Dynamics of Saturn's E ring

Dynamics of Saturn's E ring

ICARUS 58, 169--177 (1984) Dynamics of Saturn's E Ring* P. K. SEIDELMANN AND R. S. HARRINGTON U.S. Naval Observatory, 34th and Massachusetts Avenue, ...

597KB Sizes 0 Downloads 13 Views

ICARUS 58, 169--177 (1984)

Dynamics of Saturn's E Ring* P. K. SEIDELMANN AND R. S. HARRINGTON U.S. Naval Observatory, 34th and Massachusetts Avenue, NW, Washington, D.C. 20390 AND

V. SZEBEHELY University of Texas, Austin, Texas 78712 Received August 22, 1983; revised January 2, 1984 Photographic evidence for a wide outer ring of Saturn, the E ring, was first introduced in 1967 and detailed observations were obtained in 1980. The brightness profile of the E ring indicates that the ring extends from three Saturn radii, the orbit of Mimas, to at least eight Saturn radii, just inside the orbit of Rhea. The brightness profile has a peak at the orbit of Enceladus, but to the limit of the observations there is no evidence of variations in the brightness profile at the orbital distances of Tethys or Dione. It is known that there are satellites at the Lagrangian points of Tethys and Dione within the E ring. Thus the E ring is not constrained by satellites, but rather coexists at the same orbital distances as satellites which have other satellites at their Lagrangian points. Preliminary numerical and analytical investigations of the motions of the particles indicate that the majority of the particles are strictly influenced by Saturn. Particles in narrow rings at the orbital distance of a satellite move with revolution or librational motion and sometimes switch between these modes. The librational motion can be tadpole orbits at the L4 and L5 points, horseshoes including L4, L3, and L5 points and segment librations between the L4 or L5 points and the satellite.

INTRODUCTION

The Voyager 1 and 2 encounters with Saturn and the edge on presentation of the rings of Saturn to the Earth in 1980 led to the discovery, or confirmation, of new satellites of Saturn and new information concerning the structure and the extent of the rings of Saturn. There are satellites sharing orbits and there are particles in rings at the orbital distances of the known satellites, with the maximum observed intensity of the E-ring at the orbit of Enceladus. There is complex structure of the rings due to the presence of satellites, the resonance motion with the satellites, and some other causes. SATELLITES

Moving outward from Saturn through the satellite system, 1980S26, 1980S27, and * Paper presented at the "Natural Satellites Conference," Ithaca, New York, July 5-9, 1983.

1980S28 were discovered in Voyager 1 images in October and November of 1980. The satellite 1980S28 is just outside the A ring and may influence the outer edge of the ring. The satellites 1980S26 and 1980S27 are on either side of the F ring and may offer the explanation for the peculiar shape and strands of the F ring. In 1966 Dollfus and Walker observed a satellite or satellites (Dollfus, 1968). Subsequently, Aksnes and Franklin (1978) and Fountain and Larson (1977) attempted to explain the observations. No observations were available from 1966 to 1980. 1980S1 was recorded in a photographic observation with the 26-in. telescope in Washington, D.C., by Pascu (1980), indicating the dramatic change in observing conditions when the rings are seen edge on. Subsequently, many observations were made of two satellites, 1980S1 and 1980S3. Smith (1980) first suggested they were in 169 0019-1035/84 $3.00 Copyright© 1984by AcademicPress,Inc. All rightsof reproductionin any formreserved.

170

SEIDELMANN, HARRINGTON, AND SZEBEHELY

the same orbit, and their periods are now known to be approximately 16 hr 40 min and to differ by only 28 sec. Superficially, this indicates that the centers of the satellites would have been within 48 km of each other about March 1, 1982, and were in this relative position in midJanuary 1978. Either these satellites are very transitory phenomena, or there is some dynamical mechanism, such as libration, that preserves the stability of the system. The dynamics of the so-called co-orbitals were investigated by Harrington and Seidelmann (1981). This existence of two satellites in apparently the same orbit was the first indication of a new dynamical phenomenon in the Saturn system, although something similar had already been suggested for Uranus by Dermott et al. (1979). Since 1980, no observations of either of these satellites have been reported. Pascu and Seidelmann, and others, have made several unsuccessful attempts. Since the

satellite masses and the planet mass alone determine the libration period, additional observations could determine both the libration period and the satellite masses. Three more satellites have been confirmed: 1980S13 was discovered by Reitsema et al. (1980), and was shown to be at the L4 Lagrangian libration point of Tethys by Seidelmann et al. (1981). 1980S25 was discovered with the STWFPCIDTCCD camera by Pascu et al. (1980), and is at the L5 point of Tethys. Laques and Lecacheux (1980) discovered object 1980S6, which is now commonly referred to as the Dione B, located at the L4 libration point of Dione. These objects seem to obey the laws of the classical restricted problem of three bodies. There is a need for more observations of these objects reported in an observational, rather than interpreted, coordinate system. Specifically, observations need to be reported as Ao~ cos 3, A6 or Ax, Ay. Observations reported as degrees in front, or back, and in the orbit of a known

TABLE I SATURN SATELLITES Visual

Radius

mag.

(km)

Semi-major axis (1000 km)

Period (days)

137.670 139.353 141.700 151.422 151.472

0.6019 0.6130 0.6285 0.6942 0.6945

II

185.52 238.02

0.9424 1.3702

12.9 ll.7

III 1980 S13 1980 $25

294.66 294.66 294.66

1.8878 1.8878 1.8878

IV 1980 $6

377.40 377.4

2.7369 2.7391

527.04 1,221.83 1,481.1 3,561.3 12,952

4.5175 15.9454 21.2767 79.3302 550.48 R

Number

1980 1980 1980 1980 1980

$28 $27 $26 $3 S1

I

V VI VII VIII IX

(18) (15)

(15.5) (15) (14)

20 70 55 70

x x x x 110 x

10 50 x 40 45 x 35 60 x 50 100 x 80

Name

A ring shepherd (Atlas) F ring shepherd (Inner) F ring shepherd (Outer)

Coorbital (Epimetheus) Coorbital (Janus)

196 250

Mimas Enceladus

10.2 (18) (18.5)

530 17 x 14 x 13 17 x 11 x 11

Tethys L4 Lagrangian (Telesto) L5 Lagrangian (Calypso)

lO.4 17

560 18 x 16 x 15

Dione L4 Lagrangian (Dione B)

9.7 8.28 14.16 11.2 16.3

765 2575 205 x 130 x 110 730 110

Rhea Titan Hyperion Iapetus Phoebe

DYNAMICS OF SATURN'S E RING TABLE II SATURN RING DATA Feature

Distance (kin)

Distance (R~)

Equatorial radius D ring inner edge. C ring inner edge B ring inner edge B ring outer edge A ring inner edge A ring gap center A ring outer edge F ring center G ring center E ring inner edge E ring outer edge

60,330 67,000 74,400 91,900 117,400 121,900 133,400 136,600 140,300 170,000 ~180,000 ~480.000

1.000 1.11 1.233 1.524 1.946 2.021 2.212 2.265 2.326 2.8 ~3 ~8

satellite are interpreted data and of limited usefulness. Voyager 2 confirmed the existence of all these objects. Table I summarizes the information concerning the satellites. There are indications from Voyager data that there may be additional satellites yet to be discovered. RINGS

In addition to the satellites, there also exist the rings of Saturn. Table II summarizes the locations of the rings. There are apparent relationships between the dynamics of the rings and the satellites, but some of

za

o

171

these relationships are not completely understood. In this paper we will concentrate on the E ring of Saturn and not discuss the many interesting and difficult problems of the inner rings of Saturn which are presently being investigated by many others. Photographic evidence for a wide outer ring of Saturn, now called the E ring, was first introduced by Feibelman (1967). Detailed observations of the E ring were obtained in 1980 and described by Baum et al. (1981). From the 1980 observational data, it becomes evident that the brightness profile of the E ring indicates that the ring extends from three Saturn radii, or approximately the orbit of Mimas, to at least eight Saturn radii which would be just inside the orbital distance of Rhea (Fig. 1). The brightness profile has a peak at the orbit of Enceladus. To the limit of the observations, there is no evidence of variations in the brightness profile at the orbital distances of Tethys or Dione, whose orbits lie within the E ring. There is some very marginal evidence that there may be some bunching of E-ring particles in the vicinity of the Enceladus trailing Lagrangian point, L4. In the case of the E ring, we are not dealing with a ring constrained by satellites, but rather a ring that coexists at the same orbital distances as satellites which have smaller satellites at their Lagrangian points. It is a ring appar-

16

Lu

12 *B.." • •

uJ

"e 8

oo~ "o

k-x]

®

;...t @ •" ~ ;

®

®

.

r~ z 0

I

I

3

4

I " 5

""

"~J~'J~ 6

*

I 7

t

tl* 8

i 9

DISTANCE FROM THE CENTER OF SATURN IN UNITS OF SATURN RADII

FIG. 1. Mean edge-on brightness profile of the E ring in March 1980 (Baum et al.).

172

SEIDELMANN, HARRINGTON, AND SZEBEHELY ,y

ently unaffected by the presence of many satellites. The three-dimensional motion of the particles forming the E ring will be included in future analysis and will hopefully explain some of the details of the brightness profile shown on Fig. 1. NUMERICAL TEST

To understand the dynamics of the particles of the E ring, we examined the possibility that they are governed only by their gravitational interactions. To this end, the model developed by Harrington and Seidelmann (1981) was expanded to include an entire system of particles. This model includes Saturnian satellites 1 through 8, plus a J2 and a J4 term for Saturn. Solar perturbations are not considered, and neither are interactions between the ring particles themselves (thus, we have an ensemble of hundreds of integrations of single test particles). Integrations were usually run for several thousand days, enough to cover more than one complete libration period for those cases where it existed. Test particles were initially located in rings approximately at the distances of Mimas, Enceladus, Tethys, Dione, Rhea, in rings just inside of Mimas and just outside Rhea, and in rings located midway between each pair of satellites. Special test particles were also located to test the various types of librational motion that were detected. The majority of the particles appeared to be affected only by Saturn, and only those particles in a very narrow ring at the distance of one of the satellites actually were influenced by that one satellite. In the strongest case, the satellite perturbation induced eccentricity into the test particle orbit sufficient to give it a relative radial dispersion of only 6%. There are extremely narrow areas in which it is possible for the particles to be showing the kind of librational motion that is well known in the restricted three-body problem. However, because the satellites do not actually have fixed orbits, and the

L4 /+ x /

\+,

.I ./

.1

I", 1

"\

.

miz"

~'~ ~

\.,,~Jlp 0 ./~"

][~2

+\.\

L~ X

+1I'/

"\.\ "x

FIG. 2. Two-dimensional diagram indicating coordinate system with planetary and satellite masses and the Lagrangian points.

librational regions are so narrow, it is possible for a particle to show different types of motion from one revolution to the next. A particle may librate only over a rather limited range about the L4 or L5 point (Fig. 2, which shows the nonuniformly rotating and pulsating system in which the satellite of interest remains fixed); these are known as tadpole orbits. There is also the very extensive libration in which the particle stays at essentially the same distance but encompasses the three libration points L3, L4, and L5 in one synodic revolution; these are known as horseshoe orbits. This is the kind of orbit the co-orbiting satellites exhibit with respect to each other. The exact nature of the libration depends on the nature of the close approach of a particle to a satellite, and it is even possible, if the satellite has appreciable eccentricity (as does Enceladus), that a particle could miss the satellite entirely and rotate rather than librate even in the satellite-fixed system. In the presence of small satellites at the L4 and L5 points with respect to the satellites Tethys and Dione, additional types of librational motions are possible. In the case of Tethys and Dione, particles in motion at the same orbital distance can in a very small range of radial distance from Saturn have librational motion between Tethys and the satellite at the L4 point, or between Tethys and the satellites at the L5 point, or

DYNAMICS OF SATURN'S E RING

173

where by convention m2 -< ml, and theref o r e / z -< ½. The nondimensional distances are r~ and r2 and are given by

)TS."e"

rl = X/(x - /z) 2 + y2 and Ix

m2=#

0

P2(/L-I,0)

P2(#,O)

FIG. 3. Two-dimensional diagram defining the coordinates of a particle with respect to a primary (ml) and secondary (m2) body.

the particle may librate between the small satellites at the L4 and the L5 points without approaching Tethys. This is in addition to the possibility for tadpole-type libration and libration including the L3, L4, and Ls. It must be emphasized that the regions where these types of motions may take place are extremely narrow. Thus, if material was distributed in the E ring, excluding the collision process, the material could remain spread through the observed area for a long period of time. ANALYTICAL DEVELOPMENT

Having determined the basic character of the motions of particles in the E ring, it appeared desirable to investigate the motion from an analytical point of view in addition to numerical experiments. We can formulate the restricted manybody problem exactly the same way as the restricted t h r e e - b o d y problem as long as only " m a s s l e s s " particles are included in the system, besides the primary and the secondary bodies. Figure 3 explains the notation used. The dimensionless variables are obtained by dividing all lengths by the distance between the primary and the secondary, which, of course means that the distance between these bodies is one, or m l m 2 = P 1 P 2 = 1. The masses are made nondimensional by introducing the mass parameter: t~ -

m2 ml + m2

(2)

r2 = N,/(x -- /z + 1) 2 + y 2 .

m1=1 #

(1)

The nondimensional time (t) is given by (3)

t = nt*,

where t* is the dimensional time (say in seconds) and n is the mean motion in radians/second. The resulting equations of motion are / : = r ( 0 + 1)2

- (1 - /z)(r - /x cos 0) r31 /z[r + (1 - /x) cos 0] -

r~

(4)

r0 = -2k(0 + 1) + (1 - / x ) / x s i n 0

~-

.

(5)

As may be observed, these equations form a coupled, nonlinear fourth-order system with singularities at rl = 0 and rE = 0. The equations of motion may be developed for small values of the mass parameters; i.e., w h e n / x 2 may be neglected when compared to/x. For the distances we have rl ~

r--

/z C o s O,

rE ~ X/r E + 2r COS 0 + 1 [ _ /x(l+rcos0) ] 1 r 2 + 2r CO-S 0 -7- 1 " (6) The inverse cubes of these distances, appearing in the equations of motion, become rig ~ r_3 ( 1 + 3/~ cos 0) r r 2 3 --~ ( r 2 + 2 r c o s 0

+ 1) -3/2

3/z(1 + r cos 0) ] 1 + r~- ¥ ~ r c o s ~ 7 l r

(7)

Substituting these approximations in the equation s of motion (Eqs. (4) and (5)) we have

174

/: = r(/}

SEIDELMANN,

+

1) 2 -

HARRINGTON,

r - /z(r - 2 cos 0) r3

~ ( r + cos 0) - (r 2 + 2r cos 0 + 1) 3/2 r0 = - 2 k ( 0 + 1) + /z sin 0

l

,]

(r 2 + 2r cos 0 + 1) 3/2

~-5 •

[

(8)

AND SZEBEHELY

The long and short periods of the motions around the triangular libration points m a y be determined (Szebehely, 1967) by solving for the roots 0t) of the fourth-order characteristic equation 27

~4 -F )k2 + T

(tl)

]'~s(1 -- ]£s) = 0

where/Zs is the m a s s ratio of the satellites

T h e r e are nonlinearities and couplings still present. Also o b s e r v e that the singularities shifted f r o m r~ = 0, r: = 0 to r = 0 and r 2 + 2r c o s 0 + 1 = 0 which are approximations of r~ = 0 and rE = 0, as m a y be see from Fig.

I'Zs

ms/mp 1 + ms/mp

(12)

and ms is the m a s s of the satellite and mp is the mass of the planet. Since

.

In the case of a small mass satellite with a ring particle in its vicinity, we have approximately

ms mp + ms

ms/mp "~ 1,

IXs = ms/mp.

(13)

The m e a n motions, f o r small values of/Xs become r--

r~-rl~-I

SI ----- .~/2_~_7/t/'s

1 -/.~-- 1 and

sin 0 -~ 0 r2=rO=O.

S2 =

Since there is essentially no radial motion, k = 0. T h e r e f o r e Eq. (5) b e c o m e s h

= /.tr2

(')

~ 23 -- 1

27 1 - -8-/xs

(14)

The long a n d s h o r t p e r i o d s are obtained from

T1 = Ts/sl

(9)

and since r2 ~ 1, rE 3 >> 1, and therefore

T2 = Ts/s2 /x. /:2 ----- r2 2

(10)

Thus the usual equation in gravitational fields =

Table I I I lists the values of the masses used for the computations. T a b l e IV gives the (dimensionless) values of the m e a n m o t i o n s and t h e (dimensional)

/X r3 f

with a negative sign has in this case b e c o m e an equivalent equation with a positive sign indicating repulsive action rather than the usual attraction. This explains the bounce effect as the particle and satellite, or satellite and satellite a p p r o a c h each other. The libration points are defined b y the locations indicated in Fig. 2. The principal interest here is in the triangular libration points, L 4 and Ls.

(15)

T A B L E III Satellite I II

III IV v vI vii

VIII IX

Mimas Enceladus Tethys Dione Rhea Titan Hyperion Iapetus Phoebe

/xs

"

6.6 1.3 1.1 1.85 9 2.4614 : 2 1 7

× × × x × × × × ×

10 -8 10 -7 10 -6 16 6 10 -6 10 -4 10 7 10 -6 10 -1°

DYNAMICS OF SATURN'S E RING

175

T A B L E IV MEAN MOTIONS AND PERIODS OF MOTIONS OF SATELLITES AROUND THE TRIANGULAR LIBRATION POINTS Satellite

Mean motions sl

I II III IV V VI VII VIII IX

6,7 9,4 27,3 35,4 77.9 408 11,6 26.0 0.69

x × x × z x x x x

s2 10 -4 10 -4 10 4 10 -4 10 4 10 4 10 4 10 -4 10 4

1-2.23 1-4.39 1-3.72 1-6.25 1-3.04 1-8.31 1-6.76 1-3.38 1-2.37

x x x x × x x x x

10 -7 10 7 10 6 10 6 10 5 10 4 10 7 10 6 10 9

values of the long and short periods. For the largest value of/zs, associated with Titan, the errors using the approximate formulas (Eqs. (14)) are less than 10 -5. The long periods (T1) are given in days and in years, the short periods (T2) only in days. The periods of the satellites (Ts) were obtained from page F2 of the 1982 edition of the Astronomical Almanac. Satellite IX (Phoebe) is in retrograde motion around Saturn. : Table IV demonstrates that some of the long-period librational motions are indeed very long. The period is inversely proportional to the square root of the mass parameter; therefore small satellites generate long librational periods. Considering only the first six satellites, we find that Enceladus is associated with the longest period concerning the librational motion. The short-period motion is of less significance and itis to be considered a superimposed high-frequency oscillation with approximately the:same period as the orbital period of the satellite. To study the possibility of particles leaving the surface of the satellites, the escape velocities were computed using ve

V~ V Rs

(16)

where ms and Rs are the mass and the radius of the satellite and G is the constant of gravity. These escape velocities are compared

Periods of satellites (days)

Periods of mot i on at L4,5 T1d

TI y

T2d

0.94 1.37 1.89 2.74 4.52 15.95 21.28 79.33 R550.48

1403 1457 692 774 580 391 18345 30512 7.98 × 106

3.841 : 3.989 1.895 2.119 1.588 1.071 50.226 83.537 21,848

0.94 1.37 1.89 2.74 4.52 15.96 21.28 78.33 550.48

to the satellites' orbital velocities around Saturn, using the equation Us -

2~ras Ts

(17)

where as and Ts are the semi-major axes and the orbital periods of the satellites. (Note that the orbital eccentricities are small and may be neglected for the present purposes.) Table V gives the orbital and the escape velocities for the various satellites. Some of these escape velocities are not very large compared to the orbital velocities; for Mimas and Enceladus the values of the ratio vs/Ve are 90 and 63. This fact might be con-

TABLE V ORBITAL VELOCITIES OF SATURNIAN SATELLITES AND ESCAPE VELOCITIES FROM THE SURFACE OF THE SATELLITES Satellite

vs (km/sec)

ve (km/sec)

I II III IV V VI V II ,VIII IX

14.35 12.63 11.34 10.01 8.48 5.57 5.06 3,~26 R1.71

0.159 0. I99 0.409 0.530 0.923 2.538 0.389 0.329 0.037

176

SEIDELMANN, HARRINGTON, AND SZEBEHELY

sidered to support the idea that these satellites are responsible for the generation of particles. Nonlinear motions associated with the triangular libration points may be classified as tadpole orbits, horseshoe orbits, circulations, and ergodic orbits (Szebehely, 1967). Sensitivity regarding initial conditions is associated with deviation in the phase space from the libration point ( L 4 , L 5 ) and from the prescribed zero relative velocity. Large radial deviations are magnified and result in ergodic motion, while large deviations along the arc described by the libration points result in circulation or horseshoe orbits. Velocity deviations along the radial direction result in tadpole orbits while large velocity along the arc mentioned above results in chaotic motion. For large mass ratios (~ -~ 0.01) these effects take place quickly, while for small mass ratios (/x ~10 - 6 ) secular motion can be very slow in developing. Estimates for the lifetimes of horseshoe orbits show times of the order of the lifetime of the solar system for small mass ratios (10 -7) but give 100 years for larger mass ratios d0-2). The estimated widths of the horseshoe and tadpole regions are proportional with the cube root and square root of the mass ratio, resulting in large values for large mass ratios. Table VI shows the lifetimes and the widths mentioned above. The estimates are based on the following equations: [' = Ts/I.~Ss/3

Wh = 4 X 31/6 X /x~/3

(18)

(19)

and Wt = 4 X

~

(20)

where F is the lifetime of the horseshoe orbits in the same units as the satellite's orbital period (Ts), /zs is the satellite's mass parameter, Wh is the width of the region of horseshoe orbits in units of the distance between the satellites and the planet, and wt is

T A B L E VI LIFETIMES AND WIDTHS OF THE HORSESHOE ORBITS AND WIDTHS OF THE REGIONS OF TADPOLE ORBITS Satellites

I II III IV V VI VII VIII IX

Lifetimes (years)

2.39 1.12 4.41 2.69 3.18 4.52 8.52 2.17 2.73

× × × x x x × x x

109 109 107 107 106 104 109 l09 1015

Width (km x 103) Horseshoes

Tadpoles

3.601 5.792 14.614 22.233 52.659 367.915 41.605 171.062 55.244

0.156 0.280 1.009 1.675 5.164 62.623 2.163 11.630 1.119

the width of the region of the tadpole orbits in the same units. These equations were adopted from the articles by Dermott and Murray (1981a,b). Calculations based on the restricted threebody problem gave slightly different widths, but the differences were not essential. Since the horseshoe orbit rings do not overlap, they alone do not explain the existence of the E ring. On the other hand, because of the relatively large size of Mimas, it might be expected that slight perturbations or impacts will produce clouds of particles which should disperse. Such dispersions will require considerable time as well as considerable tangential velocities and/or radial displacements. The resonance effects deserve attention, since the studies based on the zero-velocity curves do not depend on resonances. The horseshoe orbit regions are completely independent of resonance effects. DISCUSSION

Particles sufficiently close to a satellite's triangular libration point will show tadpole orbits. If they approach a satellite sufficiently closely, they will "bounce" and produce an orbit that resembles a horseshoe, exactly as in the case of the co-orbiters; otherwise they will circulate past the

DYNAMICS OF SATURN'S E RING satellite without significant perturbations. In no case was chaotic motion observed in the numerical experiments, probably because of the short durations. Since radial dispersion has not been demonstrated yet, in general, it is questionable whether gravitational dynamics of the motion of the particles will provide an explanation for spreading the material through the observed E ring. Based on the calculations of the lifetimes for particles in the Saturn satellite system, it appears possible that, if the material were present at the time of formation of the Saturn system, the material would still be present as observed in the E ring. It also appears from the differences in the values of the lifetimes that this may be an explanation for the observed brightness of material at the distance of Enceladus. While it appears possible that Enceladus could be a source of material for the E ring, we have not been able to establish a means of dispersing the material from Enceladus through the range of observed material. This theory needs further development since we have not included the effects of radiation, electromagnetism, impacts, dissipative effects, and other perturbations. Also the out-of-the-plane forces have not been investigated.

CONCLUSION

The analytical investigations are confirmed by the numerical investigations. The motion of particles and satellites of various sizes as they exist around Saturn are rather stable. Librational motion by particles and satellites can exist in the form of tadpole, horseshoe, and segment orbits. The librational period involved is dependent upon the ratio of the masses only. When two particles or satellites approach each other in similar orbits they appear to experience a bounce. While the stability of the particles in the E ring can be explained, the dy-

177

namics necessary to distribute the material as observed have not been established. REFERENCES AKSNES, K., AND F. A. FRANKLIN (1978). The evidence for new faint satellites of Saturn reexamined. Icarus 36, 107-118. BAUM, W. A., T. KREIDL, J. A. WESTPHAL, G. E. DANIELSON, P. K. SEIDELMANN,D. PASCU,AND D. G. CURRIE (1981). Icarus 47, 84-96. DERMOTT, S. F., AND C. D. MURRAY(1981). The dynamics of tadpole and horseshoe orbits, 1. Theory. Icarus 48, l - I 1.

DERMOTT, S. F., AND C. D. MURRAY(1981). The dynamics of tadpole and horseshoe orbits, II. The coorbital satellites of saturn. Icarus 48, 12-22. DERMOTT, S. F., T. GOLD, AND A. T. SINCLAIR (1979). The rings of Uranus: Nature and origin. Astron. J. 84 (No. 8), 1225. DOLLFUS, A. (1968). La decouverte de 10 e satellite de Saturne. L'Astronomie 82, 253-262. FEIBELMAN, W. A. (1967). Concerning the 'D' Ring of Saturn. Nature 215, 793-794. FOUNTAIN, J. W., AND S. M. LARSON (1977). A new satellite of Saturn? Science 197, 915-917. HARRINGTON, R. S., AND P. K. SEIDELMANN(1981). The dynamics of the Saturnian satellites 1980S 1 and 1980S3. Icarus 47, 97-99. LAQUES, P., AND J. LECACHEUX(1980). IAUC 3457, March 6. LARSON, S. M., S. A. SMITH, J. W. FOUNTAIN, AND H. J. REITSEMA(1981 ). The 1966 observations of the coorbiting satellites of Saturn, SI0 and S l l . Icarus 46, 175-180. LECACHEUX,J., P. LAQUES, L. VAPILLON, A. AUGE, AND R. DESPIAU (1980). A new satellite of Saturn: "Dione B." Icarus 43, 111-115. PASCU, D. (1980). IAUC 3454. PASCU, D., W. A. BAUM, D. G. CURRIE, AND P. K. SEIDELMANN (1980). IAUC 3496. PASCU, D., AND P. K. SEIDELMANN (1981), IAUC 3603. REITSEMA, H. J., B. A. SMITH, AND S. M. LARSON (1980). A new Saturnian satellite near Dione's L4 point. Icarus 43, 116-119. REITSEMA, H. J., B. A. SMITH, AND S. i . LARSON (1980). IAUC 3466, April 10. SEIDELMANN, P. K., R. S. HARRINGTON,D. PASCU, W. A. BAUM, D. G. CURRIE, J. A. WESTPHAL, AND G. E. DANIELSON (1981). Saturn satellite observations and orbits from the 1980 ring plane crossing. Icarus 47, 282. SMITH, n. A., et al. (1980). IAUC 3483. SZEBEHELY, V. (1967). "Theory of Orbits." Academic Press, N e w York.