- The effect of falling particles on the shape and spin rate of an asteroid
- 1 Introduction
- 2 General equations
- 3 Results
- 4 Conclusions
- References

A&A 403, 413-418 (2003)

DOI: 10.1051/0004-6361:20030399

**O. Vasilkova
^{}**

Pulkovo Astronomical Observatory,
65/1 Pulkovskoje shosse,
196140
St.-Petersburg, Russian;

Isaac Newton Institute of Chile, St. Petersburg Branch

Received 31 January 2002 / accepted 25 February 2003

**Abstract**

This simulation is focused on the specific influence of the gravitational
field of a very elongated rotating asteroid on the location of zones of
the most intensive bombardment by falling particles. It is assumed that
the particles are distributed uniformly in the space surrounding
the asteroid. The asteroid shape is approximated by a triaxial ellipsoid
with semiaxes
28,12,10.5
(equal to those of asteroid 243 Ida) and by a dumb-bell of the same mass. The computations and
appropriate figures show that at a rotation period faster than approximately
9.1 hours for the triaxial ellipsoid model and 3.3 hours for the
dumb-bell one the leading sides of the asteroid receive a higher flux
of impacting particles than the trailing sides while at slower periods
the situation is the opposite. The zones of possible erosion are computed
depending on the asteroid rotation period and on the ratio of impact and
rebound velocities of particles. The contribution of all impacting particles
to the angular momentum of the asteroid is computed, which leads to the
conclusion that falling out of particles damps the asteroid rotation
at any spin period.

**Key words: **celestial mechanics - methods: numerical - minor planets,
asteroids - solar system: general

Original asteroid parent bodies are not spherical. Produced by low-velocity collisions, they could form rubble-piles of an equilibrium ellipsoidal shape or dumb-bell shaped compound planetesimals (Bell et al. 1989). This paper presents a simulation that is the first-order approximation to understand how asteroid planetesimals of such elongated shape grew during the early accretion stage. It concentrates on the effect of the nonspherical shape of the asteroid on the distribution of areas subjected to the most intensive bombardment by impacting particles.

The following assumptions were made. All particles are of the same mass and distributed uniformly inside of a spherical layer of 200 to 1000 surrounding an asteroid of a very elongated shape (triaxial ellipsoid with semiaxes 28,12 and 10.5 and a dumb-bell of the same mass). The particles are not so small (of the order of a meter in diameter) as to be subjected to gas drag, the Pointing-Robertson effect or radiation pressure. On the other hand, the mass of particles is not too large to perturb either the gravitational field of the asteroid or the motion of particles due to their mutual interactions.

The early accretionary stage is considered here when the orbits of particles and bodies were near-circular, relative velocities were low and the Jupitor core was not large enough to perturb the growing planetesimals (Safronov 1972; Hartmann 1968; Ruskol & Safronov 1998, etc.). Under these conditions the zero initial velocities of particles (with respect to the center of the asteroid) adopted in the current simulation are justified, as well as the meter sizes of falling particles which seem to be proper for the early accretionary stage when in the absence of crushing of bodies (due to very low velocities), small particles were mostly swept out by the larger bodies, and the cloud transformed into a swarm. At this point, the further process of growth consisted principally of the fall of individual bodies onto the embryos (Safronov 1958, 1972). When close to an asteroid planetesimal, the gravitational attraction and perturbations due to deviation of the body's shape from the spherical dominates the other perturbations such as solar tide, solar radiation pressure, perturbations from the planets, gas drag and the Pointing-Robertson effect, especially as the particles of the size chosen are not as sensitive to perturbations concerned with solar radiation or gas drag as the smaller ones.

In the simulation a time span of only a few days is considered that allows us to neglect the perturbations above. In a complete model monitored during a few asteroid revolutions around the sun and including a more realistic power law size distribution of falling particles, all the perturbations above should be considered. Also, interaction of the particles with each other, their own spinning, unevenness of their space distribution (e.g., enhancing of space density of particles behind the moving planetesimal, Safronov 1958), deviation of initial particles velocities from the zero, increase of asteroid mass and density during the accretion of falling particles, and gradual diminution of space density of the matter around the asteroid with time should be taken into account. Nevertheless, some qualitative results and conclusions obtained in this simplified simulation would still be applicable to some real systems. Namely, the results obtained here are valid if for every size range of particles their space distribution is nearly uniform and their velocities are nearly zero with respect to the asteroid center. Then the fall of bodies will slow down the asteroid rotation (not affecting the direction of angular momentum of the asteroid), and at a given density of the asteroid, the ratio of the number of particles impacting its leading sides to that for trailing sides will be (independent of the particles size range and their space density) determined by the value of the asteroid spin rate (of course, while the falling bodies are small enough compared with the asteroid itself). A different approach to the problem, which contains theoretical considerations on topographical changes on ellipsoidal or dumb-bell shaped asteroid surfaces due to cratering by falling particles can be found in Ronca & Furlong (1977, 1979). The authors suppose that the flux of meteoroids, as seen by an asteroid, is isotropic. This model is proper (in contrast to the model adopted in the present study) for the current solar system state (especially for tumbling asteroids).

We define a body-fixed coordinate frame in the asteroid.
The axes *x*,*y*,*z* lie along the largest, intermediate and the shortest
dimension correspondingly. The asteroid rotates uniformly around its
largest moment of inertia, thus it rotates about its shortest *z* axis.
Given a constant density
for the asteroid, there is a classical
formula for the gravitational potential of a triaxial ellipsoid with
respect to a unit mass particle (Subbotin 1949;
Duboshin 1961):

(1) |

where

(2) |

and is determined from the cubic equation

(3) |

The potential of a dumb-bell asteroid is:

(4) |

where

(5) |

(6) |

(7) |

where is rotation rate of the asteroid, and

(8) |

A particle moving is governed by equations of its motion (5)-(7) in a body-fixed frame (where the potential is computed using Eqs. (1) and (4)). The equations were solved numerically and integration was terminated when a particle had collided with any point of the surface of the ellipsoid or either of the lobes of the dumb-bell. The positions and velocities of nearly 33000 falling particles were computed at the points of collision with the asteroid surface. The triaxial ellipsoid asteroid model was chosen so that its semi-axes 28, 12 and 10.5 , density of and rotation period 4.63 hour were equal to those for asteroid 243 Ida (Beatty 1995; Matthew 2002) and the dumb-bell shaped model was chosen to be of the same mass as the ellipsoidal one. To study the dependence of distribution of zones subjected to the most intensive bombardment, the value of the asteroid spin period was varied from 0.2 hour to 365 days. Initially, particles filled uniformly a spherical layer between 200 and 1000 from the asteroid center. The space density of particles was chosen so as to provide accumulated height of a few per a year over the asteroid surface (Ruzmaikina et al. 1989).

Figures 1 and 2 demonstrate the projections of impact points
on the equatorial plane of the asteroid. It should be noticed that while for
the resting ellipsoid (Batrakov & Vasilkova 1997) the most
bombarded areas are placed around the smallest and intermediate axes ends
(where the impact velocities also are proved to be higher), the position
of such areas for the rotating ellipsoid is quite different and depends on
the value of the asteroid rotation period. This dependence is more distinct
at shorter rotation periods. Figure 1 shows the absence of
impact points at the longest ends of the trailing sides of an ellipsoidal
asteroid at a spin period of 1.1 hour. In Fig. 2 a similar picture
is drawn for an asteroid of dumb-bell shape of the same mass.
Note that the case of spin period of 1.1 hour is taken as
idealized marginal case just to demonstrate the effect of fast rotation
on the distribution of impact points. Though, the asteroids of a dumb-bell
or an ellipsoidal shape rotating so fast are not likely to exist
physically. It can be shown that at spin periods shorter than 4.17 hour the
centripetal acceleration for the lobes of our dumb-bell asteroid exceeds the
acceleration of their attraction. (These would get balanced only at the
densities higher than
(see also Burns
1975 for discussion of asteroid strengths).)

The impact points of possible erosion at short rotation periods are plotted
in Figs. 3 and 4 as projections on the equatorial plane of
the asteroids of ellipsoidal and dumb-bell shapes. While the zones occupy
only a half of the surface of the ellipsoidal asteroid, they occupy almost
the whole surface of the dumb-bell one of the same mass at the same
rotation period. At the real period of rotation of asteroid Ida (of 4.63 hour)
the erosion zones disappear on the ellipsoidal asteroid but still occupy
about a half of the surface of a dumb-bell one (Fig. 5).
The computations show that dumb-bell asteroids are subjected to erosion at
periods of rotation up to 9 hours. For the same period (where comparable),
dumb-bell asteroids are subjected to erosion more intensively.
Figure 6 shows that even for a spherical shape and idealized clean
surface, at rotation periods approaching the instability limit removal
of particles from the surface is still possible. In this case the erosion
zones are located at the band around the equator, causing poleward growth
(Hartmann 1978). To compute the erosion zones the criterion was used
described in Hartmann (1978) where shattering in a vacuum on
a clean, coherent rock target at impact velocities 30 to 50
indicates a spread of fragment velocities usually
of 0.1 to 0.7 time the impact velocity. (The similar range
of velocities was found at integration in the present simulation.
The maximum values of relative impact velocities at a spin period of 1.1 hour approached 46
for both models,
and, at a spin period of 4.63 hour, was
18.5
for ellipsoid and 28
for dumb-bell models correspondingly.)
So, choosing the best conditions to achieve erosion (which involve clean,
smooth surfaces and the maximum possible rebound velocity), we considered
the impact point as a point of possible erosion if

(9) |

where is impact velocity of a particle and is escape velocity at a point of impact

(10) |

and formulae (1) and (4) were used to compute potential *V*
at the point of collision of a particle with the surface.
But the above "best" case is unrealistic in several ways
(Hartmann 1978). The earliest planetesimals (and particles
themselves) are not the polished igneous spheres used for
experiments by Hartmann (1978). They have
regolith-covered surfaces, or consist entirely of loose
aggregates of primitive materials (Hartmann 1978;
Housen et al. 1979; Ruzmaikina et al. 1989, etc.). This regolith would
ensure additional energy loss. A regolith depth of only about 0.04 projectile diameter would cut the velocity of rebound to

(11) |

and a regolith of a depth comparable to the projectile diameter
virtually stops the projectile (as experiments by Hartmann 1978
on low-velocity impacts in a vacuum showed).
So, the value of the rebound velocity strongly depends on both
the thickness of the asteroid regolith layer and size of the projectile.

In present simulation our models gain a regolith layer of a few per a year (Ruzmaikina et al. 1989), so, we would expect them to be covered with a multimeter regolith layer. Then the velocity of rebound would be damped to nearly zero. So, for early accretionary stage considered in this study, the assumption of no rebounds at impact seems more reasonable than that of further re-ejection and redistribution of ejecta over the asteroid surface. (In particular, it means that every particle has a single point of impact and, so, a single point in Figs. 1-6). Therefore, if there is no erosion, the zones of the most intensive bombardment (Figs. 1 and 2) become regions of the most intense accretion. Of course, if rebounds were taken into account (e.g., in the case of clean non-regolith-covered surface), the regions of the most intense bombardment would be slightly shifted, especially at fast rotation.

It seems that at fast rotation the predominant shifting would occur
in the direction opposite to that of asteroid spinning, that is
(looking, for example, at the upper half of Figs. 1 and 2)
to the right of the leading side. On the other hand, at the
same time new particles coming from the left regions (including those of
the trailing sides) would partly compensate the leaving ones.
Computations show that if the condition (11) is satisfied
(that is in the case of regolith-covered surfaces), the erosion is
possible only for dumb-bell shaped asteroids and only at rotation
periods shorter than 1.1 hour.
So, as this range of spin period is not common for the asteroids
of dimensions considered, in fact, erosion was impossible for
asteroid planetesimals of both ellipsoidal and dumb-bell shapes and
all falling particles were being accumulated on the asteroid surfaces.
In Fig. 7, dependence of the fraction of particles bombarding
the leading sides of the asteroid (with respect to the total number of falling
particles) on the asteroid rotation period
is represented for 3 different cases of an attracting gravitational
field of the asteroid (from upper curve to lower):
1) it is assumed that the asteroid of ellipsoid shape attracts
like a sphere of the same mass, that is, the particles fly along
the straight line towards the ellipsoid center and the coordinates
at points of collission of particles with the rotating ellipsoid
are computed;
2) the gravitational potential of the asteroid of triaxial ellipsoid
shape is computed using Eq. (1);
3) falling of particles to the dumb-bell shaped asteroid
of the same mass is governed by the gravitational field
of a dumb-bell asteroid (Eq. (4)).
The following conclusions can be made. At first, there is a
certain value of the rotation period (at the density and dimensions
chosen in the current study, it is approximately equal to 9.1 hour
for the ellipsoid model and 3.3 hour for the dumb-bell one)
such that at shorter periods the number of particles falling to the
leading sides is greater than the number of those impacting the trailing
sides. Second, this phenomenon is the consequence of the specific
influence of the triaxial ellipsoid gravitational field only - not
the "geometrical'' result of its elongated shape (see the curve
corresponding to the case 1):
the asteroid has a triaxial ellipsoid shape but its attraction is central;
as a result, the predominance of the number of particles impacting the
leading sides with respect to trailing ones exists for all
values of the asteroid rotation period).
Figure 8 reflects the combined contribution of all impacting
particles to the angular momentum of ellipsoidal and dumb-bell asteroids
via the asteroid rotation period:

(12) |

Here

The computations described above were repeated for different sizes
of particles and at different space densities. Density of the asteroid
itself was varied also. These computations showed that:
(1) for an asteroid of either shape (a triaxial ellipsoid of any
elongation and a dumb-bell of the same mass) the flux of particles
impacting the leading sides of the asteroid is higher than that
for its trailing sides if the spin period of the asteroid is shorter
than certain value (for ellipsoid model with semiaxes 28, 12, 10.5
and density of
and for the dumb-bell model of the same mass these values are approximately
equal to 9.1 and 3.3 hour respectively).
The more the shape of the asteroid is elongated (or the density less),
the longer is the mentioned value of the period.
The shape evolution will have an asymmetrical character, forming some
excrescences at the leading sides (at areas corresponding to the zones
of the most intensive bombardment in Figs. 1 and 2)
until the period of rotation is less than the value of the period
at which leading and trailing sides receive equal flux
of impacting particles (in Fig. 7 it corresponds to a period
at which the curve *y*=0.5 crosses any other curve). At the same mass
and spin period the excrescences will be more distinct and stick out more
for ellipsoidal asteroids than for dumb-bell shaped ones, especially at fast
rotation. It seems that at a nearly uniform space distribution of particles
(for any size range) and their low relative velocities, both the ellipsoidal
and dumb-bell asteroids at short periods of rotation tend to look like
asymmetrically knitting together spheres with excrescences near
the rotation axis at the leading sides of the asteroid.
At long spin periods the zones of the most intense bombardment are
located for the ellipsoid in areas around the rotation axis and around
the ends of the small axis of the asteroid equator (see also
Batrakov & Vasilkova 1997), and for dumb-bell
asteroids, near the point of contact. Therefore, at long periods,
triaxial ellipsoids will tend to adopt the shape of ellipsoids of rotation
and further, approaching it, will tend to adopt the shape
of a sphere. So, both the triaxial ellipsoid and dumb-bell asteroids
will tend to adopt a spherical shape. (2) Independent of the space density of particles and their masses
(of course, until they are much smaller than the asteroid's),
the effect of slowing down the rotation rate of an asteroid is always
valid for uniformly distributed particles of the same mass.
The masses and space density of particles only affect the drag rate
but not the direction of rotation. Also, the shorter the period of
asteroid rotation the steeper the curve in Fig. 8 and,
therefore, the stronger the damping of asteroid rotation.
Note that the plotted value of angular momentum contribution
in Fig. 8 is a *z* component of the combined angular
momentum contribution vector, as computations show that its *x* and *y* components are nearly zero. This damping effect
is similar to a drag effect described for the model of Harris in
Davis et al. (1979) though being for the early stage of accretion.
(In the current solar system, where collisions between asteroids
are common due to high eccentricities of their orbits, it is clear
that just the rarer collisions with large bodies will determine
whether the asteroid will be spinning up, damping or changing
the sense of rotation).
(3) The computation of possible erosion zones has led to a conclusion
(not new (Harris 1977; Hartmann 1978;
Housen et al. 1979, etc.), that the largest bodies
in the swarm would accumulate regolith. The zones accreting the mass
more intensively are those where the bombardment is more intense
(see Figs. 1 and 2).

I would like to thank the referee Dr. Joseph Spitale for helpful criticism and valuable advices.

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