The resistance of the air to stone-dropping meteors

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SSS SBSISTAJTCE OS* IBS! AIR TO STOHE-BROPPIHG METEORS

N"

*u

-0^

Harry 3L„ Helson

A dissertation submitted in partial fulfillment of requirements for the degree of Boo tor of Philosophy, in the Department of Mathematics and Astronomy* In the Graduate School of the State !Salversity of Iowa

June 1950

ProQuest Number: 10311018

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uest P ro Q u es t 10311018 Published b y P ro Q u est LLC (2017). C o p y rig h t o f th e Dissertation is h eld by th e A uthor. All rights reserved. This w ork is p r o te c te d a g a in s t u n a u th o rize d c o p y in g u n d e r Title 17, U n ited States C o d e M icro fo rm Edition © P roQ uest LLC. P ro Q u est LLC. 789 East Eisenhow er P arkw ay P.O . Box 1346 A n n Arbor, Ml 48106 - 1346

I wish to express my sincere appreciation to Br. d* 0. Wjplie for the guidance and suggestions he has giron in the preparation of this thesis*

ii

tmmm

or m'fi’ s mrs Page

Part Part

I Historical Introduction

1

II derivation &f Formulas

4

Part lit

Calculations ext the Paragould Meteor

Fart

Sojamary

17

* . ♦ * . .

.........

?

» . » 20

Appendix A

Beteminat Ios of the Pressure • » • • 21

Appendix B

Petry Formula of Penetration

Appendix 0

Tim luminosity Equation

Tables

»+

22

........... 25 26

Bibliography

.46

ill

1

PART I sisTommir tsTf&XKCTZGif 0.

T. Schiaparelli^ in 1867 stated that the resistance of the

air to meteors was identical to that of its resistance to any terrestrial object. E® asserted that his results on experiments with projectiles of artillery rifles, boomerangs, and pieces of paper enabled him to predict how meteors« travelling with velocities one hundred times as great as rifle ballets, should move In the earth*s atmosphere.

Re* as well as

his successors for some fifty years, represented this resistance as a variable power of the velocity.

Different exponents in different

velocity £ones war® postulated.

Typical of such resistance functions

Is the following:^

(T is in feet per second}

▼ v^ cos V2 Also JfA st *45

and from 7able I, A - .35 .

Hence

if - 1.39 P a 1.2syOv^

whereto is la grass per cubic centimeter* v isincentimeters per second* and J£ is in dynes per square centimeter. Uhe pressure at a height of 22 kilometers where the Paragoul d was assumed to buret Is P m (1.29) (6.973) (.9423) x 107 P * 7.99 x 10? dynes per square centimeter or

P * 1159 pounds persquare

inch.

When a pro jectile of Telocity jr and cross section TT r^ penetrates a solid ’tody* it loses its energy in overcoming tfee resistance, &, offered "by the body.

Of the several types of resistance

functions that may he assumed, Poneelet chose H ® iA {a+bv®) where i is a coefficient of form* usually equal to unity® A is the cross sectional area, and a and h are empirical constants. Hence the equation of motion of a projectile of weight s w, 1st

or

w g

dy s -t& (a 4-hv3) dt

w g

dy X « ~i& {a*bv^} dx

vdv - ~i&g dx a+hr^ w Integrating 1 2b

In (a-fhv2 ) = -l&gpe4-c w

22. 0. Orans, Lehrbuch Iter Ball!atik, pp. 457-461.

23

Whet* I ? O, v s ?0 ana at -

_JL In (a +■ hv0s ) 2% In ( ^ tv*^ \ » -2hlAg x fa gL±Ja£,\ va + w ’o' y from which

35 s ■ y 3M*«

In/" a + bv*0s \ y a+'W3 V

fhe maximum penetration is, therefore# x

^

Xn(l -j-h w 02 )

Shi%

a

Fetry proposed a simplifieat ion "by assuming that & is constant for all a materials. Hence3 * £ © jr f (r0) a the nature of the material*

where jg, depends upon

g has the valuesi *94 for stone masonry 1*63 for "brick masonry 2.94 for sandy earth 3.86 for soil with vegetation 5.8? for clay soil

24

^

wh©r© v0 I© ©xpr©©s©4 in ®/©e©^h&s the following valuesi r

40

.S3

200

4.77

360

8.74

m

.72

220

5*34

380

9.15

so

1*21

240

5.89

400

9.54

100

1.76

260

6.41

420

9.93

120

2.86

2m

6.92

440

10.39

140

2*87

mo

7.40

460

10.64

160

3.68

320

7*87

480

10.98

180

4.18

340

8.31

500

11 *80

*0

* t* 0>

To

25

APFHHBXX G 1M B liTMIBOSITT

From meteor spectra it is evident that the heated surface of the meteor tody itself does not eon tribute to the visible radiation ,'33 All of the observed light most be produced b y the collision ©f atmos­ pheric molecules with the vaporised material emitted by the meteor, fhe luminous intensity* I* is assumed proportional to the energy of the mass lost per unit time.

1 » * T" m v^ ¥ For the brighter meteors* Whipple has found that

T

s ^”ov

where

Isglo = -9.0? Hence

1 ~ JCc m v 3

2 or

but

log X - -9.37 -h log

OJ.

%

(-4 ) + 3 log

w

» 24.60 -2.5 log X

Mv r 48.03 -2.5 log

(~m) -7.3 log v

2 3 . Fred X.. Whipple, Meteors and the Barth is ffpper Atmosphere, p. 2 5 4 . 2 4 . Luigi Jacchia* Photographic Meteor Phenomena and theory* p. 1 2 .

a&

I Density m 3, A Single Gone

10°

0.43 0.69

Double Cone Semi-angle

30°

0.93

40®

1.20

50°

1.52

55°

1.71

10®

0.27 0.43

30®

0.59 0.76

60®

0.95

«=«0

1.08 .53

Sphere Cube Bllipsold

a x b x e

Cylinder ParagoulA

36B x 1S.51* x 21# (91.4 x 47 x 53.3 em.)

.35

27

7ABLB II 177-161

161-145

145-129

129-113

P

2. 230 10-12

1.096 10*.11

5.248 10-11

2.541 10-10

ICT6*

1.563

1.563

1.563

1.563

10~6y

.3815

.3897

.3937

14&0*

14°S*

o< IO^v

1.609

1.611

1.611

1.812

lO**6®

6.038

6.038

6.038

6.038

io-**'/3

1.821

1.821

X.82I

1.821

-1.341

—6•429

-.06836 ty

4110

4066

4023

3981

-1.167

-3.676

-26.88

-129.0

80-74

115-97

97-86

1.521 10“9

8.606 10~9

3.024 10“8

6.915 isr6

1.563

1.563

1.563

1.583

.3977

*4016

.4042

.4056

X4°I7*

14°25t

X4?3®*

14°33*

1.612

1.613

1.614

1.614

6.038

6.037

6.034

6.028

1.821

1.821

1.821

1.820

tV

-38.08

-146.8

-279.7

-637.7

ty

3933

2646

1382

1284

-764.4

-2953

-5627

P IQ"6?

10-^r

10~2ib^

-12810

28

H A M M It (continued) I

74-68

68—62

62-56

56-50

/° 10~6x

1.308 IQ*"7

2.289 10“7

3.838 10~7

6.893 10-

1.562

1.561

1.559

1.555

.4069

.4080

.4089

.4094

14*36*

iA°ml

14°42*

14°45’

1G-®V

1.614

1.613

1.611

1.608

ID*6®

6.015

5.991

5.949

5.880

io~2® ;/3

1.819

1.816

1.812

1.8D5

i o ~6£

*

tx

-1302

-2099

-3509

-6296

1131

893.0

515.3

-223.6

-24.00

-43.00

-69.00

-122.0

50-44

44-41

41-38

38—35

1.384 ID*6

2.430 ID”6

3.768 X0~6

6.061 10

1.549

1.537

1.526

1.509

.4092

.4074

.4052

.4013

14*48f

14051*

14052*

I4°54«

10

1.602

1.590

1.579

1.561

10

5.758

5.531

5.333

5.041

10*^aiV3

1.792

1.768

1.747

1.715

ts -12.20

-11.15

-17.38

-28.14

l O ^ a v ty -1.789

-2.235

—3.889

-6.750

10“3a;r tm -227.0

-198.0

-292.0

-434.0

I /° 10~6Sr lO-^y ci

1 0 ~3av

29

f A l P 11 (eontiiraed) I

56-32

S3-30

30-38

2S-26

1*018 10“S

1.574 10*®

3.153 10“®

2.946

1.481

1.433

1.383

1.314

10“%

.3945

.3835

.869?

.3517

o