Asteroid Satellites and Tidal Moraines
                               Vernor Vinge
                    Department of Mathematical Sciences
                        San Diego State University

                           (c) 1995 by Vernor Vinge
                (This article may be reproduced for noncommercial
                    purposes if it is copied in its entirety,
                           including this notice.)    


          Loose material on the surface of asteroids is subject to
          modification by a number of mechanisms. In this paper, I
          investigate the extent of tidal influences on this regolith.
          I derive a model for near collisions between asteroids which
          depends on essentially three parameters.  This model contains a
          number of simplifying assumptions, but serves to give an overview
          of the problem space and provides estimates of what is physically
          possible as well as what is plausible.
          Based on a qualitative assessment of the model, and a number of
          simulated near collisions, I show that various interesting
          effects are physically possible (and graphics are presented for
          these situations):
             1. Surface rocks can be put into orbit about the asteroid
                they initially rested upon.
             2. Surface rocks can be put into orbit about the passing
             3. Surface rocks can be ejected to a great distance from
                both of the passing asteroids.
             4. Surface rocks can be redistributed into distinctive
                surface structures ("tidal moraines").

          Outcome 2 requires an extremely low encounter speed and/or
          unrealistically high relative density for the passing asteroid.
          Outcomes 1 and 3 also require low encounter speeds -- except in
          the case where the perturbing object is very large (Mars in this
          paper's analysis).
          Outcome 4 appears plausible for many encounters that might
          occur. If tidal moraines are observed in our future exploration
          of the solar system, their frequency, shape, size, and state of
          preservation may provide an interesting tool for investigating
          the evolution of the asteroids.
          In doing this research, I used MATLAB in several ways. MATLAB allowed
          me to develop a qualitative understanding of the encounter problem in
          an incremental fashion. MATLAB's notation made it almost trivial to
          generalize from the plane case to the three-dimensional case
          (allowing full maps of the moraines).  

     Asteroids are subject to continual bombardment and the possibility of
     catastrophic collisions. In this paper I consider the effects, not of
     collisions, but of near misses. In this case, tidal forces may still be
     sufficient to rearrange the asteroid regolith. I make a number of
     simplifying assumptions:
        o The asteroids are rigid bodies, not deformed significantly
          by the tidal forces of the encounter.
        o The asteroid rotation rates are negligible.
        o The asteroids are spherically symmetric.
        o The loose material ("regolith") is of negligible mass compared to the
          asteroid it rests on.
     (In the Feasibility section of this paper, I consider the practical
     significance of some of these restrictions.)

A Model for Near Misses:

     Figure 0 shows the notation and orientation of the encounter. The two
     objects are designated A and B. Their masses are respectively Amass
     and Bmass, and their radii are respectively Ar and Br. The center of
     asteroid A is the origin of the coordinate system. We will study the
     effect of a near miss upon loose rocks on the surface of A. (Note that
     by the term "near miss", I mean to exclude the case where the two
     objects are in closed orbit about each other. Aspects of the closed
     orbit situation are considered in [Thomas].)

     Canonical units simplify the setup: The unit distance is the periapsis
     distance (the distance from the center of asteroid A to the center of
     B, at closest approach). The unit mass is the sum of the mass of A and
     B. The unit time is the time it would take B to go one radian, if B
     were in a circular orbit at unit distance around A. (Under this
     formulation, Newton's constant of gravitation is also 1.)

     If no collision occurs (that is, if Ar+Br < 1), then B's motion along its
     trajectory is entirely determined by B's hyperbolic excess speed, vinf.
     This motion could be determined with an ODE solver, but there is (almost)
     a closed form solution [Roy, p92-98].  The polar coordinates of B at time
     t are given by:

     r = ((1+vinf^2)*cosh(F) - 1)/(vinf^2)
     theta = 2*atan(tanh(F/2)*sqrt(1 + 2/(vinf^2)))

     where F is the "hyperbolic eccentric anomaly", the solution of
     the equation:
     0 =  -F + sinh(F)*(1+vinf^2) - t*vinf^3.

     (This formulation has serious numerical problems as vinf tends to 0,
     the parabolic case. There are formulations that can handle this 
     [Bate _et al._, p191-203]; however, most near misses will be frankly

     Knowing the position of B at any time, we can now determine the net
     acceleration experienced by a small object (a loose rock, say) as B passes
     A. Let rockpos and Bpos be the positions of the rock and B, respectively,
     in the coordinate system of Figure 0. Then, using MATLAB-style notation,
     the acceleration (in an inertial frame) on the loose rock is:
              -Amass.*rockpos./(norm(rockpos))^3 ...
              -Bmass.*(rockpos - Bpos)./(norm(rockpos - Bpos))^3

     The acceleration (in an inertial frame) on A is (neglecting the
     insignificant acceleration due to the rock):
     So the acceleration of the rock with respect to the center of A is:
     netacc = -Amass.*rockpos./(norm(rockpos))^3 ...
              -Bmass.*(rockpos - Bpos)./(norm(rockpos - Bpos))^3 ...
              - (-Bmass.*(-Bpos)./(norm(-Bpos))^3);
     Using these relations, a good zero finder (to solve for the hyperbolic
     eccentric anomaly and so track B), and a good ODE solver, we can
     easily compute the trajectories of loose rocks. The script nearmiss.m
     does this for arrays of rocks initially spread across the surface of A.
     This script was used to generate all the encounter data plotted in
     the figures in this paper.

     The nearmiss.m code must cope with several complications. The most
     significant is that our "loose rock" starts out at rockpos on the
     surface of A. We can't start running our ODE solver until netacc is
     pointing above the rock's local horizon. (I neglect the possibility of
     rolling motion that might occur when the netacc vector is still
     slightly below the horizon.) nearmiss.m solves this problem in an _ad
     hoc_ and inelegant way: object B is tracked from well before the
     close encounter.  When the scalar product netacc'*rockpos is observed
     to become positive, we know we have "liftoff" and can begin tracking
     the rock by its trajectory differential equations. (MATLAB note: In
     fact, nearmiss.m makes a small effort to further approximate the
     liftoff time. However, I wasn't able to blindly use fzero for this
     purpose, since fzero was often confused by a second root corresponding
     to B's departure from the area.)

     A second complication is that, once flying, the rock may strike B or
     re-land on A. The nearmiss.m code does not account for contacts with
     the surface of B, but it does report the closest approach of each rock
     to the center of B. If a rock falls back to the surface of A,
     nearmiss.m attempts to locate the impact point accurately, and it
     turns off the trajectory solver. (This is also done in an _ad hoc_,
     inelegant way.)

Results of the Modeling:

     A number of interesting things can happen to surface rocks on A as a
     result of a close passage by object B. (Some of these possibilities are
     physically implausible for real asteroids. In the Feasibility section of
     this paper, I analyze these restrictions.)

Case 1:

     Surface rocks can be put into orbit about the asteroid they initially
     rested upon.  Such a situation is illustrated in Figure 1.  The
     orientation in this and the next two figures is the same as in Figure 0
     (so the trajectory shown in Figure 1 begins almost 180 degrees from
     the point of closest passage).  For clarity, this figure shows only
     one trajectory. In fact, this encounter raises a cloud of material
     from a large fraction of asteroid A's surface, some to orbit, some to
     escape, some to return to A, and some (perhaps) to collide with B.
     (The data for these plots can be reproduced by running nearmiss.m
     with appropriate parameters. For instance, to reproduce this first
     example, run nearmiss.m and respond with 'eg1.m' when the script asks
     you for the name of a parameter file.)

Case 2:

     Surface rocks can be put into orbit about the passing asteroid. This
     is illustrated in Figure 2. The closest the rock comes to the center
     of B is 0.8153, so achieving this orbit places very little additional
     constraint on the size of B (whose radius must in any case be less
     than 1-Ar = 0.816).

Case 3:

     Surface rocks can be ejected to a great distance from both of the
     passing asteroids. See Figure 3. In fact, at the end of this
     simulation (t = 80), the rock illustrated has more than escape
     velocity with respect to the each of A and b.

Case 4:

     Surface rocks can be redistributed into distinctive surface structures
     ("tidal moraines").

     In the preceding examples, only a small number of rocks were tracked,
     and they were all in the plane of the encounter.  This was done for
     clarity; the reader can edit eg1.m so that nearmiss.m will track rock
     swarms lifting off A.

     In fact, to illustrate Case 4, I do want to follow an array of rocks
     originally lying on asteroid A. This illustration is essentially the
     same encounter as used to illustrate Cases 1 and 3, except that now
     vinf has been increased from 1.0 to 2.0. In this situation, all the
     rocks return to A, so rather than follow their trajectories, I simply
     want to see where each rock lands. In eg4.m I set up an array of rocks
     at regular longitude and latitude intervals across the hemisphere of A
     centered on the sub-periapsis point.  Figure 4 illustrates the
     placement of these rocks after the encounter. The perspective is quite
     different from the previous figures. In Figure 4 we are in the plane
     of the encounter, looking down at the surface of A. The center of the
     figure is the sub-periapsis point.  Many of the rocks never lifted
     off; you can see the original latitude and longitude pattern in the
     layout of these rocks. As you look towards the sub-periapsis point,
     the distortion of the pattern becomes more and more extreme. In this
     example, no rock escaped, or even flew very far. We are left with an
     interestingly shaped mound of debris, surrounded by a denuded border.
     (In this example Ar = .01, so Br must be less than 0.99. In fact, this
     constraint is almost strong enough so that none of the flying rocks
     strike B; the closest any rock gets to the center of B is 0.9898.)

     At first glance, there would appear to be an additional complication
     to the analysis of tidal moraines on A -- namely the contribution of
     rocks that lift off B and impact A. It is an interesting conclusion of
     the next section that such an exchange is not possible, at least under
     our basic assumptions. In particular, if the encounter can lift rocks
     off the surface of A, then it cannot lift any rocks off the surface
     of B.

     (In this paper I do not study a possible "Case 5", namely the patterns
     that A's rocks might make in colliding with B.)

Feasibility of the Preceding Cases:

     In this model, there are essentially three free variables: Ar, Amass,
     and vinf. The following simple analysis shows that only a small part
     of this problem space is of real interest.

     Apparently, the tidal force on the surface of A is greatest at the
     time of B's closest passage and at the sub-periapsis point. If at this
     time and place, the tidal force is greater than the gravitational
     attraction of A, then at least some loose rocks will lift off; if not,
     then there will be no liftoffs at any time or place during the

     The maximum net effect, expressed as an acceleration, is:

     maxlift = -Amass/Ar^2 + Bmass/(1-Ar)^2 - Bmass/1^2
             = -Amass/Ar^2 + (1-Amass)*(1/(1-Ar)^2 - 1).

     In Figure 5, I plot 0-maxlift in terms of Ar and Amass. If [Ar Amass]
     lies to the left of the 0-maxlift contour, there will be no liftoffs,
     since maxlift is negative in that part of the diagram.  Note that:
         Bmass = 1 - Amass (by our choice of units), and
         Br < 1 - Ar (in order that B does not collide with A).
     Therefore if a particular choice of [Ar Amass] lies on the
     right side of the 0-maxlift contour -- then B's mass and radius must
     put it on the left side of the contour. Thus under our basic
     assumptions, if an encounter can lift rocks off of A, no liftoffs are
     possible from B.

     There is another quantity that may be usefully graphed in the [Ar Amass]
     Since Br must be less than 1 - Ar, as soon as [Ar Amass] is specified,
     we know the minimum relative density of B with respect to A. In fact,
        minreldens = (1/Amass - 1)/(1/Ar - 1)^3;

     This provides a significant real-world constraint, since measured
     densities for asteroids are roughly in the range 2000 to 4000 Kg/m^3
     [Millis and Dunham, p162].  (However, we might push the lower limit
     down to under 1000 Kg/m^3 if we considered icy objects. And at the
     other extreme, a solid piece of nickel-iron would be about 8000
     Kg/m^3.)  B's minimum density may be further constrained by the need
     to avoid collisions with the rocks under study.

     In Figure 5, I plot contours corresponding to minreldens 1, 2, 4, and 8.
     In the simulations I have run, the most spectacular results with
     realistic densities involved Bmass considerably larger than Amass
     (presumably because this allows the tidal effects to persist at
     greater ranges). Thus the lower portion of the diagram, near but to
     the right of the 0-maxlift contour, is a good area for realistic and
     effective encounters.

     The examples used to illustrate Cases 1, 3, and 4 all have about the
     same minimum relative density, 1.2. So for instance, if A's density
     were 2500Kg/m^3, B's would be about 3000 Kg/m^3 -- both reasonable
     values. (In the Case 2 example, minreldens is 1.0.)
     The third free variable is the hyperbolic excess speed, vinf. This is
     essentially the rate at which A and B were moving toward each other
     before their mutual attraction significantly affected their
     motion. When vinf is smaller, object B will have more time to
     accelerate rocks from A.

     The examples give for Cases 1, 2, and 3 fail when vinf is slightly
     larger than that chosen (1, 0.1, and 1 respectively). (Tidal moraines,
     on the other hand, exist to some degree wherever liftoffs can
     occur. In the example that illustrated Case 4, a striking moraine
     structure is visible at vinf = 2.0. This moraine is still marginally
     apparent if vinf is increased from 2.0 to 4.0.)

     Thus the value of vinf in non-canonical units is critical to assessing
     the real-world feasibility of these tidal effects.  From the
     definition of our canonical units, a speed of 1.0 corresponds to

          sqrt( G*Total&_Mass/periapsis&_distance )
        = Aradius * sqrt( Arho*G*(4/3)*pi*Ar/Amass )

     meters per second where G is Newton's constant, Arho is the density of
     A, and Aradius is the radius of A -- all in MKS units. (As before, Ar
     is the radius of A, and Amass is the mass of A -- both in canonical

     For the Case 1, 3, and 4 examples, taking Arho = 2500 Kg/m^3, we find
     that a speed of 1.0 in canonical units corresponds to about
          Aradius*(0.09)  meters per second.

     Thus if A is small, say 1Km in radius, vinf = 1.0 would correspond to
     about 90 meters per second. Even vinf = 4.0 would only be 360 m/s.  It's
     estimated that in our era the average collision speed of asteroids in the
     main belt is 5000m/s [Weidenschilling _et al._, p645]. These speeds are
     extremely low compared to that. (The example for Case 2 is even worse,
     since vinf was 0.1 there. I have not been able to find any situation where
     Case 2 looks as plausible as the other situations.)

     Our examples could work with larger encounter speeds, if we simply
     make A larger. However, B must scale also. So in these examples, if
     A has radius 1Km, B has radius about 99Km. The largest asteroid has
     a radius of about 500Km, which would appear to limit the Case 1 and 3
     examples to encounter speeds of 450m/s. For the Case 4 example but
     with vinf = 4.0 (barely discernible moraines), the encounter speed
     would be 1800 m/s.

     There is at least one non-asteroid candidate for object B: the planet
     Mars. If we take A's density to be 2500 Kg/m^3, then the examples
     given for Cases 1, 3, and 4 could apply to an asteroid of 37Km
     radius passing a bit over 300Km above the Martian surface. And in this
     case vinf = 1.0 corresponds to 3378m/s. 
     At the beginning of this paper, a number of simplifying assumptions
     were made. If we relax these assumptions, the analysis becomes much
     more complex. However, the resulting, more realistic, situation could
     be more favorable to the outcomes we have seen. For instance, if
     asteroid A were rotating or of an irregular shape, liftoffs would be
     easier from certain regions of the surface. 

     In fact, if the solid body of asteroid A had just moderate surface
     roughness, there could be a visible moraine feature even after a
     relatively high-speed encounter. This is illustrated in Figure 6: Here
     I assume a very plausible encounter. In MKS units, asteroid A has
     radius 30Km and B has radius 133Km. Both asteroids have density about
     3500Km and the encounter speed is a very plausible 4780m/s. Figure 6
     looks down at a small part of the sub-periapsis region of asteroid
     A. Initially, I placed "rocks" at three-degree intervals in latitude
     and longitude -- corresponding to an interval of about 1500 meters. In
     this encounter, asteroid B has only a very short time to affect A's
     regolith. The farthest that any rock flies is about 12 meters, not
     enough to make a perceptible change at the scale of Figure 6. However,
     if the solid body of asteroid A is rough, then loose material will
     tend to be removed from the "lee" of fixed obstacles and accumulate
     where obstacles are high enough to block most trajectories. The arrows
     in Figure 6 show the direction (and relative magnitude) of the shift
     in the regolith.


     Outcome 2 requires an extremely low encounter speed and/or
     unrealistically high relative density for the passing asteroid.
     Outcomes 1 and 3 also require low encounter speeds -- except in
     the case where the perturbing object is very large (such as Mars).
     As of 1995 there have been two spacecraft flybys of an asteroid, and
     one of these discovered an asteroid satellite. It would be nice to
     have a satellite-creation mechanism that could imply satellites are
     common.  A number of satellite-creation mechanisms have been proposed,
     but many require low encounter speeds or produce satellites that
     are short-lived.  Alas, the satellite-creation mechanism proposed in
     this paper has both these disadvantages: The encounters must normally
     be low speed.  Furthermore, the largest specific angular momentum that
     I have observed in my simulations corresponds to a circular orbit of
     not much more than twice asteroid A's radius. For a small satellite,
     this will often lead to a short lifetime [Weidenschilling _et al._,

     Outcome 4 appears plausible for many encounters that might
     occur. If tidal moraines are observed in our future exploration
     of the solar system, their frequency, shape, size, and state of
     preservation may provide an interesting tool for investigating
     the evolution of the asteroids.


     I am grateful to Michael Gannis, Jay Hill, F. David Lesley, and
     Michael Wester for discussions in connection with this paper. It was
     Mike Gannis who pointed out how tidal moraine structures might be
     evident even in the case of high-speed encounters.
     I found Keith Rogers' MatDraw program very helpful in the creation of
     Figure 0 and Figure 5.
     [1997: For this Web version, I used xv to convert MATLAB's
     ps files to gifs.]


    Bate, R. R., Mueller, D. D., and White, J. E., _Fundamentals of
      Astrodynamics_, Dover Publications, Inc., 1971.

    Consolmagno, G. J., and Schaefer, M. W., _Worlds Apart: A Textbook
      in Planetary Sciences_, Prentice Hall, 1994. (This is my source for
      mass and radius information for Mars.)

    Millis, R. I., and Dunham, D. W., "Precise Measurement of Asteroid Sizes
      and Shapes from Occultations", in _Asteroids II_ edited by
      Binzel, Gehrels, and Matthews, University of Arizona Press, 1989.

    Roy, A. E., _Orbital Motion_, 3rd Edition, IOP Publishing Ltd, 1988.

    Solem, J. C., and Hills, J. G., "Shaping of Near-Earth Asteroids by
      Tidal Forces", submitted to _The Astronomical Journal_. (I just came
      across the notice of this submission. I haven't seen this paper,
      but the title is very intriguing.)
      [Note added in 1997: The Solem and Hills paper models the more
      realistic situation where the asteroid is a "rock pile", and shows
      the spindle shape that arises from a close flyby of Earth.]

    Thomas, P.C., "Gravity, Tides, and Topography on Small Satellites and
      Asteroids: Application to Surface Features of the Martian Satellites",
      _Icarus_, v. 105, pp. 326-344, 1993.

    Weidenschilling, S. J., Paolicchi, P., and Zappala, V., "Do Asteroids
       have Satellites?", in _Asteroids II_ edited by Binzel, Gehrels, and
       Matthews, University of Arizona Press, 1989.