letter indicating the order of supernovae discovery. The famous SN 1987A was the first supernova discovered in 1987, and was discovered on February 23. After the rise in supernova surveys, one letter is no longer enough, and up to three letters are needed (and the convention is to use lower case letters). For example, the last supernova discovered in 2017 was SN2017jmj, which was the 7108th supernova of that year. This means that on average 19.5 supernovae were discovered per day in 2017. Note that some supernovae are also named after the telescope with which they were discovered. For example, the SN Ia SN2011fe, in the nearby galaxy M101, is also referred to as PTF 11kly. The proliferation in supernova detections have also led to discoveries of supernovae that are peculiar in one way or another; like extremely bright, or exactly the opposite. This book is not the place to discuss all the observed variations, and we limit ourselves here to the broad classes that cover the majority of the observed supernovae.

2.1 The Optical Classification of Supernovae 2.1.1 Spectroscopic Classification As stated above their are two broad classes in which supernovae are divided: core-collapse supernovae and thermonuclear supernovae, based on the explosion mechanism that causes them. However, the commonly used classification scheme is based on the observed variety in optical spectra and light curves. Most supernovae have optical spectra characterised by relatively broad absorption lines (few thousands km s−1 ) during the first months of their development. Typically the spectrum informs us about the conditions in the photosphere, which separates the outer, optically thin ejecta, from the optically thick ejecta. As the gas is expanding, more ejecta become optically thin, and the photosphere moves to deeper layers of the supernova, which have lower velocities. After a few months all the ejecta are optically thin, and the optical spectra are characterised by emission lines, rather than absorption lines. This optically thin phase is called the nebular phase. However, the spectroscopic classification is based on the early phase of the light curve development, roughly around maximum light, which occurs a few weeks after the explosion.

2.1 The Optical Classification of Supernovae

9

Fig. 2.3 Supernova classification scheme

The optical classification goes back to Rudolph Minkowski [828], who called supernovae with no hydrogen absorption lines in their spectra as Type I supernovae, and with hydrogen lines as Type II supernovae. He already noted that Type I supernovae are more homogeneous as a class, than Type II supernovae. The homogeneity of Type I is what ultimately led to their use as cosmological standard candles. The supernovae classification scheme has been many times revised, and not all schemes have passed the test of time. For example, at some point F. Zwicky recognised as many as five types (Type I–V). The current scheme, summarised in Fig. 2.3, still takes the original Type I and Type II distinction as its basis, but in the 1980s it was realised that Type I should be subdivided in Type Ia , Type Ib [362], which do not have Si II absorption lines in their spectra, but do show helium (He I) absorption lines, and Type Ic [1221], which lack both Si II and He II lines. Spectroscopically this subdivision makes sense, but it has become clear that Type Ib/c supernovae are not thermonuclear supernovae, but core-collapse supernovae. Their lack of hydrogen (and helium for Type Ic) indicates that their outer hydrogenrich envelope (and helium layer for SNe Ic) has been stripped away, either by a stellar wind or by interaction with a companion star. SNe Ic are of particular interest as long-duration gamma-ray bursts (GRBs) are associated with them. The earliest evidence for a connection between GRBs and SNe Ic was provided by the association of GRB 980425 with a peculiar, broadlined SN Ic supernova, SN1998bw [420]. The broad lines indicated velocities up to 30,000 km s−1 , and also its bolometric luminosity was larger than for normal

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Type Ic SNe. Later similar SNe have been discovered, some broad-line SN Ic are associated with other GRBs, but most are not. Given their observed properties, the explosion energies may be ten times higher than normal supernovae, and they are often referred to as hypernovae. (Never mind that the Greek word hyper is the equivalent of the Latin word super: hyper just sounds more super than super.) The close relationship between Type II and Ib/c supernovae was confirmed by supernovae that display spectra with hydrogen lines early on, but later evolve into spectra resembling Type Ib supernovae [391]. This is best explained by assuming a massive star that has lost most, but not quite all, of its hydrogen-rich envelope. This in-between type of supernovae are labeled Type IIb. An important example is the nearby SN 1993J [392]. Yet another spectroscopic subclass among core-collapse supernovae, are the Type IIn supernovae [1021]. Unlike normal Type II supernovae, which have broad absorption lines, supernovae IIn have narrow (200 km s−1 ) emission lines, most notably Hα. The likely explanation for the peculiar spectra is that the narrow lines comes from dense, circumstellar gas, caused by extensive mass loss of the progenitor. This gas is then photoionised by the supernova event itself. The differences between Type II, Ib, Ic, and IIn are mostly the result of differences in the mass-loss properties of the progenitors. The mass loss properties themselves are dependent on the main sequence mass of the progenitor (more massive stars will lose more mass through winds), metallicity, and binary interaction, in case the progenitors are part of binary system. All this variations concerns core-collapse supernovae, and makes that Type Ia supernovae stand apart, in that these are firmly associated with the aforementioned thermonuclear supernovae. As a result the name Type Ia supernova has become a synonym for thermonuclear supernova. Type Ia supernova spectra are distinguished from Type Ib/c by the appearance of silicon line in their spectra. As already noted by Minkowski, SNe Ia, are more homogeneous as a class than the various types of core-collapse supernovae. Nevertheless, as the number of detected SNe Ia has rapidly grown over the last two decades, it has become clear that there is also variety among SNe Ia. But the underlying mechanism for the variation among them is much less clear than for core-collapse supernovae. Examples of spectra of different supernova types are shown in Fig. 2.4.

2.1.2 Supernova Light Curve Classification Apart from spectroscopic differences, supernovae show a large variety in the evolution of the luminosity evolution, i.e. they have different light-curve shapes, as illustrated in Fig. 2.5). A common subdivision of basic Type II supernovae, is based on the differences in light-curve shapes, with SN IIP (for plateau) having a distinct plateau in their light curve after the peak brightness, whereas Type IIL (for linear) have a faster luminosity decline. The light curves of supernovae are largely determined by their explosion energies, providing the initial expansion, the amount of radio-active 56 Ni (and its

2.1 The Optical Classification of Supernovae

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Fig. 2.4 Left: Examples of spectra from various types of supernovae: a Type II (SN 2007aa [533]),Type Ib (SN2004gq [835]), Type Ic (SN 2004fe [835]) and a Type Ia supernova (SN 1994D). Note the P-Cygni profile of the Balmer lines of the type II supernova. Right: Nebular spectrum of SN1994D, taken about 3 month after maximum brightness. The most prominent line feature, around 4800 Å, is from Fe III. Other lines are from Fe II, Fe III, and Ca III (Data obtained from the CfA Supernova Archive)

Fig. 2.5 Examples of light curves of various types of supernovae. Note that there is some uncertainty about the absolute magnitude due uncertainties in the distance modulus and extinction, and there are uncertainties about the explosion dates (Data: SN1980K [223], SN1994D [231], SN2004et [758], SN2008ax [895])

daughter product 56 Co), which provides the most important source of energy for the light curves around peak luminosity and beyond, and the amount of supernova ejecta mass. The latter determines how fast the heat provided by radio-active elements reaches the photos-sphere. Roughly speaking, the more 56 Ni the higher the peak luminosity, and the more ejecta mass, the broader the width of the light-curve

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peak. The tail of the light curve is also powered by radio-active decay, and how easily the injected heat can escape the ejecta. On average SNe Ia have brighter peak luminosities than core-collapse supernovae, indicating a much larger mass of freshly synthesised 56 Ni. The fast decline of both Type Ia, Type IIL and Type IIb/Ib supernovae after the peak indicates relatively small ejecta masses. For Type IIL and Type Ib supernovae, this is an indication that pre-supernova mass loss was substantial. With the supernova classification scheme in mind, we will turn our attention now to the physical properties of both core-collapse and thermonuclear supernovae, focussing more on the explosion properties and what this means for the nucleosynthesis products that have an impact on the properties of supernova remnants.

2.1.3 The Supernova Rates Per Type Supernova surveys of the local Universe provide the means to investigate the supernova rate per unit galaxy mass. Often the results are expressed in SNu (supernova unit), which is the number of supernovae per 100 yr per 1010 LB, of galactic luminosity. It has been known for a long time that core-collapse supernovae require massive stars, which are usually absent in elliptical galaxies. So only SNe Ia can occur in elliptical galaxies, for which the supernova Ia rate is about 0.18 SNu [230]. For the spiral galaxies the supernova rate is about 0.7 SNu, of which also about 0.18 SNu is due to SNe Ia. In determining the division of supernovae in (sub)types care has to be taken to correct for observational biases, such as the different peak magnitudes of supernovae of different types. Figure 2.6 shows the fractions of different supernova types based on the volume-limited sample list in [722]. Shown are the overall divisions between

Fig. 2.6 The observed fractions of supernovae of different types, based on observations of nearby supernovae from the Lick Observatory Supernova Search [722]. The fractions are for the volumelimited sample. Left: overall division between SNe Ia, Ibc, and II. Right: the Type II subtypes

2.2 Core-Collapse Supernovae

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Type Ia, II and Ibc, and the subdivision of Type II supernovae. In [722] also subdivisions of Type Ibc and Ia are shown.

2.2 Core-Collapse Supernovae 2.2.1 Pre-explosion Composition Core-collapse supernovae mark the end of the lives of massive stars; that is, those stars with main sequence masses M  8 M , for a review see [1247]. Just prior to collapse the star consists of different layers with the products (“ashes”) of the different consecutive core- and shell-burning stages (see Table 2.1 for the time scales of the core burning stages). From the core to the outside one expects: – iron-group elements in the core (silicon-burning products), – silicon-group elements (oxygen-burning products), e.g. 16 O + 16 O →28 Si +4 He, – oxygen and magnesium (neon burning product), e.g. 20 Ne +γ →16 O +4 He, or 20 Ne +4 He →24 Mg +γ ), – neon and magnesium (carbon-burning products), e.g. 12 C +12 C →20 Ne +4 He), – carbon (a helium-burning product), produced through the triple-alpha reaction, – helium (a hydrogen-burning product), and, finally, an envelope of unprocessed hydrogen-rich material, if not removed by a stellar wind or binary interaction. The even elements from oxygen to calcium (i.e. O, Ne, Mg, Si, S, Ar, Ca) are often referred to as alpha-elements, as they are mostly built-up from integral units of 4 He (alpha) particles. The definition of alpha-element varies in usage, for example carbon is sometimes considered to be an alpha-element, and in optical spectroscopy titanium is considered an alpha-element, although stable titanium (the most abundant isotope is 48 Ti) is not itself an integral units of alphaparticles, but 48 Ti is the daughter product of 48 Cr, an unstable alpha-element.

Table 2.1 The consecutive core-burning stages of a star with a main-sequence mass of 15 M Stage Hydrogen Helium Carbon Neon Oxygen Silicon Core collapse

Time scale 11 Myr 2.0 Myr 2000 yr 0.7 yr 2.6 yr 18 d ∼1 s

Adapted from [1247]

Fuel H He C Ne O, Mg Si Fe, Ni, Cr, Ti

Product He C, O Ne, Mg O, Mg Si, S, Ar, Ca,. . . Fe, Ni, Cr, Ti, . . . Neutron star

Temperature (109 K) 0.035 0.18 0.81 1.6 1.9 3.3 >7.1

Density (g cm−3 ) 5.8 1390 2.8 × 105 1.2 × 107 8.8 × 106 4.8 × 107 >7.3 × 109

Luminosity (103 L ) 28 44 72 75 75 75 75

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Fig. 2.7 The average binding energy per nucleon as a function of the atomic mass number (A). The binding energy per nucleon peaks around iron and nickel. Nuclear fusion from elements lighter than iron will result in energy gains, but for elements heavier than nickel only nuclear fission will result in energy gains. Some interesting isotopes are listed, such as the relatively tightly bound 4 He, 12 C and 16 O isotopes. 56 Fe and 62 Ni are the most tightly bound isotopes

2.2.2 Neutron Star and Black Hole Formation The creation of the iron-group core, which lasts about a day, is the beginning of the end of the star, as no energy can be gained from nuclear fusion of iron (see Fig. 2.7). The core itself is supported by the degenerate pressure of the electrons, and as the core starts contracting iron integrates though reactions like 4 γ +56 26 Fe →132 He + 4n,

γ +42 He →2p + 2n. Another process is electron capture: once the core density is very large, the Fermi energy of the degenerate electrons is becoming high, and it becomes energetically favourable to form neutron-rich nuclei, or form neutrons by electron reactions with free protons. The capture of electrons deprives the core from degenerate electron pressure. Moreover, the reactions result in neutrino production, which results in energy losses to the core. So a rapid loss of pressure support leads to the collapse of the core on a free-fall time scale. The core collapses in most cases into proto-neutron star, but for the most massive stars the core may collapse into a black hole. The exact mass above which black holes are formed is not clear, and in fact may depend on the mass-loss history

2.2 Core-Collapse Supernovae

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of the star, which in turn depends on the metallicity [498, 866]. Generally it is assumed, but with some reservations, that stars more massive than 20–25 M produce black holes. A black hole can be formed directly (“direct collapse”) with all the stellar mass ending up in the black hole, in which case no supernova explosion will occur, a so-called “failed supernova”. There is some evidence, based on the mass distribution of stellar mass black holes, that most black holes are the result of direct collapse [666]. There is also observational evidence for this scenario in the form of the supernova-less disappearance of a ∼25 M star in the galaxy NGC 6946 [40]. An alternative scenario for black-hole formation is that the core of a massive star collapse into a relatively massive, rotationally supported, protoneutron star, which subsequently, after losing angular momentum or as more gas is accreted, will collapse into a black hole. In the latter case a very energetic supernova explosion may occur: a hypernova or potentially a gamma-ray burst. Note that energy injection from a newborn magnetar (see Chap. 6) is currently a more popular theory for the origin of long-period gamma-ray bursts [805, 991]. Given that the initial mass function is biased toward lower mass stars, the majority of core-collapse events will lead to neutron star formation. When the stellar core collapses, most of the gravitational energy is liberated (E ∼ GM 2 /Rns ∼ 1053 erg, with Rns the neutron star radius) in the form of neutrinos. This idea was brilliantly confirmed with the detection of neutrinos from SN1987A by the Kamiokande [539] and Irvine-Michigan-Brookhaven [169] water Cherenkov neutrino detectors.

2.2.3 The Explosion Mechanism The supernova explosion mechanism itself, which requires that 1051 erg of energy is deposited in the outer layers, is not well understood. The formation of a protoneutron star suddenly terminates the collapse, and drives a shock wave through the infalling material (the core bounce). However, most numerical simulations show that the shock wave stalls and is insufficiently powerful to eject the outer layers of the star. The most commonly used explanation for why an explosion occurs nevertheless, is that the shock wave will be re-energised by the absorption of a fraction of the neutrinos escaping the proto-neutron star, but most numerical models involving neutrino absorption are still unsuccessful in reproducing a supernova explosion [588]. Note, however, that the coupling of neutrinos with the dense matter in the interior regions of the star is complex, and computationally expensive, and therefore difficult fully incorporate into hydrodynamical simulations of core-collapse supernovae. Moreover, instabilities in the accretion flow onto the proto-neutrons can only adequately treated in three-dimensional simulations. These instabilities lead to density enhancements that may help to couple the neutrino flux more strongly to the the infalling material near the proto-neutron star [218]. One such hydrodynamical instability is the so called, non-spherically symmetric standing accretion shock instability (SASI) [184]. SASIs are also discussed a possible origin for the kick velocity of pulsars, and the origin of neutron star spin [182]. As the SASI leads

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to oscillations of the proto-neutron positions and fluctuations in dense matter of the inner material, the SASI will also give rise to gravitational waves that may be detected with a future generations of gravitational wave detectors, provided the supernova occurs within a distance of a few megaparsec. A Galactic core collapse could even be detected with the current generation of gravitational wave detectors, such as LIGO and VIRGO [452]. Given the difficulties to simulate core-collapse supernova explosions with shocks and neutrinos alone, there have been suggestions that neutrino deposition is not the most important ingredient for a successful explosion, but that amplification of the stellar magnetic field, due to differential rotation and compression, may lead to magneto-centrifugal jet formation, which drives the explosion [1222]. SASI and magneto-centrifugal models all predict deviations from spherical symmetry. In the magneto-rotational models one even expects a bipolar symmetry. Another reason why deviations from spherical symmetry has received more attention is that long duration gamma-ray bursts are associated with very energetic Type Ic supernovae (hypernovae). Given the nature of gamma-ray burst these explosions are likely jet driven. This raises the possibility that also more normal core collapse supernovae have jet components [1222]. There is indeed evidence, based on optical polarimetry, that core collapse supernovae, especially Type Ib/c, are aspherical [1196]. In Sect. 9.3.1 evidence for aspherical explosions is presented, based on supernova remnants.

2.2.4 Electron-Capture Supernovae of 8–10 M Stars Stars in the mass range of ≈8–10 M will not follow the evolution of the massive stars that is sketched above. It is thought that after carbon burning their cores consist of O, Ne and Mg, pressure supported by degenerate electron pressure, unlike more massive stars. These stars will not proceed with a contraction leading to a new core-burning cycle. Instead shell-burning brings the mass of the degenerate core dangerously close to the Chandrasekhar limit of 1.38 M . The increase in core density gives rise to an enhanced Fermi energy, which makes it energetically favourable to produce neutron-rich nuclei. So electron capture on 10 Ne (resulting in 20 F, followed by another electron capture to 20 O) and 24 Mg (leading to 24 Na and 24 Ne production) deprive the core of degenerate pressure, and a runaway process leads to the formation of a neutron star [830, 861]. The collapse to a neutron star results in a core bounce that drives a shock wave disrupting the rest of the star. Since the outer envelope will still contain hydrogen, the electron-capture supernova will be a Type II event. Unlike the situation for supernovae of more massive stars, the bounce itself may be sufficiently strong to explode the star. That is, electron-capture explosions may lead to “prompt supernova” explosions [535], although not all studies agree on this topic [658]. The explosion energies inferred from the models range from ∼ 1050 erg to 1051, so 10%–100% of the canonical supernova explosion energy. In particular, the amount

2.2 Core-Collapse Supernovae

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of ejected 56 Ni is very small compared to normal Type II explosions, giving rise to fainter supernovae [1195]. It is not completely clear whether the stars in the 8–10 M mass range will indeed lead to electron-captures and subsequent neutron-star formation, or whether the ONeMg core will create a thermonuclear runaway explosion (similar to Type Ia supernovae), partially disrupting the star, and leaving behind a low mass white dwarf [595]. Electron-capture supernovae are interesting for a number of reasons. Although the mass-range for which they occur is small, the steepness of the initial mass function of stars, means that a substantial fraction ( 30%, [1195]) of all corecollapse supernovae may be electron-capture supernovae. Since the explosions may be prompt and the exploding may have relatively low masses, the kick velocity of neutron stars created in these events may be small, and if the star is part of a binary, the binary is more likely to remain bound [924]. This may lead to a bias in neutron star masses found in binaries, as most of them may be the result of electron-capture supernovae. Finally, an electron-capture supernova origin for the historical SN 1054 has been repeatedly discussed in the literature, see for example [658, 867]. This supernova formed the Crab Nebula and its central pulsar. Based on the mass and composition of the ejecta observed in the optical, a progenitor mass of ∼ 10 M has been inferred for the Crab Nebula [289], putting it in the right mass range. Moreover, the explosion energy of SN 1054 was probably lower than for typical core-collapse supernovae, and more in agreement with some electron-capture supernova models [658].

2.2.5 Core-Collapse Supernova Ejecta Composition The ejecta of core-collapse supernovae consist primarily of stellar material, except for the innermost ejecta, which consist of explosive nucleosynthesis products, mostly Fe and Si-group elements. These elements are synthesised from protons, neutrons, and alpha-particles, which are the remains of the heavy elements that have disintegrated in the intense radiation field in the innermost regions surrounding to the collapsing core [93]. Some of the explosive nucleosynthesis products are radioactive, such as 56 Ni, and 44 Ti. In particular the energy generated by the decay of 56 Ni (τ = 8.8 days) into 56 Co (τ = 111.3 days), and finally 56 Fe, heats the ejecta, which leaves a major imprint on the evolution of the supernova light curve. The yields of these elements depend sensitively on the details of the explosion, such as the mass cut (the boundary between material that accretes onto the neutron star and material that is ejected), explosion energy, and explosion asymmetry. Since the mass of the neutron star/black hole, the location of energy deposition and the presence of asymmetries are not well constrained, the expected yields of these elements are uncertain, and vary substantially from one set of numerical models to another [262, 1109, 1246], see Fig. 2.8.

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Fig. 2.8 Left: Supernova yields for the most abundant X-ray emitting elements. The squares/black line indicates the mean yield for core collapse supernovae, whereas the circles indicate thermonuclear supernovae (the W7 deflagration model in red and the WDD2 delayed detonation model in magenta). The model yields were taken from Iwamoto et al. [585]. Right: Oxygen yield of core collapse supernovae as a function of main sequence mass. The circles and squares are the predictions of [1246], the triangles are predictions of [262], and the crosses of [1109] (Figure reproduced from [1173])

Overall the yields of core-collapse supernovae are dominated by carbon, oxygen, neon and magnesium, which are products of the various stellar burning phases (Table 2.1). The final amounts of alpha-elements is proportional to the initial mass of the progenitor (Fig. 2.8). In particular, the oxygen-ejecta mass is a good indicator for the initial mass of the progenitor. It is for this reason that a special class of supernova remnants, the oxygen-rich supernova remnants (Sect. 9.3.1) is considered to consist of remnants of the most massive stars ( 20 M ).

2.3 Thermonuclear (Type Ia) Supernovae The idea that Type I supernovae are explosions of white dwarfs dates back to the 1960s [92, 393, 490]. The two main reasons for arriving at that idea are the lack of hydrogen in the spectra of Type I supernovae and the fact that the explosions also occur in elliptical galaxies, which have an old population, and, therefore, lack massive stars. Carbon-oxygen (CO) white dwarfs are ideal thermonuclear bombs, if somehow the degenerate material in the core can be made dense enough (∼3 × 109 g cm−3 ) and hot enough (∼2 × 108 K). When this occurs, the heat produced by carbon and oxygen fusion is no longer offset by neutrino cooling and the fusion becomes a runaway process, leading to the total disruption of the white dwarf [863]. The degenerate state of the white dwarf is essential for this runaway process, as the thermonuclear energy production occurs at roughly fixed pressure, and the star will initially not expand and adiabatic cool as a result of the heat production. The question is then what causes the central density and temperature of the white dwarf to become that high. Since the 1970s it is assumed that SN Ia involve white dwarfs

2.3 Thermonuclear (Type Ia) Supernovae

19

accreting sufficient material from a companion star. As the mass of the white dwarf approaches the Chandrasekhar mass (1.38 M ), the central density will become sufficiently high to ignite CO [1223]. Indeed models of CO white dwarf explosions roughly reproduce the composition and light curves of Type Ia supernovae [863]. More direct evidence that a white dwarfs are the progenitors of SN Ia was obtained for SN Ia SN 2011fe in the nearby galaxy M101. The location of the supernova was observed before the supernova was detected, but after the explosion had occurred (as inferred from modelling the light curve). The lack of a detection of the supernova suggested that 4.5 h after the explosion the radius of the supernova must have been small, which can only be reconciled with the explosion of a compact star [185]. In order to explode a white dwarf, sufficient nuclear fuel (CO) needs to be burned in order to overcome the net binding energy of the white dwarf of about −5 × 1050 erg, which consists of the gravitational binding ∼ GM 2 /R minus the internal energy of the degenerate gas. Since the explosion energy of SN Ia are of the order of 1 − 1.5 × 1051 erg, about 1.5 − 2 × 1051 erg of energy as to be generated from explosive nuclear fusion, requiring ≈ 0.5 M of CO atoms to be burned into iron and nickel. Simulations of CO white dwarf explosions provides the following relation between explosion energy and the final mass in 56 Ni, stable iron and intermediate mass elements (Si, S, Ar, Ca) [1249]: E51 = 1.56MNi + 1.74MFe + 1.24MIME − 0.46,

(2.1)

with all masses in solar units and E51 the explosion energy in units of 1051 erg.

2.3.1 The Single Degenerate Versus Double Generate Channel There are a number of observational and theoretical problems with the view that SN Ia progenitors are CO white dwarfs accreting from a stellar companion. This companion can be a low-mass main-sequence star, or an evolved star (a red giant or asymptotic giant branch (AGB) star). This channel for producing a SN Ia is known as the single degenerate (SD) channel. From a theoretical point of view stable accretion onto a white dwarfs requires a rather narrow mass deposition rate within a factor of two of M˙ ≈ 10−7 M yr−1 [864], at which rate the accreted material will steadily burn. At lower accretion rates the accreted matter will start piling-up without burning, until a critical pressure has been built-up, and the accreted material will detonate, resulting in a classical nova. If the accreting white dwarf is not already close to the Chandrasekhar mass, more material will be ejected than has been accreted. So the white dwarf will not have a net mass gain over longer periods of time. For mass deposition rates above the stable nuclear burning regime, the accreted material may expand forming a red giant envelope with strong winds [864], or it

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may lead to an accretion induced collapse of the white dwarf into a neutron star, in a way that is similar to the formation of electron-capture supernovae [862]. The sensitivity of the fate of CO white dwarfs to the precise value of the accretion rate means that the evolutionary path of a white dwarf binary system to an SN Ia requires considerable fine tuning, in order for a white dwarf to grow from a 0.7– 1 M CO white dwarf to a 1.38 M white dwarf. A class of accreting white dwarfs that appear to accrete in this steady burning regime are so-called supersoft sources (SSSs), bright soft X-ray sources characterised by a surface temperature of kT ≈ 30 − 60 eV, and rather complex spectra (e.g. SSS Cal 83) [700]). Modelling of the spectra does indeed support the idea that SSSs could be Type Ia progenitors, as proposed in [719]. However, since stable burning requires quite some fine tuning of the accretion rate, many types of binary evolution scenarios are excluded, and results in a gross under prediction of the number of SN Ia explosions [268]. On the other hand, it has been suggested that nature has a way of regulating the accretion rate to the required stable rate by regulating the net-accretion rate through an accretion disc wind [479]. We come back to the idea that SSSs could be Type Ia progenitors in Sect. 9.2, with a not very positive outcome. There are several observational problems with the single degenerate channel. First of all, the stellar companion is likely to have a stellar wind, which should reveal itself in the form of absorption lines in the supernova spectra, or through the interaction of the supernova shock with the wind, which should result in radio synchrotron and thermal X-ray emission. Neither of these effects have been detected [266]. Moreover, the shock wave hitting the donor star should heat up its outer layers, giving rise to additional optical/UV emission [615]; for which there is generally no evidence [722], although recently an early component in the lightcurve was found that could be from the shocked stellar companion [318]. Secondly, a problem for the single degenerate channel is the lack of finding surviving donor stars inside supernova remnants of SN Ia. The donor stars of white dwarf binaries should be stars that are quite massive, ∼2 − 6 M [268], in order to be able to evolve in time and donate sufficient amount of gas to the white dwarf. Even if the donor is an evolved star, or has been stripped of its envelope by the supernova blast, a surviving donor should be bright enough to be detected inside supernova remnants for Milky-Way or Magellanic-Clouds remnants. No such surviving-donor stars have been positively identified inside SNR 0509-67.5 [1019], Tycho’s SNR (SN 1572) [640], Kepler’s SNR (SN 1604) [644], and SN 1006 [642]. Due to the observational and theoretical problems of the single degenerate channel scenario, another viable SN Ia channel has become increasingly popular: the double degenerate (DD) channel. In this scenario two white dwarfs, of which at least one is a CO white dwarf, merge, briefly creating a massive rapidly rotating white dwarf, which then explodes. Alternatively, the more massive white dwarf accretes from a lower-mass CO white dwarf [574, 1201]. The evolution up to the formation of the last white dwarf may have left the system in a very compact binary state, resulting in relatively short merging times [575], or a more detached binary may slowly lose angular momentum due to the emission of gravitational radiation. The latter would result in delay times of more than 109 yr, which would

2.3 Thermonuclear (Type Ia) Supernovae

21

be appropriate for the occurrence of SN Ia in elliptical galaxies. Clearly the DD channel result in SN Ia events that are devoid of extensive circumstellar material (with the exception of some tidal debris from just prior to the explosion), and the explosion leaves no surviving donor star, or a low-luminosity surviving white dwarf. Additional observational evidence for the DD channels comes from very bright SN Ia, with a high 56 Ni yield, and for which the total mass exceeds the Chandrasekhar limit [551, 1017]. The DD channel appears more consistent with most observational evidence, but it has not been unambiguously proven that the SD can be ruled out. For example, most SN Ia mass estimates are consistent with a Chandrasekhar mass explosion [790]. And for several individual SN Ia evidence for a SD channel origin has surfaced: there are clear absorption lines present in the spectrum of PTF 11kx [317] suggesting that the progenitor was a symbiotic nova system. Such a system consists of a white dwarf and an evolved non-degenerate star, in which the white dwarf accretes from the stellar wind rather than through Roche-lobe overflow. An equivalent of such a system is the RS Ophiuchi nova system, which is often mentioned as a system that may eventually produce a SN Ia, on account of its large white dwarf mass [1066]. Apart from PTF 11kx similar a progenitor was suggested for SN 2006X, which shows evidence for blueshifted Na I absorption lines, suggested to be caused by expanding nova shells [897]. Several other SN Ia show similar blueshifted absorption structures [1079]. Finally, in Sect. 9.2.3 we will discuss how the environment of Kepler’s SNR suggests that the progenitor system of SN 1604 points toward a binary system containing an AGB star. In reality the SN Ia population, despite its apparent homogeneity as a class, may originate from a mix of SD and DD progenitors [466]. The long delay time of SN Ia in elliptical galaxies would then imply a DD channel for these events, whereas the more prompt SN Ia channel associated with starforming galaxies [764] would then originate from SD channel systems, or or a mix of SD and DD progenitors. Note that even taking into account that SNe Ia have a mix of SD and DD progenitors, it is still difficult to match the SNe Ia rate using population synthesis codes that include binary evolution [268]. This shows that our knowledge of the evolution of binary systems toward SN Ia explosions is still incomplete.

2.3.2 Thermonuclear Explosions: Deflagration Versus Detonation A CO white dwarf near the Chandrasekhar limit has an equation of state that brings the star at the brink of self-destruction: the gas is supported by the degenerate pressure of the electrons, and the effective adiabatic index is close to γ = 4/3, which leads to large variations in the star’s radius and changes in the central density, as a function of pressure changes. Initially some central fusion of C and O may lead to expansion, moving lowering the density, and slowing down fusion reactions.

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At the same time some nuclear reactions, involving the production of a proton and 23 Na through carbon fusion, and subsequent electron capture of the protons, leads to a decrease of the electron density in the core, making the star even more unstable, as the degenerate pressure support is lowered. This process of neutronisation may continue for ∼1000 yr prior to explosion, a phase that is called the “simmering” phase [920]. Given the feeble conditions the white dwarf will be in at the onset of explosion, the trigger mechanism and location for the explosion is not clear: does it occur in the centre, or at a random point in white dwarf core, or are there perhaps multiple trigger points scattered throughout the core [1036]? In the earliest models for Type Ia explosions the trigger of the explosion was assumed to cause a thermonuclear fuelled detonation wave, which would unbind the whole white dwarf [92]. In a detonation wave the burning of material happens in the hot layer behind (downstream of) a supersonic wave. The burning increases the internal energy of the post-shock material, and drives the wave forward. However, it was found that pure detonation models do not fit the observations of SN Ia and have some theoretical problems, as well. Pure detonation models predict that most of the white dwarf material will be transformed into iron-group elements, whereas observations indicate the presence of substantial amounts of intermediate mass elements (Si, S, Ar, Ca) and unburnt carbon and oxygen. Moreover, in the core the energy output from fusion turns out to be even too low to start a detonation wave to begin with [859]. After this was realised, the standard model for explaining thermonuclear explosions became the deflagration model. One of the most widely applied deflagration models for SN Ia is the W7 model by Nomoto [863]. In deflagration models the nuclear burning takes place in a hot burning layer, in which new fuel is continuously mixed-in through convective motions. The deflagration front in the white dwarf is fast, but per definition slower than the local speed of sound, cs . In the W7 model the typical deflagration wave speed is ∼0.08cs. Deflagration models for SN Ia describe the observed nucleosynthesis yields quite well, as determined from the Galactic abundance pattern and observed SN Ia spectra. But some observational aspects cannot be explained well by deflagration models, such as the overproduction of 54 Fe in deflagration models, and the fact that the predicted line formation velocities spanned a much narrower range than observed. These problems can be alleviated by the delayed-detonation transition (DDT) model. In DDT models the explosion starts with a deflagration wave, but as the flame reaches a lower density region, the flame turns into a detonation wave [648]. Since the detonation is triggered later and at lower densities, most of the C/O is burned by the detonation wave into intermediate mass elements. Delayed detonation models have replaced deflagration models as the most popular thermonuclear explosion models for the single degenerate channel. It is, however, not entirely clear whether delayed-detonation models approach the explosion mechanism better. It is also possible that in reality both pure-deflagration and delayed-detonation explosions occur in nature [536]. The DDT models come with their own uncertainties: it is not a priori clear at what radius/density the deflagration switches to a denotation. The critical density is often treated as a free model parameter. Another uncertainty in Type Ia modelling, both pure deflagration

2.3 Thermonuclear (Type Ia) Supernovae

23

and DDT models, is the amount of mixing of the ejecta, which again is often parametrised (e.g. [1249]). Comparing the supernova yields of delayed-detonation and deflagration models with average core collapse supernova models, we see that both SN Ia models yield different nucleosynthesis patterns, but they are quite similar when compared to core-collapse supernovae (Fig. 2.8). Type Ia model predict much more mass in irongroup elements than core-collapse supernovae, and relatively similar amounts of intermediate mass elements. Carbon and oxygen is hardly produced in SNe Ia, and the ejected C and O consists mostly of unburnt outer layers of the CO of the progenitor star. Finally, there is yet another class of models, the double-detonation model [729, 860, 1245]. According to this model the white dwarf has accreted a layer of helium on its surface, which at some point ignites producing a detonation wave in the surface layer, but also a shock wave moving into the core of the white dwarf. This inward moving shock wave compresses the core material sufficiently to create a second denotation wave, disrupting the white dwarf. In this model the white dwarf does not necessarily need to be close to the Chandrasekhar mass-limit, hence the alternative name “subChandrasekhar” explosion model. The double-detonation model produces much more radio-active 44 Ti than other thermonuclear explosion models (see Sect. 2.4). Although the double-detonation model was originally conceived within the context of the single degenerate scenario, it has also been invoked for triggering explosions in the double degenerate scenario, in which case the CO white dwarfs are assumed to have a thin layer of helium, which ignites [880]. Currently the double-detonation model is not favoured, but it could potentially explain some subtypes of SNe Ia.

2.3.3 The Diversity Among Type Ia Supernova Although SNe Ia are as a class much more homogenous than core-collapse supernovae, there is still variation among them in maximum brightness, light-curve shape and spectra. That they can nevertheless be used as “standard candles”, making them cosmological yard sticks, we owe to the fact that a large part of the variation can be calibrated away with the so-called Phillips relation [915]. This is an empirical relation between the absolute magnitude of the SN Ia at peak luminosity, Mmax , and the subsequent decline in magnitude 15 days measured in the B-band, m15 (B). In an update to the original results, Phillips et al. [916] reported quadratic relations between the maximum brightness, decline rate and colours of M(B)max,1.1 =M(B)max − 0.786 [ m15 (B) − 1.1] + 0.633 [ m15 (B) − 1.1]2 ,

(2.2) M(V )max,1.1 =M(V )max − 0.672 [ m15 (B) − 1.1] + 0.633 [ m15 (B) − 1.1]2 ,

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M(I )max,1.1 =M(I )max − 0.422 [ m15 (B) − 1.1] + 0.633 [ m15 (B) − 1.1]2 , (Bmax,1.1 − Vmax )1.1 =(Bmax − Vmax ) − 0.114 [ m15 (B) − 1.1] ,

with the subscript 1.1 denoting the value for a standard SN Ia with m15 (B) = 1.1, the average value of this quantity. With these relations one can readjust the peak brightness to the value a SN Ia would have, if it would have been an average SN Ia at the same luminosity distance. Another method to address the variation in lightcurve shape, is applying a simple scaling to the time coordinate and magnitude of a standard SN Ia lightcurve, according to [444, 909]  MB (t) = MB,std

t − tmax (1 + z)s



+ 2.35(1 − s −1 ),

(2.3)

with s the so-called stretch factor, z the cosmological redshift, and MB,std(t) a template SN Ia magnitude evolution. The relation between s and m15 (B) was found to be m15 (B) = 1.7/s − 0.6 for B-band observations of nearby supernovae [909]. The physics behind the Phillips and stretch relation is not entirely clear, but one important variable is the mass of radio-active 56 Ni. However, it has been difficult to reproduce the Phillips relation completely from first principles [1249]. In the supernova community the subtyping of supernovae is often based on previous examples of similar supernovae. For example, very luminous SNe Ia are often called SN 1991T-like, after a particularly bright event. Similarly, a subclass of faint SNe Ia are called SN 1991bg-like supernovae. SN 1991T, SN 1993bg and SN1986G also are peculiar when it comes to their spectra [207], with SN 1991T showing evidence for less intermediate mass elements and more iron-group elements in their spectra, whereas SN 1993bg seems to have underproduced irongroup elements, consistent with their respective peak luminosities. More normal SNe Ia are often called “Branch-normal SNe Ia” [207]. There appears to be quite some variation in the velocity of the ejecta, and in the velocity gradients of SNe Ia, some of which are correlated with the brightness of the supernova, but even among Branch-normal SNe Ia there seems to be variation [148]. This variation can be intrinsic, and perhaps related to whether an explosion was a pure deflagration or a delayed-detonation explosion. But it is also possible that the differences are due to asymmetries in the explosion, in combination with our viewing geometry toward a particular supernova. Finally, there is a subclass of SNe Ia that follow a Phillips relation, but one that is off set in magnitude: they have lower luminosities for the same stretch factor/ m15(B), and their velocities are slower and their iron-group mass lower

2.3 Thermonuclear (Type Ia) Supernovae

25

than for normal SNe Ia [397]. Some SNe Iax show helium features in their spectra, suggesting that they may have accreted from a helium-rich companion. The prototype of this subclass is SN 2002cx, but they are usually referred to as supernova Type Iax. Their origin is not certain, but one idea is that they are SN Ia exploded by a double detonation, but with incomplete burning. Perhaps the white dwarf was not totally disrupted and a reduced white dwarf has been left behind. The relative fractions of Type Ia subtypes are displayed in Fig. 2.9, whereas Fig. 2.10 shows the variation in Type Ia light curves as compared to the historical light curve of SN 1604.

Fig. 2.9 The fractional occurrence of Type Ia subtypes, based on the same volume-limited supernova sample [722] as shown in Fig. 2.6

Fig. 2.10 Several modern light curves of SNe Ia compared to the historical light curve of SN 1604 (Kepler’s SNR) [1174]. All magnitude scales have been adjusted to a distance of 5 kpc. It shows the variation among SNe Ia. Note that the magnitudes of the very luminous SN 2009dc have been scaled down, and those of the faint SN 1991bg have been scaled up. (Reproduced from [1174])

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2.4 Detection of Radio-Active Elements from Supernovae As discussed in Sect. 2.1.2, the unstable element 56 Ni, and its daughter product 56 Co determines the peak luminosity and evolution of the tail of the supernova lightcurve. Table 2.2 shows the decay times and radioactive processes of three important radioactive elements produced during supernova explosions. Here e+ stands for beta-plus decay, and EC for electron capture. The decay of a radio-active element often leaves the daughter nucleus in an excited state, which results in hard X-ray or soft γ -ray line emission upon deexcitation. The energy released by radio-active decay is initially injected in the form of γ -ray line emission, and the kinetic energy of positrons (typically of order an MeV), in the case of beta-plus decay. Additional energy is provided by the annihilation of the positrons with electrons, releasing 2× 511 keV photon energy, either by two individual photons, or through three photon emission if first a bound electron-positron pair is formed; constituting an exotic atom called positronium. Initially, the ejecta material is opaque to γ -rays, so the radioactive energy injection is distributed over the ejecta. For the late phase of Type Ia supernovae there is some debate whether all the positrons eventually will annihilate, or whether a small fraction escapes this fate, and will eventually contribute to the faint 511 keV glow of the Galactic plane [827]. Type Ia supernovae produce much more 56 Ni (0.5 M ) than core-collapse supernovae (∼0.01–0.2 M ), with the exception of rare hypernovae [486]. The radio-active element 44 Ti has a decay time of 86 yr (corresponding to a half life of 60 yr). The production in supernovae is typically of the order of 10−5 –10−4 M for core-collapse supernovae, whereas thermonuclear supernovae are likely not producing 44 Ti, or only under certain explosion scenarios (double detonation explosions, Sect. 2.3.2). The amount of mass is small compared to 56 Ni, but due to its much longer decay time, 44 Ti will heat the ejecta for decades after the explosion. Up to 20 years after the explosion, the faint infrared and optical glow of the freely expanding ejecta in SN1987A was the result of energy injection from 44 Ti decay [702]. The production of 56 Ni is mostly the result of fusion under nuclear statistical equilibrium (NSE) situations, in which case the hot, disintegrated core material of

Table 2.2 Decay chains and most important γ -ray signatures of shortlived radioactive products from explosive supernova nucleosynthesis 56 Ni

57 Ni

44 Ti

56 Co

→

56 Fe

Decay time 8.8 d 111.3 d

57 Co

→

57 Fe

52 h 390 d

EC EC

1370 122

44 Sc

→

44 Ca

86.0 yr 5.7 h

EC e+ , EC (1%)

67.9, 78.4 1157

→

→

→

Process EC EC, e+ (19%)

Lines (keV) 158, 812 847, 1238

2.5 Light Echoes

27

the star, which is rich in 4 He, protons and neutrons, quickly builds up again massive elements, mostly alpha-rich elements, with 56 Ni as the most likely element to be formed. A relatively large production of 44 Ti occurs, if the NSE built-up of massive elements is halted, due to rapid expansion of the ejecta, leaving a surplus of 4 He. This process is called alpha-rich freeze out, and is more likely to occur in bipolar explosions [851]. The determination of the amount of 56 Ni and 44 Ti is often determined indirectly, based on modelling of the light curve. A more direct way for identifying these radioactive elements is to detect their associated γ -ray line emission. The first opportunity arose with the SN1987A explosion. Indeed, the 56 Co line emission at 847 and 1238 keV was detected by a number of balloon experiments [276, 759, 1004, 1102], and the γ -ray spectrometer (GRS) ) on board the Solar Maximum Mission (SMM) satellite [712, 783]. The observed fluxes were typically in the range (0.5 –1)×10−3ph cm−2 s−1 , indicating that only a few percent of the total ∼0.075 M of 56 Co was exposed. The detection of 122 keV line emission from 57 Co by the Oriented Scintillation Spectrometer Experiment (OSSE) on board CGRO was reported in 1992 [690]. The detection of these γ -ray lines was surprising as they were observed within 160 days after explosions, whereas models indicated that the outer layers of the envelope were still opaque to γ -rays. The implications were that about 5% of the radio-active material must have been mixed at high velocities into the outer layers of the ejecta. A more recent detection of radioactive elements in supernovae was provided by the nearby SN Ia SN2014J in M82 (3.5 Mpc). INTEGRAL-SPI detected the 56 Ni lines only 20 days after the explosion [316]. Later observations also revealed 56 Co line emission [267, 579]. The line emission of 44 Ti could in principle be detected several hundred years after the explosion for Galactic supernova remnants. So far only for two objects has the associated line emission been reliably detected: SN1987A [190] and Cas A [463, 586, 954, 1050, 1166]. See Chap. 9.

2.5 Light Echoes An important aspect of the study of supernova remnants is the reconstruction of the type of supernova explosion from which the remnant originates. In this chapter we are concerned with the supernova phenomena itself, but there is one type of observation that directly links supernova remnants with supernovae: light echoes. A light echo, as the name suggest, is caused by the reflection of light on some scattering objects, which then reaches the observer with a delay compared to the directly received light. The light echoes discussed here are caused by interstellar dust grains, and the delay times may be as long hundreds of years. Light echoes form a relatively new tool to determine the supernova origin of a supernova remnant. The first light echo observed from a supernova was the optical light echo from SN 1987A, detected a little bit more than a year after the explosion

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[281]. The light echo was used to reconstruct the circumstellar dust surrounding SN 1987A, which appeared to have an unusual asymmetric, sheet-like, morphology [282]. Light echoes from supernovae had been predicted [255, 1149]. And a light echo was in fact observed of a nova, more than a century ago: from the 1901 Nova Persei outburst [395, 611, 979]. Nevertheless, it was a surprise when an analysis of optical observations of the Large Magellanic Cloud (LMC) made from 2001 to 2004 to study microlensing, revealed multiple light-echo rings [956]. Among them around SN 1987A, but also light echoes centred on the young supernova remnants 051969.0, 0509-67.5, and 0509-68.7 (N103B), all having ages  1000 yr. A subsequent search for light echoes from Galactic supernovae revealed multiple light echoes scattered throughout the Milky Way, which, using their apparent motion in the sky, could be traced back to the positions of SN 1572 (Tycho’s SNR) and Cassiopeia A [957]; see Fig. 2.11. No light echo could be associated with the supernova remnant 3C58, which has been claimed to be the remnant of the historical supernova of AD 1181[270, 1078]. This is one reason to doubt this remnantsupernova association. The light echoes reveal themselves by subtracting two well-calibrated images in the same filter band, taken several months to years apart (Fig. 2.11). Even more spectacular is that one can obtain a spectrum of the reflected light, and correction for colour-dependent reflection and extinction properties, reconstruct the supernova spectrum. Note that the spectrum is smeared by the effect of the extent of the scattering layers, so the spectrum contains light from a time integrated part of the spectrum, but dominated by the spectrum near maximum light. The first light-echo spectrum was that of SNR 0509-67.5, which revealed itself to be a SN Ia spectrum, which best resembled the bright, SN 1991T-like subtype [958]. The light echoes of SN 1572 showed that this supernova was a normal Type Ia supernova [683],

Fig. 2.11 Left set: Images of a light echo (traced back to SN 1572): the top images show the same field separated by about 14 month. The bottom images shows the subtraction of the two images, with in the left image the direction from where the light echo comes. Right: Wide field infrared image (IRAS) with in green the fields in which light echoes were searched for. The blue line are the average directions from which light echoes appear to be coming. They turn out to center on the locations of SN 1572 (Tycho’s SNR) and Cassiopeia A. None are tracing back to 3C58, the disputed remnant of SN 1181 (Credit: Armin Rest, reproduced from [957])

2.5 Light Echoes

29

confirming the SN Ia identification based on the historical light curve and the Xray determined ejecta abundances (see Chap. 9). The light echoes associated with the Cas A supernova showed that the spectrum very much resembled SN 1993J, a Type Ib core collapse supernova [682, 957]; see Fig. 2.12. Spectra from various light echoes associated with Cas A, all of which trace back to different viewing angles toward the supernova, showed that fora certain viewing angles the He I and Hα P-Cygni profiles were blueshifted by as much as 4000 km s−1 . The light echoes, therefore, confirm the longheld view that the asymmetries of the Cassiopeia A supernova remnant can be traced back to an asymmetric explosion. The geometry of light echoes is explained in Fig. 2.13. It illustrates that all light echoes detected at a time t1 = t0 + t, with t0 the light travel time for the directly observed light, traces an ellipsoid with the transient (supernova) in one of the focal points and the observer in the other focal point. If the distance between Earth and the supernova (AB) is d = ct, with t the light travel time, and a and b are the semi-major and minor semi-axis respectively, one canshow with a simple geometric calculation that a = 12 c(t + t), but also that a = ( 12 ct)2 + b 2 . From these two expressions one can deduce that the ratio between the minor and major semi-axis is  b = a

2 t t + 1+

 t 2

t t

t

.

(2.4)

√ For t  t we see that the ratio can be approximated as b/a ≈ 2 t/t. As an example, Cas A is at a distance of 3.4 kpc (11,000 light year) [952], and the age of the supernova remnant (= t) is ≈340 yr [1114], which translates into b/a ≈ 0.25, a rather elongated ellipse. For the LMC the ellipse is even more elongated. It is, therefore, common to approximate the rays from the scattering cloud and from the supernova as traveling in parallel lines. This simplifies the ellipsoidal geometry to a paraboloid [255, 277]. In this simplification most of the scattering takes place relatively close to the supernova, which is probably a good approximation for the dust scattering echoes for the LMC supernovae, as most of the dust will be situated in the LMC itself. Light echoes studies are ideal to link the properties of supernovae to the properties of supernova remnants. Unfortunately, there are only a handful of supernova remnant/light-echo associations. For older supernova remnants the light echoes are probably too faint to observe. But there is a glimmer of hope that at some time also light echoes associated with SN 1604, SN 1006 and, perhaps even, SN 185 will be identified. For SN 1604 (Kepler’s SNR), and also for SN 1006, the problem is not so much their age, but their high Galactic latitude (and hence Galactic height), where there are few dust clouds necessary for scattering the light.

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Fig. 2.12 The light echo spectrum of the Cassiopeia A supernova compared to spectra of recent core collapse supernovae of various types [959]. The spectra have been scaled to correct for the effects of dust scattering. Clearly SN IIb, in particular SN 1993J provide the best match (Credit: Armin Rest, reproduced from [959])

2.5 Light Echoes

31

2b A

t+Δt

(SN)

2a

t t+Δt

B

(Earth)

Fig. 2.13 The geometry of a light echo

Chapter 3

Classification and Population

3.1 Morphological Classification of Supernova Remnants Ideally one would like to classify supernova remnants according to the type of explosion that caused them, but except for young supernova remnants (Chap. 9) this is not always possible. Moreover, like many fields in astronomy, the study of supernova remnants carries a history that brought about its own nomenclature. The result is that supernova remnants are traditionally classified using three types, based on the remnant’s morphology: (1) shell-type supernova remnant, (2) plerion— sometimes called filled-centre supernova remnant–and (3) composite supernova remnants. Since the late 1990s a fourth class has been added [971]: (4) mixedmorphology supernova remnant—sometimes called thermal composite supernova remnant. Figure 3.1 shows examples of each class. A shell-type supernova remnant has roughly a ring-like morphology, caused by the two-dimensional projection of spherical shell. Shell-type supernova remnants are the most common supernova remnant type. Well-known examples are Cassiopeia A, Tycho’s SNR (SN 1572), and the Cygnus Loop. The plerion and composite supernova remnant class are both classes in which the supernova remnant’s morphology has been influenced by the presence of a pulsar wind nebula, a synchrotron nebula created by the relativistic electronpositron wind from a central pulsar (Chap. 6). The difference between plerions and composite supernova remnants is that the radio and X-ray morphology of plerions is completely dominated by the pulsar wind nebulae, i.e. no shell is visible, or barely visible. In a sense one could just as well call a plerion a pulsar wind nebula, but the subtle difference is that plerions still cary some information about their supernova origin. For example, the Crab Nebula, which is the archetypal plerion, has optical and infrared filaments which consist of supernova ejecta. Moreover, around pulsar wind nebulae that are more than ∼100,000 yr old the supernova remnants has seized to exist. So a plerion may be an (almost) bare pulsar wind nebula, a pulsar wind nebula is not necessarily a plerion. © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_3

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Fig. 3.1 The supernova remnant morphological classification illustrated with examples. From top left to bottom right: (a) The Cygnus Loop, a shell-type supernova remnant (3◦ diameter), observed by the ROSAT PSPC instrument [718] (red ∼0.1–0.4 keV; green ∼0.5–1.2 keV; blue ∼1.2–2.2 keV). (b) 3C58, a plerion/pulsar wind nebula, as observed by Chandra [1057]. The long axis of this object is ∼7 . (c) The composite supernova remnant Kes 75 observed by Chandra [510], the pulsar wind nebula in the centre is powered by PSR J1846-025. The partial shell has a radius of ∼1.4 . Red: 1–1.7 keV; green: 1.7–2.5 keV; blue 2.5–5 keV. (d) The mixed-morphology supernova remnant supernova remnant W28 as observed in X-rays by the ROSAT PSPC (blue) and in radio by the VLA [342] (Image credit: Chandra press office, http://chandra.harvard.edu/photo/ 2008/w28/more.html)

The name “plerion” was coined in 1978 [1205] and derived from the Greek word pleres, which means “full”. Of the ∼300 Galactic supernova remnants listed in the Green catalogue [459], only nine are labeled as “filled-centre”, i.e. plerion: G6.1+1.2, G20.0-0.2, G27.8 +0.6, G63.7 +1.1, G65.7 +1.2 (DA 495), G74.9 +1.2 (CTB 87), 3C58, the Crab Nebula, G328.4 +0.2/MSH 11-57, but for the latter the plerion status is debated [592].

3.1 Morphological Classification of Supernova Remnants

35

Not all of these supernova remnants have been well-studied, and it is more likely that a plerionic supernova remnant will be reclassified as composite (if a faint shell is revealed, or the source is becomes better resolved), than the other way around. Composite supernova remnants are the second most common class, and it includes well-studied objects like G21.5 -0.9, Kes 79, the Vela SNR, MSH 1552/RCW 89 in the Galaxy, and SNR 0540-69.3 in the Large Magellanic Cloud. SNR 0540-69.3 was once called a Crab twin, but clearly shows a shell in both Xrays and radio [208]. Its pulsar energy-loss rate is, however, comparable to the Crab pulsar. Mixed-morphology supernova remnants were recognised as a separate class relatively late, once it became clear that several supernova remnants that have a shell-type morphology in the radio, have a filled-centre morphology in X-rays [971]. The interior X-ray emission is, however, caused by thermal X-ray emission, and, therefore, does not originate from a pulsar wind. There are about 24 mixedmorphology supernova remnants, but the class is not marked in Green’s catalogue. To make things even more complicated, some mixed-morphology supernova remnants do also harbour a pulsar wind nebula, albeit not a prominent one. Nevertheless, they could then also considered a composite supernova remnants. Well known examples of mixed-morphology supernova remnants are W28, W44, and IC443. All mixed-morphology supernova remnants are older remnants. In Chap. 10 we will discuss this class in more detail. Apart from the above four classes, one sometimes encounters also other morphological designations. One of them is barrel-shaped supernova remnants [414, 647], which are shell-typed supernova remnants, but the shell is only bright on two opposite sides, suggesting a cylindrical (barrel) shape, rather than a spherical morphology. Examples mentioned in [647] are G296.5+10 and SN 1006. What causes this shape is not entirely clear, but likely the large-scale magnetic field orientation in the local medium is involved, which can influence the particle acceleration properties at the shock. But it can also be that for perpendicular magnetic fields (i.e. the field perpendicular to the shock normal) the shocks compresses the magnetic field by a factor 4, whereas for parallel magnetic field the magnetic field is not compressed. This leads to a contrast in radio-synchrotron emissivity between shells with compressed perpendicular and parallel magnetic fields. This effect is also discussed in Chap. 12). Another idea is that the magnetic field may has a dynamical role, inhibiting the expansion in one direction more than another direction. Or finally, the morphology may reflect the wind bubble shape created by the progenitor star (note that planetary nebulae, created by less massive stars also have sometimes barrel-like morphologies). Of course, any of these explanations may apply to specific cases. For SN 1006 it has been argued that its shape reflects the magnetic field orientation in combination with particle acceleration efficiency, and its shape is probably more shield-like in three dimensions than barrel-shaped [989] (see also Chap. 12). For G296.5+10 the shape cannot be the result of an enhanced synchrotron emissivity due magnetic-field compression and orientation of the field: its morphology is similar in X-rays, which is caused by thermal emission, rather than synchrotron emission.

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We have already used here the terms “young supernova remnant” and “older remnants”, designations that are not very precise and are also not used consistently in the literature. Apart from young and old, the same remnant is sometimes labeled “mature” [311], or “middle-aged” [128], and the difference may just reflect a difference in taste. In Chap. 5 we discuss the evolution of supernova remnants, which gives perhaps some physical basis for calling a remnant young, mature, or old. Roughly speaking a young supernova remnant is younger than 1000–2000 yr, shows X-ray emission from ejecta, and has shock velocities  1000 km s−1 . Slightly older supernova remnants, somewhat in the what we will define in Chap. 5 as the adiabatic phase, one could call the supernova remnant “mature”. Once the shock velocity drops below 200 km s−1 , supernova remnants become bright in the optical—radiative shocks; see Sect. 4.4. This seems a nice physically motivated transition age for a supernova remnant to be called “middle-aged”. And toward the end of the life of a supernova remnants, the shock velocity will become comparable to the turbulent velocities in the interstellar medium, and the supernova remnant will slowly disappear, or merge with other shells, blending into a large HII region. The stage before this disappearance seems a good phase to call a supernova remnant “old”. The above is some attempt at defining the terms “young”, “middle-age”, “mature” and “old”, but it remains subjective in that many supernova remnants contain a mix of properties. For example, the Cygnus Loop consists of a mix of radiative and non-radiative shocks, like a fit middle-aged man or woman, but with gray hair.

3.2 The Galactic Supernova Remnant Population The study of extragalactic supernova rates, and dissecting it by galaxy type and galaxy mass, can be used to infer an estimate of the supernova rate in the Milky Way, which is 2.84±0.6 per century [722, 1093]. But this has some large systematic uncertainties, which includes something as basic as the actual luminosity and mass of the Milky Way. If we assume that a supernova remnant remains visible for τvis ≈ 105 yr, we may expect that the Milky Way harbours about 2000–3500 supernova remnants. This is in stark contrast to the number of 295 known supernova remnants in the Milky Way as listed in the Green Catalogue [459] (2017 version), or the 383 supernova remnants listed in SNRcat [376]. This suggests that there are over 1600 Galactic supernova remnants yet to be discovered. However, this number is quite uncertain, not only because of the systematic uncertainties in the supernova rate, but also because the average visibility time of supernova remnants is uncertain. By modelling of the population of radio detected supernova remnants in the Local Group, in particular the nearby face on spiral galaxy M33 [450], it was found that 0.33 tvis = 20–80 kyr, and varies with the local hydrogen density as tvis ∝ nH [1009]. Note that this definition of tvis may specific to the observing characteristics of M33, which is seen under an inclination of 54◦ . And visibility is not identical to the life time of supernova remnants.

3.2 The Galactic Supernova Remnant Population

37

In the Milky Way the visibility time will also fluctuate with local density, but the detectability of supernova remnants also depends on how crowded the observed field is, and even whether there are any nearby bright radio sources, contaminating the map. In particular the bright radio sources Cygnus A, Cas A and the Tau A (the Crab Nebula), are notorious for messing up radio maps of neighbouring regions. The detectability also depends on the contrast between the supernova remnant’s radio surface brightness (flux density per unit solid angle) and the local sky background. This in particular, disfavours the detection of extended supernova remnants that evolve in low density regions. Given the inherent uncertainties in life time and supernova rates, the Galactic supernova remnant population can be as low as 1000, and as high as 4000. In either case, the current number of known supernova remnants is likely to underestimate the true number by at least a factor of 2.5, if not ten.

3.2.1 Finding and Naming Supernova Remnants Most known Galactic supernova remnants have been first discovered with radio telescopes, for a variety of reasons. Firstly, there have been many wide-field surveys of Milky Way with radio telescopes [215, 423, 1225], and radio surveys have been undertaken since the late 1950s [149, 1218]. In fact, some of the commonly known names of supernova remnants come from these surveys. For example, W28, W44, and W51 refers to the catalogue number in the Dwingeloo radio telescope survey described by Westerhout [1218]; names with MSH (like MSH 14-63) refer to the catalogues by Mills, Slee, and Hill [822–824], and supernova remnants with CTA and CTB indications derive their names from list A and B of the CalTech list of radio sources [494, 1237]. A second reason most supernova remnants were first identified in the radio is that their entire life supernova remnants are identifiable radio sources. In contrast, in the optical most young supernova remnants are not bright, whereas older supernova remnants tend to be bright in the optical (Chap. 8). Finally, unlike in the optical and X-ray bands, there is little to no radio extinction of Galactic sources above 100 MHz. Some supernova remnants were long known as optically bright nebulae, such as the Crab Nebula (Messier 1), the Veil Nebula (Cygnus Loop), and IC 443 (Jellyfish Nebula). These nebulae were only recognised as supernova remnants after it became clear that they were in fact strong radio sources, as well [219]. And in case you are wondering where the designation IC comes from: it stands for the “Index Catalogue of Nebulae found in the years 1888 to 1894” by Dreyer [335]. This was an addition to the New General Catalogue (NGC), which did alreadl list several parts of the Cygnus Loop and the Crab Nebula (NGC 1952). Also the designation RCW (RCW 86, RCW 89, RCW 103) comes from an optical catalogue: “A catalogue of Hα nebulae” by Rodgers, Campbell and Whiteoak (1960) [982]. Of course all Galactic supernova remnants have also a name indicating their position in Galactic coordinates (l, b), like G184.6-5.8 for the Crab Nebula, located near the Galactic anti-centre.

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A problem for identifying new supernova remnants in radio surveys is that their shell-type morphology is very similar to those of HII regions. They can be distinguished by their different spectral indices, with supernova remnants emitting synchrotron radiation characterised by spectral indices α ≈ 0.5 (Sects. 12.1 and 13.3), defined here as flux density scaling as S ∝ ν −α , whereas HII region emit free-free emission which has α ≈ 0. Historically, many surveys were carried out at a single frequency, and α could not be immediately determined. Nowadays, with the new generation low-frequency radio telescopes like LOFAR [1160] or MWA [1119] multiple frequencies can be observed simultaneously, making it easier to immediately identify supernova remnants (e.g. [336]). Moreover, due to their steep spectral index, supernova remnants stand out more above HII regions at low frequency [215]. Discovering new supernova remnants is not the exclusive domain of radio or optical telescopes. For example, two well-known supernova remnants, RX J0852.0-4622 (Vela Jr), and RX J1713.7-3946 were discovered in X-rays with the ROSAT all-sky survey [99, 913]. These two supernova remnants are rather extended and have a low surface brightness in the radio, whereas in X-rays they are relatively bright, but peculiar in the sense that their X-ray emission is caused by synchrotron radiation (Sect. 12.2) rather than thermal X-ray emission. More recently several new supernova remnants have even been discovered at the very end of the electromagnetic spectrum, the very-high energy γ -ray domain (Sect. 12.3). These new supernova remnants are named after the γ -ray observatory H.E.S.S. with which they were discovered: HESS J1534-571, HESS J1614-518 [473], and HESS J1731347 [515].

3.2.2 Measuring Distances to Supernova Remnants A major problem for determining the physical properties of supernova remnants is the uncertainty in their distances, which are needed to convert angular sizes into physical sizes and proper motions into velocities, but it is also important for determining the three-dimensional spatial distribution in the Galaxy. The most reliable distance estimates are based on measuring both radial velocities and proper motions. For Cas A this has been done for the system of optically emitting fast moving ejecta “knots”, assuming that the optical knots are distributed in a spherically symmetric, expanding shell [209, 952, 1150]. The assumption implies that the highest proper motions measured at the edge of the projected shell correspond roughly with the highest radial velocities (either blue- or red-shifted), which should be located more toward the projected centre of the shell. The best distance estimate for Cas A is 3.4 ± 0.3 kpc [952], which is based on allowing the system of knots to not be completely spherical symmetric, but with knots on the back-side having higher velocities than on the front side. A similar geometrical estimate is based on measuring the broad-line component of Balmer-dominated shocks (Sect. 8.2). These are fast shocks, expanding in

3.2 The Galactic Supernova Remnant Population

39

partially neutral gas. As the neutral hydrogen enters the shocked region, narrow, hydrogen line emission may arise from direct excitation, but charge-exchange between shock-heated protons and neutral atoms also gives rise to broad-line emission, whose width provides a direct measurement of the post-shock proton temperature (Sect. 8.2), which is directly related to the shock velocity. If for the ˙ can be measured, one can estimate same shock region also the proper motion () ˙ the distance according to  = Vs /d, with Vs and d the shock velocity and distance respectively (using proper physical units). This method can only be used for young supernova remnants expanding into partial neutral gas. For SN 1006 [1239] this method provided a distance estimate of d = 1.85 ± 0.25 kpc [946]. The same method gives for Kepler’s SNR d = 5.1 ± 0.8 kpc [1006], 2–2.8 kpc for Tycho’s SNR [657], and 2.5 kpc for RCW 86 [504]. The caveat of this method is that the relation between proton temperature and shock velocity depends on the amount of equipartition between electrons and ions immediately downstream of the shock (see Sect. 4.3.5). In addition, efficient cosmic-ray acceleration will lower the post-shock temperatures for a given shock velocity (Sect. 11.3). Another method to measure distances is to identify stars that are located within the supernova remnant, by detecting absorption lines caused by the hot gas within the shell. Stars that are located inside the shell will only show blueshifted line absorption in their spectra, whereas background stars will show blueshifted and redshifted absorption lines associated with both sides of the shell. Foreground stars will show no line absorption. The distance to the supernova remnant corresponds then to the (average) distance of stars inside the shell. This methods works in particular well for nearby, mature supernova remnants: they have large angular sizes, and hence have many bright stars that can be targeted, and their shell plasma is relatively cool (5000–50,000 K) giving rise to optical/UV line absorption. The method has been used to measure the distances to the Vela supernova remnant (250 ± 30 pc) [244] and the Cygnus Loop (1.0 ± 0.2 kpc) [387]. There are other, more indirect, methods available for estimating distances: • HI absorption measurements in the radio (21 cm line) and linking the radial velocity measurements of the intervening neutral gas to the Galactic rotation model (e.g. [269, 792, 940]); • or related: linking the radial velocity of radiative Balmer line emission to the Galactic rotation model (e.g. [986]), or other types of line emission associated with the remnant, such as CO molecular lines [940]; • for historical Type Ia remnants with reasonably good brightness historical lightcurve measurements and compare them to those of well-calibrated extragalactic supernovae; • use the interstellar X-ray absorption as an indicator for distance [1080] • using the empirical relation between radio surface brightness () and supernova remnant diameter (D), the so-called  − D relation (e.g. [239, 269, 933]); • and, if nothing else can be used, try to establish an association of the supernova remnant with a certain spiral arm, starforming/HII region, or molecular cloud complex for which distance estimates exist.

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Technically speaking HI absorption measurements only provide a lower limit to the distance. Another problem is that in certain regions of the sky (toward the Galactic Centre for example) Galactic rotation does not result in radial velocities, in other regions the solution is not unique. Moreover, intervening gas clouds may have strong deviations from standard Galactic rotation models. The use of historical light curves, although limited to very few supernova remnants, has become more interesting now that the error on the Hubble constant has narrowed down considerably [406], and since our empirical understanding of Type Ia light curves improved drastically (Sect. 2.3). Indeed, for Kepler’s SNR (SN 1604) the distance estimate of d ≈ 5 kpc based on the historical light curve [993, 1174] agrees very well with most recent measurement based on the optical proper motion of Balmer-dominated shocks. In the past, this distance method was used the other way around: it was hoped that measuring distances to historical supernova remnants by other means could be used to constrain the maximum brightness of extragalactic SNe Ia, and thus constrain the Hubble constant [297]. Finally, the  − D has a long history as a tool to measure distances, but it is also very controversial. We will discuss it in more detail below.

3.2.3 The -D Relation The radio surface brightness, , is defined as the ratio between the radio flux density Sν (ν) and the solid angle  = A/d 2 , with A ≈ 14 πD 2 the projected physical surface area of the roughly spherical source, with D the source diameter. Since both the flux density and  scale inversely proportional to the square of the distance,  is a distance-independent quantity (in the absence of absorption, or cosmological redshifts): ν ≡

Lν (ν)/(4πd 2 ) Sν Lν ≈ 1 = 2 2. 2 /d 2  π D πD 4

(3.1)

In the 1960s it was found that older supernova remnants were in general less luminous than young supernova remnants [1048]. Moreover, for one remnant it was even measured that its luminosity was decreasing: the young supernova remnant Cassiopeia A fades with a rate of nearly 1%/yr [113, 544], as discussed in more detail in Sect. 12.1. To quantify this decline for the population of supernova remnants the relation between  and D proofed to be more reliable than Lν versus D, as the latter two quantities both require reliable distance measurements—Lν even depends on d 2 . The  − D relation suffers much less from this problem, as only D relies on the distance estimate, and in a linear way [933]. Indeed a diagram of D versus  shows a clear correlation between the two quantities of the form  = AD −β , but with quite some scatter [239, 825, 933]. See Fig. 3.2. In fact, the first  − D diagram [933] showed less scatter, probably

3.2 The Galactic Supernova Remnant Population

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Fig. 3.2 The relation between diameter and radio surface brightness ( − D) at 1 GHz for supernova remnants in the Galaxy (black points) [459], LMC (red) and SMC (blue) supernova remnants (compiled in [117], and converted to 1 GHz). The dotted line shows the best fit to all supernova remnants, with minimising the scatter in D. The solid line is the same, but now with minimising the scatter in 

because at the time only brighter supernova remnants were known (25 in total), with relatively large values for . At the time the logarithmic slope of the relation was found to be β = 2.67. The  − D diagram requires a reasonably good estimate for the distance. However, once there is a sample of supernova remnants with distance estimates, one can use it to estimate the distances to other supernova remnants: from  you can estimate D, and comparing it to the angular diameter,  gives you an estimate: d = D/. This method for measuring distances is, however, controversial. For example, in [458] the following objections to the use of the  − D relation are raised: 1. the correlation appears stronger to the eye than statistically warranted, since  ∝ D −2 ; so  and D are not statistically independent quantities; 2. there is a larger scatter in the correlation, with deviations of the order of a magnitude in D, and two orders of magnitude in ; 3. for the use of the  − D relation for measuring distances one should minimise the scatter in D, whereas often the relation is derived minimising the scatter in ; the two methods give very different values for β (as shown in Fig. 3.2); 4. the evolution of the luminosity of supernova remnants is dependent on the explosion properties and local environments; in particular supernova remnants in low density environments and wind bubbles may have intrinsically low  independent of age.

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Regarding point 1 it should be noted that the error in D (for establishing the relation) is dominated by the error in the distance estimate, and not so much by the error in the angular diameter . If the error in D was largely due to an error in  than point 1) is much more valid, as indeed both D and  are derived from it. The error in the primary distance estimate is quite large and is probably responsible for a large part of the scatter in the relation, together with the intrinsic differences in luminosity evolution of supernova remnants. There is some error in D probably of the order of 10%, largely caused by the fact that supernova remnants are rarely perfectly spherical, and  has some scatter, as well, because the radio luminosity distribution within the supernova remnants show some variation. The error in the distance can be largely ignored for supernova remnants in the Large and Small Magellanic Clouds (LMC and SMC), which are also depicted in Fig. 3.2. The  − D relations for the LMC and SMC shows largely the same trends as the Galactic one. However, inspecting the individual least square fit solutions (Table 3.1) shows that, when fitting  as a function of D, the LMC and SMC values for β are smaller (less correlation), which could be indeed related to the fact that for the LMC and SMC supernova remnants the distances are known. But there may be some selection effects as well. Another factor to be aware of is that in the SMC, and to a lesser extent in the LMC, the average interstellar medium density is smaller than in the Milky Way. One would expect the SMC surface brightness, therefore, to be biased toward lower values. However, there is no obvious trend in the  distribution between the three galaxies. At best the SMC supernova remnants seem to have even a slightly higher average of  (Fig. 3.3). Table 3.1 illustrates the point raised in [458] that it can make quite a difference whether the relation is fitted minimising  or D. For Galactic supernova remnants the preference should be for minimising the error in D, as D is more uncertain than  due to the large errors in the distance. For the Galactic supernova remnants minimising D results in a shallower relation than minimising . For the Magellanic Clouds it is the other way around, the distances are well known, and all supernova remnants are roughly at the same distance. So for the SMC and LMC the errors are probably dominated by errors in . So here the solutions in column 4 and 5 are preferred. Taking this into account suggest that for the supernova remnant populations taken together we find β = 1.9 ± 0.4. Table 3.1 Logarithmic least square solutions to the  − D relation,  = AD −β , with  in cgs units and D in pc Sample

Minimising error in D β A × 1012

Galactic shells (36)

2.45 ± 0.40

LMC (49)

2.67 ± 0.25

SMC (21)

3.04 ± 0.67

Combined (106)

3.36 ± 0.28

0.027+3.1E+05 −34.6 0.087+25.1 −0.086 0.20+8.7E+04 −0.20 0.71+82.8 −0.71

Minimising error in  β A × 1012 4.68 ± 0.77 1.87 ± 0.18 1.58 ± 0.35 1.98 ± 0.16

34.57+0.0733 −0.0195

0.0051+0.0047 −0.0024 0.0013+0.0031 −0.0009 0.0064+0.0049 −0.0028

3.3 The Spatial Distribution of Known Galactic Supernova Remnants

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Fig. 3.3 Histogram of the surface density distribution () at 1 GHz for Galactic (black) [459], LMC (red) and SMC (blue) [117] supernova remnants

3.3 The Spatial Distribution of Known Galactic Supernova Remnants Figure 3.4 shows the distribution of supernova remnants in Galactic coordinates. Not surprisingly for a distribution that follows largely the star formation in the Galaxy (i.e. for core-collapse supernovae), the galactic latitude distribution is very much confined to the Galactic plane, with a dispersion in b of 2.5◦ , but with some notable outliers. These outliers can be nearby supernova remnants, for which a large value of |b| can still correspond with a relatively small value for physical distance to the Galactic plane, but some supernova remnants are situated indeed high above the Galactic plane. Examples are SN 1006, which is situated at z ≈ 470 pc (d = 1.85 kpc), and Kepler’s SNR at z ≈ 590 pc (d = 5 kpc). Both are formed by Type Ia explosions, which should not necessarily occur close to the Galactic plane. For Kepler’s SNR there is even evidence that the progenitor had a velocity of >200 km s−1 away from the Galactic plane [134]. As for the Galactic longitude, the distribution peaks toward the Galactic Centre, as to be expected, but the histogram in l shows quite some gaps outside Galactic Centre region, which probably reflects our incomplete knowledge of the supernova remnant population. For comparison Fig. 3.4 shows the emission of radio-active 26 Al, a tracer of recent star formation. This emission has a peak at l ≈ 90◦ , the Cygnus region, whereas no peak is seen in the supernova remnant distribution. However, one should not over interpret the correlation (or lack thereof) between 26 Al and the supernova remnant distribution, since the 26 Al emission distribution

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Fig. 3.4 The longitude and latitude distribution of Galactic supernova remnants, as well as histograms in l and b, based on Green’s catalogue [459]. The binned longitude histogram is compared to the 1.8 MeV 26 Al line emission for |b| < 15◦ [665], to the peak of the supernova remnant distribution

scales with distance as 1/d 2 , biasing it to nearby emission, whereas the supernova remnant distance is reflected in a more complicated way in the distribution in longitude l. With some reserve we can also estimate the location of supernova remnants in the Galactic disk, using the  − D relation, as shown in Fig. 3.5. The distances are too poorly constrained to really find a correlation with the locations of the spiral arms. However, the figure does suggest that the currently known population is heavily skewed toward the nearby supernova remnant population (i.e. Y > 0 in Fig. 3.5); another indication that our knowledge of the Galactic supernova remnants population is very incomplete. Another striking feature is that at the far-side of the Milky Way there is a bias toward X > 0, which may reflect some observational biases (for example, there happened to be more survey covering that longitude part of the Milky Way), or may reflect large overestimates in the distances in that particular direction. In [460] a model for the Galactic longitude distribution was fitted using an underlying two dimensional surface density distribution of the form dN ∝ dA



R R

a

  R − R , exp −b R

(3.2)

3.3 The Spatial Distribution of Known Galactic Supernova Remnants

45

m

ar

s)

u gn

arm tum -Scu Crux rm aa rm o N

Carin

gita a-Sa

riu

s arm

Pe

rs eu

s

(C

ar m

y

er

t Ou

Fig. 3.5 The distribution of supernova remnants in the Galactic disk. The green dots show the positions of supernova remnants with reasonably well estimated distances, whereas the blue dots are based distances estimated with the  −D relation for Galactic supernova remnants (minimising error in D, Table 3.1). The location of spiral arms are based on the solution for a Galactic centre distance of 8 kpc, taken from [549]. The yellow dot shows the position of the Sun

with R the Galactic Centre distance (R = 8.5 kpc was adopted). In order to avoid biases only the 65 brightest supernova remnants were used, assuming that all supernova remnants above a certain surface brightness limit have been detected. Note the inclusion of the power-law here forces the density to be zero at the Galactic Centre, which is not the choice made in other studies [239]. Moreover, a recently discovered magnetar near the central black hole clearly indicates a non-zero value for the supernova rate at the Galactic Centre [837]. The best fit values are a = 1.09 and b = 3.87, which suggest that the supernova remnant surface density peaks around 2 kpc from the Galactic Centre [460], whereas a competing model, using different assumptions, produces a peak in the surface density around 5 kpc [239].

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3.4 The Supernova-Remnant Population in the Magellanic Clouds Studying the supernova remnant population in the Magellanic Clouds has many advantages over studying the Galactic population: we know their distances, 50 kpc for the LMC [917] and 62 kpc for the SMC [291], and the LMC has a disk-like structure observed almost face on (i ≈ 35◦ [1156]). Moreover, for both galaxies the optical and X-ray extinction is relatively low given the low hydrogen column toward the Magellanic Clouds of NH ≈ 1021 cm−2 . The larger distances than for Galactic supernova remnants allows us to resolve less physical details, but the Magellanic Clouds still allow sufficient details to be resolved for high spatial resolution telescopes: 1 corresponds to 0.24 pc at 50 kpc. As a result of these characteristics of the LMC and SMC supernova remnant population, our knowledge of the populations is less plagued by systematic effects than for the Milky Way population. But keep in mind that these are also populations with intrinsic properties that are bound to differ from the Galactic population: the Magellanic Clouds have smaller total stellar masses—2.7 × 109 M for the LMC [1156] and 1.8 × 109 M for the SMC [1075], compared to 6 × 1010 M for the Milky Way [723]. Also the starformation rates are different. The LMC currently has a star formation rate per unit mass that is higher than that of the Milky Way by a factor ≈ 2.7 [495, 723], with a peak in the 30 Doradus region (e.g. [1023]). In addition, the Magellanic Clouds have much lower metallicities than the Milky Way; the LMC has an abundance of ≈0.2–0.4 times solar (e.g. [559, 757, 996]), and for the SMC this is ≈0.1–0.3 (e.g. [996, 1151]). The study of the supernova remnant population of the Magellanic Clouds started in the 1960s when radio observations showed that some optical emission-line nebulae,—catalogued by Henize [514] as N 49, N 63A and N 132D—were nonthermal radio sources, and, therefore, very likely supernova remnants [774, 1220]. Subsequent radio, optical, and X-ray [733, 1038] surveys rapidly increased the list of known supernova remnants to 27 in the LMC and 11 in the SMC by 1984 [777]. Since then the list has grown even more, largely due to the “Magellanic Cloud Emission-line survey” (MCELS) [1063], radio surveys [390], and extensive X-ray observations [757, 1228]. As result, the list of known supernova remnants is currently 59 confirmed and 15 candidate supernova remnants in the LMC [204] and 23 confirmed candidates in the SMC [117]. Quite a large fraction of these have been identified as either core-collapse supernova remnants or Type Ia supernova remnants (see also Chap. 9). Figure 3.6 shows that many supernova remnants are found in the vicinity of Hα emitting regions, which are a tracer of star formation. For the supernova remnants identified as core-collapse supernova remnants (yellow circles) the association with Hα emitting regions is particularly strong. Several supernova remnants are associated with the brightest Hα region, the 30 Doradus region (also known as Tarantula Nebula). These include SN 1987A and the composite supernova remnant N157B, hosting one of the most energetic pulsars known.

3.4 The Supernova-Remnant Population in the Magellanic Clouds

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Fig. 3.6 Top: The positions of supernova remnants in the Large Magellanic Cloud as shown on a color map based on the Southern Hα Sky Survey Atlas (SHASSA) [426] (red channel), and an

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3 Classification and Population

In contrast, supernova remnants identified as Type Ia supernova remnants (green circles) appear to be mostly associated with LMC bar, a structure that dominates the visual image of the LMC. A reconstruction of the star-formation history of the LMC, based on stellar photometry, shows that most of the stars in the bar were formed during three starforming episodes, 5 Gyr, 500 Myr, and 100 Myr ago [495]. This implies that the stellar progenitors of the Type Ia supernovae were likely creating during these episodes. In an in depth study of the local starforming history around individual, well-typed supernova remnants the core-collapse supernova remnants are indeed all associated with recent star formation [119]. Surprisingly the progenitor of N49, which hosts a magnetar, was likely to have zeroage main sequence mass of less than 21.5 M , given the ages of nearby stars. This contradicts the idea that the magnetars have very massive progenitors (>40 M , see also Chap. 6.5). For the Type Ia supernova remnants the study shows that SNR N103B is associated with a local stellar population that had two relatively recent starformation peaks—50 Myr and 100 Myr ago — suggesting perhaps that the evolution of the progenitor system from binary star to Type Ia explosion was relatively rapid. In contrast, SNR 0509-67.5 (the most Northern Type Ia SNR in the figure) is found in a region outside the bar, containing mostly old stars, with an average age of 7.9 Gyr, and the most recent star formation episode around 500 Myr ago. This finding is surprising, as the light echo spectrum (Sect. 2.5) associated with SNR 0509-67.5 [958] show the supernova to have been a very bright Type Ia (SN 1991Tlike, Sect. 2.3). This seems to contradict the idea that SN1991T-like supernovae are associated with younger stellar populations. However, a caveat of the method is that progenitor systems may have migrated from their location of origin. Clearly this is the case for some Galactic supernova remnants, as previously discussed for the high Galactic latitude supernova remnants SN 1006 and Kepler’s SNR. Given the stellar mass of the LMC and SMC, and the star formation rates, one expects that the supernova rate in the LMC and SMC to be about 12% of that of the Milky Way. This translates into supernova rates of 0.2–0.4 per century for the LMC, and 0.1–0.3 for the SMC. If we assume that supernova remnants remain visible for about 50,000 yr, we expect 100 to 200 supernova remnants in the LMC and 50– 150 supernova remnants for the SMC. For both galaxies this a factor 1.5 to 3 or so larger than the number of identified supernova remnants, but the discrepancy seems much less than for the Milky Way. This suggest that the LMC and SMC supernova

Fig. 3.6 (continued) R-band image obtained with the Parking Lot camera [201] (green and blue channels). Supernova remnants that are regarded as Type Ia remnants are marked with green circles, and core collapse supernova remnants with yellow circles [757]. The blue circles are of unknown origin. Bottom: The positions of supernova remnants in the Small Magellanic Cloud (green circles) shown on an Hα image (SHASSA, red channel), and R-band (green channel) and B-band (blue channel) images from the Digital Sky Survey 2

3.4 The Supernova-Remnant Population in the Magellanic Clouds

49

remnants samples are more complete than for the Milky Way. However, the 50,000 yr life time of supernova remnants is an educated guess, and in reality probably this number depends on the local environment, varying from 20,000 yr to as much as 100,000 yr. For the LMC one can also use the number of light echoes (Sect. 2.5) and the associated supernova remnants a very rough estimate of the supernova rate. Four light echoes have been identified, associated with SN1987A, SNR 0509-67.5, SNR 0519-69, and N103B, all of which are less than 1000 yr old. This implies a supernova rate for the LMC of 0.2–0.6 per century. Remarkably, three out four of these are Type Ia supernova remnants, suggesting perhaps that the LMC has a higher fraction of Type Ia supernovae than the commonly adopted 15–20% (Sect. 2.1.3). Of course, we are dealing here with small number statistics, so some caution is necessary. On the other hand, also the abundance pattern of the LMC (and SMC) shows a relatively larger contribution from Type Ia supernovae than for the Milky Way [1131]. This could suggest that there is something intrinsic to the LMC stellar population that makes it more prone to produce Type Ia explosions, as compared to the Milky Way. Alternatively, an increase in the current Type Ia rate may be related to the star formation episodes that formed the bar, and our current epoch may be around the right time for the Type Ia supernovae to occur, from progenitors formed in the bar, 100 or 500 Myr ago. We have already discussed the  − D relation of the Magellanic Clouds supernova remnants, but an important clue to the evolution of supernova remnants is also their diameter distribution [776] shown in Fig. 3.7 and studied in detail in

Fig. 3.7 The cumulative distribution of Magellanic Cloud supernova remnants. After the work reported in [117]. The black dotted line indicates a linear relation between, whereas the cyan dotted line indicates dN/dD ∝ D 3/2 (arbitrary scaled)

50

3 Classification and Population

[117]. We will come back to the evolution of the supernova remnant sizes in Chap. 5 (Sect. 5.4.4), but note here that between ∼20 and 60 pc the diameter distribution is remarkably linear, whereas naively one would expect dN/dD ∝ D 3/2 [117].

3.5 Supernova Remnant Populations in Other Galaxies The Magellanic Clouds offer us nearly ideal conditions to study the supernova remnant population of galaxies as a whole: nearby enough to resolve details and at low column densities. However, the study of the supernova remnant population in various other galaxies are needed to asses in how far the supernova remnant population differs among various galaxies, and how the variation depends on starformation rate, the interstellar medium’s gas density, its metallicity, and what the connection is between the spatial supernova remnant distribution and the structure of the galaxy. Of course, individual supernova remnants cannot be investigated with the same amount of detail as for individual Galactic supernova remnants, or with the same level of completeness as for the Magellanic Clouds. Studies of extragalactic supernova remnants have concentrated on nearby, local group, galaxies such as M33 (Triangulum, d = 0.84 Mpc) and M31 (Andromeda Nebula, d = 0.79 Mpc), up to galaxies as far as 7 Mpc (M101, d = 7 Mpc). At 1 Mpc 1 corresponds to 4.8 pc, similar to the diameter of the young Galactic supernova remnant Cas A. So for extragalactic supernova remnants the spatial extent of young supernova remnants cannot be easily resolved, whereas for mature supernova remnants one can obtain some spatial information in the optical and Xrays. In the radio spatially resolving young supernova remnants requires (very) long baseline interferometry. For example, the population of compact supernova remnants in the starburst galaxy M82 (d = 3.5 Mpc) has been imaged with a beam size of 50 mas (corresponding to 0.8 pc) [374, 850], identifying 55 supernova remnants, with shock velocities in the range 2000–11,000 km s−1 , as measured for ten supernova remnants, and ages of 250 yr [374]. These and similar studies also show that the  − D relation is similar to that of the Milky Way and Magellanic Clouds. The nature of these studies with long baseline interferometry is confined to compact, i.e. young, supernova remnants. Most studies of extragalactic supernova remnants, however, are based on optical studies of galaxies, using spectroscopy and/or narrow-line imaging, typically targeting Hα, [S II], [N II], and [O III] line emission. See [714] for a review. According to [734], of the >1400 extragalactic supernova remnants identified so far, 95% have been detected in the optical and only about 10% in the radio and 15% in X-rays (the detections are not mutually exclusive). Since the forbidden line emission is associated with radiative shocks (see Sect. 4.4), the optical studies are biased toward mature supernova remnants, or in some rare cases young supernova remnants with clumpy ejecta, similar to Cas A (Sect. 8.1.2), such as the very young oxygen-rich supernova remnant in NGC

3.5 Supernova Remnant Populations in Other Galaxies

51

4449 [818]. Given the limitations of optical searches, which—apart from the fact that they are biased to later phases of supernova remnant evolution—also includes surface brightness limits and optical extinction, the extragalactic supernova remnant populations are not complete. For example, and extensive supernova remnant study targeting several galaxies estimate a completeness of about 25% [779]. Nevertheless, the samples are large enough to show that the supernova remnants have a spatial distribution similar to HII regions, and a clear connection to the spiral arms [779]. This is illustrated for the face-on spiral galaxy M101 in Fig. 3.8. In order to distinguish mature supernova remnants from H II regions, which are both bright in Hα and both emit forbidden line emission, supernova remnant studies typically use the flux-ratio of [S II] over Hα, as it is empirically established that I ([SII])/I (Hα) > 0.4 for supernova [172, 1065]. This relation is, however, poorly understood from a theoretically point of view [325]. The use of the [SII]/Hα

Fig. 3.8 Location of supernova remnants in M101, based on [779]. The optical image is based on the Sloan Digital Sky Survey [7], using r-,g-, and u-band filters. As quantified in [779] the supernova remnants follow largely the spiral arms

2

[Ar II] 7135 [Fe II] 7155

4

[Ca II] 7291 [O II]+[Ca II] 7325 [Ni II] 7378

(a)

[O I] 6363

6

[O I] 6300

8

He I 7065

3 Classification and Population [N II] 6548 Hα 6563 [N II] 6548 He I 6678 [S II] 6717 [S II] 6731

52

Flux (x10-15 erg cm-2 s-1Å-1)

0 0.8 0.6

(b)

0.4 0.2 0 0.8 0.6

(c)

0.4 0.2 0 0.3

(d)

0.2 0.1 0 6250

6500

6750

7000

7250

7500

Wavelength (Å)

Fig. 3.9 Spectral characteristics of supernova remnants and HII regions. Left: The top three spectra are from supernova remnants in M33, whereas the bottom spectrum is of an HII region. (Figure reproduced from [1065].) Right: The line intensity ratio of [S II]/Hα versus the ratio of [N II]/Hα for supernova remnants in NGC 300 (circles) and NGC 7793 (triangles). Black symbols denote supernova remnant candidates and red symbols HII regions. Data taken from [172]

ratio as a diagnostic tool is illustrated in Fig. 3.9. It shows both spectra of HII regions and supernova remnants, and a scatter diagram of I ([SII])/I (Hα) versus I ([NII])/I (Hα). Although remnants tend to have higher ratios for both [S II] and [N II] versus Hα, only for [S II] is there a clear dichotomy between supernova remnants and HII regions. The [SII] identification is not totally fail proof. For the example shown, there is some overlap in the line ratios between HII regions and supernova remnants for some galaxies [172]. Moreover, flux ratio measurements based on imaging with narrow line filters are not as accurate as measurements based on spectroscopy [715]. Also superbubbles can have large [S II]/Hα ratios, but superbubbles are distinguished by their sizes, which are larger than can be reasonably assumed for supernova remnants (100 pc) [401]. The [N II]/Hα line ratio is useful as a proxy for the metallicity of the local interstellar medium with which supernova remnants interact. It is used as a tool to measure metallicity gradients in galaxies. Finally, the line ratio [S II](λ = 6716 Å)/[S II](λ6731 Å) is used as a density diagnostic, and [O III] emission scales with the shock velocity [70]. In X-rays the best studied extragalactic supernova remnant populations are those of the nearby galaxies M33 and M31 [1012], with M33 offering the best opportunities to have an unbiased sample, as this relatively small spiral galaxy

3.5 Supernova Remnant Populations in Other Galaxies

53

is seen nearly face on, and with a small foreground absorption. The distance of M33, 0.78 Mpc, allows still for resolving young supernova remnants with the Chandra X-ray observatory and to perform X-ray spectroscopy for a few sufficiently bright sources. M33’s angular size of a less than a degree makes it an appealing target for surveys with the Chandra [735, 922] and XMM-Newton [425], requiring at most seven overlapping pointings to cover most of the galaxy. Through these surveys 105 supernova remnants have been detected in X-rays [425], as compared to 54 in the radio [450] and 217 in the optical [709], with the overlap between these differently identified samples shown in a Venn diagram in Fig. 3.10. This figure also show the X-ray properties of the supernova remnants in a hardness diagram, and for comparison the results of a recent optical/X-ray survey of six other nearby galaxies [715]. The Venn diagrams illustrate that our knowledge of extragalactic supernova remnant populations is very much skewed to optically identified supernova remnants, whereas for the Milky Way population most supernova remnants have been mostly discovered in the radio. X-ray

Radio 3

0

0

42 12

71 92

Optical

M33

X-ray

Radio

17 17

1

1 10

6 13 NGC 2403 NGC 3077 NGC 4214 NGC 4395 NGC 4449 NGC 5204

3

405 Optical

Fig. 3.10 Left: Hardness ratio diagram for X-ray detected supernova remnants in M33, using counts in the 0.3–0.7 keV (soft), 0.7–1.1 keV (medium) and 1.1–4.2 (hard) keV bands, as measured with XMM-Newton. N(OB) refers to the number of nearby OB stars [709] and supernova remnants N(OB) > 1 are more likely to be core collapse supernova remnants. The figure is reproduced from [425] (Fig. 7). Top right: Venn diagram showing the overlap between supernova remnants identified in the radio, optical and X-rays (adapted from [425]). Bottom right: Similar Venn diagram, but now for a sample of several nearby galaxies, as reported in [715]

Chapter 4

Shocks and Post-shock Plasma Processes

Shocks are transition layers in which the flow and thermodynamic properties of the plasma/gas rapidly change. They arise whenever gas is moving faster than the local signal speed, which for now we take to be the local sound speed:  cs =

∂P ∂ρ



 =

γ

P , ρ

(4.1)

with P and ρ, respectively the local pressure and density and γ the adiabatic index of the local gas. In the physics of supernova remnants we most commonly encounter non-relativistic monatomic gasses, for which γ = 5/3. There is a certain length scale over which the plasma properties change. But this transition layer is for supernova remnants small, and cannot be resolved. That the transition region is relatively small, is, in fact, surprising, as the length scales for particle-particle collisions are comparable to the sizes of supernova remnants. supernova remnant shocks, like many astrophysical shocks, are, therefore, called collisionless shocks. The fact that the supernova remnant shocks are collisionless has important consequences, such as potentially different temperatures for different particle species. This aspect will be discussed in Sect. 4.3. Also collisionless shocks are more prone to accelerate particles to energy in excess of the thermal energy of the bulk of the shock-heated particles. The length scale over which shocks change the flow and thermodynamic properties of the plasma can be substantially affected by cosmic rays (Chap. 11), electromagnetic radiation, and plasma waves (Sect. 4.5), propagating ahead of the shock. In this case the shock system contains a region ahead of the shock in which the plasma is already changed by the approaching shock. This region ahead of the shock is referred to as the shock precursor, whereas the actual shock is then referred to as the (viscous) subshock. In that case the word shock refers to the whole region in which the plasma properties change, including the precursor, which can have a

© Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_4

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4 Shocks and Post-shock Plasma Processes

substantial length scale (for cosmic-ray precursors even ∼1017 cm). The subshock transition region is more likely of the order of 107 –109 cm. As discussed in Chap. 5, supernova remnants may contain multiple shock regions: a forward shock, and in young supernova remnants, also a reverse shock. These shocks are responsible for heating the gas/plasma to temperatures of ∼106 – 108 K (kT ∼ 0.1 − 10 keV); temperatures at which the plasma will emit thermal X-ray radiation.

4.1 The Rankine-Hugoniot Jump Conditions The changes in plasma conditions across the shock-transition layer generally have to observe the conservation laws of mass, momentum and energy flux. Since the transition layers are usually much thinner than the curvature radii of the shocks, and because evolutionary changes are on time scales larger than the shock heating time scale, we can approximate shocks as plane parallel structures that are fixed in time when considered in a referenc frame comoving with the shock itself. We describe the shock in this comoving frame. All quantities in the unshocked region (often called upstream region) are labeled with subscript 1 and all quantities in the shocked region (downstream region) have subscript 2. So gas is entering the shock with velocity v1 , and after shock passage, the gas moves away from the shock with velocity v2 . In this comoving frame the conservation laws, the so-called Rankine-Hugoniot jump conditions, can then be expressed as: ρ1 v1 =ρ2 v2 ,

(4.2)

P1 + ρ1 v12 =P2 + ρ2 v22 ,     1 1 P1 + U1 + ρ1 v12 v1 = P2 + U2 + ρ2 v22 v2 , 2 2

(4.3) (4.4)

with ρ the density, P the pressure and U the internal energy of the gas. The combination H ≡ U + P is the enthalpy. The reason that the the enthalpy rather than energy flux enters in the relations is that, according to the first law of thermodynamics, the change in heat equals the change in internal energy plus the work done on the system: dQ = dU + P dV . Note that the above equations are not complete yet, as for now we are neglecting the pressure and internal energy associated with magnetic fields.We also neglect for now the effects of potential energy losses. The set of equations can be solved by first defining the compression ratio across the shock, χ≡

ρ2 v1 = . ρ1 v2

(4.5)

4.1 The Rankine-Hugoniot Jump Conditions

57

We can then rewrite the momentum-flux conservation equation as   1 . P2 = P1 + ρ1 v12 1 − χ For an ideal gas the relation between internal energy and pressure is U = the enthalpy is P +U =

γ P. γ −1

(4.6) 1 γ −1 P ,

so

(4.7)

Inserting (4.6)and (4.7) in (4.4) and replacing all values of v2 with v2 = v1 /χ and ρ2 with ρ2 = χρ1 , results in 

  

  1 ρ1 v12 v1 1 γ 1 γ 2 2 P1 + ρ1 v1 v1 = P1 + ρ1 v1 1 − + . γ −1 2 γ −1 χ 2 χ χ

Dividing both sides of the equation by 12 ρ1 v12 , and substituting Ms2 ≡

ρ1 v12 v2 = 12 , γ P1 cs

(4.8)

with Ms the so-called sonic Mach number, we obtain an equation with only dimensionless quantities Ms , χ and γ :  

1 2γ 1 1 1 1 1 2 + +1= 1− + 2 γ − 1 Ms2 γ − 1 γ Ms2 χ χ χ χ ⇒





(γ − 1)Ms2 + 2 χ 2 − 2 γ Ms2 + 1 χ + Ms2 (γ + 1) = 0.

This quadratic equation can be solved directly, or by making use of the fact that we know that there is one trivial solution χ = 1, corresponding to no transition in gas  properties. We  know, therefore, that the solution has the form (χ − 1)(χ − A) (γ − 1)Ms2 + 2 = 0, with χ = A the non-trivial solution. Comparing the result of this expression with the quadratic equation gives the well-known solution χ=

(γ + 1)Ms2 . (γ − 1)Ms2 + 2

(4.9)

An expression for the downstream temperature can be obtained from (4.6) by making use of P2 = ρ2 kT2 /(μ2 mp ) and P1 = ρ1 kT1 /(μ1 mp ), with μ1,2 the average

58

4 Shocks and Post-shock Plasma Processes

particle mass in units of the proton mass mp , which gives kT2 =

1 χ





 μ2 1 μ2 mp v12 . kT1 + 1 − μ1 χ

(4.10)

Note that a strong shock reduces the Mach number of the flow from supersonic (Ms > 1) to subsonic values. Since we have in general Ms2 = ρv 2 /(γ P ) we can calculated the post-shock Mach number:  Ms,2

v2 = = cs,2

  2  Ms,1 ρ2 v22 ≤ 1, = 2 (χ − 1) γ P2 χ + γ Ms,1

(4.11)

where for clarity we added the subscript 1 to the shock Mach number. For young supernova remnants the Mach numbers are very  high (> 100) compared to the sound speed in the interstellar medium (cs ≈ 15 T /104 K km s−1 ). For Mach numbers larger than Ms ≈ 5 one can use the approximation Ms → ∞, which results in the following expressions for the compression ratio, post-shock (downstream) temperature and Mach number: γ +1 = 4, γ −1   1 1 3 kT2 = 1− μ2 mp Vs2 = μ2 mp Vs2 χ χ 16 2 μ  Vs 2 ≈ 1.17 keV, 0.6 1000 km s−1   1 1 = ≈ 0.45, Ms,2 = γ (χ − 1) 5 χ=

(4.12)

where we have used γ = 5/3, and replaced v1 with the shock speed Vs ≡ |v1 |. Note that the plasma enters the shock with Vs and downstream it moves away from the shock with a speed v2 = χ1 Vs = 14 Vs . The relative difference in gas velocity across the shock is therefore   1 3

v = 1 − (4.13) Vs = Vs . χ 4

4.2 Magnetohydrodynamical Shocks In Sect. 4.1 we listed the Rankine-Hugoniot jump conditions, without including magnetic fields. For high Mach number shocks, and without taking into account

4.2 Magnetohydrodynamical Shocks

59

energy losses, the omission of magnetic fields provides accurate enough approximations for the resulting post-shock plasma properties, but for low Mach numbers the pressure and energy density associated with the magnetic field can be important. With magnetic field pressure and energy density included the jump conditions are [797, 1116]: ρ1 v1 =ρ2 v2 ,

(4.14)

v1 B⊥1 − v⊥1 B1 =v2 B⊥2 − v⊥ B2 ,

(4.15)

P1 + 

2 B⊥1 B2 2 2 + ρ1 v1 =P2 + ⊥2 + ρ2 v2 8π 8π

 2 B⊥1 1 B⊥1 B1 2 P1 + U1 + + ρ1 v1 v1 − v⊥1 = 4π 2 4π   2 B⊥2 1 B⊥2 B2 + ρ2 v22 v2 − v⊥2 . P2 + U2 + 4π 2 4π

(4.16)

(4.17)

Here the terms P and U refer to the thermal pressure only, and the magnetic field and velocities are decomposed in components perpendicular to the shock normal and parallel to the shock normal. We see that including magnetic fields makes the jump conditions more complicated, as the direction of the flow can change across the shock front, whereas with magnetic fields all plasma motions are parallel to the shock normal. Note that for purely parallel shocks (i.e. B1,⊥ = 0) the jump conditions reduce to the jumps conditions without magnetic fields. For shocks with magnetic fields, besides the sonic Mach number, another dimensionless number becomes important, the Alfvén Mach number:  MA ≡

4πρVs2 Vs = , 2 VA B1

(4.18)

with −1/2  n B H VA = √ ≈9 1 cm−3 4πρ



 B km s−1 , 5 μG

(4.19)

the velocity with which Alfvén waves propagate. Alfvén waves are are caused by the tension of the magnetic field, and are transverse waves traveling along magnetic field lines. In addition there are magnetosonic waves, where the combined gas and 2 = V 2 + c 2 . Magnetosonic waves are magnetic pressure results in a signal speed vms s A propagating perpendicular or obliquely to the magnetic field. For a shock to form the flow has to be both super-sonic and super-Alfvénic: v1 > vms . The magnetosonic

60

4 Shocks and Post-shock Plasma Processes

signal speed is associated with the magnetosonic Mach number Mms : 1 1 1 = 2 + 2. 2 Mms Ms MA

(4.20)

Shocks can only arise when the magnetosonic Mach number exceeds one. Another dimensionless number that is useful in connection to shocks in magnetised plasmas is the so-called plasma beta parameter, defined as the ratio of thermal over magnetic pressure: β≡

2MA2 Pthermal 8πnkT = = . Pmagnetic B2 γ Ms2

(4.21)

If we ignore magnetic-field amplification in the shock region, for example by cosmic-ray streaming (Sect. 11.4.3), magnetic flux is conserved, and we have B⊥2 = χB⊥1 ,

(4.22)

with χ the compression ratio, and B1 = B2 . We can simplify the equations by considering only so-called perpendicular shocks, for which B1 = 0. The thermal pressure in the shocked plasma is then   2   B⊥1 1 2 2 1 − χ + ρ1 v1 1 − P2 = P1 + . 8π χ

(4.23)

Inserting this into (4.16), and ignoring all B terms, we can solve the equation by dividing the resulting equation by 12 ρ1 v12 , and then introducing the dimensionless numbers Ms and MA (or β). This results in the following cubic equation:    



4 2 3 2 (4.24) + γ Ms χ 2 (G − 2) χ + 2G + β β

  1 2 + 2Gγ Ms χ + (2G − 1)γ Ms2 = 0, + 2G + β with G ≡ γ /(γ − 1) There is one trivial solution χ = 1; so we can write (χ − 1)(Aχ 2 + Bχ + C) = 0, and solve for the coefficients A, B, C. This reduces the non-trivial solution of (4.24) to  



 2 1 + γ Ms2 χ − (2G − 1)γ Ms2 = 0, (G − 2) χ 2 + 2G 1 + β β

(4.25)

4.2 Magnetohydrodynamical Shocks

61

Fig. 4.1 The shock compression (left) and downstream Mach number (right) as a function of magnetosonic Mach number and plasma β, assuming a perpendicular shock. The solid line corresponds to the absence of magnetic fields

which has only one physical solution: χ=



2(γ + 1)Ms2   2 1 + β1 + (γ − 1)Ms2

  

2 + 2 1 + β1 + (γ − 1)Ms2 +

.

(4.26)

 8 γ β (2 − γ )(γ

+ 1)Ms2

In the limit β → ∞ this equation approaches (4.9). Also for Ms → ∞ and β → 0 (i.e. the unshocked medium’s pressure is magnetically dominated) the overall compression ratio will still be χ → 4, as illustrated in Fig. 4.1. The reason is that P1 + B 2 /8π  ρ1 v12 ; so the internal energy in the shocked gas is dominated by the thermalised kinetic energy. The sonic Mach number of the shocked plasma can be obtained by dividing (4.23) by ρ2 v2 /γ , which results in

−1/2    1 2 χ − χ 3 + γ Ms,1 . Ms,2 = Ms,1 χ + (χ − 1) β

(4.27)

It turns out that downstream of the shock the flow may still be supersonic, but slower than the magnetosonic speed for certain values of β and magnetic-field orientation.The limiting pre-shock magnetosonic Mach number for which Ms,2 ≤ 1 is called the first critical Mach number. Depending on β and angle between magnetic field and shock normal, the critical Mach number for γ = 5/3 is in the range 1 ≤ Mms,crit < 2.76 [360]. Figure 4.1 (left) shows the downstream sonic Mach number as a function of Mms . The critical Mach number plays a role in details of collisionless shock theory [125], but is also a critical parameter for shocks with a magnetohydrodynamic precursor (Sect. 4.5).

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4 Shocks and Post-shock Plasma Processes

4.3 Collisionless Shocks We introduced shocks as “transition layers in which the flow and thermodynamic properties of the plasma/gas rapidly change”. But as we will see the transitions may not be so abrupt, and the microphysics of shock formation in tenuous astrophysical environments is complex, and still not well understood.

4.3.1 Visocity and the Shock Transition Layer Thickness The use of the Rankine-Hugoniot jump conditions In the previous sections we were only concerned with the begin and end state of a gas undergoing a shock transformation, and we treated the shock-transition layer as a box of infinitely small width. The shock transition itself causes the transformation of incoming bulk kinetic energy into thermal energy, i.e. it increases the entropy of the gas. The thermalisation is caused in most cases by some form of viscous processes, but see [1273] and Sect. 4.5 for exceptions. Within the shock transition layer additional viscous terms should be added to the momentum and enthalpy-flux conservation equations: 4 dv P + ρv 2 − μ =P1 + ρ1 v12 3 dz     dT 1 1 2 4 dv 2 −κ v = P1 + U1 + ρ1 v1 v1 , P + U + ρv − μ 2 3 dz dz 2

(4.28) (4.29)

with μ the viscosity coefficient and κ the heat conduction coefficient. The direction of v1 is in the positive z direction. In the absence of gradients, i.e. far upstream and downstream these equations are the same as (4.3) and (4.4). The viscosity coefficient is microphysically connected to the mean free path, λ, and mean velocity, v, of the particles through μ=

1 ρvλ 3

(4.30)

(assuming an isotropic velocity distribution). We can obtain a rough estimate of the thickness of the shock transition layer, lsh , by approximating (e.g. [1273]) lsh

 −1  −1    dv   dv  1     = z =  

v =   1− v1 . dz max dz max χ2

(4.31)

That is we use maximum in the gradient |dv/dx|. We have used the subscript 2 to indicate that χ2 is the final compression ratio as given by (4.9). In this section

4.3 Collisionless Shocks

63

we will reserve the symbol χ for the (variable) compression within the transition layer itself (1 ≤ χ ≤ χ2 ). The compression ratio is monotonically increasing in the transition layer, and since v is defined in the frame of the shock (v is decreasing with increasingz), the gradient dv/dz is negative.   To estimate  dv we rewrite (4.28) into dz  max

 

 dv    = 3 P1 + ρ1 v 2 − P − ρv 2 1  dz  4μ      9 1 P 1 = . ρ1 v1 − + 1− 4ρvλ γ Ms2 χ ρv12

(4.32)

In the transition layer the viscosity results in an increase in entropy, so that the gas changes smoothly between two isentropic states. The pressure in the transition zone is, therefore, enclosed between the entropic begin and end states:  P1 χ γ ≤ P ≤ P2

χ χ2

γ ,

with P2 given by (4.6). In Fig. 4.2 the initial an final states are labeled A and B, and the two isentropes are drawn with respectively solid and dashed black lines. The maximum value for dv/dz is now capped by the expression in brackets in (4.32) and the two isentropes. In Fig. 4.2 the transition of p(≡ P1 + ρ1 v12 − P − ρv) is shown as well. Using the expressions for P one can show that the maximum value

Fig. 4.2 Left: The thermodynamic changes in the shock transition region, as function of compression. The black lines indicate the lower and upper bound on the pressure (see text). The blue lines show the corresponding values of p. Right: The bounds on shock widths (black lines, normalised to λv/v1 ) as function of the sonic Mach number, and in blue, the boundaries on the maximum value of p/ρ1 v12 , corresponding to the two isentropic states. In both panels the solid lines correspond to isentropic state 1 and the dashed lines to state 2

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4 Shocks and Post-shock Plasma Processes

for p for isentropic state 1 occurs at   2/(γ +1) χ pmax,1 = min Ms , χ2 ,

(4.33)

and for isentropic state 2 at ⎛ χ pmax,2 = ⎝

γ χ2 Ms2 

1 + γ Ms2 1 −

⎞1/(γ +1) 1 χ2

⎠

.

(4.34)

Inserting these expression in (4.32) and (4.31), provides upper and lower bounds on the shock widths. The resulting expressions for the thickness of the shock transition layer are then lsh =

  1 v 4 1 λ , χ pmax 1− 9 χ2

pmax v1

(4.35)

with rather lengthy expressions for the two limiting values of pmax . We, therefore, visualise the results in the right panel of Fig. 4.2. The numerical estimates for γ = 5/3 and M → ∞ are 1.8λ

v v < lsh < 2.7λ . v1 v1

(4.36)

√ The quantity v/v1 is smaller than one, and using (4.12) we find v  6/16v1 ≈ 0.6v1 . This shows that the shock thickness for high Mach numbers is of the order of the mean free path of the particles, but it rapidly increases below Ms ≈ 2.

4.3.2 The Collisional Mean Free Path Let us now evaluate the mean free path of charged particles due to particle-particle collisions caused by Coulomb interactions (e.g. [1273]). A charged particle is deflected by ∼90◦ in the center of mass frame, if the reduced kinetic energy of the two moving charges equals the energy caused by Coulomb forces: Z1 Z2 e 2 1 m1 m2 2 , v = 2 m1 + m2 b

(4.37)

with m1 , m2 the masses of the two particles, Z1 e, Z2 e their charges, v their relative velocity, and b the impact parameter. The cross section (σCoulomb = πb2 ) for such a

4.3 Collisionless Shocks

65

deflection by a single scattering is, therefore, σCoulomb ≈ 4π

Z12 Z22 e4  m1 + m2 2 . m1 m2 v4

(4.38)

The corresponding collision time scale is τCoulomb = 1/(nσ v) ∝ v −3 ∝ E −3/2 . This shows that the cross section falls of rapidly with collision velocity. Inserting for the m1 and m2 the proton mass and a typical shock velocity of v = 1000 km s−1 , one −1 20 −1 finds τpp ≈ 1012 n−1 p s (∼32,000 np yr), and a mean free path λp ≈ 10 np cm (32 pc). These time and length scales are in reality smaller by an order of magnitude, as one should also take into account multiple small-angle scatterings [1072, 1273]. But this still implies that the mean free paths are of the order of parsecs, comparable to the sizes of supernova remnants. Clearly some other viscous processes tha Coulomb processes must operate in order to establish a shock transition layer. This problem does not only arise for supernova remnants, but for many astrophysical shocks, including shocks in the solar system, such as shocks in the solar wind caused by coronal mass ejections, and the bow shocks around planets like Earth, and large scale shocks in clusters of galaxies. Shocks for which the particle-particle collision mean free path is larger than the scale size of the shocks are referred to as collisionless shocks.

4.3.3 The Thermalisation Processes Since direct particle-particle collision rates are too small to rapidly thermalise the bulk kinetic energy in the shock transition layer, other processes must be at work. These processes are generally referred to as “collective effects”, somewhat akin to “violent relaxation” in collapsing gravitational systems [748]: the incoming particles are thermalised through interaction with plasma waves and electric fields in the shock transitions layers. But the plasma waves and electric field in turn are a result of the interaction of the interaction of charged particles with the shock transition layer. The study of collisionless shocks and the associated microphysics are rich topics with still many uncertainties. For reviews see [1123], and two recent books on the topic [125, 220]. The shock formation processes are complex and are best investigated with the electro-magnetic analogues of N-body simulations, i.e. particle-in-cell simulations (PIC). In PIC simulations a shock is established by setting up two beams of charged particles streaming against each other. The electric and magnetic fields generated by these particles are calculated on a finely meshed grid (the cells), which in turn are used to calculate the trajectories of the particles. In the shock transition regions the PIC simulation shows that a substantial fraction of the ions is first reflected back upstream, before finally being thermalised. The reflected ion create a small

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4 Shocks and Post-shock Plasma Processes

precursor in the shock region often referred to as the “shock foot”. This is associated with structures in the ion and reflected phase space; see for example [1022]. The shock transition length scale in PIC simulations is often expressed in terms of the ion inertial length scale,  c λi ≡ = ωpi

mp c 2 −1/2 ≈ 2.3 × 107 np cm. 4πnp e2

(4.39)

This is the length scale over which plasma fluctuations with frequencies close to the ion plasma frequency, ωpi , dampen out. This length scale is also found observationally, but in [123] it is argued that the shock transition length scale is better approximated with a scaling proportional to the ion gyroradius: rg =

    B⊥ −1 v cp cm, ≈ 1.0 × 109 eB⊥ 10 μG 1000 km s−1

(4.40)

with p = mv the particle momentum, B⊥ the magnetic field strength perpendicular to the particle’s motion. For the typical particle speed [123] used the shock speed v ≈ Vs . See Fig. 4.3, which illustrates that the ion gyroradius appears to be the best way to quantify the shock thickness with only one parameter. It appears, therefore, that the particle mean free path that determines the viscosity coefficient is the ion gyroradius (4.36). Interestingly, for diffusive shock acceleration (Chap. 11) it is also often assumed that the mean free path of the particle is λ ≈ rg , an approximation 5

2000–12–25/01:26:15

120

4

100 tanh(x/c)

Density vs. Distance

3 2 1 0 5

60

4

downstream

40

20

L (c/ωpi)

nf (cm–1)

80

L (vsh/Ωci)

2L

upstream

3 2 1

0 2500

3000

3500 km

4000

0 0

5

10 Mms

15

20

Fig. 4.3 Illustration of the narrowness of collisionless shocks. The left figure shows a typical Earth’ bowshock shock transition as measured by one of the ESA Cluster II satellites. The figure on the right shows the shock thickness distribution, in terms of ion gyroradius (top right), and ion inertial length (bottom right), both as a function of magnetosonic Mach number (Ms ). The ion inertial length scales provides a better description, as the shock transition region does not show a scaling with Mach number (Figures reproduced from Bale et al. [123])

4.3 Collisionless Shocks

67

that is referred to as Bohm diffusion (Sect. 11.2.2). Of course, for cosmic rays the particles are more energetic, and the corresponding mean free path is much larger than for the case discussed here. The ratio between the gyroradius and the ion inertial length scale is  rg 4πmp np Vs =v ≈ = MA , 2 λi VA B⊥

(4.41)

where we used vpart ≈ Vs and made use of (4.18) and (4.19). Whether the shock thickness for supernova remnants is best described by the gyroradius or the ion inertial length, it is clear that the length scale is much smaller than what can be resolved by telescopes now or in the near future. It is, nevertheless, relevant to certify that the length scale is indeed smaller than other relevant length scales. For example, some properties of supernova remnant shocks can be obtained by measuring the properties of neutral atoms entering the shocks. The interaction length scales of neutrals is of the order of 1015 cm. Since this length scale is much larger than the expected shock transition layer, we do not have to take into account the interactions of the neutrals within the viscous shock transition layer. On the other, the neutrals add yet another complexity to the large scale shock region, as discussed in (Sect. 8.2).

4.3.4 The Expected Post-shock Electron-Ion Temperature Ratio Superficially it looks like we should not really care about whether supernova remnant shocks are collisional or collisionless, as in the end the relationship between upstream and downstream plasma properties are in both cases governed by the Rankine-Hugoniot relations. But because Coulomb interaction length scales are very long, we can no longer assume that each particle species will immediately equilibrate temperatures within the shock transition layer; the Rankine-Hugoniot relations contain the overall pressure components and do not specify what the partial pressures of each species should be. If we treat the Rankine-Hugoniot relations completely seperate for protons and electrons, (4.12) would lead to completely different temperatures for electrons and proton temperatures downstream of the shock: kTe = 3/16me Vs2 , kTp = 3/16mpVs2 . The ratio of these temperatures is Te /Tp = me /mp ≈ 5.5 × 10−4 ! Indeed, there is ample observational evidence that in young supernova remnants the electron temperature is lower than the proton temperature. For example, all young supernova remnants appear to have electron temperatures kTe  5 keV, whereas for some supernova remnants the measured shock velocities are 3 ∼5000 km s−1 , in which case one expects kT = 16 μmp Vs2 ∼ 25 keV.

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4 Shocks and Post-shock Plasma Processes

It is not a priori clear whether the complex heating processes in the shock transition layer generally produce roughly equal electron- and ion temperatures. In the shock transition layer there may be various plasma instabilities that could couple the electrons and ions effectively, resulting in roughly equal electron/ion temperatures. An example is lower hybrid oscillations, which couple ions to electrons and has been studied in [431]. If the particles are primarily heated through interactions with magnetic field turbulence, the heating process primarily occurs through quasi-elastic magnetic scattering (not quite elastic as the scattering particles will also put energy into the plasma turbulence). Since the scattering process does not lead to energy gains of the electrons, but is primarily causing isotropisation of the available kinetic energies, the electrons are expected to be cooler than the ions. On the other hand, electric fields may temporarily build up in the transition layer, due to the different thermal velocities of the electrons and ions. Electrons traversing the electric potentials will gain energy, leading to (partial) equilibration of electron and ion temperatures. Here we discuss the approach described in [1184], which provides a lower limit on the electron/ion temperature ratio. The approach is to consider the kinetic energy of the electrons and ions separately, as two “fluids”.Note that the topic of “multifluid shocks” is quite rich and diverse. In a different context it will be used in Sect. 4.5, when dealing with a shock structure consisting of a dominant molecular (neutral) component and a small fraction of ions. In Chaps. 8 and 11 we return to the topic of multi-fluid shocks, but then with respectively ions and neutral hydrogen, and the thermal plasma and cosmic rays as separate components. For the case of separate electron and ion components, we assume that each species preserves its own enthalpy-flux conservation law (4.4). The momentumflux (4.3) can have exchanges between the electron and ion components. The reason is that elastic scatterings of the electrons with the plasma waves may affect the directions of the motions, and hence the momentum-flux, despite the fact that the kinetic energy of the electrons is preserved. Because of charge neutrality of the plasma, the final compression ratios, χ, of the electrons and ions are equal, and are given by (4.9). There is one type of coupling between the electron and ion enthalpy flux: due to the shock compression, largely powered by the energetically dominant ions, the shock perform P dV work on the electron fluid, thereby (at least) adiabatically heating the electrons. The result is that the electron pressure should at least increase according to Pe,2 = Pe,1 χ γ . If for simplicity we consider a plasma of only electrons and protons, the ratio of the electron over proton temperature is expected to be Te,2 β≡ = Tp,2



me mp



 2χ 2

2 



  χ γ +1 + Ms2 (γ − 1) χ 2 − 1    . 1 − χ γ −1 − 1 + Ms2 (γ − 1) χ 2 − 1 μmp me

(4.42)

(Note that we use the symbol β has different meaning than in (4.21).) We refer to [1184] for more elaborate equations, which take into account magnetic fields, plasma composition, and potential partial energy equilibration of electrons and ions

4.3 Collisionless Shocks

69

in the shock transition layer. In (4.42) terms with χ γ −1 represent the effects of adiabatic heating of the electrons. Asymptotically the  temperature ratio will approach Te /Tp = me /mp . For Mach numbers 2  Ms  mp /me ≈ 43 the term Ms2 (γ − 1)(χ 2 − 1) in the denominator dominates the behaviour of β. Hence, Te /Tp ∝ Ms−2 in that range. This behaviour has indeed been inferred for young supernova remnants [431]. One way to think about this model is to consider two “Mach numbers” for the electrons and the ions, c.f. (4.8):  Me ≡

γρe,1 Vs2 , Mion ≡ Pe,1



γρion,1Vs2 . Pion,1

Mion is close in value to the standard sonic Mach number. But due to the low density of the electrons the electron Mach number is much lower. Only once Me > 1 can the entropy of the electrons increase, as then the upstream kinetic energy exceeds the upstream thermal energy  of the electrons. This limit on the electron Mach number corresponds to Ms ≈ mp /me ≈ 43 (or ∼60 taking into account more massive ions). Figure 4.4 shows the relation (4.42) in comparison with measured electron-ion temperature ratios. The planetary data follow the relation well, but indicate that a 5– 10% energy exchange from the ions to the electrons occurs [1184]. The supernova remnant data [1148] are based on optical spectroscopy of non-radiative shocks (Chap. 8). These data do indeed show a proportionality of Te /Tp ∝ 1/Ms2 , but a shift in Mach number is needed to bring the data points on the model curve. This shift may indicate that the upstream sound speed is much higher than the 11 km s−1 that was used for the conversion of measured shock velocities to Mach numbers. This could imply considerable heating of the upstream plasma, perhaps as a result of the plasma heating in the cosmic-ray precursor (see Chap. 11). Indeed, there is observational evidence for pre-shock heating upstream of young supernova remnant shocks [1067].

4.3.5 Post-shock Electron-Ion Temperature Equilibration We have seen that for collisionless shocks it is not a priori clear whether the shock heating process heats the electrons and ions to the same temperatures. Moreover, observational evidence, both from solar system shocks and supernova remnants indicates that indeed the electrons are colder than the ions. In supernova remnants the temperatures within ∼1015 cm downstream of the shock can be measured for shocks in partial neutral media from optical and UV spectroscopy (Chap. 8). X-ray spectroscopy provides another opportunity to measure temperatures, but continuum and line-ratio measurements provide a means to only estimate the electron temperature. The ion temperatures can be obtained from Doppler line

70

4 Shocks and Post-shock Plasma Processes

Fig. 4.4 The solid black line shows the expected electron to ion temperature ratio, based on a model for which the electrons are only heated by thermalisation of their own kinetic energy plus some adiabatic heating [1184]. For the blue line a 5% energy tranfer from the ions to electrons is assumed. The data points are taken from the review by Ghavamian et al. [436]. For the supernova remnant data Mach numbers were estimated assuming an upstream sound speed of 11 km s−1 . The dashed line indicates a 1/Ms2

broadening measurements, but with the current X-ray spectrometers this turns out to be difficult for extended sources. This should improve in the future with new X-ray satellites like XRISM (to be launched in 2022) and ATHENA (beyond 2030) [470]. Moreover, X-ray measured temperatures are typically for regions further downstream of the shock, and one should then take into account post-shock electronion temperature equilibration. The equilibration process that slowly brings electron and ions temperatures closer together are the Coulomb collisions discussed before. The cross section for the collisions is given by (4.38), although a more detailed analysis of the problem increases the relevant collision rates if one also includes energy exchanges at large impact parameters, which gives many interactions but with smaller energy transfers [1072, 1273]. The relevant expressions for energy exchange between charged particles of various mass ratios can be found in [193] to be dkTi = 1.8 × 10−19 × dt

(mi mj )1/2Zi2 Zj2 ni ln ij i

(mi kTi + mj kTj )3/2

(kTj − kTi ) eV s−1 , (4.43)

4.3 Collisionless Shocks

71

with temperatures in units of eV and Zi , Zj the charges of the particles in units of the elementary charge e. One can recognise in this equation the dependence on charge given in (4.38) whereas the collision rate dependence of vσ ∝ v −3 explains the temperature dependence of vσ ∝ T −3/2 , since T ∝ v 2 . The factor ln ij is called the Coulomb logarithm, which is roughly similar for electron-proton interactions and for electron-electron interactions:  kT −1 e 1/2 . ln  = 30.9 − ln ne 1 keV

(4.44)

For electron-proton equilibration (4.43) implies an equilibration time scale of τep =

 kT 3/2 ln  −1 Ti ≈ 3.15 × 1011n−1 s, p (dkTi /dt) 1 keV 30.9

(4.45)

with T the mean temperature. The equilibration time scale is long as particles with a very large mass ratio do not exchange much energy in collision. The self equilibration time of electrons or protons among themselves is smaller than between two particle species. √ But since electrons move faster than protons for a given temperature (v = 2E/m), electrons have a higher collision rate and therefore thermalise faster. The associated self-equilibration time scales are relevant if there is a mixture of two electron (or proton) populations with different temperatures, or in case the initial distribution is non-Maxwellian. The electron self-equilibration time for electrons is τee ≈ 4.9 × 108 n−1 e

 kT 3/2  ln  −1 e s. 1 keV 30.9

(4.46)

 kT 3/2  ln  −1 p s. 1 keV 30.9

(4.47)

For protons this is τpp ≈ 2.1 × 1010n−1 p

Equation (4.45) shows that it can easily take 10,000 years for electrons and protons to equilibrate, much longer than the ages of many supernova remnants. Note that the equilibration time is roughly inversely proportional to np (≈ne ). A relevant parameter is, therefore, not so much the age of the supernova remnant, but the average ne t, a parameter that is discussed in Sect. 13.5.7, where it is shown that the same parameter determines the amount of ionisation equilibration. X-ray spectroscopy provides a direct way of measuring ne t, which is often called the ionisation age, thereby providing the means to estimate whether non-equilibration of temperatures is expected to be important. Figure 4.5 shows the effect of non-equilibration on the temperatures of the electrons, and various ions, under the assumption that immediately downstream of the shock the temperatures for each species is given by kTi = 3/16mi Vs2 .

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4 Shocks and Post-shock Plasma Processes

Fig. 4.5 The temperature equilibration of different charged particles, as function of time: electrons (dotted line), protons (dashed) and various ions. The calculation assumes that each species i has an 3 mi Vs2 with Vs = 3000 km s−1 . Subsequent immediate post-shock temperature given by kTi = 16 equilibration occurs through Coulomb collisions on a time scale given by ne t ≈ 1012 cm−3 s. It takes into account the time-dependent ionisation of the ions. Adiabatic or other losses are not taken into account

For the calculations time dependent ionisation is taken into account. Note that more massive ions equilibrate faster with protons than helium does, due to strong charge dependence of the Coulomb interactions. The implication is that for the case depicted in the figure (Vs = 3000 km s−1 ), for ne t  1010 cm−3 s one would measure that all ions have different temperatures, scaling with the ion mass, whereas for ne  5×1010 cm−3 s all ion temperatures are equal, but the electron temperature is lower than the ion temperature. That indeed for low ne t the ion temperatures can be very hot is clear for SN 1006. UV spectroscopy with the Hopkins Ultraviolet Telescope (HUT) of the northern part of this supernova remnant showed that the lines of He II, C IV, N V and O VI showed that all lines were equally Doppler broadened, indicating that their temperatures scaled with the ion mass [944]. For the same supernova remnant, X-ray spectroscopy with the XMM-Newton Reflective Grating Spectrometer (RGS) of a bright, small knot showed that the electron temperature was kT ≈ 1.3 keV, whereas the oxygen temperature was ≈275 keV [213, 1185]. SN 1006 evolves in a low density and hence ne t is low (∼3 × 109 cm−3 s). Note that the model in Fig. 4.5 does not take into account adiabatic cooling, which affects all particles equally through P V γ =constant. Another effect that is ignored is that post-shock ionisation results in a population of relatively cool

4.3 Collisionless Shocks

73

electrons slowly being released into the plasma. This will keep the average electron temperature cooler for a longer time, especially in metal-rich plasmas, in which most of the electrons originate from ionisation [582]. For metal-rich plasmas the average ratio between electrons and ions is expected to be much more pronounced, since 3 the average ion temperature is expected to be kTi ≈ 16 mi Vs2 , and for metal-rich plasmas mi  mp .

4.3.6 Heat Conduction Thermal conduction is a topic that every now and then enters the discussion of supernova remnants, as significant thermal conduction may alter the temperature and density structures of supernova remnants, see for example the section on mixedmorphology supernova remnants (Sect. 10.3). The role of thermal conduction in supernova remnant structure and evolution, like that of temperature non-equilibration, has never been satisfactory resolved. The reason is that thermal conduction in a plasma is likely to be anisotropic due to the inhibiting effects of heat conduction across magnetic field lines. Note that the processes of thermal conduction and temperature equilibration are related, because thermal conduction is a combination of transport and exchange of heat. Without magnetic fields, or along magnetic fields, thermal conduction is usually mediated through electrons, because they have generally higher thermal speeds and are more strongly collisionally coupled to each other. Across magnetic-field lines the particles with the largest gyroradius, i.e. the ions, will dominate thermal conduction. The process of thermal conduction is described by Fourier’s law: Fheat = −κ∇T .

(4.48)

The plasma particles can freely stream parallel to the magnetic field lines, and in this direction the thermal conductivity coefficient κ depends on the typical mean free path, λ, and the mean velocity of the particles. Since the electrons do generally have higher velocities the thermal conductivity coefficient is dominated by the transport of the internal energy (Ue ) of the electrons: κ =

1 Ue 1 λe v = ne λe vk 3 Te 2

(4.49)

√ with k the Boltzmann constant, and v = kTe /me the thermal speed of the particles. This expression is correct up to a correction factor, which takes into account that streaming of electrons from a hot part of the plasma results in a return current, ensuring charge neutrality of the plasma [1072]. For the mean free path we can

74

4 Shocks and Post-shock Plasma Processes

write λe ≈ τee v ≈ 0.2n−1 e



kTe 1 keV

2 pc,

(4.50)

with τee the self-equilibration time (4.46). Including a small correction factor, the final expression for the thermal conductivity along the magnetic field lines is [193] κ /k ≈ 3.2

 kT 5/2 ne kTe τee e = 2.6 × 1027 cm−1 s−1 , me 1 keV

(4.51)

which is independent of the electron density. It shows that thermal conduction is strongly temperature dependent. Equation (4.48) is only valid when the temperature scale length is larger than the mean free path for energy exchange. If this is not the case, then the heat flux is described by the so-called saturated heat flux [278] given by Fsat = 0.4

 2kT 1/2 e

πme

ne kTe .

(4.52)

In this case heat is transported by electrons streaming away from hot regions, but heat exchange between electrons is not taken into account, i.e. the resulting electron distribution will not be Maxwellian. The factor 0.4 takes into account that the heat flux should be electrically neutral, reducing the heat flow. Unlike the classical heat flux, the saturated heat flux is density dependent. The ratio of the classical over saturated heat flow, involving a temperature scale length parameter Rs , is (c.f. [278]): √  R −1  kT 2  ln  −1 Fheat 7.2τee kTe /me s e = = 1.5n−1 . e Fsat Rs 1 pc 1 keV 30.9

(4.53)

The saturated heat flux is, therefore, dominant for small scale sizes (Rs  1 pc). The thermal conduction across field lines, which is predominantly mediated by ions (protons), given their larger gyroradii, is [193, 1072]: κ⊥ /k =

 B  2.8np kTp 7 = 1.4 × 10 cm−1 s−1 , 2 τ 10 μG mp ωcp pp

(4.54)

with ωcp = eB/(mp c) the proton gyrofrequency, and τpp the proton selfequilibration time scale. This shows that there is an extremely large difference in conduction parallel and across field lines. Only for B ∼ 10−15 G does the perpendicular and parallel conduction become comparable (for n = 1 cm−3 ). To get an order of magnitude estimate for the importance of thermal conduction one should compare the advective enthalpy flux Fadv = v 52 nkT with the thermal conduction flux, approximating the temperature gradient with kT /Rs and assuming

4.4 Radiative Shocks

75

a typical flow speed v:  kT −5/2  R  v Fadv 5nRv s e = 0.4n ≈ , Fheat 2κ 1 pc 1000 km s−1 1 keV

(4.55)

where for κ the value along the magnetic field has been used. The fact that the numerical value is of order one suggests that in supernova remnants the issue of thermal conduction can be critical for the temperature structure. This in turn makes the topology of the magnetic field a crucial ingredient. And one could question how realistic it is to use κ in calculations of the effects of thermal conduction. Note that the advected over conductive heat √ flow ratio (4.55) is proportional to (kT )−5/2 . If in addition one assumes v ∝ kT , the advected over conduction heat flux ratio will be proportional to (kT )2 . This means that thermal conduction could in principle be more important for hotter plasmas. However, radio polarisation measurements of supernova remnants indicate that young supernova remnants, which have hotter plasmas, have more turbulent magnetic fields than old supernova remnants (Sect. 12.1). So this would argue against the importance of thermal conductivity for young supernova remnants. On the other hand little is known about the magnetic fields in the ejecta components, in particular in mature supernova remnants. This may be indeed point to the potential importance of thermal conduction for the aforementioned mixed-morphology supernova remnants.

4.4 Radiative Shocks The Rankine-Hugoniot relations may not be valid for all types of shocks. For example, if cosmic-ray acceleration is important, we have to take into account multiple constituents of the plasma, namely the thermal plasma and the population of accelerated particles. Moreover, the highest energy cosmic rays may leave the shock region, draining the plasma of energy (Chap. 11). Here we discuss another form of energy loss, namely radiation, which for supernova remnant plasmas is dominated by thermal continuum processes and line radiation (Chaps., 8 and 13). For high shock velocities (Vs  200 km s−1 ) the postshock plasma temperature exceeds 106 K, and the plasma cools relatively slowly. The cooling time scale as function of temperature can be obtained from the cooling function, or cooling curve. An example of a cooling curve is shown in Fig. 4.6. From the cooling function curve the emissivity of the plasma can be be calculated according to the relation  = ne np , with  the volumetric cooling rate, and ne and np the electron and proton density respectively. The associated cooling time scale is then obtained by dividing the internal energy by the emissivity. Since the cooling usually takes place under more or less constant pressure (isobaric cooling), it is better to use the enthalpy, U + P = γ P /(γ − 1), thereby taking into account the adiabatic heating of the plasma as the plasma loses energy through

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4 Shocks and Post-shock Plasma Processes

Fig. 4.6 The radiation cooling curve, , of an optically thin plasma, based on the calculations in Schure et al. [1028]. The blue dashed line shows the contribution of carbon to the cooling, and the red, dotted, line the contribution of oxygen

radiation, but is compressed at the same time. With this in mind, the cooling time scale is   γ n kT γ −1 np np kT τcool = ≈ 5.7 , (4.56) ne np (T ) ne (T ) with n the number densities of all ions and electrons together, and taking n/np ≈ 2.3, and γ = 5/3. This time scale is associated with a length scale, indicating the distance downstream of the shock over which the plasma will become substantially cooler than the immediate post-shock temperature: lcooling ≡ v2 τcooling ≈

1 Vs τcooling, 4

(4.57)

using v2 = Vs /χ ≈ Vs /4. The cooling time and length scales are shown as a function of shock velocity in Fig. 4.7 for a pre-shock density of np = 1 cm−3 . For the calculations (4.12) was used to link post-shock temperature to the corresponding shock velocity. In the left panel the cooling time is compared to the age of the supernova remnant using the relation between shock velocity and age as given by the Sedov-Taylor self-similar evolution model (Chap. 5). The figures illustrate that for velocities  200 km s−1

4.4 Radiative Shocks

77

Fig. 4.7 Left: The characteristic cooling time as a function of shock velocity for a pre-shock density of np = 1 cm−3 . The dotted line shows the relation between shock velocity and supernova remnant age according to the Sedov-Taylor model. Shock waves become radiative whenever the cooling time scale drops to a fraction of the supernova remnant age. Right: The length scale over which the plasma cools down

cooling becomes increasingly important, as the cooling time scale falls below the age of the supernova remnant. For Vs  200 km s−1 the cooling length scale is less than a parsec, and for even lower velocities the length scales will be of the order of 1016/ne cm [943]. Once radiative cooling becomes important, the cooling creates a runaway process: pressure balance (P = nkT ) requires the plasma to compress, but the higher density increases the emissivity even more. This process slows when the plasma has cooled down below ∼104 K, where the cooling time scale becomes long again. But more likely further compression of the plasma is halted by the pressure provided by the magnetic field, which starts to dominated at high compression ratios. The cooling process is illustrated in Fig. 4.8, which is based on the Raymond shock models [943]. It shows that at the shock the plasma temperature is that of a strong shock (4.12). As the temperature steadily declines due to radiative cooling, the near constant pressure behind the shock requires the density to steadily increase. Below 10,000 K ion recombination reduces the thermal pressure even more as it reduces the number of free electrons. The rapid cooling and the increase in density result in the formation of optically/UV bright narrow filaments, with widths of the order of the cooling length scales. The spectra of these filaments are characterised by bright Lyα and Hα line emission, and forbidden line emission from, among others, ionised nitrogen, oxygen and sulfur (Chap. 8). Shocks for which the post-shock layers cool rapidly are usually referred to as radiative shocks. Beautiful examples of the optical filaments are found in the nearby Cygnus Loop supernova remnants, as shown in Fig. 1.1.

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4 Shocks and Post-shock Plasma Processes

Fig. 4.8 The density and temperature structure behind a radiative shock, based on the model G of Raymond [943], which assumes a pre-shock density of nH = 10 cm−3 , a shock velocity of Vsh = 141 km s−1 , and a pre-shock magnetic field of B = 1 μG

4.4.1 Isothermal Shocks Radiative shocks can result in very high compression ratios, as the surrounding medium will compress the plasma, while it is losing energy. Clearly, for radiative shocks as whole, i.e. including the cooling region, the third Rankine-Hugoniot condition (enthalpy-flux conservation) is not valid. But one can still find a a solution by assuming that the cooling of the plasma stops once the post-shock temperature has reached the pre-shock temperature. These shocks are called isothermal shocks , which can be a deceiving name, as right behind the shock the plasma can be much hotter than in the unshocked medium, but only further downstream of the shock the plasma has cooled to its original temperature. One can find a solution for the maximum compression ratio by setting T2 = T1 and P2 = n2 kT1 = χn1 kT1 = χP1 , with n the particle density. Inserting this in (4.6) gives     1 1 1 1 = + 1 − P2 = χP1 = P1 + 1 − ρ1 Vs2 ⇒ χ χ γ Ms2 γ Ms2 χ ⇒ χ 2 − (γ Ms2 + 1)χ + γ Ms2 = 0.

4.4 Radiative Shocks

79

We used here (4.8) to introduce the sonic Mach number. The non-trivial solution to this equation is χisothermal = γ Ms2 .

(4.58)

This shows that without magnetic pressure support radiative shocks can easily reach compression factors that exceed a factor hundred even for modest Mach numbers of Ms = 10.

4.4.2 Magnetically Supported, Radiative Shocks As we have seen in Sect. 4.2 the perpendicular component of the downstream magnetic field is compressed linearly with the compression factor. So for high compression factors the compressed magnetic field is providing pressure support. Core collapse supernova remnants are typically found in relatively high density regions, embedded in HII regions created by the stellar winds and UV flux from the progenitor stars and neighbouring stars. Under these circumstances the plasma temperatures are ∼5000 K, whereas the magnetic fields are typically 5 μG. This corresponds MA ≈ Ms and plasma-beta (4.21) values are β ≈ 1. This means that for radiative shocks the magnetic field pressure are expected to be the dominant pressure component in radiative shocks. To calculate the final density compression ratios we solve the momentumconservation flux condition, but now including magnetic field pressure (4.16), and using again T1 = T2 . Hence, P2 = χP1 , and B⊥2 = χB⊥1 . For convenience we use a perpendicular shock B = 0. Equation (4.16) then gives χP1 +

2 B⊥1 B2 χ 2 + ρ2 v22 =P1 + ⊥1 + ρ1 v12 8π 8π

⇒ χ

1 1 1 1 1 + χ2 + = + +1 χ γ Ms2 γ Ms2 2MA2 2MA2

⇒   χ 3 + βχ 2 − 2MA2 + β + 1 χ + 2MA2 =0. For the derivation we have made use of (4.21). This cubic equation has one trivial solution (χ = 1), which is used to rewrite the equation into the form (χ − 1)(χ 2 + Bχ + C) = 0. Solving for B and C, we find one physical solution for

80

4 Shocks and Post-shock Plasma Processes

Fig. 4.9 The final compression ratio as a function of sonic Mach number for isothermal, radiative shocks, for different values of the ratio (β) of the thermal over magnetic pressure in the unshocked medium

the compression factor:  1 1 χ = − (β + 1) + (β + 1)2 + 8MA2 . 2 2

(4.59)

This shows that for large enough MA (and β  1) the compression is linear proportional to the Aflvén Mach number χ ∝ MA . In the limit (β → ∞) we 2 . This is illustrated in Fig. 4.9. So radiative shocks can still have recover χ ∝ Mms compression ratios greatly exceeding the strong, non-radiative shock compression ratio of χ = 4. But for β ≈ 1, we find compression ratios up to χ = 30. Note that for radiative shocks we have Vs  200 km s−1 , which for typical sound speeds of cs ≈ 10 km s−1 , corresponds to Ms ≈ 20.

4.5 Shock Waves Mediated by Magnetic Precursors So far we have assumed that the upstream medium of a shock is not affected by the approaching shock wave; that assumption seems even part of the definition of the word “shock”. However, in astrophysical settings the approaching shock wave is already signalling its approach to the upstream medium. At the beginning of this chapter we already alluded to this, and called the affected upstream region the shock precursor. Here we discuss specifically the shock precursor caused by large scale Alfvén/magnetosonic waves traveling ahead of the shock: a magnetic precursor.

4.5 Shock Waves Mediated by Magnetic Precursors

81

For shock formation, the magnetosonic Mach number, Mms (4.20) has to be larger than one, with Mms itself depending on the sonic and Alfvén Mach numbers. The carrier of the magnetic fields are charged particles, and hence to first order Alfvén waves and magnetosonic waves only affect the charged particles. A complication now arises if the bulk of the gas is neutral, with only a tiny fraction of ions. This is the case in HI and molecular clouds, where typical the ionisation fractions are x ∼ 10−4 , and densities and temperatures are nH = 10−100 cm−3 and T ∼ 100 K for HI clouds, and nH  100–1000 cm−3 and T ∼ 10 K for molecular clouds. The magnetic fields may be similar to the typical interstellar medium field strength B ≈ 5 μG, but could be as high as mG fields in molecular cloud cores [499]. Even for the low magnetic fields of 5 μG, the magnetic pressure in molecular clouds tends to dominate:   −1  T −1 PB nH B ∼6 . Pth 5 μG 100 K 100 cm−3 In these environments the charged particles are only weakly coupled to the neutral gas. The Alfvén velocity under these circumstances is rather high, as only charged particles participate in the waves  VA =

  −1/2 B02 B x −1/2  nH ≈ 93 km s−1 . −4 −3 4πxρ 10 μG 10 100 cm

(4.60)

In the Rankine-Hugoniot relations the Alfvén and sonic Mach numbers do not enter because of the associated sound or magnetosonic waves, but because they express relations between pressure terms and ram pressure. So for a shock to arise the relevant Alfvén Mach number is not that of the charged particles, but a similar number but for the bulk gas, which is largely neutral:  MA,n =

  −1  1/2  nH 4πρn Vs B Vs ≈ 54 . 10 μG 100 cm−3 100 km s−1 B02

(4.61)

√ The ion Alfvén Mach number is much smaller, MA,i = xMA,n , and can be smaller than one, which means that Alfvén and magnetosonic waves can travel faster than the approaching shock wave. They will then affect first the charged particles, and subsequently slowly, through ion-neutral interactions, the neutral particles. In order to understand the ensuing shock structure, we need a multi-fluid approach[252, 329, 848]. Shocks with magnetic precursor are of special interest for mature supernova remnants interacting with molecular clouds, which have relatively slow shocks (Vs  100 km s−1 ), and or of particular interest for astrochemical processes resulting from the shock heating. In Chap. 10 a special section is devoted to the observational aspects these supernova remnants, which often have associated OH masers.

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4 Shocks and Post-shock Plasma Processes

Although the neutrals do not immediately react to the magnetic precursor, on a slower time scales they are affected by it through ion-neutral collisions. The collision rate < σin v > is, surprisingly, relatively independent of the collision velocity v and for most ions and neutrals < σin v > ≈ 2 × 10−9 cm−3 s [331]. These ion-neutral collisions ensure that on a larger scale the ions and the neutrals are coupled. So a long wavelength magnetic precursor will move ahead of the shock wave with the Alfvén velocity, compress the ions and the magnetic field, and then the ions impart the disturbance to the bulk, neutral gas. Effectively this also means that the magnetic precursor will be damping out. The typical length scale over which the damping of the Alfvén wave occurs relates to the momentum exchange flux between the coupled ion-magnetic field component and the neutrals, observing the momentum-flux conservation relation   d B2 2 ρi vi + Pi + = −Fin . dz 8π

(4.62)

with Fin the ion-neutral momentum-flux exchange per unit length: Fin = ni nn μ < σin v > (vi − vn ),

(4.63)

with z defined such that the shock direction is negative, and with (vi − vn ) the drift velocity between ions and neutrals, and μ = mn mi /(mn + mi ) the reduced mass. On the left-hand side of (4.62) the first term (ram-pressure) and the thermal pressure are proportional to the ion density ρi . These are very small compared to the magneticfield pressure (we already argued that in general the overal pressure is dominated by the magnetic field in HI and molecular clouds). So we can approximate (4.62) by d B2 ≈ ni nn μ < σin v > (vi − vn ). dz 8π

(4.64)

Replacing the gradient d/dz by −1/L, and approximating |vi −vn | ≈ 12 Vs (it cannot be Vs , but if it is substantially smaller than Vs , why bother?), we obtain the following length scale for the magnetic precursor:    L ≈  8πni nn

  VA2 mn + mi 1  = 2  mi mn 1  Vs mn nn < σin v > mi +mn < σin v > 2 Vs  2  −1  −1 nn VA Vs ≈1015 cm. −1 −1 −3 100 km s 10 km s 100 cm B02

(4.65)

In order to explain some of the physics of shocks with magnetic precursors we follow here the approach of [252], which builds upon the work described in [329, 848]. However, these types of shocks can be very complex, and many effects may alter the eventual structure of the shocks. One effect is the chemistry in the

4.5 Shock Waves Mediated by Magnetic Precursors

83

Fig. 4.10 Left panel: Illustration of the effects of a magnetic precursor, with two limiting cases: (1) Mms < Mcrit , a C-type shock (black lines), and (2) Mms > Mcrit , a J-type shock (blue). The solid lines indicate the flow velocity of the neutral gas, whereas the dashed lines show the ion velocity. Right panel: The phase space of ion velocity versus neutral velocity, for various neutral Alfvén Mach numbers (after [252])

precursor, resulting from the pre-shock heating, which may alter the composition and the ionisation fraction within the precursor. The enhanced cosmic-ray density and their gradients near supernova remnants will enhance these effects. Moreover, ion-neutral collisional heating may be offset by enhanced cooling, in particular for shocks interacting with molecular clouds, as the many rotation and vibration transitions of molecules make them efficient coolants. Even if cooling within the precursor is unimportant, the shock velocities considered here are below 200 km s−1 . So the post-shock region is likely radiative, and follows the physics described in Sect. 4.4. Of special interest is that the under certain circumstances, all the heating is done adiabatically within in the magnetic precursor, and no viscous shock is needed to fulfil the Rankine-Hugoniot conditions. Such a shock is called a C-type shock [329], where the “C” stands for continuous. In contrast, shocks with a magnetic precursor and a viscous shock are labeled J-type shocks , with “J” for jump. The two cases are illustrated in the left-hand panel of Fig. 4.10. To learn more about the shock structure we resort once again to the equations of density-, momentum-, and enthalpy flux, but now in the context of a two fluid system: ions (coupled to the magnetic field) and neutrals. We already provided the expression for the gradient in the ion momentum flux (4.62), and argued that the left-hand side of (4.62) is dominated by the magnetic pressure. However, this is not true for the total momentum-flux, which includes neutrals, ions and magnetic field, but for which the ion components can be ignored:

d B2 2 2 Pn + ρn vn + Pi + + ρi vi . = 0, dz 8π

(4.66)

We have used that the term Fin in (4.62) is an exchange term: what is removed from the ion-momentum flux is added to the neutral momentum flux, and hence the right

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4 Shocks and Post-shock Plasma Processes

hand side of (4.66) is zero. The magnetic field is tied to the ions, and responds to the density of the ions. For simplicity, we assume the magnetic field to be perpendicular only, and using (4.22) we obtain B2 B2 = χi2 0 , 8π 8π

(4.67)

with B0 the magnetic field strength far upstream, and χi (= Vs /vi ) the ion compression ratio.We can now rewrite (4.66) using only dimensionless variables: χn t +

1 1 1 + χi2 ≈1+ , 2 2 χn 2MA,n0 2MA,n0

(4.68)

2  with t ≡ kTn /(mn Vs2 ). The approximation on the right-hand side holds if MA,n0 2 Ms,n,0 , which is true for magnetically supported, cold molecular or HI clouds. We can perform a similar rewriting of te enthalpy-flux conservation (4.16). Ignoring the ion thermal pressure and kinetic energy terms, we use (4.67) to arrive at

γ d 1 Pn vn + ρn vn3 =Gn + Fin vn + , (4.69) dz γ − 1 2   2 d 2 B0 (4.70) vi χi =Gi + Ge − Fin vi , dz 4π

The terms with G are energy exchange terms between the two-fluids (see [252] for details), and all energy loss terms (mostly radiative) are contained in the cooling function . Adding the two equations, integrating over direction z, and using the same dimensionless variables as used for (4.68) we finally obtain: 2

γ 2χi 2 1 = 1 + 2 − , t+ 2+ 2 γ −1 χn MA,n0 MA,n0

(4.71)

with !z ≡

∞ dz 1 3 2 ρ0 Vs



the total energy loss up to a certain coordinate (0 ≤  ≤ 1). Combining (4.68) and (4.71) by eliminating t, and ignoring losses ( = 0) we obtain  1+

2 2 MA,n0

 (1 − χi )

χn2

      γ +1 γ 1 2 −2 − 1+ = 0, 1 − χ χ n i 2 γ −1 γ −1 2MA,n0

(4.72)

4.5 Shock Waves Mediated by Magnetic Precursors

85

which has two roots: 

γ +1 χn = γ −1



    γ 1 2 1 − χi 1+ ± 2 γ −1 2MA,n0

(4.73)

⎫  2  ⎪−1  2   ⎬    γ +1 γ 2 1  2 1 − χ + (1 − χ ) . 1 + 1+ i i 2 2 ⎪ γ −1 γ −1 2MA,n0 MA,n0 ⎭

These correspond to the solutions reported in [252]. Since the ions start moving first and then impart their momentum/energy to the neutrals, we must have vi ≤ vn (note that the velocity is measured in the frame of the shock, so the velocity starts at Vs far upstream, and decreases toward the shock). This implies χi ≤ χn . The solutions are presented in the vn versus vi graph in Fig. 4.10. The only physical solutions for χn are the ones above the dotted line (neutrals move slower than the ions). However, above a critical Mach number Mcrit both solutions of (4.73) will be above the line vi = vn . The state of the gas can now jump from the upper branch of the solutions to the lower branch. This jump corresponds to a shock: by definition (4.73) correspond to gas states that the shock conservation equations. In Fig. 4.10 this jump is indicated by the dashed line for the MA,n0 = 5 case. The above picture suggests the following scenarios for shocks with a magnetic precursor. For MA,n0 > Mcrit the ions start to compress as a result of a magnetic precursor, and with some delay the neutrals follow and start compressing as well. This continuous until a critical point is reached, and a shock in the neutrals bring the state to the lower branch of the solution. For MA,n0 < Mcrit the precursor also starts compression the ions, and with some delay the neutrals, but the final state reached by the gas occurs without the gas ever going through a shock jump. Nevertheless, the Rankine-Hugoniot relations are satisfied. For γ = 5/3 the critical Mach number is 2.76, a value we already encountered in Sect. 4.2. For γ = 7/5, corresponding to diatomic molecular gas, the critical Mach number is Mcrit = 3.2. These are upper limits: for oblique magnetic fields, and β > 0 the critical Mach numbers are lower. So far, we made the approximation that cooling was not important ( = 0). The most extreme form of cooling is when the temperature cools down to zero. In this case (4.68) reduces to χn =

2 2MA,n0 2 2MA,n0 + (1 − χi2 )

.

(4.74)

This equation is depicted in Fig. 4.10 (left) by the dot-dashed line, providing an alternative final outcome, namely the crossing of this maximum cooling line with vi = vn . All possible solutions will lie in between the solid line (no cooling) and the dot-dashed line (maximum cooling). The real trajectory may even be quite complex. For example, the gas-flow may be continuous, but along the trajectory the gas state may still change from supersonic to subsonic. Such a possibility is sometimes designated as a C∗ -type shock [252].

Chapter 5

Supernova Remnant Evolution

5.1 Supernova Remnant Evolution: Four Phases After the supernova explosion about 1051 erg of kinetic energy is dispersed in the ambient medium of the supernova. The interaction of the supersonically moving ejecta with the circumstellar/interstellar medium (CSM/ISM) results in the formation of a shock wave (Chap. 4), which creates a shell of shock-heated plasma. The interaction of the outermost ejecta with the CSM/ISM marks the beginning of the supernova remnant. Note that this interaction with the CSM can start even in the first weeks after the explosion, if the supernova progenitor had a dense wind. The subsequent evolution of the supernova remnant shock wave depends largely on the density and structure of the local CSM/ISM. As more and more gas is swept up and heated by the forward shock, the energy is spread over increasingly more mass, and as a result the shock wave decelerates. In addition, energy is lost from the shell due to radiation, which becomes critical once the shock velocity becomes Vs  200 km s−1 (4.4). The evolution of the supernova remnant is often simplified by dividing it into four evolutionary phases (e.g. [1243]): (i) The ejecta-dominated phase: the ejecta mass dominates over the mass swept up by the forward shock (Mej > Msw ), and most of the explosion energy is still contained in the freely expanding, cold ejecta. This phase is sometimes referred to as the free expansion phase, but this name can be misleading as will be explained in Sect. 5.6.1. (ii) The energy-conservation phase or Sedov-Taylor phase: the swept up mass dominates over the ejected mass (Mej < Msw ), and the explosion energy is now contained in the form of internal energy and kinetic energy of the hot, expanding shell. Radiative losses are still negligible. This phase is also named the adiabatic phase, but it should be remembered that shock heating increases the entropy of a gas. So also this name is somewhat misleading. © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_5

87

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5 Supernova Remnant Evolution

(iii) The pressure-driven phase or snow-plough phase: in this phase radiative losses have become important, so one can no longer assume conservation of energy. Instead the evolution of the shock wave is governed by momentum conservation. (iv) The merging phase: once the blast-wave velocity approaches the sound speed or Alfvén velocity, the shock wave will slowly disappear and the shell will expand subsonically. This marks the end of the supernova remnant, but some of its identity will be kept for some time, as the remnant has left behind a hot plasma bubble. In this chapter we will have a closer look at these different phases. But keep in mind that the ambient medium of supernova remnants can be complex, and the description of supernova remnant evolution in terms of these four phases only partially captures reality. In some cases one can apply these phases to different parts of one singlesupernova remnant. For example, the supernova remnant RCW 86, the probable remnant of SN 185, has radiative shocks (4.4) in the southwestern part (corresponding to phase ii), and very fast shocks (∼3000 km s−1 ) in the northeastern part [1187, 1256], even emitting X-ray synchrotron radiation (12.2), putting this part in phase i or ii. These contrasts in evolutionary developments of the shocks are usually caused by strong density gradients, or linked to a specific structure of the ambient medium, such as a molecular cloud located on one side of the remnant. The strong velocity gradients in RCW 86, but also similar, less extreme contrasts in the Cygnus Loop remnant [173, 1002], are likely caused by the evolution of the supernova remnant in a wind blown bubble. Some shocks can then still move through the low density interior of the bubble, whereas other parts may have reached the dense shell surrounding the wind bubble. This aspect of supernova remnant shock evolution will be discussed in Sect. 5.8.

5.2 The Expansion Parameter As we will describe later, during the various evolutionary phases the evolution of the blast wave radius (or forward-shock radius), Rfs , can often be approximated as  m t Rfs ≈ R0 , t0

(5.1)

with t the age of the supernova remnant, and R0 and t0 some fiducial parameters. The parameter m is called expansion parameter (sometimes called the deceleration parameter). Using approximation (5.1), the shock velocity is given by Vfs =

R0 dRfs ≈m dt t0

 m−1 t Rfs . =m t0 t

(5.2)

5.3 The Reverse Shock

89

This equation shows that the expansion parameter can be measured if the shock velocity is measured and the age of the supernova remnant is known, like in the case of historical supernova remnants. One can even define m independently of the exponential evolutionary model of the blast wave as m≡

Vfs t . Rfs

(5.3)

There is a subtle difference in defining m as in (5.3), because in (5.1) m refers to the current evolution of the supernova remnant, whereas in (5.3) the shock radius may have resulted from a more complicated evolutionary path than described by (5.1). Note that for some supernova remnants one can measure shock proper motions, which can be converted into actual shock velocities if the distance is known. However, since in (5.3) both Vfs and Rfs are products of angular measurements and the distance, m itself can be determined independently of the distance, provided the age of the supernova remnant is known.

5.3 The Reverse Shock During the supernova explosion the stellar material is heated by the supernova shock and heat deposition by radioactive decay of freshly synthesised, unstable elements. However, the fast expansion of the ejecta, result in rapid adiabatic cooling. As a result, the ejecta will be cold after a few months to years. The adiabatic cooling corresponds to a rapid decline in internal pressure P . For an ideal gas we have: P V γ =constant

(5.4)

⇒  Pej = P∗

Rej R∗

−3γ

 = P∗

Rej R∗

−5

 , kTej = kT∗

Rej R∗

−2 .

with V the volume, and P∗ and T∗ the initial pressure and temperature at a radius R∗ . The fastest moving, outermost, ejecta will create a shock wave in the CSM/ISM, creating a shock-heated shell, which decelerates. The outermost, unshocked ejecta will then move faster than the shell, and will collide with it. If this collision occurs at supersonic speeds. a shock wave will form, which (re)heats the adiabatically cooled ejecta [796]. This shock wave is called the reverse shock (subscript rs), and to distinguish it from the supernova blast wave, the latter is often referred to as the forward shock (fs). The reverse shock (re)heats the ejecta, which is the prime reason for the strong X-ray line emission from metal-rich ejecta in young supernova remnants (Chap. 9). Figure 5.1 shows a schematic drawing of a young supernova remnant. It shows that

5 Supernova Remnant Evolution

d

on isc

t inu

ity

er se shoc

k

Ejecta

Con t

ard shock Fo r w

Co n

R ev

Ejecta

ac td

on isc

R ev tinu

For ward shock

ta

ct

90

erse shock

ity

Fig. 5.1 Schematic view of the forward shock/reverse shock system (after [796])

the shock-heated shell consists of two parts, roughly in pressure equilibrium: the outermost shell region consists of ISM/CSM heated by the forward shock, whereas more toward the centre is the hot ejecta, heated by the reverse shock. Inside the reverse shock is the cold freely expanding ejecta. The boundary between the shockheated ejecta and shock-heated CSM/ISM is called the contact discontinuity. As the hot ejecta and shock-heated CSM/ISM are likely to have different densities, Rayleigh-Taylor instabilities are likely to wrinkle this boundary (Sect. 5.9).

5.3.1 The Reverse Shock Velocity in the Shock- and Observer-Frame Although the name reverse shock suggests that the shock moves toward the centre of the supernova remnant, this is not necessarily the case. In fact, during most of the ejecta-dominated phase the reverse shock radius is moving outward. The use of the term “reverse shock velocity” can be misleading, as it is not always clear to what frame it refers. In the following we indicate velocities in the observer’s frame with V˜ . In this observer’s frame the reverse shock velocity is dRrs . V˜rs = dt

(5.5)

5.3 The Reverse Shock

91

This velocity will initially be positive, i.e. the reverse shocks move in an outward direction in the observer’s frame. But after at a certain age of the supernova remnant the reverse shock in the observer’s frame turns around and V˜ will become negative. The heating of the ejecta is determined by the shock velocity in the frame of the shock: Vrs = Vej − V˜rs =

dRrs Rrs − . t dt

(5.6)

We have made use here of the fact that the ejecta has been freely expanding (V = R/t) up to the moment it enters the reverse shock.

5.3.2 The Condition for Forming a Reverse Shock In order for the reverse shock to form the ejecta must have cooled down sufficiently, and the relative velocity of the newly formed hot shell compared to the ejecta has to be high enough, i.e. the Mach number in the shock frame should be larger than one. In order to quantify this condition and to be able to use the strong shock limit (4.1), we instead require Ms  5. Using (5.4) and approximating Vrs = f Vej , with f a potentially small number, as the shock-heated shell has not yet much decelerated, we can derive the condition for the formation of the reverse shock as 2 2 1 ρej Vrs2 1 μmp f Vej =  5, (5.7) γ Pej,unsh γ kTej,unsh  2  2  −2  −2 R 4 R R 1 3 μmp Vej kTej =kT∗ = = μmp Vej2 , R∗ 16 3 R∗ 3 R∗

Ms2 =

⇒ 3 Ms2 = f 2 γ



R R∗

2 =

9 2 f 5



R R∗

2  5.

We assumed here γ = 5/3 and assumed that the ejecta were initially heated to T∗ by the shock wave inside the supernova (no additional radioactive heating), and after that the ejecta freely expanded. Condition (5.7) shows that, even if initially f is small (i.e. the shell and unshocked ejecta move with nearly equal speeds), it does not require a long time for the reverse shock to form. For example, for f = 1% we obtain R ≈ 167R∗ . If we take R∗ ≈ 1014 cm and Vej = 20,000 km s−1 this would translate into a time scale of 0.26 year for the reverse shock to develop. The actual time scale depends on the size of the star at the moment of explosion (∼R∗ ) and how much circumstellar material there, is as this determines the ratio f , which itself depends on how much the forward shock has decelerate due to interaction with

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5 Supernova Remnant Evolution

the circumstellar medium. To be more precise: if initially the reverse shock in the observer’s frame expands as Rrs ∝ t m , then (5.6) implies that f ≈ 1 − m.

5.3.3 The Turning Around of the Reverse Shock Equation (5.6) allows us to obtain a rough estimate of the radius at which the reverse shock will change from moving outward in the observer’s frame to moving toward the centre of the supernova remnant. Making the assumption that the pressure behind the reverse shock is a fraction, β  1, of the pressure behind the forward shock (this is a reasonable assumption, see Sect. 5.6.1), we can write for a strong shock     1 1 Pej,1 = 1 − ρej,0. Vrs2 = βPamb,1 = β 1 − ρ0 Vf2s , χ χ

(5.8)

with subscripts 0 and 1 indicating unshocked and shocked quantities, respectively, and Pamb referring to the pressure in the shell of shocked ambient medium. At the moment the reverse shocks changes from expanding outward to moving back toward the centre we have dRrs /dt = 0. Combining this condition with (5.6) and setting χ = 4 we obtain Rrs Vrs = = t

 β

ρ0 Vfs . ρej,0

(5.9)

Replacing Vfs by using expression (5.3) shows that the maximum radius of the reverse shock is  ρ0 mRfs . (5.10) Rrs ≤ β ρej,0 This indicates that the ratio ρ0 /ρej,0 has to be of the order of unity for the reverse shock to travel backwards in the observer frame, because Rrs /Rfs  1, 25 < m < 1 and β  1.

5.4 Self-similar Solutions An important class of models to describe the evolution of the dynamics and structure of supernova remnants is the class of self-similar models. In these models the evolution of the supernova remnant is parameterised by one or more self-similar parameters, which are a combination of basic supernova remnants properties and variables, such as explosion energy, ejecta mass, ambient density, age and radius.

5.4 Self-similar Solutions

93

5.4.1 The Self-similar Sedov-Taylor Solution The best-known self-similar explosion model is the Sedov-Taylor model [1033, 1101], named after Leonid Sedov and Geoffrey Taylor, who independently came up with the model in the context of explosions in the Earth atmosphere (their research was part of the Cold War efforts on both sides of the Iron Curtain). The basic assumption is that the explosion energy is released suddenly and that all the energy is immediately transferred to the ambient medium and that there are no energetic losses: the explosion energy equals the kinetic and internal energy of the hot shell. These two assumptions apply directly to phase (ii). In the preceding phase (i) the unshocked, freely expanding ejecta constitute an energy reservoir, which are being transferred to the shell over a prolonged length of time. A further assumption is that the blast wave is in the strong shock regime M  1. This implies that for the ambient medium the pressure is assumed to be P0 = 0. The Sedov-Taylor model is an application of the Buckingham π theorem [217] according to which physical quantities depending on n parameters can be rewritten in terms of p = n − k parameters. In the case of a supernova explosion we would like to know the shock wave radius Rfs as a function of the age t of the supernova remnant (p = 2), whereas we have n = 4 dimensional quantities: the explosion energy E, the ambient density ρ0 , Rfs and t, which depend on three fundamental units, for length (cm), mass (g) and time (s). A dimensionless variable ξ can be constructed by solving [ξ ] =[Rfs ]a × [t]b × [E]c × [ρ0 ]d =cma × sb × gc cm2c s−2c × gd × cm−3d =cma+2c−d × sb−2c × gc+d . For ξ to be dimensionless we have c = −d, b = 2c and a = 5d. Setting d = 1 we see that ξ can be constructed as ξ=

R 5 ρ0 . t 2E

(5.11)

The value of the self-similar parameter for an ambient medium with adiabatic index γ = 5/3 turns out to be ξ = 2.025. The shock radius and its derivative, the shock velocity, are therefore given by 

Et 2 Rfs =1.15 ρ0

1/5 cm =

2/5 −1/5  E 1/5  t  n H pc, =5.0 1 cm−3 1051 erg 1000 yr

(5.12)

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5 Supernova Remnant Evolution

Vfs =1.15 =975

1 5



E ρ0

1/5

t −3/5 =

2 Rfs 5 t

(5.13)

2/5 −1/5  E 1/5  t  n H km s−1 , 1 cm−3 1051 erg 1000 yr

with nH the pre-shock hydrogen number density. This means that the expansion parameter (5.3) for the Sedov-Taylor model is m = 2/5.

5.4.2 An Alternative Derivation of the Sedov-Taylor Solution The above manner of deriving the Sedov-Taylor solution is somewhat abstract. To make the relation physically more intuitive we can look at it from the point of view of a thin shell containing all the explosion energy. In Chap. 4 we have seen that for a strong shock, the shock velocity and pressure in the shell are related by   1 P2 = 1 − ρ0 Vfs2 , χ

(5.14)

with χ = (γ + 1)/(γ − 1) the shock compression   ratio (χ = 4, for γ = 5/3). The 1 plasma velocity behind the shock is v2 = 1 − χ Vfs . The volume of the shell will 3 /3, as this gives an average post-shock density of ρ = χρ . be V ≈ χ1 4πRfs 2 0 We express the energy as the sum of the total internal energy U V and kinetic energy Ekin , which results in

E =U V + Ekin =

1 1 P2 V + ρ2 V γ −1 2

  1 2 2 1− Vfs χ

(5.15)

3 =Cρ0 Vfs2 Rfs ,

with C=

4π 3



4 ≈ 2.35. (γ + 1)2

(5.16)

We rewrite this as a differential equation: −1/2

Vfs =C −1/2 E 1/2 ρ0

−3/2

Rfs

⇒ 5 Rfs =

 2 5 C −1 Eρ0−1 t 2 . 2

=

dRfs dt

(5.17)

5.4 Self-similar Solutions

95

This expression is similar to Eq. (5.12), but the numerical constant ξ = (5/2)2 /C = 2.65 differs somewhat from the previous value, due to the thin-shell approximation.

5.4.3 The Sedov-Taylor Solution for a Shock Propagating Through a Stellar Wind The density profile for a stellar wind can be derived by considering mass conservation: M˙ =4πr 2 ρ(r)vw

(5.18)

⇒  −2 M˙ r = ρ0 , ρ(r) = 2 4πr vw r0 with M˙ the mass loss rate of the progenitor, and vw the wind velocity (Sect. 5.8). We can insert this relation between density and radius in (5.17), giving 1/3   3 2 −1 −2 −1 2 Rfs = C ER0 ρ0 t , 2

(5.19)

with C a different constant than in (5.17). We see that for the stellar wind case the expansion parameter is m = 2/3. This solution may apply to the young supernova remnant Cassiopeia A, which is evolving in the dense wind of its progenitor, and for which the measured expansion parameter is m ≈ 0.66 [300, 899, 1183].

5.4.4 The Expected Size Distribution of Supernova Remnants In Sect. 3.4 the diameter (D) distribution of supernova remnants in the Magellanic Clouds was discussed. Observationally it was found that dN/dt ∝ D, whereas the expectation was dN/dt ∝ D 3/2 [117]. This expectation is based on the SedovTaylor size evolution of supernova remnants. The idea being that the longest period of growth of a supernova remnant is in the Sedov-Taylor phase of the evolution. We briefly explain here the expected value, for which we assume again a selfsimilar evolution characterised by the expansion parameter m (5.1). The diameter under this condition is D/D0 = (t/t0 )m .

(5.20)

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5 Supernova Remnant Evolution

So we obtain: dN dN dt dN t0 = = dD dt dD dt mD0

 −(m−1) t . t0

(5.21)

Inserting the inverted version of (5.20) into the last equation gives dN dN t0 = dD dt mD0



D D0

 1 −1 m

(5.22)

.

Here dN/dt is the supernova rate, which is more or less constant. For the Sedov-Taylor phase we have m = 2/5, and hence dN/dD ∝ D 3/2 . The measured relation implies that m = 1/2, which is not dramatically different from m = 2/5. In [117] an explanation is given in terms of the density distribution in which supernova remnants evolve, implying a density probability function of ρ −1 .

5.5 The Internal Structure of Self-similar Explosions The internal structure of the explosion according to the Sedov-Taylor model can also be solved using similarity arguments. This involves rewriting the density, velocity, and pressure in terms of the similarity parameter  related to the radial coordinate: ρ(r, t) =Aρ0 X (),

(5.23)

r v(r, t) =B U (), t  r 2 P (r, t) =Cρ0 P(), t

(5.24) (5.25)

By choosing the conditions X (1) = 1,U (1) = 1 and P(1) = 1, observing the strong shock conditions at the shock front, and making use of the relation Vfs = 2 5 Rfs /t, the constants A, B, and C can be shown to be 2 2 3 2 γ +1 = 4, B = = , C= A= γ −1 5γ +1 20 γ +1

 2 3 2 = , 5 25

(5.26)

with the numerical values valid for an explosion in a monatomic gas (γ = 5/3). The resulting equations can be substituted into the fluid dynamic equations for conservation of mass, momentum and energy flux: 1 ∂  2  ∂ρ r ρv =0, + 2 ∂t r ∂r ∂v ∂v 1 ∂P +v + =0, ∂t ∂r ρ ∂r

(5.27) (5.28)

5.6 Self-similar Models for the Ejecta-Dominated Phase





1 2 ∂ 1 ∂ 1 2 2 U + ρv + 2 r U + P + ρv =0. ∂t 2 r ∂r 2

97

(5.29)

In Chap. 4 we encountered these equations in non-differential form for plane parallel geometry. It turns out that one can analytically solve the resulting equations in terms of X , U and P. The solutions can be written as functions of a single parameter x, which for γ = 5/3 turn out to be [1033]:  2/13  −82/195   r 10 −2/5 50 5 x x−4 − 5V = , Rsh 3 3 2 v 10 R = x , v2 3 Rsh 9/13 82/13   ρ 50 −6 5 x−4 − 5x = , − 10x) (4 ρ2 3 2  82/15   10 6/5 P −5 5 x − 5x = , (4 − 10x) P2 3 2 T P ρ2 , = T2 P2 ρ

(5.30) (5.31) (5.32) (5.33) (5.34)

with 6/25 ≤ x ≤ 3/10. The subscript 2 refer to the quantities immediately behind the shock front, which are v2 = 34 Vfs , ρ2 = 4ρ0 , P2 = 34 ρ0 Vfs2 . Figure 5.2 graphically shows the results. What is obvious from this figure is that toward the center the density drops asymptotically to zero, and most of the mass is concentrated near the shock. Since the pressure levels of at a finite value toward the centre, the temperature goes asymptotically to infinity toward the centre.

5.6 Self-similar Models for the Ejecta-Dominated Phase In order to handle the evolution in the ejecta-dominated phase, some simplifying assumptions have to be made regarding the ejecta-density distribution. For selfsimilar solutions the density structures that can be used for self-similar models are power-law distributions [254, 1128], since power laws are scale free and, in addition, computer simulations of supernova explosions indicate that the outerejecta distribution can be reasonably well approximated by a power law distribution with slopes in the range n ≈7–12 [784, 1147]. Since after the explosion the ejecta are expected to expand homologously, i.e. r = vt during free expansion, we can define the density as a function of velocity, rather than radius r. In order to obtain a finite mass it is also necessary to have a break in the power law distribution. Chevalier [254], for his well-known models,

98

5 Supernova Remnant Evolution

Fig. 5.2 The internal structure of an explosion according to the Sedov-Taylor solution. All quantities are scale to the quantity just behind the shock front. Note that the velocity has been scaled down by a factor 10

assumed that the ejecta-density distribution is constant in the core, defined by a core velocity, vcore = rcore /t, resulting in a density distribution ⎧  −3 ⎪ ⎨ρcore,0 t t ρ(v, t) =  0 −3  −n ⎪ v ⎩ρcore,0 t t0 vcore

for v ≤ vcore

(5.35)

for v > vcore

For this density distribution the ejecta mass and the kinetic energy are given by )

∞

Mej = )

4π(vt)2 ρ(v, t)d(vt) =

0 ∞

Ekin = 0

4π n 3 ρcore,0 t03 vcore , 3 n−3

n 1 4π 5 ρcore,0 t03 vcore 4π(vt)2 ρ(v, t)v 2 d(vt) = . 2 10 (n − 5)

(5.36) (5.37)

Dividing the kinetic energy by the ejecta mass shows that the core velocity can be expressed as  vcore =

10 (n − 5) Ekin . 3 (n − 3) Mej

(5.38)

5.6 Self-similar Models for the Ejecta-Dominated Phase

99

The density in the ejecta scales with 5/2

ρcore,0 t03 =

4π 3

−3/2

Mej Eej   n n−3

10 n−5 3 n−3

3/2

(5.39)

There is no valid solution for n ≤ 5, since the kinetic energy does not converge for v → ∞. Note that the kinetic energy contained by the mass within the core is Ekin,core n−5 . = Ekin n

(5.40)

This shows that for n → 5 an increasingly large fraction of the kinetic energy is contained by the ejecta outside the core region. A similar calculation for the mass gives: Mej,core n−3 . = Mej n

(5.41)

5.6.1 The Chevalier Self-similar Model for Young Remnants As long as the reverse shock has not penetrated into the ejecta core region the supernova remnant evolution can be approximated by the Chevalier model [254], which is based on the assumption of a power-law density distribution of the outer ejecta (5.35) and assumes that also the ambient medium has a power-law density distribution,  ρ(r) = ρ0

r r0

−s

;

(5.42)

for a homogenous ambient medium s = 0, whereas for a supernova exploding inside a stellar wind we expect s = 2 (5.18). There are three evolving quantities, ρ, r, and t that need to be combined to find a self-similar solution. Combining (5.42) and (5.35) that the characteristic radius of the supernova remnant should evolve as Rc = At (n−3)/(n−s).

(5.43)

In the Chevalier model the characteristic radius is taken to be the contact discontinuity, the interface between shocked ejecta and the shocked ambient medium, but since the model is self-similar the forward-shock and reverse-shock radius have the same time dependence. The expansion parameter for the Chevalier model is, therefore, m = (n − 3)/(n − s), which for s = 0 and n = 7 − 12 gives m = 0.57 − 0.75, and for s = 2 we find m = 0.8–0.9. For n = 5 we obtain the same radius

100

5 Supernova Remnant Evolution

evolution as for the Sedov-Taylor model (Sect. 5.4). The reason is that for n → 5 all kinetic energy is concentrated in the outermost ejecta. So the total kinetic energy is instantaneously transferred to the ambient medium, which is the definition of “a point explosion”. Note that from the very early stages of interaction with the ambient medium supernova remnants are not expected to expand with m = 1. Hence, calling the earliest evolutionary phase the free-expansion phase is somewhat misleading. Like in the case of the Sedov-Taylor solution one can rewrite expressions for the physical quantities, v, ρ, P in terms of dimensionless equations, with a dimensionless scaling parameter ξ = t −1 r 1/m , and inserting them in (5.27)–(5.29). There is no analytical solution to these equations, but [254] lists numerical solutions, which are depicted in Fig. 5.3. Some of the key properties of the self-similar solutions are listed in Table 5.1. Note the very different structure of s = 2 (stellar wind) models and the s = 0 (uniform ambient density) models: the s = 2 models show a more extended region of shocked ambient medium, and the density rises steeply toward the contact discontinuity, whereas for the s = 0 models the density drops to ρ = 0 at the contact discontinuity. However, the steep density gradients lead to Rayleigh-Taylor

Fig. 5.3 The normalised density, pressure, velocity, and entropy profiles for supernova remnants in the ejecta-dominated phase, according to the self-similar models of [254]. Left: For an s = 0 (uniform) ambient density profile. Right: For an s = 2 (stellar wind) density profile. Both models are for ejecta density profiles with n = 7. The radius is normalised to the radius of the contact discontinuity Table 5.1 Reverse shock/forward shock properties of the Chevalier (1982) model s 0 0 0 2 2 2

n 7 9 12 7 9 12

R1 /Rc 1.181 1.140 1.121 1.299 1.250 1.226

R2 /Rc 0.935 0.9360 0.974 0.970 0.981 0.987

ρ2 /ρ1 1.3 3.1 7.2 7.8 19 46

P2 /P1 0.47 0.55 0.60 0.27 0.33 0.37

v2 /v1 1.253 1.263 1.255 1.058 1.090 1.104

M2 /M1 0.5 0.93 1.6 0.82 1.6 2.7

5.6 Self-similar Models for the Ejecta-Dominated Phase

101

instabilities, which will in reality tend to smear out the density spikes and dips at the contact discontinuities (Sect. 5.9).

5.6.2 The Transition from Ejecta-Dominated to Adiabatic Phase For many young supernova remnants the self-similar models of Chevalier [254] are not applicable as the reverse shock has already penetrated the core ejecta layers (Rrs /t < vcore ), whereas they are still too young to be described by the SedovTaylor solutions. For example, SN 1006 has an expansion parameter of m = 0.54 [625], somewhere in between the Sedov-Taylor value of m = 0.4 and the lowest m of the Chevalier model. A special class of self-similar models aims to describe the overall evolution of the radii from the earliest phases to the Sedov-Taylor phase. This was pioneered by Truelove and McKee [1128], and later expanded to include stellar-wind profiles for the ambient medium [697, 815, 1097]. The Truelove and McKee-like models adapt the same approximation for the ejecta density structure (5.35) as Chevalier [254]. Under this assumption a set of models can be made characterised by the parameters s and n, and self-similar variables that are based upon the explosion properties Mej and Ekin , and the ambient ˙ w (5.18) for s = 2. Writing ρ = ηs r −s , density properties ρ0 , for s = 0, or M/v ˙ with ηs = ρ0 for s = 0 and ηs = M/(4πv w ) for s = 2, the characteristic variables are [1128]: Mch =Mej ,

(5.44)

1/(3−s) −1/(3−s) ηs ,

Rch =Mej

−1/2

(5−s)/(2(3−s)) −1/(3−s) ηs .

tch =Ekin Mej

(5.45) (5.46)

For example, for s = 0 the characteristic variables become 

 Mej 1/3  nH,0 −1/3 pc, M 1 cm−3 −1/2    Mej 5/6  nH,0 −1/3 Ekin =424 yr. 1051 erg M 1 cm−3

Rch =3.1 tch

(5.47) (5.48)

The age can now be expressed as t ∗ = t/tch and models can be constructed for the ∗ = R /R , forward and reverse shock radius in terms of dimensionless radii: Rfs fs ch ∗ ∗ Rrs = Rrs /Rch as a function of t . For example, for n > 5 a solution for the blast wave is [1097]: ∗ = Rfs

 −α  −α/(5−s) 1/α ς t ∗(n−3)/(n−s) + ξ t ∗2 .

(5.49)

102

5 Supernova Remnant Evolution

The parameters ς and ξ have be determined numerically for a gives n, s. One easily recognises here the expected asymptotic behaviour: for small t ∗ one obtains Rfs ∝ t ∗(n−3)/(n−s) (5.43), whereas for t ∗ → ∞ one obtains the Sedov-Taylor solution. This also shows that for s = 0 the parameter ξ corresponds to ξ = 2.025 (5.11). The parameters α and ς are tabulated in [1097] for various values of n and s. In the Chevalier model the reverse shock radius is a given fraction (lED ) of the forward shock radius. The analytical solutions by Truelove and McKee and follow up work in [697, 815] describe the development of the reverse shock. Here we give the most extensive solution derived in [815]. It divides the evolution in two phases: the early time phase when the reverse shock heats the outer ejecta layers t < tcore (corresponding to the Chevalier model) and the late phase, when the reverse shock has penetrated the ejecta core region. For the early phase (t ≤ tcore ) we have ∗ Rrs =

1 lED

*

∗n−3 vcore

3 (3 − s)2 4π n(n − 3)

+1/(n−s)

t ∗(n−3)/(n−s) ,

(5.50)

with  ∗ vcore

=

10(n − 5) , 3(n − 3)

lED =1 +

8 0.4 , + n2 4−s

 φE =[0.65 − exp(−n/4)] 1 − s/3,   s−2 1/(3−s) 1 3 (3 − s)2 lED ∗ . tcore = ∗ 4π n(n − 3) φED vcore

(5.51) (5.52) (5.53) (5.54)

The reverse shock velocity is given in this phase by Vrs = Vfs / lED . For t > tcore the evolution of the reverse shock radius and velocity is  ∗ 

∗ ) ∗ ) Rfs (t ∗ = tcore 3 − s Vfs∗ (t ∗ = tcore t = − ln ∗ t ∗, ∗ lED tcore n−3 lED tcore

∗ )  ∗  ∗ ) Rfs (t ∗ = tcore 3 − s Vfs∗ (t ∗ = tcore t Vrs∗ = − ln + 1 . ∗ ∗ lED tcore n−3 lED tcore

∗ Rrs



(5.55) (5.56)

All the reverse shock velocities here are in the frame of the observer. Figure 5.4 shows examples of the evolution of the forward and reverse shock for both s = 0 and s = 2. The input parameters have been chosen to match the measurements of the shock radii and velocities for Cassiopeia A (t ≈ 340 yr). The s = 2 model is more realistic for Cas A (e.g. [1167]), but it fails to reproduce that the reverse shock velocity is close to dRrs /dt ≈ 0 km s−1 , at least in the western

5.7 The Late Time Evolution of Supernova Remnants

103

Fig. 5.4 Examples of Truelove and McKee models for the evolution of supernova remnants [1128]. The black lines correspond to a uniform density medium (n = 7, s = 0), and the red line to a stellar wind model (n = 9, s = 2, based on [815, 1097]). The model parameters have been adjusted to obtain the correct shock radii of Cassiopeia A (e.g. [90]), but for s = 0 model the model is fine tuned to show the turnaround of the reverse shock. Left: the forward-shock (solid), and reverse-shock (dashed) radii. Right: the forward and reverse shock velocity. The dot-dashed line shows the reverse-shock velocity in the frame of the unshocked ejecta (s = 0: n = 7, Mej = 4 M , nH = 5 cm2 , Ekin = 3 × 1051 erg; s = 2: n = 9, Mej = 3 M , M˙ = 1.0 × 10−4 M /yr, vw = 10 km s−1 , Ekin = 5.6 × 1051 erg)

part [385, 1180]. A reversal of the shock velocity takes place typically rather late in s = 2 models; see the (3 − s) factor in (5.55). The failure of the Truelove and McKee model to reproduce the reverse shock properties of Cas A may be hint that the wind-loss history of Cas A was more complex than the assumed profile produced by the steady progenitor wind, or the ejecta profile deviates from (5.35).

5.7 The Late Time Evolution of Supernova Remnants At late times the assumption that all the explosion energy is still contained in the kinetic and internal energy of the supernova remnant is no longer valid. As we have seen in Sect. 4.4, radiative energy losses rapidly increase for Vfs  200 km s−1 . This corresponds to the start of the pressure driven or snow-plough phase. Instead of energy conservation the evolution is now governed by the conservation of radial momentum: MVfs =

4π 3 dRfs ρ0 Rfs = constant. 3 dt

(5.57)

Let us denote the radius, shock velocity and age at which the radiative phase starts with respectively Rfs,rad , Vfs,rad and trad . Integration of the momentum conservation equation shows that the solution is of the form 4 Rfs (t) = C1 (t − trad ) + C2 .

(5.58)

104

5 Supernova Remnant Evolution

4 By definition we have Rfs (trad ) = Rfs,rad , and hence C2 = Rfs,rad . We also require 3 that for t = trad , Vfs = dRfs /dt = Vfs,rad . This results in C1 = 4Rfs,rad Vfs,rad . Filling in these expressions and rewriting t as a function of Rfs we find [1121]

Rfs,rad t = trad + 4Vfs,rad



Rfs Rfs,rad



4

−1 .

(5.59)

Once a supernova remnant has advanced well into the radiative phase (t  trad ) one can see that the expansion parameter approaches m = 0.25. From (5.59) a shock velocity can be derived of  Vfs = Vfs,rad

Rfs Rfs,rad

−3 (5.60)

.

We can estimate the values for Rfs,rad , and trad using the Sedov-Taylor solutions (5.12) and (5.13). For Vfs,rad ≈ 200 km s−1 this gives trad =1.4 × 1012 Rfs,rad

 E 1/3 51

s ≈ 44,600

 E 1/3 51

yr nH  E 1/3  E 1/3 51 51 =7.0 × 1019 cm ≈ 23 pc, nH nH nH

(5.61) (5.62)

with nH the pre-shock hydrogen density and E51 the explosion energy in units of 1051 erg. Note that by the time that the radiative phase is reached, the forward shock 3 ≈ 1770n M . will have swept up a mass of M ≈ (4π/3)1.4mpnH Rfs H  An example of a supernova remnant that is in the snow-plough phase is Simeis 147 (S147, Fig. 5.5), which has a radius of ∼31 pc. Applying (5.59) and (5.60) gives an approximate age for this supernova remnant of 95 kyr, and a shock velocity of 81 km s−1 . The shock velocity matches well with the measured shock velocity of 80 km s−1 [656]. On the other hand, S147 is associated with a B0.5 runaway star, which could have a binary companion of the progenitor star of S147. Tracing the runaway path back to the pulsar inside S147 (PSR J0538+2817) gives an age of 30 ± 4 kyr [319], much smaller than the above estimate.

5.8 Supernova Remnant Evolution Inside Wind Bubbles So far we have dealt with the evolution of supernova remnants shocks assuming as simplified structure of the ambient medium, either we assumed a homogeneous medium with a constant density ρ0 or we assumed a stellar wind profile of ρ(r) ∝ r −2 (5.18). The ambient medium can in general be more complex, with hotter regions, molecular clouds, and low density densities, with still in them cloudlets. A more generic complexity of the ambient medium is that it is likely to have been

5.8 Supernova Remnant Evolution Inside Wind Bubbles

105

Fig. 5.5 An example of a supernova remnant in the snow-plough phase: Simeis 147 (Spaghetti Nebula), seen in Hα (red), [OIII] (green) and [SII] (blue) narrow-band filters. The shell has a diameter of about 3◦ , corresponding to ∼62 pc at a distance of 1.2 kpc [984] (Credit: Nicolas Kizilian, France (http://www.astropixels.fr))

modified by the wind of the progenitor star. This is expected to be always the case for core collapse supernovae, but there are also strong indications that Type Ia supernova remnants like SN 1604 [264, 1174], SN 1572 [1277], and RCW 86 are evolving in wind blown bubbles [212, 1229]. We concentrate here on the wind bubbles created by core collapse supernova progenitors, which have main sequence masses of MMS  8 M , corresponding to spectral classes B3 and hotter stars. During the main sequence phase, which lasts for about 90% of life time of a star, hot stars have fast winds, vw  500 km s−1 , and mass loss rates of M˙ ≈ 10−8 –10−4 M yr−1 . The mass loss is driven by the radiation pressure near the surface of the star, which couples to the gas primarily through metal line absorption. For that reason the mass loss is strongly metallicity dependent. The radiation pressure imparts a radially outward momentum to the atoms (e.g. [686, 936] for reviews). Since the winds are driven by line absorption, the modeling of the stellar winds depends sensitively on line and ionisation parameters

106

5 Supernova Remnant Evolution

in the wind, and gives rise to quite some uncertainties. According to [686] the mass loss rate M˙ due to line driven winds is Dmom M˙ = vw log Dmom



R∗ R

−1/2 (5.63)

, 

L = log D0 + x log L

 ,

(5.64)

with Dmom the modified wind momentum. For main sequence O-stars the parameters are log D0 = 19.87 ± 1.21 and x = 1.57 ± 0.21 [686]. Table 5.2 provides an overview of mass loss parameters for supernova progenitors. Note that these numbers, apart from metallicity dependencies, also have considerable uncertainty [847]. It is interesting to compare the total energy output in stellar winds (based on Table 5.2) to the typical explosion energy of a supernova of 1051 erg. Figure 5.6 shows that for main-sequences mass MMS > 7 M the total time-integrated 2.35 wind-loss energy scales roughly as Ewind ∝ MMS , whereas the typical initial −2.35 mass function of massive stars scales roughly at N ∝ MMS . The average wind energy for the mass range 8–100 M , can then be easily calculated and is ∼2 × 1049 erg, about 2% of the canonical supernova explosion energy. This shows that the mechanical power input for the interstellar medium from stars is much less than from supernovae. However, stars with MMS  100 M may have a total wind output energy approaching 1051 erg. In the earliest phases after star formation (first few million years), they may, therefore, impart comparable energy to massive starforming regions as the first supernova explosions. In addition, massive stars above ∼25 M may evolve into Wolf-Rayet stars [283], which have fast winds (1000–3000 km s−1 ) and a time integrated wind energy approaching 5 × 1050 erg. It is not quite clear yet what fraction of the most massive stars will become WolfRayet stars, also because some Wolf-Rayet stars may not evolve from the most massive stars, but from less massive stars that have been stripped of their hydrogen envelopes. Once a star evolves away from the main sequence, its outer envelope expands and its effective temperatures decreases, resulting in a a red supergiant (RSG) star. The wind velocity of an RSG star, which depends on the escape velocity from the stellar surface, is much slower than during the main sequence phase: vw  100 km s−1 . But the mass loss rate is higher, although the rates are highly uncertain. The total final mass lost by a progenitor, M, is, therefore, likely much more than indicated in Table 5.2. The final size of the wind bubble !depends on 2 dt. ˙ w the total energy transferred to the ambient medium, which is Etot = 12 Mv This depends primarily on the much more energetic main sequence wind, with the exception again for those stars that evolve into Wolf-Rayet stars. The highest mass stars may go through a luminous blue variables (LBV) phase, characterised by slow winds and even higher mass loss rates (10−4–10−3 M yr−1 ). It is not clear whether the LBV phase precedes or succeeds the Wolf-Rayet star phase [283]. To

Sp O3 O5 O6 O8 B0 B3

M (M ) 120 60 37 23 17.5 7.6

R (R ) 15.0 12.0 10.0 8.5 7.4 4.8

Teff (K) 50,000 42,000 37,000 35,000 30,000 17,000

L (R ) 1.3E+06 4.0E+05 1.7E+05 9.7E+04 4.0E+04 1.7E+03

tMS (106 yr) 2.5 4 6.3 7.9 10 31.6

vw (km s−1 ) 3200 2900 2600 1900 1500 490 29.4 28.7 28.1 27.7 27.1 25.0

10 log D mom

M˙ (M yr−1 ) 3.6E−06 7.4E−07 2.3E−07 1.4E−07 4.8E−08 1.3E−09

M (M ) 9.0 3.0 1.4 1.1 0.48 0.042

Etot (erg) 9.2E+50 2.5E+50 9.7E+49 4.1E+49 1.1E+49 1.0E+47

Rbubble ([1200]) Rbubble [256] nH (pc) (pc) (10−3 cm−3 ) 47 76 0.20 44 49 0.08 43 36 0.07 40 27 0.14 34 17 0.22 21 4 2.06

Table 5.2 Mass loss parameters for main-sequence stars, based on the model of [686] and stellar parameters obtained from [279]

5.8 Supernova Remnant Evolution Inside Wind Bubbles 107

108

5 Supernova Remnant Evolution

Fig. 5.6 The time-integrated, main-sequence wind energy of massive stars as a function of mainsequence mass. The values are based on Table 5.2. The dotted line indicates the scaling Ewind ∝ M 2.35

complicate matters even more, massive stars in binaries may follow more intricate evolutionary scenarios, due to mass transfer. This likely enhances the mass loss rates, and even cause complete hydrogen-envelope stripping. Hydrogen-envelope stripping provides another scenario for Wolf-Rayet star formation, even for stars with main-sequence masses lower than 25 M . Although the RSG (and LBV) phase may not contribute much to the total wind energy output of stars, they contribute significantly to the overall mass lost by supernova progenitors. Moreover, since the wind velocities are low, the wind density is high (5.18) during the earliest phases of the life of supernova remnants with RSG progenitors. This is probably the origin for the high density of the ambient medium in which Cas A evolves [1026, 1162]. However, the progenitor of Cas A has a rather low explosions mass of ∼4 M [682, 1181, 1235] , which suggests that (in addition) the progenitor may have been stripped as a result of binary interaction [1264]. Unfortunately, no surviving stellar binary companion has yet been found [646].

5.8.1 The Evolution of Main Sequence Wind Bubbles The evolution of a wind driven bubble can be described by a self-similar relation that is very similar to the Sedov-Taylor solution for supernova remnants [1200]. Instead 2 ˙ w of taking E as one of the parameters one can take the wind luminosity Lw ≡ 1 Mv 2

5.8 Supernova Remnant Evolution Inside Wind Bubbles

109

as one of the parameters. A dimensional analysis (or substituting E = Lw t) then 1/5 −1/5 reveals that Rbubble ∝ Lw ρ0 t 3/5 . By taking into account the work done by the wind on pushing against the inside of the bubble, Weaver et al. [1200] provide the following relation  250 1/5 1/5 −1/5 3/5 L w ρ0 t (5.65) 308π  3/5 1/5  2/5  −1/5  M˙ vw nH t =44 pc. 10−7 M yr−1 10 cm−3 107 yr 2000 km s−1 

Rbubble =

We have chosen a relatively high density, because massive stars are most likely born in over-dense regions. But even with this high density the model predicts rather large bubble sizes, as listed in Table 5.2. An alternative approach is detailed in [256]. It starts from the assumption that the ambient medium of massive stars is likely highly pressurised, as the interclump density may be nH = 10–100 cm−3 and the UV light from the massive stars heats and ionises the gas. In such an environment the wind bubble grows to a radius at which the bubble interior is in pressure equilibrium with the pressure P0 of the ambient medium. Without energy losses at the end of the main-sequence phase, the bubble radius can be inferred from   ) 1 5 P0 + P0 V = P0 V , E = Lw dt = (U + P0 )V = (5.66) γ −1 2 with U the internal energy and V the volume of the bubble. Compared to [256], we added here an addition term P V , which takes care of the work done by the bubble on the ambient medium. Using for the ambient density P0 = n0 kT0 , the corresponding final wind bubble radius is expected to be  3Lw t 1/3 ≈ = 10πP0  1/3  2/3 M˙ vw =35 10−7 M yr−1 2000 km s−1 −1/3  1/3   n −1/3 T0 t H,0 × pc, 10 cm−3 1000 K 107 yr 

Rbubble

(5.67)

with nH,0 the ambient hydrogen density, and T0 the ambient temperature. Comparing the wind bubble radii as predicted by (5.65) and (5.67) (Table 5.2) shows that for the most massive stars (5.67) predicts larger radii, but that below main-sequence masses of 40 M , which are much less rare, the model predicts considerably smaller wind bubbles. The model appears to agree quite well with the observational data of wind bubble sizes [248]. For the most massive stars (5.65) is

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5 Supernova Remnant Evolution

preferred, as a shortcoming (5.67) is that it does not take into account the time available to grow to the radius at which equilibrium is obtained. For the most massive stars, the available time is relatively short given the short life span of massive stars. For the Weaver et al. model the problem is that it does not take into account the back pressure of the ambient medium. The best approach, therefore, seems to be to take the minimum value for Rbubble as provided by the two equations.

5.8.2 Supernova Remnant Evolution of Inside Wind Bubbles The structure of a typical wind bubble for a star that ends its life as a RSG consists of a freely moving expanding wind, with a density given by (5.18), that terminates in a shock at the radius where the wind’s ram-pressure equals the pressure inside the mains sequence bubble. For this pressure we can take the ambient ISM pressure, P0 , if the bubble is in pressure equilibrium with the local interstellar medium [256]. The main-sequence wind bubble terminates at the shell created by the local ISM swept up in a shell by the main-sequence wind. The expected structure of the total wind-bubble is shown in Fig 5.7. The requirement of pressure equilibrium between RSG ram pressure and mainsequence wind bubble provides an estimate for the RSG wind termination shock radius: 2 ρ(R)vw =P0

⇒



Rts,RSG =

(5.68)

˙ w Mv 4πP0 

− 1  12  1   2 2 M˙ nH,0 − 12 vw T0 =1.3 pc. 10 cm−3 1000 K 10−5 M yr−1 10 km s−1

The supernova remnant blast wave evolution in such a medium can be approximated by(5.19), which gives suggest that the blast wave evolves inside the RSG wind for a time 

Rts,RSG t ≈ 41 pc

3  2

E 1051 erg

1  2

 12 

M˙ 10−5 M

−1

 yr

− 1

vw

2

−1

10 km s

yr. (5.69)

If Cas A, with an age of t ≈ 340 yr and a shock radius of Rfs = 2.6 pc, is indeed still located inside its progenitor’s RSG wind, both (5.68) and (5.69) suggest that the progenitor must have had a high mass-loss rate of ∼10−4 M . Once the supernova remnant blast wave has from the RSG wind bubble into the main-sequence wind bubble, the blast wave radius can be approximated by the

5.8 Supernova Remnant Evolution Inside Wind Bubbles

111

10–20

10–11

10–12

RSG WIND

10–24

10–26

10–14

10–28

10–15

10–30

10–16 0

20

40 Radius (pc)

60

80

109 Forward Shock Velocity (cm/s)

100

Forward Shock Radius (pc)

10–13

Main Sequence Bubble

Pressure

Density (gm/cc)

10–22

10

108

107

106

105

1 1000 10000 Time (years)

1000 10000 Time (years)

Fig. 5.7 Top: The density (solid line) and pressure (dashed line) of the circumstellar medium around a red supergiant. Bottom: The evolution of the shock radius and velocity for a supernova remnant evolving in a wind blown cavity surrounded by a dense swept-up shell (Credit: Vikram Dwarkadas, reproduced from [351])

Sedov-Taylor evolution (5.12), but taking into the low density inside the bubble, which can be as low as nH ≈ 10−4 –10−3 cm−3 (Table 5.2). This density is roughly equal to the total, integrated mass loss of the main sequence phase, divided by the volume of the main sequence bubble. Using the Sedov-Taylor model the time it takes for the shock wave to cross the tenuous wind bubble is  t ≈ 1.0 × 103

Rbubble 20 pc

5/2 

−1/5 nH −3 −3 10 cm



E 1051erg

 yr.

(5.70)

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5 Supernova Remnant Evolution

The shock velocity is then Vfs =

2 Rbubble 5 t

≈7.7 × 10

3

(5.71) 

Rbubble 20 pc

5/2 

−1/5 nH 10−3 cm−3



E 1051 erg



km s−1 .

This should be treated as an upper limit as the mass lost during the RSG may be as much as during the main sequence phase, so the kinetic energy is spread over more mass than assumed here. So 2000–5000 km s−1 seems more realistic, and agrees well with the numerical simulations of the blast wave radius and velocity shown in Fig. 5.7. As the shock wave reaches the steep density gradient of the dense shell surrounding the bubble, the velocity can rapidly decelerate [351, 1107]. From 2 (4.6), assuming that initially the post-shock presthe relation P2 = 3/4ρ0 Vsh sure does not change, we find that the shock velocity will drop to Vfs,shell =  nH,shell/nH,bubbleVfs,bubble. The shell consists of shocked ambient medium with nH,shell ≈ 4nH,0 . For our example of nH,0 ≈ 10 cm−3 , and nH,bubble ≈ 10−3 cm−3 , we see that the shock velocity can rapidly decelerate from several thousands km s−1 to below 100 km s−1 (c.f. Fig. 5.7). Hence the supernova remnant shock becomes radiative in a relatively short time, and ages must faster than assumed in standard scenarios [1107]. Since the supernova explosion may not necessarily happen in the centre of the bubble, or the bubble itself may not be spherically symmetric, the blast wave may interact at different times with the wind-bubble shell, creating large velocity contrasts. This may account for the different values for the shock velocity reported for the cavity supernova remnant RCW 86 [212, 504]. The bright TeV γ -ray supernova remnants RX J1713.7–3946 and RX J0852.0–4622 are also thought to be supernova remnants evolving in a wind bubble, but in their case the blast is probably still fully inside the low density bubble [153, 154]. The scenario sketched here rely on several approximations. For more accurate evolutionary scenarios, one needs to rely on numerical simulations, such as provided in [351, 1107].

5.9 Rayleigh-Taylor Instabilities So far, the treatment of the hydrodynamical evolution of supernova remnants has mostly been covered, based on considerations of spherical symmetry. Apart from extrinsic reasons for deviations from spherical symmetry (molecular clouds, ISM density gradients etc.), or explosion asymmetries, there are also intrinsic reasons for deviations from perfect spherical symmetry: hydrodynamical instabilities. These instabilities amplify small perturbations in the hydrodynamical flow, thereby break-

5.9 Rayleigh-Taylor Instabilities

113

ing the spherical symmetry, even if the initial conditions were close to spherical symmetric. The main instability to consider for supernova remnants is the Rayleigh-Taylor instability, which arises whenever the density is non-homogeneous and a lowdensity “fluid” is pressing against a high-density “fluid”, or if there is otherwise a gradient in the density in the presence of an acceleration g. In the conventional description of the instability the acceleration is caused by gravity. Think for example, about a jar with a light fluid on the bottom, and a heavy fluid on top. Clearly, this is not a situation that minimises the total potential energy, but could otherwise satisfy the criterion of pressure equilibrium. The criteria for stability versus instability of fluids of different densities, or strong density gradients, have been worked out in the classical book by S. Chandrasekhar “Hydrodynamic and hydromagnetic stability” [246]. The growth rates of the instabilities are obtained by perturbing a stable situation of hydrostatic equilibrium by allowing for excursions scaling with exp(ikx + iky + iωt). For two fluids pressuring again each other the resulting dispersion relation is  ω2 = −gk

ρ2 − ρ1 ρ2 + ρ1

 ,

(5.72)

with g the acceleration in the negative direction, and ρ2 is on top of ρ1 . The quantity between brackets is sometimes referred to as the Atwood number. If ρ2 > ρ1 the root is an imaginary number, and the situation is unstable with a growth rate ω. For the above equation we have neglected many possible stabilising effects, treated in [246], such as surface tension and viscosity, as they are not that relevant to the situation in supernova remnants. A major disadvantage is that most treatments of the Rayleigh-Taylor instability consider only incompressible fluids, although [176] shows that compressibility has a negligible effect in astrophysical contexts. We can estimate the acceleration g by using the value for the self-similar solutions, for which r ∝ t m , and, therefore, g ≈ |m(m − 1)R/t 2 |. Since the density gradients in the Chevalier models are very large we can use ρ2  ρ1 . Finally, the maximum size of perturbing wavelength should be a sizeable fraction of the radius, but not too large λ = 2π/k ∼ f R, with f < 1. The growth rate, or its inverse, the time scale of growth, is then given by  1 =t ω

f . 2πm(m − 1)

(5.73)

For example for f = 10% and m = 0.5 we obtain 1/ω ≈ 0.06t. For supernova remnants Rayleigh-Taylor instabilities have been discussed for a long time, starting with [472], in which it was pointed out that the strong density gradients around the contact discontinuity are likely locations for the instability to arise. The instability was also credited for amplifying the magnetic field through turbulent motion, which could explain the high magnetic field inferred for Cas A

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(more on that in Sect. 12.1). Since then the instabilities is often discussed both from an analytical point of view [253, 261] and studied using hydrodynamical simulations (e.g. [181, 261, 350, 377, 405]). In particular, the close proximity of ejecta to the forward shock in Tycho’s SNR (SN 1572) has been attributed to a combination of Rayleigh-Taylor instability and efficient cosmic-ray acceleration [181, 377, 1198]. Efficient cosmic-ray acceleration (Chap. 11) brings the forward shock closer to the contact discontinuity and alters the density gradient, making it easier for RayleighTaylor “fingers” to come close to the shock front. Although, the perturbation analysis in [246] is generally used for the analysis of the instability, an analysis invoking the analogy with convection in stellar layers [1029] provides better insights in which layers inside a supernova remnant are Rayleigh-Taylor unstable [261, 472]. Consider the equation of motion in a gas layer 1, which is being decelerated by an acceleration g in the negative direction. Plasma with a density ρ1 , therefore obeys the equation of motion ρ1

∂P1 ∂v1 =− − ρ1 g. ∂t ∂r

(5.74)

If now from another layer (2) gas is slowly displaced to layer 1, while adapting its inside pressure to the pressure in layer 1, is density will be given by the adiabatic invariant C2 =

P2 γ. ρ2

(5.75)

In layer 1 its density will, therefore, become ρ1 = (P1 /C2 )1/γ . It is likely that the acceleration term in (5.76) does not change, as it represent a force from the outside region on the displayed plasma. This makes the situation slightly different from the case in which we have gravity as an accelerator. So for the displaced plasma we have as an equation of motion: ρ1

∂v1 ∂P1 =− − ρ1 g. ∂t ∂r

(5.76)

Now using the relation ρ1 = (C1 /C2 )1/γ ρ1 , we find for the relation between the acceleration of the local plasma and the displaced plasma in layer 1: ∂v1 = ∂t



C2 C1

1/γ

∂v1 . ∂t

(5.77)

This means that if C1 > C2 the displaced plasma is less decelerated than the surrounding medium, and it will be advancing further in the forward direction than the surrounding plasma. Since the entropy is defined as S ≡ ln C, the criterion for Rayleigh-Taylor instability in a supernova remnant shell is that there should be a

5.9 Rayleigh-Taylor Instabilities

115

negative gradient in the entropy: 1 ∂P γ ∂ρ dS = − < 0. dr P ∂r ρ ∂r

(5.78)

Recall that this only holds in case the acceleration is in the negative direction. The gradient in pressure is much less strong than the gradient in ρ, so the gradient in the entropy corresponds roughly to the inverse of the gradient in ρ. If we look at the self-similar Chevalier model (Fig. 5.3) we see that for the s = 0 model a negative entropy gradient exists in the layer of shocked ejecta; i.e. inward of the contact discontinuity. For the stellar wind case (s = 2) a strong negative entropy gradient is present in the shocked ambient medium plasma. In both cases the strongest gradients are present near the contact discontinuity. Indeed, hydrodynamical simulations show that Rayleigh-Taylor instabilities are present in decelerating supernova remnant shells, but that for supernova remnants in a uniform density the Rayleigh-Taylor fingers are less elongated than for the stellar wind case [181, 261, 350]. This is illustrated in Fig. 5.8. The simulations show that the turbulence itself is strong enough to provide a back reaction on the reverse shock in the s = 2 case. As a result, the reverse shock becomes corrugated. Overshooting effects for the s = 0 case can mix shocked ejecta material into the shocked ambient medium. However, it requires efficient cosmic-ray acceleration to adjust the density profiles sufficiently to bring ejecta close to the shock front [377].

Fig. 5.8 Rayleigh-Taylor instabilities in two-dimensional simulations of Type Ia supernova remnants exponential ejecta density profiles, but with two different ambient density structures [350]. Left: A uniform density medium is assumed (s = 0). Right: a stellar wind profile is assumed (s = 2). The Rayleigh-Taylor fingers are more pronounced in the s = 2 case. The time and radius are normalised, but correspond to roughly 370 and 660 yr, respectively (Credit: Vikram Dwarkadas, reproduced from [350])

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5 Supernova Remnant Evolution

Note that the sides of the Rayleigh-Taylor fingers are subject to another hydrodynamical instability, called the Kelvin-Helmholtz instability (shear instability), which arises when to layers move alongside each other. This instability results in mixing of the two shearing plasmas.

Chapter 6

Neutron Stars, Pulsars, and Pulsar Wind Nebulae

The connection between supernovae and neutron stars goes back to the seminal publications by W. Baade and F. Zwicky in [109–111], in which they coined both the term “super-nova”, suggested a connection between cosmic rays and “super-novae” and postulated that a “super-nova” event “represents the transition of an ordinary star into a neutron star” [109]. A remarkably bold suggestion given that the discovery neutron was reported by J. Chadwick just 2 years before [245]. Interestingly, it was not the work of Baade and Zwicky, but similar ideas by G. Gamow and Landau [699] that provided the incentive for Oppenheimer and Volkoff [872] to perform the first theoretical calculations of the internal structure of neutron stars. Neutron stars only became an astrophysical reality with the discovery of steady radio pulsations from unresolved sources by J. Bell, A. Hewish and others [530]. These sources were dubbed “pulsars” and it was soon established that these pulsations were caused by beams of radio emission coming from rotating neutron stars [443]. Baade and Zwicky’s idea that supernovae create neutron stars was soon confirmed by the discovery of 30 Hz radio pulsations from the Crab nebula [745], an object that had been linked for a long time with the historical supernova of AD 1054 [555, 747, 787]. Soon after the discovery of radio pulsations from the Crab nebula also optical [852] and X-ray pulsations [206] were detected. These discoveries led to the rapid development of the theory behind pulsars as rotating, magnetised neutron stars by people like F. Pacini, T. Gold and P. Goldreich [443, 445, 878]. The final confirmation that neutron-star formation is connected to supernova explosions was the detection of a burst of neutrinos associated with SN 1987A [63, 169, 539]. Given that neutron stars and supernova remnants have a common origin in supernova explosions, supernova remnants form ideal objects to search for young neutron stars. Indeed, over the last 20 years neutron stars located inside supernova remnants have been instrumental in showing that the neutron star population is much more diverse than previously thought: apart from the well-known radio pulsars, there are now classes of radio-quiet, X-ray emitting neutron stars. The population of radio-quiet neutron stars is diverse as X-ray emitting neutron stars can either have © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_6

117

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

high magnetic fields (magnetars) or low magnetic fields (sometimes called “antimagnetars”). For some radio-quiet neutron stars, we do not know their magnetic field, this class owes its name to their position in the centre of supernova remnants: “Central Compact Objects” (CCOs). The purpose of this chapter is to explore the connection between supernova remnants and neutron stars. An important connection is provided by those neutron stars that create a pulsar wind nebula (PWN), which pushes from the inside to the supernova material. In the radio this will generally appear as a centre-filled supernova remnant, a so-called composite supernova remnant. We start here first with an introduction to neutron star properties, explaining in some details the physics of the magnetosphere, and then discuss pulsar wind nebulae, and finally those classes of neutron stars that do not create powerful pulsar wind nebulae.

6.1 The Internal Constitution of Neutron Stars Core collapse supernova generally leave behind a magnetised, hot (∼109 K), but rapidly cooling, neutron star. The core of the neutron star, which comprises most of its mass and volume, consist of a degenerate neutron gas, but with a proton fraction of ∼11%. The average neutron star density is about 4 × 1014 g cm−3 , similar to the density of heavy atomic nuclei. In the inner core a density of 1015 g cm−3 may be reached. It is not clear whether the inner core consists of mostly neutrons, or whether more exotic particles, like kaon condensates, or other particles containing strange quarks (hyperons), or even deconfined quarks dominate the composition [704]. The massradius relationship of neutron stars is determined by the equation of state (EOS) of neutron star matter, which is not well understood, theoretically allowing for maximum neutron star masses from 1.3 to 2.8 M and radii of ∼8 to ∼15 km. However, neutron stars masses of ≈ 2 M have now been measured [78, 304], excluding the possibility that the EOS is very “soft”, which tends to disfavour many EOS based on exotic inner core matter. The typical neutron star masses are 1.4 M , see the compilation in [704]. Since all mass estimates are based on neutron stars in binaries, there may be some observational biases. It is possible that binaries that do not unbind after a supernova explosion have a tendency to have neutron stars with relatively lower masses. For example, one mechanism that could affect the masses of neutron stars in binaries is if a significant fraction of the neutron stars in binaries have been created by the less energetic electron-capture supernovae [664]. The outer 1–2 km of neutron stars forms the neutron star crust, which contains heavy nuclei, with very neutron-rich atoms at the bottom of the core. The electrons in the crust form a degenerate gas. The envelope of the neutron star has a thickness of about 100 m and consists of 56 Fe or lighter elements. Although the envelope is provides but a tiny fraction of the mass of the neutron star, the envelope’s thermal

6.2 Pulsars

119

conduction properties determine the relation between core temperature and surface temperature [879]. The radiative properties of the neutron star do not only depend on the temperature of the surface, but also on the composition and surface magnetic field. The surface and atmosphere composition likely consists of supernova fall back material, but mass segregation may still lead to a proton-rich atmosphere. Strong magnetic field affect the quantum levels of the electrons, and even above 105 K bound hydrogen and other atoms may exists, as well as long chain molecules [492]. Depending on the surface magnetic field strength, the surface composition and temperature, the neutron star may either have a solid surface [693], or there may be an atmosphere with a scale height of H = P /(gρ) = nkT /(gμmp n) = 11(μ/0.6)−1(T /107 K) cm, with g = GM/R 2 ≈ 1.3 × 1014 cm s−2 . In case an atmosphere is present, absorption features may be present in the X-ray spectra of isolated neutron stars [478].

6.2 Pulsars The name pulsar stands for “pulsating radio source”, although now also neutron stars pulsating in other wavelength regimes are often referred to as pulsars. The radio emission from pulsars is caused by the physical processes taking place in the highly magnetised volume immediately surrounding the neutron star, the magnetosphere. This magnetosphere is also the origin of the charged particles the neutron star is spitting out into its immediate neighbourhood, which results in the formation of the pulsar wind nebula that in many cases occupies the interior of supernova remnants.

6.2.1 The Magnetic Dipole Model for Neutron Stars Observationally pulsars have a wide range of spin periods, from a few millisecond to a few seconds. It is thought that at birth all neutron star spin rapidly, perhaps even a few millisecond [347, 875]. However, observational evidence suggests longer initial rotation periods, covering a wide range from 10–500 ms. Among the known young pulsars, PSR J0537-6910 in the Large Magellanic Cloud supernova remnant N157B, is the most rapidly spinning, having a pulse period of P = 16.12 ms [1196].whereas the Crab pulsar, with P = 33.5 ms is the fastest rotating young pulsar in the Milky Way (old, “recycled” millisecond pulsars rotate more rapidly, but their rotational energy has been acquired by accretion from a companion star). The rotational energy is the primary source of energy for both pulsar radiation and particle outflows from most pulsars. For a uniformly rotation sphere the rotational energy can be expressed as Erot =

1 2 I , 2

(6.1)

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

with  ≡ 2π/P the angular frequency and I the moment of inertia. The moment of inertia depends on the equation of state, but can be approximated for many equation of stated by [705]  2

I ≈ 0.33MR = 1.4 × 10

45

M 1.5 M



R 12 km

2 g cm2 .

(6.2)

The rotational energy can, therefore, be approximated by  Erot ≈ 2.8 × 10

48

P 100 ms

−2 

M 1.5 M



R 12 km

2 erg.

(6.3)

This means that a 5 ms pulsar can in principle provide as much energy as a supernova explosion (∼1051 erg), and that a maximally spinning neutron star with P ≈ 1.5 ms has a rotational energy of ∼1052 erg. Young neutron stars have the highest magnetic fields known in the Universe of 1010–1015 G. The magnetic field line configuration is not well known, but moving away from the neutron star the higher multipole moments of the magnetic field rapidly drop off, so that to a good approximation the magnetic field of a neutron star can be described by a dipole field B(r, θ ) =

m 3(mr)r − mr 2 ˆ = 3 (2 cos θ rˆ + sin θ θ), 5 r r

(6.4)

with m the magnetic moment: |m| =

Bp R 3 , 2

(6.5)

and Bp the polar magnetic field. The energy loss of a varying dipole field is given by 2 ¨ 2. E˙ = − 3 |m| 3c

(6.6)

The time varying dipole field can be expressed as m=

1 Bp R 3 (ˆe + eˆ ⊥ sin α cos(t) + eˆ ⊥ sin α sin(t)), 2

(6.7)

with α the magnetic field obliquity, i.e. the angle between the magnetic axis and the the three unit vectors along and perpendicular to the rotational axis, and eˆ , eˆ⊥ , eˆ⊥ magnetic field axis. Combining (6.6) with (6.7) shows that the rotational energy loss

6.2 Pulsars

121

of the rotating dipole is given by E˙ rot = −

Bp2 R 6 4 sin2 α 6c3

≈ − 5.4 × 10

35



(6.8)

M 1.5 M



R 12 km

2 

Bp 1012 G

2 

P 100 ms

−4

erg s−1 .

This radiation corresponds to the Poynting-flux of the rotating dipole, and essentially consists of low-frequency radio waves. Note that |E˙ rot | is often called the pulsar’s spin-down power. The energy loss rate of the pulsar must be at the cost of the rotational energy of the neutron star, given by the derivative of (6.3):  −3  ˙  ˙ P P 2 P 34 ˙ ˙ Erot = I  = −4π I 3 ≈ −5.5 × 10 erg s−1 . P 100 ms 10−15 (6.9) Equating (6.8) and (6.9) shows that Bp2 =

3c3 1 I P P˙ . 2π 2 sin2 α R 6

(6.10)

If we now use I = 1.4 × 1045 g cm2 , R = 12 km, and sin2 α = 12 , we find that the dipole magnetic field of a pulsar can be estimated from the pulsar period and period derivative as  Bp ≈ 6.2 × 1019 P P˙ G. For historical reasons (using lower value for I , R and omitting the factor 1/2 from the expression of the dipole moment) the canonical equation that is used most often is  Bp ≈ 3.2 × 1019 P P˙ G. This equation does not take into account the possibility that the magnetic field configuration may be more complex than a dipole field, and it assumes that the period evolution is caused exclusively through magnetic dipole radiation. Equation (6.10) has the form dP /dt = C/P , which can be solved to give  P (t)2 = 2Ct + P02 = 2

Bp 3.2 × 1019 G

2 t + P02 ,

(6.11)

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

with P0 the initial spin period. If P0  P (t), we see that the age of the pulsar can be approximated by t ≈ P 2 /C = P 2 /(2P P˙ ). This estimate of the age of a pulsar defines the so-called characteristic age of a pulsar: τch ≡

1P . 2 P˙

(6.12)

Note that the equation is only valid if the rotational energy loss is solely caused by the magnetic dipole loss equation (6.9) with P0  P , and the surface magnetic field does not evolve. A comparison between estimates of SNR ages and pulsars they contain shows that there may be large discrepancies between τch and τSNR . For example, the characteristic age of the P = 65 ms X-ray pulsar inside the supernova remnant G11.2-0.3 is 24,000 yr [1122], whereas the supernova remnant has been constrained by proper motion measurements to be in the range of 960–3000 yr [1091]. This is an extreme example, perhaps caused by the fact that the current period is close to P0 . For example, the Crab pulsar has a known age of ∼960 yr, having created by SN 1054, and its characteristic age is 1274 yr. This is off by ∼25%, but still in the right ballpark. Pulsars for which the magnetic axes are aligned with the rotational axes, should, according to (6.8), not lose rotational energy. However, calculations that include the effects of conduction by the electron/positron pair plasmas inside the magnetosphere (Sect. 6.2.3) indicate that the spin-down power can be approximated by [469, 1069] E˙ rot ≈ −

 Bp2 R 6 4  2 α , 1 + sin 4c3

(6.13)

which shows a similar dependency on period and polar magnetic-field strength, but with different numerical factors and a different dependence on the obliquity of the magnetic-field axis. The associated estimate for the magnetic-field strength becomes  −1/2  Bp ≈ 2.6 × 1019 P P˙ 1 + sin2 α G.

(6.14)

Despite the fact that age and magnetic-field estimates based on P and P˙ may not be entirely reliable, they often are the only estimates available, and at least give an order of magnitude estimate of the pulsar properties. For this reason the pulsar population is often displayed in a P -P˙ (P-Pdot) diagram, which is as fundamental to pulsar astrophysics as the Hertzsprung-Russel diagram is to stellar astrophysics. Figure 6.1 shows a P -P˙ diagram of known pulsars, including lines of constant characteristic age (6.12), surface magnetic fields (6.10) and rotational energy loss (6.9). The diagram indicates that the bulk of young pulsars have periods have periods between 0.1 and 1 s, and magnetic fields of 1012–1013 G. The fastest rotating pulsars in the lower left part of the diagram are so-called millisecond pulsars, which are thought to be pulsars that have spun up after accreting from a companion star.

6.2 Pulsars

123

Fig. 6.1 The period, P , and period derivatives, P˙ , of all pulsars in the ATNF catalog∗ positioned in a so-called P − P˙ -diagram. Lines of equal characteristic pulsar age, τch (magenta), surface magnetic field Bp (blue), and E˙ rot (light blue) are indicated by dotted lines. The magnetar candidates (anomalous X-ray pulsars and soft gamma-ray repeaters) are indicated by red squares, whereas the sample of pulsars inside pulsar wind nebulae [612], are indicated by green stars. ∗ https://www.atnf.csiro.au/research/pulsar/psrcat/

For this reason they are sometimes called recycled pulsars. Given their old ages, millisecond pulsars are never associated with supernova remnants.

6.2.2 The Pulsar Braking Index A more heuristic approach to pulsar-period evolution, without assuming any specific loss mechanism, is to approximate the pulsar’s period evolution with ˙ = −Kn , 

(6.15)

with n the so-called pulsar braking index. The magnetic-dipole model corresponds in this case to n = 3. The period evolution for a given n (n = 1) is P n−1 = (n − 1)(2πK)n−1 t + P0n−1 ,

(6.16)

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

which gives a characteristic age of τch =

1 P . (n − 1) P˙

(6.17)

Equation (6.16) can be rewritten as  P (t) = P0 1 +



t

1 n−1

(6.18)

,

τch,0

with τch,0 ≡

1 1 P0 ≈ 1.2 × 104 (n − 1) P˙0 (n − 1)



P0 10 ms

2 

Bp 1012 G

−2 yr,

(6.19)

the characteristic age at the time of birth, which then does not give the age of the pulsar, but the time scale over which the pulse period significantly changes. By combining (6.18) and its derivative with (6.9) we can derive an expression for the rotational energy output of the pulsar as a function of age  E˙ dot (t) = E˙ rot,0 1 +

t τch,0

− n+1 n−1

,

(6.20)

with E˙ 0 the initial pulsar energy loss rate. For some pulsars the second order derivative of the rotational frequency can be measured, which provides an estimate of the braking index of a pulsar: ¨ = − nKn−1 , ˙  n =−

(6.21)

¨ ¨  f f¨ P P¨  = = = 2 − , ˙ n−1 ˙2 K   f˙2 P˙ 2

with f = /(2π) = 1/P the frequency of the pulsar. The measured braking indices of several young pulsars are significantly different from the magnetic dipole case, n = 3. For example, the Crab pulsar (PSR B0531+21) has n = 2.342 ± 0.001 [749], and the young pulsar B1509-58, associated with the supernova remnant RCW 89 (a.k.a. G320.4–1.2, MSH 15–52) has n = 2.839 ± 0.003 [728, 763]. There can be several causes for the pulsar spin-down evolution to deviate from the magnetic-dipole model. For example, the pulsar spin down may be governed by other rotational energy loss mechanisms, such as energy losses due to an outflow of electrons/positrons [493, 1253], and some pulsars may experience some accretion. But the most popular models focus on the evolution of the surface magnetic field. It has been argued that the magnetic field of neutron stars will slowly decay [e.g. 446], although the evidence for that for normal radio pulsars is controversial [161]. Moreover, magnetic-field decay leads to n > 3, whereas in many cases, such in

6.2 Pulsars

125

Fig. 6.2 Left: Chandra image of the supernova remnant N157B and its pulsar wind nebula powered by PSR J0537-6910 (indicated by a cross). The supernova remnant shell shows up mostly in the 0.5–1 keV band (red) and in the 1–2 keV band (green). In the 2–7 keV band (blue), only the pulsar wind nebula is visible, and consists of a small, bright ellipsoidal region (saturated here) and a fainter elongated region extending toward the Northwest. PSR J0537-6910 is the fastest spinning young pulsar with P = 16 ms and E˙ dot ≈ 5 × 1038 erg/s [771]. Right: Deviations from a regular spin down of the pulsar frequency for PSR J0537-6910, indicating frequent glitches [772]

the examples given above, n < 3. On face value for example, the dipole magnetic field of the Crab pulsar (n = 2.3) is increasing [749]. The energy-loss rate may also decrease due to a slow alignment of the magnetic-field obliquity, which appears to occur on time scales of ∼107 yr [914, 1265]. In some cases the magnetic-dipole moment may increase. For example, it has also been theorised that in young pulsars the magnetic field has been buried by late time accretion onto the neutron star after the supernova event, and it is now in the process of slowly resurfacing, resulting in a strengthening of the dipole magnetic field as a function of age [429]. Finally, it should be noted that besides a regular spin-down rate young pulsars experience every now and then sudden increases in their spin-frequency (decrease in their periods), followed by a period in which the frequency slowly decreases back to near its original value. The Crab pulsar is one of these sources that experience regular glitches [749], but the fastest spinning young neutron star, PSR J0537-6910 in the supernova remnant N157B, glitches so often that no unique measurement of the characteristic age can be made [772]; see Fig. 6.2.

6.2.3 The Magnetosphere Although the magnetic field configuration close to the neutron star follows that of a dipole magnetic field, this cannot hold at very large large radius. The reason is that magnetic fields, and the particles trapped by it, have a velocity given by vB = r, but for a certain radius this velocity will exceed the speed of light c. So the dipole field configuration breaks up for radii beyond where vB = c, which defines the

126

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

pulsar light cylinder radius, RLC ≡

  P c = 4.77 × 103 km.  100 ms

(6.22)

The cylindrical volume enclosed by the light cylinder is called the pulsar magnetosphere. Figure 6.3 shows its magnetic-field configuration of the magnetosphere. The two areas on the neutron star that connect to the space outside the light cylinder through open field lines are called the polar caps. The faster the spin of the neutron star, the shorter the light-cylinder radius. We can estimate the size of the polar cap by making use of magnetic flux conservation. The magnetic flux through the light cylinder (one hemisphere) is 2 φB,LC ≈ Bp (RNS /RLC )3 πRLC (assuming that the length of the cylinder is approximately its radius). The flux at the pole, where all the open field lines come 2 . Flux conservation (φ together, is φB,PC = Bp πRPC B,LC = φB,PC ) now shows that  RPC ≈

3 RNS = 0.60 RLC



RNS 12 NS

3/2 

P 100 ms

−1/2 km.

(6.23)

More detailed calculations are presented in [359], but give essentially the same result. P. Goldreich and W. Julian [445] showed that the magnetosphere cannot be a total vacuum. Their reasoning starts with the premiss that the neutron star atmosphere and crust are perfect conductors. This allows charges to distribute themselves on the surface and in the magnetosphere in such a way that there is no net Lorentz force on the particles, i.e.   ×r × B = 0, FL = q E + c

(6.24)

with E the electric field and v = ×r the velocity caused by the forced corotation of charged particles with the neutron star. The Lorentz forces on charged particles can easily exceed the gravitational pull on particles. As a result electrons and protons are peeled off the surface of the neutron star. The condition that there is no net force on particles in the magnetosphere (FL = 0) implies that E=−

×r × B, c

(6.25)

The charge density must obey Poisson’s equation: ∇ 2 φ = −∇E = −4πρc .

(6.26)

6.2 Pulsars

127

Fig. 6.3 A sketch of the magnetosphere of a neutron star

So the magnetic dipole field results in a charge density that is approximately ρc ≈

3 Bp RNS , 2πcr 3

(6.27)

where we have approximated B(r) ≈ Bp (RNS /r)−3 , with RNS the neutron star radius. The particles will distribute themselves according to their charges. The associated particle number density at the surface, near the pole, is nGJ ≈

Bp Bp ≈ 7 × 10−2 cm−3 . P ce P

(6.28)

The density nGJ (or ρGJ ) is called the Goldreich-Julian density. Note that (6.25) shows that there are regions for which the radial and magnetic fields vectors are aligned (B ×  = 0), so that the electric field is locally zero, and the charge density changes sign, as sketched in Fig. 6.3. Note that the magnetosphere is very dynamic, as charges are being accelerated to very high energies (of order a few GeV), which results in radiation caused by curvature radiation (related to synchrotron radiation, Sect. 13.3) and inverse Compton scattering (Sect. 13.2.2). The photons moving through the very strong magnetic field will give rise to electron-positron pair creation, and these new electrons and positrons will also produce radiation.Hence, an electron-pair creation cascade will be induced, and the magnetosphere will be filled with electrons and positrons [532, 1084], rather than by protons and electrons as initially envisaged by Goldreich and Julian. The electron/positron density is therefore expected to

128

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

exceed the Goldreich-Julian density by a large factor, often designated by the pair multiplicity parameter κ: κ≡

ne , nGJ

(6.29)

with ne the electron plus positron density. Note that the absolute number density in the magnetosphere can easily exceed the Goldreich-Julian density, because the Goldreich-Julian number density puts constraints on the difference between positively and negatively charged particles, not on the absolute number of particles. The charged particles attached to open field lines will either escape the magnetosphere or will bombard and heat the pulsar’s polar caps.

6.3 The Inner Regions of Pulsar Wind Nebulae The particles escaping the magnetosphere, together with the Poynting flux emanating from the rotating pulsar, form together the pulsar wind. The electron/positrons in the wind are moving with a relativistic bulk velocity. This flow halts where the pressure from the wind equals the external pressure, at the termination shock. At the termination shock the particles are shock-heated and a large fraction of them will be accelerated (see also Chap. 11) forming a non-thermal population of particles, with a cut-off at very high energies (TeV). The region outside the pulsar-wind termination shock is the pulsar wind nebula, which consists primarily of a nonthermal relativistic electron/positron plasma, which emits synchrotron radiation and inverse Compton radiation over many orders of magnitude in frequency, from ∼ 107 to ∼ 1024 Hz. The pulsar wind nebula itself is confined by the pressure of the inner regions of a supernova remnant, or in a later phase, by the pressure of the interstellar medium. This basic picture of a young pulsar wind nebula and its surroundings, closely resembles the composite supernova remnant G11.2.-0.3 (Fig. 6.4). The X-ray image displays X-ray synchrotron emission from the pulsar wind nebula in the centre, which is surrounded by the supernova remnant shell, emitting thermal X-ray emission. Not all pulsar wind nebulae resemble this nicely ordered layout: the most studied pulsar wind nebula, the Crab Nebula (Fig. 6.5), is evolving inside expanding supernova ejecta, but no hint of a hot supernova shell has ever been found [526]. In this section we describe in more detail the physics and evolution of pulsar winds, pulsar wind nebulae and their interaction with supernova remnants.

6.3 The Inner Regions of Pulsar Wind Nebulae

129

Fig. 6.4 The composite supernova remnant G11.2-0.3 as observed by Chandra [199, 981]. The colour correspondence is red: 1.2–1.4 keV (mostly Mg XII line emission); green: 1.7–1.9 keV (Si XIII) and blue 2.5–6.7 keV (continuum emission). The east-west elongated pulsar wind nebula is mainly visible in blue

6.3.1 The Pulsar Wind Outside the light cylinder energy is carried outward in the form of Poynting flux, S = c|E × B|/4π = cB 2 /(4π), and relativistic electrons and positrons (perhaps also ions) that have escaped the magnetosphere along the open field lines. Together, the particle flow and electro-magnetic flux emerging out of the magnetosphere are referred to as the pulsar wind. The total energy-loss rate of the wind must correspond to the rotation energy-loss rate of the pulsar (6.8) |E˙ rot | = 4πr 2 (S + pw ne me c3 ) = 4πr 2 pw ne me c3 (1 + σ ),

(6.30)

130

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Fig. 6.5 Left: Broad band multiwavelength image of the synchrotron emission from the Crab Nebula, consisting of radio emission (red, VLA 5.5 GHz [166]), a UV 290 nm image (green, XMM-Newton OM [344]) and X-rays (blue, Chandra LETGS [1210]). The bar of 1 corresponds to 0.6 pc at distance of 2 kpc. (Credits: M. Bietenholz/VLA, G. Dubner et al./A. Talavera/ESA, Weisskopf/NASA) Right: The Crab Nebula in the optical as observed by the Hubble Space Telescope’s WFPC2 instrument, using broad (blue) and narrow band filters ([OIII], red; [SII], green). The diffuse, blue-coloured, part is caused by optical synchrotron radiation (Credit: NASA/ESA/J. Hester/A. Loll (ASU))

with ne the combined electron-positron number density, pw the Lorentz factor associated with the bulk motion of the electron/positron plasma. We have introduced here the parameter σ , which is defined as the ratio between electro-magnetic energy flux (S) and particle energy flux, σ ≡

B2 . 4πpw ne me c2

(6.31)

In principle some internal energy can be associated with random (thermal) motions of the particles, but since the flow expands from a radius comparable to the light cylinder, (r ∼ 109 cm) see (6.22) to radii 1017 cm (see below), the adiabatic expansion of the plasma results in rapid cooling. So for the moment the pulsar wind can be considered to be internally cold. The pulsar wind terminates into a shock at the location where the external pressure, usually that of the pulsar wind nebula, equals the ram pressure of the pulsar wind: Ppw,ram =

˙ | |Erot = Pext , 4πf r 2 c

(6.32)

6.3 The Inner Regions of Pulsar Wind Nebulae

131

with f ∼ 1 a geometrical correction factor that takes into account the effect of a non-isotropic pulsar wind. So the termination radius is expected to be  Rts =

|E˙ rot | = 1.6 × 1017 4πf cPpwn



 12  − 1 2 1 |E˙ rot | Pext f − 2 cm. 36 −1 −10 −3 10 erg s 10 erg cm

(6.33) Like the shocks discussed in Chap. 4, the termination shock converts part of the kinetic energy of the bulk motion into internal energy. But unlike supernova remnant shocks, the pulsar wind termination shocks are relativistic shocks. In case σ is small (σ  1), the compression ratio is χ = 3, and the plasma velocity behind the shock becomes v = 13 c [638, 652]. If σ  1 then the post-shock plasma speed √ is still relativistic with an associated Lorentz factor  ≈ σ . As we will discuss in Sect. 6.3.4, observations show that the plasma in the inner regions of pulsar wind nebula has subrelativistic speeds, and, surprisingly, the situation must be that σ  1.

6.3.2 The Kennel and Coroniti Model The expected properties of the shocked plasma, downstream of the termination shock, is described in detail by the Kennel and Coroniti model, which treats the flow under the assumption of spherical symmetry [638]. This model indicates that the subrelativistic speeds observed within the Crab Nebula imply that σ is low. It is beyond the scope of this book to reproduce the Kennel and Coroniti model entirely. However, one can simplify the model once we accept that σ  1 and that the flow downstream of the termination shock is indeed subrelativistic. We start with the equation of continuity, which for a spherical geometry gives for the number density (ne ) of electron/positrons ne =

N˙ e , 4πr 2 v

(6.34)

with Ne the total number of particles passing the termination shock, r the radial distance and v the plasma velocity (c.f. the density within a stellar wind Sect. 5.18). The pressure of the relativistic electrons/positrons is assumed to be only governed by adiabatic heating/cooling, i.e. P n−γ = constant, with γ = 4/3 the adiabatic index for relativistic particles. Combining this equation with (6.34), shows that the pressure profile of particles within pulsar wind nebula should be  P (r) = Pts

ne ne,ts

γ

 = Pts

r 2v Rts2 vts

−4/3 .

(6.35)

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

The subscript “ts” refers here to the quantities immediately downstream of the termination shock, with Rts the termination shock radius (6.33). For a steady state solution the magnetic field should not vary as a function of time, i.e. ∂B(r)/∂t = 0. It is assumed that the charged particles do not feel a net Lorentz force, so FL = e(E + v × B/c) = 0. Combining this with the Maxwell equation 1c ∂B(r)/∂t = −∇ × E, gives ∂B(r) = ∇ × (v × B) = 0 ∂t

(6.36)

which shows that only a tangential field (B perpendicular to the flow) Bφ will be maintained. Using only this component, the equation becomes 1 ∂rBφ = 0, r ∂r

∇ × (v × B) =

(6.37)

which results in the following radial scaling of the magnetic field strength:  Bφ = Bφ,ts

rv Rts vts

−1 (6.38)

.

The other continuity equation to be used is the continuity of energy flux (Sect. 4.1), which for a spherical system can be written as  4πr 2 v

B2 1 γ P+ + ρv 2 γ −1 4π 2



  1 B2 + ρv 2 = constant. =4πr 2 v 4P + 4π 2 (6.39)

The electrons/positrons are ultra relativistic, whereas the flow itself is subrelativistic (0.3c). Hence the enthalpy density should be much larger than the kinetic energy density: 4P  12 ρv 2 . Since observations indicate that σ is small, we can use 4P  B 2 /(4π), and we can normalise the total energy flux to the energy flux near the termination shock 4πRts2 vts 4Pts . For a low σ downstream of the termination shock, we can neglect the energy density of the magnetic field. Moreover, the electrons/positrons are ultra relativistic, whereas the flow itself is subrelativistic (0.5c). Hence the enthalpy density should be much larger than the kinetic energy density: 4P  12 ρv 2 . Introducing now σts = Bts2 /(4πUts ) as the ratio of magnetic field energy density over the particle energy density (U = 3P ) immediately downstream of the termination shock, we can rewrite (6.39) with the help of (6.38) and (6.35) as  2

4πr v 4Pts



r 2v Rts2 vts

−4/3

 + 3σts Pts

rv Rts vts

−2 

≈ 4πrts2 vts 4Pts .

(6.40)

6.3 The Inner Regions of Pulsar Wind Nebulae

133

Fig. 6.6 The approximate solutions to the inner flow velocity and magnetic field strengths of pulsar wind nebulae for various values of the magnetisation parameter σ (Kennel and Coroniti model)

This equation can be transformed into an implicit description of the velocity profile: 

r Rts

−2/3 

v vts

−1/3

3 + σts 4



v vts

−1

=1

(6.41)

which shows that for large radii the velocity will approach a constant value v∞ = 3 1 4 σts vts ≈ 4 σts c, whereas close to the termination shock radius and for σ  1 the first term dominates, showing that v/vts = (r/Rts )−2 . Figure 6.6 displays the velocity and magnetic field profiles based on the above equations. It shows that the velocity of the plasma is steadily decreasing, whereas the magnetic field first increases, reaches a maximum and then starts decreasing. In the figure the definition of σ is that of Kennel and Coriniti, which is defined for the downstream flow (inner part of the termination shock). For σ  1 one can show that σts ≈ 4σ . For the expected post-shock flow of vts = c/3 for low σ , one finds v∞ = σ c. The Kennel and Coroniti model is the current paradigm for the density and flow properties of pulsar wind nebulae, but it is good to be aware of its limitations. An obvious limitation is that the model assumes spherical symmetry, whereas, as will be described in Sect. 6.3.3, the inner regions of pulsar wind nebulae appear to consist of a toroidal structure and two jets. Another assumption is that the relativistic electrons flow away from the pulsar without meeting any resistance from pressure of the medium outside the pulsar wind nebula. This assumption leads to an asymptotic fixed velocity depending on the σ -value in the pulsar wind. As will be described in Sect. 6.4.1 the interaction of the pulsar wind nebula with the unshocked ejecta will give rise to different properties of the expansion velocity of the boundary of the pulsar wind nebula. Finally, the Kennel and Coroniti model assumes ideal magnetohydrodynamics, but it has recently been proposed that the magnetisation of the plasma may change as a result of magnetic dissipation inside the pulsar wind nebula, as a result of hydrodynamics instabilities [930]. Despite the fact that the

134

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Kennel and Coroniti may not capture all the relevant physics of pulsar wind nebulae, the model provides a benchmark for comparing more elaborate models to, and as a paradigm to interpret observational data.

6.3.3 Wisps, Jets, and Tori The region around the termination shock has been imaged by the Chandra Xray satellite for a large number of bright pulsar wind nebulae, see [612] for a comprehensive overview. These images show that the interior regions of pulsar wind nebulae are axisymmetric, rather than spherically symmetric, as assumed in the Kennel and Coroniti model. The Crab Nebula provides the brightest and most observed example (Fig. 6.7). Most of its X-ray emission comes from a bright torus, defining the equatorial region, and jets coming out in the polar regions. Both the jets and the torus have strong asymmetric brightness distributions, indicating Doppler boosting [907] and a typical flow speed of 0.3c [907]. This value is consistent with the expectations for a low σ ( 1) shock. It may be even as low as σ ≈ 0.003, as first pointed out by Kennel and Coroniti [638]. A similarly low value of σ ≈ 0.0045 is obtained when fitting the overall spectral energy distribution of the Crab Nebula, more on that in Sect. 6.4.4. The combination of a torus-like structure with jet(s) is not only seen in the Crab Nebula, but seems to be more common [612]. Examples of other pulsar wind nebula with similar structures are the Vela pulsar wind nebula [509] (Fig. 6.7) and 3C58 [1057] (Fig. 3.1).

Fig. 6.7 Left: The Crab nebula in X-rays [529, 836, 1212] as observed by the Chandra HRCS/LETG instrument (1/2000). The stripes are instrumental artefacts. Right: The Vela pulsar X-ray nebula as observed by the Chandra ACIS-S instrument (2010) [c.f. 509]. Both pulsar wind nebulae show a torus-like structure, enhanced on one side due to Doppler boosting, and a wobbly jets, also enhanced on one side due to Doppler boosting

6.3 The Inner Regions of Pulsar Wind Nebulae

135

Fig. 6.8 Sequence of Chandra (ACIS-S) observations of the Vela pulsar wind nebula, here zooming in here on the jet. The images were adaptively smoothed. The observations were done in July and August 2011 with intervals of about 10 days. The knots appear to be moving with velocities of about 0.7c [349]

In the Crab Nebula the torus itself is build up of wisp-like structures, first identified in [1018], which, as revealed by sequences of optical (Hubble Space Telescope) and X-ray (Chandra) observations, are time variable and are moving outward with velocities of ∼0.5c [529]. The inner-most wisp has a radius along the long axis of 16 , corresponding to ≈5 × 1017 cm at a distance of 2 kpc. The jets of the Crab Nebula and Vela pulsars are not straight. In fact, a time sequence of Chandra observations of the Vela pulsar wind nebula shows that the jet is wobbling and that features are appearing and moving outward with velocities of ∼0.7c [349], see Fig. 6.8. The Crab Nebula and Vela pulsar wind nebula offer the most detailed examples of the inner regions of pulsar wind nebulae, but the torus-like morphologies appear to be common among pulsar wind nebulae [856]. The torus is likely associated with the termination shocks of an axisymmetric pulsar winds. The essential features of the morphology and even time variability has been captured by a number of magnetohydrodynamical simulations [216, 228, 668]. An example is shown in Fig. 6.9. Observationally, the typical dimensions of the tori are well in agreement with the theoretically expected radii (6.33), see Fig. 6.10 (based on [127]).

6.3.4 The σ -Problem The plasma motions of order 0.3c detected around the tori of pulsars like the Crab and Vela pulsar point toward a low value for the magnetisation parameter σ (1). This is further supported by the good match between the observed morphology of the tori and the simulated images, which are all based on relatively low magnetisation parameters. However, the theoretical expectation is that in the pulsar wind σ  1 near the light cylinder, and must remain so throughout most of the wind region. Here we explain the reasoning why we expect the pulsar wind to have a magnetisation σ > 1. First of all, note that the magnetic field in the wind, near the light cylinder, has both a radial and a tangential component. Because of magnetic-flux conservation

136

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

2

y

1

0

–1

–2

–2

–1

0

1

2

X

–1.5

–1

–0.5

0

0.5

Fig. 6.9 Synchrotron emission model based on magnetohydrodynamical simulations of the inner region of the Crab nebula [667]. The initial conditions of the pulsar wind are σ = 0.55 and γpw = 10 (assumed low for computational reasons), and the brightness scaling is logarithmic (Credit: Serguei Komissarov)

the radial component, which is divided over larger areas during the outflow, has a dependence Br ∝ r −2 . However, the tangential component is not stretched, but the associated total Poynting flux 4πr 2 |S| = 4πr 2 cBφ2 /(4π) is conserved over the whole sphere encompassing the wind. Hence, Bφ ∝ 1/r. So at sufficiently large distance the tangential component dominates the magnetic energy density. Let us assume that at the light cylinder the tangential component is a fraction fφ of the total field strength, Bφ,LC = fφ BLC . The magnetic field energy density in the wind can then be estimated to be [653] UB ≈

 −2   −2 2  6 4 fφ2 BLC Bp2 RNS 6 Bp2 RNS Bφ (r)2 r r ≈ fφ2 = fφ2 , = 4π 4π RLC 4π RLC RLC 4πr 2 c4

(6.42) where we have used that BLC ≈ Bp (RLC /RNS )−3 . Not surprisingly the associated total integrated Poynting flux (4πr 2 cUB ) yields an expression that is reminiscent of the energy loss rate for a rotating dipole (6.8).

6.3 The Inner Regions of Pulsar Wind Nebulae

137

˙ Fig. 6.10 The measured √ torus radii versus the pulsar spin-down energy E. The line indicates the ˙ The figure is based on data and analysis presented in [127] and [856] best fit with rtorus ∝ E.

The energy density of the electrons/positrons can be expressed in terms of the number of relativistic particle crossing the light cylinder through the open field lines, which is given by the particle density near the two polar caps times the speed of the particles (c): 2 N˙e = 2πRPC κnGJ c,

with RPC the polar cap radius (6.23), nGJ the Goldreich-Julian density (6.28) and κ the multiplicity (6.29). This gives N˙ e = κ

3 2 Bp RNS . ec

(6.43)

The kinetic energy density in the wind is then Ue =

3 me 2 Bp RNS pw me c2 N˙ e = κ , pw 4πr 2 c 4πr 2 e

with w the Lorentz factor of the relativistic outflow.

(6.44)

138

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Combining (6.44) and (6.42) gives [e.g. 752] 2

3

2

1 fφ eBp RNS  (6.45) κw me c 4 −2       Bp w −1 2 RNS 3 P ≈4 × 104κ −1 f . φ 100 ms 12 km 106 1012 G

σ =

We have used here a Lorentz factor for the wind of w = 106, which is commonly used for pulsar wind nebulae like the Crab Nebula [638, 810]. The Crab Pulsar has a period of 33 ms, so σ  10 for κ < 105, the upper limit that is currently theoretically supported [532, 1118]. Moreover, there are indications that w ≈ 100 [750], which results in σ  105 . So from a theoretical perspective we expect σ  1, whereas observationally it is clear that σ  1. This discrepancy between theory and observations is known as the σ -problem. Looking at the energetics, the σ -problem does indicate that the magnetic field energy density must be reduced somewhere inside the pulsar wind region. A popular idea is that the Poynting flux of the pulsar wind close to the termination shock is reduced by magnetic reconnection. For an oblique rotator, around the equator the magnetic-field polarity switches during each pulsar rotation. These regions of opposite magnetic-field polarity are frozen into the plasma, as the electrons/positrons are forced to move with the plasma waves. This configuration is referred to as a “striped wind” [653]. If the regions of opposite polarity are colliding with each other, either in the wind, or when the termination shock is reached, magnetic-field reconnection is likely to occur. This transfers energy from the magnetic field to the electrons/positrons, reducing σ in the wind or in the termination shock region itself. There are also suggestions that the conversion of magnetic-field energy density to particle density takes place immediately outside the termination shock region [138, 929]. In these models there is large non-toroidal magnetic field component in the downstream region, so violating one of the assumptions (6.37) of the Kennel and Coroniti model. Abandoning this assumption means that non-relativistic plasma flows are not necessarily implying a low σ value for the pulsar wind. Note, however, that these ideas solve the dynamical part of the σ -problem (why is the plasma moving subrelativistically?), but may not solve the estimates of σ based on observations of the synchrotron radiation and inverse Compton radiation (Sect. 6.4.4). Finally, one may reduce the σ -problem by allowing for a large fraction of ions in the pulsar wind [1070], as their rest mass results in a higher kinetic energy density of the wind, see the factor m−1 e in (6.45).

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

139

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

d

on isc

t inu

ity

er se shoc

k

PWN

Con t a ct d

on isc

R ev

tinu

Ejecta

+

ard shock Fo r w

Co n

R ev

Ejecta

For ward shock

ta

ct

On a large scale a pulsar wind nebula consists of a large volume with and internal energy dominated by relativistic electrons and positrons and magnetic field energy. Zooming out from the region near the termination shock to the pulsar wind nebula as a whole, we have a roughly spheroidal region whose pressure is dominated by the non-thermal, relativistic electron/positron plasma, producing synchrotron radiation as a result of the embedded magnetic field. The latter, as we have described before (Sect. 6.3.2) likely becomes more prominent toward the outer regions of the pulsar wind nebula. On the inside the pulsar wind nebula is bound by the termination shock, whereas on the outside there will be a contact discontinuity, which separates the pulsar wind nebula plasma from its immediate surroundings: the supernova remnant or the interstellar medium. The typical structure of a pulsar wind nebula in the early phase of its evolution is expected to resemble the schematic cartoon in Fig. 6.11. It shows the pulsar wind nebula still inside, and pushing against, the unshocked ejecta. At a later stage the outer edge of the pulsar wind nebula makes contact with the reverse shock. Depending on the pressure difference between the shocked ejecta and pulsar wind nebula, the shocked ejecta shell may crush the pulsar wind nebula (thereby brightening it, because the plasma is adiabatically heated and the magnetic field enhanced ), or the pulsar wind nebulae will prevent the reverse shock of moving inward.

erse shock

ity

Fig. 6.11 Schematic structure of a composite supernova remnant. From inside out we have: the pulsar (cross), the unshocked pulsar wind (white) bound by the termination shock, the pulsar wind nebula (often with a bipolar structure), ejecta shocked by the pulsar wind nebula’s outer shock, the unshocked supernova ejecta bound by the reverse shock, the shocked ejecta, the shocked ambient medium, bound by the forward shock

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

To complicate matters further, pulsars appear to receive a kick velocity at birth, with the average velocity being of the order of 200–300 km s−1 [97, 491], but with a large spread. This velocity is a fraction of the shock velocity of a young supernova remnant, but comparable to the shock velocity of a supernova remnant in the radiative phase (Sect. 5.7). This means that in supernova remnants older than ∼104 yr the pulsar and the inner regions of the pulsar wind nebula is likely to be displaced from the geometrical centre of the supernova remnant. If the pulsar’s kick velocity is high enough, the pulsar may even have penetrated the supernova remnant shell. The sensitivity of the dynamical behaviour of pulsar wind nebulae in supernova remnants to the energetics of both the pulsar wind (6.9) and the supernova remnant properties, brings about a large variety inpulsar wind nebulasupernova remnant systems. For example, the composite supernova remnant G11.2-0.3 (Fig. 6.4) looks not unlike the cartoon in Fig. 6.11, but the Crab Nebula (Fig. 6.5) seems to lack a supernova remnant-shell all together.

6.4.1 A Self-similar Solution for the Expansion into the Ejecta Initially the pulsar wind nebula is evolving within the unshocked, freely expanding ejecta. This drives a shock into the ejecta, but now from the inside rather than from the outside, the latter being the case for the reverse shock (Sect. 5.3). The thermodynamics of the expanding pulsar wind nebula can be understood by referring to the well -known law of thermodynamics dQ = d(U V ) + P dV (c.f. Sect. 5.8.1), with dQ/dt = |E˙ rot |, and U = P /(γ − 1), the internal energy. Since we are dealing with a relativistic gas, of low σ , one expects γ = 4/3. So the thermodynamics of the pulsar wind nebula is governed by the equation [257]   d 4πR 3 Ppwn dR + Ppwn 4πR 2 = dt dt dPpwn dR dR =Ppwn 16πR 2 + 4πR 3 . dt dR dt

|E˙ rot | =

(6.46)

√ Note that the speed of sound in a relativistic gas is cs = c/ 3, which is very fast compared to the speed of the bulk flow and supernova remnant plasma. One can therefore assume that the pressure is more or less uniform throughout the pulsar wind nebula. As indicated by (6.20), the pulsar energy loss rate is fairly constant for t < τch,0 (Sect. 6.20). So we consider here the situation that the spindown power is constant. We present a self-similar solution based on the Chevalier and Fransson model [257]. Under the assumption of self-similarity both righthand terms in (6.46) should have the same dependency on R, which requires that R 2 Ppwn ∝ R 3 dPpwn /dR. Hence, the pressure should scale as Ppwn = P0 R0−α R α , with α a parameter that we need to determine and R0 the radius at some arbitrary

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141

time t0 . Inserting this into (6.46) and integrating the differential equations shows that   4+α (3m−1)/m 1/m ˙ P0 R0−α R 3+α = 4π(m + 1)P0 R0 |Erot |t = 4π R . (6.47) 3+α We have here made use of the fact that the solution requires that R ∝ t m , with m = 1/(3 + α), the expansion parameter (Sect. 5.2) of the contact discontinuity between the pulsar wind nebula and the inner ejecta. Following [257], we can solve for m by assuming that the pulsar wind nebula initially evolves within the core region of the ejecta (Sect. 5.6), for which ρej = ρcore,0 (t/t0 )−3 (5.35). We also assume that the ejecta is swept up by the pulsar wind nebula into a relatively thin shell of gas with mass Mej,sh = 4πR 3 ρcore (R, t)/3, which moves with the velocity of the contact discontinuity between pulsar wind nebula plasma and shocked ejecta at radius R. The equation of motion for this system is now [257]   −3    R 2 d 2R t dR 2 . − Mej,sh 2 = 4πR Ppwn − ρcore,0 dt t0 dt t

(6.48)

The right-hand side gives the force (pressure times area) of the pulsar wind nebula on the inner edge of the ejecta (the contact discontinuity), but corrected for the ram pressure (ρ v 2 ) of the unshocked ejecta, with v the relative velocity between the freely expanding ejecta (R/t) and the velocity of the contact discontinuity (dR/dt). Inserting R ∝ t m and P ∝ R α ∝ t 1−3m in (6.48) shows that the left-hand side scales as t 4m−5 , whereas the right-hand side scales as t 1−m . Since the left-hand side and right-hand side should have the same scaling with t for the solution to be self-similar, it is required that m=

6 . 5

(6.49)

Inserting (6.47) and (5.39) and using dR/dt = mR/t and dR 2 /dt 2 = m(m − 1)R/t 2 , we find the following expression for the radius of the pulsar wind nebula:  Rpwn = 

1/5

|E˙ rot,0|

t 6/5

11 4π 125 ρcore,0 t0−3

125 = 33 



n n−3

Mej ≈0.8 10 M



3 n−3 10 n − 5

− 1  2

(6.50) 3/2

Esn 1051 erg

1/5 |E˙ rot,0 |

3  10

−1/2

Mej

|E˙ rot,0| 38 10 erg s −1

3/10

Esn

 15 

t 103 yr

6/5 pc,

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

with Esn and Mej the supernova explosion energy and ejected mass, respectively. For the approximation we used n = 9. The supernova energy and mass matters, as it determines the density in the ejecta: more kinetic energy and less ejecta mass results in a lower ejecta density, and hence the pulsar wind nebula ploughs more easily through the ejecta. The expansion velocity can be obtained by taking the time derivative of (6.50), which gives  Vpwn ≈ 940

Mej 10 M

− 1  2

Esn 1051 erg

3/10 

Erot,0 38 10 erg s −1

1/5 

t

1/5

103 yr

km s−1 .

(6.51) The shock velocity in the frame of the ejecta is given by the difference between the expansion velocity and the local ejecta velocity, which is the free expansion velocity: V˜s = 65 Rpwn /t − Rpwn /t = Vpwn /6. For the typical parameters used in (6.51) we obtain Vs ≈ 160 km s−1 . Interestingly, for this and lower shock velocities, the post-shock plasma rapidly cools, i.e. the pulsar wind nebula shock is a likely to be a radiative and should be bright in the optical (Sect. 4.4). This is indeed the situation for the Crab Nebula, as can be seen in Fig. 6.5 (right). Note that one of the assumptions in deriving (6.50) is that the spin-down power is entirely used to inflate the pulsar wind nebula, whereas clearly there are energy losses in the form of synchrotron and inverse Compton radiation (the optical filaments do not count as this is an effect of the work done by the pulsar wind nebula, for which we already corrected). However, for the Crab Nebula the spin-down power is ∼4 × 1038 erg s−1 , whereas the energy losses as judged from the spectral energy distribution is of the order of 1037 erg s−1 , i.e. just a few percent of the spin-down power. Hence, radiative energy losses, at least for the Crab Nebula, are not expected to have had a big impact on the expansion of the pulsar wind nebula.

6.4.2 The Appearance and Dynamics of the Crab Nebula Although the approach taken above (adopted from [257]) uses a few assumptions, many of the characteristics resemble the properties of the Crab Nebula. The theoretical prediction for the expansion parameter is m = 1.2, which means that the pulsar wind nebula is accelerating. Indeed, it is for a long time known that the Crab Nebula is accelerating [346, 1124], with the dynamical age being slightly younger than the true age, given that the supernova event occurred in AD 1054. The most recent radio expansion measurement indicates m = 1.26 ± 0.05 [164], somewhat larger, but consistent with the predicted expansion parameter. The general appearance of the Crab Nebula is surprising: there is no bright supernova remnant radio shell, and instead the ejecta that we do detect are contained in filamentary structures that are bright in optical forbidden line emission (Fig. 6.5) and through

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143

infrared emission from collisionally heated dust [e.g. 447, 1103]. These filaments are mainly found at the outskirts of the nebula, but several filaments also cross the central regions. The fact that most of the ejecta are cool enough to emit in the optical is consistent with the theory that the relative velocity between the expanding pulsar wind nebula and the ejecta is smaller than ∼200 km s−1 , so that the shocks are radiative. The filamentary structure is now generally understood to be due to the fact that the contact discontinuity between the pulsar wind nebula and the shocked ejecta is Rayleigh-Taylor unstable (Sect. 5.9), wrinkling the interface between the two plasmas [526]. The Crab Nebula is the best studied pulsar wind nebula and, therefore, much of our knowledge on the evolution and dynamics of pulsar wind nebulae is based on this object. However, the Crab Nebula is an unusual object and it is not quite clear whether our insights based on the Crab Nebula readily applies to pulsar wind nebulae in general. First of all, the Crab Nebula has an unusually high spin-down rate, 4.6 × 1038 erg s−1 , only PSR J0537-6910 (Fig. 6.2) and PSR B0540-69 have comparable rotation powers. But unlike the pulsar wind nebulae surrounding these pulsars, the Crab Nebula does not seem to have a radio or X-ray shell. For a long time it was thought that the supernova remnant is moving through and extremely tenuous circumstellar medium, resulting in a shell that is very faint in radio or Xrays. Despite several efforts [1039], no such shell has ever been detected. There is also no evidence for optical emission from photo-ionised unshocked ejecta [380]. This is all best explained by the hypothesis that the ejecta have been completely overrun by the pulsar wind nebula. The reason could be that the supernova of 1054 was a low energy explosion, with 1050 erg of energy rather than 1051 erg [384, 1262]. A particular model for a low-energy explosion is the electron-capture supernova model [865, 867] (Sect. 2.2.4).

6.4.3 Pulsar Wind Nebulae Interacting with the Reverse Shock After the phase described in Sect. 6.4.1, the outer edge of the pulsar wind nebula (and the narrow layer of ejecta swept up by it) is bound to encounter the reverse shock of the supernova remnant. Most likely at this stage of the development of the supernova remnant the reverse shock is already moving toward the centre. The start of the interaction between the reverse shock and the pulsar wind nebula will lead to a compression of the pulsar wind nebula, because the reverse shock is pushing back on the pulsar wind nebula. This results in the compression of magnetic field and adiabatic heating of the electrons/positrons. So the synchrotron radiation from the pulsar wind nebula will brighten. The crushing is halted once the pressure of the reverse shock equals the pressure of the crushed pulsar wind nebula, but this quasistatic situation may not be reached immediately [183, 1157]. Rather, the crushing of the pulsar wind nebula may overshoot, as the sound waves inside the pulsar wind nebula try to restore internal pressure equilibrium. The overshooting itself, will

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

lead to bouncing back of the pulsar wind nebula radius, which also may overshoot, etcetera. After a few reverberations the pulsar wind nebula starts evolving in a more regular fashion again, in which approximate pressure balance is maintained by the pressure of the interior region of the hot supernova remnant shell and the pulsar wind nebula. At this stage the pulsar wind nebula is expanding again, as the pulsar wind nebula is being supplied with new energy from the still active pulsar, whereas the supernova remnant interior pressure is decreasing as the supernova energy is diluted over an increasing volume. The boundary between pulsar wind nebula and supernova remnant shell is Rayleigh-Taylor unstable (Sect. 5.9), giving rise to an irregular contact discontinuity [183]. A good idea how the average radius of the pulsar wind nebula/supernova remnant contact discontinuity develops can be obtained by using the Sedov-Taylor model for self-similar evolution of the supernova remnant (Chap. 5), see reference [1157]. As shown in Sect. 5.5 the interior pressure inside the supernova remnant is more or less flat in the core of the supernova remnant, with a value   4 Esn Rfs 2 Psnr,interior ≈ 0.31 ρ0 m = 0.074 3 . 3 t Rfs

(6.52)

For this relation we have made use of (5.33) and (5.12). The pressure of pulsar wind nebula depends on the total energy output by the pulsar wind, )

t

Epsr ≡

Erot (t )dt ,

(6.53)

0

which has been used to create a plasma bubble of relativistic particles, and expand do work on its environment. In other words, we need to use the previously used equation dQ = d(U V ) + P dV (c.f Sect. 6.4.1). Assuming spherical geometry, we can write Epsr =

γ 4π 3 R Ppwn , γ − 1 3 pwn

(6.54)

with γ = 4/3 the adiabatic index of plasma. For a quasi-static evolution we require Psnr,interior = Ppwn . So we obtain  Rpwn ≈ 1.2

Epsr Esn

1/3

 Rfs ≈ 0.27

Epsr 1049 erg

1/3 

Esn 1051 erg

−1/3 Rfs .

(6.55)

If for the age of the pulsar t we have t  τch,0 , with τch,0 the initial characteristic spin-down time (6.19), then the pulsar energy loss rate is fairly constant (6.20) and Epsr ≈ |E˙ rot,0 |t. Since in the Sedov-Taylor phase Rs ∝ t 2/5 , we see that (6.55) implies Rpwn ∝ t 11/15 . On the other hand, if t  τch,0 , then for a pulsar braking index of n = 3 we have Epsr ∝ t −1 and the growth will proceed as Rpwn ∝ t 1/15 . Which of the

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

145

two situations applies does depend strongly on the initial spin-period P0 and dipole magnetic field of the pulsar. The relation (6.55) has been used to make an estimate of the initial pulsar spin period, P0 [1157]. This can be done since the total integrated energy output of the pulsar should be equal to the loss in rotational energy (6.3): 2 1  Epsr (t) =η (Erot (t = 0) − Erot (t)) = I η 20 − (t)2 2 ⎛ ⎞ P2 ⎠, ⇒ P0 = ⎝ E P2 1 + 2πpsr2 I η

(6.56)

with η an efficiency factor, which takes into account that some of the spin-down energy has been lost, for example due to radiative losses. A small η would lead to an estimate of P0 that is smaller. So using η = 1 leads to a lower limit on P0 . As shown by van der Swaluw and Wu [1157], the measured ratios of a number of composite supernova remnants shows that the initial spin periods range from 39 ms to 484 ms, assuming supernova energies of 1051 erg. This is rather slow, and contrary to the sometimes held believe that pulsars are born with periods less than 10 ms. The sample used did not include N157B, a composite supernova remnant with a current period of 16 ms (Fig 6.2), so clearly the initial spin period must have been less than 16 ms. But such a small period seems the exception, rather than the rule. If anything, the results of [1157] and the fast rotation of the pulsar in N157B shows that there seems to a be a large spread in initial spin periods. But note that the method used relies on the pulsar wind nebula to be in a quasistatic expansion and interacting with the reverse shock, and assumes the canonical explosion energy of 1051 erg. Clearly this does not apply to the Crab Nebula, as the supernova seems to have been subenergetic, and the pulsar wind nebula is interacting with the unshocked ejecta, rather than with the reverse shock.

6.4.4 The Radiation from Pulsar Wind Nebulae The particle population inside pulsar wind nebulae consists predominantly of relativistic electrons and positrons (collective referred to here as electrons), which escaped from the magnetosphere (Sect. 6.2.3). The electrons in the pulsar wind nebulae have a non-thermal energy distribution. This distribution was caused by reconnection within the pulsar wind (according to the striped-wind model, Sect. 6.3.4), or by relativistic diffusive shock acceleration (or Fermi acceleration, Chap. 11) at the pulsar wind termination shock, or perhaps a combination of the two processes. The electrons, whose energy distribution can be approximated by powerlaw distributions for broad energy ranges, emit synchrotron radiation (Sect. 13.3) in the radio to X-ray/soft γ -ray regime, and cause inverse Compton scattering of

146

6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Fig. 6.12 The broad spectral energy distribution of three pulsar wind nebulae. The emission up to ∼1021 Hz is dominated by synchrotron radiation, whereas emission above ∼1022 Hz is dominated by inverse Compton scattering. The three pulsar wind nebulae all have rotational energy-loss rates exceeding 1038 erg s−1 : the Crab Nebula [16, 51, 113, 687, 754] (black points), N157B [23, 38, 312] powered by PSR J0537-6910 (red), and PWN N158A [38, 208] (blue) powered by PSR J0540-6919 and located in SNR 0540-69.3. For the synchrotron spectrum of the Crab Nebula approximate power-law models are indicated: at low frequencies α ≈ 0.3, above 1014 Hz α ≈ 0.8, and in the X-ray regime α ≈ 1.25

photons from the local radiation fields (Sect. 13.2.2), a process that results in the γ -ray radiation detected above GeV energies. Figure 6.12 shows the spectral energy distribution (SED) of three powerful pulsar wind nebulae: the Crab Nebula, N157B and N158A, the latter two located in the Large Magellanic Cloud. All three show the broad division in a synchrotron dominated part, probably peaking in the UV-regime (although the peak itself is missed due to UV absorption) or X-rays, and the GeV to 100 TeV γ -ray regime, dominated by inverse Compton scattering. Despite the similar pulsar spin-down power for the three objects, the SEDs show remarkable differences. Part of that may be caused by different ages and spin-down evolution histories. But the differences in peak luminosities in the synchrotron regime versus the luminosities in the γ -ray regime are caused by the ratio between magnetic-field energy density and radiation energy density, see (13.32) and (13.20): (νLν )syn,peak UB ≈ . (νLν )ic,peak Urad

(6.57)

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

147

The Crab Nebula’s radiation peaks in the UV and is caused by synchrotron radiation, with a magnetic-field strength estimated to be B ≈ 100–300 μG [293, 810]. The SEDs of both Large Magellanic Cloud objects are less well covered, but the peak in the SED appears to occur in the γ -ray regime, and is caused by inverse Compton scattering. Both sources are located in the 30 Doradus star forming region, which likely provides a large infrared/optical/UV background radiation field. This explains why the SEDs of N157B and N158A peak in the γ -ray domain. However, it does not explain why N157B has a lower magnetic-field strength (B ≈ 45 μG [23]) than the Crab Nebula. If the magnetic-field strength of the pulsar wind nebula relates to the magnetic-field strength at the light cylinder (Sect. 6.3.4), N157B (powered by PSR J0537-6910) should have a larger magnetic-field strength (BLC ≈ 3.5 × 106 G) than the Crab Nebula (BLC ≈ 1.8 × 106 G), which is clearly not the case. One complication is that the frequent glitches of PSR J0537-6910 (Fig. 6.2) casts some doubt about whether the dipole field can be estimated from the spin-down rate. The best measured SED is that of the Crab Nebula. Detailed modelling of the inverse Compton emission from the Crab Nebula indicates that several background radiation fields contribute significantly to the γ -ray emission [293, 810]: synchrotron and infrared (dust) emission from the Crab Nebula itself, and the omnipresent cosmic microwave background. The dust emission from the Crab Nebula [876] shows up as a bump in the SED around 1012 –1013 Hz. The broad band radiative energy losses of pulsar wind nebulae, ultimately find their origin in the energy input by their pulsars, which is assumed to be at the cost of the pulsar’s rotational energy (6.3). It is, therefore, quite natural to compare the radiated power of the pulsar wind nebula with the pulsar’s spin-down power, using a radiative efficiency parameter defined as η ≡ Lpwn /|E˙ rot |. Typically the radiation efficiency in X-rays, which represent for several pulsar wind nebulae the peak of the SED (but not for the Crab Nebula!), lies between η = 10−4 –0.1. Figure 6.13 shows a comparison of the radiative X-ray and γ -ray efficiencies, as a function of the pulsar characteristic age. Remarkably, the X-ray radiation efficiency appears to be dependent on the age of the pulsar, with young pulsars having higher radiation efficiencies in X-rays [767, 1179]. Moreover, the X-ray emission from the pulsar itself appears to follow a very similar trend. For the γ -ray emission the efficiency shows an inverse trend: the efficiency increases as a function of age, reaching even levels of η ≈ 1. Note that because of uncertainties in the distance to the pulsar wind nebulae, η has some associated uncertainty. The high radiative efficiency of young pulsar wind nebula in X-rays can potentially be attributed to a higher electron-positron pair multiplicity κ (6.29) for young pulsars. The reason may be that pair creation requires seed photons, which are more abundant above the hotter surfaces of young pulsars. To explain the higher γ -ray efficiency for older pulsar wind nebulae, one needs to keep in mind that the electrons have energies between 100 MeV to 1 TeV, which have a much longer energy-loss time scales than the 10–100 TeV electrons, responsible for X-ray emission. We, therefore, expect that the X-ray emission from pulsar wind nebulae arises from a, more or less, steady state situation: within a small fraction of

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Fig. 6.13 The X-ray radiation efficiency η (0.5–8 keV) of pulsar wind nebulae (solid black squares) and pulsars (solid red squares) as a function of characteristic pulsar age [1179]. The open blue squares represent the efficiency of γ -ray radiation in the 0.1–100 GeV band, based on Fermi-LAT data [11]

the age of a pulsar wind nebula the energetic electrons/positrons are injected, and subsequently radiate away their energies in X-rays. For the lower energy electrons this steady state model may not apply: the radiation is more likely to correlate with the total energy content in the 100 MeV to 1 TeV electrons, which is the result of a longer built-up of ! the plasma over the life time of the pulsar wind nebula. This implies that Lγ ∝ |E˙ rot |dt ≈ Erot (t = 0), rather than ∝ |E˙ rot|. The TeV γ -ray emission follows a very similar trend to the GeV emission [475], perhaps indicating that the TeV γ -ray emission is still dominated by electrons that have not been greatly affected by radiative losses. Indeed, pulsar wind nebulae are usually larger in the sub-TeV γ -ray domain than in the X-ray domain [475, 612] (see the next section).

6.4.5 The Electron/Positron Populations in Pulsar Wind Nebulae The SEDs of pulsar wind nebulae are largely shaped by the underlying electron energy distribution, which itself reflects a combination of the acceleration processes, responsible for the non-thermal distributions, and time integrated evolutionary effects: the evolution of the spin-down power (6.20), and electron energy losses

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

149

due to synchrotron radiation and inverse Compton scattering (Sect. 13.3.3), and, in addition, adiabatic cooling. As explained before, the radiative losses depend on the energy density of the magnetic field and radiation fields. Most pulsar wind nebulae have a rather flat radio synchrotron spectral index of α ≈ 0.3. Such a flat spectrum is not expected on the basis of diffusive-shockacceleration (DSA) models for relativistic shocks, which predict particle spectral indices around q ≈ 2.2, corresponding to α ≈ 0.55 (see for example [35]). This has led to the hypothesis that acceleration occurs through magnetic-field reconnection in the striped wind [653] (Sect. 6.3.4). Indeed, reconnective acceleration is thought to produce flatter energy distributions than DSA [338, 1052]. The X-ray synchrotron spectra of pulsar wind nebulae are steeper than in the radio,  ≈ 1.5 [612], corresponding to α ≈ 0.5, q ≈ 2. The obvious explanation for this is that the very high energy electron/positron population has steepened due to radiative losses. As explained in Sect. 13.3.3 this should give rise to a break with

α = 0.5. This cannot be the explanation for the steep X-ray spectrum of the Crab Nebula. Indeed there is a steepening with α ≈ 0.5 around 1014 Hz (Fig. 6.12). This is the frequency at which a cooling break is to be expected, since for synchrotron losses in a uniform magnetic field of B = 200 μG [810] a cooling break is predicted to be located around νage ≈ 9 × 1013 Hz (13.56). But the X-ray spectrum is steeper than the radio spectrum with α ≈ 0.95, in disagreement with the cooling-break model. There is no generally accepted theory that explains the steep X-ray spectrum of the Crab Nebula, which requires a second break around 1015–1016 Hz. An explanation found in the literature is that there are two distinct electron populations (we use electron population here as a shorthand for electron/positron population). The two populations could be the result of two acceleration regimes (DSA and reconnection), or even of two different epochs of electron acceleration, with the flat radio spectrum originating from an older, “relic” electron population [103]. In such a model the freshly injected electrons only produce synchrotron radiation in the UV to X-ray regime. See also [810] for a similar model. The idea of two populations is also connected to a challenge the radio synchrotron emission poses to the canonical value of the Lorentz factor of the pulsar wind [751], which is pw ≈ 106 . See the Kennel and Coriniti model [638] and the discussion on the σ -problem (Sect. 6.3.4). A much lower wind Lorentz factor would mean that for the same mechanical wind power, we need more electrons, and hence a much higher value for the multiplicity factor κ. From a theoretical point of view we expect κ  105, and hence we obtain the canonical value, w  106 (6.45). However, a shocked electron wind with pw = 106 will accelerate electrons to Lorentz factors well in excess of e > 106 , but with a low energy cut-off corresponding to e ≈ pw ≈ 106 . In fact, it is likely that the electron distribution is a power law, but with at the low energy part resembling a relativistic Maxwellian distribution [1053] with kT ≈ pw me c2 ≈ 5 × 1011 eV. This corresponds to a synchrotron radiation frequency of ν ≈ 1014 (B/100 μG) Hz. So for pwn = 106 we expect a bump in the spectrum around 1014 Hz caused by the Maxwellian distribution and not much synchrotron radiation at lower frequencies.

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

In order to explain the non-thermal radio synchrotron spectrum well below 1014 Hz, we either have to assume that the wind Lorentz factor is much lower [652, 751] (pwn  500) in order to explain radio emission down to 107 Hz, but requiring the pair multiplicity to be κ  107 in contradiction to magnetospheric models [1118], or the radio-emitting electrons have indeed another origin. On the other hand, the idea of an older electron population does not agree with the lack of spectral index variation within the Crab Nebula [165]. One expects the older, radio emitting, population to be located at the outskirts of the nebula, which is not supported by observations. On the contrary, the radio emission appears to start also at the “wisp” similarly to the optical and X-ray emission [165]. This does not preclude the idea of two electron populations, for example caused by two acceleration mechanisms, but it goes against the idea of a “relic” population responsible for all the radio synchrotron radiation. Clearly we still lack a good model for the injection spectrum of the electrons for the Crab Nebula. For other pulsar wind nebulae similar problems likely exist, but for those objects we lack the detailed SED coverage that we have for the Crab Nebula.

6.4.6 The Frequency Dependent Sizes of Pulsar Wind Nebulae The Crab Nebula’s synchrotron spectrum extends up to ≈ 5×1021 Hz beyond which it steeply declines. This frequency corresponds to an extremely high electron energy of E ≈ 2 × 1015(B/200 μG)−1/2 eV; the Crab Nebula is a PeVatron! Recall that the synchrotron loss time scale is given by τsyn = |E/(dE/dt)| ≈ 634/(B 2E) s (13.47), which means that the highest energy electrons in the Crab Nebula retain that energy for 1 or 2 months. However, apart from radiative losses, also adiabatic energy losses need to be taken into account. In particular for the inner regions of the pulsar wind nebula. We can estimate the effect of adiabatic losses by assuming that the flow of particles is roughly radial (as is the case in the Kennel and Coroniti model, Sect. 6.3.2) and scales as  n(r) = nts

r Rts

−s ,

(6.58)

with the subscript ts denoting the quantities at the termination shock, and s a parameter of order one. For free flowing electrons s = 2 (c.f. 5.6.1), but for the inner regions of a pulsar wind nebula we have s  2. To calculate the adiabatic losses we use the adiabatic invariant En−γ +1 (corresponding to P V γ =constant), with γ = 4/3. From this we find 

dE dt

 = (γ − 1)E ad

1 1 dr 1 1 1 dn = − sE = − sE β(r)c. n dt 3 r dt 3 r

(6.59)

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

151

This shows that adiabatic losses scale with E, whereas synchrotron radiation losses scale with E 2 , see (13.46). So for high enough energies synchrotron losses will be more important. For the inner region of the Crab Nebula velocities appear to be of the order β ≈ 0.3 [1031]. The electron energies for which adiabatic losses are the dominant loss factor should therefore have  E  2.4 × 1014s

r 5 × 1017 cm

−1 

β 0.3



B 100 μG

−2 eV,

(6.60)

where we have scaled the radius to the approximate radius of the Crab Nebula’s termination shock radius. The equation shows that for high energies adiabatic losses may be more important than synchrotron radiation losses, but this depends critically on β as a function of radius, and on the magnetic-field strength. At larger radii, beyond r ≈ 3 × rts we β < 0.1 (Fig. 6.6), and there the adiabatic losses can be neglected compared to radiative losses. The overall effect of the energy losses of the high energy electrons is that the apparent size of a pulsar wind nebula should shrink with increasing energy. The radio synchrotron emitting region of the Crab Nebula is indeed much larger than the optical synchrotron nebula, which in turn is larger than the X-ray synchrotron nebula. See Fig. 6.5. A recent study with the NuStar hard X-ray telescope shows that the nebula seen in hard X-rays is also smaller than in soft X-rays [755]. In addition, the X-ray spectral index in the 0.5–2 keV band is steeper in the outer regions of the X-ray nebula [1234]. A similar gradient in the X-ray spectral index is seen for the Crab-like supernova remnant G21.5-0.9 [1056], although here the steepening in spectral index does not appear to agree with predictions by the Kennel and Coroniti model (Sect. 6.3.2) [963].

6.4.7 The Large Extent of Some Pulsar Wind Nebulae in γ -Rays The frequency dependence of the size of pulsar wind nebulae has also been observed for very-high energy γ -rays. In fact, some pulsar wind nebulae turn out be unexpectedly large in γ -rays. See [475] for an overview. The γ -ray emission at these energies is inverse Compton scattering, and the emissivity depends on the electron population and the radiation fields. Since the radiation fields are likely not dramatically changing over the scale of the pulsar wind nebula, unlike the magnetic field strength which plays a similar role for synchrotron radiation, very-high energy γ -rays gives us a more direct view of the electron population. A good example of the surprising γ -ray extend of pulsar wind nebula is HESS J1303-631 [517], which is powered by the pulsar PSR J1301-6305; see Fig. 6.14. Below 2 TeV the pulsar wind nebula is much more extended than for γ -ray energies above 10 TeV, which is likely the effect of radiative losses of the highest energy electrons. Below 2 TeV the source extent is about half a degree, corresponding

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Fig. 6.14 The extended pulsar wind nebula HESS J1303-631, as observed by H.E.S.S. in the TeV γ -ray band [517]. The different colours show the nebula in different photon energies, as indicated. The pulsar powering the nebula is PSR J1301-6305. Below photon energies of 10 TeV the pulsar wind nebula is much more extended than in X-rays (here the XMM-Newton-detected emission is indicated by white contours) (Courtesy: H.E.S.S. collaboration [517])

to 58 pc at a distance of 6.6 kpc [517]. Above 10 TeV the source is centred at the pulsar, which itself is off-centre of the nebula at low γ -ray energies. The offset either indicates a large space velocity for the pulsar, near what has been found observationally for other pulsars (∼1500 km s−1 ), or it may indicate an asymmetric development of the reverse shock of the supernova remnant, or it may be caused by strong density gradients in the ambient medium. The large extent itself is surprising given that supernova remnants rarely grow much larger in radius than 25 pc (Sect. 5.7). Moreover, at a late stage the reverse shock should have moved to the centre, crushing the pulsar wind nebula. In order to explain the large extent, one has to assume that the supernova remnant developed in a low density environment, or perhaps the pulsar has broken out of supernova remnant shell as a result of its likely large velocity (see also Sect. 6.4.8). Yet another reason can be that the total, time integrated energy input by the pulsar at some time exceeds the energy of the supernova remnant shell. This can happen late in the evolution of the supernova remnant, when the supernova remnants is no longer in the adiabatic phase (Sect. 5.7), or if the supernova remnant was subenergetic to begin with, This could happen in combination with an unusually energetic pulsar, which implies a short initial rotation period. For example, for a subenergetic supernova explosion of 1050 erg, a pulsar with an initially spin period of P < 10 ms, see (6.3), will eventually provide more energy than the supernova explosion, and its

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

153

pulsar wind nebula could in the end push the reverse shock outward and overrun the supernova remnant. Note that the characteristic age (6.12) of the PSR J1301-6305 is 11 kyr, but this seems an underestimate given the extent of the pulsar wind nebula and the large space velocity necessary to explain the large offset between the pulsar position and the centre of the pulsar wind nebula. An age of the pulsar around 50 kyr seems more likely. The large extent of some pulsar wind nebulae like HESS J1303-631 came as a surprise when TeV astronomy started to flourish. However, there are two factors that may provide the reason why it requires very-high energy γ -rays to reveal the large extent of these pulsar wind nebulae. First, imaging atmospheric Cherenkov telescopes (IACTs) like H.E.S.S., VERITAS and MAGIC (see also Sect. 12.3) have relatively large field of views (degrees), unlike imaging X-ray telescopes like Chandra or XMM-Newton. With radio telescopes the situation is even more complicated: interferometric synthesis telescopes are relatively insensitive to large scale diffuse emission. Moreover, there is a lot of source confusion in the Milky Way in the radio band. IACTS, therefore, can identify large diffuse objects more easily. Secondly, a probably more important reason why the large extent of some pulsar wind nebulae required γ -ray telescopes to be revealed, has to do with the nature of the emission mechanism: inverse Compton scattering. As already explained, inverse Compton scattering provides a more direct view of the electron population, whereas synchrotron radiation also depends on the magnetic field, which may vary across the pulsar wind nebula, and may at very large distances (> 10Rts) from the pulsar drop ( 10Rts, see Fig. 6.6). The B ∼ 200 μG for the Crab Nebula, which is a relatively compact pulsar wind nebula, is probably more an exception than the rule. In fact, modelling the SED of HESS J1303-631 gives an estimate of the average magnetic field of only 1.45 μG [517]. This means that the magnetic field energy density is lower than the radiation field of the cosmic microwave background. As a result the SED peaks in the γ -ray band, instead of in the optical/UV band. For those low magnetic fields the radiative cooling is dominated by inverse Compton scattering. The result of such a low magnetic field is that the pulsar wind nebula has a low radio surface brightness. In fact, HESS J1303-631 is so faint in the radio band that no radio counter part has yet been identified [1089]. A similarly extended pulsar wind nebula discovered in very-high energy γ -rays is HESS J1825-137 [50, 829] (pulsar wind nebula G18.0-0.7), which spans almost a degree on the sky, corresponding to a radius of 35 pc at its likely distance of 4 kpc. Also in this is the pulsar (PSR J1826-1334) offset with respect to the centre. In this case, as well as for HESS J1303-631 one has to wonder even whether the transport of the electron population is solely by advection: is the extend due to outflow of plasma, or are the electrons diffusing out (Chap. 11)? At some radius it may be that diffusion becomes a more important mode of transport, in which case one should perhaps call the extended sources “pulsar wind haloes” rather than pulsar wind nebula [726]. A combination of advection and diffusion is of course also possible. One way to test whether transport of the electrons is diffusive rather than advective is to look for energy difference in extent: diffusion is faster for the

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

highest energy particles. So we could have the situation that from the highest energy to lower energy a pulsar wind nebula first shrinks in size, due to radiative cooling, but that some energy this trends stops, as diffusion becomes important. In this context, a high profile source that received a lot of attention is the nebula (halo) around the Geminga pulsar (PSR B0656+14). Geminga is a P = 0.237 s pulsar with a characteristic age of 340 kyr, too old to still reside in a supernova remnant. It has a bow-shocked shape X-ray nebula around it [236, 932] with a total extent of about 1 . This small size forms a stark contrast to the 5.5◦ extended source around Geminga detected by the HAWC water Cherenkov telescope around 20 TeV [21]. Geminga was also identified as potential source of cosmic-ray positrons detected near Earth with PAMELA and AMS-02 experiments [25, 42], Geminga being one of the nearest active pulsars. However, the HAWC results showed that even if the γ -ray nebula is a diffusively formed halo, rather than a pulsar wind nebula, the diffusion is not fast enough to explain the positrons detected near Earth.

6.4.8 Pulsars Moving Through Hot Supernova Remnant Shells Since pulsars receive on average a kick velocity of 200–300 km s−1 (with a large spread), the pulsar has a high probability to penetrate the hot supernova shell and eventually overtake the shock. As this happens while the pulsar is still relatively young, it will continue blowing a pulsar wind, leading to an off-centre pulsar wind nebula, or even a pulsar wind nebula outside the supernova remnant shell, but with a trail of relic relativistic electrons/positrons in its wake. Only for very high pulsar velocities will the pulsar overtake the shock in the Sedov phase. Recall that the Sedov phase ends when the shock becomes radiative, which happens when Vfs  200 km s−1 (Sect. 5.7). In the Sedov phase we have Vfs = 25 Rfs /t, whereas the pulsar radial coordinate is Rpsr = Vpsr t. Combining these expressions shows that Vfs > 200 km s−1 requires that Vpsr > 500 km s−1 , for the pulsar to be outside the forward shock. Such a velocity lies in the tail of pulsar velocity distribution, corresponding to about a 5% probability [491]. In most cases the pulsar will only break out of the supernova remnant when the supernova remnant is in the radiative phase. Before that happens, however, the pulsar will first enter the shock heated shell of the supernova remnant. An important distinction in this case is whether the pulsar move supersonically or subsonically. In the supersonic case a bowshock with a shock-heated shell will form [1158]. In order to calculate whether the pulsar moves supersonically, we first assess the pulsar’s speed in the frame of the local plasma:

Vpsr =

Rfs t



 Rpsr 3v − m , Rfs 4v2

with m the expansion parameter of the shock, v/v2 the ratio of the local plasma speed (v) versus the plasma speed behind the shock (v2 ), which is itself v2 =

6.4 The Evolution and Radiation of Pulsar Wind Nebulae

155

Fig. 6.15 The sonic Mach number associated with a pulsar moving through the supernova remnant shell, based on the Sedov self-similar model for the temperature and velocity distribution in the hot shell. See also [1158]

(3/4)mRfs /t (see Sect. 4.1). The sonic Mach number of the pulsar can now be expressed as  Ms2 =

ρ v 2 γP

=

n( v)2

μmp = γ nkT2 (T /T2 )

2 3v − 4v 2   , 5 2 T 16 m T2

Rpsr Rfs

(6.61)

3 with T2 = 16 μmp Vfs2 , and T /T2 the ratio of the plasma temperature around the position of the pulsar versus the plasma temperature behind the forward shock. For the Sedov-Taylor model (m = 2/5) we have analytic solutions for the ratios v/v2 and T /T2 as a function of relative radius R/Rfs (Sect. 5.5). So one calculate the sonic Mach number of the pulsar velocity. This is depicted in Fig. 6.15, which shows that the pulsar only moves supersonically once the relative ratio of the pulsar position with respect to the forward shock is Rpsr /Rfs = 0.62. The reason is that the temperature and therefore the sound speed, rapidly increases toward the centre in the Sedov-Taylor model. Since for the Sedov-Taylor model Vfs = 2Rfs /t > 200 km s−1 , we see that the pulsar speed is required to be Vpsr  0.62Rfs/t = 0.62Vfs/m ≈ 310 km s−1 . So only the fastest pulsars are expected to have bow shocks during the Sedov-Taylor phase.

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

Fig. 6.16 Left: Three channel multiwavelength image of IC 443, with red Hα [0.9 m Kitt Peak telescope 1240], green [SII] (idem), and blue X-rays [ROSAT PSPC mosaic, see 98]. Right: Zoom in on the box in the left image. This Chandra image shows the cometary shaped pulsar wind nebula CXOU J061705.3+222127, which probably moves through the shell of IC443 [415, 870] (Credit: Middlebury Optical ATlas of Supernova Remnants (MOATS); ROSAT archive at MPE; and Chandra archive)

An example of pulsar wind nebula moving through the shell of a supernova remnant is the small, bowshock-shaped pulsar wind nebula inside IC443 (Fig. 6.16). In this case the supernova remnant itself has entered the radiative phase.

6.5 Magnetars and Central Compact Objects The Crab Nebula did not only shape our conception of the physics of pulsar wind nebulae, but its pulsar also became the archetypal young pulsar, a representative of many young radio pulsars, having magnetic fields between 1011 and 1013 G and rotating with periods of ∼10 to ∼1000 ms. However, if the Crab Pulsar or the Vela Pulsar would be truly representative of all young neutron stars, why are not all core-collapse supernova remnants composite supernova remnants like G11.2-0.3 (Fig. 6.4) or plerions like the Crab Nebula or 3C358? Indeed, in a review in 1984 Helfand and Becker wrote, after discussing the population statistics of supernovae and the lack of many supernova remnant/pulsar associations [507]: “even if all SN do result in the formation of a stellar remnant, there is a good reason to doubt that they all result in the formation of a rapidly rotating magnetized neutron star similar to the Crab”. The review appeared after the Einstein Observatory (an X-ray space observatory with arcsecond imaging capabilities) found several conspicuous X-ray point sources in the centres of supernova remnants, which did not have radio counter

6.5 Magnetars and Central Compact Objects

157

Fig. 6.17 False colour X-ray images of two supernova remnants with X-ray bright, but radio dim, neutron stars. The images are based on ROSAT-PSPC observations. Left: supernova remnant CTB 109 (G109.1-1.0) with the X-ray bright point source 1E 2259+ 586, an AXP. Right: supernova remnant PKS 1209-51/52 (G296.5+10.0) with in the centre the CCO 1E 1207.4-5209

parts, and for which the hosting supernova remnants were clearly not composite supernova remnants. Two of these X-ray sources are shown in Fig. 6.17. The point source in the supernova remnant CTB 109 (named 1E 2259+ 586) was found to have an unusually long X-ray pulsation period of P = 7 s. After several other similar X-ray pulsars with long periods (5–12 s) were found, a number of which are also located in supernova remnants, sources like 1E 2259+ 58 became known as “Anomalous Xray Pulsars” (AXPs) [1161]; the epithet “anomalous” revealing more about the bafflement of neutron star researchers, than about the nature of the sources. We now classify AXPs as a subclass of magnetars, neutron stars with magnetic fields of Bp  5 × 1013 G. The other source in Fig. 6.17, 1E 1207.4-5209, in the supernova remnant PKS1209-51/52 was only in the year 2000 discovered to have a pulse period of 0.4 s [1271]. These type of X-ray sources, with shorter periods than AXPs, or no pulsation periods detected at all, are now known as Central Compact Objects (CCOs). The word “central” here directly refers to the fact that they are found in the centres of supernova remnants. The definition of CCOs is somewhat loose, as CCOs are as much defined by what they are not (not radio pulsars or AXPs) as by their location in the centres of supernova remnants. The discovery and characterisation of AXPs and CCOs owes much to their presence in supernova remnants, so we briefly review their properties and their connection to the supernova remnants they reside in. For detailed reviews of magnetars see for example the references [616, 800, 1135], and for CCOs [455, 904].

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6 Neutron Stars, Pulsars, and Pulsar Wind Nebulae

6.5.1 Magnetars Anomalous X-ray Pulsars Versus Soft Gamma-Ray Repeaters The class of magnetars comprises the aforementioned AXPs and the so-called soft gamma-ray repeaters (SGRs). As their name indicates, SGRs are neutron stars discovered by their hard X-ray/soft γ -ray bursts. In the late 1970s the nature of gamma-ray bursts (GRBs) was not known, but in March 1979 two sources showed several gamma-ray bursts during a few days. Using the time of arrivals of the burst, as detected by the two spacecrafts, a reasonably accurate localisation of the two SGRs was made. One of them, SGR 0526-66, showed evidence for pulsations with a period of P = 8.1 s, and was found to be associated with the Large Magellanic Cloud supernova remnant N49 [789]. Its peak luminosity above 30 keV was L = 4.5 × 1044 erg s−1 , several orders of magnitude above the Eddington luminosity of an accreting neutron star (Ledd ≈ 2 × 1038(MNS /1.4 M ) erg s−1 ). Nevertheless, the pulsations and the proximity to a supernova remnant strongly suggested that SGR 0501+451 was a neutron star. The other source detected in the same month was SGR B1900+14 [788], an SGR that has so far not been linked to a supernova remnant. Several more SGRs have since been discovered since then, with SGR 1806-20 being associated with the most spectacular burst: in December 2004 this already known source had a series of extremely powerful 47 2 −1 !bursts, with a peak46luminosity of2 L ≈ 2 × 10 (d/15 kpc) erg s and a fluence of Ldt ≈ 2.5 × 10 (d/15 kpc) erg [565, 881]. The chief difference between AXPs and SGRs is their bursting behaviour, or perhaps how the source was first discovered: although AXPs were first discovered as relatively steady X-ray point sources, we now know that AXPs can also have SGR-like bursts, with the aforementioned AXP 1E 2259+586, located in supernova remnant CTB 109, showing a major outburst in June 2002 [617]. As a result the distinction between AXPs and SGRs has been blurred. In fact, SGR 0526-66 has not shown a major outburst since 1979. If it the Venera spacecraft would not have been around at the time, SGR 0526-66 would have been labeled an AXP. AXPs and SGRs are now often collectively addressed as magnetars, but it should be noted that the acronyms AXP and SGR are based on observational phenomenology, whereas the name “magnetar” is based on a theoretical concept. So the label magnetar suggest that we have now sufficient trust in the idea that the magnetic fields inferred from the timing are real, and that the long periods and strong spin-down rate are not caused by some accretion process, which was once considered a viable alternative theory to explain the phenomena associated with AXPs and SGRs, see for example [770]. The name magnetar was first coined by Duncan and Thompson [347], based on the theory that during their formation process fast spinning (proto) neutron stars (P ≈ 1 ms) may develop extreme magnetic fields as the result of an α −  dynamo operating in the convective proto-neutron star fluid. The maximum magnetic-field energy under such circumstances can approach the rotational energy

6.5 Magnetars and Central Compact Objects

159

 (6.3), which would yield interior magnetic fields of maximally B ≈ 6ErotR −3 ≈ 3 × 1017(P /1 ms) G, although they suggested that 1015 G may be more a natural magnetic field strength for magnetars. They linked their magnetar model to the behaviour of SGRs. The Duncan and Thompson hypothesis that SGRs have surface magnetic fields of the order of 1015 G was confirmed when the spin-down rate of SGR 1806-20 was measured, from which a magnetic-field strength was inferred of Bp ≈ 8 × 1014 G [675], using Eq. (6.2.1). Shortly after this measurement also the spin-down rate of SGR 1900+14 was measured, suggesting a similar magnetic field [676].

The Magnetic Fields of AXPs and SGRs In Table 6.1 all known magnetars and magnetar candidates are listed, as well as their likely association with a supernova remnant. The confirmed magnetars have pulse periods between 2 and 12 s, and occupy the upper-right corner of the P − P˙ diagram (Fig. 6.1). In many, but not all, cases their X-ray luminosity exceeds the spin-down luminosity. It was this property that made “anomalous X-ray pulsars” anomalous. It implies that for many magnetars the energy source for the quiescent radiation cannot be caused by the loss of rotational energy, as is the case for radio pulsars. Also the bursts are thought to be caused by the violent phenomena associated with the strong magnetic fields. Magnetar magnetic-fields strengths are inferred to be 1014 –1015 G. A practical definition would be to say that magnetars are neutron stars with surface magnetic fields above the quantum critical magnetic field of BQED = m2e c2 3/(h¯ e) = 4.4 × 1013 G. This is the magnetic-field strength at which the Landau energy levels are equal to the electron rest mass, affecting the physics of radiation transport in the magnetosphere. About a decade ago this was indeed a practical definition. However, more recently SGR-like outbursts have been detected from neutron stars for which the measured P and P˙ suggest magnetic-field strengths 1013 G; see for example SGR 0418-5729 [949], which has an inferred bipolar surface magnetic field of Bp ≈ 6 × 1012 G [950]. The young pulsar PSR J1846-0258 has a magnetic field of Bp ≈ 5×1013 G, lower than most AXPs and close to the critical magnetic field. Moreover, its rotational period of 326 ms is much lower than typical for AXPs and SGRs, and the pulsar has powered a pulsar wind nebula, which is also atypical among AXPs and SGRs. For these reasons it was not considered to be a magnetar, until it displayed magnetar-like outbursts [427]. This pulsar is located in the composite supernova remnant Kes 75, whose age has been recently inferred from its expansion to be ≈ 400 yr [966], making it the youngest known pulsar. Neutron stars like PSR J1846-0258 and SGR 0418-5729 have blurred the notion of what constitutes a magnetar, and what the connection is between magnetar-like phenomena and the magnetic-field strength of neutron stars. The strong magnetic fields of AXPs and SGRs are in general inferred from the spin-down rate, assuming the magnetic-field structure to be dipolar (6.10), but it is possible that for some

Swift J1822.3-1606 SGR 1833-0832 Swift J1834.9-0846

CXOU J164710.2-455216 1RXS J170849.0-400910 CXOU J171405.7-381031 SGR J1745-2900 SGR 1806-20 XTE J1810-197

PSR J1622-4950 SGR 1627-41

Name CXOU J010043.1-721134 4U 0142+61 SGR 0418+5729 SGR 0501+4516 SGR 0526-66 1E 1048.1-5937 1E 1547.0-5408

W41

CTB 37B

G333.9+0.0 CTB 33 (G337.0-0.1)

G327.24-0.13

HB 9? N49

SNR?

4.33 2.59 10.61 a/(2π).

176

7 Dust Grains and Infrared Emission

According to Draine [331] the absorption efficiency per unit wavelength can be approximated by Qabs,λ ≈1.4 × 10

−3



a 0.1 μm



λ 100 μm

−2 ,

silicates with λ  20 μm, (7.3)

Qabs,λ ≈1.0 × 10−3



a 0.1 μm



λ 100 μm

−2 ,

graphites with λ  30 μm. (7.4)

The total emission/energy loss rate of a dust grain of size a can be written as 

dE dt



) = 4πa 2

Qabs,ν (a)πBν (Td )dν = 4πa 2 < Qabs (a, Td ) > σSB Td4 ,

em

(7.5) with < Qabs (a, T ) > the frequency averaged absorption efficiency and σSB StefanBoltzmann’s constant. Typically < Qabs (a, Td ) >≈ 1.3 × 10−6 (a/0.1 μm)Td2 for silicates and < Qabs (a, Td ) >≈ 8 × 10−7 (a/0.1 μm)Td2 for graphite [331]. Assuming an MRN grain size distribution (7.1), the total emission from a population of dust grains of a certain composition can be described as 

dE dt



) =

ν,em

a2

4πa 2 Qabs,ν (a)πBν (Td,a )Ka −3.5 da.

(7.6)

a1

The dust temperature , Td,a , of the grains depend on their size, absorption properties and their heating rate. They are heated by a combination of irradiation by the ambient stellar light and collisional heating. In the interstellar medium radiative heating dominates, but in the hot plasmas of supernova remnants collisional heating dominates.

7.3.2 Collisional Dust Heating Because the dust emission properties of dust grains in supernova remnants are predominately collisional heated, we concentrate on collisional dust heating. We assume that the electrons and ions have a Maxwellian velocity distributions, albeit not necessarily with the same temperature if the plasma is not equilibrated (Sect. 4.3.5). Since the dust grains are more much massive than the electrons and ions, we neglect the velocity distribution of the dust grains themselves. For each ion

7.3 Dust Heating and Radiation

177

species or electron i we have  f (vi )dvi =

mi 2πkTi

3/2



mi vi2 exp − kTi

 4πvi2 dvi ,

(7.7)

with vi the particle’s velocity, Ti the temperature and mi the particle mass. With each collision the particle can in principle transfer all its kinetic energy 12 mi vi2 to the dust grain. The collision rate for a given velocity is vi σd , with σd ≈ πa 2 the grain’s collisional cross section. For collisions by a species i the heating rate of dust grains can, therefore, be expressed as [353] 

) Hi (a, T , ni ) = ni σd

vi

 1 2 mi vi ζ (a, Ei )f (vi )dvi , 2

(7.8)

with ni the density of a given particle i. This equation includes the dimensionless efficiency factor ζ (a, Ei ) ≤ 1, with Ei = 12 mi vi2 , which corrects for the fact that not all energy of the particle may be transferred to the dust grain: a particle may bounce off the dust grain, or simply pierce through the dust grain, only deposition a fraction of its kinetic energy. The larger the grain the more likely it is that all of the particle’s energy will be deposited onto the grain. Only a fraction ≤ mi /md will be contributing to the kinetic energy of the dust grain, with md the mass of the dust grain. This is a very small fraction, which can be safely ignored. The energy deposition fraction for electrons (subscript e) and ions colliding with uncharged grains is estimated to be [354]

2/3  ∗ E 3/2 ζ (a, Ei ) =1 − 1 − , Ei  2/3 a E ∗ =23 keV, for electrons, 1 μm   a ∗ E =133 keV, for protons, 1 μm   a ∗ keV, for Helium, E =222 1 μm   a E ∗ =665 keV, for CNO particles. 1 μm

(7.9)

Summing over all particle species, the integral of (7.8) can be written as  Hi (a, Ti , ni ) =

H (a) = i

< ζ(a, Ti ) > ni πa i

2

32π mi

1/2 (kTi )3/2 , (7.10)

178

7 Dust Grains and Infrared Emission

with the < ζ(a, Ti ) > the velocity-weighted energy deposition fraction for a given particle species. The scaling with kTi /mi can be understood as follows: the average particle will have an energy that is √ roughly kTi , which has to be multiplied by the typical thermal velocity, vi ≈ 2kTi /mi . The collisions and energy transfers are skewed to collisions with the more energetic particles, as these have a higher collision rate. The emission from a population of dust grains can be calculated by assuming steady state conditions: H (a) = (dE/dt)em ; i.e. the amount of heating for dust grains of certain size a (7.8) equals the amount of cooling due to modified blackbody emission as given by (7.5). The total emission from a population of dust particles then depends on the distribution of grain sizes, their composition, and the density, composition, and temperatures of the particles that make up the hot plasma. In the case of full electron-ion equilibration (Te = Ti = T , Sect. 4.3.5) the heating contribution of ions can be ignored, because the electrons move much faster −1/2 than the ions, which accounts for the factor mi in (7.10). For non-equilibrated plasmas ion heating may still significantly contribute. Equation (7.10) shows that for the ion dust heating to exceed the electron dust heating we need Ti  Te



ne ni

2/3 

mi me



1/3 ≈ 12

ne ni

2/3 (Ai )1/3 ,

(7.11)

with Ai the atom mass number of the ion. This means that ion heating is likely important for young supernova remnants like Cas A. For Cas A the measured electron temperatures varies between 0.7 and 3.5 keV (e.g. [1181]), whereas for the measured shock velocity of Vs ≈ 5200 km s−1 [300, 899, 1183], one would expect kTp ≈ 30–50 keV (4.12). The expected ratio Tp /Te ≈ 8–71 suggests that proton heating may be comparable or even exceeding electron heating. A similar situation is likely to exist for Tycho’s SNR [1231]. For Cas A an additional complication is that most of the dust emission appears to come from shock-heated ejecta (Sect. 7.6), which are rich in oxygen and more massive particles. Under non-equilibration conditions these particles could be as hot kTi  300 keV, but one should then also factor in that the ratio ne /ni is large (≥8 for fully ionised oxygen), and that according (7.9) a small fraction of the ion kinetic energy is transferred to the grain at these high ion temperatures. So for oxygen we can estimate that Ti /Te  121 for the oxygen ions to significantly contribute to the heating. If indeed kTi  300 keV this condition could be met, but currently we do not know if the ion temperature in the shocked ejecta of Cas A is that hot. Ignoring for a moment ion heating, it illustrative to look at the dust heating by electrons as a function of grain size, electron density and temperature, as shown in Fig. 7.3. As can be expected the dust temperature increases for increasing electron temperatures, but for Te  107 K the dust grain temperatures level off. The reason is that the energy deposition fraction ζ (a, Ei ) decrease dramatically above a certain electron energy. A similar effect can be seen for the dust temperature as a function of dust-grain size, as the threshold energies that determine ζ(a, Ei ) decrease with

7.3 Dust Heating and Radiation

179

Fig. 7.3 Dust temperatures as expected for collisional grain heating by electron collisions as function of dust size (top left, ne = 1 cm−3 ), and electron temperature (top right, a = 0.01 μm). Bottom: emission from a MRN population of dust grains as a function of electron density for Te = 107 K. The calculations assume steady state conditions and ignore the effects of ion heating

decreasing grain size. The combined effect is that for Te  107 keV the dust-grain temperatures depend much more on the electron density—or ion density in case of non-equilibration—than on the plasma temperature. Figure 7.3 also illustrates the strong effect the electron density has on the modified blackbody emission from an MRN population of dust grains (7.1). Although the overall dust grain emission exhibits a smooth modified blackbody spectrum, there are some identifiable features in the infrared spectra, which reveal the composition of the grains. For example, at 9.7 μm the spectrum of warm silicate dust grains exhibit a broad feature that is attributed to the stretching modes of the chemical Si-O bond. Another silicate-related feature is at 18 μm attributed to O-SiO bending modes. In optical extinction curves another strong spectral feature exist at 0.2175 μm, which is linked to small graphite grains.

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7.3.3 Stochastic Dust Heating So far we have discussed the temperatures and infrared emission of dust assuming steady state conditions; that is temperatures are determined by the equilibrium between collisional heating and radiative cooling. However, for increasingly small grains steady state solutions may not be valid for two reasons: (1) smaller grains, having a smaller cross section (σ = πa 2 ) have a smaller collision rate, and in between two collisions may have sufficient time to significantly cool; (2) smaller grains have a lower heat capacity, cV × 4πa 3 /3 (with cV the volume specific heat), so that a single collision may give rise to a sharp hike in temperature [352]. Although we concentrate here on collisional heating, similar effects occur for radiative heating of dust, for which absorption of individual UV photons by small grains also result in temperature fluctuations [5, 333]. The resulting fluctuations of dust temperatures complicate the calculations of dust emission properties, as one now has to calculate the probability distribution of finding a dust grain of given size in a certain temperature range. The effects are referred to as stochastic dust heating. The radiative cooling time of dust grain is defined as Td τcool ≡    dTt   dt 

(7.12)

with the cooling rate, dTd /dt, given by   < Qabs (a, Td ) > Td4 − < Qabs (a, T0 ) > T04 dTd 2 = −4πa σSB , dt (4πa 3 /3)cV

(7.13)

this expression is based on (7.5), but takes into account that there is floor to the temperature, caused by heating of abundant low energy photons. For example, the high density of cosmic-microwave-background photons will keep the temperature at least to T0 = 2.73 K. The temperature-dependent values for cV (T ) can be found in [333, 352]. The heat capacity of a grain scales with its volume (∝ a 3 ). Since, to a good approximation, Qabs ∝ a, the radiative loss rate (7.5) also scales with a 3 (and not a 2 unless the grain is very large). As a result the cooling time of a grain is to a good approximation independent of its size. Figure 7.4 (top left) shows the approximate cooling times for silicate and graphite grains, showing that typical cooling times range from several hours at low temperatures, to less than a minute for dust temperatures above 100 K. The horizontal lines indicate the collision time, τcoll = 1/(ne 4πa 2ve ), for dust particles with different sizes for kTe ≈ 1 keV and ne = 1 cm−3 . For temperatures for which τcool < τcoll the dust grains will have sufficient time to significantly cool in between two collisions. For a = 0.01 μm this is true for a relatively high dust temperature of Td  100 K. But for lower dust temperatures, stochastic heating is important for dust grains with sizes below 0.01 μm (for ne < 1 cm−3 ).

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Fig. 7.4 Top left: Dust cooling time. For comparison, dotted lines indicate the collision time scale, 1/(nσ v), for electrons with typical energy of 1 keV, and electron density 1 cm−3 , and for various dust grain sizes. Top right: Maximum dust grain temperatures as a result of a single collision with an electron of a given energy. (Based on Dwek [352], but using the optical grain properties of [701].) Bottom: Dust temperature probability function for stochastic heating of silicate grains, assuming a plasma temperature of T = 107 K and ne = 10 cm−3 (Credit: E. Dwek [352])

The second important ingredient for modelling the expected emission due to stochastic dust heating, is the contribution of temperature spikes to the emission. These spikes are caused by sharp increases in temperature after one collision. The energy needed to bring a dust grain from an initial temperature T0 to a finale temperature T1 is )

E =

T1

cV (T ) T0

4πa 3 dT . 3

(7.14)

One can invert this relation to calculate the temperature a dust particle will be heated to by a collision with a single electron (or ion) with kinetic energy ≥ E, taking into account that not all the energy of the particle may be transferred to a dust grain

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(7.9). The result of such a calculation is shown in Fig. 7.4 (top right), illustrating that the peak temperatures can be high for the smallest dust grains. The decrease in temperature beyond a certain energy is due to the fact that above a certain energy ζ (a, Ei ) < 1 . For a = 0.1 μm the maximum temperature from a single hit lies well below the equilibrium temperature of ∼50 K for ne ≈ 1 (see Fig 7.3), indicating that the steady state solution is valid. However, for a = 0.01 μm this is no longer true. For even smaller dust grains the peak temperatures may be much higher than typical equilibrium temperatures (c.f. Fig. 7.3). The resulting probability of observing a given dust particle at a certain temperature depends on the time interval, t, since the last collision. This is described by a Poisson distribution, P ( t) = exp (− t/τcoll ) /τcoll . Inserting (7.12) in this equation provides the temperature probability function for a given final temperature. An additional integration over the velocity distribution of colliding particles (7.8) needs to be done in order to assess the probability function of peak temperatures. The result of such a calculation is shown in Fig. 7.4 (bottom). It shows that the smaller the dust grain, the broader the resulting temperature probability function. For large enough dust grains (how large depends on the plasma temperature and density), the probability distribution will approach a delta function, with a temperature corresponding to the steady state solution.

7.3.4 Determining Dust Masses The close correlation between dust temperature and plasma temperature, at least for equilibrium heating, makes infrared observations of dust in supernova remnants an important diagnostic tool. In addition, it is important to obtain an estimate of the total dust mass, in order to estimate the supernova contribution to the Galactic dust production. In order to determine the total dust mass Md from infrared observations of supernova remnants one needs to measure the temperatures of the dust grains from the spectral characteristics, which can then be combined with the total the flux at a certain frequency, Sν . The final dust mass can be estimated, once the temperature distribution of the grains is known, in relatively straightforward manner. The estimate relies on the fact that to a very good approximation the dust emission coefficient Qem,ν scales linearly with dust size. Let us, therefore, express the ˜ em,ν . The total flux from an ensemble of dust emission coefficient as Qem,ν = a Q grains with grain density ρd and sizes ai is then (7.5) Sν = i

ai Q˜ em,ν 4πai2 πBν (Td ) , 4πd 2

(7.15)

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with d the distance. Since each dust grain will have a mass mi ≈ 4πai3 ρd /3, we can rewrite this as Sν =

Q˜ em,ν Bν (Td ) 4d 2

3 i

mi Q˜ em,ν Md kν Bν (Td ) =3 Bν (Td ) = Md . ρd 4d 2 ρd d2

(7.16)

We introduced here the mass absorption coefficient κν , defined as κν ≡

σabs Qabs,ν πa 2 3 Q˜ abs,ν = ≈ . 4π md 4 ρd 3 ρd

(7.17)

Typical dust densities are ∼ 3.8 g cm−3 for silicates and ∼2.2 g cm−3 for carbonaceous material [331]. A complication is that the emission comes from dust grains with a distribution of sizes and consequently a distribution of temperatures, so a spectral decomposition needs to be done to account for this.

7.4 Dust Formation in Supernova Ejecta Although there are clear indications that supernovae are needed to produce dust at high redshifts (Sect. 7.2), and there is also conclusive observational evidence for dust supernova formation from supernova remnant observations (discussed below, Sect. 7.6), from a theoretical perspective dust production in freshly formed supernova ejecta is a complex process and is not well understood. One requisite for dust formation is that the temperature of the gas is low enough that refractory elements condense onto dust grains. This is typically the case for gas temperatures  1600 K, as can be seen in Fig. 7.1, but note that the condensation temperature also depends on pressure. The first step in dust formation is the appearance of simple molecules like CO, CO2 , SiC, SiO, MgO, MgS, AlO, FeO etc. These may then form clusters, these are the nuclei from which dust particles grow. This process is called nucleation. These simple molecules typically form about 100 days after the supernova explosion. Dust nucleation is often described by borrowing from the theory of droplet formation in a water vapour, assuming thermodynamic equilibrium between the gas and the condensations [680, 1120]. However, it is not clear whether equilibrium conditions are valid for the rapidly expanding supernova ejecta. An alternative, kinetic approach to nucleation theory, based on collisions between key atoms, does, for example, predict lower dust production rates [170]. Once nucleation has occurred the dust particles grow by collisions with atoms and molecules. A simple approach is to consider the growth of a dust grain by calculating the collisional frequency and a “sticking” coefficient α ≤ 1 [1073].

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The sticking coefficient is often taken to be α = 0.1. The grain mass then grows as dmd = αvth πa 2 nr mr , dt

(7.18)

with vth the thermal velocity of the particles involved, md the dust-grain mass, and nr and mrm the number density and mass of monomers (the simple molecules) or individual atoms of refractory elements. Using md = (4π/3)a 3ρd , and ρr = nr mr , we can rewrite this as a growth rate for dust particle sizes:  1 2kT ρr da = α . dt 4 mr ρd

(7.19)

If we consider the case for the formation of silicates (ρd ≈ 3.8 g cm−3 ), then the individual particles or monomers will be oxygen (mr ≈ 16mp) or silicon (mr ≈ 28mp), or SiO (mr ≈ 44mp ). Since the atoms have a faster thermal speed than the simple molecules, let us assume that the growth occurs mostly through the capture of atoms with an average mass of mr = 20mp . We can now make some simple calculation for approximating grain growth. The calculation may not be entirely accurate, but it will be illustrative for the grain-growth dependencies on ejecta mass and explosion energy. In Sect. 5.6 it was shown that the density in the ejecta core scales as ρ(t) ∝ (t/t0 )−3 , and that the density in the inner ejecta, with velocities v < vcore is reasonably constant with vcore given by (5.38). Using the appropriate scalings given in Sect. 5.6 we find that the core density for an n = 9 ejecta density profile as a function of time is given by ρ(t) ≈ 2.4 × 10−14



Mej 5 M

5/3 

Ekin 1051 erg

−3/2 

t 100 days

−3

g cm−3 . (7.20)

We assume that in the silicon-rich layer the density of refractory elements is ρr = fr ρ, with fr ≈ 50%. The ejecta cool adiabatically according to T V γ −1 = constant. Since V ∝ t 3 we find that T (t) = T0

 −3(γ −1) t . t0

(7.21)

Adopting the model in [170] we use T0 = 6660 K for t0 = 100 days, and γ ≈ 1.43 (smaller than γ = 5/3 as it includes recombination effects).

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Inserting these expressions in (7.19) shows that the maximum grain size that can be obtained from the time of nucleation, t1 is ) amax (t) =a1 +

t t1

 da α dt = a1 + dt 4β

2kT0 fr ρ0 t0 mr ρd

   −β  t t1 −β − t0 t0 (7.22) −1/2

 α   f   A −1/2  T r m 0 0.1 50% 20 6660 K −3/2  −β    Mej 5/3 Ekin t1 × μm, 5 M 1051 erg 315 days

≈0.06

(7.23)

with β=

1 (3γ + 1) ≈ 2.64. 2

For the approximation we took t/t1 → ∞ and a1  amax , with a1 the size of the dust grain seeds. For t1 we chose 315 days, as the temperature should have dropped by then to T = 1500 K, sufficiently for condensations to start (Fig. 7.1). which corresponds to roughly t1 ≈ 315 days. We neglected here that ρr will decline as the particles get depleted, so the above estimate may be too large. Also the temperature at t0 likely depends on details of the explosions limits. On the other hand, all the parameters are order of magnitude estimates, and intended to show sensitivity of the maximum grain size on the density evolution. The dust-grain size thus estimate falls short of the 0.25 μm maximum size of the MRN model. Equation (7.22) shows that the grain size depends critically on the time nucleation starts (t1 ) and hardly on how long the process of dust formation lasts, as the density is rapidly decreasing. It also shows that the grain size will be larger for larger ejecta masses (higher densities), and lower explosion energies, as the expansion drop is quicker (Fig. 7.5). What we did not do here is how the temperature itself (7.21) depends on ejecta mass and energy. The equation nevertheless indicates that grain growth is enhanced for progenitors with large ejecta masses. So it does not favour supernovae for which the progenitors suffered significant mass loss. For example, for Mej = 10 M , we find that amax ≈ 0.2 μm, for nucleation around 315 days. If the first generation of stars produced indeed much more massive progenitors, then dust formation by the Population III supernovae (e.g. [250]) may indeed have been much more efficient than in present day supernovae. It also indicates why models of dust formation by Type IIP supernovae (see [251]), produce more dust than Type Ib/c or IIb supernovae. The reason is that Type IIP are relatively massive as the progenitors having retained an extend hydrogen envelope, whereas Type Ib/c and IIb have lost their outer envelopes, giving rise to more rapid expansion [170].

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Fig. 7.5 Maximum grain-size as a function of time of grain formation onset. The plotted results are based on (7.22) with α = 0.1 and for various ejecta masses (E = 1051 erg). The assumed temperature evolution of the supernova ejecta is shown by the dashed line, with values indicated on the right. The solid lines indicate the size a grain can maximally attain when the seed particle are in place at the indicated time. The dotted line indicates when T ≈ 1500 K, the temperature at which condensations are likely to start

The dust formation model for a Type IIb explosion [170], inspired by the case of Cassiopeia A, showed that dust formation was very limited ( ρd



md mi

 ,

(7.28)

with the sum over all impacting ions i, and md indicating the typical mass of the atoms that are sputtered off (for example oxygen or silicon for silicates). The factor < Yi vi > is the average yield times impact velocity of the plasma particles. Written in this form the similarity is clear with (7.19). The factor md /mi arises as the atoms sputtered off may have a different mass than the impacting ion. We can make an order of magnitude estimate of dust-grain erosion, by considering only the most abundant ion, namely hydrogen (i.e. protons), and T = 107 K (i.e. vth ≈ 400 km s−1 ). For protons Y ≈ 0.01 [1117]. Inserting these values in (7.28) gives a rate of da ≈ −3 × 10−7nH μm yr−1 , dt

(7.29)

which is comparable to, but somewhat smaller than, the more detailed calculations presented in [1117]. If we assume that dust grains will be emerged for 1000–10,000 yr in the hot plasma, we see that dust grains of size a = 3 × 10−4 nH micron may be completely sputtered away. Note that the collision velocity in (7.28) may not necessarily be dominated by the thermal motion of the ions. As we will explain below the dust grains can have relatively large velocities with respect to the ambient plasma. Dust

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189

sputtering caused by these large relative bulk motions is referred to as non-thermal sputtering. Dust sputtering is especially relevant for freshly formed supernova dust, which will at some point be emerged in plasma heated by the reverse shock. For interstellar dust grains that are swept up by supernova remnant most of the dust will only be swept up at a relatively late state of the supernova remnant evolution, simply because of the larger volume of old supernova remnant shells. However, in this stage the temperature of the plasma is lower (106 K) and dust sputtering is less efficient by an order of magnitude. In this late stage of the evolution grain-grain collisions are thought to be more effective. To get an idea of destruction by grain-grain collisions, consider the amount of energy needed to heat a dust grain to a temperature of ∼2000 K, the temperature at which the material is likely to complete evaporate. The heat capacity of a silicate grain at high temperatures is CV = 4π/3a 3 × 3.4 × 107 [333], so the energy needed is Q ≈ 3 × 10−4 (a/0.1 μm)3 erg. Assume that the dust grains are of equal size and that both grains need to be evaporated. The size of grains then do not matter as the kinetic energies and heat capacity of the grains are both proportional to the grain masses. This tells us that the relative velocity needs to be v  3 km s−1 —slightly larger than the thermal speed of silicon atoms for a temperature of ∼2000 K. Reality is of course more complex, as not all the kinetic energy will be transferred to heat, as the two grains may bounce of each other, leaving a residual of kinetic energy, or part of the energy is immediately radiated away. A complete theory of grain-grain collisions can be found in [594]. The critical velocities reported in [594] are indeed of the order of a few km s−1 . However, complete shattering of the grains require much higher velocities of 75 km s−1 (graphite) to even 1015 km s−1 (diamond); for silicates it is 175 km s−1 [594]. These numbers depend on the sizes of the dust grains involved. For the given numbers the projectile dust grain had a = 0.005 Å and the target a = 0.1 μm As already stated, the relative velocities of grains encountering supernova remnants may be much larger than the critical velocity of a few km s−1 , and for supernova remnants these velocities are likely to be a significant fraction of the shock velocity. The reason is that a dust grain overrun by the shock is very collisionless; its mass is much larger than the average ion mass and energy exchange is extremely slow. As a result the grain will only very slowly thermalise with the plasma, and it will have a large velocity with respect to the overall plasma velocity. Hence, the grain will move roughly with v = (1 − 1/χ)Vsh , with χ the shock compression ratio (see Sect. 4.1). This does not mean that the grain will move along straight paths, because the grains will be weakly charged and the path will be curved, but with a large gyroradius. In fact the relatively high relative velocities between the plasma and the dust grains, in combination with large gyroradii, may cause dust grains to recross the shock front again. In this way, dust grains may be entering the cosmic-ray acceleration process, providing an explanation for the relative overabundance of refractory elements in cosmic rays (see Sect. 11.1.2). The calculations by Jones et al. [594] show that in a supernova remnant eventually 70–80% of the dust grains will be broken up by shattering, greatly

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reducing the overall dust sizes. The remainder of the grains will be completely destroyed by sputtering and vaporisation. Given the large destruction rates and the uncertainties in the calculations one may wonder whether supernova remnants are net producers of dust grains or destructors of dust grains, at least in our present day Universe.

7.6 Infrared Observations of Supernova Remnants The infrared part of the electro-magnetic spectrum is often subdivided in the nearinfrared (∼0.7–5 μm), the mid-infrared (5–25μm), and the far infrared regime (25– 350 μm). Beyond 350 μm the emission band is referred to as the submillimeter band (up to ∼1000 μm). This division is not very precise, and varies from paper to paper. The near-infrared can be detected with ground-based telescopes, and has been accessible to astronomy from the 1960ies. It corresponds to the photometric bands I, J, K, L, and M in the Johnson system [591]. For the observation of dust in supernova remnants the mid- and far-infrared is of more interest, and these bands became only available to astronomy with the launch of the first space-based infrared telescope IRAS in 1983 (a joint NASA/Netherlands/UK project) [855]. Since the launch of IRAS infrared observations have become an invaluable tool for the study of dust in the Universe. IRAS main instruments performed photometry in broad bands around 12, 25, 60 and 100 μm, with which a survey of the 96% of sky was carried out with an angular resolution of 0.5 to 2 (Fig. 7.6). It detected as many as 51 supernova remnants [85]

Fig. 7.6 All sky infrared map based on IRAS observations in the 12 μm (blue), 60 μm (green) and 100 μm (red) bands (Image credit: NASA/JPL-Caltech)

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191

and established that their infrared luminosities exceeds their X-ray luminosities by factors of a few to more than 100 [357]. Infrared astronomy rapidly matured with the launch of ESA’s Infrared Space Observatory (ISO) in 1995. It carried both an imaging camera (ISOCAM) covering the 2.8–18 μm band using various filters, the ISOPHOT imaging photometer (2.5– 240 μm) and spectrometers covering the 2.5–240 μm range (the long and short wavelength spectrometers, LWS/SWS). ISO had a spatial resolution about 50 times better than IRAS, which allowed detailed imaging of supernova remnants. One of the most important findings regarding supernova remnants observations with ISO was that the infrared emission from Cas A was coming from the shocked ejecta, rather from swept-up circumstellar medium (see below). The first decade of the new century showed proofed to be a golden age of infrared astronomy with three major space-based infrared observatories: NASA’s Spitzer Space Telescope (Spitzer for short) [1216] launched in 2003, the Japanese Infrared Astronomical Mission AKARI [849] launched in 2006, and ESA’s Herschel Space Observatory (Herschel for short) [919] launched in 2009. Figure 7.7 shows four supernova remnants as observed by the Herschel PACS camera. In addition, the Wide-field Infrared Survey Explorer (WISE) [1250] provided an all sky survey improving on the IRAS survey with an angular resolution five times better than IRAS. Spitzer covered mostly the near- and mid-infrared range (although the MIPS photometer covered also the ∼160 μm range), whereas Herschel covered the midto far infrared and even part of the sub-millimeter regime, as its SPIRE spectrometer could observe wavelengths as long as 670 μm. Although we focus in this chapter on the infrared emission from dust grains, it is important to realise that the infrared band also contains line emission from mildly ionised atoms (e.g. [O I], [O IV],[Ne II], [Ne III], [S IV], [Ar II], Ar [III], [Fe II]) and from molecular transitions. Some of these transitions are responsible for the appearance of IC 443 in the infrared (Fig. 7.8). At shorter wavelengths (green and blue in the image) line transitions dominate the emission. The difference between the northern and southern half of the remnant is caused by different shock speeds, with most of the dust emission come from the Northern part, where the gas has been shocked with a shock velocity of ∼100 km s−1 whereas in the Southern half, line emission dominates from a relatively cool plasma, shocked by a C-type shock (Sect. 4.5) of ∼30 km s−1 [973].

7.6.1 Infrared Emission from Young Supernova Remnants Infrared Emission from Young Core Collapse Supernova Remnants Young supernova remnants provide important information about the properties of the supernova explosion itself: its energy, composition and mass. To this can be added that they are important to understand the contribution of core-collapse supernovae to dust formation in the Galaxy. One of the important discoveries of

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Fig. 7.7 Four 70 μm images of supernova remnants as mapped by the PACS camera on board the Herschel infrared observatory. Top left: SN 1987A (within the small circle). The blue dashed line indicates the edge of the 30 Doradus C superbubble. Top right: Kepler’s SNR. Bottom left: Tycho’s SNR. Bottom right: the Crab nebula

ISO was that it showed that the infrared emission from Cassiopeia A comes from the regions associated with the shocked ejecta, and that the dust is mostly composed of silicates [86, 327, 692]. This provided direct evidence that most of the dust grains were formed after the supernova explosion. In fact, the regions with high infrared emission follow the distribution of the optically emitting fast-moving knots (Sect. 8.1.2), indicating that a large fraction of the infrared emission comes from dust that is not so much heated by the hot 106 –108 K plasma occupying most of the shock-heated ejecta, but may be heated inside these knots, which have much lower temperatures than the main shell (∼104 K), but whose collisional heating rate is very high due to their high densities of ne ∼ 103 –104 cm−3 [327]. There is even the possibility that the dust has been formed relatively late after the explosion, namely

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193

Fig. 7.8 The western part of the supernova remnant IC 443 as seen in the infrared by the WISE observatory. Red corresponds to 22 μm, green to 12 μm, and blue to 4.6 μm. See Fig. 6.16 for an optical/X-ray image of IC 443

inside the knots after having encountered slowed down reverse shock [170]. Perhaps this can also have led to the formation of CO gas detected in the infrared by Akari through its 4.65 μm vibrational mode [974]. Alternatively, the dust grains and CO may have formed in the first 2 year after the explosion, but in high-density ejecta knots, which would be more optimal for both the formation of dust and the survival of CO gas. These high-density regions would later become the fast moving knots, after having encountered the reverse shock. The ISO and Spitzer observations revealed dust grains in Cas A that are relatively warm, from 60 K up to 500 K, and consist mostly of silicates [86, 327, 976, 978]— see Fig., 7.9 (right). In addition the spectra show narrow forbidden-line emission from low-ionisation atoms such as [O IV],[Ne II], [Ne III], [S IV], [Ar II], [Ar III]. The relatively high temperature of the dust continuum emission requires a plasma

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Fig. 7.9 Left: Long wavelength three color image of Cas A, with in red a 160 μm Herschel PACS map (cold dust and free-free emission), in green a 70 μm PACS map (“warm’” dust), and blue the 7.8 μm Spitzer IRAC map (“hot” dust). Especially the 7.78 μm emission shows a close correspondence with the optical, fast moving knots. Right: Mid-infrared spectrum of Cas A taken with the Spitzer IRS instruments, showing in blue the observed spectrum, and with various other colours the best-fit dust model components (temperatures in brackets) [978]. The bright line at 7 μm is [Ar II] and the 9 μm is [Ar III] (Credit: Adapted from Rho et al. [978])

density of n2  10 cm−3 , if they come from the hot parts of the shocked ejecta (Fig. 7.3), or from small, stochastically-heated dust grains (Sect. 7.3). The total dust mass contained of the ensemble of warmer dust grains is estimated to be between 10−4 M [327] and 0.05 M [88, 976]. These dust masses are modest, when compared to requirements that supernovae produce the dust at very high redshifts, of about 1 M per supernova [356]. It turns out that more dust mass in Cas A resides in the cooler dust component, which is mostly visible in the far infrared (the red parts in Fig. 7.9, left). This component is associated with ejecta that has not yet been heated by the reverse shock. Early estimates of the cold dust mass of Cas A, based on submm (850 μm) observations of very cold dust (18 K) with the SCUBA array on the James Clark Maxwell Telescope, even estimated 2–5 M of dust in Cas A [348]. Such a large dust mass would help to explain the amount of dust produced by supernovae at large redshifts, but would be at odds with the relatively low overall ejecta mass inferred from Cas A, of 2–4 M (e.g. [259, 1181, 1236]). And it does not agree with the idea that low-progenitor mass Type IIb supernovae should produce less dust than supernovae which retained their hydrogen envelope (Sect. 7.4). However, the detection of cold dust was soon shown to be caused by emission from the environment of Cas A [681]. Indeed, Fig. 7.9 shows that even at the shorter wavelength of 160 μm there is a lot of far-infrared emission from the surroundings of Cas A. There is still a lot of uncertainty about the dust mass in Cas A, mostly coming from different analysis techniques and modelling of the far-infrared emission, with some groups finding masses lower than 0.1 M (e.g. [88, 136]), and others still finding a much higher fraction of the ejecta mass converted to dust: 0.3 − 0.5 M [294].

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195

Although Cas A is the best example of a supernova remnant with dust emission from the ejecta component, also for other oxygen-rich supernova remnants G292+1.8 [430, 435], N132D [977, 1098] and 1E0102-7219 [975] it has been shown that some of the infrared emission is associated with the ejecta. See Sect. 9.3.1 for other aspects of oxygen-rich supernova remnants. They originate from the most massive stars, although significant pre-supernova mass-loss may make the final ejecta mass relatively small, as appears to be the case for Cas A. In contrast, a clear case for dust production by a low-mass progenitor is the Crab Nebula (Fig. 7.7). The progenitor likely kept its hydrogen envelope before it exploded, suggesting a relatively large ejecta mass, and there is evidence that the explosion itself was subenergetic [1262]. These two conditions favour efficient grain growth, according to (7.22). On the other hand, the lower core mass provides an overall lower mass of refractory elements. Herschel observations indicate a relatively large dust mass of ∼0.2–0.4 M for the Crab Nebula, with the lower estimate based on the assumption that the the dust has a carbonaceous composition [447]. The infrared emitting dust identified by Herschel is relatively cool (∼30 K) and located in the radiative filaments at the outer boundary of the nebula, but also partially submerged in the synchrotron nebula. A lower dust mass was reported in [1104], but based on Spitzer spectra at shorter wavelengths. Finally, an important object to study supernova dust formation is SN 1987A (9.3.3). It is so young that the dust content has not yet been substantially affected by dust destruction processes. The infrared emission from SN 1987A comes from two components. One is the dusty and dense circumstellar ring (ne ∼ 104 cm−3 ), which since the late 1990s has been engulfed by the blast wave and has been heated to temperatures in excess of 5 × 106 K. As more of the ring has been heated, the dust emission has rapidly increased. Modelling of the mid-infrared Spitzer spectra suggest that the emission comes from a combination of large silicate grains (∼0.2 μm) heated to ∼ 180 K and hotter (>350 K) small-sized grains [358]. The other dust component is much cooler (20–30 K) and has been freshly synthesised in the explosion. Millimeter and submillimeter observations with the Atacama Large Millimeter/ Submillimeter Array (ALMA) clearly resolve that this component comes from the inner ejecta [576]. The far-infrared Herschel data suggest that 0.4–0.8 M of dust must be present, but there is some debate as to whether this dominated by carbonaceous dust [780] or silicates [355]. Interestingly, millimetre observations of SN 1987A with ALMA also shows the presence of simple molecules like CO, SiO and HCO+ . The latter suggests some mixing of hydrogen and carbon/oxygen-rich layers [782]. The large dust mass in SN1987A suggests that a significant fraction of the available silicon may have depleted onto dust grains, providing further credence to the idea that supernovae are dust factories.

Infrared Emission from Young Type Ia Supernova Remnants Unlike for core-collapse supernova remnants, there is no evidence for dust components associated with the ejecta of Type Ia supernova remnants. There may be

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two reasons why Type Ia supernovae do not produce detectable dust masses. First, as equation (7.22) shows that the growth of grain-size depends critically on the ejecta mass, which for Type Ia supernovae is much smaller than for core-collapse supernovae. Secondly, the production of dust requires some mixing of different ejecta layers. For example, for silicates a mixture of silicon and oxygen is needed. There is some evidence that Type Ia supernova remnant ejecta are much more stratified than core-collapse supernova remnants (see for example [567, 670]). And finally, Type Ia supernovae produce much more radio-active 56 Ni (0.5 M ) than core collapse supernovae (0.1 M ). The decay of 56 Ni and its daughter product 56 Co keeps the ejecta hot for a longer time, and may thus prevent early dust condensation. Some Type Ia supernova remnants are infrared sources, but the emission comes from pre-existing circumstellar dust, heated by the hot electrons associated with forward-shock region. As such the infrared data provides a sensitive diagnostic tool to estimate the circumstellar density [1231, 1241], as explained in Sect. 7.3. In addition, the composition of the dust may provide clues to the nature of the circumstellar medium, which in itself may provide clues to the nature of the progenitor systems of Type Ia supernovae (Sect. 2.3). A case in point is the infrared emission from Kepler’s SNR. As explained in Sect. 9.2.3, Kepler’s SNR is located high above the Galactic plane (∼500 pc) and its blast wave is interacting with circumstellar material swept-up by a bow-shock, which must originate from a stellar wind coming from the progenitor or its stellar companion, most likely an AGB star [264, 1174]. The infrared spectrum of Kepler’s supernova remnant reveals that the dust grains are primarily silicates [1230]. Only relatively high-mass AGB stars, which have oxygen-rich winds, produce silicate dust [1174]. Carbon-rich winds tend to lock up all oxygen into carbon-monoxide (CO) molecules, leaving no oxygen to form silicates. Interestingly, another Type Ia supernova remnant, N103B, in the Large Magellanic Cloud, also shows evidence for the presence of circumstellar wind material, and its dust composition is very similar to that of Kepler’s SNR [1232]. Note that there is no evidence in Kepler’s supernova remnant for a surviving AGB companion star [644]. So the puzzle of the origin of the Kepler’s supernova remnant remains.

7.6.2 Observational Evidence for Dust Destruction Although there is conclusive evidence that core collapse supernovae produce dust (with some debate on how much: seconds (with A the Einstein coefficient, see Sect. 13.5). This is much longer than the typical collisional deexcitation times under laboratory conditions. That is 1/(nσ v)  A−1 . This is the handwaving explanation usually given for presence of forbidden lines in optical nebulae. But this explanation for the prominence of forbidden lines in optical nebular spectra is incomplete, because it does not explain why there are so few allowed transition detected in the optical spectra of supernova remnants, and other nebulae. If we look at the optical spectrum of a region in the Cygnus Loop (Fig. 8.3), or at the optical line emission detected (Table 8.1), it is striking that, apart from Balmer lines and a few allowed helium transitions, all other line emission concerns forbidden transitions. There are two reasons why allowed transitions are so weak in optical spectra. The first explanation is that there happen to be very few allowed transitions in the optical for most ions of interest, see [331, 874]. Immediately

Fig. 8.3 Top: Optical spectrum of a bright filament in the Cygnus Loop, concatenated from three figures in [381] (note the change in scale on the horizontal axis). Bright line emission is either due to forbidden transitions, or Balmer lines. Bottom: UV spectrum of another radiative shock filament of the Cygnus Loop, taken with the Hopkinson Ultraviolet Telescope (Adapted from Fig. 2 in [174])

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203

Table 8.1 Emission lines identified in spectra of bright filaments in the Cygnus Loop (after [381]) Designation Hδ Hβ Hα He I He I He I He I He II He II He II C II C II [N I] [N I] [N II] [N II] [N II] [O I] [O I] [O II] [O II] [O III] [O III] [O III] [Ne III] [Ne IV] [Mg I] Mg I] [S II] [S II] [S II] [S II] [S III]

λ (Å) 4101.8 5876.0 6562.9 4471.48,4471.69 4921.9 5015.7 5876.0 4685.7 5411.5 6678.2 4267.0 7236.0 5197.9 5200.3 5754.6 6548.1 6583.5 6300.3 6363.8 3726.0 3728.8 4363.2 4958.9 5006.8 3967.5 4724.17, 4725.60 4562.5 4571.1 4068.6 4076.4 6716.4 6730.8 6312.1

Designation [Ar IV] [Ar IV] Ca II [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe II] [Fe III] [Fe III] [Fe III] [Fe III] [Fe III] [Fe III] [Fe III] [Fe III] [Fe II]/[Fe III] [Fe II]/[Fe III] [Fe V] [Fe VI] [Fe VI] [Fe VII] [Fe VII]

λ (Å) 4711.3 4740.2 3933.7 4243.98, 4244.81 4287.4 4358.10, 4358.37, 4359.34 4413.78,4414.45, 4416.27 4287.4 4905.4 4973.4 5039.10, 5043.53 5111.6 5158.0, 5158.81 5261.6 5333.7 5412.64, 5413.34 4658.1 4701.6 4733.9 4754.8 4769.6 4777.9 4881.1 4985.9, 4987.2 4813.9, 4814.55 5268.88,5270.3, 5273.38 3967.5 5145.8 5176.0 5720.7 6087.0

downstream of radiative shocks atoms will be ionised once, twice or three times, before cooling reverses the trend again and results in recombination (Fig. 8.2). It happens to be the case that for many abundant atoms like nitrogen and oxygen, there simply are no allowed transitions in the optical for single and double ionised atoms. See for example Fig. 8.4 for the energy levels of twice ionised oxygen, which shows that the only transitions below 10 eV are forbidden transitions. The situation is different in the UV, where more allowed transitions are present. Indeed,

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λ4364.4

λ4960.3 λ5008.2

Fig. 8.4 Level diagram for O III, twice ionised oxygen. The only optical line transitions, between low energy levels, are forbidden lines, as indicated with red arrows. Both transitions violate the rule that l = ±1, L = 0, ±1

a UV spectrum of the Cygnus Loop (Fig. 8.3) reveals more allowed lines, but not overwhelmingly so [174]. There are some allowed transitions in the optical. For example Ca II, which has resonance lines at 3934 and 3968 Å, and Na I at 5890 and 5896 Å (the famous sodium doublet, responsible for the yellow light of sodium lamps). Of course, calcium and sodium are not among the most abundant elements. In addition, there are a few allowed optical line transitions that are not associated with transitions to a ground level. The Balmer lines are a prime example, but for example also C II has two allowed optical line transitions closely spaced around 4267 Å. This transition has been detected in optical spectra of the Crab Nebula as a weak line [380]. This brings us to the second reason why allowed line transitions are often absent from nebular spectra: resonant line scattering, in conjunction with the geometry of the filaments. Recall that resonant line scattering does not necessarily “destroy a photon”, but merely changes its direction by absorbing a photon and then emitting it again. See Sect. 13.5.9 for a full discussion. To illustrate this point, consider a supernova remnant that is roughly spherical with a radius R and emitting from √ a thin shell with a width R. √ The maximum path length through this shell is l = 2 R R(2 − R/R) ≈ 8R (Fig. 8.5). For a typical mature supernova remnant we have R ∼ 3 × 1019 cm (10 pc), whereas the radiative cooling region has a width of just R  1016 cm [322, 943]. So the path length along the line of sight through the cooling region is l ∼ 1018 cm. This length scale has to be compared with the typical path length of a resonant photon. For the absorption cross-section in a line we use (13.159). For the line broadening we can take the thermal Doppler width, assuming full electron/ion

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205

Fig. 8.5 Illustration of the effect of resonant line scattering. The left figure shows the maximum path length for a perfect spherical symmetry for a shell with width R. The right figure illustrates that forbidden line photons can come from along the full length of the filament, exhibiting strong limb brightening, whereas a resonant line photon will escape when emitted from within a distance of λmfp from the surface of the emission region (Credit for the telescope dome: Anton Pannekoek Institute, Amsterdam)

temperature equilibration  (Sect. 4.3.5). √This gives for the proton a thermal velocity dispersion of V = 2kT /mp ≈ 12 T /10,000 km s−1 . First consider the Lyα transition, which has λ = 1215.67 Å (ν = 2.47×1015 Hz) and an oscillator strength f = 0.416. The line absorption cross section according to (13.159) is σabs ≈ 2.7 × 10−13 cm2 . The mean-free path of a photon is λmfp ≈ 1/(1 − ξ )nH σabs = 3.7 × 1012/(1 − ξ )nHI cm. Here ξ is the ionisation fraction, which is 0.5 around 10,000 K and approaches ξ ≈ 1 above 20,000 K. But even for 1 − ξ = 0.001 is the mean free path smaller than the typical filament width of R ∼ 1016 cm, let alone along the full length l. So resonant line scattering is likely to be important for Lyα photons. Indeed, UV observations of the Cygnus Loop provide evidence for resonant line scattering effects [287]. Here we used Lyα as an illustration, but the above calculation applies to any resonant line transition, in particular if it concerns a transition between the ground state and an excited level, as nearly all ions will be in the ground state at any given moment. Of course, the abundance of hydrogen is much higher than of other elements. But with a mean free path of ≈ 1013 cm for a hydrogen transitions, ions with abundances nion /nH > 10−6 will still be affected by resonant line scattering for transitions from the ground state.

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If resonant line absorption is merely changing the direction of the photons, i.e. we have a case of pure line scattering, the photons will eventually escape from the emission region. A resonant photon’s last scattering will be typically λmfp from the boundary of the plasma. As a result, the resonant line over forbidden line emission ratio will depend on geometry. Forbidden line transitions will be strongest if our line of sight is along a thin emission region, and smallest if we look at such a region face on, since in absence of scattering the emission scales with the path length. On the other hand the path length for a resonantly absorbed allowed transition will be λmfp , so the length scale of the emission region hardly matters. The ratio of the resonant line over forbidden line emission will be strongest if we see a thin emission region face on. However, when observing a nearby supernova remnant one usually will aim for the brightest regions, i.e. those thin regions observed edge-on, which biases observations to those regions for which resonance line emission is strongly suppressed. However, resonantly produced photons can be “destroyed”, if the resulting excited state has multiple deexcitation options. For example, Lyβ absorption can be followed by an Lyβ photon emission, but also in the emission of an Hα photon, and then a Lyα photon. As a result Lyβ emission will be suppressed relative to Lyα emission. If we look at the list of detected optical lines in supernova remnants in Table 8.1 (see also [381]) we see that there are indeed only few non-forbidden lines among them. The allowed transitions that are detected are Hα and other Balmer lines, and several He I lines. All these transitions are from n > 2 to n ≥ 2 levels. As discussed above resonance scattering for these transitions are rarer than scattering involving the n = 1 ground state, given that most ions are in the ground state. Note that indeed, resonant transitions should be expected from Ca II and Na I in the absence of resonant line scattering (Ca II is weakly present, Table 8.1). In fact, Ca II and Na I lines are detected in absorption along lines of sights through the Vela and Cygnus Loop supernova remnants [244, 387, 1193, 1214]. The absorption lines have redshifted and blueshifted components if they are on the far sight of the remnant, and only blueshifted components, if the stars are located inside the remnant. This can be used to measure distances to supernova remnants; see Sect. 3.2.2.

8.1.2 Optical Emission from Young Supernova Remnants: Optical Emission from High-Density Clumps So far we discussed the optical emission from older supernova remnants with radiative forward shocks. Young supernova remnants can, however, also display strong optical line emission. In those cases the optical emission comes from high-density clumps overrun by either the forward shock—as is the case in Kepler’s supernova remnant, for the nitrogen-rich clumps in Cas A, the bright ring surrounding SN

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207

1987A (Sect. 9.3.3)—or from ejecta clumps heated by the reverse shock. The prime example for the latter case is Cas A, which shows numerous optically emitting clumps, some of which are bright in [O III] and some in [S II] (Fig. 9.10). The optically emission is associated with radiative shocks, which arise when shock waves penetrate the dense gas inside the clump, and decelerate. If one considers conservation of momentum through the shock (ρVs2 =constant, neglecting pressure terms), one sees that moving from region I (diffuse gas) to II (clumped gas) the shock velocity drops according to  Vs,II =

ρI Vs,I . ρII

For shock speeds of the order of 1000–5000 km s−1 one needs clump-density contrasts of 25–600 to bring down the shock velocity to below 200 km s−1 , for which shocks become radiative. Indeed, the densities inferred for the Cas A ejecta clumps are of the order of 103 cm−3 [388]. Similar density contrasts are necessary to explain the presence of radiative knots in SN1987A. For the ejecta knots the radiative properties differ from those of forward shocks associated with mature supernova remnants, because their composition is very different. For example, the ejecta knots in Cas A are hydrogen-free, and are rich in oxygen and carbon-burning products or in intermediate mass elements.

8.2 Balmer-Dominated Shocks The optical emission from most supernova remnants are associated with slow shock velocities in older remnants, or inside the dense clumps of young supernova remnants, as discussed in the previous section. However, a minority of relatively young supernova remnants have optically emitting filaments associated with the locations of the shock fronts that have spectra devoid of the forbidden line emission that is so characteristic of radiative shocks ([N II], [S II], [O III]). But the spectra do show Balmer line emission (Hα, Hβ, Hγ , . . . ). The shocks associated with these filaments are called “Balmer-dominated” shocks. Figure 8.6 shows a beautiful example of a Balmer-dominated shock; a portion of the northwestern filament of SN 1006. Balmer-dominated shocks do not exclusively emit Balmer line emission: they also emit Lyman series, and He I and He II lines, and lines from more massive elements in the UV [944]. For that reason, Balmer-dominated filaments are sometimes, more appropriately, called non-radiative shock filaments. The optical filaments tend to be relatively faint and thin, and were noticed for the first time in the late 1970s for spectra taken of optical filaments of Tycho’s SNR [655] and SN 1006 [1030]. The spectra could not be easily explained within the context of the radiative shocks, also because Tycho’s SNR and SN 1006 are young remnants. It was soon realised that the physics behind the Balmer-dominated shocks

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Fig. 8.6 This delicate structure observed in Hα emission originates from a very thin layer (1014 –1015 cm) immediately behind the shock front of SN 1006. The thickness of this layer is unresolved by the Hubble Space Telescope instruments, but the shock itself is curved and slightly undulating, given it the striking morphology of a gently sweeping ribbon (Credits: NASA, ESA, and the Hubble Heritage Team (STScI/AURA); Acknowledgment: W. Blair (Johns Hopkins University))

is quite different: instead of optical emission caused by radiative cooling, the Balmer dominated shocks emit immediately downstream of the shock as neutral hydrogen (and helium) atoms are suddenly surrounded by shock-heated ions and electrons. Before collisional ionisation leads to a population of fully ionised hydrogen and helium ions, the atoms may first experience charge-exchange interactions and collisional excitations, which give rise to the optical and UV line emission [260]. The charge exchange reactions involve collisions between hot, shock-heated ions and a beam of cold neutral atoms. As the hot ions pick up electrons from the neutral atoms, a population of newly formed hot neutral atoms is formed, often with the bound electrons in an excited quantum level. The deexcitation of these neutral atoms will lead to thermally broadened line emission, with a broad Hα component as the most easily detected feature. For very broad Hα line emission the contrast between

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209

the smeared out line emission and the spectral background may be small, and long integration times, or large diameter telescopes, may be needed for detecting these broad components. Not all young supernova remnants have Balmer-dominated filaments. Clearly, a precondition is that the unshocked gas upstream of the shock should contain neutral hydrogen. It is found that this condition is more prevalent around Type Ia supernova remnants than around core-collapse supernova remnants. See the discussion in Sects. 9.1 and 9.2 for the implications for the supernova remnant progenitors. Another condition is that the shock is indeed, non-radiative, which implies that Vsh  200 km s−1 (Sect. 4.4). Some supernova remnants display a mixture of radiative and non-radiative shocks. An example is the Cygnus Loop, which has regions with shock velocities below and above 200 km s−1 . So both radiative filaments and non-radiative filaments have been identified in this remnant [528, 718]. An even more extreme example is G315.4-2.3 (RCW 86/SN 185), for which shock velocities span a range from 200 km s−1 [987] to 1200 km s−1 [504]; in X-rays even shocks with ∼3000 km s−1 have been identified [501, 1256].

8.2.1 The Formation of the Narrow- and Broad-Line Components As can be inferred from the above description, Balmer-dominated shocks can be identified based on three characteristics: (1) absence of forbidden line emission, in particular the absence of [N II] leads to an isolated Hα line emission; (2) the optical filaments are fainter than those of radiative shocks; and (3) the line emission has a characteristic profile, consisting of a narrow-line component and a broad-line component. The third characteristic is illustrated with the Hα profile of a filament in SNR 0509-67.5 (Fig. 8.7). The spectra of Balmer-dominated shocks are shaped by the collisional processes behind non-radiative shocks. For the present discussion, we concentrate on hydrogen, but the discussion can be generalised to helium as well [1148]. Figure 8.8 shows the relevant collisional cross sections and collision rates for neutral hydrogen colliding with protons and electrons. We see that within an order of magnitude charge exchange, collisional ionisation and excitation (here only to the n = 3 level is shown) have similar cross sections of the order σ ∼ 10−15 cm2 . For a typical proton/hydrogen density of n0 = 1 cm−3 , the relevant length scale over which hydrogen is ionised, or undergoes charge exchange is therefore lH = 1/(n0 σ ) ∼ 1015 cm. So the Balmer line emission comes from a very thin filament indeed, corresponding to 0.07 at a distance of 1 kpc, which is at the resolution limit of the Hubble Space Telescope. Since shock fronts are not perfectly spherical, but may have small scale curvatures, the actually Balmer-dominated may be broader than 1015/n0 cm, as can be seen in Fig. 8.6.

8 Optical Emission from Supernova Remnants

scaled flux

210

Doppler velocity (km/s) Fig. 8.7 Left: In red, Hα emission from SNR 0509-67.5 [550], located in the Large Magellanic Cloud. The image is photomontaged on a broad band optical image, all taken with different instruments on board the Hubble Space Telescope. (Source: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)/J.P. Hughes.) Right: Spectral line shape of Hα for the southwestern region of SNR 0509-67.5, taken with the ESO VLT/FORS2 instrument. The line shape consists of a narrow, plus a broad line (the latter shown in detail in the inset). The narrow line is not resolved, due to the wide spectral slit chosen, and the broad line is blueshifted with respect to the narrow line, as a result of a slight inclination of the post-shock flow toward us (Reproduced from Helder et al. [503])

Fig. 8.8 Left: Collisional cross sections for hydrogen charge exchange (solid line), proton collisional ionisation (black dashed line), electron collisional ionisations (red dashed line), and proton and electron excitation of hydrogen to the n = 3 level (dotted lines). Note that for a plasma in electron-proton temperature equilibration the electrons have a much higher thermal speed than  the protons (ve /vp = mp /me ≈ 43). Right: Collisional rate coefficients for charge exchange reactions and ionisation (Data obtained from https://www-amdis.iaea.org (ALLADIN))

The associated time scale for hydrogen ionisation is τH = 1/(n0 σ v) ≈ 107 s ≈ 4 months for v = 1000 km s−1 . This does have the advantage that the Balmer line emission provides information about the (almost) current shock properties. In contrast, thermal X-ray emission is related to the shock-heating averaged over a time scale of several tens to thousands of years.

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The collisional excitation of the “cold” neutral hydrogen hardly affects the velocity of the hydrogen atom. As a result the observed centroid of the emission line and its width reflect the velocity and temperature of the unshocked gas. In contrast, the charge exchange process creates a population of hot, neutral hydrogen, with a temperature roughly equal to the temperature of the shock-heated protons from which they originate. As the charge exchange process leaves in most cases the newly formed hydrogen atom in an excited state, the charge exchange results in a broadened emission line. The relation between the temperature and Doppler widths of the hot hydrogen atoms is 2 kTp = mp σv,r ,

(8.1)

with σv,r = c λ/λ the Gaussian width of the line. It is customary to express the √ Doppler width as a “full-width at half maximum” (FWHM), with FWHM= 8 ln 2σv,r , under the assumption of a Maxwellian velocity distribution. 3 For a strong shock we expect kTp = 16 μmp Vs2 but this assumes that the shockheating process equally divides the available thermal energy over all particle species. The latter condition may not necessarily be the case (Sect. 4.3.4), and another extreme may be that the protons may be heated independently to other species, in which case we can set μ = 1 instead of μ = 0.6. Keeping in mind that 0.6  μ < 1 for protons and forward shocks heating gas with solar abundances, and assuming that the newly formed hot hydrogen has a temperature equal to those of shockheated protons, we expect the velocity broadening to be  σv,r =

3 √ μVs ≈ 0.43 μVs , 16

(8.2)

or √ FWHM ≈ 1.02 μVs .

(8.3)

So the FWHM is under the above assumptions almost identical to the shock velocity. The above estimate does not take into account the dependency of the charge exchange reactions on the on the relative velocity of the protons and neutral hydrogen atoms. As Fig. 8.8 shows, a relatively flat cross section is a reasonable assumption around relative velocities of 1000 km s−1 , but beyond 1500 km s−1 charge exchanges skews the reactions to lower relative velocities. It should be recalled that the neutral hydrogen atoms penetrate the shock-heated plasma unhindered with a relative velocity with respect to the plasma of v = (1−1/χ)Vs ≈ 34 Vs (Sect. 4.1). But for the total collision rates the full integration over three dimensions should be performed. So for the charge-exchange rate of “cold” hydrogen, moving

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in the z we have the following expression for the charge exchange rate: ) RCX =

∞

)

∞

)



∞

vx =0 vx =0 vz =0

vrel σCX (vrel )

mp 2πkTp

3/2

 exp −

1 vx2 + vy2 + vz2 2 kTp

 dvx dvy dvz ,

(8.4)  vx2 + vy2 + (vz − vz )2 . The velocity distribution of the neutral with vrel = hydrogen atoms after one charge exchange will be skewed toward the peak in the collision rate, i.e. toward 1000 km s−1 . Moreover, there will be a difference in the width and Doppler shift of the broad-line component observed edge-on and face-on, due to the anisotropy of the collisions. If the shock is seen edge-on, the line-of-sight velocity of the broad-component is zero (apart from velocity differences due to the local rest frame). If we see the shock completely face-on the shock-heated protons have an average velocity of v = 34 Vs along the line of sight. But due to charge exchange collision rate dependency on velocity, for Vs  1000 km s−1 the charge exchange preferentially happens for protons that happen to travel in the same direction as the neutrals. The result is that the observed radial velocity centroid will be vr > 34 Vs . The difference in the centroid of the broad-line component and the narrow-line component for shocks observed under an angle, can be seen in Fig 8.7 for the case of the Hα filament in the southwest of SNR 0509-67.5 [503]. In this case the filament is nearly edge-on, but tilted enough to result in a blueshift of the broad-line component of a few hundred km/s with respect to the narrow-line component. Figure 8.9 shows the relation between the line width and shock velocity for different values of the electron/ion temperature ratio β (Sect. 4.3.4). If the electrons and ions have the same temperature, the protons have a lower temperature—and, hence, and lower thermal velocities. For this reason β = 1 corresponds to the lowest curve in Fig. 8.9. For shock velocities Vs  2000 km s−1 the relative velocities between protons and neutral hydrogen are in the range where the charge exchange rate is relatively flat, and the relation between Vs and broad-line FWHM is nearly linear. But due to the steeply decline in charge exchange rates above

v  1500 km s−1 , the broad-line width will be skewed to lower velocities for Vs > 2000 km s−1 , resulting in a flattening of the line widths towards 3500– 4500 km s−1 .

8.2.2 The Broad- to Narrow-Line Ratio as a Diagnostic Tool As can be seen in Fig. 8.8, for relative velocities 2000 km s−1 the probability of a charge exchange is larger than for a ionisation and excitations, whereas for 4000 km s−1 the ionisation rate and excitation rates show a similar velocity dependence, and the rate coefficients for both processes are nearly constant. These different dependencies affect the ratio between the broad-line and the narrow-line

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Fig. 8.9 The FWHM of broad-line Hα emission as function of shock velocity, according to the calculations presented in [1148]. The calculations assume a hydrogen ionisation fraction of fp = 0.5. The different curves are for different levels of electron/proton temperature ratios β = Te /Tp (Reproduced from Van Adelsberg et al. [1148])

Hα component (Ib /In ) [433, 1148]. Below relative velocities of ∼1000 km s−1 the rates for producing the broad-line component rises faster than the production rate for the narrow-line component. As a result, the Ib /In ratio increases. Beyond 1000 km s−1 the trend reverses and the Ib /In ratio declines. And finally for relative velocities 4000 km s−1 the excitation rate is larger than the charge exchange rate, and the broad-line component’s intensity becomes rapidly weaker. It is worth noting that for a given temperature electrons move faster than protons:  < ve > = < vp >

 mp kTe ≈ 43 β. me kTp

(8.5)

So for β = 1 the relative hydrogen-electron velocities will be above 4000 km s−1 and contribute to the ionisations and excitations, even for modest shock velocities of Vs ≈ 250 km s−1 . However, for low β this may not be true. Note that for β = 1 the proton velocities will be about 23% lower than for β  1, which also impacts the relative rates of proton to electron collision rates. The ratio of the collision rates as a function of electron and proton velocities explains the model predictions shown in Fig. 8.10. These calculations take also into account the excitation probabilities to quantum levels n ≥ 3, and the relatively

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Fig. 8.10 The predicted Hα broad-line to narrow-line ratio for Balmer-dominated shocks under two different assumptions [1148]. Left: Case A, the shock region is optically thin to Lyβ photons. Right: Case B, the shock region is optically thick to Lyβ photons. For each case, three different values for the electron-temperature ratio are shown: β = 0.1 (blue), β = 0.5 (green), and β = 1. (solid red line) (Reproduced from Van Adelsberg et al. [1148])

branching ratio for Lyβ versus Hα emission. This branching ratio is also important for the narrow-line component, but in that case there is an additional complication: Lyβ trapping. If the transition region is optically think to Lyβ photons, which is plausibly the case for a filament seen edge on, Lyα and Lyβ photons are subject to resonant line scattering (Sect. 13.5.9). Lyα photons will eventually escape the shock transition region, although there may be a dimming of the Lyα flux edge-on, and a brightening of the Lyα-flux for a shock seen face-on (c.f. Sect. 8.1.1). For Lyβ after each resonant absorption there is a probability that the de-excitation occurs through emitting an Hα photon, rather than a Lyβ photon. As a result, Lyβ trapping results in an enhancement of the narrow-line component. The effect is very small for the broad-line component as resonant line scattering is very sensitive to the line width. The two limiting cases of no Lyβ-, and complete-Lyβ trapping on the ratio Ib /In are shown in the two panels of Fig. 8.10. Since the ratio Ib /In is sensitive to the electron-proton temperature ratio, it provides an important diagnostic on the electron/proton temperature ratio [436, 939, 1148]. The degeneracy between Ib /In as a function of shock velocity is broken by including the broad-line component’s width as an additional constraint. Electron/proton temperature ratios measured using this method [436] are displayed in Fig. 4.4. But not all measured values Ib /In are consistent with the model [1148]: for the Cygnus Loop [432] and DEM L71 [434] the Ib /In was low for the inferred shock velocity. This discrepancy between model and measurements for these two supernova remnants is not yet understood.

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215

8.2.3 Measuring Distances to Balmer-Dominated Shocks Since the FWHM can be used to measure the post-shock temperature and, hence, the shock velocity provides a means to estimate distances to supernova remnants (Sect. 3.2.2), provided that also the shock proper motion is measured, given the relation Vs = θ˙ d, with θ˙ the proper motion. Care needs to be taken to account for the fact that electrons and protons may not have the same post-shock temperature, by using the measured Ib /In ratio. This method has been used to measure distances to Kepler’s SNR (d = 5.1 ± 0.8 kpc) [1006], Tycho’s SNR (d = 2.4 ± 0.4 kpc) [657], SN 1006 (d = 1.85 ± 0.25) [946, 1239], and RCW 86 (d = 2.5 ± 0.2 kpc) [504].

8.2.4 The Shock Structure in the Presence of Neutrals In Chap. 4 we encountered the concept of multi-fluid shocks. Also the structure of Balmer-dominated shocks cannot be treated as a single fluid with a shock transition region of width corresponding to the ion inertial length scale or ion gyroradius (∼107–109 cm, Sect. 4.3.3). Instead the shock transition layer is subdivided in two regions: (1) the shock transition of the atoms that were pre-ionised, (2) a longer transition region corresponding to thermalisation/ionisation region of the neutral atoms. The latter corresponds to the collisional length scale of neutral hydrogen, which is the result of a combination of charge exchange and ionisation, with a combined length scale of ∼ 1015/n0 cm. Figure 8.11 shows the shock structure of a shock with an upstream ionisation fraction of fp = 0.5, as calculated in [1148]. Once all neutral hydrogen has been ionised, the overall shock compression ratio and downstream temperature are still given by the Rankine-Hugoniot equations, because the conservation of particle, momentum-, and enthalpy flux still applies to the system as a whole. It is just that establishing the final thermodynamical quantities is smeared out over a broader region downstream of the first shock jump.

8.2.5 Complications: Pickup Ions and Non-thermal Distributions There are few more things to consider in the context of the shock physics associated with Balmer-dominated shock filaments. First of all, the “hot” neutral hydrogen atoms—protons that have picked up an electron—are for some time dissociated from the plasma, and as a result may recross the shock boundary, back upstream, thereafter preheating the upstream gas. These hot neutrals are referred to as pickup ions [179, 840, 841, 945], and their presence

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Fig. 8.11 The spatial density profile of “cold” neutral hydrogen (ηH ), neutral hydrogen after one, two or multiple charge exchange reactions (resp. η1 , η2 , ηN ) and protons. The conditions used here are that the upstream neutral hydrogen fraction is fp = 0.5, and the shock velocity is Vs = 1000 km s−1 . All quantities are normalised to the upstream density: η = nX /(nH,0 + np,0 ). This figure has been adapted from two figures in [1148] (Credit: Matthew van Adelsberg)

upstreams results in the formation of a shock precursor. In the precursor the velocity and temperature of the gas may be altered before the actual shock arrives. The effect is somewhat similar to the preheating by cosmic rays accelerated by the shock front (Chap. 11). One predictions of pre-heating by pickup ions is the presence of an intermediate velocity component [840]. Such a component has indeed been identified in spectra of Tycho’s SNR, for which the intermediate velocity-width was measured to be ≈180 km s−1 (FWHM) [663]. A second complication concerns the assumption of Maxwellian velocity distributions for the electrons and ions immediately downstream of the shock. In Sect. 4.3.5 we showed that the self-equilibration time of electrons is around 108 –109 s (4.46), and for protons ∼1010 s (4.47). Both time scales are longer—for protons even much longer—than the time scale for charge exchange, ionisation and excitation of ∼107 s (Sect. 8.2.1). The implication is that in general we cannot assume that the velocity distributions are Maxwellian. Whether they are Maxwellian or not depends on the distribution established at the collisionless shocks. PIC-simulations [233] do indicate that the distribution function is close to a Maxwellian, but with a lower temperature than expected based on the standard shock equations and with a non-thermal tail to the distribution, caused by the initial stages of diffusive shock acceleration (Chap. 11). Below we come back to the overall effects of

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cosmic-ray acceleration on the spectra of Balmer-dominated shocks. From the observational side, there is indeed evidence for non-thermal (non-Maxwellian) velocity distributions for the broad-line Hα component for Tycho’s SNR [947]. Whether collisionless heating leads immediately to Maxwellian distributions or not, the subsequent charge exchanges and ionisations may again affect the velocity distributions in the region where most charge exchanges/ionisations occur. For example, it is known that ionisations give rise to a relatively “cold” population of liberated (secondary) electrons. The result is a strongly bimodal velocity distribution for the electrons, with shock-heated hot electrons, and a cooler distribution of secondary electrons [582]. This effect is strongest for shocks moving through a medium with a high neutral fraction.

8.2.6 The Effects of Cosmic-Ray Acceleration Efficient cosmic-ray acceleration will affect the spectra of Balmer-line dominated shock. As will be explained in more detail in Sect. 11.3, substantial cosmic-ray acceleration will lead to lower post-shock plasma temperatures. As a result, the width of the broad-line region for a given shock velocity will be narrower than expected. In particular, for a downstream cosmic-ray pressure quantified by the parameter w ≡ Pcr /(Pcr + Pth ), the post-shock temperature (4.12) will be modified to (Sect. 11.3) kT = (1 − w)

3 μmp Vs2 . 16

(8.6)

A full model taken this effect into account can be found in [841, 842], from which we display the predictions in Fig. 8.12. There have been two attempts to use the effect of cosmic-ray acceleration on the broad-line Hα width in order to measure the cosmic-ray acceleration efficiency. The first concerns the supernova remnant RCW 86. This supernova remnant has a wide range of shock velocities, from 50% [501]. However, subsequent measurements of the proper motion of the Balmer filaments indicated lower shock velocities, 1000–2000 km s−1 [504]. As a result, the cosmic-ray efficiency of the Balmer-dominated shocks in RCW 86 must be below 27%. Interestingly, subsequent X-ray proper motions indicated that the fast X-ray synchrotron emitting shock regions have Vs = 3000 km s−1 [1256], indicating a

218

8 Optical Emission from Supernova Remnants 5000

FWHM broad line [km/s]

4500

Thick lines: Thin lines:

down= down=

0.01 1.0

4000 3500 3000 2500 2000

down=0.01, down=0.01, down=1, down=1,

0.1 0.2 0.1 0.2 Present work (full calculation) Present work (Gaussian approx. for k > 2) van Adelsbergh et al.(2008)

1500 1000 500 1000

2000

3000

4000 Vsh [km/s]

5000

CR= CR= CR= CR=

6000

7000

Fig. 8.12 The relation between broad-line Hα-width and shock velocity, but now in the presence of efficient cosmic-ray acceleration. The figure is an update of Fig. 8.9, and reproduced from [842]. The parameter CR is similar to w in the text (Credit: G. Morlino)

large difference in shock velocity between Balmer-dominated shocks and other parts of the shock in the same region. Another measurement of the cosmic-ray acceleration efficiency concerns the young supernova remnant 0509-67.5 (Fig. 8.7), which has one of the fastest Balmer dominated filaments—Vs ≈ 6500 km s−1 [503, 550]. The broad-line Hα width for a southwestern filament was measured to be FHWM= 2680 ± 70 km s−1 , and for a northwestern filament a FWHM= 3900 ± 800 km s−1 was measured. In particular the small error for the SW filament was constraining, indicating that the FWHM was much narrower than the prediction in [1148], indicating a cosmic-ray acceleration efficiency of w ≈ 25% [503], or even more [842]. However, it was later argued that the measured Hα width pertains to a shock that was much slower than the average 6500 km s−1 [550], suggesting a much lower cosmic-ray acceleration efficiency. In the context of the effects of cosmic-ray acceleration, it is also worth noting that the narrow-line Hα width appears to be wider than expected: as reported in [527, 663, 1067] several supernova remnants have narrow-line width of the order of FHWM≈ 30 − 60 km s−1 . Using (8.1), we see that this corresponds to upstream temperatures of Tp ≈ 19600(FWHM/30 kms−1 )2 K. These temperatures are too hot to be consistent with a partially neutral medium, which is a precondition for the presence of Balmer-dominated filaments. It suggests that the pre-shock medium must have been heated just upstream of the shock. Additional evidence for heating in the precursor was provided by Hubble Space Telescope observations of Balmerdominated shocks in Tycho’s supernova remnant, indicating Hα emission from a region 3 × 1016 cm ahead of the shock [711]. This heating could be either due to adiabatic compression and/or Alfvénic heating in the cosmic-ray precursor [1192],

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or caused by the aforementioned heating by pickup ions [179, 945]. In both cases, charge exchanges are needed for neutral hydrogen to be broadened. However, the measured precursor length scale of 3 × 1016 cm for Tycho’s SNR is more consistent with the length scale of a cosmic-ray precursor (Sect. 11.2.4), than the 1015 cm length scale for charge exchange reactions caused by pickup ions. Narrow-line Hα emission from the precursor may have in the past been added to the narrow-line emission from the Balmer-dominated shock region itself, hence lowering the Ib /In ratio. This could be an explanation for the anomalously low for this ratios in a few supernova remnants [1148]. Somewhat contradictory to the above results are the general trend that the presence of Balmer-dominated shocks and X-ray synchrotron emitting shocks (Sect. 12.2) appear to be anti-correlated. For example, in Tycho’s SNR the Balmerdominated filaments are found on the eastern side of the remnant, whereas the X-ray synchrotron filaments are more prominent on the western side, where no Balmer-dominated filaments are found. As discussed in more detail in Sect. 12.2, X-ray synchrotron emission requires high shock velocities (3000 km s−1 ) and a high level of magnetic-field turbulence. The latter results in faster acceleration, and acceleration to higher particle energies within the lifetime of the supernova remnant. These properties are related, but not identical to the concept of cosmic-ray acceleration efficiency, which is better defined as the fraction of shock energy (or postshock pressure) being diverted to accelerated particles. It is known that a large neutral fraction dampens magnetic-field turbulence [341]. There are, therefore, two possibilities for explaining the anti-correlation: (1) the Balmer-dominated shocks are in general slower shocks (3000 km s−1 ) moving through denser gas, and these slower shocks are not associated with X-ray synchrotron radiation; (2) magnetic-field turbulence is dampened by the presence of neutral hydrogen, limiting the maximum energy of accelerated electrons, and hence suppressing X-ray synchrotron radiation. Indeed, the regions with Balmer-dominated filaments in Tycho appear to be slower (2900 km s−1 , for d = 2.5 kpc) than the western region (4100 km s−1 , for d = 2.5 kpc) [1233]. A similar contrast between Balmer -dominated and Xray synchrotron dominated shocks has been found for RCW 86 [504] and SN 1006 [1242]. But the contrasts may not be large enough to offer a full explanation. Moreover, for SN 1006 also the magnetic field geometry has been used to explain the presence of X-ray synchrotron regions in two “polar caps” [990]. So no firm conclusions can be drawn yet regarding the cause of the spatial anti-correlation between Balmer-dominated and X-ray synchrotron emitting shocks.

Chapter 9

Young Supernova Remnants: Probing the Ejecta and the Circumstellar Medium

Chapter 5 describes the evolution of supernova remnants in the context of simplified description using consecutive phases: the ejecta dominated phase, the energyconservation (Sedov-Taylor) phase, and the pressure-driven phase. In this chapter we will discuss the current observational status of our knowledge concerning young supernova remnants, supernova remnants up to 2000 yr old, which are either in the ejecta dominated, or in the Sedov-Taylor phase. However, here we are not so much concerned with actual phase the supernova remnants are in, but more with what the observed properties of the supernova remnants tells us about the supernova explosions through the composition and distribution of the supernova ejecta, and about their progenitor stars, through the imprint their wind has had on the ambient medium. An important property of the youngest supernova remnants is their ability to accelerate particles beyond 10 TeV; this aspect of young supernova remnants is discussed in Chap. 12.

9.1 Core-Collapse Versus Type Ia Supernova Remnants An important distinction among young supernova remnants is that between remnants of core-collapse (Sect. 2.2) and Type Ia (thermonuclear) supernovae (Sect. 2.3). As described in Chap. 2 the composition of the ejecta between these two classes of objects is quite distinct, with core-collapse supernovae producing mostly carbon- and neon-burning products (i.e. O, Ne, Mg), whereas Type Ia supernova remnants produce typically more than 0.5 M of iron-group elements. Both types of supernovae also produce quite a lot of intermediate mass elements (IME; Si, S, Ar, Ca). This distinction in composition is reflected in the typical spectra of young supernova remnants, and allows for efficient typing of supernova remnants, as first pointed out in [557] and illustrated in Fig. 9.1. The X-ray spectra of many Type Ia supernova remnants show a characteristic bump around 1 keV, if the spectrum is © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_9

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Fig. 9.1 CCD (XMM-Newton-MOS) and Reflective Grating Spectrometer (XMM-Newton-RGS) spectra of an oxygen-rich (1E0102.2-7219) and a Type Ia supernova remnant (SNR 0519-69.0). It shows the spectroscopic differences between Type Ia and core-collapse supernova remnant spectra, with the Type Ia supernova remnant being dominated by Fe-L emission. The also figure illustrates the better spectral resolution of the RGS instrument over CCD-detectors, even for mildly extended objects; both supernova remnants have an extent of about 30 (Figure reproduced from [1173])

measured with the medium-energy resolution CCD detectors (E/ E ∼ 15). The bump is caused by Fe-L line emission from Fe charge states Fe XVII to Fe XXIV, and it is generally referred to as the Fe-L complex (Sect. 13.5.8). If the plasma in the remnant is hot enough, kTe  1.5 keV, the Fe-L complex is accompanied by Fe-K emission, which, depending on the charge state, occurs between 6.4 keV ( 500. Many of the optical knots appear in the shell, and disappear on a time scale of ten to twenty year, suggesting that they get shocked once they enter the reverse shock, and after ∼ 15 yr the shock has traversed completely disrupted them. This is consistent with a shock passing time scale of lknot/Vs ≈ 1016 cm/2 × 107 cm s−1 ≈ 15 yr [388]. In addition to the fast moving knots , Cas A contains a network of optical knots with much lower velocities ( 200 km s−1 [608]), mainly emitting in Hα and [N II]. These are sometimes referred to as “quasi-static flocculi” . Their origin may be a shell of gas lost by the progenitor, either in the form of a red-supergiant wind, or due to stripping as caused by some form of binary interaction [1264]. The optical spectra indicate elevated abundances of nitrogen and helium [258]. In general, our best knowledge on the ejecta composition is mostly based on X-ray spectroscopy, as these target all alpha elements, and iron. A list of the known oxygen-rich supernova remnants in the Galaxy and the Magellanic Clouds is provided in Table 9.1, whereas multi-band X-ray images are shown in Fig. 9.11. The X-ray spectra of oxygen-rich supernova remnants like 1E0102.2-7219 (see Fig. 9.1) and G292.0+1.8 are characterised by bright line emission of oxygen, neon and magnesium, and a relative absence of Fe-L emission. Indeed, according to supernova explosion models oxygen-rich supernova remnants are expected to be also rich in neon and magnesium. However, the X-ray spectrum of Cas A shows a remarkable lack of neon and magnesium lines. To be sure, there is neon and magnesium line emission, but an order of magnitude lower in abundance than to

Milky Way G111.7-2.1 G292.0+1.8 G260.4-3.4 Large magellanic cloud SNR B0540-69.3 SNR B0525-69.6 Small magellanic cloud 1E 0102.2-7219 SNR B0049-73.6 SNR B0103-72.6

Identification

0.55 0.73

50 50 62 62 62

N132D

IKT 22 IKT 6 IKT 23

0.36 1.15 1.45

2.6 4.2 25

Radius (arcmin)

3.4 6 2.2

Distance (kpc)

Cas A MSH 11-54 Puppis A

Name

Table 9.1 Properties of the “oxygen-rich” supernova remnants

6.5 20.7 26.2

8.0 10.6

2.6 7.3 16.0

Radius (pc)

0.55–0.73/0.8–1.13/1.75–2.3 0.4–0.71/0.71–1.1/1.1–2.3 0.4–0.71/0.71–1.1 /1.1–2.3

0.4–0.71 0.71–1.1 /1.1–2.3 0.5–0.63/1.0–1.2/4–6.2

0.7–0.85/0.75–1.9/6.4–6.8 0.55–0.73/1.74–1.9/4.2–6.2 0.3–0.7/0.7–1.0/1.0–8.0

Colour codes Fig. 9.11 (red/green/blue in keV)

[396, 941, 1190, 1251] [1020] [1151]

[889, 1154] [128, 140, 198]

[566, 568, 952] [158, 416, 892] [343, 570, 571]

Literature

9.3 Core-Collapse Supernova Remnants 241

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9 Young Supernova Remnants: Probing the Ejecta and the Circumstellar Medium

Fig. 9.11 X-ray images of oxygen-rich supernova remnants. From left to right, top to bottom: Cas A, G292.0+1.8, B0540-69.3, N132D, 1E0102.2-7219, B0103-72.6, and B0049-73.6. All images generated from Chandra ACIS archival data by the author, except Puppis A, which is based on Chandra and XMM-Newton data (Credit: NASA/CXC/IAFE/G. Dubner et al. and ESA/XMMNewton [343].) Properties of the supernova remnants, references, and the colour coding are listed in Table 9.1

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be expected [1181], and in marked contrast to 1E0102.2-7219 and G292.0+1.8. Also in the optical there is a lack of neon emission ([Ne III]) [379]. The unique composition of Cas A suggests that core-collapse explosion models may not be uniformly valid, with perhaps some of the variations among the remnants caused by either complicated mass stripping scenarios, or bi-polar explosions; see the next section. Based on the oxygen mass estimate of 1–2 M for Cas A [1181, 1236], its progenitor main sequence mass is estimated to have been 16–20 M [259]. For 1E0102.2-7219 an estimate of the oxygen mass of ∼ 6 M the progenitor’s main sequence mass was estimate to have been ∼ 32 M [396]. A similar progenitor mass, or lower, was estimated for G292.0+1.8 [159, 448]. For N132D a progenitor mass based on optical/UV spectroscopy suggest a 25–30 M [175] progenitor, or perhaps even a 50 M progenitor [400]. On the other hand, for SNR B0049-73.6 the main-sequence progenitor mass was estimated to be only 13–15 M [1020]. Together these estimates suggest that oxygen-rich supernova remnants have progenitors with main-sequences masses  15 M . However, one should take into account the uncertainties in supernova yields, in particular since the iron yield, and for large masses, also the oxygen yield, depends on the details of the mass-loss history of the star [compare for example 1246, 1248], binary interactions, explosions asymmetries [756], and on the mass of the neutron star or black hole that is left behind. Also the unpredicted lack of neon in Cas A gives some reason not to take the yields predicted by supernova models at face value. Nevertheless, there are plenty of relatively young supernova remnants (Kes 73, for example [195, 1175]), which do not exhibit the strong X-ray emission lines observed in oxygen-rich supernova remnants, suggesting that indeed the oxygen-rich remnants originate from the massive end of the initial mass function. Most likely a large fraction of the oxygen-rich remnants exploded as Type IIb, or Type Ib/c supernovae, as indeed was confirmed for Cas A based on the light-echo spectra (Sect. 2.5).

9.3.2 Asymmetric Ejecta: Donuts, Jets, Rings and Bubbles As described in the introduction of this chapter, core-collapse supernova remnants tend to be more irregularly shaped than Type Ia supernova remnants. This can even be apparent from the images of oxygen-rich remnants in Fig. 9.11, although the supernova remnants in the Small Magellanic Cloud seem to be more “roundish” than their counterparts in the Milky Way and Large Magellanic Cloud. The deviations from spherical symmetry could reflect the inhomogeneities of the ambient medium of core-collapse supernovae, or could be caused by intrinsic asymmetries of the supernova explosion. That the small-magellanic cloud remnants appear a bit more regularly shaped may be partially due to the lower overall interstellar medium density, creating a smoother ambient medium for the stellar wind bubble, and later the remnant expanding in it. On the other hand, one could also argue that the lower metallicity of the gas in the Small Magellanic Cloud resulted in less stellar-wind

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mass loss, which caused intrinsic explosion asymmetries to be smoothed out during the explosions, as a result of the presence of a more massive hydrogen envelope. Indeed for core-collapse supernovae it has been found that explosion asymmetries, as determined through optical polarisation measurements, are stronger for more severely stripped progenitors [1222]. For Cas A and G292.0+1.8 there is strong evidence that the asymmetries in the remnant are indeed caused by the explosion itself. For G292.0+1.8 there is remarkable one-sidedness in the distribution of the Si/S-rich ejecta (green in Fig. 9.11), which appears more concentrated in the northwestern part of the remnant [158, 886, 892]. This is in a direction opposite to the location of the pulsar, which is located southeast of the centre of the remnant, potentially as a consequence of momentum conservation of the supernova-ejecta/pulsar system. As for many other properties of supernova remnants, also for studying explosion asymmetries Cas A offers a fertile study ground. Its outer shock wave is traced by X-ray synchrotron emission (see Sect. 12.2), which is surprisingly circular and centred on a point close to the optically identified explosion centre [1114]. This suggest that the ambient medium, although likely originating from the progenitor wind, does not have a large deviation from circular symmetry. However, the reverse shock of Cas A, as traced by X-ray synchrotron emission [502] and internal freefree absorption in the radio [90] is offset toward the western side of the remnant. The optical knots, which probably briefly emit upon entering the region heated by the reverse shock, show a remarkable front-back asymmetry [952]. Although (part of) the front-back asymmetry could be related to the density of the ambient medium, the perfectly circular, and explosion-centred, outer shock suggest that the shift of the reverse shock is related to explosion asymmetries. More direct evidence for explosion asymmetries comes from the distribution of metal-rich ejecta as observed in X-rays and optical. In X-rays the main shell is bright in silicon (Si XIII) line emission. The iron-line emission, which comes from a deeper layer of the explosion, however, is in some place situated outside the main shell. This is most clearly seen in the Southeast where fingers of iron-rich plasma has penetrated as far as the outer shock [560] (blue in Fig. 9.11), corresponding to average velocities as high as 7000 km s−1 . The remarkable spectral contrast between two regions, both situated in the Southeast, is shown in Fig. 9.12, with one spectrum showing an extreme example of iron-rich ejecta and the other one a silicon-rich region (showing up as bright green in Fig. 9.11). Iron in the western region is around the main shell in the West, and in the Northeast it is projected inside the main shell. However, X-ray Doppler maps based on line centroids of the Fe-K line emission reveal that iron in this region has a larger radial velocity (∼ 3000 km s−1 ) than silicon (∼ 2500 km s−1 ), and is likely also outside the main Si-rich shell [303, 1235]. Using the Doppler-shift information, the X-ray emitting ejecta of Cas A appear no so much to consist of a spherical shell, but trace-out more a “donut” shape, with the southeastern part coming toward us, and the northeastern part receding from us [546, 769, 1235]. Interestingly, a donut-like morphology was also inferred from Doppler imaging of the oxygen-rich

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Fig. 9.12 X-ray Chandra spectra of two regions in the southeast of Cas A, one extremely iron rich (blue) and the other silicon-rich (black). The iron-rich spectrum shows little evidence for emission from alpha elements with most features around 1 keV coming from Fe-L shell emission from Fe XXVI and higher, and a prominent Fe-K feature around 6.7 keV [c.f. 560, 568]

remnant 1E0102.2-7219 [396], as measured with the Chandra high-energy grating spectrometer; see Fig. 9.13. The iron distribution in Cas A does not have an apparent axis of symmetry, with some “fingers” far out in the southeast, and predominantly redshifted iron in the northeast and toward the west, inside the main shell. However, there is an obvious axis of symmetry when considering silicon and sulfur (the latter also seen in the optical). Indeed a ratio map of Si XVII line emission and Mg XI emission reveals two jet-like features on the eastern side of the shell and one diametrically position on the opposite side [566, 1168]; see Fig. 9.14. The jets are projected beyond the forward shock, and the (north)eastern jet is associated with long known streams of extremely fast moving optical knots, bright in [S II] [e.g. 380, 382], and also with an infrared counterpart [537]. Proper motions of the optical knots reveal velocities up to 15,600 km s−1 , and Doppler measurements of the optical knots reveal the jets to be fortuitously lying almost in the plane of the sky. The southwestern counterpart is more irregular, as if it was broken due to some sort of external collisions. The overall distribution of silicon-richness along the jets is not confined to the jets alone. An optical survey of outer knots in Cas A reveal a remarkable dichotomy in knots that are bright in [O II] versus knots that are bright in [S II] [382]. The latter are found in the polar directions, as defined by the jet-directions, whereas knots bright in [O II] are mostly mainly found in the more equatorial directions; see Fig. 9.14, right.

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Fig. 9.13 Chandra-HETGS spectrum of the oxygen-rich SNR 1E 0102.2-7219. The top panel shows part of the spectrum in the negative order (m = −1) as seen with the Medium Energy Grating. The supernova remnant shows up in each in individual emission lines, ordered by wavelength. Bottom panel: a comparison between Ne X Lyα images in the negative, zeroth order, and positive order. As shorter wavelength are projected toward the centre, there is a mirror symmetry in wavelength, whereas the images in individual line transitions are not mirrored. As a result, deviations from Doppler symmetry of the shell show up as image deformations that are different in the positive and negative orders, as can be seen here (Credit: Katherine Flanagan et al. [396])

The energy associated with the jet system is uncertain. Hydrodynamical simulations of the jet and the remnant suggest a jet energy of ≈ 1048 erg [1026], but a survey of optical knots was used to estimate a jet energy of  1050 erg [382]. This discrepancy could imply that the jet primarily consist of dense knots, rather than more diffuse plasma. However, it could also hint that the masses of individual knots are smaller than estimated, which could be the case if the knots are intrinsically very clumpy or “fluffy”, instead of spheres of dense, homogeneous gas. Whatever the total energy budget of the jets, their silicon-richness suggest they were created during the explosion in the oxygen-burning layer of the progenitor. It is even possible that they are the remains of a more powerful jet that drove the supernova explosion. The jet energy is too small to suggest that the explosion that created Cas A was a long gamma-ray burst [698]. But a lower energetic explosion

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Fig. 9.14 Left: X-ray map of Cas A made with NuStar in the line of 44 Sc (caused by the decay of 44 Ti) around 68 keV in blue, and an Fe-K map [566] (red) and a map of the silicon-rich jet (green) [463]. (Credit: NASA/JPL-Caltech/CXC/SAO; jet image (green) made by J. Vink.) Right: The distribution of outer lying optical knots in Cas A, projected onto a Hubble Space Telescope ASC image [382]. The blue circles indicate the positions of [N II] knots, green circles correspond to [O II] emitting knots, and red circles correspond to [S II] emitting knots (Credit: Fesen and Milisavljevic 2016)

similar to an X-ray flash, like SN 2006aj, associated with the X-ray flash XRF 060218 [791] is a possibility. Apart from their energetics, the jet also present another puzzle: if the jet originates from the inner-most regions of the explosion, why is it silicon/sulfur-rich, and not from the even deeper iron-rich layer? Were the jets perhaps initial Poynting-flux dominated and are the silicon-rich ejecta products of a layer that were dragged in its wake? Another interesting question is what the relation is between the amount and distribution of the radio-active element 44 Ti (τ = 85 yr, Sect. 2.4) in Cas A [463, 586, 954, 1166], and the explosion (a)symmetries. The inferred 44 Ti production yield for Cas A is relatively high compared to models, 1.5 × 10−4 M , which has been attributed to the idea that Cas A was a bipolar explosion, leading to a freezeout of 4 He, enhancing incomplete α-capture, resulting in a built-up of 44 Ti[851]. However, NuStar observations show that the 44 Ti is not associated with the jet [463, 465]. Most 44 Ti is still interior to the reverse shock, but some is situated in the shock-heated shell, only partially overlapping with the iron-rich ejecta (Fig. 9.14). No correlation is seen with the direction of the two jets. The impression is that the innermost, iron- and 44 Ti-rich ejecta was ejected in somewhat random directions, whereas the bipolar shape associated with the jet has affected the silicon-rich layer. There is a suggestion that the point-source in Cas A (a CCO, see Sect. 6.5) has a velocity opposite to that of the 44 Ti [465], but unlike for the 44 Ti we lack the threedimensional space velocity of the neutron star, to be completely confident about that. Moreover, it is not a priori clear why the neutron star would have a momentum vector opposite to the 44 Ti, given that 44 Ti is only partially overlapping with iron,

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which also comes from the inner-most layer of ejecta and the iron mass is expected to be larger than 44 Ti. There may, however, be a relation between the distribution of intermediate mass elements and the neutron star direction of motion [630]—but see also [545]. Finally, it is worth pointing out that there is some systematics in the layout of the optical knots, and their infrared emitting, unshocked, counterparts in the interior: the optical knots form ring-like structure, whereas in the interior the knots appear to be around relatively empty cavities [303, 819, 820], as shown in Fig. 9.15. The explanation is that these cavities were created by bubbles of radio-active 56 Ni (see Chap. 2), which were created as radioactivity heated and inflated nickel-rich ejecta, pushing non-radioactive gas into shell-like structures [180, 721].

Fig. 9.15 Three-dimensional morphology of the optical knots in Cas A based on optical spectroscopy. The blue-to-red colours indicate Doppler shifts from −7000 to 7000 km s−1 . The knots have have a patchy layout, with rings of knots, surrounding large cavities [819, 820] (credit: D. Milisavljevic and R. Fesen [819])

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9.3.3 SN 1987A: The Making of a Supernova Remnant1 SN1987A is the youngest supernova remnant in our local neighbourhood, and its development from supernova to supernova remnant has been monitored in detail by various telescopes, across the electromagnetic spectrum. As the only nearby supernova for which we have followed, and are still following, the development from the core-collapse itself to the shaping of a supernova remnant, it deserves some special attention. The Type II supernova SN1987A was discovered on February 23, 1987 in the Large Magellanic Cloud, at the outskirts of the 30 Doradus starforming region. The detection of ∼ 20 neutrinos during a short interval of ∼ 13 s by Kamiokande II [539] and IMB [169] confirmed the theory that Type II supernovae are the result of the collapse of a stellar core into a neutron star, and that most of the energy is released through a cooling population of neutrinos. The detection of neutrinos also meant that the core’s collapse was very accurately timed, and that it preceded the optical detections by about 3 h—see [94] for an early review on the supernova and its immediate aftermath. In many respects SN 1987A was a peculiar Type II supernova. Its maximum luminosity was unusually low, corresponding to MB = −15.5, compared to ∼ −18 for typical Type II supernovae. This low luminosity is the result of the compactness of the progenitor star at the moment of explosion: the progenitor was not a red supergiant, but a B3 I supergiant, identified as the star Sanduleak -69◦ 202. Such an evolved, blue giant is expected to have a fast, tenuous wind, creating a low density cavity within a red supergiant wind of a previous mass-loss phase. Instead, optical spectroscopy [404] provided evidence for dense (ne ∼ 3 × 104 cm−3 ) circumstellar material enriched in nitrogen. Later optical imaging revealed the presence of three rings [282, 1194], which was later confirmed by a, now iconic, Hubble Space Telescope image, reproduced here in Fig. 9.16. The fact that the progenitor exploded as a blue supergiant, the enhanced nitrogen abundances of its surrounding material, and the triple-ring system may be explained by the merging of a 15–20 M star with a 5 M star, about 20,000 yr prior to explosion [923]; see the hydrodynamical simulations in [843, 844]. The triple rings themselves are understood as parts of a denser, hourglass-shaped structure, with the central ring a circumstellar ring, forming the waist of the hourglass. This hourglass region appears to be embedded in much larger (few pc) bipolar shell, which has peanut-shell shape, as deduced from light-echo modelling [1086]. The density within the inner hourglass-shaped shell, and between the hourglass-shaped shell and the peanut-shaped shell is relatively tenuous. A few months after the explosion SN 1987A was detected in hard X-rays [326, 1087]. The hard X-ray emission was probably the result of multiple Compton scatterings of γ -rays generated by the decay of radioactive material. Nuclear decay-

1 This

is a revised and extended version of a section in [1173].

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Fig. 9.16 Left: The 30 Doradus region (Tarantula Nebula) with SN1987A visible as a bright star near northwest of the centre of the image, as observed with the ESO Schmidt telescope. (Credit: ESO.) Right: Hubble Space Telescope image of SN1987A [221], as observed by the WFP camera in 1994. It reveals the flash ionised material around the supernova. The supernova blast wave has in the mean time passed through the inner ring (Credit: Dr. Christopher Burrows, ESA/STScI and NASA)

line emission from 56 Co was detected suprisingly early on, indicating considerable mixing of inner ejecta toward the surface of the supernova envelope—see [712] and references therein. Moreover, the nuclear decay lines were redshifted [1004, 1102, 1134]. This is provides evidence for an asymmetric explosion, bringing to mind the evidence for an asymmetric explosion of the Cas A supernova in the form of jets and high velocity iron knots presented in Sect. 9.3. The onset of the development of a supernova remnant was marked by the detection of radio emission around 1200 days (July 1990) after the explosion with the Molonglo Observatory Synthesis Telescope (MOST) and the Australian Compact Array (ATCA) [1076]. This emission was the result of the outer blast wave sweeping up circumstellar material. Note that the first week after the explosion also some rapidly fading radio emission was detected, possibly caused by the interaction of the blast wave with the gas immediately surrounding the progenitor star [1136]. Less than a year later, around 1500 days after the explosion, X-ray emission was detected with the ROSAT instruments [496]. Since then SN1987A has been regularly monitored with radio, optical, and X-ray telescopes, revealing the gradual brightening of the radio and X-ray emission, as the blast wave started to interact with the density increase associated with the inner circumstellar ring. Some of the results are summarised in Fig. 9.17, showing images of the brightening in optical and X-rays of the circumstellar ring. Figure 9.18 shows the gradual total flux increase in the radio and X-rays.

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Fig. 9.17 Top: The evolution of the optical emission from SN1987A as imaged by the Hubble Space Telescope. Initially, the bright ring emits in the optical, because the material has been photoionised by the supernova explosion, but from mid-1990s on dense knots are lighting up as they are heated by shock wave. The central emission is coming from the freely expanding supernova ejecta, which is being heated by radio-active decay of unstable isotopes, in particular 44 Ti (Credit: NASA, ESA and R. Kirshner (Harvard-Smithsonian Center for Astrophysics and Gordon and Betty Moore Foundation) and P. Challis (Harvard-Smithsonian Center for Astrophysics)). Bottom: Similar sequence but now imaged in X-rays by the Chandra. The numbers indicate the age in days, so the first image corresponds to 1999 and the last image to 2015. The images are normalized by flux, with a square-root false color scale (Credit: Reproduced from Frank et al. [402])

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Fig. 9.18 The flux increase and expansion of SN1987A as observed in X-rays and radio (Credit: Reproduced from Frank et al. [402])

The optical and X-ray observations reveal that after an initial phase—in which the outer shock wave was moving through the, relatively tenuous, medium associated with the progenitor’s fast wind—between 1995–1999 the shock wave started to interact with the central dense ring seen in the optical. This ring has a size of 1.7 by 1.2 , consistent with a ring of radius 0.19 pc, seen under an angle of ∼ 45◦ . The dense ring itself has considerable small-scale structure, with regions of different densities, and fingers and/or knots located at smaller radii [793]. As a result the ring is not lighting up all at once, but several “hot spots” turned on; first in the northeast [222], later also in other regions. By now the blast wave has completely moved through the dense ring. The interactions with an increasingly larger fraction of the circumstellar ring resulted in a gradual brightening of the supernova remnant in the optical and X-rays. Around 2012, however, the increase in soft X-ray emission levelled off, indicating that the blast wave had penetrated almost the entire ring, leaving no more knots to be shock-heated [402]. The hard X-ray and radio emission, however, do not show a break in the flux increase, suggesting that this emission is coming from outside the ring, where the densities are lower, and the blast wave has not decelerated as much as in the ring. The evolution of the radius of SN 1987A also show difference between soft X-rays and radio emission, confirming this view. In the past two decades much attention has focussed on how the blast wave interacted with the circumstellar ring. The complexity of the interaction of the blast wave with this dense inner region do not fit into the simplified description of supernova remnant evolution, sketched in Chap. 5 [194, 793, 816]. The basic structure of a blast wave heating the circumstellar medium, and a reverse shock heating of the ejecta is still valid. But a range of shock velocities must be considered, as the encounter of the blast wave with the density enhancements in the ring and its protrusions lead to a system of transmitted and reflected shocks. The transmitted shocks heat the material of the ring, whereas the reflected shocks go back into the plasma behind the blast wave. These reflected shocks are low Mach number shocks, as they go through already shock-heated plasma. They result in additional heating

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and compression of the already hot plasma. The transmitted shocks will have a range of velocities, depending on the density of the material and the obliquity with which the blast wave hits the protrusions. The slowest transmitted shocks (VS  200 kms) become radiative (Sect. 4.4), resulting in the bright optical radiation that accounts for the bright knots that lie as beads all along the optical ring as observed by the Hubble Space Telescope [654]. Because the blast wave reached the ring in about 5–10 yr, the average blast wave velocity was initially ∼15,000–30,000 km s−1 . In contrast, when the blast wave started interaction with the ring the blast wave seemed to have slowed down to ∼ 4000 km s−1 in the radio [856] and 1800 ± 600 km s−1 in X-rays [938]. Until 2004 (6000 days after the supernova) the expansion velocity in X-rays was similar to that in the radio, indicating that the blast wave reached the denser region of the inner rings around that time. The X-ray emission also became brighter around that time [888]. The strong deceleration of the blast wave from 15,000–30,000 km s−1 to 1800–5000 km s−1 means that a strong reverse shock must have developed with shock velocities in the frame of the ejecta of ∼ 10, 000 km s−1 . Note that the difference between the radio and X-ray expansion may be partially due to the fact that the X-ray emission biases observations to the densest parts of the ambient medium, since the emission scales as ∝ n2e (Sects. 13.4 and 13.5). In contrast, the radio synchrotron emission scales probably as ∝ ne , but with some uncertainty, because synchrotron radiation also depends on the electron acceleration properties of the shock and on the magnetic-field strength; see also the discussion in [1276]. The X-ray expansion may, therefore, be more skewed toward higher density regions, than the radio expansion measurements. From 2000 to 2010 SN 1987A was observed several times with both the CCD and grating spectrometers on board Chandra and XMM-Newton. The CCD spectra are well fitted with a two component model; a soft component with kTe = 0.2– 0.3 keV, which is close to ionisation equilibrium (CIE; Sect. 13.5.7), and a harder non-equilibrium ionisation (NEI) component with kTe = 2–3.5 keV and ne t≈ 3 × 1011 cm−3 s [891, 1276]. The soft CIE component is associated with the densest regions, heated by the transmitted shock, whereas the hotter, NEI component is generally attributed to the plasma heated by reflected shocks [891, 1276]. However, it is not clear whether also reverse shock heated plasma is responsible for some of the harder X-ray emission. Note that even the reflected/reverse shock heated plasma must be relatively dense −1 as the ionisation parameter indicates ne ≈ 950 t10 cm−3 , with t10 the time since the plasma was shocked in units of 10 yr. As the supernova remnant ages, the low temperature component is becoming hotter (kTe ≈ 0.2 keV in 1999 versus kTe ≈ 0.3 keV in 2005), whereas the high temperature component is becoming cooler (kTe ≈ 3.3 keV and kTe ≈ 2.3 keV, respectively), see [891, 1275, 1276]. SN 1987A is an ideal target for the (slitless) grating spectrometers on board Chandra and XMM-Newton, due to its small angular extent and its brightness [222, 307, 512, 816, 1275], which allows for resolving the He-like triplets (Sect. 13.5.8). An example grating spectrum is shown in Fig. 9.19.

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Fig. 9.19 Combined Chandra HETGS count spectrum of SN 1987A, based on a total exposure of 360 ks [307]. The positive grating order is colored red, the negative order blue (Credit: D. Dewey)

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The earliest high resolution spectrum, taken with the Chandra HETGS, dates from 1999 [222]. It reveals prominent emission lines from H-like and He-like transitions of O, Ne, Mg, and Si, and line widths consistent with an expansion of ∼ 4000 km s−1 . HETGS and LETGS spectra from around 2008 indicate a line broadening ranging from 1000–5000 km s−1 (FWHM), but with a bulk velocity that is surprisingly low, ∼100–200 km s−1 , consistent with no major asymmetry in the velocity distribution of the shock-heated plasma [307, 1275]. This is a strong indication that the hottest component is not due to the main blast wave, for which the plasma velocity should move with 34 VS , but must result from either a reflected shock [1275] or from the reverse shock heated plasma. This would reconcile the large line widths and hot temperatures, indicating large shock velocities, with a relative low velocity in the observer’s frame (Sect. 5.3). X-ray spectra taken with different instruments do not agree in all details. For example, the Chandra CCD spectra indicate a cool component consistent with CIE [891], whereas the high spectral resolution Chandra-HETGS/LETGS spectra indicate a plasma with a higher electron temperature, 0.55 keV, and out of ionization equilibrium, with ne t = 4 × 1011 cm−3 s [1275]. The best fit XMM-Newton-RGS model gives a solution that lies somewhere in between ne t≈ 8 × 1011 cm−3 s [512]. In reality a range of temperatures and ne t values are expected, given the complexity of the ring structure. Even the lowest ne t values reported, imply densities in excess of ne ≈ 2000 cm−3 . Nevertheless, these differences in fit parameters may also result in differences in abundance determinations. In general the abundance determinations from various studies are in good agreement [512, 1275], indicating LMC abundances except for oxygen, which seems underabundant, and nitrogen, which is mildy overabundant. The RGS and LETGS/HETGS mainly disagree on the iron abundance, which is under-abundant according to fits to the RGS data [512]. A more recent study of the X-ray emission of SN 1987A tackles the issue of the electron/ion heating of shock ( Sect. 4.3.4) in SN 1987A by comparing measurements of X-ray line broadening (assumed to be thermally broadened), with expectations from a detailed hydrodynamical model of the remnant, which is needed to estimate shock velocities in different regions of the remnant [814]. The conclusion of that study is that the ion temperatures are indeed much higher than the electron temperatures. An interesting aspect of the evolution of SN 1987A has been submm and infrared emission related to the dust formation after the supernova event. For that aspect of SN 1987A we refer to Sect. 7.6.1. More than three decades of observations of SN 1987A have shown how the emission from SN 1987A changed from something that could be clearly tied to the supernova ejecta, heated by the supernova shock that disrupted the progenitor, and radio-active decay, into a supernova remnant, that is an objects whose features are more determined by the interaction of the outer ejecta layers with the ambient medium. SN 1987A in that sense shows a complex evolution, largely due to the structure of the circumstellar medium. Until recently most of the emission was associated with the interaction of the blast wave with the dense inner ring. Now that the blast wave has moved through the ring, the emission from other regions

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of SN 1987A will gradually increase in importance. In addition to radio, infrared, optical and X-ray monitoring, one can expect the γ -ray emission (Sect. 12.3) to rise, as electrons and ions are accelerated in larger quantities and to higher energies. So far SN 1987A has not been detected in TeV γ -rays [23], and at the moment of this writing, only a hint of GeV γ -ray emission has been reported [762]. On its way out, the blast wave will interact with the less dense material interior of the hourglass shape, the denser gas in the equatorial plane, and eventually, after several hundred years with the outer, peanut-shaped, shell, which has likely been created by the main-sequence wind of the progenitor [1086]. At the same time, the reverse shock will penetrate deeper into the stellar material. So far it has only encountered material from the outer hydrogen-rich envelope [513], but at some point the reverse shock will penetrate into the oxygen-rich material of the core. By that time the remnant may be just like most other young core-collapse supernova remnants, providing almost no clues about its complex initial evolution. But unlike for other supernova remnants, for SN1987A we can rely on the detailed multiwavelength observations that have been accumulated over time to understand the full evolution of a supernova remnants, from the day of the explosion to a full supernova remnant. By that time we may also more about another aspect of the explosion of SN1987A: what is the nature of the compact object that was created? The neutrino detections in 1987 proofed that a neutron star was formed, but a reliable detection has not yet been made. Is that because fall-back supernova ejecta, resulted in a latetime transformation to a black hole? Or is all the emission from the neutron star absorbed by dust near the explosion centre [781]? The neutron star does not appear to have created a powerful pulsar wind nebula that would have cleared the interior. But as we have seen in Chap. 6, neutron stars are born with a variety of spin periods and magnetic fields. We will may have to be patient to see what the properties are of the compact object star in SN1987A.

Chapter 10

Middle-Aged and Old Supernova Remnants

A couple of thousand years after the explosion supernova remnants have swept up considerably more mass than the mass of the supernova ejecta. They have been by then been in the Sedov-Taylor phase for some time—or are even (partially) in the snow-plough phase, if they their shock velocity has dropped below Vs ≈ 200 km s−1 (Chap. 5). If this stage progresses for some time, the plasma recently heated by the forward shock will be too cool to emit substantially in X-rays. Instead, they either regions may display bright filaments in the optical (Chap. 8). The interiors of these older supernova remnants should contain low-density, hot plasmas, as indicated by the pressure and density profiles of supernova remnants in the Sedov-Taylor phase (Fig. 5.2). Surprisingly, however, some older supernova remnants with slow shocks have bright interior X-ray emission. These are the mixedmorphology supernova remnants, and to be discussed in Sect. 10.3. The purpose of this chapter is to look more into the physics and phenomenology of older supernova remnants. Two prototypical supernova remnants in this regard are the Cygnus Loop (Fig. 3.1a) and the Vela supernova remnant. They are for the class of old or mature supernova remnants, what Cas A and Tycho’s SNR are for young supernova remnants, or the Crab Nebula for pulsar wind nebulae. The Cygnus Loop and Velar SNR owe this status primarily to their proximity, both being at a distance of less than 1 kpc. As a result, spatial details on small physical length scales can be studied. Moreover, there is little intervening gas, so the absorption column density, NH , is low, allowing us to observe these supernova remnants in the UV and soft X-rays, parts of the electromagnetic spectrum that are in general difficult to study for more distant supernova remnants, unless they are at high Galactic latitude. The Cygnus Loop is at a distance of 735 ± 25 pc [387], and has an angular radius of 1.3◦ , corresponding to 17 pc, with a break-out region in the south. Its age

The introduction and section on the mixed-morphology remnants are extended and revised texts from sections in [1173]. © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_10

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has been estimated to be 8000–14,000 yr [624, 716, 718], but the newly revised distance makes an age close to 20,000 yr more likely [387]. Its shock velocities are in the range of 150–400 km s−1 [171, 716, 718, 1002]. This range in shock velocities indicates that the Cygnus Loop has both radiative and non-radiative shocks (Chap. 8). In fact, the Cygnus Loop derives part of its fame from the beautiful optical nebula associated with the radiative shocks, the so-called Veil Nebula (Fig. 1.1). The large contrasts in shock properties have often been explained by the idea that the Cygnus Loop is evolving in a wind-blown cavity [247, 718, 794, 832]. But according to [387] the supernova remnant is evolving a in a low density region of the interstellar medium, but part of the shock is now interacting with interstellar clouds. X-ray spectroscopy of the Cygnus Loop has provided some peculiar results. One example is that imaging spectroscopy with ASCA, Suzaku, Chandra and XMMNewton has revealed that the abundances of the bright X-ray shell are sub-solar, with typical depletion factors of ∼5 [623, 831, 833, 853]. The low abundance of the inert element Ne shows that this is not due to dust depletion. In contrast, near the rim of the bright shell there are abundance enhancements of about a factor two [623]. This apparent abundance enhancement was later attributed to enhanced oxygen emission caused by charge exchange reactions between unshocked neutral hydrogen and O VIII [627]. In a more recent study [1142], using high-resolution X-ray spectra of a bright knot obtained with the XMM-Newton-RGS instrument, clear evidence was found for charge exchange processes, consisting of a high ratio of the forbidden to resonant line ratio (Sect. 13.5). This study again confirmed the puzzling overall low abundance of metals in the Cygnus Loop. In Sect. 10.1 the interior X-ray emission from the Cygnus Loop is discussed. One striking feature of older supernova remnants like the Cygnus Loop is the complexity of the emission, which is caused by the interactions of the shocks with density inhomogeneities (clouds, or cloudlets), and by a variety of shock velocities, giving rise to both radiative and non-radiative shocks. Moreover, these shock-cloud interactions lead to the creation of reflected and transmitted shocks [717, 902], not unlike the situation described in connection with SN 1987A (Sect. 9.3.3). A supernova remnant that has been used quite extensively to investigate the interaction of the blast wave with an inhomogeneous ambient medium is the Vela supernova remnant (Fig. 10.1). The Vela supernova remnant contains a pulsar with a rotational period of 89 ms, which is surrounded by a pulsar wind nebula [509], whose inner region is shown in Fig. 6.7. The pulsars characteristic age is ∼11,000 yr. The actual age of the pulsar and supernova remnant may be older, ∼20,000 yr [100]. VLBI parallax measurements of the pulsar show that its distance is 287 ± 19 pc [321], at which distance the angular size of the supernova remnant, 4◦ , corresponds to a physical radius of 20 pc. The supernova remnant is located in a complex, starforming region, which also contains the γ 2 -Velorum massive stellar binary. Both of them are embedded within the hot bubble/HII region that is known as the Gumnebula [100, 1088].

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Fig. 10.1 The Vela supernova remnant as observed by the ROSAT all sky survey (RASS) [100]. The blue channel represents the harder X-ray emission (0.5 keV), the green color the softer Xray emission (0.1–0.4 keV), with the total X-ray band assigned to the red channel. Hotter plasma colours purple in this image. The background image in red shows Hα emission as observed as part of the Southern Hα Survey Atlas (SHASSA [426]). The positions of the Vela shrapnels are indicated, as well as the Puppis A supernova remnant

ROSAT and XMM-Newton X-ray imaging spectroscopy of isolated regions of the Vela supernova remnant, in combination with optical narrow filter imaging, show a complex structure [188, 746, 812] that is best described in terms of three components: (1) a tenuous inter-cloud medium, which is difficult to characterise in X-rays due to its diffuse nature and low surface brightness, (2) soft X-ray emission from cloud cores, heated by slow transmitted shocks, and with temperatures around kT ≈ 0.1 keV, and (3) hotter regions surrounding the clouds, consisting perhaps of plasma evaporated from the clouds [603]; this plasma could be heated by thermal conduction from the hot intercloud medium, but may otherwise have cooled through radiation and thermal conduction with the cool cloud cores. The cloud cores have moderately high densities of n ≈ 5 cm−3 .

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10.1 The Presence of Metal-Rich Ejecta in Middle-Aged Supernova Remnants

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In Chap. 9 we learned that young supernova remnants can be typed as corecollapse or thermonuclear supernova remnants based on their X-ray spectra. This is facilitated by the fact that for young supernova remnants the ejecta mass is a large fraction of the overall mass. However, in more recent years X-ray imaging spectroscopy with Chandra, XMM-Newton, and Suzaku has also revealed X-ray spectral signatures of metal-enriched plasmas in older supernova remnants. In Sect. 10.3 we show that a majority of the mixed-morphology supernova remnants show evidence for ejecta components. Here we discuss the metal-enriched plasmas in the Cygnus Loop, which has been extensively mapped with Suzaku and XMM-Newton CCD-detectors [1138]. This has revealed that metal-rich plasma is associated with the low surface brightness interior of the SNR, which has a temperature that is considerably hotter (kT ≈ 0.6 keV [1133]) than the bright shell (kT ≈ 0.2 keV). Even within this hotter, interior region, there is some variation in abundance patterns, with O, Ne, and Mg, more abundant in the outer regions, and Si, S, Ar, and Fe more abundant in the central region [1133, 1137, 1140]; see Fig. 10.2. This suggests a layered explosion, with the more massive elements situated in the center. This is in contrast to Cas A, for which the Fe-rich ejecta has (partially) overtaken the Si-rich ejecta (Sect. 9.3.1). The overall abundance pattern of the inside of the Cygnus Loop is consistent with that expected from a core collapse supernova with an initial progenitor mass of M ≈ 15 M , but with 5–10 times more Fe than predicted by supernova explosion models [1133]. As noted in Chap. 2 the predicted Fe yield for core-collapse supernovae has considerable uncertainty.

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It is tempting to speculate that the more layered metallicity structure of the Cygnus Loop, as opposed to the more radially mixed metallicity of Cas A, has to do with the different progenitor types. The Cygnus Loop progenitor, with its relatively low initial mass, probably exploded as a red supergiant, but may still have had a significant fraction of the hydrogen envelope at the time of explosion. The Cas A progenitor was almost completely stripped of its hydrogen envelope at the time of explosion (Sect. 9.3.1), which made it perhaps easier for convective motions inside the supernova explosion to affect the layering of the ejecta. Indeed the simulations of Kifonidis et al. [650] show a qualitative difference in core ejecta velocities between Type II and Type Ib (i.e. stripped progenitor) supernovae. The metal-rich plasma in the Vela SNR is not associated with the interior, but rather with a number of protruding plasma clouds, usually referred to as the Vela “bullets” or “shrapnels” [100, 1081]; see Fig. 10.1. The name shrapnels is justified by their high abundances [1132], which indeed suggest that the clouds consist of supernova explosion products. Although it is not clear whether this also means that the clouds were ejected as shrapnels by the supernova, or whether they were formed due to some hydrodynamical instabilities at the contact discontinuity in the early supernova remnant phase. The morphology of the shrapnels are suggestive of clouds surrounded by bow-shocks, whose opening angles indicate Mach numbers M ≈ 3–4. For the hot, tenuous environment of the Vela SNR (n ∼ 0.01 cm−3 ,[1081]), this corresponds to velocities of V ≈ 300–700 km s−1 . The average shrapnel densities are ne ≈ 0.1–1 cm−3 with masses for shrapnel A and D of, respectively, 0.005 M and 0.1 M [620, 621]. Given their projected distance from the main shell of ∼1◦ , corresponding to l ≈ 5 pc, the shrapnels must have survived passage through the hot bubble for quite some time. Even for a mean average velocity difference between shock and shrapnels of

v ≈ 1000 km s−1 , the travel time is ttrav ∼ l/ v ∼ 5000 yr. This should be compared to the typical cloud-crushing time tcc [660]: tcc =

−1  R   χ 1/2 vb χ 1/2 R yr, = 980 vb 100 1 pc 1000 km s−1

(10.1)

with χ the density contrast between the shrapnel and its ambient medium. Comparison of these time scales, assuming vb ∼ v, shows that the apparent survival of the shrapnels indicates that the density contrast between the shrapnels and the ambient medium must be high, χ ∼ 1000. A higher average velocity does not help, as it lowers both the travel time and the destruction time. A higher density contrast could either mean that the initial density of the knots was higher than it is now, or that the ambient density is lower, ∼0.001 cm−3 . Either way, the Vela shrapnels may owe their survival to the low density of the ambient medium and a fortuitous timing concerning the phase in which we happen to observe the Vela SNR. There is some morphological evidence for hydrodynamical instabilities occurring in shrapnel D [620], which will lead to the destruction of this cloud. As already mentioned, imaging spectroscopy with ASCA [1132], Chandra [834], and XMM-Newton [620, 621] have shown that the clouds contain supernova ejecta.

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These studies also indicate that the shrapnels originate from different layers inside the supernova, with Shrapnel A being more rich in the oxygen-burning product Si, and shrapnel D more rich in Ne- and the carbon-burning products O, Ne, Mg (with enhanced abundances by a factor 5–10 with respect to solar). The shrapnels are standing out, due to their projected distance from the Vela SNR. However, more shrapnels may be projected unto the main supernova remnant shell. Indeed, several regions with overabundances of O, Ne, Mg and Fe were identified through XMMNewton observations [811], which can be either other shrapnels, seen under less fortuitous observing geometries, or these may be regions otherwise rich in ejecta, or even being the remains of destroyed “shrapnels”.

10.2 Interaction with Molecular Clouds Star formation preferentially occurs in the high density regions of the Galaxy, often in, or near molecular clouds. As a result, many core-collapse supernovae are expected to lie inside, or in the vicinity of molecular clouds. But it should be kept in mind that their progenitors, or other massive stars in their neighbourhoods, are expected to have significantly altered the immediate surroundings by the time the stars explode; see also the discussion on stellar wind bubbles in Chap. 5 (Sect. 5.8). Nevertheless, core-collapse supernova remnants are expected to interact with the dense medium outside the wind bubble, at some stage, late in their evolution. This probably occurs earlier for smaller wind bubbles, i.e. for progenitor stars of relatively low mass (12 M ), than for very massive progenitors. The densities in molecular clouds range from n ≈ 50 cm−3 up to 106 cm−3 , but the cloud are considerably clumpy. This in turn, will result in considerably patchy interaction of supernova remnants interacting with molecular clouds, with some part of the shock wave grinding to a halt in a dense region, whereas other parts may propagate relatively smoothly through an interclump medium. H2 is the most abundant molecule in the interstellar medium, which is not a surprise given that hydrogen is the most abundant element in the Universe. However, the perfect symmetry of this molecule, resulting in the absence of an electric dipole, makes that H2 has a low emissivity, in particular in the cold (∼10 K) cores of molecular clouds. Instead, other molecules are used to study the properties and spatial distribution of molecular gas, in particular CO (carbon monoxide) is taken as an important tracer of molecular clouds in the Galaxy, being the second-most abundant molecule (X = nCO /nH2 ≈ 10−4 ), and being radiatively efficient. See for example the Dame et al. [286] survey of CO in the J = 1 → 0 rotational transition-line at 115 GHz (2.6 mm). More on this line emission later. Since the line frequency can be measured with high accuracy, the internal kinematics of the cloud, as well as the projected rotational velocity due to the differential rotation of the Milky Way can be measured with an accuracy of ∼km s−1 . This line transition can easily become optically thick for column densities N(H2 )  2 × 1020 cm−2 [191], in which case one often resorts to the same line transition, but for the 13 CO molecule

10.2 Interaction with Molecular Clouds

263

at 110 GHz (CO in general refers to the more abundant 12 C16 O molecule). Apart from CO, other important molecules that are good tracers of the physical conditions in molecular clouds are OH (hydroxyl), CH, CS, NH, SiO, NH3 (ammonia), HCO+ , HCN, HNC, H2 O (water), H2 CO (formaldehyde), and CH3 OH (methanol).

10.2.1 Radiation from Molecules Like for atoms, the energy levels of molecules correspond to the internal degrees of freedom of the molecules, which correspond to rotations, stretching and bending of the atomic nuclei with respect to another, the electron energy levels, and the spins of the electrons and atomic nuclei. It is beyond the scope of this book to discuss the quantum physics of molecules in detail. Instead, we briefly discuss the rotational and vibrational energy transitions of diatomic molecules (molecules consisting of two atoms), which includes CO molecule, and the hyperfine transitions of OH. The most important energy levels for diatomic levels are determined by one rotational quantum number J and one vibrational quantum number v, and the electronic quantum state n:   1 + Bv J (J + 1) En (v, J ) ≈Vn (r0 ) + hν0 v + 2 Bν =

h¯ 2 , 2I

(10.2) (10.3)

with I = the moment of inertia of the molecule. For a diatomic molecule I = μr02 , with μ = m1 m2 /(m1 + m2 ) the reduced mass, and r0 the bond length. This also indicates that the energy levels of more massive molecules are lower than for lighter molecules. For rotational energy dipole transitions only J = ±1 is allowed giving EJ = 2Bv (J + 1). This means that for the J = 1 → J = 0 transition, emission is produced at a frequency ν(J = 1 → 0) = 2Bv / h, and that higher level transitions are integer values times 2Bv / h, with some deviations due to the fact that the distance r between the atoms in the molecule weakly depends on J as well (for higher J the distance increases). Molecular hydrogen (H2 ), has no dipole moment and only transitions with

J = ±2 are allowed, with hardly any interchange between species with parallel proton spins (ortho-H2), which have odd J , and anti-parallel spins (para-H2), which have even J . Note that this only applies to a symmetric molecule like H2 , not to a deuterated molecule like HD. Typical rotational energy transitions lead to line emission in the millimeter and submillimeter regime (e.g. CO J = 1 → 0, λ =2.6 mm), whereas vibrational transitions occur at much shorter wavelengths, typically a few μm (infrared). The vibrational spectrum is very rich as the transitions may involve both changes in

264

10 Middle-Aged and Old Supernova Remnants

vibration and in rotation. Since asymmetric molecules like CO are radiatively efficient, and CO couples well to H2 for densities n  100 cm−3 , molecular clouds can efficiently cool, and typically have temperatures below 10 K. At these low temperatures the vibrational transitions are not collisionally excited, and even higher J rotational transitions are rare. A special case of molecular transitions are the fine-structure lines, for which 18 cm transitions by the OH molecule are the most important example. In fact, OH was among the first molecules identified in the interstellar medium, and was first detected in absorption toward the supernova remnant Cas A [1209]. OH contains seven electrons, and, therefore, has one unpaired electron. Molecules with unpaired electrons are called chemical “radicals”, as they are highly reactive. The shells of molecules are designated with Greek letters instead of Latin letter, so instead of s, p, d,. . . shells for the l quantum numbers, the shells are indicated as , , , . . ., for the quantum number  (the equivalent to L for atoms). For the OH ground level we have, therefore, two split-states: 2 3/2 and 2 1/2 , with the superscript indicating the total number of spin states (2 + 1 = 2 for  = 1/2), and the subscript indicating the total electron angular moment Je =  − ,  + . Given that in the ground state the orbital angular momentum is Jo = 0, and that  = 1, the total angular momentum adds up either to 3/2 or 1/2 (plus additional integers for Jo = 1, 2, . . .). However, the coupling between the electron angular momentum and the non-zero spin of the hydrogen nucleus in OH, results in hyperfine splitting of the electron levels (similar to HI hyperfine splitting leading to the 21 cm emission). The hyperfine levels are indicated by the quantum number F consisting of the sum of the nuclear and electron angular momenta. The resulting energy levels of OH are shown in Fig. 10.3. The primary interest of these hyperfine levels is that the transitions of OH are easily observed with radio telescopes, and can be used to detect molecular gas either in emission or absorption (e.g. toward Cas A [163, 1209]). The hyperfine lines are sensitive to Zeeman-splitting, which has been used to infer magnetic fields in molecular clouds overrun by supernova remnants. Typical magnetic fields measured this way range from 0.2 to 4 mG [214, 271, 543]. Finally, OH radio masers are quite common, and OH maser emission in the 1720 MHz line is associated with shockcloud interactions. OH masers are discussed in more detail in Sect. 10.2.3.

10.2.2 The Interaction of Shocks with Molecular Clouds Molecules dissociate when they collide with velocities of order ∼25 km s−1 [797], which implies that for shock velocities in excess of ∼50 km s−1 molecules will be destroyed, once overtaken by shocks. However, it should be recalled that for Vs  200 km s−1 radiative cooling is important (Sects. 4.4 and 5.7). The cooling time scale is proportional to ∝ 1/n for optical/UV emission-line cooling. So the post-shock cooling time-scale for shocks passing through dense molecular clouds

10.2 Interaction with Molecular Clouds

265

OH (hydroxyl) E/k (K) 500

F J

-

3 2

5/2

400

3

+

2

+

2

F J

300

+

3 4

7/2

1

3

-

3/2

4

-

200

1 1 0

1/2

-

3

1

+

0

2

5/2

3

+

2

+

2 1

3/2

2 1 1665.4 MHz 1612.2 MHz

-

1720.5 MHz 1667.4 MHz

100

0

2

Fig. 10.3 Energy level diagram for the OH radical. The fine-structure level transitions in the radio (around λ = 18 cm) are indicated. The hyperfine level differences are exaggerated

can be very short (hundreds of years), and in the cooling regions chemical reactions will give rise to molecular reformation [854]. For interactions of supernova remnant shocks with molecular clouds one can distinguish three main regimes (e.g. [334]): (1) non-radiative shocks (Vs  200 km s−1 ) in young supernova remnants, which completely dissociate molecules; (2) radiative shocks with 50  Vs  200 km s−1 : molecules are dissociated, but can reform in the cooling zone behind the shock; and (3) Vs < 50 km s−1 , molecules are heated by the shock passage, which may be a J-type shock (Sect. 4.2), but are not destroyed. In cases (2) and (3) the molecular material is strongly heated, causing molecular line broadening (∼10–30 km s−1 ), as well as emission from higher energy rotational

266

10 Middle-Aged and Old Supernova Remnants

(J ) levels, and the excitation of vibration modes. In the submm/mm regime higher order rotational lines have been used as tracers of shock interactions. For example, the CO J = 2 − 1/J = 1 − 1 ratio has been used as a marker for shock-cloud interactions in the supernova remnants W44 and IC 443 [305, 1037]. The lack of enhancement of this ratio was recently used to suggest that Cas A was not interacting with a molecular cloud complex [1278]. Note that the shock velocity of Cas A is ∼5000 km s−1 , and molecular cloud interaction would lead to molecular dissociation. So only indirect heating through X-ray radiation or cosmic rays escaping the supernova remnant could have been expected. IC 443 has also been studied using higher levels CO transitions, as well as rotational spectra of many other molecules [306, 1159]. W44, alongside, W28, was also studied using narrow-band imaging of the 2.12 μm vibration/rotation (v = 1 → 0, J = 3 → 1) transition from H2 , in conjunction with mm/submm observations [951]. Figure 9.9 shows heated H2 vibrational emission from the supernova remnant W49B. Chemical reactions in molecular clouds are driven by fast ion-molecule reactions, where the ions are created through photo-ionisations and interactions with lowenergy cosmic rays ( ni gives rise to an increased photon production rate compared to resonant photon absorption. Hence, the radiation field for a given photon energy is amplified, and one has a maser or laser. The necessary level inversion has first to be accomplished by a process referred to as “pumping”. For astrophysical masers the pumping is either caused by infrared or UV radiation fields (so-called Class II masers), exciting the atoms/molecules to higher energy levels and followed by radiative cascade resulting in a high occupation of level j , or by collisional excitation followed by radiative decay (class I masers). Maser emission from several molecules have been identified in astrophysical settings: OH, H2 O, SiO, CH3 OH and NH3 . However, only OH masers and, more recently, CH3 OH masers have so far been found in association with supernova remnants. OH masers are quite common in a variety of environments, from shells of circumstellar material around AGB stars to HII regions. However, most OH masers are class II masers, and are bright in the 1612 MHz line (Fig. 10.3). A unique characteristic of OH masers in supernova remnants is that they give rise to bright 1720 MHz emission. The reason for this peculiarity is that in most astrophysical masers the higher level is populated due far-infrared pumping. This excites OH to the J = 1/2 state, which subsequently decays to the J = 3/2 case occupying the F = 1 levels, as the allowed transitions rules are J = ±1, F = 0, ±1. See the level diagram in Fig. 10.3. In supernova remnants, however, OH is collisionally excited (class I maser). The collisions are mostly with H2 molecules. For temperatures 60 K, n(H2 ) = 104–106 cm−3 [795]. This density range overlaps with OH masers, but methanol masers can be present at higher densities.

10.3 Mixed-Morphology Supernova Remnants Among the mature supernova remnants, one class of objects sticks out: the mixed-morphology (or thermal-composite) supernova remnants (Sect. 3.1). They

10.3 Mixed-Morphology Supernova Remnants

269

are characterised by centrally dominated, thermal X-ray emission, whereas their radio morphology is shell-like. Apart from the curious radio/X-ray morphology they have a number of other characteristics in common [280, 707, 971, 1173]. They tend to be older supernova remnants (10,000 yr) and are associated with the denser parts of the interstellar medium. Many of them are associated with OH masers (Sect. 10.2.3). And many of them are GeV, and even TeV γ -ray sources (Table 10.1), which likely arises from cosmic-ray nuclei (hadronic cosmic rays) interacting with dense gas in the supernova remnant shell, or from escaping cosmic rays interacting with unshocked parts of molecular clouds (see Chap. 11 and Sect. 12.3). The morphology of these supernova remnants is difficult to explain with standard supernova remnant evolution models (Chap. 5), but there have been several models to explain both the centrally enhanced X-ray emission, which contrasts with their shell-type radio appearances (Fig. 3.1d). The model of White and Long [1224] assumes that the central thermal emission is caused by dense “cloudlets” that have survived the passage of the forward shock, but are slowly evaporating inside the hot interior medium due to saturated thermal conduction (Sect. 4.3.6). Recall that according to Sedov-Taylor self-similar model the interior density of a supernova remnant is low, but the plasma temperature is very high (Fig. 5.2). They found a self-similar analytical model for the evolution of thermal-composite supernova remnants, which, in addition to the Sedov-Taylor solutions (Sect. 5.4), has two additional parameters: C, the ratio between the mass in the clouds and the mass in the intercloud medium, and τ , the ratio between the evaporation time of the clouds and the supernova remnant age. An alternative model proposed by Cox et al. [280] concentrates on the contrast between the radio and X-ray morphology. Their model was specifically developed to explain the characteristics of W44. According to their model the supernova remnant evolve inside a relatively high density of 6 cm−3 . The forward shock has decelerated to velocities below ∼ 200 km s−1 , resulting in a strongly cooling shell; too cool to emit X-ray emission. The density in the remnant’s interior is in this model relatively high due thermal conduction from an early stage of the evolution, leading to a different pressure/density structure than expected on basis of the SedovTaylor model. Because the shocks are radiative, the shock compression factors are large (Sect. 4.4), giving rise to strongly compressed magnetic fields, and compressed cosmic-ray electron densities. As a result the radio synchrotron emission from the shells is strongly enhanced. This explanation for bright synchrotron emission from supernova remnants shells is called the Van der Laan mechanism (see Sect. 12.1). It has been proposed that the radio emission may be additionally enhanced by the presence of secondary electrons/positrons [1143], i.e. the electrons/positrons products left over from the decay of charged pions, created due to cosmic ray nuclei colliding with the background plasma. The presence of secondary electrons/positrons may also explain the flat spectral radio indices αR in Table 10.1. An alternative model for the relatively flat radio spectra of many mixed-morphology remnants is the high shock compression [91, 871]. Yet another possibility for the flatter spectra is additional electron acceleration by magnetohydrodynamic turbulence (second order Fermi acceleration, Sect. 11.2) behind high-density shocks [225].

MSH 17-39

MSH 11-61A Kes 27

CTB 104A CTB 1 HB 3 HB 9 VRO 42.05.01 IC 443

80 34 80 140 × 120 55 × 35 45 15? 19 × 14 21 45 8×3 24

4 30 33 × 28 310 × 240 95 × 65 120 × 90

Size (arcmin) 3.5 × 2.5 48 7×5 10 35 × 27 4.5 × 2.5

65 8 45 110 7 165 0.4 42 30? 26 37 14

38 160? 8 42 120? 220

100? 310 24 20 240 22

0.65 0.61 0.6 0.64 0.37 0.36 0.6 0.4 0.5 0.4 0.4 0.4?

0.48 0.3 0.75 0.6? 0.35 0.48

2.2 8

1.5

0.8

10 6 2.8 0.8

Dist. (kpc) 0.8? 8 0.3 1.9 0.49 8.5 0.5 7.8 0.37 0.48 7.5

Sν @ 1 GHz (Jy) α

[868]

X

[187] [187, 1125] X [606] [107, 607] [249] [614]

X X X X X X X

[707] [707]

[786] [707]

X

X

X X

X X

[868]

[107, 607]

X X

? ?

X X ? [634, 1259] X

[634] [489]

[18, 523] [15, 589] [34] [34] [34] [72, 921, 953] [34] [34] [34] [34, 83] [34, 82] [37] [34] [34, 107] [34, 1252] [34] [34, 242] [34, 563]

GeV source? Ref. ? [1016] X [34] [1014] X [34, 241] [1013] [106] [634, 1141] X [19, 37] [34]

RRC/recomb.? Ref.

X X

X X

Enriched? Ref. X [1001] [1016] X [1014] ? [1013] X [1044] X [1000, 1254] X [569, 1279] X [489, 1010] X [210]

?

X X

?

X

X

OH Maser? X X X

[55]

X X

[27, 61] X

[523] [62]

TeV source? Ref. ? X [54]

[398, 1269] [1266]

[271]

[669]

[461] [210]

Ref. [1268] [271] [398] [669] [271]

Note: Size, radio flux and spectral index (α ) were taken from the 2017 version of the Green catalogue (http://www.mrao.cam.ac.uk/surveys/snrs). No entry in a column, but with a reference indicates a negative result. No reference indicates a lack of literature on the subject. This table is based on similar tables in [707] and [1173]. Based on newer literature G156.2+5.7 was removed [1139], whereas G350.0-2.0 [614] was added to the list of mixed-morphology supernova remnants

G93.7-0.2 G116.9+0.2 G132.7+1.3 G160.9+2.6 G166.0+4.3 G189.1+3.0 G272.2-3.2 G290.1-0.8 G327.4+0.4 G350.0-2.0 G357.1-0.1 G359.1-0.5

W49B W51C 3C400.2

G43.3-0.2 G49.2-0.7 G53.6-2.2 G65.3+5.7 G82.2+5.0 G89.0+4.7

W63 HB 21

Name Sgr A East W28 3C391 Kes 79 W44 3C397

Object G0.0+0.0 G6.4-0.1 G31.9+0.0 G33.6+0.1 G34.7-0.4 G41.1-0.3

Table 10.1 Mixed-morphology supernova remnants

270 10 Middle-Aged and Old Supernova Remnants

10.3 Mixed-Morphology Supernova Remnants

271

It was long thought that the X-ray emission from the interior of mixedmorphology remnants revealed a homogeneous temperature due to thermal conduction [284] and turbulent mixing [1044]. Numerical hydrodynamical models based on the model of Cox et al. [280] indeed show centrally enhanced X-ray morphologies [1043, 1164]. However, more recent observations indicate that the temperatures may not be as homogeneous as previously thought [210]. Although the nature of mixed-morphology supernova remnants is still debated, there has been considerable progress in our knowledge of them since their identification as a special supernova remnant class. For example, supernova remnant G65.2+5.7, which is clearly in the radiative evolutionary stage, shows an X-ray filled morphology. Due to the low absorption column toward this supernova remnant, ROSAT was also able to reveal a shell-like outer part in X-ray emission below 0.3 keV [1045], showing that the X-ray shell is indeed there, but would be invisible if it were not for the low absorption. This in qualitative agreement with the Cox et al. model. Due to progress in X-ray imaging spectroscopy, several other clues about the nature of mixed-morphology supernova remnants have appeared. Many of the mixed-morphology supernova remnants show evidence for metal-rich plasmas in the interior: Lazendic and Slane [706] list 10 out of a sample of 23 thermal-composite supernova remnants with enhanced abundances (Table 10.1 shows an update). Examples are W44 [1044], HB 21, CTB 1 and HB 3 [706, 883], IC 433 [1125] and Kes 27 [249]. Kawasaki et al. [634] noted another feature that may be generic for this class of supernova remnants: their ionisation state is close to ionisation equilibrium (Sect. 13.5.7), unlike young and most old supernova remnants like the Cygnus Loop. Moreover, the thermal-composite supernova remnant IC 443 [635, 1259], W49B [877], and G359.1-0.5 [868] show signs of overionisation, in the form of radiative-recombination continua (RRCs, Sect. 13.5.5) associated with metals like Si, S, and Fe (Fig. 10.5). For W449B the RRC is even responsible for most of the X-ray continuum emission [1260]. It is not clear whether overabundances are a generic feature of mixedmorphology supernova remnants, but it seems at least to be a common feature. There are several ways in which metal-richness helps to explain the characteristics of thermal-composite supernova remnants. Firstly, the uniform temperatures in their interiors are often attributed to thermal conduction [280]. A problem for this explanation is that magnetic fields limit thermal conduction across field lines (Sect. 4.3.6). However, if the interiors consist predominantly of supernova ejecta, then the magnetic fields may be very low, simply because the stellar magnetic field has been diluted by the expansion. Magnetic flux conservation gives B = B∗ (R∗ /R)2 , with R∗ the stellar radius, and R the radius of the supernova remnant interior. Taking typical values of R∗ ≈ 1013 cm, R ≈ 1020 cm, B∗ = 1 G, shows that the interior magnetic field may be as low as 10−14 G, approaching the value where conduction across field lines becomes comparable to conduction along field lines. Hence, the conditions are optimal for thermal conduction, but only up to the boundary between ejecta dominated and swept-up matter, where Rayleigh-Taylor instabilities at the contact discontinuity may have

10 Middle-Aged and Old Supernova Remnants

Counts s-1 keV-1

272

0.1

0.01

χ

4 0 -4 1

2

4

Energy (keV)

Fig. 10.5 Suzaku SIS spectrum of G359.1-0.5 [868], showing in green the Si XIV and S XVI Lyα lines and in blue the radiative-recombination continua (RRC)

enhanced the magnetic-field strengths and may have made the magnetic fields more tangled, preventing strong thermal conduction across the contact discontinuity. A complication, however, may be magnetic-field amplification though cosmicray streaming (Sect. 11.4.3). There is some evidence that this may also affect the magnetic fields around the reverse shock (Sect. 12.2.4). Secondly, metal-rich plasma emits more X-ray emission, further increasing the contrast in X-ray emission between the metal-rich interior and the cool shell of swept-up matter. An important additional aspect to consider is, whether the interiors consist of enhanced metal abundances, i.e. a mix of ejecta and swept-up matter, or of pure ejecta. For supernova remnant W49B, a rather young member of the mixedmorphology class, a pure metal abundance seems likely, but for older supernova remnants the plasma is probably best described by hydrogen/helium dominated plasma with enhanced abundances. One can illustrate the key features of the coolshell/hot-interior scenario for mixed-morphology supernova remnants with some simple numerical calculations, ignoring for the sake of argument a more rigorous analytical treatment of the supernova remnant evolution [280]. Most mixed-morphology supernova remnants appear to be in the snowplough phase of their evolution (phase iii), with Vs < 200 km s−1 , which results in cool, X-ray dim, but optical/UV-bright regions immediately behind the shock front. The interior, presumably consisting of supernova ejecta, is more or less homogeneous in temperature and, because of pressure equilibrium also reasonably homogeneous in density. Given the age of the supernova remnant and the relatively high sound speed in the interior, one can assume approximate equilibrium between the ram pressure

10.3 Mixed-Morphology Supernova Remnants

273

at the forward shock and the interior [284]. This gives: nint kTint ≈ ρ0 Vs2 ,

(10.4)

with nint the interior plasma density. If we use the values for W44 listed by Cox et al. [280], Vs = 150 km s−1 , kTint = 0.6 keV and ρ0 = 1.0×10−23 g cm−3 , we find a density of nint ≈ 2.3 cm−3 . This translates into an electron density of ne ≈ 0.8 cm−3 , in rough agreement with X-ray measurements [280]. W44 has a radius of R ≈ 10 pc, for an estimated distance of 2.5 kpc, but the X-ray emitting interior has a radius of Rint ≈ 6 pc, corresponding to an interior volume of Vint ≈ 2.7 × 1058 cm−3 . This means that the total internal energy is U = nint kT Vint ≈ 0.9×1050 erg (about 10% of the explosion energy). The X-ray emitting mass is Minterior ≈ 28 M ; somewhat higher than the expected ejecta mass of a Type II supernova, but not too far off, given the crudeness of the approximation. At least it suggests that a substantial fraction of the interior plasma consists of ejecta, and that the interior magnetic-field strength may be low due to large expansion of the ejecta gas. In case, it does contain shocked ambient medium, the magnetic-field strength may be low due to the subsequent expansion, if the interior consists of gas shocked in the early phases of the evolution. In this context, a stellar-wind density profile of the ambient medium in this early phase would help to explain that the swept-up mass was relatively high in the earlier phases, and would also have kept the expansion velocity higher for a longer time (Sect. 5.4.3). Note that if we assume that the interior consists of metals only, this does not lead to a consistent result. For example, if we assume O VII to be the dominant ion, then we have nOVII = n/8 ≈ 0.29 cm−3 and Minterior ≈ 100 M . This is clearly much more oxygen than a massive star can produce. So pure metal abundances in general for mixed-morphology supernova remnants seem unlikely, although perhaps there are some regions in the interior that have unmixed ejecta. For a discussion on the metal abundances in W44, see [1044]. A supernova remnant will reach the snowplough phase when its age is larger than trad ≈ 44,6000(E51/nH )1/3 yr. For W44 (nH = 6 cm−3 ), this corresponds to ∼25,000 yr. Hence, an ionisation age of 4 × 1011 cm−3 s is expected (Sect. 13.5.7). This is relatively high, but below the ionisation age for ionisation equilibrium (1012 cm−3 s). However, the situation is more complex, as the interior density and temperature may have been higher in the past, and, therefore, the degree of ionisation may be higher than indicated by the simple multiplication of the present day age and electron density. This could explain the discovery that most mixed-morphology supernova remnants are near ionisation equilibrium Kawasaki et al. [634], or even overionised. The high density in the interior of W44 and other mixed-morphology supernova remnants is a direct consequence of the high interstellar medium density (10.4) and, perhaps, thermal conduction, which results in a more uniform interior density with medium hot temperatures, rather than very high temperatures with very low interior densities.

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10 Middle-Aged and Old Supernova Remnants

In the context of mixed-morphology supernova remnants, it is of interest to show why some old supernova remnants are expected to have RRCs. One may expect RRCs to occur, whenever the plasma cools faster than the recombination time [868], for example due to thermal conduction [635], or adiabatic expansion [583, 1259, 1274]. However, the time scale for cooling by conduction is probably too long [1259]. So adiabatic expansion is probably the appropriate explanation. Observational evidence that adiabatic expansion is responsible for overionisation was obtained for W49B, where overionisation was shown to be correlated with the coolest plasma [1260]. The time scale for adiabatic expansion can be calculated from the thermodynamic relation T V γ −1 = constant, which means that the adiabatic cooling time scale is given by [211]: τad = −

T  T˙

ad

=−

R 1 1 V 1R 1 =− = = m−1 t, 1 − γ V˙ 3(1 − γ ) R˙ 2 R˙ 2

(10.5)

with m the expansion parameter (5.1), γ = 5/3, and t the age of the supernova remnant. For supernova remnants in the snowplough phase we can use m = 0.25 (5.59). This suggests that τad ≈ t/8. For overionisation to occur, it is first required that the ionisation age of the plasma must be large enough (otherwise there would be underionisation). The second condition is that the timescale for recombination is longer than the adiabatic cooling timescale: τad < τrec =

1 , αrec ne

(10.6)

with αrec the recombination rate. Many mixed-morphology supernova remnants show evidence for overionisation of Si XIII/XIV, for temperatures around kT = 0.2–0.7 keV. For these temperatures the recombination rate for Si XIV to Si XIII is approximately αrec = 5.9 × 10−14 cm3 s−1 [1049]. Using now that ne = 0.5nint together with (10.4), the condition for overionisation is t
1000 cm−3 [399]. But in fact, millimeter-wave and near infrared observations of W44 and W28 [951] reveal the presence of dense molecular gas. However, this dense gas has only a small filling factor. Some of the dense clumps may survive the shock passage [951], which brings us back to the scenario proposed by White and Long [1224]. The interclump densities, which applies to 90% of the volume are, however, consistent with the estimates of Cox et al. [280], nH ≈ 5 cm−3 . These infrared observations show that old supernova remnants have a complex structure, which, for a proper understanding, requires multi-wavelength observations. In addition, there is the prevalent view that mixed-morphology are core-collapse supernova remnants interacting with the molecular cloud complexes their progenitors were born from. Recently, however, this view has been challenged, first by evidence that the mixed-morphology supernova remnants W49B [1279] and 3C397 [1254] are most likely thermonuclear supernova remnants, as discussed in Sect. 9.2. Both supernova remnants are relatively young and hot compared to other mixed-morphology supernova remnants, and they are not uniformly recognised as belonging to this class. They are more metal-rich than other supernova remnants in this class. However, it is clear that from both a morphological point of view as well as from the point of view of their dense, overionised interior plasma, they do belong to this class. A question then is whether their dense environments are a mere coincidence, or whether some Type Ia supernovae have delay times short enough that the supernovae occur still in the cloud complexes their progenitors were born in. Although it is clear that most mixed-morphology supernova remnants are interacting with dense, molecular clouds, there are exceptions to this general characteristic. For example, it was recently shown that the peculiarly shaped mixedmorphology supernova remnant VRO 42.05.01 is not interacting with a dense molecular medium [89]. Also its γ -ray emission is probably not the result of pion decay, but caused by inverse Compton emission [82] (see Sect. 12.3). These exceptions show that mixed-morphology supernova remnants may share many properties, but these properties may not necessarily be the result of a uniform evolutionary path.

Chapter 11

Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

11.1 Introduction: Galactic Cosmic Rays In 1934, in the same year that Baade and Zwicky [109] first established that there exist a class of very bright novae, which they coined super-novae, they also suggested that these super-novae may be the source of cosmic rays [110]. Supernova explosions are still generally considered to provide the energy for the bulk of cosmic-ray up to energies of ∼1015 eV or even higher. However, we now think that the cosmic-ray acceleration itself, although powered by the supernova explosion, happens over a longer time, by supernova remnant shocks. The purpose of this chapter and the next is to provide an overview of cosmic rays in general, and the theory and observations of cosmic-ray acceleration by supernova remnants. Cosmic rays were discovered by Victor Hess, who showed, using measurements made during seven balloon flights, that ionising radiation increases with height above the ground, which suggested an extraterrestrial origin for this radiation [518, 519].The name “cosmic rays” was first used by Millikan [821] and stuck, despite the fact that by 1934 it had become clear that the penetrating rays were in fact highly energetic charged particles, as the intensity of the radiation correlated with the strength of the local Earth magnetic field [272, 274]. Baade and Zwicky’s idea that supernovae are the sources of the cosmic rays bombarding the Earth’ atmosphere got a boost with the discovery that supernova remnants were emitting radio synchrotron radiation [440, 1047]. This happened around the time that radio astronomy began to establish itself as a new and important branch of astronomy, capable of measuring the distribution of neutral gas in the Milky Way, but also mapping diffuse synchrotron radiation [65]. Since synchrotron emission requires the presence of relativistic electrons, this also meant that at least electron cosmic rays could now be be studied in the Milky Way and in their sources of origin, rather than having to rely on those cosmic-ray particles that reached Earth. Soon supernova remnants were known sources of synchrotron radiation, and the presence of relativistic electrons in their shells, also suggested © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_11

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that that relativistic protons and other accelerated atomic nuclei are also present. These results also pointed toward supernova remnants, rather than supernovae, as locations of particle acceleration, albeit with the energy having been provided by the supernova explosion. Since the 1950s there has been a lot of progress in understanding particle acceleration in supernova remnants. This progress has been caused by the tremendous advances in multiwavelength, observational capabilities, which enables us to observe supernova remnants using almost the entire electromagnetic spectrum, from 108 Hz to 2 × 1028 Hz, the latter corresponding to γ -ray photons with energies of ∼ 100 TeV. These energies require the presence of particles with about ten times more energy. In addition, our theoretical understanding of particle acceleration by supernova remnant shocks has greatly advanced. All this does not mean that we are certain that most cosmic rays bombarding Earth are originating from supernova remnants. As will be explained in this chapter, there are two main requirements for supernova remnants to be the primary sources of Galactic cosmic rays: 1. supernova remnants have to be able to convert 5–20% of the explosion energy to cosmic-ray energy (i.e. about 1050 erg per supernova remnant), and 2. supernova remnants have to be capable of accelerating protons to energies of at least 3 × 1015 eV (3 PeV). Accelerators that accelerate particles beyond 1 PeV are sometimes called PeVatrons. So the second requirement is sometimes rephrased as “Are supernova remnants cosmic PeVatrons?”. In this chapter and the next we will explain where these two requirements come from and what theoretical considerations and observational data tell us about whether supernova remnants can indeed be the primary sources of Galactic cosmic rays.

11.1.1 The Cosmic-Ray Spectrum The measured cosmic rays spectrum spans eleven orders of magnitude, from roughly 109 –1020 eV, see Fig. 11.1. For energies around and below 1 GeV the spectrum as observed on Earth is affected by the solar wind, and is, in fact, modulated by the variation in the solar wind properties, which varies during the 22 year cycle of solar activity. From 1010 to 1020 eV the spectrum has a near power-law distribution dN(E) ∝ E −q dE,

(11.1)

but the logarithmic scales hides some important features: between ∼1010 − 3 × 1015 eV the spectrum has a power-law index of q = 2.7, which steepens around 3 × 1015 eV to q ≈ 3.1. This steepening is usually referred to as the “knee” of

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Fig. 11.1 The cosmic-ray flux spectrum as measured by various experiments, based on the compilation of [785], and [3, 73, 77, 80]. The flux points below ∼ 1014 eV are based on proton cosmic rays only, and have been multiplied by a factor 3, in order to match the all-species cosmicray spectra at higher energies. Left: The spectrum in flux units, showing that the spectrum is nearly a power law from 1010 eV to 1019 eV. Right: The spectrum multiplied by E 2.7 , which brings out features like the “knee” and the “Ankle”

the cosmic-ray spectrum. Around 5 × 1018 eV the spectrum flattens again to q ≈ 2.7, a feature that is referred as the cosmic-ray “ankle”. And finally the cosmic-ray spectrum shows a decline around 5 ×1019 eV [4]. Since the spectrum is steeper than q = 2, the energy density of cosmic rays is determined by the low energy part of the spectrum, which cannot be easily measured due to solar-wind modulation. For a long time, however, estimates of the cosmic-ray energy density suggest Ucr ≈ 1 eV cm−3 , which has been recently confirmed by Voyager 1 data [285]. It is often assumed that the “ankle” represents the change from cosmic rays of Galactic origin, to those of extragalactic origin. This idea is based on the simple observation that the gyro-radius of a 1018 eV proton is rg ≈ 216(B/5 μG)−1 pc, which is comparable to the Galactic scale height. In other words, the Milky Way cannot contain particles which energies in excess of 1018 eV, whereas extragalactic cosmic rays with these energies can easily enter the Galaxy. This does ignore, however, some of the complexities of cosmic-ray transport, and it has been argued that the transition between Galactic and extragalactic cosmic rays occurs around a weaker feature in the cosmic-ray spectrum at ∼ 5 × 1017 eV, dubbed the “second knee” [67, 155]. The final cut-off around 5 × 1019 eV has been attributed to the Greisen-ZatsepinKuz’min (GZK) effect [468, 1270], which is caused by the interaction of cosmic-ray protons with cosmic-microwave-background photons. Such an interaction results in the creation of Delta baryon, which decays into proton, or neutron and a pion. The

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cross section for this process is such that protons with energies  5 × 1019 eV will, on average, travel not further than ∼ 100 Mpc. However, more recent composition measurements by the Pierre Auger Observatory suggest that the proton-fraction slowly decreases above the ankle, and that near the cut-off around 5 × 1019 eV the cosmic-ray particles are mostly C, N, O and Fe nuclei [1, 2, 4] . This suggests the cosmic-ray sources may not be able to accelerate protons to energies much more than 5 × 1019 eV [609]. The nature of the sources of the highest energy cosmic rays is still much debated. Most theories focus on active galaxies and starburst galaxies, but also gamma-ray bursts and large scale cluster formation shocks are considered [534, 673]. Since this a book about supernova remnant, we focus here on the origin of cosmic rays of Galactic origin, that is cosmic rays with energies up to the “knee” (∼ 3 × 1015 eV) for protons, or perhaps up to 1017 − 1018 eV for heavier atomic nuclei. As will be explained in this and the next chapter, supernova remnants are the most viable, but still problematic, candidate sources for these Galactic cosmic rays. It is important to strike a cautionary note here: cosmic-ray spectral measurements above energies of 100 GeV show indications that the cosmic-ray spectrum may not be as featureless below the “knee” as previously thought. As summarised in [727], there is evidence that the proton cosmic-ray spectrum hardens from q ≈ 2.8 to q ≈ 2.6 at a few hundred GeV, and softens again around 10 TeV to q = 2.85. However, there are some discrepancies between the various cosmic-ray experiments preventing a statistically unambiguous identification of these features. It nevertheless, provides some preliminary indications that the origin of Galactic cosmic rays may be more diverse than often assumed.

11.1.2 Cosmic-Ray Composition The most detailed knowledge about the cosmic-ray composition concerns cosmic rays with energies below ∼1013 eV, for which primary cosmic ray properties are measured using balloon and satellite experiments. For higher energies the composition of cosmic rays is more difficult to measure, as they are primarily detected and characterised by measuring the properties of the extensive air-showers on Earth. The extensive air showers are caused by a cascade of secondary particles, created by the collisions of the primary cosmic-ray particles with atoms in the Earth’ atmosphere. The Cosmic-Ray Composition Between 109 –1013 eV Balloon and satellite experiments show that cosmic rays consist mostly of protons and other atomic nuclei, which are usually collectively referred to as hadronic cosmic rays. For cosmic-ray energies around 10 GeV, 0.55% of the particles are

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electrons, a fraction that decreases with increasing energy. Cosmic-ray positrons constitute an even smaller fraction of ∼0.03% [25]. Together the electron/positron components of the cosmic rays are labeled leptonic cosmic rays, which formally includes the short-lived energetic muons that are a by-product of hadronic cosmic-ray interactions with matter. Positrons are, among others, a by-product of interactions of cosmic rays with interstellar medium atoms (Sect. 13.6). A puzzling increase in the positron-to-electron ratio above 10 GeV [25, 42] has been associated with the self-annihilation of dark matter particles [157, 265]. However, positrons from nearby pulsar wind nebulae offer a less exotic explanation for the rise in the positron to electron ratio [547], given that pulsar wind nebulae contain probably equal amounts of electrons and positrons (Chap. 6). The detailed composition of cosmic rays is important for understanding both the origin of cosmic rays, as well as the propagation properties in the Galaxy. Comparing the cosmic-ray elemental abundances to solar abundances, three properties stand out (Fig. 11.2): (1) H and He are relatively underabundant, which is not well understood, (2) odd-number nuclei are relatively more abundant in cosmic rays, (3) there is an overabundance of elements lighter than iron as compared to solar abundances. Nucleosynthesis in stellar cores and during supernova explosions essentially use 4 He as fundamental building blocks, leading to an overabundance of elements with an even number. Moreover, the even numbered nuclei are more tightly bound. The reduction of this odd-even effect in cosmic-ray abundances is caused by the spallation of primary cosmic-ray nuclei as they collide with atomic nuclei in the interstellar medium. Spallation also leads to the production of radio-active elements and positrons and anti-protons. Radio-active elements such as 10 Be, 26 Al, 36 Cl, and 54 Mn, and the presence of very rare elements such as boron—the spallation

Fig. 11.2 Cosmic-ray abundances (red) for elements up to Z = 28 (nickel) in the energy interval 600–1000 MeV/nucleon [1051], compared to solar composition abundances [731]. The abundances have been normalised to [Si] ≡ 100

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product of carbon and oxygen—can be used to put constraints on the average time a cosmic-ray particle spends in the interstellar medium and the diffusion coefficient of cosmic rays (see Sect. 11.1.3). The overabundance of elements just lighter than iron is caused by the spallation of Fe-group elements. The Enhanced 22 Ne Abundance The cosmic-ray composition offers few clues about the environments from which they originate, and which could potentially provide evidence for a supernova remnant origin of the bulk of cosmic rays. This may not be so surprising given that supernova remnants are expected to mostly accelerate particles at the forward shock, which are probably, on average, only mildly enriched with fresh nucleosynthesis products, unlike the particles encountered by the reverse shock . As will be discussed in Sect. 12.2.4, there is evidence for acceleration at the reverse shock. But the total amount of mass that passes through the reverse shock equals the ejecta mass (several solar masses), whereas the forward shock can easily process hundreds of solar masses. A question is then whether during the whole supernova remnant evolution cosmic rays are being accelerated, or whether there is a higher cosmicray acceleration efficiency in the early supernova remnant stages. In the latter case, the ratio of cosmic rays accelerated at the forward shock and at reverse shock may be more comparable. With this in mind, it important to note that there are two cosmic-ray abundance anomalies that may be linked to the typical environments from which cosmic rays are accelerated. The first clue is the isotopic ratio of 22 Ne/20 Ne, which is a factor 5 higher than the solar-system ratio [168]. The isotope 22 Ne is a neutron-rich isotope, whose synthesis depends on the presence of the neutron-rich isotope 14 N. This could mean that the cosmic rays originate from preferentially metal-rich regions, for which the initial 14 N is high,or that the environments were enriched by Wolf-Rayet stars, which have elevated 22 Ne/20 Ne ratios due to the presence of processed 14 N. It could be that the 22 Ne in cosmic rays originated directly from Wolf-Rayet star supernova remnants. But this is not very likely, since Wolf-Rayet stars are relatively rare. It is more likely that cosmic rays originate from environments pre-enriched by WolfRayet star winds. This would be the case in superbubbles, where the medium is enriched and rarified first by the most massive stars and their supernovae, before the bulk of supernovae form less massive stars occur. The enhanced abundance of 22 Ne in cosmic rays is, therefore, considered one of the lines of evidence for the importance of superbubbles for the overall Galactic cosmic-ray properties [224, 885].

The Overabundance of Refractory Elements Another compositional property of cosmic rays that offers a potential clue about the origin of cosmic rays, but also about the acceleration mechanism, is the

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overabundance of refractory elements [809]. These elements have a relatively high condensation temperature (> 1250 K, see Sect. 7.1) and, as a result, they are found in interstellar dust grains. It should be noted that refractory elements like Mg, Si, Ca, also have a low first ionisation potential (FIP), which means that they are more likely to be ionised than volatile elements like H, He, N, Ne. As a result, there may be a tendency for high FIP elements to remain neutral and penetrate the shock more deeply, preventing them from being injected into the acceleration process. However, a careful look at intermediate volatile elements and their FIPs show that the overabundance of refractory elements in cosmic rays is not the result of the low FIP of these elements, but most likely connected to the fact that these are elements that are often dust depleted [809]. This implies that the acceleration process more easily accelerates particles in dust grains [368], and also that the acceleration environments contains dust. As for the acceleration process: for the diffusive shock acceleration mechanism discussed later in this chapter, the injection into the process favours particles with large gyroradii, as it makes it more easily for the particle to return back to the unshocked medium. The gyroradius of a particle depends on its mass m and charge eZ as rg =

p⊥ c , |eZ|B

(11.2)

with p = mv the momentum of the particle, with v being initially of the order of the shock velocity. This shows that particles are more likely to return to the shock if their m/|eZ| ratio is large. Indeed, among the volatile (non-depleted) elements, more massive elements are more overabundant [809]. This makes sense if elements in the post-shock region are once ionised, leading to a mass dependent m/|eZ| ratio. Dust grains can also be ionised, if one or more of the constituent atoms are ionised. But generally the bulk of the atoms in dust grains are not ionised, so dust grains will have a much larger m/|eZ| ratio than single, once or twice ionised atoms. Once the dust grains are injected into the acceleration process the dust will be slowly sputtered, thus injecting pre-energised atoms in the acceleration process. The dusty environments needed to explain the overabundance of refractory elements were explained in [365] to be the environments through which the forward shock of the supernova remnants move. However, as we have seen in Chap. 7 dust is also associated with the ejecta of core-collapse SNe. Now in general, the amount of material processed by the forward shock is much larger than by the reverse shock. For the reverse shock M = Mej  10 M , whereas the forward shock sweeps up  1500 M of gas before entering the radiative phase of the supernova remnant evolution (Sect. 5.7). So the ratio of swept-up over supernova ejecta mass is  150, which is at face value too small for the reverse shock accelerated material to make a major impact on the cosmic-ray composition. However, note that this assumes that supernova remnants accelerate cosmic rays up to the radiative phase, and that the supernova remnant evolution follows the canonical model, which is strictly valid only for a uniform density medium. The presence of wind-blown

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cavities may alter the history of the shock velocity, and little is known of the shockacceleration efficiency as a function of shock velocity. Given the fact that the few known cases of infrared bright supernova remnants all have dust grains embedded in the shocked ejecta, rather than swept-up circumstellar dust (Chap. 7), could perhaps be a sign that cosmic-ray acceleration by the reverse shock is not as sub-dominant as sometimes assumed.

The Composition Around the Cosmic-Ray “Knee” Cosmic-rays with energies in excess of 1014 eV are primarily detected though the extensive air showers they produce. This makes that the composition of very high energy cosmic rays is more difficult to measure, as one uses secondary mass indicators, such measuring the height at which the particle cascade develops, and the composition of the secondary particles that are produced—for example the ratio of muons over electrons. This method does not provide a direct measurement of the mass of the primary cosmic ray, but rather indicates whether a cosmic ray shower is more likely to be a proton, an intermediate mass element nucleus, or a heavy particle, such as iron. Using this method, the data from the Karlsruhe Shower Core and Array Detector (KASCADE) indicate that the cosmic-ray composition is changing around the “knee” from being proton-rich up to ∼ 3 × 1015 eV, to a composition that is increasingly more dominated by massive particles [77]. The change in composition is consistent with the idea that the “knee” at 3 × 1015 eV is caused by a rigidity effect. The rigidity of particle is R ≡ pc/eZ and is defined such that the gyroradius (11.2) of a particle is given by rg = R/(Bc) (see also the previous discussion on dust acceleration). It is likely that the maximum energy that particles can be accelerated through depends on the gyroradius of the particles, and hence should be dependent on the rigidity of the cosmic-ray particles. The change in composition at the knee, therefore suggests that in the source protons can be accelerated up to energies of the knee, which implies that the most abundant heavy element, iron (Z = 26), should result in a break in the cosmic-ray spectrum of E = Z × 3 × 1015 eV = 7.8 × 1016 eV. Indeed, an extended version of KASCADE, KASCADE-Grande, measured the presence of a “knee-like” structured in the heavy element cosmic-ray spectrum around 8 × 1016 eV [79], which seems to be accompanied by a transition again to lighter elements [81].

11.1.3 Cosmic-Ray Transport in the Galaxy Being charged particles, cosmic rays move along magnetic-field lines. Since the magnetic field in the Galaxy is not fully structured, and has irregularities, the cosmic-rays do not follow ordered paths, but instead randomly wander around, diffusing away from their acceleration sites. The diffusion coefficient, D(p), depends on momentum, with particles being most sensitive to those magnetic-field

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irregularities that have length scales comparable to their gyroradius (11.2); see Sect. 11.4 for more details. In the most extreme case, when δB/B  1 on length scales of the gyroradius, the effective mean free path, λmfp , of the particles can be as small as the gyroradius. This situation is referred to as Bohm diffusion. More generally the diffusion coefficient is often parametrised using the gyroradius: D=

1 1 λmfp v = ηrg βc, 3 3

(11.3)

with η the parametrisation factor, η = 1 indicating Bohm diffusion. Since the gyroradius scales with the rigidity, another parametrisation for D that is often employed is  D(R) = D0

R R0

δ (11.4)

,

with δ typically found to be 0.3  δ  0.7 [1082]. Note that a constant (rigidity/energy independent) η corresponds to δ = 1, whereas δ = 1/3 corresponds to the Kolmogorov spectrum of magnetic-field turbulence. The magnetic-field irregularities themselves are caused by turbulence in the interstellar medium (ISM), which itself is generated by energy input from, among others, supernovae and stellar winds. The magnetic-field irregularities are associated with Alfvén waves, or its close relative, magnetosonic waves (Chap. 4), with velocities given by (4.19). The restoring force for Alfvén waves is the magneticfield pressure, which for typically Galactic values of B ≈ 5 μG is PB = B 2 /(8π) ≈ 1 × 10−12 erg cm−3 = 0.6 eV cm−3 . This is approximately equal to the local cosmic-ray energy density/pressure in the Galaxy (Sect. 11.1.1). The gas pressure is also close to the magnetic pressure. The similarities between these different pressure components suggests a symbiotic relation between cosmic rays, gas and magnetic fields in the ISM, resulting in near equipartition. The overall distribution of cosmic rays in the Galaxy is governed by the cosmicray transport equation, which can be written as (i)

(ii)

(iii)

(iv)

∂ ∂ni (E) = ∇ (D∇ni (E)) −∇vni (E) + [bi (E)ni (E)] − nγi (E) τi ∂t ∂E , , , n (E) − j >i γj τji − mρp βc j i σk,i nk +Qi (E), (v)

(vi)

(vii)

(viii) (11.5)

with ni (E) the density per unit energy of cosmic-ay species i. The different terms on the right hand side indicate respectively (i) transport through diffusion, (ii) transport due to advection with the plasma motions in the ISM, (iii) energy gains

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and losses, (iv) the decay of particle i with Lorentz-dilation-corrected time τi (for radio-active particles), (v) decay of radioactive particle j into particle i, (vi) the spallation of particle i due to interaction with the background gas, with nucleon number density nbg = ρ/mp , (vii) the gain through spallation, i.e. the break up of a primary cosmic-ray particle j resulting into a secondary cosmic-ray particle i, (viii) a cosmic-ray source term, representing the injection of new cosmic-ray particles into the interstellar medium by cosmic-ray sources such as supernova remnants. The various ingredients of the transport equation, such as densities and diffusion coefficients, vary throughout the Galaxy. To model these variations numerical simulations need to be performed, such as with the GALPROP [846] and DRAGON codes [371]. Another approach often used is simplifying (11.5) by assuming a steady state cosmic-ray density (∂ni /∂t = 0) and assuming that the diffusion and advection of cosmic rays can be approximated by an effective, energy-dependent, residence (or escape) time τesc : ∇ (D∇ni (E)) − ∇vni (E) = −

ni . τesc,i

(11.6)

This approximate model is usually referred to as the “leaky-box model”. The idea of the leaky-box model approximations is that one can think of the Galactic cosmic rays as occupying a cylindrical volume given by the area of the Milky Way disk, and a halo height H (Fig 11.3). Since it is assumed that the disk radius Rgal  H , the diffusion-dependent residence time is determined by the halo height as H =



2Dτesc .

(11.7)

R

2H

Fig. 11.3 Illustration of the leaky box model for cosmic-ray transport. The cosmic-ray particles occupy a cylindrical volume (“the box”) with height 2H and radius R  H , and diffuse within this volume for an average time τesc before escaping

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Since H is assumed to be fixed, we see that combining (11.4) and (11.6) suggests the following dependence of the escape time on rigidity R: τesc =

H2 ∝ R −δ . 2D

(11.8)

The leaky-box model ignores gradients in the diffusion coefficient and assumes that the production and spallation rates can be reasonably well described by an average background density ρ. Despite its shortcomings, the leaky-box model is often used to interpret cosmic-ray spallation products, and it provides useful insights in cosmic-ray transport in the Milky Way. As a first example of the leaky-box model, consider the spectrum of protons, and only consider the terms corresponding to diffusion from the Galaxy (11.6), and the source term, assumed to be Q(E) = KE −q . Assuming that the proton cosmic-ray density is constant in time, we obtain 0=−

np,cr + KE −q , τesc

which , together with (11.8), shows that the cosmic-ray spectrum has the form np,cr ∝ E −q τesc ∝ E −q−δ ,

(11.9)

where we used E −δ ∝ R −δ , which is strictly speaking only true for relativistic particles. Given the potential range in values for δ, the local cosmic-ray index of 2.7 implies a spectral index at the sources of q = 2.7 − δ ≈ 2.0 − 2.4. Cosmic-ray species that are mostly created through spallation (i.e. Qi ≈ 0), such as boron, can be used to constrain the average amount of gas a cosmic-ray particle moves through an escape time τesc . For example, boron is a spallation product of cosmic-ray carbon and oxygen. So using QB = 0, the leaky-box model equation (11.5) reduces to 0≈−

  ρ ρ nB − βcσB nB + βc σC,B nC + σO,B nO . τesc mp mp

(11.10)

Writing x ≡ βρcτesc , which is the column density of material a carbon nucleus travels through before escaping, one sees that x nB σC   = σ B mp nC 1 + x mp

  σC σO,B nO x   . 1+ ≈ 1.4 σ B mp σC,B nO 1+x

(11.11)

mp

We ignore here that β varies between the various nuclei involved (B,C, or O). The quantity xB ≡ mp /σB ≈ 7.0 g cm−2 is the typical “grammage” a boron nucleus encounters before being destroyed, whereas x ≈ 20 g cm−2 (at 3 GV, see Fig. 11.4) is the total “grammage” a cosmic-ray carbon nucleus encounters before escaping

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Fig. 11.4 The cosmic-ray boron-carbon ratio as a function of energy per nucleon, measured by the PAMELA space experiment [41]. The model is a simple leaky-box model with parameters x = 21.8β 3 (R/3GV)−δ g cm−2 , with δ = 0.65

the Galaxy. This “grammage” translates into a typical escape time scale of τesc ≈ 1.3 × 107 n−1 H yr. See Fig. 11.4 for the measured B/C ratio and a leaky-box model fit to the ratios. The escape time itself is more accurately estimated using radio-active cosmicray nuclei, in particular those with decay times close to the escape time, such as 10 Be, 26 Al, 36 Cl, and 54 Mn. Based on the study of radio-active and non-radioactive spallation products with energies of 70–500 MeV/nucleon, as measured by the CRIS instrument on board the Advance Cosmic-ray Explore (ACE) [1261], the escape time has been determined to be τesc ≈ (15.0 ± 1.6) × 106 yr, whereas the average ISM density and diffusion coefficient were found to be nH = 0.34 cm−3 and D = (3.5 ± 2.0)1028 cm2 s−1 . A more advanced analysis of the B/C ratio obtained by the PAMELA experiment (Fig. 11.4) and modelled with the GALPROP code, indicates a diffusion coefficient of D = (4.12 ± 0.04) × 1028 cm2 s−1 and δ = 0.397 ± 0.007 (11.4) for a rigidity R = 4 GV. The GALPROP model is more realistic than the leaky-box model used for the model in Fig. 11.4, which explains the differences in parameters. The value δ ≈ 0.4 implies that the power-law index for cosmic rays at the source is q ≈ 2.3. Considerable uncertainties in the cosmic-ray transport parameters still exist, as any realistic model needs to take into account the variation in diffusion coefficient, its anisotropy, the variation in density in the ISM, as well as the low densities in the halo; all parameters that are not precisely known. Another issue that has been discussed in the literature is the effects that enhanced magnetic field turbulence in

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Fig. 11.5 All sky map of γ -ray radiation above 1 GeV as observed by the Fermi Gamma-ray Space Telescope during its first five year of operation. The diffuse radiation from the Milky Way is caused by a combination of inverse Compton scattering and bremsstrahlung from electron/positron cosmic rays, and pion production and decay, resulting from interaction of hadronic cosmic rays with interstellar gas. (Credit: NASA/DOE/Fermi LAT Collaboration)

star-forming regions may have on the escape time and spallation products of cosmic rays. It is possible that cosmic rays spend considerable time close to their source location, before moving into the ISM with more average transport conditions. The modelling of transport through the halo poses another challenge, as little is known about the magnetic field properties there. As a result, the size of the cosmic-ray halo is not well constrained. At least for the Galactic disk we have multiwavelength data to assess the gas and magnetic field properties, and we have constraints on the cosmic-ray densities based on diffuse γ -ray measurements caused by the interaction of cosmic rays with the interstellar medium [36] (Fig. 11.5).

11.1.4 Supernova Remnants as the Dominant Sources for Galactic Cosmic Rays So why are supernovae, and their descendant supernova remnants, considered to be the most likely sources of Galactic cosmic rays? The main argument concerns the energetics of cosmic rays in the Galaxy. The local energy density of cosmic rays is Ucr ≈ eV cm−3 . This energy density is mostly determined by the lowest energy cosmic rays, since the spectrum is steeper than q = 2. As detailed in the previous section, at those low energies the average time a cosmic-ray nucleus resides in the Galaxy is τesc ≈ 1.5 × 107 yr, and the diffusion coefficient is D ≈ 4 ×√ 1028 cm−2 s−1 . We can estimate the Galactic cosmic-ray scale height to be Hcr = 2Dτesc ≈ 2 kpc—much thicker than the stellar disk!—and an effective

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

radius of the disc to be Rdisc = 10 kpc (Fig. 11.3). Approximating that the most important cosmic-ray loss term is the escape of cosmic rays from the Galaxy, one can calculate the necessary power to sustain the cosmic-ray energy density in the Galaxy by dividing the volume- integrated energy in cosmic rays by the escape time: 2 2H √ Ucr π Rdisc cr −1/2 2 = Ucr π Rdisc 2Dτesc (11.12) τesc   − 1   1  2 2 Ucr Rdisc 2 D τesc ≈1.2 · 1041 erg s−1 . −3 28 2 −1 7 10 kpc 4 · 10 cm s 1.5 · 10 yr eV cm

E˙ ≈

The kinetic power provided by supernova explosions, given a Galactic supernova rate of 2–3 supernovae/century (Chap. 2) and a canonical supernova explosion energy of 1051 erg, is E˙ ≈ 1.0 × 1042 erg s−1 . This means that supernovae provide sufficient energy to supply the cosmic rays in the Galaxy, if supernova remnants are capable of transferring ∼ 10% of that energy to cosmic rays. Models for the total cosmic-ray related radiation from the Galaxy also indicate a cosmic-ray acceleration efficiency per supernova of 5 − 10% [1083]. This efficiency is relatively high, but not unreasonably high. The availability of sufficient power for cosmic rays is the main reason why supernovae have long been considered to provide most of the energy for cosmic-ray acceleration. This gives some freedom to also consider the direct aftermath of supernova explosions, or the combined effects inside star-forming regions as the locations for acceleration. But supernova remnants have the advantaged that they many of them show evidence for the presence of accelerated particles in the form of cosmic-ray related radiation in the radio, X-rays and γ -ray, which is the topic of the next chapter. For now we can conclude that supernovae/supernova remnants satisfy criterion 1 for being the dominant sources of cosmic rays, as formulated in Sect. 11.1. But this a precondition, not a proof.

11.1.5 Other Potential Sources of Galactic Cosmic Rays To put the energetics of supernovae/supernova remnants in perspective it is worthwhile to list other high energy sources that could be potential contributors to the Galactic cosmic-ray populations. See also the summary in Table 11.1. Pulsars/Pulsar Wind Nebulae These sources have been extensively discussed in Chap. 6. The source of energy of most pulsars is the initial rotational energy of the pulsar. In principle this energy can be even larger than the typical supernova energy, if the initial spin period is  3 ms. However, most pulsars (with some exceptions, Chap. 6) appear to be born with typical spin periods of 50 ms or longer, corresponding to ≈1049 erg, two orders of magnitude smaller than a typical

Stellar winds Superbubbles Novae X-ray binaries/micro-quasars Central Black Hole

Supernova remnants Pulsars

Source type

Frequency (yr−1 ) ≈1/30 < 1/30 0. The trivial solution is again the solution with a constant particle density n2 (x, p) = n2 (p), the non-trivial solution now corresponds to an exponentially growing density. But clearly that solution is unphysical and should be ignored. The continuity of particle density at the shock requires n2 (p) = n1 (p, 0). Hence, under the assumption of a steady state, plane parallel shock, with no losses, the cosmicray density downstream of the shock is constant. In practice, however, supernova remnants shocks are neither plane parallel, nor is the shock velocity constant. So the solutions for both the upstream and downstream particle distribution breaks down if the acceleration time becomes a significant fraction of the age of the supernova

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299

remnant and/or the diffusion length scale becomes a substantial fraction of the supernova remnant radius. We will treat this in more detail below. The convection-diffusion equation (11.23) is also consistent with the powerlaw distribution for accelerated particles derived in the previous section, providing an alternative derivation of the expected momentum distribution of accelerated particles, as shown in [177]. To explain this, we consider a larger region around the shock, large enough that the gradients in density caused by diffusion can be neglected (so we ignore the precursor region), so that we can ignore the diffusion terms in (11.23). Under steadystate conditions the equation can then be approximated as v

1 ∂n ≈ ∂x 3



∂v ∂x

 p

∂n . ∂p

(11.27)

If we make the gradients more explicit we obtain v1

1 v2 − v1 ∂n2 n2 − n1 (−∞) ≈ p ,

x 3 x ∂p

(11.28)

where we cheat a little by considering the derivative in momentum in the shocked medium (region 2), whereas all other gradients are evaluated in the transition from region 1 to region. See [337] for a more precise derivation. There is a little irony in this simplification, as we removed the diffusion terms in order to derive a result about diffusive shock acceleration. But the essence is here that a change in the spatial part of the density phase-space has to be compensated for by a change in momentum space, in order to preserve phase-space conservation (Liouville’s theorem). The diffusion is part of the physics behind the phase-space evolution, whereas the derivation here puts a requirement on phase-space changes, given the boundary conditions. Rewriting (11.28) and assuming that n2  n1 (−∞)—note that n = n(x, p) is the phase-space density, of accelerated particles, not spatial density—gives us 3v1 ∂ ln n2 3χ =− = s, =− ∂ ln p v1 − v2 χ −1

(11.29)

with χ = v1 /v2 the shock-compression ratio. The momentum distribution is, therefore, a power-law function with spectral index s: n2 (p) = Kp−s .

(11.30)

For relativistic particles we have E = pc. Integrating over momentum phase space 4πp2 dp, the energy distribution of particles in the relativistic limit is, therefore, n2 (E) ∝ E −q ,

(11.31)

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

with q =s−2=

χ +2 . χ −1

(11.32)

This is the same result as derived in Sect. 11.2.1, but now using the convectiondiffusion equation. As we only made use of E = pc in the last step, this derivation also shows that we expect a power-law in momentum space, rather than in energy. Of course for relativistic particles this distinction is not important.

11.2.3 The Acceleration Time Scale DSA theory predicts that the spectral distribution of accelerated particles does not depend on the shock velocity, but only on the compression ratio of the shock. However, as we will show here the acceleration time scale does strongly depend on the shock velocity. To estimate the acceleration time scale, we first estimate how long, on average, a particle that is in the process of being accelerated resides on either side of the shock. For particles to be accelerated they have to be close enough to the shock front, namely within a length scale (11.25) )

∞

l= 0

    ) ∞ xD x exp − exp − dx = ldiff . dx ≈ v ldiff 0

Consider now a volume spanned by this length scale and corresponding to a shock surface area A, the number of particles that are in the process of being accelerated is N = n(p)Aldiff . The number of particles crossing the shock front is given by AFcross (p), with Fcross given by (11.15). Note that the diffusive flux near the shock, 14 nβc, is generally much larger than the advected flux nVs (c.f. 11.18). So the average time spend by a particle on either side of the shock front is

t = N/(AFcross ) = 4ldiff /βc. The average time for a particle to complete one cycle is the sum of the time spend on both sides of the shock: 4 4 (ldiff,1 + ldiff,2 ) =

t = βc βc



D1 D2 + v1 v2

 (11.33)

.

The average energy gain is given by (11.16), so the rate at which particles gain energy is

E dE ≈ = dt

t

4 βc

4 v c E  D1 D2 v1 + v2

3

≈

(v1 − v2 ) 3

E D1 v1

+

D2 v2

.

(11.34)

11.2 The Theory of Diffusive Shock Acceleration

301

The time needed to accelerate to a certain energy E is, therefore, tacc

3 = v1 − v2

)

E E0



D1 (E ) D2 (E ) + v1 v2



dE . E

(11.35)

This is essentially the same result as derived more rigorously in [761]. We can use the parametrisation (11.3) to gain some insight in the acceleration time. We assume that there are no energy losses and that an energy Emax can be reached in a time tacc . Furthermore, we assume that the downstream magnetic field is enhanced with respect to the upstream strength with a factor 1 ≤ χB ≤ χ, with χB = 1 corresponding to a parallel magnetic field (i.e. parallel to the shock normal) and χB = χ would be the other extreme, corresponding to a perpendicular magnetic field, which will be compressed with the same value as the shocked gas. In this case we have D2 = D1 /χB . We make the energy dependence of the parameterisation more explicit by rewriting 1 D1 = ηmax 3



E Emax

δ−1

cE , eZB1

(11.36)

with eZ the charge of the particle, and δ being a similar parameter as used in Sect. 11.1.3. As a result the acceleration time is given by tacc

 ) Emax  dE 3χ χ (11.37) = 2 D1 1+ χB E Vs (χ − 1) E0      δ  E η c χ χ max 0 =3Emax Vs−2 1 + 1− = χB χ −1 δ eZB1 Emax    χ χ D1 (Emax ) ≈3δ −1 1 + χB χ −1 Vs2    χ  −1 1 + χ −2   ηmax Emax B1 χB χ−1 Vs ≈1014 yr, 14 −1 δZ 10 eV 10 μG 8/3 5000 km s

with δ = 1 (probably δ < 1) and for the last approximation Emax  E0 . The numerical factor 8/3 is valid for χB = χ = 4. Rewriting this equation shows that Emax ∝ η−1 ZB1 Vs2 tacc . The dependence on Z is the probable explanation for why the “knee” in the cosmic-ray spectrum appears to scale with the charge of the element (see Sect. 11.1.1). We see that an energy of 1014 eV can be reached by shocks of a young supernova remnant, provided that the shock velocity remains high for at least 1000 yr and ηmax ≈ 1, i.e. close to the Bohm limit. The time scale to reach 1014 eV is the time scale for young supernova remnants; for example SN 1006 has the required age.

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However, as we will explain below, particle acceleration needs to take place on a shorter time scale than the age of the supernova remnant, in order to avoid adiabatic energy losses. To reach even higher energies than 1014 eV, which is required if supernova remnants are responsible for protons accelerated to the cosmic-ray “knee” (Sect. 11.1.1), magnetic field strengths larger than 10 μG and/or faster shock velocities are required. The acceleration time scale makes it, therefore, difficult to unequivocally associated the origin of Galactic cosmic-ray all the way up to the “knee” with supernova remnants. This conclusion was reached by Lagage and Cesarsky in the 1980ies [691], who even pointed out that acceleration up to 1014 eV required optimistic conditions (i.e. η ≈ 1). However, as we describe in Chap. 12, over the last decades the measured supernova remnant conditions appear to be close to the optimistic case.

11.2.4 The Maximum Size of the Cosmic-Ray Shock Precursor In Sect. 11.2.2 we showed that the cosmic-ray precursor falls off exponentially ahead of the shock with a characteristic scale ldiff,1 = D1 /Vs (11.26). For a planeparallel, constant-velocity shock the diffusion length scale can be arbitrary large. In reality a supernova remnant shock does not have a constant velocity and for length scales approaching the shock radius of the supernova remnant the assumption of a plane-parallel shock breaks down. We can put a constraint on the maximum precursor length scale, using our knowledge about supernova remnant evolution (Chap. 5). First of all the acceleration time can never exceed the age of the supernova remnant (tsnr ). Using (11.37) with χ = χB = 4 and δ = 1 we find that tsnr > tacc > 8

D1 ldiff,1 =8 , 2 Vs Vs

(11.38)

where the factor 8 is for the rather optimistic case of a perpendicular magnetic field (a plane parallel shock results in a factor 20). We furthermore neglected the factor δ −1 , which for δ < 1 increases the acceleration time. The shock velocity m, of a supernova remnant can be approximated during various stages as Rs ∝ tsnr with m the expansion parameter (Sect. 5.2, 0.25 < m < 1); for young supernova remnants m ≈ 0.7, and m = 0.4 for the Sedov-Taylor phase. This also implies that Vs ≈ mRs /tsnr (5.2). Inserting this approximation in (11.38) gives the following strong constraint on the precursor length scale: ldiff,1
⎨ t0 1 2 =Vsh,0 At0 ×  |2m−1|

|2m − 1| ⎪ ⎪ t0 ⎩ 1− for m < t0 + t )

2 =Vsh,0 t0 A

t0 + t

1 2 1 2

This solution is not valid for m = 12 for which the factor 2m − 1 switches sign. It shows that for m > 12 an increase in acceleration time t will always result in a larger maximum energy ( t → ∞, Emax → ∞). For m < 12 the maximum energy 2 At /|2m − 1|. For m = 1 will be limited to the asymptotic value Emax = Vsh,0 0 (free expansion, or constant shock velocity) we obtain the original Emax ∝ t as implicitly given by (11.37). The importance of this result is that after the age the supernova remnant has reached the Sedov-Taylor phase (or m < 12 ) the energy gains will be very limited. What is physically special about m = 12 is that for that value the increase of the shock radius, Rs ∝ t 1/2 , is similar to the increase in diffusion distance, ldiff = √ 2Dt ∝ t 1/2 . In other words, for m > 12 a particle diffusing ahead of the shock will always be caught up by the faster progression of the shock if enough time is allowed. But for m < 12 a particle diffusing ahead of the shock may at some time be too far ahead to ever be overtaken by the shock. As result for m < 12 , that is when the evolution approaches the Sedov-Taylor phase, the maximum energy of the accelerated particles reaches a ceiling. Later in this chapter we will describe evidence that the magnetic field strength is amplified, with faster velocities resulting in more amplified magnetic fields. If

11.2 The Theory of Diffusive Shock Acceleration

305

we for now assume that B ∝ t −a , with a some positive constant, it can be easily verified that the critical value for the expansion parameter shifts to an even larger value of m = 12 (1 + a).

11.2.7 The Escape of Cosmic Rays In Sect. 11.2.4 we showed that for active particle acceleration up to a given certain energy, the precursor length scale has to be limited to ldiff < (m/8)Rs . We can also take this constraint on the precursor length scale to see whether particles are no longer able to be accelerated. For convenience we take here a a typical value for the precursor length scale of ldiff < 0.1Rs . Again using Rs = R0 (t/t0 )m (Sect. 5.1) and (11.3), we can derive a limit on the energy above which particles are escaping: ldiff

D1 1 cE = = η Vs 3 eB

 m

R0 t0

1   m−1  0.1Rs = 0.1R0 t t0

 m t t0

(11.45)

⇔ Eesc 0.03η−1

eB1 R02 m c t0

 2m−1 t , t0

(11.46)

which again shows the different behaviour for m < 1/2 and m > 1/2, and the linear dependence on B, Vs2 , and η−1 . Rewriting the inequality, and inserting some typical values for supernova remnants, we find Eesc  0.03η−1

   −1  Rs 2 t eB1 Rs2 B1 m ≈ 45η−1 m TeV. c t 10 μG 5 pc 500 yr (11.47)

For Cas A we have Rs = 2.6 pc, t ≈ 340 yr, and B1 ≈ 50 μG (Sect. 12.2), which suggest Eesc ≈ η−1 89 TeV. This is an energy and order of magnitude below the “knee”, but higher than what was recently reported based on the modelling of the γ -ray-spectrum of Cas A [56].

11.2.8 Radiative Losses: The Maximum Electron Energy We have described so far, how the available time, shock velocity evolution, escape, and adiabatic expansion limit the maximum energy particles can be accelerated to. There is one other factor that limits the maximum energy: radiative losses.

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

For hadronic cosmic rays inelastic collisions are the most dominant loss mechanism, resulting in γ -ray and neutrino emission, as a result of pion production (Sect. 13.6). The cross-section for inelastic hadron collisions is relatively constant as a function of projectile energy, σpp ≈ 40 mbarn, and the associated time scale for collisions is much larger than typical supernova remnant ages: τpp ≈ 1/(nH σpp c) ≈ 2.6 × 107n−1 H yr, with nH the local number density of the plasma. On the other hand, for electrons and positrons radiative losses are in many cases the dominant factor limiting the maximum energy they can obtain through shock acceleration. The radiative losses are dominated by synchrotron radiation, which is described in detail in Sect. 13.3.3, where it is shown that the energy losses are strongly dependent on the energy of the electron/position: 



dE dt

≈− syn

B 2E2 erg s−1 . 634

(11.48)

During the acceleration cycle, a charged particle will spend time both upstream (region 1) and downstream of the shock (region 2). Downstream of the shock the magnetic field is likely to be compressed: B2 = χB B1 . Moreover, also the time spend is longer there (Sect. 11.2.3). Upstream we have < t >1 = D1 /v1 and downstream < t >2 = D2 /v2 = (χD1 )/(χB v1 ), using the diffusion dependence on magnetic-field strength (11.3). Using a time-weighted average of the magneticfield strength we, therefore, obtain -

dE dt

. = syn

B12 E 2 (1 + χχB )   erg s−1 , 634 1 + χ

(11.49)

χB

which we can also express in terms of B2 = χB B1 . At sufficiently large energy, an electron will gain as much energy through DSA, as it will lose during one acceleration cycle as a result of synchrotron radiation. This gives a maximum energy for the electron that we can calculate by equating (11.34) with (11.49): Ee,max =η

−1/2

≈42η

−1/2 B2

−1



  e χ −1 Vs c (1 + χχB )

B2 100 μG

−1/2 

Vs 5000 kms−1

(11.50) 

χ−1 (1+χχB )

3/17

1/2 TeV.

Note that we do not care so much here about the energy dependence of the diffusion coefficient (as long as δ < 1), as we are using here energy gain rates at a specific energy, and all the diffusion dependencies are contained here in B2 and η, the latter being specific for a certain energy/gyroradius scale.

11.2 The Theory of Diffusive Shock Acceleration

307

The reason for giving the maximum energy as a function of the downstream magnetic field, rather than upstream field, is that synchrotron radiation is strongly enhanced in the downstream region, because of the stronger magnetic field. This is, therefore, the region from which synchrotron radiation is usually detected. A value of B2 = 100 μG is typical for what has been inferred from observations of young supernova remnants (see Sect. 12.2.2). Interestingly, if synchrotron radiation losses limit the maximum energy of the electrons, then the higher the magneticfield strength, the lower the maximum electron energy is (Emax ∝ B −1/2 ). This is in contrast to the maximum energy for atomic nuclei for which Emax ∝ B (11.37). The typical photon energy for synchrotron radiation is (13.39) 

E hν = 19 100 TeV

2 

B⊥ 100 μG

 keV.

Inserting this in (11.50) shows that hνmax ≈ 3η

−1



2 

Vs 5000 km s−1

χ−1 (1+χχB )

3/17

 keV.

(11.51)

Remarkably this typical photon energy, corresponding to Ee,max , does not depend on the magnetic-field strength, as first noted in [46]. Note that around E = Ee,max the electron cosmic-ray distribution is exponentially cut-off ∝ exp[−(E/Ee,max )2 ], and, likewise, the resulting electron spectrum cuts-off exponentially, but a bit shallower: ∝ exp[−(ν/νmax )1/2 ], see [1281]. Apart from synchrotron losses, also inverse Compton scattering losses may play a role in limiting the maximum energy. Inverse Compton scattering is usually included by incorporating it in magnetic-field strength using the effective magnetic field 2 ≡ B 2 + 8πU ), where U (Beff rad rad is the radiation field energy density, which provides the seed photons for upscattering. However, one has to be careful here: inverse Compton scattering is a discrete process. For an electron the average time between collisions is τIC =

1 , nphot σT c

(11.52)

with nphot the photon number density and σT = 6.65 × 10−25 cm2 , the Thomson cross-section (13.6). The cosmic microwave background corresponds to a photon density of 413 cm−3 . So we obtain τIC ≈ 3853 yr, which is longer than the ages of what are considered to be young supernova remnants! So rather than a gradual loss of energy, inverse Compton scattering, in particular in the Klein-Nishina regime, may have a large impact on individual on a small number of individual electrons, rather than resulting in a gradual decline in energy [186]. Finally it is important to note that (11.50) and (11.51) assume that the maximum energy is indeed limited by radiation losses, but for low magnetic-field strength it

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

is possible that the maximum electron energy is still determined by the available acceleration time. In the latter case we call it an age-limited synchrotron spectrum, whereas the case discussed in this subsection is referred to as a loss-limited synchrotron spectrum [962].

11.3 Non-linear Shock Acceleration So far we considered the acceleration of charged particles, as a result of a given shock structure, with a fixed compression ratio obtained from the standard RankineHugoniot equations (Chap. 4). However, in Sect. 11.1.4 it was also shown that supernova remnants should be capable of converting ∼ 10% of their kinetic energy into cosmic-ray energy, in order to explain the Galactic cosmic-ray energy density. This is a substantial fraction of the available supernova energy, and for a cosmic-ray acceleration shock the cosmic rays themselves should have an influence on the shock structure. So instead of treating DSA with the shock structure treated as fixed structure, the shock structure is itself is modified by DSA. This mutual coupling between plasma properties and particle accelerating properties is referred to as non-linear shock acceleration, whereas the shocks are sometimes referred to as cosmic-ray mediated shocks. The opposite approach, as discussed in the previous sections, is called the “test-particle” approximation (or approach). The feedback of the accelerated particles on the shock structure, and the dependence of the accelerated spectrum on the shock properties, makes the mathematical description of non-linear shock acceleration complex. But the theory of non-linear shock acceleration has been well developed, since the first publications on the topics at the end of the 1970s early 1980s [361, 364], with both advancements in (semi)analytical models [150, 178, 232, 339, 1186], Monte-Carlo simulations [364, 1188], and kinetic codes [610]. Qualitatively the effects of non-linear shock acceleration can be understood as follows: the cosmic-ray precursor (Sect. 11.2.2) provides an additional, non-thermal, pressure component of the upstream plasma, which results in a pre-compression of the plasma and adiabatic (and perhaps non-adiabatic) heating in the precursor. This in turn results in a lower velocity and higher pressure of the plasma, just prior to entering the viscous shock. Hence, the Mach number at the viscous shock, for a given overall shock velocity, is lower than without cosmic rays. Figure 11.6 sketches the expected shock structure. The pre-compression in the precursor implies that the plasma velocity is a function of distance from the shock: far upstream the velocity in the frame of the shock corresponds to the overall shock velocity Vs , whereas close to the location of the viscous shock the velocity has decreased to Vs /χprec , with χprec the maximum compression ratio induced by the cosmic-ray precursor. The viscous shock itself is governed by the Rankine-Hugoniot relations as described in Sect. 4.1; it is often referred to as the subshock. The word “shock” then refers to the entire structure, including the cosmic-ray precursor and the subshock.

11.3 Non-linear Shock Acceleration

309

Upstream (unshocked)

Downstream (shocked)

Logarithmic scale

plasma velocity (shock frame)

gas pressure CR pressure density precompression

CR precursor

region 0 (undisturbed)

subshock

region 1 (precursor)

region 2 (shocked)

Fig. 11.6 Schematic illustration of the structure of a cosmic-ray modified shock

Since the plasma flowing into the subshock has a lower velocity than the shock velocity, and the pressure (and sound speed) have increased due to the adiabatic compression (and additional heating), the Mach number of the subshock is lower than the overall Mach number (Msub < Mtot ). As a result, the post-shock plasma temperature is lower than for shocks in the absence of particle acceleration. As explained in Sect. 11.2.1, the spectral index of the accelerated particles depends on the shock compression ratio in the test-particle approach. But for cosmic-ray mediated shocks the highest energy particles diffusively sample the plasma far upstream and far downstream, for which there is a larger velocity contrast than for the low energy cosmic rays, which diffusive not that far ahead of the subshock. The overall result of this is that the particle spectral index is variable, which at low energies a steep index (reflecting the small compression factor of the subshock), and which is flatter at high energies. So the spectrum is no longer a power law distribution, but is concave. An important aspect of non-linear shock acceleration is that the highest energy particles may diffuse too far ahead of the shock and escape. We discussed this in Sect. 11.2.4 in the context of the test-particle approach. However, since in the non-linear acceleration model the cosmic rays take up a substantial fraction of the available shock energy, escape of cosmic rays results in a loss of available shock energy, which could also affect the shock structure. A detailed description of non-linear shock structure goes beyond the scope of this book, but some salient aspects of cosmic-ray modified shocks can be described

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

using the so-called two-fluid model [339, 1177, 1186]. In this model the accelerated particles are treated as an additional fluid component alongside the normal plasma component. The Rankine-Hugoniot conditions can then still be applied, but with some modifications: instead of an upstream and downstream region (1 and 2; see Sect. 4.1) three regions are distinguished, upstream (0), precursor (1) and downstream (2); see Fig. 11.6. The continuity of mass- (4.2) and momentum-flux (4.3)—essentially pressure equilibrium—can be followed throughout all regions, but the equation of energy-flux needs to be modified. First of all, because in the precursor the cosmic-ray component of the energy-flux is originating from the downstream region. So in region 1 (precursor), there may suddenly be an excess of energy-flux compared to region 0 (far upstream). So one cannot simply assume energy-flux conservation as a function of distance to the shock. Secondly, because we have to allow for cosmic-ray escape, draining energy-flux from the entire shock system. Equation (4.4) needs, therefore, to be modified to



1 1 P0 + U0 + (1 − ) ρ0 v02 v0 = P2 + U2 + ρ2 v22 v2 , 2 2

(11.53)

with  ∈ [0, 1] the energy-flux escape fraction. For the reason already mentioned we skip over region 1. The total pressure is a combination of gas pressure Pg and cosmic-ray pressure Pcr . As a measure of the cosmic-ray acceleration efficiency we use the ratio of the downstream cosmic-ray pressure over the total pressure: w≡

Pcr,2 . Ptot,2

(11.54)

Furthermore, the cosmic-ray pressure is continuous across the subshock (Pcr,1 = Pcr,2 ), see the discussion on the precursor in Sect. 11.2.2. Just ahead of the subshock we assume that the cosmic-ray pressure has pre-compressed the gas with a factor χprec , which leads to an adiabatic pressure increase of the gas of γ Pgas,1 = Pgas,0 χprec . The sonic Mach number of the subshock then becomes 2 ≡ Mprec

ρ1 v22 2 −γ +1 = Mtot χsub , γ Pgas,1

(11.55)

with Mtot the total Mach number (the Mach number in absence of cosmic rays). Using the standard expression for the shock compression (4.9), we obtain the following expression for the subshock compression ratio: −γ +1

χsub =

2 χ (γ + 1)Mtot sub

−γ +1

2 χ (γ − 1)Mtot sub

and a total compression ratio χtot = χprec χsub.

+2

,

(11.56)

11.3 Non-linear Shock Acceleration

311

The downstream plasma pressure and temperature (kT2 = μmp Pgas,2 /ρ2 ) can be either obtained from (11.56) and the expression for Pgas,1 :   1 γ Pgas,2 = P0 χprec + 1 − ρ1 v12 , χsub

(11.57)

or directly from the continuity of momentum-flux equation (4.3), corrected for the cosmic-ray pressure:  

1 Pgas,2 = (1 − w) P0 + 1 − ρ0 v02 , χtot

(11.58)

which shows that downstream thermal pressure is reduced by a factor (1 − w). Both equations can be made dimensionless by dividing them by ρ0 v02 and 2 = P /γρ v 2 and χ inserting Mtot 0 0 0 prec = ρ1 /ρ0 = v0 /v1 . Equating these two equations gives the cosmic-ray acceleration efficiency as a function of total Mach number, and the precursor and total compression ratios:   γ 2 1− 1 (1 − χprec ) + γ Mtot χtot − χsub χprec   . ≈ w= 2 1− 1 χtot − 1 1 + γ Mtot χtot

(11.59)

The approximation holds for very strong shocks and modest precursor compression ratios. We see that for χprec = 1 (i.e. χsub = χtot ) we obtain an efficiency of w = 0, which is to be expected as this corresponds to no cosmic rays, or the test-particle approach. Finally, we can derive an expression for the required energy escape fraction from (11.53) for a fixed adiabatic index for the cosmic-ray “fluid” γcr ∈ [4/3, 5/3], 2G0 2G2 2G2  = 1+ − − 2 2 χtot γ Mtot γ Mtot χtot

  1 1 1− − 2 , χtot χtot

(11.60)

with G0 ≡

γ γ γcr , G2 ≡ w + (1 − w) . γ −1 γcr − 1 γ −1

(11.61)

For an assumed γcr and Mtot equations (11.59) and (11.60) provide a continuous set of solutions (Fig. 11.7), with χprec as the controlling variable. It shows that, in general, for a larger cosmic-ray acceleration efficiency (w) a larger escape fraction () is necessary. A larger escape fraction in turn is accompanied by a larger overall compression ratio. In contrast, the compression ratio of the subshock is in general χsub < 4. Interestingly for γcr = 4/3 (fully relativistic adiabatic index), but not for γ = 5/3, there can be one or two values for w for which the energy-flux is conserved ( = 0) and still w > 0. The highest value of w > 0 and  = 0 corresponds

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

Fig. 11.7 The effects of non-linear cosmic-ray acceleration on the total compression ratio (χtot ) and subshock compression ratio (χsub , dashed lines) (top left), the necessary energy-flux escape fraction ( top right), and the downstream temperature (bottom left). In the bottom-right panel the comparison is shown between the two-fluid model (given a continuous set of solutions), and the semi-analytical model by Blasi et al. [178], for different shock velocities and assumed maximum momentum

to a fully continuous shock, i.e. there is no subshock and all thermal pressure is caused by cosmic-ray induced adiabatic compression. This is for various reasons unphysical [340]. Moreover, this solution does not exist for Mtot > 12.27. The other solution with w > 0 and  = 0 results in an efficiency that decreases with Mach number, but always has w < 0.442. However, this requires a minimum Mach number of Mtot,crit = 5.88 for γcr = 4/3. This minimum Mach number decreases for increasing√γcr ; for γcr = 5/3 (fully non-relativistic) the critical Mach number is Mtot,crit = 5 [1177]. However, pre-existing cosmic-ray pressure changes these critical limits. The two fluid model presented here captures some of the essences of nonlinear shock acceleration, but does not self-consistently predict the power-law slope, and the corresponding adiabatic index of the non-thermal particles (γcr ), unlike some semi-analytical models [178]. The semi-analytical models on the other hand require, the choice of a maximum momentum, beyond which particles decouple [178], which is based on test-particle theory (Sect. 11.2.3), or a maximum diffusion distance [961]. Particles with higher momentum and/or larger diffusion distance

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are assumed to escape, taking away energy-flux. In these models the other input parameter is the cosmic-ray injection efficiency, i.e. what fraction of the particles will end up being accelerated by the DSA mechanism. For large enough injection efficiencies the cosmic-ray pressure fraction w can become very large w > 0.5, resulting in strongly concave spectra. One would assume them, based, on the fact that χtot  4 for w  0 that the particle index q would approach q = 0. However, it was shown in [760] that the asymptotic value is q = 1.5 even for χtot > 7. In the past it was common, by lack of alternative hypotheses, to assume that particles are injected from the population of post-shock, thermal particles. The particles are assumed to have a Maxwellian velocity distribution, which, considering one dimension, corresponds to a Gaussian with spread σ = kT /m. Some particles may, therefore, have a velocity larger than the downstream velocity, and recross the shock. This is sometimes referred to as the thermal-leakage model. However, particle-in-cell (PIC) simulations reveal a much more complex shock-heating and acceleration behaviour, with some particles being immediately reflected upon encountering the subshock, before being advected again to the subshock. It was shown that injection occurs preferentially from this population of reflected particles [234]. Non-linear shock acceleration made rapid progress from the 1980s to around 2005, and it was popular to assume that the cosmic-ray acceleration efficiency was very high, which should result in cosmic-ray spectra that at high energies have spectral indices close to q = 1.5, overall compression ratios χ  7 [298, 366] and downstream temperatures that are very low [502]. However, as will be described in Chap. 12 the observational evidence for these features is absent, or at best inconclusive. Nevertheless, non-linear acceleration effects should be present, but may not be as dominant as previously assumed. The reason that cosmic-ray acceleration efficiency may not be as high as predicted may be caused by the effects cosmicray acceleration has on the magnetic-field topology and amplification [235, 1188]. This in itself is also a form of non-linear cosmic-ray acceleration.

11.4 Particle Acceleration and Magnetic Fields In the previous section we discussed how efficient cosmic-ray acceleration alters the structure of the shock itself. Here we discuss how the accelerated particles also actively enhance the magnetic-field strength and turbulence, thereby actively creating the conditions that make acceleration to high energies possible. Recall that Emax ∝ Vs2 B/η (in absence of other limiting factors), with η a measure for magnetic field turbulence. The interaction of accelerated particles with plasma waves is an important, but also complex topic. We will provide a global overview here. It is important to know that two types of interactions are often distinguished: resonant particle interaction and non-resonant interactions.

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11.4.1 Resonant Particle-Wave Interaction A charged particle moving in a magnetic field will spiral around the magnetic-field lines, as there is Lorentz force FL = 1c eZv × B perpendicular to the magnetic-field direction. For a uniform magnetic-field strength and direction, taken to be along the z direction, the velocity of the particle is described by vz = v =constant, and vx = v⊥ cos(t + φ) and vy = v⊥ sin(t + φ) (Fig. 11.8), with  = |Ze|B/(mc) the gyrofrequency. The pitch angle, θ = arctan(v⊥ /v ), is the angle between velocity of the particle and the magnetic field direction. Assume now that an Alfvén wave (Sect. 11.1.3) passes by in the z-direction, causing an oscillation with frequency ω and wave number k in the x-direction of the magnetic field: δBx = δB sin(kz − ωt). This in turn results in a Lorentz force in the z-direction FL = 1c eZvy δB sin(kz − ωt), with z = z0 + v t the position of the particle. During the passage of the wave the particles will gain momentum in the parallel direction: )

1 eZvy δB sin(kz − ωt )dt c ) 3    4 eZv⊥ δB cos (kv − ω − )t + kz0 − φ − cos (kv −ω + )t +kz0 + φ dt. = 2c

p =

The first term in the integrand corresponds to very rapid oscillations with a frequency +ω, which is a much higher frequency than the Alfvén-wave frequency. Hence, integration of the first term over many oscillations tend to average to zero. On the other hand, the second term represents a slow oscillation. And for the resonant condition  + kv − ω ≈ 0,

θ

(11.62)

v//

v┴ B

Fig. 11.8 Sketches of the interaction between charged particles and the magnetic field. Left: The magnetic field is uniform, resulting in a regular helical path of the particle. Middle: The magneticfield is non uniform, but the deviation are on a scale larger than the gyroradius of the particle. The particles follow the magnetic field lines. Right: The particle interacts with a magnetic-field distortion comparable to the gyroradius of particle. As result the pitch angle changes, and the particle will not follow the magnetic-field lines

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315

the integrand will be finite, depending on the duration/coherence length of the Alfvén wave. Hence, the velocity v will have changed after the passage of the shock. During the passage of one oscillation, with wavelength λ = 2π/k, corresponding to t ≈ λ/v (since vA  v ) the change in parallel momentum is

p ≈

11 11 2π eZv⊥ δB cos(kz0 − φ) t = eZv⊥ δB cos(kz0 − φ) 2c 2c v k (11.63)

eZv⊥ δB eZv⊥ δBmc cos(kz0 − φ) = π cos(kz0 − φ) c c eB δB =πp sin θ cos(kz0 − φ). B ≈π

We have used here that the gyrofrequency is  = ZeB/ mc, p⊥ = p sin θ = mv sin θ , and we approximated kv − ω −  ≈ kv −  ≈ 0, which follows from vA = ω/k  v . Since magnetic fields change the momentum direction, and not the absolute value of the momentum p, we find that for small changes in momentum δp = −δ(p cos θ ) = −p sin θ δθ.

(11.64)

Equation this with the final outcome of (11.63) gives  δθ = −π

δB B

 cos(kz0 − φ).

(11.65)

The particle is subject to many interactions with various random Alfvén-wave packages, affecting only particles with gyrofrequency close to the resonant condition. This results in a steady randomisation of pitch angles. The variance of per average full phase will be < cos2 (kz0 − φ) >= 12 , so that the “diffusion” rate of pitch angles—assuming interaction times of one gyroperiod of τ = 2π/—is π ( θ )2 ≈  t 4

5

δB B

2 6 .

(11.66)

For the particle to change its original direction, we can take θ ≈ 1, which gives us a typical isotropisation time

τiso

4 ≈ π

5

δB B

2 6−1 .

(11.67)

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

Using rg = v⊥ /, the mean free path over which the particle changes direction is λmfp ≈ vτiso

4 ≈ rg π

5

δB B

2 6−1 (11.68)

.

In Sect. 11.1.3 we discussed that the mean free path is often parametrised as λmfp = ηrg , with the factor η appearing in the acceleration time and maximum particle energy (Sect. 11.2.3). We see here that 5 η≈

δB B

2 6−1 (11.69)

,

with δB the typical perturbation of the magnetic fields fluctuations satisfying the resonance condition (11.62). A more rigorous discussion can be found in [1215]. The energy density in the Alfvén waves is UδB = δB 2 /(8π) and the momentum associated with the plasma waves is pδB = (1/vA )δB/8π. As long as there is a difference between the drift velocity of the cosmic rays, vD , and the Alfvén wave, the plasma waves keep growing, until there is no net momentum transfer between the resonant particles and the plasma waves. This condition implies that for saturation the drift velocity equals the Alfvén velocity. So the net momentum transfer is p = cr mcncr (vD − vA ), and should be established in roughly an isotropisation time τis . This p is absorbed by the plasma, so the growth rate for the plasma waves is [71, 689] −1 ≈ τδB

cr mcncr (vD − vA )

p −1 1 τ =  δB −2 −1 δB 2 4 pδB iso vA 8π π B

(11.70)

π ncr vD − vA π ncr |Ze|B vD − vA = cr  = , 2 ni vA 2 ni mi c vA 2 m n , with m the typical ion mass, where we have made use of B 2 /8π ≈ 12 vA i i i assumed to be similar to the typical accelerated particle’s mass, and ni the background ion density. The ratio ncr /ni relates to the particle injection efficiency. Note that the growth rate is calculated for typical ranges of wavelength numbers, i.e. a certain dk, and that ncr /ni drops rapidly for large wavelength (corresponding to large particle momentum). √ If we now assume that initially vA  vD ≈ Vs , and use (4.19) vA = B/ 4πρ, we see that the growth rate becomes independent of B, but does depend on the density. With all numerical factors explicitly calculated: −1 τδB

≈ 8.2 × 10

−4

nH 1/2 Z 1 cm−3 



ncr /ni 10−3



mi mp

−1 

vD /vA 100



s−1

(11.71)

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317

Since 4πp2 ncr (E > Ek ) ∝ E −q+1 , the growth rate of the largest waves is smallest. For example, if we assume that particles are injected with energies around 100 keV, the accelerated particle density for E > 10 TeV is a factor 10−8 smaller (assuming q ≈ 2). This gives for the growth time scale at the largest wavelength −1/2

τδB (E  Ek ) ∼ 800nH



Vs 1000 km s−1

−1 

Ek 10 TeV

 yr,

(11.72)

which is approximately valid for ncr /ni = 10−3 at 100 keV. This time scale is comparable to ages of young supernova remnants. However, for magnetic-field turbulence on a length scale corresponding to the gyroradius of cosmic rays near the “knee” energy, one needs high velocities (∼ 5000–1000 km s−1 ) in combination with high densities (nH > 10 cm−3 ). Of course these are rough estimates, but it does show that one cannot totally rely on presence of sufficient resonant Alfvén waves for the scattering of charged particles with the highest energies. It also puts a potential limit on the maximum energy particles can be accelerated to, even when other constraints are met.

11.4.2 Streaming Instabilities and Non-resonant Processes Resonant particle-wave interactions are important for increasing δB/B and hence η (11.3) and (11.69). But in more recent years non-resonant instabilities that amplify the overall magnetic field strength, and, hence, reduce the gyroradii of accelerated particles, have received renewed attention. There are many plasma instabilities driven by the presence of anisotropies in the particle distributions, either those of the accelerated particles, but also by temperature gradients in the plasma (see [766] for an overview) . These types of instabilities are called streaming instabilities. An often discussed instability is the Weibel instability [1204], which feeds off anisotropies in the electron momentum distribution and small perturbations of the magnetic field, which will perturb the directions of the electrons. The change in momentum directions will tend to enhance the magnetic field perturbations, and lead to filamentation of the charged-particle distributions. The characteristic filaments are often seen in particle-in-cell simulations of shocks (see for example [619]), and has also been verified experimentally [564]. The structure and the length scales of the magnetic fields resulting from the Weibel instability are too small to be important to accelerate charged particles to very high energies. However, the Weibel instability is important for providing the scattering and isotropisation of charge particles necessary for the formation of collisionless shocks [619]. Moreover, the Weibel instability is capable of creating magnetic fields out of the tiny magnetic-field strength fluctuations associated with inhomogeneities of particle currents. As a result, collisionless shocks can even form when initial magnetic fields are absent.

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

11.4.3 Magnetic-Field Amplification Through the Bell Instability More recently it has become clear that another streaming instability, the so-called non-resonant Bell instability [143], is important for the amplification of magnetic fields upstream of supernova remnant shocks. The length scales of the fluctuations are smaller than the gyroradius of the accelerated particles, and, therefore, produces a non-resonant interaction. The Bell instability is driven by the cosmic rays streaming ahead in the cosmic-ray precursor, providing an electric current Jcr . The Bell instability is an magneto-hydrodynamics (MHD) instability, and is based on the equation of motion of a charge-neutral plasma fluid in the absence of external forces: ρ

∂v 1 (∇ × B) × B = −∇P + J × B = −∇P + , ∂t c 4π

(11.73)

this is a well-known ideal MHD equation [see 1074, for lecture notes]. The term 1 c J × B is the Lorentz force acting on internal currents, and it has been transformed using the Maxwell equation 4πJ + ∂E/∂t = c∇ × B. This equation is in ideal MHD supplemented with the induction equation: ∂B = −c∇ × E = ∇ × (v × B), ∂t

(11.74)

where we have used the condition that there is no large scale Lorentz-force on the plasma (force-free condition, c.f. Sect. 6.3.2): FL = eZ( 1c v×B+E) = 0, and hence cE = −v × B. For calculating the growth of Bell instability we neglect the pressure gradient term (∇P ≈ 0) and apply an additional current, Jcr , provided by the cosmic rays: ρ

∂v (∇ × B) × B Jcr × B = + . ∂t 4π c

(11.75)

The internal currents are contained in the first term on the right-hand side. We evaluate the growth of the instability, assuming for simplicity that the initial magnetic field is parallel to the shock (Bz,0 ) taken to be the z-direction, which is also the direction of the cosmic-ray current. For the undisturbed parallel field there is no net force on the plasma. However, small variations perpendicular to the zdirection do result in turbulence. To see this we insert a wave-like displacement propagating along the z-direction: v⊥ = (δv + iδv) exp [i(kz − ωt)] and B⊥ = (δB + iδB) exp [i(kz − ωt)], where we use complex numbers to separate the two polarisation directions xˆ and y. ˆ Inserting these harmonic perturbations in (11.75)

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319

and (11.74) and keeping track of the directions of the cross-products and curls gives −iωρ(1 + i)δv = +

ikδ(1 + i)δBBz,0 1 + (1 − i) Jcr δB, 4π c

iωδB(1 + i) =(1 + i)δvikBz,0 . Inserting the second equation in the first equation results in a dispersion relation: 2 ω2 (1 + i) = k 2 vA (1 + i) +

kJcr (1 − i), ρc

(11.76)

2 = B 2 /4πρ. In the absence of a cosmic-ray current, J = 0, where we have used vA cr the dispersion relation is simply that of an Alfvén wave. Note that k can be negative if the perturbation propagates in opposite direction. Depending on the polarisation mode xˆ or yˆ and the sign of k, we find that there are negative roots of the dispersion equation, γ ≡ iω, which correspond to exponential growth of the magnetic field 2 < |kB J /ρc|. It disturbance (δB exp(γ t)). The condition for growth is that k 2 vA z,0 cr can be shown [143] that (11.76) needs to be modified for currents from cosmic-ray particles with gyroradii rg  2π/k, preventing growth for these large wavelengths. The maximum growth rate can be found by solving

∂(k 2 va2 − kBz,0 Jcr /ρc) ∂ω2 = = 0, ∂k ∂k which has the solution kmax =

2πJcr , cBz,0

(11.77)

and 1 2

γmax =



4π ρ

1/2

Jcr . c

(11.78)

We can estimate the growth rate quantitatively by estimating the current Jcr = e < Z > ncr Vs and noting that ncr = Pcr / < p⊥ > c. The average momentum for a cosmic-ray spectral index of q ≈ 2 and p2  p1 is ! p2 −q 1 1 p1 pp dp 1 < p⊥ >= < p >= ! p2 −q ≈ p1 ln(p2 /p1 ). 3 3 p p dp 3

(11.79)

1

So we find Jcr ≈ e

3Pcr p1 ln(p2 /p1 )



Vs c

 ≈e

3wρ0 Vs3 , cp1 ln(p2 /p1 )

(11.80)

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11 Cosmic-Ray Acceleration by Supernova Remnants: Introduction and Theory

where we have used the definition of cosmic-ray efficiency, w, introduced in Sect. 11.3. Inserting this in (11.78) allows us to estimate the growth time and wavelength for maximum growth. The slowest growth rates are for the largest wavelengths, corresponding to the highest energy particles, which also penetrates furthest in the cosmic-ray precursor. For the growth rate of the longest wavelength instability, induced by cosmic-ray streaming of particles close to the maximum cosmic-ray energy, E  Emax , we can apply a correction factor, taking into account the reduction of the cosmic-ray flux for these energies: ncr (E  Emax ) Jcr (E  Emax ) = = Jcr (E  E1 ) ncr (E  E1 )



Emax E1

−q+1

 ≈

E1 Emax

 ,

with the approximation valid for q ≈ 2. Applying this correction factor to (11.80), using p1 = mp c, and inserting this in (11.78), we find for the growth rate of the largest structures τBell



 √ E1 −q+1 ρp1 c2 ln(p1 /p1 ) = = (11.81) γmax Emax 3wVs3  −3  w −1  ln(p /p )   n − 1  E Vs 2 2 1 H max ≈12 yr. 0.1 11.6 1 cm−3 1014 eV 5000 km s−1 1

The corresponding wavelength is 2π cBz,0 = (11.82) kmax Jcr  −3    w −1  n −1  Vs Bz,0 Emax H 15 ≈7 × 10 cm. 0.1 5 μG 1 cm−3 5000 km s−1 1014 eV

λmax =

The magnetic-field amplification is driven by the kinetic energy flux of the cosmic rays streaming ahead of the shock (∝ Ucr Vs ). The amplification is, therefore, likely to saturate when [143] 2 1 Vs Bsat ∼ Ucr ∝ ρVs3 . 8π 2 c

(11.83)

Numerical simulations in three dimensions [144] show that the Bell instability results in peculiar magnetic field structures, with “walls” of high magnetic-field strengths and large densities, enclosing low magnetic field cavities (Fig. 11.9). Note that in a follow-up paper to [143] a magnetic-field saturation value scaling as B 2 ∝ ρVs2 is also deemed possible [146]. In this section we have neglected a number of issues, such as how on a microscopic level the energy of the cosmic rays is transferred to the plasma and

11.4 Particle Acceleration and Magnetic Fields

t =2

321

t =4

(i)

(ii)

(v)

(vi)

(ix)

(x)

t= 6 (iii)

t =8 (iv)

|B| y x

(vii)

(viii)

r y x (xi)

(xii)

y x z

Fig. 11.9 The magnitude of the magnetic-field strengths and densities resulting from the the Bell instability for different e-folding times (t = 2–t = 8) (Credit: A.R. Bell [144])

what the effects are of the return current on the plasma and the interaction with the cosmic rays. Another assumption that was made is that the shock was parallel to the magnetic field, but in [144] it was shown that the instability also operates for non-parallel shocks. The instability is discussed in more detail in [144, 766, 1027]. The Bell instability has been studied in detail also by numerical simulations, and is now widely accepted as an important plasma process that amplifies the magnetic fields in the precursors of fast supernova remnant shocks. As we shall discuss in Sect. 12.2, there is strong observational evidence for magnetic-field amplification near the shocks of young supernova remnants.

Chapter 12

Supernova Remnants and Cosmic Rays: Non-thermal Radiation

12.1 Radio Observations Although Baade and Zwicky [110] first suggested that supernovae are the primary source of cosmic rays, it was the discovery of radio synchrotron emission from supernova remnants that firmly established the link between supernovae and accelerated particles. Moreover, it suggested that the particle acceleration did not take place so much during the supernova explosion itself, but afterward, by the supernova remnant shocks. The realisation that radio emission could be either thermal in nature (free-free emission, Sect. 13.4) or non-thermal was established around 1950. For the diffuse Galactic radio emission the synchrotron nature (Sect. 13.3) of the radiation was suggested by Alfvén and Herlofson [65], see also [649]. The idea that bright radio sources, such as Taurus A (the Crab Nebula) [192] and Cassiopeia A [112, 998], were associated with supernova explosions and were emitting synchrotron radiation was first presented by I. Shklovsky in [1047]. It was not immediately understood that the relativistic electrons responsible for the radio emission from supernova remnants were accelerated by supernova remnant shocks. For example, in 1954 Shklovsky [1047] still states: “Thus actual facts indicate that in the process of an outburst of Supernovae there arises a great number of relativistic electrons.” Even in his book, which was published in English in 1968, but essentially dating from 1966, he defends the assumption that “The number of relativistic electrons does not grow in the process of expansion of the nebula” [1048], but the book also contains sentences which suggests that he (sometimes) thought that particle acceleration is not confined to the supernova stage or the earliest stages of supernova remnant evolution. V. Ginzburgh (1964) unequivocally assumes “. . . that particle acceleration occurs in the shell” [438, p. 199]. This assumption was made around the time that the first radio synthesis map of Cassiopeia A appeared, showing the shell-like distribution of the relativistic electrons in surprising detail [999]. This early radio map is still very similar to more

© Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_12

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

 Fig. 12.1 Left: VLA radio map of Cas A at 6 cm. Right: The polarised intensity map ( Q2 + U 2 ) at 6 cm (Credit: courtesy of Tracy Delaney [299])

recent radio maps of Cas A (Fig. 12.1), and it sets the stage for a new era in radio astronomy. Radio astronomy has, therefore, been essential of firmly establishing supernova remnants as sources of Galactic cosmic rays. Moreover, radio astronomy is and has been important to determine the spectral index of the relativistic electrons, with its obvious link to cosmic-ray acceleration theory (Sect. 11.2.1), the energy in relativistic electrons and, through polarisation measurements, the magnetic-field topology. All these aspects we will describe below. The limitations of radio emission to study the acceleration of cosmic rays by supernova remnants is that it allows only the study of relativistic electrons, whereas cosmic rays consist for more than 99% of atomic nuclei (Sect. 11.1.2). Since the relation of synchrotron frequency to electron energy is 2 ν ≈ 0.47(B/100 μG)EGeV GHz (13.3), radio synchrotron emission informs us of electrons with energies around 109 eV, whereas we would like to learn more about the maximum energy particles can be accelerated to, which we expect to be in the 1013 to >1015 eV range.

12.1.1 The Radio Spectral Index Distribution The radio spectral index of shell-type supernova remnants, i.e. those supernova remnants without a dominant pulsar-wind-nebula component, is on average α = 0.5 (defined as S(ν) ∝ ν −α ); see Fig. 12.2. As explained in Sect. 13.3 α = 0.5 corresponds to an electron number index of q = 2 (i.e. n(E) ∝ E −q ). The average radio spectral index is, therefore, close to that predicted by the theory of diffusive shock acceleration (Sect. 11.2.1).

12.1 Radio Observations

325

Fig. 12.2 A histogram of the radio spectral index (α) distribution of shell-type supernova remnants, based on the catalogue maintained by D. Green [462]

The average radio spectral index of 0.5 is what sets most supernova remnants apart from other radio sources. In the Galaxy most extended sources are HII regions, which emit free-free radiation (Sect. 13.4), which are characterised by a spectrum of α ≈ 0.1, or even inverted (α < 0) if they are optically thick. Pulsar wind nebulae have also flatter spectra than supernova remnants, typically α ≈ 0.2–0.3. It is the rather steep spectral slopes of supernova remnants that are often used to discover new supernova remnants in radio surveys [215]. However, the spread around α = 0.5 is quite large. Part of that spread is due to measurements errors, as radio synthesis telescopes cannot always reliably measure the total radio flux of extended objects, and for not all supernova remnants the index has been determined with as much care as needed. Nevertheless, from those supernova remnants with reliable spectral index measurements it is clear that the variation in α is real. In particular it is striking that young supernova remnants tend to have steeper spectra (α > 0.5). For example, SN 1006 has α = 0.6 [69], Tycho’s SNR has α = 0.65 [674], Kepler’s SNR has α = 0.71 [301], and Cas A has α = 0.77 [113]. The radio spectrum of the youngest known Galactic supernova remnant, G1.9-0.3 also appears to be steep, but with a rather large uncertainty: α = 0.71 ± 0.26 [292]. More mature supernova remnants like CTB109, and Puppis A, on the other hand, both have α ≈ 0.5–0.55 [674, 970]. And finally, there is the class of mixed-morphology supernova remnants, mature supernova remnants with shell-type morphology in the radio, and centrally dominated emission in the X-rays, among which we find many objects with spectral indices around 0.3–0.4 (Sect. 10.3). The rather flat spectra are not well understood, but two mechanisms may be involved. One is that the diffusive shock acceleration is not very efficient in mixedmorphology supernova remnants, and the relativistic electron spectrum may be the

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

result of the compression of pre-existing interstellar medium electrons, and, perhaps in addition, turbulent reacceleration in the post shock region [1146]. Another reason for the rather flat spectra may the high post-shock compression ratios expected behind radiative shocks (Sect. 4.4). As shown by (11.22), higher compression ratios result in flatter radio spectra. The steeper spectra for younger supernova remnants is also not well understood, but again there are two possible explanations identified. One involves the theory of non-linear shock acceleration discussed in Sect. 11.3. According to this theory the subshock has a compression ratio χ < 4, whereas the overall shock compression ratio is χ > 4. As a result, the low energy cosmic-ray spectrum, for which the particles experience only the gradient in velocity across the subshock, should be steeper than q = 2 (α = 0.5), whereas at higher energies the spectrum should flatten asymptotically to q = 1.5 (α ≈ 0.25). If we apply this to Cas A, we see that α = 0.77 corresponds to q = 2.54, which, according to (11.22), implies χ ≈ 2.9 for the subshock. This is a not unreasonable value, and it would mean that nonlinear acceleration effects are only strong in young supernova remnants, which have α  0.6. Non-linear shock acceleration theory also predicts a gradual flattening of the synchrotron spectrum. Indeed, the radio spectra of Kepler’s SNR and Tycho’s SNR appear to be slightly curved [965], whereas for SN 1006 and RCW 86 curved underlying electron spectra have been inferred from the radio to X-ray synchrotron modeling [69, 1187]. An alternative theory for the steeper spectra in young supernova remnants is that the plasma waves have a preferential direction in the upstream direction [1282]. This is referred to as Alfvénic drift. In standard diffusive shock acceleration theory the charged particle scatter of magnetic-field irregularities, which are assumed to be on average stationary to the plasma. However, it is not inconceivable that the strong gradients in cosmic rays impart a preferential direction to the plasma waves upstream of the shock. In that case the relevant velocity gradient for particle acceleration is not the plasma velocity, but the average velocities of the irregularities with respect to the shock. In other words the index is then determined not by χ = v1 /v2 , but by χδB ≡ v1,δB /v2,δB . The trend that younger supernova remnants have steeper spectra may be connected with the very steep spectra sometimes measured for radio supernovae. Radio supernovae are those supernovae that in the days to years following the explosion emit bright radio emission, as the shock wave progresses through a high density circumstellar medium (for a review see [1206]). A radio supernova could as well be named a very young supernova remnant, as the emission is due to the interaction of supernova ejecta with the ambient medium of the supernova. In few cases VLBI imaging shows a rapidly expanding shell. A well-studied radio supernova is SN 1993J, whose optical spectral properties are very similar to the light-echo spectra of Cas A [682, 959]. Radio supernovae tend to have rather steep spectra, generally α ≈ 0.8–1.0, but with some having flatter, or even steeper spectra. For example, the spectral evolution of SN1993J was best described by α = 0.81 [1207]. Interestingly, the secular decrease in flux decline measured for Cas A (see below) appears to be accompanied by a gradual flattening of the spectrum of Cas A [90, 113].

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327

12.1.2 The Minimum Energy Requirement and the Van der Laan Mechanism The radio synchrotron emission of supernova remnants is caused by highly relativistic electrons. However, it is conceivable that the relativistic electrons are simply pre-existing, interstellar electron cosmic rays that are adiabatically enhanced in energy density by the shock: Ucr ∝ χ γ , with γ = 4/3. As also the magnetic field is compressed by the shock, probably by a factor χ ≈ 4 for non-radiative shocks, the passage of a shock wave will enhance the synchrotron emissivity by a large factor. As we will show below, the enhancement factor could be as much as χ 17/6 ≈ 50, or even more for radiative shocks. This mechanism for creating synchrotron sources from compressing the interstellar medium is sometimes referred to as the Van der Laan mechanism [1155]. The Van der Laan mechanism can explain the synchrotron luminosity of some, mostly older, supernova remnants. For younger supernova remnants the Van der Laan mechanism cannot explain the synchrotron luminosity. One of the earliest arguments to proof this is the so-called minimum energy requirement [439, 988], which we will explain here. In Sect. 13.3 it is shown that for a single electron the total synchrotron emitted power is dE 4 = σT cβ 2  2 UB . dt 3

(12.1)

Assuming that we have an electron power-law distribution in energy, and using E ≈ me c2 , we can write for the energy density per energy interval n(E)dE = KE −q dE, n() = K(me c2 )−q+1  −q d,

(12.2)

with K a normalisation constant. The total energy density in relativistic electrons between boundaries 1 and 2 is then ) Ucr,e =

2

me c2 n()d =

1



1 2−q 2−q K(me c2 )−q+2 2 − 1 . 2−q

(12.3)

The total energy in cosmic rays must be higher, because also hadronic cosmic rays are likely to be present. We, therefore, write for the total cosmic-ray energy density Ue = ξ Ucr . For ξ one often takes the value ξ ≈ 100, as in the cosmicray composition observed near Earth that electrons account for about 1% of all cosmic-ray particles (Sect. 11.1.2). Taking the factor ξ into account, we find that the normalisation factor is K=

(2 − q)Ucr

2−q 2−q (ξ + 1)(me c2 )−q+2 2 − 1

(12.4)

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

Calculating the total synchrotron luminosity, using (12.1), using the above expression for the normalisation factor K, and multiplying by the total volume of the supernova remnant shell V , we obtain ) Lsyn =V

2

n() 1

dE d dt

(12.5)

4 1 3−q 3−q = σT cβ 2 (me c2 )−q+1 KV UB 2 − 1 3 3−q

3−q 3−q  −  1 4 2−q 2 1

UB Ucr V . = σT cβ 2 (me c2 )−1 2−q 2−q 3 3−q  (ξ + 1) − 2

1

The relation between electron energy and frequency for synchrotron radiation (13.39) can be translated into ν ≈ 1.8 × 1018 (me c2 )2  2 B⊥ Hz.

(12.6)

Inserting this in (12.5) gives 2 − q −1/2 4 B (1.8 × 1018)−1/2 (me c2 )−2 Lsyn = σT cβ 2 3 3−q ⊥

(3−q)/2 (3−q)/2 ν2 − ν1 1

UB Ucr V . (2−q)/2 (2−q)/2 (ξ + 1) ν2 − ν1

(12.7)

Note that this shows that L/V ∝ B 3/2 Ucr . Hence an adiabatic compression may give rise to an increase in emissivity as high as χ 3/2+γ , in agreement with the earlier statement about the emissivity increase caused by the Van der Laan mechanism. Ignoring the difference between B⊥ and B, and expressing everything as a function of Lsyn , V , Ucr , ξ and B we see that Lsyn = AB 3/2 (ξ + 1)−1 Ucr V , with A containing all dependencies on constants and minimum and maximum frequencies. For the total energy density in relativistic particles and magnetic field (both of which determine the synchrotron luminosity) we find now that Utot = Ucr +

Lsyn (ξ + 1) −3/2 B 2 B2 = B . + 8π AV 8π

(12.8)

The minimum energy requirement is the requirement that the synchrotron luminosity is explained by the minimum possible total energy density in the source. This can be found by minimising the above equation with respect to B (∂Utot/∂B = 0):  Bmin,energy =

6πLsyn (ξ + 1) AV

2/7 .

(12.9)

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329

Substituting this into the expressions for the cosmic-ray energy density and magnetic-field energy density we find that Ucr = 6 UB =

−3/7

(π)

−3/7

64/7 B2 = (π)−3/7 8π 8

 

Lsyn (ξ + 1) AV Lsyn (ξ + 1) AV

4/7 ,

(12.10)

.

(12.11)

4/7

So we see that Ucr = 43 UB under this assumption. In other words, the minimum energy requirement leads to a magnetic field energy density that is close to equipartition with the relativistic particle energy. Cassiopeia A, being the brightest supernova remnant in the radio band, played a central role in the debate about whether the relativistic electrons originate from the supernova remnant or supernova itself, or whether they are simply interstellar relativistic electrons compressed by the supernova remnant shock. Its radio flux at 1 GHz for the 1965.0 epoch is 3186 Jy [113], and its radio index α = 0.77. As can be seen in Fig. 12.3, the radio synchrotron radiation has been measured from 10 MHz to about 100 GHz. At around 50 MHz interstellar freefree absorption becomes important, whereas above 100 GHz thermal dust emission (Sect. 7) becomes the dominant radiation mechanism. For now we ignore the Xray synchrotron emission from Cas A (Sect. 12.2), which does not contribute a large !fraction to the overall synchrotron luminosity, and we calculate Lsyn = 4πd 2 S(ν)dν between 50 MHz and 100 GHz. The distance of Cas A has been

Fig. 12.3 The radio spectrum of Cas A corrected to the epoch 1965.0 [113]

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

determined to be 3.4 kpc [952]. This results in a synchrotron luminosity of Lsyn = 4.6 × 1035 erg s−1 . For the above frequency limits, the numerical value for A is 5.1 × 10−8 . The volume of Cas A, which has a radius of about 2.55 pc, is V ≈ 5 × 1056 cm3 for an assumed volume-filling fraction of 25%. For the above values of Lsyn and V Eq. (12.9) results in Bmin,energy ≈ 0.3(ξ + 1)2/7 mG, which only very weakly depends on the minimum and maximum frequencies used. The associated total energy in cosmic rays is Ucr V = 43 B 2 /(8π)V = 2 × 1048(ξ + 1)4/7 erg. The energy density in cosmic rays Ucr ≈ (ξ + 1)4/72.7 × 103 (ξ + 1)4/7 eV cm−3 . This energy density is more than three orders of magnitude larger than the cosmic-ray energy density of 1 eV cm−3 in the interstellar medium (Sect. 11.1.1). So for Cas A it is clear that the origin of the relativistic particles inside the remnant cannot be accounted for by the Van der Laan mechanism. Moreover, for ξ ≈ 100 the total energy in cosmic rays is ≈ 2 × 1049 erg, or about 2% of a supernova explosion energy. This is comparable, but somewhat smaller than the acceleration fraction of 5–10% needed to explain the energy density of cosmic rays in the Galaxy Sect. 11.1.4 by supernova explosions. Moreover, the minimum energy requirements provides a conservative estimate of the total energy in magnetic fields and cosmic rays, a lower magnetic-field strength, requires a larger value for Ucr . The average magnetic field of Cas A, as based on the minimum energy requirement, is also high compared to the interstellar medium magnetic field. This was in the past attributed to Rayleigh-Taylor instabilities inside the shell (Sect. 5.9), which tangle and amplify the magnetic field [472]. However, as we discuss in (Sect. 12.2), the magnetic-field strength is also high near the shock, and therefore likely reflects magnetic-field amplification in the cosmic-ray precursor (Sect. 11.4). In Table 12.1 the minimum energy magnetic field estimates and cosmic-ray energies are listed for a number of young supernova remnants, which shows that based on this method Cas A appears to have strong magnetic fields and a large cosmic-ray energy content compared to most other young supernova remnants.

12.1.3 The Radio Evolution of Supernova Remnants Shklovsky [1046, 1048] was one of the first to point out that as a supernova remnant was expanding, magnetic-flux conservation and adiabatic cooling of the relativistic electrons should result in a decline of the synchrotron luminosity of supernova remnants. At the time, the prevailing idea was still that the relativistic electrons originated from the supernova explosion, instead of being the result of continuous cosmic-ray acceleration by the shocks. Shklovsky’s prediction of radio flux decline was soon (1961) confirmed by measurements of the brightness of Cassiopeia A in comparison to the extragalactic source Cygnus A [544]. Shklovsky predicted a flux evolution of ∼2% yr−1 , higher than the measured rate of 1.1% yr−1 . Flux measurements for Cas A are still ongoing, partially out of scientific interest, but also because Cas A is a radio calibration source [113], and Cas A is so bright that

Object G1.9-0.3 G4.5+6.8 G6.4-0.1 G34.7-0.4 G43.3-0.2 G111.7-2.1 G120.1+1.4 G189.1+0.3 G260.4-3.4 G315.4-2.3 G327.6+14.6 J0508-6843 J0509-6731 J0519-6902 J0525-6938 J0104.0-7202

Kepler’s SNR/SN1604 W28 W44 W49B Cas A SN 1572 IC 443 Puppis A SN 185/RCW 86 SN 1006 N103B 0509-67 0519-69 N132D 1E0102-7219

Name

Sν (1 GHz) (Jy) 0.6 19 310 240 38 3200 56 165 130 49 19 0.73 0.0974 0.11 5.6 0.335 α 0.72 0.71 0.71 0.37 0.46 0.77 0.65 0.37 0.51 0.6 0.6 0.44 0.73 0.44 0.7 0.7

Rsh (arcmin) 0.75 2 24 15 2.3 2.6 4.5 22 27.5 20 15 0.23 0.25 0.28 0.75 0.4

d (kpc) 8 5 5 3 10 3.4 3 1.5 2.2 2.5 2.2 50 50 50 50 60 Ltot (1033 erg s−1 ) 0.14 3.0 48.9 16.9 26.8 458 3.3 2.9 4.4 2.0 0.6 13.1 1.4 2.0 83.4 7.2

Bmin (μG) ξ =1 39 66 17 27 54 267 51 21 15 14 14 78 42 39 52 38

Ecr (1048 erg) ξ =1 0.01 0.17 20.78 2.61 1.42 1.98 0.26 0.64 1.92 1.02 0.30 0.39 0.14 0.17 5.73 0.80

Bmin (μG) ξ = 100 146 245 65 100 202 998 190 79 55 53 54 293 157 146 194 141

Ecr (1048 erg) ξ = 100 0.18 2.4 290.4 36.4 19.9 27.6 3.6 8.9 26.8 14.2 4.2 5.5 1.9 2.3 80.1 11.1

Ecr /Ecr,ISM (× 0.01) 7 20 1 3 13 330 12 2 1 1 1 28 8 7 13 7

Table 12.1 Magnetic-field and cosmic-ray energies based on the minimum-energy principle for young supernova remnants, assuming a volume-filling fraction of 25%

12.1 Radio Observations 331

332

12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

Fig. 12.4 The flux decline of Cas A around 1420 MHz, based on observations around this frequency compiled in [113, 1127], with a correction to assess the fluxes at 1420 MHz, using α = 0.77

it contaminates the radio-flux measurements of nearby sources. So an accurate flux of Cas A needs to be known in order to correct for flux contamination. The flux evolution of Cas A has been monitored for over 60 years, and appears to have declined from 1% yr−1 in the 1960s/1970s [114] to about ∼0.5% yr−1 in more recent years [908, 1127] (Fig. 12.4). Cas A is one of the few supernova remnants for which flux evolution measurements can be made, the reason is that it very bright, and that the flux decline is rather strong, as it depends on the age of the object. Consequently, only for the even younger object G1.9+0.3 [462] and for several radio supernovae similar flux evolution measurements exist (e.g. [1207]). In order to explain the flux decline we follow here first somewhat the arguments of [1048], but with more up to date information on the age and expansion properties of Cas A. We make use of (12.7), but note that there is a subtle difference between flux density (Sν ) and integrated flux, which we will ignore here. We start with Shklovsky’s original idea that the relativistic electrons are a relic of the past history of Cas A, and that there is no ongoing acceleration. Equation (12.7) shows that the synchrotron luminosity is proportional to B 3/2 , V , and Ue . If the supernova remnant expands with an expansion parameter m (Sect. 5.2)—R ∝ t m —we have V ∝ t 3m , whereas magnetic-flux conservation gives B ∝ R −2 ∝ t −2m . The relativistic electron-energy density is given by the adiabatic scaling Ue V γ = constant, implying Ue ∝ t −3mγ , with γ = 4/3 the adiabatic index for a relativistic particle population.

12.1 Radio Observations

333

Scaling all quantities to a fiducial age of the supernova remnant t0 , we infer for the flux from the source as a function of time: Ssyn (t) =

Lsyn = S0 4πd 2

 −3mγ t t0

(12.12)

The flux decline rate to be expected is thus 3mγ 1 dSsyn =− . Ssyn dt t

(12.13)

Assuming Cas A to result from an explosion around 1672 [1114] and using m ≈ 2/3 [899, 1183] Eq. (12.13) gives 0.89% yr−1 for the year 1970 and 0.78% yr−1 for 2015. Given the somewhat erratic, and poorly understood, behaviour of the flux decline, Eq. (12.13) works surprisingly well. But it cannot explain the more rapid evolution of the rate itself from the early measurements of 1% yr−1 to 0.5% yr−1 in just ∼ 50 yr. As already stated, we neglected here the contribution to the emission coming from the ongoing electron acceleration. For the forward shock we can assume that the change in luminosity per unit is due to the increase in volume per unit time of 4πR 2 Vs , combining this with (12.7) gives 

dLsyn dt



= AB −1/2 UB Ucr 4πR 2 Vs ,

(12.14)

fs

with A all the non-variable parameters, and B, UB and Ucr quantities to be determined immediately downstream of the shock front. We can estimate the variable quantities as B2 ∝ρVs2+ζ ∝ t −2ms t (2+ζ )(m−1) ∝ t −m(s−2−ζ )−(2+ζ ) 8π Ucr ∝ρVs2 ∝ t −2ms t 2(m−1) ∝ t −m(s−2)−2 R 2 Vs ∝t 3m−1 . For ζ we can take ζ = 1, assuming the expected amplification due to the Bell’s instability (Sect. 11.4.3). With all this in place, we see that for the forward-shock region we have 

dLsyn dt



= Ct −(7/4)ms+(3/4)mζ +(13/2)m−(3/4)ζ −9/2 ≈ Ct −2.75 ,

fs

with C some constant, and m = 0.67 and ζ = 1 for the numerical value.

(12.15)

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

The total flux evolution of the supernova remnant is now described by the equation Ssyn

 a  b t t =C1 + C2 , t0 t0

a ≡ − 3γ m,

(12.16) (12.17)

b ≡ − (7/4)ms + (3/4)mζ + (13/2)m − (3/4)ζ − 7/2. This equation can be solved, such that for t0 ≈ 340 yr—corresponding to Cas A—we have Sν,0 = 2282 Jy at 1 GHz, and a decline rate of r0 = 0.5% yr−1 —again ignoring the subtle difference between flux density and flux, but note that Ssyn ≈ νmax Sν . The solution to (12.16) is then C2 = S0 − C1 and C1 = (r0 − b)/(a − b)S0 , and r0 t0 = b + (a − b)C1 /S0 . This solution is shown in Fig. 12.5. For ξ = 1 the flux has a maximum close to the explosion date, which is probably a lucky coincidence as the model cannot be reliably extrapolated to early stages of the evolution. For comparison, SN 1993J, a supernova similar to Cas A (Sect. 2.5) had a flux density of about 140 mJy 1000 days after the explosion [1207]. The distance to SN 1993J is 3.6 Mpc, but scaling it to the distance of Cas A gives 1.6 × 105 Jy, lower than the ξ ≤ 1 model light curves indicate. This simple model provides some insights into the expected radio synchrotron evolution of supernova remnants, but is incomplete for several reasons. First of all, m is taken be constant here, whereas in reality it evolves (Chap. 5). Secondly, particle acceleration at the reverse shock is ignored, whereas there is strong evidence that reverse shock acceleration is important for Cas A (see Sect. 12.2) and perhaps also for other remnants. Thirdly, the model predicts an infinite flux for t = 0, if

Fig. 12.5 Radio flux evolution based on the model in Sect. 12.1.3, with various values for the downstream magnetic-field evolution parameter ξ . Left: The model is fine tuned to roughly agree with Cas A: s = 2, m = 0.67, and flux decline rate of −0.5%/yr. The magenta lines indicate the measured evolution of Cas A. Right: This model roughly corresponds to G1.9+0.3: s = 0, m = 0.8 and the evolution rate is 1.9%/yr. The dotted lines are for m = 0.4, corresponding to a more evolved supernova remnant

12.1 Radio Observations

335

we chose an input value of r0 < −0.6% yr−1 , a value that was measured in the past. Moreover, the strong evolution of the flux decline rate itself, from −1% yr−1 to −0.5% yr−1 , cannot be captured by this model. This in itself may suggest that a simple s = 2 model may not apply to Cas A. Instead it seems to suggest that Cas A underwent a brightening not long before the first radio flux measurements of Cas A were made. The early −1% yr−1 rate may, therefore, be an anomaly, and the current lower decline rate may be more typical for the general evolution of Cas A. Of course, this is speculative. Continued monitoring of the flux-evolution of Cas A may provide further insights. Besides Cas A, the only other flux-evolution measurement of a nearby supernova remnant is for the youngest known Galactic remnant G1.9+0.3, for which the estimated evolution suggest a rate of r0 = 1.9% yr−1 [462]. G1.9+0.3 is likely a Type Ia supernova remnant, and not evolving in a stellar wind profile; so s = 0. As a consequence the radio-flux evolution is different. For the model shown in Fig. 12.5 (right) we have used m = 0.8. This is not a measured value, but seems a reasonable choice for a young supernova remnant. For this choice the radio flux keeps increasing. For contrast we also show an m = 0.4 flux evolution. This corresponds to the Sedov-Taylor solution, which is applicable to later evolutionary stages. This is likely not applicable to G1.9+0.3, but it serves here to show that the flux evolution for an s = 0 model does not necessarily result in 1an ever increasing radio flux. The flux evolution (12.16) can be easily converted in an approximate surfacebrightness diameter ( − D) relation by using that D ∝ t m :  = C1 D m −2m + C2 D m −2m . a

b

(12.18)

If the first term dominates, i.e. cooling and magnetic-field decline dominate, we expect that  ∝ D −4.33 for m = 0.67 Assuming that the second term dominates, that the supernova remnant is in de Sedov-Taylor phase, and neglecting magnetic field amplification and assuming a uniform ambient medium, we find  ∝ D −3.5 . The decline in surface brightness for both cases are steeper than observed, although a slope of −3.5 is not too far off from the observed relation (Sect. 3.2.3).

12.1.4 Radio Polarisation Measurements Synchrotron radiation is intrinsically polarised (Sect. 13.3), with electric polarisation vectors informing us about the overall magnetic-field orientation—the magnetic-field orientation being perpendicular to the polarisation vectors. The measured polarisation from a region of a supernova remnant depends on the level of magnetic-field turbulence, with turbulent fields reducing the polarisation fraction; line of sight effects (emission from front and back side of the shell may have different polarisation orientations, reducing the overall polarisation fraction), and internal Faraday rotation, which can reduce the polarisation fraction.

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

Fig. 12.6 An example of a young supernova remnant with a radial magnetic field (left) and a mature supernova remnant (right) with a tangential magnetic field. Shown are the B-vectors. Left: Kepler’s supernova remnant observed with the VLA at 6 cm [301]. Right: G89.0+4.7 (HB21) as observed with the Chinese 25 m Urumqi telescope at 6 cm [488] (Courtesy T. Delaney (left) and J.L. Han)

Naively one would expect the magnetic-field orientation in a supernova remnant shell to be tangential, i.e. perpendicular to the radial vector. The reason is that the shock compresses the magnetic field in this direction, and does not affect the parallel component of the magnetic field. In other words, Bφ,1 = χBφ,2 and Br,1 = Br,2 , with the subscripts 1 and 2 referring to the upstream (unshocked) and downstream region respectively. Indeed, many supernova remnants show polarisation vectors consistent with a tangentially oriented magnetic field (e.g. [309]). In contrast, younger supernova remnants have in general radially oriented magnetic fields; see Fig. 12.6 for two examples. In addition to the polarisation direction also the polarisation fraction is found to be different between young and old supernova remnants. The polarisation fraction for young supernova remnants is in general much lower than for mature supernova remnants [310]. Figure 12.7 illustrates this by showing the polarisation fraction as a function of surface brightness. Note that older, more extended, supernova remnants have lower surface brightness than younger supernova remnants (Sect. 3.2.3). The measured polarisation fractions of several young supernova remnants are listed in Table 12.2. As discussed in Sect. 13.3, for a uniform magnetic field, the polarisation fraction of synchrotron radiation from a power-law distribution of electrons depends on the radio spectral index: =

q +1 α+1 = . q + 7/3 α + 5/3

(12.19)

12.1 Radio Observations

337

Fig. 12.7 The radio polarisation fraction as a function of the radio surface brightness, taken from [310] and based largely on the sample in [826]. Surface brightness is here a proxy for size and/or age; low surface brightness remnants tend to be larger and older. It indirectly shows that young SNRs have low polarisation fractions (few % to ∼ 20%) Table 12.2 Radio polarisation measurements of young supernova remnants Object

Typical  Peak 

Orientation B

λ

RCW 86 SN 1006

8% 17%

15% 60%

Radial Mostly radial

22 cm Some regions  < 3% [313] 20 cm Peak not in X-ray rims [969]

Remarks

SN 1572 SN 1604

7% 6%

25% 12% (?)

Radial/fine struct. (4 ) Radial/fine struct. (20 )

Cas A G1.9+0.3

5% 6%

∼20% Radial 17 ± 3% Radial

6 cm Peak at limbs 6 cm

Ref.

[314] [301]

6 cm Outer plateau  ≈ 9% [76, 209] 6 cm Faraday rotation? [292]

This compilation was taken from [1178]

For α ≈ 0.5 this  ≈ 69%. For low-surface-brightness supernova remnants the measured fraction reaches up to  ≈ 20–30%, indicating a relatively well-ordered magnetic field. However, the very bright and high-surface-brightness synchrotron emission from Cas A shows a very low fraction of polarisation  ≈ 5%, indicative of a highly turbulent magnetic field. The turbulence of the magnetic field was often attributed to hydrodynamical instabilities in the shell [472]. But presently the idea is that the turbulence is caused by cosmic-ray driven magnetic-field turbulence in the cosmic-ray precursor (Sect. 11.4). The lower polarisation fraction for young supernova remnants then implies that cosmic-ray driven turbulence is much more prevalent in the precursors of the fast moving shocks of young supernova remnants. As for the origin of the radial orientation of the magnetic field, there is no generally accepted theory. Even if cosmic-ray induced turbulence does not impose a

338

12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

net orientation on the turbulent magnetic field upstream of the shock, the shock compression would still enhance the tangential component. It could be that the radial magnetic-field orientation is due to the preferentially radial stretching of magnetic fields in Rayleigh-Taylor unstable regions (Sect. 5.9). But according to [1284] the non-resonant magnetic-field instabilities in the precursor give rise to density fluctuations, which are advected downstream of the shock, where it leads to non-radial motions of the plasma, enhancing the radial component of the field. This would imply that the radial orientation of the magnetic fields are directly connected to the presence of cosmic-ray induced magnetic-field instabilities. Yet another theory uses the idea that shock acceleration is more efficient for parallel magnetic-field orientations (see Sect. 12.2.6). In those regions the post-shock region would then accumulated more accelerated electrons, dominating the synchrotron emission [1217]. This theory, therefore, suggests the magnetic field is overall not radially oriented, but that synchrotron emission is dominated by regions with radially oriented magnetic fields. Note that the latter two ideas both are associated with the efficiency of cosmicray acceleration, but connect this to two different aspects of efficient acceleration: non-resonant magnetic field amplification [1284] and magnetic-field orientation dependent acceleration efficiencies [1217].

12.2 X-ray Synchrotron Radiation For a long time synchrotron emission from shell-type supernova remnants was synonymous to radio emission, whereas the X-ray emission from the shell was understood to be thermal X-ray emission (with the exception of synchrotron emission from an embedded pulsar wind nebula in composite supernova remnants, see Chap. 6). This commonly held view changed in 1995, when observations with the Japanese X-ray satellite ASCA (Advance Satellite for Cosmology and Astrophysics) showed that SN 1006 emits X-ray synchrotron radiation [679]. That there was something unusual about the X-ray emission from this supernova remnant was already known since the 1980s, as NASA’s Einstein satellite observed the spectrum to be devoid of line emission. Indeed, Reynolds and Chevalier [964] suggested that the continuum emission was due to synchrotron radiation rather than bremsstrahlung. However, an alternative model [483] attributed the lack of X-ray line emission to a plasma whose composition consists predominantly of carbon. The model was based on the idea that SN 1006 was the result of a Type Ia supernova, and that the reverse shock had so far only penetrated the carbon-rich outer layers of the white dwarf. This could help explain the lack of line emission, since carbon itself is almost completely ionised for electron temperatures above 0.5 keV. Moreover, carbon emission lines have energies around 0.37 keV, which can easily be missed as this part of the spectrum is affected by interstellar absorption. The discovery by ASCA was not so much the result of higher resolution spectroscopy, but caused by the emergence of X-ray imaging spectroscopy (or

12.2 X-ray Synchrotron Radiation

339

Fig. 12.8 X-ray emission from SN 1006 as observed by the Chandra ACIS detectors. The red channel corresponds to O VII line emission around 0.58 keV, the green channel to 0.85–2 keV, a combination of X-ray synchrotron emission and line emission from various Ne, Mg and Si ions, and the blue channel to emission between 3 and 6 keV, which is dominated by X-ray synchrotron emission

integral-field spectroscopy). This makes it possible to obtain spectra of many pixels simultaneously. Imaging spectroscopy revealed that the X-ray strong continuum radiation from SN 1006 originates from two crescent-like regions close to the shock front (Fig. 12.8), whereas the X-ray emission from the interior of SN 1006 was shown to be dominated by line emission, mostly from oxygen, neon, magnesium and silicon. This disproved the carbon-rich model and was in agreement with the X-ray synchrotron model. The discovery was followed not much later by the detection of hard X-ray emission from the young supernova remnant Cas A, with hard X-ray detectors on board the Compton Gamma-ray Observatory (CGRO), RXTE, and Beppo-SAX [68, 372, 1108]. Initially there was some debate about the nature of the hard Xray emission: was it synchrotron radiation or was it non-thermal bremsstrahlung (Sect. 13.4.5) from the low-energy part of the electron cosmic-ray spectrum [101, 372, 694, 1170]? However, after NASA’s Chandra provided evidence for X-ray synchrotron emission from the shock fronts of Cas A [454, 502, 901, 1176] and other young supernova remnants [572, 972, 1055], it has become generally accepted that the hard X-ray emission is caused by X-ray synchrotron radiation. This has been confirmed through hard X-ray imaging spectroscopy with the NuStar satellite, which showed that the hard X-ray continuum spatially coincides with the regions of X-ray synchrotron radiation in Cas A and Tycho’s SNRs [464, 737].

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12.2.1 The Implication of X-ray Synchrotron Radiation For the typical magnetic fields expected for supernova remnants, ranging from a few μG to 0.5 mG (Sect. 12.1.2), the emission of X-ray synchrotron radiation implies the presence of electrons with energies  1013 eV (13.39), much higher in energy than the radio emitting electrons, and closer to the “knee” in the cosmic-ray spectrum at 3 × 1015 eV (Sect. 11.1.1). More importantly, X-ray synchrotron radiation can only arise if the magnetic-field upstream of the shock is highly turbulent, resulting in a slow diffusion, which corresponds to fast acceleration. This is a consequence of (11.51), which shows that X-ray synchrotron emission above 1 keV requires Vs  3000 km s−1 , but only if η  10, and preferably even closer to η = 1 (i.e. Bohm diffusion). So the “optimistic scenario” for particle acceleration referred to by Lagage and Cesarsky [691]—i.e. η ≈ 1 (Sect. 11.2.3)—appears to be justified based just on the detection of X-ray synchrotron radiation. In deriving (11.51) we assumed that the energy gains caused by diffusive shock acceleration are balanced by radiative energy losses (i.e. synchrotron radiation losses). This is referred to as a loss-limited synchrotron spectrum. This assumption is most often used in analysing and interpreting X-ray synchrotron emission from young supernova remnants. Alternatively one could assume that the electrons producing X-ray synchrotron radiation are still in the process of being accelerated to higher energies; the so-called age-limited synchrotron spectrum. To see under which conditions one could expect age-limited X-ray synchrotron emission, we revert once more to the approximate relation Vs = mRs /tsnr . Clearly, in that case the age of the supernova remnant should be shorter than the age of the supernova remnant: tsnr = m

 −1/2  −3/2 hν Rs 634 B < τsyn ≈ 2 ≈ 5.5 yr, Vs 1 keV 100 μG B2 E

(12.20)

for which have used (13.47), and combined it with (13.39). We assume that synchrotron losses are dominated by the electrons residing downstream of the shock, since the magnetic-field strengths is largest there, and the particles dwell longer. So loss-limited X-ray synchrotron spectra require the following constraint on the downstream magnetic field as a function of radius and shock velocity: B2 > 15m

−2/3



hν 1 keV

−1/3 

Vs 5000 km s−1

2/3 

Rs 5 pc

−2/3

μG.

(12.21)

The magnetic-field limit of 15 μG corresponds roughly with the shock-compressed interstellar magnetic-field strength. So unless Vs > 5000 km s−1 , and/or Rs < 5 pc, the X-ray synchrotron spectrum is expected to be limited by radiative losses.

hνmax < 3η

−1



Vs 5000 kms−1

2 

χ−1 (1+χχB )

3/17

 keV.

(12.22)

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341

There is not much new information in this, but it does emphasise that also for agelimited X-ray synchrotron models one needs a small value for η. In other words, whatever the favourite model for X-ray synchrotron emission (age- or loss-limited), one always requires a high level of magnetic-field turbulence. A high magnetic-field turbulence is generally the outcome of either resonant or non-resonant magnetic-field amplification (Sect. 11.4). A general problem with the age-limited model is that on the one hand a low magnetic-field strength is required (12.21), whereas on the other the magnetic-field should be highly turbulent. However, there may be one exception, the young supernova remnant G1.9+0.3, which is compact and has a shock velocity of around 11,000 km s−1 [238], fulfilling the limits of (12.21).

12.2.2 The Narrow Widths of the X-ray Synchrotron Regions The ASCA satellite was first in opening up the sky to X-ray imaging spectroscopy, but had a relatively poor angular resolution of ∼1 . The next generation of large Xray telescope—ESA’s XMM-Newton satellite and, in particular, NASA’s Chandra X-ray Observatory—dramatically improved the angular resolution for imaging spectroscopy. Chandra’s resolution of ≈0.5 rivals that of the Hubble Space Telescope in the optical. The high spatial resolution of Chandra revealed that even young supernova remnants like Cas A, Kepler’s SNR, and Tycho’s SNR, whose X-ray emission is dominated by line radiation, have filaments at the shock front, which are dominated by continuum emission [454, 572]. The featureless spectra of these filaments were similar to the X-ray spectra from the crescent regions in SN 1006, and it was quickly accepted that also in this case the X-ray continuum was dominated by synchrotron radiation. For these young supernova remnants the filaments were found to be narrow, as can be appreciated in Fig. 12.9. In fact,

Fig. 12.9 Three young supernova remnants as observed by the Chandra X-ray Observatory, all with the 4–6 keV synchrotron dominated band assigned to the blue channel. From left to right: Cas A, Kepler’s SNR, and Tycho’s SNR. As can be seen the narrow (∼1 ) synchrotron emitting regions are confined to the forward shock regions in Tycho’s and Kepler’s supernova remnants, whereas for Cas A in addition there also filaments in the interior, mostly concentrated toward the Western part of the remnant

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Table 12.3 Measured X-ray synchrotron filaments width and derived magnetic field strengths and other relevant parameters, based on profiles shown in Fig. 12.11 Dist nH,0 Vs

R ldiff B2 Eel τsyn ID SNR (kpc) (cm−3 ) (km/s) ( ) (1017 cm) (μG) (TeV) (yr) 1 G1.9+0.3 (SW) 8.5 0.022 11520 3.1 2.8 66.6 33 86 2 Cas A (NE) 3.4 0.9 4773 1.1 0.4 246.5 17 12 3 Kepler (SE) 5.0 0.05 5390 1.8 0.9 137.8 23 29 4 Tycho (W) 3.0 0.5 4579 1.6 0.5 207.0 19 16 5 SN1006 (E) 2.2 0.085 4795 9.1 2.1 81.1 30 64 6 RCW 86 (NE) 2.5 0.01 3000 28.6 7.6 34.5 46 232 7 RX J1713.7-3946 1.0 0.1 2592 63.5 6.7 37.3 44 206 8 RX J0852.0-4622 1.0 0.03 3990 28.4 3.0 63.9 34 92 Data taken from [506], based on the profiles shown in Fig. 12.11. See [506] for further references on all the parameters. Updated are: density Cas A [710], the distance to Kepler’s SNR [993, 1174], the shock velocity for RCW 86 [1256]

they can only be resolved by Chandra, given their widths of the order of 1 . See Table 12.3 for a list of typical filament widths. Soon after the discovery of these filaments it was realised that their narrow widths implies relatively high magnetic-field strength [124, 131, 1176, 1191]. The simplest way to understand is to consider how electrons are advected by downstream plasma flow, which transports highly relativistic electrons away from the shock acceleration region. While these electrons are advected downstream, they lose their energy through synchrotron losses on a time scale τsyn (13.47). This is implies that at a given distance downstream of the shock the highest energy electrons will have lost so much energy that they no longer emit X-ray radiation. Since the advection flow has a velocity v2 = v = Vs /χ (Sect. 4), the width of the X-ray synchrotron filament is given by the advection length scale ladv ≈ vτsyn ≈

Vs 634 , χ B22 E

(12.23)

with E the electron energy. This equation can be rewritten using the relation between photon energy, electron energy and magnetic-field strength (13.39) into  ladv ≈ 2.2 × 10

17

hν 1 keV

−1/2 

B2 100 μG

−3/2 

Vs 5000 km s−1

  χ −1 cm, 4 (12.24)

from which we infer a downstream magnetic-field strength of  B2 = 178

ladv 1017 cm

−2/3 

hν 1 keV

−1/3 

Vs 5000 km s−1

2/3   χ −2/3 μG. 4 (12.25)

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343

For Cas A, which has a shock velocity of ≈ 5000 km s−1 , the typical filament widths of 1 imply an advection length scale of 5 × 1016 cm, corresponding to magnetic-field strengths of the order of B2 ≈ 280 μG. This is much larger than can be expected from a compressed interstellar or circumstellar magnetic field. The magnetic-field strength is comparable with what had been derived based on the radio luminosity and using the minimum energy requirement (Sect. 12.1.2). But is should be remembered that (1) the validity of the minimum-energy principle is not completely justified, and (2) the magnetic-field strength derived from (12.25) refer to the magnetic field immediately downstream of the shock, whereas the radio-based estimates concern the entire shell of the remnant. The high magnetic-field strength based on radio observation was in fact attributed to turbulent amplification due to hydrodynamical instabilities in the bright shell of Cas A [472]. So the narrow widths of the X-ray filaments of Cas A, Tycho’s SNR and Kepler’s SNR provided the first evidence that the magnetic-field strengths are large immediately downstream of the forward shock. The above explanation for the X-ray synchrotron filament widths is in principle valid even for lower photon energies, as long as synchrotron losses are important. However, for loss-limited spectra one can derive an equation that is very similar to (12.25), but which does not depend on the photon-energy, and only relies on the assumption that the emitting electrons have energies near maximum energy of a loss-limited population, i.e. as given by (11.50) [505, 1187, 1191]. There are several ways of deriving the velocity independent synchrotron filaments widths, but here we start by assuming that most of the radiative losses occur downstream of the shock.The dwell-time of an electron still in the process of being accelerated is τ2 =

D2 D2 3 3 D2 = ≈ 2, v1 − v2 v22 (χ − 1) v22 v2

(12.26)

for χ = 4 (11.35). If we now require that an electron with an energy close to the maximum energy will have a dwell time that is similar to the synchrotron loss time, we have τ2 =

D2 ≈τsyn. v2

(12.27)

We can rewrite this using the relation between diffusion length scale and dwell-time given in Sect. 11.2.2: ldiff,2 =

D2 = τ2 v2 ≈v2 τsyn = ladv . v2

(12.28)

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So we see that for electrons with approximately maximum energy the diffusion length scale roughly equals the advection length scale. This implies that for these electrons v2 = ldiff,2 /τsyn , which we can insert back into ldiff,2 = D2 /v2 to obtain ldiff,2 ≈

D2 ldiff,2

τsyn ⇒ ladv ≈ ldiff,2 ≈



D2 τsyn .

(12.29)

Using for D2 Eq. (11.3), but with subscript 2 rather than 1, and E = Emax , we find that  B2 = 110 η1/3

ladv 1017 cm

−2/3

μG,

(12.30)

which is very similar to (12.25), except that in this equation the magnetic field strength derived is independent of the observed photon energy. Downstream magnetic-field strength estimates, based on this equation, are listed in Table 12.3. The fact that for most supernova remnants the magnetic field estimates based on (12.25) and on (12.30) give very similar results [124, 506], which in itself justifies both the assumption that the cut-off synchrotron frequency is limited by synchrotron losses, and that η  10, given that (12.30) does depend on η and (12.25) is independent of η. The three young supernova remnants in Fig. 12.9 and SN 1006 are not the only Xray synchrotron emitting supernova remnants. G1.9+0.6 is the youngest and smallest known supernova remnants, and its X-ray spectrum is dominated by synchrotron radiation. Then there are a number of X-ray synchrotron emitting remnants that are quite extended: RX J1713.7-3946 (Fig. 12.10) [31, 677, 1058], RX J0852.0-4622 (“Vela Jr”) [622, 1059], RCW 86 [126, 200, 972, 1187] and HESS J1731-347 (G353.6-0.7) [129, 1115]. These objects have angular diameters larger than 30 (the diameter of SN 1006), and physical radii of more than 10 pc. Since X-ray synchrotron emission requires shock speeds in excess of ≈ 3000 km s−1 , their large radii suggest that these remnants are evolving in low-density environments. This is the most likely reason why RX J1713.7-3946, RX J0852.0-4622 and HESS J1731-347 have Xray spectra that are dominated by X-ray synchrotron emission, with hardly any Xray line emission discernible [32, 629]: line emission scales with density squared (n2 ), whereas synchrotron emission scales roughly with n (ignoring magnetic field effects). The X-ray synchrotron dominated regions for these supernova remnants are quite wide and can be resolved with XMM-Newton (∼15 ), suggesting that the magnetic fields are not as strong as for Cas A, Tycho’s SNR and Kepler’s SNR. See also Table 12.3. RCW 86 (G315.4-2.3, Fig. 12.10) a remnant linked to the historical supernova of AD 185 [1078], presents another interesting case. A large part of this remnant emits thermal X-ray emission, and some of these thermal emission regions are even associated with radiative shocks [987], implying shock velocities below 200 km s−1 (Sect. 4.4) But a region in the southwestern part and a shock region in the Northeast

12.2 X-ray Synchrotron Radiation

345

Fig. 12.10 Left: XMM-Newton/EPIC mosaic image of RX J1713.7-3946. The colours are chosen to enhance the faint regions, in this X-ray image of the 0.5–4.5 keV band [31]. (Image based on a mosaic provided by F. Acero.) Right: Mosaic of XMM-Newton and Chandra X-ray observations of RCW 86, adapted by the author from archival data [972, 1187]. The colours correspond to 0.5–1 keV (red), 1–2.0 keV (green) and 2–6 keV (blue)

are completely dominated by synchrotron radiation, whereas some faint synchrotron emission seems to come from some interior regions and from along the northern shell [126, 200, 1187]. Shock velocity measurements for the thermally emitting regions suggest velocities below 600 km s−1 [432, 732], and in the eastern region velocities of ∼1000 km s−1 have been measured [504]. These velocities are in sharp contrast to the 3000 km s−1 needed for the X-ray synchrotron emission. Indeed the northeastern shell has been measured to expand with ≈3000 km s−1 [1256]. These very contrasting shock velocities are best explained by a model in which RCW 86 is evolving in a wind-blown bubble [1182, 1229]; see Sect. 5.8. Inside the windbubble the shock velocity remains high, but as soon as the shock wave encounters the dense shell surrounding the bubble, the shock wave rapidly decelerates. For RCW 86 this probably means that some parts have penetrated the shell, whereas the X-ray synchrotron emitting parts are still within the low density bubble [212, 504]. This suggests that not too long ago, before the shock encountered the shell, RCW 86 may have been very similar to RX J1713.7-3946 and RX J0852.0-4622.

12.2.3 The Case for Magnetic-Field Amplification In Fig. 12.11 the flux profiles of X-ray synchrotron filaments of eight young supernova remnants are shown (taken from [506]), including a model based on exponentially decreasing emissivity profile, assuming a spherical projection on the sky. The relevant parameters for these supernova remnants, the fitted X-ray

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

Fig. 12.11 Overview of X-ray synchrotron emission profiles in several supernova remnants. The profiles are extracted from Chandra observations, with the exception of RX J1713.7-3946 which is based on an XMM-Newton mosaic provided by Dr. Fabio Acero [31]. The solid red line indicates a best-fit model based on a spherical projection model, with an exponential fall off in emissivity downstream of the shock (Reproduced from [506])

12.2 X-ray Synchrotron Radiation

347

synchrotron filament width ( R), and derived downstream magnetic-field strength are listed in Table 12.3. Clearly the model profiles do not fit the data perfectly, possibly reflecting that the supernova remnants are not perfectly spherical symmetric, but perhaps also indicating that the magnetic fields may not be constant downstream of the shock [926], suggesting that the synchrotron emissivity profiles may be more complex the assumed exponential fall-off. However, the fitted values give a good indication of the length scales over which the X-ray synchrotron emission falls off, and, hence, of the magnetic-field strengths downstream of the shocks (Sect. 12.2.2). For all these young supernova remnants the estimated magnetic-fields strength are larger than the compressed interstellar magnetic field strength, which should be of the order of 20 μG [124, 130, 884, 1176, 1191]. The X-ray synchrotron filaments provide, therefore, important evidence for magnetic-field amplification near supernova remnants shocks as suggested by several theories, most notably the theory about the Bell-instability (Sect. 11.4.3, [143, 146]). Additional support for cosmic-ray induced magnetic-field amplification is that it predicts highly turbulent magnetic fields, which is also required by the occurrence of X-ray synchrotron emission (Sect. 12.2.1). It has been argued that the turbulent magnetic field may decay downstream of the shock and that at least part of the width of the filaments are due to the magnetic-field damping length scale, rather than due to radiative losses [926]. If this is the case then the magnetic field estimates in Table 12.3 provide upper limits. Calculations and model fits in [960] show that in case of magnetic-field damping the derived magnetic-fields may need to be lowered by as much as 50%, but remain well above the values expected for the compressed interstellar magnetic field. A consequence of magnetic-field damping will be that the spectrum downstream of the shock will not steepen in spectral index as much as a pure advection model. One of the predictions for the Bell instability is that the magnetic field pressure upstream of the shock, in case of saturation, scales as B12 ∝ ρVs3 [143], but B12 ∝ ρVs2 is another possibility [146, 1191]. The downstream magnetic field estimates in Table 12.3 have been used to distinguish between the two cases [506, 1169, 1173]. The supernova remnant data itself, as shown in Fig. 12.12, are inconclusive, although it is clear that B 2 increases with both ρ0 and Vs . A problem is that the shock velocities of these young supernova remnants occupy a narrow range between ∼3000–6000 km s−1 , with some systematic uncertainties, as the velocities are based on proper motion studies, and for some of them the distance is uncertain by a large factor. The case for scaling as B 2 /(8π) ∝ ρVs3 is more compelling when one takes into account the magnetic-field estimates based on radio observations of the supernova SN 1993J, which evolves in a very high-density environment with a high shock velocity [1099]. However, the magnetic-field estimates for SN 1993J are somewhat more indirect and are based on a different method than for the supernova remnants, relying on modelling the evolution of the radio spectrum of SN 1993J.

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

Fig. 12.12 The downstream magnetic-field pressure divided by the upstream density (B22 /(8πρ0 )) versus the shock velocity. Labels 1 to 8 refer to the values given in Table 12.3. The labels 9 and 10 refer to the radio supernova SN 1993J and are based on the model fits in [1099], with 9 referring to day 100 after the explosion day and 10 to day 1000

Figure 12.12 shows that for supernova remnants the downstream magnetic field strength appears to be given by the approximate relation B2 ≈ 0.01ρ0Vs2 8π



Vs 4.0 × 103 kms−1

2+x ,

(12.31)

with 4000 km s−1 being close to the crossing of the two dependencies, and with x = 0 or x = 1 a parameter to distinguish between the possible scalings of the magnetic field with the shock velocity. The relation shows that the downstream magneticfield pressure is about 1% of the shock’s ram pressure. A higher value of 3.5% was reported in [1191], as the estimated magnetic-field strength were larger, the shock velocities used were lower than based on recent measurements for Cas A and RCW 86. Either way, the magnetic-field pressure is a minor fraction of the downstream pressure, which should be of the order of 75% of the ram pressure. It is also less, but perhaps comparable, to the 5–10% cosmic-ray pressure expected if supernova remnants are the dominant sources of Galactic cosmic rays (Sect. 11.1.4).

12.2 X-ray Synchrotron Radiation

349

12.2.4 Magnetic-Field Amplification Near the Reverse Shock Cosmic-ray acceleration is usually associated with the forward shock of supernova remnants. One reason is that over the life time of an supernova remnant more mass will be heated by the forward shock than by the reverse shocks (Sect. 5.3), so the forward shock is likely to dominate the number of particles being accelerated. Another reason is that the unshocked ejecta is expected to have a low magneticfield strength, as the magnetic field originates from the magnetic field of the star, but magnetic-flux conservations implies that the magnetic field has been stretched out by the expansion. Given the expectations of a low magnetic-field strength, and its importance for particle acceleration, the reverse shock is usually neglected as a source of cosmic rays. However, X-ray synchrotron emission from the southwestern region of RCW 86 (Fig. 12.10) appears to come from the interior of the remnant, suggesting that it is perhaps coming from plasma shocked by the reverse shock [972]. Since the 10–100 TeV electrons responsible for the emission are short lived (Table 12.3), the electrons must have been accelerated relatively fast by the reverse shock. Cas A provides an even clearer case for X-ray synchrotron emission associated with the reverse shock. As can be seen in Fig. 12.13 hard X-ray spectra are

Fig. 12.13 X-ray image of Cas A, enhanced to bring out the two X-ray synchrotron emitting regions: the outer shock front region, and the interior region associated with the reverse shock, c.f. [90, 502]. This image was made by plotting the ratio of two continuum-dominated images, the 5–6.3 keV band and the 3.4–3.6 keV band. Bright regions have harder spectra, and are therefore likely to be dominated by X-ray synchrotron emission (Based on Chandra observations taken in 2004)

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

associated with two regions, one is associated with the forward shock, and another roughly spherical region lies to the shell. In [502] it was shown that deprojecting the 4–6 keV X-ray continuum images showed that the emission is indeed consistent with a thin spherical shell in the interior, somewhat shifted toward the western region, as compared to the outer shock front. A radio-absorption study [90] provided another means of localising the location of the reverse shock. This indicates that the radio-determined reverse shock regions extends more toward the East than indicated by the X-ray synchrotron map, but in the western part the radio- and Xray agree on the outer location of the reverse shock. The most likely explanation for the discrepancy is that in the eastern part the reverse shock does not accelerate electrons to high enough energies to emit X-ray synchrotron radiation. This implies that either the reverse shock velocity, as given by (5.6), is smaller than the limit for X-ray synchrotron emission—≈3000 km s−1 (11.51)—or that the magneticfield turbulence is too low. There is evidence that the forward shock in the western region of Cas A has slowed down considerably [75, 502, 1183], or is even moving backward. This means that the ejecta enter the reverse shock with a relative velocity of Rrs /t ≈ 5000 km s−1 , for a reverse shock radius of 1.8 pc. Hydrodynamical models predict a reverse shock that is still moving outward with 2000 km s−1 , which may be more typical for the eastern part of Cas A [873]. With such a reverse shock velocity the ejecta may be shocked with 3000 km s−1 . Since the X-ray synchrotron emission clearly indicate that part of the reverse shock is capable of accelerating electrons, it is plausible that the radio emission from the bright radio shell of Cas A (Fig. 12.1) also originates from electrons accelerated by the reverse shock. Another, even more important consequence is that apparently the magnetic-field strength and turbulence near the reverse shock is strong enough to accelerate electrons rapidly beyond a few TeV. This suggests that magnetic-field amplification mechanisms do not require a large seed magnetic-field strength in order to create strong and turbulent magnetic field. Note in this context that the interior X-ray filaments in Cas A have widths very comparable to the forward shock, and, therefore, indicate similar magnetic-field strengths of a few 100 μG.

12.2.5 X-ray Synchrotron Flickering and Flux Decline The typical synchrotron loss time scales for the electrons producing X-ray emission ranges from ∼10 to ∼250 yr (Table 12.3). The shortest of these time scales are comparable to the life times of the X-ray telescopes XMM-Newton and Chandra. Since we observe electrons approximately near the limit where the acceleration time scale equals the radiative loss time, any local variation in acceleration properties as a function of time will result in variations in X-ray flux on an observable time scale. Indeed, time variability of X-ray synchrotron emission has been reported for small features in RX J1713.7-3946 [1145] and Cas A [898, 901, 1143]. The “flickering” time scale of several year is consistent with the high magnetic field reported for Cas A (Table 12.3), but the inferred magnetic field strength of ∼1mG

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351

for RX J1713.7-3946 is at odds with the value listed in Table 12.3. Part of the controversy regarding the nature of the γ -ray emission from RX J1713.7-3946 (see Sect. 12.3), revolves around the question what the magnetic-field strength in this remnant is. More observations and/or a critical reevaluation of the data and model assumptions may be needed to resolve this discrepancy. Long term monitoring of Cas A by Chandra also reveals a steady flux decline rate of the X-ray synchrotron emission of the order of 2 to 4%/yr [901], later revised to −0.4 to −0.6%/yr [1015]. The latter value is comparable to the radio synchrotron flux decline rate of 0.5–1%/yr (Sect. 12.1.3). But note that the X-ray synchrotron flux decline is more sensitive to the shock properties, whereas the radio flux decline is more the result of the adiabatic changes of the entire electron population and interior magnetic-field strength as a result of the expansion of the shell. The X-ray synchrotron emission is in particular very sensitive to changes in the magnetic field and acceleration speed, because the electrons producing X-ray synchrotron emission have energies around the exponential cut-off of the electron spectrum; any change in the cut-off energy will result in rather strong variation in X-ray flux. To illustrate this point we can approximate the X-ray synchrotron spectrum with 

ν FX (hν) =KB (α+1)(hν)−(α+1) exp − , νc

(12.32)

with α the spectral index (similar as in radio) and νc the cut-off frequency, K is the normalisation of the electron spectrum. Note that (12.32) is different from the overall synchrotron spectrum assumed in [1015, 1281], in that (12.32) does not contain a cooling break (Sect. 13.3.3). The view here is that the small region downstream of the shock comes from a region small enough to have no cooling break. For the overall synchrotron spectrum from the entire supernova remnant a cooling break should be included in (12.32), if one were to connect the radio to the X-ray synchrotron spectrum. The cut-off frequency in (12.32) relates to the electron spectrum cut-off frequency as νc = 1.8 × 1018Ec2 B Hz (Sect. 13.3). In Sect. 12.2.3 we showed that B 2 ∝ ρ0 Vs2+x , and for Cas A we have probably a density profile that scales as ρ0 ∝ Rs−s , with s = 2. Using also that νc corresponds Emax,e as derived in Sect. 11.2.8, we find that for the now familiar approximate supernova remnant evolution law Rs ∝ t m all the relevant quantities in (12.32) can be approximated by the following scalings: Rs t

∝t m−1 ,

(12.33)

ρ0 ∝Rs−s  B ∝ ρ0 Vs2+x

∝t −sm ,

(12.34)

∝t 2 (2−s)m−1+ 2 x(m−1) ,

(12.35)

νc ∝Vs2

∝t 2(m−1) .

(12.36)

Vs ∝

1

1

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12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

One more thing to consider is the normalisation K. We use the ansatz that K should be proportional to the number of particles entering the shock, ∝ ρVs , and to the volume occupied by the X-ray emitting electrons, i.e. ≈ 4πRs2 ldiff (12.27). So for K we assume the following proportionality K ∝ ρ0 Vs Rs2 ldiff ∝ Vs Rs2

1 νc ∝ t −3m+2sm−3x(m−1)+ 2 . BVs

(12.37)

Inserting all the above proportionalities into (12.32) we obtain    (m−1)  ν t FX (hν) ∝ t y exp − , νc,0 t0

(12.38)

with y ≡ α[ 12 (2 − s)m − 1 + 12 x(m − 1)] − 2m + 32 sm − 52 x(m − 1) − 12 and with t0 an age close to the date of observation. From this we can derive the decline rate as (c.f. [901, 1015])

 1 dFX (hν) 1 ν = . y − (m − 1) FX (hν) dt t νc,0

(12.39)

For Cas A we have α ≈ 0.77, s = 2 and m = 0.66 [899, 1183], which gives y ≈ 0.11 for x = 1 and y = −0.61 for x = 0.  Let us assume that the X-ray synchrotron spectrum around 5 keV corresponds to ν/νc,0 ≈ 3 and t ≈ 330 yr; we find that for x = 1 the expected decline rate is the −0.26%/yr and for x = 0, and −0.47%/yr for x = 1. These values are comparable to the latest measured values. There are some theoretical and observational uncertainties associated with the Xray decline rate, such as the possible role of an additional cooling-break—which is not included in  (12.32), but was included in [1015]—the value for the parameter x, the value for ν/νc,0 —Cas A’s X-ray spectrum is a power law rather than showing a clear exponential steepening—and also the deceleration rate of the shock. The value for the expansion parameter m is based on the assumption that Rs ∝ t m . But if Cas A only recently started to decelerate, the current deceleration rate of the shock may be stronger than (m − 1)Vs /t. Such a situation could correspond to an inversion of a density gradient and, hence s = 2. This may indeed happen in the (south)western region of Cas A, which is interestingly also the region with the strongest flux decline rate[1015]. Finally, the estimate decline rates are based on the forward shock evolution. Clearly the reverse shock evolution in Cas A is more complex, and not well understood. In the future we may be able to directly measure dVs /dt, which removes one source of uncertainty. The X-ray synchrotron flux decline rate is of considerable interest, as the discussion here shows that it provides an alternative manner to observe how the electron acceleration rate, and magnetic-field strength evolve as a result of changes in the shock parameters.

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12.2.6 X-ray Synchrotron Peculiarities and (Possible) Consequences As X-ray synchrotron emission comes from freshly accelerated electrons, it show us where, as far as particle acceleration is concerned, all the action is. This is especially true for SN 1006, for which the X-ray synchrotron radiation comes from two crescent shape regions in the northeast and southwest (Fig. 12.8). The remnant is remarkably symmetric in the southeast to northwest axis, as was already noted based on the radio synchrotron maps [983]. What is perhaps more surprising is that this symmetry on the sky does not seem to pertain to the three dimensional morphology. One would assume that the symmetric sky maps are the result of cylindrical symmetry. However, the lack of X-ray synchrotron emission from the interior can only be explained if the X-ray synchrotron emission comes from two “polar caps” rather than from a “barrel-shaped” emission region [990]. Ultraviolet spectroscopy, which reveals ejecta absorption lines with respect to a UV bright star behind the remnant, indicates a strong front-back asymmetry, with the front side moving slower than the back side. So both the X-ray synchrotron emission and the UV absorption studies suggest that our view of SN 1006 has very symmetric remnant is the result of a special viewing angle, rather than being an intrinsic property of the remnant. The finding that the X-ray synchrotron emission comes from two polar caps begs the question what could account for such a three-dimensional morphology. The most natural explanation is that it reveals the large scale magnetic-field configuration. The two possible large scale magnetic-field configurations are illustrated in Fig. 12.14.

Fig. 12.14 Illustration of the two possible large scale magnetic-field configurations for SN 1006. Left: the field is perpendicular for the X-ray synchrotron emitting shock region (perpendicular to the shock velocity vector). This would mean that the X-ray synchrotron region has a cylindrical morphology. Right: the field is parallel to the shock velocity vector. This configuration is preferred as X-ray synchrotron emission is lacking from the interior regions, suggesting two polar caps, rather than a barrel-like morphology

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The perpendicular magnetic-field orientation (left image) would lead to a barrelshaped emission region (if no other factors are at play), whereas a parallel magneticfield orientation for the X-ray synchrotron emitting regions, would naturally lead to two polar caps. So the X-ray synchrotron morphology of SN 1006 suggests that the highest energy electrons come from regions where the magnetic-field is parallel to the shock direction. There has been a long theoretical debate about what is the best magnetic-field angle for particle acceleration: (1) parallel, i.e. the magnetic field direction is aligned to the shock velocity vector, or (2) perpendicular, the magnetic field lines are parallel to the shock surface (and perpendicular to the velocity vector). For a parallel shock one can imagine that the particles can travel larger distances in the direction of propagation, because the particles are not inhibited by the magnetic field. This makes that particles can be more easily injected into the process of diffusive shock acceleration (Sect. 11.2.1). On the other hand, the diffusion parameter, and hence the acceleration length- and timescales are larger if the dominant magnetic-field component is parallel. So the injection may be more easy, but the acceleration process itself is less rapid. The suggestion that for SN 1006 regions with parallel magnetic field appear to be the regions of X-ray synchrotron radiation prompted a renewed theoretical interest in this matter. For example, hybrid simulations of shock acceleration, in which ions particles phase-space trajectories followed individually and electrons are treated as a fluid, revealed that (quasi)parallel magnetic fields give rise to more efficient particle acceleration [233], as suggested by the observations. Two questions may come up regarding the morphology of SN 1006 and the overall orientation of the magnetic field. One is why only SN 1006 (and perhaps G1.9+0.3) appear to have such a preferred magnetic-field angle dependence for the X-ray synchrotron emission. The likely answer is that SN 1006 is located high above the Galactic plane (b = 14◦). There the magnetic-field topology may be much more regular, with a large coherence length. In contrast, most of the other X-ray synchrotron emitting remnants are in the Galactic plane, and in regions as small as 0.1 pc or smaller, both parallel and perpendicular may alternate each other. Note, however, that in the northwest of SN 1006 there is X-ray synchrotron emitting region surrounding what looks like an ejecta knot [213]. Its location does not agree with the overall idea about the magnetic field configuration of SN 1006, and may suggest a local deviation from the large scale trend. The other question is why the large scale magnetic-field configuration matters at all, if magnetic-field amplification (Sect. 11.4.3) tends to make the magnetic-field structure highly turbulent anyway. This is a question that needs to be more thoroughly investigated, but it may be that for the initial stages of particle acceleration, before magnetic-field amplification starts, the magnetic-field structure is important. This will determine where particle acceleration will become efficient, and where magnetic fields are to be amplified [151]. In other words, the X-ray synchrotron radiation may come from regions where the initial magnetic field was parallel, and where only after some time efficient acceleration has made the magnetic field more turbulent. In this view, the shock “remembers” the initial magnetic-field configuration.

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Fig. 12.15 The eastern part of Tycho’s SNR (c.f. Fig. 12.9), as observed by Chandra in the 4– 6 keV band. It shows the peculiar horizontal features known as “Tycho’s stripes” [369]

Two other interesting X-ray results pertaining to particle acceleration concern Tycho’s SNR. The first result is the identification of peculiar X-ray synchrotron emitting structures dubbed the “Tycho stripes” [369]. In particular, the X-ray synchrotron features on the eastern side (Fig. 12.15) appear to have a quasi-regular pattern with a typical gap between the stripes of 8 . This pattern has been tentatively identified with twice the proton gyro-radius, which suggests that the protons have an energy  E ≈ 2.5 × 1015

lgap 8



B 50 μG



d 3 kpc

 eV.

(12.40)

Although the stripes are striking features, its interpretation is less straightforward. From a theoretical point of view it is not clear whether Tycho’s SNR is currently capable of accelerating up to these energies. Filling in the age of the remnant (∼440 yr), a shock velocity of 5000 km s−1 and the upstream magnetic field of ∼50 μG into (11.37) gives Emax ≈ 2 × 1014 eV for η = 1. Another issue is that for fast acceleration one needs a turbulent, isotropic magnetic field η ≈ 1, but then it is not clear why the a magnetic field pattern that emerges appears to be that regular. Finally, it is surprising that the structures appear more or less face-on, whereas most X-ray synchrotron filaments appear strongly limb-brightened. That said, two models relate the stripes indeed to anisotropic plasma waves present in the upstream medium, which are later overtaken by the shock. The structures are according to

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one model are predetermined by structures present in the ambient magnetic field [226]. According to another model the structures correspond to circularly polarised Alfvén waves generated by cosmic-ray streaming upstream of the shock [696]. These models can potentially be tested by measuring the magnetic-field orientation of the stripes with the future X-ray imaging polarisation telescope IXPE [1213]— see also [226, 296, 1178]. Finally, some words on an issue that is not entirely about X-ray synchrotron emission, but about something that connects X-ray emission properties to cosmicray acceleration in a supernova remnant. X-ray emission from Tycho’s SNR has played a prominent role in the debate about the maximum shock compression ratio. As explained in Sect. 11.3 (see Fig. 11.7), a prediction of non-linear cosmic-ray acceleration is that the shock compression ratio can be much larger than the factor χ = 4 for strong Mach number shocks. If that is the case, then the shock-heated shell will be thinner than according to standard hydrodynamical models [298]. For Tycho’s SNR this seems indeed to be the case: the ejecta contact discontinuity (Sect. 5.3) appears to be closer to the forward shock than predicted by standard models. This was first pointed out in [1198], and can be seen in Fig. 12.9 (right) the ejecta fingers (greenish) almost touch the X-ray synchrotron filaments (purplish). A caveat is the role of Rayleigh-Taylor instabilities (Sect. 5.9) in bringing “fingers” of ejecta close to the forward shock. Numerical hydrodynamical studies show that Rayleigh-Taylor instabilities, indeed, need to be taken into account. But the best explanation for the closeness of ejecta to the forward shock in Tycho’s SNR is that there is efficient cosmic-ray acceleration, with an overall shock-compression ratio of the order of eight [377].

12.3 Gamma-Rays Observations: A Window on the Hadronic Cosmic-Ray Content of Supernova Remnants The cosmic rays detected on Earth consist mostly of energetic atomic nuclei (hadronic cosmic rays), whereas the electrons and positrons (leptonic cosmic rays) comprise less than 1% of the energetic particles reaching the solar system (Sect. 11.1.2). We suspect that the cosmic-ray lepton to hadron ratio as accelerated by supernova remnants is also of the order of 1%, but this is not certain. Moreover, there may be variations in this ratio among supernova remnants. Radio to X-ray synchrotron radiation only reveal the presence of accelerated electrons, and in order to infer the energy density in leptonic cosmic rays from the radiation requires the magnetic-field strength to be known (Sects. 12.1.2 and 12.2.3). When it comes to measuring the hadronic cosmic-ray content of supernova remnants, the only signals related directly to hadronic cosmic rays are the byproducts of pion decay (Sect. 13.6): γ -rays and neutrinos. The detection sensitivity of very high energy neutrinos by current facilities like IceCube [8] is low compared to γ -ray

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observatories. So far no supernova remnant has been identified as a source of very high energy neutrinos. For the time being γ -ray observations provide the best means to probe the hadronic cosmic-ray content of supernova remnants. But, as we will discuss below, the γ -ray signals of supernova remnants are not uniquely the result of hadronic cosmic rays, as leptonic cosmic rays emit through the bremsstrahlung (Sect. 13.4) and inverse Compton scattering mechanisms (13.2.2). To distinguish γ -rays emission from leptonic radiation processes from pion decay, detailed multiwavelength models of the total non-thermal spectrum of supernova remnants are required. But there is one clear feature that uniquely identifies pion decay as a source of γ -ray radiation: the “pion bump” around 100–200 MeV (see Sect. 13.6).

12.3.1 A Brief Historical Overview of γ -Ray Astronomy The history of γ -ray astronomy goes back to rocket and balloon experiments in the 1950s and 1960s. The first source detections were made with NASA’s OSO-3 [684] and ESA’s COS-B satellite experiments [167]. A major space mission in the more recent past was the NASA’s Compton Gamma-ray Observatory (CGRO, 1991– 2000), which carried several detectors covering a wide range of γ -ray energies: CGRO-OSSE (∼100–2000 keV) [590], CGRO-Comptel (∼0.5–30 MeV) [1024], CGRO-EGRET (∼20–30,000 MeV) [1111] and the all-sky monitor BATSE [394]. None of these instruments did positively detect continuum emission from supernova remnants related to the topic of this chapter: cosmic rays. But CGRO-OSSE did detect a non-thermal tail up to ∼ 120 keV from Cassiopeia A [1108], CGROComptel detected 44 Ca nuclear de-excitation line emission caused by the decay of radio-active 44 Ti (Sect. 2.4), and CGRO-EGRET detected γ -ray emission from a region containing supernova remnants interacting with molecular clouds (γ -Cygni and IC 443) [370]. But the γ -ray emission could not be unambiguously identified with these supernova remnants. It was only in the twenty-first century that supernova remnants have been firmly established as γ -ray sources. This started with the coming of age of ground-based γ -ray astronomy using imaging atmospheric Cherenkov telescopes (IACTs). IACTs detect γ -ray photons indirectly, by measuring the bluish Cherenkov light coming from airshowers created by γ -ray photons that interact with the atoms in the Earth’ atmosphere. This technique was pioneered in the 1960s, but the first reliable cosmic γ -ray source detection, the Whipple Observatory’s detection of the Crab Nebula, was obtained in the late 1980s [1203]. The first shell-type supernova remnant to be detected in γ -rays was Cassiopeia A with the HEGRA telescope [47]. Both the Whipple Observatory and the HEGRA consisted of a single dish telescope. The sensitivity of IACTs was greatly improved when stereoscopy was introduced, in which Cherenkov flashes were detected with two or more telescopes, allowing for triangulation of the Cherenkov light cones. This results in a determination of the direction of the Cherenkov light tracks and better background event discrimination.

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Fig. 12.16 The H.E.S.S. II imaging atmospheric telescope array in Namibia. The original array (H.E.S.S.) consisted of the four 12 m telescopes. The fifth big telescope (28 m) was added to extend the energy detection range down to 200 GeV. The configuration with five telescope is known as H.E.S.S. II (Credit: Stefan Klepser, Desy and H.E.S.S. collaboration)

The background events are mainly cosmic-ray induced airshowers, which also result in Cherenkov light tracks, but with a wider cone. Stereoscopy is used by the four main IACTs currently operating: H.E.S.S. [156] (Fig. 12.16), MAGIC [122] and VERITAS [1202]. Together they have detected now more than twenty supernova remnants. IACTs cover roughly the 100 GeV to 100 TeV range of the electromagnetic spectrum, although the sensitivity range can be extended down to ∼10 GeV by employing arrays of larger dishes, which will be especially relevant for the future Cherenkov Telescope Array (CTA) [39]. CTA will start operation after 2024, and consists of both a southern and northern array, with the more extended southern array employing up to 90 telescopes of different sizes to both detect photons with energies below 100 GeV with large dishes (25 m), as well as having enough sensitivity up to 100 TeV by using 25 medium-sized telescopes (12 m) and up to 70 small-sized telescopes (4 m). Note that the effective area of an IACT is roughly the area of the atmosphere that is monitored by the an IACT array. The diameter of the telescope determines the capability to detect faint Cherenkov light, with the brightness scaling with the energy of the primary γ -ray photon. A type of γ -ray observatory that targets a similar same part of the γ -ray spectrum as IACTs are water Cherenkov telescopes, which detect extensive airshowers reaching ground levels by detecting Cherenkov light in watertanks. Cosmic-ray airshowers are discriminated from γ -ray airshowers using the size of the footprint. The technique was successfully employed by the Milagro γ -ray observatory [102], which has now been succeeded by the High-Altitude Water Cherenkov (HAWC) Observatory [22]. The angular resolution is poor compared to IACTs, but water Cherenkov telescopes can be operated day and night, and they can observe the entire sky, making them ideal survey observatories. Indeed Milagro [13] detected eight

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sources and HAWC [22] detected 39 sources, including a few supernova remnants. Several other Milagro and HAWC discovered sources are still unidentified. The other main development in γ -ray astronomy concerned the launch of two space-based γ -ray missions, the Italian AGILE mission [1100], launched in 2007, and NASA’s Fermi mission [104], launched in 2008. The main instruments on board these satellites (AGILE’s GRID and Fermi’s LAT) cover roughly the 20 MeV to 100 GeV range. Compared to IACTs the space-born detectors have a much smaller effective area, which corresponds roughly to the area of the detector. However, most sources have a power-law like spectrum with a negative slope, so the smaller effective area of Fermi-LAT and AGILE GRID are offset by the fact that the photon flux at lower energies is much higher. Moreover, the detection abilities for nontransient sources by the space-based instruments rely on the large duty cycle of observations, and the accumulation of all-sky data over multiple years. For example, the Fermi-LAT instrument has a field of view of 70◦ , as compared to a few degrees for IACTs. With the launch of AGILE and Fermi the number of γ -ray supernova remnants increased from no reliable detections in the GeV part of the spectrum [370], to over 30 γ -ray detected supernova remnants [34], with a considerable overlap with the sources detected by IACTs, as well. This is a good moment to clarify some often encountered nomenclature in γ -ray astronomy: The γ -rays in the energy range from ∼100 keV to about 30 MeV is often referred to a medium-energy γ -rays (or sometimes MeV γ -rays). In this energy range one can still expect the tail of synchrotron radiation from some sources, like Blazars and pulsar wind nebulae (as explained for the Crab nebula, Sect. 6), as well as inverse Compton scattering and bremsstrahlung. But medium-energy γ -rays also may reveal atomic nuclear lines, caused by radio-activity (e.g. 44 Ti and 56 Co, Sect. 2.4) or by direct excitation of the nuclei by collisions with energetic particles, such as C, O, Ne and Mg lines in the MeV region. The latter have been detected during solar flares [725], but no unambiguous detections of from line emission from the interstellar medium or Galactic sources has been obtained. Photons with energies in the range from 30 MeV to 100 GeV are often referred to as high-energy (HE) γ -rays (or sometimes less precisely GeV γ -rays). This is the range covered by the Fermi-LAT instrument, and it covers the “pion bump” discussed in Sect. 13.6. Photons in the energy range from 100 GeV to 100 TeV are often referred to as veryhigh-energy (VHE) γ -rays, or TeV γ -rays. There is no sharp boundary between these γ -ray regimes, except that VHE γ -rays are detected using IACTs, whereas HE are detected with space-based instruments, with some overlap between the two detection methods.

12.3.2 Hadronic Versus Leptonic Emission Although γ -ray observations provide currently the only means to gauge the hadronic cosmic-ray content of supernova remnants, firmly establishing the presence of hadronic cosmic rays, and the total cosmic-ray energy in supernova remnants has proven to be difficult, and has often lead to strong debates.

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E2 dN/dE (eV.cm-2.s-1)

E2 dN/dE (eV.cm-2.s-1)

The reason is that γ -ray emission can be caused by both hadronic processes (pion production) and leptonic radiation mechanisms (inverse Compton scattering and bremsstrahlung). The broad γ -ray spectral energy distribution (SED) for inverse Compton scattering dominated spectra and pion-decay dominated spectra are quite different. Assuming that both the electron and proton spectral index of accelerated particles have the same power-law slope q (N(E) ∝ E −q ) with q ≈ 2 (Sect. 11.2.1), inverse Compton emission will result in a photon spectral index of  = 12 (q + 1) ( ≈ 1.5 for q ≈ 2), whereas pion-production and decay will result in a spectral index of  ≈ q ≈ 2. Bremsstrahlung, which also produces  ≈ q, is often ignored as it requires relatively high plasma densities, in which case pion decay is likely to be an even more dominant γ -ray component. However, for Cas A nonthermal bremsstrahlung may be a substantial γ -ray component, also because the ratio between leptonic and hadronic cosmic rays may be peculiar [12]. Concentrating on inverse Compton scattering and pion decay, one can in principle distinguish the two processes based on the spectral slope, with  < 2 more likely to be indicative of inverse Compton scattering and  > 2 more indicative of pion decay. An even better diagnostic feature is the measurement of the “pion bump”, which consists of a peak in the spectral-energy distribution between 0.1– 1 GeV and a rapid decline in the emission below ∼200 MeV. A problem is that for some supernova remnants no reliable spectrum has been measured in the highenergy domain (10 TeV) are more difficult to produce through inverse Compton scattering due to Klein-Nishina effects (13.2.2). Based on detected rapid X-ray synchrotron variations (Sect. 12.2.5), magnetic-field strengths as high as 1 mG have been inferred for this remnant [1145]. This would be inconsistent with inverse Compton models for the γ -ray emission. In addition, potential interaction of the supernova remnant with molecular clouds seem to reinforce the idea that the γ -ray emission could be of hadronic origin [408]. However, like in the case of RX J0852.0-4622, the lack of thermal X-ray emission suggests that the ambient density is low, disfavouring the hadronic scenario [632]. X-ray synchrotron radiation requires shock velocities 3000 km s−1 (11.50), in agreement with measured proper motions for RX J1713.7-3946 of ∼3500 km s−1 [32, 1130]. Only with a low ambient density can one explain such a high shock velocity at a radius of 10 pc. Finally, although there is some narrow spatial substructure in the X-ray synchrotron emission from RX J1713.7-3946 [1144], the X-ray synchrotron emitting shell of the supernova remnant is quite broad (Fig. 12.11) resulting in a magnetic-field strength estimate that is much lower than 1 mG, and more consistent with 35 μG [505] (see Table 12.3). Full modelling of the hydrodynamics of the remnant, coupled with calculations of particle acceleration and X-ray emission, also favours a leptonic model for the γ -ray emission from RX J1713.7-3946 [367]. One would assume that the discussion could be put to rest by measuring the HE γ -ray emission with the Fermi-LAT instrument, as this would establish whether the spectrum has a spectral slope  < 2, and establish the presence or absence of a pion bump. The measurement in fact does exist [20], see Fig. 12.18. And on the face of it, the broad SED favours the leptonic scenario: the spectrum is clearly harder than can be accounted for by simple hadronic models, and there is no indication for a pion bump. But none of the inverse Compton scattering models provide a perfect fit, in particular above 1 TeV. In addition, more complex hadronic models, in which the hadronic emission comes from interstellar medium dense clumps that survived the passage of the supernova remnant shocks, cannot be excluded [410, 1283]. The idea is that the higher the energy of the cosmic-ray proton, the more deeply the protons can penetrate the highest density regions of the clumps. As a result the γ ray spectral no longer reflects the intrinsic power-law slope of the cosmic-rays, but is a combination of spectral slope of the protons modulated with the energy-dependent penetration depths. Although the case for pion decay emission from RX J1713.7-3946 is controversial, strong evidence for pion-decay emission from other supernova remnants does exist: the pion-bump has been detected in the HE spectra of the mature

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A IC 443 E2 dN/dE (erg cm-2 s-1)

10 -10

10 -11

Best-fit broken power law Fermi-LAT VERITAS (30) MAGIC (29) AGILE (31) π0-decay Bremsstrahlung Bremsstrahlung with Break

10 -12

8

10

9

10

10

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1011

1012

Energy (eV)

B

W44 E2 dN/dE (erg cm-2 s-1)

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Best-fit broken power law Fermi-LAT AGILE (19) π0-decay Bremsstrahlung Bremsstrahlung with Break

10 -12

8

10

9

10

10

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1011

1012

Energy (eV) Fig. 12.19 Spectral-energy distribution of the supernova remnants W44 and IC 443 as measured by Fermi-LAT and AGILE (Reproduced from [37])

supernova remnants IC 443 and W44 [37, 442] (Fig. 12.19). But the SEDs of these remnants also display spectral breaks at relatively low energies: ∼20 GeV for IC 443 and 2 GeV for W44 [37]. Apparently the highest-energy cosmic rays have already largely diffused away from the supernova remnants. Note that several other γ -ray spectra are also consistent with hadronic emission, including Cas A [56], HESS J1640-465 [24, 713] and W49B [523], the latter also with a break at low energies (∼8 GeV).

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12.3.3 A Few Words on Modelling Inverse Compton Scattering Modelling inverse Compton scattering from supernova remnants requires some care. First, most models employed are so-called “one-zone models”, in which one assumes a simple power-law population of electrons with an high-energy exponential cutoff. However, as we have seen in Sects. 13.3 and 12.2, electrons are much more subject to radiative losses than protons and atomic nuclei, so the high-energy cutoff is expected to vary throughout the shell, with the highest energy electrons confined to narrow regions near the shocks. An additional spectral break is expected near the cooling break (Sect. 13.3.4). So nature demands more complex models than one-zone models, but multi-zone models cannot always be meaningfully constrained. Secondly, the seed photons for inverse Compton scattering can come from multiple components. The simplest approach is to only consider Cosmic Microwave Background (CMB) photons, with a density of 410 cm−3 , and with an average photon energy of ∼6.6 × 10−4 eV. However, in the Galaxy there are other important radiation fields, caused by far infrared emission from dust and stellar radiation fields, which will vary throughout the Galaxy [928]. In addition, the supernova remnant or pulsar wind nebula may contribute itself significant to the local radiation field. For example, Cas A and N132D are bright in the far infrared (Chap. 7), and the Crab Nebula’s radiation field has a large contribution from the synchrotron emission from the nebula itself. Figure 12.20 shows a compilation of Galactic radiation fields tuned to the solar neighbourhood. It shows that in the far infrared the photon number density is a factor ∼10 smaller than the CMB photon density, but with an average Fig. 12.20 Galactic radiation fields as compiled in [1165]. By number density of photons, the cosmic microwave background (CMBR) is dominant component, followed by Galactic dust emission, the extra-galactic background light (EBL) and starlight. The CMBR and EBL components are uniform, but the other components will vary throughout the Galaxy

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photon energy that is a factor 15 larger. Optical star light has a photon density that is ∼1000 less than the CMB photon density, but with an average photon energy of around 1 eV. In order to calculate the contributions of these radiation fields to the inverse Compton scattering components one has to take into account these difference photons energies. The scattered energy will be of the order hν ≈ e2 hν (Sect. 13.2.2), with e the electron Lorentz factor. As a consequence, a 100 GeV photon can be caused by the upscattering of a CMB photon by an electron with energy of ∼6 TeV, or a far infrared photon by an electron with an energy of ∼1.6 TeV, or an optical photon by an electron with an energy of ∼16 GeV. There are many more electrons with energy of 16 GeV than there are electrons with 6 TeV energy: for an electron energy distribution scaling as E −2 there about 1400 more 16 GeV electrons than 6 TeV electrons. As a result, the contribution of upscattered optical photons to the 100 GeV γ -ray emission is expected to be comparable or even higher than the contribution of upscattered CMB photons, despite the much lower seed photon number density. This is no longer true if one considers the contribution of the different components at higher upscattered energies. Recall that the Klein-Nishina cross section for Compton scattering declines rapidly for seed photon energies hν > me c2 / e (Fig. 13.5). As a result, the contribution of upscattered star light photons rapidly declines for hν  260 GeV, see (13.26). For far infrared photons this is hν  26 TeV. Klein-Nishina effects are hardly a concern for CMB photons, as they only affect the inverse Compton spectrum above ∼ 260 TeV. Finally a few words on the connection between synchrotron radiation, inverse Compton scattering luminosity and the magnetic field. For a given background radiation field and electron population, which we assume as a distribution of KE −q exp(−E/Ec ), the inverse Compton luminosity at a given γ -ray energy well below the cutoff, depends on the normalisation K. For synchrotron radiation below the cutoff, the luminosity depends on the combination KB (q+1)/2, with B the magnetic field. So the combination of radio synchrotron emission and GeV γ -ray emission caused by inverse Compton scattering provides a direct estimate of B. A complication is of course that B and K may not be uniform throughout the source. Having also measurements in synchrotron and inverse Compton scattering around the respective cutoff energies results in more constraints: for a given cutoff energy for the electrons a larger magnetic strength shifts the synchrotron cutoff frequency to higher frequencies, whereas the cutoff photon energy for inverse Compton scattering only depends on the radiation field properties. As a result for a one zone model and a known radiation field the model is over constraint, as the magnetic field can be determined by both the radio and GeV γ -ray emission flux, as well as from the synchrotron and inverse Compton cutoff energies. A nice illustration of what can be done with modelling VHE γ -ray observations in conjunction with X-ray synchrotron observations, under the assumption of inverse Compton scattering dominated γ -ray emission, is shown in Fig. 12.21. The spectral/spatial modelling is based on a deep observation of RX J1713.7-3946 with H.E.S.S. combined with Suzaku X-ray data [522]. It shows that per sector the

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Fig. 12.21 Top row: maps of the estimated magnetic-field strength (left) and electron cutoff energy (right) in RX J1713.7-3946 based on the assumption of an inverse Compton scattering origin for the VHE γ -ray emission as measured by H.E.S.S. [522]. The contours indicate the 3, 5, 7 and 9σ H.E.S.S. detection confidence limits. The measurements are based on combining H.E.S.S. γ -ray and Suzaku X-ray data. (Reproduced from [522].) Bottom: a scatter diagram of the estimated electron cut-off energy versus the magnetic field strength, as shown in the maps. The blue lines indicate the expected relation between B and Ec under the assumption of a loss-limited electron population, and the red lines under the assumption of an age-limited electron population

average magnetic field strength, B, and electron cutoff electron energy, Ec , can be determined. Figure 12.21 also contains a scatter diagram based on these fits showing that there is a rough anti-correlation between B and Ec , as is to be expected, if the cutoff electron energy is determined by a balance between acceleration gains versus radiative losses. Indeed, overplotting the expected relation (11.50), shows a reasonable good fit for Bohm factors 4 < η < 16 and Vs ≈ 3500 km s−1 . The alternative model, an age-limited electron spectrum, for which the maximum electron energy is given by (11.37) results in different relation between cutoff energy and magnetic-field strength than indicated by the measurements. But note that the

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expected values for B and Ec are in the right range for reasonable values for the age of the supernova remnant and the age-averaged shock velocity (here taken to be Vs ≈ 5500 km s−1 ). This does perhaps indicates that the cutoff electron energy has only recently changed from being caused by age limits to loss-limits. Of course, all of this based on the assumption that the γ -ray emission is of caused by inverse Compton scattering, and not by pion decay. An assumption that is still controversial, as discussed before.

12.3.4 Gamma-Ray Evidence for Escaping Cosmic Rays It is perhaps ironic that the clearest evidence for hadronic cosmic-ray acceleration, IC 443 and W44, that the maximum energy of the particles is below 1 TeV, whereas for the supernova remnants with breaks above 1 TeV there is a more of a debate whether the γ -ray emission originates from hadronic cosmic rays. The most obvious explanation is that both IC 443 and W44 have shock waves interacting with dense molecular gas (see also Sects. 10.2 and 10.3). This molecular clouds provide a large target density for pion production. But the interaction us expected to occur for mature supernova remnants, once the shock wave has traversed wind-blown bubbles. As the shock waves interact with dense gas, the shock velocity will decelerate quickly, resulting in radiative shocks (Vs < 200 km s−1 , Sect. 5.7), which likely are no longer able to accelerate particles to high energies (Sect. 11.2.6). As shown in Sect. 11.2.7, the highest energy particles will start to escape once the supernova remnant enters the Sedov-Taylor phase, which is a fortiori the case for supernova remnants in the radiative phase for which m ≈ 0.25. When discussing the escape of cosmic rays, it is good to keep in mind that the particles are not moving away from their source of origin, in the way an escaping prisoner will try to move as far as possible from a jail. Instead the particles will diffuse rapidly, as compared to the shock velocity, but the population of cosmic rays will remain centred on the source of origin, the supernova remnant. The cosmic-ray density distribution for a homogeneous diffusion coefficient will be simply given by a three-dimensional Gaussian distribution:  2   N(E > Eesc ) 1 r N(E > Eesc , r)dr ≈ exp − √ dr (12.41) (4π)3/2(2Dt)3/2 2 2Dt If the √ size of the supernova remnant is much smaller than the diffusion length scale (r  2Dt), which is the case well above energies given by (11.45), then one can assume that the factor given by the exponent is of order one. If the source spectrum is given by a power law, N(E)dE ∝ E −q and the diffusion coefficient scales with energy as D(E) ∝ E δ (c.f. (11.4)), the resulting spectrum of cosmic rays centred

368

12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

on the supernova remnant (i.e. r = 0) scales as N(E, r = 0)dE ≈

N(E) (4π)3/2 (2Dt)3/2

3

dE ∝ E −q (2Dt)−3/2 ∝ E −q− 2 δ . (12.42)

For IC 443 the Fermi-LAT measurements indicate that above 239 GeV the spectrum the spectrum changes from a low-energy spectral slope of q = 2.26 ±0.02 to a slope q = 3.1 ± 0.1 [37]. According to (12.42) the steepening implies δ ≈ 0.5. For W44 the spectrum steepens from q = 2.36 ± 0.05 at low energies to q = 3.5 ± 0.5 above 22 GeV, implying δ ≈ 0.76. Both values of δ are consistent with what has been inferred from modelling of cosmic-ray diffusion in the Galaxy (Sect. 11.1.3), suggesting that the “escape” of cosmic rays is indeed the cause of the steepening of cosmic rays. More direct evidence for cosmic-ray escape has come from H.E.S.S. observations of two Galactic supernova remnants W28 and RX J1713.7-3946. For W28 H.E.S.S. identified four regions of VHE γ -ray emission [54] (Fig. 12.22). One region coincides with the northeastern boundary of the supernova remnant, but three other γ -ray emitting regions are located 0.5◦ south of the remnant shell. All four regions are coincident with molecular clouds. These regions have also been detected in HE γ -rays by the Fermi-LAT and AGILE experiments [17, 441]. The γ -ray observations suggest that VHE cosmic rays have diffused away from the shell of W28, and become visible where they come in contact with high-density molecular clouds. The

Fig. 12.22 Radio map at 90 cm of the supernova remnant W28 [215], with in green the H.E.S.S. 4,5,6σ significance contours [54] (Credit: H.E.S.S. Collaboration, reproduced from [54])

12.3 Gamma-Rays Observations: A Window on the Hadronic Cosmic-Ray. . .

369

process of molecular cloud illumination by escaping cosmic rays had been predicted some time ago [45, 411], and applying the theory to the observations suggests that the diffusion coefficient in the ambient medium is smaller than the average Galactic diffusion coefficient [412]. This in turn suggests that magnetic-field turbulence in the medium surrounding W28 is enhanced, either caused by the escaping cosmic rays themselves, or by the fact that W28 is embedded in a star forming region. H.E.S.S. observations of RX J1713.7-3946 provided evidence for cosmic rays that are in the process of diffusing away from the supernova remnant [522]. Figure 12.23 shows the normalised X-ray and VHE γ -ray surface brightness profiles of the southeastern region of the supernova remnant. Clearly, the VHE γ -ray emission has a larger extent than the X-ray surface brightness profile. If we assume that the maximum extent of the X-ray profile indicates the location of the shock, the γ -ray emission suggests the presence of accelerated particles far upstream of the shock. These particles can either be particles that are part of the cosmic-ray shock precursor, in which case they can realistically still be accelerated higher up in energies by recrossing the shock (Sect. 11.2.2), or they they may be so far ahead of the shock that they can be said to be escaping the supernova remnant. In region 3 the 1/e fall-of distance from the X-ray-defined shock location was measured

Fig. 12.23 The X-ray (XMM-Newton, red) and VHE γ -ray (H.E.S.S., data points) surface brightness profiles of the southeastern region of RX J1713.7-3946, showing evidence for a population of accelerated particles well upstream of the shock (Reproduced from [522])

370

12 Supernova Remnants and Cosmic Rays: Non-thermal Radiation

to be 0.11◦, or 1.9 pc. Since the radius of the supernova remnant is about 0.5◦ ,

R/Rs > 0.1Rs. According to (11.38) this is around the limit beyond which particles can be said to be no longer part of the cosmic-ray precursor. However, there are some systematic effects regarding the measurements that limit the reliability of the total extent of the precursor. So the observations do not unequivocally support that in this part of the remnant escape of cosmic rays is detected. Nevertheless, the impression is that of a population of escaping cosmic rays that have been “caught in the act”. The extent of the precursor/escaping population can be used to constrain the diffusion properties of the particles. Given that the γ -ray photons have energies around 1 TeV, the primary particles, whether hadrons or electrons, are likely to have energies of order 10 TeV. Using the expression for the diffusion length scale, √ either (11.26) or the more generic ldiff ≈ 2Dt can now be used to constrain the combination of magnetic field strength and magnetic-field turbulence. Using (11.26), the constraints are [522]    −1  B1 E Vs

r ≈ 0.36 μG. η 10 TeV 1 pc 3000 km s−1

(12.43)

This value is surprisingly low, and more consistent with an inverse Compton origin for the VHE γ -ray emission from RX J1713.7-3946 than hadronic emission. Even for the case of inverse Compton emission it indicates that η should be much larger than 1, given that the Galactic magnetic-field strength is ∼5 μG. In Sect. 12.2 we showed that the detection of X-ray synchrotron radiation from supernova remnants implies that η  10, so a large value of η is a priori inconsistent with the dominance of X-ray synchrotron radiation from RX J1713.73946. However, an increase in η over a time scale of the order of the synchrotron loss time scale for 10 TeV electrons can result in a present-day large value of η and X-ray synchrotron radiation from electrons accelerated in the past. Finally, one should consider what might cause the particles from RX J1713.7-3946 to escape in the southeastern region. One idea was already implicitly mentioned: there may have been a relatively recent increase in the diffusion coefficient, i.e. an increase in the value for η. Another reason may be that the shock encountered recently a positive density gradient, causing it to rapidly decelerate, and allowing cosmic rays to diffuse further upstream of the shock.

12.3.5 The Population of γ -Ray Emitting Supernova Remnants In the preceding subsections we have high-lighted specific observations that illustrate important aspects of γ -ray observations of supernova remnants: is the emission hadronic or leptonic? and how and when do cosmic rays escape? Clearly for young supernova remnants, which are often also X-ray synchrotron emitters, the γ -ray spectrum extends into the VHE γ -ray regime, whereas for older supernova

12.3 Gamma-Rays Observations: A Window on the Hadronic Cosmic-Ray. . .

371

remnants, often interacting with molecular clouds and showing clear indications of hadronic emission, the γ -ray spectrum seems to break in the GeV γ -ray regime. In the latter case the supernova remnant may still be bright enough to be detected in the VHE γ -ray regime, but with a relatively steep spectrum. To put these aspects into the context of the population of VHE γ -ray emitting supernova remnants, we list in Table 12.4 all VHE γ -ray detected supernova remnants, with the measured fluxes and spectral slopes. For an extensive population study that is complementary to what is shown here, see [520]. The table was compiled making use of the Chicago TeV catalogue (http://tevcat.uchicago.edu), which subdivides the TeV detected supernova remnants into shell-type supernova remnants, composite supernova remnants, and supernova remnants with molecular cloud association (like W28 discussed above). It should be noted that the difference between a shell-type SNR, and an SNR/Molecular Cloud is not always so clear cut. In one case, the supernova remnant was relabelled “shell”, N132D, as the spatial resolution is not sufficient to determine whether the γ -ray emission is coming from the nearby cloud, or from the shell [23]. Another caveat of Table 12.4 is that the fluxes and spectral parameters were not obtained in an homogeneous procedure, but instead compiled from the available literature. For example, some TeV spectral slopes were measured assuming a pure power-law spectrum, whereas in other cases the spectrum was assumed to be a power-law with an exponential cutoff. Also normalisations and spectral-energy ranges differed. Nevertheless, some striking features appear when one plots the measured γ ray indices obtained by IACTs (TeV ) against those measured by the Fermi-LAT experiment (GeV ), as shown in Fig. 12.24. The supernova remnants labeled as SNR/Molecular Cloud (red data points) appear to have steeper spectral slopes in the HE domain ( > 2). This can be best understood by assuming that the emission mechanism is pion production by cosmic rays with a spectral index q ≈ GeV  2, whereas many of the “shell” supernova remnants may have γ -ray emission dominated by inverse Compton scattering, for which GeV ≈ (q + 1)/2  1.5. One may expect that the spectrum steepens going from the HE to the VHE regime, either as a result of “escape” of cosmic rays, or due to radiative cooling. As a result most data points should lie above the line GeV = TeV . This appears indeed to be the case, but with a few exceptions. These exceptions may be for supernova remnants that are less affected by particle energy losses and/or escape, or the steepening is not present due measurement errors. Finally, is there any indication that normal “shell” supernova remnants (in the narrow definition of the TeV Catalogue) are emitting hadronic γ -rays? That is, do some “shell” supernova remnants have relatively soft spectra? The answer is yes: two young shell-type supernova remnants have GeV > 2, namely Tycho’s supernova remnant and Cas A, and both of them have ambient densities sufficiently high to expect hadronic emission. Indeed, for Tycho’s supernova remnant it has been stated that it “represents the first clear and direct radiative evidence that hadron acceleration occurs efficiently in young Galactic SNRs” [838], and most γ -ray emission models for Cas A fit the spectrum with a dominant hadronic-γ -ray component [12, 30, 47, 56]. As Fig. 12.25 shows, the latest Fermi data show the

LMC N132D

HESS J1813-178

HESS J1833-105

W41/HESS J1834-087

Kes 75/HESS J1846-029

MGRO J1908+06

LMC N132D

G12.8-0.0

G21.5-0.9

G23.3-0.3

G29.7-0.3

G40.5-0.5

SNR/Molec. Cloud?

Composite

Composite

Composite

Composite

Composite

Composite

Based on the TeVCat catalog (http://tevcat.uchicago.edu) and [474]

G54.1+0.3

HESS J1745-303

G358.9-0.6

SNR/Molec. Cloud

Shell

HESS J1731-347

G353.6-0.7

Shell

SNR/Molec. Cloud

SNR/Molec. Cloud

CTB 37B

G348.7+0.3

Shell

SNR/Molec. Cloud

Shell

Shell

Shell

SNR/Molec. Cloud

Shell

Shell

SNR/Molec. Cloud

Shell

Shell

Shell

SNR/Molec. Cloud

SNR/Molec. Cloud

SNR/Molec. Cloud

Type

G349.7+0.2

CTB 37A

G348.5+0.1

HESS J1457-593

G318.2+0.1

RX J1713.7-3946

RCW 86

G315.4-2.3

G347.3-0.5

RX J0852.0-4622

G266.2-1.2

HESS J1640-465

IC 443

G189.1+3.0

HESS J1614-518

Tycho

G120.1+1.4

G338.3-0.0

Cassiopeia A

G111.7-2.1

G331.8-0.5

SNR G106.3+2.7

G106.3+2.7

HESS J1534-571

W 51

G49.2-0.7

SN 1006

W 49B

G43.3-0.2

G327.6+14.6

W 28

G6.4-0.1

G323.7-0.92

Name

SNR

8.2

3.4

11

5

4.6

4

50

3.2

11.5

13.2

7.9

1

10

2.2

3.5

2.5

0.2

1.5

3.5

3.4

0.8

5.4

11.3

2

(kpc)

d (1033 erg s−1 )

568.7 0.9

2.64 2.36 2.3 ≡2 2.09

2.01 ± 0.19 0.94 ± 0.15 1.63 ± 0.21 1. So it is better to write ) N(E)dE =dE

tfin t =0

KE −q f (E)dt,

(13.59)

with * f (E) ≡

[1 − EA(tfin − t)]q−2 , if EA(tfin − t) < 1 0 , if EA(tfin − t) ≥ 1.

(13.60)

13.4 Bremsstrahlung (Free-Free Emission)

399

For E < Eage we are always in a regime where EA(tfin − t) ≤ 1, and we can easily integrate (13.58): ) N (E) =

tfin

KE −q [1 − EA(tfin − t)]q−2 dt

(13.61)

t=0

=

t=tfin

KE −(q+1) KE −(q+1) {1 − AE(tfin − t)}q−1 1 − {1 − AEtfin }q−1 . = t=0 (q − 1)A (q − 1)A

For E ≥ Eage we have to integrate from a lower time boundary of EA(tfin − t) = 1 or t = tfin − 1/(EA)—only populations sufficiently young are contributing to those energies: ) N(E) = =

tfin

t =tfin −1/(EA)

KE −q [1 − EA(tfin − t)]q−2 dt

(13.62)

tfin KE −q−1 KE −q−1 {1 − EA(tfin − t)}q−1 . = t =tfin −1/(EA) A(q − 1) A(q − 1)

If we now combine these two results and use that tfin = 1/(AEage ), we see that

N(E) =

⎧  KE −q−1 ⎪ ⎪ t E ⎨ fin age (q−1) 1 − 1 − ⎪ ⎪ ⎩

E Eage

q−1

, if E < Eage (13.63)

tfin Eage

KE −q−1 (q−1)

, if E ≥ Eage

This solution is not valid for q = 1. Note that for E/Eage  1, we can approximate 3 4q−1 1 − 1 − E/Eage ≈ (q − 1)E/Eage . So we for the low-energy part of the population we have N(E) = tfin KE −q = tfin N˙ (E), with a power-law index identical to the index of the injected population. But for E/Eage > 1 the powerlaw slope changes from the index at injection, to one that is steeper by one unit of E: q → q − 1. This corresponds to a change in synchrotron-spectral slope of

α = 1/2. Figure 13.9 shows for two values of initial q the resulting electron distributions.

13.4 Bremsstrahlung (Free-Free Emission) So far the radiation processes discussed are in essence caused by a constant (synchrotron radiation), or oscillating acceleration of charged particles (Thomson scattering). However, one of the most important continuum radiation process is caused by the encounters of charged particles with other charged particles, leading to an impulsive acceleration, and, hence, radiation. Like for the previous radiation processes, the emission is usually dominated by the lightest charged

400

13 Radiation Processes

Fig. 13.9 Electron spectra formed through continuous injection from a source with a power-law source spectrum, and subject to synchrotron radiation losses. The color coding is similar to that of Fig. 13.8, but unlike in Fig. 13.8 the electrons have been injected over a period of time, with the break energy (i.e. the cooling or age break) given by the cut-off energy for the oldest population of electrons

particles involved, i.e. electrons and positrons, as these accelerate strongest under the influence of an electric field. The radiation process itself is in high-energy astrophysics usually referred to as “bremsstrahlung” (German for “braking radiation”), whereas in radio and infrared astronomy the name “free-free emission” is often used. We will use here the term bremsstrahlung. Bremsstrahlung is important in ionised gas (plasma), and hence is most prevalent for temperatures with T  5000 K. For HII regions, which typically have T ∼ 10, 000 K bremsstrahlung is the dominant continuum radiation from the radio to the infrared band. One would expect that radio bremsstrahlung is also important for mature supernova remnants, for which the shock has become radiative and the plasma behind quickly cools down to 5000–20,000 K (Sect. 4.4, Chaps. 8, 10). However, even for mature supernova remnants radio synchrotron emission usually dominates over bremsstrahlung. In fact, the identification of free-free emission is sometimes used to distinguish HII regions, which have radio emission dominated by bremsstrahlung (free-free emission), from supernova remnants [508]. Bremsstrahlung is for supernova remnants a much more important radiation component in X-rays, as bremsstrahlung is often the dominant continuum radiation for X-ray plasmas with temperatures T > 106 K. Since the electrons producing the radiation are expected to have a Maxwellian velocity distribution the continuum radiation is usually referred to as thermal bremsstrahlung. As discussed in detail in Chaps. 11 and 12 supernova remnants contain accelerated particles (hadronic and leptonic cosmic rays) and these charged particles—

13.4 Bremsstrahlung (Free-Free Emission)

401

again mostly the electrons—will also produce bremsstrahlung, mostly in the γ -ray regime. Given the energy of the electrons, we are in the regime of relativistic bremsstrahlung. But depending on how far down in energy the accelerated electrons are present, one could also expect (hard) X-ray bremsstrahlung from these nonthermal electron distributions. Not surprisingly this radiation component is usually referred to as non-thermal bremsstrahlung [101, 695, 1095, 1170]. At a basic physical level, the above mentioned forms of bremsstrahlung are caused by the same radiation mechanism, but the underlying electron distributions differ. The expected emissivity can be inferred from a multiplication of the expected radiation spectrum, as function of frequency ν, expected from a single electron-ion encounter, multiplied by the collisional cross-section for that process for a given change in momentum p during an encounter: dσ dI (ν, | p|) d 2 σ ( p, ν) = . dνd| p| d| p| dν

(13.64)

We first describe the expected expected radiation spectrum dI (ν, | p|)/dν.

13.4.1 Bremsstrahlung from a Single Electron-Ion Encounter Figure 13.10 illustrates the origin of the radiation: an electron is on a trajectory to pass an ion with charge Ze, with an impact parameter b. During the passage it is slightly deflected over an angle θ . This situation is similar to the discussion on Coulomb cross-sections presented in Sect. 4.3, but there we only considered energy exchange between charged particles. Here we deduce the expected electromagnetic radiation that accompanies it.

l=vt -

θ r

b

+

Fig. 13.10 Left: Illustration of the geometry for a weak angle scattering of an electron passing an ion with charge Ze. Right: The acceleration of the electron upon passing the ion. Red is the parallel acceleration and black the perpendicular acceleration. Time is in units of v/b and acceleration in units of Ze2 /b2

402

13 Radiation Processes

For the small deflection encounter considered here, we can calculate the change in momentum by considering the two components of the Coulomb force, parallel and perpendicular to initial motion of the electron: dp Ze2 Ze2 = 2 sin φ = 2 l dt r (l + b 2 )3/2

(13.65)

Ze2 dp⊥ Ze2 b. = 2 cos φ = 2 dt r (l + b2)3/2 For a small acceleration we have l ≈ vt = βct, with t = 0 corresponding to the moment of closest approach. As shown in Fig. 13.10, the acceleration occurs during a short interval of t ≈ b/βc. The parallel component results first in a positive acceleration as the electron approaches the ion and then a negative acceleration after the nearest passage. As a result, the parallel acceleration switches sign, and for a small angle deflection the net change in parallel momentum is almost zero (Fig. 13.10). For the perpendicular component the total change in momentum is )

p⊥ = Ze2

+∞ −∞

bdt 2Ze2 , = 2 3/2 +b ] bβc

[(βct)2

(13.66)

and the mean acceleration during the encounter is 2Ze2

p⊥ = .

t b2

(13.67)

Filling this in the Larmor formula (13.2) we see that during the encounter the energy radiated is 2 e2 dE ≈ dt 3 m2e c3

   p⊥ 2 8 Z 2 e6    t  = 3 m2 c3 b4 . e

(13.68)

Note that the Z in the Larmor formula (13.2) concerns the charge of the electron (i.e. Z = 1), whereas here Z refers to the ion charge. We are not so much interested in the radiated energy per unit of time, but per unit of frequency, i.e. we need the Fourier transform of the last equation. The approximation usually taken is that a narrow pulse in time t leads to a flat spectrum up to a cut-off frequency ωmax ≈ 1/ t. The relation between power radiated as a function of time, to a Fourier transform is given by (e.g. [587])  2 ) ∞  1  dI (ω) dE 2 iωt   = |A(t)| → = 2 √ A(t)e dt  . dt dω 2π −∞

(13.69)

13.4 Bremsstrahlung (Free-Free Emission)

403

We can, therefore, approximate the spectral power by I (ω) ≈ dω



2 dE t 2π dt ωmax

=

2 e2 2 3π m2e c3 | p⊥ | ,

0,

ω  ωmax ω  ωmax

,

(13.70)

with more complicated behaviour around ω ≈ ωmax . The factor t/ωmax is a normalisation factor, ensuring that the energy emitted during an interval t equals the emission over the frequency interval 0 < ω < ωmax .

13.4.2 The Collisional Cross-Section and Total Radiation Spectrum Having now an expression for the bremsstrahlung spectrum from a single encounter, we next consider the differential cross-section for the process. We start with the differential cross-section for Coulomb scattering, the so called Rutherford formula—c.f. (4.38): d 2σ = dφd cos θ



Ze2 pβc

2

1 , (1 − cos θ )2

(13.71)

with θ the scattering angle, p and βc respectively the momentum and speed of the electron before the encounter with the ion with charge Ze. This equation can be written in terms of the electron momentum change | p|, noting that for small angle scattering the momentum components are p = p cos θ = p sin θ , and, hence, and p⊥ | p|2 = 2p2 (1 − cos θ ).

(13.72)

Using this equation to eliminate cos θ and after integrating over azimuthal angle(13.71), we find dσ d cos θ dσ = 2π = 8π d| p| d cos θ d| p|



Ze2 βc

2

1 . | p|3

(13.73)

Noting that for small angle scattering | p| ≈ | p⊥ |, we obtain the total radiation cross-section by combining this equation with (13.70), and converting to angular frequency ω = 2πν we obtain 16 Z 2 e2 d 2 σ ( p, ω) = d| p|dω 3 c



e2 me c 2

2

1 1 . β 2 | p|

(13.74)

404

13 Radiation Processes

For obtaining the average emission cross-section for an electron with a given momentum, we need to integrate the last equation over all the possible impact parameters. However, since p ∝ 1/b, we can also integrate over the equivalent range in | p|: 16 Z 2 e2 dσ (ω) = dω 3 c



e2 me c 2

2

  1 | p|max . ln β2 | p|min

(13.75)

Equation (13.72) shows that the maximum to be considered is | p|max = 2p. For a given frequency only impact parameters for which t < 1/ω will contribute to the emission. From (13.66) we see that this corresponds to pmin = 2Ze2ω/(β 2 c2 ). Using p = mβc (13.75) becomes 16 Z 2 e2 dσ (ω) = dω 3 c



e2 me c 2

2

  1 ζ me β 3 c 3 ln . β2 Ze2 ω

(13.76)

This is the classical cross-section for bremsstrahlung, valid for low-energy electrons, with ζ ∼ 1 a fine-tuning parameter, given that we made some approximations in the derivation. The quantity r0 = e2 /me c2 in this equation is the classical electron radius, which we also encountered in the Thomson cross-section. The limit | p|max = 2p used here breaks down for |p|b < h¯ , due to the Heisenberg uncertainty principle. The closest approach is for Ze/b = p2 /(2me ), which translates into b = 2Zeme /p2 . Together with | p|b ≈ h¯ , this suggest a maximum momentum transfer of | p|max = 4me Ze2 /h¯ . The two possible values for | p|max are equal for an incident momentum of p = 2me Ze2 /h¯ , which for Z = 1 corresponds to an electron energy of 55 eV. Inserting the limit for | p|max and | p|min = 2Ze2 ω/(β 2 c2 ) in (13.75) gives 16 Z 2 e2 dσ (ω) = dω 3 c



e2 me c 2

2

  1 ζ 4E ln , h¯ ω β2

(13.77)

where we have used that E = 12 mβ 2 c2 . At higher, but still non-relativistic, electron energies one needs to consider the collisional conservations laws including the momentum and energy picked up by the photon: | p|2 = |p − p − h¯ ω/c|2 and E = E + hω. ¯ Note that most of the momentum after the collision has been transferred to the ion, so one can approximate | p|2 = |p − p |2 . The maximum momentum transfer is for | p| = |p| + |p |, corresponding to the electron making a 180 degrees turnaround after the encounter with the ion. The minimum momentum transfer occurs when the electron

13.4 Bremsstrahlung (Free-Free Emission)

405

will be hardly deflected, in which case p = p −p . Using p = 

√ 2me E, we obtain

  1 p + p (13.78) ln ζ β2 p − p  √  √  2 2 ζ ( E + E − h¯ ω)2 e 1 16 Z 2 e2 = ln , hω 3 c me c 2 β2 ¯

dσ (E, ω) 16 Z 2 e2 = dω 3 c

e2 me c 2

2

√ √ √ √ √ √ where we used the relation ( x+ (x − a))/( x− x − a) = ( x+ x − a)2 /a, and again we add a factor ζ ∼ 1 to parametrise the uncertainties associated the approximations we made. For ζ = 1 this version of the bremsstrahlung radiation cross-section is known as the Bethe-Heitler approximation. Note that for E  hω ¯ the Bethe-Heitler approximation approaches (13.77).

13.4.3 Relativistic Bremsstrahlung As energy and time are Lorentz transformed in a similar way (E = E, t = t the total emitted radiation (dE/dt) does not depend on the velocity frame of the electron. However, bremsstrahlung from a relativistic electron will be strongly forward directed, with an opening angle θ ∝ 1/ . For relativistic interactions it appears that the maximum change in momentum (or change in velocity) cannot be too large, specifically | β| < 2/ , or | p| < 2me c [587]. So for the maximum momentum transfer we will use | p|max = 2me c. For determining the minimum energy transfer we make use of the relativistic identity E 2 = m2 c4 + p2 c2 , from which we can derive the approximate equation for   1 of pc ≈ E − m2 c4 /(2E). Clearly after the collision, the energy of electron must have been diminished to E = E − h¯ ω, and the minimum momentum transfer must occur when the emitted photon and electron move in the same direction: | p|min

  h¯ ω 1 m2e c4 m2e c4 m2e c3 hω ¯ |≈ E− −E + = |p − p − − hω = . ¯ c c 2E 2E 2EE (13.79)

Inserting this in (13.75) gives dσ (E, ω) 16 Z 2 e2 = dω 3 c



e2 me c 2

2



 ζ 4EE ln . me c2 h¯ ω

(13.80)

A more complete, quantum mechanical, treatment [587] gives for  → ∞ 16 Z 2 e2 dσ (E, ω) = dω 3 c



e2 me c 2

  

2  hω 2EE 1 3 h¯ 2 ω2 ¯ 1− ln − + . E 4 E2 2 me c2 h¯ ω (13.81)

406

13 Radiation Processes

For hard X-ray bremsstrahlung the electrons are only mildly relativistic and (13.78) seizes to be a good approximation, whereas (13.81) is still not valid. There are, however, good approximations to the trans-relativistic cases, see for example [497].

13.4.4 Thermal Bremsstrahlung As stated before, thermal bremsstrahlung is bremsstrahlung caused by a thermal electron population, and it is one of the most important X-ray continuum processes in supernova remnants, as well as in other sources of optically thin plasmas, such as clusters of galaxies and stellar coronae. It is customary to define √

 

Pmax 3 gff ≡ ln , π

Pmin

(13.82)

the so-called Gaunt factor. In order now to obtain the emissivity ((ω)) for thermal bremsstrahlung for a non-relativistic, isotropic, thermal electron population one needs to integrate (13.75) or (13.76) over the Maxwellian velocity distribution  f (v)d v = 3

me 2πkTe

3/2

  1 me v 2 d 3 v. exp − 2 kTe

(13.83)

The collisions rate of electrons and a specific ion species scales with σ ve ne ni , with ni the density of a given ion. For a given frequency ν = ω/2π only electrons with energies above the photon energy can result in radiation, i.e. hν = h¯ ω ≤ 12 me v 2 . Taking all this together, by summing over all ion species i and the Maxwellian velocity distribution of the electron we obtain )∞ ff (Te , ν) =2π

ne ni i



v √ v= 2hν/me

d 2 σ (ν, v, i) 4π v 2 f (v)dv dνdv

(13.84)

  hν 2π e6 exp − ne ni Zi2 g ff (Zi ) √ 3 3 m3/2 kT e kT e c i   hν −1/2 exp − ne ni Zi2 g ff (Zi ) erg cm−3 s−1 Hz−1 , ≈6.8 × 10−38 Te kTe

=

32π 3

i

where we have used the fact that the Gaunt factor is only weakly dependent on the velocity, so that we can approximate the integral by assuming a constant Gaunt

13.4 Bremsstrahlung (Free-Free Emission)

407

factor, and placing the deviations from that in a velocity integrated Gaunt factor g ff (Zi ). This equation shows that the emissivity per unit frequency is inversely √ proportional to Te and is relatively constant up to the cut-off energy hν = kTe . Integrating (13.84) over all frequencies gives a total emissivity that is propor√ tional to Te :  kTe ff (hν = 0, Te ) (13.85) h  ne ni Zi2 g ff (Zi ) erg cm−3 s−1 . ≈1.4 × 10−27 Te 

)

ff (ν, Te )dν =

i

From this a bremsstrahlung cooling time can be estimated of τff =

 , + i ni )kT kT 7 −1 ≈ 1.8 × 10 ne yr, ff (ν, Te )dν 1 keV

3 e 2 (n !

(13.86)

where we assumed the ion and electron temperatures are equal. Clearly the radiative cooling time is much longer than the lifespan of supernova remnants. Indeed, radiative cooling of supernova remnants is dominated by line radiation, which increases in strength for T < 106 K. For a given temperature the emissivity depends on the density squared , ne i ni Zi2 and on the ionic composition. The total radiation from a region of a given volume V is, therefore, proportional to the emission measure EM that can be defined as ) ) ) EM = ne (x, y, z) ni (x, y, z)Zi2 dxdydz. (13.87) V

i

Assuming a uniform composition and density the emission measure is often approximated by EM= ne nH V or simply EM= n2e V (using ne ≈ 1.2nH , valid for fully ionised hydrogen and helium). For (near-)cosmic abundances or low metallicities this is a fair approximation, as the bremsstrahlung is dominated by electrons colliding with ionised hydrogen and helium. But note that the integral is skewed toward higher density regions, so the emission is relatively sensitive to clumping. The most abundant metal has an abundance of ∼ 10−3 nH , so even for fully ionised oxygen (Z = 8) the bremsstrahlung from electron-oxygen collisions is only 6.4% that of hydrogen. However, in the shocked ejecta of young supernova remnants like Cassiopeia metals may dominated the composition of the plasma and hence may be the dominant source of bremsstrahlung. In that case care has to be taken to calculate the radiating mass from the emission measure [1181]. Since ionised metals are efficient bremsstrahlung emitters, due to the Zi2 dependence, there is a danger of overestimating the mass of the radiating plasma.

408

13 Radiation Processes

In high energy astrophysics it is common to express the emissivity in the number of photons per unit energy, which can be obtained by dividing (13.84) by hν. This gives d 2 nph ≈3.0 × 10−15 dtd(hν)



kTe 1 keV

−1/2 

hν 1 keV

−1

  hν exp − kTe

(13.88)

ne ni Zi2 g ff (Zi ) ph s−1 cm−3 keV−1 .

× i

13.4.5 Non-thermal Bremsstrahlung Many high-energy astrophysical objects, like supernova remnants, contain accelerated particles, with non-thermal energy distributions, as discussed in Chaps. 11 and 12. In supernova remnants the electron (leptonic) cosmic rays are probably energetically less important than the dominant hadronic cosmic-ray component, but radiatively the electrons are more important, and one of the ways by which leptonic cosmic rays produce radiation is bremsstrahlung. According to the theory of diffusive shock acceleration the energy distribution of particles is expected to be a power law in momentum n(p)dp ∝ p−q dp, with q ≈ 2. In the strongly relativistic regime this corresponds to ne (E)dE = KE −q dE, with K a normalisation constant. In the non-relativistic regime we have E = p2 /2m, so ne (E)dE = K E −(q+1)/2 dE, for which we can define a new slope for the energy distribution of q ≡ (q + 1)/2. Since electrons have a relatively low rest mass compared to cosmic-ray energies of interest, we can resort to the relativistic bremsstrahlung cross-section (13.81) to calculate the expected γ -ray emission. As we are interested in detecting individual photons we use the photon production rate, i.e. we divide the emissivity by hω. ¯ If we assume that the electron does not lose much energy during a collision we have E ≈ E. For the minimum electron energy to consider for a given photon energy we use Ee,min = h¯ ω. Using (13.80) we can approximate the photon emissivity as d 2 nγ (hω) ¯ ≈ d hωdt ¯

ni Zi2 i

1 1 h¯ hω ¯

1 1 16 e 2 ≈ h¯ hω ¯ 3 c =

1 16 2 e h¯ 3





)∞ c

me c 2

(13.89)

hω ¯

e2 me c 2

e2

d 2 σ (ω, E) ne (E)dE dωdE 2 ni Zi2 c i

hω ¯

2 ni Zi2

K i

 )∞  4E 2 ln KE −q dE me c 2 hω ¯

 

1 4hω ¯ −q (q − 1) ln + 2 (hω) . ¯ (q − 1)2 me c 2

13.4 Bremsstrahlung (Free-Free Emission)

409

This shows that non-thermal bremsstrahlung from an electron population with power-law slope q gives a photon spectrum with the same photon spectral energy slope bremss ≈ q, but with slight deviation caused by the additional ln(h¯ ω) factor. As discussed before the cosmic-ray spectrum is a power law in momentum, and for electrons it is not quite clear how far down in energy/momentum this can be extrapolated. Clearly at the shock front, the electron distribution will consist of a quasi Maxwellian distribution with a non-thermal tail consisting of electrons that have already picked up energy as a result of shock reflection and the initial stages of diffusive shock acceleration. This quasi-thermal/quasi-non-thermal distribution is sometimes referred to as the cosmic-ray injection spectrum. For young supernova remnants one may, therefore, expect non-thermal bremsstrahlung from electrons that have energies a couple of times the electron temperatures, giving rise to bremsstrahlung for photons with hν  10 keV [101]. In fact, non-thermal bremsstrahlung has been invoked to explain the detected hard X-ray emission from Cassiopeia A [1176], and more recently SNR W49B [1095]. However, once electrons do not interact with the shock transition anymore they will lose energy through Coulomb collisions, mainly with the population of thermal electrons (Sect. 4.3). For non-relativistic electrons the Coulomb loss cross-section scales as v −4 , so the energy exchange scale scales as vσ ∝ v −3 . As a result the lowest energy electrons that have not yet thermalised will thermalise faster than higher energy electrons, and the power-law in momentum will have some turn-over point at low energies. It depends on the density and the time since the electrons were in contact with the shock front (i.e. ne t), how much the non-thermal electron population has been effected by Coulomb losses. This will affect the detectability of hard X-ray non-thermal bremsstrahlung [1170]. Keeping in mind that the suprathermal electron spectrum may not be a simple power-law it is nevertheless instructive to calculate the expected non-thermal bremsstrahlung spectral slope in the non-relativistic limit. We assume√here that the non-relativistic relation between energy and velocity is valid, v = 2E/me .The integral over (13.78) gives a rather unwieldy result. So we use here the more approximate expression (13.77) for the bremsstrahlung cross-section, which gives d 2 nγ (hω) ¯ ≈ d hωdt ¯

≈

ni i

v

d 2 σ (ω, E) ne (E)dE dE

(13.90)

2hω/m ¯ e

1 1 16 e2 h¯ hω ¯ 3 c 

≈

)∞

1 1 h¯ hω ¯ √



e2 me c 2

e2

1 16 2 e c me c 2 h¯ 3



2

2 

c2

me 2

me K 2

2(2q

 ni Zi2 i

hω ¯ 4

−q + 1 )∞ 2

− 1) ln(4) + 2 K (1 − q )2



ln (x) x −q − 2 dx 1

4

1

−q − 2 ni Zi2 (hω) . ¯ i

This shows that for non-relativistic, non-thermal bremsstrahlung we expect bremss ≈ q + 12 = 12 q. If we assume that the electron power-law can be extrapolated

410

13 Radiation Processes

down from radio-emitting electrons to the suprathermal regime, as done in [101], we expect bremss = α + 32 , using the relation between radio spectral index α and q (13.43). There are, however, many caveats. For example, it may not be valid to extrapolated that far down in energy given that we are close in energy to the thermal distribution, and also because of the Coulomb losses discussed before. In addition, the bremsstrahlung cross-sections used here, but also in [101], are unreliable for electrons that are nearly relativistic.

13.4.6 Free-Free Absorption For low frequency radio emission the inverse process of bremsstrahlung may be important: free-free absorption. It is corresponds to absorption of radiation by a free electron that is near an ion. The expression can be obtain by using Kirchoff’s law that the emissivity cannot exceed the black body radiation limit. If that happens the rate of radiation absorption equals the rate of emission. For bremsstrahlung this means that ff = αν,ff Bν (T ) = αν,ff

2hν 3 1 , 2 c exp(hν/kT ) − 1

(13.91)

with ff given by (13.84), αν,ff the absorption coefficient, and Bν Planck’s radiation law. The expression for the absorption coefficient is, therefore, αν,ff

4π = 3





 1 1 hν 2π e6 1 − exp − √ 3 me3/2 ch kT ν 3 kT

ne ni Zi2 g ff (Zi ). i

(13.92) For hν  kT the exponential goes to zero, and we have αff ∝ ν −3 . For low frequency radio absorption we have, however, the situation that hν  kT . Expanding then the factor in brackets in a Taylor’s series we obtain αν,ff

4π ≈ 3 ≈

4π 3

 

1 1 2π e6 √ 3/2 −3 3 me ch kT ν



1 2π e6 ν −2 3 me3/2 c (kT )3/2

≈0.018T

−3/2 −2

hν +... kT

ne ni Zi2 g ff (Zi )

(13.93)

i

ne ni Zi2 g ff (Zi ) i

ne ni Zi2 g ff (Zi ).

ν

i

We see that the absorption is stronger for lower temperatures and lower frequencies.

13.5 Line Emission, Ionisation and Recombination Processes

The optical depth is τ = τ ≈ 5.6 ×



!

−2 ν 100 MHz

411

αff dr, which we can approximate with



T 100 K

−3/2 

r kpc

 ne ni Zi2 g ff (Zi ).

(13.94)

i

We see that the optical depth at 100 MHz can be substantial. But it should be remembered here that for T = 100 K most electrons are bound, so that for an interstellar medium density of nH ≈ 1 cm−3 , the electron density is only a tiny fraction of that, caused by ionisations due to the UV radiation and the cosmic-ray background. Nevertheless, for radio source at frequencies below ∼ 100 MHz there can be deviations from the power-law spectra discussion in Sect. 12.1. A very special case is (again!) Cassiopeia A, where radio absorption is most severe toward the centre of the remnant, indicating that the free-free absorption is partially due to free electrons associated with the cold ejecta interior to the reverse shock [90, 302, 618]

13.5 Line Emission, Ionisation and Recombination Processes Our knowledge of the energetics and composition of supernova remnants shells is largely the result of X-ray imaging and spectroscopy of their hot plasmas. The bulk of the shock ejecta of young supernova remnants emits X-ray emission, and for temperatures > 106 K, the temperature range expected for young and middle-aged supernova remnants, most of the alpha-elements (O, Ne, Mg, Si, S, Ar, Ca) are in the helium-like or hydrogen-like ionisation stage, with line emission in the energy band from 0.5 to 10 keV. This is the energy range accessible to imaging spectroscopy by a.o. Chandra and XMM-Newton. Iron and iron-group elements have a wider range of charge state distributions, but these have either emission lines in the 0.7–1.5 keV range (so-called L-shell emission, a term we will explain later) or in the 6–8 keV range (the K-shell line emission). A book about supernova remnants would, therefore, not be complete without some information on thermal emission from supernova remnants, in particular line formation and the processes that shapes the emission characteristics: excitation, ionisation, and recombination. Some of the information provided here will also be of importance for the lower temperature ranges at which supernova remnants emit optical/UV line emission. The thermal emission from supernova remnants is a combination of line emission, which can dominate the X-ray spectra of young supernova remnants, and continuum emission. The dominant continuum emission is often (thermal) bremsstrahlung already discussed in Sect. 13.4.4. Other sources of continuum emission are bound-free emission, and two-photon continuum. These processes are related to line emission and will be treated in this section. The thermal X-ray emission from supernova remnants is related to the thermal emission of cool stars, like the Sun, and the emission from the hot plasma of clusters

412

13 Radiation Processes

of galaxies. These are described by the “coronal model”, which is based on the following assumptions, as listed in [807]: 1. The plasma is optically thin, so that the X rays are not attenuated by the interaction with the atoms or ions in the plasma, and do not affect the populations in the bound atomic levels. 2. The gas density is sufficiently low so that the excited state populations are negligible compared to the ground state population. 3. The plasma electrons and the ions are relaxed to Maxwellian energy distributions with a common temperature,T , a free parameter controlled by external processes. 4. Radiation losses are balanced by non-radiative (mechanical) heating. 5. The gas is assumed to be in a steady state of statistical equilibrium both for the bound atomic states and for the ionisation balance. Some of these assumptions are, with some caution, also valid for supernova remnant plasmas. In particular assumptions 1–3 are generally also assumed for supernova remnant plasmas, but assumption 1 may not always be valid for certain emission lines; this is actually a point of caution for supernova remnants as well as for cluster of galaxies and stellar coronae. We will will discuss this in Sect. 13.5.9. For assumption 3 we should modify it to say that the controlling parameter is not so much the plasma temperature T , but the electron temperature Te . As discussed in Sect. 4.3.4 the electron temperature can under certain circumstances be lower than the ion temperature. Both the bremsstrahlung continuum and the ionisation/recombination and excitation processes are determined by Te rather than the overall plasma temperature. Where supernova remnant plasma differ most from coronal plasma is for the last two assumptions, both of which could be summarised as stating that the plasma is in steady-state. For supernova remnants we replace 4 by the assumption that the cooling and heating processes are relatively slow, and takes place over time scales longer than the observation time scales. Assumption 5 needs to be abandoned all together: in most supernova remnants the plasma is not in ionisation equilibrium. Often the plasma is underionised, i.e. one finds lower ionisation states than would be expected based on the electron temperature alone However, for some mixed morphology supernova remnants we find that the opposite is true: the ionisation degrees of the atoms is higher than expected given the electron temperature (Sect. 10.3). Non-equilibrium ionisation is discussed in Sect. 13.5.7. In this section we discuss several aspects of the emission from the hot supernova remnant plasmas: line transitions, the ionisation balance and continuum process. One important continuum process, bremsstrahlung, we have already discussed. Line emission is a process that is important for both the hot plasmas in young supernova remnants (T > 106 K) as well as the cooler plasmas behind radiative shocks. In both cases forbidden line emission plays an important role. To provide a better understanding of resonant and forbidden line transitions we first briefly review the quantum mechanical aspects of line radiation. For more in depth discussions of the quantum physics of line emission, in particular with a

13.5 Line Emission, Ionisation and Recombination Processes

413

view toward X-ray emission, we recommend [602], whereas a review of the more practical aspects of modelling X-ray line and continuum emission from hot plasmas is provided in [807]. For optical/UV line emission from hot plasmas the classic book by Osterbrock “Astrophysics of Gaseous Nebulae and Active Galactic Nuclei” [874] and Dopita & Sutherland’s “Astrophysics of the Diffuse Universe” [323] are recommended.

13.5.1 The Einstein Coefficients and Oscillator Strength An important aspect of the quantum physics of atomic transitions are the concepts of spontaneous transitions, stimulated transitions, and absorption, associated with the Einstein coefficients Amk , Bmk , and Bkm . Amk gives the probability (in s−1 ) for a spontaneous transition from energy level m to a lower energetic level k. For stimulated emission the probability per unit time is given by uν Bmk , with uν the spectral energy density of the radiation field. For the absorption probability we have a similar expression: uν Bmk . In case the atoms are in equilibrium with the radiation field the net rate of absorptions equals the net rate of emissions. Hence, uν Bkm = Amk + uν Bmk . From which we can derive an expression for the equilibrium radiation field energy density uν =

nk nm

Amk /Bmk , (Bkm /Bmk ) − 1

with nk /nm the population ratio for level k and m, which for an equilibration situation is given by the Boltzmann equation,   gk Em − Ek nk , = exp nm gm kT with gk , gm the statistical weights of the energy levels. As a result the energy density of the equilibrium radiation field must be uνmk =

gk gm

Amk /Bmk , (Bkm /Bmk ) exp (hνmk /kT ) − 1

(13.95)

with hνmk = Em − Ek . Since the equilibrium radiation field should be given by the Planck equation, uν =

1 8πhν 3 ,  hν  c3 exp kT −1

(13.96)

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13 Radiation Processes

we find the following relations between the Einstein coefficients: Amk =

3 8πhνmk Bmk c3

gk Bkm =gm Bmk .

(13.97) (13.98)

Note that there are several ways of defining the Einstein coefficients, differing by factors 4π/c with the above definition. The choice here is based on using the energy density rather than the intensity of the radiation. From the classical theory of a damped oscillator one can derive the following expressions for the Einstein A and B coefficient: Amk =3γ = Bmk =

8π 2 e2 2 ν me c3 mk

(13.99)

πe2 1 , me hνmk

with γ the so-called damping constant. The factor 3 is for taking into account the integration over all polarisation angles. The damping constant arises if an oscillating electron with resonant frequency νmk is forced to oscillate with an external electric field, in which case its displacement as a function of time is x(t) = exp(−γ t/2) cos(iωmk t). As we will discuss further below, the expression for the Einstein coefficients as obtained from quantum mechanics is very similar to (13.99), up to a transition dependent constant of order one, the so-called oscillator strength (fmk ): Amk =

8π 2 e2 2 ν fmk me c3 mk

(13.100)

Bmk =

πe2 1 fmk . me hνmk

(13.101)

For example, the oscillator strength for Lyα (λ = 1215.7 Å) is f = 0.41641 and for Hα (λ = 6562.8 Å) f = 0.641 [1226]. The emission of electromagnetic waves scales with |x| ¨ 2 (13.1). Taking the Fourier transform in order to get the spectral shape, it can be shown that the emission—and by virtue of (13.97) also the absorption—is proportional to φ(ν) =

1 γ /2 , π (ν − νmk )2 + (γ /2)2

(13.102)

which is called the Lorentzian profile. The normalisation here is chosen such that ! φ(ν)dν = 1. The absorption/emission profile has a full-width at half maximum of γ = A (Fig. 13.11). This width is usually referred to as the natural linewidth.

13.5 Line Emission, Ionisation and Recombination Processes

415

Fig. 13.11 The Lorentzian line profile

In quantum physics line this width is associated with the Heisenberg uncertainty relation E t ≥ h¯ /2, with t = A−1 . The discussion above involved some classical physics, but with some modifications agrees also with the quantum physics of line transitions, as will be discussed below, with some necessary background about atomic transitions.

13.5.2 Some Basic Atomic Physics Line radiation processes and related processes like ionisation and recombination are fundamentally quantum physical processes. Some of the basic aspects are reviewed here. Apart from a refresher on quantum physics of atomic transitions, this section also introduces the terminology associated with line emission from supernova remnant plasmas.

Wave Mechanics and the Schrödinger Equation In quantum mechanics wave functions (, t), which are complex functions, describe the probability of detecting a certain particle at a certain time, with a probability given by P (r, t) = |(r, t)|2 = | ∗ (r, t)(r, t)|,

(13.103)

with  ∗ indicating the conjugate of . The function  is normalised as )

(r, t)∗ (r, t)d 3 r = 1.

(13.104)

416

13 Radiation Processes

Quantities like momentum, energy or position are associated with operators, which act upon the wave function. And the expectation value of a quantity, q, associated with operator qˆ can be found by the integral ) q =< |q| ˆ >≡

3  ∗ qd ˆ r.

(13.105)

The energy operator is the Hamiltonian operator: h¯ 2 2 pˆ 2 + V (r) = − ∇ + V (r), Hˆ = 2m 2m

(13.106)

with V (r) a potential energy term and pˆ x = i −1 h¯ ∂/∂x the momentum operator (and for the total momentum squared we need to take into account all three spatial dimensions). The time evolution of the wave function is governed by the Schrödinger equation, which is i h¯

∂(r, t) = Hˆ (r, t). ∂t

(13.107)

The energy operator should return the energy of the particle, i.e. Hˆ  = E. Valid functions for which this equation holds are eigenfunctions. They correspond to appropriate values for the particle energy E. It could be that the eigenvalues are a discrete set of values Ek , as is the case with electrons bounds to an atom. Or E can consist of a continuous values; for example, in the case of a free electron. Note that if a set of eigenfunctions each satisfy the Schrödinger equation, then also an appropriately normalised superposition of eigenfunctions satisfy the Schrödinger equation. Note that eigenfunction are per definition orthogonal, i.e. if k and m are eigenfunctions than < k |m >= δmn . Inserting Hˆ  = E in (13.106) shows that valid eigenfunctions for  can be split up in a spatial and time varying part, with the time varying part simply being exp(iEt/h¯ ) (r, t) = ψ(r)e−iEt /h¯ .

(13.108)

In particular for a free moving particle (V (r) = 0) we see that (r, t) = A exp[i(p · r − iEt)h¯ ], with A a normalisation factor. A Single Electron in the Potential of an Atomic Nucleus In classical mechanics the effective central potential consist of a gravitational/Coulomb part and an centrifugal energy part L2 /(2mr 2), with L the absolute value of the total angular momentum.

13.5 Line Emission, Ionisation and Recombination Processes

417

In quantum mechanics the quantity L2 is associated with an operator Lˆ 2 , which has eigenvalue l such that Lˆ 2  = l(l + 1)h¯ 2 , whereas one component of the angular momentum (we pick the z-component) is associated with the eigenvalues m: Lˆ z  = mh¯ , with −l ≤ m ≤ l. So there are 2l + 1 possible value for m for a given l. The potential that an electron feels near an atom with charge Ze is, therefore, V (r) = −

Ze l(l + 1)h¯ 2 + . r 2me r 2

(13.109)

In reality it would be better to go to the center of mass system and write this equation in terms of the reduced mass, which for a nucleus with mass Amp would be μ = me Amp /(me + Amp ) ≈ me , but the difference is small. The potential (13.109) can be inserted into (13.106) and gives rise to eigenfunctions that can be factorised using a radial function Rn (r), which depend on a discrete value for n, and an angular function Ylm , depending on l and m: (r) = Rn (r)Ylm (θ, φ),

(13.110)

with  Ylm (θ, φ) =

2l + 1 Pl (cos θ )eimφ , 4π

(13.111)

with Pl the Legendre polynomials. The radial function Rn (r) fall asymptotically off as Rn (r) ∼ exp−Zr/na0 , with a0 ≡

h¯ 2 ≈ 0.5292 Å. me e 2

(13.112)

the Bohr radius of the hydrogen atom. In Fig. 13.12 we show the function r 2 |Rnl (r)|2 , which is proportional to the probability of finding an electron at a radius r (recall that this is proportional to 4πr 2 |R(r)|2 dr). The allowed values for l satisfy l < n, and n = 1, 2, . . . ., ∞, is called the principal quantum number. For l = 0 the energy eigenvalues are En = −

μZ 2 e4 1 1 2 2 1 Z2 α = − μc ≈ 13.6 eV, 2 n2 n2 2h¯ 2 n2

(13.113)

with α ≡ e2 /(h¯ c) ≈ 1/137, the so-called fine-structure constant. The negative sign indicates that these are bound energy states. The unbound states are associate with continuous positive eigenvalues.

418

13 Radiation Processes

Fig. 13.12 Radial probability functions for hydrogen-like atoms for the quantum numbers n = 1, 2, 3 and l = 0, 1, 2 (s, p, d)

Note the various total angular momentum eigenstates l are often represented by letters instead of by the l-numbers: Name: l=

s 0

p 1

d 2

f 3

g 4

h 5

i 6

..., ...,

where the first four letters have been labeled based on the appearance of the optical lines of alkali atom lines: sharp, principal, diffuse and fundament. After the f one keeps following the alphabet.

Atoms with Multiple Electrons: The Importance of Electron Spin A important property of particles is their intrinsic angular momentum, or the spin. For electrons the spin can have the values ± 12 h¯ , which shows that electrons are spinhalf particles, or fermions. Some particles (like the photon) are bosons which have integral spin values. For spin the same projection rules as for angular momentum apply, i.e. if the total spin is s (in units of h), ¯ the z-components can have the values ms = −s, −s + 1, . . . , s − 1, s (in units of h); ¯ i.e. there are 2s + 1 possible spin values. For a single electron s = 12 there are only two possible values ms = − 12 or ms = + 12 . Each ms enters into the wave function. Recall that the associated wave factors are ψs,+ and ψs,− , so that a total wave function can be represented as (r) = R(r)Ylm ψs . Also the combination of spin and angular momentum is an important quantum number: j = l + s, which depends on the relative orientation of angular momentum and spin axis. For atoms with two or more electrons the ensemble is described by an total wave function consisting of a multiplication of the wave functions of the individual elec-

13.5 Line Emission, Ionisation and Recombination Processes

419

tron wave functions: (1, 2, 3, . . .) = 1 (1)2 (2)2(3) . . ., where the numbers are shorthand for the different spatial coordinates of the individual particles. The corresponding Hamiltonian should be expanded with all the individual momentum operators (pˆ 12 , pˆ22 , ..) and the repulsive cross potentials of the electrons among themselves. For multiple electron systems, one has also new angular momentum , eigenvalues L = k lk , and total spin S = k sk . An important principle that governs the validity of wave function is, however, the Pauli principle, which states that bosons should be represented by symmetric wave function under permutations of identical particles, whereas fermions should have anti-symmetric wave functions under particle permutation. In other words, for bosons a combined wave function of two particles should have 1 (1)2 (2) = 1 (2)2 (1), whereas for fermions we should have (1, 2) = 1 (1)2(2) = −1 (2)2 (1). Since each (k) is factorised with different functions R(r)Ylm ψs one could make either the total factor associated with spin asymmetric, or the spatial part. A total symmetric spin function can be made in the following ways: ⎧ ⎪ ⎨ ψs,+ (1)ψs,+ (2),  sym ψs (1, 2) = √12 ψs,+ (1)ψs,+ (2) + ψs,− (1)ψs,− (2) , ⎪ ⎩ ψs,− (1)ψs,− (2),

(13.114)

The absolute value of the total spin can be obtained from Sˆ 2 ψs (1, 2) = 2h¯ 2 , whereas the z-axis projections are respectively mS = 1, 0, −1. Since we have three different combined spin-wave functions, all with S = 1, we call this state a triplet state (2S + 1 = 3 states). An asymmetric spin function can only be constructed as follows:  1  ψsanti (1, 2) = √ ψs,+ (1)ψs,− (2) − ψs,− (1)ψs,+ (2) . 2

(13.115)

The total absolute value of the spin in this case is S = 0 and consequently also mS = 0. There is only one possible configuration, so this state is referred to as a singlet state. If the spin-part of the wave function is symmetric the spatial part must be antisymmetric, or vice versa. One can show that for a symmetric wave function and anti-symmetric spin function (i.e. the spins are in opposite directions) that the electron separations are on average smaller. So electrons with aligned spin seem to repulse each other, not due to some electric force, but simply as a result of the Pauli principle. Note that the exact Hamiltonian for a multi-electron system should take into account force of the electrons, i.e. apart from , 2 the mutually repulsive , Coulomb 2 /|r − r should be included. However, as p ˆ terms, also the terms e i j i i j >i the Schrödinger equation for such a system cannot be solved, one usually tries to solve the equation using approximate central potentials for electrons in a specific

420

13 Radiation Processes

shell (see below for the shell-model). For example, for an atom with N electrons the outermost electron—if it is not in a closed shell—can be approximated by V (r) = −(Z − N + 1)e2/r for r → ∞. This assumption is that the inner electrons “screen off” the Coulomb force of the nucleus. For electrons in the lowest n levels one often tries V (r) = −Ze2 /r + C for r → 0. For example, for an element like sodium, which has one unpaired electron in the outer shell, the potential for high n levels should approximate that of hydrogen, since Z − N + 1 = 1. But for levels near the ground state the outer electron’s orbital partially overlaps with the radial functions of the inner electrons. So the screening is not complete.

Spin-Orbit Coupling The spin of an electron does not directly enter the Hamiltonian. But in the frame of the electron a magnetic field is present caused by the relative motion of the central charge. Its strength is B = −v × E/c2 = r × p|E/(me rc). As the electric field is related to the derivative of the Coulomb potential |E| = e−1 ∂V (r)/∂r, we can write B=

1 1 ∂V (r) L. me ec2 r ∂r

(13.116)

The Hamiltonian term associated with the interaction of the electron with this magnetic field is HB = −μe · B.

(13.117)

The magnetic moment of the electron is given by μe = −gs μB h¯ −1 S =

e S, me c

(13.118)

with gs ≈ 2 the electric spin g-factor, and μB = eh¯ /(2me c) the Bohr magneton. By taking into account precession of the electron orbit, HB reduces by a factor half (“Thomas’ half”). The additional Hamiltonian term becomes HB =

1 m2e c2

S·L

1 ∂V (r) . r ∂r

(13.119)

We see that this term depends on the combination L · S. Writing d J = L + S, and J2 = L2 + S2 − 2L · S, we see that L · S = 12 (L2 + S2 − J2 ). So L · S can be expressed by the quantum numbers l, s and a new quantum number j , the total angular quantum number. This quantum number is associated with a further splitting of the energy levels. The projection of J on the z-axis is indicated by mj = −j, −j + 1, . . . , j − 1, j .

13.5 Line Emission, Ionisation and Recombination Processes

421

13.5.3 The Atomic Shell Model and Electron Configurations The chemical properties of atoms, as well as their ionisation energies are determined by the number of electrons, and the subshells they fill in the ground state. The energy levels and ionisation energy of ions follow the same principles, allowing for an approximate scaling for ions with different nuclear charges, but the same number of electrons, i.e. belonging to the same isoelectric sequence. The building-up of the electron configurations of atoms and ions is controlled by the energy levels corresponding to the various (sub-)shells, governed by the n, l, m, s quantum states of the individual electrons, and the total quantum numbers L, S, J, MJ for the electrons combined. For a given number of electrons, the ground-state electron configuration follows the so-called “aufbau” (German for building-up) principle, a rule of thumb for identifying the lowest energy states (there are exceptions to the rules). One rule is that the lowest energy states correspond to the lowest value of n + l. If for two possible values of n and l the sum is equal, first the low n shell is filled. In addition the spin and orbital configurations follow Hund’s rules for open shell configurations: • For a given electron multiplicity the one with maximum S (or highest multiplicity) has the lowest energy. • For a given multiplicity the term with lowest L quantum number has the lowest energy. • For a given term, if the outermost subshell is half filled, the state with the lowest value of J has the lowest energy. If the subshell is more than half filled, the state with the highest J has the lowest energy. The highest energy for a given (excited) atom or ion corresponds to less bound configurations. The Hund’s rules assume that the effect of the spin repulsion is much stronger than the effect of orbital momentum, or the combined effects of orbital momentum and spin. This scheme is often called the LS coupling scheme or RussellSaunders coupling. For electrons in a very high Coulomb potential, i.e. Z  1 the coupling breaks down as the electrons in the inner most orbitals are becoming increasingly relativistic. Note that a completely filled subshell has all electron-spins and angular momentum projections paired, corresponding to L = 0 and S = 0. So the L and S of the ground state of an atom or ion are determined by the L and S of the subshell with the highest energy level. The electron configuration of a given atom or ion (in the ground state or excited) or ion is often expressed as 2S+1

LJ .

For the quantum state L one uses the letter designation, but then capitalised: L = 0 corresponds to S, L = 1 to P, and L = 2 to D, etc. This designation is, therefore,

422

13 Radiation Processes

determined by the outer-shell electrons only, Since inner shell electrons have L = 0, S = 0, this designation is determined by the outer-shell electrons only, To give some insights in the build-up of atoms/ions with an increasing number of electrons, we describe the filling up of the n = 1, 2, 3, 4 shells up to argon (Z = 36). The lowest energy shell (i.e. with the most negative energy) is n = 1. Since l < n only l = 0 is possible (1s shell, with the letter as specified before). As the Pauli principle only allows for two electrons of opposite spin (a singlet state) in a subshell, a neutral helium atom will have an S = 0 lowest energy level. For atoms more massive than helium, the n = 1 shell is filled with two electrons in the ground state with opposite spin. For n = 2 there are two subshells: l = 0, 1. The n = 2, l = 0 (2s) subshell again holds two electrons. The l = 1 sub=shell has three different angular momentum projections m = −1, 0, 1. It can hold two electrons per m-state, so the total is six electrons for the n = 2, l = 1 (2p) subshell. The maximum number of electrons in the n = 2 shell is, therefore, 2 + 6 = 8. The n = 1 shell and n = 2 shell together hold ten (2+2+6) electrons. This corresponds to the number of electrons of neutral neon. For n = 3, we have l = 0, 1, 2. As we have seen l = 0, 1 together hold eight electrons, whereas the l = 2 shell can contain up to 2 × (2l + 1) = 10 electrons. So the n = 3 shell can accommodate up to 18 electrons (10+8). Together with the n = 0, 1 shells, all shells up to n = 3 can contain up to 28 electrons. This number of electrons corresponds to the neutral nickel atom. However, it turns out that the aufbau principle (minimising n + l) first favours filling the n = 4, l = 0 (4s) subshell (n + l = 4), before filling completely the n = 3, l = 2 (3d) subshell (n + l = 3 + 2 = 5). So the ground state of nickel is the ground state of argon (filled up to n = 3, l = 1) plus eight electrons in the l = 2 (3d) subshell, and two in the n = 4, l = 0 (4s) subshell. Only once the n = 4, l = 0 is filled up, will the subshell n = 3, l = 2 fill up. This happens for zinc (Z = 30). The next subshell to be filled is n = 4, l = 1 (4p). This corresponds to argon Z = 36. The different subshells are designated by their n number and the letter associated with the l number (listed above): 1s, 2s, 2p, 3s, 3p,. . . The atomic number associated with completely closed subshells are: Shell: Z:

1s 2

2s 4

2p 10

3s 12

3p 18

4s 20

3d 30

4p 36

5s 38

4d 48

5p 54

6s 56

4f 64

5d 74

... ...

The complete shells with the outer electrons (valence electrons) in the l = 2 (pshell) subshell correspond to Z = 10, 18, 36, 54, . . .. These are the atomic numbers corresponding to the inert or noble-gas atoms Ne, Ar, Kr, Xe, etc. The exception is the noble gas helium having the 1s level completely filled. As the noble gasses have tightly packed shells their ionisation energies are higher than other atoms that are close charge, as can be seen in Fig. 13.13.

13.5 Line Emission, Ionisation and Recombination Processes

423

Fig. 13.13 Left: Ionisation energies of elements up to atomic number Z = 81. The local peaks in the ionisation energies (labeled) correspond to atoms with closed shells (except for Hg these are noble gasses). Right: Ionisation energies for iron ions, as a function of net charge z; z = 0 corresponds to Fe I. The ionisation energy steadily increases, except for closed-shell ions: z = 16, Fe XVII (Ne-like iron); z = 2, Fe XXV (He-like iron). For these ions the ionisation energy increases by a more sudden jump

As an illustration of the electron configuration build-up, and the ground state designation, we list here the configurations and labelling of the 11 atoms in the periodic system: H He Li Be B C N O F Ne Na

1s 1s2 1s2 2s 1s2 2s2 1s2 2s2 2p 1s2 2s2 2p2 1s2 2s2 2p3 1s2 2s2 2p4 1s2 2s2 2p5 1s2 2s2 2p6 1s2 2s2 2p6 3s

2S 1/2 1S 0 2S 1/2 1S 0 2P 1/2 3P 0 4S 3/2 3P 2 2P 3/2 1S 0 2S 1/2

Shifting our attention to ions instead of the neutral ground configuration, one notices that it takes relatively much more effort to ionise an ion with a closed shell. This is illustrated in Fig. 13.13 (right) for the ionisation energies for iron ions. The steady increase in ionisation energy as a function of charge, z, displays a relatively large jumps for z = 8 (18 electrons left), z = 16 (10 electrons left) and z = 24 (2 electrons left). These correspond to the same number of electrons as noble gas atoms.

424

13 Radiation Processes

Various ion stages are often designated with the neutral atoms with a similar electron configuration. So iron with one electron left is called hydrogen-like, and two electrons helium-like, etc. The closed shells ions for iron correspond to Fe XVII (Ne-like iron, z = 16) and Fe XXV (He-like). These ions tend to be present over relatively broad ranges in temperatures and/or ionisation age (see Sect. 13.5.7). Nalike iron (Fe XVIII) on the other hand is relatively easily to ionise. So if at a certain temperature Fe XVII is ionised, it is likely to quickly get ionised again for slightly higher temperatures (or timescales, see Sect. 13.5.7 below). A type of labelling one often encounters for the n = 1, 2, 3, 4, . . . shells are the capital letters K, L, M,. . . . In particular in X-ray astronomy theses frequently used. For example, K-shell emission lines correspond to line emission involving transitions to or from the n = 1 shell. They, therefore, typically refer to H-like or He-like ion states (but see below the discussion on inner shell ionisation). For X-ray emitting plasmas of kTe > 0.2 K the intermediate mass elements have typically one or two electrons left, resulting in K-shell line emission. Moreover, for these elements the L-shell line emission lies below 0.5 keV, which is more affected by interstellar absorption and less accessible by X-ray detectors. The L-shell corresponds to transitions of electron from or toward the n = 2 shell. The transitions involve transitions from n > 2 to the n = 2 shell, whereas the closed n = 1 shell electrons are usually not involved. As the n = 2 shell can contain many electrons and has many subshells, the number of possible transitions are large. L-shell line emission is is in particular important for iron, as for kTe  1.5 keV iron may be ionised up to, but not further than the L-shell. Since there are several ion stages with multiple electrons in the L-shell, the line emission can be rather complex, and the multitude of emission lines in the 0.8–1.5 keV band tend to blend together in the so-called “Fe-L” complex. See Fig. 9.1 for an example. M-shell transitions are of currently of less interest to X-ray astronomy. Only elements with Z > 28 have M-shell transitions in the 0.5–10 keV. Gold (Z = 79) has an M-shell ionisation energy of 2.2 keV, which can affect the calibration of Xray telescopes if they are coated with a thin gold layer. However, it may be in the future interesting to search for M-shell transitions of r-process elements in kilonova remnants, i.e. the shells created by neutron-neutron star mergers. We conclude this subsection with Table 13.1, listing the ionisation energy of ions up to nickel, with ions with the same number of electrons (so called ion isoelectronic sequences) aligned in columns. Along an isoelectric sequence the line energies and ionisation energies can be approximately scaled by the square of the screened ion charge, Zeff = z + 1; see (13.113). For example, for hydrogen the ionisation energy is 13.6 eV, and, therefore, for H-like iron (Zeff = 26) an ionisation energy of ≈ 2 = 9194 eV is expected. In Table 13.1 we see that the actual value is 13.6 × Zeff 9278 eV.

1s

config

13.6

54.4

122.5

217.7

340.2

490.0

667.0

871.4

1103.1

1362.2

1648.7

1962.7

2304.1

2673.2

3069.8

3494.2

3946.3

4426.2

4934.0

5469.9

6033.8

6625.8

7246.1

7894.8

8571.9

9277.7

10012.1

10775.4

1H

2 He

3 Li

4 Be

5B

6C

7N

8O

9F

10 Ne

11 Na

12 Mg

13 Al

14 Si

15 P

16 S

17 Cl

18 Ar

19 K

20 Ca

21 Sc

22 Ti

23 V

24 Cr

25 Mn

26 Fe

27 Co

28 Ni

Z

K

Shell

1

10288.9

9544.2

8828.2

8140.8

7481.9

6851.3

6249.0

5674.9

5128.9

4610.9

855.5

3658.3

3223.8

2816.9

2437.7

2086.0

1761.8

1465.1

1195.8

953.9

739.3

552.1

392.1

259.4

153.9

75.6

2399.3

2218.9

2045.8

1879.9

1721.2

1569.7

1425.3

1288.0

1157.7

1034.5

755.1

809.2

707.0

611.7

523.4

442.0

367.5

299.9

239.1

185.2

138.1

97.9

64.5

37.9

18.2

5.4

2s

24.6

L

1s2

3

K

2

2295.6

2119.4

1950.4

1788.7

1634.1

1486.7

1346.3

1213.1

1086.8

967.7

685.5

750.2

652.0

560.6

476.3

398.7

328.0

264.2

207.3

157.2

113.9

77.5

47.9

25.2

9.3

2s2

L

4

2130.5

1960.8

1798.4

1643.2

1495.1

1354.2

1220.3

1093.5

973.7

860.9

619.0

656.3

564.4

479.4

401.4

330.2

265.9

172.2

157.9

114.2

77.4

47.4

24.4

8.3

2p

L

5

2008.1

1844.0

1687.0

1537.2

1394.5

1258.9

1130.2

1008.6

894.0

786.3

540.4

591.6

504.6

424.4

351.3

284.6

225.0

172.2

126.2

87.2

54.9

29.6

11.3

2p2

L

6

1880.0

1724.0

1575.6

1320.3

1294.8

1165.2

1042.5

926.5

817.2

714.7

479.7

530.0

447.7

372.3

303.6

241.8

186.8

138.4

97.2

62.7

35.1

14.5

2p3

L

7

1758.0

1606.0

1460.0

1224.1

1188.0

1062.9

944.5

833.2

728.6

631.1

422.6

456.7

379.8

309.6

246.6

190.5

141.3

98.9

63.4

35.0

13.6

2p4

L

8

Table 13.1 Ionisation energies of atoms up to nickel

1646.0

1504.5

1 357.8

1133.7

1097.2

977.2

864.0

757.7

658.2

565.6

143.5

400.9

328.8

263.6

205.3

153.8

109.3

71.6

41.0

17.4

2p5

L

9

1541.0

1397.2

1262.7

435.2

1011.6

896.0

787.7

687.4

591.6

503.7

124.4

348.3

281.0

220.4

166.8

120.0

80.1

47.3

21.6

2p6

L

10

607.0

546.6

489.3

403.0

384.2

336.3

291.5

249.8

211.3

175.8

143.5

114.2

88.1

65.0

45.1

28.4

15.0

5.1

3s

M

11

571.1

512.0

456.2

343.6

354.7

308.5

265.1

225.2

188.5

154.9

124.4

96.9

72.6

51.4

33.5

18.8

7.6

3s2

M

12

495.4

441.1

392.2

314.4

296.7

254.8

215.9

180.0

147.2

117.6

91.3

67.7

47.2

30.2

16.3

6.0

3s2 3p

M

13

462.8

410.0

361.0

286.1

270.8

230.5

192.1

158.1

127.2

99.4

74.8

53.2

34.9

19.8

8.2

3s2 3p2

M

14

429.3

378.5

330.8

248.6

244.5

206.0

170.5

138.0

108.8

82.7

59.6

39.8

23.3

10.5

3s2 3p3

M

15

384.5

336.1

290.9

248.6

209.5

173.5

140.7

110.7

84.3

60.9

40.7

23.8

10.4

3s2 3p4

M

16

351.6

305.3

262.1

221.9

184.8

150.7

119.5

92.0

67.3

45.8

27.6

13.0

3s2 3p5

M

17

319.5

275.4

233.6

195.5

160.3

128.1

99.3

73.5

50.9

31.6

15.8

3s2 3p6

M

18

224.7

186.1

151.1

119.2

90.6

65.3

43.3

24.8

11.9

4.3

4s

M/N

19

193.2

157.8

125.0

95.6

69.5

46.7

27.5

12.8

6.1

4s2

M/N

20

162.0

128.9

99.0

72.4

49.2

29.3

13.6

6.6

3d4s2

M/N

21

132.0

102.0

75.0

51.2

31.0

14.6

6.8

3d2 4s2

M/N

22

108.0

79.5

54.9

33.7

16.5

6.7

3d3 4s2

M/N

23

76.1

51.3

30.7

15.6

6.8

3d5 4s

M/N

24

54.9

33.5

16.2

7.4

3d5 4s2

M/N

25

35.2

17.1

7.9

3d6 4s2

M/N

26

18.2

7.9

3d7 4s2

M/N

27

7.6

3d8 4s2

N

28

426

13 Radiation Processes

13.5.4 Electron Transition Probabilities and the Einstein Coefficients So far we did consider Hamiltonians that were fixed in time. Radiation fields or other changes to the system will induce transitions in the wave functions. This situation is treated by splitting up the Hamiltonian in a fixed and an additional perturbed Hamiltonian: H (r, t) = H0 (r) + Ht (r, t). Let us consider now a wave function as a summation over eigenfunctions corresponding to the unperturbed Hamiltonian, i.e. (r, t) =

ak (t)k (r),

(13.120)

k

, with ak (t) time dependent weighting function, such that k ak (t)= 1. We see that the changes in the wave function under the perturbed Hamiltonian is  ∂  ˆ (13.121) = H0 + Hˆ t  ∂t   ∂ψk = ak (t) Hˆ0 + Hˆ t ak (t)k . ∂t i h¯

i h¯ k

∂ak (t) ψk + i h¯ ∂t

k

k

If we assume that i h¯ ak (t)

∂k (t) ∂k (0) ≈ i h¯ ak (0) = ak (0)Hˆ 0k (0), ∂t ∂t

we see that the terms with Hˆ 0 k fall out of the equation, and we are left with i h¯ k

∂ak k = ∂t

ak Hˆ t k .

(13.122)

k

∗ and using < Multiplying both sides with the conjugate eigenfunction ψm ∗ ψm |ψk >= δmk gives

i h¯

∂ak = ∂t

∗ ˆ m H t k .

(13.123)

k

Assuming that the perturbing Hamiltonian is an oscillatory function with angular frequency ω, i.e Hˆ t (t) = Hˆ t 0 e±iωt , and using (13.108), we can integrate (13.123) over time to find  ) t )  1 1 t ∗ ˆ ak (t) = < ψk |Ht 0ψk > m Ht k dt = e(iωmk ±ω)t dt, i h¯ 0 i h¯ 0 k (13.124)

13.5 Line Emission, Ionisation and Recombination Processes

427

with ωmk = (Em − Ek )/h¯ . For t → ∞ the integral converges to δ(ωmk − ω). In other words, only a perturbed potential with a frequency matching the frequency ωmk will cause a change in quantum state. The transition rate can be calculated from |ak (t)|2 averaging from t = 0 to infinity:   i(ωmk −ω)t 2  |a(t)|2 1 2 1−e R = lim = lim 2 | < ψm |H0|ψk > | t →∞ t →∞ h t (ωmk − ω)2 t ¯ =

2π h¯ 2

(13.125)

| < ψm |H0 |ψk > |2 δ(ωkm − ω),

where we have used that |1 − exp(ix)|2 = 4 sin2 (x/2) and that lim(t → ∞) sin2 (αt)/(πα 2 t) = δ(α). This equation is sometimes referred to as Fermi’s golden rule. If we assume that the time varying term in the Hamiltonian is caused by an electromagnetic field associated with a vector potential A = Re {A0 exp[i(k˙r − ω)t]} =

1 |A0 | {exp[i(k˙r − ω)t] + exp[−i(k˙r − ω)t]} . 2 (13.126)

The associated Poynting flux S = |E0 |2 /(4π) can be obtained from E = −∂A/c∂t, which gives S(t) =

c ω2 ˆ |E|2 kˆ = |A0 |2 k, 4π 8πc

(13.127)

with kˆ the direction vector. The associated energy density is uω = |S|/c. The full Hamiltonian of the atomic system and the electromagnetic wave can be expressed as H =

1  e 2 p − A + V (r). 2me c

(13.128)

Expanding this, ignoring the higher order term A2 and assuming the Lorentz gauge, ∇A = 0, gives ieh¯ h¯ 2 2 (A · ∇) + . . . , ∇ + V (r) + Hˆ = − 2me me c

(13.129)

Inserting this into (13.125) provides an expression for the transition rate: R=

2  2π e2 h¯ 2  2 ±ikr |A | |e ∇|ψ > < ψ  δ(ωkm − ω).  0 m k h¯ 2 4m2e c2

(13.130)

428

13 Radiation Processes

For non-relativistic electrons we have |k·r|  1, which allows us to approximate < ψm |e±ikr ∇|ψk >≈< ψm |∇|ψk >. It requires some quantum-mathematical 2 | < ψ |r|ψ > |2 . Note wizardry to show that | < ψm |∇|ψk > |2 = m2e h¯ −2 ωmk m k that rˆ is related to the dipole operator d = er. If we also use uω = ω2 /(4πc2 )|A0 |2 , we can express the transition rate as R≈

2πe2 1 4π 2 e2 2 |< ψ u |r|ψ >| = uω fmk = Bmk 2πuν , ω m k me h¯ ωmk h¯ 2

(13.131)

with fmk the oscillator strength defined as fmk ≡

2me ωmk |< ψm |r|ψk >|2 , h¯

(13.132)

and Bmk the Einstein B coefficient as defined in (13.100), with the note that uω = uν /2π. As we have now a quantum physical expression for the Einstein B coefficient, we have also the related expression for spontaneous emission Amk through (13.97). , The oscillator strength satisfy the Thomas-Reich-Kuhn sum rule, k fmk = Z, with Z the number of bound electrons in the atom. The transition given by (13.131) is the so-called “allowed” electric dipole (E1) transition, as it operates on d = er. This came about as we neglected first and second order expansions of exp(k·r) in (13.130). These have a much lower transition probability, which can be shown to correspond to the interaction with the magnetic dipole of the orbiting electron (M1) or the electric quadrupole (E2). Some transitions cannot occur through the electric dipole transition, and are called “electric dipole forbidden transitions”, or forbidden transitions for short. The (E1) allowed transition lines on the other hand, are often referred to as resonant lines. Some transitions between energy levels, are forbidden according to (E1) transitions, but are allowed as (M1) or (E2) transitions, albeit at a much lower rate. Some transitions between levels are strictly forbidden, i.e. they can not occur through the emission of one photon, but the transition can occur through two-photon decay, in which the total transition energy is divided among two photons. Note that we assumed that k · r  1. However, for electrons moving in the innermost shells of a strong potential, for example for multiply ionised metals, the term k · r may start approaching one, and the (M1) and (E2) transitions will become relatively more important. This in particular of interest for the hot ∼ 107 K plasma in supernova remnants.

Selection Rules for Electric Dipole Transitions The discussion of forbidden and “allowed” transitions brings us to the selection rules for allowed transitions, as these are the transitions that in ordinary circumstances should dominate the line emission from a gas. For transitions involving a single

13.5 Line Emission, Ionisation and Recombination Processes

429

en rb idd Fo

Reso na

Two photon emission

nt

electron the rules are: l = ±1, because the photon (a spin s = 1 particle) carries away one unit of angular momentum h, ¯ and s = 0. For systems with multipleelectrons the selection rules are L = 0, ±1, S = 0, J = 0, ±1, but J = 0 → 0 is strictly forbidden. There is no restriction on the change in principal quantum number n. In the helium atom, and helium-like ions, the combined electron spin can either be in a singlet state (13.115), or in a triplet state (13.114), depending on whether the electron orbital configuration is symmetric, and the spin configuration antisymmetric, or vice versa. For an allowed transition we have S = 0. So a transition from a triplet (singlet) state can only be to another triplet (singlet) state. It is almost like there are two types of helium atoms. These two kinds of helium atoms are called ortho-helium (triplet state) and para-helium (singlet state). An example of this situation is shown in Fig. 13.14, which displays the energy levels of the O VII ion, which is prominent in a supernova remnant like SN 1006 [213, 1185]. The resonant line connects the 1s2p to 1s 2 singlet levels (i.e. l = 1, s = 0). The figure also shows the forbidden line transition (1s2s to 1s 2 ), which violates both the s = 0 and l = ±1 rule, and the intercombination line transition (1s2p to 1s 2 ), which does have l = 1, but violates the s = 0 rule. We come back to the emission of He-like ions in Sect. 13.5.8.

I

e nt

Singlet states (S=0)

rc

om

n bi

n io at

Triplet states (S=1)

Fig. 13.14 Level diagram (Grotrian diagram) of O VII (He-like oxygen), showing that the there is a separation between the singlet (S = 0) and triplet states (S = 1). Allowed transitions are indicated by green arrows. Note that the 3 P levels is split according to J = 0, 1, 2, with energy differences not discernable on this scale

430

13 Radiation Processes

13.5.5 Collisional Processes that Shape Emission Line Spectra In Sect. 13.5.2 we discussed the transition probabilities and their relation to the Einstein coefficients. This directly relates to photo-excitation and ionisation. However, ionisations and excitations in supernova remnants are dominated by collisional processes, mostly by electron-ion collisions. Bu perhaps with some ionion collisions contributions as well, in particular if there are low-energy cosmic rays present. In this section we explain some essentials of these collisional processes—and a bit photo-ionisation as well. It is important to note here that the hot supernova remnant plasmas have such low densities that collisional deexcitation can be neglected. Moreover, one can also safely assume that ions will be most of the time in the ground state. More precisely formulation: the collisional excitation time scale is much longer than the deexcitation time scale. The line emission from supernova remnants, therefore, comes from a collisional excitation, and the subsequent radiative deexcitations. Apart from direct excitation also recombination and (collisional) ionisation may result in a new ion that is not in the ground state. Before going into a more detail, the various collisional processes that shape Xray line spectra of supernova remnants are: • collisional excitation (followed by radiative deexcitation governed by Einstein coefficient A, Sect. 13.5.1): an electron changes from level m to n; • collisional ionisation (which includes inner shell ionisation): an electron is ejected from the atom as a result of a collision ; • radiative recombination: capture of a free electron by an ion; • excitation-autoionisation: a collision excites an-inner shell electron to a state with enough energy to subsequently eject an electron in another shell; • dielectronic recombination: an electron is capture by an ion, but this also results in the excitation of an additional bound electron; • charge-transfer reactions: two atoms collide and an electron from one atom is transferred to the other atom, often the resulting new ion is in an excited state (important for hydrogen emission from non-radiative shocks, see Chap. 8). The various processes are illustrated in Fig. 13.15. Collisional deexcitation is included, but as indicated before, it will not play a role in supernova remnant plasmas. Note that collisional inner-shell ionisation is just a special case of collisional ionisation, but instead of an outer-shell electron, an electron from the inner-shell (for example a K-shell) is liberated, in the presence of electrons occupying higherlevel shells (L- or K-shell). The inner-shell ionisation is then followed either by an auto-ionisation of one or multiple electrons (similar to the excitation-autoionisation process), or by a radiative deexcitation (fluoresence), as illustrated in the figure. The autoionisation process is also called an Augér process [553, 598]. The probability for a radiative transition is called the fluorescence yield. The fluorescent yield increases with atomic number/photon energy, as can be seen in Fig. 13.16.

13.5 Line Emission, Ionisation and Recombination Processes

431

n=∞

n=∞

n=∞

n=∞

2p 2s

2p 2s

2p 2s

2p 2s

1s

1s

1s

1s

Collisional excitation

Collisional deexcitation

Collisional ionisation

Inner-shell ionisation

n=∞

n=∞

n=∞

n=∞

2p 2s

2p 2s

2p 2s

2p 2s

1s

1s

1s

1s

Excitation-autoionisation

Dielectronic recombination

Radiative deexcitation

Radiative recombination

Fig. 13.15 Various collisional electron-ion processes (and radiative decay). The open circles represent the final state of the electron(s). Exceptions are inner-shell ionisation (followed by radiative deexcitation) and the excitation-auto-ionisation process, where the last step is indicated by dashed symbols. Collisional deexcitation and radiative ionisation (not shown) are not important for supernova remnant plasmas

Fig. 13.16 Fluorescence yield for K-shell (black) and L-shell (blue) transitions of neutral atoms (based on [553])

432

13 Radiation Processes

Photoionisation and the Interstellar X-ray Extinction Photoionisation (radiative ionisation) does not play a significant role in the ionisation balance of supernova remnant plasmas. However, it does affect the X-ray spectra of supernova remnants as it is the dominant component of interstellar extinction in the soft X-ray band (0.1–10 keV). Photoionisation is related to radiative transitions, except that the final state of the electron is a free electron. The crosssection for bound-free transitions is proportional to the probability | < m |Ht |k > |2 , similar to photoexcitation and deexcitation, but with m corresponding to the free electron case, ψm = exp(−ip · r/h¯ ) (c.f. 13.125). For hydrogen-like ions the cross-section for photoionisation from quantum level n is [602] σPI

64α Z 4 = √ 5 3 3n



EH hν

3 πa02 g,

(13.133)

with g a Gaunt factor of order unity, and α ≈ 1/137 the fine-structure constant, and EH ≡ e2 /2a0 = 13.6 eV the Rydberg energy. Note the sharp cross-section decline above threshold, scaling as ∝ ν −3 . For interstellar extinction by abundant atoms, the ionisations of K-shell ions provide the largest contributions. In this process, an inner 1s, electron is liberated from a neutral or low-ionisation state atom in the interstellar medium. The K-shell energies scale roughly with atomic number as EK ∝ Z 2 (more accurate relations can be found in [132]). Above the K-edges the cross-sections fall off roughly as σ ∝ E −3 , similar to the case for hydrogen-like ions. The absorbing atoms can also be locked up in dust grains, in particular the refractory elements Mg, Si, and Fe. If the dust grains are not optically thick to X-rays, the K-shell transitions also take place, but due to internal interference of the photoelectron with densely packed atoms, the K-shell edges display fine structures known as XAFS (X-ray absorption fine structure); see for example [542, 1272]. It is customary to write the optical depth in terms of a radiation cross-section per equivalent hydrogen column (NH ): τ (E) = NH σ (E) = NH

σPI,i (E) i

ni , nH

(13.134)

with i running over all atoms, with ni /nH the average interstellar abundance; e.g [1238]. Typical column densities toward Galactic supernova remnants are NH ∼ 1021–1022 cm−2 . Note that hydrogen itself still contributes somewhat to the soft X-ray absorption, but most of the absorption is caused by the photoionisation of abundant metals that have their K-shell edges in the 0.1–10 keV band, as illustrated in Fig. 13.17. Apart from photoionisation, dust scattering. and Compton scattering also contribute to the attenuation of the X-ray signal and can be incorporated in the definition of σ in (13.134). These two processes merely scatter the photon out of the line of sight, with dust scattering mostly important below 1.5 keV. Compton scattering starts

13.5 Line Emission, Ionisation and Recombination Processes

433

Fig. 13.17 The Galactic interstellar extinction crosssection, based on the model explained in [1238]. The crosssection σ is defined here as τ = σ NH . The extinction is dominated by radiative K-shell ionisations, with the K-shell (and Fe-L shell) edges indicated, but some line absorption features are also discernible

dominating above the Fe and Ni K-edges, but the effect is only severe for column densities of NH  10%/σT ≈ 1023 cm−2 . Collisional Excitation Collisional excitation occurs when a free electron or an ion passes nearby a “target” ion, and one of the electrons of the target ion changes to a higher energy level. The direct cause of this excitation is the time varying Coulomb interaction of the free charged particle with the bound electron. For an encounter at a distance, the bound electron experiences this Coulomb force for a time t ≈ 2b/v (c.f. Sect. 13.4.1, Fig. 13.10), with b the impact parameter of the collision and v the velocity of the free particle. A full calculation of the excitation rate using quantum theory, involves perturbation theory similar to what led to equation (13.125), with the incoming plane electromagnetic wave being replaced by the electromagnetic content of the Coulomb interaction ([602]), and with the spectrum extending up to frequencies ω ∼ 2π/ t. As the underlying principles are similar to radiative excitations, the same selection rules apply: allowed transitions are also more likely to occur for collisional excitations. Without going into a lot of detail, one can obtain a sense of the relevant cross-sections using a semi-classical approach [602]. Moreover, the excitation cross-sections that are used for spectral codes usually involve heuristic fitting functions to measured excitation rates [807]. In a semi-classical sense, the transfer of momentum from a free electron to a bound electron is the time integral over the Coulomb force: )

p ≈

+∞ −∞

Ze2 2Ze2 Ze2 . dt ≈

t = r2 b2 bv

(13.135)

Interestingly, one sees that the momentum transfer depends on the charge and velocity, not on the mass, of the free particle. The transferred momentum should

434

13 Radiation Processes

be comparable to the energy needed to excite the electron from level k to level m: Ekm ≈

( p)2 22 e 4 ≈ . 2me me b 2 v 2

(13.136)

For the cross-section we can now write σkm ≈ πb2 , which results in σkm ≈

2πZ 2 e4 πZ 2 e4 = , me v 2 Ekm EEkm

(13.137)

with E = 12 me v 2 the energy of the free electron, assuming non-relativistic motions. It is customary to rewrite (13.137) in atomic units using the Bohr radius a0 ≡ h¯ 2 /(m2e e2 ) and the Rydberg energy: σkm (E) ≈ 4πa02 Z 2

2 EH . EEkm

(13.138)

Finally, one defines a dimensionless collision strength i , which contains all the quantum-mechanical details of the calculations (or finetunings based on actual measurements): 2 σkm (E) ≡ πa02 EH

km , gk E

(13.139)

with gk the statistical weight of the initial level. For electric dipole transitions the collision strength is related to the oscillator strength (Sect. 13.5.1) [602]: km 8π fkm g =√ , gk 3 Ekm

(13.140)

with g a Gaunt factor. The appearance of the oscillator strengths shows the close connection between radiative and collisional excitation. Often approximate parameterisations for arbitrary cases are used in order to fit laboratory measurements. This is in particular relevant for the complex behaviour of the cross-section near the energy threshold. For example [807] advocates (U ) = A +

C B 2D + 2 + 3 + F ln U, U U U

(13.141)

with U = Ekm /kTe . For the total excitation rate coefficient one has to integrate the cross-section times velocity over the velocity/energy distribution of the free particles, ) Skm =

∞ Ekm

) f (E)vσkm (v)dE =



∞

f (E) Ekm

2E σkm (v)dE. m

(13.142)

13.5 Line Emission, Ionisation and Recombination Processes

435

For a Maxwellian distribution, f (E) =

  2E 1/2 E exp − , π 1/2 (kT )3/2 kT

(13.143)

and for a collision strength that is relatively independent of energy the electron excitation rate coefficient is approximately [807] −1/2

Skm =8.63 × 10−6 g1−1 (y)Te e−Ekm / kTe (13.144) −1  −1/2  Ekm T −1 =5.4 × 10−10 g fkm e−Ekm / kTe cm−3 s . 7 1 keV 10 K To obtain the total collisional excitation rate per unit volume (13.144) has to be multiplied by ne and the density of the specific ion ni . The numerical value shows that collisional excitations are relatively rare for typical electron densities of ne ≈ 1 cm−3 . For a given ion, the numerical value corresponds to once every ∼ 300 yr, comparable to the age of a young supernova remnant. The line emission is then given by the collision rate to a level m, and the radiative decay branching ratio to various lower energy levels. In supernova remnants collisional depopulation of excited levels can be neglected. Note that also ionisation and recombination can leave the newly formed ion in an excited state, and, therefore, contribute to the line emission of hot plasmas. This is in particular true for forbidden line emission, as further discussed in Sect. 13.5.8. For the above derivation we assumed that the excitations are caused by electronatom collisions. Excitations can also be caused by ion-ion collisions. Naively one might assume that this ion-ion collisions could even be dominant, because in the above calculation the cross-section was shown to depend not on mass of the free particle but only on the velocity as σkm ∝ 1/v 2 . And for a given temperature a more massive ion has on average a slower speed, and, hence, a larger cross-section. For the rates one has of course to integrate as indicated in (13.142), which changes the dependency on v to the advantage of lighter particles. However, the biggest difference between ion impact and electrons is not contained in the above derivation. and that is that the above derivation is only valid for E  Ekm , with all the details hidden in the Gaunt factors. Indeed for electrons, the cross-section rapidly rises at the threshold and then falls off as 1/E. For protons and other ions, the cross-section is greatly reduced for speeds v  vnl = Enl anl /h¯ , with vnl the “speed” of the bound electron, nl being the orbitals quantum numbers, and anl the characteristic radius. The following cross-section for ion-collisional excitation was proposed in [601] for particles with charge Ze: σkm =

πa02

  2  

EH 1.95 Z2 2 0.283 ln v˜ + 1 + 1.26 exp − , 2 v˜ 2 v(1 ˜ + 1.2v˜ 2 ) Ekm (13.145)

436

13 Radiation Processes

with v˜ ≡ vnl /v. This shows that the ion-impact excitation cross-section peaks at impact energies a factor mion /(me Z 2 ) larger. Nevertheless, under certain circumstances one has to take ion impact excitations (and ionisations) into account: low energy cosmic rays may cause additional excitations, and for extreme nonequilibrium ionisation, the excitation energies may be well below the typical energies of the ions (i.e. v˜  1), and, thus, dominate the excitation rates.

Collisional Ionisation Collisional ionisation is similar to excitation, except that the bound electron will transition to a free state. A natural requirement is that the initially free electron should have an energy equal or exceeding the ionisation energy (χm ) of the bound electron in shell labeled m: E > χm . For a given electron in a shell with ns electrons the direct ionisation cross-section is often approximated with the Lotz formula [743, 807]: N

σDI = πa02 k=m

*    +  E EH 2 ln(E/χm ) Ck ξk 1 − bm exp 1 − cm −1 , χk χm E/χm (13.146)

with k running over all subshells, ξk the number of electrons in the subshell, and bm and cm adjustable parameters. The factors within {} assures the correct behaviour near the threshold energy, which can be approximated by 1 for multiply ionised atoms, in which case Ck = 2.76. The integration over a Maxwellian distribution (13.142) yields an ionisation rate coefficient that is approximated by [602, 807]: SDI (T ) ≈ 1.5×10

−10

   χ −2  T 1/2 χm m exp − cm−3 s. 1 keV 107 K kTe

(13.147)

For ion impact ionisation the same considerations holds as for ion impact excitation, in that the cross-section peaks at energies a factor mion /(me Z 2 ) larger than for electron impact ionisation. The above approximation does not include yet the effect of excitation-autoionisation. The approach taken for the calculation of ionisation balances, including excitation-auto-ionisation, in spectral codes like xspec [95] or SPEX [599] is to make use of parametrised models with tabulated parameters, as a function of electron temperature. An example are the fits to the ionisation cross-sections in [96], based on the following approximation for the excitation-auto-ionisation rate coefficients: SEA (T ) = 2.28 × 10−8



T 107 K

−1/2

e−x F (x) cm3 s−1 ,

(13.148)

13.5 Line Emission, Ionisation and Recombination Processes

437

with x = EEA /kT , EEA being a suitable excitation energy (left as a free parameter in the fit to the cross-sections) and F (x) a complicated function depending on four parameters, obtained from an heuristic fit to the measured ionisation cross-sections.

Radiative Recombination (Free-Bound Transition) Radiative recombination is the process by which a free electron with energy E is captured by an ion into a bound orbit with quantum number n. It is the inverse of photoionisation. If χn is the ionisation energy of the final configuration, radiative capture leads to the emission of a single photon with energy hν = E + χn .

(13.149)

The process is the inverse of photo-ionisation (bound-free transition), with a simple relation—the so-called Milne relation—between the cross-sections of the two processes: m2 c2 v 2 gz+1 σPI = e2 2 , σRR h ω gz

(13.150)

with the gz , gz+1 ’s different statistical weights of the two ionisation levels. For hydrogen-like ions σPI is given by (13.133). To obtain the recombination rate we need to integrate σRR over the velocity and velocity distribution. For hydrogen-like ions an approximation of the recombination rate coefficient is [807, 1032]:

αRR ≈ 5.20×10−14zλ1/2 0.4288 + 0.5 ln λ + 0.469λ−1/3 cm3 s−1 ,

(13.151)

with λ ≡ 157890Z 2/T , with T in Kelvin. For a typical charge z = 9 and T = 107 K—applicable for hot plasmas in young supernova remnants, and for hydrogenlike neon—one finds αRR ∼ 2 × 10−13 cm3 s−1 .

13.5.6 Radiative Recombination Continuum (Free-Bound Emission) Apart from the role of radiative recombination in the ionisation balance of supernova remnant plasma (of which more below), it is also a, sometimes neglected, source of emission. The emission is quasi continuous with a threshold photon-energy of χn (13.149). Since atoms have many closely packed energy levels for high quantum number n, the starting energy for a given ion is close to the series limits of a given ion, for a given ion the emission sharply rises around hν ≈ χ∞ , corresponding to an

438

13 Radiation Processes

electron energy just above the bound-free threshold, E ≈ 0. The overall spectrum above the threshold for a thermal free-electron spectrum resembles the expression for thermal bremsstrahlung, with the emissivity per ion given by [602]: fb (Te , ν) = ne ni,z

gz gz+1



hν χn

3 

χ2 me c2 kT

3/2

h¯ cσPI (ν)e−hν/ kTe eχn / kT , (13.152)

with ni,z indicating the density of a certain atom ni with charge z. Since σPI ∝ ν −3 , the emissivity is constant for ν  kT , and an exponential cut-off for hν  kT . If χn  kT the shape of the continuum appears line-like, because then E  χn , and the photon energies are close to hν ≈ χn . The continuum itself is referred to as radiative recombination continuum (RRC). In the underionised plasmas associated with young supernova remnants (see below), the RRC tend to be broad, since kT > χn , and hard to identify. However, during the last two decades many more sharply line-like RRC features have been identified in mixed-morphology supernova remnants (Sect. 10.3). In these supernova remnants the plasmas are often overionised (the electrons are relatively cool with respect to ionisations states, Sect. 13.5.7), and the electrons have a relatively low energy compared to the threshold energies χn .

Dielectronic Recombination Dielectronic recombination is related to excitation-autoionisation in the sense that two electrons are at play; one free and one bound electron. The free-electron is captured by the ion, whereas at the same time the excess energy excites another bound electron. This double excited ion can now decay radiatively, producing multiple emission line photons—in which case we have ion recombination—or the excited state leads to an (auto)ionisation, in which case there was no net recombination. The dielectronic recombination rate, therefore, depends on the ratio between auto-ionisation rate S a and the radiative-deexcitation rate S r . Since S a ∝ Z and S r ∝ Z 4 [602, 807] dielectronic recombination is important for high-Z ions, most notably highly ionised iron. An approximate equation for the dielectronic recombination rate coefficient for a Maxwellian electron distribution is [807] αDR ≈ 6.55 × 10

−27



T 107 K

−3/2

Bs e−Es / kT ,

(13.153)

s

with Bs ≡

ws Aas Ars , . w1 (Aas + Ars )

(13.154)

13.5 Line Emission, Ionisation and Recombination Processes

439

The index s refers to all possible excitation states and ws = 2l(l +1) is the statistical weight of a given excited state.

13.5.7 Non-equilibrium Ionisation The discussion of thermal emission so far pertains to all coronal plasmas, whether they are found in clusters of galaxies, supernova remnants, or cool stars, with the additional complication that young supernova remnants can have very metal-rich plasmas. But there is an important difference between the emission from supernova remnants and other hot astrophysical plasmas: supernova remnant plasmas are almost always out of ionisation equilibrium. This is usually indicated with the term non-equilibrium ionisation, or NEI. The plasmas of cool stars and clusters of galaxies are referred to as collisional ionisation equilibrium, or CIE. The reason that supernova remnants plasmas are in NEI is that, for the low densities involved, not enough time has passed since the plasma was shocked, and per ion only a few ionising collisions have occurred for any given atom [581]. Let us denote an ion species with the notation ni,z , with i just an index to indicate a specific atom and z the ionisation state z = 0, 1, . . . , Z. The differential equation governing the evolution of the ionisation stages for a given electron temperature are then 3 4  dni,z = ne αi,z+1 (Te )ni,z+1 Si,z−1 (Te )ni,z−1 − αi,z (Te ) + Si,z (Te )ni,z , dt (13.155) with αz and Sz the recombination and ionisation rate coefficients, respectively— including all the different modes discussed above. The first two terms on the right-hand side show the gains to the ion density ni,z by recombination from a more ionised ion, and ionisation from a less ionised ion. The term in square brackets shows the decrease in ion density due to recombination and ionisation to respectively lower and higher ionisation states. Since there are Z + 1 ion states for a given atom, (13.155) is a system of Z + 1 differential equations. As all rates are proportional to the electron density, the evolution is governed by the combination ne t rather than t. For this reason ne t is taken as a single parameter, which is usually referred to as the ionisation age or ionisation time scale. Note that ne is not necessarily constant as function of time, as the electron density may change as the ionisation levels evolves, and because there can be external reasons for density evolution, such as expansions or compressions of the plasma. By definition, in the case of collisional ionisation equilibrium (CIE), there is no net evolution in the density of an ion, dni,z /dt = 0.The differential equation can then be unique solved for a given electron temperature Te . However, as the rate coefficients (13.151) and (13.147) indicate, for temperatures around 107 K the rate coefficients have typical values around 10−13 − 10−10 cm3 s−1 . This implies

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13 Radiation Processes

that H-like and He-like ions recombine/ionise on time scales of t ≈ 1/(αne ) or t ≈ 1/(Sne ) ≈ 1010 − 1013 s. This corresponds to several hundred to hundred thousands year for the typical low densities in supernova remnants of ne ∼ 1 cm−3 . So we do not expect the plasmas of supernova remnants to be in CIE. Equation (13.155) can be directly integrated as a function of ionisation age. Most NEI X-ray emission codes (e.g. XSPEC and SPEX) assume that the initial state corresponds to a CIE plasma at a given low temperature, and that at ne t = 0 the temperature suddenly increases to a much higher temperature. This gives in general good results, but it should be noted that in reality ne and Te are not constant after shock heating has occurred. Moreover, the supernova remnants contains plasma with a range in Te and shock histories (ne t). So a model consisting of one or a few discrete NEI components is an approximation. There are more complex X-ray spectral models that do take into account shock histories and plasma gradients— for example the vpshock model [196]. But these models require some assumptions regarding the shock-heating histories. Nevertheless, the obtained ne t parameters do give useful insights into the time scales of ionisation, which can be used to constrain the typical densities in supernova remnants and their age. In addition, the densities can also be estimated from the emission measure (13.87), allowing for tighter constraints on the age of the supernova remnant. Note that the parameter ne t also governs the time scale for electron-ion temperature equilibration (Sect. 4.3.5), and, therefore, provides an estimate on the potential importance of temperature non-equilibration. Equation (13.155) can be solved by direct numerical integration, but this is CPU intensive and not very practical when it comes to fitting X-ray spectral data. A faster approach [558, 597, 1062] is to rewrite (13.155) in matrix notation, 1 dF = A · F, ne dt

(13.156)

with F ≡ (ni,0 /ni , . . . ni,Z, /ni ) the vector with the ionisation fractions, such that its elements are normalised: z Fz = 1, A the matrix containing all the recombination and ionisation rates. One can now solve the coupled differential equations by first determining the eigenvalues and eigenvectors of A and then solve for the uncoupled equations 1 dF = λ · F , ne dt

(13.157)

with λ the diagonal matrix containing the eigenvalues, and F ≡ V−1 F, with V containing the eigenvectors. The main effect of NEI on the plasmas of young supernova remnants is that the ionisation states at a given temperature are lower than in the CIE situation. In Fig. 13.18 the effect of NEI in young supernova remnants is illustrated. As an example, consider the presence of O VIII in Fig. 13.18 (left), which peaks in CIE at kTe = 0.2 keV. Identifying a plasma in a supernova remnant with most of the

13.5 Line Emission, Ionisation and Recombination Processes

441

Fig. 13.18 The effects of non-equilibration ionization (NEI) illustrated for oxygen. Both panels look very similar, but the panel on the left shows the oxygen ionisation fraction as a function of electron temperature for collisional ionization equilibrium (CIE), whereas the panel on the right shows the ionisation fraction as function of ionisation age (ne t), and for a fixed temperature of kTe = 1 keV, assuming an initial ionisation distribution valid for f kTe = 2 eV. The ionisation fractions were calculated using the spectral code SPEX 3.05 [599]. In practise supernova remnants have found to have ne t values ranging from ∼ 109 − 1013 cm−3 s. Note the broad range in kTe and/or ne t over which He-like oxygen is present, which is caused by the resilience of the fully occupied K-shell

oxygen emission coming from O VIII, may lead to the conclusion that the plasma is relatively cool. However, as Fig. 13.18 (right) shows, the temperature may very well be kTe = 1 keV, but with a low ionisation parameter of ne t ≈ 8 × 109 cm−3 s. For temperatures relevant for young SNRs (kTe = 0.5 − 4 keV) the ionisation age needed to reach CIE is around ne t ≈ 1012 cm−3 s. This is similar to the ne t value for reaching electron-proton temperature equilibration (Sect. 4.3.5). Typical values for ne t measured in supernova remnants are between 109 − 5 × 1011 cm−3 s; i.e. the plasmas in most supernova remnants are not in CIE. Although for most supernova remnants the plasmas are underionised, a number of mature supernova remnants, in particular the mixed-morphology supernova remnants (Sect. 10.3), have been found to contain overionised plasma components. In an overionised plasma the ionisation states are higher than expected based on CIE. This situation requires that in an earlier phase of the evolution the electron temperature was high, and ionisation rapidly progressed to near CIE—probably as a result of a relatively high density. Subsequently cooling of the electrons—on a time scale shorter than the recombination time scale—resulted in a reversal of the situation regarding the ionisation states. The reason for the relatively rapid cooling can be adiabatic expansion and/or thermal conduction. The calculations for the ionisation states for an overionised plasma involve the same equations as for underionised plasma (13.155), except one usually assumes that at ne t = 0 the electron temperature is higher than the current temperature. One should, however, be aware that electron cooling and the resulting overionisation are not so well approximated by a sudden jump in temperature. For underionisation the sudden shock heating provides a convenient starting point, but for overionisation

442

13 Radiation Processes

the situation at maximum ionisation may not be well approximated by a given temperature in CIE conditions, and a sudden drop in temperature. NEI models as found in X-ray spectral codes can have various levels of refinement, both concerning the atomic data, or in the additional physical processes taken into account, such as non-equilibration of electron- and ion-temperatures (Sect. 4.3), and the related gradients in the electron temperature and ne t behind the shock front—see for example the vpshock model in XSPEC [196]. Some models [196, 482, 597] also take into account the temperature structure expected in the Sedov-Taylor phase of a supernova remnant evolution (Sect. 5.5).

13.5.8 X-ray Line Emission Diagnostics X-ray spectra of supernova remnants are characterised by emission lines from elements with atomic numbers Z = 8 (oxygen)—in a few cases also C, and N— to Z = 28 (Ni), with line emission from even elements dominating. For most of these elements the line emission comes from ions with only one (H-like ions) or two bound electrons (He-like ions). Most prominent are transitions from the n ≥ 2 to n = 1 (K-shell), and are therefore referred to as K-shell transitions. The H-like lines are often labeled similarly to the corresponding transition for hydrogen, i.e. Lyα, Lyβ, etc. For the He-like lines one often calls them Heα, Heβ, etc. The energies/wavelengths of the most prominent transitions (n = 3 → 1,n = 2 → 1) are listed in Tables 13.2 and 13.3. We have here also shown the transitions for the odd elements. The reason is that in the future, with high-resolution X-ray spectroscopy with XRISM or ATHENA we may be able to detect these emission lines as well.

The Helium Triplet As the diagram in Fig. 13.14 shows, He-like ion transitions to the ground state include forbidden and intercombination transitions. For excitations to the n = 3 level, there is always an allowed transition possible. But for transitions from the n = 2 level to the ground state, several states can only deexcite through forbidden or intercombination line emission. For He-like ions, this gives rise to four transition lines. The intercombination lines are closely spaced together, so that apparently three lines are present in the spectra. These three Heα lines are often referred to as the He-triplet. The label “triplet” only refers to the three lines, and should not be confused with a triplet quantum state. The He-triplet can be resolved with high resolution spectrometers, like the XMMNewton and Chandra gratings, and the future microcalorimetric detectors on board XRISM and ATHENA. The ratio of the three lines has often been used as a diagnostic tool to measure densities of stellar coronae, for ne ≈ 108 − 1013 cm−3 [413, 724, 807, 927]. This density regime is not relevant for supernova remnants. Nevertheless,

0.4307

0.5739

0.8536

0.9220

1.1269

1.3522

1.5983

1.8650

2.1524

2.4606

2.7897

3.1396

3.5105

3.9023

4.3153

4.7496

5.2052

5.6555

6.1804

6.7004

7.2421

7.8056

N VI

O VII

F VIII

Ne IX

Na X

Mg XI

Al XII

Si XIII

P XIV

S XV

Cl XVI

Ar XVII

K XVIII

Ca XIX

Sc XX

Ti XXI

V XXII

Cr XXIII

Mn XXIV

Fe XXV

Co XXVI

Ni XXVII

Source: https://physics.nist.gov

0.3079

CV

1.5884

1.7120

1.8504

2.0061

2.1923

2.3820

2.6104

2.8732

3.1772

3.5318

3.9491

4.4443

5.0387

5.7602

6.6480

7.7573

9.1687

11.0026

13.4471

14.5241

21.6020

28.7870

40.2678

7.7864

7.2235

6.6823

6.1629

5.6651

5.1887

4.7338

4.3001

3.8877

3.4965

3.1263

2.7771

2.4488

2.1413

1.8546

1.5888

1.3435

1.1190

0.9150

0.7317

0.5686

0.4263

0.3044

(keV)

1.5923

1.7164

1.8554

2.0118

2.1886

2.3895

2.6191

2.8833

3.1891

3.5459

3.9659

4.4645

5.0631

5.7901

6.6851

7.8038

9.2282

11.0799

13.5500

16.9443

21.8044

29.0815

40.7283

(Å)

λ

E

(Å)

λ

E

(keV)

21 P1 → 11 S0

Ion

Intercombination

23 P2 → 11 S0

Resonance

Table 13.2 Atomic transitions of helium-like ions

7.7657

7.2059

6.6676

6.1506

5.6548

5.1803

4.7269

4.2946

3.8833

3.4930

3.1235

2.7750

2.4471

2.1413

1.8538

1.5881

1.3431

1.1187

0.9150

0.7316

0.5686

0.4263

0.3044

(keV)

E

1.5966

1.7206

1.8595

2.0158

2.1925

2.3934

2.6229

2.8870

3.1927

3.5495

3.9694

4.4679

5.0665

5.7901

6.6883

7.8070

9.2312

11.0829

13.5500

16.9470

21.8070

29.0840

40.7304

(Å)

λ

Intercombination 23 P1 → 11 S0

Forbidden

7.7316

7.1734

6.6366

6.1211

5.6269

5.1539

4.7020

4.2710

3.8611

3.4723

3.1041

2.7569

2.4303

2.1413

1.8394

1.5888

1.3311

1.1078

0.9051

0.7228

0.5610

0.4198

0.2990

(keV)

E

23 S1 → 11 S0

1.6036

1.7284

1.8682

2.0255

2.2034

2.4056

2.6369

2.9029

3.2111

3.5707

3.9941

4.4972

5.1015

5.7901

6.7404

7.8038

9.3143

11.1915

13.6987

17.1525

22.1012

29.5343

41.4719

(Å)

λ

Resonance

9.1836

8.5196

7.8812

7.2683

6.6808

6.1186

5.5816

5.0697

4.5828

4.1208

3.6838

3.2716

2.8839

2.5210

2.1825

1.8687

1.5793

1.3144

1.0738

0.8575

0.6656

0.4980

0.3545

(keV)

E

31 P1 → 11 S0

1.3501

1.4553

1.5732

1.7058

1.8558

2.0264

2.2213

2.4456

2.7054

3.0087

3.3656

3.7897

4.2991

4.9181

5.6807

6.6348

7.8505

9.4330

11.5466

14.4592

18.6270

24.8980

34.9728

(Å)

λ

13.5 Line Emission, Ionisation and Recombination Processes 443

444

13 Radiation Processes

Table 13.3 Atomic transitions of hydrogen-like ions

Ion C VI N VII O VIII F IX Ne X Na XI Mg XII Al XIII Si XIV P XV S XVI Cl XVII Ar XVIII K XIX Ca XX Sc XXI Ti XXII V XXIII Cr XXIV Mn XXV Fe XXVI Co XXVII Ni XXVIII

Ly α1 E (keV) 0.3675 0.5003 0.6536 0.8275 1.0218 1.2368 1.4723 1.7286 2.0055 2.3040 2.6217 2.9624 3.3214 3.7047 4.1050 4.5315 4.9769 5.4436 5.9319 6.4417 6.9661 7.5265 8.1017

λ (Å)) 33.736 24.781 18.969 14.982 12.134 10.025 8.421 7.173 6.182 5.381 4.729 4.185 3.733 3.347 3.020 2.736 2.491 2.278 2.090 1.925 1.780 1.647 1.530

Ly α1 E (keV) 0.3675 0.5002 0.6535 0.8272 1.0215 1.2363 1.4717 1.7277 2.0043 2.3016 2.6197 2.9585 3.3182 3.6987 4.1001 4.5226 4.9661 5.4436 5.9165 6.4236 6.9520 7.5018 8.0731

λ (Å) 33.7399 24.7849 18.9729 14.9877 12.1375 10.0286 8.4246 7.1763 6.1858 5.3868 4.7328 4.1907 3.7365 3.3521 3.0239 2.7414 2.4966 2.2776 2.0956 1.9301 1.7834 1.6527 1.5358

Ly β E (keV) 0.4356 0.5930 0.7746 0.9806 1.2109 1.4656 1.7448 2.0483 2.3765 2.7292 3.1065 3.5086 3.9353 4.3870 4.8634 5.3651 5.8918 6.4436 7.0207 7.6232 8.2506 8.9047 9.5839

λ (Å) 28.4655 20.9097 16.0058 12.6437 10.2388 8.4595 7.1061 6.0529 5.2172 4.5429 3.9912 3.5338 3.1506 2.8262 2.5493 2.3109 2.1044 1.9242 1.7660 1.6264 1.5027 1.3924 1.2937

Ly γ E (keV) 0.4594 0.6254 0.8170 1.0342 1.2771 1.5457 1.8401 2.1603 2.5063 2.8783 3.2761 3.7002 4.1501 4.6266 5.1289 5.6581 6.2131 6.7949 7.4033 8.0385 8.7003 9.3896 10.1056

λ (Å) 26.9900 19.8257 15.1761 11.9882 9.7082 8.0211 6.7379 5.7393 4.9469 4.3076 3.7845 3.3507 2.9875 2.6798 2.4174 2.1913 1.9955 1.8247 1.6747 1.5424 1.4250 1.3204 1.2269

Source: https://physics.nist.gov

the line ratios still offer diagnostic power when it comes to assessing the electron temperature, the effects of NEI, and resonant line scattering. It is common to label the resonance line w, the intercombination lines x, y and the forbidden line z. The ratio G ≡ (z + x + y)/w = (f + i)/r is of interest for supernova remnants, as it is sensitive to the temperature and ionisation state of the plasma [724, 808]. For example, inner shell ionisation of the Li-like state, which has ground state configuration 1s 2 2s, will result in an excited He-like ion in the 1s2s 3 S1 state. Since de-excitation to the ground state is forbidden for electric dipole radiation ( l = 0), this gives rise to forbidden, magnetic dipole, radiation. Thus, inner shell excitation enhances the forbidden line transition, and acts as a measure for the fraction of Li-like ions. This is also related to the fact that direct collisional excitation prefers allowed transitions, which only enhances the resonance line emission. Figure 13.19 shows the G-ratio as a function of ne t, for oxygen (OVII) and silicon (Si XIII). It shows that the G-ratio is high for ne t < 3 × 109 cm−3 s for OVII and

13.5 Line Emission, Ionisation and Recombination Processes

445

Fig. 13.19 Left: The G-ratios G = (f + i)/r for He-like oxygen and silicon. Three different temperatures are show 0.5 keV (solid line), 1 keV (dotted) and 2 keV (dashed). Right: Idem, but now for the R-ratios, R = f/i. Note that these ratios are based on pure He-like ion lines only. In practice some blends may occur with Li-like satellite lines [1185], which, if not separated out, tend to increase the G-ratio and R-ratio at low ne t values. This figure is based on calculations by the author using the SPEX code [599] (Figure reproduced from [1173])

ne t < 2 × 1010 cm−3 s for SI XIII. It also shows that for higher ne t values, the G-ratio may fall below the CIE values. The reason is that Li-like ions are no longer present to feed the forbidden line, but at the same time the recombination rate to the He-like state is low, because the atoms are still underionised, and collisions are more likely to result in ionisations than recombinations. In absence of Li-like ions, recombination from H-like ions also result in forbidden line emission. Only when ne t 1011cm−3 s are the G-ratios similar to those for CIE at the same temperature. This can be expected as the plasma is getting closer to CIE. Another potential process that affects the G-ratio is resonant line scattering. Resonant line scattering only influences the intensity of the resonant line, as it scatters the resonant photon out of the line of sight. A very high G-ratio—i.e. relatively strong forbidden line emission—may, therefore, help to diagnose resonant line scattering. Its importance can best be estimated, if ne t and kTe can be estimated independently by other means. Note that, since resonant line scattering does not destroy the photon, from the other parts of the supernova remnant, in particular from the edges, there may be enhanced resonant line emission. The ratio R ≡ z/(x + y) = f/i is for CIE plasmas a sensitive diagnostic tool to measure electron densities, if the densities are in the range ne = 108 − 1013 cm−3 . For supernova remnants it is of more interest that the R-ratio is also sensitive to the ionisation age, and to a lesser extent to the temperature (Fig. 13.19, right). The reason is that the R-ratio in supernova remnants is mainly determined by inner shell ionisation, which enhances the forbidden line emission for low ne t. If the fraction of Li-like ions is negligible, and inner shell ionisations are therefore no longer important, the R-ratio becomes relatively flat. The R-ratio may help to disentangle resonant line scattering effects from pure NEI effects; resonant line scattering does not influence the R-ratio.

446

13 Radiation Processes

Iron-Line Diagnostics Iron line emission is an important diagnostic tool for the state of a supernova remnant plasma, i.e. the average electron temperature and ionisation age, ne t. This is even true for medium energy resolution spectroscopy as provided with the CCD instruments on board Chandra, XMM-Newton, and Suzaku. Because of its high fluorescence yield (Sect. 13.5.5) and high abundance, Fe K-shell emission can be observed for all ionisation states of iron, provided that the electron temperature is high enough (kTe  2 keV). The average line energy of the Fe-K shell emission provides information about the dominant ionisation state (Fig. 13.20). For ionisation states from Fe I to Fe XVII the average Fe-K shell line is close to 6.4 keV. Iron has also prominent Fe-L-shell transitions in the 0.7–1.12 keV range. The transitions involve transitions to the L-shell (n = 2), which involves usually ions with three to ten electrons, i.e. Fe XVII to Fe XXIV. Each ionisation state has its own specific line transitions, which increase on average in line energy for higher ionisation states. As these lines are densely packed, and cannot be resolved by means of CCD spectroscopy, one often refers to the broad hump of emission in this range as the “Fe-L complex”. Note that at slightly higher energies nickel L-shell transitions should also be present. Fe-L-line emission occurs for lower temperatures/ionisation ages than Fe-K, i.e. kTe ∼ 0.15–2 keV for Fe-L, and  1.5 keV for Fe-K. Taken together, Fe-L- and Fe-K-shell emission can be used to accurately determine the ionisation state of the plasma. For example, Fe-K emission around 6.4 keV could be caused by Fe XVIIXIX, or by lower ionisation states, but in the latter case no Fe-L emission should be

Fig. 13.20 Fe-K shell line emission energies, as determined theoretically (squares, Fe II-Fe XVII) [799, 882], and observationally (triangles, Fe XVIII-Fe XXV) [141] (Figure reproduced from [1173])

13.5 Line Emission, Ionisation and Recombination Processes

447

present. The presence of Fe-K-line emission around 6.7 keV indicates the presence of Fe XXV (He-like Fe), and around 6.96 keV Fe XXVI (H-like). High resolution X-ray spectroscopy greatly improves the value of Fe-line diagnostics, as it allows one to resolve the Fe-L shell emission in individual lines. For Fe-K shell emission it is possible to detect different ionisation stages individually instead of relying line emission centroids. It is worth mentioning that Kα line emission from low ionisation states of iron can be the result of dust grains embedded in hot supernova remnant plasmas [194]. Hot electrons can penetrate dust grains, giving rise to inner shell ionisations inside the grains, whereas the emitted photon can escape from small grains. In addition, dust grains are slowly destroyed in hot plasmas due to dust sputtering (Chap. 7) on a time scale of ∼ 1013 /ne s. This results in a slow release of near-neutral iron into the hot plasma. This should give rise to the presence of a broad range of ionisation stages of Fe inside the plasma. This process should also result in the release of other refractory elements like Mg and Si, although in that case the fluorescence yield of these elements is lower.

13.5.9 Resonant Absorption and Line Scattering To calculate the emission from supernova remnant plasmas it is usually assumed that the plasmas are optically thick. However, under certain conditions this may not be the case for emission from resonant lines (allowed transitions). To quantify this we have to evaluate the line absorption cross-sections for ions, for which we can make use of the expressions for the Einstein coefficients in Sect. 13.5.1. The coefficient uν Bkm (ν) (in units of s−1 ) gives the absorption rate of photons with energy hν from an electromagnetic field. This should be equal to nν σabs c, with nν the number density of photons at a given photon frequency. Using uν = nν hν we find the following expression for the atom (ion) cross-section for absorption per frequency interval: σabs,ν (ν) =

πe2 hνBkm φ(ν) = fmk φ(ν), c me c

(13.158)

where φ(ν) is the line profile and fmk the oscillator strength of the transition. For a single atom at rest φ(ν) is the Lorentzian profile, but for an ensemble of atoms with a thermal or turbulent velocity distribution, one has to convolve the Lorentz profile with the Gaussian velocity distribution, characterised by a width V , corresponding to a Gaussian spread in frequency of ν = ν V /c—we avoid using the common symbol σ for velocity dispersion here, as we use it already for the cross-section. The convolution of a Lorentzian and Gaussian function is called the Voigt function. However, for ν  Amk the core of the Lorentzian acts like a δ-function and is well approximated by a Gaussian. For a normalised Gaussian the central value

448

13 Radiation Processes

√ is φ(ν0 ) = 1/ 2π ν. The cross-section for absorption is, therefore, π 1/2 e2 1 1 σabs,ν (νkm ) = √ fmk ≈ 3.17 × 10−14 fmk

V νmk 2me



V 100 km s−1

−1 

νmk −1 . 1015 Hz

(13.159) For allowed (electric-dipole) transitions fmk ∼ 1, and one sees that the cross-section is quite large. Moreover, resonant line absorption is more likely in the optical/UV −1 than in X-rays, because of the νmk dependence. The cross-section is that for resonant absorption. However, for a resonant absorption from n = 1 to n = 2 the excited state will decay immediately by emitting a photon with the same energy. One could say that the photon was not so much absorbed as well scattered in a random direction. Hence, the name “resonant line scattering”. Note that for an n = 1 to n > 2 resonant transition photons are removed from the scattering process, because there are multiple deexcitation channels. For example, Lyβ absorption can result in a Lyβ decay, but also in an Hα transition. If the line emission from one atom is near the energy transition of another atom one talks of “Bowen” fluorescence. A well known case is the resonance between He II Lyα (λ303.78Å) and the O III 2p2 3 P2 -3d 3 P02 -3 P02 resonance line (λ303.80Å) [874]. Although resonant line scattering in X-rays is less likely than in the optical/UV. it can occur under the right conditions. Consider the Heα resonant transition of O VII (Table 13.2). The transition energy is 0.57 keV (1.4 × 1017 Hz) and f21 = 0.696. In the interstellar medium the oxygen abundance is nO /nH ≈ 0.0005, and the helium-like ionisation stage of oxygen occurs over a wide range in temperature and ionisation age, for which nOVII /nO ≈ 1 (Fig. 13.18). For the optical depth of the resonance line we therefore find  τOVIIHeα ≈ lσabs,ν nOVII ≈ 0.15nH

l 1018 cm



nO /nH 0.0005



nOVII nO





V −1 , 50 (13.160)

with l the plasma depth along the line of sight. For sufficiently high densities nH  10, and/or long lines of sight l  1018 cm, resonant line scattering is a real possibility. With future high resolution X-ray spectroscopy one can measure V directly from the line width. The default value adopted here is reasonable, given that we expect O VII for relatively slow shocks, Vs  500 km s−1 . Another option would be extreme NEI, like in SN 1006, ne ≈ 2 × 109 cm−3 s [1185], but this gives too high values for V . For line of sights perpendicular to the direction of the shock velocity, the line of sight velocity broadening is mostly due to thermal line broadening and turbulence. And for slower shocks V will be a small fraction of the shock speed. Indeed, evidence for resonant line absorption has emerged for relatively mature supernova remnants, with slow shocks: for DEM L71 [1153] and for a bright region

449

0.6

13.6 Pion Production and Decay

O VIII Lyα

O VII

Wavelength

O VII

O VII-f

C VI

O VII-r N VII

O VII

0.2

Ne X/Fe XXI Fe XX Ne IX/Fe XIX Fe XVIII Fe XVII O VIII

0.4

Fe XVII

0

Counts s-1 Å-1

O VIII

N E

15

20

25

30

35

Wavelength (Å) Fig. 13.21 Possible evidence for resonant line scattering in DEM L71, based on high resolution X-ray spectroscopy with the XMM-Newton-RGS instrument. The OVII Heα resonance line (21.6Å) can be clearly distinguished from the forbidden line (22.1Å). Left: The RGS spectrum. Right: single-line images extracted from the RGS spectrum (Reproduced from Van der Heyden et al. [1153])

in the Cygnus Loop [1142]. For the Cygnus Loop the evidence consists of an unusually high ratio of the forbidden over resonance line ratio. For DEM L71 the effect is seen in a large variation of this ratio across the remnant; see Fig. 13.21. This is consistent with the idea that resonant line scattering does not so much absorb the photon but scatters it around, until the photon escapes the shell. As a result, the overall forbidden-to-resonant line ratio should not change, but for geometrical reasons (Chap. 8) the resonant line should be repressed along the limb-brightened shell, and be enhanced in the centre of the remnant. This, indeed, appears to be the case for DEM L71.

13.6 Pion Production and Decay The radiation mechanisms discussed so far involve the change in momentum (or acceleration) of the lowest mass particles: electrons and positrons. So far, we discussed only exception: line emission associated with radio-active elements, which involve deexcitation of nucleons in atomic nuclei (Sect. 2.4). In this section we discuss an important continuum radiation mechanism that is exclusively caused by collisions of hadronic cosmic rays, i.e. atomic nuclei and cosmic-ray protons (Chap. 11), with atomic nuclei in the local gas. These collisions result in the formation of mesons, hadrons consisting of two quarks instead of three quarks, like neutrons and protons. These mesons are short-lived, and decay into photons, muons, neutrinos, and electrons/positrons—sometimes with lower-mass mesons as intermediate steps. The muons are also not stable, producing electrons or positrons and electron neutrinos.

450

13 Radiation Processes

So meson production is not only responsible for γ -ray radiation, but also for the production of energetic muon- and electron-neutrinos, which are the targets for neutrino telescopes like IceCube [6] on the South Pole (operational since 2010) and KM3NeT [631] currently being built in the Mediterranean. The secondary electrons and positrons could contribute additional radiation components, through synchrotron radiation, bremsstrahlung and inverse Compton scattering. As currently no supernova remnant has yet been detected as a neutrino source, we concentrate in this section on γ -ray emission associated with meson production. The most commonly produced mesons are the pions (π 0 , π ± ). For that reason we refer to the radiation mechanism discussed here as pion decay. But other mesons may also contribute. A list of light mesons are listed in Table 13.4. Note that apart from π 0 , only the η meson can directly decay into two photons. This is why sometimes the η is included in models for hadronic γ -ray emission [637]. Pion decay is one of the three most important γ -ray continuum radiation mechanisms, next to inverse Compton scattering (Sect. 13.2.2) and non-thermal bremsstrahlung (Sect. 13.4). Of these three, only pion decay directly informs us about the presence and the energetics of hadronic cosmic rays. The γ -ray photons produced through pion decay are typically detected between photon energies of ∼ 100 MeV to ∼ 100 TeV. This corresponds to the electromagnetic bands of the NASA Fermi-LAT satellite experiment (∼ 30 MeV to ∼300 GeV), and imaging atmospheric Cherenkov telescopes like H.E.S.S., MAGIC, VERITAS, and the future Cherenkov Telescope Array (CTA).

Table 13.4 Properties of some mesons [1094] Mass (MeV/c2 )

Life time (s)

Ethr (MeV)

Decay products

dd−uu √ 2

134.98

8.5 × 10−17

279.7

2γ

ud du us su

139.57 139.57 493.68 493.68

2.6 × 10−8 2.6 × 10−8 1.2 × 10−8 1.2 × 10−8

291.0 287.8 1119.2 1114.4

μ+ νμ μ− ν μ μ+ νμ , π 0 π + μ− ν μ , π 0 π −

ds−sd √ 2 ds+sd √ 2 uu+dd−2ss √ 6

497.61

9.0 × 10−9

1127.2

π + π − ,π 0 π 0

497.61

5.2 × 10−8

1127.2

π ± e∓ νe ,π ± μ∓ νμ ,3π 0

547.86

5.0 × 10−19

1255.7

2γ ,3π 0 ,π 0 π + π −

Name

Symbol

Content

pion

π0

pion pion kaon kaon

π+ π− K+ K−

kaon

K0

kaon

K0

eta

η

Threshold energies are for protons impacting on protons or neutrons (for negatively charged mesons)

13.6 Pion Production and Decay

451

13.6.1 Meson Production in Supernova Remnants The lightest mesons are the π-mesons or pions (Table 13.4), which are, therefore, the most common secondary particles produced in hadron-hadron collisions. Pions consist of a combination of up- (u) and down-quarks (d), with the neutral pion being a quantum superposition of up- and down quarks and their anti-particles (Table 13.4). Examples of these collisions and their products are p + p →p + p + π 0 , p + p →p + n + π + , p + n →p + p + π − , p + p →p + p + π + + π − . More generically, we can refer to the production of neutral pions as p + p →π 0 + X, X standing for any number of particles existing after the collisions, as long as the reaction observes the applicable conservation laws. For very high energies, if multiple pions are made per collision, the ratio between pions produced tends to π 0 : π + : π − ≈ 1 : 1 : 1. For collisions between atomic nuclei, the interactions are simply the interactions of the individual nucleons (or more accurately the quarks) they consist of. However, a correction factor may apply to the relevant cross-section, as discussed below. Note that pions can also be produced through photon-hadron interactions (e.g. γ + p → p + π 0 ). This mechanism is subdominant for supernova remnants, but may be important for some relativistic jets, for example in blazars. Pions decay quickly (Table 13.4) into stable particles, i.e. photons for the neutral pion and electrons, and positrons/electrons and neutrinos for the charged pions, with the dominant decay chains being π 0 →2γ , π + →μ+ + νμ ↓ e+ + νe + ν μ , π − →μ− + ν μ ↓ e− + ν e + νμ .

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Momentum conservation tells us that the two photons from the π 0 decay each have an energy of 1/2mπ 0 c2 = 67.5 MeV in the rest-frame of the pion.

13.6.2 The Energy Threshold for Pion Production The threshold energy for the production of pions can be calculated from the conservation of total relativistic four-momentum squared, a Lorentz invariant quantity:  2

2

s ≡ ημν p p = μ ν

Ei i

   − 

i

2   pi c , 

(13.161)

with pμ = (E/c, px , py , pz ) the momentum four-vector, and ημν the Minkowski metric. The energy for each particle given by the relativistic expression Ei =  (mi c2 )2 + p2 c2 . One can calculate the threshold energy required for a proton-proton collision to result in the creation of a neutral pion, by calculating the minimal value of s 2 , which corresponds to the situation in which all particles have zero momentum in the centre of mass frame. In that frame the total value for s 2 is the sum of the rest mass energies squared: s 2 = (2mp c2 + mπ 0 c2 )2 = 4m2p c4 + 4mπ 0 mp c4 + m2π 0 c4 .

(13.162)

Since s 2 is conserved and Lorentz invariant, this value should correspond to s 2 prior to the collision. In supernova remnants and in the interstellar medium the collisions typically occur between a high energy cosmic-ray proton and a nucleon at rest. For the proton at rest this means that E = mp c2 and |p| = 0, whereas for the cosmic-ray proton 2 − m2 c 2 . This corresponds to a total s 2 of we have |pcr c|2 = Ecr p 2 s 2 = (mp c2 + Ecr )2 − (Ecr − m2p c2 ) = 2m2p c4 + 2mp c2 Ecr .

(13.163)

Equating this with (13.162), we find that the threshold kinetic energy for π 0 production is Ethr,π 0 ≡ Ecr − mp c2 = 2mπ 0 c2 +

m2π 0 2mp

c2 = 279.66 MeV.

(13.164)

Table 13.4 list this energy, as well as the threshold energy for producing other mesons through proton-proton, or neutron-proton collisions. Figure 13.22 shows the total collisional cross-section for proton-proton interactions in the laboratory frame,

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Fig. 13.22 Left: Cross-sections for p-p collisions. The blue points are the total cross-sections (1 mb=10−27 cm2 ), whereas the grey points are the elastic cross-sections (p + p → p + p). Data obtained from http://pdg.lbl.gov. The solid line indicates a model for the inelastic contribution only (13.168), and the dashed line the inclusive π 0 cross-section. Right: Neutral-pion production multiplicity (ξπ 0 ) as a function of cosmic-ray proton energy, as modelled in [600], based on the GEANT Monte-Carlo code

with the target proton at rest. The figure shows that the cross-section declines, until the proton energy passes the threshold for pion production.

13.6.3 The Formation of the γ -Ray Spectrum Neutral pions decay into two photons that each have an energy of 12 mπ 0 c2 in the rest frame of the pion. However, the pion itself will typically be created with excess kinetic energy in the frame of the observer, in particular if the cosmic-ray proton had an energy well above the threshold energy. In order to calculate what the expected spectrum is for a cosmic-ray population with a certain energy spectrum and composition, several ingredients are necessary: the cosmic-ray energy spectrum above the threshold energy, the pion-production cross-section, the average number of neutral pions produced (referred to as the π 0 multiplicity—ξπ 0 —Fig. 13.22, right), and the energy distribution of the pions as a function of the impacting cosmicray particle. For a given cosmic-ray energy, the resulting pions will have a broad energy distribution, resulting also in a broad range of photon energies. As a result, sharp features in the primary cosmic-ray energy distribution will be smoothed out in the resulting photon spectrum. The details of the various steps in the γ -ray production can be found in [600, 605, 637], and are based on experimental data, and Monte-Carlo particle codes, calculating the probabilities of the various production and decay channels.

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If we consider just cosmic-ray protons impacting protons at rest, the γ -ray piondecay emissivity can be expressed as ) γ (hν) = cnp

∞

β(Ecr )ncr (Ecr ) Ethr

   dEπ 0 d  dEcr , ξπ 0 (Ecr )σpp,inel(Ecr ) dEπ 0 dhν (13.165)

with β(Ecr )c the cosmic-ray proton’s velocity, σpp,inel the inelastic proton-proton cross-section, ncr (Ecr ) the cosmic-ray density/energy distribution, np the background proton density, ξπ 0 the π 0 multiplicity. The complication is that there is a broad range in pion energies for a given primary cosmic-ray energy. So the actual calculation requires two convolutions: from primary cosmic-ray energy to pion energy distribution, and from a pion energy to final photon energy distribution. Parameterisations of these calculations can be found in [600, 605, 637]. Here, we limit the discussion to a few insights on the various elements that go into modelling pion-decay γ -ray modelling. The expression σpp,incl ≡ ξπ 0 (Ecr )σpp,inel(Ecr ), the so-called inclusive crosssection, is often used rather than applying a factorisation of both. As Fig. 13.22 shows the inclusive cross-section is much smaller than the inelastic cross-section near the threshold, as compared to the inelastic cross-section. One reason is the large contribution for π + production above the threshold for the inelastic cross-section. Above ∼ 10 GeV the multiplicity is larger than one, and π 0 production becomes more important. Equation (13.165) provides the emissivity for proton-proton collisions. Since both cosmic rays and the background consists of a mixture of atoms, one needs to correct for this. The cross-section for a proton colliding with a nucleus with atom mass number A is often approximated by σpA = A2/3σpp .

(13.166)

If both the cosmic ray particle and target particle are nuclei, an approximation that is sometimes used is 1/3

σA1 A2 = σpp (A1

1/3

+ A2

− R)2 σpp ,

(13.167)

with R ≈ 1.7 × 10−12 cm [205]. The factor A2/3 assumes that the nucleons in the atom are fully screened, i.e. it corresponds to the “surface area” of the atomic nucleus. In reality the scaling of the cross-section may be somewhere in between ∝ A2/3 and ∝ A. In addition to an increase in collisional cross-section, one also needs to take into account that the average nucleon in a cosmic-particle ray has less energy than the cosmic-ray nucleus, so there will be a different relation between projectile energy and average photon energy hν ∝ A−1 2 , with A2 the mass number of the cosmic-ray nucleus.

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455

The collisional cross-section for the production of pions have been determined experimentally. Figure 13.22 shows the total experimental cross-section for protonproton collisions. Below the neutral pion cross-section the collisions are purely elastic (no particle creation), whereas for energies above the threshold the collisions are predominantly inelastic. In [600] the inelastic cross-section has been parameterised as

σpp,inel



Ecr,kin ≈ 30.7 − 0.96 ln Ethr



 + 0.18 ln

2

Ecr,kin Ethr

 

 1−

Ethr Ecr,kin

1.9 3 mb

(13.168) with Ecr,kin = Ecr − mp c2 , the cosmic-ray proton’s kinetic energy. Note that the cross-section slowly increases with energy. In the pion rest frame the two photons resulting from pion decay will each have an energy Eγ = mπ 0 /2 = 64 MeV. But in the frame of the observer the energy can be obtained by the relevant Lorentz transformation Eγ = π 0 (Eγ −βπ 0 p c cos θ ), with  βπ 0 = v/c, and π 0 = 1/ 1 − βπ2 0 the Lorentz factor of the pion in the observer’s frame. For an isotropic distribution of velocity this results in a flat distribution of photon energies, with energy limits m 0 c2 mπ 0 c 2 π 0 (1 − βπ 0 ) ≤ Eγ ≤ π π 0 (1 + βπ 0 ), 2 2

(13.169)

which can be rewritten as mπ 0 c2 −1 m 0 c2 X ≤ Eγ ≤ π X 2 2  with X ≡

1+βπ 0 1−βπ 0 .

(13.170)

We see that logarithmically the photon energy distribution

dNγ /dE is flat and symmetrically distributed in logarithmic energies in the interval (− log X, + log X) around log(mπ 0 c2 /2) [1077]. Both the lowest and highest energy photons are, therefore, produced by the highest energy pions. For βπ 0 → 1, π 0 → ∞, the maximum photon energy approaches mπ 0 c2 , whereas the average photon energy is 12 mπ 0 c2 . Figure 13.23 illustrates how the γ -ray spectrum is built up from a superposition of flat γ -ray distributions each corresponding to a given primary pion energy. The addition of these contributions results in a characteristic “pion bump” at log(mπ 0 c2 /2). The shape of the bump is rather sharp here, as the input pion spectrum was assumed to be a power-law distribution, but this is not realistic. In the literature the meaning of “pion bump” has shifted away somewhat from the actual peak shown in Fig. 13.23 at half the rest mass energy, and is now used more to reflect the low-energy cutoff around 200 MeV that occurs with full modelling of the expected γ -ray spectrum. For cosmic-ray spectra with a steep spectral index q > 2, the low-energy cut-off also gives the impression of a bump; see Fig. 13.24.

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13 Radiation Processes

Fig. 13.23 Schematic illustration on the built-up of a pion-decay γ -ray spectrum. Left: For a given π 0 energy the photon distribution is flat, and symmetric in logarithmic energy around log mπ 0 c2 /2 (13.169). Right: The spectral-energy diagram shows that the energy output is skewed toward high energies. The pions are assumed to have a power-law distribution ∝ E −2 between 107 and 1014 eV

Fig. 13.24 The γ -ray spectral energy distribution (solid line) expected from cosmic-ray protons impacting on background protons, based on the parameterisations in [600]. The dotted line shows the input cosmic-ray energy distribution, assumed to be a power-law in momentum (Sect. 11.2.2) with a cutoff energy of 300 TeV. The coloured lines indicate the γ -ray distribution for a single proton energy of 1 GeV, 1 TeV and 1000 TeV, to illustrate the broad range of photon energies resulting from a given cosmic-ray energy (normalised to a peak at 0.2), which are less angular than for the δ-approach in Fig. 13.23

An approximation to (13.165) that is often used [538, 600, 637] is the so-called δapproximation, in which the simple approximation is made that for a given cosmicray energy one obtains a single value for the pion energy:  d  ξπ 0 (Ecr )σpp (Ecr ) = σpp,incl(f −1 Eπ 0 ), dEπ 0

(13.171)

13.6 Pion Production and Decay

457

with f ≈ 17% according to [637]. Such an approximation work reasonably well in the power-law part of the γ -ray spectrum, but give rise to large underestimates of the γ -ray emissivity near cutoff regions [600]. Despite all the various convolutions involved in calculating the γ -ray spectrum from a population of cosmic-ray protons (and other hadrons), the power-law slope of the γ -ray spectrum is approximately equal to the power-law index of the primary cosmic-ray spectrum, i.e. γ ≈ q. Compare the input cosmic-ray spectrum to the resulting γ -ray spectrum in Fig. 13.24 for the range of ∼20 GeV to ∼ 1 TeV. The deviations above 1 TeV are caused by the power-law cutoff energy of 100 TeV.

Chapter 14

Summary and Prospects

We have discussed the many aspects of the physics and evolution of supernova remnants including discussions on how the properties of supernova remnants relate to the properties of supernova explosions (e.g. Chaps. 9 and 7). The material presented shows the many discoveries made and insights that have been obtained over the last two to three decades. This period corresponds to the coming of age of X-ray imaging spectroscopy—with Chandra, XMM-Newton, and Suzaku—highenergy (GeV) and very-high energy γ -ray astronomy—with the Fermi and AGILE satellite missions and with imaging atmospheric Cherenkov telescopes (IACTs; H.E.S.S., MAGIC, VERITAS)—and three major infrared mission Spitzer, Herschel, and AKARI. Our present knowledge of supernova remnants is based to a large extent on observations done with these observing facilities.

14.1 Knowledge Gained and Outstanding Questions The major advances in observational technologies have resulted in important new insights in the physics and evolution of supernova remnant, which do not only concern their intrinsic properties and physics, but also what these properties tell us about supernova explosions and cosmic-ray acceleration in general. Here are some takeaway points from this book, provided in an order that follows more or less the chapter order: • There are 300–400 known supernova remnants in the Milky Way, which falls well below the expectation of ∼ 1500 supernova remnants (Chap. 3). • Observational evidence shows that collisionless shocks may heat electrons and ions to different temperatures, in particular for young supernova remnants (Chaps. 4 and 8). • Supernova remnant evolution is as much determined by the explosion properties as by the properties of the ambient medium (Chaps. 5 and 9). © Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2_14

459

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14 Summary and Prospects

• Supernova remnants played a crucial role in showing the diversity among young neutron stars: there is a wide variety of initial spin periods and surface magneticfield strength from Bp ≈ 1014 –1015 G—i.e. magnetars—to Bp  1011 G, measured for several young central compact objects (Chap. 6). • The low explosion energy of supernova remnants hosting magnetars suggest that magnetars are generally not born with rapid initial spin periods, as has been suggested in the past (Chap. 6). • Infrared emission from supernova remnants is caused by collisional dust-grain heating, and is mostly sensitive to the electron density (Chap. 7). • Core-collapse supernova remnants contain freshly produced dust grains in their ejecta, whereas this is not the case for Type Ia supernova remnants (Chap. 7). • Hα-spectroscopy of Balmer-dominated shocks offer unique insights into collisionless shocks, from electron-ion temperature ratios, to the effects of non-linear cosmic-ray acceleration—including the effects of preheating the gas in the cosmic-ray precursor (Chap. 8). • Young supernova remnants can be linked successfully to different types of supernovae (Type Ia or core-collapse supernovae—Chap. 9). • Core-collapse supernova remnants show evidence for asymmetric explosions (Chap. 9), which has been confirmed in some cases using light-echo spectroscopy (Chap. 2). • Among young supernova remnants only Type Ia remnants have Balmer dominated shocks, precluding supersoft sources as generic Type Ia progenitor systems (Chaps. 8 and 9). • A new class of supernova remnants, mixed-morphology supernova remnants, is recognised, which is characterised by central thermal X-ray emission, and often spectroscopically characterised by overionised instead of underionised plasmas (Chap. 10). • Mixed-morphology supernova remnants are associated with relatively dense ambient gas structures (Chap. 10). • Mature supernova remnants interacting with molecular clouds give rise to unique chemistry, which is also influenced by the presence of cosmic rays (Chap. 10). • Cosmic-ray acceleration by young supernova remnant shocks proceeds fast as a result of cosmic-ray induced magnetic-field amplification and turbulence, resulting in cosmic-ray diffusion coefficients that are close to the Bohm limit (Chaps. 11 and 12). • There is clear evidence for hadronic cosmic-ray acceleration by some supernova remnants, but for individual supernova remnants it is not always clear whether the γ -ray radiation is hadronic or leptonic in origin (Chap. 12). As important as the knowledge gained over the last two decades, we still are faced with a number of outstanding puzzles. Some of the most important open standing science issues are: • How many supernova remnants are there in the Milky Way and what are the implications for the supernova rate and average life time of supernova remnants?

14.1 Knowledge Gained and Outstanding Questions

461

• What parameters control the ratio between the electron- and ion-temperature for collisionless shocks? • The Crab Nebula has guided our view on pulsar wind nebulae for a long time, but despite decades of research, we are still facing the question why its magneticfield energy is low compared to the energy in relativistic electrons/positrons—the so-called sigma-problem. • Related to the sigma-problem: what is the Lorentz factor for pulsar winds? Traditionally it is assumed to be w ≈ 106 , but this creates problems for the particle spectrum. However, w  1000 requires an unrealistically high electron/positron production in the magnetosphere—i.e. pair multiplicity. • Are pulsar wind nebulae the main contributors to the cosmic-ray positron fraction that increases with energy, as measured in our solar system by the AMS-02 and PAMELA cosmic-ray experiments [25, 42]? • Very-high energy γ -ray observations showed that some pulsar wind nebulae are very extended; is this reflecting the expansion of these nebulae, or are their sizes indicative of diffusive transport of cosmic-ray electrons and positrons? • It is thought that dust formation in the early Universe was dominated by supernovae of (very) massive stars, but the dust production deduced from infrared observations of supernova remnants shows total dust masses falling short of the supernova dust production in the early Universe. • Balmer-dominated shocks offer the opportunity to study the cosmic-ray acceleration efficiency, but in how far does the presence of neutral hydrogen affect the acceleration efficiency? • Are all core-collapse explosions intrinsically asymmetric, or driven (partially) by jets? Or is this only true for a subset of core-collapse supernovae? • Why are the jets in Cassiopeia A silicon- and sulfur-rich, and not iron-rich? And what was their role in the supernova explosion process? • It is still unclear what the origin is of Type Ia supernovae—double- or singledegenerate binary systems or a mix of both? For a number of Type Ia supernova remnants evidence is found for ambient gas modified by stellar wind outflow, suggesting a single-degenerate systems; but the lack of surviving donor stars in supernova remnants suggest a single-degenerate scenario (Chap. 9). • Why are some mature supernova remnants mixed-morphology remnants? Is a high-density ambient medium the key ingredient or are there other parameters controlling their properties? • What causes the cooling of electrons needed for overionisation in mixed-morphology remnants? Is it adiabatic expansion, thermal conduction, or are there other processes causing overionisation? • It is doubtful that young supernova remnants are capable of accelerating cosmic rays up to, or beyond, the cosmic-ray “knee”. So which Galactic sources are responsible for acceleration up to the cosmic-ray knee? Or more popularly phrased: where (and what) are the Galactic PeVatrons? The above list is not exhaustive, but it contains some of the more prominent questions that we would to like answer in the coming decade of supernova remnant

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14 Summary and Prospects

studies. There are good hopes to make progress in these areas now that a new generation of more powerful observing facilities will emerge.

14.2 Future Observing Facilities The current observing facilities for the study supernova remnants seem to have been around us for an entire generation of astronomers: the Hubble Space Telescope, ESO’s Very Large Telescopes (VLTs), Chandra, XMM-Newton, H.E.S.S., MAGIC and VERITAS, not to speak of the Very Large Array (VLA), which has been around since the 1970s. It is true that some of these have had major upgrades, like the Hubble Space Telescope, and the VLA in the form of the Expanded VLA (EVLA). There were some important recent additions, such as Fermi and NuStar. But overall, it can be said that most of the advances listed above have come from instruments developed in the early 2000s and before. The reliance on these cornerstones of past research is about to change with a slew of astronomical facilities that will become available over the next ten years. Here we list a few of them, ordered by increasing electromagnetic frequency. Radio Radio astronomy is going currently through a major transition period, in which slowly the technology based on interferometry with radio dishes is being replaced by phased-array technology. With this technology antenna’s receive signals from the entire sky, and the localisation of sources is reconstructed computationally using the phase information. This done by computationally forming a number of beams. The total number of beams is limited by the availability of computational resources. Radio astronomers are eagerly waiting for the Square Kilometer Array (SKA). SKA will be built by a consortium of scientists and engineers from Australia, South Africa, several European countries, India, New Zealand, and Japan. It will consists of two arrays, a low-frequency array (∼5–350 MHz) to be built in Australia, and a mid-frequency array (∼0.35–15 GHz) to be built in South Africa. The midfrequency array will use telescope dishes as well as phased arrays. Low-frequency arrays, which include SKA pathfinders like LOFAR and the Murchison Widefield Array (MWA), are based on open antenna designs and phased array mapping. SKA will slowly be built up from 2021 onwards, with completion expected in the late 2020s. SKA’s observing power lies in its large collecting area, its long baselines, and the phased array technique, which allows for multiple, and wide field of views. Supernova remnant research will likely to profit from the high sensitivity of SKA together with the wide field of view coverage at low frequencies of 200 square degrees, allowing to search for the missing supernova remnants mentioned in Chap. 3. In the meantime, part of this science goal may be already partially realised with LOFAR and MWA.

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463

Submm and Infrared ALMA is not a new facility, but since it has not been around for that long, and since its large oversubscription rate has not allowed for much supernova remnant research—we mentioned in Chap. 7 observations of warm dust and molecules from SN 1987A—ALMA may still be a dominant facility to explore dust grains and molecules in supernova remnants in the coming decade. In addition, its high spatial resolution is well adapted to explore dust and molecules in extragalactic supernova remnants. Concerning a new facility in the submm/infrared domain that could have a profound impact on supernova remnant research, one should mention the longanticipated James-Web Space Telescope (JWST). JWSR will cover the near to midinfrared wavelength regime (0.6 μm–28.3 μm). Its key science topic is cosmology— in particular the epoch of reionisation—and characterising exoplanet atmospheres, but its wavelength range makes JWST ideal for studying hot dust and atomic lines from weakly ionised atoms with angular resolutions of 0.07 to 0.07 resolution [1068]. This makes it well suited to explore supernova remnants in the Magellanic Clouds, as well as obtain detailed imaging of the archetypical core-collapse supernova remnant Cassiopeia A. In particular, the low-ionisations lines from O, Ne, Si, S, Ar, and Co can be used to explore the spatial and velocity distribution of ejecta that has not yet been heated by the reverse shock. Optical Telescopes As the oldest branch of astronomy, optical astronomy has been central to the exploration of space, and this will not change in the future. What drives optical astronomy forward is the need for the deeper observations to probe the evolution of the Universe, and nowadays also the quest to find earth-like exoplanets. Supernova remnants studies are rarely defined as key science topics for new optical facilities, but as the Hubble Space Telescope has shown, supernova remnant research may well profit from the realisation of these telescopes, often resulting in iconic images. Moreover, detecting supernovae, in particular of Type Ia, are often key science objects, because of their cosmological significance. This will have also affect the study of supernova remnants. Wide-field imaging has become an important prerequisite for dark energy studies using weak lensing, the discovery of new transients, and finding exoplanet transits and microlensing events. Two near-infrared/optical facilities—the Vera C. Rubin Observatory1 (formerly LSST), and NASA’s Nancy Grace Roman Space Telescope2 (formerly WFIRST)—are indeed wide-field optical imagers, with the 8 m class Rubin Observatory telescope having a 3.5◦ field of view and the 2.4 m telescope of the Roman Space Observatory providing a 0.5◦ field of view. The latter is small compared to the Rubin Observatory, but it is two orders of magnitude larger in solid angle than the field of view of the Hubble Space Telescope [58].

1 https://www.lsst.org. 2 https://roman.gsfc.nasa.gov.

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14 Summary and Prospects

These two observing facilities will become available in the time frame from 2022 (the Rubin Observatory) to 2025 (the Roman Space Telescope). It is unlikely that they will transform supernova remnant research itself, given that their main purpose is the detection of optical transients and extragalactic research. Moreover, unlike the Hubble Space Telescope they lack the narrow-band filters that are of particular interest for nebular sources. But the supernovae detected by these facilities will have an impact on supernova remnant research. In particular, the expected early detections of supernovae will shed new light on the longstanding question about the nature of supernova Type Ia progenitor systems—for example by detecting excess flux in the early light curves of Type Ia explosions related to the impact the blast wave has on the donor star, possible pre-explosions and the statistics of these phenomena (see Sect. 2.3). For core-collapse supernovae one expects to detect shock-break outs, allowing for accurate timing of supernova explosions, as well as determining the radii of the progenitor stars. Both these types of results will have an impact on supernova remnant research, as they inform us about explosion properties and the supernova progenitor population. Another area of research that may help supernova remnant studies is the identification of light echoes, and connecting these to known supernova remnants. Light echoes are typically discovered by observing the same field a few times with weeks to months in between. The Rubin Observatory telescope has a 3.5 m field of view and it will image the entire available sky every few days, providing a rich treasure trove to search for light echoes. The Roman Space Telescope may be able to search for light echoes in nearby galaxies, given its 0.1 resolution. There are many optical facilities that will become available in the coming year, and we cannot discuss them all. But of special interest is that the 2020s will witness the rollout of 30–40 m class telescopes, such as ESO’s 39 m Extremely Large Telescope (ELT). The ELT is expected to become online around 2025. Its high sensitivity will be ideal to spectroscopically probe the faint broad-lines of fast Balmer-dominated shocks, explore the optical spectra of extragalactic supernova remnants, and to detect and provide spectroscopy of the faint thermal emission of neutron stars in the optical, which provided important information regarding the nature of their atmospheres [817]. X-Ray After some delay, the field of X-ray imaging spectroscopy, covering the 0.5–10 keV window, will finally be able to explore new grounds. One important step forward has occurred while finishing this book: the wide-field eROSITA X-ray experiment on board the Russian-German Spectrum-Roentgen-Gamma space observatory, has completed mapping the entire sky from 0.2–10 keV. The first X-ray maps of the entire sky as well as the area containing the Vela SNR, Puppis A and “Vela Jr” look fabulous, but the first science results have not yet been published at moment

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465

this book went to print.3 It is likely that the eROSITA survey will unveil new supernova remnants, following in the footsteps of its predecessor ROSAT. But due its sensitivity above 2 keV, it is much better suited to discover supernova remnants that suffer strong Galactic extinction, and to discover low surface-brightness supernova remnants that are dominated by X-ray synchrotron emission, such as RX J1713.73946 and “Vela Jr”, both discovered by ROSAT (Chap. 12.2). Two satellites to be launched in 2021/22 are promising to cover new aspects of the X-ray properties of supernova remnants. The JAXA/NASA X-ray Imaging and Spectroscopy Mission4 (XRISM) contains two types of detectors behind X-ray telescopes with arcminute resolution. Of special interest is the Resolve instrument, a calorimetric X-ray spectrometer, which provides an X-ray spectral resolution of E ≈ 6eV in the 0.3–12 keV bandpass. High spectral resolution X-ray spectrometers are not entirely new, but they relied up to now on the use of dispersive technology, such as the gratings spectrometers on board Chandra and XMMNewton—see Chap. 9 for some results for supernova remnants. However, slitless dispersive spectrometers are of very limited use when it comes to extended X-ray sources like supernova remnants. Resolve instead will provide integral field spectroscopy with an array of 6×6 pixels of 30 each. This will allow accurate measurements of X-ray line broadening and Doppler shifts, with which one can explore the kinematics of the hot plasma in supernova remnants, as well as explore thermal line broadening, which is important to measure ion temperatures (Sect. 4.3). Moreover, the high spectral resolution can be used to detect weak line emission from odd-Z elements like sodium, aluminium, phosphor, chloride and potassium, and to measure their abundances in bright young supernova remnants like Cassiopeia A. This will result in the exploration of nucleosynthesis of these odd elements by supernovae (Fig. 14.1). Although the spatial resolution of XRISM is poor, it is still good enough to distinguish the various regions in Cas A that are either dominated by oxygen/neon/magnesium, by silicon/sulfur, or by iron. This can be used to link the abundances of odd elements to different ejecta layers, as traced by the more abundant elements (Fig. 14.1). Once XRISM has given us a taste of what science integral field spectroscopy at high spectral resolution will provide, this field of exploration will fully mature with ESA’s Athena mission [135], which will deliver high spectral resolution spectra with its X-ray Integral Field Unit (XIFU) instrument [137], but with sensitivity that is much higher than XRISM, thanks to telescopes with an effective area of 1.4 m2 at 1 keV (one order of magnitude better than XRISM), and by exploring spatial details with a resolution of ∼ 5 . Athena is expected to be launch around 2031. The other X-ray satellite to be launched around 2021/22 is the NASA smallexplorer mission IXPE (Imaging X-ray Polarimetry Explorer) [1213]. As the name suggest IXPE will carry X-ray polarisation detectors, for which it will use gas-cell

3 For

updates: https://www.mpe.mpg.de/eROSITA.

4 https://global.jaxa.jp/projects/sas/xrism.

14 Summary and Prospects

0.2 0.1 0

Counts/s/keV

0.3

466

1

1.2

1.4 1.6 Energy (keV)

1.8

Fig. 14.1 Left: A simulated image of Cas A using the XRISM resolution. Despite the poor angular resolution, distinct regions rich in iron (Fe-L emission, red), silicon (1.85 keV, green), and dominated by synchrotron radiation (4-6 keV continuum, blue) can be distinguished. Right: A simulated spectrum of a 30 region of Cas A containing Na and Al K-shell line emission (sticking out in blue, see Tables 13.2 and 13.3)

detectors with imaging capabilities. The main interest for supernova remnants and pulsar wind nebulae is that one can measure the X-ray polarisation from synchrotron radiation, and deduce from that the magnetic-field topology. Synchrotron radiation is intrinsically polarised with polarisation fractions as high as ∼ 70% (Sect. 13.3). But different magnetic-field orientations within a field of view and along the line of sight, as well as magnetic-field turbulence, will reduce the measurable polarisation fraction. An X-ray polarisation fraction of 19% has been measured for the Crab Nebula [1211], but no measurements exists for shell-type supernova remnants. Radio polarisation measurements indicate a relative low polarisation fraction for young supernova remnants ( 20%), as well a largely radial magneticfield orientation (Sect. 12.1). X-ray synchrotron radiation is uniquely associated with young supernova remnants, and is only possible with high magnetic-field turbulence around the cosmic-ray accelerating shocks (Sect. 12.2). As discussed, X-ray synchrotron radiation is more confined to regions close to the shock front. An important question that IXPE will address is whether the radial magnetic-field orientation is already present in the X-ray synchrotron emitting regions in the vicinity of shock fronts, and what the level of magnetic-field turbulence may be. A priori, one could argue that the magnetic-field turbulence close to the shock front is expected to be higher than for the rest of the supernova remnant shells. Hence, the X-ray polarisation fraction should be lower than in the radio. On the other hand, the X-ray synchrotron regions are narrower than radio synchrotron emitting region. So in X-ray depolarisation due to line-of-sight effects are more limited. The question, therefore, which aspect dominates the X-ray polarisation fraction in young supernova remnants. In Fig. 14.2 we show the type of images that a mission like IXPE could provide, in this case for Cas A.

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Fig. 14.2 Monte-Carlo simulations for Cassiopeia A based on 2 Ms observation with the proposed ESA mission XIPE using photons in the 4–6 keV band. XIPE is similar in design, and uses the same type of detectors as IXPE. For the polarisation fractions (lower two images) only pixels with 3σ detection threshold are displayed (Credit: simulations by the author, published previously in [1178])

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Gamma-Ray Over the last decade Fermi, AGILE, and the current generations of IACTs will slowly be phased out. Unfortunately, in the sub-GeV and 1–10 GeV bandpasses no replacements for Fermi or AGILE are being funded, although several proposals have been put forward. Hopefully, one of the proposed missions will at some point be realised. For the very-high energy γ -ray domain the situation is much more promising. At the moment we are expecting the first results from the China’s ground-based LHAASO (Large High Altitude Air Shower Observatory) γ -ray and cosmic-ray observatory, which employs both water Cherenkov detectors and IACTs [308]. Around 2025/26 we expect the Cherenkov Telescope Array (CTA)—the successor of H.E.S.S.,MAGIC, and VERITAS—to become operational [39]. See Fig. 14.3. CTA is an IACT similar to the current generation of IACTs, but with many more telescope dishes than the two to five telescopes employed by current IACTs. CTA is an international effort, aimed at realising two observatories. One observatory (CTAN) will cover the Northern hemisphere from La Palma, one of the Canary Islands. The other observatory (CTA-S) will be built near the ESO site at Cerro Paranal, Chile. The final baseline configuration for the CTA-N is four large-sized telescopes (LSTs, 23 m diameter) and 15 medium-sized telescopes (MSTs, 12 m diameter). For CTA-S the baseline configuration is four LSTs, 25 MSTs and up to 70 small-sized telescopes (SSTs, 4 m diameter). Note, however, that the final configuration has not been settled upon, and the initial arrays are expected to consist of fewer telescopes. The LSTs are designed to detect γ -ray photons with energies below ∼200 GeV. Since these photons are more numerous (given the inverted power-law γ -ray spectra), not many LSTs are required. But the γ -ray induced airshowers are faint, requiring large diameter telescopes for their detection. On the other hand, photons with energies above 5 TeV produce bright airshowers, which can be detected with 5 m class telescopes. But since these photons are more rare, SSTs are required to

Fig. 14.3 Artist impression of CTA, showing the mix of different size telescopes (Credit: CTAO; https://www.cta-observatory.org)

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469

Fig. 14.4 CTA sensitivity (prod3b-v2) as a function of photon energy, compared to current γ -ray facilities (Credit: CTAO; https://www.cta-observatory.org)

cover a large atmospheric area, which determines the effective area of the γ -ray observatory. Apart from a factor ten time higher sensitivity compared to current IACTs (Fig. 14.4), CTA will also have a factor two better angular resolution. This combination will not only result in the detection of many more γ -ray sources, including new supernova remnants, but it will also allow better separation of sources in confusing regions of the sky, and it will clarify what are the important regions of γ ray emission within resolved supernova remnants. With current facilities, possible associations between molecular clouds and γ -ray emitting supernova remnants have been identified, based on spatial associations. But whether the γ -ray emission is indeed coming from the molecular cloud region is not always clear. Moreover, CTA will allow deep surveys of the Milky Way and Magellanic Clouds, allowing to probe the γ -ray emission properties of entire populations of supernova remnants. Another expectation related to supernova remnants is that CTA may be able to identify PeVatrons. In Chaps. 11 and 12 we identified superbubbles and very young supernova remnants/radio supernovae as alternative sources of cosmic rays above 1015 eV. CTA is much better equipped to search for TeV signatures from extragalactic radio supernovae, providing a means to probe the early phases of supernova remnant evolution in γ -rays. Another area where progress can be expected is in identifying haloes of cosmic rays diffusing away from supernova remnants. We already encountered possible examples of “escaping” cosmic rays,

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associated with RX J1713.7-3946 and W28, but CTA can explore this for many more supernova remnants, with a variety of ages. This will answer the question during what phases supernova remnants are capable of accelerating cosmic rays to certain energies, and when they are releasing these cosmic rays into the interstellar medium.

14.3 The Emergence of Multimessenger Astronomy The historical progress in astronomy is closely linked to the steady improvements in observational technologies, with as major steps the invention of the telescope (seventeenth century), the development of radio astronomy (1940s), later followed by radio interferometry (1960s), the development of satellites and spacecrafts, which resulted in the birth of X-ray and γ -ray astronomy (1960s), later followed by infrared astronomy (1980s), and more recently the development of ground-based Cherenkov telescopes for γ -ray astronomy (1980s, but really taking off since 2000). This steady progress in opening up access to the electromagnetic spectrum has now more or less been completed. Some gains may still come from opening up the radio band below 10 MHz, or from pushing γ -ray astronomy far beyond 100 TeV, but in both cases extinction of the signals in the interstellar space may prevent these extreme parts of the spectrum to be of general use to astronomy. However, astronomy’s march through the electromagnetic spectrum has now parallels in opening up other carriers of astrophysical information. This started with the discovery of cosmic rays (1910s), neutrino astronomy—starting in the 1960s for MeV neutrinos, but since the 2000s also covering neutrinos with energies 100 TeV with IceCube—and, last but not least, the first detection of gravitational waves from merging stellar-mass black holes with the Advanced LIGO detector [9]. The tighter connections, both on a technological and on a scientific level, between particle physics and astronomy has led to a new field, dubbed astroparticle physics or multimessenger astrophysics. Astroparticle physics was for some time the label for cosmic-ray and neutrino related research, but is also applied to gravitational wave research. Some aspects of supernova remnant research is related to multimessenger/astroparticle physics research. The oldest branch of astroparticle physics— cosmic-ray research—is closely linked to supernova remnants, given that supernova remnants are suspected to be the dominant sources of Galactic cosmic rays (Chap. 11). There are several ongoing cosmic-ray experiments that will explore better details of the cosmic-ray spectrum around the energy range of ∼ 1011– 1017 eV, which is crucial for understanding the contribution of pulsar wind nebulae and supernova remnants to the leptonic and hadronic cosmic-ray spectrum. Of particular interest is whether the details of the cosmic-ray spectrum reveals indeed substructure that could indicate that several source classes are contributing to the cosmic-ray spectrum above ∼ 1011 eV. See the discussion at the end of Sects. 11.1.1

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and 12.2, and reference [727]. There are several satellite, balloon and ground-based facilities that will further explore the cosmic-ray spectrum. Among these facilities are the very-high energy γ -ray facilities previously mentioned, CTA and LHAASO, which also measure Cherenkov light from cosmic-ray-induced airshowers. Chapter 13 (Sect. 13.6) explain how γ -ray radiation through pion decay has a natural counterpart in creating high-energy neutrinos. These high-energy neutrinos could in principle be detected by the antarctic ice experiment IceCube [6], and the future Mediterranean water experiment KM3NeT [631]. So far IceCube did not identify supernova remnants as neutrino sources. This is perhaps not so surprising as IceCube’s sensitivity is low below 1013 eV, where most of the supernova remnant signals are expected. Nevertheless, future upgrades of IceCube and the completion of KM3NeT, and the accumulation of data over a period of several years may result in the positive identification of supernova remnants as cosmic-ray neutrino sources. Such a signal would for once and for all establish supernova remnants as sources of hadronic cosmic rays, and shed new light on the maximum energy of the cosmic rays accelerated by supernova remnants. Supernova remnants are not gravitational-wave sources. However, the violent creation of a neutron star initiating core-collapse supernovae are expected to result in detectable gravitational waves, provided the supernova is in the Milky Way for Advanced LIGO/Virgo detectors. The next generation of gravitational wave detectors, such as the Einstein Telescope (ET) [937]—to be operational beyond 2028—will extend this to around 180 kpc [934]. For a gravitational-wave detection from a supernova we need, therefore, to have the luck of a nearby core-collapse event in the next one to two decades. If that is the case, it will have a profound influence on our understanding of core-collapse supernovae and the creation of neutron stars. Moreover, a gravitational-wave signal can be combined with the neutrino signals to determine the temperature and opacity of the contracting neutron-star. Such an event will also shape our understanding of the origin of structure in core-collapse supernova remnants, and the kick velocity of neutron stars.

14.4 The Extragalactic Transients Connection The study of supernova remnants is mostly confined to supernova remnants found in the Milky Way and the local group, except for some nearby galaxies for which optical and VLBI radio observations provide the means to study entire supernova remnant populations (Chap. 3). In contrast modern supernova studies concern distant sources—with the exception of SN1987A. One of the challenges of supernova remnant research is to connect the diversity in supernova properties to the properties of supernova remnants. Ultimately the population of local supernova remnants may be too small to connect the wide variety of supernovae and other bright optical transients to related remnants in our the Galaxy. Despite the limitations of the Galactic supernova remnant sample, it is worthwhile to explore their connection to the properties of explosive phenomena in distant galaxies in a variety of ways.

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Radio Supernovae Apart from supernovae in general, an obvious example of the overlap between supernova remnants and extragalactic transients is the class of radio supernovae—a well studied example being SN 1993J [403, 766] For one reason, radio supernovae can be considered the earliest phases of supernova remnant evolution, but for a special subset of core-collapse supernovae: those exploding inside the dense wind bubble of an red-supergiant progenitor. They are important for understanding cosmic-ray acceleration to the cosmic-ray “knee”, as already mentioned in this chapter and in Sect. 12.2. Radio supernovae have been reasonably well studied in the radio and optical, with some X-ray studies as well. But with the next generation of more sensitive X-ray telescopes (Athena) and γ -ray telescopes (CTA) it will be feasible to study their shock evolution and particle acceleration properties. In particular γ -ray studies with CTA are crucial to establish whether radio supernovae could be the long sought PeVatrons [476]. The study of radio supernovae can be considered as a natural extension of supernova remnant research, both from the point of source class and physics. More exotic transients have recently attracted a lot of attention, but their exploration has, nevertheless, connections to supernova remnant research. These exotic transients are neutron-neutron star mergers—giving rise to what so-called kilonovae—and fast radio bursts. Kilonovae It was long thought that short gamma-ray bursts (sGRBs) were caused by the merger of two neutron stars. It was later theorised that a small mass of ejected neutron-rich material would provide the ideal environment for producing r-process elements, which comprise many elements heavier than nickel [407, 1090]. The sGRB γ ray signal should be accompanied by an optical transient, powered by the kinetic energy of ejected neutron-star matter, and the decay of radioactive elements that were synthesised during the immediate aftermath of the merger event. As the optical transient was theorised be about 1000 times brighter than novae, the optical transient was dubbed kilonova [806]. The theory of kilonovae was spectacularly proved by the first LIGO-detected gravitational wave event associated with the merger of two neutron stars, GW 170817. This event was nearly coincident with a sGRB, GRB 170817A, detected by Fermi and INTEGRAL [10]. The sGRB was soon followed by an optical transient (SSS17a/AT 2017gfo) found in the host galaxy NGC 4993, located at a distance of ∼ 40 Mpc. The gravitational-wave discovery and the successful multiwavelength follow-up studies have been hailed as the birth of multimessenger astrophysics. Xray and radio follow-up studies in the weeks to months after the event revealed the expansion of a structured jet and an expanding shock wave [64, 955, 1126]. The connection with supernova remnants is that kilonovae are expected to create an expanding shell not unlike a supernova remnant. The explosion energy is comparable to that of a supernova, ∼ 1050 erg, but what is different is the low ejecta mass (10−3 –10−1 M [806]) and the likely occurrence of these explosions at the outskirts of galaxies, as binary neutron stars are associated with the halo

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population of stars. The low ejecta mass will result in a rapid evolution from the ejecta-dominated phase to the Sedov-Taylor phase (Chap. 5). For example, using Ekin = 1050 erg, Mej = 10−2 M , and nH,0 = 10−3 cm−3 , Eq. (5.47) suggests a characteristic evolution toward the Sedov-Taylor phase of just 0.14 yr. On the other hand, the expected low density of the ambient medium may result in a long lifetime. For example, Eq. (5.61) suggests that only after ∼ 200 × 105 the shock will become radiative. In other words, a typical kilonova remnant in the Milky Way is likely to be an extended source (on account of its long life time and low density ambient medium), but unfortunately also a low-surface brightness source in both X-rays and the radio. It may nevertheless be important to use SKA to search for these remnants, and follow them up with sensitive X-ray telescopes like Athena. Of particular interest will be whether in X-rays we could identify the r-process ejecta. This will be difficult for two reasons. One is that, most likely, the kilonovae that we will identify first will old remnants, which will be dominated by tens to hundreds of solar masses of swept-up ambient gas. This is in stark contrast to the expected ∼ 10−2 M of r-process products. The second reason is that the r-process elements consist of a wide variety of elements, and for each elements a multitude of ionisation stages may be present. Moreover, these ions will produce soft M-shell line transitions in the soft X-ray band, which may be even more complex than the aptly name L-shell complex of iron. So the identification of these elements will be hampered by a complex spectrum, overwhelmed by lines from shocked ambient medium. But as we are making steady progress in observational sensitivity, and extend our knowledge of X-ray spectroscopy, we may be able to still identify kilonova remnants. And perhaps among them there is a relative young, or nearby remnant. The kilonova rate is estimated to be around 100 Myr−1 [604], corresponding to a factor 2000–3000 less than the Galactic supernova rate. Given the ten times longer lifetimes of kilonova remnants, we may expect their number be a factor 200 less than the number of supernova remnants. So among the 300–400 known supernova remnants, statistically there be one or two kilonova remnants hiding. Given their expected low surface brightness, it is not likely that the list of known supernova remnants contains a kilonova remnant. But who knows? Fast Radio Bursts Another type of transients with connections to the topics in this book are fast radio bursts (FRBs); see [912] for a review. FRBs were first identified in surveys with the Parkes radio telescope [742], as transient events characterised by a very large dispersion measure.5 The large dispersion measure implied that FRBs occur at cosmological distances. This, in turn, suggested radio luminosities that cannot be accounted for by incoherent radiation processes, such as synchrotron radiation or non-Lorentz boosted blackbody emission.

dispersion measure is defined as the column density of free electrons, DM ≡ causes a frequency-dependent time delay in the arrival time of pulses.

5 The

!

ne dl, which

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For some time FRBs evaded source identification, as they were typically discovered with poor localisation. But then several repeating bursts were identified with FRB 121102 [1071], allowing the luminosity distance to be measured to be 972 Mpc [1106], and allowing for sub-arcsecond localisation of the source, using the European VLBI network [765]. The latter observation also revealed the presence of a persistent source of less than 0.7 pc in size. This discovery led to the now favoured theory that FRBs are somehow related to very young pulsars, perhaps millisecond magnetars [802]. The persistent radio source within this theoretical framework could be a young supernova remnant (a radio supernova?), or a young pulsar wind nebula, which brings us back to the topic of this book. While the writing of this book was nearing its end, another supernovaremnant/FRB connection showed up: FRB-like bursts were discovered from SGR 1935+2154 [189, 1025], one of the more rapidly spinning magnetars (P = 3.25, Table 6.1), which is located within the supernova remnant G57.2+0.8 [580]. Although the extragalactic FRBs are a few orders more luminous than their Galactic counterpart, the discovery suggest a link between FRBs and magnetars, and reasserts the possibility that FRBs are associated with young supernova remnants or, alternatively, young pulsar wind nebulae. This provides yet another perspective on supernova remnants and the exotic objects they may host.

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Index

A Accretion induced collapse, 20 Adiabatic phase, 87 Advanced Satellite for Cosmology and Astrophysics (ASCA), 338 AGB stars, 173 Age break, see cooling break Age-limited synchrotron spectrum, 308, 340, 367 AKARI mission (ASTRO-F), 191, 459 Alfvénic drift, 326 Alfvén Mach number, 59 Alfvén waves, 59, 285, 314 Alpha-elements, 13 Alpha-rich freeze out, 27 Ambipolar diffusion, 162 Amorphous dust, 172 AMS-02 (Alpha Magnetic Spectrometer), 154, 461 Andromeda Nebula, see M31 Anomalous X-ray pulsar (AXP), 157, 158 Anti-magnetar, 118 Astroparticle physics, 471 Astro-Rivelatore Gamma a Immagini Leggero (AGILE), 359, 459 Asymptotic giant branch (AGB) stars, 173 Atacama Large Millimeter/Submillimeter Array (ALMA), 195, 463 ATHENA, 70 Atwood number, 113 Augér process, 430

B Balmer-dominated shocks, 38, 199, 207, 223, 230, 231, 461 Bell instability, 318, 333, 347 BeppoSAX, 339 Beta-plus decay, 26 Bethe-Heitler bremsstrahlung cross-section, 405 Bethe (unit of explosion energy, 1051 erg), 7 Black-hole formation, 14 Bohm diffusion, 67, 285, 301, 340 Bowen fluorescence, 448 Braking index (pulsar), 124 Branch-normal Type Ia supernovae, 24 Bremsstrahlung, 289, 339, 360, 400, 411

C Carbon monoxide (CO), 262 Central compact object (CCO), 118, 166 CGRO, see Compton Gamma-Ray Observatory (CGRO) Chandrasekhar limit, 19, 21 Chandra X-ray Observatory, 227, 242, 246, 251, 339, 459 Characteristic age (pulsar), 122, 124 Charge-transfer reactions, 430 Cherenkov Telescope Array (CTA), 358, 470, 471 Circumstellar medium, 87 Classical electron radius, 383, 404

© Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2

513

514 Classical nova, 19 Collisional excitation, 430, 433 Collisional ionisation, 430, 436 Collisional ionisation equilibrium (CIE), 439 Collisionless shocks, 55, 61, 62, 65 Composite supernova remnant, 33, 118, 139 Compton Gamma-Ray Observatory (CGRO), 339 Compton scattering, 289, 384, 385 Contact discontinuity, 90, 99, 139, 141, 144, 356 Convection-diffusion equation, 297 Cooling break (synchrotron radiation), 398 Cooling curve, 75 Core bounce, 15 Cosmic microwave background (CMB), 307 Cosmic ray acceleration, adiabatic losses, 303 Cosmic-ray ankle, 279 Cosmic-ray electrons, 281 Cosmic-ray injection efficiency, 313 Cosmic-ray knee, 279, 302, 340, 461 Cosmic-ray positrons, 281 Cosmic-ray precursor, 69, 302, 337 Cosmic-ray second knee, 279 Cosmic-ray shock precursor, 298, 302 Cosmic-ray transport equation, 285 Cosmological constant, 7 Coulomb collisions, 69 Coulomb logarithm, 71 Critical Mach number, 61, 85 Crystalline dust, 172 C-type shock, 83 Curvature radiation, 127

D Deceleration parameter, 88 Deflagration models (SN Ia), 22 Deflagration model (Type Ia SNe), 22 Delayed-detonation model (Type Ia SNe), 22 Delayed-detonation transition model (DDT model), 22 Detonation model (Type Ia SNe), 22 Dielectronic recombination, 430 Diffusion length scale, 298, 302 Diffusive shock acceleration (DSA), 145, 294 Dispersion measure, 473 Double degenerate channel for SN Ia, 20, 230, 237 Double detonation model (Type Ia SNe), 23 Dust coagulation, 187 Dust depletion, 171, 172 Dust grains, 171

Index Dust heating (steady state), 176 Dust sputtering, 187 Dwingeloo radio telescope, 37

E Einstein coefficients, 413, 428 Einstein Observatory, 156 Einstein Telescope (ET), 471 Ejecta-dominated phase, 87 Electron-capture supernova, 17, 20, 118, 143 Energy-conservation phase, 87 Equation of state (neutron stars), 118 EROSITA, 465 Euclid mission, 464 Excitation-autoionisation, 430 Expanded Very Large Array (EVLA), 462 Expansion parameter, 88, 94, 141 Extensive air-showers, 280, 284 Extinction, 171

F Failed supernova, 15 Faraday orientation, 335 Fast radio bursts (FRBs), 474 Fe-L complex, 222, 424, 446 Fermi, 459 Fermi Bubbles, 294 Fermi Gamma-ray Space Telescope, 289, 359, 450, 462 Fermi gamma-ray telescope, 361 Fermi mechanism (first order), 294 Fermi mechanism (second order), 294 Fermi’s golden rule, 427 Fermi shock acceleration, 145, 149 Fluorescence yield, 430, 446 Foe (supernova energy: fifty-one erg, 1051 erg), 7 Forbidden line transitions, 199, 204, 428 Free-free absorption, 410 Free-free emission, see bremsstrahlung

G Galactic cosmic rays, 280 Gamma-ray burst, 9, 15 Glitches (pulsars), 125 Goldreich-Julian density, 127 Graphite, 173 G-ratio (helium triplet), 444 Greisen-Zatsepin-Kuz’min (GZK) effect, 280 Guest star, 5

Index H Hadronic cosmic rays, 280, 449 Hall drift, 162 Helium triplet, 442 Herschel Space Observatory, 191, 459 High-Altitude Water Cherenkov Observatory (HAWC), 359 High Energy Stereoscopic System (H.E.S.S.), 358 HII regions, 38, 50 Hopkins Ultraviolet Telescope (HUT), 72, 202 Hubble Space Telescope, 2, 200, 208, 210, 218, 240, 251, 462 Hund’s rules, 421 Hydrocarbons, 173 Hydrogen molecule (H2 ), 262 Hydroxyl (OH), 263 Hypernova, 10, 15, 164 Hyperons, 118

I IceCube Neutrino Observatory, 357, 471 Imaging atmospheric Cherenkov telescopes (IACTs), 153, 357, 459 Imaging X-ray Polarimetry Explorer (IXPE), 468 Infrared Astronomical Satellite (IRAS), 191, 459 Infrared Space Observatory (ISO), 191 Intermediate mass elements, 19, 22, 207, 221 Interstellar medium, 87 Inverse Compton scattering, 289, 384, 385 Ion gyroradius, 66 Ion inertial length scale, 66 Ionisation age (ne t), 71, 439 Isobaric cooling, 75 Isothermal shocks, 78

J James-Web Space Telescope JWST, 463 J-type shock, 83

K Kaon, 118 Kaon condensates (neutron stars), 118 Karlsruhe Shower Core and Array Detector (KASCADE), 284 Kelvin-Helmholtz instability, 116 Kennel and Coroniti model, 131 Kilonova, 473 Kirchhof’s law of thermal radiation, 175

515 Klein-Nishina cross-section, 388 Klein-Nishina effects, 362, 388 KM3 Neutrino Telescope (KM3NeT), 471 Kolmogorov turbulence, 285 K-shell transitions, 424, 442, 446

L Large High Altitude Air Shower Observatory (LHAASO), 468, 471 Larmor formula, 381, 402 Laser Interferometer Gravitational-Wave Observatory (LIGO), 471 Leaky-box model (cosmic rays), 286 Leptonic cosmic rays, 281 Light cylinder (pulsar), 126 Light echo, 28, 48 Limb brightening, 204 Line driven winds, 106 Long-duration gamma-ray burst, 9 Lorentzian line profile, 415 Loss-limited synchrotron spectrum, 308, 340, 367 Lower hybrid oscillations, 68 Low Frequency ArRay (LOFAR), 462 L-shell transitions, 424, 446 Luminous blue variable (LBV) stars, 108 Lyβ trapping, 214

M M101, 51 M31 (Andromeda Nebula), 50 Mach number, 57 Magellanic Cloud Emission-line survey (MCELS), 46 Magnetar, 118, 157, 158 Magnetic dipole model (pulsars), 119 Magnetic field obliquity (pulsars), 120 Magnetic reconnection, 138 Magnetic reconnection (pulsar wind), 149 Magnetic shock precursor, 80 Magnetite, 173 Magnetohydrodynamical shocks, 58 Magnetosonic Mach number, 60 Magnetosonic waves, 285 Magnetosphere, 119, 145, 162 Magnetosphere (pulsar), 126 Major Atmospheric Gamma Imaging Cherenkov telescopes (MAGIC), 358 Masers, 267 Mathis, Rumpl, Nordseick (MRN) dust-grain size distribution, 172, 176, 179, 187

516 Methanol masers, 268 Microquasar, 293 Milagro, 359 Millisecond pulsars, 119, 122 Milne relation, 437 Minimum energy requirement, 327 Mixed-morphology supernova remnant, 33, 461 Modified blackbody spectrum, 175 Molecular clouds, 262 Multi-fluid shock model, 68, 215 Multi-messenger astrophysics, 471 Murchison Widefield Array (MWA), 462

N Nancy Grace Roman Space Telescope (formerly WFIRST), 464 Natural linewidth, 415 Nebular phase (supernova), 8 Neutron-neutron star mergers, 473 Neutron star, 117 Neutron star crust, 118 Neutron star envelope, 119 Non-equilibrium ionisation (NEI), 223, 439 Non-linear shock acceleration, 308, 326 Non-radiative shocks, 327 Non-thermal bremsstrahlung, 401 Non-thermal radiation, 379 Nova, 5, 19, 28 Nova Persei, 28 Nuclear Spectroscopic Telescope Array (NuStar), 339 Nucleation (dust), 183

O Odd-even effect (atomic abundances), 281 OH masers, 267 Olivine, 173 One-zone models, 364

P Pair multiplicity (pulsar magnetosphere), 128, 147, 150, 461 Parallel shocks, 59 Parkes radio telescope, 473 Particle-in-cell (PIC) simulations, 65, 313 Pauli principle, 419 Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA), 154, 288, 461 P-Cygni profile, 29

Index Perpendicular shocks, 60 PeVatron, 150, 278, 374, 461 Phased array, 462 Phillips relation, 23 Pickup ions, 216, 219 Pierre Auger Observatory, 280 Pion bump, 357, 455 Pion production threshold energy, 452 Pions (π-mesons), 451 Plasma beta, 60 Plerion, 33 Polar cap (pulsar), 126 Polycyclic aromatic hydrocarbons (PAHs), 172, 173 Positronium, 26 Positrons, 281 Poynting flux, 382 P-Pdot diagram, 122, 123 Presolar grains, 174 Pressure-driven phase, 88 Pulsar, 117 Pulsar braking index, 123 Pulsar characteristic age, 122, 124 Pulsar initial spin period, 145 Pulsar kick velocity, 15 Pulsar wind, 128, 129, 145 Pulsar wind nebula, 33, 118, 119, 139 Pyroxene, 173

Q Quasi-static flocculli (Cas A), 240 Quintessence, 7

R Radiative efficiency (pulsar wind nebulae), 147 Radiative recombination, 430 Radiative recombination continuum (RRC), 437 Radiative shocks, 77, 326 Radio supernovae, 326, 377 Rankine-Hugoniot relations, 56 Rayleigh-Taylor instability, 113, 143, 144, 356 Red supergiants, 106, 173 Refractory elements, 172, 283 Residence time (cosmic rays), 286 Resonant line scattering, 204, 448 Resonant line transitions, 428 Resonant particle scattering, 314 Reverse shock, 89 Rigidity, 284 Roentgensatellit (ROSAT), 157, 465 Rossi X-ray timing Explorer (RXTE), 339

Index R-ratio (helium triplet), 445 RRC, see radiative recombination continuum (RRC) Russell-Saunders coupling, 421 S Sedov-Taylor phase, 87 Self-similar models, 92 Shell-type supernova remnant, 33 Shock precursor, 55, 80, 216 Shock width, 62 Short gamma-ray bursts, 473 SiC X grains, 174 Sigma-D ( − D) relation, 39, 40, 50 Sigma-problem (pulsar winds), 135, 138, 461 Silicates, 173 Silicon carbide (SiC), 173, 174 Simmering (Type Ia SNe), 22, 230 Single degenerate channel for SN Ia, 19, 230, 237 Singlet state, 419 SN 1987A, 463 Snow-plough phase, 88, 103 Soft gamma-ray repeaters (SGRs), 158 Sonic Mach number, 57 Spallation of cosmic rays, 281 Spectral index, 394 Spectrum-Roentgen-Gamma space observatory, 465 Spitzer Space Telescope, 191, 459 Square Kilometer Array (SKA), 462 Standing accretion shock instability (SASI), 15 Stellar mass loss, 106 Stochastic dust heating, 180 Streaming instabilities, 317 Stretch factor (s), 24 Striped-wind model, 138, 145, 149 SubChandrasekhar explosion (Type Ia SNe), 23 Subshock, 55, 308 Superbubbles, 52, 282, 292, 470 Supernova light curve, 10, 11 Supernova unit (SNu), 12 Supersoft sources, 20, 231, 460 Suzaku X-ray mission (ASTRO-E2), 229, 459 Swift, 168 Symbiotic nova, 21 Synchrotron radiation, 323, 338, 389, 395, 468

517 T Termination shock (pulsar), 128, 131, 135, 145 Thermal bremsstrahlung, 400 Thermal composite supernova remnant, 33 Thomas-Reich-Kuhn sum rule , 428 Thomson cross section, 383 Thomson scattering, 380, 382 Triplet state, 419 Two-fluid shock model, 68 Two photon emission, 428 Tycho stripes, 355 Type Iax supernovae, 25 Type Ib supernovae, 9 Type Ic supernovae, 9 Type IIL supernovae, 10 Type IIn supernovae, 10 Type IIP supernovae, 10 V Van der Laan mechanism, 269, 327 Veil nebula, 2 Vera C. Rubin Observatory (formerly LSST), 464 Very Energetic Radiation Imaging Telescope Array System (VERITAS), 358 Very Large Array (VLA), 324, 462 Very Large Telescope, 462 Violent relaxation, 65 Virgo interferometric gravitational-wave antenna, 471 Volatile elements, 172, 283 W Weibel instability, 317 White dwarf, 7, 19 Wind bubbles, 88, 104 Wolf-Rayet stars, 106, 108 W7 SN Ia model, 22 X XMM-Newton, 72, 222, 242, 459 X-ray Imaging and Spectroscopy Mission (XRISM), 70, 468 Y Ye , 230

Astrophysical objects

Symbols γ Cygni (G78.22.1), 357 1E 1207-5209 (CCO), 157 3C358, 156 3C391 (G31.90.0), 270 3C397 (G41.1-0.3), 230 3C400.2 (G53.6-2.2), 270 3C58 (G130.73.1, SN 1181), 28, 29, 34

C Cassiopeia A (G111.7-2.1), 1, 28–30, 33, 38, 50, 95, 102, 103, 110, 114, 166, 168, 174, 178, 191, 192, 207, 223, 239, 241, 242, 264, 305, 323–325, 329, 331, 332, 334, 337, 339, 341–343, 346, 349, 351, 372–376, 409, 461, 468 Crab nebula (SN 1054, G184.6-5.8), 1, 34, 37, 117, 130, 140, 142, 149, 192, 204, 239, 323, 468 Crab pulsar (PSR B053121), 119, 124 CTB 109 (G109.1-1.0), 157, 164, 165 CTB 37A (G348.50.1), 372 CTB 37B (G348.70.3), 372 CTB 87 (G74.9 1.2 ), 34 Cygnus Loop (G74.0-8.5), 1, 2, 33, 37, 77, 88, 199, 201, 202, 206, 209, 257, 449

D DEM L238, 226 DEM L249, 226 DEM L316, 226 DEM L71, 226, 449

F FRB 121102, 474

G G0.00.0, see Sgr A East G1.90.3, 331, 334, 335, 337, 342, 344 G109.1-1.0, see CTB 109 G11.2-0.3, 122, 140, 156 G111.7–2.1, see Cassiopeia A G120.11.4, see Tycho’s SNR G130.73.1, see 3C58 G180.0-01.7, see Simeis 157 G184.6–5.8, see Crab Nebula G189.13.0, see IC 443 G20.0-0.2, 34 G21.5-0.9, 151 G260.4-3.4, see Puppis A G263.9-3.3, see Vela SNR G266.2-1.2, see RX J0852.0-4622 G27.80.6, 34 G290.1-0.8, see MSH 11-61A G292.01.8 (MSH 11-54), 166, 195, 241–243 G296.510.0, see PKS 1209-51/52 G31.90.0, see 3C391 G315.4–2.3, see RCW 86 G320.4–1.2, see MSH 15–52 G327.40.4, see Kes 27 G327.614.6, see SN 1006 G328.40.2, see MSH 11-57 G34.7-0.4, see W44 G347.3–0.5, see RX J1713.7-3946 G348.50.1, see CTB 37A G348.70.3, see CTB 37B G353.6-0.7, see HESS J1731-347

© Springer Nature Switzerland AG 2020 J. Vink, Physics and Evolution of Supernova Remnants, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-55231-2

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520 G41.1-0.3, see 3C397 G53.40.0, see W51C G53.6-2.2, see 3C400.2 G57.20.8, 474 G6.11.2, 34 G6.4-0.1, see W28 G63.71.1, 34 G65.7 1.2 (DA 495), 34 G74.0–8.5, see Cygnus Loop G74.9 1.2, see CTB 87 G78.22.1, see γ Cygni G89.04.7, see HB21 Geminga pulsar (PSR B065614), 154 GW 170817, 473

H HB21 (G89.04.7), 270, 336 HESS J1303-631, 154 HESS J1303-631 (PWN), 152 HESS J1534-571, 38 HESS J1614-518, 38 HESS J1731-347 (G353.6-0.7), 38 HESS J1825-137, 154 HESS J1731-347 (G353.6-0.7), 344

I IC 443 (G189.13.0), 35, 37, 156, 191, 199, 266, 270, 331, 357, 363, 368, 372

K Kepler’s SNR (G4.56.8, SN 1604), 20, 39, 40, 43, 48, 105, 192, 196, 207, 228, 230–232, 236, 325, 326, 331, 336, 337, 341–343, 346 Kes 27 (G327.40.4), 270 Kes 73 (G27.3-0.1), 165, 166 Kes 73 (G27.3â´LŠ0.1), 243 Kes 75 (G29.7-0.3), 159, 164 Kes 79 (G33.60.1), 35, 168

L Large Magellanic Cloud, 28, 46, 224, 226, 241, 242, 249

M M101, 8 M31, 5 M33, 5, 50 M82, 50

Astrophysical objects MSH 11-57 (G328.4 0.2), 34 MSH 11-61A (G290.1-0.8), 270 MSH 14-63, see RCW 86 MSH 15–52 (RCW 89, G320.4–1.2)), 35, 124

N N103B (SNR 0509-68.7), 28, 48, 49, 196, 226, 230, 233 N132D, 46, 195, 240–243, 331, 372 N157B, 119 N49, 46, 48, 165, 200 N63A, 46

P PKS 1209-51/52 (G296.510.0), 157, 168 PKS 1209-51/62 (G296.510.0), 157 PSR B1509-58, 124 PSR J0537-6910, 119 PSR J1301-6305, 152 PTF 11kly, 8 PTF 11kx, 21 Puppis A (G260.4-3.4), 157, 166, 168, 197, 241, 242, 325, 331, 465

R RCW 86 (G315.4-2.3, MSH 14-63, SN 185), 37, 39, 88, 105, 112, 217, 234, 326, 331, 337, 344, 349, 372 RCW 89, see MSH 15-52 RS Ophiuchi, 21 RX J0852.0-4622 (G266.2-1.2,“Vela Jr.”), 38, 112, 342, 344, 346, 360, 361, 372, 465 RX J1713.7-3946 (G347.3-0.5), 38, 112, 342, 344, 351, 361, 362, 369, 372

S SGR 0526-66, 165 SGR 19352154, 474 Sgr A East (G0.00.0), 270 Sgr A∗ (Galactic Centre blackhole), 293 Simeis 147 (G180.0-01.7), 104, 105 Small Magellanic Cloud, 46, 226, 241, 242 SN 1006 (G327.614.6), 5, 20, 29, 39, 43, 48, 72, 101, 207, 222, 236, 325, 326, 338, 339, 342, 346, 353, 372, 429 SN 1572, see also Tycho’s SNR, 5, 29 SN 1604, see also Kepler’s SNR, 5, 20, 25, 29 SN 185, see also RCW 86, 5, 29 SN 1885 (S Andromedae), 5

Astrophysical objects SN 1987A, 8, 28, 49, 117, 192, 195, 207, 249 SN 1991bg, 25 SN 1991T, 25, 237 SN 1993J, 10, 29, 326, 347, 377 SN 1994D, 6, 11, 25 SN 1996X, 25 SN 2004eo, 25 SN 2004et, 11 SN 2006aj, 247 SN 2008ax, 11 SN 2009dc, 25 SN 2011fe, 8 SN 2017jmj, 8 SN 1054, see Crab Nebula SN 1980K, 11 SN 1991bg, 24 SN 1991T, 24, 28, 48 SN 2002cx, 25 SN 2006X, 21 SNR 0509-67.5, 28, 48, 49, 210, 218, 223, 226, 228, 231, 236, 237 SNR 0509-68.7, see N103B SNR 0519-69.0, 28, 222, 224, 226, 227 SNR 0534-69.9, 226 SNR 0548-70.4, 226 SNR 1E0102.2-7219, 195, 222, 241–243, 246 SNR B0049-73.6 (IKT 6), 241, 242 SNR B0103-72.6 (IKT 23), 241, 242

521 SNR B0540-69.3, 241, 242

T Tycho’s SNR (SN 1572, G120.11.4), 20, 28, 29, 33, 39, 114, 178, 192, 207, 215, 217, 222, 223, 228, 230–232, 236, 325, 326, 331, 337, 339, 341–343, 346, 354, 372, 374, 375

V Veil Nebula, see Cygnus Loop Vela Jr., see RX J0852.0-4622 Vela SNR (G263.9-3.3), 35, 206, 257, 465 VRO 42.05.01 (G166.04.3), 270

W W28 (G6.4-0.1), 35, 37, 266, 270, 368, 372 W44 (G34.7-0.4), 266, 269, 270, 363 W44 (G34.7â´LŠ0.4), 35, 37 W49B (G43.3-0.2), 234, 235, 266, 270, 409 W51C (G53.40.0), 37, 266, 270

X XRF 060218, 247