Application of Two-Variable Expansion Procedure to the Planar Restricted Three-Body Problem

APPLICATION O F THE TWO VARIABLE EXPANSION PROCEDURE T O THE

C O M M E N S U R A B U PLANAR RESTRICTED THREE-BODY P R O B L E M

T h e s i s by

R i c h a r d R. W i l l i a m s

In P a r t i a l F u l f i l l m e n t of t h e R e q u i r e m e n t s F o r t h e D e g r e e of

Doctor of Philosophy

C a l i f o r n i a Institute of Technology P a s a d e n a , California

19 66

(Submitted F e b r u a r y 21, 19 66)

ACKNOWLEDGEMENTS

The a u t h o r w i s h e s t o e x p r e s s his a p p r e c i a t i o n t o P r o f e s s o r P. A. L a g e r s t r o m for the a s s i s t a n c e given by h i m a s r e s e a r c h

a d v i s o r , a n d t o P r o f e s s o r J. K. Kevorkian, who originally s u g g e s t e d t h i s p r o b l e m and offered many suggestions throughout t h e c o u r s e of the work. Many thanks a r e due t o Mrs. Vivian Davies f o r typing t h e manuscript.

The w o r k was p a r t i a l l y s u p p o r t e d by A i r F o r c e G r a n t AFOSR 338- 65.

ABSTRACT

The n e a r l y c o m m e n s u r a b l e c a s e of the planar r e s t r i c t e d t h r e e - body p r o b l e m is t r e a t e d by application of the two v a r i a b l e expansion procedure. The polar angle of the infinitesimal body, r a t h e r than the t i m e , is taken a s the independent variable. A s e t of four coupled f i r s t

o r d e r differential equations, which govern the long-period behavior of the o r b i t a l elements, i s obtained by imposing the r e q u i r e m e n t that t h e a s s u m e d f o r m of t h e expansions must be self-consistent. The

independent v a r i a b l e i n t h e s e equations is the "slow variable". It is then found that the s h o r t - p e r i o d perturbations of the motion of the infinitesimal body do not contain s m a l l d i v i s o r s o r s e c u l a r t e r m s .

Approximate solutions f o r the o r b i t a l e l e m e n t s a r e given, f o r two different c a s e s . Both l i b r a t o r y and n o n - l i b r a t o r y solutions a r e found, depending upon the initial conditions. N u m e r i c a l r e s u l t s a r e calculated f r o m t h e s e solutions, and a r e c o m p a r e d t o n u m e r i c a l computations r e c e n t l y r e p o r t e d i n the l i t e r a t u r e .

TABLE OF CONTENTS

I. INTRODUCTION 1

11. EQUATIONS O F MOTION 6

111. METHOD OF AVOIDING SMALL DIVISORS 11

Justification f o r Use of the Two V a r i a b l e Expansion 12 P r o c e d u r e

The F o r m of the Expansions 13

Solution of the O(pO ) Equations 16

O c c u r r e n c e of S m a l l D i v i s o r s i n s I and t l 18 Explicit Inclusion of C o m m e n s u r ~ a b i l i t y i n the Expansions 20 G e o m e t r i c a l Significance of @(€I, p) 24 Dependence of the Orbital E l e m e n t s on p. 26

The O(p) Equations 2 9

S e r i e s Expansion of the P e r t u r b i n g T e r m s 3 0 R e m o v a l of Resonant P e r t u r b i n g T e r m s 3 4

IV, BEHAVIOR OF THE ORBITAL ELEMENTS 42

1. Equations f o r the Orbital E l e m e n t s 2. Use of the J a c o b i I n t e g r a l

3. Approximate Solution f o r e(9, p) e o

N

% A

4. Approximate Solution f o r e(9, p) = p e(8,p) 58 5. C o m p a r i s o n of R e s u l t s with Calculations by Schubart 64 APPENDICES

VI. NOTATION

VII. REFERENCES

1

I. INTRODUCTION

The p l a n a r r e s t r i c t e d three-body p r o b l e m m a y be s t a t e d a s follows: Two bodies move i n c i r c u l a r o r b i t s about t h e i r common c e n t e r of m a s s , and a r e a s s u m e d t o be point m a s s e s . A t h i r d body having infinitesimal m a s s moves i n t h e o r b i t a l plane of the two l a r g e m a s s e s , u n d e r t h e i r combined gravitational attraction.

The above problem, although highly idealized, p r o v i d e s an approximate m a t h e m a t i c a l model of s e v e r a l a c t u a l p r o b l e m s which occur i n c e l e s t i a l mechanics. One s u c h p r o b l e m i s the motion of a n a s t e r o i d ( m i n o r planet) about the sun. The m a s s of a n a s t e r o i d is sufficiently small, i n c o m p a r i s o n t o the m a s s e s of t h e s u n and m a j o r planets, that the effect of the gravitational pull of the a s t e r o i d upon the motion of t h e s e l a r g e r bodies m a y be neglected.

The two l a r g e s t planets i n the s o l a r s y s t e m a r e J u p i t e r and Saturn, the m a s s of S a t u r n being approximately 0.299 that of Jupiter.

( T h e next l a r g e s t planet, Neptune, h a s a m a s s only 0.053 t h a t of Jupiter. ) The orbit of J u p i t e r l i e s much c l o s e r t o the o r b i t s of the a s t e r o i d s than does the orbit of Saturn. The r e f o r e , the p e r t u r b a t i o n s of the motion of a n a s t e r o i d c a u s e d by the g r a v i t a t i o n a l a t t r a c t i o n of J u p i t e r a r e much l a r g e r than those c a u s e d by any other single planet.

The o r b i t of J u p i t e r a r o u n d the s u n is n e a r l y c i r c u l a r , its e c c e n t r i c i t y being approximately 0.0482. The o r b i t a l inclinations of many of the a s t e r o i d s , with r e s p e c t t o the s u n - J u p i t e r plane, a r e only a few d e g r e e s . F o r the above r e a s o n s , a solution of the p l a n a r r e - s t r i c t e d t h r e e - b o d y p r o b l e m m a y be expected t o provide a n approxi- mation t o the motion of a n a s t e r o i d around the sun.

The m a s s of Jupiter, although being l a r g e i n c o m p a r i s o n t o the m a s s e s of the other planets, i s only about 1/1047 that of the sun. T h i s suggests the application of a perturbation p r o c e d u r e t o obtain a n

approximate s olut'ion of t h e problem.

. Another instance i n which the planar r e s t r i c t e d three-body p r o b l e m m a y be u s e d a s a n approximate model i s t h e motion of a n a r t i f i c i a l e a r t h s a t e l l i t e i n the o r b i t a l plane of the e a r t h - m o o n system.

In t h i s c a s e the motion of the a r t i f i c i a l satellite about the e a r t h is p e r t u r b e d by the gravitational a t t r a c t i o n of the moon.

A s e r i o u s difficulty o c c u r s i n the c l a s s i c a l v a r i a t i o n of con- s t a n t s solution of the problem, f o r those c a s e s where t h e period of the infinitesimal body i s c o m m e n s u r a b l e with that of the perturbing body. This difficulty will be briefly described, following a d i s c u s s i o n by Brouwer and Clemence. (1)

The equations of motion f o r t h e infinitesimal body a r e solved by the method of variation of constants. The f i r s t approximation yields a Keplerian orbit t h a t m a y be d e s c r i b e d i n t e r m s of four o r b i t a l elements. The p e r t u r b a t i o n s caused by the gravitational a t t r a c t i o n of the body of m a s s a r e t a k e n into account i n the next approximation, and a s e t of four f i r s t - o r d e r equations is obtained f o r the v a r i a t i o n of the constants of integration; i.e. for the behavior of t h e o r b i t a l elements. F o r example, the equation f o r da is a s follows:

with the notation ^'

s e m i m a j o r a x i s of the orbit of the infinitesimal body time

m a s s of the perturbing body

m e a n motion of t h e infinitesimal body m e a n motion of the perturbing body

m e a n longitude of the infinite sirnal body

a f @

longitude of the p e r i c e n t e r of the infinitesimal body coefficients depending only on a and e ( f o r the planar r e s t r i c t e d t h r e e -body p r o b l e m )

i n t e g e r s which a r e s u m m e d over

and where ao, no, eo, woP and E a r e the corresponding unperturbed

0

Keplerian values. The s e r i e s on the r. h. s. of the above equation c a n be a r r a n g e d i n i n t e g r a l powers of the e c c e n t r i c i t y e

.

Equations s i m i l a r i n f o r m t o the above a r e obtained f o r

-

de d t

' -

dw and

-

d t'

dE T h e s e equations a r e integrated by neglecting the v a r i - d t '

ation of the o r b i t a l e l e m e n t s of the infinitesimal body on t h e r. h. s.

,

a s is indicated by t h e u s e of a n o and eo i n s t e a d of a, n, w, andc.

0' 0' 0'

The following r e s u l t is obtained f o r the s e m i m a j o r axis:

where

The solutions f o r de, do, and 6~ a r e s i m i l a r i n f o r m t o that f o r da.

If

the m e a n motions n and n1 a r e approximately c o m m e n s u r -

0

able, t h e r e will e x i s t a p a r t i c u l a r p a i r of i n t e g e r s jl= J1 and jg= J 3 n 0

f o r which

(J3$ J,

- 7 )

=

⁰

.

The e x p r e s s i o n s f o r da, de, do, and 66

n n

will then contain t e r m s which a r e divided by t h e s m a l l d i v i ~ o r ( ; r j + J , ~ ) . 0

n

F o r c a s e s i n which t h e s e s m a l l d i v i s o r s occur, the above

solution is not valid. This is because the o r b i t a l e l e m e n t s a, e, a, and E a s given above undergo l a r g e oscillations having amplitude p r o - portional t o (J3 t J,

3)-',

i n violation of the approximation that was used i n integrating the equations f o r da

_{dt' dt9}

de

_dt'

dw and

.%.

_{dt '} ^{This is}

known a s the "difficulty of s m a l l divisors1'.

The difficulty of s m a l l d i v i s o r s a l s o o c c u r s i n t h e v a r i a t i o n of constants solution of the non-planar r e s t r i c t e d t h r e e -body problem, a s well a s i n the m o r e g e n e r a l p r o b l e m where the orbit of the perturbing body is taken a s elliptic r a t h e r than c i r c u l a r . However, i n o r d e r t o investigate the b a s i c f e a t u r e s of the difficulty of s m a l l d i v i s o r s , with- out becoming u n n e c e s s a r i l y encumbered by a l g e b r a i c detail, i t is reasonable t o consider the s i m p l e s t p r o b l e m w h e r e the difficulty o c c u r s -the planar r e s t r i c t e d three-body problem.

A qualitative method of treating the p r o b l e m of s m a l l d i v i s o r s h a s been given by ~ o i n c a r 6 ' ~ ) for the c a s e where the m e a n motions a r e i n the r a t i o J'l

-

^{with J} a positive integer. The t i m e is taken a s

J

the independent variable, and a l l the s hort-period p e r t u r b a t i o n s a r e neglected. Two approximate i n t e g r a l s of the long-period motion a r e obtained, because the Hamiltonian then contains neither t h e t i m e nor the s h o r t - p e r i o d angular variable. However, only the g e n e r a l f o r m of the Hamiltonian is given, without specifying the e x p r e s s i o n s f o r those t e r m s which a r e multiplied by the perturbing m a s s . Hence the t i m e - dependence of the motion is not t r e a t e d i n a s a t i s f a c t o r y manner.

~ a ~ i h a r a ( ~ ) l a t e r extended P o i n c a r 6 I s method t o t h e c a s e

where the m e a n motions a r e i n t h e r a t i o

kK,

J ^J and K being positive integers. Higher powers of the e c c e n t r i c i t y a r e r e t a i n e d i n t h e p e r - turbing t e r m s . However, i n t r e a t i n g the time-dependence of the

motion, s e v e r a l important perturbing t e r m s have i n c o r r e c t l y been neglected, a s the r e s u l t of not having o r d e r e d t h e s m a l l quantities i n a s y s t e m a t i c manner.

chuba art'^)

h a s published the r e s u l t s of extensive n u m e r i c a l computations for the n e a r l y commensurable c a s e of t h e r e s t r i c t e d three-body problem. In h i s work, the s h o r t - p e r i o d perturbations a r e removed by a n u m e r i c a l averaging p r o c e s s , and only the long-period effects a r e included i n the o r b i t a l elements. These r e s u l t s provide considerable insight into the qualitative and quantitative f e a t u r e s of the motion f o r a wide range of initial conditions.

The purpose of the work d e s c r i b e d i n t h i s t h e s i s i s t o demon- s t r a t e how the two v a r i a b l e expansion p r o c e d u r e m a y be u s e d t o obtain a solution which i s f r e e of s m a l l divisors. This method e s t a b l i s h e s the p r o p e r t i m e - l i k e v a r i a b l e f o r the long-period motion, and c l a r i f i e s the dependence of the amplitudes of the o r b i t a l e l e m e n t s on the s m a l l p a r a m e t e r i n the problem. Both the s h o r t - p e r i o d and long-period p e r t u r b a t i o n s of the motion of the infinitesimal body c a n be determined.

EQUATIONS OF MOTION

The p l a n a r r e s t r i c t e d t h r e e - b o d y p r o b l e m will be non-

dimensionalized by choosing t h e units of m a s s , length, and t i m e a s follows: the unit of m a s s i s c h o s e n i n s u c h a way that the l a r g e r of the two m a s s i v e bodies h a s m a s s 1 - p , and t h e s m a l l e r one h a s m a s s p,, where 0

<

^{p, 4} f o r a l l c a s e s ; the unit of length is c h o s e n s u c h t h a t the c o n s t a n t d i s t a n c e between the two m a s s i v e bodies, a s t h e y revolve i n t h e i r c i r c u l a r o r b i t s , i s e q u a l t o 1; the unit of t i m e i s c h o s e n s u c h that the constant angular velocity of the two l a r g e bodies about t h e i r c o m m o n c e n t e r of m a s s i s e q u a l t o 1.

The c e n t e r of m a s s will lie on t h e line joining t h e two l a r g e bodies, a t a d i s t a n c e p f r o m the body of m a s s 1-p. The c e n t e r of m a s s i s a s s u m e d t o be moving a t constant r e c t i l i n e a r v e l o c i t y with r e s p e c t t o a n i n e r t i a l f r a m e of r e f e r e n c e .

L e t the non-rotating X-Y coordinate s y s t e m have i t s o r i g i n fixed a t the c e n t e r of m a s s . T h i s f r a m e of r e f e r e n c e will be a n i n e r t i a l one. The line of c e n t e r s will r o t a t e about t h e m a s s c e n t e r with unit a n g u l a r velocity. Choose the angular o r i e n t a t i o n of the X-Y s y s t e m i n s u c h a way that the positive X a x i s c o i n c i d e s with the position of m a s s p a t t i m e t

=

0. The line of c e n t e r s then m a k e s a n angle t with t h e positive X axis.

The g e o m e t r i c a l s i t u a t i o n i s shown i n F i g u r e 1.

MASS I - !

F i g u r e 1. B a r y c e n t r i c Coordinate S y s t e m

* *

L e t t h e X

-

^Y s y s t e m be a non-rotating r e f e r e n c e f r a m e c e n t e r e d a t t h e body of m a s s 1-p. A s s e e n f r o m the i n e r t i a l f r a m e

* *

X - Y , the o r i g i n of c o o r d i n a t e s of the X

-

Y s y s t e m will move a t constant a n g u l a r velocity i n a c i r c l e of r a d i u s p about the c e n t e r of

*

^sg

m a s s , a n d hence the X

-

Y f r a m e i s not a n i n e r t i a l one. L e t the

x*-

Y* s y s t e m have the s a m e fixed a n g u l a r o r i e n t a t i o n a s does the X-Y s y s t e m . The positive X - a x i s will t h e n p a s s through t h e

*

position of m a s s p a t t = 0. T h e r e f o r e t h e line of c e n t e r s w i l l m a k e a n angle t with the positive X -axis. *c

The g e o m e t r i c a l s i t u a t i o n i n the

x*-

Y

*

s y s t e m i s shown i n F i g u r e 2.

F i g u r e 2. H e l i o c e n t r i c Coordinate S y s t e m

Let r denote the distance of the infinitesimal body f r o m the

$

*

origin of the X

-

Y s y s t e m , and let 8 denote the angle f r o m the positive X

*

a x i s to the r a d i u s v e c t o r of the infinitesimal body. The distance between the infinitesimal body and the body of m a s s _p i s

&

then equal t o [l+r2-2r c o s ( 8 - t d

.

The equations of motion of the infinitesimal body m a y e a s i l y be derived in t e r m s of r and 8 , c o n s i d e r e d a s functions of the t i m e t.

d r

They a r e a s follows ( w h e r e r = etc. ) :

In applying the two v a r i a b l e expansion p r o c e d u r e that will l a t e r be used t o solve t h e s e equations, a different s e t of v a r i a b l e s i s m o r e useful. The new f o r m of the equations will make i t e a s i e r t o t r e a t i n a p r o p e r m a n n e r the t e r m s which would otherwise produce s m a l l divisors.

Introduce the v a r i a b l e

Then t r a n s f o r m t o 8 i n s t e a d of t a s the independent variable, s o that s = ~ ( 8 ) ~ t

=

t(8)

.

This may be done by rneans of t h e following relations:

The equations of motion f o r the planar r e s t r i c t e d three-body p r o b l e m then a s s u m e the following f o r m :

It i s s e e n that both the time t(8, p) and the independent variable 8 a p p e a r explicitly i n the equations of motion, i n the t e r m s which involve sin(8-t) and cos(6-t). The p r o b l e m i s t h e r e - f o r e non-autonomous.

* *

Because of the manner i n which the orientation of the X

-

Y axes was specified, the initial condition on t i s a s follows:

where 0, i s the initial angle between the r a d i u s vector t o the infini- t e s i m a l body and the line of c e n t e r s of the two m a s s i v e bodies.

The t e r m s which involve

[

1

+

s 2 - 2 s c o s ( 9 - t )

I ^-

^{2 3'} lead to the o c c u r r e n c e of s m a l l divisors. These t e r m s r e p r e s e n t the gravitational a t t r a c t i o n of the body of m a s s y upon the infinitesimal body. The

dt on the r.h.s. of eq. ( 5 ) o c c u r s a s a r e s u l t of having t e r m - y ( s

I

chosen l-yI instead of 1 , f o r the m a s s of the l a r g e r body. The r e - maining t e r m s on the r.h. s. of eqs. ( 4 ) and ( 5 ) a r e "apparent f o r c e s "

which r e s u l t f r o m the f a c t that the X*

- ^Y*

s y s t e m i s not a n i n e r t i a l r e f e r e n c e f r a m e . T h e s e "apparent f o r c e s " do not l e a d to s m a l l divisors.

Eqs. (4) and ( 5 ) a r e a n exact m a t h e m a t i c a l r e p r e s e n t a t i o n of the planar r e s t r i c t e d three-body problem, valid f o r a l l values of

O < y

< $ .

T h e s e equations p o s s e s s one exact integral, the well-known J a c o b i integral: ^~

where

C

depends only on the initial conditions.

In the r e m a i n d e r of t h i s work, i t will be a s s u m e d that ~<p.<<i.

The quantity y m a y then be t r e a t e d a s a s m a l l p a r a m e t e r i n the equations of motion.

METHOD AVOIDING SMALL DIVISORS

The o c c u r r e n c e of s m a l l d i v i s o r s in the v a r i a t i o n of constants t r e a t m e n t of the p r o b l e m r e s u l t s f r o m having neglected the v a r i a t i o n

of the m e a n motion, and the other o r b i t a l e l e m e n t s , while c a r r y i n g out the integration of the p e r t u r b a t i o n equations. The s m a l l d i v i s o r s a r e produced by the integration of t e r m s whose p e r i o d i s v e r y l a r g e c o m p a r e d t o the o r b i t a l p e r i o d of the infinitesimal body. This

suggests the existence of a second t i m e scale, the "slow-time" scale, over which i m p o r t a n t changes occur i n the o r b i t a l elements.

The physical r e a s o n f o r the o c c u r r e n c e of the difficulty i s the f a c t that the p e r t u r b i n g f o r c e i s n e a r l y r e s o n a n t with the motion of the infinitesimal body. This n e a r - r e s o n a n c e a s p e c t of the motion will now be d i s c u s s e d briefly.

A s s u m e t h a t the infinitesimal body moves i n a n e l l i p t i c a l o r b i t about the l a r g e r m a s s 1-p. T h i s e l l i p t i c a l o r b i t will be p e r t u r b e d by the gravitational f o r c e e x e r t e d by the m a s s p. The distance between the infinitesimal body and the p e r t u r b i n g body will be a p p r o x i m a t e l y a periodic function of time, s o that the p e r t u r b i n g f o r c e i s a l s o n e a r l y periodic. If the o r b i t a l p e r i o d of the infinitesimal body i s approx- i m a t e l y a r a t i o n a l f r a c t i o n of the o r b i t a l p e r i o d of the p e r t u r b i n g body, the perturbing f o r c e o s c i l l a t e s with a n e a r l y r e s o n a n t frequency. The i m p r o p e r m a t h e m a t i c a l t r e a t m e n t of this n e a r - r e s o n a n c e l e a d s t o the o c c u r r e n c e of s m a l l d i v i s o r s .

The p r o b l e m a t hand is t o d e r i v e a s e t of equations which gives a n adequate d e s c r i p t i o n of the behavior of the o r b i t a l e l e m e n t s ,

in the p r e s e n c e of the n e a r l y - r e s o n a n t perturbing f o r c e s .

1. Justification for Use of the Two Variable Expansion P r o c e d u r e The two v a r i a b l e expansion procedure has been d i s c u s s e d i n the l i t e r a t u r e by Cole and ~ e v o r k i a n , ' ~ ) and by Kevorkian. ( 6 ) 1t is a

syste'matic method of constructing an expansion, of the solution of a n o r d i n a r y differential equation containing a s m a l l p a r a m e t e r , which r e m a i n s valid f o r l a r g e values of the independent variable. T h i s method is e s p e c i a l l y useful i n p r o b l e m s where a s m a l l perturbing f o r c e produces important effects which occur over a t i m e s c a l e that is l a r g e c o m p a r e d t o the t i m e s c a l e of the m a i n f e a t u r e s of the motion.

In applying the two v a r i a b l e procedure, it is a s s u m e d that t h e exact solution m a y be r e p r e s e n t e d by a n expansion which depends explicitly upon two different t i m e ( o r t i m e - l i k e ) v a r i a b l e s , a "fast t i m e " v a r i a b l e and a "slow t i m e " variable. The use of two different v a r i a b l e s introduces a n indeterminacy into the v a r i o u s t e r m s of the expansion. This indeterminacy is r e m o v e d by r e q u i r i n g t h a t t h e a s s u m e d f o r m of the expansion m u s t he self-consistent.

When t h e two v a r i a b l e expansion p r o c e d u r e is applied t o the planar r e s t r i c t e d t h r e e -body problem, t h e o r b i t a l e l e m e n t s will exhibit only long-period effects. Short-period p e r t u r b a t i o n s will be taken into account by the second t e r m of t h e expansion. However, it is p r e c i s e l y i n the long-period effects t h a t the fundamental difficulty of t h e p r o b l e m lies. Thus the u s e of the two v a r i a b l e expansion p r o c e d u r e l e a d s d i r e c t l y t o a study of the b a s i c difficulties of t h e p r o b l e m

The v a r i a t i o n of constants approach yields both s h o r t - p e r i o d and long-period e f f e c t s i n the o r b i t a l elements. The s h o r t - p e r i o d effects m u s t be r e m o v e d before the fundamental difficulty of the p r o b l e m c a n be studied.

2. The F o r m of the Expansions

F o r p<<i, the t e r m s on the r. h. s. of eqs. ( 4 ) and ( 5 ) m a y be t r e a t e d a s s m a l l perturbations, provided that [1+ s 2 - 2s cos(8- t)] 3/2 does .not become a r b i t r a r i l y small. T h i s i m p l i e s that the infinit- e s i m a l body m u s t not make a "close approach" t o the body of m a s s p.

Close a p p r o a c h e s cannot occur f o r o r b i t s which lie e n t i r e l y withinthe orbit of the p e r t u r b i n g body; i.e, f o r o r b i t s having s(8, p)>1 for a l l 8.

F o r c a s e s where s(8, p)<1 during p a r t of the orbit, the p e r t u r b a t i o n s will r e m a i n s m a l l only if [ l + s 2 - 2 s cos(8-t)] 3/2 r e m a i n s bounded away f r o m zero. '

Orbits f o r which [ l + s 2 - 2 s cos(8-t)] 312 a p p r o a c h e s 0 will not be c o n s i d e r e d i n t h i s work.

The solution of eqs. ( 4 ) and ( 5 ) will be sought by u s e of the two variable expansion p r o c e d u r e i n the following f o r m :

where the slow v a r i a b l e i s

( 9 )

8 = ~ 4 e

The e s s e n t i a l f e a t u r e s of the difficulty of s m a l l d i v i s o r s occur

N N

i n the t e r m s of 0 ( p ) ; i.e. i n the solutions f o r s1(f3, 8, p) and t (0, 0, p).

1

Hence, f o r the purpose of resolving the b a s i c difficulty, the t e r m s of higher o r d e r i n p m a y be neglected.

Derivatives a r e to be calculated by the r u l e

The following expansions a r e obtained by applying t h i s derivative r u l e t o expansions ( 8 a ) and (8b):

Applying the derivative r u l e again,

T h e s e expansions m a y be used t o e x p r e s s the 1. h. s. of eqs. ( 4 ) and (5). retaining a l l t e r m s of O(pO), O(p 1/2 _), and O(p).

It is now n e c e s s a r y t o d i s c u s s the manner i n which the p e r - turbing t e r m s on the r.h. s. of the equations of motion m a y be expanded

4'

i n p o w e r s of p. Since only the t e r m s of O(pO), 0 ( p 2 ), and O(p) a r e

to be retained, i t i s sufficient t o use the O(p 0 ) approximation t o the quantities i n b r a c e s on the r. h. s. of eqs. ( 4 ) and (5).

The t e r m s which involve powers of s d s d t

3

and

TiB

m a y be ex- panded a s above. The only r e m a i n i n g t e r m s a r e t h o s e which involve s i n ( 8 - t ) and c o s ( 8 - t ) .

By the expansion f o r t(8, p.) we have

The two v a r i a b l e expansion p r o c e d u r e will be used t o make tl(O, 0, H P)

N

a bounded function of 0. T h e r e f o r e pt1(0. 0, p) will r e m a i n a quantity of O(y), and m a y be dropped f r o m eq. (13), s o that

(14)

&(e-t) = & [ ( e - ~ ) -AC +&@I

= ACvz(e-to)

^f

B(U$)

= A ( e - $ ) + ^B,+)

Similarly,

(15)

a(*-t) ⁼ m ( e - t ) + 8 ( ~ )

The following expansion i s t h e r e f o r e valid f o r the t e r m s on the r. h. s. of eq. (4):

A s i m i l a r expansion i s v a l i d f o r t h e t e r m s on the r. h. s , of eq. (5).

Thus t h e p e r t u r b a t i o n t e r m s of O(p) involve only t h e quantities

N N

so(O, 0, p) and to(O, 0, p) and t h e i r d e r i v a t i v e s . However, t h i s

N

a p p r o x i m a t i o n will be valid only if i t c a n be shown that t (8, 1 0, p) a n d

N

s (8, 8, p) a r e indeed bounded functions of 8, 1

3. Solution of the O(pO) Equations

The t e r m s multiplied by p 0 i n t h e equations of motion l e a d to the following equations:

T h e s e a r e ' t h e equations of K e p l e r i a n motion. That i s , i f the pertu-rbing m a s s p w e r e e q u a l t o zero, the i n f i n i t e s i m a l body would d e s c r i b e a n u n p e r t u r b e d K e p l e r i a n o r b i t about t h e l a r g e m a s s .

In t h i s work only d i r e c t o r b i t s will be c o n s i d e r e d . That i s , i t will be a s s u m e d t h a t both the i n f i n i t e s i m a l body and the p e r t u r b i n g m a s s p r e v o l v e about t h e l a r g e m a s s i n a counterclockwise d i r e c t i o n ( s e e F i g u r e 2).

Eqs. (17) and (18) will be solved, r e g a r d i n g 0 a n d a s being two e n t i r e l y different v a r i a b l e s . Eq. (17) h a s the solution

where

N

a = a(8, p) = s e m i m a j o r a x i s of the orbit of the infinitesimal body e = e(0, p) = e c c e n t r i c i t y of the orbit of the infinitesimal body

~ q . (19) defines the angular momentum of the orbit. F o r r e t r o g r a d e orbits, eq. (19) would be r e p l a c e d by s, 2 2 -1/2

W =

-a-V2(1-e )

.

Only e l l i p t i c a l o r b i t s ( 0 ,< e

<

1) will be c o n s i d e r e d here. P a r a - bolic and hyperbolic o r b i t s ( e 3 1) do not produce the difficulty of

s m a l l d i v i s o r s , because the motion of the infinitesimal body is not periodic i n t h e s e c a s e s .

Eq. (18) becomes

The g e n e r a l solution of this equation i s

where A and B , a r e a r b i t r a r y functions. In t e r m s of the K e p l e r i a n o r b i t a l e l e m e n t s , t h e s e functions a r e

where

w = w(8,

-

p) = longitude of p e r i c e n t e r of the o r b i t of the infinitesimal body

Therefore,

s o that

(23)

H

The quantity tO(O, 8, p) m a y be obtained f r o m the r e l a t i o n

where

w

Eq. (22) i s s t i l l s a t i s f i e d if a n a r b i t r a r y function of 0 is added t o

If one expands the integrand on the r. h. s. of eq. (23) i n a

w

Taylor s e r i e s about e = 0, and then holds 0 fixed while c a r r y i n g out the i n t e g r a l w. r. t. 6, the following e x p r e s s i o n i s obtained:

sinusoidal functions

,

multiplied by e, e2, e3,

where T(;, p) i s a n a r b i t r a r y function which defines the position of the infinitesimal body i n i t s orbit.

4. O c c u r r e n c e of S m a l l Divisors i n s , and t ,

N

The unbounded p a r t of to(O, 8, p) i s e n t i r e l y contained i n the quantity [ T

+

^2/20]. T h e r e f o r e

s h o r t - p e r i o d s i n u s oidal functions (25)

@-c) ⁼

^=(/-)B

^{-7- f}

^{{of 0,} multiplied by e, e2, e3, etcLf

It follows that

( 2 6 ) s h o r t - p e r i o d sinusoidal functions

&, ,, (*-t)= [~-$)@-d

^{f- {of 0,} multiplied by e, e2, e3, etc]

A s i m i l a r expansion would be valid f o r c o s ( 8 - t ).

0

N

T h e r e f o r e , if eq. ( 2 4 ) w e r e used f o r to(O, 8, p) i t would be

N N

found that the equations f o r sl(O, 0, p) and tl(8, 0, p) would contain f o r c i n g functions which would involve sin[ (1-a 3/2 )8- T] and

3/2 ^N

cos [ (1-a )8- T]

.

^{Since 0} i s held fixed during the i n t e g r a t i o n s w. r. t.

8, the quantity (1-$I2) would appear a s a constant frequency. In combination with other f r e q u e n c i e s which a r e p r e s e n t i n the p e r t u r b i n g t e r m s , t h e s e t e r m s would produce sinusoidal functions of 8 having f r e q u e n c i e s c l o s e t o z e r o and o t h e r s with f r e q u e n c i e s c l o s e t o 1, f o r c e r t a i n values of a312. Upon integration w.r.t. 0, t h e s e t e r m s would produce s m a l l d i v i s o r s i n s1 and t

1 '

By e x p r e s s i n g the perturbing t e r m s a s functions of 0 and the o r b i t a l e l e m e n t s a3I2, e , w , T, and then expanding i n periodic s e r i e s 1 t o d e t e r m i n e which f r e q u e n c i e s occur, i t m a y be shown that s m a l l

N

d i v i s o r s would o c c u r i n sl(O,

z,

p) and tl(6. 8, p) f o r d i r e c t e l l i p t i c a l o r b i t s i n those c a s e s where the s e m i m a j o r a x i s h a s a value s u c h t h a t

where n and m a r e r e l a t i v e l y p r i m e positive i n t e g e r s , with n > rn

.

It m a y a l s o be shown that the perturbing t e r m s which a r e multiplied by the f i r s t power of the e c c e n t r i c i t y would produce s m a l l d i v i s o r s only f o r c o m m e n s u r a b i l i t i e s with m = 1; the p e r t u r b i n g

t e r m s multiplied by e 2 would produce s m a l l d i v i s o r s f o r both the m = l and m=2 c a s e s ; those multiplied by e 3 would produce s m a l l d i v i s o r s f o r the m=l, m=2, and m=3 c a s e s ; etc. Correspondingly, one would expect the behavior of the o r b i t a l e l e m e n t s t o be somewhat d i f f e r e n t f o r the v a r i o u s values of m.

F o r brevity, this a n a l y s i s will not be c a r r i e d out here. How- e v e r , i t should be mentioned that the o c c u r r e n c e of s m a l l d i v i s o r s in the above f o r m i s equivalent t o the corresponding difficulty encountered in the v a r i a t i o n of constants t r e a t m e n t of the problem.

Although r e t r o g r a d e (clockwise) elliptical o r b i t s will not be d i s c u s s e d h e r e , s m a l l d i v i s o r s would occur f o r c e r t a i n c a s e s where

2/2

i s the r a t i o of two positive i n t e g e r s . T h e s e s m a l l d i v i s o r s could be avoided by a method s i m i l a r t o that which will be d i s c u s s e d i n the next section.

5. Explicit Inclusion of Commensurability i n the Expansions

A s d i s c u s s e d above, s m a l l d i v i s o r s would occur if the s e m i - n- m m a j o r a x i s i s s u c h that a3/2(& p) i s n e a r one of the values -,

n This suggests that the n e a r - c o m m e n s u r a b i l i t y should be taken into account f r o m the outset, and that the s e m i m a j o r a x i s should be ex- panded i n the f o r m

The corresponding derivative i s

N

The e x p r e s s i o n f o r to($, 0, p) m u s t now be r e - e x a m i n e d , taking into account expansion (27). The e x p r e s s i o n given i n eq. ( 2 4 ) w a s ob- tained by holding the s l o w v a r i a b l e

-

8 fixed while c a r r y i n g out the i n t e g r a t i o n w. r. t. 0. Such a p r o c e d u r e is v a l i d f o r t h e t e r m s which d o not givk r i s e t o unbounded quantities p r o p o r t i o n a l t o 8. T h e r e f o r e

s i m i l a r s h o r t - p e r i o d s i n u s o i d a l functions of 6, multiplied by e3, e4,

- - -

T h e r e i s no non-uniform a p p r o x i m a t i o n t o the unbounded p a r t of

-

3 4

to(B, 0, p) c a u s e d by dropping the t e r m s multiplied by e

,

e

, - - - .

s i n c e the i n t e g r a l s of a l l s u c h t e r m s w.r.t. 0 a r e bounded.

Using eq. (27) f o r a , p , one obtains

If the i n t e g r a l on the r.h.s. of eq. ( 3 0 ) c a n be e x p r e s s e d a s a

M N

function of 8 alone, r a t h e r t h a n a s a function of both 0 a n d 0, i t will be p o s s i b l e t o d i s t i n g u i s h between the unbounded behavior of

H

to(8, 0, p) which i s p r o p o r t i o n a l t o 0 and t h e unboundedness which i s

N

p r o p o r t i o n a l t o 0

.

T h i s will make i t possible t o avoid the o c c u r r e n c e

N N

of s m a l l d i v i s o r s i n s l ( O , 0, p) and tl(O, 0, p)

.

- ^]/2

T o a c c o m p l i s h t h i s i t i s n e c e s s a r y t o u s e the r e l a t i o n 0=p 6 when c a r r y i n g out t h e i n t e g r a l on the r.h.s, of eq, (30). T h e r e f o r e

Introduce the notation

Eq. ( 2 9 ) m a y now be written a s follows:

+ 3 % s i m i l a r s h o r t - p e r i o d sinusoidal of 0, m u l t i p l i e d b y e 3 , e 4 ,

- - -

F o r brevity, the following notation will be used, whenever it is con- ve nie nt:

The corresponding derivative i s

Eq. (33) t h e n b e c o m e s

s h o r t - p e r i o d sinusoidal functions by e3, e$

- - - . 3

This e x p r e s s i o n will be u s e d f o r t f r o m this point on.

0

The t e r m

--

( n - m ) 9 r e p r e s e n t s the unbounded behavior of t

n o

H

which is proportional t o 6,and (P(9, p) r e p r e s e n t s a possible unbound- edness of to on the

;

^{scale. A} g e o m e t r i c a l i n t e r p r e t a t i o n of @ will be given l a t e r .

Having e x p r e s s e d t by eq. (36) i t i s n e c e s s a r y t o e x p r e s s a t

0 a t o

the de rivative s

a 3

^{and -NI} 0 i n a self-consistent manner. The

ae

f o r m e r i s given by

By the derivative r u l e (10) we expect that

F o r m a l l y applying the derivative r u l e t o eq. (33), i t is found that

(38b)&(t)= mim+~'(&+iy+

of s h o r t - p e r i o d t e r m s

3

a

derivatives of s h o r t - p e r i o d t e r m s

3

F r o m eqs. (37), (38a), and (38b) i t follows that

( 3 9 )

.& - ^dr +

(&derivatives of s h o r t - p e r i o d

dG -a

-

^&-;%

+

- ^{di3 a}

derivatives of s h o r t - p e r i o d t e r m s

3

The quantity T(;, p) should be r e g a r d e d a s the fourth o r b i t a l

N

element, The quantity @(8, p) i s completely defined i n t e r m s of

T and

g3I2

^{by eq. (34).}

6. Geometrical Significance of @(;, p.) Using the approximation

i t follows that

s i m i l a r s hort-period sinusoidal functions + {of

6 ,

multiplied by e 3

,

^{e 4}

., - - -

The quantity ( 8 - t ) r e p r e s e n t s the angle f r o m the line of c e n t e r s of the two l a r g e m a s s e s t o the radius vector of the i d i n i t e s i r n a l body.

The g e o m e t r i c a l situation is shown i n F i g u r e 3.

Y*

F i g u r e 3. Geometry of the Orbit

M N

The e l e m e n t s a(8, p ) and e(8, p) specify the s i z e and shape of

U

the slowly-varying elliptical orbit. The longitude of p e r i c e n t e r 4 0 , p) specifies i t s angular orientation. The quantity

4(?,

^p) specifies the position of the infinitesimal body i n i t s orbit.

- Consider the g e o m e t r i c a l situation which o c c u r s e v e r y n t h time the infinitesimal body i s a t p e r i c e n t e r . Between two s u c h o c c u r - ences, the i n f i n i t e s i m a l body will have completed exactly n revolutions i n i t s elliptical orbit, and the m a s s y will have completed approx- imately ( n - m ) revolutions i n i t s c i r c u l a r orbit. At e a c h s u c h instant,

0 = w ( ~ A ) + & ~ & T

; p a non-negative integer

s o that eq. (41) becomes

The simple f o r m of eq. (42) r e s u l t s f r o m the fact that e a c h of the s h o r t - p e r i o d t e r m s i n t o vanishes when 8 = a S p * 2 n n. The geo- m e t r i c a l situation when the infinitesimal body i s a t p e r i c e n t e r is shown i n F i g u r e 4.

F i g u r e 4. G e o m e t r i c a l Significance of (- m _n

w - 4 )

Thus the quantity ( m - ) i s equal to the angle between the p e r i c e n t e r of the infinitesimal body and the position of the m a s s p., m e a s u r e d e v e r y n t h t i m e the infinitesimal body i s a t p e r i c e n t e r . 7. Dependence of the Orbital E l e m e n t s on p

N

The e c c e n t r i c i t y i s a s s u m e d t o depend on 8 and p. i n the following m a n n e r :

(43)

e(q~() = eo +A% (q4)

; e o a constant The corresponding derivative i s

In c e r t a i n c a s e s i t will be possible to u s e the approximation

e = e o t O(p.

l/z

) However, if e o i s sufficiently s m a l l , i t i s n e c e s s a r y t o r e t a i n both t e r m s on the r. h. s. of eq. (43).

N

The quantities w and r a r e both unbounded functions of 8, i n

N

general. They will be a s s u m e d t o depend on 8 and p i n the following manner :

(45

1 w ( q ~ ) =

W , + A ' ~ ( ~ M ) ; a. a constant (46) T ( ~ A )

= 7; +~'fel/)

; T O a constant

The corresponding d e r i v a t i v e s a r e a s follows:

It i s not n e c e s s a r y t o a s s u m e i n advance that eo9w

,

and 70

0

a r e constants. However, i f one begins with eqs. (43), (45), and (46) d e o -

i t wil-1 be found t h a t ^--7;-

-

0, = 0, d T O

- -

⁰ ~y a s s u m i n g

d 8 do do

eo,w

,

and 'rO t o be constants f r o m the outset, t h e s e u n n e c e s s a r y

0

calculations a r e avoided.

A

-

^{A N}

The quantities ~ ( 6 , p) and ~ ( 8 , p) will be unbounded functions

N

0

of 8 i n general. Hence i t i s not c o r r e c t to write o = w

+

O(p

J/z

) o r

T

= r o t

O(p 1/2 ) Both t e r m s on the r. h. s. of eqs. (45) and (46) m u s t be retained.

By substitution of the expansions (11) and (12) i n t o eqs. ( 4 ) and (5), the following equations a r e obtained f r o m the t e r m s which

v 2 a r e f o r m a l l y of O(y ):

It will now be shown that because of the f o r m of the expansions 3/2 de

da dw d r

0 I;;

, ,

and -

,

:the t e r m s which occur i n eqs. (49)

d o do do d0

and (50) a r e actually of

0(d2),

i n s t e a d of O(p 0 ). By eq. (19).

F r o m eq. (21), it follows that

By c a r r y i n g out the indicated derivatives i n eq. (391, and

2 - 2 2 - 2 2

then multiplying b y s = a ( l - e ) [ 1 t e c o s ( 6 - a ) ]

,

the following

0

r e s u l t i s obtained:

z

at,

4

-9 _~2p+(wl

(54)

4

_=A5

[(&$-&)g+(%+~(@$e&]+M kz&

^8-a2

4

e2

di?

t e r m s multiplied

b y e 3 , e 4 ,

- - - I

Note: T h e r e is no equation (53).

By differentiation of eq. (54) w. r. t. 0, i t follows that

Thus, e a c h t e r m which o c c u r s in eqs. (49) and ( 5 0 ) i s v 2

actually of 0 ( p ) r a t h e r than O(yO). T h e s e t e r m s m u s t t h e r e f o r e be included i n the O(p) equations. Hence t h e r e a r e no O(y

]/z

) e q - uations t o solve.

8. The O(p) Equations

By u s e of eqs. 1 1 , ( 2 , (1 6 , ( 4( 5 0 , and (55) i t m a y be shown that the O(y) t e r m s of the equations of motion l e a d to the foll- owing equations :

+

- 2 e A w

& ,-A

emu

da%+

r&e?

d'&

[a(,-@)

diF 3

- a]

.a$

2

at.

"'@%)[@gf 2dkI@+& G ~ ~ ~ ) ]

f ~ ~ @ ~ c & i ( * - ~ )

-

^(A-

=-

[/-~~-c.-t;)f-g$&(e-tJ [I+A,Z-~A.C~~('-~,I%

The quantity is of o(ELO), a s m a y be s e e n f r o m eq.(54).

1 2

The notation

7

^{( 6}

-)

i s m e r e l y a convenient way of writing t h i s

)L O

a 5

t e r m .

Note: T h e r e i s no equation (56).

9. S e r i e s Expansion of the P e r t u r b i n g T e r m s

In o r d e r t o e x p r e s s the perturbing t e r m s which involve sin(8-to) and cos(8-t ) i n a useful f o r m , i t i s n e c e s s a r y t o expand these quan-

0

t i t i e s i n powers of e. The amount of a l g e b r a i c labor that is r e q u i r e d i n c r e a s e s v e r y rapidly a s higher powers of e a r e retained. F o r t h i s

3 4

reason, a l l t e r m s multiplied by e ,

,

^e

, ---

will be neglected i n the r e m a i n d e r of t h i s work. F o r o r b i t s with s m a l l e c c e n t r i c i t i e s , t h i s should yield a reasonable approximation. The approximation could be improved i n a s t r a i g h t f o r w a r d manner, me r e l y b y retaining the higher powers of

e.

Using eq. (36) f o r to, the quantity sin(0-t ) m a y be expand-

0

e d i n powers of e a s follows:

(59)

-(*-c) ⁼ .din($e-4)

-I-

~ P e k ( B - ~ ) ~ & e - + )

-~a%~!~&?(e-w)~~@$?+-(6)+{

s i m i l a r sinus oidal functions

f

of 8, multiplied by e

The quantity c o s ( 8 - t o ) m a y be expanded i n a s i m i l a r form.

The perturbing t e r m s on the r.h.s. of eqs. (57) and (58) m a y then be expanded i n powers of e. F o r example,

s i m i l a r sinus oidal functions f {of 8, multiplied by e3, e4,

- -1

3

Similar expansions can be made f o r the t e r m s -s:

(%)

s i n ( 8 - t o ) and

The expansions of

and (s: %)I [1-so cos ( 9 - t o i n

powers of e a r e quite lengthy, and a r e t h e r e f o r e given i n the appendix.

The r.h.s. of eqs. ( 5 7 ) and ( 5 8 ) have now been e x p r e s s e d a s functions of 6 and the o r b i t a l e l e m e n t s 2/2.e,w, and

4.

^However,

t h e integration of t h e s e equations cannot be c a r r i e d out explicitly with the r.h.s. i n its p r e s e n t form.

A convenient way t o c a r r y out the integration is t o e x p r e s s the v a r i o u s periodic functions of 9 i n t h e i r F o u r i e r s e r i e s ' expansions, and then t o i n t e g r a t e t h e s e s e r i e s 1 t e r m w i s e . The use of F o u r i e r s e r i e s 1 identifies the v a r i o u s frequencies which occur i n t h e p e r t u r b - ing t e r m s , t h e r e b y making i t possible t o identify and r e m o v e the t e r m s which would otherwise produce quantities proportional t o 8 i n

s l and t l .

T h e r e a r e s e v e r a l ways i n which the F o u r i e r expansions could be c a r r i e d out. The one that will be used h e r e is convenient when one wishes t o determine the n u m e r i c a l values of the F o u r i e r coefficients.

It i s sufficient t o u s e the following t h r e e F o u r i e r s e r i e s expansions:

The F o u r i e r coefficients a r e given by

N

f o r k = 0,1, 2,

- - - .

The value of a(@, p) i s held fixed i n c a r r y i n g out these integrations with r e s p e c t t o x.

If a l l the perturbing t e r m s multiplied by e 3 w e r e retained, it

2 - 9 / 2

would be n e c e s s a r y t o e x p r e s s the quantity [I+ a -2a C O S ( ~ D - $ I ) ] n

i n i t s F o u r i e r expansion. In general, one additional F o u r i e r expansion of the above type i s r e q u i r e d f o r e a c h additional power of e t h a t i s retained i n the perturbing t e r m s .

The s e r i e s r e p r e s e n t a t i o n of e a c h perturbing t e r m c a n be ob- tained f r o m the above F o u r i e r expansions, by t e r m w i s e multiplication.

F o r example,

S i m i l a r expansions can be made f o r e a c h of the perturbing t e r m s . These F o u r i e r coefficients m a y be e x p r e s s e d i n t e r m s of the hyper g e o m e t r i c function. F o r example,

S i m i l a r e x p r e s s i o n s a r e valid f o r Bk(a) and Ck(a). They m a y a l s o be e x p r e s s e d i n t e r m s of the complete elliptic i n t e g r a l s of the f i r s t and

second kinds, K(a) and E ( a ) , respectively. The r e c u r s i o n r e l a t i o n s f o r the hypergeometric function m a y be u s e d t o prove c e r t a i n relation- s h i p s between the F o u r i e r coefficients.

I n o r d e r t o obtain r e s u l t s r e l a t e d t o the behavior of the o r b i t a l e l e m e n t s f o r a specific n u m e r i c a l value of y , i t is n e c e s s a r y to know the n u m e r i c a l values of the F o u r i e r coefficients. T h e s e c o

-

efficients could be calculated d i r e c t l y f r o m the definitions i n eqs. (62a), (62b), and (62c), by n u m e r i c a l integration over the r a n g e 0 6 x ,( 2 n.

However, these values m a y a l s o be obtained f r o m extensive t a b l e s published by Brown and Brouwer (7) T h e s e t a b l e s give n u m e r - i c a l values of G ( ~ ) (a), G ( ~ ) (a), and G ( ~ ) ( a ) f o r 0.0 6 a S 0.845,

3/2 5/2 7/2

where

f o r k = 0,1, 2,

- - - .

The quantities G3/2, G5/2. ^{( k ) ( k ) and G} ( k ) a r e 7/2

known a s Laplace coefficients.

10. Removal of Resonant P e r t u r b i n g T e r m s

2 a t l at, 1 2 a t

The quantity

[

^{(so -t 2sos1 -t (So}

g)]

m u s t be P

known explicitly i n t e r m . s of 8 before eq. (58) c a n be solved. Hence eq. (57) will be solved f i r s t . After e x p r e s s i n g e a c h of the p e r t u r b i n g t e r m s a s d i s c u s s e d above, eq. (57) c a n be w r i t t e n i n the following f o r m :

where the bounded function h i s composed of t e r m s of the following 1

types :

0 2

( a ) s e v e r a l infinite s e r i e s 1 which a r e multiplied by e

,

^{e, e}

,

etc. and which contain sinusoidaJ functions of 8, whose f r e q u e n c i e s a r e independent of 8.

_{- -}

T h e s e infinite s e r i e s

'

r e s u l t f r o m the ansion of the t e r m 2 1

- 2 s o c o s ( 8 - t o )

I

s i n ( @ - t ) i n powers of e.

o

( b ) sinusoidal functions of

6

which r e s u l t f r o m the expansion 3 a t o 3

of - s o

( w )

s i n ( @ - t ) i n powers of e.

0

In c a r r y i n g out the integration of eq. (65) w. r. t. 0, the slow v a r i a b l e will be held fixed. T h e r e f o r e any t e r m which depends only

N

on 8 (i.e. which i s independent of -

8)

would produce a n unbounded t e r m proportional t o 8 i n the quantity 2 "o). T h i s would lead t o the o c c u r r e n c e of s i m i l a r unbounded t e r m s i n

s j Q <

^p)

and tl(8,

g,

^p), c o n t r a r y t o the assumptions of the o r i g i n a l two v a r - iable expansion,

S e v e r a l t e r m s which a r e independent of 0 will occur i n the infinite s e r i e s 1 . These a r e the t e r m s which produce s m a l l d i v i s o r s i n the v a r i a t i o n of constants solution. F o r example, i f the i n t e g e r s m and n have values such that t h e r e e x i s t s a non-negative integer k

n n

such t h a t

- -

^{1 = k ,} then the (--1)th t e r m of s e v e r a l of the infinite

m m

s e r i e s ' will contain the quantity

Each of the s e r i e s t will contain one o r m o r e t e r m s of the above type, depending upon the values of m and n. By a c a r e f u l inspection of the s e r i e s ' which occur on the r.h.s. of eq. ( 6 5 ) , the s u m of a l l s u c h t e r m s m a y be determined.

F r o m t h i s point on, only the c a s e m = 1 will be d i s c u s s e d i n detail. This i s the most important c a s e for c o m p a r i s o n of the r e s u l t s with the motion of a s t e r o i d s .

In o r d e r that 2

will not contain a t e r m proportional t o 0, the s u m of a l l t e r m s on the r.h.s. s f eq. (65) which a r e independent of 0 must vanish. This r e q u i r e m e n t yields the following equation:

++

^{e2dinr @-a#)}

⁺

s i m i l a r t e r m s multiplied by e 3 , e 4 ,

- - - J

The quantities Q, and

p

a r e functions of a 3/z only, and a r e de

-

fined i n the appendix. They a r e the s u m of s e v e r a l of the F o u r i e r

coefficients, e a c h multiplied by s o m e power of a 3/2

.

F o r the c a s e rn = 2, the r. h. s. of eq. (67) would not contain a t e r m multiplied by e ; the leading t e r m would be multiplied by e 2

.

F o r m = 3, the leading t e r m would be multiplied by e 3

,

^etc.

After the t e r m s which a r e independent of 0 have b e e n r e m o v e d by means of eq. (67), eq. (65) c a n be integrated with r e s p e c t t o 8, holding

5

^fixed. The r e s u l t will be f r e e of s m a l l divisors, but will not be written out explicitly here.

The e x p r e s s i o n f o r the i n t e g r a l of eq. (65) c a n then be sub- stituted into eq. (58). The r e s u l t will be a s follows:

where t h e bounded function h2 contains t e r m s of the following types:

0 2

(a) s e v e r a l infinite s e r i e s 1 which a r e multiplied by e , e, e etc.

,

and which contain sinusoidal functions of 0 whose frequencies a r e independent of

g.

These s e r i e s f r e s u l t f r o m the expansion of t h e quantity

+xi-

i n powers of e, and a l s o f r o m the

ae

( b ) sinusoidal functions of 8 which r e s u l t f r o m the expan

-

s i o n of the quantities s cos(0-t ) and so

88

0

*sin($-t ) i n powers of e, and a l s o f r o m the c o r r e s p o n d -

0

a

to 1 s2

ing t e r m . contained i n

[(< ₊

_2s04

_{a8 +}

_{T ( o P}

G)].

2

If a t e r m i n s i n 8 or cos 8 w e r e t o occur on the r.h. s. of eq.

(68), the r e s p o n s e t o this t e r m would contain the unbounded quantity 8 s i n

6

o r 9 c o s 8. This would c l e a r l y be a resonance effect, and

N

would violate the assumption that psl(8, 8, p) r e m a i n s a s m a l l quantity

S e v e r a l s u c h t e r m s i n s i n 8 and cos 8 a r e contained i n the infinite s e r i e s ' . F o r example, if the i n t e g e r s m and n have values s u c h that t h e r e e x i s t s a non-negative integer k f o r which

-

2n _m ^-1

⁼

^k, the (- 2n -1)th t e r m of s e v e r a l of the infinite s e r i e s 1 will contain the

m quantity

E a c h of the infinite s e r i e s 1 will contain one o r m o r e such t e r m s , provided that m and n have the n e c e s s a r y values. By a

a2S-

c a r e f u l inspection of the r.h.s. of the equation for

3

1

+

^sl, the s u m of a l l t e r m s i n s i n 8 and cos 8 may be determined.

N

In o r d e r for s (8, 8, p.) not t o contain a t e r m proportional t o 8, 1

the s u m of the t e r m s i n s i n 0 and cos 8 m u s t vanish, for a l l values of

z.

This r e q u i r e s that the coefficients of s i n 8 and c o s

8

m u s t

IY

vanish s e p a r a t e l y , for all values of 8. This l e a d s t o the following equations, f o r the c a s e m = 1:

t e r m s

by e

,

^e

, - - -

The quantities K ~ ~ P , yn,dnsqn, and

en

depend only on a 3/2

.

They a r e defined i n the appendix.

A f t e r the t e r m s i n s i n 6 and c o s 8 have been r e m o v e d f r o m eq. ( 6 8 ) by m e a n s of eqs. (70) and (71), the solution f o r s l will be a s follows:

where the bounded function h3 contains t e r m s of the following types:

0 2

(a) s e v e r a l infinite s e r i e s ' which a r e multiplied by e

,

e, e

,

e t c . , and which contain sinusoidal functions of 8. T h e s e infinite s e r i e s ' do not contain any s m a l l divisors.

(b) sinusoidal functions of 6 which a r e multiplied by e,

d@ dQ d+

sinw, cos ^W, s i n n @, c o s n @, LNL,

,

and

--.;;.

d8 d6 d9

The d e r i v a t i v e s ^{--;5 d'}

, -

^{dD and}

-

^d+ m a y be e l i m i n a t e d f r o m

d8 d z

'

dg

the equation f o r s a f t e r the e x p r e s s i o n s f o r t h e s e d e r i v a t i v e s have 1

been found i n t e r m s of a 3/2

,

e, o, and

$I.

The r e s u l t i n g e x p r e s s i o n

N

f o r ~ ~ ( 8 , 8, p) will be f r e e f r o m s m a l l d i v i s o r s .

The quantity

-T a a

m a y be e x p r e s s e d a s follows:

a t

When the e x p r e s s i o n s for s 1 and

C(

i n t e g r a l of eq. ( 6 5 ) ] - i 6 z

G)]

~2

a e

-I

a r e substituted into eq. (73), the following equation i s obtained:

where the bounded function h contains t e r m s of the following types:

4

0 2

( a ) s e v e r a l infinite s e r i e s ' which a r e multiplied by e

,

e, e

,

etc. and which contain sinusoida2 functions of 8, whose f r e q u e n c i e s a r e independent of 8.

( b ) sinusoidal functions of 8 r h i c h a r e multiplied by e,

d2 dw d?

s i n up cos W, s i n n @, --;Z

,

^--;=

,

^and

-= .

d8 do d 8

In c a r r y i n g out the integration of eq. (74), the s a m e c o n s i d e r - ations that w e r e d i s c u s s e d i n r e l a t i o n t o the integration of eq. (65) will apply. The s u m of a l l t e r m s on the r.h.s. which a r e independent of 8

N

m u s t v a n i s h f o r a l l 8. T h i s r e q u i r e m e n t yields the following equation:

The quantities )I, hn, r, and

5

depend only on a3I2, and a r e defined .n

i n the appendix.

After the t e r m s which a r e independent of 8 have been r e m o v e d f r o m eq. ( 7 4 ) by m e a n s of eq. (75), eq. (74) m a y be i n t e g r a t e d w.nt.

8, holding fixed. The r e s u l t i s of the following f o r m :

where the bounded function h5 contains t e r m s of the following types:

0 2

( a ) s e v e r a l infinite s e r i e s ' which a r e multiplied by e

,

^{e, e}

,

etc. and which contain sinusoida2 functions of 8, whose f r e q u e n c i e s a r e independent of 8. These s e r i e s ' a r e f r e e f r o m s m a l l divisors.

(b) sinusoidal functions of 8 which a r e Amultiplied,,by e,

d$ d o d r

sinw, cosw, s i n n 4 , c o s n @ ,

, ,

and

. :

d8 ' d8 d8

d$ dG

The d e r i v a t i v e s ^--;2,

, :

and

-

dTA m a y be eliminated f r o m the ex-

d8 do dZ

p r e s s i o n f o r t by u s e of eqs. (67), (70), (71), and (75).

1

Thus the a s s u m e d f o r m of the two v a r i a b l e expansions given in eqs. ( 8 a ) and (8b) has been shown t o yield a self-consistent approx- i m a t i o n t o the solution of eqs. (4) and (5), provided t h a t the o r b i t a l e l e m e n t s s a t i s f y the four f i r s t - o r d e r differential equations (67), (70),

N N

(71), and (75). The p e r t u r b a t i o n t e r m s ~ s ~ ( 8 , 8, p.) and p.tl(8, 8, p), a s given i n eqs. (72) and (76), will be f r e e f r o m s m a l l d i v i s o r s and

will r e m a i n s m a l l quantities of O(y).

If t h e p e r t u r b i n g t e r m s of O ( y 2 ) w e r e taken into account, the r. h. s. of eqs. (67), (70), (71), and (75) would a l s o contain O( y) t e r m s involving a, e, o, and (b, The s h o r t - p e r i o d p e r t u r b a t i o n s would be

IY N

accounted f o r b y t e r m s p2 s Z (0, 0, y) and y 2 t 2 (8, 8, y), similar in nature t o s, and t l

.

T h e r e f o r e a n approximate solution f o r the motion of t h e infinit- e s i m a l body, which r e m a i n s valid f o r l a r g e v a l u e s of 0, h a s been obtained f o r t h e c a s e of n e a r l y c o m m e n s u r a b l e m e a n motions. The difficulty of s m a l l d i v i s o r s h a s been avoided i n t h i s solution by r e q u i r i n g t h a t the o r b i t a l e l e m e n t s m u s t s a t i s f y a s e t of f o u r f i r s t o r d e r d i f f e r e n t i a l equations.