Physical and Dynamical Properties of Asteroid Families

Home , Asteroid family, Eos family, Flora family, Maria family, Themis family

Zappalà et al.: Properties of Asteroid Families 619

V. Zappalà, A. Cellino, A. Dell’Oro Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Torino

P. Paolicchi University of Pisa

The availability of a number of statistically reliable asteroid families and the independent confirmation of their likely collisional origin from dedicated spectroscopic campaigns has been a major breakthrough, making it possible to develop detailed studies of the physical properties of these groupings. Having been produced in energetic collisional events, families are an invaluable source of information on the physics governing these phenomena. In particular, they provide information about the size distribution of the fragments, and on the overall properties of the original ejection velocity fields. Important results have been obtained during the last 10 years on these subjects, with important implications for the general understanding of the collisional history of the asteroid main belt, and the origin of near-Earth asteroids. Some important problems have been raised from these studies and are currently debated. In particular, it has been difficult so far to reconcile the inferred properties of family-forming events with current understanding of the physics of catastrophic collisional breakup. Moreover, the contribution of families to the overall asteroid inventory, mainly at small sizes, is currently controversial. Recent investigations are also aimed at understanding which kind of dynamical evolution might have affected family members since the time of their formation. In addition to potential consequences on the interpretation of current data, there is some speculative possibility of obtaining some estimate of the ages of these groupings. Physical characterization of families will likely represent a prerequisite for further advancement in understanding the properties and history of the asteroid population.

1. INTRODUCTION observed families, and to estimate the typical times needed to progressively erode freshly formed families in such a way The improvements of the techniques of family identifi- to make them no longer identifiable at later epochs. The cation, leading to the establishment of a database of about most recent investigations of this particular aspect have been 20 statistically reliable families recognized during the last published by Marzari et al. (1999). decade (see Bendjoya and Zappalà, 2002), and the subse- Another essential aspect concerns the information that quent confirmation of the cosmochemical self-consistency can be obtained about the physics of the breakup events of these groupings from dedicated spectroscopic campaigns from which families originated. Understanding how aster- (see Cellino et al., 2002) has introduced new exciting per- oids react to mutual collisions typically characterized by spectives for the physical studies of asteroids. relative velocities around 5–6 km/s (Farinella and Davis, Since the early times of the first family discoveries by 1992; Bottke et al., 1994) is an essential piece of information Hirayama at the beginning of the past century, these group- for any attempt at modeling their overall history, as well as ings have been interpreted as the observable outcomes of their internal properties. Apart from the purely theoretical energetic collisional events, leading to complete disruption issues, understanding the internal structure of asteroids is a of a number of original parent bodies, and to dispersion of necessary prerequisite to developing any reasonable strategy the collisional fragments. In this respect, asteroid families of mitigation of the near-Earth asteroid (NEA) impact hazard. are nothing but nice examples of the collisional phenomena In the past, most of our understanding of the way aster- that have marked the history of the solar system, although oids behave when they are collisionally disrupted came generally at much smaller energy regimes with respect to from laboratory experiments, based both on hypervelocity major events involving large planetary bodies (origin of the impacts and explosive charge techniques [for reviews of this Earth’s Moon, tilt of the spin axis of Uranus, etc.). subject, see Fujiwara et al. (1989) or Martelli et al. (1994)]. In the framework of asteroid studies, families are cru- The major source of uncertainty, in this respect, was the cially important in many respects. In general terms, they scaling problem. In other words, it was not clear how to represent a constraint for any attempt at modeling the over- extrapolate the results of laboratory tests involving centi- all collisional history of the asteroid belt. In other words, meter-sized targets, to the astronomical situation of kilo- the models should be able to justify the number of currently meter-sized bodies, including relevant gravitational effects

619 620 Asteroids III

(see Holsapple et al., 2002). In this context, the availabil- Moreover, 951 Gaspra, also observed by Galileo, might be ity of a number of reliable families placed at the disposal a member of the Flora family, although it has not been listed of theoreticians a number of breakup experiments that na- as such in the most recent family classification by Zappalà ture performed for us, directly in a range of masses and et al. (1995). It is known that for the Flora family the nomi- energies absolutely beyond current and future laboratory ca- nal membership has been likely underestimated due to the pabilities. severe statistical criteria used for identification (Migliorini As a consequence, a great effort has been made in recent et al., 1995). The important results of the Galileo observa- years to derive from the observed properties of family mem- tions of these two objects, including analyses of the size bers clues about the physics and the kinematics of the events distributions of impact craters on their surfaces (conse- from which families originated. These studies have been quences of their collisional histories), are extensively dis- able to obtain relevant results, whose interpretation is still cussed in Chapman (2002) and Sullivan et al. (2002). somewhat debated, as we shall see in the following sections. The family spectroscopic data (item 3 in the above list) We should still mention the problem that the presence deserve a separate analysis, and are the subject of Cellino of random interlopers represents for physical studies of et al. (2002). In what follows, we then focus our attention on families. The likely numbers of objects sharing purely by the properties mentioned in items 1 and 2 of the above list. chance the same orbital proper elements of “true” family members have been predicted on the basis of pure statis- 2.1. Ejection Velocity Fields tics by Migliorini et al. (1995) as a function of size. At the same time, spectroscopic observing campaigns have been The structure of the families in the space of proper ele- able to identify in several cases a number of likely inter- ments are used to derive information on the ejection veloci- lopers, as described in Cellino et al. (2002). Interlopers are ties of the fragments in family-forming events. This is pos- more “harmful” when they are relatively big, since they can sible because we can interpret the differences in orbital affect the overall behavior of the family size distributions, elements in terms of differences in ejection velocity from and the reconstruction of the original ejection velocity fields the original parent body. The conversion from velocities to of the fragments (see below). However, we can now be orbital elements or vice versa is given by the well-known confident that nearly all the biggest interlopers have been Gaussian formulas, which can be written as follows, under identified in most cases. Thus, the results described in the the assumption (verified for families) that the ejection ve- following section are not expected to be appreciably affected locities are much smaller than the orbital velocity of the by the presence of these undesired guests. parent body

2. AVAILABLE DATA AND COMMON δ = 2 ++ a/a (] 1 efVefV cos )TR ( sin ) ] INTERPRETATIONS na()1 − e2 1/2 Physical studies of dynamical families are based on anal- (12−+e2 )1/2 ef cos + efcos2 δe = VfV+ (sin ) yses of essentially three pieces of information: (1) the co- na 1 + efcos TR ordinates of the family members in the space of the proper −+2 1/2 ω elements a', e', and i' (proper semimajor axis, eccentricity, δ = (1 e ) cos (f ) I VW (1) and inclination respectively); (2) the size distributions of na 1 + eefcos family members; and (3) spectroscopic and spectrophoto- metric properties of the members. where na is the mean orbital velocity, and VT, VR, and VW In addition to the items listed above, we should mention are the components of the ejection velocity vector along the that some photometric data are also available. The major direction of the motion, radial, and normal to the orbital problem for this kind of investigation is certainly the faint- plane respectively. The parameters f and ω are the true ness of most family members, apart from the generally few anomaly and the argument of perihelion of the parent body largest members of the different groupings. A few light- at the epoch of its disruption. These angles are a priori curves are not sufficient for any relevant statistical inference. unknown, and this fact has long prevented any attempt at For this reason, no definitive indications have yet come from reconstructing the original ejection velocity fields from the photometry. The only exception is a tentative clue of a likely presently observed locations of the members in the proper young age of the Koronis family, based on the observed elements space. distribution of lightcurve periods and amplitudes (Binzel, By means of extensive numerical simulations, however, 1988). This investigation found also evidence that in the case Zappalà et al. (1996) showed that very reasonable recon- of the Eos family the distribution of spin periods seems to structions of the velocity fields in family-forming events are suggest collisional relaxation and a likely old age. possible when the real velocity fields were not completely Another important source of information has come from random, but were characterized by a wide variety of possible direct in situ explorations by means of space probes. In field structures, including spherical, ellipsoidal, conical particular, the Ida-Dactyl binary system observed by the (cratering), and more complicated combinations of the Galileo probe is a nominal member of the Koronis family. above fields, in which some symmetry property is (even Zappalà et al.: Properties of Asteroid Families 621

weakly) satisfied. The basic idea was to use some adimen- of nonfamily asteroids in different regions of the main belt sional parameters, given by some combinations of the distri- (Cellino et al., 1991; Klaçka, 1992). butions of the velocity components of the family members, For a long time such behavior has been essentially mis- to estimate the most likely values of the unknown f and ω understood. General models based on an assumed power- angles. Of course, in practical cases the reconstruction can law trend coupled with simple requirements of conservation be attempted only when a sufficiently large number of mem- of the total mass of the parent body led Petit and Farinella bers is available, because of statistical reliability. In many (1993) to make predictions that were able to fit satisfacto- situations a reconstruction of the fields was found to be pos- rily only a minor fraction of the whole set of known family sible, as in the cases of Vesta, Dora, Merxia (Zappalà et al., size distributions. In most cases, the slopes of the observed 1996), and Maria (Zappalà et al., 1997). The reconstructed power laws were much steeper than the predictions. A pos- fields turn out to be generally symmetric and similar, in sible explanation was developed by Tanga et al. (1999), terms of general structure, to what is observed in laboratory based on a very simple concept first proposed by Paolicchi experiments involving much smaller targets. These results et al. (1996). The idea was that the fragmentation processes could be obtained in spite of the difficulty related to the fact must satisfy other constraints in addition to conservation of that families are identified in the space of the orbital proper the parent body’s mass. In particular, the fragments are elements, whereas the Gaussian formulas describe the ex- formed in a finite volume, and cannot mutually overlap. A pected behavior in the osculating elements space. Accord- simple model taking into account these geometric con- ing to Bendjoya et al. (1993) the overall structure of the straints was developed by Tanga et al. (1999), and was ejection velocity field is conserved in the transformation found to give excellent fits of the observed size distribu- from osculating to proper elements. The major effect of the tions of a large number of families (see Fig. 1). As a by- transformation was found to be a parallel displacement of product of this technique, some estimates of the original the whole set of family members, leading to conservation parent body size and of the actual ratio mLR/mPB between of the overall velocity field structure. Of course, one can the mass of the largest remnant and that of the parent body expect that over long timescales the stability of the proper were also obtained. More recently, the likely role of geo- elements can change, making a reconstruction of the veloc- metric constraints in fragmentation phenomena has been ity fields harder to obtain. On the other hand, one can also also recognized by Campo Bagatin and Petit (2001), al- expect that over long timescales families tend in any case though the authors did not find any real proof that the to disappear due to progressive collisional erosion (Marzari modeled geometric effects are in any way related to the et al., 1995, 1999). actual physics of collisions. An indication that aging of the proper elements is likely These above-mentioned results have important and de- present in several cases is given by the fact that the result- bated implications concerning the overall inventory of the ing values of the computed f and ω angles turn out not to asteroid population, since they suggest that family members be homogeneously distributed within their natural range of might dominate the asteroid population at small sizes, at variation. This is related to the well-known fact that most least if we can believe in the reliability of extrapolations of families appear to be more elongated in proper eccentricity the observed size frequencies to small sizes (around 1 km). and inclination with respect to semimajor axis (Zappalà et This will be discussed more thoroughly in the following al., 1984). section. Another problem is that the computed ejection velocities Another recent result that must be mentioned concerns of family members resulting from their differences in or- the relationship found between size and ejection velocity bital proper elements turn out to be considerably large (see of the fragments in family-forming events. This problem has section 3.3). been analyzed by Cellino et al. (1999). The result was that a relation exists between the sizes of the fragments and their 2.2. Size Distributions and Size-Velocity maximum possible ejection velocities. In particular, fami- Relationship lies generally fit the predictions of a model that assumes that the maximum amount of kinetic energy achievable by The study of the fragment size distributions is another fragments in family-forming events is a constant. As a con- major field of investigation. Asteroid sizes are generally not sequence, the maximum possible ejection velocity as a func- directly measured, but they are derived from knowledge of tion of size follows a power-law trend, and is represented absolute magnitude and albedo. Families have been found by a straight line in a log-log plot of size vs. velocity, as to be characterized by very homogeneous albedo distribu- shown in Fig. 2. In particular, the following relation holds tions according to the available radiometric data, and this under the assumptions mentioned above, and is found to fit fact has also been used to derive an estimate of the limit of satisfactorily the observational data for most families completeness of the current asteroid inventory in terms of Zappalà and Cellino, d 2 apparent magnitude ( 1996). log=− logVK − ' The size frequencies of all the major families have long D 3 been known to exhibit a power-law trend, characterized by steep slopes, much larger than the typical slopes of samples where d and D are the diameters of the ith fragment and of 622 Asteroids III

0.11 10 100 1000 0.11 10 100 1000 4 4

Gefion Koronis

3 3 D)] D)]

2 2 Log[N( Log[N(

1 1

0 0 –10 1 23 –10 1 23 Log(D) (km) Log(D) (km)

0.11 10 100 1000 0.11 10 100 1000 4 4

Maria Themis

3 3 D)] D)]

2 2 Log[N( Log[N(

1 1

0 0 –10 1 23 –10 1 23 Log(D) (km) Log(D) (km)

Fig. 1. The size distributions of four prominent families identified in the asteroid main belt are shown, with corresponding error bars due to the uncertainties in the sizes of the single family members. The lines represent fits of the size distributions obtained by Cellino et al. (1999) using their model. Three different curves (practically identical) are shown for each family, corresponding to different choices of some random seeds determining the results of different simulations. The simulations show a very good agreement with the data. Also shown are the limits of completeness for each family, due to the fact that the available inventory is not complete at small sizes due to the corresponding faintness of the smallest objects.

the parent body respectively, and V is the maximum pos- kinetic energy of the fragments, and A is an a priori un- sible value for the ejection velocity of the fragment of size d. known parameter, describing the maximum fraction of the The K' parameter is given by total kinetic energy of the set of fragments that can be de- livered to fragments of any size. 1 Apart from the most technical issues, an important re- K' =− log (2Af E/M) 3 KE sult of the analysis has been to recognize that not only a fixed slope around (–2/3) is found to actually fit the behav- where E/M is the specific energy of the impact (defined as ior of most families, but also the value of the intercept K' the ratio between the kinetic energy of the projectile and the turns out to be a linear function of log mLR/M (mLR being mass of the impacted body), fKE is the so-called anelasticity the mass of the largest family member, and M the mass of parameter, defined as the fraction of E that is converted into the parent body). This prediction is respected by most fami- Zappalà et al.: Properties of Asteroid Families 623

–0.6 –0.4 –1

Eos Erigone Eunomia K' = 1.42 –0.6 K' = 1.60 K' = 1.65 –1.2 ) ) ) –0.8 PB PB PB –0.8

–1.4 –1 lg(d/D lg(d/D lg(d/D –1

–1.2 –1.6

–1.2 –1.4 –1.2 –1 –0.8 –0.6 –1.5 –1 –0.5 –0.8 –0.6 –0.4 –0.2

log(vej) (km/s) log(vej) (km/s) log(vej) (km/s) –0.2 –0.8 –0.4 Flora Gefion Koronis –0.4 K' = 1.32 K' = 1.32 –1 –0.6 ) ) ) K' = 1.50 PB PB PB –0.6 –1.2 –0.8 –0.8 lg(d/D lg(d/D lg(d/D –1.4 –1 –1 –1.6 –1.2 –1.2 –1 –0.5 0 –1.5 –1 –0.5 –1.5 –1 –0.5

log(vej) (km/s) log(vej) (km/s) log(vej) (km/s) –0.2 –1.6

–0.4 Maria –0.8 Themis Vesta K' = 1.30 K' = 1.45 –1.7 K' = 2.10 ) ) ) –0.6 PB PB PB –1.8 –1 –0.8 –1.9 lg(d/D lg(d/D lg(d/D –1 –1.2 –2 –1.2

–1.4 –1.4 –2.1 –1.5 –1 –0.5 0 –1 –0.8 –0.6 –0.4 –0.2 –0.6 –0.4 –0.2

log(vej) (km/s) log(vej) (km/s) log(vej) (km/s)

Fig. 2. Size-ejection velocity plots for nine of the most prominent families in the asteroid main belt. The straight lines shown in each plot 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 the intercept corresponding to the best-fit line are also written in each plot. From Cellino et al. (1999).

lies for which mLR/M < 0.8. This fact, if fully confirmed, 3. IMPLICATIONS AND OPEN PROBLEMS has some important, although not yet fully explored, implications for our understanding of the physics of cata- The observational facts mentioned above are generally strophic breakup processes. In particular, it would suggest self-consistent, and suggest some kind of coherent scenario. that the currently observed ejection velocity fields were In particular, it turns out that families were formed by events determined mostly by the impact energies (directly related that produced ejection velocity fields having “reasonable” to the mLR/mPB ratio) and that the original size-velocity structures (when we compare them with the outcomes of trend has been scarcely affected by subsequent dynamical laboratory experiments), although the resulting velocities and/or physical evolution, being still visible today. Such a were generally high. The fragmentation of the parent bod- conclusion may be still premature, however, and what can ies produced huge swarms of fragments, with size frequen- be said at this stage is only that the Cellino et al. (1999) cies characterized by steep power-law trends, still recog- size-velocity model seems to be in good agreement with nizable today. However, several facts must be taken into available data for a large number of known families, and account before accepting literally all the above conclusions. also with some laboratory experiments. Moreover, the exact implications of the above-mentioned 624 Asteroids III scenario must be adequately stressed and discussed. We properties of the surface, on the orientation of the spin axis, touch here on a number of very delicate points affecting and on the obliquity angle (the angle between the spin axis our overall understanding of the properties and history of and the normal to the orbital plane). In its “diurnal” vari- the asteroid population. ant, the Yarkovsky effect can cause the orbital semimajor In what follows, we give (in no special order) a list of axis to drift either inward or outward, depending on the open problems and general implications of the results com- sense of rotation of the body. In addition, there is a “sea- ing from analyses of the family data at our disposal. sonal” variant, due to the periodic preferential heating of different hemispheres during the orbital motion. The net 3.1. Dynamical and Physical Aging of the Families effect of this variant is a systematic decrease of the orbital semimajor axis. It is certain that, over time, asteroid families undergo an An estimate of the expected variation of the semimajor evolution modifying their originally observable features. axis due to the Yarkovsky effect has been recently computed This evolution is expected to affect both the physical prop- by Spitale and Greenberg (2001). The effectiveness of this erties of the members and their location in the proper ele- kind of nongravitational force is stronger for smaller ob- ments space. jects, and should be negligible for bodies larger than kilo- In particular, the orbital proper elements a', e', and i' meter-sized (the exact value depending on many poorly cannot be assumed to be really constant over time (see also known parameters). Moreover, another kind of relevant Knezevic et al., 2002). They are expected to vary slowly long-term dynamical effect that is expected to affect the over timescales on the order of 10–100 m.y. An estimation orbital semimajor axes of all asteroids, including family of the range of fluctuation of the proper elements was given members, is that of mutual close encounters with the mas- by Milani and Knezevic (1992, 1994). As a consequence, sive Ceres, Pallas, and Vesta objects (Carruba et al., 2000). the inferred kinematical properties of families (including In summary, dynamical aging can increase the disper- a reconstruction of their original ejection velocity fields) sion of the eccentricities and inclinations of family mem- should slowly vary as a function of the time elapsed since bers, while the Yarkovsky effect and close encounters can family formation. This does not prevent us from identifying also increase the dispersion of the semimajor axes. Dynami- today a number of around 20 “robust” families in the space cal aging does not depend on the size of the body, whereas of proper elements, and to derive information on the origi- the Yarkovsky effect is certainly negligible for objects larger nal collisional events from which they originated. However, than, e.g., 10–20 km. it is reasonable to expect that the apparent kinematical prop- The interplay of dynamical aging and the Yarkovsky erties of families can be somewhat “aged” with respect to effect can be important, because during a Yarkovsky-driven the original configurations at the epoch of their formation. drift in semimajor axis the objects can be trapped into some Not all the proper elements behave in the same way. The of the many dynamically unstable regions crossing the aster- semimajor axis seems to be the most stable parameter ac- oid main belt (Nesvorný and Morbidelli, 1998; Morbidelli cording to available dynamical theories (see Knezevic et al., and Nesvorný, 1999), with the effect of being eventually 2002). Dynamical aging of the proper elements affects removed from the family. Of course, the timescales of dy- mainly eccentricities and inclinations. This may influence namical and Yarkovsky-driven evolutions must be compared the determination of the unknown f and ω angles in the with typical collisional lifetimes for bodies of comparable Gaussian equations when the techniques of reconstruction sizes. We touch here on some key problems that will be of the ejection velocity fields are applied. As shown by the likely the subject of extensive analyses in the near future. simulations performed by Zappalà et al. (1996), this effect In fact, collisions independently produce a “physical” can produce a spurious clustering of the inferred values of aging of the families. Collisional lifetimes of main-belt f around 90°. In turn, this can affect the derived ejection asteroids are size-dependent, and range between 107 and velocities, since an enhancement of the dispersion of the 109 yr. The exact estimates are model-dependent, since they eccentricities and the inclinations produces a spurious in- are derived from estimates of both the impact strength of crease of the radial and normal ejection velocity compo- the objects and the number of existing projectiles capable nents, leaving the transversal component essentially un- of disrupting an object of a given size. For a 100-km aster- changed (as can be easily seen from equation (1)). oid, the average collisional lifetime should be on the order Another possible source of variation of the proper ele- of 109 yr. As a consequence, and according to Marzari et ments is the so-called Yarkovsky effect (Vokrouhlický and al. (1995), the smallest family members tend to be colli- Farinella, 1998; Farinella and Vokrouhlický, 1999; Vokrouh- sionally disrupted over shorter timescales with respect to lický, 1999; Bottke et al., 2000; Bottke et al., 2002). Un- the bigger members. As a consequence, collisions not only like the dynamical aging processes mentioned above, the modify the size distribution of families, but also produce Yarkovsky effect produces changes of the orbital semima- an erosion in the space of the orbital proper elements, lead- jor axis, and is size dependent. Several parameters, whose ing to progressive disappearance of the families as recog- values are currently uncertain, determine the effectiveness nizable groupings (Marzari et al., 1999). of the Yarkovsky effect, and the range of sizes for which Moreover, the possibility of secondary events involving it is really important. The effect depends on the thermal first-generation family members must also be taken into Zappalà et al.: Properties of Asteroid Families 625

account, since this may potentially produce very compli- sults by Dohnanyi (1969, 1971). The convergence should cated structures in the space of proper elements, as possi- take place over timescales depending on the size, the small- bly in the cases of the so-called “clans,” such as the Flora est members being expected to run into relaxation in shorter family (Farinella et al., 1992). times (Marzari et al., 1995). The problem in this scenario All the facts mentioned above must be seriously taken can be that the Dohnanyi-predicted value of the slope was into account when interpreting the data at our disposal. model-dependent (the outcomes of the collisions were as- Moreover, we observe today families that likely have dif- sumed to be independent of size), and it seems to be un- ferent ages, and show different stages of evolution. Under- confirmed by general analyses of the overall size distribu- standing the complicated interplay of the different evolution of main-belt asteroids (Cellino et al., 1991; Jedicke and tionary tracks will certainly help in understanding the prop- Metcalfe, 1998). In particular, it turns out that nonfamily erties of families, and will possibly help in determining their asteroids, for instance, exhibit a size frequency much shal- current ages, with important implications for the studies of lower than the predicted Dohnanyi slope. the collisional evolution of the asteroid belt. Apart from causing some uncertainty about the exact value of the slope to which families should be expected to 3.2. Size Distributions and Asteroid Inventory converge, the above fact should also be taken into account when another controversial consequence of the steepness In many cases the slopes of the power-law distributions of the family size distributions is discussed. The subject here fitting the size frequencies of asteroid families turn out to is that of the possible predominance of family members in be very steep. According to Tanga et al. (1999), the result- the general asteroid inventory at small sizes. According to ing exponents of the cumulative size distributions reach Zappalà and Cellino (1996), family members increasingly often values beyond –3. In these cases, it is easy to show dominate the asteroid population going toward smaller that integrations of the size distributions would lead to in- sizes. This conclusion is a consequence of both the steep finite reconstructed masses for the corresponding parent slopes of the family size distributions and the apparently bodies. For this reason, it is clear that the size distributions shallow slopes of the nonfamily population. The situation must change slope somewhere at small sizes. A difficult is controversial also because family members should be problem is thus determining the exact value of this transi- expected to become dominant at values of size (~1 km) for tion size, and whether this is generally constant or varies which the current asteroid inventory is severely incomplete. for different families. Some general indications come from Many authors tend to dislike the idea that 99% of the main- different kinds of evidence. belt asteroids larger than 1 km should be family members First, some families in the inner region of the main belt (Zappalà and Cellino, 1996). On the other hand, avoiding are complete down to sizes on the order of 5–6 km. This this type of conclusion is not trivial, if we can believe in means that any change in slope of the size distributions must extrapolations (even moderate in some cases) of the ob- occur at smaller sizes for these particular families. Of served family size frequencies. Allowing for some impor- course, this does not imply that the same is necessarily true tant change of slope in the size frequencies of families for all the families in the main belt. Second, Campo Bagatin somewhere between 1 and 5 km is reasonable. However, and Petit (2001) explored some generalizations of the Tanga even in this case, families could hardly become so shallow et al. (1999) model, and found analytical reasons to pre- as to not give a predominant contribution to the main-belt dict that the value of the power-law exponent should as- asteroid inventory at small sizes. ymptotically converge toward a fixed value, found to be Only observations will definitely clarify the real situa- around –2.8, for any model based on a similar treatment tion. Models of the general asteroid inventory based on the of the geometric constraints affecting the production of dominance of families down to sizes of 1 km are currently fragments. The authors stressed that the above result can being developed. Some definite predictions on the number be a pure artifact of the assumed geometric model, and there and apparent magnitudes of the asteroids that should be is not any proof that the model is really representative of found in any given sky field can be made, and observations the actual physics of the phenomena of catastrophic disrup- will soon confirm or rule out these models (see, e.g., Tedesco tion. If proven to be adequate to represent the real world, and Désert, 1999). the above result would in any case prevent the possibility of obtaining distributions diverging to infinity in terms of 3.3. Ejection Velocities and Comparison reconstructed mass. with Hydrocodes Another very important consideration is that further collisional evolution experienced by the freshly formed The results mentioned in sections 2.1 and 2.2 concern- family members should lead to changes of the size frequen- ing the derivation of the original ejection velocity fields of cies. According to Marzari et al. (1995), the family size families, and the discovery of a defined size-velocity rela- distributions should be expected to converge progressively, tionship, can certainly in principle be affected by subse- starting from the low-size end of the distribution, toward quent evolutionary effects experienced by family members the equilibrium value expected for a collisionally relaxed (section 3.1). However, it seems difficult to reconcile the population, being equal to –2.5 according to classical re- expectation of very important evolutionary changes with the 626 Asteroids III

2 2

1.5 1.5 Log d (km) Log d (km)

1 1

2.9 2.95 3 3.05 3.1 3.15 0.05 0.1 a (AU) e

1.5 Log d (km)

0.14 0.16 0.18 0.2 0.22 sin i

Fig. 3. Proper semimajor axis, eccentricity, and sinus of inclination plotted as a function of the logarithm of diameter for the Eos family (1645 members identified). The data refer to a new, yet-unpublished analysis for the identification of asteroid families based on a 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-ejection velocity relationship. The same trend is recognizable in a large number of currently identified families, in spite of the existence of dynamical and physical mechanisms that are expected to progressively modify the original locations of the family members in the space of the proper elements. conservation of regular patterns such those that are observed interplay of the residence times in the resonances experi- for the velocity field structures and the size-velocity rela- enced by bodies of different sizes, and the relative rate of tions of the most robust families. Figure 3 shows a typical variation of the proper elements under the influence of the example concerning the Eos family. The existence of de- size-dependent Yarkovsky effect and the size-independent fined, “triangular” trends in the diameter vs. proper semi- resonant behavior. It seems fairly difficult to expect that the major axis, eccentricity, and inclination plots can be easily final result of the evolution should be such a regular be- recognized in spite of any level of noise possibly affecting havior like the one shown in Fig. 3. the proper elements. Although conclusions are certainly premature at the In particular, we should note that in principle a size- present stage, it seems that if the current interpretation of dependent spreading in semimajor axis can be interpreted in the data at our disposal is not grossly wrong, we should terms of the Yarkovsky effect. What seems more difficult to conclude that either the evolutionary effects are weaker than explain, however, is the size-dependent spreading in eccen- currently believed, or the families themselves are relatively tricity and inclination. For this, effects like Yarkovsky can young, and still poorly affected by the above effects. Of be only indirectly effective, by moving objects into narrow course, this conclusion would have important consequences resonant zones. The resulting distribution of the objects in for our understanding of the collisional evolution of the the proper elements space should be due to a complicated asteroid belt (see Davis et al., 2002). Zappalà et al.: Properties of Asteroid Families 627

From the point of view of the physics of collisions, the 3.4. Formation of Binaries ejection velocities that are derived from application of the Gaussian formulas constitute another debated subject. Gen- Strictly related to the modeling of family formation is the erally speaking, the problem is that according to the most problem of the possible formation of binaries. This subject sophisticated hydrocode models of catastrophic breakup is currently very hot, after the recent discoveries of several processes (Love and Ahrens, 1996; Nolan et al., 1996; binary systems among both near-Earth and main-belt aster- Asphaug et al., 1998), the impact energies that are appar- oids (see Merline et al., 2002). Interestingly enough, the ently needed to disperse the family members with the ob- first convincing discovery of an asteroid satellite came from served speeds should have been sufficiently high as to the Galileo flyby of the Koronis family member 243 Ida. completely pulverize the family parent bodies. More recently, the asteroid 90 Antiope, a member of the In particular, the reconstructed velocity fields of fami- Themis family, has also been found to be an outstanding lies indicate that ejection velocities well beyond 100 m/s example of a binary system with components of comparable are not rare for the smallest fragments. As a comparison, size (Merline et al., 2000; Weidenschilling et al., 2001). It typical speed values in laboratory experiments are 5–10× is clear that the formation of binaries in collisional processes smaller, velocities of 100 m/s being only typical of pulver- is an interesting field of research for which we can expect ized fragments produced very close to the impact point to have important results in the near future. This subject is (Martelli et al., 1994). This is a well-known problem. When discussed extensively in Paolicchi et al. (2002). trying to reproduce the reconstructed ejection velocity field of the Maria family using the Paolicchi et al. (1996) semi- 3.5. Collision Probabilities empirical model (SEM) of catastrophic disruption events, which is generally able to give a good fit to laboratory The formation of big asteroid families is not only an experiments, Zappalà et al. (1997) found that SEM gives interesting phenomenon per se, but it also has a number of maximum velocities a factor of 5 too low to reproduce the important consequences. Dell’Oro et al. (2001) have found family data. that a long-term increase of the collision probabilities in the According to hydrocode results, the observed velocities region of the main belt swept by the orbital motion of the are not possible unless we assume that the observed fam- newly formed family members must be expected. Again, ily members are essentially reaccumulated rubble piles. In this is due to the huge numbers of kilometer-sized mem- this scenario, all asteroids larger than some hundreds of bers that are apparently produced by these events. When a meters should be rubble piles. Pisani et al. (1999), how- large family is produced, its members will cross the orbits ever, made some quantitative tests, and concluded that of the majority of the main-belt population, apart from reaccumulation cannot be sufficiently efficient to justify the objects located far enough in terms of semimajor axis and observed size distributions of several known families. In eccentricity (mostly the low-eccentricity asteroids orbiting particular, important phenomena of reaccumulation were close to the inner and outer edges of the belt). In this re- found to be possible for the largest remnant alone, but cer- spect, it is not unlikely that major collisional events like tainly not for a large number of objects. Moreover, prelimi- those that produced the big Eos and Themis families can nary results of the in situ exploration of the near-Earth trigger the subsequent collisional regime over substantial asteroid 433 Eros, a 31 × 13 × 13-km object, thought to be timescales. This is just another example of family-related a collisional fragment from the disruption of a larger par- mechanisms that should be taken into account by modern ent body, indicate that it is very unlikely that this asteroid models of the overall collisional evolution of the asteroid can be a rubble-pile [for the meaning of terms like “rubble belt (see Davis et al., 2002). pile,” see Cheng (2002) and Richardson et al. (2002)]. Another fact that should be mentioned is that according Of course, a possible dynamically induced spreading of to recent investigations (Dell’Oro et al., 2002), the very families in the proper elements space might artificially in- early times after formation of any given family are charac- crease the apparent ejection velocities of family members. terized by a larger impact probability among members of Taking this into account one could conclude that the real the same family, with respect to the typical values control- ejection velocities at the epoch of formation were lower, as ling the steady collisional regime in the main belt. Although required by hydrocode results. However, there is not yet any these epochs of enhanced interfamily collisions were found conclusive evidence of a dominating role played by post- to be short, and were on average characterized by low impact formation dynamical evolution, as mentioned above. It is velocities, there is still the possibility that the freshly formed true that when observations do not fit theory, scientists tend surfaces of family members can have been marked in a to believe that observations are likely wrong. But at this significant way, with consequences on the cratering record stage we think that the self-consistency of much observa- observed by in situ exploration by space probes (243 Ida tional data is very notable, and would tend to indicate that and 951 Gaspra were observed by Galileo). We should also something can be wrong in current theoretical models of note that, according to Weidenschilling et al. (2001), the collisional processes. Of course, this is another field in which existence of binary asteroids like 90 Antiope (a member of new observational and theoretical activities are needed to the Themis family), characterized by components of com- achieve some reliable interpretation of what is observed. parable size, might also be explained by the occurrence of 628 Asteroids III relatively mild interfamily collisions likely occurring dur- 3:1 ing the very early history of a newly born family.

4. FAMILIES AS POSSIBLE SOURCES OF 0.5 NEAR-EARTH ASTEROIDS

Families have been found to also play an important role in the production of near-Earth asteroids and meteorites. The first indications were found by Morbidelli et al. (1995), 0 who convincingly showed that a number of well-known families (such as Themis) are sharply cut by the edges of (km/s) Velocity 30 km important mean-motion resonances with Jupiter. This quali- 20 km tatively showed two important things: First, the formation 10 km of families in several cases led to the immediate injection –0.5 of substantial numbers of sizeable fragments into resonant orbits. Many of them (mostly in the case of resonances located in the inner belt, like the 3:1 mean-motion reso- –0.5 0 0.5 nance or the ν secular resonance; a smaller fraction in the 6 Velocity (km/s) case of resonances located in the outer belt) achieved near- Earth-like orbits. This process was quantitatively analyzed by means of numerical integrations of the orbits by Gladman Fig. 4. Reconstruction of the ejection velocity field of the frag- et al. (1997). The results showed that the transfer times are ments produced in the disruption of the parent body of the Maria extremely short for resonances like 3:1 and 5:2, on the order family. The location of the edges of the 3:1 mean-motion reso- of a couple of million years. Based on these results and on nance with Jupiter are also indicated. The family turns out to be a preliminary assessment of the numbers of objects actu- sharply cut by the border of the resonance. From Zappalà et al. ally injected by several families at the epochs of their for- (1997). mation, Zappalà et al. (1998) found that in the past several episodes of paroxysmal near-Earth-asteroid production, with consequent enhancements in the collision rate with the terrestrial planets, took place, with durations on the order Eos family members on the basis of their orbital elements, of several million years in most cases. This also indicates but it was straightforward to assume that these objects might that the inventory of near-Earth objects is not steady, and be original members of this family, observed during the can be strongly influenced by the occurrence of important early stages of a dynamical evolution that will decouple collisional events in the main belt. them from the asteroid main belt. This conjecture was fully Second, family-forming events left groupings of objects confirmed by subsequent observations by Zappalà et al. that are very close to the borders of several important reso- (2000) (see also Cellino et al., 2002). nances. This fact is important, because it can be expected Another fact that should be mentioned when speaking that the normal collisional evolution experienced by these about the relation between near-Earth objects and families objects can likely produce a steady rate of production of is the following: If direct collisional injection into some of small fragments directly injected into the most efficient the dynamically unstable regions in the main belt (such as ν dynamical routes to the inner solar system (see also Zappalà the 3:1 resonance with Jupiter, the 6 secular resonance, and and Cellino, 2001). A good example is given by the Maria the region of Mars-crossing) is assumed to be the most family, whose reconstructed ejection velocity field, sharply effective mechanism for supplying kilometer-sized NEAs, cut by the 3:1 resonance with Jupiter, is shown in Fig. 4 some constraints must be respected. In particular, direct (Zappalà et al., 1997). injection cannot be accepted if it can be shown that it nec- An important result obtained by Morbidelli et al. (1995) essarily predicts the formation of too many observable fami- was the discovery that several objects are currently located lies. The reason is simply that the number of known families into the 9:4 mean-motion resonance with Jupiter. This rela- in the main belt is limited. This is a strong constraint for tively thin resonance cuts across the big Eos family, and any mechanism of steady NEA supply, mainly for sizeable was shown by Gladman et al. (1997) to give a modest but objects having diameters beyond 1 or 2 km. According to not fully negligible contribution to the near-Earth popula- Zappalà et al. (2002), it does not seem trivial to explain the tion. The reason is that the dynamical evolution inside the currently observed number of NEAs larger than 2 km only by 9:4 resonance is relatively slow, with eccentricity being means of steady collisional injection into chaotic regions in pumped up over timescales sufficiently long to allow Mars the main belt. Many candidate parent bodies exist (the most to capture some objects before they become dominated by effective ones have been listed in Table 1), but the ranges Jupiter (with subsequent ejection from the solar system). of “permitted” impact energies that satisfy the nonfamily Most of the 9:4 resonant objects cannot be recognized as formation constraint for most of them are fairly narrow. 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 Class ID Number a' (AU) e' sin i' Region (km) Tholen (1994) Bus (1999) ν 304 2.404 0.122 0.257 6 68 C XC 313 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 S 753 2.329 0.214 0.153 MC 24 S L 877 2.487 0.142 0.059 3:1 38 F — ν 930 2.431 0.121 0.265 6 36 — Ch 1080 2.420 0.243 0.086 MC 23 F — 1715 2.400 0.249 0.181 MC 23 — X 2143 2.281 0.194 0.130 MC 19 — —

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