Life near the Roche limit: Behavior of ejecta from satellites close to planets

ICARUS 42, 422--441 (1980)

Life near the Roche Limit: Behavior of Ejecta from Satellites Close to Planets A. R. D O B R O V O L S K I S * AND JOSEPH A. BURNS Cot'nell University. Ithaca, New Yorl~ 14853 Received March 6, 1980: revised April 22, 1980 The distribution of ejecta from impact craters significantly affects the surface characters o f satellites and asteroids. In order to understand better the distinctive features seen on Phobos, Deimos, and Amalthea, we study the d y n a m i c s o f nearby debris but include several f a c t o r s - planetary tides plus satellite rotation and nonspherical s h a p e - - t h a t complicate the problem. We have taken several different approaches to investigate the behavior of ejecta from satellites near planets. For example, we have calculated numerically the usual pseudoenergy (Jacobi) integral. This is done in the framework of a restricted three-body problem where we model the satellites as triaxial ellipsoids rather than point m a s s e s as in past work. lso-contours of this integral s h o w that Deimos and Amalthea are entirely enclosed by their Roche lobes, and the surfaces of their model ellipsoids lie nearly along equipotentials. Presumably this was once also the case for Phobos. before tidal evolution brought it so close to Mars. Presently the surface of Phobos overflows its Roche lobe, except for the regions within a few kilometers of the sub- and anti-Mars points. T h u s most surface material on Phobos is not energetically bound: nevertheless it i~" retained by the satellite because local gravity has an inward c o m p o n e n t everywhere. Similar situations probably prevail for the newly discovered satellite of Jupiter (JI4) and for the several objects found just outside S a t u r n ' s rings. We have also examined the fate of crater ejecta from the satellites of Mars by numerical integration of trajectories for particles leaving their surfaces in the equatorial plane. The ejecta behavior d e p e n d s dramatically on the longitude of the primary impact, as well as on the speed and direction of ejection. Material thrown farther than a few degrees o f longitude remains in flight for an appreciable time. O v e r intervals of an hour or more, the satellites travel through substantial arcs of their orbits, so that the Coriolis effect then b e c o m e s important. For this reason the limit of debris deposition is elongated toward the west while debris thrown to the east e s c a p e s at lower ejection velocities. We display s o m e typical trajectories, which include m a n y interesting special effects, such as loops, c u s p s , "'folded" ejecta blankets, and even a temporary satellite of Deimos. Besides being important for understanding the formation of surface features on satellites. our work is perhaps pertinent to regolith development on small satellites and asteroids, and also to the budgets of dust belts around planets.

INTRODUCTION

Phobos, Deimos, and Amaithea each reside in a distinctive dynamical setting and exhibit unusual physical properties, two aspects of the satellites which may be related. Let us first describe the moons dynamically and then outline their associated physical character. The above-mentioned three satellites [as well as the newly discovered satellite of Jupiter (J14) and those * N o w on leave at C N E S - G R G S , 18 Ave. E.-Belin, 31055 Toulouse, France.

recently found near Saturn's rings] travel along nearly circular and equatorial orbits relatively close to their respective primaries (see Table I). All three are in synchronous rotation as a result of tidal dissipation in their interiors. Thus each keeps its axis of least inertia pointing roughly toward its primary, while its axis of greatest inertia is perpendicular to its orbital plane. Since Phobos orbits faster than Mars rotates, tides in the body of the planet are causing its orbit to shrink measurably (Shor, 1975; Burns, 1978). Due to the currently large (and increasing) tidal stress on it, Phobos 422

0019-1035/80/060422-20502.00/0 Copyright ~ 1980 by Academic Press. Inc. All rights of reproduction in any form reserved.

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TABLE 1 SATEI.I.ITE ORBITStl Satellite

Primary

Phobos Deimos Amalthea (J5) JI4

Mars Mars Jupiter Jupiter

Semimajor axis 9,378 23,459 181,300 128,000

km km km km

= ~ =

2.76r~a~, 6.90r~t.r~ 2.54rj.. 1.79rju.

Orbital period (hr)

Eccentricity

7.7 30.3 12.0 7.1

0.015 0.00052 0.003 .-0

a See Burns (1977), Jewin et al. (1979), and R. J. Greenberg (private communication, 1980).

today lies near or inside its Roche limit and could be on the verge of disruption, depending upon its internal strength. One might expect that the proximity of Phobos and Deimos to Mars, as well as of Amalthea to Jupiter, may have influenced their surface morphologies. In fact their shapes are distinctly triaxial, although all three are heavily cratered. At one time it was suggested that the global forms of these moons might be determined by tides (Soter and Harris, 1977a; Morrison and Burns, 1976) but the notable cragginess of their shapes makes this less attractive now that better images of the satellites are available from Viking and Voyager (Veverka and Burns, 1980). Phobos' surface is distinguished by being crisscrossed with several sets of nearly straight, subparallel grooves which may be related to either impact fracturing (Thomas et al., 1979), tidal strains (Soter and Harris, 1977b), or a combination of these two (Weidenschilling, 1979), or possibly even to deep-seated fractures produced in a drag-assisted capture event (Pollack and Burns, 1977). These grooves are definitely absent on Deimos (Thomas, 1979; Thomas and Veverka, 1979). On the other hand, Deimos exhibits an amount of debris mantling not seen on Phobos and shows both streamers and filled craters. Presently available surface resolution on Amalthea and JI4 does not permit a direct comparison with the Martian satellites. Since, as we shall see, escape velocities on both Phobos and Deimos are less than 10 m/sec, one might anticipate on the basis of hypervelocity impacts into basalt (Gault

and Heitowit, 1963; Gault et al., 1963) that ejecta from a primary impact on either satellite are not easily retained. There are. however, two important qualifications that could allow ejecta to be trapped by Phobos, Deimos, and Amalthea. First, Soter (1971) has pointed out that, although ejecta readily leave the Martian satellites, debris seldom escapes from the p l a n e t ' s gravity. Most satellite ejecta remain in orbit around Mars and therefore eventually reimpact the source moon. These subsequent collisions occur at velocities lower than that of the primary impact, and hence would generate impact craters that would be intermediate between primaries and secondaries but probably not distinguishable from either; ejecta from these more gentle recollisions may not so easily depart the satellites. Second, experimental studies (Strffler et al.. 1975) demonstrate that impacts into unconsolidated material produce ejecta with most particles having speeds measured in meters per second, two orders of magnitude less than when craters are carved out of solid rock (Gault and Heitowit, 1963; Gault et al., 1963). Since Phobos and Deimos are observed to have somehow accumulated regolith to appreciable depths (100-200 m on Phobos and at least 10-20 m on Deimos: Thomas et al., 1979; Thomas and Veverka, 1980), they probably retain a substantial fraction of primary ejecta. Escape velocities rescaled to Amalthea reach 100 m/sec, but its unusual surface properties (Gradie et al.. 1979) suggest that the fate of its ejecta may not be certain. By studying the motion of ejecta near

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these satellites we hope to elucidate the relationship between surface characteristics and the m o v e m e n t of debris. We wish to understand the connection, if any, between the disparate surface features of Phobos (e.g., grooves and lack of loose surface debris) and Deimos (e.g., streamers and filled craters), and their nearness to Mars. We also desire to learn whether material is added to, or withdrawn from, any dust belts of Mars (Soter, 1971) in a characteristic m a n n e r by the satellites. A true understanding of the dynamics of ejecta from craters on Phobos, Deimos, and Amalthea will probably have important implications for the d e v e l o p m e n t of regoliths on asteroids (Housen et al.. 1979), objects whose sizes and rotation rates e n c o m p a s s those of the satellites we study: indeed the regolith problem for the minor planets is a simpler one since asteroid ejecta a b o v e a certain speed are lost forever. The p h e n o m e n a that we describe are also pertinent to the question of whether debris leaving the surfaces of small unseen satellites might account for the narrow rings of Uranus, and perhaps even of Saturn and Jupiter (Burns, 1979: Dermott e t a l . , 1980). This paper will be organized in the following manner. We will first show how surfaces of pseudoenergy can be defined in general for the motion of a massless particle traveling near a h o m o g e n e o u s , triaxial ellipsoidal body which itself circles a point mass: these contours are equivalent to the Hill (or " ' z e r o - v e l o c i t y " ) curves in the circular restricted three-body problem. Such surfaces are plotted and discussed for the satellites under consideration. We then provide the reader with a qualitative description of how trajectories are modified when rotational effects and shape b e c o m e appreciable: in particular we d e m o n s t r a t e the effect of rotation on escape velocities. With this as background, we display some characteristic trajectories for particles leaving Phobos and Deimos with velocities near the local e s c a p e speed. Throughout we discuss the implications of our numerical results for

the formation of surface features on satellites orbiting near planets. HiI.L CURVES Mathematical

Formalism

To study the motion of ejecta near satellites, we first examine a particle's behavior in the f r a m e w o r k of a circular restricted three-body problem (cf. Szebehely, 1967); our approach here is distinguished from previous work in that it treats the satellite as a triaxial ellipsoid instead of the usual point mass. A rotating Cartesian coordinate system (see Fig. 1) has its origin at the center of the " s e c o n d a r y " (Phobos, Deimos, or Amalthea) while the primary (planet) is located on the negative x-axis. The north pole of their mutual orbit lies along the positive z-axis, while the ),-axis completes a right-handed triad and is in the direction of the orbital motion of the secondary. Since all three satellites are in s y n c h r o n o u s rotation, this system also constitutes a satellite-fixed reference frame. The small eccentricities of the satellite orbits, as well as the observed libration of Phobos (Duxbury, 1977), have little effect on the validity of the approximation. In these uniformly rotating coordinates, the equations of motion for a particle take the form -

2ny = n 2 (x + d k

m)

-f~ d

G M ( x + d) R :~

0 U, Ox

(I)

Porhcle/• R," d~~ Plone~ FIG. I. A Cartesian coordinate system is shown fixed to a rotating ellipsoidal satellite as it orbits the planet M.

LIFE

GMy R3

+ 2n.f = nZy

-

- GMz R3

NEAR

0 U,

Oy

0 U. Oz

THE

(2)

(3)

Here G is the Newtonian constant o f gravity, M is the mass of the primary, m the mass of the secondary, d the distance between them, and n is the mean motion of their mutual orbit; the dot signifies differentiation with respect to time. These constants are very accurately related by Kepler's third law: n2d 3 = G ( M + m ) . Since we are mainly concerned with motions in the vicinity of the satellite, the primary is regarded as a point mass at a (great) distance R = [(x + d)" + yZ + z2]tt2 from the particle. On the other hand, the gravitational potential of the secondary is written as U, in order to account crudely for any nonsphericity in the satellite. The terms -2nS' and 2nx appearing in (I) and (2) above represent the Coriolis effect, while n2[x + d - (m/M)d] and n~y are the centrifugal accelerations. The system (1), (2), (3) above can immediately be integrated once to give

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425

of force. Owing to the Coriolis terms on the left-hand sides of (1) and (2), the entire acceleration of a particle cannot be derived from the gradient of the potential in (4); nevertheless the pseudoenergy ~ti2 + J is conserved since the Coriolis force always acts at right angles to the velocity, and therefore does no work. The requirement that this energy integral be constant restricts permitted motions and allows us to draw various semiquantitative conclusions concerning the behavior of ejecta. For simplicity, Phobos, Deimos, and Amalthea are assumed to be triaxial elliposids homogeneous in density. The adopted values for the principal semiaxes a > b > c, lying along the x , y , z - a x e s , respectively, are listed in Table II. Each body has sizeable departures from its ellipsoidal figure, but these have only minor effects on their global gravity fields. Nevertheless this lumpiness can be important in determining the local motion of particles. The gravitational potential exterior to a homogeneous ellipsoid (Danby, 1962, pp. 98-107) is given by U = -~Gm[Ut

- xZU~

-

yZU b -

z"-Uc],

½v2 - constant

(5)

= ½n"[x + d - ( m / M ) d ] z + ~n.2y.2 + G_____MM =_ _ j R-U

where (4) U~ =

where v = (.,3 + 3~ + z2)tt2 is the speed of the particle. The quantity J defined by (4) is the well-known Jacobi integral, or pseudopotential, corresponding to the given field

-S'

U,, =

dt U,. = f x ~ (c" + t)A

Ut, = f ~ (/F dt + t)A'

TABLE

(a'-' + t)_X'

(6) il

PRINCIPAL SI-MIAXES AND CORRESPONDING

INTEGRAI.S °

Satellite

a (m)

b (m)

c (m)

/./1.(0) (m -t)

U,,(O) (m -a)

Uo(O) ( m -a)

U,(O) (m -a)

Phobos Deimos Amalthea J 14

13,500 7,500 275,000 <20,000

10,800 6,100 205,000

9,400 5,500 155,000

1.79 × 10 -4 3.15 x 10 -4 9 . 5 0 × 10 -n

3.78 × 10 ,a 2.12 × 10 l~ 5.15 × 10 - I t

4 . 9 7 x I0 ~a 2 . 7 4 x 10 -tz 7 . 4 6 x 10 - t r

5 . 8 4 × I0 ' ' a 3.09 × 10 -12 1.03 x 10 -"~

" T h e s i z e s o f P h o b o s a n d D e i m o s a r e f r o m D u x b u r y (1977), w h i l e A m a l t h e a ' s d a t a c a m e f r o m P. C. T h o m a s ( p r i v a t e c o m m u n i c a t i o n , 1979).

426

DOBROVOLSKIS AND BURNS

with A(t) = [(a 2 +

t)(b" +

/)(C2 +

point on the model ellipsoids representing Phobos, Deimos, or Amalthea.

t)]'':

t is a d u m m y variable and h(x,y,z) is the greatest root of the cubic equation .xe y" z'-' a" + ~ + bz +--------~+ ~ +~-----~- 1.

(7)

Differentiating (5) then gives simply

OU=3 i).---r 5 Gm xU,,, 0 3 O--y U = ~ G m y Ut,, 0 3 0--~ U = -~ Gm zU,.

(8)

(Danby, 1962, p. 107). Along the surface of the satellite, A = 0 according to Eq. (7). The corresponding values of the integrals evaluated numerically from (6) are also listed in Table II. Combined with (5) and the equations of motion, these allow us to find the net potential, normal gravity, and the gradient of the potential (and thereby the " s l o p e " ) at any

~

General Considerations: The Example o f Phobos Contours of constant values for the Jacobi integral in the vicinity of Phobos are depicted in Fig. 2, using the mass of 9.6 × 10~s kg (Tolson et al., 1978) which corresponds to a density of about 2.0 g / c m :~. The stippled central ellipse represents Phobos. The contours (Hill curves) drawn outside Phobos are sections through the xy-plane of surfaces having constant pseudopotential J. Inside the ellipse, the curves represent the intersections of these equipotentials with the three-dimensional surface of Phobos; these intersections are projected onto the equatorial plane. The roughness of these curves is caused by numerical limitations in our plotting routine. Successive contours are separated by a constant contour interval of 1 m"/sec" relative to the value at the inner Lagrangian point L2, values of

~ii!ii!i!i

i:~!!i!!i:!i~ili~!!i~i~i:i:i iili~iiiii!i~iiiiilililiii~ !iiii~ii!!!iiiiii~ii~!i ii::i::~PHOBOS:

i;ii!',ili',iii i:iii,illiiiiii ii!!li!iiiiiiLii i

iiii~i!::~iiiii;iiiiiii?i!ili~,ii~;:iili iiiiiiil~i~:i':i',~',i

FIG. 2. Zero-velocity curves drawn in the equatorial plane for Phobos (shown stippled) at its current distance from Mars (2.76 Re). Curves interior to Phobos are projections onto the equatorial plane of the intersections of the zero-velocity curves with the surface of Phobos. All contours are separated by I mZ/sec2 Dashed contours are at higher "'energy" values than the inner Lagrange point L2; thus Phobos is overflowing its Roche lobe and m u c h o f its surface is not energetically bound. The scale o f the drawing is set by the distance between L2 and the center of the satellite.

LIFE NEAR THE ROCHE LIMIT J greater than J ( L 2 ) , corresponding to higher pseudoenergies, are indicated by the dashed curves and crosshatching. Plots of the Jacobi integral are helpful in understanding the dynamical setting in which the satellites reside. For example, the isocontours are the surfaces that would be filled if varying quantities of a m a s s l e s s fluid were placed on Phobos. [By m a s s l e s s we mean the fluid does not contribute to the satellite's field and therefore, if it changed location, would not alter that field; at the same time the fluid has mass in the sense that it responds to the existing field and thus is capable of flowing to any gravitational lows.] Therefore these curves suggest crudely the form that a satellite would assume if its surface layers had no strength (cf. Soter and Harris, 1977a). Additional surface roughness superposed on these ellipsoids does not change substantially the shape of the equipotentials since gravity is a long-range force; instead irregularities principally just displace a particle relative to these contours. That is to say, loose debris at the top of an isolated peak on Phobos would find it preferable energetically to be at the base of that peak and, given an opportunity to do so, will move there. A particle with a given pseudoenergy E = ½v2 + J can reach the Hill curve for J = E only just as its speed vanishes. For this reason, equipotentials of the Jacobi integral are often termed " z e r o - v e l o c i t y " curves. In addition such a particle cannot go beyond the Hill curve corresponding to J = E, since then its speed would be imaginary. Thus for a given initial condition, say a particle ejected from the north pole o f Phobos with a given velocity, the corresponding zero-velocity surface divides space into " a c c e s s i b l e " and " f o r b i d d e n " regions. The nature of the equipotential surface therefore provides useful information. L1 and L2 are evidently saddle points of J since their potentials are lower than those of any other positions along the direction of the orbit, but higher than those of all other

427

positions along the line connecting the centers o f Phobos and Mars. As saddle points of J, the two Lagrange points LI and L2 are positions of equilibrium. In fact these equilibria are unstable for any values of the mass and principal axes of the secondary (Lebrun and Robe, 1964). The Hill curves passing through L2 have a special significance, however. Notice, that the zero-velocity surfaces passing through L2 cut space into four separate domains: the "interior lobe" surrounding the primary (off to the left of each figure), the " o u t e r reaches'" (off to the right of each figure) extending to infinity from the neighborhood of LI, the "'Roche l o b e " surrounding the secondary, and the crosshatched " h o r s e s h o e " region filled by the dashed curves in the figures; the name of the latter region comes from the overall shape of the Hill curves in the planet-satellite system. • It is clear that Phobos is currently overflowing its Roche lobe (except for the regions within a few kilometers of its suband anti-Mars points); indeed the innermost Martian satellite probably even exceeds its Roche lobe in volume. The same is presumably also true for the recently found J 14 as well but we cannot be sure since its size and shape are essentially unknown. Of course, Phobos does not move to fill its Roche lobe because of its internal strength. Consider a particle resting on the surface o f a satellite, at a position (inside the Roche lobe) corresponding to an initial potential J0. If the particle is ejected at a speed v0, the pseudoenergy is ~v0z + J0. If this is less than J(L2), the particle can n e v e r traverse the associated Hill surface and therefore is forever restricted to the vicinity o f the satellite. Thus the Jacobi integral sometimes provides a sufficient condition for nonescape of ejecta. We can make this more quantitative by employing the definition of E and the concept of the zerovelocity curve in order to place a lower bound on the minimum speed necessary to transfer regions:

428

DOBROVOLSKIS AND BURNS Vmin

:>

[2J(L2) - 2J011'-';

(9)

this bound ignores the finite target size of Phobos and does not include the Coriolis acceleration. To illustrate this bound, we c o m p u t e from Fig. 2 that a particle at the sub-Mars point of Phobos would need more than - ( 2 x 5) "2 ~ 3 m / s e c to escape by consideration of " e n e r g y . " The actual escape speed is found numerically (below) to be about 3.5 m / s e c from this position when ejection occurs at 45 ° to the vertical in the prograde sense but 6.5 m / s e c in the retrograde case. The "midriff bulge" of Phobos cuts its Roche lobe into two disjoint volumes, so that material near the anti-Mars point must achieve a pseudoenergy at least equal to J ( L I ) in order to escape. But J ( L I ) and J ( L 2 ) are very nearly equal, to order a / d , as can be seen by the slight difference in the contours near L ! and L2. Because these volumes are disjoint, ejecta from craters forming on one tip o f Phobos are unlikely to land near the other end of the satellite since to get there would require an energy greater than escape energy: later trajectory calculations will support this statement. Unfortunately this line of reasoning does not delimit the motion of a surface particle lying outside the Roche lobe. Since its rest potential already exceeds the " e s c a p e ene r g y " J(L2), it is not energetically bound to Phobos and its accessible region may formally extend to infinity. The Hill curves have yet another interpretation. In this formalism the acceleration of a stationary particle is - V J , the gradient o f the pseudopotential. Thus, the spatial density of the equipotential surfaces indicates the local ~'gravity" felt by a motionless particle. This varies from a m a x i m u m o f 0.58 c m / s e c z on the poles of Phobos through 0.52 c m / s e c z at the leading and trailing edges to a minimum of 0.29 c m / s e c 2 at the tips. F u r t h e r m o r e , the normals to the contours show the direction o f " g r a v i t y " : it is easy to verify

from Fig. 2 that " s u r f a c e g r a v i t y " on Phobos always has an inward c o m p o n e n t . in fact, this attraction is greatest outside the Roche lobe! Thus loose particles on the surface o f Phobos will be retained, much as is an object on a tabletop, even though that object could attain a lower energy if it were only able to drop to the ground. Figure 2 also shows a gravitational slope towards the sub- and antiMars points (where gravity is weakest), in agreement with previous results (Goguen and Burns, 1978). Surface material might therefore be expected to creep, or to be gradually jostled, downhill, and thereby to accumulate in these areas. H o w e v e r , such deposits are not evident in photographs of Phobos ( T h o m a s and Veverka, 1980) and their absence m a y restrict the acceptable range of orbital evolution histories or of models for regolith formation (Y. Langevin, private c o m m u n ication, 1980). Note that at positions with x ~> L I or x ~< L2, the direction o f " g r a v i t y " points away from Phobos. Thus, for example, a mass placed in one o f these regions and tethered, like a balloon, to kilometers " a b o v e " the surface. It is important to keep in mind that the dynamical environment for satellites with tidally evolving orbits is not unvarying. Presently Phobos is evolving closer to Mars at a rate o f - 4 c m / y e a r (Shor, 1975: cf. Burns, 1978) under the influence o f tidal dissipation in Mars. When Phobos reaches a distance of 7000 km ~ 2.08 R~, the Lagrangian points L I and L2 will touch this moon of Mars. Then, loose material at the sub- and anti-Mars points will float right off the satellite's surface. Thus Phobos will be progressively denuded of its regolith blanket until it crashes onto the surface of Mars unless it is first disrupted by internal tidal stresses. Even before this happens, when Phobos reaches a distance of 8800 km 2.61 R e , the rate of advance of its periapsis will b e c o m e equal to the mean motion of Mars; the resulting eccentricity-type reso-

LIFE NEAR THE ROCHE LIMIT nance m a y drastically alter the subsequent orbital evolution o f Phobos. This question is currently under study. In the past Phobos w a s - m u c h farther from Mars than it is today. It is important to study the motion o f ejecta during this era because, p r e s u m a b l y , it was then that most extant craters were formed. To illustrate the past situation, Fig. 3 shows the Hill curves for Phobos in a circular orbit at d = 15,000 km ~ 4.45 R~. This position would have been passed by Phobos some 600 million years ago, if S h o r ' s (1975) relatively rapid secular acceleration m e a s u r e m e n t is correct and if the Martian tidal Q has been constant. In any case, Phobos was n e v e r much beyond this distance (in particular, at no time was Phobos e v e r outside the synchronous orbit located at d = 20,500 km 6.01 R~), provided its orbital eccentricity has always been small, which is admittedly questionable (cf. L a m b e c k , 1979; V e v e r k a and Burns, 1980). Figure 3 d e m o n s t r a t e s that, at 4.45 R,~, Phobos is completely enclosed by its Roche lobe. Thus all surface material would then

429

have been energetically bound to Phobos, requiring by Eq. (9) a speed o f at least (2 x 20) ~jz - 6 m / s e c (see below) to escape. In addition, the sub- and anti-Mars points were then the gravitational high points, contrary to the present situation, whereas the poles were low! Note, h o w e v e r , that the figure of Phobos would have been much closer to an equipotential. Finally we alert the reader that the contour spacing in Fig. 3 is 2 mZ/sec 2, twice that of Fig. 2, so that the potential well is even deeper than it at first seems. We also note that the scale of these figures is fixed by the distance between L2 and the center of the satellite, accounting for the apparent change in size of Phobos between Figs. 2 and 3.

Deimos

Results c o m p a r a b l e to those just mentioned for ancient Phobos apply to Deimos at its present, relatively unchanging (see L a m b e c k , 1979), position. Figure 4 shows the Hill curves for Deimos, assuming the same density as Phobos (Duxbury and

ilI!!iNENT

~+?~i !~ili:' !~?/~ i ~i~i~:v~~

FIG. 3. Zero-velocity curves for Phobos when at 4.4.5R except the contour spacing is now 2 m2/sec2.

d .

Nomenclature is the same here as in Fig. 2

430

DOBROVOLSKIS AND BURNS

Fl(;. 4. Zero-velocity curves for Deimos at its current distance (6.01 R') See caption to Fig. 2. Veverka, 1978). These curves totally envelope Deimos, even though the outer satellite is about one-fifth as massive as Phobos because Deimos is more than twice as distant from Mars as is Phobos; since the tidal perturbations decrease as d -3, the proximity o f Mars has about one-third the effect on its Hill curves. Nevertheless, the minimum escape speed works out to be about 4.5 m / s e c , which is coincidentally similar to that on Phobos (see below). According to these results for our triaxial ellipsoidai model, D e i m o s ' poles are gravitational lows while the long dimension again tends to " s t i c k o u t " to higher potentials. H o w e v e r , Deimos is only crudely ellipsoidal and a better fit would be a series of facets separated by prominent ridges (Thomas, 1979; V e v e r k a and Burns, 1980). Since the tops of these ridges are at local gravitational highs, it is not surprising that the regolith appears thinnest there, as well as on crater rims, the other local highs. Occasionally "'streamers," suggesting downslope m o v e m e n t off these high areas, are visible in high resolution images of Deimos (Thomas and Veverka, 1980). Some small craters on Deimos have substantial fill (Thomas, 1979), that T h o m a s

and Veverka (1980) believe is ballistically emplaced. In contrast to the situation on Deimos, there appears to be no evidence for the e m p l a c e m e n t and m o v e m e n t of debris on Phobos. One must wonder whether the difference in the surfaces of the two satellites reflects the fact that Deimos has forever been entirely enclosed by its Roche lobe whereas much of Phobos pokes through its Roche surface today. Arguing against this interpretation are the facts that the escape velocities are actually similar (notwithstanding these energy arguments), that P h o b o s ' current location close to Mars is a recent occurrence (Burns, 1978), and that ejecta which escape Phobos still may reimpact it (Soter, 1971), unless the initial orbits, or their subsequent modification by radiation forces (Burns et al., 1979), causes such debris to be lost from the system. Amalthea

Assuming A m a l t h e a ' s density to be 2.0 g / c m 3, the relation between this small Jovian satellite and its Roche lobe, as depicted in Fig. 5, is topologically similar to that of Deimos. The format here is the same as that in Figs. 2, 3, and 4, except that the

LIFE NEAR THE ROCHE LIMIT

431

/ FIG. 5. Zero-velocity curves for Amalthea. The caption of Fig. 2 applies except the contour spacing is 1000 m2/sec2' distance scale is naturally greater, and the Hill curves are now separated by a contour interval of 1000 mZ/sec 2. The corresponding threshold for escape is at least 100 m / s e c . For our choice of principal semiaxes (P. Thomas, private communication, 1979) and of mass, the surface of Amalthea lies close to an equipotential (although the poles tend to be low points)• This may be significant for interior models of the Jovian satellites. However, we note that Amalthea, as glimpsed by Voyagers I and 2, is even more irregularly shaped than the Martian satellites (P. Thomas, private communication, 1979). O t h e r Close Satellites

We have not carried out calculations for any other satellites but we mention that the parameters that are important in producing nonclassical results (like those found here) are rapid rotation and irregular shape. J14 and any satellites within, or just beyond, the ring systems o f the outer planets orbit quickly, and probably have highly irregular shapes since they are small and presumably would be heavily cratered. Thus our results are most likely pertinent to these objects as well.

ESCAPE FROM A ROTATING SPHERE While the Jacobi integral often supplies a useful lower bound to the escape speed, such an approach cannot always be profitably applied (e.g., it is not presently useful over most of the surface of Phobos). Moreover, as we now illustrate by a simple example, the energy constraint can be misleading since the escape velocity is made anisotropic by the satellite's rotation• Consider an isolated spherical body• As viewed in an inertial frame, any ejecta leaving such a body follow (portions of) Keplerian orbits. A test for escape is thus the (very elementary) energy criterion: any particle departing the surface with an inertial speed greater than a specified, and everywhere constant, v,. will never return• This escape velocity is given by v~ = ( 2 G m / r ) "2 = (87rGp/3)"2r = (2gr) "2, where m is the mass of the body, p its mass density, r its radius, and g = G m / r e is the acceleration of gravity at its surface. The picture is considerably less clear when studied from a reference frame fixed to the body which rotates now with an angular velocity co. And yet we know that the attractive gravitational force exerted on

432

DOBROVOLSKIS AND BURNS

the particle by the spherical body remains p r e c i s e l y the same. The complication oc-

curs since any near-surface particle moving at a given speed v with respect to the rotating frame a p p e a r s to be subject to centrifugal and Coriolis accelerations (relative to the body) of toZr and 2toy, respectively. The associated centrifugal " f o r c e " equals gravity for to = ( g / r ) 'rz ~- v,./r, the latter being the angular velocity at which the inertial speed of the rotating surface is the escape velocity. For such rapid rotations, the Coriolis effect also b e c o m e s important for ejecta speeds near g/to (gr) 1~2 =

These "fictitious" accelerations produce ballistic trajectories that are nonintuitive when viewed from the body-fixed frame (see the later section on trajectories and Figs. 7-12 therein). Nevertheless, it is easy to answer the yes-or-no question of escape simply by transforming the velocity of ejection back into an inertial frame, as illustrated in Fig. 6. The sheaf of vectors in Fig. 6a represents various velocities of ejection, seen in the body frame of reference, for debris from a primary impact on a small body. This pattern is taken to be symmetric about the vertical, regardless of the direction of the incoming projectile, in accordance with the e x p e r i m e n t s of Gault et al. (1963). Figure 6b shows the velocities of the

,,' ,

same particles as measured relative to inertial space; a constant horizontal velocity of 3 m / s e c to the left (representing the rotational speed of the surface) has been added to each vector. As shown, some speeds relative to the inertial frame are increased by the motion of the surface while others are reduced. In one case the direction of ejection is reoriented to be essentially vertical, whereas in another it is even reversed; the rather narrow and symmetric core of ejecta velocities of the left figure is grossly broadened and skewed toward the direction of rotation. Even though the (once-simple) cone of ejecta is distorted as viewed in the sidereal frame, it is in this frame that the criterion for escape is trivial. Any particle with an inertial speed greater than v,, escapes; all the rest are retained. These two domains are separated by an "'escape h e m i s p h e r e " centered at the point of primary impact: this is represented in Fig. 6 by the dashed semicircle at an escape speed arbitrarily selected to be 6 m / s e c . When transformed back into the body frame, this surface is displaced off center in the direction contrary to rotation; nevertheless it is easy to see from the figure that the same particles escape in either case. Thus we expect the escape velocity off a rotating body to be an anisotropic aitazimuthal function, favoring

\~Escope

jP~Surfoce ve/

,

,"--~\

//

vs

~C,l,~ ,~Sur face

=3m/s

tm.,~

(a) Body Fixed Frame

(b) Inertiol Frame

FtG. 6. A heuristic illustration that particles thrown in the direction of a surface's motion are more likely to escape. On the left a symmetrical distribution of velocities, as might be produced in an impact event, is shown relative to the b o d y ' s surface. In Fig. 6b, on the right, these are transformed to velocities measured with respect to inertial space, assuming that at the moment of ejection the surface moves to the left with velocity t:~ of 3 m / s e c . The dotted semicircle in Fig. 6b shows an escape speed t,,. of 6 m / s e c measured in inertial space: velocity vectors beyond this ci(,de are those of particles that will leave the body. This escape surface is transformed back into the body frame in Fig. 6a; here we notice that particles thrown against the motion are preferentially retained.

I J F E NEAR THE ROCHE LIMIT the retention of debris ejected in the antirorational sense. The relative influence of rotation in modifying the escape velocity (whether the particle is thrown in, or against, the direction of motion) is measured by tor/v~ = to(~TrGp) U2, the ratio of the rotational speed of the surface to the e s c a p e velocity. Note that this quantity is independent of the b o d y ' s radius but proportional to its angular velocity to. Related to this m a y be the interesting fact that (on the average) to ~ 2~'/(10 hr) for minor planets regardless of size (Burns, 1975): with such a spin ratio is about .~. For planetary satellites in s y n c h r o n o u s rotation, like those considered here, rotational effects b e c o m e more important the closer the satellite is to its planet. DYNAMICS RELATIVE TO A ROTATING ELLIPSOIDAL SATELI.ITE We have just learned that the e s c a p e of particles from Amalthea and the Martian satellites will be appreciably affected by any satellite rotation. The general motion of any debris is similarly influenced because, in the inertial frame, prograde (eastward) particles get an extra boost o v e r retrograde particles from the eastward surface velocity. F u r t h e r m o r e , noninertial accelerations must be taken into account since we are interested in the m o v e m e n t of particles relative to the (rotating) surfaces o f the satellites. In addition, the actual motion of debris from these satellites is further corn-

433

plicated by the shapes o f the satellites and by their proximity to Jupiter or to Mars. Prior to computing some ejecta orbits, we will now estimate the relative importance of various forces and accelerations with the aim o f developing some insight into the factors that influence the dynamics o f satellite debris. After discussing these, we will tabulate their magnitudes for the several satellites under consideration. All terms that enter the equation of motion will be written as accelerations and will be c o m p a r e d in Table llI against the surface gravity o f an equivalent spherical satellite, g = ( G m / r 2) = ( 4 r r G p r / 3 ) 1~2. The centripetal acceleration a,. = nZr, where the spin rate to of the satellite can be taken equal to n, the satellite's mean orbital motion, since tides have caused close satellites to rotate synchronously; note that, because of the satellite's triaxial shape, a,. varies by n 2 (Ar) o v e r the satellite's equator. The Coriolis acceleration, 2m × v, is largest for particles with the fastest velocities and is tabulated for v - v~., the escape speed, when it equals 2(2ga,.) t'z i f a spherical body is considered. This acceleration results from parts o f the satellite to the east of the source appearing in inertial space to drop a w a y from the particle while those to the west rush to meet it; since numerical solutions show that the escape velocity in the prograde direction is less than for retrograde particles, the m a x i m u m Coriolis acceleration will also be proportionately smaller. The Coriolis acceleration will be

TABLE

III

ACCELERATIONS (IN c m / s e c 2) SUFFERED ON TIlE SURFACE OF CURIAIN CLOSE SATEI.LI rES Acceleration Gravitational Centripetal Coriolis Tidal Nonspherical shape

Expression

Phobos

Deimos

Amalthea

J 14

toZr = a,. 2tnv,. ~ 2a;2(ga,.) ''2 4 G M , r / R 3 = 4a,.

0.7 0.06 0.4 0.2

0.4 0.002 0. I 0.01

14 0.5 6 2

I 0.1 1 0.5

g Ar/r

0. I

0.07

4

?

GMs/r 2 = g

434

DOBROVOLSKIS AND BURNS

influential for any problem in which the free fall time is c o m p a r a b l e to the rotation period. The tidal acceleration caused by the planet peaks at the sub- and antiMars points: it is a differential acceleration of a p p r o x i m a t e m a x i m u m magnitude 2(GM/R:~)(2r) ~ 4n2r = 4a,. and thus at the tips of the satellite it is always larger than the centrifugal acceleration. The nonspherical nature of the satellites has two separate consequences: on the one hand, various surface particles are not at the same distance r from the sateilite's center but, on the other hand, the gravity field is no longer that of a sphere. Each of these effects is of order g(Ar/r) (see Burns, 1975). The values of all the accelerations are presented in Table Ill for the satellites under study. It is instructive to examine the strengths of these accelerations relative to gravity. The Coriolis acceleration ratioed to g r a v i t y ' s attraction is a constant times oJ/(Gp) ti', whereas both the relative tidal, and the relative centrifugal, accelerations are other constants multiplied by toZ/Gp, where all the coefficients are of order I. We note that: (i) these ratios are independent of r or, to put it another way, a near-surface particle suffering strong perturbations with respect to its satellite does so regardless of the satellite's size: a corollary of this is that satellites of all sizes having a given density will lose sections of their surface debris for spins faster than a particular ~ (i.e., w h e n e v e r the satellite is within a certain R): this ignores higher-order tidal terms and considerations of shape: and (ii) within p = 1.5 g . c m -:~, toZ/Gp ~> I for P ~< 6 hr, pointing out that Phobos, J14, and hypothetical ring satellites are particularly susceptible to these effects. At the Roche limit, tidal accelerations approximately cancel the effects of selfgravitation (4nZr ~- g) by definition. Thus, at the same location, for such synchronously rotating satellites, the Coriolis effect and the centrifugal force are c o m p a r a b l e to the tidal force. Combined with the essentially

nonspherical gravitational fields of these four small irregular bodies, these accelerations mean that all these satellites, but most especially Phobos and J14, reside in an unique dynamical regime. TRAJECTORIES RELATIVE TO THE MARTIAN SATELLITES The preceding two sections d e m o n s t r a t e that (i) it is not easy to intuit precisely how debris m o v e s with respect to the surfaces of satellites, and (ii) even the pivotal question of escape versus retention (for the dynamical regime in which Phobos, Deimos, and Amalthea live) requires some thought before answer. And yet each of these issues is of paramount importance since, after all, on the first, only traces of the relative motion might ever be seen in photographs of the m o o n ' s surfaces and, on the second, the answer could tell how regoliths, and even planetary rings, develop. Yet no analytical constants of the motion, besides the Jacobi integral, exist for the restricted three-body problem, and so the only way to determine the actual fate of ejecta from a close satellite is by numerical integration. We have thus calculated a variety of trajectories for material ejected from the Martian satellites. (Amalthea and J 14 were not done because o f their uncertain densities and shapes.) Numerical trajectories similar to ours, but with the particle influenced only by point masses, have been presented by, a m o n g others, Fang et al. (1976), Smyth and McElroy (1977), as well as Macy and Traflon (1980), in regard to the motion of material spailed off Io: in addition a few analytical results (see, e.g., Berreen, 1979) have been put forth for the trajectories of a space probe ejected from a space station. Three-dimensional plots like ours for the Martian satellites have also been calculated by Housen and Davis ( 1978, 1980). Our c o m p u t e r program applies a standard subroutine ( D V E R K o f the International Mathematical and Statistical Library) to solve the equations of motion (1) and (2). Calculations were carried out in double

LIFE NEAR THE ROCHE LIMIT precision, and the invariant p s e u d o e n e r g y ½v2 + J was evaluated at each step as a check on accuracy; this quantity was typically c o n s e r v e d to eight digits. Throughout, for simplicity, the initial positions and velocities of the ejecta are chosen to lie in the equatorial plane of Phobos or Deimos. Then Eq. (3) indicates that z = 0 for all time by virtue of the s y m m e t r y of the gravitational model. Thus Eq. (7) b e c o m e s a simple quadratic in h, and the trajectories can be plotted in the xy-plane without any projection effects. Even though we have attacked only the two-dimensional problem, the three-dimensional motion can be partially anticipated. Rotational effects are most significant in distorting the trajectories and these effects do not directly influence the z motion. Therefore our solutions can crudely be projected out of the equatorial plane and then wrapped back around the satellite. This basic view of how to extend our results has received support from Housen and Davis (1978, 1980) who have found numerically the intersections of t h r e e - d i m e n s i o n a l ejecta sheets with the satellite surfaces, employing a model similar to ours. We now show an array of plots that illustrate typical ejecta motion in the vicinity o f Phobos or Deimos. Figure 7 shows the c o n s e q u e n c e s o f a recent cratering

435

event at the anti-Mars point of Phobos. Ejecta are sprayed out with a distribution of speeds in both prograde (easterly) and retrograde (westerly) directions, at an angle of 45 ° from the local normal, typical of laboratory experiments (cf. Strffler et al., 1975). The path of each particle is labeled with its initial speed in m / s e e relative to the surface; only velocities near the escape speed, and lower, are illustrated. The particle's successive positions are separated by IO0 sec in elapsed time, and are connected by dashes in alternating pairs. T h e s e timesteps permit one, for example, to keep track of elapsed time to reimpact, to identify p l a c e s - - s u c h as near the crest of the westward 6 m / s e e t r a j e c t o r y - - w h e r e the particle stands almost stationary as seen by an o b s e r v e r on the satellite, and to confirm that the eastbound 6 m / s e e particle leaves the satellite vicinity more quickly than any other shown. The anisotropy of fallback, as well as escape, is very evident in this figure, as it will be in later ones. For c o m p a r i s o n , Fig. 8 shows the patterns for two impacts at longitudes of 45 ° (near the large crater Stickney) and 270 ° west of the sub-Mars point. This demonstrates that the results depend on the point of primary impact, as might have been expected, since the various accelerations and forces vary with particle location. Nev-

x x

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FIG. 7. The motion o f ejecta from the anti-Mars tip o f Phobos. The debris is ejected at 45 ° to the vertical at initial velocities given on each curve in meters per second. The dashes are 100 sec o f flight time. Notice the ejecta pattern is skewed to the west and there is an overlap in the ejecta blanket.

436

DOBROVOLSKIS AND BURNS \ \

\

\

\ \

\ \

\ \ \

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ertheless, because of the n e a r - s y m m e t r y of the equation of motion and the small gradient in tidal terms, ejecta patterns are virtually indistinguishable for impacts diametrically opposite each other: no such pairs are therefore shown. Notice on the Mars side of Stickney how a slight change in velocity from 3 m / s e c to 4 m / s e c (actually between 3.2 and 3.4 m / s e c from other calculations) separates escape from retention. This abrupt cut-off when e s c a p e speed is attained is a general feature of our solutions and is of particular interest here since Stickney's ejecta blanket terminates abruptly (Thomas, 1979) to the east of the giant crater. Our calculations also indicate some dependence on the cone angle of the ejecta. The results of Fig. 9. where the ejecta cone is 30 ° from the vertical, should be c o m p a r e d against the trajectories shown in the bottom o f Fig. 8 for a 45 ° cone angle. This plot shows the existence of a narrow " e s c a p e w i n d o w " at 7 m / s e c in the prograde direction and, for greater initial speeds, the unexpected folding o f the ejecta blanket

back over the point of primary impact! The importance of cone ejection angle is further d e m o n s t r a t e d in Fig. 10 by plotting trajectories for 30 and 60 ° , the limiting ejection angles we investigated. As mentioned earlier, the motion from the sub- and anti-Mars sides would be the same to within (a/d) if the initial ejecta cones were identical. As might have been suspected, debris with the same speed that leaves with a larger vertical c o m p o n e n t more readily e s c a p e s in this case. Notice, however, that a larger horizontal c o m p o n e n t favors escape from the 270 c west position (compare Figs. 8 and 9). This difference is partly understandable in terms of the zero-velocity curves of Fig. 2. Particles at 0 or 180° need to be given enough energy to reach LI or L2, but those at 90 or 270 ° already have more than enough energy to escape if only they can avoid colliding with the satellite. Similar investigations were also made for impacts on Deimos, where, as evident in Table III, the nonstandard accelerations are relatively smaller and therefore e j e c t a trajectories are markedly more symmetri-

LIFE NEAR

THE ROCHE

LIMIT

437

\ \ \ \

\ \ \ \ /

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\ \ \

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ii

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I

I

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I

I

k

/ I

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,

',

"--

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to"

FIG. 9. The motion o f debris leaving the trailing side of Phobos with an initial cone angle of 30 ° to the vertical. See caption to Fig. 7.

very interesting special case, h o w e v e r . N o t e that the particle ejected with a (45 ° ) prograde speeds of 5 m / s e c , close to escape velocity, m a k e s over t w o complete revolu-

cal, as exemplified by Figs. 11 and 12. The latter figure, which s h o w s the paths o f ejecta from an impact at a longitude o f 270 ° west along the D e i m o s equator, includes a \

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FIc;. 10. The importance of the cone ejection angle on trajectory shapes. The trajectories on the two sides o f Phobos would be essentially the same for similar ejection angles. Sub-Mars debris is thrown out at 60 ° to the vertical and the anti-Mars ejecta at 30 °. See caption to Fig. 7.

\\ \

438

DOBROVOLSK1S

AND

BURNS

\\x'_ " .J

F~(;. 1 I. Ejecta paths relative to Deimos for a sub-Mars impact. See the caption to Fig. 7.

tions o f D e i m o s in 7 × 104 sec; that is, it circles D e i m o s about three times in inertial space, before landing near its point o f origin. Thus we have found a temporary satellite orbit o f D e i m o s , and a nearly periodic one at that. There are undoubtedly others, but we expect them to be similarly short-lived.

INTERPRETATIONS

The computed conditions for escape are summarized in Table IV. The first column gives the name of the satellite and remarks. Columns 2 and 3, respectively, provide the longitude o f primary impact (measured westward from the sub-Mars point) and the

/ '\

,,

2

FIcl. 12. Trajectories of debris from a trailing side crater on Deimos. Notice that the particle thrown in the prograde direction with 5 m / s e e b e c o m e s a temporary moon of Deimos. orbiting it twice before impact.

LIFE NEAR THE ROCHE LIMIT

439

TABLE IV SUMMARY OF NUMERICAl. RESULTS Satellite (remarks)

Longitude of impact (°)

Cone angle 0 (°)

Prograde escape speed (m/sec)

Retrograde escape speed (m/sec)

Jacobi threshold (m/sec)

~(m/sec)

Limits of deposition

Phobos

180 (~0) 45 (~225)

45

3 < v < 4

6 < t' < 7

3.0

4.36

10°E-35°W

0

3.85

30°E-27(FW

270 (~90) 315 ( ~ 135) 0

45

5 < v < 6

8 < c < 9

0

3.48

45CE-10YW

45

7 < v < 8

6 < v < 7

0

3.85

60

4 < v < 5

v :> 10

3.0

5.34

15°E ( f o l d e d to 35°W)-90°W 2 f i E - ~" 8 5 ° W

60

4 < r < 5

9 < v < 10

0

4.26

25°E-200°W

30

3 < v < 4

4 < t' < 5

3.0

3.08

5"E-20¢W

30

8 < t' < 9

0

2.46

45

5 < r < 8, t" --'- 10 6< v< 7

45

4 < ~,' < 5

45

5 <~ 1,' < 6

Phobos (Stickney) Phobos Phobos (folded) Phobos

45

3.2<

v < 3.4

I1.0<

t' < 11.2

(~ 180) Phobos

270

(~90) Phobos Phobos (double) Phobos (at 5 . 4 5 R : ) Deimos

180 (~0) 270 ( ~ 901 0 ( ~ 180) 0

5 5 ° E ( f o l d e d to

:-190°W)-140~W v>

10

6.2

2.15

35°E-;>II5°W

5 < c' < 6

4.5

0.61

6YE-120~W

5 < v < 6

4.7

0.50

745°E-140<~W!

(~ 180) Deimos (subsatellite)

270 (~90)

cone angle 0 o f the ejecta (with respect to the local normal). The next two columns list the speed o f escape in the prograde and retrograde directions, as bracketed by the results from the integrated trajectories. This tabulation shows how dangerous it is to draw generalizations, but we are obliged to try. In most cases, escape requires a greater initial velocity in the retrograde direction than in the prograde, as expected from our earlier arguments. Also, not surprisingly, the difference between prograde and retrograde e s c a p e is much less for Deimos than tbr Phobos. Material thrown off the sub- or anti-Mars points seems to escape most readily from Phobos. The column labeled "'Jacobi t h r e s h o l d " in Table IV lists the lower limit on escape speed obtained from the Hill curves by the energy argument given by Eq. (9); this bound is always respected, though not always useful. The next column gives the p a r a m e t e r • = 2for sin0, where co is the

rotation rate of the satellite and r is its local radius at the point o f primary impact. This quantity is the difference between prograde and retrograde escape speeds in the simple model of Fig. 6, and accounts for the major effects of rotation. The actual tabulated values of qb for impacts on Phobos and Deimos show fair-to-poor agreement with the anisotropy of the calculated escape speeds. Thus the perturbing presence of Mars, the nature of the moving target, and the nonsphericity of the satellites can destroy the accuracy of this estimate. The final column of Table IV gives the a p p r o x i m a t e extent of s e c o n d a r y impacts, measured from the primary impact site in both directions along the equator. In all cases but one (the t e m p o r a r y satellite of Deimos), the ejecta blanket from a given crater extends further in the westerly direclion (retrograde) than towards the east (prograde). These results concur with those of Housen and Davis ( 1978, 19801, whose three-

440

DOBROVOI~SKIS AND BURNS

dimensional calculations d e m o n s t r a t e that this p h e n o m e n o n also occurs at other latitudes. In general this effect can be understood qualitatively. Material ejected at speeds considerably less than escape velocity reimpacts within less than an hour, and produces nearly concentric "'bulls-eye'" patterns within about a steradian of the primary (cf. Housen and Davis, 1978, 1980). Any higher-velocity ejecta remain in flight for a longer period: during this time the Martian satellites travel through substantial arcs of their orbits, so that the Coriolis effect b e c o m e s pronounced. As seen from the satellite, the Coriolis acceleration is --to × v, where v is the velocity measured with respect to the moon. This is directed in toward the surface for ejecta moving westward, whereas it appears to lift eastbound debris offthe satellite. Thus for a given initial speed, prograde (eastbound) ejecta generally travels farther than retrograde (westbound) material before secondary impact and so the contours of ejecta deposition for a fixed initial speed are skewed to the east. On the other hand, particles thrown to the east also tend to escape at lower velocities, leading to the counterintuitive result that the limit of de-

bris deposition is elom,,ated toward the west/ The same reasoning ought to apply to J I4 and, on a larger scale, to Amalrhea. Apropos impacts at other latitudes (Housen and Davis, 1978, 1980). it is obvious that trajectories outside the xy-plane are subject to a considerable torsion. Even in the simple model of an isolated spherical body, where ejecta follow (fractional) Keplerian orbits relative to inertial space, the longitude of the node on the equator regresses rapidly (i.e., at the b o d y ' s rotational rate) as seen in the body-fixed frame? Precessional effects would be still more complicated for J14, Amalthea and the Martian satellites. This suggests that extensive ejecta blankets and ray patterns on these objects may be twisted into unusual, or even unrecognizable, arabesques. [The

celebrated grooves of Phobos are notably straight, evidence against their formation as chains of secondary craters (Head and Cintala, 1979: cf. Housen and Davis, 1979. personal communication): m o r e o v e r , the reimpact velocities (which can be computed from the trajectory plots) are comparable to the ejecta speeds, i.e., less than 10 m / s e c , far too low to gouge out the deep and c o m p l e x grooves.] Finally, it is amusing to realize that a vehicle driving (or a person walking) over the surface of Phobos in excess of the local "'speed limit" would lift right off the ground? This problem is most severe at the sub-Mars point, where takeoff velocity is about 4.3 m / s e c = 16 k m / h r in the westerly direction, but only 0.39 m / s e c = 1.4 k m / h r (a leisurely stroll) towards the east. This leads us to conclude that Phobos must be covered with a network of one-way streets with different speed limits, surely accounting for the purpose of the grooves? ACKNOWLEDGMENTS The authors have profited by discussions with P. Thomas. J. Veverka. J. Goguen. K. Houscn, and D. Davis. An earlier version of the manuscript was improved by comments from J. Goguen and P. Thomas. Thorough refereeing by D. R. Davis and S. J. Peale further strengthened our work. This research was supported by NASA grants under the Viking Guest Investigator and Mars Data Analysis programs.

REFERENCES

BERREEN, T. F. (1979). The trajectories of a space probe ejected from a space station in a circular orbit. ('el. Mech. 20, 405-431. BURNS, J. A. (1975). The angular momenta of solar system bodies: Implications for asteroid strengths. h'arus 25, 545-554. BURNS, J. A., Ed.(1977). Planetary Satellites. Univ. of Arizona Press, Tucson. BURNS, J. A. ([978). The dynamical evolution and origin of the martian moons. Vistas Astron. 22, 193210. BURNS, J. A. (19791. Source of Jupiter's ring'.) An unseen satellite by Jove'. Late paper. I lth DPS meeting Bull. Amer. Astron. Soc. 12, 435. BURNS. J. A., LAMV, P. 1. . . . '~ND SOU~R, S. (1979). Radiation forces on small particles in the solar system, h'arus 40. 1-48.

LIFE NEAR

THE ROCHE

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