the three-dimensional motion of trojan asteroids

T H E T H R E E - D I M E N S I O N A L M O T I O N OF

A S T E R O I D S

T R O J A N

B A L I N T I~RDI

Department o.f Astronomy, L. E6tv6s University, Budapest, Hungary

(Received 15 November, 1977)

Abstract. The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian points L4, L5, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.

1. Introduction

The problem of the Trojan asteroids has always attracted considerable attention in celestial mechanics. The large number of different theories for the Trojans beginning with the work of Brown (1925) to Garfinkel (1977) clearly demonstrate this fact. One of the interesting contributions was made by Kevorkian (1970) using a two-variable expansion procedure (1966). Under the assumptions of the circular restricted three- body problem he pointed out some new features of the motion of these asteroids.

Kevorkian's method was applied for the Trojan asteroids in the plane elliptic restricted problem of three bodies by this author (1977). It was shown that to derive a solution to second order without secular terms a three-variable asymptotic expansion is needed. This three-variable expansion procedure is applied in this paper for the three-dimensional motion of the Trojan asteroids. The aim of the following considerations is to derive the main perturbations of Jupiter. According to this the problem will be considered under the following assumptions: the asteroids are only influenced by the gravitational forces of the Sun and Jupiter, and the orbit of Jupiter around the Sun is a fixed ellipse.

2. The Equations of the Motion

Let the Cartesian coordinate system S X Y Z be specialized as follows" the system is centered at the Sun (see Figure 1), the S X Y plane is Jupiter's orbital plane, the S X

axis is directed to the perihelion of Jupiter's orbit. To give the position of the asteroid the dimensionless coordinates r, z and the angle will be used. These are related to the rectangular X, Y, Z coordinates of the

Celestial Mechanics 18 (1978) 141-161. All Rights Reserved. Copyright�9 1978 by D. Reidel Publishing Company, Dordrecht, Holland.

142 BALINT ERDI

Z

/\

As TERO I D

X v

JUPITER

Fig. 1. T h e c o o r d i n a t e sys tem.

asteroid by the relations

X = R j r cos (c~ + v)

Y = R a r sin (o~ + v)

Z = R j z

R j ~- aj (1 - e~)

1 + ej COS v

(1)

where v is the true anomaly and Rj the radius of Jupiter, aj the semi-major axis and e j the eccentricity of Jupiter's orbit.

Introducing v as an independent variable the following equations of motion may be derived

d 2 r - - r (d~v)2 - - 2 r dcz 1 E l - / , - r r +

dv 2 dv 1 + e j cos v R31

COS o~ - -

3 + / z R 2

r

d ( da ) > r s i n a ~ dv r2 r 2 = 1 dv t- 1 -7- &cos v (2)

d2z z E1 I - /, dv 2 + z = 1 + ej cos v R 3

1

THREE-DIMENSIONAL MOTION OF TROJAN ASTEROIDS 143

where

R1 = ~ / r 2 n t- 2 2, R 2 = %/1 + r 2 - 2r cos c~ + z 2

and t~ is the mass of Jupiter divided by the total mass of the Sun-Jupiter system.

Equations (2) may be regarded as the equations of the elliptic restricted three-body

problem. In the case of the Trojan asteroids we look for the solution of Equations

(2) in the following three-variable expansion form.

r = l + N

r u, r) + 0(~ N +'), n=l

o~ = ~o(U) +

N

e ~ . ( , , . , ~) + o(~ ~ +'), n----1

(3)

Z __

N ~__~ ~n+l]2Zn(V , U, T) -[- 0(~N +3/2),

n=O

where

e = @ ~ , , (4a)

u = r ~o) , (4b)

= ~2(~ _ ~o) . (4c)

It is also assumed that

es = eel, (4d)

where el is a constant not very large compared to unity.

The above form of the solution for ?' and ~ differs from that used in the author's

paper on the planar motion of the Trojan asteroids (1977). This alteration results in

some simplification in the solution.

Substituting the expansions (3) into Equations (2) and equating the coefficients of

the same powers of s, the following systems of differential equations can be obtained

f o r Z0, ?'1, 0~1, ZI , ?.2, 0/'2, Z2, ?'3, 0~3

~2Z 0 + Zo = O, (5a)

cqv 2

e oh ~2rl 2 ! = 3rl + 3z2 (5b) ~o 2 ~ c~u / o,

Ov 2rl -t 0v = 0, (5c)

144 B,~LINT i~RDI

~2 z1

819 2 l - Z 1 + 2 ~2Z 0

8v 8u - (3r~ + 3z2)z o, (6a)

cq2r2 802

c32ri / ~ s l } - 2 b b r~ t- 2 8v 8u 8v au L oo ~u ao

=

+ S u /

3 2 = 3 r2 + ~ r l + 3ZoZx l S ( 2 q + Z2) 2 v(3rl 71- 3 2 8 + 1 - e l cos ~-Zo) +

+ r l ( 3 r t + 3 2 2 ) - - COSSo - - 2 - 3 / 2 ( 1 - - COSSo) -'/2, (6b)

8 Er~ q_ 2re q_ 2ra ( 8 ~ " k)~o'~ ()~2 ()~1]

B 1 ~)u 2rt -1 ~v r 8u_J + sin So[1 - 2 - 3 / 2 ( 1 -- COS So)- 3/2],

82Z2 ~2Z 1 82Z0 82Z0 , S(2rl + Z~)2 802 Jr- Z 2 -+- 2 8v 8u t- 2 80 8r r 8U 2 -- [3r2 + 3-r2 + 3ZoZ~ 8 +

(6c)

+ 1 - el cos v(3q + 3 2 0],~ 12., -3/2(1 -- COS , ~Zo)]Zo + [3rt + 23-z - Zo2 SO) - 3 / 2 (7a)

82r3 82r2 82r'~ +,82rl -2(/~--~-~ ~- 8So~ (8c~2 ~ 8sx ) ( 8 S , _ r , al)2 l- 2 8v 8u + 28v cr 8u a \ 8v ~ / 8v 8u 8v

~So I 2 8u

~ SS 3 8S 2 8S 1 - 2 Ov ~ ?u t 8 r t -r l ?-~v t ~u + r 2 8v r }--s

= 3r3 + 3rxr2 + 3 2 ~Z 1 -Ji- 3ZoZ2 ' S(2r, + 2~) ( r 2 4 1 + 2r2 + 22o21) + + ; o 13__~ (2r, ,.,2)3 - -

3 2 [3r2 + 3 2 - 3rl - ~Zo - e~ cos v ~ r I + 3ZoZ~ 15( 2 r , 8 + z2) 2 + l ] +

+ q[3r2 + 3r2 + 3ZoZl ,S (2r + Zo 2)2 8 l + 1 - e l cos v(3rt + 3,.,2~ >w] +

+ r 2 1 3 r l + 3 2 3 2 ~Zo] + e2(�89 + -} cos 2v)(3rt + ~Zo) +

- 1 2 + 3" 2-s/2(1 - cos So) 3 / 2 [ r t + ~ z o - - rl cos So + sl sin So] -

-- 2 - 3 / 2 ( 1 -- COS S o ) - 3 / 2 [ S l sin So + 1"1 + el cos v(cos So - 1)] +

+ ea cos v cos So + s i sin So, (7b)

6S 1 8v 2r3 + 2r~r2 + (r'i + 2r2) ~v

8So) ( 8s2 8 S l ) 8S3 8S 2 q ?,u + 2q ?v t 8u -t 8v t c0u t a~ A

8 E (~S 1 ~So/ 80~ 2 ~)Sl] 8 ( 8Sl' ~ = r~ + 2r2 + 2rj q + . - - r 2rt + +

+ 3 "2- s/2(1 - cos So)- 5/2[rl + 1_2 > o - q cos O~o + o~ sin O~o] sin O~o +

+ [1 -- 2 - 3 / 2 ( 1 -- COS SO)-3/2][S 1 COS O~ 0 "-[- r~ sin So - el cos vs in So]. (7c)

To obta in a complete solut ion for Zo, rl, Sl, Equa t ions (6) and (7) must be considered in a d d i t i o n to E q u a t i o n s (5).


3 . T h e F i r s t O r d e r S o l u t i o n

The solution of Equat ion (5a) is

Zo = Ao cos (o + Vo), ( 8 )

where A o and Vo do not depend on v.

From Equation (5c) it follows that

~O~ 1 - P l - 2 r l , ( 9 ) c%

where pl .dOes not depend on v. Substituting Equations (8) and (9) into Equat ion

(5b) the following solution for r~ can be obtained.

ri = p~ cos (v + r 2p~ + 2 ~ao gu - c o s +

where pl and r are the functions of u and ~-. From Equation (9),

~0~ o ~0~1 2pt COS (V + r + 2'~0 COS -- - 4 3~2 . (11) _ _ ! 2 2 ( v + v o) 3 p l - - 2 8o c~u

In order to avoid secular terms in ~l after the integration of Equation (11) P l is taken

a s

P l -" 4 1 2

- ~A o. 3 ~u

Thus the solution for rl and ~1 from Equations (10) and (11) is

2~O~o 1 2 ~:Ao rl = Pl cos (v + r - ~}A~ cos 2(v + %) 3 c~u (12)

O~l = --2pl sin (o + r + [A2 sin 2(v + Vo) + q,, (13)

where q l does not depend on v.

The solution for Zo, rt and al contain the six unknown functions Ao, Vo, pl, ~'~,

OO~o/~U, ql. Here Ao, Vo, p~, r are the functions of u and T. F rom Equat ions (6) these

functions can be determined as the functions of u and from Equations (7) as the func-

tions of ~-. The quantities ~o and ql might also be the functions of u and r. However,

it can be seen that ao and q l are the functions of u only. F rom Equat ions (6) an

equation can be derived for the determination of ~o and from Equations (7) another

equation for ql.

Having known A o, Vo, pl, r as the functions of u and r, and C~o, ql as the functions

of u, the first order solution is complete.

4 . T h e S e c o n d O r d e r S o l u t i o n

Substituting Equations (8) and (12) into Equat ion (6a) an inhomogeneous differential

equat ion is obtained for Zx.

146 BALINT I~.RDI

t~2Z1

69u2 [- Z 1 ---- 3/~0p I COS (20 -t- 110 "4- ~bl) 4- 23-/~OPl COS (110 - - ~/1) "4-

+ 2 ~Ao sin (v + 11o) + 2Ao( c~11~ ~ao'~ / cos + yo). (14)

It can be seen from this equation that the homogeneous solution for z~ is a harmonic

oscillation in v with unit frequency. Therefore, to avoid secular terms in the general

solution for zl, all terms with unit frequency in the right hand side of Equation (14)

must be eliminated. This condition provides the following equations for A o and 11o.

~Ao - 0 ~3u

110 ~0~0

cOu t3u (15)

From Equations (15) it follows that Ao is the function of ~ only and

110 = 0~0 "at- 1100, (16)

where 11oo depends on r only.

Taking into consideration Equations (15) we obtain from Equation (14)

Z1 -- /~1COS(U -al- 1/1) - - 1/~0p 1 COS (2V + 11o + ~bl) -ll- 23-/~oP1 c0s(11o -- ~/tl), ( 1 7 )

where A1 and vl are the functions of u and r.

Let us turn to Equation (6c) now. Calculating the right hand side of this equation with Equations (12), (13) and (15) we obtain

r~ ~ ~u / ~v ~ ~u_l c%

1 ~20~ 0

3 Ou 2 + sin ao[1 - 2 - 3 / 2 ( 1 - COS 0~0)-3/2] . (18)

As the right hand side of this equation does not depend on v, to avoid secular terms

in the solution for r2 and (E 2

~20~ 0 ~- 3 sin ao[1 - 2 - 3 / 2 ( 1 - - COS 0~0) - 3 / 2 ] - - 0 (19) 8 u 2

is needed. This equation gives ao as the function of u.

From Equation (18) it follows now

~0~ 2

~gv - P 2 - r 2 - 2r2 - - 2r1\ ~v ~ ~u ] 0 u '

where P2 does not depend on v.


Substituting Equat ion (20) into Equat ion (6b) and applying the solution for Zo, rx,

at , the following equation can be derived.

~32r2 F r2

~.)v 2 = 2 8pl sin (v + $ , ) + 2p, - ( ~ b l

8u \ 8u

+ 2 e 1 ~

COS (13 + ~b 1) -~- eu ]

~ao 3 2 2(v + ~ , ) + COS v + ~-Pl COS

1 ~ o i ~ ~uu A~ cos 2(v + % ) + ?6 h~ cos 2(v + %) -

3 (2v 2elPl c o s

- 3A2p~ cos (3v

+ ~bl) -J- 3/~o/~ 1 COS (2v + Vo + v,) -

I s~4 4(v + Vo) + + 2Vo + ~ 1 ) + ~-~,,o cos

+ 2p2

+ I - cos ~o - 2- 3/2(1 - -

I + kp + . . . o + ] _31_ 5/~2 ~(x0 Obu I

COS 0~0)--1/2 31.-

+ ~'~o'~ c o s ("o - "~) ~elp~ c o s ~ l " (21)

As in the case of Equat ion (14) it follows again that the terms with unit frequency

must be set equal to zero.

2 ?___21ou sin (v + ~b,) + 2 p i \ au ~uu/cos (v + ~i) +

f

O(X o + 2 e l cos v = 0.

8u (22)

This provides two equations for pl and fix on the basis that the coefficients of cos v

and sin v must vanish. After some transformations we obtain

0 cqoL o Ou[p 1 COS ( ~ , - - O~o)] - - - - e l ~u sin %

0 c9o~ o 8u [pl s in (~1 - %)] = - el 8u ~ C O S ~o.

The solution of this system is

pl cos ~bl = el + Pl0 COS (0~ 0 -]- ~blO )

pl sin ~hl = plo sin (ao + ~1o), (23)

where plO and r are the functions of r. F r o m Equat ion (21) with the condition (22) one can determine r 2. After this from

Equat ion (20) ~2 can be obtained. The quanti ty P2 must be selected so that the solu-

tion for ~2 does not contain secular terms. Without the details the solution is

148 BAI..INT ~RDI

r2 = P2 cos (v + @2) 1p21 cos 2(0 2 1 0 ~ A ~ cos 2(0 + Vo) - + ']q) - 6

1 A4o c o s 2 ( u - [ - ]"o) -1" �89 cos(2v + ~h~) - 1 A o A s cos (2v + v o + v,) + ~ 1 - - 6

3 2 (30 + 2Vo + ~hop~ cos + r ~4A~ cos4(o + Vo) +

5

+ + g\ Ou] I &Xo A~

- 63-4A~ 6 gu - ~ ( 1 - cos ~o) +

+ 1"2-1/z(1 - COSO~o) - 1 / 2 + �89 cos ~h, -

- 1 , ~ o ~ c o s (~o - ~ , ) 2 8qs

3 c~u (24)

O~ 2 ~--- --2p2 sin (v + if2) 4 gO~o 3 8u p' sin (o + ~hl) lA~p, s in (o + ~h,) - 2

3 2 - ~ho01 sin (o + 2Vo - 4~1) - 2e~ f

bu sin o +

s 2 1 bo~ o + ~01 sin 2(0 + ,~) q ~ ~uu A~ sin 2(0 + 1,,0 ) + S 4 ~A o sin 2(v + Vo) -

�89 pl sin (20 + ~1) + �89 sin (20 + Vo + v,) -

'A~0 t sin (3v + 2Vo + V~,) + 3~A*o sin 4(0 + Vo) + q2, 2 ( 2 5 )

where P2 and ~2 are the functions of u and z, while q2 may be the function of u only. The second order solution contains the unknown functions A 1, v s, p2, $2,

Oql/8u, q2. From Equations (7) A1, vs, p2, r and qs can be determined as the functions

of u. To obtain As, vl, p2, I//2 a s the functions of r and q2 as the function of u, the equations for za, r4, cx4 should be derived and solved. This question however will

not be treated here.

5. The Complete First Order Solution

Now Equations (7) are used to obtain Ao, Vo, ps, $s as functions of ~- and As, I,'S,

02, $2, qs as functions of u. Calculating the right hand side of Equations (7a) we conclude again that all terms

with unit frequency must vanish. Thus the following equation is obtained.

~'c~ ~ c~A o gAl sin (v + vl) + Uu(VS ~o) As cos (o + vt) + ~r gu sin (v + Vo) +

gVo 1 c02~ o ~ql r ~9-~ ho cos (v + Vo) 4 ~ ff~u2 ho sin (o + Vo) tu - - h o cos (o + ,'o) +

+ �89 - 2 - 3/2(1 - - COS cX0)- 3 /2]A 0 c o s ct 0 c o s (0 -[- Vo) = O.


Separat ing in this equat ion the coefficients of sin v and cos v, after some t ransforma-

tions we obtain

t~ ~2CE 0 t~h 0 [~ cos (~ - ~o)] = - � 8 9

60U 9/,/2 ~'t"

c~ c~qx ~Vo [A1 sin ( v l - Vo)] - A o ~ Ao

8u ~u ~r lao[A~ + ~ ( u ) ] , (26)

where

A 3 -t- B 3 ( u ) - - [1 - 2 - 3 / 1 ( 1 - - COS 0~ 0 ) - 3 / 2 ] COS ~0 (27)

a n d A3 is a c o n s t a n t .

Equat ions (26) may be written as two systems

t~2CX0

8u 2 =-[~ cos (~, - ,,o)] = 1~o 8 u

[A, sin (v, - Vo)] = Ao aql t3u Ou

1AoB 3 (28)

and

~Ao - 0 (29)

~V o

~'r - - - - 21-A3"

The solution of Equat ions (28) is

1A 0 /~1 COS V 1 - - 2 ~0~ o

�9 cos Vo - ho q lsin Vo + 1Aob3 sin Vo + 9u

+ h~o cos (~to + ~o)

~0~ o m 1 ~0 A1 sin vl - 2

c3u sin Vo + Aoql cos Vo 1 h o b 3 c o s v o -[- 2 (30)

+ A I o sin (vl o + Vo),

where Alo and Vlo are functions o f t and

b 3 - - f B 3 ( u ) d u .

Considering Equat ion (16) the solution of Equat ions (29) is

(31)

Ao = constant

1 Vo = a o - - ~A3~" + VOl~ (32)

where Vol is constant.

150 BALINT IERDI

Now calculating the right hand side of Equat ion (7c) and taking into considerat ion

that the terms independent of v must be set equal to zero, an inhomogeneous second

order differential equat ion can be derived for q l.

1 82ql 4 8O~o 820~0 3 8u2 9 8U 8U2

~- {cos o~ o + (3 -I- cos O~o)2-5/2(1 -- COS o~0)- 3/2}q I =

__ [1 q_ 2-9/2(1 _ COS a:o)- 3/2 _ 3" 2-9/2(1 -- COS a~o)- 5/2]A~) sin ao. (33)

The solution of this equation is

ql 4 8ao 8ao

= ~ ao 8---u + C28u 2

t- 3C, ~ Ou ] du -

3 2 t~O~o f h o 8u cos O~o [ 1 - 2 - 3/2(1 cos ~o)-~/~] ~u / du, (34)

where C1 and C2 are constants.

At last Equat ion (7b) is considered. After some laborious but s traightforward

calculation we obtain the following equat ion f rom the condition, that the terms with

unit frequency on the right hand side must vanish.

2t3P2 8u

sin (v + 4~2) + 2 - ao) p2 cos (v + 62) +

( 8pi 6q2~176 ) E + 2 ~ {" Cq/,/2 /91 sin (v + 4'~) + 2 b~z t- 2cosc~o +

/ + (5 cos ~o + 9)2- 5/2(1 - cos ~o)- 3/2 pl cos (v + 4~1) -

I~ 20~ 0 - 8u 2 t- 2sin~o{1 - 2-3/2(1 -- COS 0~0)- 3/2}]e 1 sin v +

-']- 21/~2 ~_ 2 8 u j e 1 coso + 52_e1,~ ~a:0c~.____u cos (v + 2Vo) = 0.

This equat ion can be separated into the following two systems


0 bu [P2 sin (~2 - =

1 020~0 2 ~U 2

plO sin ~ o + Ou Bo)pl o cos Cto +

1 +e~ -4 3~)u I c%~~ sin Sol + �88 z gO~o - - t3U o ~ COS o~ o

2 ~0~0 - ~ho Ou cos (2Vo - C~o) + BI}

0 Ou ~ [ P 2 COS (t/I 2 - - (XO) ] - -

"2 1~O~o 2 bu 2 p l o cos 4'~ o _ Bo)pl o sin ~'1 o +

1 ?IO~Xo +e~ 3Uu c~u

1 ~2~)Cg 0 ~ c o s O~o + ~"o 7uu sin O~o +

t~OL 0 --I-- 5 2 m gho

c~u sin (2Vo - So) + B2} (35)

and

0 [P lO sin ~blo] =

c~r -AoPlo c o s ~10 + elA1

cSr [P lO COS ~blO ] = AoPlo sin ~ 1 0 -[-" elA2, (36)

where

Ao + Bo(u) = cos ~o + (5 cos So + 9)2-7/2(1 - cos Cto)- 3/2

A 1 -+- Bl (U ) = - I A o + Bo 7 COS o~ o 3 \ c~u] _l

A2 + B 2 ( u ) = - [Ao + Bo 3 ( eu ] sin CXo (37)

and Ao, A~, A2, are constants.

The solution of Equat ion (35) is

p2 sin ~h2 - - /920 sin (~o + ~h2o) 1 ~ o 2 au

pl o sin (ao + ~h, o) +

+ (ql - bo)plo cos (O~o + r +

_5_ 2 + e l [ - 8,Xo sin 2Vo + b I cos o~ o -Jr- b2 sin ~o] (38)

152 B/~LINT I~RDI

p~ c o s 4'~ = p~o cos (0,0 + r 1 c%~ o

p~ o c o s (C~o + 4'~o) - 2 ~u

- (q, - bo)pjo sin (So I 1 ~ao

+ ~1o) + el --3 au .1_ 2 4/~0

5 2 - ~'~o cos 2 Vo bl sin O~o + b2 cos o%1 ,

where P20 and 1fi20 a re functions of r and

bo = f Bo(u) du, b, = f B,(u) du, b~ = f B ~ ( u ) du. (39)

The solution of Equations (36) is

P,o sin ~1o = - p, , sin (Ao~" + ~1,) A2

- e l A o

AI P,o cos ~,o = P,, cos (Aor + ~b,l) + e"" o'A

(40)

where pll and ~11 are constants.

6. Libration Around L4

We start the discussions of the above results with Equation (19) giving So as the function of u. An approximate solution can be derived in the form

C ~ o = _ + 2_} 4 + 3 23 28

+/cos4-(_~12 q 2~__~33214) cos24 _1 - (~59613

25X/314 2732 COS 44, +

1283 / 5 COS 5 4 -i- O( /6)

212.3.5

65 is' ~ cos ~.77 ,} 3 r

(41a)

with

4 - "V/~-( 1 - ~ 12 - 9 ~ 1 , ) . u -I- a,

where I and 3 are the constants of integration. Equation (41a) describes a long-period libration around

period is

T~ T• - ? Z r ) '

L4. The

(41b)

approximate

(42)

where Ts is the orbital period of Jupiter.


Table I gives the amplitudes and periods of this libration for different values of l with Tj = 11.862 tropical years and e = 0.030 885.

TABLE I Librational amplitudes and periods around L4

l ct 0 [degrees] Tl [years]

0 60 147.8 0.1 54.5-66.0 148.1 0.2 49.5-72.5 149.0 0.3 44.9-79.6 150.5 0.4 40.8-87.3 152.9 0.5 37.2-95.6 156.3

Equation (41a) may be used for the calculation of the constants Ao, -41, A z, A a defined by Equations (37) and (27).

27 1291z _ 8714 A 0 = 23 + 26 27 + O(16 )

27 141 109514 _ O(16 ) (43) A1 = 24 26 12 2 la

27X/3- 45X/3-/2 711V/3/4 } 4 -- 0 ( / 6 ) A2 = 24 26 211

O( /6 ) .

7. The Main Perturbations of e and

The perturbations of the orbital elements may be derived by computing the coordinates X, Y, Z and the velocity components d X/dt, d Y/dt, dZ/dt with the solution for r, ~, z and then using the standard formulations of the two-body problem to obtain

the osculating orbital elements. Omitting the detailed derivations the following results are obtained for the eccen-

tricity e and for the longitude of the perihelion ~o of the Trojan asteroids

e = ePlo + 0 ( g 2 )

~ 4, o( ) t o - - 7r m i o -a t- ~ . (44)

Substituting Equations (44) into Equations (40)

obtain _ A z

e sin o~ = - epll sin (AoT + ~11) -- es Ao

and remembering Equation (4d) we

+ O ( e : ) (45a)

1 5 4 B/~LINT I~RDI

e c o s to = A 1

- - ~ P l l c o s (Ao r + qJ11) - es Ao (4519)

From the relations (43) it follows that

A1

Ao 1 17 12 329 14 ?. 2*3 2-'~ 3' + 0(/6) 06)

A2 m

Ao

"k/~ ~ t ~73~//3/2 - 6233V/3/'~a-~-g - - O ( ] 6 ) .

Neglecting the terms of O(l 6) in Equations (46) it can be seen, that A 1 / A o and A2/A o

are negative quantities. Let the following notations be introduced

A2 Al a = - e S A o , b = - es--;-,Ao C -'- E;PI1, X "-- AOT -[- r (~tT)

where a and b are positive quantities and c may also be supposed to be positive. TlaOS from Equations (45) we obtain

e sin 7o = a - c s i n x

~ 08) e cos to = b - c cos x.

Equations (48) give the main perturbations of e and ~. From Equations (48) it follows that

e = v i a 2 + b 2 + C 2 - 2c(a sin x + b cos x). 09)

I t can be seen that for

sin xl = a b

%/a2 ~_ b 2 ' c ~ - % / a 2 + b2

e has its minimal value

e,,,i. = [%/a 2 + b 2 - c] (50)

and for

sin X 2 : a b

V a 2 + b 2 ' COS X 2 - -

~ / / a 2 --]- b 2

e has its maximal value

t

ema x -- V a 2 + b 2 + c.

Between these limits e varies periodically with the period

T H R E E - D I M E N S I O N A L M O T I O N O F T R O J A N A S T E R O I D S 1 5 5

T A B L E II Periods of the eccentricity, the perihelion and

the ascending node

1 T~, 5 [years] TO [years]

0 3685 0.1 3663 1659 x 10 a 0.2 3600 416 x 10 a 0.3 3502 185 x 10 a 0.4 3379 105 x 10 a 0.5 3241 67 x 103

T, Te. ~ - - A o ~ 2 .

(52)

This is also the period of ~o. Table II gives Te,5 for different values of l.

The variations of 70 may be determined from the equation

tg ~ = f ( x ) - a - c s i n x

b - c cos x (53)

obtained from Equations (48). Investigating the function f ( x )

may be distinguished depending on the coefficients a, b and c.

the following cases

Case I: V a 2 -3 t- b 2 > C. (54)

C a s e A : b < c.

For X l given by the equations

s i n x l = ac + b . v /a 2 + b 2 - c 2

a 2 + b 2 C O S X 1 =

bc - a ~ / a 2 + b 2 - c 2

a 2 + b 2 (ss)

f ( x ) has a local minimum

f ( x , ) a = c , v / a 2 + b 2 - - C 2 - - a b

C 2 w b 2 (56)

For Xz given by the equations

s i n x 2 "--

ac - b , v / a 2 + b 2 - - C 2

a 2 + b 2 C O S X 2 =

bc + a ~ / a 2 + b 2 - c 2

a z + b z (57)

f ( x ) has a local maximum

f ( x2)a = c ,v /a 2 + b 2 - c 2 + ab

C 2 w b 2 (58)

156 B~d.INT I~RDI

5

t

xl I 2 - x >

/

t

Fig. 2. Case A (curves 1, 2, 3), Case II (curve 4) and Case 1II (curve 5): (b = 0.03, c = 0.05;

curve 1: a = 0.08, curve 2: a = 0.05, curve 3: a = 0.046, curve 4: a = 0.04, curve 5: a = 0.02).

It can be seen t h a t f ( x z ) ~ < 0 a n d f ( x 2 ) ~ < f ( x a ) A. It can be p r o v e d tha t :

(1) f ( x l ) a > 0, when a > c. T h e n f ( x ) has no zero value. (See F igure 2, curve 1.)

(2) f ( x l ) A = 0, when a = c. T h e n x l = rr/2. (See F igure 2, curve 2.)

(3) f ( x ~ ) A < O, when a < c. F o r x3 and x4 given by the equa t ions

a ~ / ( a ) Z sin x3 - , cos x3 = 1 -

r

a ~ / ( a ) sin x~ - , cos x4 = - 1 - C

2 (59)

f (x) has two zero values. (See F igure 2, curve 3.)

F o r xs and x6 given by the equa t ions

sin xs = 1 - , cos x5 = - r (60)


sin x6 = - 1 - , cos x6 - - r

f ( x ) has s ingula r i t i es . I t c an be seen t h a t

f ( x ) > - o o , w h e n x > xs , x < xs

f ( x ) > + oe, w h e n x > xs , x > x5

f ( x ) > + oe, w h e n x > x6, x < x6

f ( x ) > - o e, w h e n x > x 6 , x > x 6.

Case B: b = c. F o r x l g iven by the e q u a t i o n s

sin xa = 2ab

a z + b 2 ' COS X 1 =

b 2 _ a 2

a 2 + b 2 (61)

f(x) has a l oca l m i n i m u m

a 2 _ b 2

/ ( X l ) 8 = 2ab (62)

4

r F i g . 3. C a s e B : (b = c = 0 . 0 5 ; c u r v e l : a = 0 . 0 8 , c u r v e 2 : a = 0 . 0 5 , c u r v e 3 : a = 0 . 0 2 ) .

It c a n be s een tha t

(1) f(xl)~, > 0, w h e n a > b. T h e n f ( x ) ha s no z e r o va lue . (See F i g u r e 3, c u r v e 1.)

(2) f ( x l ) ~ - 0, w h e n a - b. T h e n x l = rr/2. (See F i g u r e 3, c u r v e 2.)

(3) f ( x l ) B < 0, w h e n a < b. F o r x3 a n d x4 g iven by E q u a t i o n s (59). f ( x ) has

t w o ze ro va lues . (See F i g u r e 3, cu rve 3.)

158 aALINT ERDI

For x5 - 0 and x6 = 2zr, f ( x ) has singularities. It can be seen that

f ( x ) ~ + oe, when x > x5, x > xs

f ( x ) ~, + oe, when x > x 6 , x -Q x 6.

Case C: b > c.

For xl given by Equat ions ( 5 5 ) f ( x ) has a local min imum

f ( X l ) c =

ab - c,v/a z + b 2 - c

b 2 __ C 2

2

(63)

For x2 given by Equat ions ( 5 7 ) f ( x ) has a local maximum

. f (X2)c ab + c%/a 2 + b 2 - - C 2

[.)2 __ C z (64)

5

I

5

X

Fig. 4. Case C: (b = 0.07, c = 0.05; curve 1" a - - 0 . 0 9 , curve 2" a = 0.05, curve 3" a = 0.01).

It can be seen that f ( x 2 ) c > 0 and f ( x l ) c < f(x2)~. It can be proved that

(1 ) f ( X l ) c > 0, when a > c. Then f ( x ) has no zero value. (See Figure 4, curve 1.)

(2) f ( x l ) ~ = 0, when a = c. Then xl = rr/2. (See Figure 4, curve 2.)

(3) f ( x l ) ~ < 0, when a < c. For x3 and x4 given by Equat ions ( 5 9 ) f ( x ) has two

zero values. (See Figure 4, curve 3.) In this case f ( x ) has no singularity.

C a s e H : -v/a 2 + b 2 = c . (65)


Introducing fl as

a b sin/3 = - , cos/3 = - (66)

C C

from Equation (53) it follows that

x+t f ( x ) = - ctg (67)

2

when x -# /3. For x - /3 Equat ion (53) has a singular point, but it can be proved that

f ( x ) > - ctg fl as x >/3. (For the representation of Case II see Figure 2, curve 4.)

Case III: v ia 2 + b i < c. (68)

In this case f ( x ) has no finite minimum or maximum (see Figure 2, curve 5). For

x3 and x4 given by Equations ( 5 9 ) f ( x ) has two zero values. For x5 and x6 given by

Equations ( 6 0 ) f ( x ) becomes singular. It can be seen that

f ( x ) > + oo, when x > xs, x < x5

f ( x ) -~- - oo, when x ~ xs, x > x5

f ( x ) > + m, when x > x6, x < x6

f ( x ) > - 0 0 , when x - ~- x6, x > x6.

Knowing the behaviour of the function f (x), the variations of oJ can be determined.

It can be seen that Case II separates two different types of the mot ion of the perihelion.

In all the three cases of Case I there is a region in w h i c h f (x) does not take any values

and thus ,5 is restricted to some interval between 0 and 27r. In this case the perihelion

librates. In Case III f (x) takes every values from - oo to + ~ and thus oJ varies from

0 to 2~r. In this case the perihelion circulates.

In Case A the libration takes place between the limits

arctg f (x i)a < ~ < arctg f (Xz)a + ,r. (69)

F rom Equations (56) and (58) it follows that f ( x l ) A >f(x2)A when c , v / a 2 + b 2

and thus the difference between the two limits tends to ,r. Equations (47), (56), (58)

and (69) give the following extreme limits for oo

A 1 ,,,, A 1 - a r c t g < o~ < ~ - arctg , (70a)

A2 A2

where according to Equations (43)

_ 31A//312 A2 3 332

2143V/3 14 _ O(16). (70b) + 3426

In Case B, ~ librates between the limits

arctgf(xl)B-<_ o7, < ,r/2. (71)

160 B.~LINT I~RD!

From Equations (43), (47), (62) and (71) it follows that

21 14 31 3 lZ q + 0(16 arctg < 7_o < rr/2. (72)

In Case C the limits of the libration are

arctg f(Xa)c < /o < arctg f ( X 2 ) c .

It can be proved that Case B is the limiting case of Case C when c Equation (63) it follows that

a 2 _ b 2

f ( X l ) c > 2ab ' when c ~- b

(73)

> b. Namely, from

and from Equation (64) it can be seen that

f(x2)c > + oe, when c -~- b.

Thus the extreme limits for o~ are

arctg

~

31~/3/2 , +

3 3 2Iv/3 / 4 + 0 (16 ) < ~ < - .

25 2 (74.)

T A B L E l l I

E x t r e m e l i b r a t i o n a l l i m i t s o f the p e r i h e l i o n

! C a s e A C a s e B

0 - 30~176 300.0-90 ~

0.1 - 3 0 . 4 - 1 4 9 . 6 2 9 . 1 - 9 0

0 .2 - 3 1 . 7 - 1 4 8 . 3 2 6 . 5 - 9 0

0.3 - 33 .9 -146 .1 2 2 . 2 - 9 0

0 .4 -- 3 7 . 0 - 1 4 3 . 0 1 6 . 1 - 9 0

0.5 - - 4 1 . 0 - 1 3 9 . 0 8 . 6 - 9 0

Table III gives examples for the libration of the perihelion obtained from Equations (70) and (72) for different values of 1.

As can be seen from Equations (45) the libration of the perihelion is due to the orbital eccentricity of Jupiter.

From Equation (51) and from the condition of the libration given by (54) it follows

that

e m a x < 2,v/a 2 + b 2. (75)

Remembering the notations (47) and making use of Equations (46) we obtain the following condition for the libration of the perihelion

I 5203 4 ] e ma x < e j 2 6 Iz -~ ~q-~-gl - O ( / 6 ) . ( 7 6 )

T H R E E - D I M E N S I O N A L MOTION OF TROJAN ASTEROIDS 161

8. The Main Perturbation of

For the longitude of the ascending node 12 the following relation is obtained

7r

I2 = ~o - vo 2 f- O(e). (77)

From Equations (32) and (43) it follows that

s = - l 2 3 l, - O(/6 ) r Vo + O(e) (78) ,

It can be seen from this equation that in the case 1 4= 0 the ascending node circulates

in retrograde direction with the period

To = �9 (79) ( i t - A t , ) ,

Table II gives To for different values of I.

9. Summary

In this paper an asymptotic solution has been derived for the three-dimensional motion of the Trojan asteroids. The solution is obtained completely in the first order and partly in the second order. This solution may be used to investigate the perturbations in the orbital elements. As an example the main perturbations of the eccentricity, of the perihelion longitude and of the longitude of the ascending node

have been discussed. It has been shown that the eccentricity and tlae perihelion longitude undergo periodic

variations with periods between 3200 and 3700 years and depending on the eccentricity the perihelion may librate. The ascending node circulates in retrograde direction with periods more than 60 000 years.

A more detailed theory for the perturbations of the orbital elements is being developed now and it will appear together with the applications for the actual Trojan

asteroids in a subsequent paper.

The

Acknowledgement

author wishes to thank Professor V. Szebehely for his valuable comments.

References

Brown, E. W.: 1925, Trans. Yale Obs. 3, 1-47, 87-133. l~rdi, B." 1977, Celest. Mech. 15, 367-383. Garfinkel, B. : 1977, Astron. J. 82, 368-379. Kevorkian, J.: 1966, 'The Two-Variable Expansion Procedure for the Approximate Solution of

Certain Nonlinear Differential Equations', Space Math. 3, A.M.S. Kevorkian, J. : 1970, 'The Planar Motion of a Trojan Asteroid', in G. E. O. Giacaglia (ed.), Periodic

Orbits, Stability and Resonances, D. Reidel P-ablishing Company, Dordrecht, Holland, pp. 286-303.

the three-dimensional motion of trojan asteroids

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