3

SECOND-ORDER PLANETARY THEORY PART I S. E. HAMID

Smit hsonian Astrophysical 0 bservatory SPECIAL REPORT 302

https://ntrs.nasa.gov/search.jsp?R=19690030023 2020-08-04T03:16:43+00:00Z

Research in Space Science

SA0 Special Repor t No. 302

SECOND-ORDER PLANETARY THEORY

Part I: Outline of the Method

S. E. Hamid

July 1, 1969

Smithsonian Institution A s t r o p hy s i c a1 Ob s e rva t o r y

Cambridge, Massachusetts 02138

902-53

TABLE OF CONTENTS

Section Pan e

ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 THE EQUATIONS OF MOTION. .................... 3

3 THE DEVELOPMENT O F GZx, GZy, G Z z . . . . . . . . . . . . . . 7

4 THE DECOMPOSITION O F 6x2, by2, d e 2 . . . . . . . . . . . . . . 23

5 NUMERICAL APPLICATION . . . . . . . . . . . . . . . . . . . . . . 31

6 NUMERICAL RESULTS. . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 81

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

iii

LIST OF TABLES

Table - Page

Four ie r representation of d x 2mj k (periodic part) . . . . . . . . . . 36

Four i e r representation of byZmjk (periodic par t ) . . . . . . . . . . 52 Four ie r representation of d z (periodic par t ) . . . . . . . . . . 68

2mjk Four i e r representation of bxZ (mixed par t ) . . . . . . . . . . . . . . 72

73 Four ie r representation of b y 2 (mixed pa r t ) . . . . . . . . . . . . . . Four ie r representation of dz2 (mixed par t ) . . . . . . . . . . . . . . solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

75 Comparison of numerical integration with the analytical

iv

ABSTRACT

The analytical procedure for computing second-order perturbations in

rectangular coordinates, according to Brouwer's theory of planetary motion,

is given. Single- and double-harmonic analyses and the multiplication of

Four ie r s e r i e s with numerical coefficients a r e used in the computations. In

the se r i e s multiplication, a variable tolerance is considered, enabling us to

avoid the difficulties arising f rom a smal l divisor.

Also presented is an example computing that par t of the second-order

perturbation of Mars containing the masses of Jupiter and Saturn.

analytical solution of this perturbation is compared with the numerical inte-

gration of the differential equations defining this perturbation.

numerical integration covered the interval f rom 0 to 40,000 days.

parison shows an agreement within 1 x 10

The

The

The com- -9 .

RESUME

L a procldure analytique de calcul des perturbations de second ordre en

coordonnees rectangulaires, d ' ap r i s la theorie de Brouwer des mouvements

p l a d t a i r e s , est exposie.

la multiplication de se r i e s de Four ie r avec des cogfficients numeriques sont

u t i l i s ies dans les calculs.

d'une tol i rance variable qui permet d'&iter les difficultls provenant d'un

petit divis eur.

r r

Des analyses d'harmoniques simples et doubles e t r r

Dans l a multiplication de sgr ies on tient compte

U n exemple de calcul de la perturbation de second ordre de Mars due aux

m a s s e s de Jupiter e t Saturne est Lgalement p r i sen t l .

de cette perturbation est compar4e avec l ' inthgration numkrique des &quations

difflrentielles qui d;finissent cette perturbation.

couvre l ' intervalle de 0 A 40. 000 jours. -9 concordance de 1 X 10 .

La solution analytique

L'intkgration numirique

L a comparaison montre une

V

v i

SECOND-ORDER PLANETARY THEORY

P a r t I: Outline of the Method

S. E. Hamid

1. INTRODUCTION

The author has successfully applied Brouwer's theory of general per tur-

bation in rectangular coordinates to obtain a f i r s t -order planetary theory for

a l l the principal planets except Pluto (Hamid, 1968). Brouwer's theory over other planetary theories is its convenience when higher

order perturbations are considered.

The advantage of

In this report , the adaptation of the theory in the computation of second-

o rde r perturbations is discussed.

developed f o r the computation of the different second-order t e rms of plane-

ta ry perturbations. These programs have been applied for the planet Mars

to compute the second-order perturbations factored by the product of the

masses of Jupiter and Saturn. The numerical results obtained have been

tested successfully against the numerical integration of the differential equa-

tions satisfying these perturbations.

General computer programs have been

This work was supported in par t by grant NGR 09-015-002 f rom the National Aeronautics and Space Administration.

2. THE EQUATIONS OF MOTION

Consider a s e t of rectangular axes, the x axis corresponding to the

direction f rom the sun to the perihelion of the orbit of the perturbed planet

at a given epoch, and the z axis perpendicular to that orbital plane a t this

epoch, Then, the perturbations 6x, by, 6z in the rectangular coordinates

satisfy the following set of differential equations:

3P Y o -tPk--- (XO 6 x f yo 6y) = G

d26 y

dt2 r 3 5 Y ’ 0 r O

6 2

0

2 d 6z

dt2 r t p T = G .

Z

z ) a r e the coordinates of the planet, with its unper- 0’ YO’ 0 The quantities (x

turbed orbit assumed a t epoch, while r

of the planet.

denotes the heliocentric distance 0

The functions Gx, G G can be separated into different par ts of Y’ z

descending order of magnitude, the first par t giving r i s e to f i r s t -order per-

turbations, the second par t to second-order perturbations, and s o on. W e

denote these par ts by G lx’ Gly’ G l z ; G2x’ G 2y’ G2z; . . . , and le t 6 x 1’ 6Y1’ 6z1; 6x2’ by2, 6z2; . . . be the corresponding perturbations in the rectangular

coordinates.

The f i r s t -order perturbations 6 x by1, 6z1 will satisfy equations (1) 1’ G are put. equal to Glx, Gly, Glz, and x’ Gy’ z when the values of G

s imilar ly for higher o rde r perturbations.

3

W e consider

2 d 6x2

dt2

2

dt2

2 d 6 z 2

dt2

t

d 6Y2 t

t

the second-order perturbations, which a r e written as

The solution of equations ( 2 ) , given by Brouwer and Clemence (1961), takes

the following form:

+- G2x t - ag, G 2y )dt

G2x +q G 2y )dt2 ,

G 2 x f a L 0 G 2y )dt

+q ayo[(>G2xtaygG TO arlo 2 Y )dt -%/ ( ? G 2 x t s G 2y )dt

t -

4

F o r the definition of the different partial derivatives of the coordinates xo,

y 0 (1 961).

and of the quantities q 1, q2, in equations (3) , see Brouwer and Clemence

5

3. THE DEVELOPMENT OF GZx, GZyJ GZz

The expressions fo r G G GZz have the following form: 2x' 2y'

a2 R~ a' R~ k 6 2 -

G2x -- 2 k axkaYk 6yk aXkaZk

k ax

+ P

a2 R~ a2 R~ a2 R~ 'yk ay aZ 6zk 6Xk t -

k k 2 ayk

G = 2Y aYkaXk

7-

a2 R~ d2 Ro

k b x t - G2z - aZ k k ax dzkdyk

j=1

In equations (4), we have

x , y., z = the rectangular coordinates of the disturbing planet j, j ~ j

with i t s unperturbed orbit assumed at epoch;

xk, yk, zk = the rectangular coordinates of the disturbed planet k,

with its unperturbed orbit assumed a t epoch. Note that - Xk = xO> yk - yo, Z k = O;

6xk, byk, 6zk = the perturbations in the rectangular coordinates of

the disturbed planet k;

6x 6y 6 z = the perturbations in the rectangular coordinates of j’ j’ j

the disturbing planet j ;

2 p = k (1 t m ), where k is the gaussian constant and m k k is the mass

of the disturbed planet k;

Ro = the well-known disturbing function of the different disturbing

planets on planet k, given by

8

2 x k j x . t y k j y . + z 3 Ro = k mj (* -

r j j#k kJ

where A the heliocentric distance of planet j .

is the mutual distance of planets k and j , and r . is k j J

We note that c represents the sum over all thk disturbing planets j. F o r

example, if we consider the theory of Mars , then we have k = 4, and j will

take the numbers 1 ,2 ,3 ,5 ,6 ,7 ,8 , corresponding to the effects of Mercury,

Venus, Earth, Jupiter, Saturn, Uranus, and Neptune. In this report , we shall

exclude the effect of Pluto.

j#k

F o r the perturbations 6xk, byk, 6zk and 6 x j , byj, 6 z we shall consider the values derived f rom the f i r s t -order theory. The 6xk, byk, 6zk a r e com- posed of different parts, owing to the perturbations of the different disturbing

planets.

j’

If we let Flk(jk), FZk( jk ) , F3k( jk) be, respectively, the f i r s t -order per-

turbations in 6 x k, byk, 6zk due to the disturbing planet j , then,

j#k

j f k

6 z k = F3k(jk) . j#k

Similarly ,

i#j

9

6z = F3j(ij) . j

if j

The coefficients of 6xk, 6yk, 6zk in equations (4) can be written as follows:

where

2 3(x. - Xk) 2 +xl(jk) = k m - -

j A 3 A 5 [ 'kj' " k j

J

10

+z3(jk) = k 2 mj (-$-+$-) kj kj

The coefficients of 6 x byj, bz. in equations (4) can be written j* J

where

11

6 (jk) = k 2 mj [- 9

r x 3

2 2 2 Finally, the coefficients of (6x ) , (6yk) , (6zk) , 6\6yk, 6%6zkJ k 6yk6zk in equations (4) can be rewrit ten as follows:

xl

12

Y3 'k

'k

yk $ (k) = p 3- 22 5 .

13

With the above definitions of the various coefficients in equations (4), we

have the following:

j#k j#k j f k j f k

I- -

r 1 2

- j f k J

j f k j fk j f k j f k

j#k j f k

14

+ +,I

r F1

j #k j f k j #k j f k

j fk j #k

I- 1

I- 1

Let us now look m o r e closely at the different t e r m s defining the quan-

t i t ies GZx, Gzy, GZz.

Four ie r s e r i e s i n the mean anomalies 1

the disturbing planet j.

special numerical values of + and 8 for different combinations of equidistant

values of the mean anomalies I I t o double-harmonic analysis.

The + and 0 t e r m s can be represented by double

I of the disturbed planet k and k’ j These s e r i e s can be obtained by computing the

These special values are then subjected k’ j’

1 5

The $ t e r m s can be represented by Four i e r s e r i e s in one argument,

the mean anomaly I

obtained by computing the special numerical values of L/J for different

equidistant values of the mean anomaly I

values to single-harmonic analysis.

of the disturbed planet k. These se r i e s can be k

and then subjecting these k

In other words, by expressing the +, 8, and $ t e r m s as Four ie r s e r i e s

in the mean anomalies, we can avoid analytical expansions.

and single-harmonic-analysis techniques can be applied.

have done in the present work.

constructed to have a s output the Four i e r representations of the different

+, 8 , and $ t e r m s for any given values of j , k.

Only double-

This is what we

In fact, a general computer program can be

The terms Flk(jk), F (jk), F (jk) have already been obtained in the 2k 3k f i r s t -o rde r theory. It should be remembered that these perturbations in

rectangular coordinates a r e composed of two parts: the periodic and the

secular.

mean anomalies I

(measured f rom the given epoch) and a single Four ie r s e r i e s in the mean

anomaly I Le t the periodic par t be denoted by flk(jk), fZk(jk), f3k(jk),

and the secular part, by t Sljk(k), t S2jk(k), t S3jk(k).

The periodic par t is represented as double Four ie r s e r i e s in the

lk, and the secular par t by the product of the t ime t j’

k’ Hence,

where i = 1,2,3.

Le t us consider the par t c +xl(jk). F r o m the above remarks, this par t

is represented by the summa ion of different Four ie r se r ies , and each series .p is represented in the mean anomalies I , and I,-. F o r example, i f we are

K J considering the theory of Mars, we have k = 4, and c $xl(jk) will be com-

i f k posed of the sum of seven Four ie r ser ies : the first $&ies in 11, 14, the

m e a n anomalies of Venus and Mars ; and so on. Similarly, the par t c Flk (jk) j f k

is composed of the sunn of different Four i e r ser ies , and each se r i e s is

m e a n anomalies of Mercury and M a r s ; the second s e r i e s i n I 2’ 14, the

16

expanded in the mean anomalies P In addition to these Four ie r se r ies ,

this pa r t contains a t e r m t multiplied by a Four ie r se r ies in single argument

lk, the mean anomaly of the disturbed planet.

I k.

In fact, for i = l , Z J 3 ,

j f k j f k

where

j f k

We note that t Slk(k), t SZk(k), t S

rectangular coordinates of all the disturbing planets on planet k.

(k) a r e the total secular perturbations in 3k

Similar considerations apply for the different par t s of equations (1 3 ) .

Hence, Gzx, GzyJ G Z z can be represented by the following equations:

"

j f k

17

j f k

where G(k), G(jk), and G(mjk), appearing on the right-hand side of equations

(16), denote, respectively, Four i e r s e r i e s in one argument, the mean

anomaly I in two arguments, the mean anomalies l

arguments, the mean anomalies lm, lj, lk. The double summation c c means that m and j take the values corresponding to all the disturbing

planets, excluding m = j and avoiding double counting.

lk; and in three j' k;

j m

Following a r e the expressions for the different G(mjk) in equations (16):

3

(mjk) =x[+ .(jk) fik(mk) t (p .(mk) fik(jk) t 8 .(jk) f..(mj) 2qmjk q1 q1 q1 1J

G

i= 1

where q = x, y, and

18

The t e r m s G(jk) (not multiplied by t) in equations (16) a r e expressed as

follows:

where q = x, y, and

The t erms G(jk) (multiplied by t) take the following forms:

19

where q = x, y, and

2 F o r the t e rms with coefficient t in equations (16), we have

where q = x, y, and

In equations (17) to ( Z O ) , al l the different te rms on the right-hand side

We l k * a r e expressed in Fourier s e r i e s in the mean anomalies lm, l i ,

have already outlined how we obtain these ser ies . We a r e now in a position

to evaluate the Four ie r representations in mean anomalies of the functions

G 2qmjk(mjk)> G2qjk(jk)S G2qjkt' G2qktt, where q denotes the parameters

x, y, and z .

(mj k) , who s e exp r e s s ions a r e 2qmjk Let us consider, for example, G

given in equations (17).

appearing in the right-hand side of equations (17), we can, by the technique of

multiplying Four ie r se r ies , obtain the Four ie r s e r i e s representing

In this case, we did not r e so r t to triple-harmonic analysis G because it would have been excessively laborious.

Since we have the Fourier s e r i e s for all the t e r m s

(mjk). 2qmjk In fact, we constructed

2 0

a general computer program that has as input the numerical values of m, j ,

and k and that will give as output the Four i e r representations of G 2qmjk (mjk) B (q denotes the values x, y, and z). m’ ‘j’ k in the mean anomalies B

(jk) and GZqjk(jk) sjkt In computing the Four i e r representations of G2

f o r q = x, y, z , we can use the double-harmonic-analysis approach or the

multiplication-of-series approach.

of G2qktr(k) for q = x, y, z, we can very conveniently use the single-

harmonic- analysis technique.

To compute the Four ie r representations

In our work, we have a general program that, fo r given j , k as input,

produces as intermediate output the Four ie r representations of G 2 jkt(jk) , ..-

(k) fo r q = x, y, z. qktt G (jk), and G2 2 qjk

21

4. THE DECOMPOSITION OF bx2, 6y2, 6z2

In the previous section, we developed the different components of the

functions G2x, GZy, G2j.

posed of the summation of the following ser ies :

We found that these functions a r e generally com-

A. B.

C.

D.

Fourier s e r i e s in three arguments.

Fourier s e r i e s in two arguments.

Fourier s e r i e s i n two arguments, multiplied by the t ime t.

Fourier s e r i e s in one argument, multiplied by t . 2

in equations 2x’ G2y’ G2z By substituting the general expressions of G

will be composed of the following y2’ %2 (3), we can see that b X 2 , b different parts, where q takes the values x, y, z :

A.

B.

C.

D.

E.

l?.

G.

H.

Fourier s e r i e s in three argument (1 ma ‘j’ 1 k ), denoted by

bq2mjk (mjk).

Fourier s e r i e s i n two arguments (1

Four ie r s e r i e s in two arguments (1 t, denoted by 6qZjkt(jk).

Fourier s e r i e s i n one argument (1 k) denoted by 6q2k(k).

Fourier s e r i e s in one argument (1 ) multiplied by t, denoted

1 ), denoted by 6q2jk(jk). lk) multiplied by the t ime

j’ k

j’

k by 6q2kt (k)*

2 Fourier s e r i e s in one argument (1 ) multiplied by t , denoted by

3

k 6q2ktt (k)* Four ie r s e r i e s i n one argument (1 ) multiplied by t , denoted

4

k by &q2kt-t (k)* Fourier s e r i e s in one argument (I ) multiplied by t , denoted k by ‘q2ktttt (k)*

23

We mus t remember that I

I is the mean anomaly of the disturbed planet, and k

I . a r e the mean anomalies of the disturbing planets m, j . m’ J

In order to present m o r e conveniently the equations defining the various

par t s of 6x2, 6y2’ 6z2, l e t u s put

t - G ) d t , x a c 0 2 Y

fo r q = x, y. With this abbreviated notation, we have

(mjk) t -

where q = x, y, and

24

The integrands on the right side of equations (22) can now be developed in

Four ie r ser ies . Integrating these Four ie r representations, we obtain other

Four i e r representations of the integrals.

representations by the Four ie r s e r i e s representing the coefficients aq/aa,

-3 p. Lo (aq/awo), q2, -q l , and adding the different results, we obtain the

Four i e r representation 6 x

the constant coefficient in the Four ie r s e r i e s representing these different

integrands, i. e . , the coefficients of the argument 0, will, when integrated

once, give r i s e to a numerical coefficient multiplied by t; when integrated 2 twice, it will give r i s e to a numerical coefficient multiplied by t .

the final representations of 6q (mjk) will contain, besides the purely

periodic t e r m s given by the Four i e r representations in three arguments,

Multiplying these Four i e r

2 -4

(mjk), 6yZmjk(mjk), 6 Z2mjk (mjk). We note that 2m jk

Hence,

2mjk

mixed t e r m s composed of the time t multiplied by Four i e r s e r i e s in one

argument and, i n the case of q = x, y only, the square of the time ( t ) 2

multiplied by Four ie r s e r i e s in one argument.

added to the perturbations 6 qZkt(k), 6 qzktt(k).

These mixed t e rms will be

A computer program has been constructed with the ser ies G 2xmjk (mj k) , (mjk) as input and, a s output, Four ie r representations

6z2mjk (mjk) and the corresponding mixed t e rms 2 ym jk(mjk) ’ G2 z m jk

X2mjk

G

(mjk),

in 6X2kt(k)> 6Y2kt(k)> Z2kt(k)> 6X2ktt(k), 6Y2ktt(k)‘

F o r the evaluation of 6q2jk(jk), 6q2jkt(jk), for q = x,y, z , we mus t

r e call the following relations :

Itf d t = t j f d t -

(s t f dt2 = tJJf dt2 - 2 JJs f dt3 ,

where f is any function of time t.

given by

The equations

(23)

defining 6q (jk) will be 2 jk

2 5

Through the multiplication- of- s e r i e s approach or the double-harrnonic-

analysis technique, we can develop the Four ie r representations of all the

integrands appearing in the above equations and then evaluate the Four ie r

representations of 6 q We note again that the constant

t e r m s in the various harmonic representations of the above integrands will (jk) for q = xa ya z . 2jk

give rise to mixed t e rms with coefficients t and t 2 in the expressions for

6qZjk(jk), f o r q = x, y, z . Mixed t e r m s with coefficient t 3 will also appear in

the cases for q = x, y because of the presence of tr iple integrals.

various mixed t e r m s appearing in 6 q

tions 642kt(k)a 6q2ktt(k)a and 6q 2kttt

These

(jk) will be included in the perturba- 2 jk (k) f o r q = X, ya Z.

26

The equations defining 6q (jk) will be given by 2 jkt

where q = x, y, and

Again, through the double-harmonic- analysis technique or the multiplication-

of-ser ies approach, we can get the harmonic representations of 6q2 jkt(jk)

f o r q = x,y, z. Also, we expect mixed t e r m s with coefficient t in 6q2jkt(jk)

fo r q = x, y, z and with coefficient t2 in the case of q = x, y. Since 6 q2 jkt( jk)

is already multiplied by t, these mixed terms will have coefficients 2 3 t and t . As before, these mixed t e r m s will be included in the perturbations

q2ktt(k)' q2kttt(k)'

Finally, for the evaluation of 6q2kt(k), 6q2ktt(k), 6q2kttt(k) fo r q = x, y, z

and 6q2ktttt(k) for q = x, y, we must recal l the following relations:

j t 2 f dt = t ' j f d t - 2 t j J f dt2 t 2 j j j f dt3 ,

where f is any function of time t.

27

We mentioned ea r l i e r the contributions to 6qZkt(k). 6qZktt(k), 6 qzkttt(k)

obtained while we were deriving expressions f o r 6q 2mjk (mjk), 6qZjk(jk), (jk). In addition to these contributions, we have the following: ti q2 jkt

where q = x, y;

dt3 ,

( 2 7 )

where q = x, y; and

(k) dt . Z2ktt(k) = q2 q1 G2zktt (k) dt - q1 5.2 G2zktt

3 4 and t We note that mixed t e r m s with coefficients t will appear f rom the

expressions of 6q 2kt(k), 6qZk,(k) given in equations (27) and (28). can be added t o those defining 6qZkttt(k), 6qZktttt(k).

periodic and are expressed in Four i e r series in one argument I

and are given by 6 q (k), where

These terms T e r m s that a re purely

will appear k

2k

2 8

f o r q = x, y, and

6z2k(k' = q2 111 q1 G2zktt (k)dt3- q1 ( I (q2 G2zktt (k) dt3 ,

4 Again, the t e r m s may give r i s e to mixed terms with coefficients t3 and t . These will be added to 6q2kttt(k) fo r q = x, y, z and to 6 (k) for q = x, y. q2ktttt

A computer program THEORY 2 has been constructed to compute

(jk)> &q2jkt(jk)> 6q2k(k), 6q2kt(k)' * * 6q2ktttt (k). The input of this 6q2jk program is j, k. perturbations.

analysis methods; we did not apply the multiplication-of-series technique.

The-final output is the Four i e r representations of these

In this program, we followed the double- and single-harmonic-

29

5. NUMERICAL APPLICATION

In the previous section, we outlined the method followed for computing

We have two ma in com- the second-order perturbations in 6x, 6y, and 62;. puter programs. By use of the harmonic-analysis approach, program

THEORY 2 computes the periodic and secular perturbations expressed in

Fourier s e r i e s in the two mean anomalies 1 lk of the disturbing and the

disturbed planets and also in one mean anomaly 1 j’

of the disturbed planet. k

The second main program computes the periodic perturbations expressed

in Fourier s e r i e s in the three mean anomalies: 1

of the disturbing planets, and lk, the mean anomaly of the disturbed planet.

l the mean anomalies m’ j’

We used the multiplication-of-series approach, which required carrying

the multiplication to a cer ta in tolerance.

rectly proportional to the divisor when we compute the integrand that will

be integrated once.

twice, we take the tolerance to be directly proportional to the square of the

divisor (the constant of proportionality is 10

device will a s su re us that there has been no lo s s of any significant digits

owing to the small divisor.

This tolerance is taken to be di-

F o r the case of the integrand that will be integrated

-13 ). This variable tolerance

The author will soon publish the details of these two main programs and

the different subroutines associated with them.

31

6. NUMERICAL RESULTS

In this section, we present the results of the computation of the second-

o rde r perturbation of Mars containing the rnasses of Jupiter and Saturn;

according to the notation given previously, we give the results of 6q2mjk

(q = x, y, z ) , where m = 6, j = 5, and k = 4.

i n another paper.

The other resul ts will be given

Tables 1, 2 , and 3 give the periodic par t of the Fourier se r ies repre-

The mixed terms arising f r o m 2mjk’ and 6z 2mjk’ 6Y2mjk’ sentation of 6x the evaluation of these perturbations are given in Tables 4 ,5 , and 6 . The

-1 9 coefficients in these mixed t e r m s are computed up to a tolerance of 1 0

As a check, i t would be interesting to compare the se r i e s we obtained for

these perturbations with the results obtained f rom numerical integration.

.

The above se r i e s representations (periodic and mixed) a r e simply the

analytical solution of the set of differential equations ( 2 ) , where G 2x’ GZv’ That set of differ- and G2zmjk‘ Zxmjk’ G2yrnjk’

and G2z a r e replaced by G

entia1 equations has been solved numerically, using Cowell’s method of numerical integration.

neglected and the interval of integrations is taken to be 1 0 days.

values of the numerical integration are chosen such that 6x, 6y, 6z at

t = 90 days and t = 100 days are given by the analytical solutions of 6x, by, 6 2 .

In applying this method, the tenth difference has been

The initial

The evaluation of GZxmjk, G2ymjk’ G2zmjk in the numerical integration

of the differential equation was car r ied out by use of the original definition

of these G’s as given by equations (17).

out up to t = 40, 000 days. The integration has been car r ied

33

When we compare the resul ts of the numerical integration with the

analytical representation, deviation is found between the two. The deviations

found in the comparison of the perturbations in x and y a r e periodic in char-

acter , with the amplitude increasing with time. The amplitude reaches

5 X 1 0-7 around t = 20, 000.

perturbation in z is again periodic, with smaller amplitude.

reaches 1 X The disagreement between the numerical solution and the

analytical representation of the perturbation in x, y is very alarming.

The deviation found in the comparison of the

The amplitude

However, we must expect a satisfactory agreement if the starting values

used in initiating the numerical integration a r e given to a great accuracy.

These start ing values have been obtained, as mentioned ear l ier , f rom the

analytical solution. In obtaining the analytical solution, we carr ied the -13. evaluation of the different integrals involved up to a tolerance of 10

t e r m s with absolute values l e s s than have been neglected. These

t e r m s may add up, causing the accuracy of the evaluation of the integral to

be more than

multiplied by the partial derivatives ax /8wo, ayo/aWo, 8xo/8Lo, ayo/8L0, . . . . The coefficients of the harmonic representation of these partial derivatives

amount to 10 . Thus, the accuracy of the evaluation of the periodic repre-

sentation of the perturbation in x, y, and z may amount to 10 o r even

, i.e.,

We must remember, also, that these integrals must be

0

2

-1 1

-10 10 .

Numerical integration of the differential equations defining the second-

order perturbation is very sensitive to the start ing values, which we have

just found m a y be in e r r o r to within 10

we can apply differential corrections to the starting values, such that the

deviation between the numerical integration and the analytical solution is

minimum, in the least- squares sense.

4. -1 0 *a.

to lo- ' ' . To mee t that situation,

.L *P

The author owes this idea to Prof. G. M. Clemence.

34

We have applied differential correct ions where our equation of condition

corresponded to deviations at t = 200, 1800, 3400, . . . , 19,400 days.

resu l t s of this follow:

The

At t = 90 days,

A (6x) = -8.61 98240974 X 1 0-'A

A ( 6 ~ ) = -1. 3774726475 X 10 -1 0

A ( 6 z ) = t-3.2748239038 X 10 -10

At t = 100 days,

A (6x) = -3.6212757640 x

A (by) = -2.4084833490 X 10-l '

-10 A ( 6 z ) = t 3 . 5234609123 X 10

When we apply these corrections to the start ing values of the numerical

integration, the agreement between the analytical solution and the numerical

integration improves appreciably. The deviation, after the integration is car r ied to 40,000 days, never exceeds 4 X 10 -10 in x, y and 1 X 10 -9 in z ,

an excellent agreement indeed. This comparison is shown in Table 7.

35

cc)

I 0

w

v) c,

I-(

4

0

i i d

a, k Id v) c, d a, 0 w w a, 0 u a, c E-l

.r(

.r(

.A

h ct k 6 a u r(f 0 k a, a

.A

.,-I

v

A

'E" 8" Lo

w 0

.r( ct cd c, d a, v) a, k c4 a, k k a,

.A

$

I-(

a, 4

2 E-l

U .*

m 0

* VI

Q

U .r( ID

I

8

7

VI

Q

U .*I 0

m 0

* VI

36

37

38

C 0

0

c

39

~ ~ ~ 0 1 - ~ - o o o o ~ m u m m ~ ~ ~ - o ~ - a o - ~ - ~ ~ ~ ~ ~ ~ m n m m m - a m ~ n m ~ ~ m m u m ~ n n w n ~ ~ u ~ a ~ ~ I 1 1 I N U ~ N I I - - 0 4 - 1 ~ N I I N N I ~ N U I I Q - O - I I P U ~ N t u n r n u ~ n i A

N N 1 - - 1 I 1 - 1 - 1 I 9 0 a m I 1 - I

." 22

40

tn

t I

41

42

4 3

44

B ' N W 8 - N N I

m n c - 4 - 4

I

45

.B U - D Q - - Q Q ~ ~ N ~ Q N Q N ~ N Q ~ ~ Q - ~ N N ~ ~ ~ ~ ~ U I N N ~ ~ N N ~ U I ~ F P N Q - N U I N - - N Q ~ N N O ~ ~ ~ Q ~ ~ I 1 1 4 - I I i m s a a IUI m I I I I I I IUI - - U I ~ N O P N I m N - t - f - 1 m ~ i

I - ! n - I - - 1 0 1 m 1 - - 1 I N N

m I l l

m 0 U

c - ~ - ~ ~ n - - ~ m m n ~ ~ ~ ~ m ~ ~ u ~ - ~ ~ ~ m ~ ~ o o o ~ ~ o - o ~ - m ~ ~ m r m ~ n m - m - m m m ~ ~ m m m - - o ~ ~ .- -o . - -u I i - m m m m ~ I - N I I ~ - I I I N Q F S Q N I I I ~ m m ~ I

I I i m m I N N I

1 1 - 1 1 I 1

d m P O m Q n u m ~ - O ~ ~ m u n Q r m P o ~ o P m r ~ n u m ~ ~ o ~ ~ m u ~ ~ r m ~ c - ~ - o ~ m r ~ n u m ~ - c - ~ m u n ~

VI ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ N N N N N N N N N N N N N N N N N N N N N N N - - - - - - - - - - - - - - - - - - 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 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

- - - - - I t I I I I I I I I I 1

- 1 1 I 1 I I I - - - 1 1 1 1 1 1 1 1 1

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

46

-# ~ - ~ m u ~ - ~ m u - ~ - ~ m u ~ - ~ - ~ m u ~ - o - ~ m u ~ o ~ ~ m ~ o ~ ~ ~ o ~ ~ m ~ o ~ ~ ~ ~ - o - ~ m ~ - c ~ ~ ~ - o I I 1 I I I I I I I 1 0 1 I 1

0

m a m N u I i m a o r - I m m i m

u - 0 0 o r e- - F-

m u u m I m ~ - I

Q U O O C I 1 1 -

u t I

W N e U Q ~ Q u m ~ Q U o m u I I U P ~ - U Q N N ~ m I I N ~ U ~ W N O N

I N O - u m m i - m - m u I - m--

I I N N O N - 4 - 4

I 1

-# c m ~ o ~ m r ~ n u m ~ - ~ ~ ~ m u n ~ m ~ o o w r ~ n u m ~ ~ ~ ~ ~ m u n ~ r w m ~ - o ~ m r ~ n u m ~ - o - ~ m u n ~ ~ ~ - - - A I I I I I I I I I

I 1 1 1 - - - 1 1 1 1 1 1 1 1 1 I l l

- 1 1 1 I 1 I 1 I I

VI ~ ~ ~ ~ 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ N N N N N N N N N N N N N N N N N N N N N N I l l 1

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN~NNNNNNNNNNNNNNNNNNNNNNNN

47

48

p N m N m u m e - p1 m N 4 m u N nu m n NN m o e o c .t Q m e m o N Y) e+ PO 4- m m rn- ... NN m n- u N N m - o N - o 1 - 4 I 1 I 1 1 1 1 1 1 1 I - N 4-

1 1 1 1 I 1 I 1 I 1 P d U m Q N N I I L n l N U N N N l I I 1 . 4 - 1 1 I I

49

fi

50

m 0 4 Q Q N O 4 4 0 N N N 0 I I N d - N I

I IUN I

m r - ~ n * m u m ~ - o I I I I I I - - - - -

1 1 1 1 I

d( Q I - ~ D O - U ~ N ~ O

lcI m m m m o m r - r - r - r - r - 1 1 1 1 1 1 1 1 1 1 1

.n m m m m m m m m m m m

- - l l l l

51

yr)

i 0

n 0

4

4

0 k a,

.rl

3 '

.rl c1

c, rd

s 10 a, k a a, k

k a,

.A

2

Q m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m

o ~ ~ m ~ n b ~ ~ m ~ n ~ m ~ ~ ~ ~ ~ m ~ n ~ o ~ ~ w r ~ n ~ m ~ ~ o ~ ~ m ~ n ~ ~ ~ ~ ~ m ~ - o - ~ ~ m ~ n ~ m ~ ~ ~ ~ m ~ - o 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

LO 0 0 0 0 0 0 o ~ - - - - ~ ~ ~ ~ ~ ~ ~ - - - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ m m m m m m m m m m m m m m ~ ~ ~ ~ ~ ~ ~ ~ ~

l l l l l l l l - 1 1 I 1 I I 1 I 1

Q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

52

53

q u m - ~ ~ m ~ ~ n u m ~ - ~ - ~ m u n ~ r - o m m ~ ~ n ~ m ~ ~ o ~ ~ m u m ~ ~ ~ o m a ~ ~ n u m ~ - o - ~ m u n o m m r ~ n

m o o - - - - - - - - ~ - - - - - - - - - - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ m m m m m m m m m m m m m m m m m u u u ~ u u

- 1 I I I I -- 1 I I I I I I I I 1 1

4 - 1 I I I I I I I I I 1

-- I I I I 1 I I I I I 1

a m m m m m m m m m ~ m m m m m m m m m m m m m m m m m m ~ m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m

54

0 0

55

I3 .d m

m 0 0

C N - 0 N

0 - o u ~ - n - u e - o - n ~ ~ e a u o ~ o e o w n ~ o r m m ~ - O ~ ~ ~ N - W N I I I - - N I I - U ~ Q O ~ ~ ~ N N N - m ~ e ~ I I I n r u I I I I I ~ N U U - N ~ Q N

8 mare-e- I N Y I N W

1 - - 1 1 N e

- r o I 1

QN e *

56

57

d( m u m ~ r o ~ m r ~ n u m ~ - o - ~ m ~ m ~ o ( F m r ~ m ~ m ~ - o ~ ~ m - ( F m r ~ m u m ~ - o - ~ m u ~ - o ( F m r ~ n u m ~ ~ - - - I I I I 1 I I I I I I 1

- l l o l l l l 1 - 1 1 1 1 1 1 1 4 1 - 1 1 I 1 1 I 1 1 I

58

k

m ~ r - ~ ~ m u ~ o - - ~ ~ ~ ~ ~ ~ o n ~ n ~ m m u m o ~ ~ ~ m ~ ~ o ~ m m ~ ~ ~ o r - r - o ~ + + m o o ~ ~ o ~ m ~ o m o o - ~ n m u I 1 I I I I I I 0 I d I l l N N 4 I 4 - 0 - 1 I I I I I I I I I I I I I ~ N N I I - I N I U N N ~ M

59

6 0

6 1

e i n m w ~ ~ ~ s u r m ~ m ~ r ~ ~ ~ ~ u ~ u e m m e r r n ~ o m ~ n s r u ~ ~ - ~ o m m ~ m - ~ ~ ~ - o m ~ e e ~ m o - - ~ m w m - m I I I I i m m m m i I I I NN

m~ N N ~ I A d 4 I i m m I I 1

.C( in N U I m W W d I I *N N O N I I I I N 4 I I I I I I

N N n W N I I - - - 1 1 1

", 0 e r . ~ o o e - e m

I s r ~ I Ins I n

m m u - o u- I 1

d ~ ~ w r s n u m ~ - ~ ~ ~ m u n s r w ~ o - o e w r s n u ~ ~ - o ~ ~ m u n s ~ w e o ~ ~ ~ o e m r s n u m ~ - o - ~ m u n s

tn U U ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ N N N N N N N N N N N N N N N N N N N N N N N - - - - - - - - - - - - - - - - - - l 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 l 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

- - - - - I I I I I I l l 1 I 1

- 1 1 I I I I 1 I - - - 1 1 1 1 l l 1 1 1

9 NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN~NNNNNNN

62

6 3

h

c,

u Y

fi I

m - a ~ m m u t n u m 1 1 - I I

u ~ ~ u u m ~ - ~ ~ o ~ o o - m - ~ o - m m n n m ~ - m o - o u a ~ a m I I I I I I I I ~ m o o u I i ~ m a m n u o ~ n m

I N O ~ - O N I I 4 I I I N A Q I - U d Y \ d N

U N

I N N N N I I N N I

I 1

~ ; ~ m - m ~ - - ~ ~ - - ~ - ~ - m m m a m a m m o m m u ~ u u o m - u - ~ ~ m ~ ~ m o m o r - o m n m o m o a r u ~ m ~ c o I I I I I I I t m u a m I ~ I C I I ~ N O I I I I I I I I I I I i a m a - ~ i m a u o r - n ~ t - m

t m m c I I I I I 1 I NV) I N - a d N II-N O N I I

I N N I I 1

I 1

0

w Y) ac

- m~

m o o m e a m u m I J t J 1 1

N O U ~ ~ Q Q N O N F ~ P N I i - ~ o o m u m a m a In- I - 1

N

n a ~ - m o o m r - o n u - i l # l l I

0 - - I 1 1000eNI-

I N - C U

1 - 0 1 -

I -o rno . - 1 - m m

o m r a t n u 1 1 1 1 1 1

r n a u r - r n m ~ u ~ n e n m m ~ i I 1 - 1 0 1 Y)*N I In O N I I N I

mu n a - o ~ m c a - - I I I I 1 1

VI m m m m m m m m m m m m m m m m m m m u u u u u u u u u u u u u u u u u n n n n n n n n n n n n n n a a a a a a a a a a a a a

N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N

64

N

~ ~ ~ m m ~ ~ a ~ n ~ o ~ ~ u m u n + m + ~ m ~ ~ m ~ o ~ + u m o ~ o + m o ~ n m ~ ~ o a ~ m o ~ ~ ~ ~ - - ~ - u - ~ - m - m ~ m I 1 I I - - - I u I o a m n m ~ m I I u m ~ + I I I I l l

3 - 1 -- I I 1 1 I " I l l 1 I l l I " I I

6 5

fi m ...

m 0

66

E 4 4 O N N N

* N N 4 - N

6 7

.,-I c, Q N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

c1 (d

NNuNN I N C C Y i

- -< or- I- ~ r u u u - i m r o ~ r - I I i u o 4 IN^- ~r I I I 4 I I N N N - I urn-

1 - m N m a m - I - N I 1 I I I m- a, k

r - I I

68

69

7 0

h

c; F: 0 u

v ~ - ~ m - ~ - ~ m u - ~ - ~ m u ~ ~ ~ ~ ~ m ~ ~ ~ o ~ ~ m r m ~ - o ~ ~ m u m ~ - o - ~ m ~ m I I * I 1 1 1 1 1 1 I 1

f f i ~ - - - w r w - n - - ~ o ~ m 1 - ~ 0 m 0 1 ~ r - ~ w ~ ~ o m ~ o ~ 0 1 o o - m s n w ~ Q N - r m - c m~ I I ~ ~ N N O Q - I I N ~ ~ N I I N I - O N r m ~ a r n I - m o m * - I l ~ v u m

Q ~ I I N - I - 1

.- I I i n m m . ~ ~ I - 0 4 I 1 1 N n W l 1 1 1 1 - t m 0 - N I *

7 1

Table 4. Four ie r representation of 6x2 (mixed par t )

T measured f r o m epoch in days,

The coefficients are in units of 10 . 4

-1 9

The argument is k14, l mean anomaly of Mars .

k cos s i n ~

1

2

0

1

2

3

4

5

6 7

8

9

0 T2

0 T2

313774 T

10849 T -102779 T

-14383 T

-1786 T

-217 T

-26 T

-3 T

O T

O T

-26 T2 2 -2 T

O T

-78999480 T

-8840577 T

-977485 T

-110676 T

-12768 T

-1493 T -176 T

-21 T

-3 T

72

Table 5. Four ie r representation of 6y2 (mixed part)

T measured f r o m epoch in days.

The coefficients are in units of 1 0 . -19

The argument is kB L mean anomaly of Mars . 4’ 4

k cos s in

1

2

26 T2

2 T2

0 T2

0 T2

0 4512938 T O T

1 78686060 T 20653 T

8823289 T

976183 T

110563 T

12758 T

1492 T

176 T

21 T . 3 T

-102322 T

-14351 T

-1784 T

-216 T

-26 T

-3 T

O T

O T

73

Table 6. Fourier representation of Sz (mixed par t ) 2

T measured from epoch in days.

The coefficients a r e in units of 10 . -19

The argument i s k l l mean anomaly of Mars. 4’ 4

k cos s in

0 658698 T O T

1 -4692988 T 873448 T

2 -218295 T 40658 T

3 -1 5234 T 2838 T

-1260 T

-115 T

-11 T

-1 T

235 T

21 T

2 T

OT

74

D 0 N Q

I d

75

- 2 a

- D

- E 9

76

- 5 2

77

- P

- D P

~ ~ ~ ~ ~ o O O O O ~ ~ ~ O O O O o - - - - o o o o o o o o o o o o o o o o o o o o o o ~ o o o o ~ " ~ o ~ - - ~ o o o o o ~ , I I I I I I I I I 1 l 1 1 I l 1 1 l 1 1 1 . I 1 1 1 1 1 1 t 1 I I l l I I

78

- - E e

7 9

- 2 a

0 O N O O r n r 3 . 4 N d O - d 3 4 N 3 . 4 I * I l l 1 . 1 1 1 1 1

d 0 0 0 4 3 0 3 ~ 3 4 O N 3 0 ~ 3 . 4 ~ I I l l I l l I

8 0

7. ACKNOWLEDGMENTS

I wish to acknowledge the continuous interest and encouragement given

by Drs. F. L. Whipple and C. A. Lundquist. I a lso express m y gratitude to

Dr. G. Clemence for his valuable advice and his numerous illuminating dis-

cussions. I wish to extend m y thanks to Drs. P. Musen, R. Broucke, and

M. Davies for clarifying and constructive discussions.

Mrs. Yun-Ying Fang, f rom the SAO's programing department, who pro- gramed the multiplication-of-series program used in this work. I thank

Drs. A. Allison and B. Marsden for discussing various aspects of the problem

with me.

My thanks also to

81

REFERENCES

BROUWER, D., and CLEMENCE, G. M.

1961. Methods of Celestial Mechanics. Academic Press, New York,

598 pp. (2nd printing, 196 5).

HAMID, S. E.

1 968, F i r s t -o rde r planetary theory, perturbations in rectangular

coordinates, Hansen's variables, longitude, and distance.

Smithsonian Astrophys. Obs. Spec. Rep. No. 285, 35 pp.

83

BIOGRAPHICAL NOTE

SALAH E. HAMID received h is B. S. C. f rom Cairo University in 1944

and his Ph. D. i n astronomy f rom Harvard in 1950.

D r . Hamid was an assis tant professor of astronomy on the Faculty of

Sciences, CairoUniversity, and has a l so held the position of d i rec tor of the

Operation Research Center of the National Planning Institute of Cairo, the

U. A. R. He joined SA0 in 1961 as a celestial mechanician.

NO TICE

This se r ies of Special Reports was instituted under the supervision of Dr . F. L. Whipple, Director of the Astrophysical Observatory of the Smithsonian Institution, shortly after the launching of the first artificial ear th satellite on October 4, 1957. Contributions come f rom the Staff of the Observatory.

First issued to ensure the immediate dissemination of data for satel- lite tracking, the reports have continued to provide a rapid distribution of catalogs of satellite observations, orbital information, and prel imi- nary results of data analyses pr ior to formal publication in the appro- priate journals. The Reports a r e also used extensively for the rapid publication of prel iminary o r special resul ts in other fields of as t ro- physic s .

The Reports a r e regularly distributed to all institutions partici- pating in the U. S. space r e sea rch program and to individual scientists who request them f rom the Publications Division, Distribution Section, Smiths oni an As trophy s i c a1 Ob s e r va t o r y , Mas s a chus e tt s 02 138.

C amb r i d g e,