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.)
Abstract
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.)
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
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.
Conclusions:
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.
Acknowledgments:
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.]
References:
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.