Smithsonian

Contributions to Astrophysics

VOLUME 6

RESEARCH IN SPACE SCIENCE

SMITHSONIAN INSTITUTION

Washington, D.C.

1963
Publications of the Astrophysical Observatory

This series, Smithsonian Contributions to Astrophysics, was inaugurated

in 1956 to provide a proper communication for the results of research conducted at the Astrophysical Observatory of the Smithsonian Institution.

Its purpose is the "increase and diffusion of knowledge" in the field of astrophysics, with particular emphasis on problems of the sun, the earth, and the

solar system. Its pages are open to a limited number of papers by other

investigators with whom we have common interests.

Another series is Annals of the Astrophysical Observatory. It was started

In 1900 by the Observatory's first director, Samuel P. Langley, and has been

published about every 10 years since that date. These quarto volumes, some

of which are still available, record the history of the Observatory's researches

and activities.

Many technical papers and volumes emanating from the Astrophysical

Observatory have appeared in the Smithsonian Miscellaneous Collections.

Among these are Smithsonian Physical Tables, Smithsonian Meteorological

Tables, and World Weather Records.

Additional information concerning these publications may be secured from

the Editorial and Publications Division, Smithsonian Institution, Washton, D.C.

FEED L. WHIPPLE, Director,

Astrophysical Observatory,

Smithsonian Institution.

Cambridge, Mass.

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402 - Price $2.
Contents

Page

Preface v

Contributors vn

Orbital Results for Satellite 1957 pi: L. G. Jacchia (Smithsonian Astrophys. Obs. Spec. Rep. 13, 8 pp.) 1

The Descent of Satellite 1957 (il: L. G. Jacchia (Smithsonian Astrophys.

Obs. Spec. Rep. 15, 12 pp.) 5

Orbital Acceleration of Satellite 1958 fi2: L. G. Jacchia and R. E. Briggs

(Smithsonian Astrophys. Obs. Spec. Rep. 18, pp. 9-12) 13

A Flashing Satellite for Geodetic Studies: C. A. Whitney and G. Veis

(Smithsonian Astrophys. Obs. Spec. Rep. 19, pp. 9-19) 17

The Earth'8 Gravitational Potential as Derived from Satellites 1957 pi and

1958 $2: L. G. Jacchia (Smithsonian Astrophys. Obs. Spec. Rep.

19, pp. 1-5) 25

The Diurnal Effect in the Orbital Acceleration of Satellite 1957 (31: L. G.

Jacchia (Smithsonian Astrophys. Obs. Spec. Rep. 20, pp. 5-8) . . 29

An Empirical Formula for Ephemerides of Satellites Near the End of

Their Lifetime: L. G. Jacchia (Smithsonian Astrophys. Obs. Spec.

Rep. 20, pp. 1-4) 31

The Structure of the High Atmosphere: I. Linear Models: C. A. Whitney

(Smithsonian Astrophys. Obs. Spec. Rep. 21, pp. 1-12) 35

The Structure of the High Atmosphere: II. A Conduction Model: C. A.

Whitne y (Smithsonia n Astrophys . Obs . Spec . Rep . 25 , 7 pp. ) .. . 4 3

On the Effects oj the Sun and the Moon upon the Motion of a Close-Earth

Satellite: Y. Kozai (Smithsonian Astrophys. Obs. Spec. Rep. 22,

pp. 7-10) 47

The Earth's Gravitational Potential Derived from the Motion of Satellite

1958 $2: Y. Kozai (Smithsonian Astrophys. Obs. Spec. Rep. 22,

pp. 1-6) 51

Solar Effects on the Acceleration of Artificial Satellites: L. G. Jacchia

(Smithsonian Astrophys. Obs. Spec. Rep. 29, 15 pp.) 55

Note on the Secular Motions of the Node and Perigee of an Artificial

Satellite: Y. Kozai (Smithsonian Astrophys. Obs. Spec. Rep. 30,

pp. 14-15) 67

Anticipated Orbital Perturbations of Satellite 1959 82: Y. Kozai and

C. A. Whitney (Smithsonian Astrophys. Obs. Spec. Rep. 30, pp.

1-8) 69

On the Effects of Image Motion on the Accuracy of Measurement of a

Flashing Satellite: J. A. Hynek (Smithsonian Astrophys. Obs.

Spec. Rep. 33, 4 pp.) 73

The Effect of a Variable Scale Height on Determinations of Atmospheric

Density from Satellite Accelerations: L. G. Jacchia (Smithsonian

Astrophys. Obs. Spec. Rep. 46, pp. 1-4) 77

mmm SEP ^4 1964
CONTENTS

Page

A Second-Order Solution oj Vinti's Dynamical Problem: I. G. Izsak

(Smithsonian Astrophys. Obs. Spec. Rep. 52, 54 pp.) 81

Effects oj Solar Radiation Pressure on the Motion oj an Artificial Satellite: Y. Kozai (Smithsonian Astrophys. Obs. Spec. Rep. 56, pp.

25-33) 109

The Effect oj Radiation Pressure on the Secular Acceleration oj Satellites:

S. P. Wyatt (Smithsonian Astrophys. Obs. Spec. Rep. 60, 16 pp.) . 113

Experimental and Theoretical Results on the Orbit oj Echo I: P. E.

Zadunaisky, 1.1. Shapiro, and H. M. Jones (Smithsonian Astrophys.

Obs. Spec. Rep. 61, 22 pp.) 125

The Atmospheric Drag oj Artificial Satellites During the October 1960 and

November 1960 Events: L. G. Jacchia (Smithsonian Astrophys. Obs.

Spec. Rep. 62, 13 pp.) 133

Effect oj the Diurnal Atmospheric Bulge on Satellite Accelerations: S. P.

Wyatt (Smithsonian Astrophys. Obs. Spec. Rep. 63, 17 pp.) . . . 139

The Motion oj Satellite 1958 Epsilon Around Its Center oj Mass: G. Colombo (Smithsonian Astrophys. Obs. Spec. Rep. 70, 26 pp.) . . . 149

On the Accuracy oj Measurements Made upon Films Photographed by

Baker-Nunn Satellite Tracking Cameras: K. Lassovszky (Smithsonian Astrophys. Obs. Spec. Rep. 74, 16 pp.) 165

Density oj the Heterospere Related to Temperature: M. Nicolet (Smithsonian Astrophys. Obs. Spec. Rep. 75, 30 pp.) 175

Effects oj the Earth's Ionosphere on HF Radio Astronomy jrom Artificial

Satellites: M. D. Grossi, K. M. Strom, and S. E. Strom (Smithsonian Astrophys. Obs. Spec. Rep. 76, 16 pp.) 189

Short-Periodic Oscillations in the Drag oj Satellite 1958 Alpha: L. G.

Jacchia and J. Slowey (Smithsonian Astrophys. Obs. Spec. Rep. 77,

11 pp.) 199

The Analysis oj Gravity: H. Jeffreys (Smithsonian Astrophys. Obs. Spec.

Rep. 79, 12 pp.) 205

The Stabilization oj an Artificial Satellite at the Inferior Conjunction

Point of the Earth-Moon System: G. Colombo (Smithsonian Astrophys. Obs. Spec. Rep. 80, 13 pp.) ; . . . 213

References ; . . . 223

Special Reports of the Smithsonian Astrophysical Observatory ; . . . 231
Preface

The Special Reports of research in space science were initiated with the

Smithsonian Astrophysical Observatory's research program on artificial earth

satellites in 1957, and they have provided an avenue of immediate publication

which has proved indeed critical to proper elaboration of the field. By early

1963 the 113 Special Reports listed on pages 233 to 244 had been produced and

distributed to all institutions participating in the U.S. space-research program

and to individual scientists requesting them. The Special Reports have thus

become valuable tools for the space scientist.

A number of the purely observational Special Reports are so highly specialized that only the few investigators presently using the data are likely ever

to need the information available in them. It is possible that selections from

these observational tables may be reproduced elsewhere, but many will not.

On the other hand, certain of the Reports record important scientific research

that should be available in the general literature.

Hence, the first 10 Special Reports were published as volume 2, number

10, of the Smithsonian Contributions to Astrophysics. Here we have selected

29 contributions from Special Reports 11 through 80 as meriting a permanent

record. We present them, with such minor revisions as the authors have found

necessary to bring them up to date and to clarify them, as volume 6 of the

Contributions to Astrophysics. Other Special Reports have been published elsewhere in the literature.

To those members of the Observatory staff, and especially to Drs. Jacchia

and Whitney, who helped to choose the papers that appear in this volume, I

am grateful, as I am to the members of the editorial staff who saw the volume

through the press.

FRED L. WHIPPLE, Director,

Astrophysical Observatory,

Cambridge, Mass., April 2, 1963. Smithsonian Institution.

Contributors

R. E. BKIGGS, mathematician, Smithsonian Astrophysical Observatory.

GIUSEPPE COLOMBO, University of Padova, Padova, Italy; celestial mechanician, Smithsonian Astrophysical Observatory.

M. D. GROSSI, staff engineer, Raytheon Corporation, Bedford, Massachusetts;

electronics engineer, Project Celescope, Smithsonian Astrophysical Observatory.

J. ALLEN HYNEK, director, Dearborn Observatory, Northwestern University,

Evanston, Illinois; former associate director, Satellite Tracking Program,

Smithsonian Astrophysical Observatory.

IMRE IZSAK, supervisory astronomer, Smithsonian Astrophysical Observatory.

LUIGI G. JACCHIA, physicist, Smithsonian Astrophysical Observatory.

SIR HAROLD JEFFREYS, Cambridge University, England; consultant, Smithsonian Astrophysical Observatory.

H. M. JONES, physicist, Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Massachusetts.

YOSHIHIDE KOZAI, astronomer, Tokyo Astronomical Observatory, Tokyo,

Japan; former astronomer, Satellite Tracking Program, Smithsonian

Astrophysical Observatory.

KAROLY LASSOVSZKY, late astronomer, Smithsonian Astrophysical Observatory.

MARCEL NICOLET, director, Institut Royal de Meteorologie, Brussels, Belgium.

1.1. SHAPIRO, physicist, Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Massachusetts.

K. M. STROM, student, Radcliffe College, Cambridge, Massachusetts.

S. E. STROM, physicist, Project Celescope, Smithsonian Astrophysical Observatory.

GEORGE VEIS, professor of surveying, National Technical University, Athens,

Greece; consultant, Smithsonian Astrophysical Observatory.

CHARLES A. WHITNEY, physicist, Smithsonian Astrophysical Observatory.

STANLEY P. WYATT, professor of astronomy, University of Illinois Observatory,

Urbana, Illinois.

PEDRO E. ZADUNAISKY, director de investigaciones, Instituto de Calcul6,

Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,

Argentina; former astronomer, Smithsonian Astrophysical Observatory.

VII

Research in Space Science

636-014—63 2

Orbital Results for Satellite 1957 pi

By Luigi G. Jacchia

This report covers the complete lifetime of

Satellite 1957 /31. Since all the observations

have been reduced anew with improved elements, it supersedes the first set of orbital data

(Jacchia, 1958a) that covered the lifetime of the

satellite to Feb. 10, 1958.

Method of reduction

Tables 3 and 4 summarize the results of an

analysis of approximately 2,800 optical and

radio observations, reduced by means of an

improved version of the author's subsatellite

program (Jacchia, 1958d) which contains the

following new features:

Continuously varying perigee distance.—The

relation g,=g(a) between perigee distance and

major axis is determined by fitting the solution

of the differential equation dq/da=f(q,a) through

well-observed points, and then is fed into the

machine program in the form of an approximating equation with 10 numerical coefficients.

A special high-latitude program.—Observations at low and middle latitudes yield independent determinations of the right ascension

and of the time of the ascending node. For

observations made at high latitudes, near the

orbital apex, the effect of observational errors

is greatly magnified in the independent solutions. If, however, the position of the node is

accurately known for the time of observation,

an accurate time for the equatorial crossing can

still be obtained. The new feature in the subsatellite program provides for an optional

switch to the latter type of reduction when the

subsatellite point is within a specified number of

degrees of latitude from the apex.

For the computation of the subsatellite

points, the following orbital elements were used:

Inclination: i=65?29.

Argument of perigee: &>=58?0—0?3938(*— T)

-2?50 X 10-4(*-T)2-3?lX10-7 (t-T)*

(T=1957 Nov. 6.0 U.T.; t in days).

The major axis of the orbit was derived from

the nodal period, obtained by differentiation of

the interpolating equations that follow:

Ta=1957 Nov. 4.41000+0*072082571—1*19

X10-67i2-0*03820(e°-°°17n-l) +0*01700

sin (0?237n-128°), (0<TI<1800); (1)

TQ=1958 March 9.51267+0*066453(TI-1800)

—2*56X10-6(n—1800)2-0*00125927

feo.oo9«»-i8oo,_1] (l800<n<2300); (2)

TO=1958 April 10.95583+0*062547(n-2300)

—8*78X10-6(n-2300)2-8*316X10-4

[e0.097(n-2300)_1] (ft>2300). (3)

The residuals from these equations never

exceed 0*002, and the errors in the computed

period and heights are negligible in the computation of subsatellite points, nodes, and

equatorial crossings.

The relation between the major axis and the

perigee distance is given in table 1. Values

of dg/da were computed from Sterne's (1958a)

equations with the Smithsonian Interim Atmosphere (Sterne, Folkart, and Schilling, 1957)

and integrated, starting from the normal point

2=1.03302 for or—-1.13. The integration was

performed by Jack Slowey, using a special

program on the IBM-704 Calculator.

For the reduction of the high-latitude observations, the following equations were used for

the position of the ascending node.

aQ=108?2-2?6079(*-Nov. 6.0)-l?656X10~8

(*-Nov. 6.0)2-2?08X10-6(*-Nov. 6.0)3 (4)

(for dates before 1958 Mar. 11).
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Assumed relation between semimajor

axis and "perigee distance

a

1.15

1.14

1.13

1.12

1.11

1. 10

1.09

1.08

1.07

1.06

1.05

1.04

1.03

1.02

<Z

1. 03360

1. 03333

1. 03302

1. 03268

1. 03230

1. 03187

1. 03137

1. 03079

1. 03008

1. 02920

1. 02805

1. 02639

1. 02370

1. 01869

e

0. 1012

.0936

.0858

.0780

.0700

.0619

.0538

.0456

.0373

.0291

.0209

.0131

.0061

.0013

TABLE 2.—Normal values of aa and observed nodal

precession

aa daa/dt

a; Semimajor ails of orbit In units ol the earth's equatorial radius.

q: Perigee distance.

e: Orbital eccentricity.

ao=112?l-3?175(*-Mar. 11.0)-l?5X10-4

(<-Mar. 11.0)3 (5)

(for dates after 1958 Mar. 11).

Results

Normal values of aa at 5-day intervals, together with the observed precession, are given

in table 2.

Table 3 gives normal values of To, the time

of the equatorial crossing of the satellite from

south to north, at intervals of 25 revolutions.

The numbering of the revolutions is the same

as in Jacchia U%8a); the first crossing in the

I00O ISOO

NUMBER OF REVOLUTIONS

1957

Nov.

Dec.

1958

Jan.

Feb.

Mar.

Apr.

6.0

11.0

16.0

21.0

26.0

1.0

6.0

11.0

16.0

21.0

26.0

31.0

5.0

10.0

15.0

20.0

25.0

30.0

4.0

9.0

14.0

19.0

24.0

1.0

6.0

11.0

16.0

21.0

26.0

31.0

5.0

10.0

14 0

108? 2

95.0

81. 7

68.3

54. 8

41. 5

28.0

14 4

0.7

346. 9

333. 1

319. 1

305. 1

291.0

276.7

262.3

247.9

233. 3

218.6

203.9

189.0

174 0

158. 8

143.4

127.8

112. 1

96. 3

80.2

63.8

47. 1

30. 1

12. 7

358.2

-2?64/day

-2.65

-2.66

-2.67

-2.68

- 2. 70

-2.71

- 2. 73

— 2. 74

-2.76

- 2 . 78

-2.80

-2.82

— 2. 84

-2.86

-2.88

-2.90

— 2. 92

-2.94

-2.96

-2.99

-3.02

-3.05

-3.09

- 3 . 13

- 3. 16

-3.20

-3.25

-3.31

-3.37

-3.45

-3.57

FIGURE 1.—Rate of change of the orbital period as a function

of the number of revolutions («). For a conversion of n to

time, see table 2.

ao: Right ascension of the ascending node.

daajdU Nodal precession in degrees per day.

list (n=0) was actually the 16th crossing after

the launching of the satellite. Also given

are the nodal period, Pa, and the orbital

acceleration dPfdt, in seconds per day; these

quantities were obtained by numerical differentiation of TQ.

The last fifty revolutions are tabulated in

greater detail (at intervals of 5 revolutions)

in table 4. A plot of dP/dt against n, covering

the first 2100 revolutions, is given in figure 1.

Due to the slow motion of the perigee for the

particular inclination of this satellite, there is

no difference, within the tabular accuracy,

between the accelerations of the nodal and of

the anomalistic period.
ORBITAL RESULTS FOR SATELLITE 1957 01—JACCHIA

n

0

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

525

550

575

600

625

650

675

700

725

750

775

800

825

850

875

900

925

950

975

1000

1025

1050

1075

1100

1125

1150

1175

1200

1225

1250

1957

Nov.

1Q57 •L «7U f

Dec.

1958

Jan.

TABL E 3.—Orbital values

ta{U.T.)

4 39469

6. 19415

7. 99206

9. 78842

11. 58327

[13.37664]

[15. 16860]

[16.95918]

[18.74831]

[20.53596]

[22. 32214]

[24 10679]

25. 88993

27. 67149

29. 45146

1. 22989

3. 00673

4 78202

6. 55577

8. 32796

10. 09857

11. 86764

13. 63516

15. 40117

17. 16566

18. 92860

20. 68992

22. 44952

24 20735

25. 96338

27. 71753

29. 46977

31. 22002

1. 96825

3. 71436

5. 45850

7. 20068

8. 94090

[10.67911]

[12.41528]

[14 14928]

[15.88102]

17. 61056

19. 33801

21. 06338

22. 78666

24 50798

26. 22734

27. 94476

29. 66027

31. 37387

Pa

(0? 072009)

. 071947

. 071885

. 071824

. 071764

[. 071706]

[.071651]

[. 071594]

[. 071536]

[. 071477]

[. 071417]

[. 071356]

. 071294

. 071231

. 071168

.071105

.071042

.070981

.070919

.070856

. 070794

.070732

.070671

. 070610

. 070549

. 070486

.070419

. 070349

. 070278

. 070204

.070128

.070050

. 069970

. 069887

.069804

. 069726

. 069648

. 069569

[. 069488]

[. 069404]

[.069315]

[.069225]

. 069139

. 069056

. 068973

. 068892

. 068814

. 068735

. 068658

. 068582

. 068505

dP/dt

sec/day

-2.97

-2.92

[-2.84]

[-2.71]

[-2.66]

[-2.79]

[-2.85]

[-2.86]

[-2.93]

[-2.95]

[-3.05]

-3.07

-3.04

-3.06

-3.03

-2.99

-3.05

-3.06

-3.03

-3.00

-2.97

-2.96

-3.04

- 3. 17

-3.37

-3.48

-3.57

-3.68

-3.78

-3.90

- 4 05

- 4 12

-3.96

-3.84

-3.91

-3.97

[-4 08]

[-4 33]

[-4 51]

[-4 39]

- 4 20

-4 18

-4.15

— 3. 98

-3.92

-3.91

-3.84

-3.85

-3.90

n

1275

1300

1325

1350

1375

1400

1425

1450

1475

1500

1525

1550

1575

1600

1625

1650

1675

1700

1725

1750

1775

1800

1825

1850

1875

1900

1925

1950

1975

2000

2025

2050

2075

2100

2125

2150

2175

2200

2225

2250

2275

2300

2325

2350

r

1958

Feb.

1958

Mar.

1958

Apr.

FABLE 3.—Orbital values

ta(U.T.)

2. 08553

3. 79525

5. 50294

7. 20851

a 91190

10. 61300

12. 31171

14 00809

15. 70225

17. 39420

19. 08389

20. 77124

22. 45618

[24 13870] [.

[25. 81897] [.

[27. 49706] [.

[1. 17279] [.

2. 84608

4 51683

6. 18491

7. 85025

9. 51273

11. 17220

12. 82844

14 48130

16. 13073

17. 77672

19. 41925

21. 05828

22. 69375

24 32557

25. 95362

27. 57771

29. 19766

30. 81321

1. 42396

3. 02941

4 62906

6. 22238

7. 80859

9. 38686

10. 95582

12. 51318

14 05088

Pa

068428

068349

068266

068180

068091

067996

067901

067810

067722

067633

067541

067446

067349

067225]

067168]

067077]

066981]

066881

066777

066669

066557

066440

066315

066183

066046

065909

065771

065631

065490

065346

065198

065044

064882

064712

064529

064327

064106

063863

063596

063296

062957

062547

062008

06050

dP/dt

sec/day

3 94

— 4 10

- 4 28

- 4 42

- 4 67

- 4 85

- 4 75

- 4 54

- 4 50

-4 61

-4 80

-4 97

-4 94

[-4 62]

[-4 52]

[-4 80]

[-5.06]

-5.27

-5.50

-5.69

-5.92

-6.28

-6.72

-7.05

-7.14

-7.21

-7.28

-7.37

-7.51

-7.71

- & 00

-8.38

-8.82

-9.40

-10.28

-11.34

-12.48

-13.80

-15.3

-17.3

-20.2

-25.3

-38.2

-430. :

Values enclosed in brackets are uncertain because of scarcity of observations,

n —

taP odP/dt -

Numbe r of revolutions elapsed.

Tim e of ascending-node crossing.

Nodal period in days.

Orbital acceleration in seconds per day.
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 4.—The last 50 revolutions of Satellite 1957 01

n

2300

2305

2310

2315

2320

2325

2330

2335

2340

2345

2350

tQ(U.T.)

1958 April

10. 95582

11. 26830

11. 58032

11. 89184

12. 20281

12. 51318

12. 82285

13. 13175

13. 43970

13. 74637

14. 05088

Pa

0? 062547

. 062543

. 062355

. 062251

. 062136

. 062008

. 061859

.061692

. 061477

.06117

. 06050

dP/dt

sec./day

-25.3

-26.8

-28. 0

-30. 4

-33. 6

-38. 2

-43.0

-52. 0

-69. 0

-120. 0

-430. 0
The Descent of Satellite 1957 jSl

By Luigi G. Jacchia

The present paper

 x covers the final part of

the last revolution of Satellite 1957 /31. The

observations that are at the basis of this report

can be divided into three categories—those from

the northeastern United States (table 1), those

from ships at sea (table 2), and those from the

Caribbean Islands and the northern coast of

South America (table 3).

While reliable observations, including altazimuth estimates of sightings, had been received

at the Smithsonian Institution Astrophysical

Observatory from ships at sea, only isolated

and mostly descriptive observations were available from land areas. Therefore, at the suggestion of Dr. F. L. Whipple, Director of the

Observatory, the author undertook, at the end

of May 1958, a trip to the Caribbean area to

collect additional information. In the course

of this trip, he visited Antigua, Martinique,

Barbados, Trinidad, British Guiana, and

Surinam. The information collected is included in table 3.

Reduction of the observations

The observations in table 1, all made at nearly

the same time from localities on both sides of

the satellite path, provided the means for the

computation of an excellent normal point in the

trajectory. Its numerical values were:

Longitude: 74?00 west

Latitude: 41?40 north

Height: 101 km above sea level

Time: lh45m258 U.T. (14 April 1958)

A set of orbital trajectories was computed by

numerical integration starting from this normal

point, with different initial radial-velocity conditions and different (constant) drag parameters, with the ARDC model atmosphere

1

 Essentially an extension of the preceding paper.

(Minzner and Ripley, 1956), to compute the

drag at these low altitudes. For simplicity,

these integrations were performed with polar

coordinates used in the orbital plane. One of

these integrations, whose results are given in

table 4, and represented graphically in figure 1,

fitted the observations in the Caribbean area

within their estimated errors and was chosen

as representing the most probable trajectory of

the satellite in its final plunge.

The initial conditions and the drag parameter

used in this integration were:

Orbital inclination=65?29

Total velocity=7.737 X105 cm/sec

Radial velocity (dr/dt) = — 1.28X103 cm/sec

CDA/m=K=0.0n (e.g.)

In the last quantity, CD is the drag coefficient;

A is the presentation area of the satellite; and

m is its mass. The atmosphere was assumed

to be rotating solidly with the earth and to

have the same density at the same height above

sea level, irrespective of latitude. The earth's

oblateness and the precessional effects on the

orbital plane were taken into account.

In the other experimental trajectories, radial

velocities up to — 2X103 cm/sec and values of K

between 0.02 and 1.0 were tried. As should be

evident, the effect of such different conditions

on the trajectory itself is to vary to a considerable extent the position of the end point, although

the subsatellite track is not greatly affected by

the changes. The assumption of a constant drag

parameter is, of course, a little unrealistic, since

the drag coefficient CD must have been decreasing considerably in the course of the trajectory

with the formation of an air cap, while the areamass ratio A/m must have been increasing due

to the continuous shedding of molten metal.

Since, however, these two variations must have

compensated to a certain extent, the assumption
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Observations over northeastern United States

Station

Millbrook,

New York

New Haven,

Connecticut

Bryn Athyn,

Pennsylvania

Merrow,

Connecticut

(Mr. R. D.

House)

Pittsford,

New York

(Mr. R. E.

Jenkins)

Longitude

0

73

73

75

72

77

1 n

37 27

00 30

04

19.0

37 41

Latitude

°

+ 41

+ 41

+ 40

+ 41

+ 43

' "

51 30

12 35

08

49.4

15 03

(U.T.)

14 April

1958

h m s

1 45 08

1 45 35

1 46 04

1 45 47

1 45 21

1 45 46

Moonwatch observers at

a

b m

5 05

10 06

12 10

17 15

14 39

5

+ 45

0

+ 23 25

+ 66

+ 13 57

Isolated Observers

1 46

1 46 05:

1 46 25:

14 24

14:

+ 20

-10 :

Z

180°

NW

W

s

h

40°

25°

45

15

Remarks

Magnitude+1. Tail of tiny

particles about 1° long,

seen more or less all the

time; white.

Blue-white when first seen;

magnitude +1 ; at lh45m42*

got brighter (mag. —1)

and turned red.

Magnitude +3 to — 1; reddish; tail 10' to 20' long

Magnitude 1.5 at maximum

brightness; red; faint,

transparent tail 3° to 5°

long, visible only through

binoculars.

Magnitude +3.

is perhaps justified, especially since the agreement between the resulting trajectory and the

observations is quite good.

Phases of the trajectory and end point

There is general agreement among practically all

observers concerning the developments in the

appearance of the object. When seen in the

northeastern United States, the satellite already

had a faint tail in which sparklike particles

could be seen through binoculars. Its visual

magnitude, reduced to a standard distance of

100 km (absolute magnitude), was then about

+ 1.0 and the tail could be followed through

binoculars for 6 km behind the head, according

to the Merrow, Connecticut, observation.

After crossing over Long Island, the satellite

went unreported for about 5 minutes. When

sighted again by ships in the Caribbean, it was

at latitude 23° north and had become a spectacular sight. Its tail was 60 km long as seen

from Antigua, 80 km long at the latitude of

Martinique, and nearly 100 km long as seen

from Barbados and Trinidad. (Of all the Caribbean Islands, Barbados was the closest to the

satellite's path, about 120 km from the subsatellite track.) When the object was at the

latitude of Barbados (13° N), its head, according

to the best estimates, had an absolute magnitude of about — 7 or — 8, and the total light emitted by head and tail must have been close to

—9 or —10.

The color of the head was generally described

as white with tinges of blue or green, while the

tail was described as white, or white-yellowish,

near the head, and degrading to a deeper yellow

and orange—even red—toward the far end.

The "globules" described by all observers close

enough to see them were obviously drops of

molten metal shed by the object; their observed

splitting (A. H. C. Campbell, Barbados) confirms this explanation. Attention is called to

the small, semiperiodic light fluctuations

observed by Mr. Hart in Barbados, since they

might have some bearing on similar phenomena

observed in bright fireballs.

Several observers saw a bright flash when the

satellite reached 11° or 10° of north latitude

and spoke of a "shower of fragments" or "complete disintegration" of the object, but there is
DESCENT OF SATELLITE 1957 /31—JACCHIA

TABLE 2.—Observations from ships

Ship

1. Volvula

2. Maron

3. California

4. Mitra

5. Duivendijk

6. Mormick Sea

7. Eirmil

8. Ells

9. Mobil Brilliant

10. S. Africa Star

11. Regent Springbok

12. Regent Hawk

13. K. G. Lohse

14. Sunkirk

15. Triton

16. Omnium Freighter

17. Rio Atuel

Position of ship

Long.(W) Lat.

o r o /

65 00 +20 34

62 12 +19 54

60 10 +17 55

64

61

53

55

65

56

62

59

60

52

58

64

61

59

10

04

04

38

34

30

50

00

10

29

03

43

33

47

•t-azimuth (from North, positive to East)

+ 17

+ 17

+ 16

+ 16

+ 15

+ 15

+ 15

+ 14

+ 12

+ 12

+ 12

+ 12

+ 10

+ 10

46

21

46

20

53

30

20

30

37

28

25

06

37

22

14 April 1958

U.T.

1 53

1 49 (a.)

1 50

1

1

1

1

1

2

1

1

1

1

1

2

1

55

48 (a.)

52 (a.)

20

55

10

50

50

53

53 (a.)

51 (a.)

00

55

: ft—altitude above horizon.

Duratio

of obs.

2 min

3-4 min

> 3 min

2 min

2 min

3 min

30 sec

3 min

2-3 min

20-30 s«

4 min.

1 min

35 sec

6 min

Appearance*

Moving in direction 328° to 148°.

Moving NNW to SSE.

Appeared at 2=330°, disappeared at

2=150°.

Appeared in North, h =10°; h max 45° in

East; disappeared at 2=140°, h=10°.

Appeared at 2=350°, disappeared at

2=160°; h max=30°.

h=10°; disappeared at 2=210°, h=10°.

Green flame moving NW to SE.

Appeared at 2=30°, A=15°; disappeared

at 2=130°, h=10°.

At 2=246°, h=17°; disappeared at

2=205°, h=2° to 4°.

Moving from NNW to SSE, A=10°.

Passed overhead; appeared to disintegrate

at h=15°.

20-30 sec At 2=34°, A=16°; disappeared at 2=94°

Appeared at 2=290°, disappeared at

2=225°, h=3° to 5°.

Passed overhead.

Appeared at 2=55°, disappeared at

2=95°, h=5°.

h max=18°.

h max=20°; disappeared in SE at A=8°.

no unanimity on this point among the Trinidad

witnesses. It seems, however, that when the

satellite was disappearing from view in Trinidad,

much of the fiery tail had faded out, and only

an isolated object (the head, or what remained

of it) was still proceeding on its course. This

object was seen from British Guiana, at closer

range than from Trinidad, as a conspicuous but

not spectacular object, which faded out while

still well above the horizon in the northeast.

This rapid fading out in midair was reported by

all observers who managed to follow the course

of the object until that point, i.e., the three

observers in British Guiana, the witnesses on the

vessels K. G. Lohse and Rio Atuel, and two

observers in Trinidad, G. R. Robson and J.

Saunders. The disappearance was seen from

various directions and corresponded to a point at

about 57° of longitude west and 9° north latitude, at a height of some 40 km above sea level.

It appears quite probable that fragments

originating in the burst, including the object

seen from British Guiana, fell unseen into the

Atlantic Ocean along an arc of 100 km or so

on the subsatellite track, when their velocity

became too low to sustain the light-producing

mechanism. The "impact point" recorded in

table 4 should not have much more than academic significance, although it is expected that

no major fragment succeeded in traveling

much more than 1° beyond that point.

Remarks on Table 2

(1) Volvula: Long tail.

(2) Maron: Many short flames (very bright)

behind main body, followed by a long-pointed

tail (less lighted).

(4) Mitra: Light of flame was equal to that

of full moon, brilliant white with long tail of

sparks and smoke.

(7) Eirinil: Big green flame. Its head was

very intensively lighted and its tail faintly.

Along the tail there were smaller, bright heads.

(8) Ells: Several times brighter than Venus.

(9) Mobil Brilliant: When object was ahead

(Z=246°, h=l7°), its size was larger than that

of Venus, with a tail 11° long. Parts of the

head were continually breaking off and following the lower part of the tail, creating a

bulge below the tail. Each part was a minia-
Place Observer

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 3.—Observations frovi land

Longi- Duration

tude U.T. 14 April of

(W) Latitude 1958 sighting Appearance*

St. Thomas, 1. J. H. Jouett 64 57 +1 8 21 l^O-1 — l''45m 2 min Appeared at z=40° , disappeared at

V.I.

Antigua,

B.W.I.

2. H. Merrifield 61 48 17 10 l

h

45TM

z = 110°.

4 min Appeared at z=20°, h=7°; h max=

21°; disappeared at z=150°,/i=10°

(obstacles).

2 min h max=30°.

When at z=90°, h=21°; disappeared

at z=133°, very low.

Disappeared at z=155°, very low.

>1 min Appeared at z=10°, A=15°; h

max =18°; disappeared at 119°

behind clouds.

h max=20°.

20 sec

2.5 min Appeared at z=345°, h=5°; at

z=50°, h=22°; atz=110°,/i=21°;

disappeared at 145°, h = 6°, behind

trees.

35 sec Appeared at z=6°; at z=67°, h=

29°; disappeared at z = 128°,

h = 13°, behind clouds.

^2 min Appeared at z=10°, /i=12°; h

max=21°; disappeared at z=100°,

/»=15° (behind house).

h max=25°-30°.

Additional observers in Barbados: 15. J. Babb; 16. A. H. C. Campbell; 17. L. Hassell; 18. T. Stoute; 19.

C. Walcott.

3. W. P. Novinsky.

4. H. Laing

5. Y. Carty

Martinique, 6. R. de Reynal

F.W.I.

7. R. de la

Coste

8. Pere Tneon

9. Abb6 Beaubrun

St. Lucia, 10. Mrs. N.

B.W.I. Moffat

Barbados, 11. R. Parris

B.W.I.

12. A. Hart

13. D. Rufolo

14. F. C. Goddard

61

61

61

•61

61

60

59

59

59

59

48

48

48

04

04

55

37

35

34

27

17

17

17

14

14

13

13

13

13

13

10

10

10

36

36

49

15

04

04

06

Trinidad, 20. G. R. Rob- 61 27 10 19

B.W.I. son

21. J. Saunders 61 27 10 19

l

h

55m

22. E. H. F.

Beadon

61 39 10 39

30 sec Appeared at z=70° , fc=8° + 3°.

30 sec Appeared at z=70° , A=5°30'±30';

disappeared at z=100°, fc=l°-20.

Appeared in north; disappeared at

z=60°, h=Z°, behind mountains.

Observers in Trinidad, who gave rough estimates of h max: 23. A Hinkson (6°); 24. T. R. Ivison (18°); 25. N.

McClean (10°-15°); 26. W. Orebaugh (15°); 27. E. Skeen (10°-15°); 28. W. Scott (5°); 29. H. Sookoo (5°-7°);

30. D. St. Aubyn (10°-15°); 31. D. Taylor (5°-10°). Other observers: 32. J. Jackson, 33. R. G. Lovell; 34. M.

Tulloch.

Appeared in north, h=30° (unchecked), disappeared in NE, h—

15° (unchecked).

Appeared in north, h=15°; disappeared in ENE at A=10°.

*2=azimutb (from north, positive to east); A=altitude above horizon.

British

Guiana

35.

36.

37.

H

A.

E.

. Scott

Persaud

van

Lewin.

57

58

58

46

14

36

6

6

6

34

29

36
DESCENT OF SATELLITE 1957 £1—JACCHIA

ture replica of the larger, but lasted only a

very short time. The maximum number at

one time must have been about 15. A flash

was observed when the object appeared;

following the flash, the object emitted a series

of four other flashes that were bright but most

probably less bright than the first. After

closest approach, the tail became foreshortened,

and all the object became fainter until it gave

one last bright flash; a second afterwards it

disappeared, at an altitude of 2 to 4 degrees.

(10) S. Africa Star: Tail 20° to 30° long,

colored bright orange, silver-blue and green.

Crossed (astern of) vessel, appeared to be losing

height, and then disintegrated.

(11) Regent Springbok: Bright comet-like

shape with an associated firepath of smaller

glowing bodies. The colorful trail, many miles

long, was described as a peacock's tail, each

particle glowing through the spectrum from

white to deep blue in magnificent display,

finally fading away. Sinking in southern sky,

spearhead was seen to disintegrate, appearing

to do so at 15° above the horizon. Deck

and sea around were bathed in pale light

as mass crossed overhead.

(12) Regent Hawk: When closest, object had

tail about 15° long (another report says 20 to 25

times the diameter of the main body) and about

1°.5 broad, in which burning particles were

erupting. The main body appeared to have a

blue-white head, then a short dark space before

the glowing orange-yellow tail. Twenty-seven

separate particles were actually counted as

they appeared in the main plume. Each

followed the main body, and on leaving it

developed its own glowing tail. These particles

were visible for about one second, starting as a

bright white and rapidly darkening to a deep

red. The main body was estimated to be about

three or four times as bright as Venus at its

brightest—while the offshoots were each twice

the brightness of Sirius. One observer who saw

it in the early stages of its approach states that

it was round on top and bright blue-white, but

the lower hah* seemed to be flattened and was

more reddish in color, and "sparks" appeared

to be coming from the lower half.

(13) Karl Guntker Lohse: (account translated

and summarized from German): When it

appeared, the object had a comet-like shape

and was brighter than Jupiter. Toward the

end of its trajectory, the body brightened up

brilliantly and disintegrated into several glowing

parts. A few seconds later all had disappeared.

(14) Sunkirk: Was at first believed to be a

powerful white searchlight as from an aircraft

flying at low altitude. A trail of blue and red

fragments could be clearly seen in its wake.

Reflection from the object covered a wide area

in the sky and a section of the sea was also

illuminated by its brilliance.

(16) Omnium Freighter (in Port of Spain

harbor; account translated and summarized

from German): At first the object appeared

like a fireball, except that the white glowing tail

persisted longer. Tail was "25 to 30 nautical

miles long"; from white it degraded into dark

red and then into a black smoke trail, where a

few isolated sparks still glowed. Through

binoculars the body was seen to brighten up

several times. The bright light was observed

for about 35 seconds, the fainter for some 5

seconds, and the black smoke trail for about

15 seconds, "without seeing the end of it.''

(17) Rio Atuel: "Satellite appeared like enormous planet, white in color, very bright in head

and with reddish trail with spots of lesser intensity as if detached parts from the central

nucleus ended in secondary, bluish trail.

Disappeared at some 8 degrees altitude after 6

minutes of observation." (This description was

sent to the Argentine Association "Amigos de la

Astronomia" as a further elaboration on a

telegram, in which the vessel had announced

it had observed the "total disintegration of

Sputnik II.")

Remarks on Table 3

(The numbers in parentheses refer to the

observers listed in table 3.)

St. Thomas, V.I.—(1). "Gave appearance of

airplane at an altitude of 2500 ft. about 5

miles away. Object seemed dark with cherryred color at body, changing to almost

white at rear end or tail. The cherry-red

body was symmetrical, but became uneven

aft of the tail."

Antigua, B.W.I.—When first noticed in the

north-northeast it looked like a diifuse object,

without tail, with diameter not greater than

one-tenth of that of the moon (2). As it ap-
10 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

proached, a tail was perceived, which at

closest approach reached a length of 20° (2)

or 30° (3). The object was brightest in

front, where the color was yellowish (2).

Sparklike particles became detached and

were seen especially in the lower part of the

tail (2, 3). The central part of the tail

appeared of a blue-grayish color; shortly

after the main body a dead region (less

luminous) could be seen in the tail (2). The

end of the tail was mainly composed of

particles (2, 3) like flying embers (2). The

tail appeared streaked. The greatest intensity was in the upper and lower part, but

the lower was brighter (2). The total

brightness of the object, including the tail,

was comparable with that of the moon at

first quarter, but the brilliance of the head

was much greater than that of the moon

(2). The "sparks" were quite bright at

start, but would quickly burn out, in 8-10

seconds; no tail was seen attached to individual sparks (2).

Martinique, F.W.I.—Made its appearance

as a white (8, 9) or greenish (7) globe, onethird of the moon in diameter (7); the

apparent diameter may not have been real,

but due to optical illusion (6). At closest approach the main body was brighter than Venus, some 10 times brighter than Jupiter (6,

7), probably of magnitude —6. The globe

was sharply delimited; the tail was attached,

but a little separation could be discerned between body and tail (6, 7). At closest

approach the length of the tail was 20° (8, 9)

or 30° (6, 7) and its color was reddish (6, 7,

8, 9); none of the observers noticed sparks

or detached particles, nor any light

fluctuations.

St. Lucia, B.W.I.—"Appeared as a blazing

orange-yellow ball, about one-fourth the

size (in diameter) of the full moon, in the

wake of which streamed a long, wide, bright

trail of light and sparklike fragments, these

appearing as large as an average- or mediumsized star in the sky."

Barbados, B.W.I.—At closest approach the

object was cometlike, with a dazzlingly

brilliant head, at least as bright as Venus, and

a long tail which, as a whole, was several

magnitudes brighter than the head, although

TABLE

Latitude

41°. 40

24.0

22.0

20.0

18.0

16.0

14.0

12.0

10.0

9.0

8.6

4.—Most •probable trajectory of descent of

Satellite

Longitude

(W)

74°. 00

63. 20

62.23

61. 29

60.41

59.54

58.70

57.88

57. 12

56.8

56.6

1957 &1

Height

(km MSL)

101

88

86

83

80

76

71

63

50

37

Time

(U.T.)

l

b

45m25»

1 50 14

1 50 47

1 51 20

1 51 52

1 52 26

1 53 00

1 53 37

1 54 21

1 55 01

(Impact point)

less brilliant (12, 19). The head seemed to

have a definite diameter, a little less than

one-fifth of the diameter of the moon (12);

all witnesses asserted that it did not look

like a point source of light (16). Its color

was described as white with a bluish (12, 16,

19) or blue-greenish (13) tinge. The tail

consisted of a bright, flamelike appendage,

white (19) or yellowish (13) in color, gradually

fading into a more reddish tinge at greater

distance from the head (19). Numerous

brilliant particles or globules were observed

in the tail, especially in the lower part (all

observers); their color was white at the beginning, but became reddish as they were left

behind, falling, and faded out (12, 13, 19).

Each of the globules was followed by a small

tail, simulating the shape of the main body

(all observers); there were as many as 12 to

18 such "offspring" visible at any time, and

some of them subdivided forming two little

cometlike objects (16). The length of the

flamelike tail was estimated at 10° (13) or

20° (12, 19); some dying-out offspring could

be traced as far as 40° from the head. Small

fluctuations in brightness were observed in

the head, every three seconds or so (12).

Most observers saw the object disappear low

in the southeast, behind a cloud bank, which

became illuminated at the edges. Mr. Hart

(12) noticed flashes behind the cloud; there

was at least one big flareup after the body

disappeared from sight, like a distant flash

of lightning. Mr. Walcott (19), who observed the phenomenon from a point several

miles away, saw the tail fade out before
DESCENT OF SATELLITE 1957 /31—JACCHIA 11

^GEORGETOWN

PARAMARIB

1

 /' FRENCH

IGUIANA

68°

62° 60" 58*

WEST LONGITUDE

FIGURE 1.—End path of Satellite 1957 /Si in the Caribbean area. Times of arrival and heights above sea

level, in kilometers, are marked on the subsatellite track. The black triangles represent localities

from which quantitative positional observations were given; qualitative observations came only from

the localities marked with open triangles. (Chart reproduced through courtesy of Sky and Telescope.)

the object vanished and then saw five or six

red pieces fall toward the horizon (hidden by

a distant ridge).

Trinidad, B.W.I.—When it appeared in the

north-northeast, it was not brighter than an

average bright star (30); it rapidly increased

in brightness as it approached, and a tail

became visible. When it was in the eastnortheast it was brighter than any star or

planet (all observers agree). The head was

variously described as "metallic blue-white"

(30), "vivid intense green" (22) or "whitish

as a welding flame" (33). The tail was

definitely redder than the body (practically

all observers), speckled with starlike sparks,

at least 10° long (25); some estimated the
12 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

maximum length of the tail at 50° (30) or

even 60° (29). Some observers (23, 24)

mention a definite "burst" at the end of the

phenomenon; in other cases it is difficult to

decide whether what was described as a

burst resulting in a shower of fragments

(27, 31) was really an explosion or simply the

coming into view of the tail with the sparklike objects in it. Mr. T. R. Ivison (24),

who was watching the object together with

Mr. W. Orebaugh (26), American Consul

General in Trinidad, saw it give out a big

white flare, like a magnesium flash, after

which it quickly disappeared. Mr. Orebaugh

did not mention the flash. Mr. Hinkson

(23), a newspaperman who saw the phenomenon from the Piarco Airport, also saw an

explosion some 3 seconds before he saw the

object disappear. It is remarkable, however,

that other observers who had an especially

good view of the phenomenon did not

mention any burst. Two observers (30, 34)

reported that toward the end the fiery tail

faded and only the "leader" of the procession

of fragments remained visible, only to

disappear shortly afterwards.

British Guiana—A. Persaud (36) saw the object

at the Atkinson Airport and describes it as

brighter than any star or planet, but not

spectacular; it had a tail 1° long and seemed

to fly parallel to the horizon, from north

(where it was first sighted) to northeast.

Before it disappeared, in the east-northeast

direction, it seemed to drop closer to the

horizon. It disappeared quite suddenly while

still high (10°, checked by the writer) above

the horizon. The color was described as that

of an electric bulb. No sparks or fragments

were visible during the entire course. H.

Scott (35), from a place 20 miles east of

Georgetown, describes the object as "white

like a white electric light and bright as a

bright star." He saw no bursts or fragments.

It was like a plane traveling, descending very

fast, to about 15° (unchecked) above the

horizon, where it disappeared. Steady white

light all the time. Mr. E. van Lewin (37)

from a boat on the Essequibo River near

Bartica describes the color as reddish and

saw it "grow smaller until it diminished from

view."

Acknowledgments

It is a pleasure to record the enthusiastic

help received by the author from the U.S.

consular authorities in the various places

visited by him to collect information on the

demise of Sputnik II. By giving advance

publicity to his arrival and arranging for eyewitness interviews at the U.S. Information

Libraries or in other suitable locales, they

enormously facilitated his task. In particular,

the author extends his thanks to Mr. Walter

Orebaugh, American consul general in Trinidad; Mr. Jesse M. MacKnight, American

consul in Surinam; Mr. A. John Cope, Jr.,

American consul in British Guiana; Mr. Knox

Lamb, American consul, and Mr. Lindsay,

consular aid, in Barbados; and Mr. Hopkins,

American consul in Martinique. Invaluable

aid was given to the writer by the Barbados

Astronomical Society and its President, Dr.

Harry Bayley (it was shocking to receive the

news of Dr. Bayley's death, which occurred

on June 14, 1958, only a few days after the

writer's visit to the island). A particular

acknowledgment is extended to Pere Pinchon

of the Seminaire-College in Fort de France, for

his efforts to secure eyewitnesses in Martinique.

The author expresses his indebtedness to

Miss Jeannie R. B. Carmichael for her help in

computing many of the exacting hand integrations which led to the final trajectory.
Orbital Acceleration of Satellite 1958 02

By Luigi G. Jacchia and R. E. Briggs

The present study is based on an analysis of

1,450 Minitrack observations of Satellite 1958

/32 made between March 17 and September 16,

1958. The observations, kindly made available

by Project Vanguard at the Naval Research

Laboratory, were reduced by use of the subsatellite-point program (Jacchia, 1958d, 1958f),

with the assumption of the following values of

the elements:

^=34?255

g=1.1028 earth's equatorial radii

an =158? 53—3?0126 (t—1958 March 16.0)

w=121?58+4?4027 (*-1958 March 16.0)

(t in days, U.T.).

The resulting times of perigee passage, the

anomalistic period Pr, and the orbital acceleration dP/dt, are given in table 1; the last two

quantities are shown graphically in figure 1.

It appears that the acceleration varies

rhythmically, with cycles of the length of 24

to 37 days; the mean period is close to 30 days.

During each of these cycles the acceleration

varies over a range of 50 to 100 percent of its

mean value; the mean value itself seems to be

subject to slower fluctuations. Of particular

interest is the sharp rise in the acceleration during the second half of August, when it increased

by a factor of 4 in just over two weeks.

A periodicity of approximately 25 days was

barely discernible, according to Cornford (1958),

in the acceleration of Satellite 1957 /31; he

mentioned that "this had provoked speculation

that the tide-raising force of the moon may play

some part," although several other possible

sources for the irregularities were envisaged.

The acceleration curve (Jacchia, 195Sf) does

not bear out this periodicity; it seems rather to

point to 19-day cycles or to double oscillations

with a period of 37 days.

NUMBER OF REVOLUTIONS

500 I0O0 1500

APRIL MAY JUNE JULY AUGUST SEPT.

1958

FIGURE 1.—Time variation of anomalistic period and orbital

acceleration of Satellite 1958 02.

The fluctuations in Satellite 1958 02, which

appear to be too irregular to be explained by

tidal phenomena, seem to suggest semiregular

changes in the atmospheric density such as

could be caused by variable solar radiation—

for which one would expect to find occasional

periodicities of the order of 27 days and possibly

a correlation with geomagnetic activity. A

preliminary comparison of the observed accelerations with geomagnetic planetary indices

appeared rather inconclusive, although the

sharp acceleration maximum near September 1

13
14 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Orbital elements for Satellite 1958 02

No.

revolutions

n

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

1800

1850

1900

1950

Times of perigee

passage

r.(U.T.)

1958

Mar. 17. 61053

22. 27335

26. 93614

31. 59889

Apr. 5. 26160

9. 92425

14. 58685

19. 24941

23. 91193

28. 57441

M ay 3. 23682

7. 89917

12. 56146

17. 22370

21. 88590

26. 54806

31. 21017

June 4. 87222

9. 53420

14. 19614

18. 85805

23. 51993

28. 18176

July 2. 84355

7. 50530

12. 16702

16. 82872

21. 49038

26. 15202

30. 81362

Aug. 4. 47517

9. 13666

13. 79810

18. 45950

23. 12083

27. 78208

Sept. 1. 44322

6. 10424

10. 76515

15. 42597

Anomalistic

period

Pr (days)

040932561

554

546

536

525

516

508

500

489

476

464

453

444

436

427

416

403

392

385

379

371

362

354

347

342

336

330

324

315

304

293

284

273

258

239

216

193

173

0. 0932157

Orbital

acceleration

dP/dt

(days/day)

- 1 . 7X10-7

-1.6

-2.7

- 2 . 1

-1.7

-1.7

-1.6

- 3. 1

-2.6

-2.6

- 2 . 1

-1.7

-1.7

- 2 . 1

-2.6

- 3 . 1

-1.6

-1.3

-1.2

-2.2

-1.7

-1.7

-1.3

-0.8

-1.8

-0.8

-1.7

- 2 . 1

-2.6

- 2 . 1

-1.6

- 3. 1

-3.3

-4.8

-5.3

-4.7

-3.9
ACCELERATION OF SATELLITE 1958 /32—JACCHIA AND BRIGGS 15

followed two weeks of strong geomagnetic

activity.

It must be emphasized that the accelerations

determined from positional observations of

artificial satellites cannot be compared, day by

day, with geomagnetic or other phenomena,

since they are the second-time derivatives of the

observed function. At best, the computed

accelerations will always be a smoothed-out

version of the true accelerations, in which each

individual value is the average of a few days.

We may add that the synodic period of

precession of the line of apsides for Satellite

1958 j82 amounts to approximately 9,000 revolutions, so that any effect arising from the motion

of the perigee from night to day and vice-versa

should give origin to fluctuations with a periodicity of this order. This is clearly not true in

the present case; in particular, the rapid rise of

the acceleration during the second half of August occurred at a time when the perigee was

not far from the subsolar point. In a previous

report (Jacchia, 1958c), a value log p= —15.45

(g/cm3) was computed for the atmospheric

density at perigee height (656 km) under the

assumption of an acceleration of — 1.95X10"7

days per day. The present analysis shows that

this value was quite close to the average acceleration in the interval from March to August,

1958. Since, however, the acceleration has

shown variations from — 1.1X10"7 to — 5.3 X

10~7 days per day, the computed atmospheric

density must be considered to be variable

between log p= —15.0 and log p= —15.7.

A Flashing Satellite for Geodetic Studies

By Charles A. Whitney and George Veis

This paper is intended to provide basic

recommendations for an engineering study and

design of a flashing geodetic satellite. The

study assumes a weight limit of approximately

100 lbs., and a launching time within the near

future. On this assumption we have attempted

to frame our recommendations as a compromise

between what the geodesist would like as an

ideal and what the engineer can reasonably be

expected to produce within a year. For this

reason, we do not discuss the long-range aspects

of geodesy or the additional goals that might be

attained with a more generous weight limit.

In the first section we outline the aims, the

precision required, and the techniques of the

geodetic applications of a close satellite; in

the second section, we discuss how these factors

influence the choice of a satellite orbit; and in

the third, we discuss various problems of

design and operation. The last section provides

a summary of recommendations.

The geodetic aspects

According to the Merriam-Webster unabridged

dictionary, geodesy is "that branch of applied

mathematics which determines the exact position of points and the figures and areas of large

portions of the earth's surface, or the shape

and size of the earth, and the variations of

terrestrial gravity."

A satellite provided with a mechanism producing flashes of short duration would be

highly useful for geodesy.

Accuracy required.—The existing geodetic

nets are based on optical (or radio) triangulation

for the planimetry and leveling for the altimetry. They are generally accurate to one

part in 100,000, which we shall denote as an

accuracy of 10~5. The Baltic Geodetic ring is

perhaps the most accurate existing net, with

an accuracy of about 10~6.

If triangulation techniques are employed to

connect these nets, the relative accuracy of the

connections will be of the same order as that

of the typical base line employed, i.e., 10"5.

Therefore, if the absolute accuracy (in meters)

of the connections is to equal that existing

within the nets, the triangulation scheme must

not involve connections over distances significantly greater than the sizes of the nets themselves. If this restriction is observed, triangulation techniques should provide a worldwide

geodetic net with an absolute accuracy of about

20 to 30 meters.

In principle, this restriction to tie distances

comparable to the length of the base line might

be overcome by direct radio-ranging techniques

combined with optical triangulation. However, the advantage of such a technique is

more apparent than real since, because of the

nature of the data, the need to evaluate the

propagation velocity and its variations, for

example, would probably reduce the relative

accuracy to that of the baseline employed,

i.e., 10"5.

We adopt the following requirement for the

accuracy of measurement of a geodetic satellite:

The angles observed must be measured to an

accuracy of 10~5 radians or 2 seconds of arc.

This will generally produce an ellipsoid of

uncertainty with a radius of about 15 meters.

For strength in the geometric solutions,

observations should be made so that the different lines of sight intersect at nearly right angles.

This will usually necessitate restricting observations to elevation angles greater than 30°

above the horizon.

On the other hand, since the velocity of the

satellite will be about 8 km/sec, to obtain a

positional accuracy of 15 meters we must have

an accuracy of relative timing better than

±0.002 second.

17
18 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Geodetic techniques.—Although the number of

possible detailed procedures of observing a

satellite for geodetic purposes is very great,

there are, in principle, three distinct methods

of approaching the problem.

First, a purely geometric procedure can be

employed for geodetic connection if the flashes

can be observed simultaneously from both of

the nets to be connected. Triangulation from

each net will give the position of the satellite

relative to that net and thus provide the desired

three-dimensional connection.

A second method involves fitting an orbit

through observations of a set of flashes from

one net and of a second set of flashes from

another net. We call this the orbital method

since we use the orbit as an extrapolation device.

Because of uncertainties in the gravitational

(and air density) parameters, the method could

be used only for flash sets separated by about

one orbital revolution or less.

With these two methods of geodetic connection we contrast a third type of analysis, a

purely dynamical method in which the coefficients of terms describing the gravitational

field are treated as unknowns.

Available data on satellite orbits have already

substantially improved the assumed parameters

of the earth's gravitational field, through application of what is essentially the dynamical

method in which the station positions are

treated as knowns. However, data of a rather

different nature, such as would be provided by

a flashing satellite, are requisites for an efficient program of station-fixing, or geodetic

connection.

Choice of the satellite orbit.—There are several

differences between the requirements of the

geometrical method and those of the orbital

method.

Perhaps the most fundamental difference is

that the orbital method requires millisecond

timing of the flashes, while the geometrical

method requires timing just accurate enough to

locate the celestial sphere at the time of observation (±0.05 second). The problem of timing

will be discussed later.

Orbital inclination.—The two methods have

similar requirements in regard to orbital inclinations. Geodetic connections will be needed

primarily in the west-east direction, because the

existing nets are generally more extensive in the

north-south direction and because transpolar

geodetic connections will only rarely be allowed

by visibility restrictions. Thus a polar orbit can

be ruled out for the general purposes of a geodetic satellite. Polar orbits may be useful in

the future, however, for special problems.

A satellite orbit at a height of 500 miles will

have a segment more than 30° above the horizon

for at least four hours per day from all latitudes

less than i-\-6°, where i is the orbital inclination.

This would suggest that the inclination could

be as much as 6° less than the greatest latitude

of the geodetic points to be employed. However, the geometric solutions are greatly

strengthened if the satellite does, indeed, pass

occasionally to the poleward of these high-latitude stations, or at least through their zeniths.

These considerations, and the geography of

northern Europe and Asia, suggest an orbital inclination between 50° and 60° for the first

satellites.

We do not feel it advisable (see p. 22) to aim

for the critical inclination itself, 64?3, in the

neighborhood of which the latitude of perigee

changes very slowly. Also, since the high-latitude regions of primary interest to us lie in the

northern hemisphere, it would seem advisable

to launch the first satellite during the fall season

to ensure maximum nighttime visibility in the

north during the early, and therefore perhaps

more reliable, months of operation.

It is important to note that the success of the

geometric method does not require observations

to be made in both the northern and southern

hemispheres. In other words, the satellite need

be observed only from the stations that are to

be connected geodetically.

Satellite height.—The height requirements of

these two methods differ in the following ways.

Ideally, the geometrical procedure would require that the flashes be bright enough to be

observed over ground distances of 1,500 to 2,000

miles in order that nets 2,000 miles apart can be

connected. Also, the satellite height must be

great enough so that, even at this distance, the

observed elevation angles will be sufficient (30°)

to provide strong trigonometric solutions.

These idealized requirements, which demand a

satellite height of approximately 2,000 miles, are

perhaps rather severe and could be reduced so
FLASHING SATELLITE FOR GEODETIC STUDIES—WHITNEY AND VEIS 19

that a height of 1,000 to 1,500 miles would

suffice.

Height requirements for the orbital method of

geodetic connection are more modest, since the

satellite need not be observed over such large

distances. The height would be chosen so as to

keep within tolerable limits the orbital perturbations that result from variations of atmospheric drag and gravitational departures from

an ellipsoidal field. So little is known about the

exact origin and nature of these smaller-scale

gravitational anomalies that it is difficult to predict the rate at which their effect will decrease

with height. Through an application of the

dynamical method of analysis, determination of

this rate will, indeed, be one of the ultimate objectives of observing a geodetic satellite. Thus,

considerations of gravitational perturbations do

not place a clear-cut minimum on the allowed

orbital height.

Atmospheric drag demands a more tangible

lower limit on height. For many reasons, the

orbital method will probably not be useful in

connecting observations separated in time by

more than one or two orbital revolutions. We

therefore adopt the limiting requirement that

the mean atmospheric drag should not displace

the satellite from its "undragged" position by

more than a few meters during one revolution.

This criterion will give an absolute lower limit

on perigee height for a satellite of geodetic value.

Analyses of satellite behavior indicate that

variations of atmospheric drag, when averaged

over 100 revolutions, generally are comparable

in magnitude to the mean drag itself. For

lack of evidence we may assume that, over

intervals of one revolution, the drag variations

are of the same magnitude.

The orbit itself will probably have an eccentricity slightly greater than 0.05. Apart from

launching difficulties, reasons are given below

(p. 22) for an eccentricity of this order. Therefore, we may invert the simplified expression

developed by Sterne (1958a) for inferring perigee

density from observed changes, and estimate

the expected displacements for various orbits.

We adopt the Smithsonian Interim Atmosphere

No. 2 (Sterne, Folkhart, and Schilling, 1958)

for values of average density and employ the

drag coefficient (6rx>=2) used in deriving this

model from satellite behavior. For satellites

of mean heights between 200 and 800 miles,

the following relation is accurate to within a

factor 2, which is adequate for present purposes:

«I—§3— 1016,

where SL is the displacement, in meters, produced along the orbit by the average atmospheric drag during one revolution; p is the

perigee density (gm/cm3); e is the orbital eccentricity; and A/m is the ratio of effective crosssectional area to mass of the satellite (cm2

/gm).

If we have a sateUite with e=0.05, m=50 kg,

.A=0.25 m2 and 5L<5, we find a maximum

tolerable perigee density of 2X10~15 gm/cm3.

This requires the perigee height to be greater

than 300 miles, although this figure is not

precise since the atmospheric models are rather

uncertain at these heights and the density

gradient is low because of the high temperature.

For comparison, the hypothetical satellite

considered above has a specific gravity (0.58)

about 10 percent less than that of the six-inch

Vanguard sphere (1958 /32), and an area-mass

ratio one-third as great, neglecting antennae.

Because of uncertainties and possible fluctuations of atmospheric density at 300 miles, it

would seem advisable to aim at a specific

gravity somewhat greater than unity, and a

minimum perigee height of 400 miles.

We conclude, therefore, that a moderately

dense satellite with a perigee height of 400 miles

and an apogee height of 1,000 miles would be

well suited to the geodetic problems. If the

launching vehicle and batteries permit, a more

desirable orbit might have a perigee of 500 to

600 miles and an apogee of 1,000 to 1,500 miles.

Operational problems and satellite design

The required accuracy of angular measurement

suggests the use of photographic techniques in

which the satellite position is measured relative

to a star background. Cataloged stellar positions are typically accurate to ±0?5 of arc,

which is entirely adequate.

Parallactic refraction.—Measurement relative

to a star background has the advantage that,

to the first order, atmospheric refraction is

eliminated since the starlight is also refracted.

However, there is a small effect, which might be

called parallactic refraction, arising from the
20 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

fact that the satellite is much closer than the

stars. This effect is most severe for low satellites observed near the horizon; for a satellite

with .#=500 km it amounts to 1T5 of arc when

the elevation angle is 30°. Since the effect

can probably be evaluated to within 10 percent

for each observation, the restriction to elevation

angles greater than the 30° that is required

by geometrical considerations is expected to

keep the uncertainties due to parallactic refraction well below timing and measuring errors.

Telescopic focal length.—The position of the

center of a circular image on a photographic

plate can, with care, be measured to one-fiftieth

of its diameter, or about one micron for most

photographic plates and telescopes of quality

adequate for providing satellite measures. In

principle, therefore, a telescopic focal length of

0.5 meter would be sufficiently great to provide

positions of the background stars to within

0?5 of arc. In practice, a one-meter focal

length will be more nearly compatible with

the required aperture.

Duration of flash.—Achieving an angular accuracy of 1 second of arc for the moving satellite image is not easy, for the satellite's apparent

angular motion can reach 3,000 seconds of arc

per second of time. If tbe time profile of the

light flash were square, the photographic image

would become elongated in proportion to the

duration of the flash, but it would remain

symmetric. Measurements could still attain

the required accuracy if the duration of the

flash were less than about 0.01 second.

However, a distortion of the satellite image

arises from the asymmetry of the light curve of

the flash pulses. The light curve shows a

rapid increase (a few microseconds) followed

by a sharp peak and an exponential decline.

If we represent the light curve by an instantaneous rise to peak intensity /„, followed by

a decline with a half-life T, in seconds, then the

instantaneous intensity I(t), at time t, following

the peak is given by the expression,

The nominal exposure time tn of a flash is

taken by convention to be that value of t for

which 7(0// P=l/3, so that <n«r.

In a typical camera-plate combination the

light from a point source is diffused over an

area with a diameter of some 20 microns, and

the instantaneous spatial profile of light intensity on the plate is well represented by a

Gaussian curve. If the photographic plate

responded linearly to light, the image could be

considered as the sum of a large number of

Gaussian curves, arrayed according to the satellite motion, along a line on the plate; the height

of the succeeding Gaussians would decrease

exponentially. This concept will be very nearly

true for images not heavily exposed.

The resulting image will show a well-defined

peak of density that will very nearly correspond to the onset of the flash. The observer

can be instructed to measure the position of

this peak. However, if the image is badly

overexposed, the image will saturate, and the

peak will become less well defined and will

move forward from the instant of onset.

Positional errors thus introduced by varying

degrees of image saturation will probably not

greatly exceed the distance moved by the light

during its nominal exposure time. Thus, these

errors can probably be eliminated by keeping

tn less than .001 second and properly diaphragming the observing instruments to avoid heavily

overexposing the image.

Atmospheric "tremble" will produce highfrequency angular displacement of the optical

image of the satellite. Although the amplitude

and frequency spectra of these "seeing excursions" are still under investigation, amplitudes

are known to exist as high as 1 second of arc

with frequencies greater than 10 to 100 per

second. For the stellar background, which will

be given exposures of several seconds, these

excursions will average to a rather small mean.

However, the image of the short flash can be

expected to show excursions as great as 1 second

of arc even at moderately high elevation angles.

The effects of "tremble," plus the fact that

the cost of measuring the photographic plates

increases essentially in proportion to the number

of plates, suggest that flashes should be arranged

in short bursts, or groups, to provide a number

of images on each exposed plate.

Brightness of flash.—The brightness of the

flash required for detectability by a given

telescope depends primarily on the aperture of

the telescope objective, the distance to the

satellite, the amount of atmospheric absorption,
FLASHING SATELLITE FOR GEODETIC STUDIES—WHITNEY AND VEIS 21

and the limiting image density considered

usable.

A telescopic aperture between 5 and 10

inches would seem optimum in view of the

light-gathering power, availability, cost, and

ease of installation. Speeds in the region of f/7

would give focal lengths adequate for the

desired accuracy of measurement.

The following specification seems to be

generally acceptable as a basis for computing

the necessary flash brightness: A one-second

exposure with a 10-inch aperture gives a workable image of a lOth-magnitude star of solar

color at the zenith if fast panchromatic film is

used. The xenon discharge tubes do, indeed,

have very nearly solar color, and we shall

assume the use of panchromatic plate of A.S.A.

speed 200.

A lOth-magnitude star as observed at the

zenith corresponds to a source of 2.0X10"4

candlepower at a distance of one kilometer and

with no absorption. With these quantitative

conditions, and the inverse-square law of

illumination, and the reciprocity law for photographic response, we can easily construct the

relation between R, the satellite distance in

miles; P, the output in candlepower per second

of the flash; A, the aperture of the telescope

in inches; and 5M, the atmospheric absorption

in magnitudes at the zenith distance of observation minus the absorption at the zenith. The

relation is given by the equation,

TABLE 1.—Visual atmospheric absorption

log P= 2 log R-2 log 5M-1.19.

We adopt the values given in table 1 for the

atmospheric absorption as a function of zenith

distance (Allen, 1957), noting that they apply

to a dry, clean atmosphere.

Adopting a reasonable value of 3 candlepower per watt for the efficiency of the discharge

tubes, and letting W be the required watt per

second rating of the flash unit, we obtain the

relation,

log W=2 log R—2 log A+0A8M—1.70.

We display the photometric results in table 2,

in which satellites at heights of 500 miles and

1,000 miles are treated. For telescopic apertures of 10 inches, the required value of W is

given for various values of D, the distance in

Zenith

distance

0°

20

30

40

50

60

70

Absorption

(mag.)

0.21

0.22

0.24

0.27

0.32

0.42

0.61

SM

(mag.)

0.00

0.01

0.03

0.06

0. 11

0.21

0.40

miles along the earth's surface to the subsatellite point. Angular elevations, h, are also

tabulated.

In examining these tables, we must keep

several points in mind. 1) Atmospheric absorption increases rapidly with decreasing

angular elevation when the elevation becomes

less than 40° to 50°. This reduces the efficiency

of the flash.

2) If, for reasons of reliability, the purely

geometric method of analysis is to be employed

for the first satellites, the project will more

nearly fulfill its geodetic purpose if stations

separated by at least 1,000 to 1,500 miles can

simultaneously observe the satellite at an

angular elevation greater than 30°. Thus, D

should be 750 miles for h—30°. This means

that the height must be 600 miles at some part

of the orbit.

3) Although table 2 is constructed for use

with a telescope of 10-inch aperture, the BakerNunn satellite-tracking cameras have an

aperture of 20 inches, while a much larger

number of cameras in the range 5 to 10 inches

could be available. These smaller cameras

have the advantage of being easily transportable

from one site to another.

4) Table 2 is based on general astronomical

experience and on a reasonable, though perhaps

conservative, value for the luminous efficiency

of the flash.

Although further engineering study and

testing may show that the power requirements

given in this table are larger than necessary,

the following comments will probably remain

valid: Values of connection distances D up to

680 miles are much more economically obtained

for a 500-mile height than for a 1,000-mile

height. However, the connection distance can-
22 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.»

TABLE 2a.—Required flash output

(H=600 miles)

Subsatellite

distance

(D)

0°

2

4

6

8

10

Omi

136

271

408

545

680

Angular

elevation

w

90°

72.5

57

45.5

36.5

29

Satellite

distance

(R)

500 mi

525

585

670

775

895

Energy

(watt/sec)

(W)

51

60

72

98

140

200

TABLE 2b.—Required flash output

(H=1,000 miles)

Subsatellite

distance

(D)

0°

2

4

6

8

10

12

14

Omi

136

271

408

545

680

815

950

Angular

elevation

(A)

90°

80

70

62

53

47

41

35

Satellite

distance

(«)

1000 mi

1015

1050

1105

1175

1260

1360

1475

Energy

(watt/sec)

(W)

200

210

220

260

300

350

420

510

not exceed 700 miles unless the satellite height

is greater than 500 miles. Increasing D to

1,000 miles by elevating the satellite to 1,000

miles requires an increase in the flash output

by a factor of nearly three.

On the whole, it seems probable that a flash

output of 200 to 300 watts per second would be

adequate.

The orbital eccentricity.—We have suggested

placing the satellite in an orbit with a perigee

height of at least 400 miles and an apogee

height of 1,000 miles. These are really minimal

figures and lead to a mean height of 700 miles

and an eccentricity of 0.06. If the orbital

inclination differs from 64?3 by at least 5°,

the latitude of perigee will vary between positive and negative values numerically equal to

the inclination with a total period of some three

months. At each observing station, the height

of the satellite will thus oscillate between 400

and 1,000 miles. Therefore, depending on the

equipment and specific geodetic problem at

hand, the observers may choose that observing

time at which the height is optimum. With

this configuration, the line-of-sight distance to

the satellite will vary from 400 miles (perigee

at zenith) to some 1,500 miles (apogee at

minimum allowable elevation angle). The

most "typical" line-of-sight distance will be

about 600 miles with an atmospheric attenuation by a factor 0.80.

Flash programming.—The flashes should be

grouped in bursts of four to six flashes. The

maximum duration of a burst is set by the

following considerations. The flashes will be

photographed with a telescope that tracks the

background stars at the sidereal rate. Background stars are catalogued accurately only

to the 9th or 10th magnitude; above this

brightness there is an average of 5 stars per

square degree. Thus, there is no need for a

background exposure going beyond 10th magnitude, which corresponds to 1 second with a

10-inch aperture. In fact, if the background

exposure is much greater than this, accuracy

may be lost because overexposure and errors of

tracking may produce enlargement of the stellar

images. Also, the satellite will cross a 5-degree

telescopic field in from 15 to 5 seconds; therefore, the burst must last less than 5 seconds.

Flexibility of telescopic angular field and

allowable background exposure would seem to

demand that the burst duration be less than

oue second. However, there is no apparent

reason for reducing the duration much below

one second.

The mean speed of the satellite will be about

250 miles per minute; therefore, a maximum

of 1.5 minutes or 375 miles may reasonably be

set on the interval between bursts. Provision

for a variable interval between flash bursts

would be advantageous. From the observers'

point of view a 1.5-minute interval would give

about three or four usable bursts per passage.

Clearly, the flash need not operate over

ground illuminated by the sun. However, a

simple sun sensor to cut off the flash when the

satellite itself is illuminated would be too

restrictive, since for a considerable portion of

the orbit, a satellite at 1,000-mile height will

be illuminated when ground conditions are

favorable for photography. Indeed, this is

just the portion of the orbit during which inert

satellites are now photographed.
FLASHING SATELLITE FOR GEODETIC STUDIES WHITNEY AND VEIS 23

We come finally to the intriguing question

of how to provide for timing the flashes and

what possibilities exist for ground control over

the programming.

We distinguish two types of flash-timing and

give them names, for convenience. "Geometric" timing indicates a system by which a

simple clock on the satellite activates the

bursts at times well enough determined so

that observers can photograph the flashes,

identify them for the purpose of simultaneous

photography from more than one station, and

determine the position of the celestial sphere

against which the satellite image is measured.

An accuracy of 0.05 second would be adequate.

"Orbital" timing, on the other hand, means

that the satellite carries a clock whose stability

and calibration are adequate for the determination of the difference in time between bursts,

with an error not to exceed 0.001 second. This

clock can be calibrated from the ground.

Operating the satellite.—The simplest and

hence the most reliable method for operating

the flashing satellite would equip the satellite

with a simple clock, whose sole function would

be geometric timing. The clock would operate

continually, day and night, and would weigh

only a few pounds.

In principle, the clock might be made stable

enough to allow orbital timing as well, although the requirements for weight and temperature control would be greater. The clock

could be calibrated by photoelectric timing

through a telescope, although not easily; the

data could be analyzed at the individual

observing sites or at a central agency, which

would publish an ephemeris of flashes. This

use of master timing stations would require

clock stability sufficient to keep clock errors

below 0.001 second per two-hour period, since

the time interval between clock checks would

be on the order of an orbital period.

The second possible method would include a

transmitter on the satellite to emit a radio

pulse at the time of the flash. If sufficiently

powerful, these pulses could be detected at

the observing sites. This technique does not

require extreme clock stability. In view of

its rewards, this use of an onboard transmitter

to provide orbital timing seems highly advisable. If the transmitter failed, the geo636-014—63 3

metric timing would still be available or

photoelectric timing could be attempted.

If power requirements dictate a weak pulse,

a stable clock could be put on board, and

several master timing stations with sensitive

receivers could be established to calibrate the

clock and prepare an ephemeris of flashes.

This second procedure could be elaborated

to enhance greatly the flexibility and efficiency

of power utilization. A clock stable to 0.001

or 0.002 second in two hours and operated

continually would cause a radio transmitter

to emit a pulse, or group of pulses, every 1.5

minutes. When commanded from one, or

two, or three master ground stations, a program would be initiated within the satellite to

produce a light flash with each radio pulse.

The light flashes would be emitted for 100

bursts, or for two and one-half hours, and then

stop, although the radio would continue. It

might be safer, in view of possible failure, to

use the command to suppress the flashes

rather than to activate them. By this scheme,

the observer could wait for favorable weather

and orbital configurations at the observing

sites.

One final point concerning the flash programming should be mentioned. If the times

of flashes are determined at master stations,

rather than at each observing site, it will be

necessary to provide a means of identifying

the flashes observed at the geodetic stations.

The identification of the burst will present no

difficulty if the bursts are separated by at

least a minute of time. Also, one or two

members of a burst might fail to register, due

to self-occultation or plate size. If the individual members of the burst are spaced at

stable, but unequal, time intervals, they could

be identified. For example, a simple pattern

of intervals such as 0.2, 0.4 second between the

members, would allow a unique determination

of the missing member if only three flashes

were observed.

Self-occultation could be eliminated at the

expense of slightly reduced efficiency, if the

satellite had two lamps that flashed simultaneously to illuminate opposite hemispheres.

Edgerton's suggestion (personal communication) of employing four lamps, two in each

side of the satellite, and letting each lamp
24 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

produce one member of the burst, would seem

an advantage.

Flash lifetime.—It is difficult to place exact

limits on the required lifetime of the flash. If

the flash operates continually, a minimum lifetime would be about three months, during which

the orbital node would have regressed sufficiently to provide visibility at all stations with

latitude less than the orbital inclination. This

suggests a program lifetime of some 2,000

hours. If, on the other hand, the flash is

allowed to operate only during revolutions in a

favorable orbit, the efficiency is greatly improved. A fairly complete coverage could

probably be obtained with 500 hours of program

time. These numbers should be considered

lower limits.

Summary

We may summarize our recommendations as

follows:

Launch the satellite during the fall months, for maximum visibility in the northern hemisphere. Provide

for the operation of the flash, either continuous or

intermittent, over a period of at least three months, to

allow a complete rotation of the orbit in space.

The best period of operation for the flash program is

difficult to estimate, but it should probably be on the

order of 500 hours if the command receiver is aboard.

If it is not aboard, the time will have to be at least three

months or somewhat more than 2,000 hours.

Design the satellite so that (1) a minimum number of

flashes is lost by self-occultations; (2) the area-mass

ratio is less than 1/20 (cm2/gm); and (3) the flash output

is in the range 200 to 300 watts per second.

Place the satellite in an orbit of inclination 55° ±5°

with a perigee height of 400 miles and an apogee height

of 1,000 miles.

In addition to the flash, provide the satellite with the

following equipment:

(1) A tracking beacon of Minitrack-type to allow

acquisition of the satellite and orbital prediction

immediately after launching.

(2) A clock stable enough to give millisecond accuracy over periods of one orbital revolution

(nominally, two hours).

(3) A clock-pulse transmitter to emit calibration

pulses every ten seconds.

(4) A mechanism to cause a burst of four programmed

flashes with every 10th clock pulse (i.e., every 100

seconds).

(5) A command receiver to suppress the flashing for

1,000 clock pulses (3 hours), and thus allow

power conservation.

We do not regard the millisecond clock

stability as an absolutely essential part of this

device, since provision can be made for timing

the clock pulses with radio receivers at the observing stations. Such stability would, however, greatly simplify the ground operation and

the analysis^of data, since it would mean that

the timing could be done at two master stations

and that analysis of the clock would be allocated

to one central group.

The command receiver to allow suppression

of the flash is an important part of the device

if dry cells are to supply the flash power, since

it would quarter the flash power required. Of

course, if engineering studies demonstrate the

feasibility of using solar energy or a radioactive

power supply, the command receiver would be

unnecessary.

Acknowledgments

This paper is based, in part, on extensive discussions with J. G. Baker, G. Van Biesbroeck,

R. J. Davis, H. E. Edgerton, L. G. Jacchia,

R. E. McCrosky. To Fred L. Whipple, Director of the Smithsonian Astrophysical Observatory, we are particularly grateful for thoughtful

criticism.
The Earth's Gravitational Potential as Derived

From Satellites 1957 §1 and 1958 02'

By Luigi G. Jacchia

An excellent set of 68 orbit determinations of

Satellite 1957 01 (Sputnik II), covering the

interval from Nov. 9, 1957 to Apr. 13, 1958,

has been published by the Royal Aircraft

Establishment, Farnsborough, England (Cornford, 1958). These data permit an accurate

interpolation of all orbital elements during the

lifetime of the satellite. Since the orbital

inclination of this satellite differs by 31° from

that of Satellite 1958 02 (Vanguard)—which,

thanks to its high perigee and continuous radio

performance, yields highly reliable values for

its secular perturbations—it was thought advisable to investigate the possibility of using the

two satellites for the independent determination of the second- and fourth-order coefficients

of the earth's gravitational potential.

The secular precession tl of the right ascension of the ascending node, when all distances

are expressed in units of the earth's equatorial

radius, can be written as follows:

Q——n*a~3/2p~

2

 cos i J-\—5 ( 77: sin2 i—'.

~^( 7 sin2z-4)l (1)

14 p J

Here a is the major axis of the orbit; n* is the

mean motion of a satellite revolving around a

point mass equal to the mass of the earth in

an orbit with a=l; p is the orbital parameter

[p=a(l— e2)]; i is the orbital inclination; Jand

K are, respectively, the coefficients of the

second and fourth harmonics in the earth's

gravitational potential. In the above expression, all terms containing J3

, K2 and higher

' Manuscript received Nov. 1, 1958.

powers of J and K, as well as their crossproducts, have been neglected. We have used

n*=6135?58/day.

For Satellite 1958 02 we have used the following mean elements, obtained by R. E.

Briggs from Naval Research Laboratory orbital data and corrected on the basis of Minitrack observations :

Epoch: *O=1958 June 18.858

Anomalistic

period: Px=0*0932379

Eccentricity: e=0.19023

Inclination: t=34?253—0°.00010(t—to)

Right ascension

of ascending

node: Q=25?976—3?0126(«—«0)

Argument of

perigee: «=179?2+4?4027(«—<0)

(t in days, U.T.).

No departure from linearity can be observed

in the motion of the node in the interval from

1958 March 17 to October 10. The estimated

error in fi is ±0?0002. From the anomalistic

period, using first-order perturbation theory,

we can compute a= 1.361527.

For Satellite 1957 01 we have interpolated

at 10-day intervals, from the data of Royal

Aircraft Establishment, the orbital elements

in table 1. The nodal precession derived from

the data in the last column is tabulated in

table 2, together with the individual values of

J computed from them, with if=1.0X10~5.

These values of J were computed only to test

the inner consistency of the data.

To obtain mean values of J and K over the

observed interval for Satellite 1957 01, we have

25
26 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Orbital elements for Satellite 1957 j3i interpolated from RAE data

t

(OhU.T.)

1957

Nov.

Dec.

6

16

26

6

16

26

1958

Jan.

Feb.

Mar.

Apr.

5

15

25

4

14

24

6

16

26

5

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

1.

a

14601

14236

13869

13494

13106

12694

12238

11721

11193

10676

10138

09547

08877

08085

07113

05750

e

0. 0981

0. 0953

0. 0926

0. 0897

0. 0867

0. 0835

0. 0800

0. 0759

0. 0719

0. 0678

0. 0635

0. 0588

0. 0534

0. 0470

0. 0392

0. 0284

i

65? 37

.36

.34

.33

.32

.31

.30

.29

.28

.27

.26

.25

.24

.22

.21

65.20

CO

59? 3

55.2

51. 1

47.0

42.9

38.7

34.5

30.3

25. 8

21.6

17.3

12.8

8.3

3.7

358.9

353.7

(108? 3)

81.69

54.93

27.95

0.69

333. 09

305. 11

276. 72

247. 91

218. 67

188. 97

158. 76

127. 96

96. 45

64.03

30. 41

integrated equation (1) between the time limits

tx and £2, and we thus obtain:

where

(2)

-

2

 cos idt=

Values of Ju j 2 , and f3 are given in table 2.

Integrating from ^=1957 Nov. 26.0 to *2=1958

March 16.0, we obtain (for Satellite 1957 £1):

-318?48=-l?9624X105

t

/

-4?91X104J

2

+6?10X104K (3)

For Satellite 1958 j82, on the other hand, we have

-3?0126 = — 1853?74J+536°<72—411°K (4)

Solving equations (3) and (4) simultaneously,

we obtain

J=0.0016244; (5)

In the interval from ti to U, the inclination of

Satellite 1957 /31 changed by — 0?12. Assuming that the perturbing force which caused

this change had an exactly equal effect on the

node, i.e., caused an extra precession of the node

AQ=— 0?12, we find for the result:

J=.OO16241; (6)

The values of K that have been derived from

gravity measurements and from different hypotheses concerning the departures of the geoid

from a rotational ellipsoid vary between

9.0 X10"6 and 11.4X10"6. If we wish to adopt

the higher of these two values, then we must

postulate an extra precession Afi=—0?47 in

the node of satellite 1957 /31; we then obtain

J=0.0016234; K= 11.4X10-°. (7)

The fact that the rate of change of the orbital

inclinations is negative shows that the force

responsible for it must be acting in a transverse

direction along the general eastward motion of

the satellite. Such a force would, if anything,

have caused a westward (negative) motion of

the nodes; it would seem then that the possibility of a positive excess in the nodal precession

due to this cause can be excluded. We can

thus consider the value of 6.9X10"6, obtained

under the assumption of AQ=0, as a lower limit

for K. It is reasonable to assume that the

actual value of K is somewhat smaller than

10X10""6. As to J, it is clear that, no matter

what final value is adopted for K, it must be

not far from 0.0016240, with an estimated error

of ±10X10~7.

The coefficients J and K are related to the

"flattening," a, of the spheroid, by the equations

=*~ P~\ «

2

+^ (8)

(9)

In these equations p=i?a>2/&e

2

, where R is the

equatorial radius of the earth, « is the earth's

angular velocity of rotation, and ke is the

"geocentric" gravitational constant; we have

adopted p= 0.00346149. The parameter K is

associated with the depression of the spheroid.

If we assume that the equipotential surface of

the earth is an exact spheroid, we have K=0 ;

estimates of K vary from zero to 6.8X10"7.
EARTH'S GRAVITATIONAL POTENTIAL, FROM SATELLITES 1957 £ 1 AND 1958 /3 2—JACCHIA 2 7

TABLE 2.—Observed values of nodal precession (from RAE data) and individual values of J for Satellite 1957 fil

t

(OhU.T.)

1957

Nov. 26

Dec. 6

16

26

1958

Jan. 5

15

25

Feb. 4

14

24

Mar. 6

16

h

(degrees per

day)

- 2 . 686

- 2 . 711

- 2 . 742

- 2 . 778

- 2 . 818

— 2. 860

- 2 . 902

- 2 . 946

— 2. 994

- 3 . 049

- 3 . 113

- 3. 191

U

(degrees)

-1652. 8

-1670. 9

—1690. 0

-1710. 5

-1733.7

-1760. 5

-1788. 5

-1816.6

-1846.6

-1880.3

-1919.6

-1967.6

u

(degrees)

-399

-405

-412

-420

-428

—438

—448

-458

-469

—482

-497

-516

U

(degrees)

+495

503

512

521

531

544

556

569

583

599

618

641

103 J

(K= 1.0X10-*)

1. 6275

1. 6248

1. 6249

1. 6265

1. 6278

1. 6270

1. 6250

1. 6242

1. 6238

1. 6240

1. 6242

1. 6243

For J=0.0016240 we obtain a= 1/298.26

under the assumption of /c=0, and a = 1/298.30

assuming K=6.8X10~ 7. The corresponding

values of K are 8.8X10"6 when K=0, and

.EL=11.2X10- 6 when K=6.8X10~ 7. Since, to

first-order, d(-)= 5 dJ and dK=6adJ, we

find that a change of +10X10"7 in J will

result in a change of —.09 in -> but of only

—2X10~8 in K; (that is, K as derived from

equation (9) is unaffected by any reasonable

error in J). We are thus led to the following

most probable values with their estimated

errors:

J=0.001624±. 000001

2£=9(±2)X10-6

l/a=298.28±.ll

K=3(±3)X10- 7.

The value of I/a given above is in essential

agreement with the values of 1/298.0 ±0.3 and

1/298.38±.07 derived by the U.S. Army Map

Service (O'Keefe, Hertz, and Marchant, 1958)

from Satellites 1958 Alpha and 1958 /32. It

should be noted, however, that this flattening

derived from dynamic data does not necessarily

have the same meaning as the flattening determined from geodetic measurements on the

earth's surface. As pointed out by G. Veis

(private communication), the "dynamic" flattening is the flattening of a fictitious, equipotential spheroid containing all the earth's

mass, which will give rise to the observed

gravitational potential. This fictitious spheroid does not necessarily coincide with the

spheroids obtained from geodetic observations,

which need special assumptions to correct for

the effect of masses left outside the geoid.

The Diurnal Effect in the Orbital

Acceleration of Satellite 1957

By Luigi G. Jacchia

A tabulation of the orbital acceleration of

Satellite 1957 /SI has been given in Jacchia

(1958c). At the time the report was written

there was still some uncertainty—at least in

the writer's mind—about the origin of the

fluctuations in the observed accelerations.

Since then observations (Jacchia and Briggs,

1958) of Satellite 1958 02 (Vanguard) have

made it clear that the fluctuations are not

caused by a change in the effective presentation

area of the satellite, but are of atmospheric

origin. It seems appropriate, therefore, to reappraise the results for Satellite 1957 /31 in

this light. In particular, it may be of some

interest to check the acceleration curve for the

presence of a diurnal effect, which should manifest itself as the perigee changes its position with

respect to the sun.

Table 1 gives, at 10-day intervals, the geocentric angular distance \p between the perigee

and the sun, together with other basic quantities : the argument of perigee a>; the right ascension of the ascending node aa] the right ascension a,; and the declination 5T of the perigee.

The number of revolutions n corresponding to

each date has the same origin as in Jacchia

(1958c). The observed accelerations are plotted

in figure 1, at the bottom of which the angle \p

is graphed for comparison.

There seems to be little doubt about the

presence of the diurnal effect. Apart from

shorter, superimposed fluctuations, the departures of the observed acceleration from a smooth

curve are very closely in phase with the \p curve.

The effect is gradual, and there is no evidence of

a discontinuity at the time when the perigee

crosses from light into shadow, or from shadow

into light. The observed semiamplitude of

the diurnal effect is roughly 10 percent of the

acceleration. Since the corresponding semiamplitude in ^ was 60°, we can say that the

magnitude d of the diurnal effect can be expressed as

where A is the undisturbed acceleration and

\p—\f/0 is expressed in degrees; y^ is a reference

angle comprised between 90° and the zenith

distance of the horizon at perigee height—depending on the assumption concerning the effective height at which the diurnal effect operates.

During the time interval covered by table 1,

the height of the geometric horizon at perigee

height decreased from 104?6 to 103?3.

Number of revolution*

500 1000 1500 2000

MAR.

1958

APR.

1958

1

 Manuscript received Dec. 15.1958.

FIGURE 1.—Orbital acceleration of Satellite 1957/51, compared

with the geocentric angular distance $ between the sun and

perigee. A smooth curve has been drawn for reference

through the jagged acceleration curve (top). When ^ is

approximately 104° (bottom), perigee is at the divide

between light and shadow.

29
30 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Position of perigee with respect to sun for Satellite 1957 f}l

*(U.T.)

1957

Nov. 6. 0

16.0

26.0

Dec. 6. 0

16.0

26.0

1958

Jan. 5. 0

15. 0

25.0

Feb. 4 0

14.0

24.0

Mar. 6. 0

16.0

26.0

Apr. 5. 0

n

22

162

302

442

583

725

867

1012

1157

1303

1450

1598

1747

1898

2051

2206

01

59? 3

55.2

51. 1

47.0

42. 9

38.7

34 5

30. 3

25.8

21.6

17.3

12.8

8.3

3.7

358.9

353.7

108? 3

81.7

54. 9

28.0

0. 7

333. 1

305. 1

276.7

247.9

218. 7

189.0

158. 8

128.0

96.5

64. 0

30.4

143? 6

112.8

82.4

52. 2

22.0

351. 7

321.2

290. 5

259.4

228. 1

196.5

164. 3

131. 5

98. 1

63.5

27. 7

5,

+ 51? 4

48. 2

45.0

41. 6

38.2

34,6

31.0

27.3

23.3

19.5

15.7

11.6

7. 5

+ 3.4

-1.0

- 5 . 7

94? 6

122. 4

150. 2

154 5

126. 5

93. 6

63. 7

48. 8

62.8

94 5

132. 2

172. 3

145. 6

102. 9

59. 2

18. 3
An Empirical Formula for Ephemerides of

Satellites Near the End of Their Lifetime1

By Luigi G. Jacchia

In a post-mortem analysis of Satellite 1957

01, searching for a simple function that would

represent the period variation of the satellite

during its last few hundred revolutions, the

author found a relation between the period and

the number of remaining revolutions. This

turned out to be extremely useful in making

predictions during the last week of the lifetime

of Satellite 1958 SI. Since some interest has

been expressed concerning prediction techniques

at the end of a satellite's lifetime, we think we

are justified in presenting a brief discussion of

this formula, in spite of its purely empirical

character.

Let n be the number of revolutions of a satellite, counted from an arbitrary origin, and n*

be the number of the last revolution in such

count. Moreover, let P be the orbital period of

the satellite at the n

th revolution, and P* be a

critical period, constant for each satellite.

For Satellite 1957 01, the author found that

during the last 100 revolutions the relation

dlog (P-P*)

d log (n*— n) (1)

was almost rigorously satisfied when P* was

put equal to 0*06030.

For Satellite 1958 51 the same relation held

equally true, with P* again 0*06030. Since the

area/mass ratio was very similar for the two

satellites, k itself was nearly identical in the two

cases (0.406 for 1957 01, and 0.403 for 1958 51).

A plot of log (P—P*) against log (n*—n) for

both satellites is shown in figure 1.

It will be noticed that the critical period P*=

0*0603=86.8 minutes is just about the value of

i Manuscript received Dec. 15,1958.

636-014—63

the period that a satellite reaches in the course

of the last revolution, when it starts in its final

descent arc; the corresponding height in a circular orbit is 120 km above the equator. This

critical height should not be too different from

one satellite to the other, so it stands to reason

that P* can always be taken equal to 0*0603,

thus eliminating one unknown parameter in

equation (1).

From equation (1) we have, by integration

log (P-P*)=log b+k log(n*-n), (2)

or

P=P*+b(n-n*)k, (3)

where 6 is a constant.

obtain

Integrating again, we

-n)-j^-T (n*-n)*+1, (4)

where T is the time of ascending-node crossing

(or any other crossing from which the period

P is determined), and a is another constant.

A comparison of the observed instants of

ascending-node crossings with instants computed by equation (4) with suitable parameters

is given in tables 1 and 2, for both Satellites 1957

/31 and 1958 51. It will be seen that the residuals in both cases amount to only a moderate

number of seconds. The equations used are:

For Satellite 1957 01:

T=1958 Apr. 8.02102+0.0603(nfl-2250)

-0.0003275(2350—nB)

um

.

For Satellite 1958 51:

(5)

T=1958 Nov. 27.163164-0.0603(nD-2790)

—0.00032894 (2897-ft*,)1-

403

. (6)

31
32 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

-2.6

-2.8

«L

Q.

~-3.0

o

-3.2

-X A

1

—

—

O

o

o

1

1

o

•

1

1

o

o

•

1

.*

—

-

0.5 1.0

log(n*-n)

1.5 2.0

FIGURE 1.—Plot of log (P—P*) against log (n — «*) for Satellites 1957 01 (open circles) and 1958 51 (dots).

Counted from an arbitrary origin, nB and nD

represent the number of revolutions for each

of the two satellites. The quantity An tabulated for 1958 61 is the difference between nD

and the value of nB for which the first satellite

had the same period. As can be seen, the two

satellites were out of phase by 544 revolutions

at the start of the tabulation, but the difference

increased slowly, at a nearly constant rate.

This circumstance made it possible to have an

independent estimate of n*—i.e., of the time

of demise—for 1958 81. A similar procedure

could have been used, presumably, even if the

area/mass ratio of the second satellite had been

widely different; in that case, however, a

scaling factor in n would have been advisable

before computing An.

It should be clear that for a determination

of the parameters k and n* it is simpler to use

the observed periods P, rather than the crossing

times T. If we write log (n*—n)~z, and log

(P—P*)=y, and select three corresponding sets

of x and y (subscripts 1, 2, 3), the procedure is

to find by trial and error a value of n* that will

satisfy the relation,

(7)

x3—x2 y3—y2'

which derives from equation (1). Once n* is

found, we have

x3—xi k=- (8)

after which a and b can be determined from

two observed values of T.

It should be obvious that equation (1)

cannot be expected to hold throughout the

lifetime of a satellite, especially in the case of

those with highly eccentric orbits, so that the

formula does not appear suitable for the

computation of long-range life expectancies of

satellites. It does appear, however, that toward the end of a satellite's lifetime, when

formulas based on the orbital eccentricity are

bound to fail, this empirical formula may offer

a distinct advantage.

TABLE 1.—Comparison of the last revolutions of Satellite pi with ephemerides

2250

2275

2300

2310

2320

2330

2340

2345

2347

2348

2349

2350

ta obs.

Apr. 7. 80859

9. 38686

10. 95582

11. 58032

12. 20281

12. 82285

13. 43970

13. 74637

13. 86853

13. 92947

13. 99028

14 05088

O-C

. 00000

+. 00010

-. 00004

-. 00013

-. 00012

— . 00007

+.00002

. 00000

— . 00006

— . 00008

-. 00011

- . 00014

P

4063296

. 062957

.062547

. 062355

. 062136

. 061859

.061477

.06117

. 06099

. 06088

. 06072

(. 06050)

log (n*-nB)

(n* = 2350)

2. 0000

1. 8751

1. 6989

1. 6021

1. 4771

1. 3010

1. 0000

0. 6990

0. 4771

0. 3010

0. 0000

log (P-P*)

(P* = .O6O3)

7. 4765-10

7. 4244

7. 3516

7. 3128

7. 2639

7. 1928

7. 0708

6. 940

6.839

6. 763

6.623

[last revolution]
EMPIRICAL FORMULA FOR SATELLITE EPHEMERIDES—JACCHIA 33

TABLE 2.—Comparison of the last revolutions of Satellite SI with ephemerides

nD

2790

2800

2810

2820

2830

2840

2850

2860

2870

2880

2890

2895

2896

2897

ta obs.

Nov. 26. 93177

27. 56463

28. 19623

28. 82651

29. 45533

30. 08256

30. 70811

Dec. 1. 33186

1. 95348

2. 57255

3. 18814

3. 49371

3. 55456

3. 61530

O-C

. 00000

+. 00010

+ . 00016

+. 00019

+. 00015

+ . 00003

-.00009

-.00013

-.00016

-.00009

+.00003

-.00007

-.00007

+•00005

P

4063347

. 063224

. 063096

. 062957

. 062804

. 062641

. 062472

. 062272

. 062046

. 061756

. 061304

. 06090

. 06080

(. 06067)

log (n*-nD)

(n* = 2897)

2. 0294

1. 9868

1. 9395

1. 8865

1. 8261

1. 7559

1. 6721

1. 5682

1. 4314

1. 2304

0. 8451

0. 3010

0. 0000

log (P-P*)

(P*=.O6O3)

7. 4839-10

7. 4660

7. 4465

7. 4244

7. 3986

7. 3694

7. 3369

7. 2949

7. 2420

7. 1632

7.0017

6.778

6.690

[last revolution]

An

544

545

545

545

546

546

546

546.0

546.4

546.8

547.0

547.2

547.4

The Structure of the High Atmosphere:

I. Linear Models

By Charles A. Whitney

This paper is the first of two on the structure

of the terrestrial atmosphere above 100 km as

inferred from data on artificial satellites.

Orbital accelerations derived by analyses performed at the Smithsonian Astrophysical Observatory will be employed to infer atmospheric

densities near the perigee heights of the

satellites.

Despite remaining uncertainties in chargeaccumulation effects on the drag parameters

(Schilling and Whitney, 1958), computations

neglecting these effects give the most reliable

data available on densities above 100 km.

The principal aim of this first paper is the

presentation of a homogeneous set of mean

satellite data and the fitting of three simple

models to the inferred densities. These models,

being based on sectionally linear distributions

of molecular temperature, must not be considered as more than smoothing operators for

the satellite data. On the other hand, these

models provide an excellent basis for a study of

the range of models that will fit the satellite

data. They allow a demonstration of the

uncertainties inherent in the derivation of a

temperature distribution from data on the

density distribution.

The nature of the high atmosphere

Although I shall defer detailed discussions of

the physical nature of the high atmosphere to

later papers, several outstanding features must

be mentioned at the outset. The reader is

referred to the literature (Kuiper, 1952; 1954)

for more detailed discussions and further

references.

First, the chemical composition of the atmosphere above 80 km is not well known. Molecular oxygen undergoes dissociation somewhere

between 90 and 200 km; the details of the

transition evidently undergo significant temporal and geographic variations. Auroral spectra indicate the presence of molecular nitrogen

to 500 km at least; and although it must be

partially dissociated at such heights, the

working approximation that nitrogen is molecular is sufficient for discussions of mean

molecular weight.

Also, diffusional separation of molecules of

differing weights is significant above 120 to 150

km.

The variation of molecular weight resulting

from these effects makes it convenient to

employ a molecular temperature rather than

true temperature in establishing preliminary

models. A molecular temperature, Tm, may be

defined as

M (1)

where M and T are the true molecular weight

and temperature, and Mo is a reference molecular weight, conveniently taken as the

molecular weight at the ground or at the

greatest height for which it is well known.

Although radiative effects are important at

lower levels, it seems fairly certain that conduction plays the major role in heat balance

above 300 km (Spitzer, 1952). This statement

may not be valid during times of large ultraviolet excess in solar radiation. Chapman's

(1957) suggestion that the high atmosphere is

heated by conduction from the solar corona is

clearly of basic importance in this regard.

The concept of an exosphere has been

rediscussed by Spitzer (1952). The base of the

35
36 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

exosphere, or the critical level, is defined as that

level at which "a fraction 1/e of a group of very

fast particles moving upward . . . will experience no collision as they go to infinitely greater

heights." The atmosphere above the critical

level is essentially isothermal, and diffusional

separation must take place.

For neutral particles the number density at

the critical level is given by

ne= (2)

where <r is the typical atomic cross section

(6.8X10~16 cm2) and J f is the scale height.

Nature of the data

Orbital accelerations employed in this paper

for deriving densities have been obtained from

second differences of the times of perigee passage.

In practice it has been found necessary to

employ intervals of at least 30 to 50 orbital

revolutions in deriving accelerations, so the

data are averaged over several days.

The accelerations thus derived have shown,

for all satellites, semiperiodic fluctuation with

mean periods around 28 to 30 days. Although

I suggested (Whitney, 1958) that for 1958

Alpha this could be explained by systematic

variations in geometric effective cross section,

this interpretation is no longer valid. Jacchia

(1959e) has shown a remarkable correlation

between the variations experienced by Satellites

1958 02 (Vanguard) and 1958 51 (the rocket of

Sputnik III). Further examination removes

all doubt that this correlation is ubiquitous, all

satellites showing virtually simultaneous maxima and minima of acceleration. Although

these variations are only semiregular, they show

mean semiamplitudes of 15 percent to 50 percent; Satellite 1958 /S2, the highest satellite,

shows the greatest amplitude.

Priester (1958) has recently discovered a

surprisingly close positive correlation between

the daily averages of 20-cm radiation and the

variable acceleration of 1957 /32 as derived by

Jacchia for the interval Nov. 1,1957, to Feb. 10,

1958. Priester notes that Elwert's work (1956)

indicates that these atmospheric fluctuations

may be produced by 6 A to 30 A radiation from

coronal condensations.

The satellite data show an increase with

perigee height of the amplitude of the acceleration variations. In discussion, Jacchia and I

noted that this indicates a change of density

gradient over a rather large depth interval in

the atmosphere, since the mean density gradient

probably decreases with increasing height.

The discoveries of Jacchia and Priester will

be of tremendous importance in studying the

high atmosphere, since they imply the existence

of worldwide, synchronous variations of density. Jacchia has also shown that there is a

term in the acceleration of Satellite 1958 51

that can be correlated with the zenith distance

of the sun at the subperigee point.

At this time it seems best to proceed by

averaging out these periodic fluctuations and

deriving densities from mean accelerations over

several months. I have taken the data, insofar

as possible, from the spring and early summer

months of 1958 in order to eliminate possible

difficulties from longer-period terms.

Also, rocket data provide evidence for rather

considerable latitudinal, short-period or irregular fluctuations of atmospheric density at high

elevations. However, an earlier analysis has

shown that there is no evidence in the data for

Satellite 1958 Epsilon of any severe latitudinal

changes of mean density at Z=260 km. Therefore, I have combined data from different latitudes in the present study.

It is hoped that the mean models thus obtained will provide a basis for a "differential

corrections" study of the fluctuations observed.

The satellite data

Using Sterne's formula (1958c) and applying

also the major portion of his correction for the

assumed solid-body rotation of the atmosphere,

I have derived atmosphere densities from

accelerations. (I am indebted to Dr. Sterne

for providing me with this correction prior to

publication.) The data pertinent to the use of

these formulas are assembled in table 1.

For two reasons, I have taken all geometric

perigee heights, Zp, in km as measured from a

sphere of radius 6378 km. First, the motions

of the lines of apsides were, in most cases,

rather large in the intervals covered; and second,

the mean accelerations could not be defined so

well that a few kilometers would be significant.
STRUCTURE OF HIGH ATMOSPHERE: LINEAR MODELS—WHITNEY

TABLE 1.—Satellite data

37

Satellite

1957 a.1

1958 Alpha

1958 02

1958 Gamma

1958 Epsilon

Height of perigee

Geocentric

(km)

220

360

656

185

257

Geopotential

(km)

213

341

594

180

247

Semimajor

axis

a

1.086

1. 2276

1. 3619

1.211

1. 1883

Eccentricity

e

.0482

. 139

. 190

.153

. 125

Acceleration

P

(sec/day)

-2. 4

- 0 . 455

-0.017

-12.6*

-1.7 2

Mass-area

ratio

m/A

(gm/cm *)

24. 1

5.5

3.97

5.6

6.4

Length

(cm)

58

200

15

200

200

•Derived from curve communicated by J. Siry.

The quantity HP is the geopotential height of

perigee derived from the relation

(3)

Geopotential height, like molecular temperature, is an artifice used to facilitate the integration of the equation of hydrostatic equilibrium.

Since Sterne's formula for the density at

perigee actually gives pv^~in where 3^ is

the scale height just above perigee, it is necessary to have an approximate model before

densities can be derived. I have, therefore,

proceeded by successive approximations. In

table 2 are given the densities as derived from

scale heights given by models 4 and 5 described

below. The densities derived from the scale

heights of model 4 are the values used in constructing models 4 and 5, and represent the

results of iteration. Actually, the process had

converged after only one adjustment, since the

densities are insensitive to the model employed.

Thus, although model 5 is not rigorously selfconsistent, further adjustment would produce

only a trivial change. Table 2 also includes

the drag coefficient, CD, used in deriving the

densities. These values are consistent with

the densities of model 4 and their derivation is

discussed later (see p. 41).

The convergence of such an iteration process

can be speeded through a technique which was

apparently first employed at the Vanguard

Computing Center and reported by J. W. Siry

(1960; see also King-Hele, 1959).

TABLE 2.—Atmospheric densities from satellite

accelerations

Satellite

1958 Gamma

1957 a2

1958 Epsilon

1958 Alpha

1958 j82

180

213

247

341

594

CD

1. 2

1.9

1.9

2.0

2.0

logio p

Model 4

-11.86

-12.37

-13.00

-13.74

-15.47

(gm/cm*)

Model 5

-11.91

-12.40

-13.02

-13.75

-15.45

Sterne's formula for the satellite acceleration

may be written

In this form, f(e) is a known function of the

orbital elements whose relative error is much

less than the relative error of P, the observed

acceleration. The quantity p* is the true

value of density at perigee, and ^ f * is the true

value of scale height at perigee.

If Sterne's formula is employed with an

assumed scale height, ^ which is not the true

value, then the derived density pv will differ

from the true density. In fact, the relative

error of the derived perigee density is one-half

the relative error of the assumed scale height.

Suppose, however, we modify Sterne's formula and use it to derive the density at some

height above the perigee. Denote this height

by htf, where h is dimensionless. The essence

of the present technique is to choose k so that

the derived density falls nearly on the true
38 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

density curve regardless of moderate errors in

•5^f Siry has called h the isopycnotic height.

Explicitly, we first write Sterne's formula in

terms of the assumed scale height and the

derived density at perigee,

TABLE 3.—Percent errors of derived densities

Denote as p(HP-\-h3f) the derived density at

a height h^f above the perigee, whose height is

Hp. Then

Therefore, the density derived for the

height HP+hj>t may be written

On the other hand, the true density at the

assumed height HP-\-h^f is

The isopycnotic height, hP, is that value of h

for which

It may be found by equating the expressions

above, i.e., from

e ~

h

p_

Clearly h depends on Jf/jf*, the error in the

assumed value of , but fortunately the dependence is not strong.

For small errors, i.e.,

Jt : - l

we find

When the error of M" cannot be assumed small

we must proceed numerically and choose that

isopycnotic height that is appropriate to the

estimated uncertainty of ^?.

1.0

1.1

1.2

1. 3

1. 4

1. 5

h

0

0.0

- 4 . 6

- 8 . 7

-12. 3

-15. 5

-18. 3

0.43

0.0

-0. 5

-0. 5

-0. 2

+ 0. 4

+ 1.2

0.5

0.0

+ 0. 2

0.9

1.9

3.2

+ 4.8

1.0

0.0

+ 5.4

11.5

18.4

26. 1

+ 34.6

In a qualitative way the basis of the technique

is the following. If the assumed scale height is

too large, the derived density will be too small.

However, this density will be applied at too

great a height. When h—hp, these errors cancel.

In table 3 we illustrate the power of this technique by listing the percent error in derived density as a function of c^f/Jf * and the adopted

value of h.

On the basis of this table, it appears that the

appropriate choice, if the atmosphere were truly

exponential, would lie in the range

0.5>h>0A3,

the upper limit being favored when the uncertainty of $P is small.

Therefore, we conclude that when Sterne's

formula is written in the form

the derived densities will be virtually unaffected

by errors of up to 50 percent in the assumed

value of Jf?

An earlier linear model

Sterne, Folkart, and Schilling (1957) derived

three tentative model atmospheres from a

density at Z=200 km evaluated from the

acceleration of Satellite 1957 e*2. They adopted

the formula,

p(220)=4.5X10"13 gm cm"5,

and forced the temperatures and densities of

these models to fit the ARDC model (Minzner

and Ripley, 1956) at Z' = 80, 90, and 100 km,

by adjustment of the constant temperature
STRUCTURE OF HIGH ATMOSPHERE! LINEAR MODELS—WHITNEY 39

gradient above Z'. These models were designated Interim Atmospheres Nos. 1, 2, and 3,

respectively. The defining relations are:

Model 2.

^=-603.3+9.0167^ ,

log p=2.588-4.79 log Tn.

In these relations, and in those to follow,

logarithms are to the base 10, densities are in

gm/cm3, and heights are in kilometers. The

molecular temperature for this model is defined

by equation (1), and Mo=28.966.

The densities given by this model are plotted

against geometric height in figure 1. Considering that this model, which the authors preferred over models 1 and 3, was based on

data from a single satellite, it is remarkably

good even at 500 km. However, since the later

data seem to deviate systematically, I have

considered it worth while to construct further

models.

Models 4 and 5

I have extended the calculations of Sterne et al.

(1957) by adjusting the height Z'', at which the

-11.00

-12.00

7* -13.00

£

i uiB Sf-14.00

-15.00

-icnn

1 1 1 1 1

- \ \

VI 1958 GAMMA 2

V. 1957 a 2

- Vyl958 EPSILON

\S^I958 ALPHA

- \\^- Mode l

Model 4—-*V\V - W\

1958

1 i l i i

l i

-

2 —

Model 6

\ -

0 100 200 300 400 500 600 700 800

Geometric Height (km)

FIGURE 1.—Atmospheric densities given by models 2, 4, and

6, plotted against geometric height.

ARDC model is fitted, and L the temperature

gradient above Z', to produce a fit to the satellite

data of table 2 at heights of 260 and 656 km.

Explicitly the model fits, by construction, the

data at

Z'=88.4, log P=—8.29, Tm= 196.9.

The defining relations are:

Model 4-

Tn= -465.6+7.6410

log p=4.311—5.4712 log Tm.

The densities given by this model are plotted

in figure 1; they seem to be a significant improvement over those of model 2. Furthermore, the excellence of the over-all fit of model

4 to the data indicates that the assumption of a

constant temperature gradient above 100 km

is adequate.

In order to estimate the range in models

permitted by satellite data, I have constructed

a fifth linear model based on rather different

assumptions. Models 4 and 5 are in no way

limiting cases on acceptable models. However,

being based on independent boundary conditions at the bottom and on rather different

temperature assumptions, they might be considered as independent samples of acceptable

models. In this sense, the differences between

them give a reasonable, rather than an extreme,

estimate of the remaining uncertainties.

I have constructed this fifth model by forcing

the density through three points and the

temperature through one. For the lowest

point I took r=150°K, M= 28.57, log p=

—8.66 at #=9 0 km. These values are nearly

those given by Nicolet's tabulation (1958a).

The height of 90 km was chosen as the maximum height at which uncertainties in the dissociation and diffusion are insignificant.

The atmosphere was assumed to have constant, but different, temperature gradients

above and below 150 km. The dividing height

of 150 km was chosen arbitrarily as representing, roughly, an elevation at which the nature

of the atmosphere changes significantly (in

relation to diffusion and O2 dissociation). Also,

it is a convenient round number near the

lowest satellite perigee.
40 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The model is described by the following

formulas, whose derivation is standard (see

Sterne et al., 1957):

90<#<150.

Tm=150+60i7+L"(fl-150),

log p=log

»(150),

fl>150.

In these relations Lf and L" are the temperature gradients (°K/km) below and above 150

km, and Tm is related to the true temperature

through equation (1), with MO=28.57.

After several trials, a pair of L' and L" was

found that fit the required densities. These

lead to the defining relations:

Model 5.

log p= -0.56-3.723 log Tn J

log P=6.275-6.039log Tn

-H>150.

As a measure of the uncertainties within the

framework of this particular model, I have

evaluated the following partial derivatives for

heights of 260 and 656 km:

Taking A log p= ±0.1 as a reasonable upper

limit on the uncertainty in the basic data,

we see that the temperature gradients for this

particular model are reliable only to approximately ±0.5° K/km.

The actual uncertainties, however (see below),

are considerably greater.

Model 6

Considerable interest is attached to the

temperature gradient above 300 to 400 km.

Bates (1951) has shown that absorption of

ultraviolet solar radiation appears insufficient

to maintain a significant positive temperature

gradient above the F layer. If such a gradient

indeed exists, it will probably be explained by

conduction from above, as outlined by Chapman

(1957).

Unfortunately, the satellite data give only

two mean-density values above 300 km, so this

question probably remains an open one. As an

experiment I have constructed model 6 in the

following manner. From model 4, I derived

densities at Z=300 km leads to Tm=2527° K

(Mo=28.966) above this height. To fit the

model to lower levels I assumed a constant

temperature gradient and forced the temperature to T=150°K at H=90 km (Nicolet,

1958a). It was not necessary to assume a

density at heights other than Z=300 and

600 km.

The resulting model is described by the

following equations:

Model 6.

r90<#<287.

log p=1.274-4.411 log TmJ

Tm=2Y&K \n^2R7

logp=11.404-.0069591flj ^

The densities given by this model are graphed

in figure 1 and they seem to fit the data quite

well. Also, at H— 90 km, this model gives

log p=—8.32, in excellent accord with log

P= —8.31, given by the ARDC model.

Table 4 compares the densities and molecularscale temperatures given by models 4 and 6.

The results of this comparison are neither

surprising nor encouraging. Although the temTABLB 4.—Densities (log p) and temperatures (Tm)

given by linear models

Height

Z(km)

200

300

400

500

600

Model 4

Tm

1017

1727

2407

3080

3722

logp

-12. 14

-13. 40

-14. 19

-14.78

-15.22

Model 6

Tm

1192

2123

2123

2123

2123

logp

-12.30

-13.40

-14.02

-14.63

-15.22
STRUCTURE OF HIGH ATMOSPHERE: LINEAR MODELS—WHITNEY 41

peratures agree well at low heights and near

350 km, as they must, they differ considerably

at heights greater than 400 km. I would not

say that it is possible at present to choose

between these two nonphysical models.

The base of the exosphere

Equation (2) has been used to evaluate Ze, the

geometric height of the critical level, or base of

the exosphere. For model 2, Zc=630 km; for

model 4, Zc=500 km. If the numerical

coefficients of equation (2) are correct, the

value of Zc as given by model 4 is probably

accurate to 20 km for the mean atmosphere

during the early summer of 1958.

At Z=500 km, model 4 gives 7^=3080° K,

and model 6 gives TOT=2123° K. If, as seems

likely from Nicolet's (1958a) discussion and

further results to be presented in Paper II

(p. 43), we may assume the gas at this level

to be pure oxygen, then these molecular temperatures lead to true temperatures of 1700° K

and 1200° K, respectively.

These values are lower than 2250° K, the

lower limit derived by Nicolet for the required

escape of helium, but the discrepancy may not

be serious. Further, the present data do not

exclude the possibility of a rise to 2000° K

somewhat below 500 km, since these data

determine only the mean temperature and not

its detailed distribution.

The drag coefficient

Dr. R. M. L. Baker, Jr., of Aeronutronic

Systems, Inc., Glendale, Calif., kindly made

available, prior to publication, his formula for

the drag coefficient CD in the transition region.

The expression is of the form,

TABLB 5.—Transitional drag coefficient (Co)

— l ) exp (-<

where a is a dhnensionless atmospheric density;

CD(j=CD(<r=l); and C is related to the ratio

of mean free path to body size.

Employing Baker's formula and the atmosHeight

Z(km)

160

180

200

220

240

260

280

300

320

340

Drag coefficient for satellite of linear

dimension d (meters)

lm

0.99

1.30

1.60

1. 79

1.89

1.93

1.96

1.98

1.99

2.0

3m

0.92

.96

1.21

1.48

1.68

1.81

1.88

1.92

1.95

1.97

10m

0.92

.92

.92

1.04

1.26

1.48

1.65

1.77

1.75

1.90

30m

0.92

.92

.92

.92

.95

1.06

1.24

1.43

1.59

1.69

pheric densities of model 4, I have computed

CD for the transition region. I have adopted

the velocity v=8 km/sec for the satellites, and

MCT,=8000° K, where Mc is the dimensionless

molecular weight of the particles "emitted"

from the satellite and T, is the surface temperature of the satellite. Table 5 gives representative values of CD; Z is the geometric height of

the satellite, and d is its "typical" linear dimension.

Since the algebraic representation of CD is

rather arbitrary, the details of the transitional

CD are not too reliable. However, the heights

of the transition regions of various satellite

sizes are probably fairly accurate.

Baker's formula leaves little doubt that variations of CD from the conventional value of two,

employed heretofore in deriving satellite densities, are significant. There may, of course, be

an increase of CD due to plasma effects.

The possible implications of Baker's results

for analyses of satellite data are manifest. I

shall defer a discussion of some of these possibilities to later papers, and merely state that I

have used these values in computing the

densities for table 4. The changes introduced

by values of CD<i2 are important only for the

two lowest satellites considered in the present

analysis.

The Structure of the High Atmosphere:

II. A Conduction Model

By Charles A. Whitney

In the first paper of this series (Whitney,

1959), I summarized the values of atmospheric

densities as derived from satellite accelerations.

In an attempt to smooth the data, I constructed

three models with sectionally constant temperature gradients. Such models, although fitting

the data quite well and providing a useful

basis for discussion, are admittedly nonphysical.

Nicolet (1958b) has outlined several aspects

of physical models for the high atmosphere.

In this paper I shall construct a conduction

model based on his discussion.

Algebraic description of the conduction

model

The total heat flux F carried downward by

conduction across a geocentric sphere of radius

r is given by the equation

dr (1)

where BT1'

2

 is the coefficient of conduction.

On the assumption that radiative losses

may be neglected, i^is a constant, and equation

(1) may be integrated directly. It is convenient

to measure heights from a reference level whose

geocentric distance is r0. Let z be the geometric height measured from this level and To be

the temperature at 2=0. If Th is the temperature at z=h, then the temperature T at the

general level is given by the equation

7^3/2 f 3/2 ,,

(2)

Also, if r=ro-\-z, the flux per unit area, E,

follows from the relation

r0rh2

(3)

If diffusive equilibrium holds above 2=0,

and NQ(M) and N(M) are the number densities

of particles of molecular weight M(gm) at

height 2=0 and at the general level, respectively, we find the following approximate relation, valid to within several percent in N(M):

, N(M) , T losWM)=

~

lOST

-1.3029^0 ( g

where

_QM

Also <7o=980.7 cm sec-2, R is the radius of the

earth, and all distances are measured in centimeters. Logarithms are to the base 10.

The density p at any level is then derived

from the sum of the partial densities,

P=2MN(M), (5)

and the mean molecular weight M follows from

the relation

M= p (6)

Construction of the model

I have assumed, following Nicolet's suggestion, that the important atmospheric constituents are N2 and O. Also, I have adopted

the value T0—5Q0° at 2=0, where z is measured

from a geometric height of 140 km, and

n=

A^(O)^5.60X101 0 ^ 1

AT(N2) 4.15X1010 at 2=0.

43
44 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Atmospheric properties (model no. 7)

Geometric

height z

(km)

140

175

200

250

300

350

400

450

500

600

T(°K)

560

677

754

898

1026

1146

1258

1364

1465

1653

2V(O) (cm-')

1. 39X10"

4. 01X1010

1. 94X1010

5. 58X109

1. 94X10°

7. 80X108

3. 44X108

1. 85X108

8. 30X107

2. 47X107

2V(N2) (cm-3)

1. 03X1011

1. 40X1010

4. 26X109

5. 47X108

9. 54X10'

2. 10X107

5. 36X109

1. 92X109

5. 00X105

6. 55X10*

log p

(gm cm"3)

-11.07

-11 . 76

-12. 15

-12. 76

-13. 25

-13. 66

-14. 03

-14. 30

-14. 65

-15. 18

21. 10

19. 11

18. 16

17.07

16. 56

16.31

16. 18

16. 12

16.07

16.03

(erg cm"1 sec"1)

2. 29X10"7

6. 62X10-8

3. 20X10-8

9. 21X10-*

3. 21X 10-9

1. 29X10-»

5. 67X10-10

2. 71X10"10

1. 37X10-10

4. 08X10"11

The value of TA(A=360 km) and iV0(O)

were adjusted to fit the satellite data, and the

results were the following: At 2=0,

iVo(O)=1.39X10", iV0(N2) = 1.03X10".

These concentrations are 2.48 times those suggested by Nicolet. At ^=360 km (a real

height of 500 km above the earth),

#=0.27 erg cm-3,

F= 1.609 X1018 erg/sec.

Table 1 lists some physical properties of this

atmosphere as functions of height Z from the

earth's surface.

The mean molecular weight ju is defined by

M=M/1.673X10~24. The column headed 0t

gives the rate of radiation (erg cm"3 sec"1)

from atomic oxygen according to Nicolet

(1958):

'=1.65X10-18iV(O). (7)

The densities of model 7 are plotted against

geometric height in figure 1 for comparison

with the satellite data. The over-all fit to the

data is quite adequate, and I conclude that the

form of the density profile predicted by the

conductive atmosphere is not inconsistent with

present data.

The temperature of model 7 is plotted as the

solid smooth curve in figure 2. The temperature curves given by models 4 and 6 (Whitney,

1959), included for comparison, were derived

\ 1958 GAMMA

\ 1957 CM

Xj958 EPSILON

>I958 ALPHA

MODEL 7 ^" v

• i i i i i

-

-

-

O 958)92

i i

-10

- I I -

-12 -

§-13

E

-15 -

— i r

100 200 300 400 500 600 700 800 900

FIGURE 1.—Densities of model 7 plotted against geometric

height.

from the molecular-scale temperature by use

of the mean molecular weights of model 7

(table 1). The spread of temperature above

250 km is rather large, considering how well

these models fit the satellite data; this spread

results from the coarseness of the data and

the nature of the equation of hydrostatic

equilibrium.

Domain of failure of the conductive

atmosphere

The temperature equation (2) is based on

the assumption that the global conductive flux

F is independent of height. If there are radiative sources and sinks of energy, the transfer

equation may be written:
br

STRUCTURE OF HIGH ATMOSPHERE! CONDUCTION MODEL—WHITNEY

4), (8)

45

where £% and A are the rates of emission and

absorption per unit volume per second. In

general, the forms of A and & will require

numerical integration of equation (8).

Introducing the explicit form for F and performing the indicated differentiation, we see

that equation (8) becomes

?. ±_dT /dT\-ld?T

r^2T dr^Kdr) dr2

F(r)

It is convenient to put this equation into

dimensionless form by expressing the temperature in terms of the temperature, To, at a

reference level of geocentric distance r0.

Writing

r=xr0,

and introducing

T=rTOt

r0 dT dr

y

~¥0 Wdx

Z =

(10a)

(10b)

(Ha)

(lib)

I find the following pair of first-order equations

in y, z and x:

.^.4. dy 4*ro*(&-A)

~2z~r dx 47 '

y=2z dz

dx

(12)

(13)

This pair can be integrated downward from

the following boundary conditions:

At x=l(r=r0): Z2=T=1, y=y0.

The value of y0 follows from the physical

conditions at r0 through the following relation,

Jj1 faj. JJ27 3/2y (14)

where Fo is the assumed global flux at the

reference level. Alternatively, y0 may be

evaluated from the value To and an assumed

temperature gradient, with the use of equation

(lla).

If the value of 01—A is known from an initial

model as a function of height, equations (10)

through (14) suffice to determine T(r) in the

400

2000

^1600

*

"5 1200

a.

E

* 800

400

-

-

-

- " "

i I i i i i

•

•

•

7 -

-

-

i i i i

100 200 300 400 500 600 700

FIGURE 2.—Temperatures of models 7,6, and 4, plotted against

geometric height.

presence of conductive and radiative heat

transfer.

As a simple test of the region of failure of

the conductive model, I have performed a

numerical integration of these equations, setting A=0 and adopting equation (7) for the

radiative loss due to oxygen. Of course, in

reality the radiative absorption A is finite, and

will to a large extent compensate the radiative

loss. For this reason, the present test should

give a sure indication of the level below which

the conductive model fails.

The results of this test are shown in table 2,

which compares the temperatures and global

fluxes obtained with and without the radiativeloss term. The tabulation indicated quite

clearly that, within the framework of model 7,

radiative-emission terms are not significant

above 200 km. It should be borne in mind,

however, that radiative-absorption terms may

significantly perturb the temperature distribuTABLE 2.—Effect of radiative loss

Height

(km)

500

400

300

200

150

T(°K)

1465

1258

1026

754

595

T(°K)

1465

1259

1046

851

859

F/Fo

1.000

0.982

0.880

.446

.400
46 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

tion below 300 km. The temperatures computed with radiative losses are plotted in figure

2 as the dashed branch of model 7.

Conclusions

In summary, it appears that through adjustment of parameters, the conductive model can

be made consistent with present satellite data.

The total flux required, F= 1.61 X1018 erg sec"1,

or E=0.27 erg cm"2 sec"1, is comfortably below

2.4 XlO19 erg sec"1, the amount of solar coronal

heat estimated by Chapman (1957) to be available for such conduction in the earth's upper

atmosphere.

As indicated by figure 2, the temperature

distribution above 250 km is still open to considerable uncertainty insofar as direct evidence

from satellite data is concerned. However, the

data are of such a quality that within the

framework of a particular theoretical or semitheoretical model the temperature distribution

can be determined with a fairly small uncertainty.

I shall repeat two comments made in the first

paper. First, the satellite data show a definite

correlation with solar activity. For this reason

considerable care must be exercised in combining data from different epochs. The present

data represent an average over several months

during 1958, with the exception of the data

from Satellite 1957 a2. Second, although satellite data give no evidence for a considerable

latitude-dependence of density, the nature of

the data tends to obscure such a dependence.

Evidently the present type of data derived

from the acceleration of dense satellites is of

principal value in deriving a global picture of

the density structure, and not for fine structure

of either a temporal or a spatial nature. It is

to be anticipated that balloon satellites of small

mass-area ratio will offer an improved picture

of the more rapid variations.
On the Effects of the Sun and the Moon upon

the Motion of a Close-Earth Satellite

By Yoshihide Kozai

In the present paper I shall treat the lunar

and solar perturbations of a close-earth satellite

whose radius is very small compared with that

of the moon.

Since the disturbing functions of both the sun

and the moon have similar forms, only the

method of deriving the perturbation for the

moon will be described here.

If we denote the geocentric radius vector of

the satellite and of the moon by r and r',

respectively, and expand the disturbing function

R into a power series of r/r', a small quantity,

we obtain the expression,

Gm' (\ r2

(1)

where Q is the constant of gravity; m' is the

mass ratio of the moon with respect to the earth ;

and St is the Legendre polynomial of the ith

order; that is,

s=rr'/rr'

(2)

We can omit the first term, which does not

depend on the orbital elements of the satellite.

Since we cannot expect any secular contributions from the odd-order terms, I will derive

the perturbation produced from the secondorder term.

Adopting geocentric coordinates, with z-axis

directed towards the equinox and 2-axis towards

the north pole, we have the following three

conponents of r (by using the conventional

orbital elements):

or 7

-=cos (Z-f- 12) -J-2 sin2 „ sin Z sin 12,

-=sin (Z+12)—2 sin2 ~ sin Z cos 12,

Z . . . T -=si n % sin L

r

(3)

where L is the argument of latitude. We derive

similar expressions for the moon, using primed

letters to represent elements referred to the

equator. Then s is expressed as follows:

n_xx'+yyf

 +zzr

rr

=cos (Z+Q—Z'—0')

- 2 sin21 sin L sin (Z' + Q'—Q)

—2 sin2 \ sin U sin (Z+Q— Q')

i i' +4 sin2

 jr sin2 -^ sin L sin L' cos (12—12')

+sin i sin i' sin L sin L'. (4)

It is convenient to express the disturbing

function by mean longitudes X and X' and by

other orbital elements. Dropping all terms depending on the mean longitude of the satellite,

which have little effect on the satellite's motion,

we have as the principal terms of the disturbing

function:

cos (X'-«'—

-4e'sin(X'-a/-i2')

(5)

47
48

where

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

-^ sin 2i sin 2i'cos (12—12')

ID

16

sin2 i sin2 i' cos 2(12-12')

|sin 2i' (l-|sin 2i \ cos 2(X'-12')

+ | sin2 i cos4 ^ cos 2(X'-Q)

- § sin 2i sin *' cos2 \r cos (2X' —Q-Q')

| sin2 i sin4 ~ cos 2(X'-212' + 12)

-f sin 2i sin i' sin2 -^ cos (2X' + Q—30'),

#=cos 41 cos4 ^- cos 2(X'—a)—12)

+« sin2 i (1 — 5 sin2 i' j cos 2u

+ 5 cos41 sin2 i' cos 2(w+Q- Q')

1 sin si

4

 ^ sin2 i' cos 2(w—Q+0')

Z 2

K sin i sin2 ^ sin 2i' cos (2co— Q+Q')

+sin i sin2 ^sin i' cos2—cos (2X' +2w— Q—Q')

—312')

i i

intsin2^sini'sin2—cos(2X'—3Q'—

By selecting all terms depending on X' and

by replacing cos by sin, we can also derive

expressions for A' and B', from A and B.

The variations of e and i are obtained from

the equations:

[-e 2

d e _ di2

eft na2e do>'

di cos i di2

—e2sin i dw

• (6)

By using the variations de and 5z, we can

derive 8ca and 5fi from the formulas

+sin4 i cos4 j cos 2(o>-Q+X/)

+cos4 ^ sin4 ^- cos 2(co+a+X'—2J2')

+sin i

4

 5 sin4 ^- cos 2(X'—2Q'-«+0)

+ o sin2 i sin2 i" cos 2(X'—0'—«)

o

+ | sin21 sin2 i' cos 2(X'+w—0')

X X

+sin i cos2^sin i' cos2 -5- cos (2X'—Q'—2CJ— Q)

1 z

—K sin i cos2 5 sin 2i' cos (2w+Q— Q')

cost

dt — e

2

 sin

dR -\/l — e

2

dR

i i ^ 2

na ?>e 2e

. doi. . rfw .

+

Tede+diH>

—e2 sin

(7)

where

4—5 sin2

12= In cos iIt is remarkable that these disturbing

functions do not affect the semimajor axis.

In the righthand sides of the equations, i',
EFFECTS OF SUN AND MOON ON SATELLITE MOTION—KOZAI 49

the inclination of the lunar path to the equator,

is changing gradually, but we may regard it

as constant during one year or so.

As for the secular terms of £2 and o>, we have

dQ. 3n'2

 . cos ^

—— m

dt 4 n yl — e*

do) Zn'2 , 1

dt 4 n VI — <

x

(2- | sin2 i+l - | sin2

•(8)

where

sin i

2

 i/=zn sin2 J{\ +cos2 e)+sin2 6 cos2 J

+o sin 2e sin 2J cos A^

—~ sin2 J sin2 e cos 2iV.

Here «7 and N are the lunar inclination and the

longitude of the ascending node referred to the

ecliptic; and e is the obliquity. We can find

the values of J, N and X' in the American

Ephemeris.

If we set m/= l, and J=0 , we can derive the

solar perturbations from the same equations.

The Earth's Gravitational Potential Derived from

the Motion of Satellite 1958

By Yoshihide Kozai

O'Keefe and Eckles (1958) recently reported

that the long-period (80-day) variations in the

eccentricity of the orbit of Satellite 1958 /82

(Vanguard) can be explained by a north-south

asymmetry in the gravitational potential of

the earth. That is, the axial symmetry of the

field is retained, but the third harmonic in

latitude is added. Explicitly, it is assumed

that the potential of the earth is

GM A2(\ As/5

at a point whose declination is S and whose

geocentric distance is r. In this expression,

A2, A3, and AA are constant, G is the constant of gravitation, and M is the mass of

the earth.

Expressing U3, the part of the third harmonic, in terms of the conventional orbital

elements, rather than declination and radius

vector, and eliminating terms depending on

mean longitude, I find the long-period part of

U3 to be as follows:

X«(l-e 2)-2sin«, (2)

where i is the inclination of the orbital plane

to the equator, a is the semimajor axis, and

w is the argument of perigee.

We note that no secular variations of the

orbital elements are produced by this odd harmonic in the potential and that the coefficient

of cos 3o> is zero.

The equations for the variation of the elements produced by the third harmonic are the

following:

dSa

dSe

na

2

e

dhi_ cos i

dt na

2

*sj\—e2 sin i

bU3

— e ?+%*+£**, 2 sin i dt di de

dSo)=_ cos i

dt na?ijl—e2smi di na

2

e de

. da .. . do)

where n is the mean motion, Q is the longitude

of the ascending node, and

il= | n cos i,

A2 4—5 sin2 i

n

p=a(l-e2). (4)

Integrating these equations we have the

following results:

. 3 A3 . . . 8e=-r -T-2

- sm i sin o>,

4 A&

3 A3

di=—- -r-1 e cos i sin w,

4Ap

3 A3 sin2 i—e2 cos2 % 1 bu=- -r

1

 :—: cos

4 A2p sin % e

rn 3 A3 cos i

5Q= - —-i- r « COS O>.

4 A2p sin t

(5)

51
52 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

To confirm the suggestion of O'Keefe and

Eckles, I analyzed the orbital elements of

Satellite 1958 02 from June 19, 1958, to Jan. 29,

1959; they were furnished by the Vanguard

Computing Center.

At first I subtracted the effect of air-drag as

given by the following formulas,1 under the

assumption that the perigee height is not

changed through this period:

Ae= Aa,

TABLE 1.— Variations in orbital elements

4—5 sin2 i F7—

L

AM= An dt (6)

where Aa, An and Ae are deviations from the

values for Oct. 16 12»> 27m (U.T.) 1958.

Then I also subtracted solar and lunar

perturbations (Kozai, 1959c) as follows:

«eX10*= —1.7 cos 2(a>+Q-Xo)

+0.5 cos (2«+2Q—3\©+T©)

+0.1 cos 2(w+£2—Xc),

5i=0?003 cos fi,

5OJ=0?011 sin 2(co+fl—Xo)

—0?062 sin (2co+2ft—3Xo+iro)

+0?003 sin (2cj+fi-2Xo)

+0?005 sin Q+0?055 sin iVc,

5Q=— 0?001sin2Xo+0?028sin2(aj+fi—Xo)

+0?001sin2(Xo-fi)

+0?032sin (2w+2fi—3Xo+7ro)

-0?005 sin J2-O?O39 sin AT

C, (7)

where Xo is the mean longitude; TTO is the longitude

of perigee of the sun; X^ is the mean longitude of

the moon; and A7^ is the longitude of the ascending node of the lunar path referred to the ecliptic.

Then assuming that the periodic inequalities

are due to the third harmonics we obtain the

variations given in table 1.

1

 K. Squires, private communication to Dr. C. A. Whitney.

Inequality

8e

hi

3o>

50

Observed

(0.43±0.02)X 10-'sin w

-(0?007±0°001) sinw

(0?106±0?010) cosw

(0°018±0°003) cos w

Computed

0.42 X10-8

-0?007

0?122

0?012

Figure 1 shows the observed values plotted

against theoretical values for this same period,

June 19, 1958, to Jan. 29, 1959.

Assigning to each observed amplitude the

weight that is reciprocally proportional to the

square of the probable error, we have as the

coefficient of the third harmonic,

4r=(2.20±0.08)X10-6.

The predicted amplitudes from this value are

also shown in figure 1.

The values of the secular motions of the node

and the perigee, given in table 2, correspond to

the following quantities:

7i=3862?640, anomalistic mean motion per day,

e=0.190 00,

i=34?250.

To compare with my results,

A2 f A2n 4 .

1

(

A* /6 3 . . A/, . 11 A

——t n U—2

 sl n %) ( !+ y e ) co s

 *»

=f n (4-5

/4(16-6 2 sin2 i+49 sin4 i)

+ e2(328-1244sin2i+973sin4i) V, (8)

TABLE 2.—Secular motions of node (£2) and perigee (w)

Motions

Observed (per

day)

Solar part

Lunar part

Corrected

Node il

-3?015 07±4

-0.000 13

-0.000 28

-3.014 66

Perigee w

4? 404 62 ±10

0.000 18

0.000 39

4. 404 05
EARTH'S GRAVITATIONAL POTENTIAL, FROM MOTION OF 1958 /3 2—KOZAI

150 175 200 225 250 275 300 325 350 375

53

+ .00040

Se -ooooo

-.00040

+?20

8w .oo

-.10

-?20

+?040

+.020

Sii -00

-.020

-?040

+!020

+ .010

Si .00

-.010

-?020

/ \

1958

-\J

150 175 200 225 250 275 300 325 350 375

Epoch (number of days after January 0.0,1958)

FIGURE 1.—Observed values (dots) and theoretical values

(solid lines) of variations in orbital elements due to the

third harmonic. Each point represents 7 days.

we must subtract solar and lunar perturbations

from the observed values. The solar and lunar

parts have been derived from the following

equations:

Here m<^ is the mass ratio of the moon to the

earth; J is the inclination of lunar path to the

ecliptic; and e is the obliquity. If we set

w<£=n©, m<£=l and J=0, we can also derive

the solar secular perturbation from these

equations.

Then from fi and « I derived the following:

4

2

=1.6208X10-3,

X -j=L= (2- | sin* i+5 A (9)

where

1 . =pr sin2

 «7(l+cos2 e)+sin2 e cos2 J. 2

I am grateful to Drs. C. A. Whitney, L. G.

Jacchia and G. Veis for their valuable discussions.

Solar Effects on the Acceleration

of Artificial Satellites

By Luigi G. Jacchia

Slow fluctuations connected with variable

solar radiation

The atmospheric drag that artificial satellites

experience around perigee causes them to lose

energy and to fall into orbits with smaller

major axes and shorter periods of revolution.

The change in period that the satellite undergoes from one revolution to the next is called

the "secular acceleration"; in this discussion

the term "acceleration" will refer to the nondimensional quantity dP/dt, when P is the

period of revolution, and t is the time.

Erratic changes in the acceleration of a

satellite were first detected by this writer in

an analysis of the observations of Satellite 1957

Beta (Jacchia, 1958e; 1958a). At first it was

not clear whether these changes were owing to

variations in the presentation area of the

satellite or to density variations in the atmosphere, but when their presence was detected also

in Satellite 1958 j32 (the spherical Vanguard) no

doubt was left about their atmospheric origin.

The presence of a 27-day periodicity pointed to

variable solar radiation as the cause of the

atmospheric fluctuations (Jacchia and Briggs,

1958). The discovery that the accelerations of

Satellites 1958 02 and 1958 51 varied more or

less in unison (Jacchia, 1959a; 1959e) and that

those of the other satellites followed the same

rhythm proved that the atmospheric fluctuations at their origin are truly global.

In a communication dated December 18,

1958, Dr. W. Priester of Bonn, Germany,

pointed out a remarkable similarity between

my acceleration curve of Satellite 1957 /SI

(Jacchia, 1958b) and the 20-cm solar-flux curve

in the interval from November 11, 1957, to

February 10, 1958. I had no 20-cm solar data

63G-014—03 H

at hand, but had 10.7-cm data from the National Research Council in Ottawa; a comparison of the satellite curves with these data,

extended over a whole year, showed a correlation that could be classified as little short of

perfect. The amplitude of the fluctuations

increased with the perigee height of the satellite. They amounted to about 20 percent in

typical, well-defined 27-day cycles in the acceleration of Satellite 1958 51 (perigee height

200 km), but to about 70 percent in corresponding cycles of Satellite 1958 /32 (perigee

height 650 km).

Transient fluctuations connected with magnetic storms

The 10.7-cm radiation is closely correlated with

the sunspot numbers, so an excellent correlation

also exists between satellite accelerations and

sunspot numbers. Searching for further clues

about the nature of the radiation that interacts with the upper atmosphere, I reduced

anew all the available observations of Satellites

1958 /32 (about 2500) and 1958 51 (about 9000)

with the best elements at hand and obtained

more accurate accelerations at twice the original

resolution. The results are plotted in figure 1,

together with the 10.7-cm solar-radiation curve

(Jacchia, 1959b). As can be seen, the correlation with the solar radiation is remarkably good,

even in details, for the high-flying, spherical

Satellite 1958 02 (Vanguard), for which accurate, well-distributed Minitrack observations were available throughout. For Satellite 1958 51 the observations are mostly optical,

less accurate, and more irregularly distributed,

with occasional periods of near-invisibility;

these facts, together with the elongated shape,

05
56 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

may have contributed somewhat to the poor

correlation for 1958 51.

The procedure for computing the accelerations was the following. Residuals of individual

nodal (for 1958 51) or perigee (for 1958 02)

passages from empirical equations were plotted

and ordinates on the residual curves were read

off at intervals of 25 revolutions; the resulting

table was then differenced and accelerations

were computed from the differences. The

residual curve for 1958 51 is a succession of long,

smooth waves of great amplitude that are

mainly due to the 27-day fluctuations. At only

two spots in the whole curve is the smooth

succession of long waves interrupted by a transient, short-lived but unmistakable secondary

90

-16

120 150 180

1958 Year Days

210 240 270 300 330 360 390

x 10

-5

-14 -

-12 -

-10

-8

-6

-4

-2

Satellite 1958 Si

Orbital acceleration dP/dt

- 8

xlO

- 7

- 6

-4

-2

Satellite 1958/32

Orbital acceleration dP/dt

300

x 10

-22

200

100

Solar flux at 10.7-cm. wave length

v«.

,vv

V

FIGURE 1.—Secular accelerations of Satellites 1958 51 and 1958 02 compared with the 10.7-cm solar flux. The dates of the two

great geomagnetic disturbances, July 8-9 and September 4, 1958, are marked (1) and (2), respectively. Accelerations were

computed at intervals of 25 revolutions.
SOLAR EFFECTS ON ACCELERATION OF ARTIFICIAL SATELLITES—JACCHIA 57

oscillation. The dates of these disturbances

are July 8 to 9 and September 4, 1958; they are

exactly coincident with the only two great

geomagnetic disturbances that occurred during

the satellite's lifetime. Both magnetic storms

followed, at the usual one-day interval, the

appearance of a flare of importance 3 + on the

sun.

Although the original accelerations had a

resolution of 25 revolutions, it appeared possible to obtain accelerations with a resolution of

10 revolutions around the critical dates, and

these—computed at intervals of 5 revolutions—

are plotted in figure 2 together with the threehourly geomagnetic planetary indices Kp (Jacchia, 1959b). It is quite evident that in both

the July and the September events, there was no

immediate response at the time when the flare

appeared, and that the atmospheric disturbance—an increase in density—proceeded nearly

1958 Jujy 4

- 4

8 10 12 14 16

X 10

- 5

- 3 -

- 2

1 1 1 1 1 1

Flare

• *

• •• • * • *•

• * * • •J J • • • * © •

1 1 1 ••'•*!• | |

1 1 1

*a »• • •• •

1 1 1

1 1

dP/dt

• * • • • •

Kp

* • • • a •

• • • a • • •

"l "" ' 1

1

•

• a a . .

10

5

0

1958

- 5

X lO" 5

- 4

Aug. 30

-

.'•• a . •

1

Sept 1

i i

• •

a

a #

3

1 1 1 .

Flare

1 •

a

..f | ,

5

•

•

" V.

i i

7

• •

i

9 1

i i i

dP/dt

• • •

• •

Kp

10

- 5

0

FIGURE 2.—Secular accelerations of Satellite 19S8 51 computed with a resolution of 10 revolutions around the dates of two great

geomagnetic disturbances and compared with the three-hour geomagnetic planetary indices Kv. The instants of the flares

that preceded the magnetic storms are marked on the diagram.
58 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

synchronously with the magnetic disturbance.

The increase in the acceleration amounted to

40 percent in the July 8 event and to a little

more than 30 percent in the event of September

4. These are actually lower limits, inasmuch

as the limited resolution may have smoothed

out the peaks (with a resolution of 25 revolutions the peaks are highly reduced, as can be

seen in figure 1, where the two disturbances are

marked (1) and (2) respectively.) Since the

amplitude of the 27-day oscillations did not

exceed 20 percent for this satellite (1958 51),

it appears that the corpuscular radiation

capable of causing a first-rate geomagnetic disturbance has a greater, although more transient,

effect on the atmospheric density at the 200-km

level, than the ordinary fluctuation in solar

radiation associated with the 27-day period.

That this latter radiation might also be corpuscular in nature appeared possible, in view of

the failure of the atmospheric density to react

promptly to the stream of electromagnetic

radiation that accompanies the appearance of a

flare. Wave radiation could not be ruled out,

however, since the duration of a flare is much

shorter than that of a magnetic storm, and the

effect of a transient drag perturbation on the

satellite position is proportional to the square

of the duration.

No disturbance could be detected in the residual curves for Satellite 1958 /32 around the two

critical dates in July and September. This

could be due to the much smaller value of the

acceleration for this satellite, which would make

a short-lived perturbation difficult to detect.

Otherwise, we should conclude that during a

violent magnetic storm the atmospheric layers

in the 200-km zone are more strongly affected

than those at a height of 600 to 700 km. Judging by the increase of the 27-day fluctuations

with height, it appears that in ordinary conditions the opposite situation prevails, with the

higher levels affected more than the lower. It

might be of interest to remark that at the time

of the July disturbance the latitude of perigee

of Satellite 1958 51 was +35° ; at the time of the

September disturbance it was +15°; the corresponding geocentric angular distances between sun and perigee were 110° and 77°,

respectively.

Diurnal effects

The existence of a "diurnal" effect, i.e., a

variation of the acceleration with the geocentric angular distance between the sun and

the satellite's perigee, was announced by

this writer for Satellite 1957 /31 (Jacchia,

1959c). The published amplitude of the effect—about 20 percent—is almost certainly too

large, much of the fluctuation being probably

due to a slow variation of solar radiation

superimposed on the shorter, erratic fluctuations. This slow variation is quite evident in

the 10.7-cm flux curve (fig. 3), but was unknown to the author when the paper was

written. Only a bare suggestion of a diurnal

effect is discernible in the acceleration curve

of Satellite 1958 51.

A much larger diurnal effect is suggested by

the acceleration curves of the higher-perigee

Satellites 1958 Alpha and 1958 02 (fig. 4),

but the picture here seems to be complicated

by a major perturbation of the atmosphere that

started in the second half of August 1958, and

continued for two or three months, and possibly

longer. This perturbation occurred when the

perigee of Satellite 1958 Alpha was in daylight

and raised the acceleration level to extremely

high values, which were never reached again

when the perigee returned to the same position

with respect to the sun the following year

(April to August, 1959). The global nature

of the perturbation is evidenced by the extremely rapid and perfectly synchronous rise

in the accelerations of both satellites between

August 17 and August 27, 1958. The curve

of the 10.7-cm solar flux does not offer any

clear-cut clue to the cause of this perturbation.

For some time the writer was tempted to

speculate about the effect of stored corpuscular

radiation following major magnetic storms,

but it must be admitted that the evidence for

such a mechanism is not very convincing.

Due to the presence of the large atmospheric

perturbation, it is difficult to draw definite

conclusions regarding the amplitude and the

phase of the diurnal effect in Satellite 1958

Alpha and 1958 j82. Although the 1958 parts

of both curves suggest a phase lag of some

30° to 50° in air—ao (ax=R.A. of perigee, a o =

R.A. of sun), the 1959 portion of the 1958 Alpha
SOLAR EFFECTS ON ACCELERATION OF ARTIFICIAL SATELLITES—JACCHIA 59

300

1957 Year Days

350 400 450

xlO

-5

10

8

0

90°

180°

400

—22

xlO

300

200

1957

Orbital acceleration

(days per day)

Geocentric angular distance sun-perigee

10.7-cm solar radiation

•V

.-

 v

. r.

NOV. Dec. Jan. Feb. Mar

1958

Apr.

FIGURE 3.—The diurnal effect in Satellite 1957 /3l. Accelerations (top) are compared with angular distances sun-perigee and with

the solar flux at 10.7 cm. The smooth line in the acceleration diagram was drawn as a visual aid.
60 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

100 200

1958 YEAR DAYS

300 400 500 600

X10"7

- 8

- 6

- 4

- 2

0

30°

60°

90c

120°

150

180°

-10

xio

- 3

— 6

- 4

0

30°

60-

90

120°

150°

180

3C0

XHT"

200

100

1956 32

dP

dt

1958 a

dt

SOLAR FLUX AT 10.7-cm.

•ft* •• *»••»>".

5 6

1958

10 11 12 1 2

1959

FIGURE 4.—Acceleration of Satellites 1958 j32 and 1958 Alpha, compared with angular distances sun-perigee ^ and with the solar

flux at 10.7 cm. Values of the difference in right ascension between the satellite's perigee and the sun (a,-ao ) are marked on

the horizontal line corresponding to ^=90° .

curve seems to be reasonably well in phase.

The ratio of mean maximum to mean minimum

acceleration due to purely diurnal effect should

be about 1.4 or 1.5 for 1958 Alpha (perigee

height 350 km) and 7 or more for 1958 02

(perigee height 650 km). The mean maximum

and minimum values are those obtained by

passing a mean curve through the monthly

oscillations connected with the solar flux.

Such a large variation of the effect with height

indicates that the density profile and the scale

height of the atmosphere above 300 km differ

markedly in the bright and in the dark hemispheres. Preliminary computations show that

to account for the effect, the scale height must

vary by a factor of about two at the 650-km

level.

A remarkable feature that emerges from

figure 4 is the disappearance of the monthly

oscillations when the perigee comes from daylight into night. The phenomenon is present

in both satellites (1958 Alpha and 1958 /82).

Since atmospheric variations with constant

amplitude are reflected in the acceleration

curves with amplitudes proportional to the

mean acceleration itself, part of the variation

in the amplitude of the fluctuation observed in

1958 /82 is only apparent. If, however, we
SOLAR EFFECTS ON ACCELERATION OF ARTIFICIAL SATELLITES—JACCHIA 61

compare the curves of 1958 /32 in the period

April to July, 1958, and in May, 1959, when

the mean acceleration was about the same

(dP/dt~—2X10-7), we can notice the difference in behavior between the first interval,

when the perigee was in daylight, and the

second interval, during which the perigee

was crossing into the dark hemisphere. The

10.7-cm solar flux was actively oscillating during both intervals, and the oscillations are

clearly visible in 1958 Alpha also during the

second interval, at which time this satellite

had its perigee in daylight.

The disappearance of the oscillations in the

dark hemisphere would be expected if the

variable radiation responsible for them was

wave radiation. In view of the association of

the 10.7-cm flux with sunspots and with ion

density in the F layer, the association of this

type of atmospheric oscillation with wave

radiation would seem to make good sense.

We should not forget that for the low-perigee

Satellites 1957 Beta and 1958 51, the monthly

oscillations persisted also when the perigee

was in the dark hemisphere. A look at the

acceleration curves for these two satellites

shows, however, that during such periods the

amplitude of the oscillations was definitely

smaller.

Summary and conclusions

Summarizing, we find that four distinct types

of fluctuations can be distinguished in the

acceleration of satellites.

(1) Fluctuations that follow the rhythm of

the solar flux at 2800 me (10.7-cm wavelength).

These fluctuations increase in amplitude with

height and become smaller or disappear when

the perigee is in darkness. Very probably

these fluctuations reflect variations in atmospheric density caused by variable short-wave

solar radiation (extreme ultraviolet).

(2) A slow fluctuation connected with the

position of the perigee with respect to the subsolar point ("diurnal effect"). This effect is

small at the 200-km level, but becomes very

large at heights above 350 km. This effect is

intimately connected with (1) and should have

the same primary agent (ultraviolet absorption)

as its origin. It reflects a difference in the

density profiles of the bright and the dark

hemispheres of the earth.

(3) Transient fluctuations accompanying

magnetic storms. These perturbations are

seemingly of corpuscular origin and should

reflect a heating of the atmosphere through

some interaction with corpuscular radiation.

(4) Erratic fluctuations of unexplained origin,

such as the perturbation of August and September, 1958. A comparison with conditions in the

radiation belts may provide a clue to this effect.

Explanation of tables

For Satellite 1958 Alpha the accelerations are

given at equal time intervals, at 0h U.T. For

1958 /32 and 1958 51 the accelerations are

tabulated at equal intervals of revolutions (N).

Uncertain values are in parentheses.

In Tables 2a, 2b and 2c, the tabulated angle

^ is the geocentric angular distance between

sun and perigee; aT is the right ascension of

perigee; and aQ is the right ascension of the sun.

Acknowledgments

The accelerations of 1958 (82 were obtained by

Mr. R. E. Briggs, using the author's "subsatellite point program," in which each observation yields a time of perigee or nodal passage.

The writer is indebted to Mr. Pedro

Zadunaisky for communicating to him, in

advance of publication, the results of 109

orbital determinations for Satellite 1958 Alpha.

These orbits were computed by the differentialcorrection method of Dr. George Veis, from

Minitrack and Baker-Nunn camera observations. The orbits were computed, whenever

possible, at 3-day intervals, although in some

sections intervals of 6 days and more became

necessary. Each orbit yielded a value of the

mean motion of the satellite for a central epoch,

and accelerations were derived from these mean

motions.

Mr. Stephen Maran assisted in the preparation of most of the tables.
6 2 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.«

TABLE la.—Secular accelerations of Satellite 1958 Alpha

Date WdP/dt Date 10s dP/dt Date 10« dP/dt Date 105 dPjdt

1958 1958 1958 1958

Feb. 16 -4. 9 Aug. 5 -6. 6 Jan. 17 -4. 9 May 12 -6.36

21 -4. 2 10 -6. 3 22 -4. 9 14.5 -6.74

26 —4.1 15 -6. 2 27 (-5.0 ) 17 -7.26

Mar. 3 -4. 7 20 -7. 3 Feb. 1 (-5.0) 19.5 -7.22

8 -5. 3 25 -9. 0 6 (-5.0) 22 -7.10

13 -5 . 7 30 -9 . 7 11 -4. 9 24.5 -6 . 78

18 -5 . 9 Sept. 4 -9 . 7 16 -4 . 8 27 -6.32

23 (-6.0) 9 -9.6 21 (-4.7) 29.5 -5.96

28 (-6.1) 14 -9.6 26 (-4.6) June 1 -5.76

Apr. 2 (-6.2) 19 -9.6 Mar. 3 -4.5 3.5 -5.70

7 (—6.2) 24-9. 3 8 -4.4 6 -5.86

12 (-5.8) 29 -9.0 13 -4.3 8.5 -6.14

17 -5.4 Oct. 4 -8.7 18 -4.7 11 -6.48

22 -5.4 9 -8.4 20. 5 -5.06 13.5 —6.78

27 -5 . 7 14 -8 . 4 23 -5.48 16 —6. 94

May 2 -6.1 19 -8.6 25.5 -5.76 18.5 -6.98

7 -5.9 24 (-8.7) 28 -5.82 21 -6.76

12 -5.3 29 (-8.7) 30.5 (-5.70) 23.5 -6.46

17 -4.7 Nov. 3 -8.6 Apr. 2 (-5.48) 26 -6.16

22 -4. 5 8 -7 . 5 4. 5 -5.36 28.5 —5. 94

27 (-4.9) 13 -6.3 7 -5.46 July 1 -5.76

June 1 (-5.0) 18 -6.0 9.5 -5.78 3.5 —5.56

6 (-4.7) 23 -6.0 12 -6.26 6 -5.42

11 (-4.4) 28 -5.9 14.5 -6.52 8.5 -5.46

16 (-4.3) Dec. 3 -5.9 17 -6.60 11 -5.80

21 -4. 6 8 -5 . 9 19. 5 -6. 72 13.5 —6.34

26 -5. 1 13 (-5.6) 22 -6.94 16 —6.80

July 1 -5. 7 18 (-5.5) 24.5 -7.20 18.5 -6.70

6 -6. 1 23 (-5.4) 27 -7.32 21 -6.18

11 -5. 6 28 (-5.2) 29.5 -7.00 23.5 -5.74

16 -5. 0 1959 May 2 -6.36 26 —5.50

21 -5. 4 Jan. 2 (-5.0) 4.5 -6.08 28.5 —5.26

26 -6 . 2 7 -4. 9 7 -6.06 31 —5. 14

31 -6 . 6 12 -4 . 9 9. 5 -6 . 10

TABLE 1&.—Secular accelerations of Satellite 1958 02

N Date 107 dP/dt N Date 107 dP/dt N

1958 1958

0 Mar. 17. 61 500 May 3. 24 —3. 1 1000 June

25 19.94 —2.4 525 5.57 —2.8 1025

50 22.27 —2.1 550 7.90 —2.4 1050

75 24.60 —0.9 575 10.23 —2.4 1075

100 26. 94 —1.0 600 12.56 —2.2 1100

125 29.97 —1.5 625 14.89 —1.9 1125

150 31.60 -2. 4 650 17.22 —1.4 1150 July

175 Apr. 2.93 —2.7 675 19.55 —1.5 1175

200 5.26 -3. 1 700 21.89 —1.7 1200

225 7.59 -2. 2 725 24.22 -2. 2 1225

250 9.92 —1.2 750 26.55 —2.2 1250

275 12.26 —1.4 775 28.88 -2. 4 1275

300 14.59 —1.2 800 31.21 -2. 7 1300

325 16.92 -1. 7 825 June 8.54 -2. 7 1325

350 19.25 —1.7 850 4.87 —2.2 1350

375 21.58 -1. 9 875 7.20 -2. 1 1375

400 23.91 —2.1 900 9.53 —2.1 1400

425 26.24 -2. 6 925 11.87 -2. 1 1425

450 28.57 -3. 4 950 14.20 —1.4 1450

475 30.91 -3. 3 975 16.53 -1. 2 1475 Aug.

Date

1958

18.86

21. 19

23.52

25.85

28.18

30.51

2.84

5.17

7.51

9.84

12. 17

14.50

16.83

19. 16

21.49

23.82

26. 15

28.48

30.81

2. 14

107 dP/dt

-1. 2

-1. 5

-1. 0

-1. 0

-2. 2

-1. 9

-1. 2

-1. 9

-1. 4

-1. 5

-1. 5

-1. 0

-0. 5

-0. 9

-1. 0

-1. 9

-1. 9

-1. 9

-2. 6

-3. 4
SOLAR EFFECTS ON ACCELERATION OF ARTIFICIAL SATELLITES'—JACCB2A 63

N

1500

1525

1550

1575

1600

1625

1650

1675

1700

1725

1750

1775

1800

1825

1850

1875

1900

1925

1950

1975

2000

2025

2050

2075

2100

2125

2150

2175

2200

2225

2250

2275

2300

2325

2350

2375

2400

2425

2450

2475

2500

2525

2550

2575

2600

2625

2650

2675

2700

2725

2750

2775

2800

2825

Aug.

Sept.

Oct.

Nov.

Dec.

Date

1959

4 48

6. 81

9. 14

11.47

13.80

16. 13

18.46

20.79

23. 12

25.45

27.78

30. 11

1.44

3.77

6. 10

8.43

10.77

13. 10

15.43

17.76

20.09

22.42

24.75

27.08

29.41

1.74

4 07

6.40

8.73

11.06

13.39

15.72

18.05

20.38

22.71

25.04

27.37

29.70

1.03

3.36

5.69

8.02

10.35

12.68

15.01

17.34

19.66

21.99

24.32

26.65

28.98

1.31

3.64

5.97

TABLE 16.—Secular accelerations of Satellite 1968 02-

107 dP/dt

-2. 7

- 2 . 4

- 2 . 1

- 2 . 2

-1. 9

-2. 4

- 2 . 4

- 2 . 4

-3. 6

- 4 1

- 4 8

- 4 5

— 4 3

— 5. 3

- 4 8

- 4 1

— 4 1

-5. 0

-5. 0

— 5.0

-3. 8

-3. 6

-4. 5

-3. 6

- 4 1

- 4 3

- 4 5

-5. 5

- 4 6

- 4 8

-6. 0

-5. 8

-7. 4

-7. 4

-8. 9

-7. 0

-5. 5

-5. 5

-6. 3

-5. 7

-5. 2

— 5.0

- 4 8

-3. 8

— 3. 8

-4. 3

-3. 9

-5. 0

- 4 8

-6. 0

-6. 3

-6. 9

-6. 9

-5. 5

N

2850

2875

2900

2925

2950

2975

3000

3025

3050

3075

3100

3125

3150

3175

3200

3225

3250

3275

3300

3325

3350

3375

3400

3425

3450

3475

3500

3525

3550

3575

3600

3625

3650

3675

3700

3725

3750

3775

3800

3825

3850

3875

3900

3925

3950

3975

4000

4025

4050

4075

4100

4125

4150

Date

1958

Dec. 8.30

10.63

12.96

15.29

17. 62

19.95

22.27

24 60

26.93

29.26

31.59

1959

Jan. 2. 92

5.25

7.58

9.91

12.24

14 56

16.89

19.22

21.55

23.88

26.21

28.54

30.87

Feb. 2. 19

4 52

6.85

9.18

11.51

13.84

16. 16

18.49

20.82

23.15

25.48

27.81

Mar. 2. 14

4.46

6.79

9.12

11.45

13.78

16. 10

18.43

20.76

23.09

25.42

27.75

30.07

Apr. 1.40

3.73

6.06

8.39

107 dP/dt

-5. 2

- 4 8

-5. 0

-5. 3

- 4 5

-3. 9

-3. 4

- 4 3

- 4 6

- 4 5

-5. 2

-5. 2

-5. 3

-5. 7

-5. 2

- 4 3

-3. 6

-3. 9

- 4 8

- 4 8

-5. 0

- 4 8

-5. 2

- 4 5

- 4 3

-3. 9

-3. 6

-3. 6

-2. 9

-3. 4

-3. 5

-3. 5

-3. 3

- 4 5

- 4 1

—4.0

-3. 4

-3. 6

-3. 6

-3. 8

-3. 4

- 4 0

-3. 8

-4. 3

-4. 3

- 4 5

- 4 3

- 4 1

-3. 8

—3.6

-3. 8

-3. 6

-Continued

N

4175

4200

4225

4250

4275

4300

432?

43'5t

4375

4400

4425

4450

4475

4500

4525

4550

4575

4600

4625

4650

4675

4700

4725

4750

4775

4800

4825

4850

4875

4900

4925

4950

4975

5000

5025

5050

5075

5100

5125

5150

5175

5200

5225

5250

5275

5300

5325

5350

5375

5400

5425

5450

Apr.

May

June

July

Aug.

Date

1959

10.71

13.04

15.37

17. 70

20.02

22.35

24.68

27.01

29.34

1.66

3.99

6.32

8.65

10.97

13.30

15.63

17.96

20.29

22.61

24 94

27.27

29.60

31.92

3.25

5.58

7.91

10.23

12.56

14 89

17.22

19.54

21. 87

24 20

26.53

28.85

1. 18

3.51

5.84

8. 16

10.49

12.82

15. 15

17.47

19.80

22. 13

24.46

26.78

29. 11

31.44

2.77

5.09

7.42

107 dP/dt

- 4 1

- 4 0

-3. 8

-3. 8

-3. 3

-3. 6

-3. 4

-3. 4

-3. 4

-2. 9

-2. 2

-2. 4

-2. 6

-2. 2

-2. 8

-2. 4

- 2 . 1

-1. 9

-1. 9

-1. 4

-1. 2

-0. 9

-1. 0

-0. 9

-1. 0

-1. 0

-0. 9

-1. 4

-1. 0

-1. 2

-1. 5

-1. 2

-1. 2

-1. 7

-1. 4

-1. 9

-1. 2

-1. 2

-1. 4

-1. 4

-1. 5

- 2 . 1

-1. 7

- 1 . 4

-1. 2

-0. 5

-0. 7

-0. 9

-0. 7

-0. 7

-0. 7

-0. 7

636-014—63-
64 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE lc.—Secular accelerations of Satellite 1968 S2

N Date 1O5 dP/dt

1958

0 May 16. 02

25 17. 86 —3. 13

50 19. 69 -2 . 98

75 21.52 —2.87

100 23. 36 -2 . 73

125 25. 19 -2 . 70

150 27.02 -2.7 3

175 28. 85 -2 . 76

200 30. 67 -2 . 63

225 June 1.50 —2.56

250 3. 32 —2. 44

275 5.15 -2.4 1

300 6.97 -2.5 0

325 8. 79 —2. 46

350 10.61 -2.4 2

375 12.43 -2.3 8

400 14.25 -2.3 8

425 16.06 -2.3 9

450 17.88 -2.3 5

475 19.69 -2.4 5

500 21. 51 -2 . 65

525 23.32 -2.7 3

550 25. 13 -2 . 87

575 26.94 -2.9 0

600 28.75 -2.9 9

625 30.56 -2.8 5

650 July 2.36 -2.5 9

675 4.17 -2.5 9

700 5. 97 -2 . 70

725 7. 77 -2 . 84

750 9. 58 -3 . 11

775 11.37 -2.5 6

800 13.17 -2.5 9

825 14.97 -2.5 6

850 16.77 -2.5 4

875 18. 56 -2 . 74

900 20.36 -2.8 6

925 22.15 -3.0 8

950 23.94 -3.0 9

975 25.73 -3.4 7

1000 27.52 -3.6 6

1025 29.30 -3.5 8

N 10s Date dP/dt

1958

1050 July 31.09 —3.63

1075 Aug. 1.87 -3.5 9

1100 3.66 -3.5 1

1125 5. 44 -3 . 37

1150 7.21 -3.4 4

1175 8. 99 -3 . 53

1200 10.77 -3.6 3

1225 12.54 -3.5 6

1250 14. 31 —3. 57

1275 16.08 -3 . 71

1300 17.85 -3.9 6

1325 19.62 -4.0 3

1350 21.39 -4.3 1

1375 23.15 -4.6 8

1400 24.91 -4.9 6

1425 26.67 —5.01

1450 28.43 -4.8 7

1475 30.18 -4.6 0

1500 31.94 -4.2 5

1525 Sept. 2.69 -4.1 1

1550 4.44 -4.8 5

1575 6.19 -4.2 7

1600 7.93 -4.3 1

1625 9.68 -4.2 5

1650 11. 42 —4. 14

1675 13. 16 -4 . 33

1700 14.90 —4. 58

1725 16. 63 -5 . 12

1750 18. 37 -5 . 34

1775 20. 10 -5 . 13

1800 21. 83 -5 . 37

1825 23.55 -5.5 4

1850 25.58 -6.1 9

1875 27.00 -6.3 5

1900 28. 72 -5 . 56

1925 30.44 -5.3 4

1950 Oct. 2. 15 -5 . 22

1975 3.86 -5.3 0

2000 5. 57 -5 . 34

2025 7. 28 -5 . 39

2050 8.99 -5.4 7

2075 10. 69 -5 . 68

N Date 105 dP/dt

1958

2100 Oct. 12.39 -5.57

2125 14.09 -6.17

2150 15.79 -6.82

2175 17.48 -7.66

2200 19. 17 —8.20

2225 20.86 —8.59

2250 22. 54 -9 . 38

2275 24. 22 -9. 71

2300 25. 89 —8. 61

2325 27. 56 -8 . 58

2350 29.23 —8.40

2375 30. 89 -8.06

2400 Nov. 1.55 —7.99

2425 3.21 -8.46

2450 4. 86 —8 . 60

2475 6. 52 —8. 61

2500 8. 16 -8. 85

2525 9. 80 -9. 12

2550 11. 44 -9 . 55

2575 13.08 -10. 26

2600 14 71 —10. 88

2625 16.34 -12.31

2650 17. 96 —13. 36

2675 19.58 —14.42

2700 21.19 -15.28

2725 22. 79 -15. 49

2750 24. 39 -16. 58

2775 25.98 -18.01

2800 27.56 -20.15

2810 28. 20 -21.4

2820 28. 83 —23.0

2830 29. 46 -25.0

2840 30.08 -27.0

2850 30. 71 -29. 1

2860 Dec. 1.33 -32.

2865 1. 64 -37.

2870 1. 95 -41.

2875 2. 26 -46.

2880 2.57 —52.

2885 2.88 -70.

2890 3.19 -101.

2895 3.49 -160.
SOLAR EFFECTS ON ACCELERATION OF ARTIFICIAL SATELLITES—JACCHIA 65

Date

1958

Feb. 6

16

26

Mar. 8

18

28

Apr. 7

17

27

May 7

17

27

June 6

16

26

July 6

16

26

Aug. 5

TABLE 2a.—Relative positions of sun and perigee for Satellite

CtT—CtQ if/

159.0

161. 4

178.7

191.6

195. 6

215. 7

225.7

231. 3

252.6

259.0

267. 1

287.2

290.4

302. 1

319. 1

321.4

337.4

351.2

354.5

159. 8

144.2

138. 1

160.8

153.7

132.0

133. 1

129.0

111.6

99.9

82.8

66.5

75.0

78.7

57.6

37.0

23. 1

11.7

39.6

Date

Aug. 15

25

Sept. 4

14

24

Oct. 4

14

24

Nov. 3

13

23

Dec. 3

13

23

1959

Jan.

Feb.

2

12

22

1

11

14.3

24.9

31.5

53.5

59.8

70.8

91. 1

94.7

110. 9

125.5

128.5

142.4

155.6

162. 4

182.8

186.2

199.3

216.3

219.4

47.8

28.8

36.0

56. 1

59.9

71.6

88.3

97.2

115.6

127.2

114 1

116.0

124.7

163.2

176.9

148.3

123.9

138.4

142.0

1958 Alpha

Date

Feb. 21

Mar. 3

13

23

2

12

22

2

12

22

1

11

21

1

11

21

31

10

20

Apr.

May

June

July

Aug.

239.7

250.7

257.9

280.6

286.4

300.6

319.2

323. 1

343.5

354. 1

1. 1

22.8

27.0

41.6

58.0

62.0

84 1

92.8

103.6

121.4

109.6

99.0

80.7

73. 1

59.9

39.7

50.3

52.1

24 2

7. 1

20.9

42.7

68.9

66.8

57.2

75.9

93. 1

ioa i

TABLE 26.—Relative positions of sun and perigee for Satellite 1968 02

Date

1958

Mar. 21

31

Apr. 10

20

30

May 10

20

30

June 9

19

29

July 9

19

29

Aug. 8

18

28

Sept. 7

17

292.7

290.6

291.4

302.7

311.5

309.7

309. 1

318.6

327.0

325. 1

323.3

332.0

341.5

340.7

339.0

347.3

358.7

359.4

358.0

68.9

69.2

74 6

70.8

59.5

51.6

47. 1

38.3

30.3

41. 1

58.4

62.6

49.2

28.5

20.3

23.7

16.5

1.9

21.9

Date

Sept. 27

Oct. 7

17

27

Nov. 6

16

26

Dec. 6

16

26

1959

Jan. 5

15

25

Feb. 4

14

24

Mar. 6

16

26

5.9

18. 1

19.4

17.4

23.3

34.6

35.8

36.0

36.0

47.0

48.2

44 3

47.7

59.4

62.9

60. 1

63.3

76. 1

81.3

32.9

28.5

19.4

34 6

53.7

58.2

46.0

34 7

33.0

42.2

47.5

55.8

68.4

74 3

67.5

59.0

64 1

77. 1

82.0

Date

Apr.

May

June

July

Aug.

Sept.

Oct.

5

15

25

5

15

25

4

14

24

4

14

24

3

13

23

2

12

22

2

12

etv—cto

79. 1

82.0

94.5

100.3

97.6

9& 9

110.1

116.0

113.0

113.0

123.8

130.9

128.6

130.0

140.0

149.0

14a 0

14a 2

153.2

16a 6

*

7a 3

7a 4

86.8

95.0

99.7

107.3

118.6

120.3

108.4

98.0

102.9

117.6

127.9

132.2

137.4

147.4

147.3

140.0

140.2

160.8

TABLE 2C.—Relative positions of sun and perigee for Satellite 1958 61

Date

1958

May

June

July

15

25

4

14

24

4

14

««-«©

0.6

320.8

281.2

242.0

202.3

162.9

123.6

34.1

42.4

66.0

92.3

112.9

117.9

102.8

Date

July

Aug.

Sept.

24

3

13

23

2

12

22

84 5

45.6

6.8

328.5

289.8

250.9

211. 6

75.6

42.9

10.2

31.3

68. 8

107.4

146.9

Date

Oct.

Nov.

Dec.

2

12

22

1

11

21

1

av-ao

171.7

131. 1

89.7

47.3

3.7

318.2

270.6

+

171.2

131. 1

89.6

47.5

9.6

40.8

83.5

Note on the Secular Motions of the Node and

Perigee of an Artificial Satellite

By Yoshihide Kozai

i) | ^ (l—

The values of the coefficients of the second

and the fourth harmonics of the earth's gravitational potential, which I derived (Kozai, 1959a)

from the motion of the node and perigee of

Satellite 1958 /32, the first Vanguard, are quite

different from those obtained by other authors.

Recently I discovered an error in my formula

for the secular motion of the perigee as it

appeared in the previous paper. Instead of

we must read

This change should give more reasonable

values for the coefficients. In the meantime,

however, I have derived more complete formulas for the secular motions of the node and

perigee, which are applicable to any satellite

even if its eccentricity is not small. They are

of closed form as far as the terms of order

of Alfa*.

The new formulas are:

Here n is the anomalistic mean motion; and

p=a(l—e2); i and e are their mean values

over an entire orbital period; and a is determined from the equation

rra*= (

I used the following data of Satellite 1958 £2

for Oct. 16, 12h 27m (U.T.), 1958, as given in

Special Report No. 22 (Kozai, 1959):

fi=—3?014 66 per day,

ci=4?404 05 per day,

n=3862?640 per day,

g=0.190 00,

t=34?250,

and derived the following values:

t

 Y 2

-prn co s l (j~2sm V V+

2

—[=0.94X10~s.

These values are reasonable and are not very

much different from those obtained by other

authors.

67

Anticipated Orbital Perturbations of Satellite

1959 62

By Yoshihide Kozai and Charles A. Whitney

The orbit of Satellite 1959 82, the "PaddleWheel," is significantly affected by lunar and

solar perturbations. Thus, Satellite 1959 82 is

unique among satellites launched to date, and

we believe it worthwhile to publish the present

analysis to provide a basis for anticipating its

orbital behavior.

Starting from the orbital elements provided

by the National Aeronautics and Space Administration for September 3, 1959, we have

carried out numerical integration to predict the

future behavior of this satellite. These integrations, based on a variation of parameters,

are preliminary in nature. Techniques for

a more precise analysis of the orbit are in

preparation.

The equations employed are outlined below.

The technique of integration was a simple one,

taking the semimajor axis as independent

variable.

The perturbation equations

Solar and lunar perturbations.—Variations of

the orbital eccentricity due to the moon and the

sun are expressed by the equation (Kozai,

1959c),

de_ Vl—e

2

bR , .

dt na?e do/

where the principal terms of R are:

R=— e

2

a

2

 n% cos4 \ cos4 ^ cos 2 (Xo—a>—fi)

1 / 3 \ 2)) sin2

il 1—^sin2«Jcos 2«

1

 2 / •

2 "'C 'c \

i

sin2e cos4Kcos2 (co+fl—^

—sin i cos2 ^ sin 2e cos (2w+Q—S

2

5in2ecos4jr cos 2 (w+0)

z

sin icos2^sin2«cos (2a)+fi) V

sin i cos2 - sin e cos2 ^

COS (2A^—2w—! (2)

Here, no is the mean motion of the sun; n^, that

of the moon; e, the obliquity; XG, the mean

longitude of the sun; m^, the mass of the moon

(the unit is the mass of the earth); and fl<£, the

longitude of the ascending node of the moon

with respect to the earth's equator. On the

right side of equation (2), i, Q, and o> may be

considered unaffected by the moon and the

sun.

As the semimajor axis does not change

rapidly due to the moon and the sun, the variation of the perigee distance q is given by the

equation

dq

Jt

de

a

di' (3)

Oblateness.—The rates of change of the argument of perigee and the right ascension of the

ascending node were computed from the usual

first-order equations,

do} J ,n o c * S «\

M=-2 (2-2.5 arft),

dQ J

7\7 = 2 co s

dN p

2

(4)

where N is tbe revolution number, and p—

G9
70 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

a(l—e

2

). The value J= .0016230 was employed, as well as the equation,

P=.0586745a3/2, (5)

relating the orbital period in days to the semimajor axis in units of 6378.388 km.

Atmospheric Drag.—The effects of atmospheric drag on the orbital energy and the perigee height were introduced in the following

manner.

The effect on perigee height was computed

from the approximation

dq=H

dQ ae

(6)

where H is the atmospheric scale height. This

relation can be transformed to the equation

da Kae+l' (7)

where K=H~1. This approximation is valid

for orbits of high eccentricity, but clearly it is

very poor for low eccentricities. We do not

feel that the present calculations would be

significantly affected by a more precise computation of dqjda.

The drag effect on orbital period was evaluated with a formula derived to be valid for

a wide range of eccentricities.

The loss of energy due to drag may be written

as

=-^ ApuPds, (8)

where CD is the drag coefficient; A is the satellite's effective cross-sectional area; p is the

atmospheric density; w is the orbital velocity;

and ds is the differential of distance along the

orbit.

For the orbital velocity we substituted the

value at perigee,

\AJ It. , -

a{l — e)

For ds we used the equations,

ds

dr sin v

dv l-\-e cos v

On the assumption that v, the true anomaly, is

much less than one radian in the region of

significant drag, we derived the equation,

ds=a(l-e) (10)

We further assumed that the atmospheric

density above perigee can be represented by

an exponential function of height, writing

p(»)=p(g) exp (-K(q){r(v)—q)).

We then found that

Integrating over true anomaly and employing

the relation

da=—fa du, mk2

 '

where m is the satellite mass, led to the following expression for Aa, the change of semimajor

axis per revolution:

A

m

(12)

We note that the integration around an orbit

is performed with an exponential atmosphere.

However, because of the wide range of perigee

heights produced by the lunar and solar

perturbations, it was not sufficiently accurate

to assume that the entire atmosphere was

isothermal.

We employed the following formulas for

p(g) &nd K(g):

p(q) =7.94X10-" exp [—161.55(g—1)

+2.3029 exp (—138.46(2-1)+2.8)],

X(2) = 161.55+318.86

exp (-138.46(2-l)+2.8), (13)

where p is in gm/cin3; g is in units of 6378.388

km; and K is in km""1.

Table 1 lists the values of log10 p derived

from this model and from the Smithsonian
ANTICIPATED ORBITAL PERTURBATIONS, SATELLITE 1959 8 2—KOZAI AND WHITNEY 71

TABLE 1.—Atmospheric densities

Height

(km)

150

180

210

240

270

logio p(gm/cm3)

Approximation

— 11. 12

-11.7 5

—12. 24

-12 . 65

—13. 02

Model 4

-11.0 8

-11.7 7

— 12.29

-12.7 3

—13. 09

Astrophysical Observatory Interim Model Atmosphere No. 4 (Whitney, 1959).

Numerical results

The perturbed orbit has been calculated from

the following initial orbit:

Epoch 1959 September 3.150 GMT

a=4.3446

e=.76O4

i=47?10

w=41?66

fi=55?62

The results are summarized in figures 1, 2,

and 3. The arrows indicate the date of

satellite launching. These figures show integrations for two values of Ajm, in order to

indicate the sensitivity of the orbit to variations

of atmospheric density and effective area of the

satellite.

The semiannual variation of perigee distance

is produced by the solar perturbation and does

.04

.03

.02

n i

:Ait \

—_ i

\

\w

 A / \ \

/

/

A / M

1

\

\

\

-.20

1

A/M.

\y

V

\

\

\

1

1

1

1

t

-

.05

• v

\

1

—

-

O 100 200 300 400 500 600 700

DAYS FROM

SEPTEMBER 3,1959

FIGURE 1.—Expected perigee distance of 1959 52 measured in

units of earth radii.

10

10.

100 200 300 400 500

DAYS FROM

SEPTEMBER 3,1959

FIGURE 2.—Expected acceleration of Satellite 1959 52 given in

terms of the decrease of semimajor axis (in earth radii)

per revolution.

2.00

_

- t

-

-

1

—•^ .

A/M

A/M«

«.20 \

r

.05

\

\

I

_

| -

-

I

100 200 300 400

DAYS FROM

SEPTEMBERS, 1959

500 600 700

FIGURE 3.—Expected semimajor axis of 1959 52 measured in

earth radii.

not drastically affect the satellite lifetime.

The lunar perturbation produces the rapid

drop of perigee around 600 days after September

3, 1959 and ends the life of the satellite.

The two "acceleration" curves of da/dN,

plotted logarithmically, are essentially mirror

images of the curve of perigee height.

From observations of this satellite during the

first month of flight, Dr. Don Lautman has

derived an acceleration of

w=-1.5X10- 4 earth radii.

dN

This value falls between the plotted curves.

On the Effects of Image Motion on the Accuracy

of Measurement of a Flashing Satellite

By J. Allen Hynek

Artificial satellites carrying flashing lights

that can be triggered by internal programming

or by command from earth have frequently

been proposed as a means of increasing the

geodetic usefulness of the satellites. Individual

flash durations of less than a millisecond, or a

pattern of such bursts, have been suggested.

According to this proposal, such flashes would

allow us to fix the linear position of a typical

close satellite to within 25 feet, and simultaneous observations of a given flash from two

or, preferably, more stations would not only

permit us to fix the position with a high degree

of accuracy, but would also obviate the necessity

for precision timing of the observations.

These suggestions, insofar as they concern

positional accuracy, overlook salient astrometric facts relating to atmospheric unsteadiness and image motion, which can cause the

instantaneous photographic position of the

satellite image to differ from its "true" position

by easily as much as 2 or 3 seconds of arc.

Observers have long known, for example,

that if exposures are too short—under a minute

of time—the accuracy of measures of stellar

parallax is apt to suffer. The mean position

of the wandering image will not be the same as

the centroid of an image formed by a longer

exposure. Instantaneous visual measures will

likewise suffer. It has also been recognized

that asteroid positions obtained from relatively

long exposures are more accurate than those

made visually by filar micrometer, since the

photographic method integrates the random

 1

motion of the image, while the visual observation does not.

The motion of a stellar image, apart from

its scintillation (defined solely as changes of

brightness with time, independent of image

size or motion), can be measured in a number

of ways. For measurements in the focal plane,

motion pictures of an image and the associated

focal-plane reticle can be made, or the images

can be allowed to trail on the film by diurnal

motion or by motion of the film itself. The

stellar image can also be photographed extrafocally, preferably by rapid-sequence photographs. The successive frames then show the

motion of the motley array of bright-image

elements, each arising from a separate portion

of the incoming wavefront, warped by atmospheric irregularity in the neighborhood of the

telescope. From the motion of the elements

of the extra-focal image, the motion of the

centroid of the focal image can be derived.

Observations of the "dance" of stellar images

have been made by many astronomers. As

part of a research program on fluctuations of

starlight, carried out several years ago by a

group associated with the writer at the McMillin

Observatory,2 R. Hosfeld (1954) made quantitative measures that apply directly to the

problem of flashing satellites. It should be

pointed out that the numbers obtained are

characteristic of seeing conditions in the Ohio

Valley region but may not be representative for

the entire country.

It is particularly instructive to study the motion of the two small, extra-focal image elements

formed by admitting starlight to the telescope

through two relatively small, equal apertures;

Hosfeld used two circular apertures of 3-inch

diameter whose centers were separated by 9

i Private communication from J. Ashbrook.

> Work sponsored by Geophysics Research Directorate, Air Force

Cambridge Research Center (Contract No. 19(604)-*l).

73
74 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

inches. The separation of these two elements is

related to the motion of the focal image, in that

the latter motion is one-half the motion of the

two extra-focal images as measured by the difference between their maximum and minimum

separations.

It is also instructive to observe focal-plane

images directly. Motion pictures of the movement of focal images of stars reveal a surprising

amount of change of image structure, and of the

motion of the "centroid" of the image. Deviations of an image from a mean position can also

be recorded by allowing an image to trail on a

photographic plate, or by having a film or plate

move Linearly while being exposed. These and

successive-frame photography of extra-focal im3"

ages showed that the average image motion is

2.2 seconds of arc during night hours, and 3.1

during the daytime. The random motion of

Capella during the daytime is shown in figure 1.

The consecutive points are separated in time by

%2 second; points can be as far apart in

distance as 5 seconds of arc.

The image of a satellite flash, therefore, may

appear at a position several seconds of arc away

from the "true" position of the satellite. Simultaneous observations from two or more stations

will not necessarily tend to cancel the effect; if

the errors are of opposite sign, they can yield a

position error as great as 10 seconds of arc.

A pattern of bursts of individual short flashes

would, in the mean, tend to produce a reliable

toO

I CO

xo

I -Z ~.U €CP 0

ztu

CO

- I '

- 2"

- 3"

/

1

 — ..

— .

/

A

—

--7

/

f

/

^ ^

'

¥/—1

Aya

it

P

/

v

f 1

!1/

/

/

7

r

/

—

W

ahK/

/

.

/

7

- 31

- 2" 0" I"

EAST-WEST

SECONDS OF ARC

2"

FIGURE 1.—Random walk of Capella during two seconds of time.
EFFECTS OF IMAGE MOTION ON MEASURING FLASHING SATELLITE—HYNEK 75

position, although the position of each individual flash would be subject to large error. It

would appear unfortunately, therefore, that as

long as satellites must be observed through an

appreciable atmosphere, the use of very short

flashes for geodetic purposes will tend to increase rather than to decrease the positional

error, if accuracies of a second of arc are required.

The study of image structure and motion

carried out by the Ohio State group and others

shows that such motions are to be ascribed to

atmospheric disturbances in the general neighborhood of the telescope, and not to the disturbances at the heights of the order of the height of

the tropopause that cause scintillation. That

is, the positional errors under discussion are not

introduced by stellar twinkling, as might at

first be thought, but by local wavefront changes.

Since these atmospheric disturbances are local,

a proper location of the observing sites would

help to minimize the effects of the disturbances.

Several independent lines of evidence indicate

that stellar scintillation does not cause shifts in

image position. The immediate cause of scintillation is the play upon the telescope objective

of the interference pattern induced by the atmosphere. One can easily observe this by placing his unaided eye at the focal plane of a telescope of moderate size that is trained on a

bright star. The entire telescope objective is

then seen to be illuminated. If the observer

uses his eye as a field lens, he discerns the

rapidly moving and changing pattern of starlight.

A more slowly moving pattern of light, suggestive of a viscous flow, is apparent if the

image is examined at a point slightly beyond the

focal plane of the telescope. These relatively

slow-moving light patterns represent deformations of the wavefront responsible for image

motions and explain the fact that elements in an

extra-focal image tend to persist in form and

position for large fractions of a second, quite

unlike the kaleidoscopic changes seen on the

objective itself which even successive photographic exposures of 0.01 second are generally

unable to "stop."

The distinction between image motion and

stellar scintillation can be illustrated in many

ways (Hosfeld, 1954). For example, the images

of double stars show random scintillation but

coordinated motion; with increasing zenith distance, scintillation increases more rapidly than

does image motion; we can induce image motion

artificially, but not scintillation except by interfering with the beam at relatively large distances.

One might suppose that since the images of

components of a double star move coordinately,

as do indeed the images of stars in a field nearly

a degree across, the flashing satellite and the

stars would exhibit the same image motion.

This supposition, of course, does not hold, since

the stellar exposure would obviously be many

times as long as that for the satellite flash, and

the image of each star would therefore represent

the integral of many excursions from the mean

position.

In long exposures of faint stars, a larger

number of developable emulsion grains accumulate near the mean stellar position than

elsewhere. The rapid motion of images to

points away from the mean position contribute,

at most, to the slight halo around the stellar

image. For the average stellar image, 30 to

40 microns in diameter and corresponding to

many seconds of arc, the position can be

measured within an accuracy of about 2

microns. Because the centroid of a stellar

image corresponds to the true position of the

star, such accuracy is meaningful; in the case

of the image of a short flash from a satellite,

it is apt to be misleading.

The Effect of a Variable Scale Height on

Determinations of Atmospheric Density

from Satellite Accelerations

By Luigi G. Jacchia

The anomalistic period P of an artificial

satellite changes under the action of atmospheric drag. The rate of change can be

expressed very approximately (Sterne, 1958 a;

King-Hele, Cook, and Walker, 1959) by the

equation

dP cos

The symbols are defined as follows:

C.D=drag coefficient,

A=effective cross section of satellite,

m=mass of satellite,

a=semimajor axis of satellite's orbit,

e=orbital eccentricity of satellite,

E= eccentric anomaly,

p=atmospheric density.

Equation (1) assumes a stationary atmosphere; the effect of atmospheric rotation

has been evaluated by Sterne (1959). The

only approximation made in equation (1) is

that the orbit can be defined by a set of Keplerian elements in the course of one revolution;

in the general case, therefore, the error arising

from this approximation is entirely negligible.

Useful formulas for computing atmospheric

densities from satellite accelerations can be

derived from equation (1) on the assumption

of a spherically symmetrical atmosphere in

which the density varies exponentially with

height (Sterne, 1958b; Groves, 1958; KingHele, Cook and Walker, 1959). The procedure consists in replacing p by the expression

=pp exp (2)

(where pP is the atmospheric density at perigee;

r the geocentric distance; q the perigee distance;

and H a constant, the density scale height), and

in expanding the integrand as a power series of

e. The integral can then be evaluated in terms

of Bessel functions of the first kind of imaginary

argument In(x), with z=ae/H. For small eccentricities, the Bessel functions can be replaced

by their expansion at the origin; for larger

eccentricities, use is made of their asymptotic

expansions (King-Hele, Cook and Walker,

1959).

Formulas of the type just described have been

used by all investigators in deriving atmospheric densities from satellite accelerations.

If the density scale height H varies with height,

instead of being a constant, the use of such

formulas causes a systematic error. It is the

purpose of this note to evaluate the size of this

error.

Let us assume, for simplicity, that H varies

linearly with height, and that Hp is the value

of H at perigee; we shall then have

(3)

and, since by definition

H

1 dp

"d? p

we obtain, by integration,

(4)

The gradient /3 of the density scale height is

nondimensional quantity. Recent atmo s77
78 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

pheric results (Jacchia, 1960a) have shown that

at the height of 400 km /3«+0.1 in the dark

hemisphere and |8«+0.2 in the center of the

diurnal bulge. The density scale height itself is of the order of 55 km at night and 72

km in the diurnal bulge, for the same height

of 400 km above the geoid.

Replacing p in equation (1) by the expressions of equations (2) and (4) respectively, and

using the same values for pP and HP in both

cases, we obtain two values of -p, whose ratio

R is 1 for j8=0 and > 1 for positive values of 0.

The explicit value of R is

R=

where

(l-ecosE)1

It should be apparent that R— 1 is the relative

error that is committed when pp is determined

from equation (1) under the assumption that

0=0.

The values of R to be found in table 1 were

computed by numerical evaluation of the

integrals of equation (5), with the value,

Hp=0.01 earth's radii=63.78 km. Extensive

use was made of the "logarithmic" finitedifference method of integration introduced by

the author (Jacchia, 1955), which is highly

advantageous when the integrand is a nearexponential function.

It has been assumed at the outset that for

values of Hp different from 63.78 km, R must

differ from the values given in table 1. Sample

integrations with #,,=31.89 km seem, however,

to reproduce table 1 to the last digit. The

reason for this fact is not obvious to this writer.

No serious attempt has been made to prove that

di?/d#p=0; the problem is hereby left to an

investigator endowed with greater persistence

or deeper mathematical insight.

It will be noticed that, while 5—1 is nearly

proportional to 0 for any given value of «, its

behavior is quite different when its variation in

function of e is considered for a given 0 (figures

1 and 2). For e=0, we must obviously have

R=l, irrespective of 0; even an extremely

small eccentricity, however, will make R

considerably different from unity when 0?^O,

and for any given value of 0, R reaches a maximum for e«0.02. For greater values of e, R

becomes a little smaller and rapidly approaches

a nearly asymptotic value, which is practically

reached for e=0.2.

It is a pleasure to acknowledge the expert

help of Miss J. R. B. Carmichael, who performed most of the numerical integrations.

TABLE 1.—Values of R for Hv=63.78 km

e

0.00

0.01

0.02

0.05

0. 10

0.20

0.40

0.60

1.00

0 = 0

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

1.000

Values of R

0=0.1

1.000

1.032

1.050

1.044

1.040

1.039

1.039

1.039

1.039

0=0.2

1.000

1.061

1. 100

1.095

1.086

1.083

1.082

1.082

1.083

1.00

+ 0.2

.00

FIGURE 1.—Variation of R (see equation (5)) in function of the

scale-height gradient 0 for different orbital eccentricities.

The relative error of densities determined using formulas

that assume 0=0 is given by R— 1. Scale height at perigee,

ffB=63.78 km.
EFFECT OF VARIABLE SCALE HEIGHT ON DETERMINING ATMOSPHERIC DENSITY—JACCHIA 7 9

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

FIGURE 2.—Variation of R in function of e for /3= +0.1 . (For R see legend to fig. 1.)

Scale height at perigee, Hv= 63.78 km.

A Second-Order Solution of Vinti's

Dynamical Problem

By Imre G. Izsak

The accurate description of the motion of artificial satellites has presented a new problem in

celestial mechanics. It is true that satellites of some other planets—for instance, the five inner

satellites of Jupiter and the five inner satellites of Saturn—have quite significant perturbations

caused by the oblateness of the parent planet, but the obtainable planetocentric accuracy of the

observations did not demand a highly sophisticated theory. In the case of artificial satellites of

the earth, however, the accuracy of the topocentric observations made by the Baker-Nunn cameras

amounts to a few seconds of arc. That means we have a geocentric accuracy of about one second

of arc, while the time record is accurate to a few thousandths of a second. It would be very difficult

to construct an analytical theory that would represent the motion with such an accuracy, either

for the computation of exact ephemerides or, what is even more important, for the best possible use

of the observations. The oblateness perturbations are only a part of the factors that influence the

satellite motion, of course. For instance, we know that low-orbiting satellites suffer large perturbations of an "irregular" character caused by atmospheric drag, and that high-orbiting satellites are

affected by luni-solar perturbations. Solar-radiation pressure is another factor complicating the

situation, especially in the case of satellites with small mass/area ratio. The present paper deals

with the motion in the gravitational field of the earth.

Although a great many papers devoted to this problem exist in the recent literature, the most

comprehensive investigations were presented in papers by Brouwer (1959), Garfinkel (1959), and

Kozai (1959b), all of which appeared in the celebrated November 1959 issue olAstronomicalJournal.

These authors treated, by different methods, the first- and second-order secular perturbations, as

well as the first-order long-periodic and short-periodic perturbations of the orbital elements, where

order refers to the oblateness parameter. Less related to the present subject is the very powerful

semianalytical method for the computation of gravitational perturbations developed by Musen

(1959).

The distinction among the different kinds of perturbations has significance not only from the

theoretical but also from the practical point of view. In theoretical work, the secular, long-periodic,

and short-periodic terms are usually obtained by different techniques. In practical work, for

instance in an orbit improvement program, the short-periodic terms are used for the reduction of

the real motion of the satellite, as reflected in the observations, to a certain mean elliptical motion.

The secular and long-periodic variations are still present in this mean elliptical orbit and can be

obtained with a high degree of accuracy by the use of many revolutions of the satellite.

If the coefficients of these terms are expressed as functions of the orbital elements and the geophysical constants, the latter can be determined. The next step beyond the work of Brouwer,

Garfinkel, and Kozai would be to obtain the second-order short-periodic perturbations. As far as I

can see, this would be a rather tedious undertaking with their methods, because it involves several

multiplications of Fourier series. In this paper I shall adopt a quite different approach, based on a

remarkable approximation to the actual gravitational field of the earth, originated by Vinti (1959).

81
8 2 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 VOL

- «

Vinti's dynamical system

On the assumption that the earth is an axially symmetric oblate body, its potential can be developed

into the series

where /i=/w«=398.618±.003 Mm3 ksec"2 (O'Keefe, Eckels, and Squires, 1959),

/ is the gravitational constant,

mE and as are the mass and the equatorial radius of the earth,

the J% denote geophysical constants,

r and 5 are the geocentric distance and declination of the satellite,

and the P»(sin 5) are Legendre polynomials.

According to the latest determination1 by Kozai (1961b), we have the following values of the Jn:

J2= (1.08219 ± .00002) X 10~3, J3= - (2.29 ± .02) X 10~6,

J 4 = — (2.12±.04)X10-6, Js=-(2.3±.2)X10- 7.

Thus J3, Jt, and J5 can be considered as of the second order with respect to J2.

Vinti has shown that if the relations

Jtn = (-1 )

 n+1Jn

2, Jf*+l = (-1) Vi J5

are valid, the dynamical system f = — VV can be solved exactly; or to be more precise, the integration

of the differential equations can be reduced to quadratures. It is expedient to note here that the

representation of the coordinates as functions of time or of another independent variable may still

be a difficult problem, involving the solution of a system of simultaneous equations. This is sometimes called the inversion problem and will play an important role in this paper. From the values

of the Jn given above it follows that for the gravitational field of the earth Vinti's relation JK=—J\

is not satisfied; instead we have

=-. 95 X10"6.

On the other hand, J\ is practically zero; hence it can not be used to generate the coefficients of the

odd harmonics. Nevertheless, the potential function

(2)

containing only the even harmonics, is a very good approximation to the potential function (1), the

difference

. . A

being of the second order in the oblateness parameter J2. We call the differential system, r = — VK,

Vinti's dynamical system. If this has been solved, the perturbations due to V— V can be treated by a

first-order method, that is, without multiplications of Fourier series.

1

 Note added on this appearance of paper: Recent values of the geophysical constants as determined by Kozai

(1962) are:

J,= (1.08248±.00004) X 10"» J3=-(2.562±.007) X 10-«

J4=-(1.84±.O9) X 10-« J,= -(6.4±.7) X 10-«

J.= (3.9±.9) X 10-7 J 7=_(4.7 ± .l) x 10-*.

It follows that D4= —.67 x 10~»; therefore, Vinti's approximation is better than expected two years ago.
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 83

Oblate-spheroidal coordinates

At this point it is necessary to introduce oblate-spheroidal coordinates, which are defined by the

equations

x= *)Jp2-\-c2 -yjl — a

2

 cos a

y=^Jp*+c2 T/1 —a

2

 sin a r= VP 2+ C 2( 1 — a

2

)

Z=pcr. (3)

It follows that

The coordinate a is the right ascension of the satellite, and the surfaces a=constant are meridian

planes.

The equation of the surfaces p=constant in rectangular coordinates is

x*+y2 z

2

p2+c 2-t- p 2 *•

Consequently, these equations describe oblate spheroids, the intersections with the meridian planes

being confocal ellipses of linear eccentricity c, and semiminor axis p.

Similarly, for the surfaces a=constant we have the equation

-a

2

) c

which describes one-sheet hyperboloids of revolution, the intersections with the meridian planes being

confocal hyperbolas of linear eccentricity c. The geometrical meaning of the coordinate a is the sine

of half the angle between the asymptotes.

In the present application the constant c is meant to take care of the earth's oblateness, and is

necessarily small relative to the equatorial radius of the earth. This means that, far enough from the

origin, the coordinate surfaces p=constant and o-=constant are nearly spheres and cones, respectively. Solving the equations (3) for p2

 and a

2

 we get

2

the developments in spherical coordinates,

p=r-l 1— k(^) (1— sin2

 5)—1(~) (1—sin2 5)(1—5 sin2 5)—... \ ,

<r=sin &< l+jr(-) (1—sin2 5)+-(- ) (1—sin2 5)(3—7 sii

are easily derived from the preceding relations.

A

Vinti obtained the potential (2) by asking himself: What is the most general solution V{p, a) of

the Laplace equation

dp
84 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

for a gravitational potential which makes the Hamilton-Jacobi equation

1 c2 \f i

separable in these oblate-spheroidal coordinates? Requiring further that the solution should have no

singularities except a pole at the origin, and that as well as axial symmetry it should also possess

equatorial symmetry, he found that V(p, a)= , , .,) where b is a constant. In order to see that

P "r 1

this is the same as equation (2), let us expand it into a series of spherical harmonics.

Obviously, bptf+ct^-^bReiip+ica)-1}- But

(P+ic<7)2=(p2-c2o

2

)+2tcp(T=(r2-c2)+2tc2=r2{l—2(—ic

so that by the definition of the Legendre polynomials

sin icr"1

)

2

},

sin 8+(-

n-0

(-icr-1)»PB(sin 8).

Taking the real part of this equation we get

2B(sin 8) \

J

and comparing it with the equation (2) we see that we must choose

The above value of J 2 and the value a£=6378.388 km give c=209.828 km.

Separable dynamical systems of Stackel's type

Vinti's dynamical system belongs to an important type of solvable problem in analytical dynamics,

known as dynamical systems of Stackel's type. The essentials of these systems can be summarized

as follows. Let

011 (<Zl) • • • 01n(<Z]

be a nonsingular matrix, in which the elements of the jth row depend upon the ,7'th coordinate only.

We call the inverse ^=* - 1 of such a matrix a Stackel matrix, the elements of which depend on all

the coordinates &>••• > qn, in general.

As was shown by Stackel (1891) (see Charlier, 1927), a Hamilton-Jacobi equation of the form

can be solved by the method of separation of the variables if, and only if, the coefficients a}{q) constitute the first (or any) row of a Stackel matrix such that a/<z)=iAi/<z)> a11^ the potential energy can

be represented in the form

where the functions X]{<L)) depend on the jih. coordinate only. Such a dynamical system is said to
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 85

be of Stackel's type. For our present purposes it is sufficient to see that the separation follows from

these conditions.

Indeed, the function TF(<z|a) defined by the equations

VS^^)i(^) q} (4)

contains n arbitrary constants, cti=h, a2, . . . , an, and is a complete solution of the HamiltonJacobi equation:

2

According to Jacobi's theorem, the solution of the canonical system p>=—->—> 2=5-— is then given

ogj opj

implicitly by the equations

bW(q\a)

where the fij are the other n constants of integration.

We can also say that a dynamical system with the Hamiltonian function

/-I

has a system of n quadratic integrals

\ S ^(2)^+ S Mq)XJ(qJ)=ai, (i=l, . . ., n)

I i=i j=i

in involution.

The separability of the variables is a formal property, and it occurs only in appropriate coordinate systems.

In the most important practical applications the functions of qj under the square root sign in

equations (4) must all be positive inside closed intervals It and must vanish in the first order at their

endpoints. Then the solutions qj=qj(t\a\(i), pi=pj(t\a\fi) of equations (5) can be represented

for — 00 <£ < + 00 by multiple Fourier series (at least in general). This is the ideal of celestial

mechanics. However, in the much more difficult problems with which it deals, such a representation can be only an approximation, valid only in a finite interval of time.

I shall now describe briefly an important special case of Stackel-type dynamical systems. The

natural context for the treatment of these dynamical systems of Liouville's type seems to be the

apparatus of the Lagrange equations:

Y-—=0

It can be shown (Whittaker, 1959) that if the Lagrangian function has the structure

(6)
86 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

where w(g)=S ut(([{), and the functions w<(g<), Vjiq

only, the Lagrange equations possess the "integrals"

depend on the indicated coordinate

(7)

Here h means the energy constant, the cu . . ., cn_x are independent constants of integration, but

n-l

Cn=— 1=1

Actually, in Whittaker's book we find the not-quite-fortunate specialization Vj(gj) = l; however, it is

easy to generalize his procedure.

The importance of this type lies in the fact that the inversion problem associated with it can be

much simplified. Indeed, if we introduce a new "independent" variable T by the differential relation

dt u(q)

where T denotes a conveniently chosen constant, the "integrals" (7) take the form,

r

2

(8)

These equations can be integrated again immediately, giving the relation

(9)

where the d} represent n new constants of integration. Each of the equations (9) contains only a

single q]} instead of containing them all as in equations (5) in the case of a general Stackel-type

dynamical system. In other words, a dynamical system of Liouville's type of n degrees of freedom

splits up into n dynamical systems of one degree of freedom. After the coordinates q} have been

obtained as functions of the parameter T, the time dependence of the motion is given by the integral

of equation (8); that is,

S (TUi(qi(T))dr=T(t-t0)-

i=lJo

The Hamiltonian function belonging to the Lagrangian function (6) is

H=l {u(q)\

It is easy to see that to write it in Stackel's form we have to choose:

U3V3

l\

0

0

0

0

0

0

V*

0

0

0

0

0

0

0 »„_,

UJD —vn
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 87

and Xi=

For n=2, Stackel's type is not more general than that of Ldouville.

The integration of Vinti's dynamical system

Let us now proceed with the integration of Vinti's dynamical system. To convince ourselves that the

Hamilton-Jacobi equation

is of Stackel's type, let us put

1 c

2

2(P

2

+c 2) 2(p*+c2)

2

c?o* 1 1

1-a2 2(l-<r2) 2(l-a2)

2

0 0 \

,and X2=O

(11)

Then, det *=

4(p2+c2) (1- except for polar orbits, and

P

2

+c2

1-CT 2

0

P

2

+cV

0

(rThe first line in this Stackel matrix agrees with the coefficients in the Hamilton-Jacobi equation (10).

The second line permits us to write down immediately the quadratic integral,

For <r=0, that is, for an equatorial crossing, the expression on the left side is positive. This is why

we are permitted to write the constant of integration as C2. The value of C is close to that of the

magnitude of the nonconstant angular momentum, the difference being 0(c2).

A

The third line corresponds to the square of the linear integral Pa=0, which results from the fact

that a is an ignorable coordinate; 6 is the 2-component of the angular momentum, and is positive or

negative according as the motion is direct or retrograde.

In order to obtain a complete solution of the Hamilton-Jacobi equation (10), substitute the

expressions of <£« and Xj given by equations (11) into the general formulas (4). By a suitable choice

of the lower limits of integration we get

(12)

where P(P)=2V+2MP 3-(C 2-2C 2 A)P 2+2C 2 MP-C 2(C' 2- & 2)

and Q(a) = — 2c2haA— (C2—2c2A)o-2+(C2 — G2) are two quartics fundamental in this theory.

636-014—6S
88 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. a

The qualitative nature of the motion depends on the character of the roots of the quartic equations P(p)=0 and Q(<r)=O. There are several cases to distinguish according to different values of

the constants of integration h, C, G. In this paper, however, we are interested only in the kind of

motion for which p(t) oscillates between two real roots pi and p* that are bigger than the equatorial

radius of the earth, and <r(t) oscillates between two real roots —ax and -\-ai that are smaller than 1.

The condition pi=P2, that is, p(t) =pu characterizes "circular" orbits, and the condition o-x=0, that is

<r(t)=O corresponds to orbits in the equatorial plane of the earth.

Because for c=0 (two-body problem) the motion would be elliptic, parabolic, or hyperbolic

according as the constant of energy is negative, zero, or positive, it is reasonable to assume that in

A

our problem h<CO. Further, we must have Q((T)^0 during the motion, especially for <r=0, and

this condition gives CP^G2. Then it is clear that the equation Q(a)=0 will have four real roots

*u —<ri, <*2, —°2 with the following properties: 0^<TI<1 , O-2^>1. For c=0 the equation P(p)=0

would have two positive roots and zero as a double root. The roots for small values of c can differ

from these roots only by small amounts; that is, we shall expect two finite roots pi and P2, corresponding to "perigee" and "apogee" respectively; furthermore, we shall expect two infinitesimal roots

Pa and pt, which are in general complex but become real in the case of very small inclinations. We

shall have opportunity to return to this point later. It would hardly be worth the labor to obtain the

exact, and necessarily rather complicated, algebraic conditions needed for the characterization of all

these circumstances.

Let us return to the complete solution (12). Differentiating with respect to h, C, G, we should

consider the lower limit pi as a function of the quantities pi(h, U, G). But because the integrand is

zero for p=pi, the result is the same as if pi were independent of h, C, G. We obtain the following

implicit equations for the determination of p(t), a(t), a(t):

=t-t, (13)

- « , (14)

The sign of the square root of P(p) has to be considered as positive if p is increasing from pt to pi,

but negative if p is decreasing from ps to p2. A similar requirement holds for the square root of

Q(o), the limits being — o\ and + <ri. The symbols t, w, and Q denote three constants of integration

whose meanings are related to the time at perigee, the argument of perigee, and the right ascension

of the node in a Keplerian motion, respectively. The caret above the letters serves to distinguish

the canonical set of elements h, C, G; —t, w, Q from other elements to be introduced later, which will

be more convenient in the development of the theory. The chief merit of the canonical elements

(automatically introduced in the method of separation of the variables) lies in the fact that the variations of these elements in the case of a perturbed motion are described by a canonical system of

differential equations.

To represent the coordinates as functions of time we would have to solve equations (13) and

(14) for p and a, then substitute p(t) and <r(t) into equation (15). This is a very hard task to carry

out and, by the way, an impractical procedure. Instead, we relate p and a to another "independent"
SECOND-ORDER SOLUTION OF VINTTS DYNAMICAL PROBLEM—IZSAK 89

variable v, analogous to the true anomaly in a Keplerian motion, as follows. If we put

dp

J PI =v, (16)

where r is a constant (depending on h, C, Q) to be defined later, equation (14) becomes

Jo

For a certain p=pi we will havez;=0, while <r=0 will correspond to vo= — Fw/C. If we introduce and

use Jacobian elliptic functions, we can easily invert relations (16) and (17) to obtain p and a as

functions of v. Then equations (15) and (13) take the form

67 p dv c*G r< dv

t-t=± p

2

dv+% o*dv. (19)

1

 JO A Jr 0

The first of these equations gives a as a function of v; the second is equivalent to Kepler's equation in a Keplerian motion.

Without going into details, we remark that the above simplification of the inversion problem is

made possible by the circumstance that a is an ignorable coordinate. If we first carried out the reduction to two degrees of freedom by using the integral #a=(p2

+c 2)(l—o^a^Q, we would then obtain a

dynamical system of Liouville's type.

Introduction of elliptic functions

 2

We propose next to reduce the elliptic integrals of the first kind in equations (16) and (17) to Legendre's normal form.

For this purpose let us seek a linear transformation with real coefficients of the complex p-plane,

<

20>

such that the roots pi, p2, p3, and p4 of the equation P(p)=0 are mapped onto the points £i=l, €2= —1,

£3=i/m, and ^i=—i/m, respectively. The quantity TO is considered to be real or pure imaginary,

according as the roots p3 and p4 are (conjugate) complex or real. Because three points and their

images already determine a linear transformation, this number TO cannot be chosen arbitrarily.

Indeed, the cross ratio is an invariant of the linear transformation; that is,

Pi—Pi. P2—P3=ti—t3. {2—&

Pi—Pi'p2—p* £1—£4 "£2—$4 \l—imj

We have to determine p, d^, e% and m as functions of the roots p1; pj, p3 and p<; or if the latter

are written in the form

p1=a(l—e), p2=a(l + e), p3=a(/c—iX), p4=a(x+iX), (21)

as functions of a, e, K, and X. The first two of these equations give the definition of the orbital

3

 Detailed references with regard to elliptic functions seem unnecessary. The book of Whittaker and Watson (1958) can serve as a general reference.
90 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.*

elements a and e; although the orbit is not an exact ellipse, we can still call them the semimajor axis

and the"eccentricity. It will be shown later how K and X are determined as functions of a, e, and s.

The relations Pj=?T J (i= 1>

 2

»

 3

»

 4) ^ h tn e

 values of £, given above can be written in

the form:

p+d* —Pig*=pi,

p—d*+P2e*=f>2,

—imp+rf*—P3«* = — wip3,

tmp+d*—p4g* =impA. (22)

Solving the first two of these equations for p and d# we obtain

p .ffiZfiW(^-«). (23)

Also, transformation (20) becomes

In order to obtain convenient relations for the determination of em and m, take the sum of the products

of equations (22) by —1, 1, 1, 1, and use the factors im, im, 1, —1 in the same way. These combinations yield:

{(PI+P2) — (P3+ PO } <?* = {(pa—Pi) + (P4—Pa)im}

and

} 2—fh)im\ e*= {(P1+P2) — (P3+P4)} im,

or, taking relations (21) into consideration, we obtain:

(1—K)e*=e—\m, and (X+em)«* = (l—K)m. (24)

Eliminating em or TO between these equations, we obtain second-degree equations for the determination

of m or et, which can be written as

———2=7

 e

 — (25)

and

1 + 4 (l-K)2+(

Having the choice between two possible values of m and e+, we require that m and 0* be close to 0

and e, respectively. For e=0 both m and e* vanish.

Now let us apply the linear transformation (20) to the elliptic integral (16). The radicand can

be written in the formP(p)=2A n (p—pj), where

and

A=petl—d^=ae(l—el). (28)
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 9 1

As a consequence of our choice of the %Jf we have

4 t.—t n

so that

On the other hand, dp——. . >2 <Zf. Consequently, the integral (16) takes the form

^ T fg <ft . , , _. , / -2AA2

-

1

* Jo V(i-i2)(i+m2a~"'

 Wltn ^*- V (1-4X4

which is still not Legendre's normal form. That form, however, can be obtained at once by putting

'0 V(I—v Ml —

where

/

. ./ - 2

^ " V (l -

and

(30)

The inverse functions TJ(P) and ^f») are now given by the definition of the Jacobian elliptic functions

with modulus k as

fl|£M and£=cn (— »

This is the point where we can determine the constant T by demanding that the period of the

functions £(») should be 2x. Because the real quarter-period of cn{z\k2) is

K

-5{1+(s)'*+(ri)'*+•••}• <31>

we must have £—cn ( — v\k? y that is,

r-£A (32)

The reduction of the elliptic integral (17) is quite immediate, as the quartic Q(a) contains

merely even powers of a. Let us write

and apply the transformation <r/oi=f. When we introduce the notation <rx/<r2=Z, equation (17)

becomes

(Vc2—^2

 \ —P (p—vo)\l2

it where r0

— TwfC as in equations (18) and (19).

means
92 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. a

It is advisable to introduce some new notations. First, because a oscillates between the values

— 0i and <ri, we can think of ax as the sine of the "inclination" / of the orbit; let us denote it by

«=sin /. Second, we shall need the real quarter-period L belonging to the modulus I, that is,

Third, we define an " argument of perigee" 3 as a linear function of v by the equation

2L , . _ T](>-G2 . v — (»+S)=—^ — (v—v0).

Then if we put &>=€»+«, the small quantity

"21 r«

will be the motion of perigee, and

e =^ r = 1 (34)

(35)

a constant close to the canonical element w.

Summarizing the results of this section, we have obtained the explicit representation of the

coordinates p and a as functions of the true anomaly v in the following form:

#+<4cn ( — vlk2)

P= )gfe ( (36)

l+e*cn (—vlk 2)

\ * /

and

/or \

(37)

These formulas correspond to r=——— and sin d=s sin (»+«) in a Keplerian motion.3

* 1 + 6 COStJ v • / r

Determination of the constants e#, fe2, I2, and e

In the preceding section we gave the definitions of the orbital elements a, e, and s, which can be

A A

thought of as three independent constants of integration replacing the canonical elements h, C, and

G. The other fundamental constants e+, k2,12, and e must be expressed as functions of these orbital

elements.

According to the equations (25), (26), and (30), we first of all have to determine the quantities

K and X in terms of a, e, and s. The coefficients and the roots of the quartic equation P(p)=Q are

related to those of Q(a)=0 in the following manner:

1

 Just before finishing the manuscript I received a copy of a very interesting paper on the present subject by Davis and Stein (I960), in which the

relevant elliptic integrals were reduced by the same transformations that I was using. In other aspects, however, the goal of these authors is quite different

from that of my paper. They were primarily concerned with the development of a satellite orbit computation program, and not with the construction of

an analytical theory in the spirit of celestial mechanics. In particular, Davis and Stein do not seem to be interested in the explicit representation of the

several fundamental constants as functions of the orbital elements.
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 93

a*(l -e 2) C^+X2) = -c 2( &2- &")/2X=eV(«7P). (38)

A

After introduction of the dimensionless, small parameter p—c/a, the elimination of A between the

first and third equations gives (1— e

2— V2

)K-\- (K2-{-\2)—P2.

Similarly, eliminating s

2

/l2 between the second and fourth equations we get

-4p2s

2K+(l-e2

-p

2

s

2

)(K

2+\2

)=p2s

2

(l-e2-p

2

8

2

).

These two equations determine the roots P3=CL(K—tX) and p4=a(ic+iX) in terms of the orbital elements o, e, and s. We obtain first

K

~(l-e2-p2)(l-e2-p2s

2

)+4p2s

2

and

or, developing in powers of v

2

,

«=^+^0-V-e-)+... (41)

and

^ ^ i e2]+ ... . (42)

(Although the present theory is confined to the order of p*, the last i>9 term will be needed in the computation of «, the motion of perigee to the order of v*.) Then

This expression shows that for very low inclinations X becomes imaginary, i.e., ps and p« become real,

as stated before. The value of the inclination for which X becomes zero is approximately given by

the formula $2=p

2

/(l—e2)

2

. For artificial satellites of the earth, v

2

 is of the order 10~3 and the

eccentricity is small, in general; hence this inclination is about 2°.

Returning now to our original purpose, we remark that the last of equations (38) written in the

j,2g2 y2s2

form 12=TZTz XTT2

 an d equation (40) give immediately

/2 rV Q.-e2

-p

2

)(l-e2-p

2

s

2

)+4p2s

2

1-e2 (l-e 2- v

2

)(l-e 2- '

for the parameter of the elliptic function in equation (37). Or, developing in powers of p

2

, we obtain

. . . (44)

The quantities e+, m, and k are given as algebraic functions of the orbital elements by equations

(25), (26), (30), (39), and (40). However, since their closed expressions are too complicated to be

useful in numerical computations, we instead seek to represent them by power series in p

2

. Putting

1 -\-/> e*= e \l+p 2

x+p*y + . . . } , w e hav e -yj^=(l-he 2)~p 2(l-e 2)x+p*[x 2-(l-e 2)y]-\ - ... . O n th e

httnd , (l-*) ^
94 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. «

p* From equations (41) and (42) it follows that (1-€ 2

)K-(K 2+ X 2)=V 2(1-2S 2)+^ = ^T- 2 (1-S 2)

(I_8s 2-e 2) + ...,and [(1—«

2

) K -(K 2 + X 2)]IC=^2(1-S2)(1-2S2). Then the reciprocal of equation (26)

becomes -v

2

(l-e2)x+p*[x2-0—e2)y]+ . . . = -r 2(l-2s 2) -, _ (l-s 2)[(l-5s 2)-(l-s 2)e 2] + . . .,

and, determining x and y by comparison of the coefficients, we obtain:

(45)

The equation (25) for the determination of m as a function of e, K, and X, contains also the

first power of X which, because of the branch-point at s

2=v2

/(l— e2)

2

+ • • •, cannot be represented as

a power series in p

2

. In other words, this branch-point is also a singularity of m. Therefore, we shall

take the square of equation (25). Working with power series, we can determine immediately the

parameter A:2=m2/(l+wi2) of the elliptic functions in equation (36). Putting k 2=e

2

 \v2x-\-v*y-{-. . . },

we have ( 1 -m*)2= (1-2PV =F( 1 + 3 F + • • -) = «

2

 {u'x+p^Se^+y)^-. . .} . On the other hand,

eX

From equations (41) and (42) it follows that

1 _ ^ =(13^ 2 [(2-3 S

2

j+s 2«

2

]+^£^- 4 [(l- 7 s

2

+6 S *)-(l-3 S

2

+2 S *)e 2] + • • •, (46)

and using equation (43) we get

A comparison of the coefficients gives x and y, and we obtain

T#2/>2 ,,4 .,2

fSV] + (47 )

The computation of e is somewhat more tedious. It is given by equations (34) and (32), or,

what is the same, by

(48)

The quarter-periods K and L can be considered as already known quantities, expressed in terms

of the orbital elements by use of equations (31), (33) and (47), (44). Let us transform the expression

of A as given by equation (29) into a more convenient one. Recalling equations (21), (27), and

(28), we can write

2 .. __ _ _ A £3—£4 2Aim

or

(\-e%)(el+m2) " m

Then equations (48) and (29) yield l+c=^ / ^ 2

 x

 G* JB * K. But from the last of equa-
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 95

A A

/^2 /T2 2 I \ 2 I 2

tionsj(38) we have *—= (1 —e2) ——> and equation (25) gives — —/•,_ \2__/ 2_X2y

 nence>

1

 / 1-e 2

•\(l-t)!i-((2_

L e t us develop this expression into a power series of the small parameter v*. The first radicand

is given b y equation (42), and the third can be written as-J\—2k 2, b y use of equation (30). Developing the second radicand, we use equation (46), as follows:

1 — e

2

 1

(1-K) 2-(« 2- X 2) , 2K- (^-X2)

1-e 2

_ , , 2K-(K2+X2)

The square roots of this series and of equation (42) are:

V

I—e

2

(l-K) 1-(e*-X«) =

^ 2w [(20-92s2+75s4)-(8-36s2+34s4)e2+3s4e

4

]+... (50)

and

/ O,,4

Furthermore, in virtue of relations (31), (33), (44), and (47), we have

= 1 ~4(l-e 2

)

2

 (s2+

2g2g2) +64(l-e 2

)

4

 [(64*

2

-

69*4

)+(48-544s2+614s4)e2-8s4e

4

]+.... (52)

Multiplying the series (50), (51), and (52), we obtain the motion of perigee as:

€ = 4(l-e 2

)

2 (1 2

Development of Fourier series for the right ascension and the generalized Kepler's equation

By now, the coordinates p and a are known functions of the true anomaly v and the orbital elements

a, e, and *. These coordinates were represented by the closed expressions (36) and (37), and the

relevant constants were given by the series (45), (47), (44), and (53). Our next task is to elaborate

the equations (18) and (19) to expedient formulas for the right ascension of the satellite and a generalized Kepler's equation, respectively. Since these equations contain elliptic integrals of the

second and third kind, we cannot hope to obtain concise expressions for them. In the spirit of

celestial mechanics, we propose to deduce Fourier series well adapted for numerical computations.

We would like to point out, however, that the role of these Fourier series is not the same as in the

customary constructions of satellite theories, where Fourier series are used from the very beginning

636-014—63 8
96 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.8

to obtain successive approximations of different orders to the solution. The exact solution of Vinti's

dynamical system has already been denned by equations (36), (37), (18), and (19); our problem is

"only" to bring the latter two equations to an explicit form. This can be done by developing

"known" functions into multiple Fourier series.

To begin with, let us consider the first term in equation (18), which, taking into consideration

equation (37), and using the abbreviated notation

(54)

we can write

G Ca

C" dw (55)

) Jo , 2 2{

2L i ; 2V

/ •/• 1—g2Sn2/ W\l2)

where the integral is essentially of the standard form of the elliptic integral of the third kind, and

can be expressed by the Jacobian 0-function with (in our case) imaginary argument. It is instructive, however, to deduce directly the Fourier series of the integrand by the theory of residues,

and then to integrate it as follows.

The function 1/[1—s2sn2(2|Z2)] of the complex variable 2 is doubly periodic with periods 2L and

2iU, regular at all points of the fundamental period-parallelogram 0, 2L, 2L-\-2iL', 2iL', except at

two simple poles zx and 22, where the denominator becomes zero. A little consideration shows

that, for Z<s<l, these poles must be of the form Z\=L-\-iy, Za=L-\-2iLf—iy, with 0<y<27.

Now let us determine the residues of the above function at these poles by seeking the first terms in

its Laurent series. Since the Taylor series of sn2s near zj(j=l, 2) begins with sn

2

(zi-\-8)= sn

2

z,

+ (2sn z, en z, dn z/)d-hO(S2) and 1—^sn2 Zj=0, we obtain {l—s

2

 sn2 (zj-\-8)}-1 =— {2s2sn zf cn z}

dn 2^}~15~1-f 0(1). But at the pole 21 the values of the Jacobian elliptic functions aresn z1^lfs, cn zy=

—i^l—s^js, dn 2i = •yjl — l2lsL; at 22, on the other hand, sn 22=l/s, cn Z2=—ijl—s2/s, dn z%=

—Vl — Pis*. Consequently, the residues of the function 1/[1—s

2

 sn2(2|Z2)] at the poles zt and 22 will be

and u ...9 u i9< 9J and, changing the argument from 2 to — w, those of the

integrand in expression (55) will be:

—irr/4:L , iir/4L ,„_.

k

a n d

VI— /= , /= gf5 - (56) s \1—l/s

The coeflficients of the Fourier series

l-s 2s n 2(^w|Z 2)

are given by the expression

<hn cos 2nw (57)

1 f* exp (2inw) ,

=

* /2T \dw'

f JO 1 9 9 / *'*-' 119 1

1—^snM—w\l2j

If we extend the path of integration over the entire period-parallelogram, then from the periodic

/2L \ properties of the elliptic function sn I — w\l2

) and exp (2inw) it follows that

r fr fr+TiL'lL friL'/L /»0 fr

9= + + + =(l-22n) -ra-fl^o*., . (58)

J J0 J •* Jr+riL'tL JriL'lL JO

where

g=exp (-xL'/Z). (59)
SECOND-ORDER SOLUTION OF VINTTS DYNAMICAL PROBLEM—IZSAK 9 7

The only singularity of the function exp (2inw) being at infinity, the residues of the integrand are the

values (56) multiplied by exp (2inwi) and exp (2inw2) respectively. Here Wi and w2 are the points

corresponding to zx and 22, namelyKT (L-\-iy)=^-\-iT, and ^y (L-\-2iL/—iy)=^-\-iiry-—iY, where

F=xyy , so that using the definition (59), we see that exp (2m«1 )=(-1) " exp (—2nT) and

exp (2inw2) = (—l)ng

2n exp (2nY). Then, according to the theorem of residues we have

±-. <j>=- Res (Wl)+ Res (w2)^^—J^= (-1)" {exp (-2nY)-<?» exp (2nY)} .

Comparison of this result with relation (58) gives, for

a2 (~l)n

with

r=exp (—Y)=exp(—~y\ (61)

This procedure cannot be applied to the determination of the coefficient a0, because for n= 0

the divisor 1—q

2n

 becomes zero. But substituting ?/>=0 in equation (57) we see that the constant

term in our Fourier series will be represented by the infinite series

c r o =l—2S% - (62)

n=l

A consideration of the merits of the method of residues leads to the following remarks. Besides

unquestionable elegance, the chief advantage of the method consists in giving an expression for all the

coefficients of the Fourier series. In a satellite theory, though, we represent these coefficients in

terms of the orbital elements as power series in the small parameter v

2

, and this is not easy if we base

this development on equations (60) and (61).4 For example, we had first to develop the quantity y,

then F and T=exp (— Y), the latter being approximately the tangent of half the inclination. Also,

some simple transformations of the expressions (60) and (62) should be made to show that their finite

parts (not containing v

2

) are identical with formulas for a Keplerian motion. In the case of the

p-integrals in equations (18) and (19) we had to handle even more complicated expressions. These

difficulties make it advisable to seek for a more elementary process that will yield the first coefficients

of the Fourier series in a rather immediate way. From the theoretical point of view, the arguments

v and w are the most natural ones: to the equations (36) and (37) there correspond Fourier series

whose arguments are multiples of the single arguments v and w, respectively; and equations (18)

and (19) give rise to the sum of two such series. In other words, none of these Fourier series contains

trigonometric terms with arguments of the form av+Pw; in particular, there are no long-periodic

terms whatsoever. The more elementary process we are going to follow now will make use of other

arguments, which are not linearly connected with each other but will permit us to deduce relatively

simple formulas well adapted for numerical computations.

Let us consider again expression (55), and introduce a new variable rf/f the so-called amplitude of

2L

—•K w by the definition

sn I —w\ I2 l=sin ^. (63)

* Note added on this appearance of paper. A consistent application of the method of residues seems now to me the preferable procedure.
98 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Simultaneously we will have

cn I —i/;|t )=cos y and an ( —w\ I J=Y1 — t sin*^;

moreover,

dw=7rr

so that the expression (55) becomes

r(l+e)

As shown in the theory of elliptic functions,

where

g=exp (_^7Z)=_^l + _

or, with sufficient accuracy for our purposes,

^=w+-g ( 1 + 2")

 sin 2 + i

jr_ C* d+

2ij o (l-^sin^jVl-^sin2^ '

the inverse relation of which is

I

2

 / V\ 3 Z4

ID^}[/—~-— I 1 -|—— i si n 2\p-\-j:— si n 4np— ... . (66)

The development of the integrand in expression (64) into a Fourier series turns out to be simpler,

G if we include in it the constant factor . . 7rr • First of all, according to equation (34) we have

and from the second and fourth of equations (38) it follows that \—C"""

T(l+e)2L JfrZf

• Q2 Q2 I g2

„ , J or what is essentially the same, A — A-=—J - (1— J2/*2)- Hence expression (64)

l

2

fr CPG2 s

can be brought to the form

c* yi—$2vi—p/s2<fy

Jo (1—s

2

 sin2 ^) VI — I2si

j £/s Z21 s

2

 sin2 \^

Proceeding now to its development, we put 1 _ p s m 2^, aBl ~^2 1 — p s m 2 ^ from which we obtain

V

l—P/s2 __1 ^ 1—s2_sin2jA__l /I2 l-g 2sin 2 A

2

1— / 2sin 2^ 2«21—Z2sin2^ 8 \s21 - 1

2

 sin2 ^ / * "
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK

so that the integral becomes

C* •\ll—s2-\l\—l2ls2dyp _ C* VI— s

2

d\p VI2 , I*

Q/4

1—s2sin2*+

The elementary integral on the right side is familiar to us from the theory of Keplerian motion:

C* /17*yya-tan-» (Vl=? tan*).

Jo 1—s2sin2*

There is not much to say about the second integral in equation (19). Its value is needed only

up to 0{v2), which turns out to be:

r u s2 p 2f

2L i; 2 ^ «2

 «• c* sin2 ^

JH l+«Jo \«- / l+e2Zj 0 Vl— J2si

To develop the second integral in equation (18), we define a new variable <f>, the amplitude of

2 K

— v by the equation

7T

/2K \

sn I — v\kr )=sin 4», (69)

\ ir /

which is analogous to equation (63); the integral to be evaluated is then

d<t>

Jo P2+C*-2K)0

where

Again, we try to obtain a Fourier series in the argument 0 by developing first the integrand according

to powers of v

2

, which is the only reasonable method we can think of.

Anticipating the not surprising result, that the coefficient of a trigonometric term with

argument n<f> will contain e

n

 as a factor, we shall use here the notations

so that
100 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

and

/i u • 2 l+|sin^+.. .

VI—Arsin2^

 z

If, as always, we neglect terms of 0(v6), we have

.. .W|Y

which upon substitution of

and

(j\ =v{{l-\-U cos 4+6e2 cos2</>+4e3 cos3

becomes

cos 0

cos3 -('»' 1 + ^ e4

 cos20+ .. A

Expressing the powers of the cosine by cosines of multiple angles, and integrating, we obtain

C* n ,

Jo

e sin *+§ [Q-S^-S^l)-^2] «

2

 sin

In order to deal with the first integral in equation (19), that is,

Jo 2K

we have to introduce some analogue of the eccentric anomaly in a Keplerian motion. It seemed to

us that the following procedure is a quite natural one for this purpose.

The above integrand can be represented up to O(y*) as a linear combination of the functions

D~2, D~l, 1, D, D2 with constant coefficients, where the abbreviation £>=l+g#cos <f> stands for the

denominator of p. The integrals of the 1, D, and D2 terms (all representing "perturbations") can

immediately be expressed by <f>, sin <f> and sin 2<f>:

(*d<t>=<t>,

Jo

I Dd4>=<f>-{-e^ sin tf>,

Jo

(73)
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 10 1

Furthermore, if we define the eccentric anomaly E by

fr 7 8^ (74)

so that -j—=^—r > then we have

a0 l+ecos £

Jo Kyi — el)

f*D-*dt=-JL. (75)

Jo yl—e%

As a consequence of this definition of the eccentric anomaly and equation (70) we can write

p=a(l—e cos E), (76)

which is the same expression as that for the radius vector r in a Keplerian motion.

Turning now to the execution of this development we put

il^ JZ^j (l-2sV]+ . . ., (77)

and

^ ^ sV]+ . . ., (78)

to give (1— ee*) + (e*—g)cos 0=(1—e^)(l—a2)+52I> and (1—4)Fsin2 0=— &2[(1—4)— 2D+Z?2].

Furthermore we have

52 )32 (l-^)-^- 1+5l(l- e |)- 2 (79)

and

yl—Psin20

+ (l-^f e (l-^ ) Z> - | [(1-e* 2)-^- 2 (3-e|) ] D 2- ^ ? (4Z^-Z?*)+ .... (80)

Multiplication of equations (79) and (80) yields the integrand in question in the required form:

—52)52+Ar2 (l-35 a ~^) ]

Now all we have to do is to apply the rules of integration (73) and (75); we obtain

- « 2 ) - | (i-«*-x)] e sin ^

| | . . . . (81)
102 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

Let us remark, in addition, that the above expansion features first the powers of the eccentricity

e* as a factor of the trigonometric terms. These were converted to powers of the eccentricity e by

using the relation g=e*(l—52). If we neglect quantities of O(v6), this transformation makes a

difference only in the coefficient of sin E.

The explicit form of the solution

For purely practical purposes, we have already solved the proposed problem, because equations (18),

(19), (36), (37), and the Fourier series (67), (68), (72), (81) of the integrals, together with the definitions of the relevant constants, permit us to compute the position of the satellite for a given time.

This form of the results, however, is not completely satisfactory from the theoretical point of view,

until we express all the appearing constants explicitly as functions of the orbital elements. No difficulties of principle are involved in carrying out this program, just a few multiplications of power

series, of which only terms up to 0(v*) are required. As a by-product, we shall have opportunity

to define and express the motion of the node and the mean motion of the satellite in an appropriate

way.

Let us rewrite equations (18) and (19) corresponding to the results reached in the previous

chapter, taking into consideration also the definition (32) of the constant T:

Jo (1—s 2sin2#Vl-J 2sin2

(1—s ^ A Jo

2

sin2#Vl-J 2sin2^ A Jo (p

2

+c

2

) Vl -

These formulas indicate that we still have to determine the expansions of the constants a

2

/A up

to OW), and d/A and ^ up to O(v2).

According to equations (28), (29), and (30) we have

A

and the first of equations (38) gives —2h=—^—r-

= V e 2

(l-4) ' (84)

It was found convenient to define a t this point another semimajor axis and an auxiliary mean

motion by the Keplerian relations

and n=J%-: (85)

V a3 - 2A

also, we define a "mean anomaly" hj M=n(t—t). Instead of equation (84) we can then write

a

2 A a 1 1 /gZij^ n e2\

71 ~A==TTZ \ * a/-, —*\> or remembering the definitions (77) and (78) of the quantities k2 and 52,

we have

w ^ =

 1 ^ 1 + ^2

 . (86)

Preparing for the derivation of our final results we obtain the following developments immediately

from the definitions of the relevant quantities:

- • •.} (87)
SECOND-ORDER SOLUTION OF VTNTI'S DYNAMICAL PROBLEM—IZSAK 103

and

,-=!-; [(12-48s2

+39s4)-(8-16s2+7s4)e2]- (88)

Let us come to the determination of the constant UjA. From the second and fourth of equations

(38) we have

-2ka2

= (1—S2)(S2P

2

/12—V2), which, combined with equation (84), gives

-j=Vl—s2^s2v

2

jl2—v

2

 -\/i 2+k2/e2. The development of this quantity can be obtained by use of

equations (44), (45), and (47) as

Finally, the third constant to consider becomes simply

TTIT /1A 7 2 \ ,,2

2(1—e2)

2

'

(89)

(90)

How to express equations (82) and (83) in an explicit form is now obvious. In formulas (67),

(68), (72), and (81) we have to substitute the developments (44), (71), (47), (77), and (78) of the

figuring constants, and carry out the multiplication by the constants (89), (86), and (90). The

details of these elementary but somewhat lengthy operations will not be given here; rather we

confine ourselves to giving the results, which are:

£ A i / R

-J2=tan- 1 (Vlll- i

^

f 2 /i 2 4 / i o2

and

^(3+e2)}}** s siin

sin *

(vS\-e2 y

4Vl —

\4(l-c 2) 8(l

x,2\3

[(6-7s2)-(2-3«2)e2] U2

vs si siin )• (92)
104 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

The connection between the arguments <f> and ^ is expressed according to equations (54) and

(66), and the analogous equations for the argument v, by the relation

-». (93)

It seems appropriate to comment on the developments in this paper with the same reminder

Dr. Brouwer used in the conclusion to his paper (1959): "Anyone who has ever carried out developments of the type presented here knows that extreme caution is necessary if errors are to be avoided,

even if the method used is straightforward."

All the necessary computations I performed at least twice, and in two different ways whenever

possible. Unfortunately, such a procedure in itself cannot guarantee that there is not a single

mistake in the formulas.

I make some further remarks concerning the derivation and form of the results. Certain

simplifications in the formulas suggest that by handling the constants in a more efficient way it should

be possible to arrive at a shorter derivation of expressions (91) and (92). (For instance, due to the

definition (85) of the quantity n, the coefficient of the eccentric anomaly E becomes 1. This is in

agreement with earlier, unpublished results of the present writer. Furthermore, the coefficient of

e sin E'\s simply —1/(1+#c), at least up to O(v*).) Admittedly, it would be more elegant to use the

linearly connected arguments v and w in the developments instead of <f> and rp.

A rather pleasant job has still to be done, namely, the determination of the motion of the node

and that of the anomalistic and draconitic mean motion of the satellite.

Since we have not yet mentioned the "plane" of the orbit, we have had no chance to define its

ascending node on the equator, whose mean rate of change should be obtained now. The idea of the

motion of the node comes originally from perturbation theories based on the concept of osculating

orbital elements. In a theory like this one, which uses a pure coordinate method, the motion of the

node can nevertheless be defined in a formal way.

Let us consider again the expressions (91) and (92), in which the fundamental arguments are <i>

and rp, the first being a " true anomaly" and the second an "argument of latitude." As known from

the theory of Keplerian motion, the following expansions hold true:

tan"1 (Vl-s 2 tan $ =iH S

where s=sin/, r=tan -, and £"=0+2 ]£} tz^L s i n j<f>, where <?=(1 — VTrei)/e*=4±(l+j+ . . A

This means that equations (91) and (92) could be written in the forms

(94)

and

=M, (95)

where 8, 6, 0, and 7 are constants of Oiy1), and the Latin capitals denote Fourier series in the indicated arguments. We are not going to substitute series expansions for closed expressions in any

practical application, of course. All we are interested in is the possibility of these expansions, and

the fact that we can read off the coefficients of the secular terms immediately from equations (91)

and (92); we shall need them soon. Moreover, according to equation (93), the relation between the

arguments 0 and ^ is of the form

w, (96)
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 10 5

where, as before, e is the motion of perigee given by equation (53), and the Fourier series G(4>) and

H(#) are of 0(*»).

When the argument ^ increases from 0 to 2x, i.e., the satellite makes a complete revolution with

respect to the equator, the right ascension a changes by 2v minus a quantity of Oiv2). This small

amount can be thought of as the regression of the node, yet it is in general not the same for consecutive revolutions. But equations (94) and (96) imply that lim — ^——'-^- exists, its value

a

being (l+5)+0/(l + e). We can call the small quantity V=^-\-TJT ^ e motion of the node. Comparison of equation (91) and (94) yields the values of the constants 8 and 0, and e is already known.

In this way we obtain for the motion of the node the formula

(97)

To determine the anomalistic and draconitic mean motion of the satellite we proceed as follows.

Ignoring their physical meaning in connection with the motion of a satellite, let us consider equations

A

(95) and (96) as relations between two pairs of variables, partly <f> and yj/, partly M and w. We are

interested in periodicity properties of the functions defined by these equations. First of all, it is

A A

clear that for any integers i and j the identities M(<f>-\-2iri, \l/-\-2irj)—M(<j>, ip)=2iri(l-\-8)-\-2irjy,

and cc(<f>+2iri, \f/-{-2ir})—«(</>, ^) = — 2Ti(l+€)-\-2Trj are valid. To simplify them, we shall perA

form the linear transformation M=(l-\-8)M-\-yN, w= — (l+e)M-\-N, the inverse of which is

A A

±i£—*vco {_ 1 ~\~ c )xVz ~\— 1 1 ~4~ 8 ) co

M=, . o\,/, • \ > N= ,. i o\_i_/ii \— The periodicity properties of the functions M(<f>, \f/)

and N(<f>, \j/) wHl be expressed by the identities M(<f>-\-2in, rl/+2irj)—M(<t>, ^)=2xi and

N(<t>-\-2iri, $+2Trj)—N(<l>, \f/)=2irj. Then a little consideration reveals that, if we assume the

existence of uniquely determined inverse functions </>(M, N) and #(M, N), their periodicity properties must be the same, that is,

/)-tf(M, N)=2iri

)-t(M, N)=2wj. (98)

As we see, the inversion problem of the basic equations (13), (14), and (15) has been changed to that

of the equations (92) and (93); in this altered form, however, the problem is almost trivial.

Now, equations (98) mean that the functions <A(M, N)—M, and ^(M, N)—N can be expressed

A

as Fourier series in the two arguments M and N. Consequently, if we again consider M, M and N

as linear functions of the time, and define the anomalistic and draconitic mean motion of the satellite

by the limits n, = lim

 yv " ^ VK*', n#= lim rv " ^ yw

, then we have

(99)

and

n,. (100)

It is interesting to observe that up to Oiv3) the mean motion n^ depends only on the total energy

h of the system.
106 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

Summary

The second-order solution of Vinti's dynamical problem is represented in the following form, well

adapted for numerical computations.

The radius vector and the sine of the declination are given by the formulas

r=aV(l—e cos E)2+v2(l-s2sin2t), sin S=

I . 2 1—

"V1"

1"1

' (1- ecosE)2

Actually, these expressions are exact, i.e., valid to any order of approximation. They can be

thought of as representing a transformation from the coordinates r and 5 to the angular variables

E and ^. Then the right ascension of the satellite and the generalized Kepler's equation are found

to be the Fourier series with secular terms, (91) and (92). The relation between the true anomaly <t> and

the argument of latitude f is expressed by equation (93), and equation (74) is the definition of the

eccentric anomaly E.

A A

The orbital elements a, e, s=sin/, t, Q, and « serve as the fundamental constants of integration. Other important constants are:

a second eccentricity, e*;

the parameters P and I2 of the elliptic functions encountered in this theory;

the motion of the perigee, e;

the motion of the node, JJ;

the auxiliary mean motion of the satellite, h

the anomalistic and draconitic mean motions of the satellite, n^, and n^=(l + «)w^.

These constants are given as functions of the orbital elements by the expansions (45), (47),

(44), (53), (97), (87), and (99).

No small divisors appear in this theory; neither c, which becomes 0 in the case of the critical

inclination 63°26', nor e, which is troublesome in some methods using orbital elements instead of

coordinates. The absence of « as a divisor reflects the fact that the solution of Vinti's dynamical

problem can be exactly represented by (rapidly) convergent trigonometric series. The absence of

e as a divisor lends itself to a discussion of the case of very small eccentricities.6 This problem and

A

the perturbations due to the potential V— V will be treated later in another paper.

Appendix

For the sake of completeness, I give here the expressions for the right ascension of the satellite and

the generalized Kepler equation, in case one prefers to use the arguments v and w=(l-\-e)v-\-a

instead of <$> and \f/:

e sin v

* After the completion of tbis paper, I found that a very Interesting Investigation Into the geometry of satellite orbits has appeared in several recent

papers by Strnbel, the latest of which (Strubel, 1960) seems to be the most complete. He paid special attention to the problem of very small eccentricities, and to that of the critical inclination.
SECOND-ORDER SOLUTION OF VINTI'S DYNAMICAL PROBLEM—IZSAK 107

and

V 8i n 2v

{

v

2

-<Jl—e2

4(l- e») ~

sin 16(l-e2

)

3

 ^

This form of the

results is easily derived from equations (91) and (92) by substituting into these equations the

expansions

The differences between Q* and S2 and between M* and M are constants of O(v?).

s i n

 ^

sin 2<^=

s i n s i n

sin

sni 2^= 6 ) sin

If the inclination is near 90°, the right ascension does not seem to be a proper coordinate of the

satellite, because its finite part becomes a discontinuous function of w for s=l. To avoid this

unessential singularity, we could obtain another interesting and useful form of the results by introducing a slowly rotating reference plane, with constant inclination /, and right ascension of the

ascending node i2=

Effects of Solar-Radiation Pressure on the

Motion of an Artificial Satellite

By Yoshihide Kozai

The effects of solar-radiation pressure on

the motion of an artificial satellite have been

studied by several authors (Musen, 1960;

Parkinson, Jones, and Shapiro, 1960). As

they predicted, the orbit of Satellite 1960

i 1 (Echo I) has been greatly affected by the

solar-radiation pressure (Shapiro and Jones,

1960). And even for Satellite 1958 /82 (Vanguard I), which is of moderate size, Musen,

Bryant, and Bailie (1960) found that the discrepancy between the observed and computed

values of perigee height could be explained

by the solar-radiation effect.

When the author (Kozai, 1959a, 1961b) derived several constants of the earth's gravitational potential from the motion of artificial

satellites, he did not take this effect into consideration. Although the effect is very small

for the average satellite, it must be considered

in the future in the reduction of observations.

In the present paper the author wants to

study this problem in order to reduce the

observations of satellites of moderate size.

The analytical expressions for the perturbations of the first order are easily obtained;

however, the two limits of integration are

derived by numerical methods.

Disturbing functions

The equations of variations to be solved are the

following:

da 2na3 „ f

di na

2

 Tm*r r

-rr= . WF- cos L, dt ji—2

 a

. . dQ,

sm % -rr— dt

dw

, =—cos i -jT+na2 - F \ —S(v) cos v

+T(v)(l+-\smv~\,

dt

where the conventional notations are used for

the orbital elements, L=v-\-u, p=a(l—e2);

n

2

a*FS(v), n

2

a?FT(v), and n

2

a*FW are three

components of the disturbing force due to the

solar-radiation pressure in the direction of the

radius vector of the satellite, in the direction

perpendicular to it in the orbital plane, and in

the normal to the orbital plane; and F is a

product of the area-mass ratio, solar-radiation

pressure, and a reciprocal of GM.

To derive the expressions of S(v), T(v), and

W, we make the following assumptions:

(1) The distance of the sun and the satellite is

infinite; that is, the parallax of the sun is negligible.

(2) The solar flux is constant along the orbit of the

satellite if there is no shadow.

(3) There is no re-radiation from the surface of the

earth.

Then the expressions are the following:

S(v) = — cos2 K cos2 K cos (X©—L— Q)

-sin21 sin21 cos (Xo+G-Z)

109
110 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

—2 sin i sin £ {cos (X©—Z)

—cos (—X©—L)}

—sin2 - cos2 2 cos (Q—X©—Z)

o 6

—cos2 5 sin2 ^ cos (—X©—L— Q),

W=sin i cos2 -x sin (X©—12)

—sin i sin2 _ sin (X©+fi)

—cos i sin « sin X©, (2)

where X© is the longitude of the sun, and c is

the obliquity. The expression of T(v) is obtained if cos in S(v) is replaced by sin except

for the trigonometrical terms with an argument

i, e, i/2, or e/2.

Solutions

There is a very important difference between

the solar-radiation force and the gravitational

force. The solar-radiation force is sometimes

a discontinuous function of time because when

the satellite enters the shadow, its motion is

free from this effect.

Suppose that the satellite exits from the

shadow at a point where the corresponding

eccentric anomaly of the satellite is Eu and

enters the shadow at E2. If the force is continuous, the integral of the short-periodic effect

can be neglected. However, for this case there

is a possibility that this effect will be cumulative to a certain amount during a long interval

of time. Therefore, the short-periodic terms

must be kept in the solutions.

By use of the eccentric anomaly E as the

independent variable, the perturbations of the

first order after one revolution can be derived

in closed forms as follows:

5a=2a3F (S cos E+ TVl — e

2

 sin E)

+T (—2e sin

-e

2

 cos 2E

^ sin

5i=a2^ -pSL fl / (1+e2) sin E

•yl—e2L\ v.

—jsin 2E vcos «

+Vl — «

2

 (cos E—| cos 2S j sin w

£2_3

B, 2 e

sin •i5S2=a2

JF

- 4 sin 2£

4

vsin a

— Vl-e 2 ( cos E~ cos 2£;

' J COS W

3w=—cos'iSQ

(e sin E+^ sin 2E\

—e

2

(e cosE

= - | r'-dM-JT 2 Jo a

—y/l — e

2

 cos

- | sin 2E) -TV ^

(3)

where the limits of integration are Ex and Ei

unless other values are written; S and T are

the expressions of S(v) and T(v), in which L is

replaced by w; that is,

S=S(0),

T=T(0). (4)

If the satellite does not enter the shadow

during one revolution, the terms depending

explicitly on E vanish, and in particular, 5a

vanishes.
EFFECTS OF SOLAR-RADIATION PRESSURE ON SATELLITE MOTION—KOZAI 111

In the expressions of So? and 612, indirect

effects of the solar-radiation pressure through

w and J2 must be considered as follows:

(5)

Shadow equation

If the geocentric angular distance between the

sun and the satellite is denoted by a, the

shadow boundary is expressed by the equation,

d8o

~dt =

i8Sl_

dt ~

do

de

dil

de

i da ..

'

+TiSl

. dh..

+

dih

, do

+

 da

,dQ

+

 da

r sin a=a®, (6)

where a© is the radius of the earth and is assumed to be constant. The following relations

hold among r/a, E, S, T and a:

- cos a=—&(cos

-=l—e cos E.

a

— T-y/l — e2 sin E,

(7)

If o, Q, and Xo are assumed to be constant

during one revolution of the satellite, S, T and e

can also be regarded as constant; then E can

be derived from equation (6).

However, as the equation is of the fourth

degree with respect to sin E or cos E, it is very

difficult to get general analytical solutions

except for a circular orbit. If the orbit is

circular, the terms of odd powers of sin E or cos

E disappear, and the equation becomes quadratic for cos2

 E or sin2 E. And it is almost

impossible to expand the solutions into power

series of the eccentricity because of slow convergency. Even if the series is convergent,

it is possible that the equation has no real

root even though the equation has two real

roots for the circular orbit, and vice versa.

Therefore, the author thinks that equation (7)

must be solved numerically for every revolution.

There are four roots; however, under a condition

that cos a must be negative, the number reduces

to two, at most. If there is no real root, the

satellite does not enter the shadow. And if

there is only one real root, the satellite touches

the shadow at one point.

Numerical examples

The author has devised a program to calculate

these effects on the IBM-704 computer. This

program computes the inequalities of the orbital

elements due to the solar-radiation pressure,

if the approximate expressions of the orbital

elements are known.

As an example, inequalities of the orbital

elements for Satellite 1958 /3 2 during the

period 36526-36615 (Modified Julian Days)

are plotted in figure 1. This computation is

based on an estimated acceleration of 9.7X10"8

cm/sec2 (Musen, Bryant, and Bailie, 1960).

The semimajor axis is expressed in earth equatorial radii, and the mean anomaly is in revolutions. The author (Kozai, 1961b) earlier used

the same observations to derive geodetic

constants; the effect of the solar-radiation

pressure was partly taken into account in

that reduction.

The perturbations for Satellite 1960 tl have

been roughly computed by this program, and

have been checked with observed values.

As the eccentricity is small (order of 10~2), the

change of the argument of perigee is very rapid

due to the solar radiation. For this case it is

36525 550 575 600 SO

0

8a

-IxlO5

or0004

8c

8x10*

Se

o

or 04

8M

o

o

8X1

oroi

36525 550 575 600 SO

FIGURE 1.—Inequalities of the orbital elements due to the

solar-radiation pressure for Satellite 1958 02.
112 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

2.0 xlO2

I.5xl62

I.OxlO2

0.5 xlO2

-

-

V

1 1

/

if

/

\

\

60 70

SD

80

FIGURE 2.—Variations of {' and »j for Satellite 1960 tl with

observed values represented by dots. Epochs are given

in 37100 (MJD).

better to use, instead of e and a, $=e sin «,

17=6 COS co.

The following expressions are derived under

an assumption that the satellite never enters

the shadow, and they are compared with observations in figure 2:

£=£'+0.00066,

€'=0.04409 sin (91?99+2?965J)

+ 0.03579 sin (AQ—Q) + 0.00188 sin (Xo+fi)

—0.00519 sin Xo,

77=0.04409 cos (91?99+2?9650

+ 0.03373 cos (Ao—fi)—0.00200 cos (Xo+8)

—0.00169 cos Xo, (8)

where 0.00066 in £ appears because of the odd

harmonic of the earth's potential; 0.04409 is the

so-called proper eccentricity; and other terms

are due to the solar-radiation pressure. The

time t is measured from 37171.0 (MJD) in days.

The proper eccentricity and the phase angle

91?99 are determined from observations by the

method of least squares. The acceleration is

also determined to be (4.470 ± 0.024) X10~3.

Although the satellite entered the shadow

after 37171 (MJD), the above expressions can

follow the actual variations quite well.
The Effect of Radiation Pressure on the Secular

Acceleration of Satellites

By Stanley P. Wyatt

The secular perturbations of a satellite orbit arising from solar-radiation pressure have been

discussed recently (Musen, 1960; Musen, Bryant, and Bailie, 1960; Parkinson, Jones, and Shapiro,

1960). In particular, the variations of 1 or 2 km in the perigee height of Satellite 1958 /32 (Vanguard

I) predicted by Musen and his collaborators, when combined with the gravitational effects of the

sun and moon, agree very well with the observed changes during the first two years in orbit. Very

recently, much larger variations of eccentricity and perigee height of Satellite 1960 il (Echo I)

during its early life have been observed and found to be in excellent accord with theory (Jastrow

and Bryant, 1960; Shapiro and Jones, 1960).

The problem to be considered here concerns the short-term secular variations in period to be

expected from solar-radiation pressure. The secular acceleration of a satellite as a function of the

time can often be derived from observation with considerable accuracy. As is well known, these

accelerations, AP/P, may then be employed to deduce values of P,T/H{, where pq and Ht are the

density and scale height of the atmosphere at the locations of perigee. The question naturally

arises how high in the atmosphere it is legitimate to deduce these parameters from the observed

period changes. Are there other perturbing forces, in addition to atmospheric drag, that will produce

finite values of AP/P? If so, how can their effects be eliminated in order to avoid erroneous conclusions about the structure of the thermosphere? In what follows it will be shown that the effect

of solar radiation on the week-to-week variation in period is negligible when a satellite is continually

in sunshine. Later, however, because of the motion of sun, node, and perigee, the satellite must

spend some of its time passing through the earth's shadow; the secular acceleration due to the force

of sunlight may then exceed that due to atmospheric drag at heights above 800 km or so.

The perturbing acceleration due to radiation pressure

A satellite of average physical cross-section A and mass m at distance r o = 1 a.u. from the sun intercepts energy at the rate LQA/4:irT%, where Lo is the total power output of the sun. Hence the

momentum gained per unit time, or repulsive force, is of amount LQAJA.TCT3®, where c is the velocity

of light. If the incident energy is reflected specularly or is absorbed and re-emitted isotropically,

virtually no net momentum is carried away. We therefore assume that the radial acceleration

has a magnitude

f=(A/m)LQ/4ircri

e. (1)

We also assume for simplicity that during the few hours a satellite spends revolving once around the

earth the vector f is a constant relative to the satellite's orbit. We thus ignore a variety of small

effects: (1) possible variations of the solar constant; (2) the minuscule change in the solar distance;

(3) the slight motion of the sun in right ascension and declination; (4) the motion of the satellite's

node; (5) the motion of the satellite's perigee; and (6) the Poynting-Robertson drag. The magnitude

of f is probably constant to good accuracy. The change in direction of f, through items (3), (4),

and (5), amounts at most to a few tenths of a degree during one typical orbital period. Item (6)

113
114 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

modifies the direction of f by less than one minute of arc. The influence of all these factors on the

secular acceleration of a satellite is ordinarily much smaller than that due to the earth's shadow*

and we shall not consider them here.

A more important effect appears to be re-radiation of sunlight from the earth. When the sun

is overhead, a satellite at a typical height experiences an upward push due to the reflected component

of sunlight amounting to at least 20 percent of the downward push of direct sunlight. When the

sun is at larger zenith distances, the effect is less important, but is complicated by the fact that the

repulsive force is no longer quite radial from the center of the earth. Over the entire earth the

magnitude of the mean outward acceleration due both to the reflected sunlight and to the infrared

radiation by the surface and atmosphere is less than 20 percent of that due to direct sunlight for

satellites with perigee heights greater than 800 km. Although it is desirable that the influence

of terrestrial re-radiation be calculated, we shall not attempt to do so in this study.

The secular acceleration of a satellite

The instantaneous time rate of change of semimajor axis of an earth satellite is given (e.g., Moulton,

1914; Smart, 1953) by

da Pe sin 0 „ . P(l-\-e cos 6)

dt

where P is the orbital period, e the eccentricity, 6 the true anomaly, R the component of f directed

radially away from the center of the earth, and S the component in the satellite's orbit plane at right

angles to the radius vector and making an angle less than 90° with the velocity vector of the satellite.

To change this expression to the rate of change of period with true anomaly, we make use of the law

of areas, the polar equation of the orbit, and the derivative of Kepler's third law. Substitution gives

dP

de~-

dPdadt_

'da dt dd~

3Pa2(l-e2)

6M9

Re sin 6+S(l-\-e cos 0)

(l+ecos0)2

•}

(3)

The components of the disturbing acceleration may be deduced from figures 1 and 2. The a^-plane

coincides with the orbit plane of the satellite, Q is the direction of perigee, and P the instantaneous

position of the satellite. The direction of the sun is S, inclined by an angle i'=ZS from the orbit

normal. The direction J defines the direction^of the z-axis and is the intersection of the orbit plane

fsini'

FIGURE 1.—The celestial sphere, showing the direction of the

orbit normal Z, of the sun S, of perigee Q, and of the instantaneous satellite position P. The direction of the

radiation vector is from S toward the origin.

FIGURE 2.—The satellite's orbit plane, with symbols as in

figure 1. The component of the perturbing acceleration

due to solar-radiation pressure is / sin i' and is directed

toward the negative x-axis.
EFFECT OF RADIATION PRESSURE ON SECULAR ACCELERATION—WYATT 11 5

and a perpendicular plane which contains the sun. The instantaneous true anomaly of the satellite is

6=QP; we define the angle 0=JQ. Figure 2 shows that the total magnitude of the perturbing

acceleration in the orbit plane is j sin i'; the radial and transverse components are therefore

R= —/sin i' cos (0+0),

S= + / sin %' sin (0+0). (4)

Substituting equations (4) in equation (3) and integrating around the orbit, we find that the secular

acceleration of an earth satellite due to the pressure of sunlight is

AP_2 f2T dP ,fl_3a2(l—e2)/ sin i' CW

P~PJo ddm~ GM® Jo.*,

[cos 0 sin 0+sin 0(e+cos 0)]

ecos0)2

Again, it should be stressed that the assumptions implicit in this formulation are that f has a constant

magnitude as the orbit is described and also that the angles i' and 0 are constant during this interval.

The limits of integration on the right side of equation (5) account for the fact that in the general case

a satellite will enter the earth's shadow when the true anomaly is Oi and emerge when it is 02. During

this time, of course, the perturbing acceleration vanishes. The first term in the integrand is

readily evaluated, while the second term is found on substituting the well-known relation tan (6/2)

= (l+e)1/2(l —e)~m tan (E/2), where E is the eccentric anomaly. The value of the integral is

e"1 cos /3+sin 0 sin 0

1 +e cos 0

«"2* e"1 cos /3+sin 0 sin 0 |9>

0,0, 1+ e cos 0 U

cos 0 . cos (0+0) 1

e 1 + e cos 0J

* = cosj0+0 )

1+e cos 0 (6)

When equations (1) and (6) are substituted in equation (5), the expression for the secular acceleration

of a satellite due to solar-radiation pressure becomes

APjZ(A/m)Lea

2

(l—e

2

) sin %' [~cos (0+0)11 * . .

P 4vcr'oGMe Ll+ecos0_|L'

 (J)

It is convenient for numerical purposes to describe the physical characteristics of the satellite itself by

a dimensionless quantity D, such that (A/m)=D, cm2/gm, and also to express the ratio of perigee

distance to the earth's equatorial radius by another dimensionless quantity K such that

qJR@=a(l—e)/R®=K^l. Then equation (7) simplifies to

AP=3D,LoR'eK

2

(l+e) sin i' [~cos (0+0)"]

P~ 47Tcra

o6

rM©(l-e) Ll+«cos0j

—e)

sin (8)

The earth's shadow

To evaluate the bracket in equation (8) we must know the values of the true anomaly, 0\ and

0a, a t which the satellite enters and leaves the earth's shadow. To keep the problem tractable we

assume, without appreciable error, tha t the shadow is a circular cylinder of radius R@, with axis of

course in the anti-sun direction. The intersection of this cylinder with the orbit plane is a semiellipse, as shown in figure 3. Its equation in the coordinate system already defined is

z

2

 cos2 i'+if=R%, x£0. (9)
116 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Q FIGURE 3.—The general geometrical relation of a satellite's

orbit and the semi-ellipse of the earth's cylindrical shadow

projected onto the orbit plane.

Transforming to polar coordinates and equating to the expression for the satellite's orbit, we find the

values of di and 02 are the two solutions of

(10)

1—sin2 i' cos2 (0+0) (1+e cos 0)2'

If there are no solutions in the second and third quadrants, the satellite is in sunshine all around the

orbit; if there is one solution, 0i=02, the satellite touches the shadow at only one place and spends

none of its time in darkness. As before, we may substitute q/R®=K^ 1; then 6\ and 02 are the two

solutions of

(1+e cos 6)2=K?{\+e)2[\—sm2 i' cos2 (fl+B)), (ii)

The specific values of 0X and 02 for a given high satellite can be found by computing i' and /3 for

every few days and solving equation (11) by graphical or other approximate methods. The angle %',

between the orbit normal and the sun, is given by

cos i'=cos i sin 5©+sin % cos 8© sin (ay—a©), (12)

where i is the inclination of the orbit plane to the equator, a© and 5© are the right ascension and

declination of the sun, and ay is the right ascension of the ascending node. The angle /3 can be

found from

cos 0 sin i'=sin 5© sin fig+cos 5© cos SQ cos (OQ—a©), (13)

where OQ and 8Q are the right ascension and declination of perigee. When the angles of entry and

exit, 0i and 0j, have been found for various epochs they may then be used in equation (8) to compute

predicted secular accelerations as a function of the time.

When we abandon consideration of a specific satellite and ask for a general solution of equation

(11) to be substituted in equation (8) for AP/P, the problem is formidably complex because the angles

of entry and exit depend on four arbitary parameters. Let us therefore first consider some specific

applications that will perhaps elucidate the effect of radiation pressure on the period changes of a

high satellite and then proceed to develop a quasi-general solution as a power series in the eccentricity.

(For objections to this procedure, see Kozai, 1961a, especially page 30.)

Special cases of orbital orientation and shape

(a) Orbit normal pointing toward the sun.—Here i'=0° and therefore the right side of equation (8)

is zero. Physically, no satellite can spend a finite fraction of its time in the earth's shadow, even if

K=l.

(b) Satellite in sunshine all around the orbit.—If i' is sufficiently small, the satellite will see the

sun above the earth's horizon continually, as was the case with Echo I (1960 tl) during its first two
EFFECT OF RADIATION PRESSURE ON SECULAR ACCELERATION—WYATT 117

weeks aloft in August, 1960. Under such conditions, even if %' is not zero, AP/P=0. Mathematically,

the left side of equation (6) is to be evaluated from 0 to 2x, and the result is zero. Regarded physically, the perturbing effects cancel on opposite sides of the orbit. The null result here is like that of

the solar gravitational perturbation of a satellite, where there is never a "shadow" in which to hide:

the secular acceleration is zero when f is regarded as a constant vector relative to the satellite orbit.

(c) The circular orbit, i' arbitrary.—To find the period change of a satellite in a circular orbit

and spending time in the earth's shadow, set e=0 in equation (8) and note that, by symmetry,

O3+02)=27r— (/9+0i). Evaluating the bracket, we see that AP/P=0. Because of the symmetry

the momentum loss while the satellite is moving toward the solar hemisphere is just balanced by

the momentum gain when it is moving toward the opposite hemisphere.

(d) The angle /3=0° or 180°, i' arbitrary.—By symmetry, 02=2TI—0X. The bracket in equation

(8) is therefore zero and hence AP/P=0 for this case also, the interpretation being similar to that of

case (c).

(e) The asymmetric case with i'=90°, &=90° or 270°.—It may be thought from all the foregoing

that radiation pressure has no effect at all on satellite accelerations. The present example is intended

to show otherwise. It is also one that can be evaluated without great difficulty. When /3=90°

the solution of equation (11) gives cos 6i=[K(l-\-e)—e]~1 and cos 02= — [K(1 -\-e) +e\~l. Substitution

in equation (8) shows that for i'=90°, /S=90°, the secular acceleration is

AP ,e),

U(K, «)= (14)

the period decreasing with the time. If j8=270° the effect is equal and opposite, the period increasing

secularly. Although I have not tried to prove it, this special example probably reveals about the

maximum secular acceleration to be expected from radiation pressure. For one thing, with i'=90°

the full force of sunlight is in the orbit plane; for another, with /3=90° or 270° the effects of asymmetry are large. Table 1 presents numerical values of U(K, e) for several relevant values of the

eccentricity e, and the perigee distance K=qJR®.

TABLE 1.—Values of U(K, e) for use with equations (14)

K

1.00

1.05

1.10

1.15

1.20

1.30

1.40

1.50

e=0. 00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.20

0.06

0.05

0.05

0.04

0.04

0.04

0.04

0.02

0. 29

0. 12

0.10

0.09

0.09

0.08

0.08

0.08

0. 05

0.48

0.27

0.23

0.22

0.21

0.20

0.20

0.20

0. 10

0.74

0.49

0.45

0.42

0.41

0.41

0.41

0.42

0.15

0.98

0.71

0.66

0.63

0.62

0.62

0.63

0.65

0.20

1.22

0.93

0.88

0.86

0.85

0.86

0.88

0.91

0.30

1.78

1.45

1.39

1.37

1.37

1.40

1.45

1.51

0.40

2.49

2.10

2.04

2.04

2.05

2. 11

2. 19

2.29

0.50

3.46

2.99

2.94

2.94

2.98

3.09

3.22

3.38

Inspection of the table reveals several points. First, the secular acceleration is zero for circular

orbits of all sizes, as expected. Second, for any fixed value of the perigee distance, U(K,e) increases

monotonically with e, because of the increasing asymmetry of the passage through the earth's

shadow. Third, for any fixed value of the eccentricity, U(K,e) has a relative maximum for the

smallest possible orbit (K=l), falls to an absolute minimum at some intermediate value of K, and

then rises once again. Interpreted physically, for a fixed e there is maximum asymmetry for K=l,

while there is none at all as K—>» and the earth's shadowing effect becomes infinitesimal. With

increasing K, however, the decrease in asymmetry is compensated and then overtaken by the
118 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

y

FIGURE 4.—The construction for a nearly circular orbit. The

semi-ellipse of the projected shadow is shown, as are also

the orbit and its inscribed circle of radius equal to the

perigee distance. Notation as in case (f).

decreasing gravitational control of the earth on the satellite. When U{K, e) is differentiated partially

with respect to K, it is found that the absolute minimum occurs at a perigee of

(15)

For any eccentricity in the range 0.0 ^ e ^ 0.5 the minimum value of U(K, e) may be found by adopting

the smallest value in any given column of table 1, because the absolute minima for all these eccentricities occur within the range of K that is tabulated. Thus, for this particular orientation and

independently of K, the magnitude of AP/P ranges from zero for a circular orbit to at least 0.5 X 10~7

Dt at «=0.10, at least 1.1X1O~7.D, at 6=0.20, and at least 4.1X10"7 Ds at 6=0.50.

(f) The orbit of low eccentricity.—A quasi-general solution for the secular acceleration may be

developed as a power series in e. In equations (8) and (11) make the substitution <£=/3+0, where r

and <f> are polar coordinates in the orbit plane as defined by the coordinate system of figure 2, such

that x=r cos </», y=r sin <f>. Next circumscribe a circle of radius q around the center of the earth,

as shown in figure 4. In the figure, ZJCA=<f>i&nd ZJCB=<j}2. Define <£i = <£oi+'Mi, #2=002+^2, where

Ui and U2 are small quantities if the eccentricity is small and zero for a circular orbit. As can be seen

from figure 4, <£Oi = ZJCAQ and must lie in the second quadrant, while 0oo= ZJCB0 and must lie in

the third quadrant. The solution of equation (11) with e=0 gives

COS $01=COS 002=—

K sin i'

s i n <fcn=—sin </>02=-

Ksini' (16)

Now set

 iMi=a16+o2e2

+ . . .,

 <

u2=61€+62e

2

+ • • •> substitute each in equation (11), and then equate

coefficients of like powers of e in order to obtain expressions for au a2, . . ., bu b2, . . . . Then

return to equation (8) for the secular acceleration and expand it similarly as a power series in e,

a series which will of course contain at, a?, . . ., bu b2, . . . . Substitution of the explicit expressions for these coefficients, already found above, then gives the secular acceleration as a power series

in e to any degree of accuracy. Unfortunately, of course, the greater the desirable degree of accuracy,

the greater is the undesirable degree of complexity in working out the coefficients. I find the quasigeneral solution, as far as order e

2

, to be

^-=-I.4OXIO- 7Z>,FCK:, 6,/M'),

(17)

where n and v are given by equations (16).
EFFECT OF RADIATION PRESSURE ON SECULAR ACCELERATION—WYATT 119

This expression has both merits and demerits. It vanishes, as it should, for a circular orbit.

For a finite eccentricity it is zero for /3=0° or 180°, as expected from case (c). Also, if we set i'=90°

and 0=90° in equation (17), it is found to agree with the exact equation (14) when the latter is

expanded to the second order in e. The acceleration predicted by each formula for this special

situation is

0 772a

^=-1.40X10- (.18/

Although equation (17) is consistent with some of the foregoing special cases, it arouses suspicion

on at least two grounds. First, consider a time when n=0 and therefore by equation (16) 0oi=<£o2

= 180°. The final term in equation (17) suggests an infinite secular acceleration under these conditions. Actually, this particular situation occurs when the circumscribed circle of figure 4 touches

the projected shadow at one and only one point—on the anti-sun axis. It is readily seen, therefore,

that no satellite traveling on any orbit whose inscribed circle is such that /x=0 can spend a finite

fraction of its period in darkness. The difficulty here is a mathematical one rather than a physical

one, and as has already been pointed out AP/P=Q on all occasions when equation (11) has less than

two solutions. A second, and legitimate, suspicion is aroused when #= 1 in equation (17). It is

not enough to dismiss this problem with the comment that any satellite with K— 1 is itself in grave

trouble; we deal with this problem as case (h).

(g) The orbit of low eccentricity, i'~90°.—Specializing equations (16) and (17) for those times

when the radiation force lies fully in the orbit plane, we find that

F(#,g,/3,9O°)=eF(#,/S)

, 2#2 sin 0

f K2—2 . COS &^K2—1\ .

(19)

Table 2 gives values of V(K, /3) for several combinations of # and 0 and may be used to estimate

quickly the leading term of equation (19). The absolute value of the second term is less than 20

percent of the leading term for all values of /S at #=1.0 5 if e<0.022; at #=1.10 if e<0.048; at

#=1.15 if e<0.077; at #=1.20 if e<0.109; at #=1.30 if e<0.184; at #=1.40 if e<0.270; and at

#=1.50 if e<0.211. The minimum absolute contribution of the second term occurs in the neighborhood of # = -y/2 and then rises again with increasing perigee distance until for very large orbits 20

percent contribution occurs at e=0.100. The sign of the second term is usually negative for #<y f2

and is negative for all values of /? provided #<1.27.

From this specific example it appears likely that equations (16) and (17) constitute an adequate

approximation to the practical estimate of secular accelerations due to radiation pressure, at least

for a fair variety of orbits. First, as will be shown in the next section, the effect of sunlight on

TABLE 2

K

1.05

1. 10

1. 15

1.20

1.30

1.40

1.50

— Values of V(K, ,fl) for use with equations (19)

0 = 0°

0.00

0.00

0.00

0.00

0.00

0.00

0.00

15°

1.78

1.37

1. 21

1. 12

1.05

1.04

1.04

30°

3.44

2.64

2.33

2. 17

2.03

2.00

2.01

45°

4. 87

3.73

3. 29

3.07

2.88

2.83

2. 85

60°

5.96

4.57

4.03

3.76

3.52

3.46

3.49

75°

6.65

5. 10

4.50

4. 19

3.93

3.86

3.89

90°

6.89

5. 28

4.66

4.34

4.07

4.00

4.02

636-014—6
120 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

AP/P is swamped by the effect of drag if .K<O.12 approximately, and it is therefore unnecessary

in practice to be concerned about the nature of the solution when K is very close to unity. Second,

whenever 10*1-12 the contribution of the second term of equation (17) is moderately small if e

is not too large. Presumably the higher-order terms converge rapidly. Although an extension

of the power series beyond 0(e2) would be useful for more eccentric orbits, it has not been attempted

in the present work.

(h) The nearly circular orbit, i'=90°, K=l.—When perigee is at or very near the earth's surface, the approximation leading to equation (17) breaks down. Geometrically, when K=l, we

have <£oi=9O° and #02=270°, and even when the eccentricity is small the angles U\ and tt2 are of

rather good size, and therefore terms beyond 0(e2) are needed to obtain an adequate approximation. Alternatively, it is straightforward enough in the present special case to solve equation (11)

and substitute the results in equation (8) as a power series in e. The appropriate formula, to order

3/2 in e, turns out to be

(20)

This expression vanishes for e=0 and also for 0=0° or 180°, as we are entitled to expect from previous illustrations. Table 3 contains W(e, /S) for a few selected pairs of e and /S. The approximate

equation (20) with 0=90° is identical with the exact equation (14) when K=\ and the function

U is expanded to order 3/2 in e. When 0=90°, equation (20) is good to 1 percent if e<0.08, and

to 10 percent if e<0.28.

The competition of radiation pressure and atmospheric drag

It is well known that the instantaneous tangential acceleration of a satellite moving through a

stationary atmosphere is given by

T=-(A/m)(CD/2W, (21)

where CD is the dimensionless drag coefficient, p is the atmospheric density at the point in question,

and v the speed of the satellite there. When the magnitude of this perturbing acceleration is compared with that due to solar-radiation pressure as given by equation (1), the ratio of the two is

(22)

a quantity that is independent of the characteristics of the satellite itself. Setting CD=2 , P

=f/X10~18 gm/cm3, adopting for v the circular velocity at 800 km, and inserting the other constants,

we have

R^l.2J. (23)

TABLE 3.—Values of W(e, 0)

e

0.00

0.01

0.02

0.03

0.04

0.05

0=0°

.000

.000

.000

.000

.000

.000

15°

.000

.037

.054

.067

.078

.089

30°

.000

.074

.107

.133

.155

.176

for use with equations (20)

45°

.000

.110

. 158

.196

.229

.260

60°

.000

.144

.206

.256

.300

.340

75°

.000

.175

.251

.312

.365

.414

90°

.000

.203

.291

.362

.424

.481
EFFECT OF RADIATION PRESSURE ON SECULAR ACCELERATION—WYATT 12 1

Thus, outside the earth's shadow the two forces are of equal magnitude at a height near 800 km for a

mean state of the atmosphere (Nicolet, 1960c). When the sun is active and when it is close to midday

at the perigee of a satellite, the high atmosphere is distended and the level of equal magnitudes is

above 800 km.

In equation (23), J decreases more or less exponentially with height. Therefore, atmospheric

drag is all-important for low satellites and less so for high ones. For example, Vanguard I (1958 02),

with D,=0.21, should by equation (14) and table 1 show a maximum secular acceleration due to

radiation pressure of ±0.25 X10~7. Referring to equation (8), the quantity sin i' passes through a

maximum on the average every 45 days, which is the time taken on the average for (aN—ao) to

regress through 180°; the semi-amplitude of this fluctuation is about 0.1X10"7. A second periodic

variation is that of the angle 0, which passes through a complete cycle every 2.4 years. This interval

is the length of the "day" at perigee, the average time taken for (<XQ—ao) to advance by 360°. The

semi-amplitude of this long-run variation is about 0.25 XI 0~7. When the anticipated effect of radiation pressure is compared with the observed accelerations of Vanguard I (Jacchia, 1959d; Briggs,

1959), it appears that the period changes can confidently be attributed to drag. The effect of

radiation pressure is small, and about equal to the precision with which the accelerations can be

determined; it is less than the effect of drag by a factor ranging from about 5 when perigee occurs at

night to about 50 when it occurs nearly under the sun. Thus for satellites as low as Vanguard I

radiation effects are to be found by analyzing such elements as perigee height (Musen, Bryant, and

Bailie, 1960; Musen, 1960) rather than orbital period. The latter may be employed to deduce the

structure of the thennosphere.

At greater heights the situation is more delicate. During its first two weeks aloft, Echo I (1960

il) had a perigee height near 1500 km, several scale heights above the reference level of 800 km. For

example, if the scale height in this layer averages about 110 km, corresponding to T=2000° K and

a composition of atomic oxygen, equation (23) then gives i?<1.2e~'<0.01, so that the relative

magnitude of the drag force is small.

Echo I of course spends some of its time in sunshine all around the orbit. As shown in case

(b) of the previous section, the effect of radiation pressure on period changes is then zero within

the framework of our assumptions. On such occasions the characteristics of the thennosphere

can be studied without accounting for the comph"cations of solar radiation. There is, however,

the limitation that at these times i'^0° and we are therefore confined to examining the twilight

zone of the high atmosphere. If the diurnal sunward bulge persists at these great heights, its

properties can only be deduced when Echo I has the sun near its zenith in part of its orbit and

thus is in darkness hah* a period later.

When a very high satellite periodically transits the earth's shadow, radiation pressure then

has the upper hand. The preliminary mean value of AP/P for Echo I during the first twelve

weeks of passage through the umbra was near —4XlO~*. The sign accords with expectation because the angle fi was in the first and second quadrants during this interval, and the amount is

roughly consistent with prediction. The open circles of figure 5 give weekly predicted accelerations and the filled circles the observed accelerations. The first seven observed points are weekly

means of the day-to-day values computed under the direction of P. E. Zadunaisky; the last five

observed points are relatively rough graphical accelerations by the author and are preliminary

only. In order to obtain the best vertical fit of predicted and observed points we must adopt

Dt^270 in equation (17), whereas a diameter of 100 feet and a mass of 60 kilograms gives D,^120.

Apart from this, the two curves are in reasonable agreement.

The factor of 2 or 3 by which the observed period changes exceed the predicted ones could

arise from several effects. First, an error in the announced value of D, could either aggravate

or reduce the discrepancy. Second, Echo I during this time may have been passing more or less

centrally through a diurnal atmospheric bulge, which would reduce the discrepancy. A third

possibility is nonspecular reflection from the balloon surface; complete back reflection will double the magnitude of the perturbing acceleration of equation (1), whereas isotropic reflection by
122 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

9

8

7

6

xlO6

 4

3

2

1

0

-

-

-

-

o

1 1

•

o

o

o o

8 o °

> * 2 S " #

1 1 1 1 1 1 1 1 1 1 1

FIGURE 5.—Comparison of observation and theory. The

filled circles are preliminary observed secular accelerations

of Echo I during its first three months of transiting the

earth's shadow; open circles are the predicted values due to

solar radiation pressure.

20 27 3 10 17 24 I 8

AU6 SEPT OCT

I960 DATE

15 22 29 5 12

NOV

each surface element into its outward hemisphere will increase equation (1) by a factor of 4/3.

One definite reason why the observed negative accelerations during these three months should be

greater than the predicted values is the effect of terrestrial re-radiation. Reflected sunlight was

acting during these weeks to amplify the negative secular acceleration. In addition, if the infrared radiation from the surface and atmosphere is nonisotropic in the sense of being stronger on the

daylit hemisphere, this component was also acting during September and October of 1960 in the

same fashion. We can, of course, unscramble some of these effects more easily when Echo I has

been aloft for a longer time. Perhaps it is not too much to hope that the radiation effects can

be assessed well enough so that residuals may then permit deductions on the structure of the highest

atmosphere even when Echo I is encountering the earth's shadow.

Remarks and conclusions

Studies of the atmosphere above about 800 km are made difficult because extraneous effects

rival that of drag on the period changes of satellites. The competing effect of solar-radiation

pressure can be evaluated and eliminated by the use of equations (16) and (17), provided the

orbital eccentricity is not too large, that Dt is well known, and that it is possible to estimate a

factor by which the satellite deviates from a specular reflector. The effect of terrestrial re-radiation has not been taken into account quantitatively in this study, although its role relative to

direct solar pressure may be appreciable. Two further investigations are therefore suggested:

(a) extension of the power series of equation (17) for more accurate assessment of the effect of

direct solar-radiation pressure; and (b) calculations on the influence of re-radiation from the earth

on the orbit of a satellite.

I am grateful to Dr. F. L. Whipple, Director, for the opportunity to work at the Smithsonian

Astrophysical Observatory during the summer of 1960, when this study was first conceived; to

Dr. L. G. Jacchia of Smithsonian, whose comments helped get the study started; to Mr. P. E.

Zadunaisky of Smithsonian, whose acceleration data helped complete it; and to Dr. P. Musen of

the National Aeronautics and Space Administration and Drs. H. M. Jones and I. I. Shapiro of

Massachusetts Institute of Technology, Lincoln Laboratory, for furnishing me information on the

work being done at these institutions on radiation pressure effects.

Abstract

Satellite accelerations play a crucial role in determining the structure of the high atmosphere, and it is therefore

important to assess and eliminate the effect of perturbing forces that compete with air drag and that may therefore

confuse our picture of the thermosphere. In particular, this study evaluates the effect of solar-radiation pressure

on the secular acceleration of earth satellites. For perigee heights less than about 800 km the period changes due to

radiation pressure are minor compared with those due to atmospheric drag. At greater heights and lower air

densities, radiation pressure becomes increasingly important. When a satellite is in sunshine all around its orbit,
EFFECT OF RADIATION PRESSURE ON SECULAR ACCELERATION—"WTATT 123

the period change arising from the pressure of sunlight is zero. But during the weeks or months it is penetrating

the earth's shadow and is therefore exposed to a photon wind only part of each circuit, the secular acceleration may

attain substantial values, positive or negative depending on the orientation of the orbit relative to the sun. Several

special cases of orientation are discussed, and a general formula for computing secular accelerations due to radiation

pressure is derived as far as terms in the square of the eccentricity.

Experimental and Theoretical Results on the

Orbit of Echo I

By Pedro E. Zadunaisky, Irwin I. Shapiro, and Harrison M. Jones

In this paper we compare experimental and

theoretical determinations of the orbital elements of the Satellite 1960 i 1 (Echo I) for

every day from August 13, 1960 (one day after

launching), through March 1, 1961. This

comparison shows that the strikingly large

variations of the eccentricity and geocentric

perigee distance can be attributed almost

wholly to the effects of sunlight pressure on the

balloon, in accordance with our theoretical

predictions (Parkinson, Jones, and Shapiro,

1960; Shapiro and Jones, 1960, 1961).

We have also inferred preliminary atmospheric densities for the altitude range (about

950 to 1500 km above the International

Ellipsoid) traversed by the perigee of Echo.

In this calculation it is necessary to consider

the energy gain (or loss) of the satellite from

the solar-radiation field. After subtracting

the variations of the rate of change of period

owing to this latter effect, we find that the

remaining fluctuations correlate with those

of the solar flux at 10.7-cm and 20.0-cm wavelength and can be attributed almost entirely

to air drag. In addition, the air drag shows a

large increase during the major solar flare of

November, 1960.

From the time variation of eccentricity we

deduced approximate values for the rates of

gas leakage from the balloon. Our conclusions

are only tentative because many other imprecisely known quantities (such as the reflection characteristics and shape of Echo as a

function of time) influence our results.

Our present (highly speculative) estimate

of the lifetime of Echo I is that, barring a

substantial change in its shape, it will perish

in the spring of 1964.

Description of experimental orbit computation

From observed positions of the satellite we have

derived the mean orbital elements by a process

of successive differential corrections. That

process has been described in detail elsewhere

(Zadunaisky, 1960), and we shall therefore

give only the main features of the present

computations.

The observations used came mainly from

the Smithsonian Baker-Nunn photographic

tracking stations. Some radio measurements

from the Minitrack stations of Goddard

Space Flight Center were included for the time

during which the satellite's beacon was operating. In addition, we included a very large

number of excellent optical observations sent

from the Observatory of Paris at Meudon,

France.

The word "mean" hi reference to the orbital

elements must be understood here in two senses.

First, it indicates that the first-order shortperiod perturbations due to the second harmonic

of the earth's gravitational potential have been

computed analytically and then subtracted from

the observations before the differential correction of the elements was performed. Second,

the elements are mean in the sense that each

set represents observations distributed along a

two-day arc, which corresponds to approximately 25 revolutions of the satellite.

A quadratic polynomial in tune was used to

represent each element, except the inclination;

for the latter we found that a linear expression

was accurate enough. Then by a process of

successive differential corrections we improved

only the constant term of each polynomial to

obtain a best least-squares fit of the orbit to

125
126 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

the observations made during the two days,

one before and one after the epoch of the

elements. In the polynomial representing the

mean anomaly we always corrected all three

coefficients in order to take into account the

rapid and unpredictable changes due to atmospheric drag. The coefficients of the linear

and quadratic terms of the other polynomials

were kept constant for periods of several weeks

and were then readjusted. As elements were

determined for each day, the observations used

in two successive determinations overlapped

for one day; we were thus assured of a better

continuity in our results.

Description of theoretical orbit computation

Our theoretical orbit computation is based on

the well-known first-order differential equations

relating the rate of change of the orbital elements to the perturbing accelerations (Moulton, 1914, chap. 10). We solve for the time

dependence of the elements by first integrating

these equations over an orbital period—keeping

the elements constant during the integration.

The integration can be carried out analytically

in closed form for all perturbing accelerations

except that due to air drag. The new values

for the elements are then used in determining

the changes in the elements during the next

orbital period, and so forth. This iteration

procedure is carried out on an IBM-7090

digital computer. The initial values used

for the elements are those determined from

the observations of the satellite; all future

values are calculated by the iteration method.

This method yields a very accurate estimate of

the long-period and secular contributions to

the changes in the elements. However, the

contributions of the short-period terms are

averaged out.

At present, our computer program includes

perturbations due to the following phenomena:

(a) direct solar-radiation pressure, including the

effects of the earth's shadow;

(b) neutral atmospheric drag.1 The atmospheric

density is assumed to be spherically symmetric and

constant in time, but this all-too-simple model is now

being generalized;

(c) the second through the fifth harmonics of the

earth's gravitational field;

(d) Solar and lunar gravitational fields.

1

 Charge drag is quite small for Echo I and has been neglected.

(e) Solar radiation reflected from the earth.2 Our

model for the reflection characteristics of the earth

involves an arbitrary, but uniform, mixture of diffuse

and specular reflection. However, only specular reflection is now included in our program.

It is clear that (aside from the approximate

nature of our physical model and the errors

accompanying observations) our theoretical

elements and our experimental elements differ

even in principle. Our comparison of the two

sets of elements is based on the assumption that

the differences that arise purely from the differing principles of element definition are small

and can be neglected in this analysis.

Comparison of experimental and theoretical

results

In calculating a theoretical orbit for Echo I in

order to compare it with results derived from

the observations, we must first consider the

physical characteristics of the balloon. Soon

after launch, the satellite closely approximated

a sphere 100±l ft in diameter. It was constructed from half-mil Mylar, externally coated

with an aluminum layer approximately 0.2/*

thick. Its initial weight was 156.995 lb, including 33.34 lb of sublimating powders. The

powders were of two kinds: the first (weighing

10 lb) was highly evaporative, while the second

had a much lower vapor pressure.

The magnitude of the acceleration due to the

pressure of sunlight is K(A/M) (I/c), where

(A/M) is the cross-sectional area-to-mass ratio

of the satellite (initally 102 cm2/gm); / is the

solar energy flux; c is the velocity of light; and K

is a scattering constant (0<K<2) whose value

depends on the reflection characteristics of the

surface. The constant c is known very accurately. For the solar constant we have used

the value 2.00 cal/cm2 min, which is quoted in

the American Institute of Physics Handbook

with a probable error of 2 percent. The ratio

A/M, on the other hand, is not accurately

known. Small holes that were introduced

* The infrared radiation from the earth, if it were uniform, would only

have the effect of changing the earth's mass by an entirely negligible

amount, provided that the satellite always presented the same crosssectional area towards the earth. (This last requirement is satisfied for

the case of the spherical Echo I.) Preliminary investigations of the infrared radiation from the earth (Intermountain Weather, Inc., Final

Report under Contract AF 19 (604) 2418) Indicate that at 1000 km this

radiation is within ±10 percent of uniformity. Since the re-radiated

power at 1000 km is only a small fraction of that of direct radiation, the

effect of the former on the orbit is probably quite small.
RESULTS ON THE ORBIT OF ECHO I—ZADUNAISKY, SHAPIRO AND JONES 127

before launching and meteoric punctures will

permit gas to escape at a rate almost impossible

to predict accurately. Hence, since 21 percent

of the initial satellite mass was in the form of

sublimating powders, it is difficult to determine

purely theoretically the accurate time dependence of the satellite mass. The value of K is

also uncertain: for specular reflection from a

perfect sphere, K=l; however, small irregularities in the shape or any diffuseness in the

reflection of sunlight will tend to increase K.

In view of these difficulties, we have considered

K(A/M) (lie) to be a (partially unknown)

parameter and have allowed it to vary in a

reasonable manner so as to obtain the best

agreement between the theoretical and observed variations in the orbital elements.

In figure 1, we have plotted the residuals of

the eccentricity e from a polynomial expression

obtained by the method of least squares. The

experimentally deduced values are indicated by

points, and the theoretical values by open

circles. The rather close agreement between

the two sets was obtained by assuming that

K=l and that the total mass of the satellite

decreased at the rate of 0.64 lb/day for the first

13 days, and then decreased at 0.16 lb/day.

According to this model, only a negligible

amount of the gas remained in the balloon after

January 15, 1961. The decrease by a factor

of four in the rate of mass loss, in spite of the

expected increase in the meteorite holes, may

possibly be due to the escape of the more

volatile of the two powders.3 These rates of

mass loss are somewhat arbitrary; other rates

may lead to equally good or better agreement

with observations. However, the uncertainty

in our value of K(A/M) (lie) at any given time is

probably not more than a few percent. Further improvement in our estimate of this

parameter apparently will require consideration of other physical effects, such as a possible

diurnal bulge of the atmosphere (see discussion

below of the argument of perigee). Since we

infer K(A/M)(I/c), as well as an air-density

model, from the data, we may possibly be

«The slow final rate of mass loss could not be measurably Influenced by

an accumulation of air molecules penetrating only one surface of the

balloon, since It Is possible to show that the balloon had so far collided with

only about one pound of air. Accretion of mass through collisions with

meteoritic dust probably amounted to much less than a pound.

636-014—63 10

"covering up" the effects of some neglected

physical perturbations.

That the satellite did gradually lose a substantial portion of its mass (or, more precisely,

that K(A/M) increased gradually) may be

inferred from comparisons of the observed

maximum value of the eccentricity (emax=

0.0788) with the following theoretical calculations:

(a) By assuming that K(A/M) maintains its

initial value of 102 cm2/gm, we find emax=0.0705.

(b) By assuming that the satellite lost all of

its gas soon after launching while K remained

unity [K(A/M) = 130 cm2/gm], we obtain

«max=0.0885.

We have also plotted in figure 1 the residuals

(from a polynomial) of the experimental and

theoretical values of the geocentric perigee

distance q. The densities used in computing

the theoretical curve were obtained by matching approximately a second-degree polynomial

in altitude to the logarithm of the experimentally determined densities. The relatively low

values of the theoretical perigee distance in the

neighborhood of January 1, 1961, are due

mainly to the correspondingly high values of

the eccentricity. On the other hand, the

relatively low theoretical values that persist

after the middle of January are related to the

"theoretical" density values being somewhat

greater than the average "observed" values.

This discrepancy could be largely eliminated by

a small change in the density values used in

the theoretical computation. However, our

present air model is independent of time.

Hence, for a given altitude, we cannot represent properly both the average air density during November-December and that during January-February when solar activity was relatively

much lower (see fig. 3).

The perigee distance reached its first minimum value on December 29, 1960, about two

days later than the eccentricity reached its

maximum value. This difference is due mainly

to the decrease in semimajor axis caused

by air drag.

In figure 2, we have plotted the residuals

(from polynomials) of the experimental and

theoretical values of the argument of perigee

w, the right ascension of the ascending node 12,

and the orbital inclination i.
128 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

.04

.02

2

o -.02

UJ

-.04

-.06

.010

.008

.006

.004

.00

-,002

-.004

GEOCENTRIC PERIGEE

DISTANCE

.* RESIDUALS FROM THE EQUATION

q= 7.5133-.3765 «102(T-T0 )*.24988 x

To ' 37238 *NOV. 0

COMPUTED FROM OBSERVATIONS

COMPUTED FROM THEORY

RESIDUALS FROM THE EQUATION

e -.0562+.4243 x 10"3(T-T0)-.3106 x lO

T0*37328=N0V. 0

- ECCENTRICITY

37150 170

AUG. 12,60 SEP 0

190

OCTO NOV. 0*T0 DEC. 0

290 310

JAN. 0,61 FEB.0

I I

350 |

MAR. 0

FIGURE 1.—Comparison of experimental and theoretical values for orbital elements

of Echo I.

8

4

0

- 4

.4

Q

-.4

04

0

.04

e

6

-

R.A.

a

-

RESIDUALS FROM THE EQUATION

U) - 257°47 + 3°4060 (T-To) +.1834 x IO"Z (T-To)

2

-.79 x

ARGUMENT OF PERIGEE To -37238

Vl. ...~-X-"r1 ^ _ C--°~ "SI

*-ti.«^"<f' RESIDUALS FROM THE EQUATION "*'*

fl-9°56O2 -3°13386(T-T0 )-.399xl0' 3 (T-T0)

2

+.559x106

OF ASCENDING NODE To»37238

o o o o

...

O

...

O

...

O

..°..-..° ° ° ° o

'*©. o-o ? °-. ..••**"•"*** *"**'-.. ° o o o o o

RESIDUALS FROM THE EQUATION

INCLINATION i - 47.B2520 + .2996 x 103(T-To)

-

-

_

-

i

37150

T o -3723 8

•o-o-"- .o o o o ?..•?•

-t> • • o • o . ... •• °

•«V - - ° " ° - • r o°-'" b0 , 0 ° " '" " Q"

o . . • •• o -o ^ . .o.. . ••/• .•.«•.•-. ' "•

o °" •

I I I 1 1 1 1 1 1 1

170 190 |210 230 | 250 270 290 310

0"5 (T-To

o °-

(T-To)

3

1 1

3 3 0

)

3

o

o

0 °

o ° .• •

o

• o .

• °>i

1 1

350 |

FIGURE 2.—Comparison of experimental and theoretical values for orbital elements

of Echo I.
RESULTS ON THE ORBIT OF ECHO I—ZADUNAISKY, SHAPIRO AND JONES 129

We see that for co the two curves are generally

in good agreement until about February 1,

1961, at which time they start to deviate

systematically. Part of this discrepancy would

be eliminated by reducing the rate of mass loss

of the satellite during the middle of December,

1960, since the contribution to d> due to radiation pressure is positive from 40 days after

launch through March 1, 1961.4 (The combined effect of these changes would reduce the

differences between the theoretical and the

experimental values of the eccentricity.) We

are also investigating the consequences of a

possible diurnal bulge in the atmosphere

(Jacchia, 1960a). Such a bulge in the equidensity contours will be effective in changing

o) when it intersects the orbit plane in a region

asymmetrically placed with respect to the

apsidal line. To estimate crudely the magnitude of this change in «, we consider the

density to have a 8-function contribution

superposed on the otherwise uniform density

of the atmosphere along the satellite path.

Limited by the average, over-all density deduced from figure 3, we find that co can change

at a maximum rate of approximately 0.1

deg/day.

The theoretical and experimental residuals

of fi are in good agreement until the end of

September, 1960. The systematic deviations

that occur after that date would be substantially eliminated by correcting the average

densities. The relatively low theoretical air

density values before January 1, 1961, caused

an increase in the residuals, whereas the relatively high subsequent values reduced the

residuals. The experimental and theoretical

values of i are in close agreement except for

a short period near the end of September,

I960.5 The oscillations in the inclination angle

during the first few months after launch are

due primarily to effects of sunlight pressure as

predicted elsewhere (Shapiro and Jones, 1960).

In figure 3 we have plotted the contributions

* Except for the first week after launch, the change in u due to radiation pressure is negligibly small until early October, 1960. However,

we note that sunlight pressure contributes about 15 percent of the 3.4

deg/day mean rate of change of a.

> The change in i due to a rigid rotation of the atmosphere was not

Included in our theoretical computations. Its cumulative value on

March 1, 1961, would be about —0.01 degree. A more refined analysis

will be necessary to establish the presence of this effect on the orbit

of Echo.

to the rate of change of period (P) due to

atmospheric drag and to solar-radiation

pressure. We computed first the total

"observed" dP/dt; then, using the theory

outlined in the previous section, we found the

rates of change due to radiation pressure

(P(R)). These were then subtracted from

dP/dt to obtain the values P(D), which we

attribute to air drag.6 As can be seen by

comparison with the upper graph in figure 3,

the variations in P(D) seem to correlate with

the variations of the solar flux at the 10.7-cm

wavelength. In addition, —P(D) exhibits a

marked increase at the time of the major solar

flare on November 12, 1960 (Jacchia, 1960a).

(The wide "scatter" of P(D) values in early

December, 1960, makes it difficult to discern

a correlation with the major flare that occurred

during that period.)

We note that radiation pressure can have

no effect on the period if the orbit is circular.

However, if the orbit is noncircular and is

partly in shadow, the satellite can enter and

leave the shadow region at different distances

from the center of the earth. The resulting

net gain or net loss of energy from the radiation

field is proportional to the difference of the

projections of these distances along the

earth-sun line.

From figure 3, we also see that the radiationpressure contribution to dP/dt is, in general,

of the same order of magnitude as that of air

drag at these altitudes. Paradoxically, when

perigee height is near its minimum value, the

change in energy induced by radiation pressure

is still comparable to that caused by air drag,

despite the increased air density. This fact

can be understood in a qualitative way from

the following argument. If the air density

depends only on height, then as a function

of eccentricity, the energy change in one

revolution owing to air drag is represented by a

constant term plus terms in even powers of

the eccentricity. On the other hand, when

the qualifications given above are satisfied,

the energy change per revolution caused by

radiation pressure is represented by a linear

• A portion of this drag may be attributed to a possible dust belt around

the earth. However, if one assumes that the air density is correlated

with the 10.7-cm wavelength solar flux and that the density of the dust

belt is not so correlated, then It appears that at most a small fraction

of P(D) is due to dust drag.
130 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

140 • 10

MO

JOO

140

140

120

1 0 0

-p J

- j

-17.•

-IT.l

-ITO

1900

1*00

IJOC

I20C

MOO

IOOC

-

-

-

- ••••**M«

•'•'• •

•. . • "*** fL*«f

BATE OP CMAHttC Of THE PERIOD (in toy%/4

MM

* • •

(Seal* MifM o1«imM H • 25 0 km)

HEIGHT OF PERIGEE ABOVE THE INTERN!

»>) . "

TIONAL ELLIPSOID (km)

ITO| 7 aro MO 2to soo no MO «O MO JM

JU0.il r

•AH 0

FIGURE 3.—Effects of solar activity on the motion of Echo I.

term plus higher powers in the eccentricity.

Hence, the energy change owing to radiation

pressure can increase more rapidly with increasing eccentricity than can the energy change

owing to air drag when the atmospheric scale

height is reasonably large.

We note that Echo I lost energy to the solarradiation field until the end of December,

1960; then the satellite stayed in sunlight

throughout its orbit for about two weeks.

In January, 1961, the satellite gained energy

from the solar radiation, in the manner described above.7

 In fact, during the latter part

of January and much of February, 1961, it

gained more energy from the solar-radiation

field than was lost to air drag. This marked

the first time that a passive artificial satellite

exhibited an actual increase in period.

Through March 1, 1961, the cumulative loss

of orbital energy of Echo I was manifested in a

decrease of about 50 km in its semimajor axis.

Of this total, approximately 0.1 km can be

7

 In general, a satellite will lose energy to the radiation field when

sunlight pressure causes its perigee height to fall, and will gain energy

when this pressure causes the perigee height to rise.

attributed to the Poynting-Robertson drag

effect. Unfortunately, this effect is hopelessly

masked by the large and rapid fluctuations in

the atmospheric drag.

Computation of atmospheric densities

It is generally assumed that the neutral air

drag acceleration is given by —%CD(A/M)v2p,

where CD is a dimensionless aerodynamic drag

coefficient; v is the velocity of the satellite

relative to the atmosphere; and p is the atmospheric density. Using this assumption, one

obtains a relation between P(D) and the air

density pp at the height of the perigee of the

satellite's orbit. This relation is

P(D)=-fCD(A/M)p,, (1)

where / is given by a rather complicated function of the orbital eccentricity, the oblateness

of the earth, and the (assumed) velocity of rotation of the atmosphere (Sterne, 1959). In

the derivation of this formula it was assumed

that the atmospheric density at an altitude
RESULTS ON THE ORBIT OF ECHO I—ZADUNAISKY, SHAPIRO AND JONES 131

AZ above perigee can be expressed by

p=Pp exp (—AZ/H), (2)

where H is the density scale height.

There are several difficulties associated with

inferring atmospheric densities from equation

(1). First, it is necessary to use the appropriate P. With the mean anomaly represented

by a polynomial like M=M0-\-Mi(t—to)-\-M2

(t—t0)

2

, where t$ is the epoch of the elements,

it is usual to consider the rate of change of the

period to be given by P=—2M2fM\. This

form, however, leads to difficulty because the

mean anomaly is measured from perigee, the

angular motion of which is subject to relatively

large variations owing to the solar-radiation

pressure. On the other hand, equation (1) was

derived on the assumption that the perigee

position is stationary. Therefore, in order to

use an equivalent P, we corrected M2 for the

contribution due to the mean acceleration of a>.

We then calculated P(D) by subtracting P(R)

from the "corrected" values of P.

Second, it is not clear what value should be

used for CD. The concepts usually employed in

calculating CD may require considerable modifications for the high altitudes traversed by Echo I.

If, however, a future analysis indicates that a

different (but constant) value of CD should be

used, then it will be quite simple to correct our

density determinations.

Third, it is difficult to determine a meaningful

value for the scale height since densities at different altitudes have been calculated for different periods of time. Since the density at each

altitude suffers large fluctuations in time, comparing densities at different altitudes at different times is inadequate for determining the relation between densities at different altitudes

at the same time. From figure 3 it would seem,

as has been shown at lower altitudes, that the

solar activity must be considered when determining scale heights. A further difficulty is

caused by the possible diurnal bulge in the atmosphere which, among other things, can invalidate equation (2). In any case, the rather

large fluctuations in our values of P(D) seriously limit the accuracy of scale-height calculations.

In our computation we assumed that the ratio

{AjM) varied with time (as determined from

our theoretical match with the experimental

variations in eccentricity), and that CD maintained the constant value 2.5 (Stirton, 1960).

Air densities corresponding to five possible scaleheight values (ranging from 50 to 450 km) were

then computed. Some of our results are given

in table 2 of Smithsonian Special Report No.

61. For illustrative purposes, we present in

figure 3 only the densities derived from the

assumption of a 250-km scale height. Of course

it should be understood that the scale height

is probably not constant over this altitude (and

time) range.

Because of the uncertainties in our values of

the drag coefficient and the scale height, the

densities we have inferred from the orbit of

Echo I must be considered provisional. Certainly a more refined analysis of the data is

needed to improve bad determinations of P(D).

Lifetime of Echo I

Because our knowledge of air densities and

the variation of drag coefficient with altitude is

inadequate, we find it almost impossible to predict accurately the lifetime of Echo I. We do

know, however, that the perigee altitude and

the eccentricity oscillate with a peak-to-peak

period of about 300 days.8

In view of the large amplitude oscillation of

perigee height, it is most likely that the satellite

will perish soon after one of these regular dips of

perigee into the atmosphere. Hence, one can

estimate the month of "death" almost as accurately as the year. The best estimate we can

make, using air-density values determined during the first few months after the launch, is

that the balloon will stop orbiting shortly after

its fourth dip into the atmosphere in July, 1963,

if its shape is not substantially changed before

then.9 However, in view of the decrease in air

density to be expected in the next few years (in

accordance with the correlation between sunspot activity and upper atmosphere densities),10

the balloon will probably survive at least until

its fifth dip in the spring of 1964. In figure 4 we

* More accurately, the period from maximum-to-maximum value of

perigee altitude Is closer to 310 days, while the minimum-to-minimum

period varies.

• Cross-section measurements made on Echo I by the M.I.T. Millstone Hill Radar Indicate that little change occurred In the shape of the

balloon from the first few days after launch until January 11,1961.

'• For this and other useful suggestions, we thank Dr. L. O. Jacchia.
132 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

MEAN ALTITUDE

JAN 0

200H 1961

, I •

MJD 37,150 350

JAN 0

1962

JAN 0

1963

JAN 0

1964

JAN 0

1965

550 750 950 38,150 350 550

TIME (days)

FIGURE 4.—Time variation of perigee altitude and mean altitude of Echo I.

750

present the time variation of perigee altitude

and mean altitude for several different extrapolations of our density model. These curves

illustrate various possibilities for the behavior

of perigee distance near the end of the balloon's

life.

Acknowledgment

We express our gratitude to Mrs. Beatrice

Miller for her excellent work in calculating the

theoretical and experimental residuals of the

orbital elements.

Abstract

In this paper we compare theoretical and experimental determinations of the orbit of Satellite 1960 i 1 (Echo I).

The experimental orbit was deduced from Baker-Nunn photographic data.

The observed variations of the Echo orbit—due primarily to the effects of the pressure of sunlight—are in excellent agreement with our theoretical results. The perigee altitude has an oscillation of large amplitude (approximately 600km) and long period (approximately 300 days), which has a decisive influence on the lifetime of Echo I.

Our present best estimate is that the balloon will perish in the spring of 1964.

We have also estimated the rate of escape of gas from the balloon by comparing the observed and the calculated

variations in orbit eccentricity.

From the changes in orbital period we have inferred air densities over the altitude range 950 to 1500 km, making

corrections for the rather large changes in period that are due to a net loss or net gain of energy from the solar-radiation field. The atmospheric drag is strongly correlated with the flux of the 10-cm and 20-cm solar radiations, as

well as with a major solar flare.
The Atmospheric Drag of Artificial Satellites

During the October 1960 and November

1960 Events

By Luigi G. Jacchia

Accurate accelerations dP/dt were derived in

the interval from November 4 through 24, 1960,

for six satellites with perigee heights ranging

from 350 to 1121 km.

The accelerations were determined by the

following procedure. Using the Differential

Orbit Improvement (DOI) program of the

Smithsonian Astrophysical Observatory, we

computed the orbits at suitable intervals. We

then fitted, by means of least squares, empirical time functions to all the elements in

the 20-day interval, allowing no systematic

residuals except in the mean anomaly M, in

which the residuals were permitted to rise to

db 10~8 revolutions in exceptional cases (the error in the major axis resulting from such residuals is smaller than 50 meters). The empirical

functions representing the elements were then

fed into the DOI program modified to yield

residuals in the mean anomaly for individual

observations when the orbital elements and

their time variations are given. We then plotted these residuals, and added the second time

derivative of the resulting curve to the second

time derivative of the empirical equation for

M; allowance was made for any acceleration resulting from the motion of the perigee. The resulting values of d2M/dt2

 were then transformed

to dP/dt.

The results of these computations are shown

in figure 1, in which the acceleration curves

are arranged in decreasing order of perigee

height zP. At the bottom we have added the

acceleration curve derived by G. V. Groves

(1961) for Satellite 1959 e2. The accelerations

of Satellite 1960 tl have been corrected for the

effect of radiation pressure, according to the

data of Zadunaisky, Shapiro, and Jones (1961).

All these curves are rather faithful images

of the geomagnetic-index curves shown in

the lower part of the diagram. Since the

acceleration curves are drawn in natural

rather than in logarithmic scale, the correspondence should be best with the aP index, of which

Ap is a daily mean. The two major magnetic

perturbations, with maxima at November 13.4

and November 16.1 (U.T.) are reflected in

all the acceleration curves with no appreciable

time differential and with relative intensities

comparable to those of the peaks in the av

curve. The beginning and end of the perturbations are essentially coincident with the

beginning and end of the magnetic storm,

confirming previous results by the author

(Jacchia, 1959b). The weaker perturbation

of November 21 is clearly visible in the acceleration curve of Echo I (Satellite 1960 il) and

can be traced in those of Satellites 1959 a\

and 1960 fl.

The figures outside the right margin of

figure 1 give the mean value A of dP/dt for

each satellite before and after the perturbations

of November 12 through 17^ The plotted

points represent the ratio A/A of the instantaneous acceleration A to this mean value A.

The relative intensity of the perturbation for

each satellite can thus be estimated directly

from the amplitude of the individual curves—

i.e., by reading the maximum ordinate on

the scale on the left. Other pertinent data—

the declination of the satellite's perigee 8P

and the geocentric angle \f/ between the perigee

133
134 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

I960 NOVEMBER

15

A

A 4

3

2

3

2

I

2

I

2

I

400

300

200

100

0

1958 ALPHA

(EXPLORER I ) V -- 3-

1959eZ

(DISCOVERER! CAPSULE)

(GROVES)

205 km

'..V7* • •—...»-- , -f-j -i —..* •

io.7-cm flux

• .if . . t . . t i ' i I I I I L

5 10 15 20 24

I960 NOVEMBER

FIGURE 1.—Atmospheric drag of seven artificial satellites during the November 1960 events, compared

with the geomagnetic planetary index av and the solar flux at 10.7 cm.
400

300

200

100

0

200

t

.d"

100

25

ATMOSPHERIC DRAG OF ARTIFICIAL SATELLITES—JACCHIA

I960

135

'.-.•,~* ••••% .W* V-i\

10.7-cm. SOLAR FLUX

I

September

10

October

15 20

FIGURE 2.—Atmospheric drag of two artificial satellites during the magnetic storm of October 6-7,1960,

compared with the geomagnetic planetary index av and the solar flux at 10.7 cm.

and the sun—are given inside the right margin

of the diagram. The arrows at the top and

at the bottom of the diagram mark the instant

corresponding to the appearance of 3+ flares

on the sun.

It should be apparent that the amplitude of

the perturbation shows a general tendency

to be greater at greater heights, although

other factors probably complicate the picture.

The two lowest satellites (zp = 205 and 305

km, respectively) show a jump by a factor of

2 in the drag during the November 13 perturbation, not much greater than the amplitude

observed for Satellite 1958 61 (2p«220 km)

during the magnetic storms of July 9 and

September 4, 1958 (Jacchia, 1959b). A correlation of the amplitudes with 8P and ^ is not

immediately apparent. It is true that Satellite

1960 £1, with a large amplitude, had its perigee

in relatively high latitudes (+49°); on the

other hand, Satellite 1958 /S2, with the largest

amplitude of all, has an inclination of 34°,

which keeps the satellite at low latitudes

throughout its orbit. The smaller amplitude

of the perturbation for the Echo I satellite

could be explained by the much greater scale

height prevailing at that height.

More clues seem to be necessary to understand the mechanism through which corpuscular

radiation interacts with the atmosphere.

As an addendum we present in figure 2

the accelerations of Satellites 1960 il and

1958 Alpha, during the magnetic storm of

October 6-7, 1960. The increase of the atmospheric drag amounted to a factor of a little

over 2 for both satellites; again the development of the perturbation was simultaneous

with that of the magnetic storm.

All satellite accelerations are tabulated in

tables 1 and 2.
136 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Accelerations P of six satellites during the

November 1960 events

(Pu is the acceleration due to radiation pressure, according to Zadunaisky,

Shapiro, and Jones (I960).)

Nov. 1960

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

11.8

12.0

12.2

12.4

12.6

12.8

13.0

13.2

13.4

13.6

13.8

14.0

14.2

14.4

14.6

14.8

15.0

15.2

15.4

15.6

15.8

16.0

16.2

16.4

16.6

16.8

17.0

17.2

17.4

17.6

17.8

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

25.0

1960

10»P

-6. 0

-6. 0

-5. 5

-6. 2

-6. 2

-6. 2

-6. 7

-7. 3

-7. 1

-7. 3

- 7 . 1

-7. 6

-8. 0

-9. 4

-10.8

-15.7

— 19.2

-14.1

-11.0

-9. 6

-7. 4

-7. 2

-6. 6

-6. 2

-6. 0

-6. 5

-6. 7

-7. 2

-7. 2

-8. 3

-8. 8

-7. 6

-7. 5

-7. 3

-6. 6

-6. 3

-6. 4

-6. 4

-6. 1

-6. 2

-6. 3

-6. 2

-6. 4

-6. 0

-6. 4

-6. 8

-7. 4

- 7 . 1

- 7 . 1

-6. 4

-6. 4

—6.6

-6. 6

-6. 6

il (Echo I)

io«pfi

-2.9

-3.0

-2.9

-2.9

-2.9

-2.9

-2.9

-2.9

-2.9

-2.9

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.8

-2.9

-2.9

-2.9

-2.9

-3.0

-3.0

-3.0

1 0 «(p-/

- 3 . 1

-3.0

-2.6

-3.3

-3.3

-3.3

-3.8

-4.4

-4.2

-4.4

-4.3

-4.8

-5.2

-6.6

-8.0

-12.9

-16.4

-11.3

-8.2

-6.8

-4.6

-4.4

-3.8

-3.4

-3.2

-3.7

-3.9

-4.4

-4.4

-5.5

-6.0

-4.8

-4.7

- 4 5

-3.8

-3.5

-3.6

-3.6

-3.3

-3.4

-3.5

-3.4

-3.6

-3.2

-3.6

-4.0

- 4 6

-4.2

- 4 2

-3.5

-3.5

-3.6

-3.6

-3.6

TABLE 1.—Accelerations P of six satellites during the

November 1960 events—Continued

1958 02 (Vanguard I)

Nov. 1960 107P

4.0

6.0

8.0

10. 0

11.0

11.5

12. 0

12. 5

13.0

13. 5

14.0

14. 5

15.0

15.5

16.0

16. 5

17.0

18.0

19.0

20.0

21.0

22.0

24.0

26.0

-0.61

-0.61

- 0 . 59

-0.65

-0.69

-0. 8

- 1 . 1

-2. 5

-3. 6

-4. 2

-2. 2

-0. 4

- 0 . 1

-0. 4

-0. 8

-0. 4

- 0 . 1

-0. 3

-0.35

-0. 4

-0. 4

-0. 4

-0. 5

-0. 5

1969 al (Vanguard II)

Nov. 1960

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

12.5

13.0

13. 5

14.0

14. 5

15. 0

15.5

16.0

16. 5

17.0

18.0

19.0

20.0

21.0

22.0

23.0

24.0

WP

-1. 5

-1. 8

-1. 7

-1. 7

-1. 6

-1. 8

-1. 9

- 1 . 9

-1. 7

-3. 8

-7. 7

-8. 3

-3. 7

- 3 . 1

-1. 3

-1. 2

-2. 7

-2. 7

-1. 4

-1. 4

-1. 4

-1. 8

-1. 7

-1. 7

-1. 4

- 1 . 1
ATMOSPHERIC DRAG OF ARTIFICIAL SATELLITES—JACCHIA 137

TABLE 1.—Accelerations P of six satellites during the

November 1960 events—Continued

1959 Eta {Vanguard III)

TABLE 1.—Accelerations P of six satellites during the

November 1960 events—Continued

1960 £1 (Explorer VIII)

Nov. 1960

4.0

6.0

8.0

10.0

11.0

11.5

12.0

12.5

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

22.0

24,0

107P

-4. 9

-5. 0

- 5 . 1

-5. 4

-5. 6

-5. 6

-5. 9

-7. 6

-9. 5

-9. 9

-6. 9

-5. 0

-5. 3

-4. 7

-6. 0

-6. 7

-5. 7

-5. 7

-4. 7

- 4 . 1

-3. 7

-3. 7

-3. 4

-3. 5

-3. 6

1960 $1 (Explorer VIII)

Nov. 1960

6.0

7.0

8.0

9.0

10.0

11.0

11.5

12.0

12.5

12.75

13.00

13.25

13.50

13.75

14.00

14.25

14.50

14.75

15.00

15.25

15.50

15.75

16.00

16.25

16.50

16.75

10«P

-0.37

-0.39

-0.41

-0.45

-0.50

-0.56

-0.58

-0.61

-0.75

-0.92

-1. 2

-1. 7

- 2 . 15

-2. 0

-1. 5

-1. 0

-0. 7

-0. 5

-0. 5

-0.45

-0. 5

-0. 6

-0. 8

-0. 7

-0. 7

-0. 6

Nov. 1960

17.00

17.25

17.50

17.75

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

23.0

24.0

10«P

-0. 6

-0. 5

-0. 5

-0.3 5

—0.37

-0.3 6

-0.3 5

-0.3 0

—0.40

-0.4 4

-0.5 0

-0.4 6

—0.47

—0.44

-0.4 2

1958 Alpha (Explorer I)

Nov. 1960

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

12.2

12.4

12.6

12.8

13.0

13.2

13.4

13.6

13.8

14,0

14.2

14.4

14.6

14.8

15.0

15.5

16.0

16.5

17.0

17.5

18.0

19.0

20.0

21.0

22.0

23.0

24.0

106P

-5.1 4

-5.1 1

— 5. 16

-5.3 7

-5.5 9

-5.6 7

-5.7 9

—6. 10

-5. 9

-6. 1

- 6 . 1

-5. 9

-6. 5

-7. 0

-10. 3

— 10.7

-10. 3

-7. 6

-6. 4

-5. 6

-5. 2

-5. 2

-5. 2

-5.3 1

-5.3 8

— 6. 16

-5.8 4

- 5 . 17

— 4.56

—4.48

-4.5 7

-4.5 8

-4.5 0

—4,56

-4.5 7

-4.5 5
138 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 2.—Accelerations P of two satellites during the

October 1960 events

(p is the acceleration due to radiation pressure, according to Zadunaisky,

Shapiro, and Jones (I960).)

Oct.

Sep. 1960

25.0

26.0

27.0

28.0

29.0

30.0

1.0

2.0

3.0

4.0

5.0

5.50

5.75

6.00

6.25

6.50

6.75

7.00

7.25

7.50

7.75

8.00

9.0

10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

19.0

20.0

21.0

22.0

1960 il {Echo I)

108/>

-3.55

-3.55

-3.26

-3.07

-2.87

-2.84

-2.74

-2.52

-2.1 1

-1.89

-2.46

-2. 8

-2. 9

-3. 0

-3. 8

- 4 . 1

-4. 7

-4, 5

-3. 8

-3. 3

-3. 3

-3. 2

-3.04

-2.97

-3.57

-4.08

-4.22

-4.23

-4.79

-4.87

-4.79

-4.99

-4.95

-4.55

—4.79

-4.71

10° PR

-1.63

— 1.56

— 1.49

-1.36

— 1.22

-1.05

-0.88

—0.71

-0.56

-0.50

—0.51

-0.56

-0.58

-0.61

-0.64

-0.68

-0.7 1

-0.75

-0.79

-0.82

-0.86

-0.90

-1.05

- 1 . 19

-1.32

-1.44

-1.45

-1.62

-1.70

-1.78

-1.84

-1.91

-2.00

-2.06

—2. 13

—2.21

1958 Alpha {Explorer I)

Sep. 1960

25.0

26.0

27.0

28.0

29.0

30.0

106£

-3.5

-3.3

-3.5

-3.5

- 3 . 1

-3.2

10° {P-PR)

-1.9

-2.0

-1.8

-1.7

-1.65

-1.8

-1.9

-1.8

-1.6

-1.4

-1.95

-2.2

-2.3

-2.4

-3.2

-3.4

-3.95

-3.8

-3.0

-2.4

-2.4

-2.3

-2.0

-1.8

-2.25

-2.6

-2.8

-2.6

- 3 . 1

-3.1

-2.95

-3.1

-2.95

-2.5

-2.6

-2.5

TABLE 2.—Accelerations P of two satellites during the

October 1960 events—Continued

1958 {Alpha Explorer I)

Oct. 1960 10« P

1. 0

2.0

3.0

4.0

5.0

5. 5

6.0

6. 2

6. 4

6.6

6. 8

7.0

7. 2

7. 4

7. 6

7. 8

8.0

8. 2

8.4

8.6

8.8

9.0

9. 5

10.0

10. 5

11. 0

12.0

13. 0

14.0

15.0

16.0

17.0

18.0

19. 0

20.0

21.0

22.0

-3. 7

-3. 6

- 3 . 1

-3. 0

-3. 4

-3. 6

-3. 8

-3. 8

-4. 8

-5. 6

-7. 2

-8. 3

-7. 3

-5. 8

-5. 0

-4. 8

-3. 9

-3. 6

-3. 8

-3. 8

-3. 9

-3. 8

-3. 9

-3. 9

-4. 0

-4. 5

-4. 7

— 5.0

-5. 0

-5. 2

-5. 3

-5. 4

-5. 3

- 5 . 1

-4. 8

- 4 . 7

- 4 . 5

Acknowledgment

The assistance of Mr. Jack W. Slowey is gratefully acknowledged.
Effect of the Diurnal Atmospheric Bulge on

Satellite Accelerations

By Stanley P. Wyatt

The spherically symmetric isothermal atmosphere

Several authors have derived expressions for the secular acceleration of a satellite moving through

the high atmosphere (Sterne, 1958b; Groves, 1958b; King-Hele, Cook, and Walker, 1959). The

simplest and most tractable assumptions one can make are that the terrestrial atmosphere is spherically symmetric, stationary, and of constant scale height throughout those strata traveled by the

satellite in question. Within the framework of these assumptions, the secular acceleration is given by

AP A qPa

P~ 6LDmJl^e

Here, AP/P is the dimensionless change of period per period; CD the drag coefficient; Aim the ratio

of the average geometrical cross-section of the satellite to its mass; q the perigee distance and e

the eccentricity of the satellite; pq the atmospheric density at the level of perigee; E the eccentric

anomaly of the satellite; and e the base of natural logarithms. The dimensionless constant c is

defined by c=qe/H(l —e), where H is the scale height of the atmosphere and is assumed to be constant.

Equation (1) can be integrated by two techniques, depending on the magnitude of c and e

for the satellite in question. For present-day satellites, the value of q is between approximately

6600 and 8000 km, and of e between 0.00 and 0.25. The scale height of the high atmosphere lies in

the approximate range 50 to 100 km. Thus q/H is a large number, between 60 and 160, and the

relevant values of c for existing satellites are between 0 and 50. For a satellite orbit of moderate

eccentricity, the drag is significant only near perigee. Using the formulas of Sterne (1959), one can

change the variable of integration in equation (1) through the substitution 1—cos E=y2/c, making

the integrand the product of e""2 times a power series in y

2

fc. The upper limit of integration can be

extended to infinity because of the negligibly small air density at large values of y. The secular acceleration for such satellites is given by Sterne's equation (10), with d=l. I have confirmed his

result; in a spherically symmetric, stationary atmosphere, the secular acceleration of a satellite in

an appreciably eccentric orbit is

where

== . . > (3a;

a nd

This formula is accurate to one percent for values of c ^2.5 and therefore can be used for all satellites

with e>0.04.

139
140 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 V0L

- •

For small eccentricities, the atmospheric drag is appreciable all around the orbit; consequently,

equations (2), (3a), and (3b) are not valid. Instead, equation (1) can be expressed as the

product of ee cos * times a power series in e cos E. The integral can then be evaluated in terms of

Bessel functions of imaginary argument. If n is a positive integer, we have

r

Jo

/

cos nEAB=f/,(c)=xS ,, , w (4)

To order c3, Sterne (1959) finds and I confirm that within the framework of the stated assumptions,

the secular acceleration of a satellite is

A P A

~=-3irCD £ Opt[e-e/o(c)F.+««-Ji(c)V,], (5)

IT ifh

where a is the orbital semimajor axis and where

V0 = l+§«'-£ (6a)

and

Tables of <rclo(c) and e~e/i(c) are available in Watson's (1944) treatise on Bessel functions. For a

circular orbit, equation (5) reduces to

^p-=-?*CD^aPa. (7)

Some complicating effects

Modifications of these equations have been derived for several effects. One interesting generalization is the assumption that atmospheric density is a function of height above the oblate-spheroidal

earth rather than above a sphere (Groves, 1958; Sterne, 1959). Thus when the perigee of a satellite

is located at high latitude, the predicted drag is less than when it is later located near the equator.

For a polar satellite, the secular acceleration arising from such an effect would vary approximately

15 percent either side of the mean during rotation of perigee for a scale height of 50 km, and approximately 10 percent for a scale height of 100 km. For low-inclination satellites, the variation would

be less. As we shall see later, however, this effect is masked by others for high satellites.

A second modification of the drag equation takes account of the rotation of the atmosphere with

the earth (Sterne, 1959). A satellite moving in a direct orbit experiences a "headwind" of smaller

magnitude than does one moving in a retrograde orbit. For equatorial orbits, neglect of atmospheric

rotation leads to errors of approximately 10 percent in the secular accelerations. For orbits of

higher inclination the error is smaller.

A third modification accounts for the increasing scale height of the atmosphere with height

above ground (Jacchia, 1960b). From data at heights near 400 km, Jacchia finds that neglect of

this variation leads to overestimates of the acceleration by approximately 5 percent in the nighttime

atmosphere, and up to 10 percent in the daytime atmosphere. The maximum error occurs near

e=0.02; of course, the error must be zero for a circular orbit.

The diurnal bulge of the atmosphere

The assumption of a spherically symmetric atmosphere at heights of several hundreds of kilometers

has been shown to be untenable by several analyses (Jacchia, 1959c; 1960a; Priester and Martin, 1960;

Wyatt, 1959). The high atmosphere bulges toward a point in the sky some 15° to 30° east of the

sun. For a fixed index of solar activity, the observed accelerations of Satellite 1958 #2 (Vanguard
EFFECT OF DIURNAL ATMOSPHERIC BULGE ON SATELLITE ACCELERATIONS—WYATT 14 1

I) indicate that the air density at 665 km is about 10 times as great when perigee passage occurs an

hour or two after noon as when it occurs during the night. It is also clear from observations of

several satellites that the scale height of the atmosphere increases with height at all times of day.

Interpreting these results physically, Nicolet (1960a) finds that the density of the high atmosphere

is governed largely by the absorption of solar ultraviolet radiation below 200 km. The influx of

energy fixes the temperature gradient of the atmosphere between 200 and 300 km and also the

temperature of the nearly isothermal atmosphere at greater heights. Because the ultraviolet input

at any place depends both on the time of day there and on the general level of solar activity, the

high-atmosphere density variations depend on both of these parameters, as observed. Above

approximately 300 km the time scale of heat conduction is short and any vertical column of air is

isothermal, provided that the injection of heat from the Chapman corona is small. Although T is

presumably constant at these levels, the scale height, H=kTlgm, increases with height. The acceleration of gravity decreases, of course, as the inverse square of the distance from the earth's center, thus

contributing to the increase of H. The mean molecular weight w also decreases with height, because

above approximately 150 km the air is in diffusion equilibrium; each type of molecule is sorted out

according to its mass and thus the concentration of N2 relative to O decreases with height. This

factor contributes importantly to the observed increase of scale height with height. It should be

added that, although a vertical column above 300 km is spatially isothermal at any one moment,

there is a temporal variation of about 500° K between day and night, a variation arising from the

varying injection of solar radiant energy at the lower atmospheric levels.

The fundamental drag equation in a bulging atmosphere

Jacchia (1960a) stresses that the diurnal bulge will distort the motions of satellites from the motions

they would have if the atmosphere were spherically symmetric. The chief problem addressed in the

present paper is the derivation of a fundamental drag equation for a bulging atmosphere in which

the scale height increases with height, followed by a comparison with the spherical approximation

with H constant. At the expense of some added calculation, the new equation should permit the

derivation of more precise atmospheric densities and other parameters of the high atmosphere.

The most tractable assumption is that the atmosphere is axially symmetric, and this assumption

is not at odds with observations to date. I shall adopt it here and, with Jacchia (1960a), shall

further assume that the atmospheric bulge always points toward the same declination as the sun,

but with a lag angle X. Thus the right ascension of the symmetry axis is ao+X. Previous analyses

indicate that 15° ^X^30° , so that at any given level of the high atmosphere the peak density occurs

between one and two hours after local noon. Jacchia's analysis of the accelerations of Satellites 1958

Alpha, 1958 02, 1958 62, and 1959 al shows that between 200 km and 700 km the atmospheric

density can be well represented by

p=Po(z)F2O{l+O.19[exp (0.0055s)-1.9] cos9 (*'/2)}, (8)

where

log po(3) = —16.021—0.0019852+6.363 exp (—0.00262). (9)

These equations are in cgs units except that z, the height above ground, is in kilometers. The angle

yf/' is the geocentric angle from the axis of the bulge. Equation (9) gives the density as a function of

height for the nighttime atmosphere at a moment when the daily-mean 20-cm solar flux of F20

(Priester and Martin, 1960) is unity. Unit flux density is defined as 10"20 Wm~2(c/s)~1. As can

be seen from equation (8), Jacchia finds that the density at a given point is proportional to the first

power of the solar flux, other factors being equal. In the brackets of equation (8), the dependence on

^' indicates the sharpness of the bulge as found empirically and the dependence on z shows that the

diurnal effect increases with height. Equation (8) is of such a form that it includes the effect of

increasing scale height with height. Jacchia's two formulas are remarkably successful in representing

satellite observations to date. Although they are empirical in nature, I propose to adopt them as a

basis for computing the drag equation in a bulging atmosphere.
1 4 2 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 V0L

-

6

For any satellite in an orbit of moderate eccentricity, the maximum drag occurs not necessarily

at perigee passage, but nevertheless fairly near it. For another satellite in a more circular orbit,

the maximum drag may occur at any value of the true anomaly. In either case, the height above

perigee, z— zQ, at which significant drag occurs is never very large. Let us therefore express equations

(8) and (9) in terms of s=z—za. Also, for economy we shall deduce the drag equation for a constant

level of solar activity, setting -F2o=l. Equation (9) for the nighttime atmosphere, $' = *, becomes

0026 ^S2

). (10)

The approximation here introduced is needed in order for us to integrate the equations. It underestimates the density given in equation (9) by about one percent two scale heights above perigee and

by about 20 percent four scale heights above perigee in the range 200 km^zff^600 km; thus, it is

an adequate representation for any given orbit. Equation (8) becomes, without approximation,

5

<

i

'-1.9] cos6(*72)}. (11)

The general density function we shall employ is obtained by substituting equation (11) in (10) and is

P(S, *0=P(0 , TT) {[1-K COS6 (^/2)]6-°'+Z COS6 (+'/2)r**} (1+ Us*), (12)

where

#=.361,

i=.19cOO55%

^(.OSSOge--0026^.004571) km"1,

R= (.038096" •°°26^—.000929) km"1,

U= .00004952c-

 00262

« km"2.

These quantities are either constant or are functions of perigee height alone and can be tabulated

once and for all. We shall consider the physical significance of some of them later.

Having chosen equation (12) for the density function, we find that the secular acceleration of

a satellite becomes

^j' 1l p s« (f £)«-*•} (l+U*)dE. (13)

To progress further, we must express both $' and s as functions of E, the eccentric anomaly.

First let us find the relation between \j/r, the instantaneous geocentric angle between the bulgeaxis and the satellite, and E. Figure 1 shows the geometry, with A the north celestial pole, V the

vernal equinox, N the ascending node of the orbit, Q the direction of perigee, P the instantaneous

position of the satellite, and B the instantaneous direction of the atmospheric bulge. We wish to

evaluate ^'=Bp. In triangle ABP we have

cos ^'=sin 8B sin SP-f-cos 8B cos 8P cos (aP—aB), (14)

where aB, 8B; aP, 8P are the right ascension and declination of the bulge and satellite, respectively.

By hypothesis aB=ao-\-\ 8B=8Q; and since these coordinates change slowly we shall regard them

as effectively fixed over an interval of a few days. Next let us express the equatorial coordinates

of the satellite as functions of its slowly varying orbital elements and of its rapidly varying eccentric anomaly. In the spherical triangle NFP, we have NF=aP—ay, where aN is the right ascension

of the ascending node; FP=8P; NP^OJ+O, where w is the argument of perigee and 6 is the true

anomaly; ZFNP=i, the inclination; and ZNFP=90°. With the aid of the relations

sin 5i>=sin i sin

cos (co+0)=cos 8P cos (aP—aN),

tan (aP—as) =cos i tan (o>+0), (15)
EFFECT OF DIURNAL ATMOSPHERIC BULGE ON SATELLITE ACCELERATIONS—WYATT 143

FIGURE 1.—The celestial sphere, showing the interrelationship of the locations of the

atmospheric bulge axis B, of satellite perigee point Q, and of the instantaneous satellite position P.

the desired formula comes out to be

cos ^'=M cos 6-\-v sin 0,

M=sin 8B sin i sin u-f cos 5B[cos (aN—aB) cos w—cos i sin (aN—aB) sin w],

j»=sin 8B sin i cos a—cos 8B[cos (aN—aB) sin w+cos % sin (aN—aB) cos «].

(16)

Next, the true anomaly is related to the eccentric anomaty and e, the eccentricity, by the equations

cosE—e

cos 6=- ^ 1 — e cos k,

sm Q=

ll-e2 sin E

1 — e cosE

It thus turns out that the required angular separation of the satellite and the bulge axis is

,. M(COS E— ej+vVl — e2sin E cos^=!

-

i

^

(17)

(18 )

1-e cos E

An alternative formula for $' can be found if we consider in figure 1 the triangle BQP. We have

cos ^'=cos \f/'t cos 0+sin ty'q cos x sin 6, (19)

where ^/'q=QB is the geocentric angle between perigee and the bulge axis; x= £BQP; and 6 again

is the true anomaly. Comparison with equations (16) shows that

i>=sin ^'i cos x-

(20)
144 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 V0L

-

6

We next wish to obtain s as a function of E. We readily find that

s=z—zQ=r— q=a{\—e cos E)~a(l — e)=±-^-z -• (21)

Substitution of equations (18) and (21) in equation (13) yields the equation for the secular acceleration of a satellite passing through a Jacchia-type atmosphere, the integrand now being a function

of E and of miscellaneous "constants" of the atmosphere and the satellite orbit.

Finally, instead of expressing the acceleration in terms of the nighttime density, P(0,TT), we find

it more useful to give it as a function of the density and scale height at the perigee point at the time

in question, since the latter quantities are the ones that can be deduced from the observations.

From equation (12) we find that

\~K) cose(^/2)}. (22)

Next, we define the density scale height (Jacchia, 1960b) by

1 __ 1 dp (23)

The density scale height is identical with the ordinary pressure scale height, kT/gm, if the temperature gradient and molecular-mass gradients are zero separately or if Tfm is independent of height.

Under actual conditions, with dT/ds^O and d7w/ds^0, the density scale height is somewhat smaller

than the ordinary scale height. Differentiating equation (12) and evaluating the right side of

equation (23) at s=0, \f/'=}f,'t> we obtain

H(0 , n - (l + (-E-g)cos(^/2)}

K

 '*a)~~Q{l-Kcos* (t'/2)+RLcos6 (fJ2)} K '

In particular, we may now interpret the constant Q, since equation (24) shows that H(0,ir) = Q~1.

Thus Q is the reciprocal of the nighttime scale height at perigee. The perigee scale height on the

bulge axis is H(0,0) ={ 1 + {L-K)}{Q(l -K)+RL} ~K

The equation for an eccentric orbit

Let us define a dimensionless variable of integration, y

2

, by y

2=s/H, where for simplicity in the

sequel we shall write H(0, \f/'g) =H. It follows from equation (21) that

lcosff^ . . (25)

ge

This is the same substitution effected in the first section of this paper, but it should be stressed that

we are here concerned with the scale height at a perigee point located at a particular height and at a

particular angular distance from the sunward bulge. Our next task is to transform the integrand

of equation (13) into a function of y alone. First, the three factors involving s transform very

simply, and exactly, by substitution of s=Hy1. Second, the three factors explicitly containing E

can be expressed as power series in (H/q)y2. In particular,

(.1— e cos

and

(!+«)'" r H(2-e) 3ffl "1

(l-e)1/2 L 2(1+e)

 V ^222

(l+«)2 V " 'J

, g(l-g ) . • 3H 2(l-e) 2 ^ , "I ,

-^- y + 32gv ^ • • Jdy-
EFFECT OF DIURNAL ATMOSPHERIC BULGE ON SATELLITE ACCELERATIONS—WYATT 14 5

The product of these factors is

^- {1+eyUy |l+ f My>+^Ny*+ . . . 1 (28)

qe L 2 T J

where M and N, given by equations (3), are functions of e alone. Physically, these terms occur

because atmospheric drag exerts a tangential perturbing force; they appear independently of any

particular choice of atmospheric model. Third, we need to express ty' in terms of y. We have

cos8 (^'/ 2)=f (1 + cos rp')z. Substitution of equation (18) then gives us an equation for ^' as a function of E. With the transformation equation (25) we then obtain ^'as a power series in -J(H/q)y.

Finally, note that the upper limit of integration of equation (13) becomes y

2=rr/^_ \' Since we

are here dealing with orbits of appreciable eccentricity we may extend the upper limit to infinity

without appreciable error. All integrations involving odd powers of y then vanish, and therefore

#' can be expressed for our purposes as a power series in (H/q)yz. The formula turns out to be

cos6 (*'/2) = u-v — y

2+w ^-V*- (29)

The coefficients u, v, and w are constants for a given orbit; they are given by

| (30)

When we substitute equations (21), (22), (25), (28), and (29) in equation (13), the secular

acceleration becomes

(31)

Equation (31) can be integrated term by term after multiplication of the several brackets. Note

that the three power series in the first two brackets of the integrand have been truncated at

(H2/q2)y* in order to keep the calculation manageable. The coefficients of terms of higher order

decrease approximately in the ratio H/qe, which is less than about 0.1 provided the eccentricity is

not very small. The justification for terminating the final bracket at order y* was described in

the previous section.

Comparison with the simpler drag formula

Instead of writing down the general result of integrating equation (31), it is simpler to give the

ratio of this acceleration to that given by equation (2) for the spherically symmetric atmosphere of

constant scale height. One reason, of course, is that the coefficients are nearly identical. A second

reason is that the ratio is independent of F20, the solar flux at 20 cm. Let us call this ratio J. To

state its meaning more precisely, consider a date on which the perigee of a satellite lies zt km above
146 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 VOL

-•

the earth's surface and at a geocentric angle ^ from the bulge. The quantity J is then the secular

acceleration of the satellite as it describes one orbit through the Jacchia atmosphere divided by the

secular acceleration of the same satellite as it describes the same orbit through a spherically symmetric atmosphere in which the scale height is constant and in which the density and scale height

agree with the Jacchia atmosphere at the perigee point (but not necessarily anywhere else along

the orbit).

I have integrated equation (31) and computed values of t/for perigee heights ranging from 200

km to 600 km, for eccentricities from 0.1 to 0.3, and for four specific orientations of the orbit with

respect to the bulge. It turns out that the values of J are never much less than unity and that no

term involving either M or N contributes more than 0.007 to the end result. We may thus drop all

such terms in both equations (2) and (31) without significant error. The ratio then becomes

(32)

where we have set R = QZ* and where all other quantities have been previously defined.

In table 1 the values of J are computed from equation (32) for several perigee heights, eccentricities, and orientations of the orbit relative to the sunward bulge. The values are given to three

decimal places to indicate better the trend of the numbers. Two points should be kept in mind,

however, when the entries are examined. First, the final calculations have been much simplified by

leaving out all terms in M and N. Second, the convergence of the terms in equation (32) is very

rapid in all situations except where the perigee height is large (zff^600 km) and the eccentricity

rather small (e^O.l). In this extreme situation the atmospheric distortion is a maximum, the

scale height is large, and at very low eccentricities the drag may be significant at all points around

the orbit. Calculating and integrating the terms in y

6

 in equation (31), I find that the true value

of e/in the most extreme case of table 1 is even more exaggerated than the tabulated entry of 1.52;

it is greater by about 3 percent. Truncation of equation (29) at order y* affects other entries by

smaller amounts.

TABLE 1.—Values of J for several types of orbit

z/km)

200 300 400 500 600

0. 10

0. 15

0.20

0.25

0.30

Perigee coincident with bulge (p

1.024

1.030

1.033

1.035

1.035

1.016

1.028

1. 035

1.039

1.041

1.005

1.026

1.037

1.045

1.050

= 1, *=0)

0.990

1.022

1.040

1.052

1.060

Semi-latus rectum of orbit coincident with bulge G»=0, >

0. 10

0. 15

0.20

0.25

0.30

1.044

1.040

1.039

1. 038

1.037

1.076

1.067

1.062

1.059

1.057

1. 145

1. 122

1. 109

1. 101

1.096

1.280

1.230

1.203

1. 185

1. 173

0.977

1.018

1.044

1.060

1.072

'=±1)

1.520

1.435

1.382

1.346

1.322

Orbit normal coincident with bulge 0*=0, »=0)

All e 1.032 1. 042 1.057 1.084 1. 132

Apogee coincident with bulge G»=—1, r=0)

Alle 1.030 1.035 1.040 1.045 1.049

Interpretation of the results

Generally, the entries of table 1 show that the secular acceleration in a bulging atmosphere of

increasing scale height is greater than in the same orbit when the atmosphere is spherically symmetric
EFFECT OF DIURNAL ATMOSPHERIC BULGE ON SATELLITE ACCELERATIONS—WYATT 14 7

and of constant scale height. This means that use of the simpler formula tends to produce overestimates of the values of p^H from the observed accelerations, although not always.

The dominant reason for «/>l when either the orbit normal or the apogee coincides with the bulge

is the increase of scale height with height. Reference is made, for these and subsequent remarks,

to figure 3 of Jacchia's paper (Jacchia, 1960a), which is helpful in visualizing these effects. When

the normal to the orbit plane coincides with the bulge axis, the satellite is entirely ignorant of the

bulge because it always moves 90° away from it; when apogee coincides with the bulge axis, J is

not significantly different from unity because of the cos6 (\f/f/2) dependence. For both orientations

the values of J increase with increasing perigee height because of the gradient of the scale height.

When perigee coincides with the bulge, the gradient of the scale height governs the run of J at

low values of z8; the bulge is not pronounced at these levels. At 500 and 600 km, however, the

interplay of the two effects is stronger. At the higher eccentricities a satellite climbs steeply away

from perigee, and in the limiting case of an outbound radial orbit it is oblivious of the bulge and

the value of J is governed solely by the effects of scale height. At low eccentricities, however,

a satellite climbs uphill from perigee rather slowly and encounters equidensity contours that are

trending downhill with increasing distance from the bulge axis. The values of J for these cases

are slightly less than unity.

Finally, when the semi-latus rectum of the orbit coincides in direction with the bulge axis, the

asymmetry is marked, particularly for large perigee heights and low eccentricities. In this case a

satellite encounters maximum density at some point displaced 10° or 15° from the perigee point

toward the bulge axis; the secular acceleration is therefore significantly greater, up to 50 or 60

percent, than that given by the usual simpler formula.

The equation for a circular orbit

A general formula for the drag in a bulging atmosphere can in principle be derived for orbits of

small eccentricity, as before, in terms of Bessel functions of imaginary argument. Tentative work

indicates, however, that the resulting expression is of formidable complexity and provides little

or no insight into the physical effects. Accordingly, this paper will not be concerned with finding

an analytic expression for the acceleration in orbits of very small, but nonzero, eccentricity. Qualitatively, however, it is apparent that the more circular an orbit is, the more indifferent the satellite

is to the atmospheric density and scale height at perigee. Maximum drag may occur at any point

on the orbit.

In the limiting case of a circular orbit, the greatest drag occurs when \f/f is a minimum. The

minimum geocentric angle between satellite and bulge axis, f^, is given by tLB=irl2—i', where

%' is the angle between the bulge axis and the orbit normal, with 0 ^ir ^ ir/2. The density function

to be employed for a circular orbit is given by equation (12) with s=0, and is

p(0, yf,') = p(0, *•) {1 + (L—K) cos8

 (^'/2)}• (33)

In this particular case, moreover, 1]/' is given by

cos ^'=sin i' cos E, (34)

and the secular acceleration turns out to be

f (T xr\ r* o ~i^

(35)

Since no satellite passes through the point ty'=v unless i'=x/2, and since all satellites must pass

through ip'=ir/2, we shall find it more useful to express equation (35) in terms of p(0, ir/2), which

can be determined from equation (33). The result is then

*?- Q n A rn w/i 1 3(L-K) sin*i'\ f .
148 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 v01

-«

Notice that this result is identical with the simpler equation (7) when i'=0. The reason is, of

course, that when i'—Q and e=0 (and under no other conditions) a satellite sees constant density

all around the orbit, just as in the spherically symmetric approximation. When AP/P is observed

for a satellite moving in a circle with i's^O, the atmospheric density deduced from equation (7)

will be greater than p(0, x/2) by a factor equal to the bracket of equation (36). Alternatively

interpreted, this factor is the ratio of the secular acceleration at i' to that at t'=0. It is greater

than unity because of the sharpness of the bulge: density excesses in the daytime hemisphere more

than compensate density deficiencies in the dark hemisphere. When t'=ir/2, the satellite passes

squarely through the bulge; in this case the bracket is a maximum, ranging from 1.04 at 200 km

to 1.22 at 400 km and up to 1.56 at 600 km.

Conclusions

It should be emphasized, with Jacchia, that the density function here employed is not a necessary

consequence of high-atmosphere physics, but rather a product of numerical analysis of satellite

observations. For this reason it does not seem necessary at this stage to work out the exceedingly

complex higher-order approximations to equation (32) or the drag equation for orbits of very small

but non-zero eccentricity. The results of table 1 for orbits of moderate eccentricity and of section 8

for circular orbits indicate, without more elaborate calculations, that use of the drag formula for a

spherically symmetric atmosphere of constant scale height leads to a somewhat erroneous evaluation

of the structure of the high atmosphere.

The formulas developed here should, at the expense of some added calculation, make it possible

to improve our picture of the Jacchia-type atmosphere by refining the values of its constants. Qualitatively, the results of table 1 indicate that at heights of 500 km or so the atmospheric density on the

bulge axis is a bit greater than given by the usual drag formula, while the density 90° around from

the bulge axis is substantially less. We may therefore tentatively conclude that the sunward bulge

is even sharper than previously thought.

Acknowledgments

I am happy to thank Dr. F. L. Whipple, Director, for the opportunity to work at the Smithsonian

Astrophysical Observatory during the summer of 1960; Dr. L. G. Jacchia, whose comments suggested

this work and whose generous interest and advice have been most helpful; and others of the staff for

assistance in a variety of ways.

Abstract

Formulas are developed to express the secular acceleration of a satellite on passing through an atmosphere which

bulges in the sunward direction and in which the scale height increases with height, these two properties of the high

atmosphere having previously been established from satellite observations. Comparison of the new formulas with

those for a spherically symmetric atmosphere of constant scale height indicates that deduced atmospheric densities

may be systematically incorrect by up to 50 or 60 percent at heights of 500 to 600 km when the earlier and simpler

equations are used.
The Motion of Satellite 1958 Epsilon around

Its Center of Mass

By G. Colombo

This report presents a preliminary study of the motion of Explorer IV (Satellite 1958 Epsilon)

around its center of mass. The body is, to a good approximation, symmetric around its longitudinal

axis and, therefore, the tangential motion can be considered a regular precession, at least during one

orbital revolution.1 I shall refer to a precessional velocity ^, and to an angular velocity around the

longitudinal axis, <P, the angular velocity vector <a of the body being the resultant of these two components. Many good observations (Naumann, 1961) of the motion, especially of the variation of

<p, were made during the first 60 days after launch, and the variation in the direction of the precessional axis was computed. Although the dererminations are not precise, they definitely indicate

an angular displacement of the precessional axis of at least 10° per day in the first 30 days after

launch. Also, it is certain that the angular momentum of the body remained almost constant during

the whole period under consideration.

I have derived a method for computing the angular perturbation of the angular momentum.

Computation of the angular variation of the precessional axis due to the atmospheric drag, using a

nutation angle of 84°, yields a maximum value of 5° or 6° per day. For the gravitational torque,

the same perturbation has a maximum value of 2° per day. The perturbation due to the action of

the earth's magnetic field on an intrinsic magnetic dipole of the satellite amounts to 2° per day if

the maximum torque is of the order of 300 dyne-cm, since the effective value of the torque is the

value averaged over one precessional period. In addition, an evaluation of a magnetic dipole

induced in the ferromagnetic shell of the satellite has been tentatively done. The corresponding

theoretical torque would give an important contribution also for a nutation angle of 90°. It seems,

however, very difficult to choose a realistic value for the magnetic permeability because (1) the field

is very weak and the experimental determinations are made only for stronger fields, and (2) the actual

value of the permeability is related to components of the field which are changing sinusoidally. In

any case, we give the formula needed to compute the total perturbation per revolution, which

depends on the velocity vector of the center of mass at perigee, the position of the earth's magnetic

dipole with respect to the orbital plane, and, naturally, on the orbital elements.

This study also treats the problem of the rapid decrease in <P during the first 30 days after launch.

The decrease in the spinning of the satellite is not easily explained by the hypothesis that it is caused

by the action of an external torque, since the modulus of the angular momentum remains almost

constant. The most probable cause of this decrease is an internal vibrational motion that causes

the transfer of angular momentum from the axis of symmetry to a transverse axis. The observed

variations are in good quantative agreement with the law for the decrease in <p.

The question does not seem to be closed, since I definitely think that the problem of the motion

of a ferromagnetic body of any shape in a varying weak magnetic field has yet to be solved. With

torques as weak as those with which we are working, any effects neglected and perhaps undetectable

in the usual experimentations may have a significant influence on a satellite.

> Some Information (Lundqulst, Naumann, and Fields, 1961) about the physical structure of the body of Explorer IV was received just after I completed this report.

149
150 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

1. Coordinate systems

Let us associate the satellite S with three coordinate systems: first, with a moving coordinate system

(i, j , k) whose axes lie along the principal axes of inertia of the satellite and whose origin is the center

of mass G of the satellite (k is the unit vector of the gyroscopic axis oriented toward the satellite's

nose); second, a coordinate system (d, c2, £2) with £2 oriented along the precessional axis; and third, a

coordinate system (X, Y, Z) with origin at the perigee of the orbit of G; the X-axis is oriented toward

VQ (the velocity of G at perigee), the F-axis is normal to the orbital plane, and the Z-axis is oriented

toward the radius vector of the orbit at perigee. Let \p, <p and 8 denote the Eulerian angles of the

system (i, j , k) with respect to the system (eu c2, Q,). Let Ko denote the angular momentum of S with

respect to the center of mass G (note that KG is in the same direction as $2, since the tangential motion

is a regular precession). Also let 0 denote the angle between fi and the X-axis; 0x the angle between Q

and the F-axis; \ the angle between the projection of £2 on the XZ-plane and the Z-axis; and X the

angle between the projection of £2 on the FZ-plane and the Z-axis (see figs. 1 and 2).

FIGURE 1.—Definition of angles. FIGURE 2.—Reference systems.

2. Perturbation of the precessional motion

The elements that characterize the regular precessional motion may suffer perturbations due to the

torques exerted by the following: (a) atmospheric drag, (b) gravitational field of the earth acting on

the nonspherical shape of S, (c) interaction of the earth's magnetic field with an intrinsic dipole of S,

(d) interaction of the earth's magnetic field with an induced dipole of S, (e) energy dissipation due to

internal vibrational motion. Let Pa, Ph, Pe, Pd, and Pe denote the perturbations due to a, b, c d,

and e, respectively. Assume, as usual, that the total perturbation of the tangential motion is the

sum of the perturbations Pa, Pb, Pe, Pd, and Pe evaluated over the unperturbed motion.

To begin, we note that the average torques of Pa, Pb, Pe, Pd, and Pe that affect the motion in

one revolution are always perpendicular to Ka and that for perturbation Pe the external torque is

zero. Therefore, the modulus of the angular momentum of S during the motion may be considered

constant. The second-order effect of the drag is not considered since it is insignificant. In the

motion of Explorer IV (Satellite 1958 Epsilon), this property of the angular momentum was well

maintained during the first 30 days after launch. Only for this period do we have good observations.

2.1. Perturbation Pa.—First, the torque due to the atmospheric drag is computed. The

instantaneous torque Mo can be described by the expression

Mo=c*p(G)\vo\voXk=(C0-C2 cos2 7+Ci cos3 y)p(€f)\va\faXkt (1)

where y is the angle k makes with vo; p(G) is the atmospheric density at G; vo is the velocity of

G; and Co, C2, and C3 are the coefficients that characterize the shape of S.2 We arrive at this expres1

 More exactly, the coefficient c* should be expressed as c*=Co+Ci cos 7—Cj cosJ -H- Ct cos3 y— & cos4 y, as we will see later, but the second and

fifth terms of this expression play the same role as the fourth and third terms respectively. Therefore, we will have a good approximation if we consider

only equation (1). We shall return to this argument in section 3.1.
MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 15 1

sion by assuming that the center of pressure Q' is always along the gyroscopic axis of the satellite

and that this center can vary in its position with a change in y. The computation of Co, C2, and

C3 in the case of Explorer IV will be made in section 3.1. Finally, we assume that the velocity

of the atmosphere relative to the orbit, compared to the velocity of 0 over the orbit, is small.

Next, the average value of Mo in one precessional period is computed. Let vx, vv, and vt

denote the components of vo with respect to the system (ci, c2, Q). Thus,

\va\ cos y=vo-k=(vx sin \[t—vy cos $) sin 8-\-vt cos 5,

k=(c! sin \p—c2 cos ^) sin 5+Q cos 8. (2)

Averaging over $ from 0 to 2ir, we obtain

k=cos 512,

cos27k=cos 8 ["sin2 5+^- (2-5 sin2 5)1 2,

cos37k=r-r^3 ( | sin4 5+cos4 8)-|•£-. sin2 5(sin2 5-4 cos2 5)1 2. (3)

Finally, the average value of Mo in one revolution is computed. Let

v%2Io=^ £ yo\vo\ P(G)dt,

£ I {G)dt (4)

where T is the orbital period. Hence,

s 5 (Co-C2 ^ ^ IO-C72 cos 5 (l- | sin2 5^ I24-C3 (J sin* 5+cos4 8) I','

—I C3 sin2 5(sin2 5—4 cos2 8)lz\ XQ; (5)

thus, the modulus of Ko will remain constant for this perturbation (Ma is perpendicular to Ko)

and also for the other perturbations, as we shall see in sections 2.2 and 2.3.

If the moment of inertia around the gyroscopic axis is designated by C, and the initial velocity around this axis by r0 (assuming that the initial motion is approximately a rotation around

the gyroscopic axis only), then Ka=Cr0Q. Therefore, equation (5) becomes

-JT=~yif- ~{ co s 5 ( Co— C2 ) Io—C2 cos 5 ( 1— s sin2 5 H2

+C3 ^ | sin4 5+cos4 5) I'3

f— I C3sin2

 5(sin2 5-4 cos2 8)%|xa (6)

636-014—63 11
152 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

Equation (6) differs from the analogous form used by Beletsky (1960) for two reasons. First,

Beletsky does not consider the possibility of the displacement of the center of pressures along the

gyroscopic axis. This displacement is certainly true for Explorer IV, which is very asymmetric

with respect to the plane normal to the gyroscopic axis passing through G. Second, the Beletsky

equation contains a factor cos2 8, which can be misleading. Actually, there should be a factor of

cos 8 only. In a quasi-tumbling motion, the factor cos 8 is very small, and, as we shall see, it plays an

important role in the discussion. Therefore, in one revolution, the atmospheric drag causes a rotation of the precessional axis around a well-determined axis. This axis is dependent on the vector

given by equation (4) and, naturally, on the nutation angle. Also, it is important to note that the

terms in I3 and %' are not zero if cos 8 is zero.

Note that the rotation axis of 8 varies from one revolution to another since it depends on the

orbital elements of G, of which the most important are the secular variation of the argument of

perigee (the motion of perigee) and the right ascension of the ascending node (the regression of the

node).

2.2. Perturbation P&.—To compute the torque corresponding to Pb, we refer to the article by

Beletsky (1960). We can write the expression

^£2=5 Z\A-C) cos 0t (l-|sin2s) NX8, (7)

Cub £ \ £ /

where N is the unit vector normal to the orbital plane and parallel to the F-axis, and A is the moment

of inertia with respect to an axis normal to the axis of symmetry through G. Moreover,

% (8)

where h is the characteristic constant of the earth's gravitational attraction, and r is the geocentric

distance of G. If, as before, K6=O0 2, then

^7=1 7ZT (A~

 c) co s ei (l

 ~% sin2 5) N X 8. (9)

at & L/TQ \ & j

Therefore, the gravitational torque causes a rotation of 8 around an axis normal to the orbital plane.

This axis of rotation can be considered constant during one revolution. Also, the angular displacement of 8 due to perturbation Pb in one revolution can be computed by use of equation (9).

2.3. Perturbation Pe.—The intrinsic magnetic moment of S can be expressed as

(10)

and the earth's magnetic field as

H=Hxc1+Httc2+HzSt. (11)

Thus, the corresponding torque acting on the body is

M£°=/*XH. (12)

Assume that, to a good approximation,

where m^n and nr is the period of precession. If ip is much greater than ^, equation (13) can be

written with a value of m that is not so large.3 Also suppose that H remains constant along the arc

of the orbit covered by G in the period mnr. In this way,

M

(

a

m)

 = M3 cos 5QX H. (14)

1 The assumption is not necessary If MI—0.
MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 15 3

If m=n, the average value of M&"0 over one precessional period is

M e 0^ ^ 1"

0 0 8

 5)HXCi+/i3 cos 5J2XH. (15)

The last case will be excluded for the present, but we shall return to it in section 4.1, at the end of

this paper. Therefore, equation (14) is retained.

We should average MS"5 over one orbital period. To compute this average, we assume that the

earth's magnetic field is, in the usual form,

where u is the unit vector of the earth's dipole; r is the unit vector of the direction from the earth's

center to G; and fiB is the magnetic moment of the earth's dipole. The mean value of Mo is

-j-dv, (17)

dv ' v '

where v is the true anomaly of G over the orbit. Since

dv 27ra2

 /r—jj a(l-e*)

-77= rn-t V I — * »

 r==:T~i 7 \ '

dt Tr2 v 1+e cos (p— w)

we obtain with the usual notation

ft^ f (19)

where H should be expressed as a function of v. For this perturbation, we should remember that

the earth's dipole makes one rotation around the polar axis in one day. Since the angular displacement of a is about 9° during one revolution, we cannot consider a constant when integrating

equation (19).

We must write u=0.95Hi+0.29[n cos Ta(M-\-a0—Q) + UiXn sin Td(M+ao—Q)], where u, is

the unit vector of the earth's polar axis; n is the unit vector of the ascending node of the orbit;

Ta is the orbital period in days; M is the mean anomaly of Q; ao is the right ascension of u at the

time of perigee passage of G; and Q is the right ascension of the ascending node. The computation

of this perturbation is not easy; it would be preferable to postpone it until we have exact information

about the internal circuitry and the eventual permanent magnets in the payload.

Let im denote the inclination of the orbit over the equatorial plane, rm, of the earth's magnetic

dipole, and nn the radius vector of the ascending node of the orbit in this plane. To obtain an idea

of the magnitude of this perturbation, we consider nm and u as constant during one revolution.*

Thus, from equation (19), we have

tf(i-y<* [(~Isin 2 *-)

 u+ ls i n

 *•cos

and finally,

'

 nX L 0 ) a + 2 Sm

 *"

 COS

From equation (21), the rotation of $2 due to perturbation Pe can be determined. Therefore, in one

revolution Q rotates around an axis that depends on im and nm.

In each of the three perturbations just discussed, we have found that Ko • —fr=Q> which means

that the modulus of Ko is not affected by these perturbations, at least if MI=0-

-> ->

* This assumption is justified, since the small change in the orientation of H along the orbit due to tbe variation of u is negligible when the angle

-> -> -»-» -*

between Q and H is large. When the angle Q H is small, tbe variation In the direction of Q is also small.
154 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

2.If. Perturbation Pd.—Let us consider the perturbation of the angular momentum due to the

interaction between the earth's magnetic field (Fischell, 1961) and the ferromagnetic components of

the satellite. To give a possible explanation of the angular rotation of Q, and also for an angle 5

very near to 90°, we suppose that the satellite has a ferromagnetic component in the shape of an

elongated cylinder as the shell of the satellite. For such a component, the magnetization vector is

almost always oriented as is the axis of the cylinder. Thus,

1=0*—l)(H-m)m, (22)

where m is the unit vector of the axis of the ferromagnetic component (with m=mii+fn3k), and p

is its magnetic permeability. If we neglect the small hysteresis effect, the torque acting on the body

can be written in the following form:

M S W 0 =(M-1 ) ^ (m.H)mXH=(m.H)mXM*H, (23)

where V is the volume of the ferromagnetic component. Averaging as in the computation of perturbation Pe, and assuming (if mi5^0) that <p^>^>\p, we obtain

Mr>=M* [ ^ (1- 3 cos2 5)+ ^ (2-3 sin2 5)1 £2X(H.$2)H. (24)

We suppose, as before, that a and um are constant during one revolution. Then, averaging over one

orbital period and neglecting any small terms in e

2

, we find that

3 /9 \ 1 9

+2 cos im sin in I - sin2 im— 1 JuXnOT-8 + - sin2 ^(n^S) ^

j~27

+ — sin2 im cos im(u-nmXG cos im+u-Q sin im)

L o

- | u-Q sin in cos vjuXnTM V[ ^ (1-3 cos2 3)+ ^ (2-3 sin2 5)1 (25)

A/2

For an approximate computation, let us suppose that sin im=cos im=^-> with cos 5=0 and

sin 5=1. Hence

a

6

(l-c 2)

9

9 F27 *? ~\ "\

(Q)+ [ X2+ | QjX m V- (26)

This torque is zero if a, €1, and nm are coplanar and if nm and Q are parallel, or if m\=2m%, and,

therefore, these are all exceptional cases.

2.5. Perturbation Pe.—The observations of Explorer IV definitely indicate that the angular

momentum maintains a quasi-constant modulus during the first 30 days after launch. In this

period, a diminution of the order of 0.1 of the initial value occurs. Thus, the great diminution

of the kinetic energy (the final value is about 1/80 of the initial value) cannot be ascribed to

external torques, and therefore must be ascribed to internal dissipation. In this section, we shall

study the possibility of a transfer of angular momentum from the motion about the axis of

symmetry to the motion about the precessional axis, resulting from internal dissipative vibrational motion sustained by the precessional rigid body motion around the center of mass. Note
MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 15 5

that the aerodynamic torque (without the second-order effect) and the gravitational torque are,

a priori, coplanar with the gyroscopic axis, and, therefore, cannot be responsible for the observed

variation of •*. Not even the torque due to the second-order effect of the aerodynamic drag can

completely explain such diminution. Indeed, the perturbation due to the aerodynamic torque

will also cause an equal damping of the precessional component of the angular velocity (Notni

and Oleak, 1959; 1960). We should also examine the possibility that this dissipative action,

which does not change the modulus of the angular momentum, can result from the interaction

between the earth's magnetic field and the satellite. A simple computation of the average work

done by the magnetic field acting on the intrinsic magnetic moment gives a value of zero, if

In the usual notation, the expression for the kinetic energy of S is

[A($+ft sin2 5)+<?(«>+* cos 5)2]. (27)

Moreover, in any instant t of the precessional motion

r

 A cos 5

Suppose that initially the motion is a quasi-rotation about the axis of symmetry of the body with

velocity r0. Let P denote a point mass with a very small moment of inertia relative to the inertial

characteristics of S. Let Po denote the equilibrium position of P when the body is at rest, where

(zb, 0, s0) are the coordinates of Po with respect to (i, j , k). Assume that, if P is not at Po, then

both an elastic force with center at Po and a dissipative force acting in the same direction as the

velocity of P relative to (i, j , k) and proportional to the velocity act on P. If S has a component

with small mass elastically connected to the rigid body, the dissipation can be due to elastic

hysteresis. The motion of P with respect to the moving frame (G, i, j , k) is governed by the

equation

m ^=E+D+F £ +F^+R , (29)

where E is the elastic force; D is the dissipative force; Fc is the centrifugal force; F^is the Coriolis

force; and finally R is the linkage force. Since we assume that the displacement of P is very small,

the centrifugal force that acts on Po can always be substituted for the centrifugal force that acts

on P. A simple computation gives us the centrifugal force as the sum of two forces. The first

of these two forces is almost constant with respect to (i, j , k):

F'c= -m S (2<ft, cos8+<?)zok-fck+^Q)2(xoi+ z<>k) +** (j£ sin2 d+z0 cos2 s\ k \ ; (30)

the second is pulsating:

F "= —m/f2^ 0 sin <p sin 8+ft ^ sin 25 sin «T1 k+ ^ [~—j sin2 8 cos 2*

+2 z0

 s m

 25 sin ^> i-f^2 ^ xo sin2 5 sin 2^+jr z0 sin 25 cos <p j r (31)

The Coriolis force is not computed since we assume that the point P moves in only one direction,

with direction cosines au a2, a3 relative to (i, j , k). The equation of motion of P is

8+2Ds-\-os(s—so)=— ( 2x<^ sin <p sin 5+ ^ ~ sin 25 sin <p J a3

+^* f —y sin2 8 cos 2^+2 z0 sin 25 sin <p J ax-\-^ l-i 0 sin2 5 sin2 <p+^ z0 sin 25 cos <p J a2 > (32)
156 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 V0L

-«

where s is the distance from P to Po along the direction of motion, and sQ is the coordinate of the

new equilibrium position after the addition of the centrifugal force to the elastic force. Naturally,

s0, a?o, and ZQ depend on 5, <p, and \p; however, in order to make a simple, approximate computation,

we assume that s0 is so small that the evaluation of the right-hand side of equation (32) is exact

enough. In the specific case of Explorer IV, the constant force involved is so small that this

approximation is fully justified. We definitely can write that the equation of motion of P relative

to S is, with the above approximation, in the form

s-\-2Ds-\-a2s=—2a>i^\f/sin <psiu 8—\l?cb2 sin 28 sin {(p—951)—IJ^CLZ sin2 5 cos (2<^>—v^)' (33)

For this perturbation, the motion of S is characterized by the equations

|KG|=constant, (34)

and

where Ec is the total amount of the energy dissipated in one period of rotation of the body around the

axis of symmetry. To evaluate Ec, we assume that <f> can be considered constant during one rotation

of the body around the axis of symmetry. Also, <p=<pt. Equation (34) can be written

cos 8)2+Aty sin2 8=0*7%. (36)

Therefore, from equations (28) and (36), we have

<p-\-^ cos 5=7*0 cos 5, $~~r' (37)

and finally,

A n

<p=v= , rQ cos 5. (38)

Thus, equation (33) can be written in the following form:

s-\-2Ds+a2s=A0 sin 25 sin (pt—<pi)-\-Ai sin2 5 sin (2v—<p2). (39)

The energy necessary to sustain a forced vibration of the oscillator P is

„ ZM2,sin225-j'2 , 4ZL42.sin45-J'

2

 /A~

tLc=- - g 1 -~2 > (4U)

where

(41)

It is natural for us to assume at this point that <r2>>^2, and also that (r 2»16D V (these assumptions exclude the presence of free masses in the interior of the satellite, which is a possibility if any

breakage occurs during the launching of the satellite). Hence, the energy dissipated for one cycle is

Ee=El sin2 25 cos2 5+Ef sin* 5 cos2 5, (42)

and the energy dissipated per second is

En=E\ sin2 25 cos3 5+£f sin4 5 cos3 5. (43)

Now, if the modulus of the angular momentum is assumed constant during the motion, then &~

depends on 5 only. Then,

cos 2 h+2A' (44 )
MOTION OP SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 15 7

If

we have

and the equation

becomes

cos 8

cos 8=->

V

dm (45)

(46)

if (A—(7)>0. Let us consider the solution of equation (46) corresponding to the initial conditions

p(0) = l+«, with *>0 and arbitrarily, e>0. Since -£ >0 , p is an increasing function of t for anyone value of the parameters Ef2

. The sign of d?p/dt2 is, for £>0, the same as that of (22<22—I??8)

P

2+2(E?-E?), which is positive for

2{E?-E?)_ Ef

^

The curve in figure 3 shows the trend of the function p(f) if Efl^>2Eti, where

p**=

(47)

(48)

This deduction is based on the hypothesis that Dfa* remains constant while 8, <p, and ^ vary during

the motion. Since, however, the characteristics of the oscillator change during the motion, they

are functions of S. If, for example, we assume that -1=X2v~

<l

, equation (46) becomes

(49)

Also, the trend of the function p(t) can change since there are two points of inflection. The curve in

figure 4 shows the trend of the function p(t) if 0<eZ<l and Ef

V

FIGURE 3.—Variation of the precessional period with time

(Case I).

FIGURE 4.—Variation of the precessional period with time

(Case II).
158 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

3. Body motion of Explorer IV

In this section, we shall compute the extent to which the phenomena considered above can explain

the observed motion of Explorer IV during the first 30 days after launch.

3.1. Torque due to the atmospheric drag.—We shall now evaluate for the specific case of this

satellite the coefficients introduced above. To begin, we note that the variation of the modulus of

the angular momentum during the first 30 days is, to a very good approximation, not more than 3

percent of the initial value. The external forces acting on the body cause a first-order variation in

the direction of the momentum, and a small second-order variation in the magnitude of the momentum. The ratio of the magnitude of the total variation of the momentum to the variation of modulus of the angular momentum is only a few percent. An exact computation of the aerodynamic

torque would be very time-consuming. Assume that the satellite moves in free molecular flow

and that the re-emission of the molecules from the body is completely diffuse.

Since we need only a good approximation of the function C0—C2 cos2 y—C3 cos3 y in order to

have equation (1) represent the terms of the atmospheric drag in the best manner, only the coefficients Co, C2, and C3 will be evaluated now. First, we compute the modulus of the momentum for

7=ir/2. The total force of the atmospheric drag acts in the direction opposite to that of vo, while

the resultant of the reactions due to the re-emitted molecules is in the same direction as vo. The

drag is ^^VQL, where 2 is the maximum projected area, and ^"is equal to 2.2 for the hypothesis of

free molecular flow and diffuse re-emission. The point of application of the resultant drag is on

the axis of symmetry at a point G' where GG'=—/Ok, and / 0>0.

Next, let us consider the situation for y^ir/2. The computation is now more complicated

since the moments due to the impinging molecules should be distinguished from the moments due

to the diffusely re-emitted molecules. If we substitute for the satellite's body an equivalent cylinder

with plane bases normal to the axis of symmetry, the resultant force of the impinging molecules is

^fiP(So|cos7|+S|sin7|) -£} where 2 is the same as above; ^=2 ; and 20 is the transverse area.

The center of pressure, G'', is independent of y and is the same as above. The moment, with respect

to G, of the atmospheric drag due to the impinging molecules is

^ 0K|. (50)

Now we evaluate the component of the torque due to the diffusely re-emitted molecules. As

computed by Cook (1959) ,

fi

 the total force acting on the lateral surface of the cylinder is perpendicular

to the axis of symmetry in the plane parallel to vo, and the modulus of the total force is

where sr is the "molecular speed ratio of re-emission."

 6 Since the center of pressure of the total

force is also the same as before, and since the force is perpendicular to the axis of symmetry, then

^K| . (51)

' On page 26 of bis paper, Cook evaluates only the component of the total drag due to re-emission, In the direction of TO.

* Mr—*-\j—, where « - r—SI?

eer10

 ^ ^ *? r> T-absolute temperature of the atmosphere, and T,-absolute temperature of the gas re-emitted

f Tr most probable molecular speed

from the surface of the satellite.
MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 15 9

We note that the torque due to re-emission from the bases of the cylinder is an order of magnitude

smaller than the aerodynamic torque and that if the bases are plane, or if the cylinder is open, this

torque is zero. Therefore, for any one 7 we obtain, with good approximation,

M G = | [2(20|cos 7|+2x|sin 7|)+0.221]k/0Xvo|»o|. (52)

We must now compute the influence of the asymmetry of the body with respect to a plane normal to the axis through the center of pressure. Let us suppose that the momentum of the re-emitted

molecules of the nose of the satellite, not computed for the equivalent cylinder with plane bases, is

at least one order of magnitude smaller than the total momentum for very small values of 7. We

will evaluate the asymmetry of the body by substituting / 0 + / i cos3 7 for /0, where 0 < /1 < /0- This

assumption is very approximate, but we shall see later that a more exact evaluation is not worth the

effort required. The relation /i> 0 depends only on the fact that none of the molecules impinging

on the exhaust nozzle is diffused out of the body. Thus

M o =|[2(S | 0|cos7l+2x|sin7|)+0.22,] (/„-/, cos37)kXvoM, (53)

where 20=200 cm2, 2,=3000 cm2, / 0=14 cm, and /,= 7 cm.

We made the approximate evaluation of /, by considering position I of the satellite corresponding

to the angle 7^x/6, and position II corresponding to the angle 7^5TT/6; these positions are diagrammed in figure 5. The momentum of the impinging molecules is the same for the two positions,

but the momentum of the diffusely re-emitted molecules is not the same. If only a small fraction

of the (p20tfo cos 7) molecules impinging on the exhaust nozzle are re-emitted from the body in posiV3

tion II, then these molecules are diffusely re-emitted with a momentum equal to 0.2 p20flo \- L in

position I, where L (the distance from the nozzle to G) is about 100 cm. This value for the momentum is the difference in the momentums corresponding to the two positions. If we evaluate this

difference by using equation (53), then /,= 7 cm.

Finally, we evaluate the coefficients Co, C2, and C3 by identifying the value of Mo from equation

(53) with the value of Mo from equation (1) for the same value of 7(7=TT/2, 0, ir). We obtain

C0=4.2X104cm3, C2=3.5X10*cm3, C3=2X103 cm3.

To finish, we must evaluate for Explorer IV the vectors defined in equation (4). In figure 6

the ratio (/WQ)/(P*VO2) has been plotted against the true anomaly of Q over the orbit, where p* is the

density and t% is the velocity at perigee. We assume that the scale height is about 40 km.

The vectors in equation (4) can be evaluated from the graph in figure 6. If we consider only the

contribution of the drag in the interval (—15°, 15°) around perigee, and if vo is assumed constant

and equal to y%, then

T 1 vo T cos2

 6 ya r cos 0 vo T,,_cos3 0 vq

12 \vo\' ^ 12" |»«|* *3 ~ 12

FIGURE 5.—Asymmetric property of the satellite.

636-014—63 12
160 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Substituting Co, C2, C3> Io, Is, Is and l'a' in equation (6), we obtain this differential equation:

{ 2- 1 ) cos>> , (l- | sin*

+2 103

 ( | sin* 5+cos4 sVlO3 • |• cos 0 sin2 3(sin2 5-4 cos2 5) |^|xn . (55)

Thus, the direction of the angular momentum rotates around •$. Using equation (55) with 8=84°,

0=45°, t>*=8 km/sec, p=1.8X10"13 grams/cm3, ro=2ir/12 sec"1, C=4.7X107 grams/cm2, we obtain

— ^0?45. Note that the motion of the satellite is a quasi-tumbling motion corresponding to a roll

period of 1 second.

i

1 in.

0.754

0.323

1 >

10°

1

20° 30° 40°

FIGURE 6.—Relative atmospheric density versus true anomaly

along the orbit.

Hence, ^ =

' rev

We observe also that the value of p* was taken from tables from Jaccbia (1960a). This is a representative value, close to the mean for the epoch. Since, according to the same source, erratic

fluctuations in p* by a factor of 2 are in order, we can expect that the variations of Q due to the

atmospheric drag were similarly affected.

8.2. Gravitational torque.—In this section, we shall compute the magnitude of the perturbation

due to the gravitational torque, Pb. Using the orbital elements of Explorer IV, we find that equation

(8) becomes ^=l(r 6 rad2/sec2, and equation (9) becomes ^=1.62X10"6 cos 0X (l— | sin2

cos 0, sin 0X (l— | sin2 A V 2 /

3.3. Magnetic torque.—To compute the perturbation due to the magnetic torque, Pe, we assume

that the earth's magnetic dipole is /*B=8.1X1025 emu; thus, 3,

 Mg

 2.3/2^0.25 gauss. Let us put

a ^1 e )

tw=50°, —-=1.4X103^

3

-cos5( —j/i+jii^Xn )XQ. For example, we notice that if the intrinsic

rev 07*0 \ 4 4 /

magnetic moment has, parallel to the axis of the body, a component of the order of 0.5 amp-m2

(corresponding, for example, to a solenoid parallel to the axis of symmetry, with 10,000 turns and

1 cm in diameter, in which there is a current of 0.5 amp), then ^=0?01. This action corresponds

rn rev

to a maximum torque of 12.5 dyne-cm.

Measurements carried out by the Radio Corporation of America (private communication) on

Tiros I (Satellite 1960 /32) showed the presence of a disturbing magnetic torque of about 200 dyne-cm

owing, probably, to interaction between the earth's magnetic field and the component of the intrinsic

magnetic dipole parallel to the spin axis. Perhaps 70 percent of the torque comes from the residual

magnetism of the high permeability components in the satellite; the remaining 30 percent comes

from current loops in the electrical circuitry of the satellite's instrumentation.

Grumman (private communication) has estimated the disturbing magnetic torque expected

for the Orbiting Astronomical Observatory proposed for launch in 1963. The major part of the

torque expected in this satellite will result from its permeable structure; the contribution of the
MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 16 1

internal complex circuitry will be minimized by proper electrical design. The amount of torque

predicted is many hundred dyne-cm. Since we do not know if this minimization has been done for

Explorer IV, the value 17.5 may be too small.

Finally, let us make a crude evaluation of the torque, averaged over one precessional period,

due to the induced magnetic dipole that we obtained in section 2.4.

If we assume that wii=0, 7713=1 and Q=u ^—n m —-»

then

BW-+.X-.

Finally,

M£»'> = +7XlO-^M-l^uXn ^ dyne-cm. (57)

The shell of the body of Explorer IV and the casing of the last-stage rocket left attached as

part of the satellite were both made of 410 stainless steel, whose initial permeability is 110. If we

assume that there is no premagnetization

 7 to be considered, then equation (57) becomes (using

y=2000 cm3 and M=H0 )

M^I'

)

 =+154nXn«, dyne-cm,

which corresponds to a variation of 1?3 per revolution. The chosen value of 7 may be far from the

actual value for the reasons given in the introduction. This amount is quite sufficient to explain

the variation of £2 during the motion, and even during the period of tumbling motion.

sec

3

2

1

0

-

5

1

10 15

y

20 25 day*

FIGURE 7.—Observed variation of the precessional period.

4. Results and conclusions

4.1. The period of rotation around the gyroscopic axis.—The history of the period of rotation

around the gyroscopic axis is traced in figure 7. Fields (Lundquist, Naumann, and Fields, 1961)

determined the period from the modulation of the counting rate of the directional counter. The

observations seem to have been made very carefully. Thus, the observed curve can be compared

with the theoretical one8 that was obtained from the energy transfer due to the internal vibrational

motion (see figs. 3 and 4). The existence of the first point of inflection is evident. The explanation given earlier of the definite second point of inflection, which corresponds to the 23rd day after

launch, may seem artificial. However, this second point of inflection can be explained in another

way. Perhaps the decrease in the probably significant torque due to the interaction between the

earth's magnetic field and the ferromagnetic components of the satellite plays an important role.

The rapid increase in the period on the 31st day after launch corresponds to the first time in

the history of the satellite that <p—$. Perturbation Pt may also be the explanation of this

phenomenon. Note that this particular case has been excluded for reasons of simplicity.

»If there is any Important premagnetization (Lundquist, Naumann, and Fields, 1961), this may become a predominant factor with respect to the

induced magnetization, and therefore must also be considered in the perturbation P,.

* We observe that *> + 4> cos $ s ^ and also that p sx n/y during the whole period of motion.
162 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

 V°L

-«

The rapid decrease in the period on the 33rd day after launch is very striking. Several causes

sufficient to explain such a phenomenon can be found quite easily. The necessary torque is so

small, 10 dyne-cm in comparison to the 200 dyne-cm which are needed to explain the decrease in

the first 5 days, that any electromagnetic phenomenon may be responsible for this exceptional

decrease. Consequently, we would need to know the internal ferromagnetic components to evaluate

their interaction with the earth's magnetic field. This torque can at least partially explain the

decrease in lp due to energy leakage by magnetic hysteresis in the ferromagnetic components.

Because of the shape of the body, we prefer to consider this action more than the influence of the

eddy currents. While we exclude the possibility that the initial decrease of lp can be explained

only by the eddy currents, the action of these currents and the magnetic hysteresis in the ferromagnetic components consistent with the variation of the earth's magnetic field along the orbit

can contribute to the decrease during the whole period of motion.

4-2. Possibility of the determination a priori of the body motion.—There are two phenomena of

first-order magnitude in the body motion of Explorer IV: the decrease of lp and the rotation of $2,

with the modulus of the angular momentum assumed to be constant in the first-order approximation.

The first phenomenon was explained in section 4.1. In this section, the change in the direction of

Q will be discussed.

If we exclude the possibility of handling the global problem of the motion of G and of the motion

of S around G, then we should know the function 8(i) and the dependence of the orbital elements

of G on time. From equations (6) and (9), and from the equation attained by averaging exactly

equation (19), we can write the global perturbation equation

^=R(8,d}d1,y,\1,\2)XQ, (58)

which has the first integral Q2=l. If we project this equation over the inertial frame of reference,

taking into account the variability with time of the X, Y, Z axes, we obtain the equations

^-/.(Qi, Q» t), ^-2=/2(Qx, 02,1), (59)

where J2i is the right ascension and Q2 is the declination of Q. In this way, if exact observations of

the variation of 12 and also of dQ/dt are available, we can compute the additional torque that is

necessary to explain the total variation of Si. The experimental determination of the axis of tumble

is not so exact as the determination of the angular velocity lp. Therefore, such a computation

cannot give exact enough information about this additional torque. Indeed, the determination

of the axis of tumble has a possible error of 10°, an error that is very much greater than we can

tolerate in a complicated computation, considering the work involved.

4.8. The displacement of Q during one revolution.—The only numerical computation which can

be made easily is that of the average diurnal variation of Q. The variation observed in the first 30

days after launch was evaluated as more than 10° per day. Of this amount, 5° or 6° can be

explained as a consequence of the perturbations Pa, Pb, and Pe. That is, 60 percent of the variation

which was evaluated from the recorded signal strength observations correlated with the satellite's

antenna radiation pattern (Naumann, 1961), from the variation of the angle of the axis of tumble

with v% (Zadunaisky, 1961), and from the recorded variation of the temperature of the body correlated with the position of the sun.

The rapid change in the direction of Q during the first day after launch is easily explained if

we assume that the factor cos 8 is very important for the perturbations Pa and Pe and that the atmospheric drag can cause a variation of 10° or more per day. We compute the angular variation of

Q for 5=84° in order to obtain an idea of the order of magnitude of the perturbations Pa, Pb, and

Pe in the period of quasi-tumbling motion.
MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS—COLOMBO 16 3

The gravitational torque may be responsible for a variation of 0?l per revolution around the

orbital plane, the drag torque for an average variation of 0?4 per revolution, and the magnetic torque

for an average variation of 0?2 per revolution, if we assume that the satellite has an intrinsic magnetic moment corresponding to a maximum torque of 300 dyne-cm, as has been provided for the

other satellites. If we take into account the fact that the axes of the rotations due to Pa, Ph, and Pe

are not parallel, we can compute the average variation. The result is not much more than 0?45 per

revolution, or 6° per day, which is the maximum amount.9

We note, finally, that the perturbation Pd due to the interaction between the earth's magnetic

field and the ferromagnetic components of the satellite is not easily computed, because of three

chief problems: the direction of the induced magnetic moment, the evaluation of the permeability,

and the hysteresis phenomenon.

Acknowledgment

I express my appreciation to Dr. Mario D. Grossi, Mr. Imre G. Izsak, Dr. Luigi G. Jacchia, Dr.

Charles A. Whitney, and Mr. Pedro Zadunaisky for their help in the preparation of this paper.

••

9

 It is easy to show that the total amount of the angular variation of Q in one day is less than the sum of the variations per revolution, because of the

variation of the orbital elements caused mostly by gravitational effects and by air drag, and because of the variation in the orientation of the earth's magnetic dipole.

On the Accuracy of Measurements Made Upon

Films Photographed by Baker-Nunn Satellite

Tracking Cameras

By Karoly Lassovszky

At present the Smithsonian Astrophysical

Observatory operates twelve Baker-Nunn satellite tracking cameras throughout the world.

The photographs taken by these cameras are

reduced at the photoreduction division of the

Observatory's headquarters in Cambridge,

Massachusetts.

The introduction to the first catalog of

precise satellite positions (Lassovszky, 1960)

mentioned that we had earlier begun an investigation to determine the accuracy of the measurements of the satellite positions on Baker-Nunn

films. These studies included consideration of

personal measuring errors and similar problems.

Some preliminary results appeared in the first

catalog. In the present paper, we give further

details.

Personal measuring errors

The focal length of the Baker-Nunn camera is

50 cm. Although the exposure time is usually

very short, varying from 0?2 to 3*2 when the

camera is stationary, a fast-moving satellite

produces on the film a trail that may be several

centimeters long. If the exposure is made near

the equator, we do not obtain perfect point

images even of the stars. Plate 1 shows a typical

exposure. Since a rotating barrel shutter interrupts the exposure five times, the photographic

trail of the satellite is chopped into six segments

divided by short breaks. (The star images are

too short to show breaks.) The length of each

break corresponds to the time interval during

during which the shutter was closed. The interval may be as much as 0*16; and the length,

several millimeters. At the instant the third

(the central) break is made, the time of the

exposure is photographed on the film. To

determine the position of the satellite at this

time, we measure the center of the central

break and the centers of the reference-star

images.

Generally, the satellites are so faint that, in

order to obtain measurable images, we must

have the camera move to track the satellite.

If the tracking rate is precisely the same as the

rate of satellite motion, we obtain a point image

of the satellite with star trails in the background

(see plate 1). We then measure the point image

of the satellite and the central breaks of the

reference stars.

Even if we obtain a point image of the satellite on some frames, we do not generally find

such images on all frames of the same film, since

the angular velocity of the satellite is not the

same at different altitudes. The image may be

more or less elongated, or even appear as a

short trail. If the image has an oblong shape,

we measure its center. If the image is actually

a trail, we obtain the position of the center by

measuring both ends of the trail.

If the camera tracking rate differs substantially from that of the satellite, the satellite

trail may be so long that shutter breaks become

visible in it as well as in the star trails. In that

case, the central breaks in the trails of both the

satellite and the stars are measured.

The accuracy of the measurements on trails

or on breaks is, of course, less than that of the

measurements made on point images, and we

may expect that the greater the length of the

trail or break, the less the accuracy of the

165
166 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

measurement. We shall discuss now some personal errors that may affect the accuracy of the

measurements.

Settings at different orientation angles.—During

the measurement, the filmholder is rotated

until the north-south direction on the film is

parallel to the F-axis of the measuring machine.

(All the measurements discussed here were made

on Mann two-screw comparators.) Since the

camera may track in any direction on the sky,

the trails or breaks can form various angles

with the X- or F-axis of the comparator. We

might expect that the accuracy of the setting of

the image on the reticle lines would be affected

by the angle. To answer this question, we

measured two different breaks at different

orientation angles.

In one case, the length of the break was IOOJU.

The film was rotated through ten degrees in

orientation angle between each set of 30 independent settings made on both the X- and Freticle lines. The standard error of all the

settings made on the X-reticle line was 3.31/x,

and the standard error of all the settings made

on the F-reticle line was 3.33/*. Since the

standard errors were almost the same, we conclude that the accuracy of the measurements

on the average is the same whether the measurer makes the settings on the horizontal (X)

or on the vertical (F) line. The standard

errors of the 30 measurements on each reticle

line made separately at different orientation

angles varied between 2.6 and 4.0M on the inline and between 2.5 and 4.8/x on the F-line.

Neither of these values, however, shows any

systematic change with the orientation angle,

in either the Jf-axis or the F-axis. Thus, we

believe that the accuracy of the measurements

is not a function of the orientation angle.

A second set of measurements carried out by

another person supports this conclusion. In

this case, the length of the break was 4V; the

film was rotated five degrees between each set

of measurements; and 12 independent settings

were made on both the X- and F-reticle lines

at each orientation angle. Figure 1 shows the

standard errors of the -ST-measurements obtained at each angle. This figure and three

others not given here indicate that the standard

errors have a random distribution and show no

sign of dependence on the orientation angle.

FIGURE 1.—Standard errors of the measurements of a break

at different orientation angles.

Since the accuracy of the measurement

seems to be independent of the orientation

angle, we restricted our other investigations

of personal errors to a single angle. We chose

0°; that is, we placed the film in the comparator

in such a position that the trails or breaks

were parallel to the horizontal reticle line

(X-axis), and the settings were made on the

vertical line (F-axis).

Frequency distribution oj settings.—Seventeen

measurers made settings on trails and on

breaks of different lengths, turning the screw

of the measuring machine until the reticle

line, according to the measurer's estimate,

was centered on the image. Each such series

contained about two dozen settings. Since

some persons measured more than one trail

or break, the total number of measurement

series was 34.

We find great variety in the frequency distribution of the settings made by the different

measurers. Of course the distribution also

depends strongly on the length of the image.

Moreover, we might expect that the distribution depends also on whether the measurements were made on trails or on breaks.

Of the 34 series of images, 28 were trails

and 6 were breaks. The trails varied in

length from 0.052 to 3.955 mm; and the breaks

from 0.031 to 0.410 mm. The frequency

distributions of the settings vary greatly with

the persons measuring and with the image

length; also, not many settings were made on a

single image. Therefore, in order more easily

to compare the results obtained on trails and
LASSOVSZKY PLATE 1

0 1 2 3 4 5

1 I I I I I

B.D. +32 101

t 1958 «,

1 Aug. 1958

0 1 2 3 4 5

I I I I I r

Photographs of satellites. Top, when camera is stationary; bottom, when camera is tracking.

ACCURACY OF MEASUREMENTS MADE ON BAKER-NUNN FILMS—LASSOVSZKY 167

on breaks, we determined the frequency distribution from all the settings made on the six

breaks and independently determined the

distribution from all the settings made on

the six shortest trails. The mean length of

the breaks was 0.151 mm, and that of the trails

was 0.154 mm; that is, they were practically

equal. The two frequency distributions are

given in figure 2; the length of each vertical

line indicates in percent the number of settings

made at different distances (expressed in

microns) from the zero point, which is the

mean of all the settings. (Because of the

"magnitude error," to be discussed later in

the paper, this point is not necessarily the

center of the break or the trail.)

Although the scattering of the settings on

the breaks seems to be slightly smaller, considering the inhomogeneous and scanty material, we cannot definitely state that there is

any substantial difference between the two

distributions. On the contrary, from their

similarity we can draw the conclusion, at

least on the basis of the material at hand,

that the accuracy of the measurements is

not essentially affected by whether the measurement was made on a break or on a trail. There1 II.

_ BREAKS

jl =0.151 mm

II,

fore, in the following discussion, we shall

disregard whether the image was a break or

a trail and use all the data together. In any

case, the measurements made on breaks were

too few in number to be discussed separately.

The lengths of the images varied from

0.031 to 3.955 mm. If we arrange the observation series according to the increasing length

of the image, we notice a progressive flattening

of the frequency distribution curves. However, continuity is strongly disturbed because

of the different accuracies of the measurements

made by different persons; this personal

effect fades out if we combine the material into

several groupings. From the groups formed

in this way, we show in figure 3 the frequency

distribution curve of the group having an

average image length of 0.080 mm, and that

of the group having an average image length

of 2.886 mm. If we add to these curves those

obtained for an intermediate image length

(shown in figure 2), we can see by comparison

that the frequency distribution curves flatten

progressively as the length of the image

increases.

1 ill!

Jt • 0.080 mm

III.

30 25 20 15 10 15 20 25 30

30 25 20 15 10 15 20 25 30 fX

1 1 ill

_ TRAILS

^f'0.154 mm

III.......

30 25 20 15 10 10 15 20 25 30

FIGURE 2.—Mean frequency distributions of settings made on

breaks and trails.

lllll.lll

, *. 2.886 mm

him! ..., i . .

I I I I I I I I I I I I I

30 25 20 15 10 5 0 5 10 15 20 25 30 fl

FIGURE 3.—Frequency distributions of settings made on

images of different lengths.
168 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The "magnitude error."—In measuring, we

set the reticle line at the point that, in our

judgment, is the center of the break or the

trail. In the frequency distribution curves,

the zero point represents the mean of these

settings. This does not mean, however, that

the zero point is really the center of the image,

for two reasons. Besides the scattering of the

observation errors, an effect that may be

lessened by increasing the number of observations and taking the mean, our procedure

involves a systematic error whose effect cannot

be reduced even by increasing the number of

observations; it is well known that we usually

do not measure the center of the image exactly

even if it is a perfect disk since, for physiological

reasons, we do not stop turning the screw at the

exact instant when the reticle line is centered

on the image. This systematic error differs

for different observers; futhermore, it varies

even for the same observer according to the

size of the image disk, which in turn depends

upon the brightness of the star. For this

reason, it is often called the "magnitude error."

We usually eliminate this error by first making

a direct measurement of an image and then,

after rotating the film or plate through 180°,

making a reverse measurement of it.

When we measure breaks or trails instead

of disks, a similar error may occur which

depends upon the width of the break or the

length of the trail, as well as upon the person

who makes the measurement.

To determine the magnitude error, we made

settings not only at the point that the observer

estimated to be the center of the image but

also at both ends of the oblong image. The

difference between the settings at the two ends

gives the length of the image, and their mean

gives the center of the image. We can assume

that the values for the center and the lengths

computed by this method may also be affected

by systematic error for some physiological

reason, since we may perhaps make these

settings differently when the end of the image

is on the left side or on the right side, or when

it is above or below. The results of our

statistical investigations, however, indicate

that this systematic error is insignificant

compared to the "magnitude error."

The Mann comparators we use are provided

with a projection system. When the X-screw

is moved clockwise, the X-readings increase,

and the image projected on the screen passes

through the vertical cross line from left to right.

Of the 34 observation series discussed here, in

21 cases the mean values of the estimated

centers were less than the values of the centers

computed from the settings made on the two

ends of the images; in 11 cases they were more,

and in 2 cases the centers coincided.

Of the 17 observers who made the measurements, on the average 9 stopped turning the

screw only after the center of the image passed

the reticle line, 7 stopped before it passed the

line, and one did it both ways. It must be

emphasized that this statement is valid only

for the means of the settings. In fact, the

scattering of the accidental errors is so great

that many accidental errors are larger than the

magnitude error. Perhaps the best way of

stating this is that a given person systematically

makes more settings on one side of the center

than on the other side.

The magnitude errors, that is, the differences

between the estimated and the computed

centers obtained for the images, varied from

—39/* to +65fi. Arranging all the magnitude

errors in order of increasing absolute value and

indicating for each error the corresponding

length of image, we can expect to find a relationship between the two quantities. The

magnitude errors are, of course, greatly influenced by the systematic personal effects,

but, since our data are scarce, we are obliged

to use all of these heterogeneous data. Plotting

the magnitude errors against the image lengths,

we find that although there is a great scattering,

the expected relationship is certainly apparent.

Forming 8 groups from the material, we get

the relationship represented in figure 4. The

scattering of the individual values is conspicuous only at the longer lengths, but these long

images were measured only for the purpose of

this study. In practice, the trails are usually

less than 0.2 mm and the breaks smaller than

0.1 mm.

The effect of the distances of the reference

stars on the accuracy

For the determination of the satellite positions,

we used the linear plate-constant method. For
ACCURACY OF MEASUREMENTS MADE ON BAKER-NUNN FILMS—LASSOVSZKY 169

LENGTH OF IMAGE IN MILLIMETERS

FIGURE 4.—The relationship between the length of image and

the "magnitude error."

the computation of the six plate constants, three

reference stars would theoretically be sufficient.

However, we have persisted in using six stars,

employing the least-squares method to compute

the six constants from twelve equations. When

there were large residuals, one or two reference

stars were sometimes omitted. If large residuals still remained, we repeated the measurements, never using fewer than four reference

stars.

We further decided that the reference stars

should lie not farther from the satellite image

than 20 mm on the film, a distance corresponding to 2?3 in the sky, and that the reference

stars should be symmetrical about the satellite

image, so that the image will always lie within

the configuration even if some of the reference

stars must later be omitted.

Among the different sources of error that

influence the accuracy of the reduced position,

distortion of the emulsion may cause serious

problems. Another source of error may be the

1

_

I

0.9 I

\

»0.7

optical distortion of the field of the camera.

When we began to use the linear plate-constant

method, we assumed that, within the admissible maximum area of configuration of the

reference stars, the distortion did not appreciably influence the accuracy. We were aware,

however, that this assumption needs proof

since, if it were not valid, our method would not

be wholly reliable.

To find out how the position obtained is

affected by use of reference stars at different

distances, we chose various star groups, each

containing six stars, around the object whose

position was to be determined. The average

distances of the groups differed, but, in a

given group, the distances of the stars were

nearly the same. On a chart of the area

surrounding the object, rings were constructed

around the central image; stars were chosen so

that the reference stars of a given group were

all inside the ring and properly distributed.

Since the width of the frame was only 54 mm,

the exterior radius of a complete ring cannot be

larger than 27 mm even when the center object

was exactly in the middle. However, we could

go even farther from the center object by choosing three reference stars in the left part and

three in the right part of an incomplete ring

with an exterior radius larger than 27 mm.

We chose three stars on a frame and determined their positions by using reference stars at

different distances. One of the stars was

approximately in the center (II) of the frame;

of the other two, one was in the center of the

left (I) hah0 and one was in the center of the

right (III) half of the frame. In field I, four

groups of reference stars were available around

the chosen center star; in field II, seven groups;

and in field III, six groups. The greatest dis-

-t -i

FIGURE 5.—The shifts of the positions of three different stars on the same frame when

reference stars at different distances from the center star are used. The numbers

at the positions give the mean distances of the reference stars in centimeters.
170 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

tance of the groups in field I was approximately

2 cm; in field II, 3 cm; and in field III, 2.5 cm.

The position of each center star was determined

separately from each of the groups around it;

then the mean position of each center star was

computed from all the stars in its field. Choosing the mean position as origin, we can see in

figure 5 how the position of the center star

changes when different groups are used for the

reduction. For every position of the central

star, the chart indicates the mean distance of

the group of reference stars used to determine

that position. No relationship seems to exist

between the position and the distance, at least

in the area examined. This result may be

regarded as favorable since the scattering of the

mean positions is not very much greater than

that to be expected even if these positions had

been obtained from the reiteration of the

measurements using the same reference stars.

We also tried to find out, from another

graphic representation of these results, how the

distances of the reference stars influence the

computed positions. Combining the results

obtained for the three centers we took the mean

of the positions obtained from the groups that

were nearer to the center star than 1.4 cm,

then took separately the mean of the positions

from groups that were farther than 1.4 cm from

the center star. The distributions of the

individual positions around the two mean positions are illustrated in figure 6. Although the

number of observations is not very large, one

notices at first glance that the scattering of the

positions derived from the groups of reference

stars nearer to the center star than 1.4 cm is

undoubtedly smaller than the scattering at

distances larger than 1.4 cm. However, even

•f

1

0

?

•

r < 1.4 cm

• <

•

_ J I

2"

1

•4 ot 0

-1

- 2

•

•

l.4cm < r < 3.0cm

iff

• •

1

•

#

- 2 " -I

FIGURE 6.—The scattering of the positions obtained when

reference stars at different mean distances are used.

in the latter case, all the positions are in a

4 " x 4 " square; that is, they are in an area not

larger than we can expect from the measurements made with the Mann comparators on the

Baker-Nunn films.

In practice, we of course choose reference

stars that are as near as possible to the satellite

image, even though sometimes, in areas poor in

stars, we are obliged to use reference stars that

are farther than 1.4 cm from the satellite image.

On the basis of our investigation, however, we

seem justified in using the linear plate-constant

method, at least in an area with a diameter of

5 cm (5°.8).

The accuracy of the measurements

We may regard the figures given in the preceding section as a graphical illustration of the

accuracy that we obtain in the reduction of

Baker-Nunn films. Making use of the numerical values of the measurements, we can also

express this accuracy numerically; that is, we

can compute the standard error of a position

determined on a Baker-Nunn film.

Before we made the investigations discussed in

the preceding section, we had already obtained

some information about the relative accuracy of

the measurements made with both the Van

Biesbroeck goniometers and the Mann

comparators.

In SAO Special Report No. 41, we described

the methods used to measure the position of a

satellite, and recounted the sources of errors

that we could expect. The accidental personal

errors cannot be avoided, but we can lessen their

influence by using a greater number of reference

stars, or by increasing the number of settings on

them, or by repeating the whole measurement.

We assume that the so-called magnitude error

is eliminated (or at least lessened) when both

direct and reverse measurements are made.

At the beginning of the photoreduction project at the Smithsonian, two Van Biesbroeck

goniometers were chiefly used. The goniometer,

which in fact consists of two independent instruments, a theodolite and a filmholder, may itself

be a source of errors. Since the two parts of

the machine are not stably connected, changes

may occur, in time, so that the axis of the film

holder may not pass exactly through the center

of the theodolite, and the vertical axis of the
ACCURACY OF MEASUREMENTS MADE ON BAKER-NUNN FILMS—LASSOVSZKY 171

theodolite may not be perpendicular to the axis

of the filmholder. The smallest readable unit

on the Van Biesbroeck goniometer is 1"'.

On a screw comparator, there are two chief

sources of errors: the periodic and the secular

screw errors. The smallest readable unit of the

Mann comparators is 1/z, corresponding to 0"A

on the Baker-Nunn films. According to the

manufacturer, the periodic error of the Mann

machines is less than In, and the secular error

within the measuring area of the reference stars

is less than 0.5M- We can regard both values as

satisfactorily low.

In the photoreduction division, each computer

from time to time measures the same image

twice to check his own consistency. (Averages

of the differences obtained from a large number

of such double measurements give characteristic

information of the efficiency of the measurers.)

Figure 7 shows the results of 54 such double

measurements made with the Van Biesbroeck

goniometers, and of 90 double measurements

made with the Mann comparators. The coordinates of each dot give the differences in a

and in 5 of the two positions obtained by the

double measurement in question. The distance

D of the dot from the origin represents the

angular separation of the two positions. The

much smaller scattering of the dots shows at a

glance that the measurements made with the

comparators are of higher accuracy than those

made with the goniometers.

From all the double measurements, the

following standard errors have been derived

for a single determination of position:

Number of double

measurements made with

Standard errors

M* MJ MD

Van Biesbroeck 54

goniometer:

Mann comparator: 90

3T99 3T21 5T72

1. 10 1. 12 1. 82

These values show that the accuracy obtainable

with the Mann comparators is more than three

times greater. These results differ only slightly

from those given in Special Report No. 41

(p. 8), which was based upon fewer observations.

The positions obtained from the measurements made with goniometers seem to have on

the average a larger error in a than in 5. The

difference is small, however, and after the

omission of a few double measurements in

which the agreement between the two positions

is poorest, the difference becomes insignificant.

On the other hand, the positions obtained with

the Mann comparators showed the same standard error in both a and 5. As we shall see, the

same conclusion was reached from other measurements made later. This result does not

seem to confirm earlier conclusions that the

error in right ascension is generally larger.

(This larger error in right ascension was explained by the fact that most of the satellites

whose positions were measured at that time

were traveling in a west-to-east direction, and

errors in the direction of motion are somewhat

greater than errors perpendicular to the line

of motion.)

We must not forget that for each double

measurement the same reference stars were

used; therefore, the effect of any catalog

errors is the same in both measurements.

Thus, the catalog errors are not reflected in

the position errors of the satellite. To examine

the effect of catalog errors, we made special

measurements. We determined the positions

of seven satellites when they were passing

through areas rich in stars. Thus we could

determine the position of each satellite using

four, and in one case even five, independent

configurations of reference stars. The configurations used for each satellite were chosen

so that their mean distances from the satellite

were approximately the same. (This was done

in order to eliminate the occasional effect of

the distance of the star from the satellite image.)

The measurements were made by five different

persons. The agreement between the positions

obtained from different configurations was

satisfactory. From the 29 independent positions, the following standard errors were derived for one position:

29 1T00 1T01 1T50

These values differ only slightly from those

determined by the double measurements (11'0,

IT 12, l!82). Although we might have expected to obtain larger values when different

reference stars were used, in fact the values

are slightly smaller. We can conclude that

the catalog errors have practically no effect

on the accuracy that can be obtained with the

Mann comparator on the Baker-Nunn films.
172 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

A<J

15"

10"

5"

5"

VAN BIESBROECK

10"

A at cos 8

15" 20"

20'

Arf

I51

10"

5"

MANN

5" 10" 15" 20"

A a cos 8

FIGURE 7.—Differences in a and 8 of double meaurements made with Van Biesbroeck goniometers and with Mann comparators-
ACCURACY OF MEASUREMENTS MADE ON BAKER-NUNN FILMS—LASSOVSZKY 173

In addition, we have some other measurements that can be used for the determination

of accuracy. As mentioned in the first paragraph of this section, we can also use to determine the accuracy the measurements made to

find out how the distance of the reference stars

from the satellite image influences the determination of the satellite's position. We made

two groups of positions, one group derived

from the reference star configurations that

were nearer to the center star than 1.4 cm, and

the other group where the distances varied

between 1.4 and 3.0 cm. From these two

groups, the following standard errors were

found for one position:

r n p* MJ ftp

r<1.4 8 0f83 0T66 1T03

1.4<r<3.0 9 1.4 1.12 1.60

The standard errors obtained from the first

group are, of course, smaller. The standard

errors obtained from the second group are

approximately the same as those from the many

previous measurements. This fact may seem

unexpected since the average distances of the

reference star configurations are generally

smaller for the usual measurements than they

were in the second group of these special

measurements. This favorable result can perhaps be explained by the facts that for these

special measurements a better film was used,

and the center object was not a satellite but a

star image.

If we combine all the results discussed here

but discard the eight positions obtained from

the measurements of the reference star configurations nearer to the center star than 1.4

cm, we get as the final result:

1T09 1T11 1!77

If restricted to one decimal, we can say that

the standard error of a position determined

from the measurements made with Mann comparators on Baker-Nunn films is ±1*1, both

in right ascension and in declination.

Acknowledgment

I express thanks to Mr. G. Kirklin for helping

with some of the reduction work.

Density of the Heterosphere Related to

Temperature

By Marcel Nicolet

An analysis of the behavior of the heterosphere, i.e., of the terrestrial atmosphere where

the mean molecular mass cannot be taken as a

constant parameter, requires a theoretical study

to supplement observational results from which

it is not yet possible to obtain all the parameters needed for a complete picture of the physical conditions.

If a general consistent picture of the vertical

distribution of density in the heterosphere at

heights between 200 km and 1500 km is obtained from acceleration data derived from

earth-satellite orbits, the analysis leads sometimes to unjustified conclusions concerning the

physical conditions. The sharp increase of

molecular mass, the gradient of temperature

increasing with height, the inflection of density

curves, and the correlation of density with unusual indices of solar activity are among other

physical anomalies that are introduced into this

analysis.

In this paper, an improved method for calculating physical parameters of the heterosphere

is applied, in which the temperature is selected

as the essential parameter, and diffusion and

heat conduction are introduced.

Formulas for density

The vertical distribution of the upper atmosphere density deduced from the rate of change

of the period of the motion of a satellite in its

orbit is generally represented by the formula

p=Poexp (— (1)

in which p denotes density at height z, and po

at s=0 ; and Hp is a parameter of the vertical

distribution of density, which must be defined

by

ldp

pdz

with the condition

dHp/dz=0.

(2)

(3)

These formulas cannot lead to correct conclusions concerning the temperature or the mean

molecular mass in the heterosphere.

To provide a basis for an analysis of the vertical distribution of the temperature and mean

molecular mass, we must use at least the general

formula representing conditions of a perfect gas

and of hydrostatic distribution.

The general form to be used is (Nicolet, 1954)

dp dn.dT dz (4)

where p is the total pressure; n is the total

molecular concentration; T is the absolute temperature; and H is the local atmospheric scale

height. The atmospheric scale height H is

defined by

kT H=—, (5)

mg '

where m is the mean molecular mass; g is the

acceleration of gravity; and k is Boltzmann's

constant. The variation is, therefore, related

to the variation of T, m and g; that is,

dH=dT dm dg

H~T m g' (6)

By using equation (6), and introducing the

gradient of the atmospheric scale height

(7)

175
176 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

we find that the general equation (4) leads to

pdz H'

and

ldp

pdz

1+0

" H

(8)

(9)

Thus, from equations (2) and (9), the relation between the local atmospheric scale height

H and the density parameter Hp is

H= (1+0)HP. (10)

The values of Hp, which are deduced from the

slope of the density-versus-altitude curve,

cannot represent real atmospheric conditions.

The use of Hp may lead to wrong conclusions

about the structure of the thermosphere,

especially where the gradient of the atmospheric

scale height is large.

If the height interval is small enough, 0 can

be assumed constant for calculation purposes.

If equations (8) and (9) are written in the form

dp=_ldH

V & H'

1+0 dH

and

dpg 1

~P9=~~ 0 H'

the integration with /3=constant leads to

and

Po \H{

(11)

(12)

(13)

P9 =

Po9o

The expansion of equations (13) and (14) gives

Po

-^—exD /

The second term in brackets is less than 0.01

at z=H0 if 0<O.42 and at z=\ Ho if /3<0.84.

Neglecting terms less than 0.01 before unity,

we can reduce equation (16) to

-^-=expf —

Po9o x

(17)

which is sufficiently accurate when an analysis

is made in a height interval less than one scale

height.

With a constant gradient of the scale height,

equation (10) leads to

(18)

and equation (12) becomes

dpg_ \^d

Pg~ JP~

(19)

If the same terms are neglected as were for

expression (16), equation (19) becomes after

integration,

(20)

Equation (15) must be compared with expressions (1) and (20). Since an analysis of

the physical conditions of the heterosphere

cannot be made without allowing for the effect

of the gradient of the scale height, it is useful

to have available a variety of atmospheric

models for which computations have been

carried out for combinations of the scale-height

gradients associated with the variations of

temperature and of mean molecular mass.

It will be noticed that the difference between

the parameter of density Hp with its gradient

/3P and the local atmospheric scale height H

with its gradient 0 increases with increasing

values of j8.

Method of determining atmospheric models

Inspection of density values at 200 km obtained

bv calculation reveals that the scale-height

gradient below 150 km plays the leading role

in determining the density at higher altitudes

(Nicolet, 1960b). In other words, if boundary

conditions are assumed in the neighborhood of

120 km, the atmospheric density between 200

km and 250 km is fixed by the scale-height

gradient between 120 km and 150 km but is

not much affected by the values of the scale-
DENSITY OF HETEROSPHERE RELATED TO TEMPERATURE—NICOLET 177

height gradient above 150 km. The net

result is that the vertical distribution of density

at high altitudes depends on the temperature

at the level of the thermopause, and the variation of the scale-height gradient above the

thermopause is due essentially to the decrease

of molecular mass. Thus, the atmospheric

conditions are defined by a diffusion distribution and related to the time of conduction

(Nicolet, 1960c).

From an analysis using various boundary

conditions and scale-height gradients, it appears

that the temperature must be considered the

most important parameter in deducing the

vertical distribution of atmospheric density.

Consequenlty, the vertical distribution of

temperature is related to the conditions of

heat conduction, and the heterosphere can be

assumed to be in diffusion equilibrium.

Average conditions lead, therefore, to consistent atmospheric models in which the temperature and its vertical gradient can be determined

when diffusion equilibrium is assumed.

The diurnal variation of the density is represented by a change of the temperature and

a variation of its vertical gradient. In the

same way, the effects of solar activity can be

interpreted by a variation of the ultraviolet

radiation that affects the temperature gradient

and leads to variations of the temperature of

the isothermal layer. In fact, it should be

possible to predict how the density must vary

during a solar cycle when the boundary conditions are known.

Finally, the magnetic storm effects can also

be included since Jacchia (1961) has shown

that the atmospheric-drag perturbations during

geomagnetic disturbances have a worldwide

distribution. The interpretation must be that

a general heating occurs, corresponding to an

increase of the temperature at all latitudes.

It is certain that atmospheric conditions

change in the E layer and involve a variation

of temperature that is an important parameter

to determine boundary conditions for atmospheric models of the heterosphere. We cannot

present here all the possibilities, and average

conditions have been adopted in order to lead

to a density of the order of 4 X10"12 gm/cm3 at

200 km. It must be noticed that for the same

gradient of temperature, a variation of ±50°

K at 120 km leads to a variation of ±50 percent

of the density at 200 km.

The average conditions adopted here are as

follows:

Density p(120 km) =3.5X10-" gm/cm1

Temperature T(120 km) = 325° K

Concentration n(0) = 7.6X1010 oxygen atoms/cm*

Concentration n(02) = 1.2 X10" oxygen molecules/

cm3

Concentration n(N2) = 5.8X10" nitrogen molecules/cm8

Scale height H= 10.37 km

Atmospheric conditions from 150 km to 650

km

The vertical distribution of density between

120 km and 150 km is practically unaffected

by the form of the variation with height of the

scale-height gradient, and the density in the

region of 150 to 160 km is almost a constant

for the same boundary conditions at 120 km.

The values of density to be expected at 150 km

for temperatures varying from 877° K to 642°

K are given in table 1. Such a range of temperatures corresponds to equal intervals of

times (t—0, t=10) for a cooling by conduction

of the whole heterosphere considered as a

plane-parallel atmosphere subject to diffusion.

The 12 models of table 1 correspond to the

following temperatures at 150 km and their

associated gradients between 150 km and 160

km:

Model T(°K) fdT\

no. \dz/

0

1

1.5

2

3

4

5

6

7

8

9

10

877

873

863

852

829

803

777

751

726

699

671

642

18

15

14

13

11

10

9

8

7

6

5

4

In tables 1 to 10, T refers to temperature in

°K; H, the atmospheric scale height in km; /3,

the gradient of H for the height interval indicated in the table; Hp, the density parameter

deduced from formula (10); p, the density in

gm/cm3 with (-12) for 10~12; p, the pressure in

mm Hg with (-6) for 10~6; M, the mean
178 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

molecular mass with the mass of atomic oxygen

M—16; n, the total concentration in cm"3 with

(+10) for (1010).

The essential character of the data of table

1 is that the parameters Hp and M are almost

constant, while the atmospheric scale height

and its gradient vary. The density is almost

constant, (2.5±0.1)X10~12 gm/cm3 for temperatures higher than 700°K. Below 700°K,

boundary conditions at 120 km are affected.

Since Hp= (20 ±1) km for the whole range

of temperatures, it is clear that density data

deduced from satellite air-drag may lead to

wrong conclusions about the atmospheric structure. For example, a constant scale height is

sometimes deduced from an analysis of satellite

data even though there is in fact a strong

gradient of the atmospheric scale height. When

an association was made between the apparent

scale height and the temperature, the wrong

conclusion was reached that the atmosphere

was almost isothermal.

Consequently, table 1 reveals that the variation of the atmospheric scale height and its

gradient can be very important, while other

parameters, and particularly the density parameter Hp, show no practical variation. Such

a variation of the atmospheric scale height

must lead to the variation of all physical

parameters at higher altitudes.

No.

0

1

1.5

2

3

4

5

6

7

8

9

10

No.

0

1

1.5

2

3

4

5

6

7

8

No.

0

1

1.5

2

3

4

5

6

7

8

T

877

873

863

852

829

803

777

751

726

699

671

642

T

1540

1414

1358

1305

1210

1126

1051

985

925

867

T

1791

1613

1532

1458

1327

1215

1120

1037

965

896

H

29.8

29.6

29.3

2& 9

28. 1

27.3

26.4

25.6

24.7

23.8

22.9

21.2

H

56.0

51.8

49.6

47.8

44.3

41.6

39.0

36.8

34.8

32.8

H

68.4

62.2

59.3

56.7

52.0

48.2

44.9

42. 1

39.7

37.4

TABLE 1.—Physical parameters at 150 km

0150-180

0.67

0.56

0.53

0.48

0.44

0.38

0.35

0.31

0.28

0.28

0.25

0.20

H,

20.0

20.7

20.9

21.0

21.0

21.0

21. 1

20.8

20.4

20. 1

19.8

19.2

p

2. 41(-12)

2.43

2.45

2.48

2.54

2.59

2.62

2.65

2.61

2.35

2. 14

1. 97(-12)

V

5. 0(-6)

5.0

5.0

5.0

5.0

5.0

4.9

4.8

4.5

3.9

3.4

3. 0(-6)

TABLE 2.—Atmospheric conditions at 200 km

0200-210

0.32

0.27

0.25

0.23

0.20

0. 18

0. 15

0. 13

0. 13

0. 12

H,

42.7

41.3

40. 1

39.6

38. 1

36.2

34.5

32.0

31.3

30.4

p

4. 07(-13)

4. 14

4. 17

4. 17

4. 16

4. 10

3.98

3.84

3.59

3.03

V

1.6(-6)

1.5

1.4

1.4

1.3

1.2

1. l(-6)

9. 8(-7)

8.6

6.9(-7)

TABLE 3.—Atmospheric conditions at 240 km

#2«0-i«0

0.25

0.21

0. 19

0. 18

0. 16

0. 14

0. 13

0. 12

0. 11

0. 10

58.8

54.2

52.3

49.7

49.0

44.2

41.8

39.4

37.0

35. 1

p

1. 77(-13)

1.72

1.70

1.66

1.58

1.47

1.35

1.23

1. 09(-13)

8. 64(-14)

V

8. 3(-7)

7.3

6.9

6.4

5.6

4.8

4.2

3.5

2.9

2. 2(-7)

M

26.2

26.2

26.2

26.2

26. 1

26. 1

26. 1

26. 1

26. 1

26. 1

26.0

26.0

M

24.8

24.7

24.7

24.6

24.5

24.4

24.3

24. 1

24.0

23.8

M

23.9

23.7

23.6

23.5

23.3

23.0

22.8

22.5

22.2

21.9

N

5.6( + 10)

5.6

5.6

5.7

5.8

6.0

6.0

6. 1

6.0

5.4

4.9

4. 8( + 10)

N

9. 9( + 9)

1. 0( + 10)

1.0

1.0

1.0

1.0( + 10)

9. 9( + 9)

9.6

9.0

7.7( + 9)

N

4. 5( + 9)

4.4

4.3

4.2

4. 1

3.8

3.6

3.3

2.9

2.4(+9)
DENSITY OF HETEROSPHERE RELATED TO TEMPERATURE NICOLET 179

At 200 km (see table 2), the gradient of the

atmospheric scale height is about 50 percent of

its value at 150 km; even for a range of 500°

K in the temperature, the density remains

practically constant; p=(4.1±0.1)X10~13 gm/

cm3 for 1050°K<T< 1550°K. This shows

again how important are the boundary conditions that are chosen in the E layer; what is

needed is a very small gradient to reduce the

200-km density by an appreciable percentage.

However, the variation is more apparent in the

pressure, which decreases with decreasing

temperature.

Between 250 km and 650 km, the gradient

of the atmospheric scale height is generally

between /3=0.2 and 0.1, as can be seen from an

inspection of the figures in tables 3 to 8. Since

the gradient is relatively small, the difference

between the atmospheric scale height H and

the density parameter Up becomes less important, i.e., 20 percent to 10 percent. Nevertheless, the effect of small values of /S between

0.1 and 0.2 cannot be neglected since it corresponds to the decrease of the molecular mass in

the isothermal layer.

Inspection of figures of physical parameters

from 200 km to 650 km shows how their

amplitude increases with altitude. For example, at 400 km (see table 6), the density

varies by a factor of 10, and the pressure by a

factor of 20, and the mean molecular mass

decreases from M=20.6 to M=16.6 when the

No.

0

1

1.5

2

3

4

5

6

7

8

T

1975

1752

1650

1556

1393

1261

1150

1057

978-1(*)

903-1

TABLE

H

81.3

73.3

69.6

66.3

60.5

55.8

52.0

48.7

46.0

43.3

4.—Atmospheric conditions at 300 km

/S300-320

0. 18

0. 16

0. 15

0. 14

0. 12

0. 12

0. 11

0. 10

0. 10

0.09

H,

73.7

65.9

63.3

60.6

55.5

51.7

48.8

46. 1

43.3

41.0

p

6. 86(-14)

6. 15

5.80

5.44

4.76

4.08

3.45

2.89

2.35

1.71 (-14)

P

3. 7(-7)

3.0

2.7

2.4

1.9

1.5

1.2(-7)

9.4(-8)

7.2

5.0(-8)

M

22.6

22.2

22.0

21.8

21.4

21.0

20.4

20.2

19.8

19.4

N

1. 8(+9)

1.7

1.6

1.5

1.3

1.2

1.0( + 9)

8.6(+8)

7.2

5.3( + 8)

'The symbol I means the temperature at the thermopause.

TABLE 5.—Atmospheric conditions at 340 km

No.

0

1

1.5

2

3

4

5

6

7

8

No.

0

1

1.5

2

3

4

5

6

7

8

T

2036

1796

1683

1583

1408

1271

1155-1

1059-1

978-1

903-1

T

2086

1826

1707

1597

1412-1

1272-1

1155-1

1059-1

978-1

903-1

H

sa 0

79.4

75.3

71.6

65.2

60.3

56. 1

52.6

49.7

46.8

11

96.9

87.3

82.8

78.6

71.6

66.2

61.5

57.6

54.2

52. 1

P34O-380

0. 16

0. 14

0. 13

0. 12

0. 12

0. 11

0. 10

0.09

0.08

0.09

H,

80.0

72.4

68.6

65.8

61.0

57.2

53.6

49.8

47.3

44.8

P

4. 00(-14)

3.40

3. 13

2.86

2.36

1.92

1.54

1. 23(-14)

9. 55(-15)

6. 61 (-15)

V

2.3(-7)

1.8

1.6

1.4

1.0(-7)

7.7(-8)

5.7

4.3

3. 1

2. 0(-8)

TABLE 6.—Atmospheric conditions at JfiO km

/3<00-420

0. 14

0. 13

0. 12

0. 11

0. 10

0.09

0.08

0.07

0.08

0.09

H,

41.2

82.3

74.6

73.6

67.7

62.7

59.2

55.4

52. 6

50.4

p

1. 93(-14)

1.53

1.36

1. 19

9. 11(-15)

6.89

5. 16

3.86

2.81

1. 82(-15)

p

1.2(-7)

8. 7(-8)

7.3

6. 1

4.2

3.0

2. 1

1.4(-8)

9. 9(-9)

6. 0(-9)

M

21.8

21.3

21.0

20.8

20.3

19.8

19.4

18.9

18.3

18. 1

M

20.6

20.0

19.7

19.4

18.9

18.4

18.0

17.6

17.3

16.6

N

1.1(4-9)

9.6(4-8)

8.9

8.3

7.0

5.8

4. 8

3.9

3. 1

2.2

N

5.6(4-8)

4.6

4. 1

3.7

2.9

2.2

1.7

1.3(4-8)

9.8(4-7)

6.4(4-7)
180 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

temperature range is between 2100°K and

900° K, corresponding to a variation of H from

97 km to 52 km. Such a general result shows

that an arbitrary choice of the mean molecular

mass cannot be made as it was for the COSPAR

International Reference Atmosphere (1961).

It is unwise to attempt to specify atmospheric

conditions with an arbitrary molecular mass,

since it may lead to inconsistent information

relative to the thermal or diffusive equilibrium

of the upper atmosphere. Wrong conclusions

indicating a decrease of temperature between

400 km and 500 km, such as obtained by

Lundback (1961), reveal that the analysis of

observational data is not correct since the atmosphere at these altitudes is at least isothermal.

It should also be borne in mind that diffusion

must adapt the molecular weight mg to the

temperature conditions prevailing in the isothermal region, and there is no reason, therefore, to expect some sharp decrease of molecular

mass at a particular altitude.

When the molecular mass is more than 14,

the gas is presumed to consist essentially of

nitrogen and oxygen, and it should be remembered that molecular nitrogen behaves almost

in the same way, both for mixing and for

diffusion conditions. Its mass M=28 does not

differ very much from the mean molecular

mass of the air, M=29. Thus, the vertical

distribution of molecular nitrogen is closely

related to the vertical distribution of temperaNo.

0

1

1.5

2

3

4

5

6

7

8

No.

0

1

1.5

2

3

4

5

6

7

8

No.

0

1

1.5

2

3

4

5

6

7

8

T

2123

1837-7

1711-7

1598-7

1412-7

1272-7

1155-7

1059-7

978-7

903-7

T

2128

1837-7

1711-7

1598-7

1412-7

1272-7

1155-7

1059-7

978-7

903-7

T

2131-7

1837-7

1711-7

1598-7

1412-7

1272-7

1155-7

1059-7

978-7

903-7

77

113.4

101.6

96.4

91.6

83.6

77.6

72.3

68.1

65.0

63.6

77

118.3

105.9

100.4

95.4

87.1

80.9

75.8

72.0

69.7

70.2

77

128.8

114.8

109.0

103.8

95.5

90.1

86.8

86.0

88.3

97.6

TABLE 7.—Atmospheric conditions at 520 km

&20HM0

0. 13

0. 12

0.11

0. 10

0.09

0.08

0.09

0. 10

0. 11

0.15

77,

104.0

95.2

90.9

86.5

79.4

74.1

6a 6

64.2

60.6

58.0

p

5. 45(-15)

3.85

3. 19

2.62

1.75

1. 17(-15)

7.76(-16)

5. 15

3.35

1. 92(-16)

V

3. 9(-8)

2. 15

1.9

1.5(-8)

9. 2(-9)

5.7

3.5

2.2

1.4(-9)

7. 7(-10)

TABLE 8.—Atmospheric conditions at 560 km

0MO-SM

0.12

0.11

0.10

0.10

0.09

0. 10

0.10

0.12

0.16

0.22

77,

104.2

100.2

95.2

90. 1

83.3

76.9

70.9

66.7

63.5

60.4

p

3. 72(-15)

2.54

2.06

1.66

1. 06(-15)

6. 85 (-16)

4.36

2.78

1. 74(-16)

9. 65(-17)

V

2.7(-8)

1.7

1.3(-8)

9. 8(-9)

5.7

3.4

2.0

1.2(-9)

7. 6(-10)

4. 2(-10)

TABLE 9.—Atmospheric conditions at 650 km

0MO-7OO

0. 10

0.10

0.10

0.10

0.11

0.12

0.19

0.26

0.36

0.70

H,

124.2

109.9

104 8

99.3

90.6

84.3

7a 2

74.5

72.6

74.9

p

1. 66(-15)

1. 07(-15)

8.25(-16)

6.32

3.70

2. 19

1. 28(-16)

7. 55(-17)

4.42

2.36

V

1.3(-8)

7.4(-9)

5.4

4.0

2.1

1.2(-9)

6.7(-10)

3.9

2.4

1.4(-10)

M

18.6

17.9

17.6

17.3

16.8

16.3

15.8

15.4

14.9

14. 1

M

18.0

17.4

17.1

16.8

16.3

15.8

15.3

14.8

14.1

12.9

M

17.1

16.5

16.2

15.9

15.2

14.6

13.7

12.7

11.4

9.5

A

1.8( + 8)

1.3

1.1 ( + 8)

9. l( + 7)

6.3

4.3

3.0

2.0

1.4(+7)

8.2( + 6)

N

1.2( + 8)

8. 8( + 7)

7.3

5.9

3.9

2.6

1.7

1. K + 7)

7.4( + 6)

4.5(4-6)

N

5.9(+7)

3.9

3. 1

2.4

1.5(4-7)

9.1(4-6)

5.6

3.6

2.3

1.5(4-6)
DENSITY OF HETEROSPHERE RELATED TO TEMPERATURE—NICOLET 181

ture in the homosphere and heterosphere. Its

concentration at 400 km varies according to

temperature conditions defined by table 6

from 2X108 molecules/cm8 to 5X10* molecules/

cm3, i.e., a factor of the order of 40 when its

concentration at 150 km remains constant,

No.

0

1

1.5

2

3

4

5

6

7

8

1.

No.

0

1

5

2

3

4

5

6

7

8

T

2133-7

1837-7

1711-7

1598-7

1412-7

1272-7

1155-7

1059-7

978-7

903-7

T

2133

1837

1711

1598

1412

1273

1155

1059

978

903

77

156.5

144.2

140.8

140. 1

145.3

159.6

180.6

201.2

216.6

224.9

TABLE

z

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

1000

1250

1500

2000

TABLE 10.—Atmospheric conditions

/3»oo-iooo

0. 14

0.20

0.24

0.30

0.44

0.58

0.61

0.53

0.39

0.23

77,

at 900 km

P V

147.6 2. 52( —16) 2. 2(-9)

133. 0 1. 29 1.0(-9)

126.9 8.86(-17) 7. 0(-

122. 5 6. 06

118. 6 2. 88

4.8

2.4

122.8 1.47(-17) 1. 3(-

135.0 7.93(-18) ai( -

154. 8 4. 73

179. 2 3. 11

5.4

3.8

205.3 2. 18( —18) 2. 8(-

-10)

•10)

-11)

ID

11.—Atmospheric data between 1000 km and 2000 km

H

171

232

352

666

164

257

412

642

165

278

434

612

170

303

446

579

189

339

435

516

216

347

405

466

241

336

371

423

254

316

342

387

255

294

316

358

247

273

292

330

p (total)

1. 28(-16)

2. 82(-17)

8. 20(-18)

1. 79(-18)

6. 08(-17)

1. 19(-17)

3. 68(-18)

1. 05(-18)

4. 03(-17)

7. 73(-18)

2. 59(-18)

8. 14(-19)

2. 68(-17)

5. 24(-18)

1. 94(-18)

6. 49(-19)

1. 24(-17)

2. 79(-18)

1. 22(-18)

4. 10( —19)

6. 51 (-18)

1. 79(-18)

8. 48(-19)

2. 64(-19)

3. 78(-18)

1. 24(-18)

5. 93(-19)

1. 67(-19)

2. 48(-18)

9. 04(-19)

4. 17(-19)

1. 05(-19)

1. 78(-18)

6. 71 (-19)

2. 94(-19)

6. 65(-20)

1. 34(-18)

4. 99(-19)

2. 06(-19)

4. 11 (-20)

p(He)

6. 2(-18)

4. 1(-18)

2. 8(-18)

1. 4( —18)

5. 3(-18)

3. 4(-18)

2. 2(-18)

9. 9(-19)

4. 8(-18)

2. 9(-18)

1. 9(-18)

7. 9(-19)

4. 5(-18)

2. 6(-18)

1. 6(-18)

6. 4(-19)

3. 7(-18)

2. 0(-18)

1. 1(-18)

4. 1(-19)

3. 0(-18)

1. 6(-18)

8. 2(-19)

2. 6(-19)

2. 5(-18)

1. 2(-18)

5. 9(-19)

1. 7(-19)

2. 0(-18)

8. 8(-19)

4. 2(-19)

1. 1(-19)

1. 6(-18)

6. 7(-19)

2. 9(-19)

6. 7(-20)

1. 3(-18)

5. 0(-19)

2. 1(-19)

4. l(-20)

V

1. 2(—9)

3. 4(-10)

1. 4(-10)

5. l(-ll)

5. 5(-10)

1. 6(-10)

7. 3( —11)

2. 9(-ll)

3. 6(-10)

1. l(-10)

5. 4(-ll)

2. K-ll)

2. 5(-10)

8. l(-ll)

4. 2( —11)

1. 6( —11)

1. 3( —10)

4. 8( —11)

2. 6( —11)

9. 0(-12)

7. 7( —11)

3. 2( —11)

1. 6( —11)

5. 2(-12)

5. 0(-ll)

2. l(-ll)

l.K-ll)

3. 0(-12)

3. 5(-ll)

1. 5( —11)

6. 8(-12)

1. 7(-12)

2. 5( —11)

l.O(-ll)

4. 5( —12)

1. 0( —12)

1. 8( —11)

7. 0(-12)

2. 9(-12)

5. 8(-13)

M

14.2

11.2

7.9

4.2

12.7

8.7

5.8

4.2

11.8

7.5

5. 1

4. 1

10.7

6.4

4.6

4.0

8.5

5. 1

4.2

4.0

6.7

4.4

4. 1

4.0

5.4

4.2

4.0

4.0

4.7

4. 1

4.0

4.0

4.4

4.0

4.0

4.0

4. 1

4.0

4.0

4.0

M

15. 1

14. 1

13.4

12.6

10.7

8.8

7.1

5.8

5.0

4.4

AT (total)

5.4(4-6)

1. 5(4-6)

6.3(4-5)

2.3(4-5)

2.9(4-6)

8.3(4-5)

3.8(4-5)

1. 5(4-5)

2.0(4-6)

6.2(4-5)

3.0(4-5)

1. 2(4-5)

1.5(4-6)

4.9(4-5)

2.5(4-5)

9.7(4-4)

& 8(4-5)

3.3(4-5)

1.8(4-5)

6. 1(4-4)

5.9(4-5)

2.4(4-5)

1.2(4-5)

4.0(4-4)

4.2(4-5)

1.8(4-5)

8.9(4-4)

2.5(4-4)

3.2(4-5)

1. 3(4-5)

6.2(4-4)

1.6(4-4)

2.5(4-5)

1. 0(4-5)

4.4(4-4)

1. 0(4-4)

1.9(4-5)

7.5(4-4)

3.1(4-4)

6.2(4-3)

N

1.0(4-7)

5.5(4-6)

4.0

2.9

1.6

1.0(4-6)

6.8(4-5)

4.9

3.8

3.0(4-5)

AT (He)

9.3(4-5)

6.2(4-5)

4.3(4-5)

2.1(4-5)

8.0(4-5)

5.0(4-5)

3.3(4-5)

1.5(4-5)

7.3(4-5)

4.4(4-5)

2.8(4-5)

1.2(4-5)

6.7(4-5)

3.9(4-5)

2.4(4-5)

9.6(4-4)

5.5(4-5)

3.0(4-5)

1.7(4-5)

6.1(4-4)

4.6(4-5)

2.3(4-5)

1. 2(4-5)

4.0(4-5)

3.7(4-5)

1.8(4-5)

& 9(4-4)

2.5(4-4)

3.0(4-5)

1.3(4-5)

6.2(4-4)

1.6(4-4)

2.4(4-5)

1-0(4-5)

4.4(4-4)

1.0(4-4)

1.9(4-5)

7.5(4-4)

3.1(4-4)

6.2(4-4)
182 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

namely 4X1010 molecules/cm3. Such a large

variation affects the mean molecular mass since

the corresponding variation in density is the

order of a factor of 10.

Presence of helium

It should be noticed from the figures of tables

7 and 8 that the mean molecular mass becomes

less than Af=16. This is due to the introduction of helium. Diffusion leads to an important concentration of helium atoms at the

upper levels (Nicolet, 1961a). Physical conditions such as that represented by the already

analyzed models lead to an almost constant

value of n(He) at 500 km. The values of

helium concentration do not vary by more

than ±30 percent from an average value for

temperatures between 2100° K and 750° K.

The absolute value is, however, related to the

level of the beginning of diffusion.

Starting from the normal mixing ratio

n(He)/n(N2), the concentration of atomic

helium at 105 km is

n(He)ioskm=3.OXlO7 atoms/cm3. (21)

If diffusion begins at that level, the concentration at 500 km is

w(He)5ookm=(1.8±0.5)X106 atoms/cm3, (22)

when the temperature at 500 km varies from

2133° K (model no. 0) to 733° K (model no.

10). If, instead of beginning at 105 km, the

diffusion starts at 110 km or 115 km, the concentration given by equation (22) must be

decreased by a factor of 2 or 4. A difference

of about 5 km of the altitude of the diffusion

level corresponds to a difference of a factor of

2 in the concentration of helium.

In the present calculation, we have adopted

105 km as the level where diffusion of helium

begins; a correction is easily introduced if

diffusion is considered to start at other

altitudes.

Table 9, which gives the atmospheric conditions at 650 km, shows how the effect of

helium may be important at such altitudes

when the temperature is sufficiently low. For

example, if T<1000°K, corresponding to

nighttime conditions at the beginning of 1961,

2000 -

1500 -

1000 -

500 -:

I00O 1500

TEMPERATURE (K)

FIGURE 1.—Altitudes of surfaces of equal density as a function of the temperature.

2000
DENSITY OF HETEROSPHERE RELATED TO TEMPERATURE—NICOLET 183

the total density decreases to less than 5X10~17

gm/cm3, while the density of helium reaches

about 5X10~18 gm/cm3. The presence of helium leads to a molecular mass less than 12.

The effect of helium has also an important

consequence. The atmospheric scale height

never decreases below 85 km even if the temperature at 650 km varies by about 1000° K.

The effect of helium is particularly apparent

at 900 km, where there is a large variation of

the mean molecular mass (see table 10). The

mean molecular mass, which is still of the

order of 15 near 2000° K, may reach such a low

value as 5 for temperatures less than 1000° K.

Since the perigee heights of Satellite 1960 i 1

(Echo I) are in the range of 900 to 1600 km,

a set of atmospheric data is given in table 11

for altitudes between 1000 km and 2000 km.

The density and concentration of helium are

included in the table to allow a comparison

with the total density and concentration.

A change in the level of diffusion of helium can

easily be introduced by an adequate correction.

From an analysis of data on the orbit of Echo

I, Zadunaisky, Shapiro and Jones (1961) and

Romer (1961) have deduced densities that

cannot be explained by an atomic oxygen

atmosphere or atomic hydrogen atmosphere,

but are in accordance with an effect of helium.

An atomic mass M= 16 would require a

temperature too high to fit observational data

between 500 km and 700 km. An effect of

atomic hydrogen on the mean molecular mass

would lead to too high concentrations. The

atmospheric-drag perturbations as observed by

Jacchia (1961) during geomagnetic disturbances

show that the drag of the neutral constituents of

the atmosphere has still the most important

effect even at altitudes above 1000 km. The

figures of table 11 show how helium can be

introduced in the analysis of Zadunaisky,

Shapiro and Jones (1961) with scale heights

greater than 300 km at 1500 km, and then 150

km at 900 km. The presence of helium is unavoidable above 1000 km.

1500

-15 - 1 3 -12 - 1 0

IOOO -

5 0 0

1

 V \ ' \ \ ' 'WY\

\\\

\\\

\\

i ii i

\ \ '\ \' \ V \

\\\\ \ \ \

\\\\ \ V \W\\\ \ \\\ \\ \ x

AW \\ \ l

i i i

\ '

°\

\ \ A \ \ \ \ \\\ \ \ \ \ \ 11 1 ^N4 .

TE

CURVE 0 =

1

1.5

2

3

\ 4

If

1 i '

MPERATURE (°K)

2131

1837

1711

1598

1412

1272

5 1155

6 1059

7 978

8 903

9 837

10 773

-

-15 -14 -13

LOG PHI / 2

- 1 2

FIGURE 2.—Vertical distribution of log pH1'

2

 as a function of temperature.

636-014—63 13
184 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

General tables of p, ptf"2, and H

Table 12 summarizes the vertical distribution

of density deduced according to the physical

conditions described in the preceding sections

and, therefore, the results obtained for the

various models already analyzed by their

various parameters in tables 1 to 11. It can be

noticed that the levels of constant density

increase regularly with increasing temperature.

A picture of constant density levels between

500 km and 2000 km is provided in figure 1.

There is a linear relation between the altitude

and the temperature, and the various slopes for

curves of constant density show the increasing

effect of the temperature when the absolute

value of the density decreases. Such relations

between temperature and variation of density

with height must be related to tbe diurnal

bulge described by Jacchia (1960a) and must be

also associated with the results of his (1961)

accurate analysis of the atmospheric-drag

perturbations.

From comparison of the density data of

table 12 with published data on densities

deduced from satellite air-drag, a picture of

the temperature variation during the years

1958 to 1961 can be provided (Nicolet, 1961b).

The maximum density obtained in October

1958 corresponds to a temperature between

2000° and 2100° K, while the average value

from September 1958 to January 1959 may

correspond to a temperature between 1700°

and 1800° K. The two density distributions

given by Martin et al. (1961) correspond to

about 1750° K and 1150° K. As far as the

density distributions given by King-Hele and

Walker (1961) are concerned, they represent

average conditions showing the variation from

1958 to 1960, namely 1750° K to 1350° K

for late-1958 and late-1960 daytime values, and

1050° K to 950° K for late-1959 and late-1960

nighttime values. Even if a systematic error

is involved in such a determination, the variation of the temperature shows the general trend

of the diurnal variation and the solar activity

effect (Nicolet, 1960c).

In order to make available a consistent table

of parameters more closely associated with

data deduced from satellite observations, table

13 gives the values of pH1'

2

 and H from 150 to

2000 km. The value of H1'

2

 is not subject to

variation at 150 km for a range of temperatures

of about 125° K with defined boundary conditions at 120 km, and a small variation is

apparent near 200 km. The maximum variation occurs in the region of 600 to 700 km. At

1000 km (see fig. 2), the variation for 750 < T<

1850° K corresponds to the variation obtained

at 500 km. It should be noticed that pH112 and

H do not necessarily vary in the same way, and,

therefore, p and pW2 may have different

variations in the region where different constituents are involved.
Height

(km) No. 0

77 2133° K

1

1837

1.5

1711

2

1598

TABLE 12.—Computed atmospheric densities

Model number and temperature

3 4 5 6

1412 1272 1155 1059

7

978

8

903

8

150

160

170

180

190

200

210

220

240

260

280

300

320

340

360

380

400

420

440

460

480

500

520

540

560

580

600

650

700

750

800

850

900

1000

1250

1500

2000

-11.618

-11.836

-12. 009

-12. 154

-12. 280

-12. 390

-12.492

-12. 583

-12.752

-12. 900

-13.036

-13. 164

-13. 284

-13.298

-13.507

-13.613

-13.714

-13.810

-13.907

-14. 000

-14.089

-14. 178

-14. 264

-14.347

-14.430

-14.513

-14.592

-14.780

-14.955

-15. 126

-15. 289

-15.446

-15.599

-15. 893

-16.550

-17.086

-17.747

-11.614

-11.824

-11.996

-12. 139

-12.268

-12.383

-12. 488

-12. 585

-12. 765

-12.925

-13.072

-13.211

-13. 343

-13.469

-13.588

-13. 703

-13.815

-13.921

-14.026

-14. 127

-14.266

-14.321

-14.415

-14. 506

-14.595

-14. 682

-14.765

-14.971

-15. 168

-15. 358

-15.541

-15.719

-15. 889

-16.216

-16.925

-17.434

-17.979

-11.611

-11.818

-11.987

-12. 134

-12. 264

-12. 380

-12.488

-12. 588

-12. 770

-12.900

-13.096

-13.237

-13.374

-13.505

-13.631

-13.752

-13.867

-13.983

-14.090

-14. 196

-14.299

-14.398

-14.496

-14.592

-14. 686

-14. 777

-14.867

-15.084

-15.291

-15.492

-15. 684

-15.873

-16.053

-16.395

-17. 112

-17.587

-18. 089

-11.606

-11.813

-11.983

-12. 130

-12.261

-12.380

-12.490

-12.592

-12. 780

-12. 955

-13. 113

-13. 264

-13. 408

-13. 544

-13.676

-13.801

-13.925

-14.042

-14. 157

-14. 268

-14.376

-14. 480

-14.582

-14. 682

-14. 780

-14.876

-14.971

-15. 199

-15.418

-15.629

-15. 833

-16. 029

-16.218

-16.572

-17.281

-17.712

-18. 188

-11.595

-11.801

-11.971

-12. 123

-12.257

-12. 381

-12. 495

-12. 602

-12.801

-12.987

-13. 159

-13.322

-13.479

-13.627

-13.770

-13. 907

-14.041

-14. 169

-14.294

-14.416

-14. 533

-14. 646

-14.757

-14.867

-14. 975

-15.079

-15. 182

-15.432

-15. 672

-15.903

-16. 126

-16. 338

-16.541

-16.907

-17.554

-17.914

-18. 387

-11.587

-11.793

-11.967

-12. 120

-12. 259

-12.387

-12.507

-12. 622

-12.833

-13. 029

-13.214

-13.389

-13.558

-13.717

-13.873

-14.020

-14. 162

-14.300

-14. 434

-14.564

-14.690

-14.813

-14. 932

-15.049

-15. 164

-15.277

-15.388

-15.660

-15.917

-16. 166

-16. 403

-16.625

-16.833

-17. 186

-17.747

-18.072

-18. 578

•11.582

-11.788

-11.967

-12. 123

-12.266

•12.400

-12.526

-12. 644

-12.870

-13.078

-13.275

-13.462

-13.640

-13.813

-13.975

-14. 134

-14. 287

-14. 434

-14. 578

-14.717

-14. 854

-14.983

-15.110

-15.237

-15.361

-15.483

-15.602

-15.893

-16. 170

-16.433

-16.678

-16.900

-17. 101

-17.423

-17. 889

-18. 227

-18.777

-11.577

-11.785

-11.967

-12. 128

-12.277

-12.416

-12.548

-12. 674

-12.910

-13. 131

-13.340

-13.539

-13.728

-13.910

-14. 085

-14.252

-14.413

-14.570

-14.724

-14. 870

-15.016

-15. 151

-15.290

-15. 423

-15.556

-15.686

-15.813

-16.122

-16.413

-16.684

-16.928

-17. 143

-17.325

-17.606

-18. 044

-18.380

-18. 979

-11.583

-11.796

-11.979

-12. 146

-12.301

-12.445

-12.583

-12.714

-12.963

-13. 197

-13.419

-13. 629

-13. 830

-14.020

-14.203

-14.381

-14.551

-14.717

-14.876

-15.034

-15.188

-15.330

-15.475

-15.618

-15.760

-15.896

-16. 032

-16.355

-16.654

-16.921

-17. 154

-17.348

-17.507

-17.750

-18. 173

-18.532

-19.177

-11.629

-11.845

-12.033

-12.206

-12.367

-12.519

-12.662

-12.801

-13.064

-13.311

-13.545

-13.767

-13.979

-14.180

-14.374

-14.561

-14.740

-14.910

-15.079

-15.244

-15.405

-15.552

-15.717

-15.867

-16.016

-16. 159

-16.299

-16.627

-16.917

-17. 165

-17.367

-17.529

-17.662

-17.873

-18.302

-18. 686

-19.386

•1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1
681 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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RESEARCH IN SPACE SCIENCE—NICOLET 187

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CO 00 S i

Effects of the Earth's Ionosphere on HF Radio

Astronomy from Artificial Satellites

By M. D. Grossi, K. M. Strom, and S. E. Strom

Ground-based radio telescopes are known to

be able to conduct reliable observations only

within the band roughly defined by 10 mc/s and

10,000 mc/s. The lower limit is established by

ionospheric refraction effects, and the upper

limit by atmospheric behavior.

Airborne and balloon-borne radio telescopes

can extend the upper limit above 10,000 mc/s

with high-altitude observations. For the frequency band below 10 mc/s, a satellite-borne

radio telescope appears a suitable means of

observation.

Recent literature (Haddock, 1960) has

pointed out the desirability and the feasibility

of a satellite-borne HF

 l radio telescope orbiting

outside the ionospheric layer of maximum electron density. Among others, the Harvard College Observatory is presently involved in the

design of such a satellite (Space Radio Project).

The satellite will be able to perform observations not affected by the limitations imposed by

the ionosphere.

Many relevant experiments can be performed

by such orbiting telescopes. Among these are

(Haddock, 1960; Burke and Franklin, 1955;

Davies, 1954; Piddington, 1951; Herbstreit and

Johler, 1948): (a) measurement of integrated

radio flux from our galaxy at frequencies below

the F2 layer cutoff; (b) measurement of the

dynamic spectra of solar bursts well below

30 mc/s; (c) measurement of the dynamic

spectra of planetary radio bursts well below

30 mc/s.

General

The purpose of the present report is to present

i Considered as the frequency range 1-10 mc/s.

an analysis of some effects of the ionosphere on

HF observations from satellite-borne radio

telescopes. We deal primarily with a search

for focusing effects of the ionosphere on incoming cosmic noise in the HF band. The

analysis covers a variety of cases of interest

for satellites in various orbital altitudes and

with a wide range of observation frequencies.

In this preliminary analysis, we have considered the ionosphere to be a nonhomogeneous, nonisotropic, magnetoionic medium. The

electron-density profile used in the calculations

is illustrated in figure 1. We assume the earth's

magnetic field to be dipole in nature. We have

taken into account ionospheric irregularities

and discontinuities only for cases in which

these disturbances are much larger in size than

are the wavelengths considered.

We compute the effects of the ionosphere on

incoming radiation by employing a Hamiltonian

optics approach. We discuss the necessary

numerical procedures involved in the raytracing in relation to a program suitable for

use in an IBM-7090 computer.

Rays are computed and plotted for sources

located at infinity and for frequencies in the

band 1-30 mc/s.

We basically consider refraction, reflection,

and resultant focusing effects of the ionosphere

for frequencies higher and lower than critical,

and for satellites in conjunction and in opposition with respect to the source and the earth.

In cases of special interest, we compute the

equivalent aperture of the ionosphere, considering the ionosphere to be analogous at radio

frequencies to a spherical shell lens, with the

observing satellites located at the focal region.

189
190 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

10

h (km.)

\

I

:

:

:

ELECTRON DENSITY

(tl/ce)

FIGURE 1.—Electron-density profile derived from averages

of IGY data.

Hamiltonian ray-tracing in the earth's

ionosphere

In this section, we consider plane waves incident

on the ionosphere from a radio source effectively

at infinity. We disregard the specific nature

and position of the source, and the effect of the

interplanetary medium.

The effect of absorption in the earth's ionosphere is not included in the equations as

written, although the inclusion of this effect

would involve the addition of only one equation

to the system. It is reasonable to neglect

absorption because the collision frequencies

encountered in most cases considered here are

extremely small. However, the Hamiltonian

ray-tracing method is suitable for use in quite

general cases of propagation in a magnetoionic

medium.

The method of ray-tracing follows from the

application of the principles of Hamiltonian

optics. Mathematically, this ray-tracing problem reduces to solving simultaneously the

following six differential equations (Haselgrove,

1955; Wong, 1960).

(i)

(2)

i _

^

s i n 0;

(3)

(4)

2/2=- [- ^- cos 0J;

(5)

These are the Hamiltonian equations for a

ray path in a general magnetoionic medium.

They are analogous to the Hamiltonian equations in classical mechanics. The ray is described in terms of the associated wave normal.

The quantities in equations (1) through (6)

are defined as follows: r, 0, $ are the spherical

coordinates of a point in the earth's ionosphere;

ylt y2, y3 represent the components of the wave

normal with respect to a Cartesian system

whose origin is placed at r, 9, <£. This system

is illustrated in figure 2. The yi direction is

along r; y2 along 0; and y3 along 3>. The system

is chosen to be right-handed and orthogonal;

r, 0, etc., are the derivatives of these variables

with respect to time.

The phase refractive index /u is given explicitly by a form of the Appleton-Hartree

formula (where collisions between electrons

and air molecules or positive ions are neglected):

N(r, 0, 3>)

1.24X104/

2

1—

(7)

& cos*

where /=frequency of radiation in mc/s;

N(r, 0, <$) is number of electrons/cm3 (the electron

density, which is, in general, dependent upon

the position in the ionosphere as well as the

height of the point under consideration);

JH{T, 0, $)=the gyrofrequency in mc/s; and M

is given by

W

1—1.24X104

/

2

(^=the angle between the wave normal and the

earth's magnetic field). If this field is taken to

be dipole in nature, and if the polar axis of the

spherical coordinate system coincides with the

dipole axis, then:

1

yi cos 0+~ 2/2 sin 0

cos \f/= {

cos

{ -7 sin2

 0 4

(9)
HF RADIO ASTRONOMY FROM SATELLITES—GROSSI, STROM AND STROM 19 1

WAVE

NORMAL

FIGURE 2.—Spherical coordinate system used in ray tracing.

Note that in a general magnetoionic medium

the direction of the wave normal is not coincident

with the direction of the ray. We must therefore make the distinction between these two

directions when considering a magnetoionic

medium (such as the earth's ionosphere) that is

nonisotropic. The electromagnetic energy is

propagated along the ray direction. However,

there are two distinct phase velocities, one along

the ray and the other along the associated wave

normal. Equations (1) through (6) are written

for a nonisotropic medium; thus we need to

include the wave normal components, yu y2, y3,

and their derivatives.

Note also that an electromagnetic wave

incident in a magnetoionic medium splits into

two components. The energy is split into two

636-014—63 14

characteristic modes of propagation called the

extraordinary rays and the ordinary rays, which

are generally distinct entities. In equation (7),

the positive sign corresponds to the ordinary ray,

and the negative to the extraordinary ray.

For our case of a radio source effectively at

infinity, we consider the incoming rays as parallel.

These rays are incident on the ionosphere. The

boundary of the ionosphere is chosen to be at

about 8,000 km from the earth's surface, where

the electron density reaches the assumed interplanetary value of 600/cm3

. The initial conditions for the ray-trace are easily assigned by

examination of the geometry of the situation.

The geometry is illustrated in figure 3, where r0

is chosen as 14,378 km; 0O and f>0 vary for the

individual ray trace; and yu y2, and y3 are then
192

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

y^-COSO-a )

y2= SIN(6-a )

LIMIT OF EARTH'S

IONOSPHERE

FIGURE 3.—Geometry for assigning initial conditions for ray-tracing.

assigned according to the convention in figure 4.

If we want, we can set y3=constant for each

ray trace. Then y3 can be varied for each set of

ray traces. For our computations, we found

this to be the most systematic way of assigning

initial conditions. For simplicity, in the cases

considered here, we assumed ^3=0. After the

initial conditions are assigned, the rays are

automatically traced by the computer. The

results of these computations are illustrated in

figures 5, 6, 7, 8.

y, »ACO S A

ya = /i COS B

integration. It is possible to estimate^the

truncation error per step of integration (Hildebrand, 1956). But in practice it is extremely

difficult (for the present functions) to approximate the total rounding-off error for this method;

therefore, there is an uncertain factor in the

solution. However, for nearby paths, the errors

are systematic. A detailed analysis of the

method can be found in Hildebrand (1956),

Bennett, Milne, and Bateman (1956), and

Ralston and Wilf (1960).

The boundary conditions are the initial

values of r, d, $, yu y2, y3, with

WAVE NORMAL

2/x=M cos A,

y2=n cos B,

y3=M cos C, (10)

FIGURE 4.—Convention for assigning wave normal components.

The method chosen for numerical integration

of equations (1) through (6) is the fourth-order

Runge-Kutta method. The advantage of this

method lies in the fact that we need only to know

the values of the variables r, 8, f>, ylf y2, y3 at

t=to- In other words, only one set of initial

conditions is needed with which to begin the

in accordance with the notation shown^in figure 4.

We assign the position, and we assume the

direction of the ray and the wave normal

initially coincident. The program for solution

of the differential equations (1) through (6)

is written in FORTRAN (Graham, Grossi,

Strom, and Strom, 1960).

Numerical solutions of the equations follow

from an application of the Runge-Kutta

method. The initial conditions and the frequency are read into the machine. Subroutines
HF RADIO ASTRONOMY FROM SATELLITES—GROSSI, STROM AND STROM 193

for electron-density profiles and the necessary

derivatives are interchangeable for various

ionospheric models, including different electrondensity profiles and discontinuities. Both ordinary and extraordinary ray paths are then

computed. At each integration step, the

present calculated values of r, 0, and $ are

printed out. Calculation time for ray-path

length 10* km is approximately 15 seconds on

the IBM-7090 computer. One can see that, at 0

and 0=180° (i.e., at the magnetic poles), there

are discontinuities in equations (1) through (6).

These discontinuities do not significantly affect

the computation of the ray paths, provided that

the path is a straight line over the magnetic

poles. However, if the ray is undergoing a

change in direction (refraction or reflection) in

the vicinity of the pole, the integration errors

increase. In fact, the certainty to which focal

points may be predicted is greatly reduced.

Significant errors are also introduced when the

ray enters a region in which the gradient of

electron density is great, i.e., at about 300 km

of altitude (see fig. 1). These errors can be reduced by use of smaller integration steps.

The illustrated ray paths avoid these discontinuities so that the total errors are small.

The search for focal regions

The method outlined in the previous section

has been applied to the cases of 5, 10, 12, 18,

and 22 mc/s sources at infinity. For simplicity,

we restricted the analysis to cases in which the

rays remained in the (r, 6) plane. Thus, 3^=0

for all cases considered.

In figures 5, 6, 7, and 8, we provide an

example of the plotting performed for the 5, 10,

12 mc/s cases. The incident rays are initially

equidistant and parallel. Owing to the close

proximity of these rays on the scale of figures

6, 7, and 8, we have illustrated the incident rays

by thick black lines. The rays are then plotted

separately after the point of significant divergence.

We notice there are extended focal regions

along the line containing the center of the earth

and the source in opposition. Haselgrove,

Haselgrove, and Jennison (1961) postulated

the presence of focal regions in a situation of

conjunction. Figure 5 illustrates our tracings

for this case. The source here is considered

extended. Figures 6, 7, and 8 illustrate our

calculations for the situation of opposition and

contradict the common belief that the ionosphere would create a shadow in situations of

opposition with respect to a source at infinity.

The rays illustrated in these figures are ordinary

rays.

It is important to note, however, that a

satellite-borne radio telescope located at a

point of the focal region will not experience a

continuous intensified reception of the cosmic

radio background. Owing to the multipath

structure of the ionospheric propagation, alternate maxima and minima of the intensity

received should be observed.

Calculation of the geometric gain for the

ionosphere

With the data obtained, we are able now to

calculate the geometrical gain of the ionosphere.

We again assume the ionosphere to be analogous to a spherical shell lens. The gain as

computed is Ai/A2, where areas Ax and A2 are

illustrated in plate 1; At is the area intercepted

by the incident rays at the boundary of the

ionosphere, and A2 is the area of the circle of

least confusion, where the observing satellite is

placed.

Taking advantage of the symmetry of the

earth's magnetic field, we have Ai=2irr2 sin ddd,

where r and 6 obey the convention in figure 2.

The value of A2 follows directly from the

determination of the radius of the circle of

least confusion. For the case illustrated in

figure 6, the value of the geometrical gain lies

between 3 and 10, depending on the rays

chosen for calculation. Figures 7 and 8 illustrate cases in which the value of the gain lies

between 5 and 20.

The results obtained are based on idealized

assumptions concerning the electron density

and magnetic field. However, the focal properties of the actual ionosphere are affected by

irregularities and discontinuities of various

causes; consequently, the focusing behavior

can be predicted only in statistical terms.

To obtain a preliminary idea of the magnitude of these effects, we have considered a nonspherically symmetric distribution of electron

density. We retained the N(r) curve as illustrated in figure 1, but we imposed the simple
194 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

0 10 20 30 40 50

100

110

120

180 170 160 150 140

f = 5mc/sec

130

FIGURE 5.—Ray-tracing for a situation of conjunction with/=5 mc/s (distances are measured from the

center of the earth).

model of day-to-night variation shown in

figure 9. The minimum value of electron

density corresponds to the center of the night

side of the earth. Computations show that

the effect of this model of day-to-night variation on the geometrical size of the focal region

is negligible.

Extension of the method and further search

for interesting focal regions

We plan to refine our analysis by adopting a

fully three-dimensional model of the ionosphere and by including various realistic

models of irregularities and discontinuities

that are large with respect to the wavelength.

These models will include electron clouds,

ionospheric winds, seasonal variations, solar

bursts, and magnetic storms..

Absorption, when not negligible, will also be

computed.

With our refined analysis, we will study the

effect of the ionosphere on the incoming cosmic
GROSSI, K. STROM, AND S. STROM PLATE 1

Geometry for computing the geometrical gain.

HF RADIO ASTRONOMY FROM SATELLITES—GROSSI, STROM AND STROM

-50 -40 -30 -20 -10 0 10 20 30 40 50

- 6 0

- 7 0

- 8 0

-100

-110

-120

YY/M&

r

sy^cv\x^

x s\1 \ //J - -.-*"*' 1 l000° l5| °°

2 0 000^-

-A

195

60

^ 7 0

80

90

100

110

120

-130 -140 -150 -160 -170 180 170 160 150 140 130

f » 5 me/sec

FIGURE 6.—Ray-tracing for a situation of opposition with / = 5 mc/s (distances are measured from the

center of the earth).

noise, taking into account known radio sources,

both concentrated and diffused. We will plot

cosmic noise distributions on spheres that are

concentric with the earth and that have various

radii.

In the search for interesting focal regions,

we will investigate the focusing properties of the

ionosphere for frequencies higher than the

critical frequency of the F2 layer.

For these frequencies, the optical behavior

of the ionosphere below the F2 layer will also be

examined by using a Luneberg lens model

(Luneberg, 1944), with the source at infinity

and the focal region on the boundary of the

spherical refractor.

The search for the existence of a possible

focal region of the earth's ionosphere on the

surface of the moon is also planned. Data

will be collected to determine the feasibility

of a moon-based HF radio telescope exploiting

the focusing properties of the earth's ionosphere.

Acknowledgments

We wish to thank Dr. R. J. Davis for his encouragement, his sustaining interest, and

guidance in performing this research; and Dr.

A. E. Lilley for his valuable criticisms and

suggestions.

The computer program was developed, in

part, at Raytheon Missile and Space Division

in conjunction with Independent Research

Program 71-1587-010. Computations relevant

to the radio astronomical cases were performed

on the IBM-7090 of the Smithsonian Astrophysical Observatory's computation and analysis division.
196

- 6 0

- 7 0

- 8 0

- 9 0

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

-50 -4 0 -3 0 -20 -10 0 10 20 30 40 50

-100

-110

100

110

120

-130 -140 -150 -160 -170 180 170 160 150 140

f=IOmc/sec

130

FIGURE 7.—Ray-tradng for a situation of opposition with/= 10 mc/s (distances are measured from the

center of the earth).
HF RADIO ASTRONOMY FROM SATELLITES—GROSSI, STROM, AND STROM 197

-50 ^40^ -30 -20 -10 0 10 20 30 40 50

- 6 0

- 7 0

- 8 0

-140 -150 -160 -170 180 170 160 150 140

-100

-120

100

110

120

FIGURE 8.—Ray-tracing for a situation of opposition with/= 12 mc/s (distances are measured from the

center of the earth).

Nirl

.79

SNIrl

.2 5

n

-

-

-

1 1 1

r,e,»)-N(r)/IO*9Ntr> C0S(6,/E) COS(*/2) /I0

IS DEPENDENT UPON THE POSITION OF THE SUN

\

\

/

\ / v

V

i

/

i

i

-

-

-

i

FIGURE 9.—Day-to-night variation of electron density.

Short-Periodic Oscillations in the Drag of

Satellite 1958 Alpha

By Luigi G. Jacchia and Jack W. Slowey

The authors have derived accurate orbital

accelerations over extensive time intervals for

several artificial earth satellites; they are now

in the process of converting the accelerations to

atmospheric densities and then analyzing them.

The procedure for deriving the accelerations,

which has been briefly described by Jacchia

(1961), involves the plotting of residuals in

mean anomaly for all individual satellite

observations. A survey of such plots for

Satellite 1958 Alpha (Explorer I) from January 1958 to April 1961 has revealed the existence of persistent, periodic drag oscillations

that can easily be recognized over intervals of

several weeks when the observations are

sufficiently numerous and well distributed.

The usable observational material for this

satellite consists almost exclusively of fieldreduced positions from photographs taken with

the Baker-Nunn cameras; unfortunately, the

time intervals during which the number and

distribution of observations were favorable to

the detection of the oscillations are not very

numerous. Whenever the oscillations could

be detected, they had a period of about 3.5

days, except toward the end (March 1961),

when the period increased to 4.2 days.

Table 1 gives values of the orbital accelerations -j - during four time intervals during

which oscillations were rather obvious; these

values are plotted in figure 1. Although most

of the observed oscillations are tightly grouped

in a few such intervals, scattered oscillations

with an unmistakable periodicity of 3.5 days

can be recognized here and there, providing,

we think, sufficient continuity for the numbering

of the individual cycles over the whole time

interval here considered. The time in the

tables and diagrams is expressed in Modified

Julian Days (MJD), i.e., in Julian Days minus

2 400 000.5.

Table 2 gives a list of all observed times of

maximum of the short-periodic drag oscillations,

together with then* residuals from a linear reference equation: max=MJD 36280.0+3.52n,

where n is the serial number of the maximum.

The data of table 2 are plotted in figure 2.

Instantaneous periods, derived from the

data of table 2, are given in table 3 and plotted

in figure 3.

The drag oscillations we have described must

be interpreted as due to a systematic variation

of the effective presentation area of the satellite,

caused by the precession of the angularmomentum vector of the satellite body in its

tumbling motion.

If the satellite is a cylinder of length L and

base radius r=eL, its total area A is given by

A=2treL2{\-\-t). Let a be the angle that the

angular-momentum vector makes with the

direction of motion. The mean presentation

area <S will be a function of a; let us call So the

value that S assumes when a=0 , and Si the

value when a=90°. We shall have

and, always,

So>S>St.

S 2 The ratio ^r=—(-« represents the maximum

factor by which the satellite drag can vary

owing to change in a, provided the period of the

tumbling is very small compared to the orbital

199
200 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

UJ

in

«> »»• <M

I • I

I I 1 ">

i r

o

UJ

1 1 1

•

•

•

•

•

•

1

•

•

•

<

1 1

•

•

>

1

1 1

•

<

•

1 1

1

I

Q

-3

•o l-o

i

2

i

• • » «

I

TJ '"O

I
SHORT-PERIODIC OSCILLATIONS IN DRAG OF 1958 ALPHA—JACCHIA AND SLOWEY 20 1

Ao n io °

 2o ° 3o °

+ 20

o-c

+10

-10 1

__-V^—

1

i

1 1

/

36 750 37000

MJD

FIGURE 2.—Residuals from the equation Max= 36280.0+3.52n.

4.5

Period

4.0

3.5

.

•—•—-^ - y

3.0

36 200 400 600 800 37000 200 400

MJD

FIGURE 3.—Period of the short-periodic drag oscillations in 1958 Alpha.
202 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 1.—Accelerations -rr of Satellite 1958 Alpha in

(XL

MJD -

SECTION I

selected

10« ^

(MJD 36305-36349)

36305. 0

05.5

06.0

06.5

07.0

07.5

08.0

08.5

09.0

09.5

10.0

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14 0

14 5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19. 5

20.0

20.5

21.0

21. 5

22.0

22.5

23.0

23.5

24 0

24.5

25.0

25.5

26.0

26. 5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

30.5

31.0

31.5

5.0

5. 1

5.2

5.0

4 6

4.7

4 5

4.5

5.6

6.2

6. 5

6. 1

5. 1

5.2

5.0

4.7

5.3

5.2

5.4

4.9

4. 5

4.0

4 5

4.8

6.3

7. 1

6.4

5. 1

4 6

4 9

5.4

6.4

6. 9

7.7

7.2

6.5

5. 2

5. 2

7.2

7.5

6.7

6.2

5.2

5.0

4 7

6.0

7.7

6. 4

6.2

5.6

5. 3

6.0

6.0

6.0

sections

MJD

36332. 0

32. 5

33.0

33. 5

34 0

34. 5

35.0

35. 5

36. 0

36. 5

37.0

37. 5

38.0

38. 5

39.0

39. 5

40. 0

40. 5

41. 0

41. 5

42.0

42. 5

43. 0

43. 5

44 0

44. 5

45.0

45. 5

46.0

46. 5

47.0

47. 5

48.0

48. 5

49.0

49. 5

SECTION

6.9

6. 2

4.8

5. 1

4 8

5. 2

5. 5

5. 1

5. 1

5. 0

5. 0

5. 1

5. 1

5.0

4. 7

5. 1

4. 4

4. 0

3. 5

4. 5

5. 7

5. 4

4 7

3. 9

3.3

4 0

5. 2

5. 3

5.0

4. 7

4.2

4 8

5.4

6. 0

5. 7

4 8

II

(MJD 36668-36711)

36668. 0

68.5

69.0

69.5

70.0

70.5

71.0

71.5

72.0

72.5

73.0

73. 5

74.0

74. 5

75.0

75.5

76.0

6. 9

7.0

7.4

6.8

5.4

5.4

6. 1

6. 6

7. 1

6.9

5.6

5.4

6.2

7. 1

7.4

7.4

6.2

MJD

36676. 5

77.0

77.5

78.0

78.5

79.0

79. 5

80.0

80.5

81.0

81. 5

82.0

82.5

83.0

83. 5

84.0

84. 5

85.0

85. 5

86.0

86. 5

87.0

87. 5

88.0

88. 5

89.0

90.0

91.0

92.0

93.0

94.0

95.0

96.0

97.0

98.0

98.5

99.0

99.5

36700. 0

00.5

01.0

01. 5

02.0

02.5

03.0

03.5

04 0

04 5

05.0

05.5

06.0

06. 5

07.0

07. 5

08.0

08. 5

09.0

09. 5

10. 0

6.3

6.3

6.6

6.8

7.4

7.4

6. 5

5.8

5.6

6.6

7.4

8.3

7.9

7.9

7. 5

6. 8

7.3

6. 9

7.4

8. 1

7.9

7. 5

7. 5

7. 1

7. 1

6.6

6.3

5.5

5.6

6.5

6.0

5. 1

5. 8

6.3

6.2

6. 1

6.3

6.8

6. 5

7.2

7.4

7.2

7.2

6.7

5.7

6.7

7.8

8. 1

7.8

6.4

6.4

6. 1

7.8

7. 8

7. 5

6.3

6.8

6. 9

8. 2

MJD

36710. 5

11.0

11. 5

dP

dt

7. 0

7.0

6. 7

SECTION III

(MJD 37191-37213)

37191. 0

91. 5

92.0

92. 5

93.0

93. 5

94 0

94.5

95.0

95. 5

96.0

96. 5

97.0

97. 5

98.0

98. 5

99.0

99. 5

37200. 0

00. 5

01.0

01.5

02.0

02.5

03.0

03.5

04.0

04 5

05.0

05. 5

06.0

06.5

07.0

07.5

08.0

08. 5

09.0

09. 5

10.0

10.5

11.0

11. 5

12.0

12.5

13.0

3. 8

4 0

4. 5

4 3

4 3

4 1

3. 9

4 2

3.7

4. 4

4 4

5. 4

5. 6

5. 2

3. 4

3. 2

2.8

3. 4

4. 1

4 6

4 4

4 2

3. 1

3. 1

2.9

3.6

3. 1

3.8

3.6

3.4

2.3

3.0

3.2

3.5

3.9

4 0

3. 7

3. 5

2.2

1.8

3. 1

4 0

4 5

3. 4

3.0

SECTION IV

(MJD 37356-37379)

37356. 0

56. 5

57.0

57. 5

58.0

58.5

1. 5

1.6

1. 4

1.4

1.6

1.8
SHORT-PERIODIC OSCILLATIONS IN DRAG OP 1958 ALPHA—JACCHIA AND SLOWEY 203

TABLE 1.—Accelerations -TT- of Satellite 1958 Alpha in

selected sections—Continued

MJD MJD

37359. 0

59.5

60.0

60. 5

61.0

61. 5

62.0

62.5

63.0

63. 5

64 0

64 5

65.0

65.5

66.0

66.5

67.0

67. 5

6a o

68.5

69.0

1. 8

1. 6

1. 8

1. 3

1.3

0.9

0.9

1. 1

1. 5

2.0

2.0

1.8

2.0

1.8

1. 3

1. 1

1. 6

1.6

1.8

2.4

2.0

37369. 5

70.0

70. 5

71.0

71.5

72.0

72. 5

73.0

73. 5

74 0

74 5

75.0

75.5

76.0

76.5

77.0

77.5

78.0

78.5

79.0

1. 6

1. 1

1. 4

1. 6

1.4

1.8

1. 8

1.8

2. 1

1.6

1.2

1.6

1.6

1.7

1.9

1.7

1.7

1.7

1.7

1.7

TABLE 2.—Satellite (1958 Alpha: Observed drag maxima

expressed in MJD and residuals from the equation

Max=86280.0+S.52n

n

0

1

2

9

10

11

12

13

14

15

16

17

18

19

20

94

95

96

103

104

111

112

i

36280. 2

283. 7

287.3

310.0

313.6

317.5

321.4

324.4

328.2

331. 6

(335. 1)

338.6

342.2

345.4

(348. 5)

608. 1

611.4

614. 8

640.0

643.3

669.0

672. 1

o-c

+ 0.2

+ 0.2

+0.3

-1.7

-1.6

-1.2

-0.8

-1.4

- 1 . 1

-1.2

(-1.2)

-1.2

-1.2

-1.5

(-1-9)

-2.8

-3.0

- 3 . 1

-2.6

-2.8

-1.7

- 2 . 1

n

113

114

115

116

118

120

121

122

123

191

192

193

195

200

207

209

210

228

229

230

231

232

248

249

260

261

262

263

264

265

271

272

273

274

303

304

305

306

307

l

36675. 2

678.7

682.5

686. 1

(693. 0)

700.9

704.5

707.0

(710. 0)

951.0

954.6

958.0

965.0

36982. 3

(37007. 5)

013.5

(017. 2)

079.4

083. 0:

086.0

090. 0:

093.6

37150. 0

153.5

193.0

197.0

200.7

204.5

208.3

212.0

233.5

(237. 5)

241.8

245. 6

(359. 6)

364.0

368.2

372.4

376.5

o-c

-2.6

-2.6

-2.3

-2.2

(-2.4)

-1.5

— 1.4

-2.4

(-3.0)

-1.3

-1.2

-1.4

-1.4

-1.7

(-1.1)

—2.2

(-2.0)

-3.2

-3.0:

-3.6

- 3. 1:

-3.0

-3.0

-3.0

-2.2

-1.7

-1.5

-1.3

-1.0

-0.8

-0.4

(+0.1)

+ 0.8

+ 1. 1

+ 13.0

+ 13.9

+ 14.6

+ 15.3

+ 15.9

TABLE 3.—Periods of the short-periodic drag oscillations

observed in Satellite 1958 Alpha, deduced from the

data in table 2

MJD

36330

620

690

37000

090

150

220

370

Period

3.43 days

3.55

3.55

3.47

3.50

3.58

3.72

4.2
204 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

period (if not, the departure from the statistical

mean involved in the computation of Si could

become important).

For Satellite 1958 Alpha we have e=0.0375,

which gives So/Si = 1.48. Although most of the

observed oscillations appear to fall within this

amplitude, a few cycles seem to exceed it,

reaching in the amplitude a factor of nearly 2.

If we consider that the plotted accelerations

are the second derivatives of a curve that is

often determined by only a few points, we

would not, at this stage, attribute much significance to these large amplitudes but rather

consider the over-all distribution of amplitudes as falling reasonably well within the

computed limits.

A variation of the drag by the maximum

factor SofSi would imply a variation of a close

to 90° in half the precessional period of 3*5, i.e.,

values of -j- of the order of 40° to 50° per day.

Colombo (1961) found that the interaction of

the earth's magnetic field and the ferromagnetic

components of Satellite 1958 Epsilon (Explorer

IV) can easily explain precessional perturbations of the angular momentum of 17° per day

for that satellite, while the gravitational torque

and the drag torque cannot account for more

than 6° per day. It may be necessary to

invoke similar magnetic interactions to explain

the much larger precessional motion in Satellite

1958 Alpha.

The data of table 3 need to be corrected if

they are to be interpreted as periods of the

precessional motion of the angular-velocity

vector referred to a fixed coordinate system.

The period is obviously referred to the perigee

of the satellite's orbit, which had an average

daily precessional motion of +2?1 in 1958 and

-+-2?4 in 1961. This would result in corrections

of ±0*07 to ±0*10; unfortunately, it is impossible to specify the sign of the correction, unless

the direction of the precessional motion of the

angular-velocity vector is known.

In a forthcoming paper we shall give a

complete listing of the accelerations of Satellite

1958 Alpha, together with those of the other

satellites involved in the atmospheric investigations. This preliminary note is intended

primarily to call to the attention of atmospheric

investigators the errors that can arise when

drag data of nonspherical satellites are used.
The Analysis of Gravity

By Harold Jeffreys

Properties of gravity

Gravity over the surface of the earth is a continuous function of latitude and longitude. A

little more can be said, because it is an integral

based on the distribution of density, which at

worst has only finite discontinuities. We can

form an idea of the rate of convergence of an

expansion in spherical harmonics as follows.

In one variable, if / , a function of bounded

variation in (0, 2T), is analyzed in a Fourier

series, the coefficients must decrease like 1/n

and their contributions to fj2dx like l/n?. For

the integral of such a function the decreases

will be like n~

2

 and TIT4. Over a sphere we

should expect the sums of contributions to

SSPdS from all harmonics of degree n to

decrease similarly, and if all harmonics were

normalized to make J'J%S

2

nmdS=4:ira2

 the expectations of the coefficients of Snm would

decrease like n~

5/2

. As the derivation of gravity

from the potential introduces a factor n— 1,

the coefficients in the potential should decrease

like n~

in

. Then a rough calculation shows that

if the major variations are for n=0 , 1, 2, we

should expect the contributions to fffdS

from TO=4 to be about % of those for TO=3,

and those for TO>4 to be about % of those for

n=3 . For a smoother function than the integral of a function arising from simple discontinuities in density the decrease would be more

rapid. We should note also that if we simply

assume ffpdS to exist, it is necessary that

the double sum of the squares of the coefficients

a>nm, bnm shall converge. Hence

must decrease faster than I/TO. If it decreases

like I/TO2 the coefficients will on the whole

decrease like n~

s/2

. This tolerates greater

irregularity in/tha n we have just considered.

But in the series

the first few terms are 1, 0.25, 0.11, 0.06, and

2/3 of the excess over 1 is already taken into

account. So it is worth while, even for the

study of gravity alone, to proceed by examining

the coefficients of spherical harmonics for TO=0,

then for TO=0 and 2 (those in TI=1 being

theoretically absent), then for TO=0, 2, 3, then

for TO=0, 2, 3, 4, and so on. This consideration

is even more striking for the external potential

on account of the extra factor -, and also

71—1

because there is a factor (a/r)n+1 in the potential,

which falls off rapidly as we recede from the

Earth's surface.

There is one apparent objection to this

analysis, that there are special narrow regions,

especially mountains and ocean deeps, where

the height of the rock surface above sea level

is anomalous for a region of the order of 100

km in width. This is the distance such that a

spherical harmonic of degree 200 or so keeps

the same sign, and in a harmonic representation

such regions would require many harmonics of

such orders to represent them, the terms adding

up in the anomalous regions but nearly canceling outside. Then the rapid decrease of the

coefficients might not start until TO=200 or so.

However, such regions, on account of their

limited area, cannot contribute much to

J\f(fdS. The essential point of the spherical

harmonic analysis is to consider a classification

of gravity anomalies into widespread ones,

keeping the same sign over distances comparable

with the radius of the Earth, and local ones that

205
206 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

change sign within much smaller distances.

The analysis can investigate the former. The

local anomalies will need special treatment,

but the best treatment must have a standard of

comparison, and the best standard would be a

function that best fits the Earth as a whole,

that is, one based on the lower harmonics.

If only we knew gravity everywhere, the

problem would be easy. If

?=ES(4 , cos n\+Bn, sin n\)P'n(sm<f>), (1)

then on account of the orthogonality of the

spherical harmonics

JJ <7 cos <£(cos ri\, sin 7iX)P£(&^==JJ (P"n)

2

(Ans cos2 n\, Bn> sin2 ri\) cos <fxl\, (2)

taken over the surface. Thus the determination of all An,, Bni would be reduced to simple

integration. For conveniently spaced values

of g the formulas could be replaced by numerical

summation. However, this does not work.

Gravity is observed only at isolated points and

must be interpolated before we have the data

for the integrals. In some attempts linear

interpolation has been used right across major

oceans, and this is certainly invalid. What we

have to do is to choose the coefficients to fit

the observed values as well as possible; any

interpolation will introduce errors.

The harmonic analysis of a continuous function, such as the separation of a periodic term

from random noise, is a difficult statistical problem, which has not been satisfactorily solved

even for one variable. The ordinary method

of least-squares depends entirely on the errors

of separate observations being independent,

and this is nearly true for errors of measurement. For gravity the errors of measurement,

judged by different observations at the same

place, are nowadays not more than 1 mgal;

even early in the 19th century most of them

were not much over 10 mgal. But differences

between stations within 1° or 2° of each other

are normally 20 mgal or so, and cannot be errors

of observation. In fact the observations depart from any smooth representation of gravity

by a short-range variation, which is mainly

genuine; and such a variation of gravity must

be a continuous function of position. But the

correlation between neighboring values of such

a function tends to + 1 when the separation

tends to 0, and the basis of the method of leastsquares breaks down.

Can we arrange the analysis so as to retain

the advantages of the method of least-squares?

For some separation the correlation must

become negative, and hence there is a separation

such that it is zero. Then there is no point in

using a lot of observations very close together—

they all say the same thing. If we can find a

distance such that the correlation is small and

can form summaries for observations covering

such a range of distance, there is some hope of

an approach to independence.

Analysis of gravity observations

I should say at this point that the external field

is completely determined if we know free-air

gravity defined by gf— g ( 1 -\— V where g is

gravity at height h above sea level (Jeffreys,

1959). A formula correct to orders h2 and eh

is available. All I have to say really concerns

free-air gravity, which is constant on a vertical

ascent; the correction 2g/a is 308 mgal/km.

Our problem is to estimate the low harmonics

in g}. The five harmonics Pu P\ (cos X, sin X)^

PI (cos X, sin X) are theoretically absent from

Some principles that can guide us are as follows. First, it is known that heavy grouping

of data can give a very accurate determination.

If we have observations of a periodic function

of x uniformly spaced from 0 to 2ir, then comparison of the means over the ranges — \v to

\TT, and f TT to fir, determines the coefficient of

cos x with a standard error only 1.1 times that

obtained by using the whole of the observations; and similarly for the coefficient of sin x.

Now a harmonic containing cos 3X keeps the

same sign over a range of 60° in longitude.

If we used only the central half of such ranges

we should get a good determination for cos 3X,

and the end halves would give sin 3X. Further,

whether higher terms are present or not, there

will be little correlation between the errors produced in the means. Thus least-squares would

be expected to give a good approximation if

the data are summarized for ranges of 30° in
THE ANALYSIS OF GRAVITY—JEFFREYS 207

latitude and longitude. The problem is to fill

in the gap between individual stations and 30°

squares.

In my analysis (Jeffreys, 1941, 1943), I saw

no advantage, and some disadvantages, in trying to use more than one observation in a 1°

square. I took the most northerly observation

in whatever list I was using. The 1° squares

were then combined into 10° squares. It was

found that the free-air anomaly (the departure

from the International Gravity Formula of

Cassinis, which was used as the standard) regularly showed an increase with height over land,

and a decrease with depth of the sea bottom

over sea. This is a consequence of any form

of compensation, but study of compensation

and its imperfections was not the object at this

stage. The important thing was that here was

a clearly systematic effect, and if the result for

a 10° square was to be representative of the

square it must be evaluated for the mean height

of the square. Thus to the data in a 10° square

I fitted a formula a-\-bh by least-squares and

reduced to mean height by evaluating a-\-bh,

where h is the mean height interpolated from

Prey's calculation. Similar methods were used

for sea squares and squares that were partly

land and partly sea (the values of b for sea

squares were generally less than for land

squares). The standard errors T0 of the estimates for the mean height were evaluated as

usual.

The next question was, how much would the

10° means be expected to differ from the 30°

means? We have no information other than

the gravity data themselves; and harmonics in

a range about w=18 would produce a fluctuation in the 10° means and largely cancel in the

30° means. I grouped the 10° squares into 30°

squares, so that a 30° square might have nine

10° squares in it. There was a scatter larger

than the apparent uncertainty of the 10° means

(treated as having independent errors) could

explain, and this implied a variation that could

keep the same sign for 10°, with an additional

uncertainty which I expressed by a standard

error TX of 20 mgal. This, it must be understood, is not the uncertainty of the 10° means

themselves, but of the probable departure of

the true means over 10° squares from a smooth

formula that would fit the 30° squares. It

was the latter that I was trying to find.

Combining the uncertainty TX with the crude

uncertainty T0 I derived means and standard

errors for the 30° squares. Again the actual

variation was greater than the uncertainties

taken into account so far would explain, and

I introduced a further variation with root mean

square T 2 = 1 2 mgal to take this into account;

T2 would include contributions from all harmonics of degree <6. Then I formed the

normal equations, neglecting all nondiagonal

terms, to see what harmonics might have coefficients substantially more than their standard errors. Apart from the constant and P2

terms, whose existence was not in doubt, three

showed up, namely Pi cos X, Pi cos 2X, and

Pi cos 2X. Normal equations were then formed

fully for coefficients of Po, P2, and these three

terms, and solved. As the T2 variation would

include these variations, their mean square was

subtracted from rf and the result was used to

estimate uncertainties. Residuals were formed

and reanalyzed, and a term in Pi sin 3X was

detected. (Two slight departures were made

from this treatment, but they need not delay

us here.)

If I were doing the work again I should make

two modifications. In the first approximation,

T2 included the variation I was trying to estimate, and this was removed in the second. A

more usual practice would have been to analyze

from the 10° squares directly and look for outstanding correlations between the residuals at

different separations, increasing the uncertainties correspondingly if any showed up. Also, the

analysis was carried out piecemeal, groups of

harmonics whose values were likely to influence

estimates of one another being taken together.

Now that electronic calculators are available,

it would be easy to do the direct analysis for all

possible harmonics to degree 3. I should not,

however, expect either change to affect the

results greatly.

Zhongolovitch's analysis

After my work an analysis resembling it in

many respects was made by I. D. Zhongolovitch

(1952). He determines systematic corrections

for some of the older observers very much as I
208 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

did, and also some corrections to base station

values for more recent ones. My 10° "squares"

were of course not square except near the

equator, the range of distance in longitude

being less than in latitude. So long as the

uncertainties were evaluated I do not think

that this could make much difference. Zhongolovitch used regions more nearly square.

Each extended over 10° of latitude, but the

numbers along each zone of latitude were

chosen so that the areas would be approximately

the same as for equatorial squares. Thus they

dropped from 36 in the equatorial zone to 3 in

each polar zone. The advantage of this is, I

think, more than compensated by the fact that

it is no longer possible to group nine 10° squares

into one 30° square and get an estimate of the

Ti variation at an early stage in the analysis.

He estimates b and b' as I did, but gives two

sets of solutions, one (first variant) using the

simple mean for the stations in each square, the

other (second variant) reducing to the mean

heights of the squares as I did. As the second

variant corrects for an obvious systematic

error, I think the first needed no consideration.

In my work I considered the possibility that

for forms a-\-bh, a

/Jrb'h' applied to the squares

that were partly land and partly sea, the values

of a, a' for the land and sea portions might be

different. I took all the 18 squares where a and

a' were well determined and found a'^>a in 9

cases, a'<C.a in 7, and a'=a in 2, and decided

that there was no systematic difference. Zhongolovitch, however, used 46 squares and finds

a mean a'—a=19 mgal and allows for it. The

respective numbers are a'^>a, 32 cases, and

a'<a , 14 cases. As his total number of squares

is not very much different from mine, I think

that the difference between 18 and 46 cases

must be due to his having retained squares

where I should not have considered a and a'

well determined.

My comparison was with the International

Gravity Formula, which can be written 979770

+3446.0 P2 (sin tf>) + 5.3 P4 (sin <f>). Zhongolovitch's is with Helmert's formula 978.030

(1+0.5302 sin2 <*>-0.000007 sin2 2^=979754.85

+3454.40 F2+6.26 P4.

However, his solutions take no account whatever of either T0 or rt

. The first, T0, will be

arge if there are few observations or if their

values vary irregularly within a square. But

he forms normal equations for the coefficients

of the harmonics using the data for the separate

squares as independent estimates, treating all

as of equal weight. This is clear from his

table 6; if any allowance had been made for

the uncertainty of the reduction to the mean

height of the region, the diagonal coefficients

would be less for the second variant than for

the first, but in fact they are the same. Thus

excessive weight has been given to squares

with scanty or irregular data. Standard errors

are derived from the scatter of the data, but

they may be too low on account of correlation

between adjacent errors due to the presence

of harmonics of degrees about 6; and as the

values for squares near the poles have especially

large uncertainties T0, if there is any systematic

error in them it will have biased some of the

estimates, especially those of the coefficients

of P2 and P3.

In table 7 Zhongolovitch (1952) gives 15

different solutions from his data for each

variant. Here Anm, Bnm are departures from

the coefficients in Helmert's formula. The

normal equations are for coefficients up to Au,

Buf with certain multipliers, but in the solutions some coefficients are either taken as zero

or given values taken as not subject to error.

I state the coefficients estimated.

Column 1. .Aoo, A20 only.

2. Aoo, A20 (.A40 taken equal to difference between Helmert's and the

International value).

3. Aoa, A2o, -"30-

4. .4oo, A20, Am (An treated as initially unknown).

5. Am, A20, AM, A4Q.

6 . AQQ, A2o, -"22> X>22-

7. Aoo, A20, An, Azi, A32, B33.

8. All to £33.

9. All to Bu.

10,11,12. A30, Aw re-estimated as for 3, 4,

5 with values of ^00 and A20 from

column 1.

13. All to B33; coefficients up to Bn

taken from column 6.

14. All to Bu', coefficients up to Bzs

taken from column 13.

It is column 14 that Zhongolovitch quotes

in a later paper (1957). It practically halves
THE ANALYSIS OF GRAVITY—JEFFREYS 209

A30 and Azi in comparison with column 9.

Column 15 is based on the hypothesis that the

same gravity formula holds in the unoccupied

squares as in the occupied ones. This is

reasonable, but is applied as follows. Values

for the unoccupied squares are calculated from

column 13; then the whole of the coefficients

up to Bgg are calculated by numerical integration over the whole surface. Naturally this

makes only slight changes in the coefficients up

to #33. No uncertainties are given, and in

fact the nearest approaches to estimates with

valid uncertainties axe given by columns 8 and

9. Column 15 fits the observed squares with

standard deviation 18 mgal, the unobserved

ones with standard deviation 5 mgal. This

of course is too good to be true. We really

have no idea how the contributions from

intermediate harmonics vary in the unobserved regions; even if a range of 180° in

longitude is available, it cannot adequately

separate a harmonic of degree n from one of

degree n + 1.

Itfappears that if harmonics above the second

degree are considered at all, only columns 8 and

9 are genuine least-squares solutions and all the

others must underestimate the uncertainties.

As indicated above, even columns 8 and 9 are

doubtful on account of correlation of errors and

of treatment of all squares as equally well determined. Nevertheless the comparison is of

some interest. Column 7 estimates the same

harmonics as I found, and should agree with

mine if the method is free from bias.

Comparison of the solutions is made easier

in terms of the mean squares of the contributions of the various harmonics to gravity. The

mean squares of Pnm (cos mX, sin mX) are as

follows:

n

0

1

2

3

4

m

0

0

1

0

1

2

0

1

2

3

0

1

2

3

4

Mean square

1

1/3

1/3

1/5

3/5

12/5

1/7

6/7

60/7

360/7

1/9

10/9

20

280

2240

From column 9, foreach'entry of the form a±a

we take a

2

, a

2

 times the mean square of P^

(cos mX, sin mX). An extra figure was kept in

the calculation. (See table 1.)

Ato is not included in the summation because

some departure from the international formula

in this coefficient is to be expected on any hypothesis concerning the earth's internal state.

Am

Atl

B3l

Aa

Ba

TABLE 1.

Estimate

+ 15. 42 ±6. 05

+8. 23±3. 14

+0.32 ±2. 35

+ 0. 23±0. 80

-1.93 ±0.68

+0. 57±0. 32

+0. 47±0. 31

+ 2.27 ±6.28

-4.96 ±2.46

-2.74 ±1.93

—0. 07 ±0. 43

+ 0.65 ±0.46

+ 0.21 ±0. 12

+ 0. 09 ±0. 12

+ 0.033 ±0.041

+0.061 ±0.042

Mean square

from estimate

34.0

58.0

0.9

0.4

31.9

16.7

11.4

153.3

27.3

8.3

0. 1

8.4

12.3

2.3

2.4

8. 3

Mean square

from uncertainty

5.2

8.5

4.7

5.5

3.9

5.2

4.8

38. 1

6.7

4. 1

3.7

4.2

4.0

4.0

3.8

3.9

69.5 34.6
210 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

An

A2i

Atl

Bs.

Au

B32

A33

B33

I/a

Column 8

+ 22. 32 ±1.79

-15. 80 ±4. 86

+ 3. 54±1. 30

+ 0.06 ±1.29

+ 12. 37 ±5. 24

+4. 25 ±2. 16

-0.96 ±1.95

+ 1. 08±0. 66

-1.32 ±0.60

+ 0.95 ±0.25

+ 0. 65±0. 25

296.0 ±0.6

TABLE 2.

Column 9

+ 22. 80±l. 83

-17. 53 ±5. 24

+ 5. 27±1. 52

+ 0. 46±1. 39

+ 15. 42 ±6. 05

+ 8. 23±3. 14

+0.32 ±2. 35

+ 0. 23±0. 80

- 1 . 93±0. 68

+ 0. 57±0. 32

+ 0. 47±0. 31

295. 8 ±0. 7

Column 7

22. 33 ±1.83

-11 . 87±4. 37

+4. 43 ±1.22

+4. 32 ±2. 24

+ 1.07±0. 68

+ 0. 62±0. 25

296.6 ±0.6

H.J. (1943)

+ 2. 47±1. 93

- 6 . 1 ±5.0

+4. 01 ±1.42

+ 4. 24±2. 43

+ 1. 30±0. 68

+ 0. 46±0. 26

296.2 ±0.7

Taking the mean-square contributions from the

uncertainties as estimates of the probable contributions to the mean squares of the estimates

from the outstanding variation, we have from

the third harmonics 5.4, and from the fourth 4.3.

The value for x

2

 would be 28.1 on 7 degrees of

freedom for the third harmonics, 16.4 on 8 degrees of freedom for the fourth. Both would be

highly significant if the estimates were independent, but this is doubtful, and nearly half

of the sum 69.5 for the fourth harmonics comes

from A4l, which is just over twice its standard

error. In my preliminary analysis Aa was

+ 1.4±2.0.

If we subtract the contributions from the

random errors, we are left with 115.2 mgal2

from the third harmonics and 34.9 mgal2 from

the fourth. From our rough preliminary consideration these are in about the ratio that we

should expect, although it is rather surprising

that so much of the contribution comes from

one harmonic. Also the decrease cannot continue at the rate suggested, since it would imply

rf+r|<l/ 8 of the contribution from n=3;

actually it is about ten times that contribution.

In my solution the mean-square contribution

from 7 third harmonics was 46 mgal2, of which

27 mgal2 was accounted for by random errors;

thus my terms are substantially less than

Zhongolovitch's.

On the face of it the third harmonic coefficients that can be determined without serious

doubt as to their signs are ^30, A3U Bw, and

possibly ^33. Those that I found were A3U A&,

and .B33, thus agreeing only for A31.

Comparing column 9 with column 8, in which

the fourth harmonics were not estimated, we

find the following (table 2). I also give my

solutions and Zhongolovitch's column 7.

Three of the changes exceed the standard

errors in column 9, and five exceed those in

column 8. The most striking is for A3i. The

changes are surprising, seeing that only one

coefficient of a fourth harmonic in column 9

exceeds twice its apparent standard error. The

difference between his Aw, -^20 and mine is

mostly due to the fact that he has compared his

data with Helmert's formula, whereas I used

the International formula. The estimates of a

(the eUipticity) are comparable. For the

harmonics that I found worth estimating the

agreement between my results and column 8 is

quite good. However, I found no indication of

a significant A30, which is striking in both

column 8 and column 9; and my A32 is in good

agreement with column 8 but not with column

9, and I failed to find B& or A&.

Comparison with results from artificial

satellites

Comparison with results from artificial satellites

shows further anomalies. The zonal harmonics

in the field are well determined because they

produce effects that accumulate secularly or

over long periods. If the potential is taken in

the form

=$ [l+± ± (£)' P- (Cnm cos s\

1 L n=»2 m-0 W

+Snm sin sX)

determinations for m=0, n=0 to 5 have been

found. Terms with ray^O give effects that

accumulate for much shorter times, and hence

are more difficult to evaluate. The coefficient

in g is nearly n— 1 times the corresponding one

in U. Then we have, chiefly from the work of

Kozai (1961b,c):
THE ANALYSIS OF GRAVITY—JEFFREYS 211

I/a

10«C22

106Cso

106C31

10«S31

10«C32

10«£32

1O°C3 3

108<S33

10«C4O

lO'Cw

298. 20 ±0.03

+ 0. 60±0. 28

- 2 . 24±0.28

+ 2. 29±0. 02

+ 2. 99±0. 36

+ 1. 18±0. 31

+ 0. 20±0. 20

+ 0. 36±0. 20

-1.71 ±0.29

+ 0. 84±0. 27

+ 2. 12 ±0.041

+ 0.232 ±0.021

^ 2 2

B22

AM

A3l

Bzi

Ai2

B32

A33

B33

Ai0

Am

+ 0.60 ±0.28

- 2 . 24 ±0.28

+4. 58±0. 04

+ 5. 98±0. 72

+ 2. 36±0. 62

+ 0. 40±0. 40

+ 0.72 ±0.40

- 3 . 42±0.58

+ 1. 68±0. 54

+ 6. 36±0. 12

+ 0.93 ±0.08

Comparing with the values from gravity

(allowing for the n—1 factor), we see that the

best established term from artificial satellites,

(72O, leads to a value of the ellipticity a differing

from the gravity value by about three times the

standard error. Again Cm would correspond to

A30= +4.6±0.04. As my value (1943) was

+ 1.6±5.0 (Table VI, p. 63), it is consistent;

but Zhongolovitch's values deviate by about

1.5 and 1.7 times their apparent standard

errors. With regard to these terms I am sure

that the artificial satellite values are right.

Helmert's formula, used by Zhongolovitch for

comparison, corresponds closely to the artificial

satellite value of a. The sum of his A20 and A^

is small, and P20 and P& keep the same sign

through most of the northern hemisphere. It

looks as if the separation of these harmonics

must depend mostly on the scanty observations

in latitudes south of —30°, and the difference

between his solution and mine for A30 is due to

his having given too much weight to them. I

am inclined to think that the error in my A20

is due to errors in the comparison of base

stations. As many of these have now been

improved, and there are now many more

observations in the North Pacific and in the

southern hemisphere, it should now be possible

to make a much better determination.

For m=0, A3i is reasonably consistent with

my value and with Zhongolovitch's column 8,

and A32f B33 might be consistent. But A22, B22,

-B31, B32, ^33 are in serious disagreement. Here

it seems to me that the values from gravity and

from satellites are both under suspicion. The

error in A2Q indicates at least that there is some

systematic error in gravity, which may have

affected the estimates of the other coefficients.

Kozai (1961c, p. 10), on the other hand, remarks that the true standard errors of his

values are much larger than those quoted; if I

understand him, nonorthogonality in the normal equations had not been allowed for.

Possibilities of improvement

I think that in the long run gravity should give

much better determinations for m^O than the

artificial satellites. The number of 10° squares

would be 542 if all were occupied; the polar

circles of radii 10° are each treated as one square.

A full gravity survey would not eliminate rlt

which will contribute a mean square departure

in the neighborhood of 400 mgal2. The error of

representation, the departure of the true mean

over the square from the estimated mean,

would be about the same, but would be divided

by about 3 if we had one observation for

every 5° square. Thus, if we had an observation in every 5° square, the mean square contribution to any harmonic arising from the

uncertainties could be brought down to about

1 mgal2. This is to be compared with about

5 mgal2 from the present gravity data. Kozai's

results as they stand give less than 1 mgal2 up

to B3i, 1.4 mgal2 for A32 and £32, 15 mgal2 for

^33 and .B33. He also gives values for coefficients up to Biif some of which have greater

uncertainties still. Since his uncertainties are

said to be probably too low, it seems that even

if artificial satellites may give the better values

^or m= l and possibly m=2, gravity survey will

still be needed for higher m.

It may appear that Zhongolovitch's work

would have used many more observations than

mine, but this does not seem to be the case.

He has 410 squares altogether, of which 204

contain observations. I had 542, of which 217

had observations. For the 30° squares I had

49 occupied out of a possible 62, but if we take

only those where the standard error is <15

mgal the number sinks to 33, which do not

include any centered on latitudes +90°, —60°,

and —90°. Zhongolovitch also comments on the

extreme asymmetry of the distribution with

regard to the equator.

Acknowledgments

This work was done while I was acting as a

consultant at the Smithsonian Astrophysical

Observatory. The translation of the relevant

parts of Zhongolovitch's work was done by my

wife, to whom my thanks are due.
2 1 2 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TML. 6

Abstract

Methods of analysis of gravity to estimate the low harmonics are discussed. The work of Zhongolovitch is

described in some detail, as it does not appear to have been translated and is the most extensive analysis made so far.

There are serious inconsistencies between his results, mine, and those derived from artificial satellites, and suggestions

are made for future work in the hope that these may be reconciled.
The Stabilization of an Artificial Satellite

at the Inferior Conjunction Point

of the Earth-Moon System

By G. Colombo

It is obviously important to be able to keep an artificial satellite stable at the inferior conjunction

point (Li) of the earth-moon system, while using the minimum amount of momentum. The fact

that the inferior conjunction point is an exact solution of the four-body problem (earth-moon-sunsatellite) (Colombo, 1960; Klemperer and Benedikt, 1958, p. 25), if we neglect only high-order

perturbations, is an advantage of this solution that makes it preferable in some ways to the triangular

Lagrangian solution (Z,4) of the restricted three-body problem. On the other hand, the larger

degree of instability of position Lx compared to that of Lt is a disadvantage. It is, however, necessary

to note that the perturbing action of the sun on Z-4 is of the same order of magnitude as that of the

moon.

In the discussion that follows, we shall first compare the order of approximation of the solutions

Zj and L4 to indicate their characteristics. Second, we shall study some devices to stabilize a satellite

in a region around Li without using momentum. The devices considered are a solar sail of relatively

small dimensions, and a mechanical autonomous device. It is also possible to obtain this stabilization with a jet system that uses a minimum momentum.

Each stabilization system needs a built-in position-detection device of high precision, which

may become possible in the next few years. But the final goal of this paper is to indicate the

smallness of the acceleration needed to keep a satellite in a region of 50 km around Lly and to examine

the possible stabilization systems, assuming that the position-sensors could detect displacements

of the order of 2 km. This discussion, even though theoretical at present, may be useful in the future

for designing a device to stabilize a satellite at the inferior conjunction point.

1. Collinear solution

The symbols used are defined as follows:

E, center of the earth;

M, center of the moon;

S, center of the sun;

T*, inertial reference system;

TE, reference system centered at E and oriented as T*;

a(#), acceleration of a moving point Q with respect to rg

A(Q), acceleration of a moving point Q with respect to r*

As(<?), acceleration due to the action of all the perturbing bodies S, V, M, J

A-B(Q), acceleration due to the action of the earth

Aj/(Q), acceleration due to the action of the moon

mi, mass of the earth

m2, mass of the moon

m3, mass of the sun

213
2 1 4 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

F, force on E due to M

\M, velocity of the moon with respect to TS

The fundamental hypothesis may be stated as follows: AS{Q) is a linear function of Q, at any

time t, in a spherical region around E. With this hypothesis we shall neglect small terms, which

we will evaluate at the end of this section. (We shall also neglect the perturbations due to the oblateness of the earth and to the possible oblateness of the moon.)

Let us define P with the following equation:

EP=PEM. (1)

Thus, for the above hypothesis we can write

As(P) = (l-P)As(E)+pAs(M). (2)

Next, we write the equations of motion of M and P with respect to TE, and the equation of motion

of E with respect tor*:

> = -A*(£)+A*(P)+AM(P)+A5(P), 1

(3)

Consequently,

miTM2

 \ (4)

z(P)=~+As(P)+AM(P)+As(P)-As(E).

Finally,

The accelerations A*(P) and AM(P) and the force F are parallel; therefore, we can solve the equation

in p:

A

/ D \ I A fT>\ ^2( 1 P) PW] -p ,~N

B \*J I "-At \±) = = * • \O)

If we take, for example, the solution pu satisfying the conditions 0<p1<l, we will have

a(P)=P la(M). (7)

This solution corresponds to the inferior conjunction point Lu If we put a satellite at point Li

with an initial velocity v(P) equal to PiVjir, the satellite will always satisfy the equation

EP=P lEM. (8)

The motion of P will be homotetic to the motion of M with respect to E.

To obtain a rough estimate of the degree of approximation, let us consider the most unfavorable

condition for the perturbation. Assume that the moon and sun are in conjunction and, for simplicity,

neglect the smaller action of the other perturbing bodies. We have for any Q of EM, under these

conditions, the equation

AJG)—AJE) C2\EQ\

\ES\* \ \ES\
STABILIZATION OF AN ARTIFICIAL SATELLITE COLOMBO 215

We showed that the linear term does not cause relative perturbations, which can never come from

the total quadratic term, but only from the difference,

1.5X10-8 cm/sec2. (10)

This perturbation is of the same order of magnitude as the solar pressure for a body with an areato-mass ratio of 10~4

 cm2/gram.

2. Triangular position

The difficulty of defining exactly a triangular solution for the real earth-moon system is well known.

We choose to define the point L4 with the equation

(11)

where

n=|EMXvI

I think this is the most natural definition of the libration point Li, but it is not an exact solution of the

real earth-moon-sun-satellite problem, even if we neglect the parallactic term as we did in the previous

section.

We shall now evaluate the acceleration needed to keep a satellite at Lt, taking into account only

the sun's action. Let us put, as in the fundamental hypothesis given above,

As(P)=As(S)-aEP+3a ^|^ ? ES, (13)

where

From equation (3), taking into account equation (13), we obtain the following equations of motion for

P and M:

a(P)=A,(P) + AM(P) - AM(E) -«EP+3a

F«5 FM

Ag(M)-AM(g)-aEM+3« j^j 2 ES.

Now, if we define the operator,

e

<9a=a cos 0+nXa sin 6, (16)

we will have for Z4:

EL«=e"/3EM. /

The acceleration needed to keep P at L4 is:

{(ES.EM)nXES-(ES.nXEM)ES}+:f ^XT^+^^XEM . (18)

The intensity of the acceleration (18) is of the order of 10~3 cm/sec2. Note that in this position the

acceleration due to the action of the moon is of the same order of magnitude, that is to say, 3X10~a

cm/sec2. Therefore, the question is: since the triangular Lagrangian solution is a solution of a

schematic model of the real problem, made by neglecting a force of the same order of magnitude as

636-014—63 15
216 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

the force that we do consider, what information will we gain from this model about the real motion of

a satellite in a region around the point i4?

The perturbing acceleration (18) may be considered an almost-periodic perturbation. We

denote by nm and n, the mean motions of moon and sun respectively. The quasi-period of A(<) is

—r- This acceleration may cause a forced oscillation, but it is necessary to observe that if we

w)

take into account all the terms of the same order of magnitude, we have to work with a nonlinear,

nonautonomous system. Moreover, this system seems to be very close to a harmonic or subharmonic resonance condition. In fact, it is very easy to write the vectorial equation for the

motion of P in a small region around L^ We put EP=EL4+L4P=EIi4+x, and we obtain as a first

approximation:

TM < mE\L-6 \ELi\

2V~mu]'X

x]ML,

[(EL4.ES)EL4-(EM. ES)EM]}+A<«>=0. (18')

A preliminary crude analysis of equation (18) makes the opinion stated above almost certain. We

will not waste time by indicating the difficulties of the problem of the secular stability of the motion

around Z4. We will only refer the reader to the work of Brown and Shook (1933) on the Trojan

group, and observe that the case we are studying is more complicated than the Trojan case for the

following reason: the earth-to-moon mass ratio is much smaller, and the effects of the perturbing

acceleration due to the sun are very much greater than are those due to Saturn in the case of the

Trojan group.

If the problem were only to study the short-period stability, it would be enough to perform a

numerical integration. I think, however, it is necessary to begin with a solution defined precisely as

we have defined it. Starting as we suggested above, we can evaluate carefully all the forces of the

same order of magnitude acting on the satellite.

3. Inequalities for the acceleration in a region around L}

To study the possibility of stabilizing a satellite in position L1} we must first study carefully the

gravitational acceleration of a point P in a region around Lx. Taking into account only the action of

the sun and neglecting the other celestial bodies, we have from equation (3):

a(P)=A 5 ( P ) - 5 ()+*( ) + *( )

=-«EP+3 a ^ +^H^ )

We know that

EP*=EL1=PlEM (20)

is an exact solution of equation (19). Now we put

EP=EP*+P*P (21)

and we obtain the equation

P*P _/dAg(P)\ (*kM(P)\

dt* V 5P /P.

 r+ \ dP A
STABILIZATION OF AN ARTIFICIAL SATELLITE COLOMBO 217

denoting by O2(P*P) terms of two orders of magnitude less than the other terms, in a spherical

region R around Lx of radius 150 km. We will write equation (22) in the following form:

dt2

\EM\*

OL2 =

htn2

\MP

(23)

Now we consider the reference system T, centered at P* with the Z-axis normal to the ecliptic,

and the X-axis directed toward the point T • We define a fictitious moon M* and a fictitious earth

E*, in the following manner: M* and E* move in circular orbits centered at P* in the XF-plane

with radii (1 — px)a and pxa, where a is the semimajor axis of the real moon. The true longitude of

M* is the mean longitude of the real moon, and E* is always in the opposite direction to M* with

respect to P*. We note that it is possible to choose the initial position of M* in such a way that

the angle between the directions P*M and P*M* is never greater than 8°. Now we write equation

(23) in the form:

E*M*+Aa+O2(P*P),

\M*P*

(24)

We will evaluate Aa when P moves in the rotating orbit we shall define soon. We take a reference

system T(r)(x, y, z) centered at P*, the 2-axis coincident with the Z-axis, and the z-axis directed

always toward M*. We suppose P moves in an ellipse e with respect to T(r), with principal diameters

in the x and y directions and semidiameters p and q, respectively. The components of the explicit

part of the acceleration at the right-hand side of equation (24) are

2(a*l+a%)x, - (25)

If we take an auxiliary reference system T[T) centered at P*, £-axis directed toward M, f-axis normal

to the osculating plane of the moon's orbit, the components of the explicit part of the acceleration,

we have at the right-hand side of equation (23) are

Now we put

and we obtain

|Aa| = (a*+«2*) {4(\£-

(26)

(27)

(28)

Let us denote by i*>0 the maximum of the angles z£, yrj, z{. On the arc of the ellipse e where

z>0, y>0, we have

0< £< x cos i*+y sin i*,

0<! 17^ x sin i*-\-y cos i*,

0< f < (xt+y2)

1

'

2

 sin i*.

(29)
218 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Let X* be the maximum value of the function \(t) we defined in equation (27). If we put

x=p cos 0,2/=2 sin 0, 05s 0<TT/2, we will find:

|Aa|< (c$+c$) {cos2 0[3p2(X* cos i*-l) 2+ p 2(X*+l) 2-2XV cos i*]+sin2 6[Zp2 sin2 i*

+g2(X*2+l)—2XY cos i*]+2 sin 0 cos 0[3pg(X* cos i*—1) sin i*-2\*pq sin i*]}1/2. (30)

Or if we take into account that i* is less than 8° and 0°< 0< 90°, X*<3, we obtain

|Aa| <(<*?+<*£) {p2 cos2 0[4(X*-l)2+sin2 i*]+g2 sin2 0[(3+X*) sin2 i*+(X*-l)2]}1/2. (31)

Let us put g_=8p; then we have finally,

|Aa|<8p(a?+a2*){(X*-l)2+(3+X*) sin2i*}1/2. (32)

4. A forced orbit around Lx

Now let us consider a particle P moving with the fictitious acceleration with respect to T,

E*M*+S

*F*

\S*E*\' (33)

where SJ>0 is constant, and S* is the mean sun. Let us choose the unit value of time in such a way

that the mean motion of the moon is 1 and the unit value of length is a. We write the equation of

motion of P with respect to r(r), and we find:

-|flf| cos (1 -

sin <l -

d?z_ .

dt2~ *'

(34)

where n is the mean motion of the sun (ri^l/13 for the unit value we chose) and

(35)

We note also that for the unit value we chose A^Q. This system has a periodic solution in the form

x=p cos (1—n)t,

y=q sin (1— n)t, - (36)

2=0.

For our purpose we need only a crude evaluation of the solution. We have approximately

P=-ftS> q^—S. (37)

If we take p^20 km we need a constant acceleration |#|=4.2X10~4 cm/sec.2

5. Stabilization on the forced orbit

Now we go back to equation (23) and project it onto the reference system 7(r), taking the same unit

value of time and length we used in the preceding section. We will have

x—2y=[l+2A+all(t)]x+aa(t)y+au(!t)z-\S\ cos (l—rfit+X-nit)

y+2x=[\-A+a22{t))y+a2l{t)x+a23(t)z+\S\ sin (\-n)t+Y-^{t), )- (34')

s=[—A+an(t) ]z+a3, (t) x+Ov (t) y+Z—^t).
STABILIZATION OF AN ARTIFICIAL SATELLITE—COLOMBO 219

We added three distinct accelerations to the right-hand side. The first is the constant acceleration

S we introduced in the previous section. The second is the result of a stabilizing control device; we

suppose that this control device supplies the following acceleration:

X= — (1+2A+8f)[x-p cos (1—n)t]—8\[x+(l—ri)p sin (l—n)t],*\

Y=-fi[y-(l-n) q cos (l-n)t], (38)

The coefficients are related to the intensity of the answer and the sensitivity of the instrumentation

for detecting deviations from the prescribed motions. The third acceleration (<pi, <&, &) is the acceleration we need to ensure that equation (34') still has the exact solution of equation (36). For this

purpose the component must be

<pi=an(t)x+a{2(t)y+ai3(t)z+O^(P*P), (39)

where 02{)(P*P) are the components of the accelerations that are two orders of magnitude smaller

than those we put in 02(P*P) in equation (23). We have

{ S <p\t)2}

 m< |Aa|+O2(P*P). (40)

If we take into account equation (32) and we observe that in our case i*<8°, \*<l+3 e (e is the

eccentricity of the moon's orbit), we will have

\Aa\<0.5\S\, (41)

when the point P moves in the proposed orbit around Lt in equation (36). No doubt, if we are able to

achieve the accelerations in equations (34) and (35) to be added to the constant acceleration S, the

satellite will be stable in the orbit in equation (36). The degree of stability is related to the coefficients 8i, S2, ft, 7X entering into the components X, Y, Z. We can use as much as 0.2 \S\ for the

purpose of stabilization; this means (if p=20 km) 1X1O~* cm/sec2. If we want to have a damping

factor of 1 or more, we need a device capable of detecting a variation of 2 km/hr in the velocity—

that is, a high-sensitivity device. In any event, we need to be able to detect a variation of 2 km in position to have a sufficiently good stabilization. Figure 1 gives a diagram of the acceleration needed to

keep the satellite stable in the orbit in equation (36). The needed acceleration vectors have their

origin at the point 0; their ends are in the shaded region. The maximum angle between the direction

mean sun-satellite and the required orientation is less than arc sin 7/10 or 45°. Therefore, it would

be possible to obtain the acceleration we need from the solar pressure over a system of sails or flat

orientable balloons that are linked to each other and to the satellite in such a way that the configuration can be altered from within the satellite.

To obtain a crude evaluation of the maximum extent of the surface needed to provide the

required external force we shall proceed as follows. We suppose the satellite to be at the geometric

center of a plane, orientable, reflecting surface. The maximum extent of the surface will be computed in such a way that the corresponding acceleration of the complex satellite-sail, due to the

radiation pressure, is 1.8- \S\, when the normal N to the plane is parallel to the direction SP. If the

orientation of the sail is changed so that N makes an angle #( < | ) with the direction SP, the

acceleration will have the direction of N and its intensity will be 1.8 • |S| • cos2 0. Thus diminishing

the extent of the surface and varying the orientation, we will be able to supply to the complex all

the acceleration vectors with the ends inside the region bounded by the closed curve OQRQ'O and

therefore all the acceleration vectors with the ends inside the shadowed region. We can now

evaluate the ratio Ajm from the equation (Garwin, 1958: Sohn, 1959) 1.8|S|=0.8X10~M./TO in the

c.g.s. system. If we assume, as before, that p=20 km we will find ^l/m=9.5. This means that for
220 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

-r

FIGURE 1.—Diagram of the accelerations.

a 200-kg satellite we need a maximum reflecting area of 19X105 cm2 or 190 m2. I think we are not

very far from the actual possibility of achieving an automatic control device (including directional

and positional sensors, computer, and servomechanism) that can arrange at any time the required

configuration of the system of sails or flat balloons; the solar pressure over the complex will provide

both the external force and the external torque needed for the adjustment of both the position and

the direction of the complex.

We shall not discuss the technical questions involved, since this is not the purpose of the present

study. Nor shall we discuss in detail the possibility of providing the same stabilization with very

small rockets. The preceding sections make it clear that it would be possible to program the action

of such rockets so that they could act from time to time for a short period, when the control device

detects some variation in position and velocity, in such a way that the amount of consumed momentum is a minimum. However, to make this computation we need first a careful analysis of the

gravitational motion of a nonsailed satellite around the libration point and, second, an exact

evaluation of the sensitivity of the instrumentation.

6. Autonomous stabilization of a system of two points

Since we are considering only the theoretical possibilities, at least for the present, we should like to

discuss one other method for stabilizing a satellite at the inferior conjunction point without consuming momentum. For simplicity, we shall deal with the classical restricted three-body problem;

and to stay close to the problem of the preceding sections we shall use the same reference systems

and the same unit values for the dimensions as in section 4. Let us consider the motion of two

points P^Tt, yu 2,) and P2(x3, y2, z2) of equal mass, and suppose that P, and P2 are constrained by

the following holonomic constraint:

PH[6() 2 (42)
STABILIZATION OF AN ARTIFICIAL SATELLITE COLOMBO 221

where (xi+x2) is the x coordinate of the center of mass Q of the two bodies. To build this constraint

we need only a position sensor and an internal device to change the distances between the points as

necessary. We can change the exterior force that acts on the system by changing only (with the

interior force) the distance PiP2. We will need only energy without loss of mass for the system.

Now we write the Lagrangian equations of motion of the system:

*i-2&=(l+2^)z1+X0r1-:r2)-4X&(z1+z2) {

*i-2&= (1 +2A)x1+\(r2-x1) -

x2= (1—A)y2+\(y2—

zl-\-Ail=\{zi—z2)

(43)

Let us consider the static solution,

3/1=2/2=0,

x

A

(44)

and study the linear stability of the solution. We put

a

»2=— o

(45)

x=x*+x,, J

and we will obtain from equations (42) and (43) the following system:

(yi—th)t

(y2-yi),

(Zl-z2),

(46)
222 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.6

It is very easy to see that if

4a6>l, (47)

we have linear stability.

It would be possible to study some nonholonomic constraint of the form IP1P2I =/(Pi, P2, P\, P2, t),

which can give a more realistic and reliable stabilization device, by taking into account the actual

possibility of detecting deviations in position and velocity as well as the time lag between the detection of the deviations and the suitable variation of the distance computed and accomplished by the

interior mechanisms. To gain an idea of the reliability of the stabilization complex, we will suppose

a=10 km. This means a= 10/380000 and, from equation (47), we need 6>9500. If we choose

6=19000, for a displacement of G in the ^-direction of 1 km we need to vary the distance PiP2 from

10 km to 9.5 km. Obviously, we need a highly sensitive control complex. It is interesting to note

that it is possible, in principle, to change the exterior gravitational force acting on the complex by

using only an interior force and information from outside.
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224 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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226 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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1958. Selenoid satellites. Astron. Acta, vol. 4, pp. 25-30.

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1959a. The earth's gravitational potential derived from the motion of

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1961a. Effects of solar radiation pressure on the motion of an artificial

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1961. Atmospheric properties up to 500 km concluded from circular

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1944. Mathematical theory of optics. Brown University, Providence.

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1914. An introduction to celestial mechanics, 2nd ed. Macmillan Co.,

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MUSEN, P.

1959. Application of Hansen's theory to the motion of an artificial

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1960. The influence of the solar radiation pressure on the motion of an

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1960. Perturbations in perigee height of Vanguard I. Science, vol. 131,

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1961. Directional dependence of counting rates from Explorer IV.

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1954. Dynamic effects in the high atmosphere. In G. P. Kuiper, ed.,

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228 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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1959. New method of solution for unretarded satellite orbits. Journ.

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1952. The external gravitational field of the earth and the fundamental

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Special Reports of the

Smithsonian Astrophysical Observatory

Numbers 1 through 113

No. Date Title Author

1 October 14, 1957 (re- *Preliininary Orbit Information for G. F. Schilling and T. E.

issued December 2, USSR Satellites Alpha One and Sterne

Alpha Two

•Additional Orbit Information for J. S. Rinehart and G. F.

USSR Satellites 1957 Alpha One Schilling

and Beta One

*Some Preliminary Values of Upper T. E. Sterne and G. F.

Atmosphere Density from Observa- Schilling

tions of USSR Satellites

•Glossary of Astronomical Terms for J. Ashbrook, G. F.

the Description of Satellite Orbits Schilling, and T. E.

Sterne

•Soviet Orbit Predictions and Orbital G. F. Schilling and E. S.

Information for USSR Satellites Fergusson

1957 Alpha One, Alpha Two and

Beta

•Visual Observations of Alpha One L. Campbell, Jr. and

Made by Moonwatch Stations Dur- J. A. Hynek

ing Lifetime of the Object

*An Interim Model Atmosphere Fitted T. E. Sterne, B. M.

to Preliminary Densities Inferred Folkart, and G. F.

from USSR Satellites Schilling

•Soviet Orbit Information for USSR G. F. Schilling

Satellites 1957 Alpha Two and Beta

One

•Basic Orbital Data for Satellite 1957 L. G. Jacchia

Beta One

•Processed Observational Data for R. M. Adams, N.

USSR Satellites 1957 Alpha and McCumber, and M.

1957 Beta Brinkman

Status Reports on Optical Observa- G. F. Schilling, ed.

tions of Satellites 1958 Alpha and

1958 Beta

Preliminary Results from Optical J. A. Hynek and

Tracking of the U.S. Earth Satel- F. L. Whipple

lites

1957)

2 November 5, 1957

3 November 15, 1957

4 November 30, 1957

5 December 4, 1957

6 December 17, 1957

7 December 31, 1957

8 January 31, 1958

9 February 21, 1958

10 March 1, 1958

11 March 31, 1958

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 2, no. 10, 1958.

636-014—63 16 231
232 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

No.

11 (cont'd.)

Date

12 April 30, 1958

issued August

1959)

Title

Optical Satellite Observations

The Network of Precision Photographic Satellite Tracking Stations

Moonwatch Observations of Satellites 1958 Alpha, 1958 Beta and

1958 Gamma

Scientific Results

The Orbit and Variable Acceleration of Satellite 1958 Alpha

The Density of the Upper Atmosphere

Life Expectancy of Satellite 1958

Alpha

Use and Distribution oj Satellite Predictions

Program for Determination of

Geographic Sub-Satellite Points

Predictions for Crossings of Given

Latitude Parallels—APO Ephemeris 5

Predictions for Photographic

Satellite Tracking Stations—

APO Ephemeris 4

Program of Spot Predictions for

Specifying Observing Sites—

APO Ephemeris 3

Charts of Predicted Satellite

Positions

Harvard Announcement Cards

(re- Miscellaneous Information on the

20, Artificial Earth Satellites

The U.S.S.R. Satellites

Soviet Orbit Information for

Satellite 1957 Beta

Moonwatch Observations of the

Fall of Satellite 1957 Beta One

A Precision Measurement of the

Brightness of Satellite 1957

Alpha One

Note on the Mass-Area Ratios of

the U.S.S.R. Satellites

The U.S. Satellites

Moonwatch Catalogue

Author

K. G. Henize

L. Campbell, Jr.

C. A. Whitney

T. E. Sterne

L. G. Jacchia

R. M. Adams

L. G. Jacchia

J. Gaustad

C. H. Moore and

D. A. Lautman

R. Briggs

J. B. Fairman and

G. Veis

J. A. Hynek and G. F.

Schilling, eds.

J. B. Clarke and D. L.

O'Hara

E. P. Bullis and L.

Campbell, Jr.

G. S. Hawkins

G. F. Schilling and J. S.

Rinehart

E. P. Bullis and L.

Campbell, Jr.
SPECIAL REPORT LISTING 233

No.

12

Date Title

(cont'd.)

Author

C. A. Whitney

13 May 21, 1958 (reissued December 1,

1959)

14 July 15, 1958

15 July 20, 1958 (reissued March 28,

1960)

16 July 25, 1958

17 September 5, 1958

(out of print)

18 October 4, 1958

The Acceleration of Satellites 1958

Alpha and Gamma

The Secular Perturbations and L. G. Jacchia

Orbital Acceleration of Satellite 1958 Beta Two

Improvements in the Prediction R. M. Adams

Program for Crossings of Given

Latitude Parallels

Satellite Characteristics and Scientific Results

Densities of the Upper Atmosphere G. F. Schilling and

Derived from Satellite Obser- T. E. Sterne

vations

Technical Parameters of the Arti- G. F. Schilling

ficial Satellites

•Orbital Results for Satellite 1957 Beta L. G. Jacchia

One

Reports and Analyses of Satellite Observations

Moonwatch Catalogue—May through

June 1958

Status of the Photographic Satellite

Tracking System

Timing Satellite Observations

Preliminary Note on the Mass-Area

Ratios of Satellites 1958 51 and

1958 52

*The Descent of Satellite 1957 Beta One

Positions of Satellite 1957 Beta One

During the First 100 Revolutions

Positions of Satellite 1958 Alpha During the First 1400 Revolutions

Satellite Data and Analyses

Technical Parameters of Satellites

1958 Delta and 1958 Epsilon

Communications Center of the Optical Satellite Tracking Program

•Orbital Acceleration of Satellite 1958

Beta Two

Atmospheric Densities from Explorer

IV

Moonwatch Catalogue—July and

August, 1958

G. F. Schilling, ed.

E. P. Bullis and

L. Campbell, Jr.

K. G. Henize

R. J. Davis

G. F. Schilling, C. A.

Whitney, and B. M.

Folkart

L. G. Jacchia

R. M. Adams, R. E.

Briggs, and E. K. L.

Upton

R. G. Teske

G. F. Schilling, ed.

J. B. Clarke

C. M. Peterson

L. G. Jacchia and

R. E. Briggs

G. F. Schilling and

C. A. Whitney

E. P. Bullis and

L. Campbell, Jr.

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 6, 1963.
234 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

No. Date

19 December 6, 1958

20 January 5, 1959

21 February 27, 1959

22 March 20, 1959 (reissued March 3,

1961)

23 March 30, 1959

24 April 9, 1959

25 April 20, 1959 (reissued May 5,

1961)

26 May 21, 1959

27 June 30, 1959

Title

*The Earth's Gravitational Potential as

Derived from Satellites 1957 Beta

One and 1958 Beta Two

A Satellite Meteor-Trap

*A Flashing Satellite for Geodetic

Studies

Operational Information

*An Empirical Formula for Satellite

Ephemerides Near the End of Their

Lifetime

•The Diurnal Effect in the Orbital Acceleration of Satellite 1957 Beta One

Progress Report on the Planning of an

Artificial Satellite Containing an

Astronomical Telescope

A Suggested Rocket Experiment for

Determination of Atmospheric Densities and Winds at Extreme Heights

Operational Information

Moonwatch Catalogue—September

1958

*The Structure of the High Atmosphere—I. Linear Models

Moonwatch Catalogue—October, November and December, 1958

Operational Information

*The Earth's Gravitational Potential

Derived from the Motion of Satellite

1958 Beta Two

*On the Effects of the Sun and the Moon

upon the Motion of a Close-Earth

Satellite

The Orbit of Satellite 1958 Zeta

Catalogue of Satellite Observations for

January and February 1959

*The Structure of the High Atmosphere

II. A Conduction Model

Author

L. G. Jacchia

G. S. Hawkins

C. A. Whitney

and G. Veis

L. G. Jacchia

L. G. Jacchia

R. J. Davis

R. E. McCrosky

E. P. Bullis and L.

Campbell, Jr.

C. A. Whitney

Y. Kozai

Y. Kozai

G. Veis

R. G. Albert

and R. M. Adams

C. A. Whitney

Catalogue of Satellite Observations for R. G. Albert

March and April 1959

An Iterative Method of Orbit Determination from Three Observations of

a Nearby Satellite

Determination of the Orbit of Satellite

1958 Beta One

and R. M. Adams

R. E. Briggs and J. W.

Slowey

A. S. Leonard

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 6, 1963.
SPECIAL REPORT LISTING 235

No. Date

27 (cont'd.)

28 September 16,1959

28A September 23, 1959

29 September 21, 1959

30 November 12, 1959

Title Author

31 January 18, 1960

32 January 29, 1960

33 February 1, 1960

34

(C-8)

35

(C-9)

36

(010)

37

(C-ll)

38

39

40R

Cumulative Table of Contents, Special

Report Numbers 11 through 27

Catalogue of Satellite Observations for R. G. Albert

May and June 1959

Catalogue of Satellite Observations:

Addenda and Corrigenda, Station

Coordinate Lists

*Solar Effects on the Acceleration of L. G. Jacchia

Artificial Satellites

*Anticipated Orbital Perturbations of

Satellite 1959 Delta Two

February 2, 1960

February 5, 1960

February 8, 1960

February 10, 1960

January 15, 1960

March 30, 1960 (reissued January 30,

1961)

May 24, 1960 (revised

June 30, 1960)

A Table of the Times of Perigee Passage for Satellite 1959 Beta Two

•Note on the Secular Motions of the

Node and Perigee of an Artificial

Satellite

Catalogue of Satellite Observations for

July and August 1959

Osculating Elements

Catalogue of Satellite Observations

*On the Effects of Image Motion on the

Accuracy of Measurement of a Flashing Satellite

Catalogue of Observations of Satellite

1958 5 2 for the Period August 1-22,

1959

Catalogue of Observations of Satellite

1958 5 2 for the Period August 23-31,

1959

Catalogue of Additional Observations

of Satellite 1958 52 for the Period

Mayl-May29,1959

Catalogue of Observations of Satellites

1958 52,1958 Epsilon, and 1958 Zeta

Orbit Determination from Simultaneous Doppler-Shift Measurements

A Variable Atmospheric-Density Model

from Satellite Accelerations

Orbital Elements for July and August,

1959

Satellite 1958 Alpha

Satellites 1958 £1 and /S2

Satellite 1958 52

Satellites 1959 al and a2

Satellite 1958 Gamma

Relative Positions of the Sun and Perigee of an Artificial Earth Satellite

Y. Kozai and

C. A. Whitney

R. E. Briggs

Y. Kozai

R. G. Albert

Y. Kozai

R. G. Albert

J. A. Hynek

R. G. Albert

R. G. Albert

R. G. Albert

R. G. Albert

I. G. Izsak

L. G. Jacchia

B. Miller

C. A. Martin

Y. Kozai

R. C. Nigam

I. G. Izsak

P. E. Zadunaisky

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 6, 1963.
236 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

No.

41

Date

May 25, 1960

(out of print)

42

(0-12)

43

(C-13)

44

(0-14)

45

46

May

May

May

July

July

25,

25,

25,

11,

11,

1960

1960

1960

1960

1960

47

(C-15)

48

(C-16)

49

(C-17)

50

51

September 9, 1960

September 9, 1960

September 9, 1960

October 3, 1960

October 17, 1960"

52 November 21, 1960

53 December 5, 1960

Title

The Catalogue of Precise Satellite

Positions

Preliminary Time Reduction for the

Determination of Precise Satellite

Positions

The Star Chart Project

Explanation of Codes Used in the

Catalogue

Shutter Correction in Time for the

Baker-Nunn Camera

Catalogue of Satellite Observations

Catalogue of Satellite Observations

Catalogue of Satellite Observations

List of Coordinates of Stations Engaged in the Observation of Artificial Satellites

*The Effect of a Variable Scale Height

on Determinations of Atmospheric

Density from Satellite Accelerations

Comment on the Paper Entitled

"Symmetry of the Earth's Figure"

Catalogue of Satellite Observations

Catalogue of Satellite Observations

Catalogue of Satellite Observations

Author

K. Lassovszky

E. Weston

P. A. Pardue

E. P. Bullis

P. E. Zadunaisky

D. V. Mechau

D. V. Mechau

D. V. Mechau

D. V. Mechau

L. G. Jacchia

C. A. Whitney

D. V. Mechau

D. V. Mechau

D. V. Mechau

The Orbit of Satellite 1958 Alpha P. E. Zadunaisky

(Explorer I) During the First 10,500

Revolutions

Satellite Orbital Data D. V. Mechau

Satellites 1958 01 and 1958 02 B. Miller

Satellites 1958 52 and 1959 il Y. Kozai

On the Orbital Elements of Satellite Y. Kozai

1959 d

*A Theory of Satellite Motion About an I. G. Izsak

Oblate Planet: I. A Second-Order

Solution of Vinti's Dynamical Problem

The Orbits and the Accelerations of R. C. Nigam

Satellites 1959 al and 1959 o2

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 6, 1963.
SPECIAL REPORT LISTING 237

No. Date

54 December 19, 1960

(0-18)

55 December 19, 1960

(0-19)

56 January 30, 1961

57 March 3, 1961

(C-20)

58 March 3, 1961

(0-21)

59 March 3, 1961

60 March 10, 1961

61 March 20, 1961

62 May 26, 1961

63 May 29, 1961

64 July 7, 1961

65 July 14, 1961

Title

Catalogue of Satellite Observations:

Satellite 1958 Alpha, 1958 01, 1958

02, 1958 52, 1958 Epsilon, for June

1-Aug. 31, 1960

Catalogue of Satellite Observations:

Satellites 1959 al, 1959 «2, 1959 Eta,

1959 il, for June 1-Aug. 31, 1960

A Method of Analysis for Lens and

Mirror Systems

A Determination of the Ellipticity of

the Earth's Equator From the

Motion of Two Satellites

Effects of Solar Radiation Pressure on

the Motion of an Artificial Satellite

Catalogue of Satellite Observations:

Satellites 1960 01 (carrier rocket

Tiros I), for Apr. 1-nJune 1, 1960;

1960 02 (Tiros I), for Apr. 2-Aug. 31,

1960; 1960 7I (carrier rocket, Transit

IB), for Apr. 13-June 3, 1960; 1960

72 (Transit IB) for Apr. 14^July 25,

1960

Catalogue of Satellite Observations:

Satellites 1960 tl (Echo I), and 1960

i2 (carrier rocket, Echo I), for Aug.

12-Aug. 31, 1960

The Positions of the Baker-Nunn Camera Stations

•The Effect of Radiation Pressure on the

Secular Acceleration of Satellites

•Experimental and Theoretical Results

on the Orbit of Echo I

•The Atmospheric Drag of Artificial

Satellites During the October 1960

and November 1960 Events

•Effect of the Diurnal Atmospheric

Bulge on Satellite Accelerations

The Revised Orbit of Satellite 1958

Zeta

Atmospheric Drag on Non-Spherical

Artificial Satellites

Author

D. V. Mecahu

D. V. Mechau

R. J. Davis, S. E. Strom,

and K. M. Strom

I. G. Izsak

Y. Kozai

D. V. Mechau

D. V. Mechau

G. Veis

S. P. Wyatt

P. E. Zadunaisky, I. I.

Shapiro, and H. M.

Jones

L. G. Jacchia

S. P. Wyatt

R. C. Nigam

P. E. Zadunaisky

•Reprinted in Smithsonian Contributions to Astropyhsics, vol. 6, 1963.
238

No. Date

66 July 17, 1961

(C-22)

67 July 17, 1961

(C-23)

68 July 17, 1961

(C-24)

69 July 17, 1961

70 July 18, 1961

71 July 24, 1961

72 August 9, 1961

73 August 10, 1961

74 September 18, 1961

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Title

Catalogue of Satellite Observations:

Satellites 1958 Alpha (Explorer I),

1958 j81 (carrier rocket, Vanguard

I), 1958 Epsilon (Explorer IV), 1959

01 (Vanguard II), 1959 «2 (carrier

rocket, Vanguard II), 1959 Eta

(Vanguard III), and 1959 d (Explorer VII), for Sept. 1-Dec. 31,1960

Catalogue of Satellite Observations:

Satellites 1960 d (Echo I), and

1960 i2 (carrier rocket, Echo I), for

Sept. 1-Dec. 31, 1960

Catalogue of Satellite Observations:

Satellites 1960 01 (carrier rocket,

Tiros I), for Sept. 14-21, 1960; 1960

02 (Tiros I), for Sept. 2-Oct. 15,

1960; 1960 yl (carrier rocket, Transit IB), for July 7-27, 1960; 1960 T2

(Transit IB), for July 26-Nov. 7,

1960; 1960 rjl (Transit 2A), for June

26-Dec. 29, 1960; 1960 V2 (Greb),

for June 22-Dec. 23, 1960; 1960 i?3

(carrier rocket, Transit 2A, Greb),

for June 23-Dec. 31, 1960; 1960 £1

(Explorer VIII), for Nov. 4-Dec. 30,

1960; 1960 £2 (carrier rocket, Explorer VIII), for Nov. 19-Dec. 24,

1960; 1960 Omicron (Discoverer

XVII), for Nov. 13-Dec. 31, 1960;

and 1960 Sigma (Discoverer XVTII),

for Dec. 8-31, 1960

List of Coordinates of Stations Engaged

in the Observation of Artificial Earth

Satellites

*The Motion of Satellite 1958 Epsilon

Around its Center of Mass

Elements of the Orbit of the Satellite

1959 Eta (Vanguard III) During the

First Year after Launching

Tesseral Harmonics of the Potential

of the Earth as Derived From Satellite Motions

Differential Orbit Improvement with

the Use of Rotated Residuals

*On the Accuracy of Measurements

Made Upon Films Photographed by

Baker-Nunn Satellite Tracking

Cameras

Author

D. V. Mechau

D. V. Mechau

D. V. Mechau

D. V. Mechau

G. Colombo

P. E. Zadunaisky and

B. Miller

Y. Kozai

I. G. Izsak

K. Lassovszky

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 6, 1963.
SPECIAL REPORT LISTING 239

No. Date

75 September 19, 1961

76 October 2, 1961

77 October 24, 1961

78 October 25, 1961

79 October 30, 1961

80 November 1, 1961

81 November 24, 1961

82 November 30, 1961

(P-D

83 January 31, 1962

84 February 9, 1962

85 February 12, 1962

(P-2)

86 February 21, 1962

Title

•Density of the Heterosphere Related

to Temperature

•Effects of the Earth's Ionosphere on

HF Radio Astronomy From Artificial

Satellites

•Short-periodic Oscillations in the Drag

of Satellite 1958 Alpha

Satellite Orbital Data:

Satellites 1958 Alpha, 1958 /SI and

1959 il, for May-Dec. 1960

•The Analysis of Gravity

•The Stabilization of an Artificial Satellite at the Inferior Conjunction Point

of the Earth-Moon System

The Orbits of the Satellites 1959 cd

and 1959 a2 and the Perturbations

on the Perigee Distance of 1959 a l

Catalog of Precisely Reduced Observations: Satellites 1959 al, for Feb. 17-

June 30, 1959; and 1959 Eta, for

Sept. 18-Dec. 31, 1959

Project Celescope

Author

M. Nicolet

M. D. Grossi, K. M.

Strom, and S. E.

Strom

L. G. Jacchia and J.

W. Slowey

D. V. Mechau

B. Miller

H. Jeffreys

G. Colombo

R. C. Nigam

G. Veis

R. J. Davis and Celescope staff

L. G. Jacchia and

J. W. Slowey

Preliminary Analysis of the Atmospheric Drag of the Twelve-Foot

Balloon Satellite (1961 51)

Catalog of Precisely Reduced Observa- G. Veis

tions: Satellites 1959 al, for July 1-

Dec. 31, 1959; 1959 a2, for Mar. 6-

May 31, 1959; 1960 i2, for Sept. 10-

Dec. 31, 1960; 1960 Omicron, for

Nov. 13-Nov. 16, 1960; and 1960

Sigma, for Dec. 8-Dec. 10, 1960

Satellite Orbital Data I. G. Izsak

Satellites 1958 Alpha, for Jan. 1- B. Miller

July 1, 1961; 1958 01, for Jan. 1-

July 27, 1961; 1960 $1, for Nov. 4,

1960-July 3, 1961; 1961 51, for

Feb. 16-July 2, 1961; and 1961 wl,

for April 28-Sept. 1, 1961

Satellite 1959 d, for Dec. 31, 1960- J. Weingarten

July 1, 1961

•Reprinted in Smithsonian Contributions to Astrophysics, vol. 6, 1963.
240 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

No. Date

87 February 23, 1962

(C-25)

88 February 23, 1962

(C-26)

89 February 23, 1962

(C-27)

90 March 14, 1962

91 April 20, 1962

(P-3)

92 April 23, 1962

(E-l)

93 May 4, 1962

Title

Catalogue of Satellite Observations:

Satellites 1958 Alpha (Explorer I),

1958 01 (carrier rocket, Vanguard

I), 1959 ol (Vanguard II), 1959 a2

(carrier rocket, Vanguard II),

1959 Eta (Vanguard III), and 1959

il (Explorer VII), for Jan. 1-June

30, 1961

Catalogue of Satellite Observations:

Satellites 1960 il (Echo I) and 1960

i2 (carrier rocket, Echo I), for Jan.

1-June 30, 1961

Catalogue of Satellite Observations:

Satellites 1960 £1 (Explorer VIII),

for Nov. 3, 1960-June 30, 1961; 1961

51 (Explorer IX), for Feb. 16-June

30, 1961; and 1961 ul (Explorer XI),

for April 27-June 30, 1961

On the Critical Inclination in Satellite

Theory

Catalogue of Precisely Reduced Observations: Satellites 1958 52, for Dec.

1-17, 1958; 1959 o2, for Jan. 1-Dec.

31, 1960; and 1959 Eta, for Jan. 1-

June 30, 1960

Satellite Orbital Data

Satellite 1958 52, for Dec. 7-14, 1959

Satellite 1959 a\, for Feb. 21-Dec.

30, 1959

Satellite 1959 a2, for Mar. 19-May

28, 1959

Satellite 1959 Eta, for Sept. 23-Dec.

30, 1959

Satellite 1960 i2, for Sept. 11, 1960-

Mar. 12, 1961

Satellite 1960 Omicron, for Nov. 13-

15, 1960

Satellite 1960 Sigma, for Dec. 8-10,

1960

Satellite Orbital Data

Satellite 1958 Alpha, for July 1,

196 Wan. 1, 1962

Satellit Jan. 1e 195 , 19692 d, for July 1, 1961-

Satellite 1960 £1,

Jan.1, 1962

Satellite 1961 51,

Jan. 5, 1962

for July 1, 1961-

for July 2, 1961-

Author

D. V. Mechau

D. V. Mechau

D. V. Mechau

I. G. Izsak

I. G. Izsak

Y. Kozai

P. Stern and

M. Gutierrez

P. Stern

R. Nigam and

P. Stern

I. Izsak and

J. Weingarten

J. Weingarten

J. Weingarten

I. G. Izsak

B. Miller

M. Gutierrez

M. Hall

J. Weingarten
SPECIAL REPORT LISTING 241

No. Dale

94 May 23, 1962

95 June 18, 1962

(P-4)

96 June 25, 1962

(C-28)

97 June 25, 1962

(C-29)

98 June 25, 1962

(C-30)

99 July 16, 1962

100 July 30, 1962

101 July 31, 1962

102 August 27, 1962

(P-5)

103 August 28, 1962

104 September 10, 1962

(P-6)

105 September 28, 1962

106 November 1, 1962

(P-7)

Title

On the Motion of Explorer XI Around

its Center of Mass

Catalog of Precisely Reduced Observations: Satellites 1959 Eta, for July

1-Dec. 31, 1960; and 1960 i2, for

Jan. 1-June 30, 1961

Catalogue of Satellite Observations:

Satellites 1958 Alpha (Explorer I),

1959 al (Vanguard II), 1959 Eta

(Vanguard III), and 1959 d (Explorer VII), for July 1-Dec. 31,1961 ;

and 1959 al (carrier rocket, Vanguard II), for July 1-27, 1961

Catalogue of Satellite Observations:

Satellites 1960 il (Echo I) and 1960

i2 (carrier rocket, Echo I), for July

1-Dec. 31, 1961

Catalogue of Satellite Observations:

Satellites 1960 £1 (Explorer VIII)

and 1961 51 (Explorer IX), for July

1-Dec. 31, 1961; 1961 v\ (Explorer

XI), for July 1-Sept. 28, 1961; and

1961 o\ (Transit 4a) and 1962 o2

(Injun Solar Radiation), for June 29-

Dec. 31, 1961

Chemical Analysis of 643 Particles

Collected by High-Altitude Aircraft

and Balloons

Accurate Drag Determinations for

Eight Artificial Satellites; Atmospheric Densities and Temperatures

Numerical Results from Orbits

Catalog of Precisely Reduced Observations: Satellite 1959 al for the

Entire Year 1960

Satellite Orbital Data

Satellite 1959 Eta (Vanguard III),

Sept. 1, 1960-Dec. 31, 1961

Satellite 1960 il (Echo I), Jan. 1-

Dec. 31, 1961

Catalog of Precisely Reduced Observations: Satellite 1961 SI from Launch

Feb. 16^June 30, 1961

The Trajectory of Tektites

Catalog of Precisely Reduced Observations: Satellite 1959 al From Jan.

1-June 30, 1961; Satellite 1959 ijl

from Jan. 1-June 30, 1961

Author

G. Colombo

K. Haramundanis

B. Miller

B. Miller

B. Miller

F. B. Riggs, Jr.,

F. W. Wright,

and P. W. Hodge

L. G. Jacchia and J.

W. Slowey

Y. Kozai

J. MacDonald, K.

Haramundanis et aL

I. G. Izsak

B. Miller

B. Miller

J. MacDonald et al.

G. S. Hawkins and S. K.

Rosenthal

P. Stern
242 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

No. Date

107 November 9, 1962

108 November 20, 1962

109 December 21, 1962

110 December 14, 1962

111 December 15, 1962

112 January 21,1963

113 January 23, 1963

Title Author

On Some Singular Orbits of an Earth- G. Colombo and D. A.

Moon Satellite with a High Area- Lautman

Mass Ratio

On the Libration Orbits of a Particle G. Colombo, D. LautNear the Triangular Point in the man, and C. Munford

Semirestricted Three-Body Problem

Re-Entry and Recovery of Fragments C. A. Lundquist, R. C.

of Satellite 1960 cl Vanderburgh, W. A.

Munn, D. Tilles, E. L.

Fireman, and J.

DeFelice

Project Celescope, an Astrophysical R. J. Davis, ed.

Reconnaissance Satellite

Possible Contributions of Space Experi- P. Swings

ments to Cometary Physics

On the Secular Decrease in the In- R. C. Nigam

clination of Artificial Satellites

Satellite Orbital Data I. G. Izsak

Satellite 1958 Alpha, for April 1- B. Miller

July 1, 1962

Satellite 1959 al, for March 31- M. Gutierrez

June 30, 1962

Satellite 1959 Eta, for March 31- M. Hall

June 30, 1962

Satellite 1959 il, for March 31-June M. Gutierrez

30, 1962

Satellite 1960 {1, for April 1-July 1, M. Halt

1962

Satellite 1961 81, for March 31- J. Weingarten

June 30, 1962