A statistical study of impact ejecta distribution around Phobos and Deimos

ICARUS 90, 237--253 (1991)

A Statistical Study of Impact Ejecta Distribution around Phobos and Deimos M.

BANASZKIEWICZ* AND

W.-H.

IPt

*Space Research Centre, P1-00716 Warsaw, Poland; and tMax-Planck-lnstitut fiir A eronomie, D-3411 Katlenburg-Lindau, Germany Received August 4, 1990; revised N o v e m b e r 9, 1990

turbs the circum-satellite motion of the ejecta. Second, the irregular shapes of the satellites lead to appreciable deviations from the Newtonian gravitational potential. As a better approximation, the potential of a triaxial ellipsoid should be considered instead of that due to a sphere of equal mass. Finally, the rotation of the satellites, which is synchronous with their orbital motions, gives rise to inertial forces, i.e., centrifugal and Coriolis forces, that further affect the ejecta dynamics in a coordinate system rotating with the satellite. All these effects on the motion of individual particles around Phobos and Deimos have been considered previously by Dobrovolskis and Burns (1980) as well as by Davis et al. (1981). In both papers, the dynamics of the ejecta have been modeled in the framework of a restricted three-body problem with the gravitational field of the satellite being approximated by a uniform triaxial ellipsoid. Dobrovolskis and Burns (1980) have analyzed Hill curves or contours of constant values for the Jacobi integral in the equatorial planes of the satellites. These contours limit the regions accessible for particles ejected with a given launch velocity. Dobrovolskis and Burns have also studied the configurations of orbits in the equatorial planes. T h e y found that the escape velocity from the satellites depends strongly on the particle's starting position and elevation angle at launch. They have discussed how the reimpact distribution of ejecta could be affected by the various perturbing forces. Davis et al. (1981) extended the analysis to the threedimensional situations. T h e y calculated the variation of the effective gravitational force over the surfaces of the satellites. They found that the maxima are at the rotational poles and the minima at the sub-Mars and anti-Mars points where tidal and centrifugal forces act against the gravitation of the satellite. For Phobos, the differences between the maximum and the minimum values of the effective gravitational force are s u b s t a n t i a l - - m o r e than a factor of 2. As for Deimos, which is farther away from Mars and rotates more slowly, the effective gravitational force varies by 15% across the satellite; and this is mainly because

Impact-ejecta dynamics around Phobos and Deimos are considered for dust particles of the order of 1 mm and greater. Only the gravitational forces of the planet and the satellite as well as the centrifugal and Coriolis forces are taken into account. In the case of Phobos the ejecta generally either reimpact the surface in tens of minutes after their launch or escape rapidly away from the gravitational attraction of the satellite. Deimos, being subjected to a relatively smaller perturbation force from Mars, can keep its ejecta on reimpacting and temporary satellite orbits for hours. Phobos shows large variability in its escape velocity as a function the launch point and the direction of launch. The statistical properties of large numbers of particles moving around the satellites are determined in a series of Monte Carlo simulations. Two factors, important for the regolith-formation process, are investigated: the escape probability from a given site on the surface and the reimpact probability. Both of these functions vary appreciably over the surfaces of Phobos and Deimos. The differences of the escape probabilities between adjoining regions are greater for Phobos than for Deimos, while the reimpact probability variations are substantial for both satellites. The density patterns for Phobos exhibit an interesting dependence on the particles' launch velocities. For small velocities (4-5 m sec- 1) we observe two curved tubes of enhanced density that start from the sub-Mars and anti-Mars points on the surface. The density tends to become isotropic for greater launch velocities. Deimos shows a rather regular, ellipsoidal dust halo. The overall density of dust particles--with the meteoroid flux intensity, cratering laws, and velocity distribution of ejecta taken into account--is estimated to be of about a few hundred particles per kilometer cubed close to the surface of Phobos and drops down to about tens of particles per kilometer cubed at a distance of 40 km from the satellite's center. The density of dust around Deimos

is two to three times larger.

© 1991AcademicPress,Inc.

1. INTRODUCTION The two Martian satellites, Phobos and Deimos, are continuously bombarded by interplanetary meteoroids. The ejecta produced by impacts would move along trajectories quite different from the usual Keplerian orbits due to at least three factors. First, the close proximity of Phobos to Mars results in a tidal force that strongly per237

0019-1035/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

238

B A N A S Z K I E W I C Z A N D 1P

of the nonspherical shape of Deimos. For Phobos, the tangential component of the gravitational force reaches maximum at medium latitudes and is directed toward the sub-Mars or anti-Mars points. We find a different behavior for Deimos, with the tangential component directed toward the poles. Davis et al. discussed the effect of the tangential component on the movement of unconsolidated material on the satellite surfaces giving special attention to the ejecta distribution from the largest impact event on Phobos, i.e., that which formed the crater Stickney. They found that the loci of the reimpact points are not correlated with the observed pattern of grooves which are centered on Stickney. Finally they addressed the problem of different surface characteristics of the two satellites. According to the observations of the Viking orbiters (Thomas 1979, Thomas and Veverka 1980), the surface of Deimos is covered by a layer of loose material. Amid this dust layer, large boulders are buried. On the other hand, Phobos does not show these features, its surface being not as smooth. This difference could be caused by the different ways that the ejecta escape and reimpact these two satellites. The problem ofimpact-ejecta recycling was first considered by Soter (1971) who found that most of the material escaping from the satellites would move in elliptical orbits around Mars and eventually would collide with the parent body. Davis et al. explored further the reaccretion process and discussed the model that might qualitatively explain the difference between Phobos and Deimos appearances. They assumed that the satellites differ slightly in their material strength so that the velocity distributions of ejecta are different. If Phobos has greater material strength, its velocity distribution would be shifted toward higher velocities compared with that of Deimos. As a result, more ejecta would escape from Phobos. However, after performing more detailed, quantitative analysis based on the best estimates of ejecta reaccretion times, Davis et al. came to the conclusion that their model could not account for so many unlike surface characteristics of the satellites. In our paper we shall follow the analysis started in the above-mentioned papers. First, we study the main patterns of the three-dimensional orbits of particles around Phobos and Deimos. We also systematically analyze the escape velocities of the ejecta as a function of the impact position and the velocity vector. In the present work, the velocity azimuthal angle in the local horizontal plane is varied while the elevation angle is kept constant. Then a series of Monte Carlo simulations are performed to model the dynamics of particulate matter. Two issues are of basic interest. The first concerns the statistics of escaping and reimpacting trajectories of ejecta as a function of their launch point. Our results point at the possibility of nonuniform regolith formation on the satellite surfaces. The second issue is the possible existence of a

dust halo surrounding the satellites. We calculate number densities and column densities of the impact ejecta in the equatorial and polar planes of the satellites as functions of the initial velocity of the ejecta using the particle-in-abox method. In the present work, we focus our attention mainly on the question of the dust halo while some preliminary remarks will be made on regolith formation. Finally, we estimate the density of Phobos ejecta in the dust disk extending between the planet and the satellite's orbit, and compare it with the upper limit on the density of the supposed disk obtained from the recent analysis of Viking images (Duxbury and Ocampo 1988). In the last section we discuss some of the implications of our work and consider the possible applications. 2. EJECTA TRAJECTORIES AND ESCAPE VELOCITIES Following Davis et al. (1981), we place the origin of the coordinate system in the center of a satellite with the x-axis directed toward Mars, the y-axis anti-parallel to the orbital velocity and the z-axis perpendicular to the orbital plane. The equations of motion are: /a

2=

V~V+ co(2~; + o~x)+ G M ~ I r ~ - x

5~= V,V+ ~0(2~i - ~o>,)+ GM(IrM-~v~)

~2)

(I)

(2) (3)

where G is the gravitational constant, M is the mass of Mars, ~o is the angular velocity of the orbital motion of the satellite, d is the distance to Mars, r~ = (d, 0, 0) is the vector to the centre of Mars, r = (x, y, z) is the position of a particle ejected from the satellite surface, and V is the gravitational potential of the satellite in question. We use the exact formula for V (Dobrovolskis and Burns 1980) instead of the series expansion method proposed by Davis et al. (1981). The latter is inaccurate close to the satellite where higher-order multipoles are still important. Assuming that the satellite is a triaxial ellipsoid with semimajor axes (a, b, c) and with mass m we have V~V -

OV Ox

~GmxU.

(4)

V.,.V-

OV Oy

~GmyUh

(5)

~GmzU,.,

(6)

V_V -

OV

az

-

PHOBOS AND DEIMOS EJECTA where

239 TABLE I Satellite Data"

U, =

Ub =

(a 2 + t)A

f] f~

Uc = where

dt

(7)

(b E + t)A

(8)

dt (c 2 + t)A

(9)

A = X/[(a 2 + t)(b 2 + t)(c 2 + t)]

(10)

m [kg] a [km] b [km] co~[kin] [sec- 1] d [km] 0 [g cm-3]

Phobos

Deimos

9.984 × 10t5 13.5 10.7 9.6 2.280 × 10-4 9337.7 2.0

1.997 x 10t5 7.5 6.0 5.5 5.760 × 10_5 23459. 2.0

" Mars mass is 6.441 × 1023 kg. and X is the greatest root of the cubic equation X2 y2 Z2 a2 +--------~+ b2 +-------~+ c2 +-----~- 1.

(11)

In Table I we present the parameters for the two satellites as used in the calculations. Examples of trajectories from the numerical integration of Eqs. (1)-(3) are presented in Figs. 1 and 2. We use west longitude ~ and colatitude O to describe the starting position of a particle, called the impact point, on the surface of a satellite. The launch velocity vector of ejecta is described by its modulus V~, elevation angle with respect to the local horizontal plane O and azimuthal angle in the horizontal plane qb measured clockwise from the direction to the south pole. Our three-dimensional trajectories are generally similar to the two-dimensional orbits presented by Dobrovolskis and Burns (1980). The conclusions we derive from the present calculations are as follows: (a) The dynamical conditions in the vicinity of Phobos are more unstable than for Deimos because of larger perturbing forces. Generally particles either quickly escape or reimpact the surface after a short flight. More complicated looping trajectories can be found only for launch velocities close to the local escape velocity from the impact point. Particles ejected at the same point but with different azimuthal angles could follow quite different trajectories (Figs. la, lb). A set of 12 trajectories is shown on each of these figures. In the local coordinate system with the z~ axis directed along the normal to the surface and the x~ axis pointing along lines of longitude toward the south pole, the elevation angle of the velocity vector is taken to be 45 ° for all cases. Eight particles have a launch velocity o f 7 m sec ~ and the remaining four are ejected with a velocity of 9 m s e c - l . The azimuthal angle of the velocity vector, measured from the x~ axis, changes from 0° to 360 ° with a step o f 45 ° for the slow particles and 90 ° for the fast ones. T w o sets (a and b in Fig. 1) of trajectories have been computed, differing in the launching point of ejected particles. The colatitudes of the launch points are always the

same at 60 °, and two different longitudes (0 ° and 90 °) are considered. The set of trajectories with the launching longitude of 0 ° (Fig. la) is a symmetric image of the set with the longitude 180 °, i.e., one of them could be obtained from the other by a rotation of 180 ° around the center of the satellite. In the first set, eight escaping particles move toward Mars. The remaining four travel on reimpacting orbits. A different ratio of reimpacting to escaping orbits is found for the second set of trajectories (Fig. lb). Here, only three particles escape, all of them in the direction of Mars. We can expect a symmetric figure for this set with a launching longitude of 270 ° . (b) There are two preferred directions of escape from Phobos for particles with low and medium launch velocities (4-10 m s e c - l ) ; see Fig. 1. T h e y lie along the (x, - y ) diagonal and point in opposite directions. In the above velocity range, the emitted particles tend to migrate toward and to move along the equatorial plane. Particles with larger launch velocities are not so tightly confined to the equatorial plane; they also tend to escape more easily in directions other than along the (x, - y ) diagonal. (c) In the vicinity of Deimos, the particles move much more slowly than in the neighborhood of Phobos. Their escape directions lie in a broader range than for Phobos; however, directions toward and away from Mars are still favorable. The a s y m m e t r y between the motions in the equatorial and polar planes, so pronounced for Phobos, is greatly reduced here (Fig. 2a). Particles presented in Fig. 2a have a launch velocity of 5.5 m sec-J or 6 m sec J. The elevation angle of the velocity vector equals 45 ° and the azimuthal angle varies with a step size of 45 ° for the slow particles and 90 ° for the fast ejecta. All particles start from the point with a colatitude of 60 ° and a longitude of 180°. We can identify here escaping as well as reimpacting trajectories and, in a few cases, even temporary satellite orbits, i.e., particles described by " E " and " L . " All but one escaping particle move in the anti-Mars direction. H o w e v e r , they do not keep so close to the equatorial plane as the Phobos ejecta. If we change the launching

240

BANASZKIEWICZ

J

a

c

/"

/

t=

/

/

/

)'

,,,,,,,

A N D IP

B

oR

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~c

I=

p

,,,=

,,~ ,,,,,~H ,,~ F C

/

/

I

//

,,m

•

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:,, K',/

,,,/

,,m

"=Jm ~=

J

N

EQURTORIRL Iq:

v=

.7.0,0=

•

E:

v=

"7.0o0=tE]0

1 :

v=

9.0,0=

0

PLFqNE

POLRR

B:

v=

.7.0,0=

4S

F:

,,=

"7.0,@=22

C

J:

v=

9.0,0=

c; 9~

G: I~:

0

b

V = 7.O,0= 90 V----- 7 . 0 , 0 = 2 7 0 V= 9.@,0=tfl0

C

C

7

PLRNE O:

v=

"/.0,0=13B

H: L:

v=

"7.0,0=315 9.0,0=270

v=

d

'I

p P P

7

p

TM~

FPP

q q

p

p

\

FFP

"'.

E(~URTORIRL R:

v=

.7.0,@=

E:

v=

"7.0,@:

I :

v=

9.QI,@=

•

PLRNE

El:

v=

.7.0,0=

I.E]TM

F:

v=

'-/.0,0:22S

I~

J:

v=

9.0,@=

POLRR 4El 90

C:

v=

"7.0,@=

13:

v=

K:

v=

90

/ /

PLFqNE D:

v=

"7.0,@=13S

7.0,@=270

H:

v=

7.0,@=31..%

9.~,0=190

L :

v=

9.~,0=27171

FIG. 1. (a,b) T r a j e c t o r i e s o f P h o b o s e j e c t a in the e q u a t o r i a l and p o l a r p l a n e s of the satellite. The l a u n c h i n g point of e j e c t a is g i v e n by c o l a t i t u d e 0 a n d l o n g i t u d e st. T h e v e l o c i t y o f e a c h p a r t i c l e is d e s c r i b e d by its m o d u l u s v = VI in m sec t, the e l e v a t i o n a n g l e ® = 45 ° a b o v e the h o r i z o n t a l p l a n e a n d the a z i m u t h a l a n g l e qb in d e g r e e s , m e a s u r e d c l o c k w i s e in the h o r i z o n t a l p l a n e f r o m the d i r e c t i o n t o w a r d the s o u t h pole. N d e n o t e s the n o r t h p o l e a n d the a r r o w i n d i c a t e s the s e n s e o f r o t a t i o n of the satellite. E a c h s e g m e n t o f a d a s h e d line c o r r e s p o n d s to 100 sec of p a r t i c l e m o t i o n . C r o s s e s d e n o t e t h a t the p a r t i c l e is, at t h e s e p o i n t s , b e l o w the e q u a t o r i a l plane. (a) L a u n c h point: st = 0 °, e) - 60 °. (b) L a u n c h point: ,~ = 90 °, e) -

6 0 o.

241

PHOBOS AND DEIMOS EJECTA K

a

K

o.-" ,/

/

oo O~

/

B0

/-/...----,.

"%

..o"

K

q~

\

s

.o - "s*"

\ /

e

/

/

/

/ ,'/

\

\

',

"%

"

/

f

,/

.° oO,"

,) • .'"

""

j

....,;

................." r.,":] ""':.,,, I

R

B

v: v=

S.S,@=18E S.0,@: •

C

//,):S ,': i

R

J

PLFINE

EQURTOR I RL E: I:

":"1'

•,--""

F: J:

v= v=

S.S

5.0

POLRR 2

C: O: K:

v= v= v=

c

IB

S ,S,@= 90 S . S , @=2"70 S . • • @= 180

J

PLRNE 0: H" L:

v= v= v=

S.S,0=I"qS S.S,@:31S F~.01,@=2"70

b

TMRRS

TMRRS

"°% -o~"•

/ /, ./

I

EQURTORIRL

PLRNE

POLRR c':

v=

PLRNE 5.S,0=180

F I G . 2. (a,b) Trajectories o f D e i m o s ejecta in the equatorial and polar planes of the satellite. The definitions for O, @, and v are as in Fig. I. (a) L a u n c h point: ~p = 180 °, O = 60 °. (b) L a u n c h point: ~p = 180 °, ® = 60 °.

242

BANASZKIEWICZ AND IP

point to the 90 ° longitude, f e w e r particles will escape. T h o s e that m o v e along reimpacting orbits will hit the surface far f r o m their launching point. (d) The dynamical e n v i r o n m e n t of Deimos permits the t e m p o r a r y trapping of impact ejecta in satellite orbits. T h e y could be quite long-lived, as illustrated by the trajectory shown in Fig. 2b. After 3 days, this particle would still be in orbit around Deimos. The projection of the trajectory onto the equatorial plane has a very regular rosette-like shape. The height of the orbit a b o v e the surface of Phobos ranges from about 2.5 to 7 kin. In the polar plane the particle bounces between the extreme points of its orbit with an average amplitude of about 10 kin. With the knowledge gained from the previous work (Dobrovolskis and Burns 1980, Davis et al. 1981) on the range of e s c a p e velocities for Phobos, we have started a systematic search for the values of this variable. We have used the bisection method with the lower and upper limits of 3 and 18 m sec J, respectively, and an accuracy of 0.1 m sec ]. A particle would be considered as escaping when it reached a velocity of 20 m sec ~(a value greater than the global m a x i m u m of the escape velocity from the surface of Phobos) or has traveled to a distance of 20 satellite radii (i.e., the largest s e m i m a j o r axis) a w a y form Phobos. We have chosen three values of the elevation angle ® (0 °, 45 °, 90 °) and in each case have kept it constant while varying three other p a r a m e t e r s : the starting longitude and colatitude and the azimuthal angle of the velocity vector. The results for ® = 45 ° and for some chosen vales of ¢ and O are shown in Fig. 3. The smallest escape velocities, depending on @, are found at the sub-Mars point (0 = 90 °, ~ = 0°), and the largest, at the trailing edge of the satellite (t9 = 0 °, ¢ = 270°). Intermediate values are reached at the pole. The change of escape velocity with varying azimuthal angle • is found to have strong dependence on the location of particle launch. For example, at (O, ~) = (71 °, 210 °) the escape velocity of a particle directed toward the south pole equals 4,6 m sec ~but when @ changes by 45 ° the escape velocity j u m p s to 11.8 m sec - ~(Fig, 3). We have looked at the trajectories of these particles and have found that the j u m p resulted from the change of final e s c a p e directions. The south-pole directed particle e s c a p e s in the anti-Mars direction, while the second particle m o v e s toward Mars when it leaves the satellite only after a long travel o v e r the P h o b o s ' surface. This a p p e a r s to be a kind of polarization mechanism. The largest e s c a p e velocity o v e r the grid of 109 positions and 12 values of qb is found for (O, cp, @) (60 °, 345 °, 135 °) and is almost equal to 14 m sec -~. A word of caution should be said about the meaning of escape velocity in such complicated dynamical situation as in the environment of the Martian satellites. A particle ejected radially at the pole with a velocity much a b o v e the range consid=

G~

•~

I<

I

I

~--

~

/

i

VH: ~l !~ 1

Vft 3 !,4

f~ is ^

I <-

-71

f~VR=

VA : "1 . 1S E~ 12 !

4 i I pc

4 U 12

4 i I1~

"7 .34\ I~VR=

I

/

n b

] .29\ d A~I~ \VR=6 . 1 9

40'

8~

I<-''| L

T

225

/

- "I"

1-

/

21~

1-

"/

J

igS

LONGITUDE

FIG. 3. The variation of escape velocity for Phobos. The elevation angle of the velocity vector at the launch point ® is equal to 45°. The grid is determined by launching positions on the northern hemisphere. The north pole and the equator correspond to 0 and 90°, respectively. The sub-Martian and anti-Martian points are at longitudes 0 and 180°. The trailing edge of Phobos has a longitude of 270°. Direction of an arrow indicates the azimuthal angle of the velocity vector in the horizontal plane. The south pole is downward. Small triangles point in the direction of Mars. The length of an arrow corresponds to the local escape velocity value. Dashed circles indicate values of the escape velocity: 4, 8, and 12 m sec ~. VA is an average value of the escape velocity at a given point.

ered here must reimpact the satellite it left when its Keplerian orbit around Mars crosses again the circular orbit of the satellite. We therefore expect some cases when a particle cannot avoid collision with its parent body due to geometrical rather than dynamical reasons. For Deimos the range of escape velocity is much smaller (4.5-7 m sec ~). Changes in p a r a m e t e r s such as starting position and angle @ are smoother, according to the singleparticle trajectory studies we summarized above. 3. EJECTA STATISTICS With the a b o v e information on systematic variations of the escape velocities f r o m the satellites we can now approach the p r o b l e m s of the escape and reimpact statistics as well as the dust halos around Phobos and Deimos. In the Monte Carlo model of the ejecta dynamics studied here, the impacting flux of meteoroids o v e r the satellite surface is a s s u m e d to be uniform so that the starting points for the ejecta trajectories can be chosen randomly in ¢ and cos O o v e r the northern hemisphere (the problem is s y m m e t r i c with respect to the equatorial plane). In each of the numerical e x p e r i m e n t s the initial velocity of the

PHOBOS AND DEIMOS EJECTA

ejecta is assumed to be constant. The velocity range considered is 4-15 m sec -I with a step size of I m sec I for Phobos, and 4-8 m s e c - i with a step size of 0.5 m sec ] for Deimos. The elevation angle ® of the velocity vector is 45 ° which is the most probable value found in cratering experiments (Housen 1981). We have performed test runs with the discrete elevation-angle distribution. A grid of seven values of ® ranging from 0 to 90 ° has been placed at intervals of 15° . The distribution function has reached a value of 0.4 at 45 ° and decreased steeply toward the grid end-points where it is equal to 0.03. The results obtained were qualitatively similar to those with the elevation angle equal to 45 ° for all ejecta. The azimuthal angle q~ is distributed randomly in the interval [0, 2¢r]. The standard number of particles in each experiment is equal to 10,000, but in some cases we calculate 50,000 or even 300,000 trajectories to obtain better statistics. The trajectories of ejecta are integrated numerically with a constant time step until one o f the following criteria is fulfilled: (i) the particle hits the surface, (ii) the particle is at a distance greater than 20 major semiaxes from the center of the satellite, (iii) the particle velocity reaches a value of 20 m sec ~, or (iv) the n u m b e r of integration steps reaches the limiting value. In the first case the particle is in a reimpacting orbit, in the second and the third it is considered to have escaped from the satellite, while in the fourth we assume that a temporary satellite orbit has been found.

3.1. Escape Probability To find the escape and reimpact probabilities, the grid of points is set on the surface of the ellipsoid. Due to s y m m e t r y of the equations of motion with respect to the (equatorial) xy-plane we consider only particles ejected from the surface of the upper (northern) half of the ellipsoid assuming that each one of them has its counterpart ejected, at the same time, from the symmetric point in the southern part of the ellipsoid. Therefore, the grid points, placed at 10° intervals in longitude and equidistant in cos(O), belong to the northern half-ellipsoid and are given by the formulas:

Oi = a r c o s ( ( i - I)/NT)

i = 1. . . . . N T +

~oj = 2 ~ - ( j -

j = 1 . . . . . NF,

1)/NF

1

(12) (13)

where N T = 9 and N F = 36. We have then 324 sectors determined by the neighboring grid points: (~j, ~Pj+l) × (Oi, Oi+l). The areas of

243

different sectors would be equal if the considered surface were a sphere. In the case of ellipsoid the areas differ. For Phobos and Deimos the smallest sectors are close to the poles and the largest ones can be found on the equator, nearby the sub-Mars and anti-Mars points. The escape probability is simply calculated as the fraction of escaping particles out of the total number of particles ejected from this sector. The results for the escape probability from Phobos for the starting velocities of 6, 8, and 10 m sec 1 are shown in Figs. 4a, 4b, 4c. In the case of Deimos the escape probability for 5.5 m s-J is illustrated in Fig. 5; we do not present the results for other values for two reasons. First, the range of escape velocities for Deimos is very narrow; all the interesting things happen between 5 and 6 m sec ~ Second, the escape probability patterns for different velocities are much more uniform than for Phobos, and Fig. 5 is a good representation of all of them. The results show that the maxima (100% of escaping ejecta) are at the sub-Mars and anti-Mars points, and the minima are between these points at the equator. The differences in the escape velocity are quite large across the surface of Phobos. For a launch velocity of 6 m sec-~ (Fig. 4a) we observe large regions from which particles do not escape, but there are also some spots on the equator where the escape probability equals I00%. For larger launch velocities, the regions with zero probability of escape cease to exist. H o w e v e r , the differences between maximum (100%) and minimum escape probabilities are still substantial. The minimum escape probability for 8 m sec -~ particles is about 20% (Fig. 4b), and for 10 m sec-J particles, 35% (Fig. 4c). The positions of the two antipodal regions with 100% escape probability vary with the launching velocity, but they are always around the equator. Their longitudes are about 170 and 350 ° in the case of a launch velocity of 6 m sec 1, but they shift to 150 and 330 ° when the velocity changes to l0 m secAn interesting effect can be seen for regions will small probabilities of escape. They form a bow-like structure, especially p r o n o u n c e d for V~ = 8 m sec -j. In this case, the centers of the bows are placed at longitudes 1 l0 and 290 ° . Deimos shows much smaller differences in the escape probability. For the launch velocity of 5.5 m sec-~ (Fig. 5) the difference is not greater than 60%. The shift of the maximum escape probabilities from the longitudes 0 ° and 180 ° is very small, if any. A summary of the numerical experiments is given in Fig. 6 where the escape probability from the whole surface is shown as a function of VI. The escape velocities for Phobos range from 3 to 18 m sec -1 with the median at 7 m s e c - I . For Deimos the range of escape velocities is much smaller, extending from 5 to 7 m sec-] with the median at about 5.5 m sec-1.

244

BANASZKIEWICZ

O @ ......

~-

.......... L_

_

_*

[

.........

AND

J

............

IP

L 00

'

a 90 80

70 ¢o

60

50 I O

40 g 1

3o,.~ 2fl

i

1 i{)

-tL k

180 i ...... 1 [30

r

........ i......... 90

T

r 0 t,ongitude

r

T 270

t I{~0

100

t

90 80

70 ~, '-m 60 "-u

:S

5o p)


I 0

40

20 ):~ IO

180

90

0 l,ongil.ude

270

180

i 180

90

0 l,ongitude

270

I~0

F I G . 4. (a,b,c) T h e v a r i a t i o n of e s c a p e p r o b a b i l i t i e s f r o m P h o b o s . C o l a t i t u d e s 0 °, 90, a n d 180° c o r r e s p o n d to the n o r t h pole, e q u a t o r , a n d s o u t h pole, r e s p e c t i v e l y . T h e s u b - M a r t i a n p o i n t is at l o n g i t u d e 0 °. The s u r f a c e is d i v i d e d into 748 a p p r o x i m a t e l y e q u i a r e a l s e g m e n t s . (a) V m= 6.0 m / s e c . (b) Vi = 8.0 m / s e c . (c) Vm = 10.0 m / s e c .

245

PHOBOS AND DEIMOS EJECTA

I00 90

J

30-

8O 70

60-

~,

60 e ~J

5o ~

~ 90-

4o ~r

0

1802 0 ;'~

150-

10

180180

90

0 Longi[ude

270

180

FIG. 5. The variation of escape probabilities from Deimos. (V I = 5.5 m/sec). Colatitudes 0, 90, and 180 ° c o r r e s p o n d to the north pole, equator, and south pole, respectively. The sub-Martian point is at longitude 0 °. The surface is divided into 748 approximately equiareal s e g m e n t s .

3.2. R e i m p a c t Probability The average n u m b e r of particles launched from a grid sector is NS = N I / ( N T * NF); where NI is the total number of particles ejected in a given numerical experiment. The actual n u m b e r of particles ejected from the i-th sector NSAi differs from N S due to statistical fluctuations with rms. error equal to X/NS. The average surface density of launch points in the i-th sector, o i = N S / s i depends on the sector area si. Therefore, to obtain the same surface density of ejecta in each sector, which is a reasonable assumption in the case of Phobos and Deimos, each NSAi must be weighted with a factor s/(s); (s) is the average area of a sector. Similarly, the number of reimpacting

100 Phobos

80

~

6o

~-

40

~

20

_

_

_

I II

Delmos I

0

~

I

0

5

10 VELOCITY[M/S]

I

I

15

20

F I G . 6. The variation o f the escape probability f r o m Phobos and Deimos as a function o f launch velocity V I .

particles in t h e j - t h sector NRAj is proportional to sj. The surface density of reimpact points cry, which is an unbiased characteristics o f r e i m p a c t process, is equal to NRAj/ s3. In the case of uniform distribution of reimpact points across the surface it would be approximately equal (within the limit of statistical fluctuations) to the average value (o-r) = N R T / ( N T * N F ) / A T ; N R T is the total number of particles in reimpacting orbits and AT is the area of all sectors. F o r Phobos and Deimos, however, one can expect substantial deviation of or3 from (o-$ due to anisotropic force field in which the particles move. Therefore, the deviation from the mean Ao-~ = (o-~ - (o-$)/(o-$ is introduced as a measure of nonuniformness of the density of reimpact points on the surface. To obtain reliable estimates of Ao-~ as many particles' orbits as possible should be calculated. Statistical fluctuations are of about 20% for 10,000 particles, 10% for 50,000 particles, and only 3% for 300,000 particles. Deviations from the mean reimpact density are presented in Figs. 7a, 7b for I/1 = 4 and 8 m sec-~ in the case of Phobos and in Fig. 8 for Deimos for Vl = 5.5 m sec-J. To improve the statistics in the case of Phobos, we have used 300,000 particles for Vt = 4 m sec -I and 50,000 particles for V~ = 8 m sec-J. One can see large deviations from the mean value of 0% for V, = 4 m sec -~ in the vicinity of the sub-Mars point and its antipodal point. They are very much above the RMS error (3% in this case) and it can be explained in terms of a competition between the escape flux and the flux of reimpacting particles that are pulled toward Mars or in the anti-Mars direction. " E q u i l i b r i u m " exists over the rest of the surface because these are short-range trajectories and not very different from the Keplerian ones. For VI = 8 m sec

246

BANASZKIEWICZ

A N D IP

-80~

-100~ ie0 .

.

.

.9o .

.

6 Longitude

2+o

le0

160 140 ~

lzo 100 80 ~"

6o ~ 40 ~ 20 o O~ -40 -80 ~ -81/, ~ m - 100 180

90

0 Longitude

270

t 80

FIG. 7. (a,b) T h e r e i m p a c t s t a t i s t i c s for P h o b o s . D e v i a t i o n from the m e a n v a l u e o f r e i m p a c t s is s h o w n for e a c h s e g m e n t . T h e c o o r d i n a t e s y s t e m a n d the grid o f s e g m e n t s are the s a m e as in Fig. 4. (a) V I = 4.0 m / s e c . (b) 'v"I = 8.0 m / s e c .

(Fig. 7b) we can see the well-defined structures of maxima and minima of the reimpact density. Comparing these two cases, one can see that the differences between the maximum and the minimum deviations grow with increasing launch velocity. The distance between the extrema increases with the velocity as well. For Vt = 6 m sec ~it is equal to 35 ° , but for V~ = 8 m s e c - I it increases to about 55 ° . Both minima and maxima shift toward smaller longitudes when the velocity is increased. An interesting feature is obtained for V~ = 8 m s e c - ~. At the longitudes of 135 and 315 ° new maxima begin to appear. The results for the greater velocities show patterns similar to that found for VI = 8 m s e c - J. H o w e v e r , due to the decreasing number of reimpacting particles, statistical errors grow and the random variation of the deviation function eventually o v e r c o m e the regular structures.

The profile of reimpact density for Deimos (Fig. 8) shows greater variations of the deviation from the mean than those for Phobos. Instead of the pronounced minimum we obtain a plateau of small values of the deviation function e v e r y w h e r e over the surface with the exception for two narrow bow-like structures centered at 90 and 270 ° longitudes. Such structures were not found for the escape probability, so they must be attributed to the large increase of the reimpact density in these regions as compared with the mean value. 4. DENSITY AND COLUMN DENSITY We calculate densities of particles within a cubical volume of 80 km x 80 km × 80 km which is centered on Phobos or Deimos. The whole volume is further divided

247

PHOBOS AND DEIMOS EJECTA

160 140

lzo tO0 8O g 0

8o 4o 20 ~

I 0 O

0,.., -20 - 4 0 ,.., -60 -80~ -I00~ 180

0 Longitude

90

180

270

FIG. 8. The reimpact statistics for Deimos. (V I = 5.5 m / s e c ) , Deviation from the mean value of reimpacts is shown lbr each segment. The coordinate system and the grid of segments are the same as in Fig. 5.

into bins l km × 1 km × 1 km in dimension. The trajectory of a particle ejected from a randomly chosen point is calculated and the position of this particle is found every At = 50 sec. The bin containing the particle is identified and the number of counts in it is increased by one. This procedure is performed for all N = 10,000 particles. Assuming stationary flux of impacting meteoroids and hence stationary flux of ejecta one can find the density of ejecta in a cubical bin with a size Al = 1 km as a product of three factors: the number flux of ejected particles I, the

a

40

~

30 20

E

10

v

.~

0

average time of residence of a particle inside the bin T, and l / A l . If the ejection of N particles from an area A of the satellite is repeated at At intervals then we find that I = N / ( A t A ) . Since I is constant for fixed values of N and At, the only factor determining the density that can vary from bin to bin as well as from run to run is T. It is proportional to the number of counts in the bin and gives a simple means to find the relative density of ejecta as a number of counts per 1 km 3. This technique has been used before in many simulations (Fang et al. 1976, Ip 1984).

b

J

40

I

J

J

I

J

l

l

*

J

J

l

l

l

J

30 /Oo ,. , .

20 10 .~

0

X [

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-20

-20

-3O

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•

-40

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-10 0 10 X-axis (Km)

20

30

40

',

i

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i

-20

,

i

I

i

-10 0 10 X - a x i s (Kin)

i

20

x

i

30

40

F I G . 9. (a,b) The density distribution of ejecta in the equatorial and polar plane of Phobos. The density is expressed as a number of counts in a cube 1 km on a side. Dashed and dotted contour lines correspond to densities 2 × 10" a n d 5 × 10", r e s p e c t i v e l y . (a) V I = 8.0 m / s e c . (b) Vi = 8.0 m/sec.

248

B A N A S Z K I E W I C Z A N D IP

a

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X-axis (Km)

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X axis (Km)

( a , b ) The c o l u m n density distribution of Phobos ejecta in the equatorial and polar planes. (a) V I = 4 . 0 m / s e c . (b) V I = 4 . 0 m / s e c .

F I G . 10.

The c o l u m n density is obtained by summing up counts in 80 bins that belong to the same column perpendicular to the equatorial or polar plane of a satellite. Densities in the equatorial and the polar plane of P h o b o s for 8 m sec are s h o w n in Fig. 9. The results of the column-density calculations are presented in Figs. 10, 11, and 12 for P h o b o s and in Fig. 13 for D e i m o s . The dust halo around Phobos e v o l v e s from something that looks like a pair of

a

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0

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spiral arms at V~ = 4 m sec 1 to an almost isotropic distribution for V~ = 12 m sec '. For V~ = 4 m sec ', reimpacting orbits prevail and w e observe a layer of large density o f the order of 1000 particle counts per kilometer cubed close to the surface of Phobos. The density drops quickly as a function o f vertical distance from the midplane. At a height of 2 km w e find no ejecta. Inside the spiral arms, a c o l u m n density on the order of 10 counts

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(a,b) The column density distribution of Phobos ejecta in the equatorial and polar planes. (a) V I = 8.(1 m / s e c . (b) V t = 8 . 0 m / s e c .

PHOBOS

a

40

I

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EJECTA

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F I G . 12.

(a,b)

i

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i

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I l l

0

-10

20

30

40

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-10

+ - .........

-----""

-40

-30

0

10

20

30

40

The column density distribution of Phobos ejecta in the equatorial and polar planes. (a) V I = 12.0 m / s e c . (b) V I = 12.0 m/sec.

I

~

I

I

I

J

I

+

I

i

I

I

L

would extend along the diagonals, but scaling the isodensity contour from V~ = 8 m sec-J suggests a distance of about 100 km from the center of Phobos. The column density around Deimos is almost an order of magnitude greater compared to that of Phobos. The central region of the maximum density of -> 1000 counts per kilometer cubed (Fig. 13) is a very regular oval in the polar and equatorial planes. The isodensity contours corresponding to smaller values of the column density

+

4O

I

l

l

l

l

t

l

l

l

l

l

L

i

t

¢

b 30

30

20

20

+......... +ooo ~

-20

-

X - a x i s (Km)

km -3 is observed as far as 30 km away from the surface of the satellite. In the case of V~ = 8 m s - ' a thin, dense layer of reimpacting ejecta still exists, but now it is surrounded by a vast region of appreciable density of 100-1000 counts km -3 that extends 40 km to either side the center of the satellite in two opposite diagonal directions. For Vj = 12 m sec -~, a relatively large density of - 100 counts per kilometer cubed is found inside the whole cubic volume (see Fig. 12). It is difficult to say how far it

a

...............

40

I

10 X-axis (Kin)

-20

I00.0

.

......

:::Jill

-" .

.

.

.

.

.

. ..-+

.

zoo~o

.

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'"--.

--.

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--

--10 "-,

.

-._

.-

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30

-40

~ -30

.

, -20

,

, -10

i

i

0

10

X - a x i s (Km) 13.

......

_.--'"

....

/. ,

_

...

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:-s-

%-:5.--',_2

,_,---;

.....

-30

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FIG.

.'"

.

-....

-20

-20

(a,b)

r

20

-40

i

30

40

-40

, -30

, -20

,

, -10

.

r 0

, 10

, 20

, 30

40

X - a x i s (Km)

The column density distribution of Deimos ejecta in the equatorial and polar plane. (a) V] = 5 . 5 m / s e c . ( b ) V I = 5 . 5 m/sec.

250

BANASZK1EWICZ

show a similarity to the isodensity lines around Phobos, being parallel to the equatorial plane and exhibiting a tendency to turn toward the (x, - y ) diagonal. The higher particle concentration around Deimos can be partly explained by the fact that in our calculations the same number of particles are launched from the two satellites while Deimos' surface area is three times smaller than that of Phobos. Taking this into account we still can expect 2-3 times greater density of ejecta in the vicinity of Deimos. This effect is a direct result of the longer time of residence of particles orbiting around this satellite. So far we have discussed the shape and the pattern of isodensity contours for each launch velocity independently. The units used are number of counts per I km 3. If we want to know the absolute densities for different values of VL, we should calculate real fluxes of the impact ejecta and then rescale the values obtained in our numerical experiments. The ejecta flux depends on the impacting flux of meteorites and the amount of mass produced in collision events. The latter is difficult to determine as it depends on the material strength of the satellites' surface regolith, the value of which is not known yet. M o r e o v e r the experimental data on mass ejection during hypervelocity impacts are sparse. This means that, at best, only a crude estimate of the expected density could be made. The surfaces of the Martian satellites are covered by regolith layers of various depths. Impacts of small particles produce craters in a thin layer of loose, crushed material which may be similar in its mechanical properties to the upper layer of lunar regolith. Large meteoroids affect deeper, more consolidated regions of the satellites. The material strength of the target Y can therefore vary from a sand value of 104 dynes c m 2 to a chondriticmaterial value of about l 0 7 dynes cm 2 In spite of differences in the material strength of a target for small and large craters, all of them are produced in the strength regime as opposed to the gravity regime which is valid for very large craters or strong surface gravity g. The transition from strength- to gravity regime occurs for a crater diameter s, = Y/pg, where p is the target density. For Phobos and Y = 107 d y n e s c m 2 we find that s, = 10 kin. According to crater formation laws (Housen 1981) the total ejected mass can be expressed as Mei = 0.02 pK~( Y)Ek, where E k is the kinetic energy of an impacting meteoroid, a density p = 2 g cm 3 and KD(Y) = 0.019 Y-o.092 is a function derived from the values given by Housen. The impact velocity of meteoroids depends on their origin. Small particles that form the interplanetary dust complex and are produced mainly by comets, move with an average velocity of 15 km sec J at the Mars' orbit (GriJn et al. 1985). Asteroids and meteoroids derived from asteroids approach Mars with a velocity of only 10 km s e c - i (Hartmann 1977) even if the gravitational accelera-

AND

1P

tion of Mars is taken into account. In our analysis we use the former value because we are interested in a quasicontinuous flux of ejecta produced by impacts of micrometeoroids rather than in rare collisions with asteroids. Taking a material strength of 10 7 dynes cm -2, we obtain finally M U m = 3.61 × 103, where m is the mass of an impacting meteoroid. The mass spectrum of particles produced during an impact into bedrock ranges from very small debris to grains about 103 heavier than the projectile (Dohnanyi 1978). Calculations of ejecta flux as a function of mass, material strength, and structure of a target are rather complicated and will be considered in a future paper. For a preliminary estimation of a possible spatial density of ejecta we assume that all ejected particles have the same mass which is equal to the mass of the impacting meteoroid. The approximate number of ejected particles during one impact thus will be equal to Ne9 = 3610. The cumulative interplanetary meteoroid flux at the orbit of Mars is equal to I m = 1.95 x 10- ~0m-2 sec - 1for particles greater than 10 -4 g (Grt~n et al. 1985). The total mass flux of these particles is about 3.54 x 10- ~4 g m--2 sec- l giving 1.8 × 10 4 g as the average mass of the meteoroids. Davis et al. (1981) introduced a simple formula for the cumulative velocity distribution of ejecta parametrized by the massvelocity excavation coefficient Ce~: if v > vo

f(u, ~ U)= i~,elU 9/4

(14)

if v -< v0, where Cejvo 9/4 = I. It follows from their considerations that the plausible value of C~9 for regolith on the Martian satellites is about 10 6 if v is expressed in cm sec ~. The cut-offvelocity v0 is then equal 4.66 m sec i. One can find from scaling laws (Housen et al. 1983) that this value corresponds to a material strength of about l 0 7 dynes cm--', which agrees with our previous estimates. The flux of ejecta (particles cm 2 sec i) in the velocity range [ v , v + A v] and for v _> v0is

Iej

=

imNej ~,~eiu '3 F . -

13/4

Av.

(15)

In our computations the fluxes of ejected particles were equal to I~i -

N2 AtA

,

(16)

where N is the number of particles in a numerical experiment, At is the time interval between counts and A / 2 is the surface area of the northern half-ellipsoid from which particles are ejected. Fluxes of ejecta lei as well as the renormalizing factors Iej/l~j in the case of Phobos for different velocity intervals are presented in Table II. For

251

PHOBOS A N D DEIMOS EJECTA

TABLE II Phobos: Number Flux of Ejecta I~j and Normalizing Factors Iej/I~j for Different Velocity Intervals [v, v + Av]" v [ m sec i]

le~ x 1013 [cm -2 sec -I]

lej/lnj

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

200. 115. 72. 48. 33. 24. 18. 14. 11. 8.

0.75 0.43 0.27 0.18 0.13 0.09 0.07 0.05 0.04 0.03

a Ao = 1 m s e c - I .

Deimos with a velocity range 5.0-6.0 m sec-~ we have

Iei/I~i = 0.75. Since the ejecta with small values of v (-< I0 m sec-~) should dominate the total flux coming from the satellite surfaces, we find that, in our present work, the scaling factors for the computed number density and column density should be about 0.1-0.75. We have calculated fluxes of ejecta using a high value of Y, which is appropriate for compact carbonacious material. If we assume that the regolity layer on the surface has a material strength of sand (Y = 104 dynes cm-2) then the total n u m b e r of ejecta will be large--Nej = 24,500, but only 1% o f them will have velocities greater than 5 m sec- 1. Therefore, in this case the flux of ejecta Iej will be less than that for Y = l 0 7 dynes cm 2 by an order of magnitude. Particles that have escaped from the satellites with an ejection velocity of the order of 10 m sec-I move on almost circular orbits around Mars. According to Soter (1971), they would form tori surrounding the planet. There are at least three mechanisms removing material from these tori: collisions with the satellite, catastrophic breakup caused by interplanetary meteoroid impacts, and spiraling toward Mars due to the P o y n t i n g - R o b e r t s o n effect. Sorer (1971) estimated that for particles ejected with a velocity of 25 m s e c - ~ the lifetime against collisions with their parent satellite is about t~ = 30 years for Phobos and ts = 5000 years for Deimos. F o r smaller ejection velocities the corresponding lifetimes will be even lower. Gr0n et al. (1985) calculated the mean collisional lifetime of meteoroids at 1 AU. Extrapolating their results to the orbit of Mars one can obtain lifetimes in the range o f t m = 104-105 years for particle masses between 10 - 4 and 102 g. Finally the P o y n t i n g - R o b e r t s o n effect removes particles of I mm size on a timescale o f tpR = 4 x 10 6 years as determined from the formula given by Mignard (1984). Smaller particles are r e m o v e d faster.

We can estimate the upper limit of the particle-ring density around Phobos, assuming that the only mechanism acting is the P o y n t i n g - R o b e r t s o n effect. Ejecta will then have a long time to accumulate in the region between the satellite and the planet. Taking the mean production rate of particles as equal to ~r = InjA = 400 sec-J and the P o y n t i n g - R o b e r t s o n lifetime for a particle with an average mass of 1.8 x 10 4 g to be tpR = l 0 6 years we obtain for the steady-state conditions: /VtpR = 7r(r2h - r~l)Ns,

(17)

where N~ is the surface number density of particles contained in a ring that extends from the planetary surface r M to the orbit of Phobos rph. We neglect the influence of the Martian atmosphere on the motion of particles. Comparing the formulas for the gas drag force (Sterne 1960) and the P o y n t i n g - R o b e r t s o n force (Mignard 1984) one can find that in the case of Mars both forces will be equal if the concentration of atmospheric constituents is about l 0 6 cm -3. According to photochemical models of the atmosphere of Mars such concentration is reached at an altitude of h = 400 km (McElroy et al. 1977). Therefore the error produced by using r M instead r M + h in (17) should not be greater than 5%. One can find easily from (17) that N S = 5 x 10 -3 cm which translates to an optical depth of about ~" = 1.4 x 10 4. If we assume that the ring thickness is similar to the thickness of the ejecta layer in the vicinity of Phobos which is equal to about 50 km, as found from our calculations, then the average spatial density of particles of millimeter size will be equal to about I0 9 cm-3. Recently D u x b u r y and Ocampo (1988) have analyzed images of the Martian surroundings taken from the Viking Orbiter in 1980 in an attempt to identify rings or additional satellites. T h e y found no evidence of rings and hence set an upper limit on the quantity of possible ring material. According to their estimates, the optical depth of a possible Martian ring should not be greater than 5 x 10 -5. This value is only three times less than our theoretical upper limit estimate in which we assumed very long lifetimes of ring particles. Note that smaller and more abundant ejecta could contribute appreciably to the ring density if they were not effectively r e m o v e d from the ring system either by the P o y n t i n g - R o b e r t s o n effect or by electromagnetic forces. 5. C O N C L U S I O N S A N D D I S C U S S I O N

In a series of numerical simulations, we have shown that the unusual dynamical environment of the Martian satellites leads to interesting patterns in the spatial distribution of ejecta. Both the reimpact probability and the density of ejecta are subject to the influence of forces that act on dust particles: gravitational attraction of a

252

B A N A S Z K I E W I C Z A N D IP

nonspherical body, the tidal force due to Mars, and centrifugal and Coriolis forces. The first one introduces an asymmetry in the escape probability of particles, favoring some locations on the surface over others. The tidal force further increases this asymmetry as it pulls particles away from a satellite in the Mars and anti-Mars directions and pushes ejecta toward the surface along the y (the orbital direction of the satellite) and z (the normal to the satellite orbital plane) axes. The inertial forces tend to keep the ejecta moving parallel to the equatorial plane and in curved trajectories. As a result we obtain a sort of spiral structure for the density in the equatorial plane; the overall distribution of ejecta is confined in a disk configuration. Such effects are much stronger for Phobos than for Deimos. It is more difficult to interpret the reimpact statistics. Some regions on the surface are preferred as reimpact sites and others show deficiencies of reimpacting particles. The pattern varies with the launch velocity of ejecta. Qualitatively we may describe the situation as follows: If the target body is a sphere and the only acting force is its gravity, loci of particles ejected with the same initial velocity and elevation angle from a give point will form a circle around this point. Now, when we introduce perturbations (gravity due to an ellipsoidal shape, tidal force, etc.) the loci will deform to ovals for small launch velocities. For larger velocities particles ejected in some directions will be able to escape, while the others will still move on reimpacting orbits. We will then obtain arc-like structures for the loci of the reimpact points (see Davis et al. 1981 for more details). If we now vary the launch points of ejecta and trace their reimpact loci, we will obtain regions with different densities of the loci, corresponding to regions where the deviation of reimpact probability from the mean value is positive or negative, respectively. In the case of Phobos, the computed density distributions for ejection velocities less than 10 m sec-~ differ appreciably from the distributions one would expect in the case of an unperturbed sphere. When analyzing the formation of dust bands around Mars, Soter (1971) assumed larger ejection velocities than considered here. He neglected the satellite gravity and obtained a picture of a toroidal distribution of ejecta around the orbit of a satellite. He also analyzed the possible relaxation of such initial, toroidal distribution of ejecta to a disk structure. Our results show that, for small velocities of ejecta, a disk-like distribution will result, with a vertical thickness slightly greater than the polar diameter of the satellite at the very beginning. There are several second-order effects that we have not considered. We have assumed a uniform interplanetary meteoroid flux over the surfaces of the satellites and a constant velocity of meteoroids. In fact we should rather

expect a certain distribution of the velocity and impact direction of meteoroids. Such a distribution could, in principle, be obtained from the distribution of meteoroids in the phase space of elements (a,e,i), which has an advantage of being easily linked to the possible sources of meteoroids: comets and asteroids. Constraints on the distribution in the (a,e,i) phase space can be obtained from the observed density of zodiacal dust (Leinert et al. 1983). The satellite motion itself and the proximity of the planet influence the flux of impacting particles. Three factors are worthy of attention: the relative velocity of the satellite with respect to the planet, the gravitational focusing effect due to Mars, and shadowing of some part of the satellite surface by the planet. Each of these effects should change the interplanetary meteoroid flux by less than 10% for Phobos and by an even smaller percentage for Deimos. The list of problems to investigate in the future is long. Three of them are in our opinion of major importance; dynamics of small particles, regolith formation, and longterm evolution of ejecta. In this paper we have limited our studies to dust ejecta with masses greater than 0.1 g which implies diameters above 1 mm. For smaller particles electromagnetic forces must be included. They are charged by solar wind ions and electrons and influenced by magnetic fields. If we add the Lorentz force to the gravitational and inertial forces, the already complicated problem may become even less tractable. As for regolith formation, this problem has been carefully considered by Housen (1981) for uniform spheres. He has used a Monte Carlo approach to model the changes in regolith depth as a result of competition between cratering events that remove regolith from the points of impact and blanketing by the ejecta from neighbouring craters. As we have shown in this paper that the second process strongly depends on dynamical environment of a body and can be quite different from that found for a sphere if the considered object is very irregular, which is often the case for asteroids (Farinella et al. 1981, Catullo et al. 1984). Finally, our study is a good starting point to study the evolution of ejecta that escape from Phobos and Deimos. This problem has been addressed qualitatively by Soter (1971). With our approach his considerations could be extended further. Starting from the initial distribution of particles in phase space one can calculate the long-term motions of particles, which are determined by the gravitational attraction of Mars, radiation forces, and, for smaller grains, perhaps by the Lorentz force. Collisions with the satellites and the planet are unavoidable for the major part of the ejecta. It would be very interesting to determine the reimpacts distributions among the three bodies, and establish the steady-state nature of these distributions.

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HARTMANN, W. K. 1977. Relative crater production rates on planets. Icarus 31, 260-276.

This paper was written during a visit of one of authors (M.B.) to the Max-Planck-Institut for Aeronomie in Lindau. The hospitality of MPAe is gratefully acknowledged.

HOUSEN, K. R. 1981. The stochastic evolution of asteroidal regolith and the origin of brecciated and gas-rich meteorites. Ph.D. thesis, University of Arizona.

REFERENCES

HOUSEN, K. R., R. M. SCHMIDT, AND K. A. HOLSAPPLE 1983. Crater ejecta scaling laws: Fundamental forms based on dimensional analysis. J. Geophys. Res. 88, 2485-2499.

CATULLO, V., V. ZAPPALA, P. FARINELLA AND P. PAOLICCH1 1984. Analysis of the shape distribution of asteroids. Astron. Astrophys. 138, 464-468. DAVIS, D. R., K. R. HOUSEN, AND R. GREENBERG 1981. The unusual dynamical environment of Phobos and Deimos. Icarus 47, 220-233. DOaROVOLSKIS, A. R., AND J. A. BURNS 1980. Life near the Roche limit: Behavior of ejecta from satellites close to planets. Icarus 42, 422-441. DOHNANYI, J. 1978. Particle Dynamics. In Cosmic Dust (J. McDonnell, Ed.), pp. 527-606. Wiley, Chichester.

IP, W.-H. 1984. The ring atmosphere of Saturn: Monte Carlo simulation of ring source models. J. Geophys. Res. 89, 8843-8847. LEINERT, C., S. ROESER, AND C. BUITRAGO 1983. How to maintain the spatial distribution of interplanetary dust. Astron. Astrophys. 118, 345-357. MCELROY, M. B., T. Y. KONG, AND Y. L. YUNG 1977. Photochemistry and Evolution of Mars' Atmosphere: A Viking Perspective. J. Geophys. Res. 82, 4379-4388.

FANG, T.-M., W. H. SMYTH, AND M. B. MCELROY 1976. The spatial distribution of long lived gas clouds emitted by satellites in the outer Solar System. Planet Space Sci 24, 577-588. FARINELLA, P., P. PAOLICCHI, E. F. TEOESCO, AND V. ZAPPALA 1981. Triaxial equilibrium ellipsoids among the asteroids? Icarus 46, 114-123.

MIGNARD, F. 1984. Effects of radiation forces on dust particles in planetary rings. In Planetary Rings (R. Greenberg and A. Brahic, Eds.), 333-366. Univ. of Arizona Press, Tucson. SOTER, S. L. 1971. The dust belts of Mars. Cornell Center for Radiophysics and Space Research Report 462. STERNE, T. E. 1960. An Introduction to Celestial Mechanics. Interscience Publishers, Inc., New York. THOMAS, P. 1979. Surface features of Phobos and Deimos. Icarus 40, 223-243.

GR~3N, E., H. A. ZOOK, H. FECHTIG, AND R. H. GIESE 1985. Collisional balance in the meteoritic complex. Icarus 62, 244-272.

THOMAS, P., AND J. VEVERKA 1980. Downslope movement of material on Deimos. Icarus 42, 234-250.

D u x a u a v , T. C., AND A. C. OCAMPO 1988. Mars: Satellite and ring search from Viking. Icarus 76, 160-162.