Application of the two variable expansion procedure to the commensurable 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

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.

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

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)

=

64

VI. NOTATION

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 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

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

.

de 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

-

d t

'

-

-

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

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,

=

0

.

n n

0

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)-',

dt'

dt9

.%.

dt'

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

-

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 -

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

kK,

K

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'^)

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

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

<

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

=

m a k e s a n angle t with t h e positive

X

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

-

*

*

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

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

*

m a s s , a n d hence the X

-

x*-

*

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

=

*c

a n angle t with the 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 n the

x*-

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

*

origin of the X

-

*

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}

&

.

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 ,

They a r e a s follows ( w h e r e r

=

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

=

=

.

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.

* *

X

-

where 0, i s the initial angle between the r a d i u s vector t o the infini-

The t e r m s which involve

[

+

I

-

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*

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 <

<

$

.

J a c o b i integral: ~

where

C

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

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 .

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 )

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.

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

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 )

1/2

and

(5).

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' 2

0

to be retained, i t i s sufficient t o use the O(p ) 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

3

and

TiB

s i n ( 8 - t ) and c o s ( 8 - t ) .

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

H

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, 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)

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

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

0

The t e r m s multiplied by p 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

where

N

a

=

=

e

= e(0,

= 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

2

orbits, eq. (19) would be r e p l a c e d by s, 2 -1/2

W =

.

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

<

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

=

=

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

=

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

,

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

+

It follows that

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

&,

,,

(*-t)=

[~-$)@-d

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

3/2

f o r c i n g functions which would involve sin[ (1-a )8- T] and

3/2 N

cos [ (1-a )8- T]

.

0

8,

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,

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

2

t e r m s multiplied by e would produce s m a l l d i v i s o r s f o r both the

3

m = l and m=2 c a s e s ; those multiplied by e 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

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

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,

-

- -

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

-

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

.

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

Introduce the notation

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

+ 3

%

of

0,

- -

-

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

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 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

;

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 0

the de rivative s

a 3

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+

3

a

3

( 3 9 )

.& -

dG

-a

dr

+

-

&-

;%

+

-

di3

a

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

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

,

,

.,

- -

-

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*

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 ,

specifies i t s angular orientation. The quantity

4(?,

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

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

n.

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

w - 4 )

n

m

Thus the quantity ( - ) 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

following m a n n e r :

(43)

e(q~()

=

eo

+A%

(q4)

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

l/z

e

= e o t

O(p.

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 ~ ) =

(46) T ( ~ A )

=

7;

+~'fel/)

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

,

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;-

-

-

-

d 8 do do

eo,w ₀

,

calculations a r e avoided.

A

-

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

N

0

J/z

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

+

1/2

T

= r o t

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

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

da

dw

0 I;;

,

,

,

d o do do d0

0

and (50) a r e actually of

0(d2),

(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,

2 - 2 2 - 2 2

then multiplying b y s = a ( l - e )

[

,

r e s u l t i s obtained:

z

at,

4

-9

~2p+(wl

(54)

4

=A5

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

kz&

4

di?

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

]/z

be included i n the O(p) equations. Hence t h e r e a r e no O(y ) 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

da%+

d'&

[a(,-@)

diF

-

a]

.a$

at.

"'@%)[@gf

2dkI@+&

G ~ ~ ~ ) ]

-

=-

[/-~~-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

-)

)L O

a 5

t e r m .

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 _,

,

,

---

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)+{

f

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

3

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

(%)

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

4.

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

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

N

f o r

k

- - -

.

3

If a l l the perturbing t e r m s multiplied by e 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

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

(7)

t a b l e s published by Brown and Brouwer 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

=

- - -

.

7/2

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

[

g)]

P

known explicitly i n t e r m . s of

8

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

,

,

8,

-

-

'

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

( b ) sinusoidal functions of

6

of - s o

( w )

In c a r r y i n g out the integration of eq. (65) w. r. t.

0,

N

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

8)

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

<

and tl(8,

g,

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

-

-

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

=

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)

++

e2dinr

@-a#)

+

by e 3 , e 4 ,

- -

-

J

3/z

The quantities Q, and

p

-

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

.

F o r the c a s e rn

= 2,

2 a t e r m multiplied by e ; the leading t e r m would be multiplied by e

.

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

,

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

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.

,

g.

f r o m the expansion of t h e quantity

+xi-

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

-

88

*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

[(<

+

2s04

a8

+

T ( o

G)].

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

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

2n

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

-

=

k,

m 2n

the (- -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-

1

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

3

+

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.

8

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

t e r m s

by e

,

,

- -

-

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

en

.

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 )

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

,

,

(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,

,

--.;;.

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

,

-

-

'

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

3/2

been found i n t e r m s of a

,

o,

$I.

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 .

a

The quantity

-T

a

a t

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

C(

( 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

,

,

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

,

,

-=

.

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

N

The quantities )I,

hn,

5

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

,

,

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 @ ,

,

,

.

:

d8 ' d8 d8

d$ dG

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

,

:

-

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),

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,

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

.

IV. BEHAVIOR OF THE ORBITAL ELEMENTS

In s e c t i o n 111 it was shown that the difficulty of s m a l l d i v i s o r s c a n be avoided 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 of the infinites- i m a l body m u s t s a t i s f y a s e t of four coupled f i r s t - o r d e r equations, having the independent v a r i a b l e

8"

=

&

0

1. Equations f o r the O r b i t a l E l e m e n t s _{- ,}

d8'/~

Eq. (67) gives one r e l a t i o n between -7and

--;;.

d0 d0

r e l a t i o n m a y be obtained by multiplication of eq. (70) by - a ( l - e 2 ) c o s o and multiplication of eq. (71) by a(1-e2)sino, followed by addition of the r e s u l t s :

%

Multiplication of eq. (67) by -2a e ( l - e 2 ) % , followed by addition of the r e s u l t t o eq. (77) yields

Similarly, multiplication of eq. (70) by a(1-e2 )sin o and eq. (71) by a(1-e2)cos a, followed by addition of the r e s u l t s , yields the

following:

Eq. (75) then yields the following equation, a f t e r dropping a l l t e r m s in e 3 , e 4 , etc:

Since the angular quantity (w-n$) o c c u r s frequently i n the

N

above equations, i t s behavior a s a function of 0 will be of c o n s i d e r - do

able importance. Using the e x p r e s s i o n s f o r

-=

9

d0 d z

previously, one obtains

The second t e r m on t h e r.h.s, of eq. (82) i s w r i t t e n s e p a r a t e l y f r o m

1/2 Y2

the other t e r m s of 0 ( p ) because if e ( a p ) is s m a l l of O ( p ) t h i s t e r m will become O(pO).

If the p e r t u r b i n g t e r m s of O(p2) f r o m eqs. ( 4 ) and (5) had been retained, equations (78)-(82) would contain additional t e r m s of higher o r d e r i n p on the r.h.s. T h e s e additional t e r m s would involve

3/2

a

,

.

Having obtained the equations f o r the behavior of t h e o r b i t a l e l e m e n t s , i t is useful t o distinguish between those t e r m s which occur on the r.h. s. of eqs. (78)-(82) because of the n e a r l y c o m m e n s u r a b l e periods, and those which would a l s o occur i n the non-commensurable case. E a c h t e r m which contains a sinusoidal function of (w-n@) is solely the r e s u l t of the commensurability. In the non-commensurable c a s e t h e s e t e r m s would not occur. The t e r m s which involve the co- efficient s p , ~ , and w a r e not the r e s u l t of the commensurability, and would t h e r e f o r e occur in the non-commensurable c a s e a s well.

1

s

This i m p l i e s that 8

=

A h e u r i s t i c explanation of why the angle ( a - n @ ) will tend t o

oscillate about the value 0" will now be given, f o r the c a s e m=l. This explanation is based on the c r u d e approximation that the t o t a l effect, produced by t h e m a s s y on the motion of the infinitesimal body during one complete orbit, will be qualitatively the s a m e a s the effect e x e r t e d n e a r the point of c l o s e s t approach t o the perturbing body.

F o r the c a s e m=l, t h e point of c l o s e s t a p p r o a c h o c c u r s once during e v e r y n revolutions of the infinitesimal body i n i t s orbit. If ( a - n @ ) 0°, the point of c l o s e s t approach o c c u r s e v e r y nth revolution at approximately the time of p e r i c e n t e r passage.

Let 8=8, designate a n instant when the infinitesimal body is N

a t p e r i c e n t e r , s o that

= o ( B l ,

H

@ ( e l , y ) designate the value of @ a t t h i s s a m e instant. A s s u m e that ( w l - n o l ) = 0'. After n additional complete revolutions i n its

orbit, t h e infinitesimal body will again be a t p e r i c e n t e r , s o t h a t

N

€12 = 2 n n t w ( 8 2 , y ) ~ 2 n n t o 2 . However, w 2 will differ slightly f r o m w l , s o that the i n f i n i t e s i m a l body will have made slightly m o r e o r l e s s than n complete revolutions about the l a r g e m a s s , m e a s u r e d i n the non-

*

*

rotating

X

-Y

q2

@ Since a

%

z

-

'

(n-1) complete revolutions about the l a r g e m a s s (1-p).

s m a l l but

>

N

the next s u c h i n t e r v a l O2 d 8 s

=

0(e3

W

instant when 8 = €$ the angle (-

-

s o that

( 3 -

<

( 2 -

Thus if (

-

>

(2

-

$1).

n

the s a m e qualitative manner a t the end of e a c h n revolutions, s o W

long a s (-

-

>

-

n n

and the r e s t o r i n g f o r c e will change sign. That is, when

(E

-

s m a l l and

<

0

(

-

2

F r o m the definition of ( p ) it follows that a change i n

--

d B

r e q u i r e s a change i n

2 3 X ( ~ , p ) .

:I2

n-1 c l o s e t o

-

.

n

2. Use of the J a c o b i I n t e g r a l

r I The t e r m s on the r.h.s. of eq. ( 8 3 ) which appear t o