The dynamics of tadpole and horseshoe orbits

ICARUS 48, 1-11 (1981)

The Dynamics of Tadpole and Horseshoe Orbits I. Theory STANLEY F. DERMOTT AND CARL. D. MURRAY Center for Radiophysics and Space Research, Space Sciences Building, Cornell University, Ithaca, New York 14853-0355 Received June 23, 1981; revised August 28, 1981 The properties of the tadpole and the horseshoe orbit solutions of both the circular and elliptic restricted three-body problem with small mass ratio are examined using analytical and numerical methods. We show how the trajectory of a particle in a near-circular orbit is critically dependent on the initial radial separation between the particle and satellite orbits. This separation can also be used to derive an approximation to the minimum particle-satellite distance. A useful relation between the shape of a particle's path in a reference frame in which the perturbing satellite is stationary and the shape of the particle's associated zero-velocity curve is presented. We also determine the circumstances in which horseshoe paths rather than the more common tadpole paths are to be expected. By numerical integration of a number of horseshoe orbits we have investigated the effects of changes in eccentricity and longitude of pericenter over repeated satellite encounters. Such changes will determine the long-term stability of particles in horsehsoe orbits.

INTRODUCTION

The restricted three-body problem has been extensively studied for more than 200 years and an impressive amount of knowledge has been accumulated (see, for example, Szebehely, 1967). In most of the work the emphasis has usually been placed on the "tadpole" solutions since these describe the motion of the Trojan asteroids which librate about one of the Lagrangian equilibrium points, either L4 or Ls, of Jupiter. Recently, however, the neglected "horseshoe" solutions in which objects librate on paths which encompass L3, L4, and L~ have been used in attempts to account for the various features of narrow ring systems (Dermott, et al., 1979; Dermott et al., 1980). This paper describes the general properties of the tadpole and horseshoe orbits, with particular emphasis on the horseshoe solutions. In a subsequent paper we extend this work to the case in which perturbations due to the smallest body are not negligible, and investigate the coorbital satellites of Saturn.

THE CIRCULAR PLANAR RESTRICTED THREE-BODY PROBLEM

Consider a system of three bodies: a planet of mass M, an orbiting satellite of mass ml and a particle of mass rn~, where M > ml >> ms. The planet and satellite are constrained to move in circular orbits about their common center of mass, while the particle moves under the gravitational influence of the planet and the satellite in the plane of their orbits but has negligible effect on their motion. The problem of describing the motion of the particle is the circular planar restricted three-body problem. Since the planet and the satellite are moving with constant angular velocity and have a constant separation, it proves convenient to consider the motion of the particle in a frame centered on the planet and rotating with the line joining the planet and satellite centers. If we take M as the unit of mass and the separation of the planet and the satellite as the unit of distance, then the mean motion n of the satellite is n 2 = 1 + m l,

(I)

0019-1035/81/100001-11502.00/0 Copyright© 1981by AcademicPress,Inc. All rightsof reproductionin any formreserved.

2

DERMOTT AND MURRAY

and, as Darwin (1897) has shown, the equations o f motion of the particle are Y¢ -

(2)

2 n ~ = Ol'~/x

(3)

+ 2n,~ = O [ l / O y ,

with the integral 2 2 + ~ 9 2 = V2 = 2 1 " 1 - C

(4)

2 I I = 2 / r + r 2 + ml(2/A + A2)

(5)

where

and C is the Jacobi constant. The positive direction o f the x axis passes through the satellite, r 2 = x 2 + y2,

(6)

and A is the distance o f the particle from the satellite. Substitution o f J~ = j~ = x = y = 0 into Eqs. (2) and (3) and solving for r and A yields the colinear Lagrangian equilibrium points L1, L2, and L3 and the triangular equilibrium points, L4 and L5 (see Fig. 1). The latter correspond to solutions where r = A and by applying a first-order perturbation analysis to objects displaced from L4 and L5 it can be shown (see, for example,

Darwin (1897)) that the subsequent librations are stable if ml < 0.040. These are the so-called " t a d p o l e " solutions of the restricted three-body problem and represent stable oscillations of objects about the Lagrangian equilibrium points 60° ahead of and 60° behind the satellite. The Trojan asteroids are known to be moving in tadpole orbits about the L, and L5 points of Jupiter, and recent ground-based observations of the Saturnian satellite system have led to the discovery o f an object librating about the leading Lagrangian equilibrium point L4 of the satellite Dione (Reitsema, e t al., 1980), while Harrington e t al. (1981) consider that objects may also be librating about both L4 and L5 of the satellite Tethys. Solutions o f Eqs. (2) and (3) also allow for libration of objects on paths which encompass L4, L5, and L 3. These are the " h o r s e s h o e " orbit solutions, which until recently have been somewhat neglected. Such orbits, however, were used by Dermott e t al. ( 1 9 7 9 , 1980) in attempts to account for various features o f narrow ring systems in which small satellites ( m l / M <<10 -9 ) were postulated to maintain ring particles in stable horseshoe orbits. In this paper we describe the properties of these horeshoe orbit solutions and in the subsequent paper we show how they can be applied to the coorbital satellites of Saturn. TYPES OF ORBITS

,?;o,o,,,,o

FIG. 1. Schematic diagram showing the Lagrangian equilibrium points and the critical zero-velocity curves. The critical horseshoe curve actually passes through L~ and L2 and the critical tadpole curve passes through La. Horseshoe orbits will exist between these two extremes. The rectangular coordinate frame is centered on the planet and rotates with the satellite.

Since the particle velocity V must be real, V 2 in Eq. (4) must be positive and the particle is confined to move in those regions where 2tl > C. The bounds of these regions, the so-called zero-velocity curves, obtained by setting 21-1= C and solving for r and A, are particularly useful in our problem since there is a close correspondence between the shape of a particle's path in the rotating reference frame shown in Fig. 1 and the shape of its associated zero-velocity curve. Brown (1911) has shown that in the region where tadpole orbits are possible

TADPOLE AND HORSESHOE ORBITS I C = 3 + tim

(7)

3-
(8)

where

and m = m l / M , and that in the region where horseshoe orbits are possible C = 3 + a m 213 + O ( m ) ,

(9)

where 0 -< a -< 3 ~3.

(10)

A schematic diagram of the critical zerovelocity curves which separate the circulation region f r o m the horseshoe region, and the horseshoe region from the tadople region is s h o w n in Fig. 1. It follows f r o m Eqs. (5), (7), and (8), that in those regions where, depending on the value of r, either a tadpole or a horseshoe orbit is possible we must have 2v 2 -

1-< As-< 2

(11)

and 3 -< (2/A~ + Az2) = h -< 5,

which describes the associated zero-velocity curve. D e r m o t t e t al. (1980) have found by numerical integration o f particular cases that the actual path that a particle follows on encountering the satellite is determined by a, that is, if a m 2/a ->m and A ,~ (21/2 - 1) on encounter, then the shape o f the path in a reference frame rotating with the satellite is determined by ot alone and scales as m v3. In Fig. 2 we show some typical results for the case m = 10 -~. F o r high values of a (~>1) the particles either strike the satellite or are scattered but for small values oft~ (< 1) the particles are repelled by the satellite and motion in horseshoe orbits is p o s s i b l e - - f o r a simple discussion of the dynamics involved see D e r m o t t e t al. (1979). The nature of the path changes dramatically as ot is reduced and for very small values of a (~<0.2) the path is almost perfectly s y m m e t r i c about the y axis. If we write a = 1 + Aa,

(12)

where subscript z refers to the coordinates o f a point on the zero-velocity curve. We now use polar coordinates (r, 0) and write

3

(18)

where a is the semimajor axis o f the particle orbit, and

[Aa0[ -[Aa~[

-- ___m"

(j = 1, 2),

(19)

where the subscript refers to the n u m b e r of consecutive encounters with the satellite that the particle has experienced, then we and find that n increases to values <0.7 (j = 1) r = 1 + 8r . (14) and - 1 . 2 (j --- 2) as a decreases to -<0.2 (see Fig. 3). Thus for small values o f a the In our problem, the m a s s ratio m is exceedorbits are nearly periodic and Aa0 and Aa2 ingly small (-<10 -8) and we have 1 -> m 1/a ,> are equal and opposite to order m. This m112 >> m, 8r .¢ 1 and, e x c e p t near those r e m a r k a b l e s y m m e t r y is not a p r o p e r t y o f turning points where 0 but n o t / . is zero, I/.[ circular orbit case alone. Figure 3 IrO[ and the particle orbits are nearly the shows that there is no substantial differcircular. H e n c e , ence b e t w e e n the circular- and the eccenV --- r0 = - ~ S r , (15) tric-orbit cases: in both cases [Aal appears to be an adiabatic invariant of the and Eqs. (4) and (5) b e c o m e motion. In those regions in which the particle ~Sr 2 = 3 + 3~r 2 + h m - C , (16) paths are s y m m e t r i c a b o u t the r = 1 line, there is a simple relation b e t w e e n the width which describes the particle path, and W o f the region bounded b y the particle 0 = 3 + 38rz 2 + h m - C , (17) path and the width W~ o f the region V 2 = i.2 + (r0)2,

(13)

4

DERMOTT AND MURRAY

PARTICLE TRAJECTORIES

x(xlO-3)

(Z

2.502.00-

..

1.50t. 2 5 ~ - ~

0.75 :'_ .,,"~

....

(

3'.o " " - )z:o

J / / ~ / / / /

• '"(o)

, .

.N.

" " ......

"'" " .

t 'o "

-,

Mass Ratio: 4(] 9

Fla. 2. Particle paths in a frame centered on the planet and rotating with the satellite. The ratio of the satellite and planet masses is 10-9 and the satellite orbit is circular. The satellite radius corresponds to that of a satellite of mean density 2 g cm -s at a distance of 2 planetary radii from a planet of density 1.2 g cm -3. For high (~1) values of a (an impact parameter), the particles are either captured or scattered by the satellite, but for small values of a, motion in horseshoe orbits is possible.

b o u n d e d b y the a s s o c i a t e d z e r o - v e l o c i t y c u r v e (see Fig. 4). I n the h o r s e s h o e o r b i t c a s e , f r o m E q s . (9) a n d (16), we o b t a i n 3 + ~Sr 2 + h m

= 3 + a m 213 + O ( m )

(20)

a n d , f r o m E q s . (9) a n d (17), 3 + 38rz 2 + h m = 3 + a m 213 + O ( m ) . (21)

p e r f e c t l y s y m m e t r i c a l l y d i s p o s e d with res p e c t to the r = 1 line, this v e r y close corr e s p o n d e n c e b e t w e e n a p a r t i c l e ' s path a n d its a s s o c i a t e d z e r o - v e l o c i t y c u r v e p r o b a b l y a c c o u n t s for the difference b e t w e e n the two c u r v e s d e s c r i b i n g [Aa0l - I Aall a n d [Ha0[ - [Aaz[ s h o w n in Fig. 3. If we write

H e n c e , s i n c e rn 2t3 >> m , W = 2Wz = 4 ( a / 3 ) v2 m va.

(22)

In the t a d p o l e o r b i t c a s e , f r o m E q s . (7) a n d (16), w e o b t a i n 3 + ~¢Sr2 + h m

= 3 + Om

(23)

a n d , f r o m E q s . (7) a n d (17), 3 + 38rz 2 + h m = 3 + 0 m . Hence,

(27)

rint = 1 - 8rin t < 1,

(28)

w h e r e the s u b s c r i p t s ext a n d int refer to the exterior and interior branches of a given z e r o - v e l o c i t y c u r v e , t h e n f r o m Eq. (5) we obtain 2/rext

(24)

rex t = 1 + 8rex t > 1,

+

text 2 =

2/rin t

+ t i n t 2 q-

O(m)

and 8rext - 8rint = ~Sr 2 + O ( m ) / S r ,

W = 4[( 0 - h)/3lV2m v2

(25)

and, again, (26)

S i n c e the z e r o - v e l o c i t y c u r v e s are n o t

(30)

where 8r = ½(Srint + 8rext).

W = 2Wz.

(29)

(31)

H e n c e , since in the h o r s e s h o e o r b i t r e g i o n 8r - m i/3,

TADPOLE AND HORSESHOE ORBITS I i

I

'

I

,

5

These are effectively Hill's equations (see, for example, B r o u w e r and Clemence, 1961, p. 337) which were first used in connection with lunar theory. Consider a particle initially moving in a circular orbit prior to e n c o u n t e r with the satellite. If A >> rn l~a, then, from Eq. (34),

1.2

3'o = -~Xo = -~Aao,

0.8

(36)

[cf. Eq. (15)] and, from Eq. (35),

n

JAao[ = 2(ot/3) 112m va

(37)

[cf. Eq. (22)]. I f a ~< 0.2 and the path o f the particle is symmetrical about the y axis then when x = 0, )' = 0, ~t = ~tma~, and y = Ymin, where, from Eq. (35),

0.4

o

,

0

I

,

0.4

I

3Crnlng = 2m/Ymi n

,

0.8

t .2

FlG. 3. n is a m e a s u r e o f the s y m m e t r y of the horseshoe path in (a, 0) space a b o u t the line a = i [see Eq. (19)]. The circular points refer to c h a n g e s in the semimajor axis a o f the particle orbit after a single e n c o u n ter with the satellite: el, the circular orbit case (es = 0, where es is the eccentricity o f the satellite orbit); O, the elliptical orbit case (es = 0.01). Triangles indicate the total change in a after two c o n s e c u t i v e e n c o u n t e r s : &, es = 0; A, es = 0.01.

(32)

and it would follow that the value o f n corresponding to JAa0] -- JAa~[ should not e x c e e d ~ . Furthermore, if the particle path does track the associated zero-velocity curve, then it would follow that the change in a on the next encounter compensates for the a s y m m e t r y o f the previous e n c o u n t e r and thus the path o f the particle in (a, 0) space remains closed to order m. If we now transform Eqs. (2) and (3) to a frame centered on the satellite (as in Fig. 2) and only retain terms o f order m ~/a we obtain J~ - 2y = x(3 - rn/Aa), y~ + 2.2 = - y m / A

3,

-

a m ~ls.

(38)

m z/3,

(39)

(2/Or) 1/2 m 1/8,

(40)

3Cmtn 2 <~

yo 2 ~

hence Ymin = or

Aa02) m.

(41)

Since, when x = 0 we have ~t~ + ~ ~ m 2/8, it follows from Eq. (35) that on passing the axis o f s y m m e t r y the path o f the particle grazes its associated zero-velocity curve. Two c o m m e n t s can now be made con-

Path~ Satellite

Particle

i ~~ - U

-

Associoted~

(33) (34)

and the integral of motion becomes ±2 + j,2 = 3 x 2 + 2 m / A

otto 2la.

We have found by numerical integration o f particular cases that

Y m i n -----" (8/3

18r~t - 8rind ~ m 213

--

(35)

FIG. 4. Schematic diagram showing a particle path o f width W and its associated zero-velocity c u r v e o f width Wz.

6

DERMOTT AND MURRAY

cerning the type o f orbit favored b y a coorbital particle and the stability of that orbit. Since the width of the tadpole region is - m 112 [see Eq. (25)] and the width of the h o r s e s h o e region is - m ~/a [see Eq. (22)], the ratio R o f these widths is ~ m 1/6 and decreases, albeit slowly, as rn decreases. For m - 1 0 -a we have R - 0.3, but for rn - 1 0 -9, R is as small as 0.03. This alone suggests that horseshoe orbits are more likely w h e n m is very small. Secondly, the lifetime of a particle in a horseshoe orbit m a y be finite. I f Eq. (19) was sufficient to describe the effect o f a particle-satellite encounter, then we would e x p e c t a particle to be lost due to a r a n d o m walk o f the quantity [Aa0[ - ]Aa~[. This would o c c u r on a time scale

F ~ T i m 5ta,

(42)

where T is the orbital period o f the satellite. It m a y be of significance that F increases m a r k e d l y as m decreases: only very small satellites m a y be able to maintain other coorbital satellites moving in horseshoe orbits. F o r Jupiter m --- 10 -3 and T = 12 years, hence F - l0 s years and an object in a coorbital horseshoe with Jupiter would be short-lived. It is p r o b a b l y significant that none o f the Trojan asteroids are o b s e r v e d to have horseshoe paths. For Dione m --- 2 × 10 -~ and T ~ 10 -2 years, hence F ~ 107 years. Again it is p r o b a b l y significant that the coorbital satellite o f Dione, Dione B m o v e s in a tadpole orbit. H o w e v e r , for 1980S 1, the larger of Saturn's coorbital pair rn - 6 × 10 -9 and T = 10 -a years, hence F 101° years and its greater than the age of the solar system. ORBITAL ECCENTRICITIES AND A L I G N M E N T OF PERICENTERS

Orbital stability is not simply determined by the changes in [Aa[ on encounter, since each encounter also results in changes in the orbital eccentricities e and the longitudes of pericenter &. Using the same subscript notation e m p l o y e d in Eq. (19) we define the eccentricity gradient g produced

by the gravitational interactions on encounter b y g

_ e0-

e~.

a0-

ax

(43)

We have determined how g depends on ot by numerical integration of particular cases. In our experiments the satellite m o v e d in an orbit of semimajor axis as and eccentricity es = 0.0078 and the initial eccentricity e0 of the test particle was chosen such that e0a0-

es _ 0.62, as

(44)

while Aa0, and hence a0, was obtained from a through Eq. (37), and the pericenters of the initial orbits were chosen to be aligned. Since the satellite orbit is eccentric, the results of encounter depend on the longitude of the satellite with respect to the longitude of the pericenter of its orbit at encounter and this is determined by the initial longitude of the satellite ~.s and the initial longitude o f the test particle h0. That is, the results vary with the initial phase ~b of the test particle and thus all values of ~b need to be investigated. We define ~b by

6 = I(xo- x , ) / a n o [ - I,

(45)

An0 = -~Aa0,

(46)

where

and the integer I is chosen such that 0 < ~b ---t. For the very low mass ratios (<-10 -9) used in our experiments, we have found that if[ho - ks[ >~ 10° and An <~ 1, then I is large and the results are insensitive to the value of I and depend only on ~b. Values o f g as a function of ~bfor various values of a for both exterior (Aa0 > 0) and interior (Aa0 < 0) initial orbits are shown in Fig. 5 and the corresponding changes in &, A& are shown in Fig. 6. The magnitudes of g and A& are clearly a function of a . The highest values are obtained for large values of a since it is those orbits which pass closest to the satellite [see Eq. (40)] and which suffer the largest perturbations. The large o~

TADPOLE AND HORSESHOE ORBITS I +0,61

~

I

0

.o

•

0

0

•

0000

For lower values o f a we find lower values o f g and A& and a more well-behaved s y s t e m . T h e s y m m e t r y o b s e r v e d be-

o.J •

o

•

o

00

t w e e n e x t e r i o r a n d i n t e r i o r o r b i t s for low v a l u e s o f ~t in Figs. 5 a n d 6 c a n be p a r t l y a c c o u n t e d for b y a n e x a m i n a t i o n o f G a u s s ' f o r m o f the p e r t u r b a t i o n e q u a t i o n s :

0o

0.0

o o

" =t.t

-0.6

7

d a / d t ~ 2T/n,

(47)

d e / d t = (1/na) (R s i n f + 2T cos f ) ,

+0.6 o

O0

0 O0

0

. • ~°.

g 0.0

___._0_ _ _4_ .e..~ 0

0

d& = l_l__ ( - R c o s f + 2T sin f ) ,

O.o

0

•

dt

0

•

(49)

nae

0

w h e r e R a n d T are the radial a n d t a n g e n t i a l

•

I

50* Exterior

Q =0.8

-0.6

(48)

•

I I

+0.6

o

o

..:_;

.

0° 0

o°OeOeOeOeo

0.0

g.o..o.B..o.~,~, oe._. . . . . . . . . . . . .

0

._e_.

0.0

0 0

O

°= = t.t

Interior

-0.6

0

O

O

I

50 °

a =0.6 I

0.5

t .0

¢ FIG. 5. Measurements of the eccentricity gradient g(= (e0 - el)/(ao - al)) as a function of the initial phase ~b for various values of a. Test particles were placed on orbits either exterior (O) or interior (©) to that of the satellite and the changes in eccentricity and semimajor axis were measured after encounter. Although the behavior is regular for small values of a it becomes increasingly chaotic as a increases. However, there is always a strong tendency for g to be positive. e n c o u n t e r s also e x h i b i t the l a r g e s t variat i o n s o f g a n d A& w i t h initial p h a s e ~. S o m e particles are p l a c e d in c i r c u l a t i n g o r b i t s while o t h e r s pass too close to the p o i n t m a s s satellite for the n u m e r i c a l r e s u l t s to be m e a n i n g f u l w i t h o u t r e g u l a r i z a t i o n o f the p r o b l e m (see S z e b e h e l y , 1967): for a satellite o f finite d i m e n s i o n s s u c h o r b i t s w o u l d i m p a c t the satellite s u r f a c e . W e n o t e that for b o t h e x t e r i o r a n d i n t e r i o r initial o r b i t s g is p r e d o m i n a n t l y p o s i t i v e ,

Exterior

ZX~'0 o

• •

Interior

0 @ 0

0

0

o o o o o

o

o o

o

o o

o

-50 ° 30 o

00

o

o°oo

o=0.8 I

k F'xterior

I

zoo-oo-2--ioooo-ooo-ooo-o,

-30 ° 0.0

Interior

a =0.6 I

0.5

t0

¢ FIG. 6. The change Ac~ in longitude of pericenter o~ as a function of initial phase ~bfor various values of the impact parameter a. The test particles were started on orbits either exterior (O) or interior (O) to that of the satellite. Acbis small and its behavior regular for low a, but increases and becomes more chaotic as a increases.

8

DERMOTT AND MURRAY I

3~

i

I

0

o -30:

o o

,

0.2

I

0.4

,

I

Off

,

I

,

0.8

I

,

t.2

t.0

FIG. 7. The maximum observed change in longitude of pericenter over all phase angles ~b for both exterior (O) and interior (©) initial orbits. The large values of ~ m a x for high a imply that such orbits would be highly unstable.

forces on the particle due to the satellite and f is the particle's true anomaly. The radial force R is m u c h smaller than T and changes sign during e n c o u n t e r and so we neglect it. For exterior orbits T > 0 while for interior orbits T < 0. H e n c e , one might e x p e c t the values of A& for exterior and interior initial orbits to have opposite signs, w h e r e a s since ti and e are both proportional to T one might expect the sign o f g to be invariant and this p r o v e s to be the case. The one clear conclusion from these results is that high a orbits are highly unstable: a single particle could not remain in such an orbit for long without either impacting the satellite or being scattered into a circulating orbit. F o r small values o f a (<~0.2), h o w e v e r , not only is IAal an adiabatic invariant of the motion, the changes in e and & also a p p e a r to be nearly periodic: as we have found for I Aa I, the a s y m m e t r y o f one e n c o u n t e r is to a large extent compensated for b y the next. The decrease o f At~ with decreasing a is s h o w n in Fig. 7. (Note, foi" small values o f a (<0.4) appreciable changes in the orbital elements o c c u r e v e n w h e n the separation o f the particle and the satellite is as large as 10O. Therefore, in this and all other numeri-

cal integrations with a < 0.4, initial and final separations o f 180° were used.) The magnitude of the changes in e, which are best described by the equation eo -

ej =

(2" = 1, 2)

+_m n

(50)

[cf. Eq. (19)] are shown in Fig. 8. For the circular orbit case (e0 = es = 0) there is little difference b e t w e e n the first and second encounters. F o r j = 1 and t~ ~< 0.2 we have n > 1 e v e n though Iaa0l and laa,I are not equal to order m (see Fig. 3). On e n c o u n t e r the eccentricity o f the particle orbit does increase appreciably (see Fig. 9), but after e n c o u n t e r the gravitational forces on the particle act to recircularize its orbit. This interesting p h e n o m e n o n can be partly accounted for by an analysis of Hill's equations of motion. When the particle is distant from the satellite we can assume that it m o v e s in an unperturbed Keplerian ellipse, and if m is so small that 1 >> A >> m '~a, then we can write x = Aa + e sin

nt

+ O(e~).

(51)

Hence, since n 2 = 1 + rn~ = l, Jc = e cos

nt

(52)

TADPOLE AND HORSESHOE ORBITS I '

I

I

i

i

1.2

08 n

tx

o

o

o

~zx z, °1

oo

~

04 ~

,

I 0.4

O eo

,

go

I

i

0.8

ct

FIG. 8. n is a m e a s u r e o f the c h a n g e in e c c e n t r i c i t y o n e n c o u n t e r w i t h the s a t e l l i t e [see E q . (50)]. The c i r c u l a r p o i n t s r e f e r to c h a n g e s in the e c c e n t r i c i t y e o f the p a r t i c l e o r b i t a f t e r a s i n g l e e n c o u n t e r w i t h the satellite: @, the c i r c u l a r o r b i t c a s e (es = 0, w h e r e es is the e c c e n t r i c i t y o f the s a t e l l i t e orbit); O , the e l l i p t i c a l o r b i t c a s e (es = 0.01). T r i a n g l e s i n d i c a t e the t o t a l c h a n g e in e a f t e r t w o c o n s e c u t i v e e n c o u n t e r s : A , e~ = 0; A, es = 0.01. T h e c o r r e s p o n d i n g c h a n g e s in s e m i m a j o r a x i s are s h o w n in Fig. 3.

and, f r o m Eq. (33), p = - ~ A a - 2e sin nt.

(53)

J a c o b i ' s integral, Eq. (35), then gives

9

where, from Fig. 3 for a -< 0.2 a n d j = 1 we have n - 0.72. Thus, for a ~< 0.2 a n d j = 1 we would not expect n in Eq. (50) to e x c e e d 0.7. In fact, we find that n e x c e e d s unity. F o r the eccentric orbit case (e0 = 0.01 and es is such t h a t g = 0.62), the b e h a v i o r o f e is similar to that o f IAal. For ot < 0.4, there is a m a r k e d difference b e t w e e n the j = 1 and t h e j = 2 cases (see Fig. 8). F o r j = 1, n increases to 0.6 as a --~ 0, but f o r j = 2 as a ~ 0 we find n ~ 1. Thus, as for [ Aa[, the a s y m m e t r y o f one e n c o u n t e r is c o m p e n sated for b y the next and the motion is nearly periodic. H o w e v e r , although the s y m m e t r y bet w e e n exterior and interior orbits shown in Figs. 3, 5, 6, 7, 8, and 9 suggests that l o w - a horseshoe orbits are inherently stable, and although our results suggest that stability should increase as the mass m 1 o f the satellite decreases, we do not claim that these limited numerical integrations are actually sufficient to determine the long-term stability o f these orbits. Deprit and Delie (1965) showed that the periodic horseshoe orbits obtained b y Rabe (1961) were linearly unstable. We do not c o n c e r n ourselves here with truly periodic orbits, since we realize that in our applications e x a c t periodicity would be short-lived due to the small perturbations o f other satellites. It is our intention to investigate the factors which deter-

¼Aa 2 - e 2 = constant = a m 2/3. (54)

2

I

I

%.

I

I

356

358

The same result can be obtained directly from T i s s e r a n d ' s relation: l / a + 2[a(1 - e 2 ) ] u~ = C

+O(m)

(55) x

(Danby, 1962, p. 189). Thus ~ A a o 2 -- e o 2 =

¼Aal 2 -

el 2

(56)

and if IAa~l --~ IAaol, then e~ = eo = 0. H o w e v e r , this is only a partial explanation o f the observations. F o r if eo = 0, then, since Aa - m 2/3, we have e ? - (laa01 -

t A a , I ) m 2/3

m "+2/s,

(57) (58)

0 0

2

4

360

0*

FIG. 9. The v a r i a t i o n o f the e c c e n t r i c i t y e o f the p a r t i c l e o r b i t w i t h a n g u l a r s e p a r a t i o n 6 o f the p a r t i c l e a n d the s a t e l l i t e f o r m = 10 -a a n d a = 0.2. The p a t h is s y m m e t r i c a l a b o u t the line # = 180°.

DERMOTT

10

AND MURRAY

mine the length of time that a particle can remain in libration before being lost from the system. We must also stress that our integrations, although extensive, have been limited to a number of special cases. Firstly, we have only fully investigated those cases in which the pericenters of the eccentric orbits of the satellite and the test particle are initially aligned. In Fig. 10 we show that for m = 10eg, CY= 0.4 and e, = es = 0.0078, small departures (~4”) of 10, - Li,J from zero have little effect on n in Eq. (19), but as IO0 _ i&l increases then there is a marked decrease in n and this could imply a loss of stability. Secondly, we have only fully investigated those cases in which g, as defmed by Eq. (43), is < 1. The latter restriction is appropriate for narrow eccentric rings (Dermott et ul., 1979), but not for coorbital satellites. In Fig. 11 we show that for m = 10wg, a! = 0.4 and W0 = W,, tI decreases as e, - e, increases and, again, this could imply a loss of stability. a=04

FIG. 1 I. II is a measure of the symmetry of the horseshoe path in (a, 8) space about the line N = 1 [See Eq. (19)]. In this integration m = IOeg, e, = 0.0078, O0 = (js and a = 0.4 and we show the effect on n of increasing the initial difference in the eccentricities of the satellite and particle orbits, c0 - e,.

I

FIG. 10. n is a measure of the symmetry of the horseshoe path in (u , 0) space about the linen = 1 [see Eq. (19)]. In this integration m = loss, e, = r, = 0.0078 and a = 0.4 and we show the effect on n of increasing the initial separation of the pericenters of the satellite and particle orbits, & - &.

SUMMARY

While the general two-body problem can be solved analytically the restricted threebody problem permits no closed analytic solution for physically meaningful cases. However, by means of the Jacobi integral, which is equivalent to the energy integral of the two-body problem, it is possible to define regions which are inaccessible to the particle. The boundaries of the regions are called zero-velocity curves. In this paper we have investigated the tadpole and horseshoe solutions of the restricted three-body problem for the case where m 1’2 < m 1’3 and where the eccentricity of the particle remains small (50.01). Under these conditions we have shown that particle trajectories in the rotating frame are closely related to the shape of their associated zero-velocity curves: the radial width of the region bounded by the particle path is twice the width of the region bounded by the particle’s associated zero-

TADPOLE

AND HORSESHOE

velocity curve. In the case of larger particle orbit eccentricities, where L is not negligible, equivalent statements can be made concerning the guiding center of the particle’s trajectory although this has not been discussed in this paper. We have demonstrated that particles close to the orbit of the satellite (a 5 0.02) move in near-periodic orbits with changes in eccentricity and longitude of pericenter produced by one satellite encounter being effectively cancelled out by those produced at the following encounter. We have derived an approximate expression for the minimum angular separation between the satellite and a particle in a horseshoe orbit. The lifetimes of particles in horseshoe orbits increase with decreasing mass. This could explain the absence of Trojan asteroids with horseshoe orbits and suggests that small-mass satellites (m 5 1O-8) could maintain ring particles for the age of the solar system. The relative widths of the horseshoe and tadpole regions is also a function of the mass ratio, with the horseshoe region increasing relative to the tadpole region as m decreases. Thus it seems likely that objects in horseshoe orbits in the solar system will be associated with small For the satellite-planet mass ratios. 198OSl-198OS3 system, m - 6 x lo+ and these objects move in a horseshoe configuration. This system will be investigated more fully in the next paper. This paper has been concerned with the properties of the tadpole and horseshoe solutions of the restricted three-body problem

ORBITS I

11

in which the particle mass, m2 is considered to be negligible.. In the following paper we will discuss the more general case where the mutual perturbations of the three bodies are considered. The results will then be applied to an investigation of the various coorbital satellites of Saturn. ACKNOWLEDGMENT This research was supported by NSF Grant AST8024042. REFERENCES

BROUWER, D. AND CLEMENCE, G. M. (1961). Methods of Celestial Mechanics. Academic Press, New York. BROWN, E. W. (1911). On a new family of periodic orbits in the problem of three bodies. Mon. Not. Roy. Astron.

Sot.

71, 438-454.

DANBY, J. M. A. (1962). Fundamentals of Celestial Mechanics. Macmillan, New York. DARWIN, G. (1897). Periodic orbits. Acta Muth. 21, 99-242.

DEPRIT,A., AND DELIE, A. (1965). Trojan orbits. 1. d’Alembert series at L,. Icarus 4, 242-266. DERMOTT,S. F., GOLD, T., AND SINCLAIR,A. T. (1979). The rings of Uranus: Nature and origin. Aswon. J. 84, 1225-1234. DERMOTT,S. F., MURRAY,C. D., AND SINCLAIR,A. T. (1980). The narrow rings of Jupiter, Saturn and Uranus. Nature 284, 309-313. HARRINGTON,R. S., PASCU, D., AND SEIDELMANN, P. K. (1981). IAU Circular No. 3583. March 11, 1981. RABE,E. (1961). Determination and survey of periodic Trojan orbits in the restricted problem of three bodies. Astron. J. 66, 500-513. REITSEMA,H. J., SMITH, B. A., AND LARSON, S. M. (1980). A new Saturnian satellite near Dione’s L, point. Icarus 43, 116-l 19. SZEBEHEL~,V. G. (1967). Theory of Orbits. Academic Press, New York.