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Zappalà et al.: Properties of Asteroid Families 619
619
Physical and Dynamical Properties of Asteroid Families
V. Zappalà, A. Cellino, A. Dell’OroIstituto Nazionale di
Astrofisica,
Osservatorio Astronomico di Torino
P. PaolicchiUniversity of Pisa
The availability of a number of statistically reliable asteroid
families and the independentconfirmation of their likely
collisional origin from dedicated spectroscopic campaigns has been
amajor breakthrough, making it possible to develop detailed studies
of the physical properties ofthese groupings. Having been produced
in energetic collisional events, families are an invaluablesource
of information on the physics governing these phenomena. In
particular, they provideinformation about the size distribution of
the fragments, and on the overall properties of theoriginal
ejection velocity fields. Important results have been obtained
during the last 10 years onthese subjects, with important
implications for the general understanding of the collisional
historyof the asteroid main belt, and the origin of near-Earth
asteroids. Some important problems havebeen raised from these
studies and are currently debated. In particular, it has been
difficult so farto reconcile the inferred properties of
family-forming events with current understanding of thephysics of
catastrophic collisional breakup. Moreover, the contribution of
families to the overallasteroid inventory, mainly at small sizes,
is currently controversial. Recent investigations are alsoaimed at
understanding which kind of dynamical evolution might have affected
family memberssince the time of their formation. In addition to
potential consequences on the interpretation ofcurrent data, there
is some speculative possibility of obtaining some estimate of the
ages ofthese groupings. Physical characterization of families will
likely represent a prerequisite forfurther advancement in
understanding the properties and history of the asteroid
population.
1. INTRODUCTION
The improvements of the techniques of family identifi-cation,
leading to the establishment of a database of about20 statistically
reliable families recognized during the lastdecade (see Bendjoya
and Zappalà, 2002), and the subse-quent confirmation of the
cosmochemical self-consistencyof these groupings from dedicated
spectroscopic campaigns(see Cellino et al., 2002) has introduced
new exciting per-spectives for the physical studies of
asteroids.
Since the early times of the first family discoveries byHirayama
at the beginning of the past century, these group-ings have been
interpreted as the observable outcomes ofenergetic collisional
events, leading to complete disruptionof a number of original
parent bodies, and to dispersion ofthe collisional fragments. In
this respect, asteroid familiesare nothing but nice examples of the
collisional phenomenathat have marked the history of the solar
system, althoughgenerally at much smaller energy regimes with
respect tomajor events involving large planetary bodies (origin of
theEarth’s Moon, tilt of the spin axis of Uranus, etc.).
In the framework of asteroid studies, families are cru-cially
important in many respects. In general terms, theyrepresent a
constraint for any attempt at modeling the over-all collisional
history of the asteroid belt. In other words,the models should be
able to justify the number of currently
observed families, and to estimate the typical times neededto
progressively erode freshly formed families in such a wayto make
them no longer identifiable at later epochs. Themost recent
investigations of this particular aspect have beenpublished by
Marzari et al. (1999).
Another essential aspect concerns the information thatcan be
obtained about the physics of the breakup eventsfrom which families
originated. Understanding how aster-oids react to mutual collisions
typically characterized byrelative velocities around 5–6 km/s
(Farinella and Davis,1992; Bottke et al., 1994) is an essential
piece of informationfor any attempt at modeling their overall
history, as well astheir internal properties. Apart from the purely
theoreticalissues, understanding the internal structure of
asteroids is anecessary prerequisite to developing any reasonable
strategyof mitigation of the near-Earth asteroid (NEA) impact
hazard.
In the past, most of our understanding of the way aster-oids
behave when they are collisionally disrupted camefrom laboratory
experiments, based both on hypervelocityimpacts and explosive
charge techniques [for reviews of thissubject, see Fujiwara et al.
(1989) or Martelli et al. (1994)].The major source of uncertainty,
in this respect, was thescaling problem. In other words, it was not
clear how toextrapolate the results of laboratory tests involving
centi-meter-sized targets, to the astronomical situation of
kilo-meter-sized bodies, including relevant gravitational
effects
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620 Asteroids III
(see Holsapple et al., 2002). In this context, the availabil-ity
of a number of reliable families placed at the disposalof
theoreticians a number of breakup experiments that na-ture
performed for us, directly in a range of masses andenergies
absolutely beyond current and future laboratory ca-pabilities.
As a consequence, a great effort has been made in recentyears to
derive from the observed properties of family mem-bers clues about
the physics and the kinematics of the eventsfrom which families
originated. These studies have beenable to obtain relevant results,
whose interpretation is stillsomewhat debated, as we shall see in
the following sections.
We should still mention the problem that the presenceof random
interlopers represents for physical studies offamilies. The likely
numbers of objects sharing purely bychance the same orbital proper
elements of “true” familymembers have been predicted on the basis
of pure statis-tics by Migliorini et al. (1995) as a function of
size. At thesame time, spectroscopic observing campaigns have
beenable to identify in several cases a number of likely
inter-lopers, as described in Cellino et al. (2002). Interlopers
aremore “harmful” when they are relatively big, since they
canaffect the overall behavior of the family size distributions,and
the reconstruction of the original ejection velocity fieldsof the
fragments (see below). However, we can now beconfident that nearly
all the biggest interlopers have beenidentified in most cases.
Thus, the results described in thefollowing section are not
expected to be appreciably affectedby the presence of these
undesired guests.
2. AVAILABLE DATA AND COMMONINTERPRETATIONS
Physical studies of dynamical families are based on anal-yses of
essentially three pieces of information: (1) the co-ordinates of
the family members in the space of the properelements a', e', and
i' (proper semimajor axis, eccentricity,and inclination
respectively); (2) the size distributions offamily members; and (3)
spectroscopic and spectrophoto-metric properties of the
members.
In addition to the items listed above, we should mentionthat
some photometric data are also available. The majorproblem for this
kind of investigation is certainly the faint-ness of most family
members, apart from the generally fewlargest members of the
different groupings. A few light-curves are not sufficient for any
relevant statistical inference.For this reason, no definitive
indications have yet come fromphotometry. The only exception is a
tentative clue of a likelyyoung age of the Koronis family, based on
the observeddistribution of lightcurve periods and amplitudes
(Binzel,1988). This investigation found also evidence that in the
caseof the Eos family the distribution of spin periods seems
tosuggest collisional relaxation and a likely old age.
Another important source of information has come fromdirect in
situ explorations by means of space probes. Inparticular, the
Ida-Dactyl binary system observed by theGalileo probe is a nominal
member of the Koronis family.
Moreover, 951 Gaspra, also observed by Galileo, might bea member
of the Flora family, although it has not been listedas such in the
most recent family classification by Zappalàet al. (1995). It is
known that for the Flora family the nomi-nal membership has been
likely underestimated due to thesevere statistical criteria used
for identification (Miglioriniet al., 1995). The important results
of the Galileo observa-tions of these two objects, including
analyses of the sizedistributions of impact craters on their
surfaces (conse-quences of their collisional histories), are
extensively dis-cussed in Chapman (2002) and Sullivan et al.
(2002).
The family spectroscopic data (item 3 in the above list)deserve
a separate analysis, and are the subject of Cellinoet al. (2002).
In what follows, we then focus our attention onthe properties
mentioned in items 1 and 2 of the above list.
2.1. Ejection Velocity Fields
The structure of the families in the space of proper ele-ments
are used to derive information on the ejection veloci-ties of the
fragments in family-forming events. This is pos-sible because we
can interpret the differences in orbitalelements in terms of
differences in ejection velocity fromthe original parent body. The
conversion from velocities toorbital elements or vice versa is
given by the well-knownGaussian formulas, which can be written as
follows, underthe assumption (verified for families) that the
ejection ve-locities are much smaller than the orbital velocity of
theparent body
δ
δ
a/ana e
e f V e f V
ee
na
e f
T R( )
( cos ) ( sin ) ]]
( ) cos
=−
+ +
= − +
2
11
1 2
2 1/2
2 +++
+
= − ++
e f
e fV f V
Ie
na
f
T Rcos
cos(sin )
( ) cos ( )
2
2 1/2
1/2
1
1
1δ ω
ee fVWcos
(1)
where na is the mean orbital velocity, and VT, VR, and VWare the
components of the ejection velocity vector along thedirection of
the motion, radial, and normal to the orbitalplane respectively.
The parameters f and ω are the trueanomaly and the argument of
perihelion of the parent bodyat the epoch of its disruption. These
angles are a prioriunknown, and this fact has long prevented any
attempt atreconstructing the original ejection velocity fields from
thepresently observed locations of the members in the
properelements space.
By means of extensive numerical simulations, however,Zappalà et
al. (1996) showed that very reasonable recon-structions of the
velocity fields in family-forming events arepossible when the real
velocity fields were not completelyrandom, but were characterized
by a wide variety of possiblefield structures, including spherical,
ellipsoidal, conical(cratering), and more complicated combinations
of theabove fields, in which some symmetry property is (even
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Zappalà et al.: Properties of Asteroid Families 621
weakly) satisfied. The basic idea was to use some adimen-sional
parameters, given by some combinations of the distri-butions of the
velocity components of the family members,to estimate the most
likely values of the unknown f and ωangles. Of course, in practical
cases the reconstruction canbe attempted only when a sufficiently
large number of mem-bers is available, because of statistical
reliability. In manysituations a reconstruction of the fields was
found to be pos-sible, as in the cases of Vesta, Dora, Merxia
(Zappalà et al.,1996), and Maria (Zappalà et al., 1997). The
reconstructedfields turn out to be generally symmetric and similar,
interms of general structure, to what is observed in
laboratoryexperiments involving much smaller targets. These
resultscould be obtained in spite of the difficulty related to the
factthat families are identified in the space of the orbital
properelements, whereas the Gaussian formulas describe the
ex-pected behavior in the osculating elements space. Accord-ing to
Bendjoya et al. (1993) the overall structure of theejection
velocity field is conserved in the transformationfrom osculating to
proper elements. The major effect of thetransformation was found to
be a parallel displacement ofthe whole set of family members,
leading to conservationof the overall velocity field structure. Of
course, one canexpect that over long timescales the stability of
the properelements can change, making a reconstruction of the
veloc-ity fields harder to obtain. On the other hand, one can
alsoexpect that over long timescales families tend in any caseto
disappear due to progressive collisional erosion (Marzariet al.,
1995, 1999).
An indication that aging of the proper elements is likelypresent
in several cases is given by the fact that the result-ing values of
the computed f and ω angles turn out not tobe homogeneously
distributed within their natural range ofvariation. This is related
to the well-known fact that mostfamilies appear to be more
elongated in proper eccentricityand inclination with respect to
semimajor axis (Zappalà etal., 1984).
Another problem is that the computed ejection velocitiesof
family members resulting from their differences in or-bital proper
elements turn out to be considerably large (seesection 3.3).
2.2. Size Distributions and Size-VelocityRelationship
The study of the fragment size distributions is anothermajor
field of investigation. Asteroid sizes are generally notdirectly
measured, but they are derived from knowledge ofabsolute magnitude
and albedo. Families have been foundto be characterized by very
homogeneous albedo distribu-tions according to the available
radiometric data, and thisfact has also been used to derive an
estimate of the limit ofcompleteness of the current asteroid
inventory in terms ofapparent magnitude (Zappalà and Cellino,
1996).
The size frequencies of all the major families have longbeen
known to exhibit a power-law trend, characterized bysteep slopes,
much larger than the typical slopes of samples
of nonfamily asteroids in different regions of the main
belt(Cellino et al., 1991; Klaçka, 1992).
For a long time such behavior has been essentially
mis-understood. General models based on an assumed power-law trend
coupled with simple requirements of conservationof the total mass
of the parent body led Petit and Farinella(1993) to make
predictions that were able to fit satisfacto-rily only a minor
fraction of the whole set of known familysize distributions. In
most cases, the slopes of the observedpower laws were much steeper
than the predictions. A pos-sible explanation was developed by
Tanga et al. (1999),based on a very simple concept first proposed
by Paolicchiet al. (1996). The idea was that the fragmentation
processesmust satisfy other constraints in addition to conservation
ofthe parent body’s mass. In particular, the fragments areformed in
a finite volume, and cannot mutually overlap. Asimple model taking
into account these geometric con-straints was developed by Tanga et
al. (1999), and wasfound to give excellent fits of the observed
size distribu-tions of a large number of families (see Fig. 1). As
a by-product of this technique, some estimates of the
originalparent body size and of the actual ratio mLR/mPB betweenthe
mass of the largest remnant and that of the parent bodywere also
obtained. More recently, the likely role of geo-metric constraints
in fragmentation phenomena has beenalso recognized by Campo Bagatin
and Petit (2001), al-though the authors did not find any real proof
that themodeled geometric effects are in any way related to
theactual physics of collisions.
These above-mentioned results have important and de-bated
implications concerning the overall inventory of theasteroid
population, since they suggest that family membersmight dominate
the asteroid population at small sizes, atleast if we can believe
in the reliability of extrapolations ofthe observed size
frequencies to small sizes (around 1 km).This will be discussed
more thoroughly in the followingsection.
Another recent result that must be mentioned concernsthe
relationship found between size and ejection velocityof the
fragments in family-forming events. This problem hasbeen analyzed
by Cellino et al. (1999). The result was thata relation exists
between the sizes of the fragments and theirmaximum possible
ejection velocities. In particular, fami-lies generally fit the
predictions of a model that assumesthat the maximum amount of
kinetic energy achievable byfragments in family-forming events is a
constant. As a con-sequence, the maximum possible ejection velocity
as a func-tion of size follows a power-law trend, and is
representedby a straight line in a log-log plot of size vs.
velocity, asshown in Fig. 2. In particular, the following relation
holdsunder the assumptions mentioned above, and is found to
fitsatisfactorily the observational data for most families
log log 'd
DV K= − −2
3
where d and D are the diameters of the ith fragment and of
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622 Asteroids III
the parent body respectively, and V is the maximum pos-sible
value for the ejection velocity of the fragment of size d.The K'
parameter is given by
Af E/M)KElog (= −1
32'K
where E/M is the specific energy of the impact (defined asthe
ratio between the kinetic energy of the projectile and themass of
the impacted body), fKE is the so-called anelasticityparameter,
defined as the fraction of E that is converted into
kinetic energy of the fragments, and A is an a priori un-known
parameter, describing the maximum fraction of thetotal kinetic
energy of the set of fragments that can be de-livered to fragments
of any size.
Apart from the most technical issues, an important re-sult of
the analysis has been to recognize that not only afixed slope
around (–2/3) is found to actually fit the behav-ior of most
families, but also the value of the intercept K'turns out to be a
linear function of log mLR/M (mLR beingthe mass of the largest
family member, and M the mass ofthe parent body). This prediction
is respected by most fami-
Gefion Koronis
Maria Themis
00
1
2
3
4
–1 1
Log(D) (km)
2 3
10.1 10 100 1000
Log[
N(
D)]
00
1
2
3
4
–1 1
Log(D) (km)
2 3
10.1 10 100 1000
Log[
N(
D)]
00
1
2
3
4
–1 1
Log(D) (km)
2 3
10.1 10 100 1000
Log[
N(
D)]
00
1
2
3
4
–1 1
Log(D) (km)
2 3
10.1 10 100 1000
Log[
N(
D)]
Fig. 1. The size distributions of four prominent families
identified in the asteroid main belt are shown, with corresponding
error barsdue to the uncertainties in the sizes of the single
family members. The lines represent fits of the size distributions
obtained by Cellinoet al. (1999) using their model. Three different
curves (practically identical) are shown for each family,
corresponding to differentchoices of some random seeds determining
the results of different simulations. The simulations show a very
good agreement with thedata. Also shown are the limits of
completeness for each family, due to the fact that the available
inventory is not complete at smallsizes due to the corresponding
faintness of the smallest objects.
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Zappalà et al.: Properties of Asteroid Families 623
lies for which mLR/M < 0.8. This fact, if fully confirmed,has
some important, although not yet fully explored, im-plications for
our understanding of the physics of cata-strophic breakup
processes. In particular, it would suggestthat the currently
observed ejection velocity fields weredetermined mostly by the
impact energies (directly relatedto the mLR/mPB ratio) and that the
original size-velocitytrend has been scarcely affected by
subsequent dynamicaland/or physical evolution, being still visible
today. Such aconclusion may be still premature, however, and what
canbe said at this stage is only that the Cellino et al.
(1999)size-velocity model seems to be in good agreement
withavailable data for a large number of known families, andalso
with some laboratory experiments.
3. IMPLICATIONS AND OPEN PROBLEMS
The observational facts mentioned above are
generallyself-consistent, and suggest some kind of coherent
scenario.In particular, it turns out that families were formed by
eventsthat produced ejection velocity fields having
“reasonable”structures (when we compare them with the outcomes
oflaboratory experiments), although the resulting velocitieswere
generally high. The fragmentation of the parent bod-ies produced
huge swarms of fragments, with size frequen-cies characterized by
steep power-law trends, still recog-nizable today. However, several
facts must be taken intoaccount before accepting literally all the
above conclusions.Moreover, the exact implications of the
above-mentioned
Fig. 2. Size-ejection velocity plots for nine of the most
prominent families in the asteroid main belt. The straight lines
shown in eachplot have a fixed slope equal to –2/3, and correspond
to a computed best-fit value, plus or minus its uncertainty. The
value K' of theintercept corresponding to the best-fit line are
also written in each plot. From Cellino et al. (1999).
log(vej) (km/s)
lg(d
/DP
B)
–1.2
–1
–0.8
–0.6
–1.2 –1 –0.8 –0.6
log(vej) (km/s)
lg(d
/DP
B)
–1.6
–1.4
–1
–1.2
–0.8
–1 –0.5 0
EosK' = 1.42
Flora
K' = 1.50
log(vej) (km/s)
lg(d
/DP
B)
–1
–1.2
–1.4
–0.8
–0.4
–0.6
–0.2
–1–1.5 –0.5 0
MariaK' = 1.30
log(vej) (km/s)
lg(d
/DP
B)
–1.4
–1
–1.2
–0.8
–0.6
–0.4
–1.5 –1 –0.5
log(vej) (km/s)
lg(d
/DP
B)
–1.2
–1
–0.6
–0.4
–0.8
–0.2
–1.5 –1 –0.5
ErigoneK' = 1.60
GefionK' = 1.32
log(vej) (km/s)
lg(d
/DP
B)
–1.2
–1.4
–1
–0.8
–0.8–1 –0.6 –0.4 –0.2
ThemisK' = 1.45
log(vej) (km/s)
lg(d
/DP
B)
–1.6
–1.4
–1.2
–1
–0.8 –0.6 –0.4 –0.2
log(vej) (km/s)
lg(d
/DP
B)
–1.2
–1
–0.6
–0.4
–0.8
–1.5 –1 –0.5
EunomiaK' = 1.65
KoronisK' = 1.32
log(vej) (km/s)
lg(d
/DP
B)
–2
–2.1
–1.7
–1.8
–1.9
–1.6
–0.6 –0.4 –0.2
VestaK' = 2.10
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624 Asteroids III
scenario must be adequately stressed and discussed. Wetouch here
on a number of very delicate points affectingour overall
understanding of the properties and history ofthe asteroid
population.
In what follows, we give (in no special order) a list ofopen
problems and general implications of the results com-ing from
analyses of the family data at our disposal.
3.1. Dynamical and Physical Aging of the Families
It is certain that, over time, asteroid families undergo
anevolution modifying their originally observable features.This
evolution is expected to affect both the physical prop-erties of
the members and their location in the proper ele-ments space.
In particular, the orbital proper elements a', e', and i'cannot
be assumed to be really constant over time (see alsoKnezevic et
al., 2002). They are expected to vary slowlyover timescales on the
order of 10–100 m.y. An estimationof the range of fluctuation of
the proper elements was givenby Milani and Knezevic (1992, 1994).
As a consequence,the inferred kinematical properties of families
(includinga reconstruction of their original ejection velocity
fields)should slowly vary as a function of the time elapsed
sincefamily formation. This does not prevent us from
identifyingtoday a number of around 20 “robust” families in the
spaceof proper elements, and to derive information on the origi-nal
collisional events from which they originated. However,it is
reasonable to expect that the apparent kinematical prop-erties of
families can be somewhat “aged” with respect tothe original
configurations at the epoch of their formation.
Not all the proper elements behave in the same way. Thesemimajor
axis seems to be the most stable parameter ac-cording to available
dynamical theories (see Knezevic et al.,2002). Dynamical aging of
the proper elements affectsmainly eccentricities and inclinations.
This may influencethe determination of the unknown f and ω angles
in theGaussian equations when the techniques of reconstructionof
the ejection velocity fields are applied. As shown by
thesimulations performed by Zappalà et al. (1996), this effectcan
produce a spurious clustering of the inferred values off around
90°. In turn, this can affect the derived ejectionvelocities, since
an enhancement of the dispersion of theeccentricities and the
inclinations produces a spurious in-crease of the radial and normal
ejection velocity compo-nents, leaving the transversal component
essentially un-changed (as can be easily seen from equation
(1)).
Another possible source of variation of the proper ele-ments is
the so-called Yarkovsky effect (Vokrouhlický andFarinella, 1998;
Farinella and Vokrouhlický, 1999; Vokrouh-lický, 1999; Bottke et
al., 2000; Bottke et al., 2002). Un-like the dynamical aging
processes mentioned above, theYarkovsky effect produces changes of
the orbital semima-jor axis, and is size dependent. Several
parameters, whosevalues are currently uncertain, determine the
effectivenessof the Yarkovsky effect, and the range of sizes for
whichit is really important. The effect depends on the thermal
properties of the surface, on the orientation of the spin
axis,and on the obliquity angle (the angle between the spin axisand
the normal to the orbital plane). In its “diurnal” vari-ant, the
Yarkovsky effect can cause the orbital semimajoraxis to drift
either inward or outward, depending on thesense of rotation of the
body. In addition, there is a “sea-sonal” variant, due to the
periodic preferential heating ofdifferent hemispheres during the
orbital motion. The neteffect of this variant is a systematic
decrease of the orbitalsemimajor axis.
An estimate of the expected variation of the semimajoraxis due
to the Yarkovsky effect has been recently computedby Spitale and
Greenberg (2001). The effectiveness of thiskind of nongravitational
force is stronger for smaller ob-jects, and should be negligible
for bodies larger than kilo-meter-sized (the exact value depending
on many poorlyknown parameters). Moreover, another kind of
relevantlong-term dynamical effect that is expected to affect
theorbital semimajor axes of all asteroids, including
familymembers, is that of mutual close encounters with the mas-sive
Ceres, Pallas, and Vesta objects (Carruba et al., 2000).
In summary, dynamical aging can increase the disper-sion of the
eccentricities and inclinations of family mem-bers, while the
Yarkovsky effect and close encounters canalso increase the
dispersion of the semimajor axes. Dynami-cal aging does not depend
on the size of the body, whereasthe Yarkovsky effect is certainly
negligible for objects largerthan, e.g., 10–20 km.
The interplay of dynamical aging and the Yarkovskyeffect can be
important, because during a Yarkovsky-drivendrift in semimajor axis
the objects can be trapped into someof the many dynamically
unstable regions crossing the aster-oid main belt (Nesvorný and
Morbidelli, 1998; Morbidelliand Nesvorný, 1999), with the effect of
being eventuallyremoved from the family. Of course, the timescales
of dy-namical and Yarkovsky-driven evolutions must be comparedwith
typical collisional lifetimes for bodies of comparablesizes. We
touch here on some key problems that will belikely the subject of
extensive analyses in the near future.
In fact, collisions independently produce a “physical”aging of
the families. Collisional lifetimes of main-beltasteroids are
size-dependent, and range between 107 and109 yr. The exact
estimates are model-dependent, since theyare derived from estimates
of both the impact strength ofthe objects and the number of
existing projectiles capableof disrupting an object of a given
size. For a 100-km aster-oid, the average collisional lifetime
should be on the orderof 109 yr. As a consequence, and according to
Marzari etal. (1995), the smallest family members tend to be
colli-sionally disrupted over shorter timescales with respect tothe
bigger members. As a consequence, collisions not onlymodify the
size distribution of families, but also producean erosion in the
space of the orbital proper elements, lead-ing to progressive
disappearance of the families as recog-nizable groupings (Marzari
et al., 1999).
Moreover, the possibility of secondary events
involvingfirst-generation family members must also be taken
into
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Zappalà et al.: Properties of Asteroid Families 625
account, since this may potentially produce very compli-cated
structures in the space of proper elements, as possi-bly in the
cases of the so-called “clans,” such as the Florafamily (Farinella
et al., 1992).
All the facts mentioned above must be seriously takeninto
account when interpreting the data at our disposal.Moreover, we
observe today families that likely have dif-ferent ages, and show
different stages of evolution. Under-standing the complicated
interplay of the different evolu-tionary tracks will certainly help
in understanding the prop-erties of families, and will possibly
help in determining theircurrent ages, with important implications
for the studies ofthe collisional evolution of the asteroid
belt.
3.2. Size Distributions and Asteroid Inventory
In many cases the slopes of the power-law distributionsfitting
the size frequencies of asteroid families turn out tobe very steep.
According to Tanga et al. (1999), the result-ing exponents of the
cumulative size distributions reachoften values beyond –3. In these
cases, it is easy to showthat integrations of the size
distributions would lead to in-finite reconstructed masses for the
corresponding parentbodies. For this reason, it is clear that the
size distributionsmust change slope somewhere at small sizes. A
difficultproblem is thus determining the exact value of this
transi-tion size, and whether this is generally constant or
variesfor different families. Some general indications come
fromdifferent kinds of evidence.
First, some families in the inner region of the main beltare
complete down to sizes on the order of 5–6 km. Thismeans that any
change in slope of the size distributions mustoccur at smaller
sizes for these particular families. Ofcourse, this does not imply
that the same is necessarily truefor all the families in the main
belt. Second, Campo Bagatinand Petit (2001) explored some
generalizations of the Tangaet al. (1999) model, and found
analytical reasons to pre-dict that the value of the power-law
exponent should as-ymptotically converge toward a fixed value,
found to bearound –2.8, for any model based on a similar
treatmentof the geometric constraints affecting the production
offragments. The authors stressed that the above result canbe a
pure artifact of the assumed geometric model, and thereis not any
proof that the model is really representative ofthe actual physics
of the phenomena of catastrophic disrup-tion. If proven to be
adequate to represent the real world,the above result would in any
case prevent the possibilityof obtaining distributions diverging to
infinity in terms ofreconstructed mass.
Another very important consideration is that furthercollisional
evolution experienced by the freshly formedfamily members should
lead to changes of the size frequen-cies. According to Marzari et
al. (1995), the family sizedistributions should be expected to
converge progressively,starting from the low-size end of the
distribution, towardthe equilibrium value expected for a
collisionally relaxedpopulation, being equal to –2.5 according to
classical re-
sults by Dohnanyi (1969, 1971). The convergence shouldtake place
over timescales depending on the size, the small-est members being
expected to run into relaxation in shortertimes (Marzari et al.,
1995). The problem in this scenariocan be that the
Dohnanyi-predicted value of the slope wasmodel-dependent (the
outcomes of the collisions were as-sumed to be independent of
size), and it seems to be un-confirmed by general analyses of the
overall size distribu-tion of main-belt asteroids (Cellino et al.,
1991; Jedicke andMetcalfe, 1998). In particular, it turns out that
nonfamilyasteroids, for instance, exhibit a size frequency much
shal-lower than the predicted Dohnanyi slope.
Apart from causing some uncertainty about the exactvalue of the
slope to which families should be expected toconverge, the above
fact should also be taken into accountwhen another controversial
consequence of the steepnessof the family size distributions is
discussed. The subject hereis that of the possible predominance of
family members inthe general asteroid inventory at small sizes.
According toZappalà and Cellino (1996), family members
increasinglydominate the asteroid population going toward
smallersizes. This conclusion is a consequence of both the
steepslopes of the family size distributions and the
apparentlyshallow slopes of the nonfamily population. The
situationis controversial also because family members should
beexpected to become dominant at values of size (~1 km) forwhich
the current asteroid inventory is severely incomplete.Many authors
tend to dislike the idea that 99% of the main-belt asteroids larger
than 1 km should be family members(Zappalà and Cellino, 1996). On
the other hand, avoidingthis type of conclusion is not trivial, if
we can believe inextrapolations (even moderate in some cases) of
the ob-served family size frequencies. Allowing for some impor-tant
change of slope in the size frequencies of familiessomewhere
between 1 and 5 km is reasonable. However,even in this case,
families could hardly become so shallowas to not give a predominant
contribution to the main-beltasteroid inventory at small sizes.
Only observations will definitely clarify the real situa-tion.
Models of the general asteroid inventory based on thedominance of
families down to sizes of 1 km are currentlybeing developed. Some
definite predictions on the numberand apparent magnitudes of the
asteroids that should befound in any given sky field can be made,
and observationswill soon confirm or rule out these models (see,
e.g., Tedescoand Désert, 1999).
3.3. Ejection Velocities and Comparisonwith Hydrocodes
The results mentioned in sections 2.1 and 2.2 concern-ing the
derivation of the original ejection velocity fields offamilies, and
the discovery of a defined size-velocity rela-tionship, can
certainly in principle be affected by subse-quent evolutionary
effects experienced by family members(section 3.1). However, it
seems difficult to reconcile theexpectation of very important
evolutionary changes with the
-
626 Asteroids III
conservation of regular patterns such those that are observedfor
the velocity field structures and the size-velocity rela-tions of
the most robust families. Figure 3 shows a typicalexample
concerning the Eos family. The existence of de-fined, “triangular”
trends in the diameter vs. proper semi-major axis, eccentricity,
and inclination plots can be easilyrecognized in spite of any level
of noise possibly affectingthe proper elements.
In particular, we should note that in principle a size-dependent
spreading in semimajor axis can be interpreted interms of the
Yarkovsky effect. What seems more difficult toexplain, however, is
the size-dependent spreading in eccen-tricity and inclination. For
this, effects like Yarkovsky canbe only indirectly effective, by
moving objects into narrowresonant zones. The resulting
distribution of the objects inthe proper elements space should be
due to a complicated
interplay of the residence times in the resonances experi-enced
by bodies of different sizes, and the relative rate ofvariation of
the proper elements under the influence of thesize-dependent
Yarkovsky effect and the size-independentresonant behavior. It
seems fairly difficult to expect that thefinal result of the
evolution should be such a regular be-havior like the one shown in
Fig. 3.
Although conclusions are certainly premature at thepresent
stage, it seems that if the current interpretation ofthe data at
our disposal is not grossly wrong, we shouldconclude that either
the evolutionary effects are weaker thancurrently believed, or the
families themselves are relativelyyoung, and still poorly affected
by the above effects. Ofcourse, this conclusion would have
important consequencesfor our understanding of the collisional
evolution of theasteroid belt (see Davis et al., 2002).
2.9
1
1.5
Log
d (k
m)
2
2.95 3 3.05
a (AU)
3.1 3.15
0.14
1
1.5
Log
d (k
m)
2
0.16 0.18
sin i
0.2 0.22
1
1.5
Log
d (k
m)
2
0.05 0.1
e
Fig. 3. Proper semimajor axis, eccentricity, and sinus of
inclination plotted as a function of the logarithm of diameter for
the Eosfamily (1645 members identified). The data refer to a new,
yet-unpublished analysis for the identification of asteroid
families based ona dataset of more than 30,000 asteroid proper
elements computed by Milani, and available at the AstDys Web site
(http://hamilton.dm.unipi.it/cgi-bin/astdys/astibo). These plots
show the general existence of a well-defined trend, possibly
related to the size-ejectionvelocity relationship. The same trend
is recognizable in a large number of currently identified families,
in spite of the existence ofdynamical and physical mechanisms that
are expected to progressively modify the original locations of the
family members in thespace of the proper elements.
-
Zappalà et al.: Properties of Asteroid Families 627
From the point of view of the physics of collisions, theejection
velocities that are derived from application of theGaussian
formulas constitute another debated subject. Gen-erally speaking,
the problem is that according to the mostsophisticated hydrocode
models of catastrophic breakupprocesses (Love and Ahrens, 1996;
Nolan et al., 1996;Asphaug et al., 1998), the impact energies that
are appar-ently needed to disperse the family members with the
ob-served speeds should have been sufficiently high as tocompletely
pulverize the family parent bodies.
In particular, the reconstructed velocity fields of fami-lies
indicate that ejection velocities well beyond 100 m/sare not rare
for the smallest fragments. As a comparison,typical speed values in
laboratory experiments are 5–10×smaller, velocities of 100 m/s
being only typical of pulver-ized fragments produced very close to
the impact point(Martelli et al., 1994). This is a well-known
problem. Whentrying to reproduce the reconstructed ejection
velocity fieldof the Maria family using the Paolicchi et al. (1996)
semi-empirical model (SEM) of catastrophic disruption events,which
is generally able to give a good fit to laboratoryexperiments,
Zappalà et al. (1997) found that SEM givesmaximum velocities a
factor of 5 too low to reproduce thefamily data.
According to hydrocode results, the observed velocitiesare not
possible unless we assume that the observed fam-ily members are
essentially reaccumulated rubble piles. Inthis scenario, all
asteroids larger than some hundreds ofmeters should be rubble
piles. Pisani et al. (1999), how-ever, made some quantitative
tests, and concluded thatreaccumulation cannot be sufficiently
efficient to justify theobserved size distributions of several
known families. Inparticular, important phenomena of reaccumulation
werefound to be possible for the largest remnant alone, but
cer-tainly not for a large number of objects. Moreover,
prelimi-nary results of the in situ exploration of the
near-Earthasteroid 433 Eros, a 31 × 13 × 13-km object, thought to
bea collisional fragment from the disruption of a larger par-ent
body, indicate that it is very unlikely that this asteroidcan be a
rubble-pile [for the meaning of terms like “rubblepile,” see Cheng
(2002) and Richardson et al. (2002)].
Of course, a possible dynamically induced spreading offamilies
in the proper elements space might artificially in-crease the
apparent ejection velocities of family members.Taking this into
account one could conclude that the realejection velocities at the
epoch of formation were lower, asrequired by hydrocode results.
However, there is not yet anyconclusive evidence of a dominating
role played by post-formation dynamical evolution, as mentioned
above. It istrue that when observations do not fit theory,
scientists tendto believe that observations are likely wrong. But
at thisstage we think that the self-consistency of much
observa-tional data is very notable, and would tend to indicate
thatsomething can be wrong in current theoretical models
ofcollisional processes. Of course, this is another field in
whichnew observational and theoretical activities are needed
toachieve some reliable interpretation of what is observed.
3.4. Formation of Binaries
Strictly related to the modeling of family formation is
theproblem of the possible formation of binaries. This subjectis
currently very hot, after the recent discoveries of severalbinary
systems among both near-Earth and main-belt aster-oids (see Merline
et al., 2002). Interestingly enough, thefirst convincing discovery
of an asteroid satellite came fromthe Galileo flyby of the Koronis
family member 243 Ida.More recently, the asteroid 90 Antiope, a
member of theThemis family, has also been found to be an
outstandingexample of a binary system with components of
comparablesize (Merline et al., 2000; Weidenschilling et al.,
2001). Itis clear that the formation of binaries in collisional
processesis an interesting field of research for which we can
expectto have important results in the near future. This subject
isdiscussed extensively in Paolicchi et al. (2002).
3.5. Collision Probabilities
The formation of big asteroid families is not only aninteresting
phenomenon per se, but it also has a number ofimportant
consequences. Dell’Oro et al. (2001) have foundthat a long-term
increase of the collision probabilities in theregion of the main
belt swept by the orbital motion of thenewly formed family members
must be expected. Again,this is due to the huge numbers of
kilometer-sized mem-bers that are apparently produced by these
events. When alarge family is produced, its members will cross the
orbitsof the majority of the main-belt population, apart
fromobjects located far enough in terms of semimajor axis
andeccentricity (mostly the low-eccentricity asteroids
orbitingclose to the inner and outer edges of the belt). In this
re-spect, it is not unlikely that major collisional events
likethose that produced the big Eos and Themis families cantrigger
the subsequent collisional regime over substantialtimescales. This
is just another example of family-relatedmechanisms that should be
taken into account by modernmodels of the overall collisional
evolution of the asteroidbelt (see Davis et al., 2002).
Another fact that should be mentioned is that accordingto recent
investigations (Dell’Oro et al., 2002), the veryearly times after
formation of any given family are charac-terized by a larger impact
probability among members ofthe same family, with respect to the
typical values control-ling the steady collisional regime in the
main belt. Althoughthese epochs of enhanced interfamily collisions
were foundto be short, and were on average characterized by low
impactvelocities, there is still the possibility that the freshly
formedsurfaces of family members can have been marked in
asignificant way, with consequences on the cratering recordobserved
by in situ exploration by space probes (243 Idaand 951 Gaspra were
observed by Galileo). We should alsonote that, according to
Weidenschilling et al. (2001), theexistence of binary asteroids
like 90 Antiope (a member ofthe Themis family), characterized by
components of com-parable size, might also be explained by the
occurrence of
-
628 Asteroids III
relatively mild interfamily collisions likely occurring dur-ing
the very early history of a newly born family.
4. FAMILIES AS POSSIBLE SOURCES OFNEAR-EARTH ASTEROIDS
Families have been found to also play an important rolein the
production of near-Earth asteroids and meteorites.The first
indications were found by Morbidelli et al. (1995),who convincingly
showed that a number of well-knownfamilies (such as Themis) are
sharply cut by the edges ofimportant mean-motion resonances with
Jupiter. This quali-tatively showed two important things: First,
the formationof families in several cases led to the immediate
injectionof substantial numbers of sizeable fragments into
resonantorbits. Many of them (mostly in the case of
resonanceslocated in the inner belt, like the 3:1 mean-motion
reso-nance or the ν6 secular resonance; a smaller fraction in
thecase of resonances located in the outer belt) achieved
near-Earth-like orbits. This process was quantitatively analyzed
bymeans of numerical integrations of the orbits by Gladmanet al.
(1997). The results showed that the transfer times areextremely
short for resonances like 3:1 and 5:2, on the orderof a couple of
million years. Based on these results and ona preliminary
assessment of the numbers of objects actu-ally injected by several
families at the epochs of their for-mation, Zappalà et al. (1998)
found that in the past severalepisodes of paroxysmal
near-Earth-asteroid production,with consequent enhancements in the
collision rate with theterrestrial planets, took place, with
durations on the orderof several million years in most cases. This
also indicatesthat the inventory of near-Earth objects is not
steady, andcan be strongly influenced by the occurrence of
importantcollisional events in the main belt.
Second, family-forming events left groupings of objectsthat are
very close to the borders of several important reso-nances. This
fact is important, because it can be expectedthat the normal
collisional evolution experienced by theseobjects can likely
produce a steady rate of production ofsmall fragments directly
injected into the most efficientdynamical routes to the inner solar
system (see also Zappalàand Cellino, 2001). A good example is given
by the Mariafamily, whose reconstructed ejection velocity field,
sharplycut by the 3:1 resonance with Jupiter, is shown in Fig.
4(Zappalà et al., 1997).
An important result obtained by Morbidelli et al. (1995)was the
discovery that several objects are currently locatedinto the 9:4
mean-motion resonance with Jupiter. This rela-tively thin resonance
cuts across the big Eos family, andwas shown by Gladman et al.
(1997) to give a modest butnot fully negligible contribution to the
near-Earth popula-tion. The reason is that the dynamical evolution
inside the9:4 resonance is relatively slow, with eccentricity
beingpumped up over timescales sufficiently long to allow Marsto
capture some objects before they become dominated byJupiter (with
subsequent ejection from the solar system).Most of the 9:4 resonant
objects cannot be recognized as
Eos family members on the basis of their orbital elements,but it
was straightforward to assume that these objects mightbe original
members of this family, observed during theearly stages of a
dynamical evolution that will decouplethem from the asteroid main
belt. This conjecture was fullyconfirmed by subsequent observations
by Zappalà et al.(2000) (see also Cellino et al., 2002).
Another fact that should be mentioned when speakingabout the
relation between near-Earth objects and familiesis the following:
If direct collisional injection into some ofthe dynamically
unstable regions in the main belt (such asthe 3:1 resonance with
Jupiter, the ν6 secular resonance, andthe region of Mars-crossing)
is assumed to be the mosteffective mechanism for supplying
kilometer-sized NEAs,some constraints must be respected. In
particular, directinjection cannot be accepted if it can be shown
that it nec-essarily predicts the formation of too many observable
fami-lies. The reason is simply that the number of known familiesin
the main belt is limited. This is a strong constraint forany
mechanism of steady NEA supply, mainly for sizeableobjects having
diameters beyond 1 or 2 km. According toZappalà et al. (2002), it
does not seem trivial to explain thecurrently observed number of
NEAs larger than 2 km only bymeans of steady collisional injection
into chaotic regions inthe main belt. Many candidate parent bodies
exist (the mosteffective ones have been listed in Table 1), but the
rangesof “permitted” impact energies that satisfy the
nonfamilyformation constraint for most of them are fairly
narrow.
Fig. 4. Reconstruction of the ejection velocity field of the
frag-ments produced in the disruption of the parent body of the
Mariafamily. The location of the edges of the 3:1 mean-motion
reso-nance with Jupiter are also indicated. The family turns out to
besharply cut by the border of the resonance. From Zappalà et
al.(1997).
–0.5
–0.5
0
0.5
0.50
Velocity (km/s)
30 km
3:1
20 km
10 km
Vel
ocity
(km
/s)
-
Zappalà et al.: Properties of Asteroid Families 629
TABLE 1. A list of the most effective NEA candidate parent
bodies found by Zappalà et al. (2001).
Proper Elements Transfer Diameter Taxonomic ClassID Number a'
(AU) e' sin i' Region (km) Tholen (1994) Bus (1999)
304 2.404 0.122 0.257 ν6 68 C XC313 2.376 0.236 0.201 MC 96 C
—495 2.487 0.118 0.045 3:1 39 — —512 2.190 0.199 0.142 MC 23 S S753
2.329 0.214 0.153 MC 24 S L877 2.487 0.142 0.059 3:1 38 F —930
2.431 0.121 0.265 ν6 36 — Ch1080 2.420 0.243 0.086 MC 23 F —1715
2.400 0.249 0.181 MC 23 — X2143 2.281 0.194 0.130 MC 19 — —
5. FUTURE DEVELOPMENTS
From the facts discussed in the previous sections, it iseasy to
see that the physical properties of asteroid familiesplay an
important role in our current understanding of thecollisional and
dynamical evolution of the asteroid popu-lation. The interpretation
of the data at our disposal is notstraightforward, and a
coordinated observational and theo-retical effort will be needed in
order to confirm or rule outa number of conclusions that are
suggested by the data thatare currently available. A determination
of the times neededto collisionally erode and dynamically disperse
a newlyformed family will certainly be a primary field of
investiga-tion in the near future. At the same time, trying to
under-stand how families can achieve their steep size
distributions,and how long they can keep them unchanged in the
rangeof sizes covered by current datasets, will be another
majorfield of research. If families evolve rapidly, and those
thatare currently observed can be shown to be really young,this
will deeply influence current ideas of the collisionalevolution of
the main belt, and will provide new importantconstraints in
addition to those currently recognized, likethe preservation of the
basaltic crust of 4 Vesta; the exist-ence of metallic meteorites;
the scarcity of taxonomic typesdirectly interpreted in terms of
mantle material from dif-ferentiated parent bodies; the observed
cratering records ofIda, Gaspra, Mathilde, and Eros, etc. In
general, it will bepossible to obtain a better and more reliable
estimate of theasteroid inventory at small sizes. Two hundred years
afterthe discovery of the first asteroids, we are still far from
fullyunderstanding these interesting objects. As the direct
out-comes of mutual collisions, families certainly represent avery
important piece of the asteroid puzzle.
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