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 asteroid. 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 hyperbolic.) 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): -Bmass.*(-Bpos)./(norm(-Bpos))^3 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.)
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.)
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).
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.
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 encounter. 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] space: 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 units.) 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._, p646)]. 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.