Paolicchi et al.: Side Effects of Collisions
517
Side Effects of Collisions: Spin Rate Changes, Tumbling
Rotation States, and Binary Asteroids
P. Paolicchi
University of Pisa
J. A. Burns
Cornell University
S. J. Weidenschilling
Planetary Science Institute
Energetic collisions in the asteroid belt produce several observable effects, altering their size
distribution and rotational properties. Moreover, interesting objects or systems may be created:
dynamical families, or binaries, and asteroids in unusual rotation states (fast rotators or tumbling bodies). While the evolution of the size distribution and properties of families as well as
binaries are discussed in separate chapters, the other relevant effects of collisions will be dealt
with here. During collisions, both the projectile and the target bring their own spin and orbital
angular momenta to the system. In the collision the fragments may acquire a “breakup spin”
connected to the breakup process. The “breakup spin” is related to the ejection velocity and size
of the fragment; fast and small ejecta can achieve rapid spin rates. Usually, freshly created fragments leaving a collisional site do not spin around their axis of maximum momentum of inertia;
thus, tumbling (observed also in laboratory experiments) is “natural.” The presence of dissipative
processes damps the wobbling, with a timescale that is usually short compared to the age of the
solar system. Observations of main-belt asteroids suggest a lower limit of ~2 h to their spin
periods. Very fast rotators, which must be monolithic bodies held together by internal strength,
are present only among the smallest observed asteroids (all of which are NEAs, owing to obvious
selection effects). Several asteroids have very long rotational periods, which remains puzzling.
Theoretical arguments give preliminary and qualitative explanations for the observed properties.
Asteroidal binary systems are an expected outcome from catastrophic collisions, through the
effect of the mutual gravitational interaction or as a consequence of later bursting fission.
1.
INTRODUCTION
Collisions have shaped many of the observable properties of the asteroids. The most interesting and relevant cases
are those in which catastrophic disruption or high-energy
cratering processes create a very large number of sizable
fragments from a parent system (the two bodies that impact
each other are typically referred to as the smaller projectile
and the larger target). Throughout the history of the solar
system, essentially all asteroids (see Davis et al., 2002) have
been influenced by such processes.
These kinds of events have been widely analyzed
through laboratory experiments (Holsapple et al., 2002) that
have unfortunately involved bodies smaller by many orders
of magnitude than corresponding asteroidal systems. Analysis of possible observable consequences of such processes
among asteroids suggests various peculiar objects and systems, as well as the overall properties. In particular, it is
usually assumed that the so-called dynamical families
(Zappalà et al., 2002) are typical outcomes of catastrophic
breakup.
Despite the difficulties connected with scaling from the
decimeter-sized targets used in experiments to the bodies
involved in astronomical events (Housen and Holsapple,
1990, 1991; Holsapple, 1993, 1994), the outcomes of laboratory experiments and astronomical observations may be
interpreted in terms of a unique, qualitatively consistent
description of the process (Paolicchi et al., 1996; Doressoundiram et al., 1997; Zappalà et al., 1996; Cellino et al.,
1999; Tanga et al., 1999). The basic features of this description (discussed below) provide a sort of a “minimal standard
model,” useful, at least as a logical framework, for the topics
discussed here.
Nevertheless, some recent indications from theoretical
studies cast serious doubt on the validity of the standard
model in the size range of asteroids. Several numerical computations have been performed, using hydrodynamical
codes adapted to deal with processes typical of solid bodies,
such as plastic processes, creation and growth of fractures,
and breakup (see Asphaug et al., 2002). These simulations
have proved accurate enough to reproduce many of the
results of laboratory experiments (Melosh et al., 1992; Ryan
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Asteroids III
and Melosh, 1998). The extension of the same models to
cases involving astronomical systems leads to controversial
results only partially consistent with observations (Benz and
Asphaug, 1999; Love and Ahrens, 1996). The possibility of
reconciling the results of “hydrocodes” and the whole body
of observational data [in particular those concerning families, as discussed by Pisani et al. (1999)] is one of the main
challenges facing asteroidal science. Perhaps a different
interpretation of the data obtained from astronomical observation has to be given. For example, families could be outcomes of peculiar collisional processes. This might be seen
in the preliminary computations by Michel et al. (2001),
which reproduce some families as resulting from impacts
with high specific energy, moderate relative velocity, and
high transfer of angular momentum. In this case, most of
the observable features of asteroids should be due to the
post-event gravitational reaccumulation, rather than properties of individual fragments, and the physical meaning of
the “minimal standard model” would be seriously limited.
Alternatively, perhaps the existing hydrocodes are not yet
sophisticated enough to reproduce impact processes involving bodies endowed with a complex internal structure, such
as the asteroids certainly are, with that structure itself produced by collisional history. That is, the result of a collision
could depend on several unknown parameters: internal texture of target, shape, impact point and direction, etc. Some
numerical simulations (Asphaug et al., 1998) give a qualitative indication in this sense. In this latter case, the “minimal
standard model” should be a tool allowing a qualitative and
empirical — but general — description of outcomes. Here
we decided to keep to the standard model for a qualitative
description of the processes, guessing that several features
will remain valid even if, in the future, a consistent paradigm
and a related general consensus on it becomes available.
2. “STANDARD MODEL” OF COLLISIONS
The basic features of the standard model of collisions,
derived from the analysis of laboratory experiments and of
astronomical observations (mainly involving dynamical families) are as follows.
1. The largest remnant(s) of a catastrophic process originate far from the impact point. Several experiments (Giblin
et al., 1994b, 1998a) suggest the formation of an antipodal
cap as the largest remnant, surrounded by a few other large
fragments; according to other experimental findings (Fujiwara et al., 1989), the largest remnant is a central core. Experiments performed at low impact velocities generate a
conelike fragmentation, with a set of large fragments, again
formed not too close to the impact point, and elongated
approximately along the impact direction. The numerical
models support these ideas, even if they show, when applied
to astronomical sizes, that fragmentation might be rather
fine throughout the entire target. Since fragments close to
the impact point are ejected at higher velocities, the possible
gravitational reaccumulation should be more effective for
fragments formed at distances farther from the impact point.
2. Some of the original mass is dispersed into very fine
debris, which cannot be recovered in laboratory experiments
and cannot be observed at all as independent astronomical
bodies. Formation of debris is the rule in the region around
the impact point, but small fragments can come out also
from different regions of the target, even from the region
eventually dominated by large fragments: These last, in
fact, allow only an approximate reconstruction of the parent
body, as if it were a jigsaw puzzle. Some interstitial (missing) mass is usually lost from between the large fragments.
3. The fragments have different shapes. If we define the
shape of an irregular fragment in terms of an equivalent
ellipsoid [for a discussion see Verlicchi et al. (1994); La
Spina and Paolicchi (1996)], the shape distributions from
experiments usually peak at ratios among the c:b:a axes,
close to 0.5:0.7:1 (Giblin et al., 1994a,b, 1998a). A similar
shape distribution can be found to hold for the asteroids
(Capaccioni et al., 1984). Nevertheless, this shape similarity
does not necessarily require that the asteroids are fragments
created in some impact-disruption process; in fact, their
shapes are also consistent with a rubble-pile model for
bodies with internal stresses (Holsapple, 2001) that originated directly from the collision or resulted from later
reaccumulation. Note that because of frictional and other
effects, it is not required for a rubble pile to assume an equilibrium shape, such as the Maclaurin or Jacobi ellipsoids,
typical of a “fluid” rotating body.
4. The fastest fragments are usually small (Martelli et
al., 1994). The largest remnant and the other big fragments
(if present) usually leave at moderate velocities. In general,
the ejection velocities depend strongly on the degree of
fragmentation. The correct scaling of velocities to processes
involving asteroids remains a major open problem, as thoroughly discussed in Zappalà et al. (2002).
5. Fragments always rotate. The initial total angular
momentum of the parent bodies, including both spin and
orbital contributions, is partially distributed to invisible
debris. Thus the angular momentum may not appear to be
“conserved” when we compare the rotational properties of
the final observable bodies with the initial conditions. Of
course, individual fragments may develop spins even when
the total angular momentum before and after the breakup
is zero, as shown by laboratory experiments in which the
initial impacting bodies are not rotating and the collision
is central. This “breakup spin” is obviously imparted in different directions (Giblin et al., 1994a,b) with a geometry
that can be easily understood if we assume that the kinematical properties of fragments may be explained in terms
of a “breakup velocity field” for which the ejection velocities are larger close to the impact point and decrease regularly from the impact to the antipodal direction. The gradient of the velocity field results in rotation of the fragments.
If one assumes also [as in the semiempirical model by
Paolicchi et al. (1996)] that the sizes of the fragments are
somehow connected to the gradient of the velocity field,
then the observed correlation between sizes and velocities,
and a similar one between sizes and spins (smaller bodies
Paolicchi et al.: Side Effects of Collisions
rotate faster), comes out naturally (Giblin et al., 1994b;
Martelli et al., 1994).
No physical mechanism exists such that the original
direction of the spin of a fragment should be correlated with
its principal axes. According to laboratory impact experiments, the axis of rotation is not usually close to the c axis,
corresponding to the maximum moment of inertia (Fujiwara
and Tsukamoto, 1981), even if the consequent tumbling is
not experimentally found as often as one might expect
(Giblin and Farinella, 1997). This behavior may be understood in terms of semiempirical modeling (A. Verlicchi, personal communication, 1998) and is consistent with results
from hydrocodes, applied to fragmentation of the kilometersized asteroid 4769 Castalia (Asphaug and Scheeres, 1999).
Fragments that are not initially rotating about their axes of
maximum moment of inertia will eventually evolve to that
state by internal dissipation of energy while conserving
angular momentum. Before that happens, they will tumble,
and experience time-varying internal stresses that may cause
delayed breakup into smaller fragments (Giblin et al., 1998b).
This secondary splitting could result in formation of bound
pairs, i.e., binary asteroids.
Several aspects of the standard model can be understood
with the aid of Fig. 1, based on the semiempirical model
by Paolicchi et al. (1996) and adapted from a similar figure
by D’Abramo et al. (1999). In the figure we represent the
original starting points of the fragments produced in a catastrophic process. The vertical line passes through the impact
100.0
Boundary of parent body
Escaped fragments
Reaccumulated fragments
Fast rotators (T< 2 h)
Binaries
z (km)
50.0
0.0
50.0
100.0
0.0
50.0
100.0
150.0
200.0
r (km)
Fig. 1. Representation of the original position within the parent
body (projected onto a semicircle, the vertical diameter corresponding to the one passing through the impact point, while the
abscissa represents the distance from this line) of fragments with
different fates (reaccumulation, formation of a binary, free escape,
free escape with a period smaller than 2 h). The empty region on
the top corresponds to the region very close to the impact point,
at which very small fragments are formed; these are not dealt with
in the model. Data adapted from D’Abramo et al. (1999).
519
point (at the top) and the center of the parent body, while
the abscissa represents the distance of a fragment from this
line. Thus the parent body (assumed to be spherical) is projected into a semicircle.
Different symbols indicate the fragments that are later
reaccumulated by mutual self-gravity (or give rise to binary
systems; see the discussion in section 5), according to the
results of an N-body integration (see Doressoundiram et al.,
1997), and those that definitely escape from the system
(some of them rapidly spinning). Note that usually the fragments that undergo reaccumulation are larger than those that
freely escape. For this reason the former ones are less
“dense” in the figure. The sizes of the various regions, from
which are derived reaccumulated bodies, escaping fragments,
and so on, depend on the model, i.e., the impact energy and
strength and size of the target. They may be larger or
smaller, but the qualitative pattern remains the same.
3.
ROTATIONAL PROPERTIES
A theoretical understanding of the rotational outcomes
of catastrophic collisions is far from complete, in spite of
recent valuable efforts (Love and Ahrens, 1997; Asphaug
and Scheeres, 1999). On the other hand, our ideas on rotational properties, derived from impact experiments, and
accepted by the “minimal standard model,” cannot be
readily incorporated into a general model of collisional
evolution of asteroids. In contrast to what happens in laboratory experiments, the rotational properties of the fragments are not always dominated by the “breakup spin.” The
contributions of the original spin angular momenta of target
and projectile and of the transferred orbital angular momentum are dominant for large and intermediate-sized bodies.
There is no satisfactory model of the collisional evolution of asteroids that includes spin properties. After Davis
et al.’s (1989) description of the state of the art, few attempts have been made to further refine the model (Farinella et al., 1992). The problem is difficult mainly because
several critical parameters are not known. Other noncollisional processes, such as radiative torques (Rubincam,
2000) or close encounters with other asteroids or planetary
bodies (Scheeres et al., 2000; Richardson et al., 1998), may
also play a role in the evolution of spin rates. The creation
of a robust theoretical scenario for the evolution of rotational properties of asteroids is a rather ambitious task facing future researchers.
Critical uncertainties include the scaling of catastrophic
processes to astronomical sizes, and the precise distribution
of angular momentum among the fragments (experiments
and empirical arguments suggest a faster rotation for smaller
bodies, but the steepness of the rotation vs. size relation is
not well known). Moreover, and primarily, we do not know
what fraction of the original angular momentum of the parent system (arising out from spins of both projectile and
target and from the orbital angular momentum) is imparted
to the macroscopic fragments. According to some evidence
from experiments, a significant fraction of this angular mo-
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mentum is dispersed by the cloud of debris, which is formed
close to the impact point; nevertheless, as previously mentioned, the astronomical case is different and has been only
preliminarily analyzed by means of numerical simulations
(Love and Ahrens, 1997; Asphaug and Scheeres, 1999). If
one assumes the simple conservation of angular momentum, unrealistically rapid spins would result for most asteroids, with the exception of the few largest bodies.
The observed rotational properties of asteroids are discussed in Pravec et al. (2002) (see also Fig. 2 below). Here
we only sketch those features whose interpretation involves
collisional processes.
1. For asteroids larger than ~40 km diameter, the shortest spin periods are about 4 h. The observed distribution is
consistent with a Maxwellian distribution (Binzel et al.,
1989), and the absence of shorter periods may be due to
the small number of bodies in this size range. However, this
limit also corresponds to the fastest rotation allowed for
weak, self-gravitating bodies in hydrostatic equilibrium, for
plausible densities of 2–3 g/cm3 (Weidenschilling, 1981).
Any addition of angular momentum results in slower rotation, by formation of a Jacobi ellipsoid or binary. Thus, the
observed limit is also consistent with predictions of collisional evolution models (Davis et al., 1989) that most large
asteroids have been shattered.
2. In the intermediate size range (~10–100 km), the
mean rotation period is slightly longer than for smaller or
larger bodies. This effect has been explained as the result
of “angular momentum drain” (Dobrovolskis and Burns,
1984; see also Davis et al., 1989), the preferential escape
of ejecta in the prograde direction following cratering impacts. A similar mechanism, with similar consequences, has
been shown to work for larger catastrophic collisions; this
has been called “angular momentum splash” (Cellino et al.,
1990).
3. Among very small asteroids, those less than a few
hundred meters in diameter (all NEAs due to observational
selection), a high proportion are fast rotators, with periods
of <2 h [the shortest known is only a few minutes (Pravec
and Harris, 2000)]. These cannot be bound by gravity, and
their interiors must be in tension; i.e., they are monolithic
bodies. Their internal stresses can be estimated to be ρr2ω2,
generally in the range ~105–106 dynes cm–2, which can be
sustained by unfractured meteoritic material. Their rapid
spins are probably acquired during breakup of parent bodies disrupted by catastrophic collisions, although radiative
torques may also be a factor.
4. Asteroids smaller than a few tens of kilometers but
larger than ~150 m (main-belt objects and most NEAs)
show a lower limit to the observed spin period of about
2.1 h. It seems unlikely that this limit corresponds to a tensile bursting strength of competent bodies; the stresses
would be only ~102–103 dynes cm–2. However, the observed
limit corresponds essentially to the surface orbit period for
a sphere of plausible density (2.5 g cm–3), i.e., the rate at
which loose material would not be bound to the surface by
gravity at the equator of a spherical body. It is likely that
bodies in this size range are not monolithic, but not so
completely shattered as to have equilibrium shapes; they
may consist of a core or a few large components covered
by a layer of regolith. Such a body may have a naturally
regulated rotational “speed limit” imposed by the loss of
material from its equatorial region.
An elementary and rough argument may be useful for
exploring this process and understand its potential consequences. Let us assume that a homogeneous spherical body
of radius r rotates at a rate ω0[= (4πGρ/3)1/2], for which
the equatorial centrifugal force exactly equals gravitational
attraction. The mean specific angular momentum of the
body is
L 0 = 2/5r2ω0 = (2/5)(GMr)1/2
(1)
0.01
100-m.y. damping timescale
1-b.y. damping timescale
4.5-b.y. damping timescale
Confirmed or suspected tumbling asteroids
Rotation Period (h)
0.1
1
10
100
1000
0.01
0.1
1
10
100
1000
Diameter (km)
Fig. 2. Spin rate plotted vs. diameter for all (~1000) asteroids
with high-quality spins as of March 31, 2001. Open circles identify
those asteroids interpreted to be tumbling. Lines of constant damping timescale td for wobble (equation (6)) are shown. Figure courtesy of Alan W. Harris, JPL.
Suppose that the body gains an increment of angular
momentum that tends to increase its spin rate. Some material at the equator is no longer bound to the surface and is
removed. It carries off the local specific angular momentum
at the equator, which is 5/2× the mean value. This loss of
mass must continue until either the equatorial rotation speed
decreases to the surface orbital velocity or the supply of
loose regolith material at the equator is depleted. If the body
is initially spherical, loss of material from the equatorial
region would also alter the mass distribution so that its
rotation is no longer about the axis of maximum moment
of inertia. The asteroid would tumble until dissipation produced a spin state about a new principal axis with larger
moment of inertia and a lower angular velocity.
In general, angular momentum would be added by an
impact that would also tend to remove mass from the target. More gradual spinup might also occur due to radiative
torques (Rubincam, 2000). Material shed by these processes
would accrete into a satellite. In the aftermath of a disrup-
Paolicchi et al.: Side Effects of Collisions
tion of a parent body, a rapidly rotating fragment may also
experience rotational bursting (Giblin et al., 1998b). Any
of these processes may produce a binary system. The angular momentum is divided between the spins of the components and the orbital term. The latter may be dominant,
especially for a tidally synchronized system with a mass
ratio not too close to unity. The orbital specific angular
momentum is expressed in terms of the separation R by
Lf = x(1 – x)(GMR)1/2
(2)
where x is the fraction of the total mass in the smaller component and M the total mass. The final orbital frequency,
which is equal to the rotation rate of tidally synchronized
components, is given by
Ω = (GM/R3)1/2
(3)
By equating L0 and Lf, one obtains the final value of Ω
 5x(1 – x)  3
Ωf = ω0 

2


(4)
We see that Ωf < ω0 for all values of x.
5. A few very slow rotators are observed, among bodies
not too large. These rotations cannot be original: It is almost impossible to obtain a quasizero rotational rate as the
result of several processes transferring angular momentum
in a stochastic way (randomly oriented vectors behave as
in a random walk process; it is difficult to go back close to
the origin). The above-discussed fission processes may have
something to do with their origin [as qualitatively suggested
by Farinella et al. (1982), and recently pointed out by A. W.
Harris at the Asteroids 2001 Conference], even if, according to the discussion of Pravec and Harris (2000), it is not
easy to explain in this way the slow rotation of asteroids
such as 253 Mathilde (Mottola et al., 1995; see also Cuk
and Burns, 2001).
4.
TUMBLING ASTEROIDS
Almost all asteroids with measured lightcurves are in
pure spin; that is, they rotate at a constant rate around a
direction fixed in space and in the body’s reference frame
(Pravec et al., 2002). This condition requires that the body’s
angular momentum L and angular velocity ω (each measured relative to an inertial reference system at the body’s
center of mass) are parallel to one another. This will only
occur if L and ω lie along one of three principal inertia axes
of the body, whether it be the maximum, minimum, or intermediate axis. In the most general rotation state L and ω
are not aligned with one another or with the body’s principal axes. Then we will say that the body “tumbles” or
“wobbles” as ω moves. In the case of axially symmetric
bodies, various authors describe this motion as Chandler
wobble or free precession or nutation.
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When an asteroid is in pure rotation about its ith axis,
its rotational kinetic energy is T = Iiω 2i /2, or, correspondingly, T = L2/2Ii, where Ii is the principal moment of inertia about the ith axis. As long as only internal forces act, L
is constant regardless of how the rotation state evolves. Thus
the body’s kinetic energy is an absolute minimum (vs. maximum) for rotation about the principal axis of maximum (vs.
minimum) inertia. Hence the observed pure spin of typical
asteroids can be most simply understood as indicating that
those objects have damped down into their minimum rotational energy well. Alternatively, one may think of tumbling
bodies as being in a “heated” or excited energy state. This
minimum-energy condition is remarkable since asteroids
continually suffer mutual collisions that usually misalign
the body’s principal axes from L’s direction.
Considering the mutual collision between a small (projectile) asteroid with a much larger (target) asteroid, angular
momentum is transferred from the orbital motion of the projectile to the spin of the target. Moreover, impacts often
cause some mass and angular momentum to escape the target. Each of these effects has the capacity to knock L out
of the parallelism it has when the asteroid rotates purely
about one of the body’s principal axes, here presumably the
axis of maximum inertia. Since this bombardment is surely
occurring, but apparently is not effective in producing a
“heated” rotation state, some process must be draining rotational energy from the asteroid just as fast as collisions
can impart it. To give a feel for the ease of misalignment,
we mention that, for typical collisions between typical asteroids, a “bullet” of relative mass ~10 –5R, where R is the
asteroid target’s radius in kilometers, carries roughly the
same angular momentum as the target originally possessed
in its spin. We caution, however, that some fraction, perhaps a large fraction, of the angular momentum will be removed in the collisional ejecta.
Processes other than collisions may also cause tumbling.
When an asteroid passes closely by a planet, that planet will
gravitationally tug on the asteroid’s irregular shape (Richardson et al., 1998; Black et al., 1999). Since flyby times
are usually a few hours, they are comparable to spin periods,
meaning that asteroids present roughly fixed orientations
during flybys so that the transmitted torque has a constant
sign. Similar physics allows impact ejecta escaping an asteroid to exert gravitational torques on that body, and to significantly alter its spin state (Scheeres et al., 2000). Radiation
forces, whether the direct solar flux (Paddack, 1969) or the
(Yarkovsky) recoil from reemitted thermal radiation (Rubincam, 2000), may generate torques that induce nutation.
Mass redistribution — for example, produced as craters are
gouged out or as ice sublimates — will usually reorient the
body’s principal axes and will commonly produce wobble
(Peale and Lissauer, 1989).
Laboratory experiments (e.g., Giblin et al., 1998a), albeit at relatively tiny scales, show how impacts generate
wobble as mentioned above. Moreover, Giblin and Farinella (1997) concluded that some tens of percent, as determined both from lab experiments and estimated in a simple
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Asteroids III
TABLE 1.
Asteroid
253 Mathilde
288 Glauke
1689 Floris-Jan
3288 Seleucus
3691 Bede
4179 Toutatis
4486 Mithra
13651 (1997BR)
1999 GU3
1999 JM8
2000 WL107
Tumbling asteroids.
Class*
a (AU)
e
D (km)
2π/ω (h)†
MB
MB
MB
A
MX
A
A
A
A
A
A
2.64
2.75
2.45
2.03
1.77
2.52
2.20
1.34
2.08
2.72
2.90
0.27
0.21
0.21
0.46
0.29
0.63
0.66
0.31
0.51
0.65
0.65
58
32
14.3
1.9
3.3
2.7
2.4
0.8
0.3
2.9
0.037
417
1200
145
75
227
130
days
33.6
216
137
0.322
References‡
Mottola et al. (1995)
Harris et al. (1999); Ostro et al. (2001)
Harris (1994); Pych (1999)
Harris (1994); Harris et al. (1999)
Pravec et al. (1998)
Spencer et al. (1995)
Ostro et al. (2000)
Pravec et al. (1998)
Pravec et al. (2000)
P. Pravec, personal communication (2001)§
P. Pravec, personal communication (2001)§
*MB = main belt; MX = Mars-crosser; A = Apollo-Amor.
† Rotation periods are often poorly determined because of the complex lightcurve.
‡ Exhaustive references are not given, but rather just enough to allow an introduction to the data.
§ Unpublished lightcurve data (available on line at http://sunkl.asu.cas.cz/~ppravec/neo.html).
model, will display visible tumbling initially. In a numerical
simulation of breakup, Asphaug and Scheeres (1999) notice
that all disrupted fragments leave in a state of complex rotation, with widely varying spin rates and very little memory
of the parental target’s spin.
The rotational states of real asteroids can be ascertained
from an analysis of their brightnesses, usually observed at
unevenly spaced times (Burns and Tedesco, 1979). Fourier
techniques are used to produce a lightcurve and extract the
rotation period from these brightness measurements. Lightcurves that exhibit two periods indicate a precessing object
(e.g., Kaasalainen, 2001) or perhaps an eclipsing pair (e.g.,
Pravec and Hahn, 1997), while complex (nonrepeatable)
lightcurves suggest tumbling. Radar returns (e.g., Ostro,
1993, Hudson et al., 1997) may more directly indicate complex rotations and shapes.
The rotation rates for all objects with clear determinations are plotted against size in Fig. 2, similar to diagrams
in Harris (1994) and Pravec and Harris (2000). Of the
nearly 1000 asteroids with well-resolved rotation periods,
just 11 are known to wobble, with 4179 Toutatis the first
asteroid to be identified as a complex rotator (Spencer et
al., 1995). Since that time, an increasing number of tumblers (Harris, 1994; Pravec et al., 1998, 2000; A. W. Harris,
personal communication, 2001) have been discovered (see
Table 1).
Most of the tumblers known today are small and/or
slowly rotating, as seen in the lower left corner of Fig. 2.
A sizable fraction of them (7 out of the 11) are Earth-crossing objects and another is a Mars-crossing object, but it is
uncertain whether this result occurs because most small
objects with measured lightcurves are predominately Earth
crossers or because planet-crossing events themselves may
engender the wobble (Black et al., 1999). Among the very
fastest-spinning asteroids, only 2000 WL107, a 20-m-radius
NEA turning in ~20 min, is tumbling (P. Pravec, personal
communication concerning unpublished observations by
himself, C. Hergenrother, and S. Mottola, 2001).
For an oblate body with moments of inertia (C, A, A), the
wobble timescale is τω = (P/∆)(C/A), where P is the spin
period and ∆ = (C – A)/C (Burns and Tedesco, 1979). If
such a motion can be detected, the relative difference in the
moments of inertia is determined, and this constrains density
inhomogeneities in the asteroid’s interior. Unfortunately, for
the more likely case when the body’s inertia-ellipsoid is
triaxial, the dynamical motion becomes much more complex and often indecipherable (cf. Kaasalainen, 2001).
The mechanism that dissipates rotational energy, and
thereby damps wobble, can be simply stated, but it is much
more difficult to compute in a general circumstance. Internal friction (or anelasticity) occurs in all real materials: Due
to processes occuring at the atomic or grain level, energy is
lost whenever materials undergo cyclic stress-strain (recall
the warmth produced when a paper clip is rapidly flexed).
Each element of a tumbling body is subject to variable stress
and strain as that element’s acceleration changes due to the
rotation axis moving through the body. Thus, energy is lost,
and the only available source to supply it is the rotational
energy. Of course, once the body moves into pure spin, the
internal stresses no longer vary and energy loss ceases.
One can estimate the timescale for damping from dimensional analysis alone. It is
τd = K1µQ/(ρR2ω3)
(5)
where µ, Q, ρ, and ω are the object’s rigidity, anelasticity
(or quality) factor, mass density, and rotation rate; K1 is a
nondimensional scaling coefficient. Since Q is defined to
be E/dE, where dE is the energy lost over a cycle with E
the maximum stored strain energy, it is a nondimensional
quantity; thus it does not arise from the dimensional analysis mentioned above and should be strictly incorporated in
Paolicchi et al.: Side Effects of Collisions
K1. The functional dependence given in equation (5) has
been derived by numerous researchers in many contexts
using several different approaches. For example, to argue
why so few asteroids wobble and to estimate the likely
nutation angles for mutually colliding minor planets, Burns
and Safronov (1973) computed the strain energy stored in
bending stresses and the polar bulge. Yoder and Ward
(1979), in investigating Venus’s spin state, approached the
problem somewhat similarly. Peale (1973) derived nutational damping times for planets and satellites applying a
Hamiltonian formulation. Purcell (1979), to study the alignment of interstellar dust, attacked the problem with classical rotational dynamics and conventional elasticity theory.
Lambeck (1980) calculated the alignment time for the
Earth’s Chandler wobble by considering the Earth as a Maxwellian material and using the Liouville equations of rotational dynamics.
A recent series of papers (e.g., Efroimsky and Lazarin,
2000; Efroimsky, 2001; hereafter referred to as EL) has
sharply criticized these earlier works as missing important
dynamics, failing to satisfy boundary conditions perfectly
and using unphysical choices for the parameters in equation (5). Nevertheless, all the previous solutions come to
similar conclusions and have the same functional dependence, differing only in K1’s magnitude. Using an approach
similar to Purcell’s, EL derive a comparable expression but
maintain that the actual damping time is 2–6 orders of
magnitude shorter than earlier researchers had estimated.
Burns and Safronov (1973) assert that K1 will depend on
the asteroid’s shape; EL concur and also point out that τd
may change depending on the magnitude of nutation. Because of its relevance to many dynamics problems for small
solar system bodies, this problem is being reconsidered by
several groups. Most maintain that EL’s claims are overstated; accordingly, for the purposes of further discussion,
we will select the numerical values chosen by Harris (1994).
Sharma et al. (2001) compare damping times calculated by
various authors and discuss the reasons for the slight differences in expressions, before deriving their own result.
The damping time may be alternatively expressed as
τd = K2P3/D2
(6)
where τd is given in billions of years, P is the rotation period in hours, D is the asteroid’s diameter in kilometers and
K2 = (1/17)3 to within an order of magnitude (Harris, 1994,
Pravec et al., 2002). To obtain equation (6) from (5), Harris
(1994) chose µQ = 5 × 1012 and ρ = 2.5, each in cgs units;
Burns and Safronov (1973) had assumed a much stronger
and harder material.
Various damping times, corresponding to different (P, D)
combinations in equation (6), are shown on Fig. 2. We see
that, according to this criterion, almost all asteroids with
well-known spin periods damp in times much less than the
solar system’s age. In contrast, all the identified tumblers —
except for the smallest, fastest one — have damping times
that are comparable to or longer than the age of the solar
523
system. Hence, by this standard, regardless of when these
wobblers last received a significant angular momentum
kick, they should be still wobbling, just as they were found
to be. Note also that, whereas some apparently nontumbling
objects lie below the 4.5-Ga line, in many cases their rotations are not well enough determined as yet to rule out tumbling (Harris, 1994).
Of all the wobblers, only the fast-spinning 2000 WL107
has a damping time much less than the solar system’s age.
In fact, on Fig. 2 its damping time is ~5 m.y., a time comparable to, but shorter than, the estimated collisional age
[5–20 m.y. (Davis et al., 2002)] of such a minor planet in the
main belt where this object presumably originated. Thus it
is no surprise that this NEA is wobbling. On the other hand,
if the much shorter scale of EL were chosen, the damping
time would be a mere 5 × 10 4 yr or so, making it highly
improbable to discover this object tumbling.
5. BINARIES
There is now ample observational evidence for the existence of binary asteroids. A full discussion of observations
and possible mechanisms for their origins is given in Merline et al. (2002). Here we mention only those aspects of
binary origins relevant to disruptive collisions.
Hartmann (1979) suggests that, in the collisional disruption of a parent body, individual fragments might have sufficiently low relative velocities to be gravitationally bound;
if their relative angular momentum were large enough, they
would remain in orbit about each other. Weidenschilling et
al. (1989) point out that due to purely geometrical effects,
adjacent fragments moving fast enough to escape from the
parent body would also tend to have relative velocities large
enough to escape from each other. That argument depends
on the assumption of a smoothly varying radial velocity
field during the disruption, however. Durda (1996) performs
numerical simulations of disruptive events and shows that
a small dispersion of radial velocities increases the fraction
of binaries among the fragments (as well as reaccumulation
of pairs in contact). Even with no imposed velocity dispersion, some (<1%) bound pairs were produced, apparently
by nonradial motions and nonuniformities due to jostling
of fragments during the early stage of disruption. Similar
results, based on the implementation of a N-body algorithm
into the semiempirical model by Paolicchi et al. (1996), are
obtained by Doressoundiram et al. (1997) and Paolicchi et
al. (1999). According to them, binary systems are formed
within the overall process of partial reaccumulation of fragments, due to self-gravity. As discussed by D’Abramo et al.
(1999) (see Fig. 1), the reaccumulation involves a localized
region from which the fragments originated; the size of this
region depends on the amount of total reaccumulation. The
binary systems are formed close to the boundaries of this
region. According to these computations, binary systems are
not very numerous.
The limited number of binaries in both models is to some
degree due to the limited range of sizes in the simulations;
524
Asteroids III
if more numerous, smaller fragments were considered, there
would be a higher probability of bound pairs having very
unequal masses. A system such as Ida-Dactyl fits properly
within this scenario.
On the other hand, if the ejection velocities of the fragments were larger (as suggested by different considerations;
see the discussion in Zappalà et al., 2002), little self-gravity-driven reaccumulation would take place, and the formation of binaries through the above-discussed process might
be hindered [note that a possible channel for the formation
of binaries may be through the formation of the so-called
“jets” observed in some experiments (Martelli et al., 1993)].
Nevertheless, in the case of high ejection velocities, many
bodies rotating faster than 2 h are presumably created. Since
such bodies are absent, they (if formed) are subject to fission immediately after formation or later, triggered perhaps
by a further minor collisional event. Thus a widespread formation of binary systems follows. In this case, the mass
ratio between the components should depend on unknown
initial conditions or parameters with no preferential value
(Paolicchi et al., 1999).
The potential binary systems that develop in a catastrophic collision are represented in Fig. 1 (adapted from
D’Abramo et al., 1999). Among the fragments reaccumulated by self-gravitation, a few give rise to binary systems.
Moreover, numerous fragments that freely escape rotate
faster than 2 h and are thus subject to a later bursting fission,
eventually ending up as binary systems. The size of the
various regions (reaccumulation, escape, fast rotation) depends on the model; nevertheless, when we have a small
amount of reaccumulation (and thus few “reaccumulated”
binaries), we usually have many fast rotators (and thus
many “bursting” binaries) and vice versa.
properties is usually possible only for a relatively short time
after the causative event.
4. As a result of catastrophic disruption, binary systems
may be formed; a few different channels may lead to this
outcome. The uncertainties of the physics affect the efficiency of different channels and thus the expected detections. Nevertheless, several binaries are probably always
formed among the numerous fragments, although their relative proportion is low.
A more quantitative analysis of the above-presented features (and the related development of a general model for
the collisional evolution of asteroids) will require understanding a few critical points. Among them the most relevant
open problem is how large bodies react to energetic impacts,
i.e., how strong or weak they are and how large are the ejection velocities of the fragments. Another question is of paramount importance for the problems we have discussed:
How effective is the transfer of angular momentum during
the impact, or, in other terms, what percentage of angular
momentum is carried away unnoticed with the debris?
The above-discussed properties (and the related open
problems) are relevant also for other aspects of asteroidal
science. For example, the rotational properties influence the
diurnal Yarkovsky effect; thus the collisional evolution of
rotational properties plays to strengthen/weaken this effect,
which in turn is relevant for the overall evolution of the belt
and for the observed properties of asteroid families.
Acknowledgments. We are grateful to A. Cellino, A. Dell’Oro,
and Ishan Sharma for useful discussions. We are also grateful to
the referee Petr Pravec. P.P. was supported by a special grant from
the University of Pisa. J.A.B. and S.J.W. received funding from
NASA’s PG&G program.
6. OPEN PROBLEMS AND CONCLUSIONS
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