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 r a t h e r than 0. In t h i s section, some approximate solutions of t h e s e equations will be given.

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

d8'/~

^d$

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

--;;.

^{A second}

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

F r o m eq. ( 6 7 ) i t then follows t h a t

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

-=

do ^and

9

defined

d0 d z

previously, one obtains

Using eqs. (80) and (81) t h i s becomes

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 a 3/2

,

^{e,w, and} @

.

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

Thus, if ;=p12 0 w e r e used a s the slow v a r i a b l e f o r the non- c o m m e n s u r a b l e c a s e 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 problem, it would be found that

s

This i m p l i e s that 8 = ye is the c o r r e c t slow v a r i a b l e f o r the non- c o m m e n s u r a b l e case.

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 , y) wl. (See F i g u r e 4, ) L e t

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-

* *

^N

rotating X

-Y

system. Also, @(Bz,y)

q2

will differ slightly f r o m

@ Since a

% z -

n-1 the m a s s p. will have made approximately n

'

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

If a t the end of the above interval, the angle

(% - 9 , )

is

s m a l l but

>

0°, the infinitesimal body will be slightly displaced counterclockwise f r o m the m a s s p.. The perturbing f o r c e a t the point of c l o s e s t approach will then a c t i n a clockwise direction. T h i s f o r c e will tend t o d e c r e a s e the counterclockwise angular velocity of the infinitesimal body. Since t h e m a s s p moves at constant angular velocity, i t will begin t o "catch up" with the infinitesimal body during

N

the next s u c h i n t e r v a l O2 d 8 s = 4n7t t

0(e3

,p). T h e r e f o r e , by the instant when 8

=

€$ the angle (- W

-

^{@ )} will have d e c r e a s e d somewhat,

n s o that

( 3 -

n $,)

< ^{( 2 -}

^{n $,).}

Thus if (

-

⁾ i s s m a l l but

>

0' a " r e s t o r i n g f o r c e n c o m e s into play n e a r the point of p e r i c e n t e r passage, and t h i s r e s t o r i n g f o r c e tends t o d e c r e a s e the value of

(2 - $1).

Th i s situation will r e c u r i n

n

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

-

⁾ is s m a l l and

>

^0'. F i n a l l y (

-

⁾ will become <0°,

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

_n

-

^{@) is}

s m a l l and

<

0" t h e r e s t o r i n g f o r c e will tend t o i n c r e a s e t h e angle ( 0

-

⁾ t o w a r d t h e value 0'.

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

2 --

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

2 3 X ( ~ , p ) .

Hence oscillations of ( o-n@) about 0' will be accompanied by oscillations of

:I2

(z, t . ~ ) about some fixed value c l o s e t o

-

n-1 _n

.

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

The J a c o b i i n t e g r a l (7) will now be e x p r e s s e d i n t e r m s of the two v a r i a b l e expansions. Using eqs. (11) it m a y be shown that

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

Y2

"/z

O(y ) a r e actually of O(p), since

%

^{and a r e of O(p ).}

a e a e

F o r t h i s s a m e reason, s e v e r a l t e r m s involving

!kQ

which appear

3/2

a'ii

t o be of O(p.) o r of O ( y ) a r e actually of O(p2). It m a y be shown that

The Jacobi i n t e g r a l may t h e r e f o r e be written a s follows, i n t e r m s of the two v a r i a b l e expansions being used:

r-O&") c

w h e r e C is a constant which depends only on the i n i t i a l conditions.

It m u s t be r e m e m b e r e d t h a t the t e r m s i n eq. (86) which involve

%

a t "/2

and

4

^{a r e of O(p),} r a t h e r t h a n O(p ) ^,

a e

By f o r m a l differentiation w.r.t. 8, followed by u s e of eqs. (57) and (58) t o eliminate t e r m s i n s l y tl

, 9, %,

^etc.

^,

^{it m a y be}

shown t h a t

N

The r.h.s. of eq. ( 8 7 ) is independent of 8, and depends a t m o s t on 8.

However, the quantity in b r a c e s on the 1.h. s. d o e s not contain a n y t e r m s which a r e p r o p o r t i o n a l t o 8. T h e r e f o r e the

a

d e r i v a t i v e of t h i s quantity cannot produce a t e r m which i s independent of 8. T h i s i m p l i e s t h a t t h e r. h. s. m u s t vanish; i.e. that

s o that

Eq. (89) r e p r e s e n t s one of the four g e n e r a l i n t e g r a l s n e c e s s a r y t o determine the behavior of t h e o r b i t a l elements. It is valid for a l l values of the i n t e g e r s n

>

m

>

0.

By u s e of expansion ( 2 7 ) it may be shown that

Eq. (89) then becomes

N N

Eq. (90) m a y be u s e d t o e x p r e s s a(8, p) i n t e r m s of e2(8, p) and the initial conditions.

N

3. Approximate Solution f o r e(8,p)

g

e o

Approximate solutions of e qs. (78)-(82) will now be investigated by neglecting the v a r i a t i o n of e on the r.h. s. That is, the approxi-

(91)

e(4.l)=

e . t i 3 i i Y ) ;

e,

constant

will be u s e d on the r.h.s. of the equations. T h i s approximation i s not valid f o r e x t r e m e l y s m a l l e o , since the v a r i a b l e p a r t of e is then not negligible.

The coefficients ( $ a K,),

( z

1 a y n), etc.

,

m a y be expanded i n powers of p about a =

- %.

E q s . (78)-(82) then become,

( n ) respectively,

The coefficients ( t a ~ , ) , (ayn), etc., on the r.h.s. of eqs. (92)-96) depend only on n.

Since the angle (a-n@) will be unbounded i n many c a s e s , i t is

1

n e c e s s a r y t o include t h o s e t e r m s of O ( p h ) on the r.h.8. of eq. (96) which would contribute to a possible s e c u l a r behavior of (w-n@). L e t n b designate the ,constant p a r t of the O(p 1/2 ) t e r m s i n eq. (96). Then

s o that

Division of eq. (98) by eq. (93) yields the i n t e g r a l

( 9 9 )

(~~-&!)'= s,

+ ~ ~ c ~ ( w - ~ ) + q , s ~ ~ ~ ~ ~ - ~ ) with

The value of cos(w-no) m u s t r e m a i n s u c h that

(23'2

- p % b ) 2 2 0.

By computing the n u m e r i c a l values of /3 and yn a t

a = ( ) it m a y be shown that S3

<

0 f o r the c a s e s n

=

2, 3,4. It should be noted that the r , h. s. of eq. (99) would contain t e r m s multi- plied by e:, e:,

- - -

if the corresponding t e r m s i n e3, e 4 ,

- - -

^had

been r e t a i n e d i n eq. (79).

N

The behavior of c o s ( o - n o ) a s a function of 8 will now be determined. F r o m eq. (98),

Using eq.

(99)

and writing

5

i n place of cos(w-n$), t h i s becomes

The r.h.s. m a y be f a c t o r e d a s follows:

w h e r e

Depending upon the initial conditions, P1 and P2 will be e i t h e r both real-valued, o r complex conjugates. Since S3

<

⁰ it follows that P23P1 when Pl and P2 a r e real-valued. Also, P1d5dP2 i n o r d e r

that (c%

-

^{p'b) be B 0.}

If the r o o t s P1, P2, 1, -1, a r e a l l distinct, the value of

5

^will

not c r o s s any of them, because t h i s would make

(3; <

^{0. Thus}

d;

cos(w-n@) will oscillate between two fixed limits. If Pl and P2 a r e c o m p l e x conjugates,

5

will oscillate between the v a l u e s 1 and -1, corresponding t o a monotonic i n c r e a s e o r d e c r e a s e of (w-n@).

LU

T o exhibit t h e explicit dependence of cos(w-n@) on 0 for a typical c a s e w h e r e ( a - n @ ) o s c i l l a t e s between two fixed l i m i t s ,

a s s u m e t h a t the i n i t i a l conditions

23/2

^{( z o ,1"),} e o , and (wo - n @ o ) a r e s u c h that -1

<

Pl<

5, _<

1

< ^P2 .

Then cos(w-n@) will o s c i l l a t e

between the two values P and 1. F r o m eq. ( 9 7 ) i t is s e e n t h a t

-!k%!&

= 0 only when ( a AX

-

p% b) = 0; that i s , only when dG'

c o s ( o - n @ )

=

PI

.

T h e r e f o r e ( a - n @ ) will o s c i l l a t e about 0" between t

-

¹

the l i m i t s

- I

^{c o s} Pl

I .

T h i s type of motion i s known a s libration.

Keeping i n mind t h a t S3

<

^{0, eq.} (100) b e c o m e s

The solution of eq. (103) m a y be e x p r e s s e d i n t e r m s of elliptic functions.

F o r t h e s a k e of definiteness, a s s u m e t h a t [a/2(60,p)-yXb]<0 and c o s - ' ~ ~

<

(wo - n q o )< 0"

.

Then _{0 and --;} d

S I_>$ ^.

After r e p l a c i n g

6

by d0 €I=@,

cos(w-n@), the solution of eq. (103) b e c o m e s

w h e r e

I

a n d (105)

The function s n is the Jacobian elliptic function, T i s the oscillation period of t h e angle (w-n@), and K(k) i s the complete elliptic i n t e g r a l

N N

of the f i r s t kind. When 0 = ^€I0 f T the motion will begin t o r e p e a t itself.

The quantity

(8% -

p% b ) will oscillate between the values

t

?4

-

^{(S, +S2} eo tSS e t ) attaining i t s m a x i m u m and m i n i m u m values a t (a-n@)= 0". Hence a3I2 o s c i l l a t e s about the value pb with the

n

2 %

amplitude py2 ( s ~ t S 2 eo +S3 e o )

.

N 1

Since c o s ( o - n @ ) i s periodic i n 8 with p e r i o d ^i; T, the constant b i s given by

The t e r m s i n cos(o-n@), cosZ(w-n@), and cos3(w-n@) m a y be

N

e x p r e s s e d a s functions of 8 by u s e of eq. (104). Since Pl,

PZ,

and T a r e independent of b c o r r e c t t o O(y

X

) the value of b m a y be calculated f r o m eq. (106), t o within t e r m s of O(y

'/z

).

An a n a l y s i s s i m i l a r t o t h e above m a y always be u s e d t o

N

d e t e r m i n e cos(w-n@) a s a n explicit function of 8, when the r o o t s PI and

PZ

a r e real-valued. After t h i s h a s been done, the e x p r e s s i o n s

A ^N

f o r

$(gP),

^w ($,I"), and Q(8, p) c a n be obtained by integration of the

N

known r. h. s. of eqs.

( 9

2), (94), and (95) w. r. t. 0

.

However, c a n be obtained d i r e c t l y f r o m eqs. (99) and

^?Amp%

^b)

(104). C a r e m u s t be taken t o choose the p r o p e r sign f o r ( a

when taking the s q u a r e r o o t of eq. (99). After ^2'3$ has been determired, the e x p r e s s i o n f o r e m a y be obtained by u s e of the i n t e g r a l A (89).

u s i n g expansions (27) and (43) f o r

a%

^{and e,} it m a y be shown that

By eq. ( 8 9 ) , t h i s i m p l i e s that

Specializing t o the c a s e m=l, and evaluating the constant of integration, this b e c o m e s

Thus e r e m a i n s bounded if e o A # 0. The value of the e c c e n t r i c i t y a t any instant is then given by

e&,4 = e, +~@(%-ql

F r o m eq. (104) it is s e e n that if t h e initial conditions a r e s u c h that Pl

=

1, cos(w-n#) will have the constant value t1. T h i s c o r r e s - ponds t o the condition (w-n@) = constant

=

0". F r o m eq. (102), the condition PI

=

1 i m p l i e s that

s,e,' + ^&e, +q

^{= O}

Evaluating S1 f o r the initial condition (w-n@o)=OO, t h i s r e q u i r e s

2*'~&>.iU!, ^{=A 4A}

If PI is slightly l e s s than 1, ( a - n @ ) will undergo infinitesimal oscillations with the p e r i o d T, i n a c c o r d a n c e with eq. (104). The value of T i s given by eq. (105) with PI= 1. Values of T calculated f r o m eq. (105) a r e given below f o r s e v e r a l v a l u e s of e,, using the value y =048' r for the c a s e n=Z. It should be r e m e m b e r e d t h a t t h e p r e s e n t approximation (91) is not valid f o r e o ^-t 0.

1 P e r i o d 612 y e a r s 5 79

576 594 644

The difference between the n u m e r i c a l values calculated f r o m

eq. (105) and those given by Schubart ( s e e F i g u r e (4) 5 ) is l a r g e l y due t o

3 4

the neglect of t e r m s i n e o , e o , etc. f r o m eq. (99). The a g r e e m e n t could be i m p r o v e d by inclusion of the higher p o w e r s of eo i n the c a l - culations, although a g r e a t d e a l of additional a l g e b r a i c l a b o r would be r e q u i r e d e v e n t o d e t e r m i n e the coefficient of e,, 3

.

Since the magnitudes of the n u m e r i c a l values of a n,

fin, vn,

^Kn, etc. i n c r e a s e with n, the influence of the higher p o w e r s of eo is r e l a t i v e l y g r e a t e r f o r l a r g e r n.

Consider now the c a s e i n which PI and

PZ

a r e complex con- jugates. T h i s i m p l i e s that

(2%

-p,%b) does not vanish, and is t h e r e f o r e

N

of constant sign. The angle (w-n@) will be a monotonic function of 8, d e c r e a s i n g if % (;

-&

b)> 0 and i n c r e a s i n g if

(kg

^{-p,& b)< 0.} Eq. (100) b e c o m e s

F o r the s a k e of definiteness a s s u m e that

p%(t0

, t ~ ) - p .

$ 1

b

<

0 and that sin(wo -n@, )> 0. Then d(o-n(b) N N

> O

and

% I m

^{< O .} A f t e r r e -

d; d 0 = 0 0

placing $ by cos(w-n@) the solution of eq. (108) is

w h e r e

e;=4-f-

^fie. I

^(-%I "fi-r'fi+r'~f-e)~-~j i ^d/

-I

and (110)

and

The function c n is the J a c o b i a n e l l i p t i c function, and T is the p e r i o d of (w-n4).

The constant b i s e a s y t o evaluate f o r t h i s c a s e . Since ( a - n 4 ) v a r i e s f r o m ( ~ ~ - n # ~ ) t o ( ~ ~ - n $ ~ ) + 2 n during one period, the c o n t r i - bution t o b f r o m e a c h of the t e r m s i n cos(w-n@), cos2(a-n+), and cos3(w-n@) vanishes. T h e r e f o r e

N

The quantity

2%

m a y be obtained a s a n explicit function of 6 f r o m eqs. (99) and (lO9), choosing the ' I - " sign when taking the

2

s q u a r e r o o t of eq. (99).

(2%

^-p,'b) will oscillate between the values

and

with the p e r i o d T. The amplitude of % (: - p b)

X

is t h e r e f o r e

" A "

After

2%

has been obtained a s a n explicit function of 0, e (8,p) c a n be

A " A "

obtained f r o m eq. (107). Also, a(@, p) and ~ ( 8 , p) c a n be e x p r e s s e d

N

a s the i n t e g r a l s of the r.h.s. of eqs. (94) and (95) w.r.t. 8.

"

4. Appr oximate Solution f o r e(0, p)=

O(S,

p,)

The solutions d i s c u s s e d i n the previous section a r e not valid when e o =0, because the v a r i a b l e p a r t of e is then not negligible. In-

stead, eq. (43) becomes

Eqs. (78)-(82) become, respectively,

where

4%)

d(2-a

4 = d ( a * = y

3

9A

A s i n the p r e v i o u s section, the coefficients ( i a ~ , ) , (3an*a K,), etc.

.-

n-l)g and t h e r e f o r e have been expanded i n powers of pX"aX about a=(- n

depend only on n.

E q s . (113) and (114) have the i n t e g r a l

Y 2

Hence the oscillations i n a r e O(p ) for t h i s c a s e .

Since (w-n$) will be unbounded i n many c a s e s , it is n e c e s s a r y t o include those t e r m s of O(p 1/2 ) on the r.h.s. of eq. (117) that would contribute t o a possible s e c u l a r behavior of (o-n@), The remaining

Y 2

t e r m s of O(p ) will only produce bounded t e r m s i n ( a - n q ) . L e t

Eq. (117) then b e c o m e s Fk~r

Eqs. (113) and (120) have the i n t e g r a l

with

Since cos2(w-n+) m u s t be G l , the value of e m u s t always s a t i s f y A

the condition

Eq. (113) m a y now be w r i t t e n a s

The quantity on the r. h. s. m a y be f a c t o r e d a s follows:

z ^A+ t AZ

-6

e

+(1-2~4)$~-4* = -4

( C - ~ ) ( $ : ~ J 20 where

T h e r e f o r e

d 2

If Q 1 and Q2 w e r e complex conjugates, -7would never

ff

vanish, and

G 2

would be a n unbounded function of 8. The p r e s e n t

ff

approximation (112) would not be valid, a s e(8, p) would become large. T h e r e f o r e the approximate equations (113)-(117) will be valid only if Ql,Qz a r e real-valued; i.e. only if

F o r real-valued Q 1 and Q,, it is s e e n that Q 1 3 Q z .

>

In o r d e r that 0 throughout the motion, i t is n e c e s s a r y that

A 2

Since the s i g n of

$

c a n change only when $ Z = Q , or a 2 = ~ , , the d e

value of

g2

will o s c i l l a t e between t h e s e two limits. Correspondingly,

the value of cos(w-n@) will oscillate between two limits. Using i n - equality (125), i t m a y be shown t h a t Qz >, 0. In f a c t Qz vanishes, and e consequently p a s s e s through A 0, only f o r the s p e c i a l c a s e

F o r t h i s c a s e the m a x i m u m value of

G2

is

It is s e e n f r o m eq. (123) t h a t Q, -co when ~ ' ( & , p ) - p K c

.

T h e r e - fore, e will r e m a i n of o r d e r unity only A i f k3/z(~0,p) is O(pO). T h i s is i n a g r e e m e n t with the well-known fact that t h e r e a r e no periodic o r b i t s of t h e f i r s t kind a t c o m m e n s u r a b i l i t i e s with m=1.

By calculation of t h e n u m e r i c a l v a l u e s of the F o u r i e r coefficients and it m a y be shown t h a t K ~ < 0 for n=2, 3,4. T h i s is p r e s u m a b l y a l s o t r u e f o r n > 4 .

The solution of eq. (124) is

with

The value of cos-' { } should be chosen s u c h that

The e x p r e s s i o n f o r

8%

then follows f r o m ' e q . (118). The p e r i o d of the oscillations of A e i s

Since e A 0, it follows f r o m eq. (121) t h a t

N

Since

2

and cos(w-n$) a r e periodic functions of 8 with period T,

The constants

R 1 , R 2 , Ql,

Q2, and T a r e independent of c, c o r r e c t

X

^N

t o O(p ) A f t e r e x p r e s s i n g the integrand i n t e r m s of 8 by u s e of eqs. (126) and (128), c m a y be evaluated f r o m t h i s integral, c o r r e c t t o O(p

X 1.

N

Having d e t ~ r m i n e d

8

and cos(w-n$) a s functions of 8, the

A

-

^{A "}

e x p r e s s i o n s f o r w(8, p) and ~ ( 8 , p) m a y be obtained by i n t e g r a t i o n of t h e r. h. s. of eqs. (115) and (116).

The angle (w-n$) will a t t a i n a m a x i m u m o r m i n i m u m only if its derivative vanishes. By eq. (120) t h i s can only o c c u r when

which c o r r e s p o n d s t o

(130)

Ri,

- 2 ( ~ - m ~ )

= WR, ^; g2= -

6

d(w-n ) If

%

l i e s outside the r a n g e Q,

< G 2 < a l ,

the derivative

3

R1 d9

will never vanish, s o that ( a - n @ ) will i n c r e a s e o r d e c r e a s e mono- tonically. However, f o r c a s e s i n which (w-n@) o s c i l l a t e s between two fixed l i m i t s , eq. (130) m a y be used t o d e t e r m i n e the amplitude of

oscillation, a s a function of the initial conditions.

It follows f r o m eqs. (126) and (128) that

2

and (w-n$) a r e

constant, corresponding t o infinitesimal oscillations, i f

Q l

= Q2 z 1 4 ~

This c o r r e s p o n d s t o

- 1

- = - ^aK, _R<s

23

2 f i

[~*(Z~>A)-M%]

9

The choice of the sign follows f r o m the f a c t t h a t $(<o,CL) 2 0. Since K,< 0, eq. (121) yields

+ I 7 a q N ) + Y % w(w-m.&)

=

&(%-&*)

=

-/ ²

;%&>&)>A&

It follows t h a t i n f i n i t e s i m a l oscillations with e=CL%$ a r e possible only about ( a - n @ ) = 0" when 23/2(G0 , j - ~ ) <

&

c and only about (a-n$) = 180' when

$.)I2

(Go, t ~ )

> c'

^c.

In the d e r i v a t i o n of eqs. (67), (70), (71), and (75) it was

dw ^m

a s s u m e d t h a t the angular r a t e

-

a 9 of the p e r i c e n t e r angle w(9,p) is

s m a l l in c o m p a r i s o n t o the angular r a t e

aT=

1 of the infinitesimal body. However, t h i s a s s u m p t i o n would be i n c r e a s i n g l y violated i n the c a s e of infinitesimal oscillations of ( o - n @ ) with e = p,

% g

if it w e r e

A

-

attempted t o make calculations f o r e(OO ,p,) v e r y s m a l l . Accordingly the a c c u r a c y of the p e r i o d s calculated f r o m eq. (127) is not expected

A

-

t o be v e r y good f o r e x t r e m e l y s m a l l values of e ( e O , p,).

5. C o m p a r i s o n of R e s u l t s with Calculations by Schubart

Extensive n u m e r i c a l calculations have been c a r r i e d out by Schubart ( 4 ) f o r 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 r e s t r i c t e d three-body problem. The following v a r i a b l e s ( w r i t t e n i n t e r m s of the p r e s e n t notation) a r e u s e d i n his work:

The disturbing function o r Hamiltonian i s

w h e r e rnl i s the m a s s of the perturbing body and (1

+

m l ) % is i t s m e a n motion.

The s h o r t - p e r i o d t e r m s involving A a r e ljsmoothed out" f r o m F by a n u m e r i c a l averaging p r o c e s s , and the r e s u l t i n g quantity is denoted by

.

Only long-period t e r m s a r e r e t a i n e d i n

Fe

The

following two i n t e g r a l s a r e then valid f o r the long-period effects:

u = &

F = d

T h e first of t h e s e i s equivalent t o eq. (90), t o within t e r m s of O(p).

Following a suggestion by ~ o i n c a r 6 ' ~ ) . t h e v a r i a b l e s

= (2s)%w4.

a r e introduced and the r e s u l t s of the calculations a r e g r a p h i c a l l y p r e - sented i n the f o r m of c u r v e s F(x, y, U)

-

= constant, d r a w n i n the x - y plane f o r a fixed value of U. These c u r v e s bring out the nature of the behavior of (w- "9) and the e c c e n t r i c i t y e, f o r a wide r a n g e of

m

i n i t i a l conditions, and a r e t h e r e f o r e useful i n obtaining a n intuitive understanding of the motion. Since the n u m e r i c a l averaging p r o c e s s

-

used t o c o n v e r t F t o F was c a r r i e d out on a n e l e c t r o n i c computer, without the n e c e s s i t y of expanding F i n p o w e r s of e, the calculations a r e valid f o r o r b i t s of a l l e c c e n t r i c i t i e s 0 ,( e

<

l.

Although t h e y c l a r i f y the qualitative nature of the motion, the

-

c u r v e s F = constant do not provide information about i t s t i m e

dependence. However, r e f e r e n ~ e ' ~ ) gives the p e r i o d of i n f i n i t e s i m a l l i b r a t i o n s of (a-n$) about 0" a s a function of the e c c e n t r i c i t y , c a l - culated by m e a n s of a n u m e r i c a l v a r i a t i o n a l theory. This w a s c a r r i e d out f o r the c a s e s n = 2, m

=

1 and n

=

3, m

=

1, using the n u m e r i c a l value rnl 71047 t o c o r r e s p o n d t o the s u n - J u p i t e r - a s t e r o i d problem.

T h e s e v a l u e s a r e plotted i n F i g u r e 5, The p e r i o d of i n f i n i t e s i m a l

( 4 ) librations f o r v e r y s m a l l e c c e n t r i c i t i e s is not given i n r e f e r e n c e

.

However, t h i s c a n be calculated f r o m eq. (127). The r e s u l t s a r e shown i n F i g u r e 5. The a c c u r a c y of the r e s u l t s d e c r e a s e s f o r

A

-

l a r g e v a l u e s of e ( Q O , p.) because the approximation (112) b e c o m e s unrealistic. S c h u b a r t r s r e s u l t s f o r n

=

2 indicate that the p e r i o d d e c r e a s e s f o r v e r y s m a l l e, and the values calculated f r o m eq. (127) c l e a r l y show t h i s . Although the c u r v e s f o r n

=

3 do not f i t together a s well a s do those f o r n

=

2, a m a r k e d d e c r e a s e i n p e r i o d is indicated f o r s m a l l e c c e n t r i c i t i e s .

In o r d e r t o exhibit a typical c a s e of finite-amplitude l i b r a t i o n s of ( a - n @ ) about 0 ° , the l i b r a t i o n amplitude h a s been calculated f r o m eq. (102) f o r the c a s e n = 2 and using y =

-

₁₀₄₈ 1 _' I n eq. (102) t h e value

A)

of e o is t a k e n e q u a l t o e

-

e(OO, p.). T o facilitate c o m p a r i s o n of initial-

the p r e s e n t r e s u l t s with those given i n r e f e r e n ~ e ' ~ ) , the l i b r a t i o n is a s s u m e d t o s t a r t f r o m the i n i t i a l condition (wo - n q o ) = 0°, and t h e initial condition

g3I2

^($

^,

^p.) h a s been adjusted f o r e a c h value of einitial i n s u c h a way t h a t the r e l a t i o n

U

= aG[2-(1-e2 )y2] = .8000 is

maintained.

The c o m p a r i s o n of r e s u l t s is shown i n F i g u r e 6, and the a g r e e m e n t i s good f o r l a r g e amplitudes of (w-24). F o r t h e l a r g e r values of einitial, the neglect of the higher p o w e r s of e o c a u s e s eq. (102) t o yield amplitudes which a r e somewhat too small. The

a g r e e m e n t could be i m p r o v e d by r e t e n t i o n of a l l t e r m s i n e t h r o u g b u t 3

the calculations.

F i g u r e 6. L i b r a t i o n Amplitude of (0-2@).

1

n

=

2 ; a h[2-(l-e2)%]

=

.8000

APPENDIX 1. Numerical Values of Commensurabilities

A P P E N D I X 2, Expansions in Powers of e

+ {

similar terms i n e3, e4,

- - -3

similar terms in e 3 , e 4 ,

- -

A P P E N D I X 3. E x p r e s s i o n s f o r Coefficients

= (&,%)A& + ^(a- $ 4 ~ ~ +

^(-.h

+ $&?)A,,

+-

6 4 a 2 + 9 4 k .

m

2 3 6

*

f-#a+~&3-glL)ki&+'

+(++pa

-.a)%-,

+-(F- ^fa3g+*

,2/ 2 2/ 5 H r 1s

+ /s-

7

+(ra+Ta)g-r"&

^a'p ' p ) ~ ,

+

^(+%a

+ %'-

$5&")&+/

+ (- Fd- ^435a3-

^Bit,?~,,

_r ^+(-$G2+/f

^a*

_{- r} 01 '

^$*+c

/5-9 /

+ (y'+f a'-$%3~, ⁺ (e ^-

^{+YG+S C&O 1 b 8 4 ,}

*

^&-3

+ (-gp3-c gay c_,

Note: The coefficients Cn-,

,

C

++, ^,

-

etc., a r e t o be included only

when (n-3 ) 3 0, (-nts ) P 0, etc. The summation on k in the F o u r i e r s e r i e s 1 does not include k < 0 .

NOTATION

The symbols which a p p e a r m o s t often in the text a r e l i s t e d below:

IJ m a s s of the perturbing body divided by t o t a l m a s s of the s y s t e m

t t i m e

r, 8 polar coordinates of the infinitesimal body; s e e F i g u r e 2.

1 1

S

-

r

s o ? S 1 leading t e r m s of the two v a r i a b l e expansion f o r s

to* t l ^{I1 I1 I1 1) II I f l r 'I} t

5

the slow v a r i a b l e

($=

p% Q)

9 0 9 5 initial values of 8 and

5

^{a t t} = 0

a s e m i m a j o r a x i s of the o r b i t of the infinitesimal body e e c c e n t r i c i t y II II II 11 rl 11 II

w longitude of p e r i c e n t e r

" "

^{1' II 11}

a

T quantity which defines the position of t h e i n f i n i t e s i m a l body i n i t s o r b i t

n, m positive i n t e g e r s which specify the p a r t i c u l a r

..I c o m m e n s u r a b i l i t y being c o n s i d e r e d

~ o s ~ o * ~ o *

9%

A A A

'

v a r i o u s t e r m s i n the expansions of t h e o r b i t a l e,W, ⁷ e l e m e n t s

@

a slowly-var ying angular quantity which defines the position of the i n f i n i t e s i m a l body i n i t s orbit;

s e e eq. (34).

Ak(a), Bk(a),Cda) F o u r i e r coefficients u s e d t o expand the p e r i o d i c p e r t u r b i n g t e r m s

k s u m m a t i o n index

an, Sn,

yn, ~ ~ , p , 6*. v a r i o u s combinations of the F o u r i e r coefficients

r. h. 6 . ; 1.h. s. r i g h t - hand s i d e ; left-hand s i d e