A numerical study of Saturn's F-ring

ICARUS 52, 526-544 (1982)

A Numerical Study of Saturn's F-Ring MARK R. SHOWALTER AND JOSEPH A. BURNS Cornell University, Ithaca, New York 14853 Received April 9, 1982; revised June 24, 1982 We study the short-term effects of "shepherding" satellites on narrow rings, in the general case where all bodies move along eccentric orbits. We do this by following numerically a ring of test particles as their orbits evolve under the gravitational perturbations of the shepherds. Planar motion is assumed. Our numerical scheme vastly improves (by a factor of - 104) the computation speed over conventional orbital integration methods by constructing a table of the perturbation integrals and then utilizing it over and over. The approach is applicable to any narrow ring with a nearby satellite, such as a ring confined by the shepherding mechanism of P. Goldreich and S. Tremaine [Nature 277, 97-99 (1979)]. We arrive at results for a variety of orbital configurations, and then apply these to the F-ring of Saturn. Several features of the numerical integration are reminiscent of the kinks and clumps observed by Voyager. If the ring-to-satellite distance changes significantly due to eccentricities, then the ring can break up into periodic clumps in an azimuthal domain which trails the satellite. This region may lag somewhat in longitude. The perturbations may also cause the ring to vary significantly in width, being narrowest near the point of closest approach of the shepherd and widest at the opposite side. It is as yet unclear whether this effect is, or could be, observed in the Voyager images. And finally, the perturbations of the shepherds can impart a significant, but probably time variable, eccentricity to the ring. The short-term tendency is not toward alignment of ring and satellite apsides; longer time effects have not been explored.

I. INTRODUCTION

The discovery of Saturn's narrow F-ring and its two attending satellites provided dramatic demonstration of the "shepherding" process predicted by Goldreich and Tremaine (1979). The ring particles are confined through the long-term gravitational influence of many circumnavigations by each satellite which, when coupled with interparticle collisions, can repel ring material. This obvious theoretical success was, however, upstaged by several troublesome features in the Voyager images, variously described as braids, kinks, and clumps. One proffered explanation again involves the shepherding satellites, this time in secondary roles as the F-ring sculptors. In this view the mysterious features are once more due to satellite perturbations, but now only their short-term effects, those produced during the last few passes of each satellite, are relevant. It is this possibility upon which we elaborate.

Dermott (1981) first proposed that the braids are caused by numerous passes of one of the shepherds, which happens to have a single first-order resonance within the ring. Particles on opposite sides of this resonance oscillate radially 180° out of phase, like any oscillator driven near resonance. Hence they periodically separate and rejoin to generate the strands. Unfortunately this appealing idea is now known to be inconsistent with the observations; the ring is sufficiently wide and the shepherds sufficiently close that many overlapping first-order resonances are present. Hence the satellite's effect could not be so straightforward. In addition, Dermott's simplifying assumption that the ring and both shepherds are on circular orbits, though necessary for analytic results, is now known to be invalid for Saturn's F-ring (Synnott, 1982). In this paper we include the effects of eccentricities on the ring-satellite interactions. A computational approach is de526

D019-1035/82/120526-19502.00/0 Copyright© 1982by AcademicPress, Inc. All rightsof reproductionin any formreserved.

NUMERICAL STUDY OF SATURN'S F-RING scribed which makes ring evolution calculations extremely efficient numerically, and provides added insight into the nature of this two-body interaction. After some analysis, the procedure is applied to several possible configurations of ring and shepherd orbits. We discover features which may be relevant to the F-ring's structure, but it appears that a complete explanation will involve the interplay of other effects as well. Possible complications, which our treatment ignores, include (i) inclined particle and satellite orbits; (ii) electromagnetic effects (Burns and Showalter, 1981; Smoluchowski, 1981; Hill and Mendis, 1981): (iii) discrete collisions within the ring; and (iv) the presence of relatively large objects within the ring (cf. Burns et al., 1980). II. O U T L I N E OF A P P R O A C H

At the heart of our numerical approach is the fact that encounters between ring particles and the satellites are relatively quick, lasting on the order of one rotation period, whereas the interval between such encounters is perhaps -100 times longer. Hence, ignoring collisions, each particle spends most of its time traversing a mundane unperturbed orbit, with only an occasional "kick" to modify its orbital elements. We assume throughout that the satellite perturbations are small. This allows us to make a wave approximation, in which the kicks from successive passes of the satellite add linearly. The gravitational attraction of the F-ring shepherds is of order - 1 0 -5 of Saturn's, so this superposition approximation is reasonable. In the simplest possible scenario, a shepherding satellite follows a circular orbit near a narrow cirular ring (Fig. la). As viewed from a rotating frame fixed with the satellite, each ring particle is in turn seen to drift by and be scattered gravitationally onto a new, slightly eccentric orbit. In this frame it is thereafter observed to follow a sinusoidal path, with a period equal to its orbital period and wavelength 3"rrd, d being the radial separation of the two orbits

(a)

527

Circular • Satellite

-~" d

(b)

Eccentric Satellite

FIG. 1. Perturbation o f a ring particle trajectory by a nearby satellite. (a) The satellite is on a circular orbit at a distance d from the ring particle's orbit. The reference frame is fixed with respect to the satellite. The overall direction of motion is counterclockwise, i.e., toward the left. (b) The satellite is now on an eccentric orbit, with a rotating reference frame tied to the guiding center of its epicycles. The ring particle's deflection depends on parameters d and 4, the satellite's

mean longitude(measuredfrom pericenter) at the moment the ring particle passes its guidingcenter. In this rotating frame, + is equal geometricallyto the angle shown. (Dermott, 1981). The perturbation received in this circular problem depends only upon this one parameter d. The universe is rarely so accommodating. The problem can be entangled one step further by placing the perturbing satellite on an eccentric orbit, with its semimajor axis unchanged. When observed in the same rotating reference frame, the shepherd is now seen to follow epicycles about its guiding center (Fig., lb). Here the kick received by a ring particle depends on the additional parameter O, the longitude of encounter. To be precise, this is the longitude at which the ring particle overtakes the satellite's guiding center, i.e., at which their mean longitudes match. In this discussion longitudes are measured with respect to the shepherd's pericenter. In the most general planar situation the ring is eccentric as well. Then we require two longitudes O~,O0 to describe the encounter, each measured with respect to its body's pericenter. However, in the absence

528

SHOWALTER AND BURNS

of significant relative precession (caused by Saturn's oblateness and perturbations from other satellites), these longitudes differ by a constant and so only one is truly independent. The differential precession rate between the ring and its shepherds is utterly negligible over the time scales we will be considering, and so can be ignored. The precise change in a ring particle's orbital elements are ascertained by integrating a set of perturbation equations. However, since any encounter depends upon just two p a r a m e t e r s (qJ,d), a simpler approach is possible. We can first tabulate the excursions in orbital elements for an array of relevant parameter values. Then, for any particular encounter, we can deduce the corresponding kick by interpolating among tabulated values, rather than by solving the same differential equations over and over again. This method produces a substantial increase in computational efficiency. It is developed more rigorously in the section to follow.

tion equations. The ring overall is described by its mean elements (a0 -~ 1, h0, k0), so each individual particle has elements

h=ho+h', k=ko+k'

and 0.

We assume in the remaining discussion that a', h', and k' are small quantities. The Gauss perturbation equations of celestial mechanics (Danby, 1962, pp. 242-243) specify how orbital elements change when an object is acted upon by R and T, the radial and tangential components of any perturbing force per unit mass:

da ~/ a 3 d----t= 2 1 - e2 [Re sin f + dh dt

T(1 + e cos f ) ,

We begin by selecting a convenient set of units for the problem. The semimajor axis of the ring can be chosen as the linear scale, so we define a0 -= 1. We also set GMsaturn --= 1, so masses of the shepherds will be given as fractions of a Saturn mass. In these units the mean motion of a ring particle is no = 1, and hence its orbital period P0 = 2"rr. The position and velocity of a planar ring particle can be defined by four parameters--semimajor axis a, eccentricity e, longitude of pericenter to, and mean longitude 0. We define mean longitude 0 to be that of its epicyclic guiding center, so, in the absence of perturbations, 0 increases uniformly in time. In place of elements (e,to) we define a more convenient set (h,k), where (1)

k -= e cos to. This pair leads to better-behaved perturba-

dk dt

(3a)

~/a(1 - e 2) [ - R cos g L

( sing+tf)] + T sing + 1 + ecos

III. P R O C E D U R E

h -- e sin to,

(2)

a = a0 + a',

'

(3b)

~/a(1 - e 2) [ R s i n g

]]

(

cos g +/,

+ Tcosg

+ 1 + ecosff]'

(3c)

dO W~a [-2R(_l - e 2) ~/1 - e 2 d--t= L 1 + ec-os-f + 1 + ~/1 - e z

( - R e c o s f + Te s i n f (2 + e c o s f ) ~ ] + e cosf/J' (3d) w h e r e f i s a ring particle's true anomaly and g =-f + to is its true longitude. F o r our purposes R(t) and T(t) are the appropriate components o f the particle's gravitational acceleration produced by either satellite at time t. If we assume that changes developed in the elements by an encounter are small, then these coupled differential equations can be blindly integrated, as if the elements on the right-hand side were constant. During one encounter (3a) becomes,

NUMERICAL STUDY OF SATURN'S F-RING for example, Aa = 2

I - e2

f~,~2[Resinf+

T(1 + e c o s J ) ] d t ,

(4)

where

'r =- 2"rr/lni -

n[

and n; is the mean motion of the ith shepherd. Thus ~ is the synodic period of a shepherd with respect to the ring particle. During the integration period the ring particle drifts from 180° behind the shepherd to 180° ahead. Clearly the major share of the kick occurs at roughly the midpoint of this interval, near t = 0, but to be complete the entire period is included. The values of the integrals such as (4) are functions of t~ and d, and may be evaluated for an appropriate array of choices. Once we have tabulated Aa, Ah, Ak, and A0, the problem is essentially solved. The need to do any more tedious integrations has been eliminated. We then use a simple procedure to study the orbital evolution of any specific ring particle. From an initial time, we extrapolate ahead to the next passage of a shepherd, assuming an unperturbed orbit. The parameters (0,d) of the encounter are thereby determined, and interpolation from tables yields the changes in orbital elements. From these, the new ring particle orbit is established, and the process may then be repeated. In this way, the orbit's evolution can be traced for an arbitrary length of time. The savings in computer time of this method is quite substantial. A standard numerical integration scheme might require -100 time steps per orbital period, or - 1 0 4 steps per synodic period of a shepherd. Our method reduces that to one step, while the accuracy of the method is constrained only by the size of the table one cares to build. For the given problem a table of 64 0-steps by 3 d-steps was sufficient to reach the de-

529

sired level of accuracy, three to four significant figures in each interpolated value. The major limitation of this numerical study is that it excludes particle interactions. Since ring particle orbits are evolved only in sequence, we have no opportunity to consider pairwise interactions. However, several crude attempts were made to include their statistical effect. For example, the net consequence of collisions must be to damp ring particle eccentricities back toward a mean value for the ring; in some integrations we modeled this as an exponential decay. Several analogous methods for the damping of a' were also studied. None of these led to results substantially different from those to be presented in Section V; we consider sufficiently short-term integrations that damping in the ring will not alter our findings. IV. ORBITAL CONFIGURATIONS

The integrated kicks Aa, Ah, Ak, and A0 received by ring particles can now be tabulated as functions of 0 and d. This mandates selection of elements for the satellite (al, el, too and for the mean ring (a0, e0, tOo). It is useful and instructive to consider how many of these quantities are physically significant. We are free to define a distance scale and longitudinal reference arbitrarily. This leaves four physically meaningful parameters upon which our tabulations rely. The actual orbits of Saturn's F-ring and its shepherds are not well determined and also vary slowly in time, so we do not have the privilege of considering only a specific set of values. We are left with a distressingly large parameter space in which to be wandering freely. In desperation we turn to a simple discussion of Keplerian orbits for guidance. The primary effect of the ring and satellite elements is through the acceleration components R(t) and T(t), though e0 also appears weakly in the coefficients of the perturbation equations (3). The forces depend upon the precise motion of shepherd and ring particle. To find their relative posi-

530

SHOWALTER AND BURNS

tions, we describe each b o d y ' s orbital motion as r(t) = a -

ae

cos(nt - 00) + O(e2),

(5)

sin(nt - o~) + O(e2),

l(t) = ant + 2ae

where r and l are the radial and tangential position components. Alternately we can replace variables (e,o~) by ( h , k ) , whence r(t) ~ a -

cos(nt) - a h sin(nt),

ak

l(t) ~ ant + 2ak

(6)

sin(nt) - 2 a h cos(nt).

The separation (Ar,A/) between bodies is then given as approximately Ar(t) ~ (al - ao) - ( a l k l cos n i t - aoko cos not) ( a l h l sin n i t - aoho sin not), -

A l ( t ) -~ a o ( n I -

(7)

no) t

+ 2(alkt sin n i t - aoko sin not) - 2(alhl cos n i t - aoho cos not). Note that the mean longitudes match at time t = 0, earlier defined to be the m o m e n t of encounter. The vast majority of the shepherd's kick c o m e s in the vicinity of t = 0, because R(t) and T(t) are negligible at other times. Since nl ~ no, we have near encounter cos n i t = cos n o t + O [ ( n l - n0)t],

(8)

sin n i t = sin n o t + O [ ( n l - n0)t]. For Saturn's F-ring we expect (nl - no) 10 -z, and the satellite perturbations are only appreciable for a unit time interval, so errors of order 1% are expected in this approximation. The order of these errors is equivalent to ( a l - a o ) / a o . To this a c c u r a c y it is therefore reasonable to write (7) as A r ~ ( a l -- ao) -- ( a l k l --

(alhl -

a0ko)cos not

a0ho)sin not,

A l -~ a o ( n l - n o ) t + 2 ( a l k l -- aoko)sin n o t -

2(alhl -

aoho)cos not.

(9)

We notice in (9) that the separation and accordingly the force c o m p o n e n t s are only

functions of old p a r a m e t e r s al and a0, plus the new ones H =- a l h l -

aoho,

K =- a l k l -

aoko.

(10)

The two quantities (H, K) here replace the previous four (hi, kl, h0, k0), so we have relegated two to lesser importance. It is now only necessary to vary four parameters (al, a0, H, K) to study the full range of effects of a shepherd on a ring particle. Only small discrepancies ( - 1 % in our examples) will arise in perturbations generated by different orbital configurations (hi, kl, h0, k0) which nevertheless have the same H and K values. This abstruse observation can be conceptualized if we define an " e c c e n t r i c displacement v e c t o r " E with rectangular coordinates (ak,ah) or polar coordinates ( a e , c o ) . This is distinguished f r o m the standard eccentricity vector by our inclusion of the factor of a; its magnitude is simply the maxim u m radial displacement out-of-round that any orbital body attains. N o w we can describe any fin#satellite configuration by al, a0, El, and E0 (Fig. 2a). In the previous discussion we simply o b s e r v e d that the effect of satellite on ring depends primarily on the vector difference E1 - E0 = (K, H), as well as on al and a0. This being established, we can always choose to set E6 = 0 and El = El - E0, in effect assigning all of the eccentricity to the satellite (Fig. 2b). The effect of this " p r i m e d " satellite on its ring is essentially the same as that in the f o r m e r unprimed scenario. F u r t h e r m o r e , at this stage we have retained the freedom to choose a reference longitude. The effect of a satellite with an orbit of given eccentricity on a circular ring is always the same, aside f r o m a trivial rotation. We can choose, for example, to orient the satellite's orbit El' such that co~'= 0 (Fig. 2c). The three configurations of Fig. 2 all lead to essentially the same tabulations of satellite perturbations on ring particles.

NUMERICAL STUDY OF SATURN'S F-RING oh

(o) ,,,E~-Eo

=3k

(b)

ah

='ok

531

needed. F r o m the six elements with which we began, we have reduced our task to a problem of exploring the effects o f only two. To summarize, we can reduce the effect of any satellite (at, el, tol) on any ring (a0, e0, too) to that of another eccentric satellite (a'~ = al/ao, er, toI' = 0) on a circular ring (aft = 1, ef = 0, to~' = 0). In the latter description only the two p a r a m e t e r s (a~, e~') are needed. In terms of this new p a r a m e t e r space, we can transform our tabulated kicks via the relations: Aa(O) = Aa*(O -- d~), Ah(O) = Ah*(O - 0 ) c o s ~b + Ak*(O - ~b)sin d~,

(I1)

oh

A k ( , ) = -Ah*(t~ - ¢)sin d# + Ak*(* - +)cos 4,

(c)

a0(~0) = ~:

Qk

a0*(¢

-

+),

which are straightforward to derive. H e r e

TABLE I FIG. 2. T h e t r a n s f o r m a t i o n o f v e c t o r e c c e n t r i c i t i e s . (a) A g i v e n r i n g / s a t e l l i t e c o n f i g u r a t i o n is p l o t t e d . E0 refers to the ring a n d Et to the s a t e l l i t e . (b) T r a n s l a t i o n o f origin, to m a k e E~ = 0. T h e e f f e c t o f t h i s n e w satellite o r b i t o n the c i r c u l a r ring is t h e s a m e as in (a). (c) A r o t a t i o n o f a x i s t h r o u g h a n g l e d#. T h i s m e r e l y s e t s the r e f e r e n c e l o n g i t u d e to t h a t o f s a t e l l i t e p e r i c e n t e r .

In Table I we illustrate the numerical equivalence of the configurations shown in Fig. 2. Integrated values (Aa, Ah, Ak, A0) based on expressions like (4) are presented for selected values of e n c o u n t e r longitude 4, and for the ring/satellite configurations graphed in Fig. 3. It is clear that all four have the same H and K values, so we expect the tabulated integrals to be similar. The satellite has orbital radius 1.01 and mass 2 × 1 0 - 9 M s a t u r n . Inspection of the table reveals discrepancies generally of order 1%, as advertised. Once we define the ring radius a0 - 1 and orient the system properly, only the satellite p a r a m e t e r s al and El = ( H 2 + K2) l/z are

A COMPARISON OF SAMPLE TABULATED VALUES ( A a , Ah, Ak, A0) FOR THE CONFIGURATIONS OF FIG. 3 (THROUGHOUT, SATELLITE PARAMETERS ARE a I = 1.01 AND M1 = 2 × 10 -9 ) t~ (o)

Config.

Aa

Ah

Ak

A0

× 10 +8

x 10 +8

x 10 ÷s

x 10 +8

0

ref. a b c

0. 11. 0. 60.

--21,597. -21,574. -21,633. -21,611.

0. 56. 0. -184.

--33,922. --33,581. -34,600. -34,295.

22½

ref. a b c

9,669. 9,641. 9,727. 9,699.

- 6,244. -6,144. -6,288. -6,525.

12,108. 12,051. 12,281. 12,129.

- 14,872. -14,841. - 15,078. -14,780.

45

ref. a b c

5,542. 5,522. 5,575. 5,567.

-490. -429. -518. -650.

6,935. 6,896. 7,035. 6,953.

-6,652. -6,683. -6,679. -6,538.

90

ref. a b c

2,071. 2,064. 2,083. 2,082.

1,404. 1,433. 1,387. 1,336.

2,650. 2,635. 2,685. 2,659.

-2,916. -3,006. -2,940. -2,897.

180

ref. a b c

0. 0. 0. 1.

1,748. 1,755. 1,732. 1,739.

0. 0. 0. 0.

xlO -s

xlO 8

xlO-8

-

1,914. 1,934. 1,874. 1,894.

×10 8

532

SHOWALTER AND BURNS oh

oh Reference

I

I

(o)

0.006 l=ok

Ring ~tellile

I

I

~

~k R

S

1 0 - 9 orbits at radius al = 1.01. Three values are considered for its orbital eccentricity-el = 0.00594, 0.00330, 0; these yield variations in ring/satellite separation by factors of 4, 2, and 1, respectively. For the special case of a shepherd on a circular orbit, the curves are easy to interpret. First we observe that Aa = 0 for all encounters. This is to be expected; such a satellite exerts no net tangential force on

~ok

-fSO° I

I

',

-9(~

1 " 0 ~ ~ _ ~

~'a

!-'-ok

-1.0

FIG. 3. An illustration of the translation property. Configurations (a), (b), a n d (c) are identical after trans-

lation of the origin to the reference situation. -t

O*

the rotation angle ¢ (see Fig. 2b) is defined by -2.0

sin ¢ = H / N / H E + K 2, COS qb - - K / N / H E + K 2,

(12)

where H and K are defined in (10). The eccentricity e~' of the equivalent satellite is found by solving (ale'S) 2 = H z + K z.

-t

"."..........

-1.0

(13)

As compared to the real problem, errors arising out of this different representation will be of order [(al - ao)/ao, e0]. V. R E S U L T S

We are at last prepared to assess in general the short-term sculpting skills of shepherding satellites. In Fig. 4 we present graphs of the excursions (Aa, Ah, ak, A0) induced in a ring particle trajectory as functions of encounter longitude ¢ during one complete pass of the satellite. Their dependence on radial separation d over the limited range considered is very minor and is not shown. The overall ring is circular, a convenient choice which, as just argued, represents no loss of generality. The satellite of mass M1 = 2 x

Ak

. . . .

~'""~"~

~

-1.

"""

. . . . . .

:...-...: \ /

°, FIG. 4. The integrated excursions of a r i n g particle's orbital elements during a satellite's complete passage, as a function of encounter longitude ~. The satellite has parameters a~ = 1.01 and M~ = 2 × 10 9; three values are considered for its eccentricity. The solid line corresponds to a satellite varying by a factor of 4 in its distance from the ring, the dashed line to one varying by a factor of 2, and the dotted line to a circular satellite orbit, invariant in its ring separation. ¢ = 0 is also the reference longitude, with respect to which h and k are oriented.

NUMERICAL STUDY OF SATURN'S F-RING the ring over a complete passage, and therefore does no work. The deflection in eccentricity Ae = (Ah 2 + Ak2) 1/2 is seen to be constant, and is consistent with analytic results (Dermott, 1981). The sinusoidal forms of its components Ah and Ak are due to the additional phase information they carry. The angular deviation A0 is caused by a reduction in the net radial force on a ring particle as the satellite passes outside its orbit. This temporarily reduces its angular velocity, and so a negative angular displacement results. By rotational symmetry, it must be constant for the satellite on a circular orbit. Eccentric satellites have a drastically different effect. As expected, orbital element kicks are largest in the vicinity of to = 0, where the satellite approaches closest to the ring. Most notable is the behavior of Aa, with its abrupt change of sign at to = 0. Recall that Aa depends upon the net tangential force ]" received by a ring particle. For an encounter precisely at pericenter (to = 0), /~ vanished by symmetry. But consider, for example, a small positive tO. In Fig. lb the satellite would lead the ring particle throughout the time of closest encounter, so its net tangential force would be substantial. This leads to a large increase in orbital energy, and a correspondingly large positive jump Aa. For to slightly less than zero, the ring particle experiences a drag force and so its jump Aa is in the opposite direction. H e n c e the rapid switch of sign. The consequences of these four curves are not intuitively clear, but their nature is revealed in the numerical integrations to follow. Figure 4 contains the dependence of the perturbation curves on satellite eccentricity. Changing the distance d to the satellite results in no features qualitatively different from those shown; only changes in amplitude and peak width are ever generated. Indeed, the action of satellites on a narrow ring exhibits a much smaller variety of functional forms than might have been anticipated.

533

The curves of Fig. 4 are, however, specific to an external satellite. An interior satellite generates curves which are simple reflections of these about the horizontal axis, provided longitudes are now measured from the satellite's apoapse. In switching from an outer to an inner satellite, both R(t) and T(t) change sign; consequently so do the four curves. We will now define an exemplary ring/ satellite configuration which exhibits all of the important features observed. The ring of test particles is circular, with width I0 -4. It is confined at the outside by the satellite having the greatest e c c e n t r i c i t y - - e l = 0.00594 in Fig. 4. An inner shepherd of the same mass but zero eccentricity is placed at radius a2 = 0.993. This choice enables us to contrast the effects of circular and eccentric satellites. Initial conditions are shown in Fig. 5a. Here the ring is made up of three separate strands of 800 particles each. Both satellites begin at 3 o'clock on the diagram, which is also the pericenter of the eccentric shepherd. To improve visibility, radial distances from the ring center are expanded in the diagrams; see figure captions. The initial conditions chosen are admittedly quite specialized, but the features we present are not contingent upon these particular circumstances. The ring's appearance is shown in Fig. 5b, after 30 ring periods have elapsed. The shepherds have drifted in opposite directions, and each has distorted its associated ring section into periodic waves. The inner shepherd on circular orbit imparts smooth sinusoidal waves onto the ring as expected. The outer, eccentric shepherd generates waves of a profoundly different character. The waves are kinked, due to the fact that neighboring ring particles are now travelling with slightly different mean motions. As the top panel of Fig. 3 shows, eccentric satellite orbits will alter a, and hence orbital speeds. However, it is more notable that clumps are formed. These can be understood by

534

SHOWALTER AND BURNS

(o) 7%31.'..-~. .... :.-i.'.~i.,!:::i. * .

~!:i?i!!:~::. "7:?::.

.:;;?

/

,:;!;?

~x. x

I ÷ "1

~!ii

:~.fi:

/

:;iiY ....ili!iii:;:

:~iii!:r. .:;5" S"

"r~!ii:~. I

.

.

.

:;~!.4!;'::~

FIG. 5. Sample integrations of shepherding satellites, perturbing a narrow ring. (a) Initial conditions. A circular ring is made up of three strands (of 800 particles each), spaced radially by 0.5 x 10 4. Small crosses mark the satellite positions, and the central circle represents the planet. The outer satellite has the orbit described by the solid curves of Fig. 4. It is initially located at pericenter in the diagram. The inner satellite has the same mass M2 = 2 x 10-9 and follows a cirular orbit at radius a2 = 0.993. In the figures all radial distances from ring center have been expanded to improve visibility, ring particles by a factor of 200 and satellites by a factor o f 20. (b) Same configuration, after 30 ring periods have elapsed. (c) Same configuration, after 120 ring periods have elapsed. Here the 2400 particles were initially scattered randomly in the same radial range, to eliminate any illusory features. Also, the ring's radial expansion factor here is 100, half the previous value.

considering the rapid switch in the sign of Aa. Particles with opposite signs of $ have their mean motions changed in opposite directions, and so a gap begins to open. Clumps are spaced by the characteristic wavelength h -- 3"rrd. Prominent clumps can only be produced in a sufficiently narrow ring. If the maxi-

mum radial kick A amax is smaller than half the ring width, then the necessary bimodal distribution of mean motions will not be produced, and a gap will not form. Some short-lived density enhancements, which might be viewed as clumps, could still appear. Figure 7a is a plot of Aamax as a function of the shepherd's eccentricity.

NUMERICAL STUDY OF SATURN'S F-RING

535

(b) ~-:::!:~~?~.:i:%"~::~?:~::'~~!!:."~~:"~ !ii-.~:!ii:. ~::i.~:_.,{I~:.-~':: •. S "

...'.).,;.;;

+

:.,~..: :. .... .:..?

.....

..,

I}~:L

.,~

+

%~i!:'

F I G . 5--Continued.

All of these clumps have a finite lifetime, because they are composed of particles travelling with different mean motions. They tend to dissipate in a region which trails the eccentric shepherd. This dissipation longitude, measured from the perturbing satellite, is approximately d2 L ---- ---270° a0Aama-------~ ,

(14)

where the plus sign refers to an exterior shepherd, and the minus sign to an interior one. This expression is derived by considering the longitude at which the fastestdrifting particles will have traversed a full

wavelength, h, relative to the mean ring. The value of L is only approximate, because of the imprecision in defining when a clump vanishes; it tends to be an underestimate, because it is based upon the motion of the fastest particles. Initial ring width is also a weak factor in determining clump lifetime. The formation of clumps takes a finite time, as we wait for particles pushed inward to drift ahead, and those pulled outward to lag behind. Clumps are most pronounced at about half of the dissipation angle L. In Fig. 5c we allow the ring to evolve for a greater period of time, 120 rotations. The

536

SHOWALTER AND BURNS

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same clumping effect is seen, now throughout the full azimuthal extent of the ring. Using current numerical parameters in (14), we would expect clumps to be visible for 270 ° following the outer shepherd; in truth they are just beginning to dissipate in this region. Note that even when significant gaps have not yet opened, periodic density features are observed, e.g., near 9 o'clock. Here we also see a noticeable variation in the width of the ring, it being much narrower near the outer satellite's pericenter at 3 o'clock. This can be understood as follows. Each encounter acts to kick the ring panicle from one orbit onto another, having

slightly altered elements. Nevertheless, since the satellite encounter is quick, the old and new orbits always intersect at a point close to the location of encounter. The kicks of greatest amplitude are received near the eccentric shepherd's closest approach, a region which stays fixed in inertial space (neglecting precession). They scatter ring particles onto quite different orbits, but ones which nevertheless always return to the region where the scattering took place. Hence the ring broadens least near t~ = 0, and this region stays anomalously narrow. Since the narrow region remains tied to the satellite's pericenter, the

NUMERICAL STUDY OF SATURN'S F-RING

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FIG. 6. The induced width variation and eccentricity in a narrow ring. The orbital trajectories o f 32 ring particles are traced, after one encounter with the satellites of Fig. 4. Initially all particles were at unit radius, had zero eccentricity, and were spread uniformly in longitude of encounter ~. Satellite

eccentricities are (a) e~ = 0.0000, (b) et = 0.0330, (c) e~ = 0.0594. individual ring particles must alternately disperse and regroup once during each orbit. The p h e n o m e n o n of ring width variation is illustrated in Fig. 6. Orbital tracks of 32 ring particles are traced after one passage of the eccentric shepherd. All began at unit radius, and were spaced uniformly in longitude of encounter. The figure can be viewed as a time-smeared photograph of the ring particles, unlike the snapshots of Fig. 5. The structure is subsequently unwound into a straight line, so eccentric orbits appear as sinusoidal tracks. The graphs are repeated for each of the three eccentricities considered previously in Fig. 4. For zero satellite eccentricity panel above, final trajectories have constant eccentricity and are distributed uniformly in pericenter. However, as satellite eccentricity grows, the width variation becomes more and more pronounced. Trajectories tend to overlap near 0 = 0 °, but are widely scattered near 0 = 180°. Here intersecting trajectories should not be construed as evidence for frequent collisions; we have eliminated all information about where each particle resides along its path at any given time. Figure 7b is a graph of the width difference A W induced in the ring during one

passage of the satellite, as a function of its eccentricity. A W can indeed be quite large if the satellite approaches very close to the ring. Note that this width variation is induced during a single ring circumnavigation; it will continue to grow from each passage. H o w e v e r , competing with this process are collisions, which tend to reduce any width variations. The final ring state is determined by a balance of these two effects. Finally, Fig. 6 shows another unexpected feature---the overall sinusoidal shapes of the curves imply that the ring has been given a small mean eccentricity. Pericenter is oriented 90 ° behind the point of closest approach of the shepherd. This stems from the fact that the mean A h delivered to the ring is nonzero, although the mean Ak vanishes by symmetry (Fig. 4). In more physical terms, a satellite encounter can be regarded as a primarily radial kick, so a perturbed orbit tends to reach maximum radial displacement roughly 90 ° afterward. The changes in mean ring eccentricity are dominated by the largest kicks, which are delivered near the satellite's closest approach. Hence the observed 90 ° misalignment ensues. As in the case of the width variation, the

538

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mean eccentricity will continue to grow on successive passes. Again this enhancement will be moderated by internal ring processes, although damping of the ring's overall eccentricity takes much longer. The process is additionally complex because, as the ring eccentricity grows and the orbits slowly precess, its point of closest approach to the shepherd will slowly shift. Thus consecutive passes of the shepherd will not be identical. All we know for certain is that the resultant ring eccentricity can be significant, and that these short-term effects tend to push the ring and satellite apsides out of alignment. This is not to say that apsides cannot be aligned, but only that they will be aligned less often than would be expected under free precession. VI. SATURN'S F-RING Now we turn our attention exclusively to the F-ring of Saturn, which resides at radius a0 = 140,185 (---30) km, with eccentricity e0 = 2.5(---0.6) x 10 -s and longitude of pericenter to = 230°(± 15°) (Synnott, 1982). The longitude here is referenced to an epoch near the time of Voyager II encounter. The three components in the Voyager I images were each - 2 0 km wide (Smith et al., 1981), giving the ring an overall breadth of less than 100 km. Voyager II's more sensitive cameras revealed fainter material,

spread out over a radial extent of - 5 0 0 km (Smith et al., 1982). The Voyager II photopolarimetry experiment showed an F-ring 60 km wide in its brightest region, a value consistent with Voyager I results. It also revealed fine structure down to a scale of less than 0.5 km (Lane et al., 1982). The ring is confined by two shepherding satellites, with 1980S26 outside and 1980S27 inside. The former has orbital elements al = 141,700 km, e] = 4.3(-+2.0) × 10-3, tol = 20°(-30°), and the latter a2 = 139,353 km, e2 = 2.5(-+0.6) x 10 -3, ~2 : 173°(-+20 °) (Synnott, 1982). Masses of these irregularly shaped objects may be crudely determined by fitting their dimensions to triaxial ellipsoids, and assuming a mean density p = 1.2(---0.2) g/cmL Axial diameters are (I 10 -+ 15, 85 - 10, 65 - 10) km for 1980S26 and (140 -+ 10, 100 ± 15, 75 -+ 15) km for 1980S27 (Thomas et al., 1982). These yield masses of 0.67 × 10 -9 and 1.16 X 10 -9, respectively, with uncertainties of order 50%. It should be noted parenthetically that the inner shepherd seems to be simultaneously closer to the ring and more massive. This appears to be inconsistent with the idea that the ring occupies an equilibrium position, where the confining torques from both shepherding satellites balance (see Dermott, 1981). One would expect the ring to be farther from the more massive satel-

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lite. Perhaps the outer shepherd's larger eccentricity helps it to compensate for this handicap. The F-ring clumps appear to be uniformly distributed around the ring, with typical spacings of -9000 km. This distance, perhaps not coincidentally, is also the approximate crossover spacing for the F-ring braids (Smith et al., 1982). It compares favorably with the inner shepherd's associated wavelength h = 3~rd = 7800 (---300) km. The outer shepherd's associated wavelength, 14,300(---300) km, is rather too long. Photometric analysis (Smith et al., 1981; Pollack, 1981) reveals that most of the observed F-ring material is submicron in size. This material typically has optical depth -0.10 (Lane et al., 1982). However, Pollack (1981) finds photometric evidence for a

small population of larger bodies as well, with optical depth -0.01. Since electromagnetic forces may significantly alter the dynamics of the submicron population (Burns and Showalter, 1981; Hill and Mendis, 1981), we interpret our results as applicable mainly to the larger population. It is hoped that the aura of visible particles in some way reflects the distribution of these larger bodies. For 0.01 normal optical depth, collisions occur on the order of once every -100 periods, a time comparable to the shepherds' synodic periods (62 periods for the outer shepherd, 112 for the inner). This limits the length of our integrations, beyond which our results are no longer meaningful. In Fig. 8 we present graphs comparable to those given for our hypothetical shepherd (Fig. 7), here using the masses and

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FIG. 7. The effects of a satellite as functions of its eccentricity. Here we fill in the continuum between the three discrete eccentricities of Figs. 4 and 6. The dashed line is the limiting eccentricity, beyond which the satellite must actually pass through the ring. (a) The maximum semimajor axis kick Aamx received by a ring particle. (b) The induced rms width variation after one satellite passage. Ring width is plotted near satellite pericenter W(0°) and near apocenter W(|80°), as well as the difference A Wof these two. The curious dip in 14/(0°) represents a configuration where the kicks in semimajor axis and eccentricity almost perfectly cancel; it seems to have no other significance. (c) The change in overall eccentricity during one circumnavigation of the ring by the shepherd.

orbital radii o f the F-ring shepherds, as n o w determined. Observed eccentricities and their error bars are also shown. T h e s e are actually values relative to the ring, after the

transformations o f Section IV have been accomplished. N o t e that the curves o f Figs. 8b and c, which represent secular changes in the ring, are n o w scaled to the ring rota-

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tion period. It a p p e a r s that both shepherds have c o m p a r a b l e effects; the inner one seems to be s o m e w h a t m o r e potent, but un-

certainties in eccentricity bar a definitive conclusion. Using these eccentricity values, the inner

542

SHOWALTER AND BURNS

shepherd produces radial displacements of up to ~10 km (Fig. 8a), and therefore is capable of generating clumps in a 20-kmwide ringlet. This is comparable to the widths of the brightest F-ring strands, and so we expect clumps spaced h = 7800 km in these radial domains. The outer shepherd is not able to generate clumps in a region wider than ~ 10 km, so its effect should not be observed. According to (14), clumps produced by this mechanism will persist for approximately 150° trailing the inner shep-

herd. This is consistent with the sample integration of Fig. 9. Two aspects of the observed clumps are not quite consistent with our mechanism. First, they are reported to cover the full azimuthal extent of the ring, whereas our clumps span only half that. However, considering the uncertainties in masses and eccentricities involved, this factor of 2 discrepancy is not particularly worrisome. Second, the clumps are reported as not quite uniformly spaced, which is difficult to

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NUMERICAL STUDY OF SATURN'S F-RING understand under this pure satellite mechanism. Perhaps some internal ring process acts to modify this primary perturbation. In any case, a closer inspection of the Voyager images is warranted. These satellites generate width variations of approximately 0. I km per rotation period (Fig. 8b). This will grow continuously, but is moderated by collisions on a time scale of --100 periods. Width variations of order 10 km are therefore to be expected in response to each shepherd. Unfortunately at the time of the Voyager II encounter the shepherds approached closest to the ring at widely separated longitudes; they were acting to widen the ring in different places, and so their efforts are partially in conflict. Nevertheless, some width variation is apparent in the narrow ringlet of Fig. 9. This may be observable in the Voyager images, but has not been reported. We note in passing that this effect should be highlighted periodically, during the occasional antialignment of shepherd apsides as they precess under Saturn's oblateness. The ring eccentricity will also grow in response to the shepherds, though the effect is quite small. The inner satellite generates an eccentricity growth of - 8 x 10-8 per ring rotation (Fig. 8c), equivalent to a mere 10-m excursion on the ak/ah plane. For comparison we consider orbital precession, the other relevant process. The differential precession period between the F-ring and 1980S27 is 2 × 10 4 rotations, which leads to ring steps of -100 m with respect to the satellite in this same time. Therefore, the eccentricity growth process described is only about 10% of the story. It should not be observable in the Voyager data set. In light of this work, perhaps the most puzzling F-ring feature is its fine radial structure, on scales of less than 1 km, according to the occultation experiment (Lane et al., 1982). This is much smaller than the typical excursions in semimajor axis taken by individual ring particles during one encounter. Thus such fine structure will have to be radically time variable. It is

543

unclear how it can exist at all, unless the occultation profile simply indicates the occasional presence of larger objects. We do expect the F-ring to be rather broad, due to the repeated radial kicks from the shepherds. The more diffuse material, spread out over a -500-km radial extent (Smith et al., 1982), may be understood in this way. The spreading by shepherd encounters will supercede that caused by interparticle collisions. Dermott (1981) argued that the final F-ring width is caused by an equilibrium between collisional spreading and the shepherding mechanism of Goldreich and Tremaine (1979); his conclusion that the ring must be composed of numerous 40-m-sized bodies is therefore nullified. We have shown that shepherding satellites have very significant short-term effects. Those effects are nevertheless limited in nature. For example, we have observed no tendency toward braid formation, so that phenomenon must result from the interplay of other effects. However, the near-coincidence of the braid length with that expected from the shepherds still tempts one to believe that the satellites are somehow involved. In summary, Saturn's F-ring remains an enigma. ACKNOWLEDGMENTS We thank J. N. Cuzzi, B. A. Smith, and S. P. Synnott for illuminating discussions of the Voyager results. We thank S. Dermott, R. Greenberg, R. Harrington, J. Lissauer, C. Murray, and an anonymous referee for helpful suggestions and criticisms. This research was supported by Ames Interchange Agreement NCA2-ORI75-001 and NASA Grant NAGW310. REFERENCES BURNS, J. A., M. R. SHOWALTER,J. N. Cuzzl, AND J. B. POLLACK (1980). Physical processes in Jupiter's ring: Clues to its origin by Jove! Icarus 44, 339-360. BURNS, J. A., AND M. R. SHOWALTER (1981). The puzzling dynamics of Saturn's F-ring. Presentation at the United States-Brazil Symposium and Workshop on "The Motion of Planets and Natural and Artificial Satellites," S~o Pauio, Brazil, December 14-18, 1981. To be published by the University of Silo Paulo.

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DANNY, J. M. A. (1962). F u n d a m e n t a l s o f Celestial Mechanics. Macmillan, New York. DERMOTT, S. F. (1981). The "braided" F-ring of Saturn. N a t u r e 290, 454-457. GOLDREICH, P., AND S. Tt~MAINE (1979). Towards a theory for the Uranian rings. N a t u r e 277, 9799. HILL, J. R., AND D. A. MENDIS (1981). On the braids and spokes in Saturn's ring system. M o o n Planets 24, 431-436. LANE, A. L., et al. (1982). Photopolarimetry from Voyager 2" Preliminary results of Saturn, Titan and the rings. Science 215, 537-543. POLLACK, J. B. (1981). Phase curve and particle properties of Saturn's F-ring. Bull. A m e r . Astron. Soc. 13, 727.

SMITH, B. A.,

et al. (1981). Encounter with Saturn: Voyager I imaging science results. Science 212, 163-191. SMITH, B. A., et al. (1982). A new look at the Saturn system: The Voyager 2 images. Science 215, 504-537. SMOLUCHOWSKI, R. (1981). The F-ring of Saturn. Geophys. Res. Lett. 8, 623-624; erratum G R L 8, 946. SYNNOTT, S. P. (1982). Orbits of Saturn's F-ring particles and the shepherding satellites. Presentation at the Saturn meeting, Tucson, Arizona, May 11-15, 1982. THOMAS, P., J. VEVERKA,D. MORRISON,T. JOHNSON, AND M. DAVIES (1982). Saturn's small satellites: Voyager imaging results. Submitted.