On the oldest asteroid families in the main belt
Abstract
Asteroid families are groups of minor bodies produced by highvelocity collisions. After the initial dispersions of the parent bodies fragments, their orbits evolve because of several gravitational and nongravitational effects, such as diffusion in meanmotion resonances, Yarkovsky and YORP effects, close encounters of collisions, etc. The subsequent dynamical evolution of asteroid family members may cause some of the original fragments to travel beyond the conventional limits of the asteroid family. Eventually, the whole family will dynamically disperse and no longer be recognizable.
A natural question that may arise concerns the timescales for dispersion of large families. In particular, what is the oldest still recognizable family in the main belt? Are there any families that may date from the late stages of the Late Heavy Bombardment and that could provide clues on our understanding of the primitive Solar System? In this work, we investigate the dynamical stability of seven of the allegedly oldest families in the asteroid main belt. Our results show that none of the seven studied families has a nominally mean estimated age older than 2.7 Gyr, assuming standard values for the parameters describing the strength of the Yarkovsky force. Most “paleofamilies” that formed between 2.7 and 3.8 Gyr would be characterized by a very shallow sizefrequency distribution, and could be recognizable only if located in a dynamically less active region (such as that of the Koronis family). Vtype asteroids in the central main belt could be compatible with a formation from a paleoEunomia family.
keywords:
Minor planets, asteroids: general – celestial mechanics.1 Introduction
Asteroid families are born out of collisions. They “die” when the fragments formed in the collisional event disperse so far because of gravitational or nongravitational forces that the family is no longer recognizable as a group. The time needed to disperse small groups has been the subject of several studies (from our group, see for instance Carruba et al. 2010a, 2011, and 2015). Less attention has been given to the dispersion of larger families. Recent studies (Brasil et al. 2015) suggested that no family could have likely survived the Late Heavy Bombardment (LHB hereafter, at least 3.8 Gyr ago), in the jumping Jupiter scenario, which set an upper limit on the maximum possible age of any asteroid family. In this work we focus our attention on the families whose estimated age, according to Brož et al. (2013) could possibly date from just after the LHB, and whose existence, if confirmed, could provide precious clues on the early stages of our Solar System.
Brož et al. (2013) identified 12 families whose age estimate might have been compatible with an origin during the LHB, or just after: the Maria, Eunomia, Koronis, Themis, Hygiea, Meliboea, Ursula, Fringilla, Alauda, Sylvia, Camilla, and Hermione families. The Sylvia family and the proposed longlost groups around Camilla and Hermione in the Cybele region were recently studied in Carruba et al. (2015), and will not be treated in detail here. That work found that, while all asteroids in the Cybele region were most likely lost during the jumping Jupiter phase of the model of planetary migration of Nesvorný et al. (2013), some of the largest fragments ( km, with D the body diameter) of a hypothetical postLHB Sylvia family may have remained in the Cybele region, but the identification of these groups could be difficult. Due to local dynamics, any Camilla and Hermione families would disperse in timescales of the order of 1.5 Gyr. The Fringilla family is a rather small group (134 members, according to Nesvorný et al. 2015) in the outer main belt. Since we are already studying the larger Themis, Meliboea, Alauda and Ursula families in the outer main belt, and since the group is not large enough for the techniques used in this work, we will not further investigate this family in this paper. The Hygiea family was studied at length in Carruba et al. (2014a), that found a maximum possible age of 3.6 Gyr, just a bit younger than the minimum currently believed epoch of the last stages of the LHB, i.e., 3.8 Gyr (Bottke et al. 2012; also the age of the Hygiea family should be most likely younger than 3.6 Gyr, because of the longterm effect of close encounters with 10 Hygiea and of the stochastic YORP effect of Bottke et al. 2015, this last effect not considered in Carruba et al. 2014a). Age estimates for the other families listed in Brož et al. (2013) are however known with substantial uncertainty, and were obtained before Bottke et al. (2015) modeled the socalled stochastic YORP effect, in which the shapes of asteroids changes between YORP cycles, and whose effect is generally to reduce the estimates of the given asteroid family age. Understanding which, if any, of the proposed “Brož seven” oldest groups (Maria, Eunomia, Koronis, Themis, Meliboea, Ursula, Alauda) could have survived since the LHB, and how, is the main goal of this work.
This paper is so divided: In Sect. 2 we identified the seven “Brož” families in a new space of proper elements and SDSSMOC4 slope and colors, and use the method of Yarkovsky isolines to obtain estimates of maximum possible ages of these groups. In Sect. 3 we use Monte Carlo methods (see Carruba et al. 2015) to obtain refined estimates of the family age and ejection velocity parameters of the same families. Sect. 4 deals with the dynamical evolution and dispersion times of fictitious simulated “Brož” groups since the latest phases of the Late Heavy Bombardment, when the stochastic YORP effect (Bottke et al. 2015) and past changes in the solar luminosity (Carruba et al. 2015) are accounted for. Finally, in the last section, we present our conclusions.
2 Family identification
In order to obtain age estimates of the seven Brož families, we first need to obtain good memberships of these groups. Any method used should aim to reduce the number of possible interlopers, objects that may be part of the dynamical family, but have taxonomical properties inconsistent with that of the majority of the members. DeMeo and Carry (2013) recently introduced a new classification method, based on the BusDeMeo taxonomic system, that employs Sloan Digital Sky SurveyMoving Object Catalog data, fourth release (SDSSMOC4 hereafter, Ivezić et al. 2002) slope and z’ i’ colors. In that article the authors used the photometric data obtained in the five filters , and , from 0.3 to 1.0 , to obtain values of colors and spectral slopes over the , and reflectance values, that were then used to assign to each asteroid a likely spectral type. Since the authors were interested in dealing with a complete sample, they limited their analysis to asteroids with absolute magnitude higher than 15.3, which roughly corresponds to asteroids with diameters larger than 5 km, and for which the SDSSMOC4 dataset is supposed to be complete. Since Carruba et al. (2015) showed that the vast majority of asteroids that could survive 3.8 Gyr of dynamical evolution in the postLHB scenario should have diameters larger than 5 km, we believe that this choice of limit in could also be suitable for this research. For completeness, we also included asteroids with with known spectral types from the Planetary Data System (Neese 2010) that are not part of SDSSMOC4. These objects were assigned values of the slope and colors at the center of the range of those of each given class. We refer the reader to DeMeo and Carry (2013) and Carruba et al. (2014a) for more details on the procedure to obtain slope and colors, and on how to assign asteroids spectral classes based on these data.
Once a complete set of data with proper elements, obtained from the AstDys site (http://hamilton.dm.unipi.it/cgibin/astdys/astibo, accessed on April , 2015 (Knežević and Milani 2003), taxonomical and SDSSMOC4 data on slope and colors has been computed for asteroids in each given asteroid family region, family membership is obtained using the Hierarchical Clustering Method (HCM) of Bendjoya and Zappalá (2002) in an extended domain of proper elements and slope and colors, where distances were computed using the new distance metric:
(1) 
where is the standard distance metrics in proper element domain defined in Zappalá et al. (1995) as:
(2) 
where is the asteroid mean motion; the difference in proper and ; and are weighting factors, defined as = 5/4, = 2, = 2 in Zappalá et al. (1990, 1995). and are difference between two neighboring asteroids slopes and colors, is a constant equal to (see also Carruba et al. 2013 for a discussion on the choice of this constant) and is a normalization factor to account for differences among the mean values of differences in slope and colors (different values of in the range 0.050.001 have been used without substantially affecting the output of the method). We assigned families to their regions, defined as in Brož et al. (2013) as the central main belt, the pristine region, the outer main belt and the outer highly inclined region, and compute proper elements and slope and colors for all asteroids in each given region. We then computed nominal distance cutoff and stalactite diagrams with the standard techniques described in Carruba (2010b), for all the seven Brož families, and then assigned values of diameters and geometric albedos from the WISE catalog (Wright et al. 2010) to family members for which this information is available (other objects were assigned the values of geometric albedo of the largest object in the family, and diameters computed using equation 1 in Carruba et al. 2003). Finally the method of the Yarkovsky isolines (see also Carruba et al. 2013) is applied to obtain an estimate of the maximum possible age of each given family. In the next subsection we will discuss in detail the method for families in the outer main belt, results are similar for other regions.
2.1 The outer main belt: the Themis, Meliboea, and Ursula families
The outer main belt is defined in Nesvorný et al. (2015) as the region between the 7J:3A and 2J:1A meanmotion resonances and . Three possible old Ccomplex families have been proposed in this region: the Themis, Meliboea, and Ursula groups. We identified 5472 asteroids with either taxonomical or SDSSMOC4 information in the area, 3524 of which with data in the WISE dataset, and 2896 (82.2%) with . The value of the minimal distance cutoff was of 157.5 m/s and the minimal number of objects to have a group statistically significant was 25.
Fig. 1, panel A, displays a stalactite diagram for this region. The families of Themis, Meliboea, and Ursula were all visible at the nominal distance cutoff , but not for lower values of . 137 Meliboea merges with the local background at m/s. The family of 375 Ursula merges with Eos at cutoff of 250 m/s, and with other minor groups at m/s, so we choose to work with a cutoff of 220 m/s. Other important families in the region were those of 221 Eos and 10 Hygiea. Panel B of Fig. 1 displays a contour plot of number density of asteroids in the plane. Whither tones are associated with higher values of local number density. For our grid in this domain, we used 30 steps of 0.01 AU in , starting from AU, and 34 steps of 0.01 in , starting from . The members of the identified Themis, Meliboea and Ursula families, after taxonomical interlopers were removed (there were none in these cases), are shown as red, green, and blue circles, respectively.
Finally, we also apply the method of Yarkovsky isolines (Carruba et al. 2013) to the Themis, Meliboea, and Ursula families, using standard values of the parameters of this force for Ctype groups from Brož et al. (2013), Table 2. Results are shown in Fig. 2 for the Themis family. The red and blue lines are the expected displacement of family members over 3.8 and 4.6 Gyr because of the Yarkovsky effect and close encounters with massive asteroids, assumed equal to 0.01 AU for simplicity (Carruba et al. 2014a). All asteroids were assumed to be initially at the family barycenter. This method yields maximum ages of 4.6 Gyr for the three Themis, Meliboea, and Ursula families. Information on number of confirmed members and interlopers is given in Table 1, among other quantities.
Family  Orbital  Number of  Spectral  Number of tax.  Number of dyn.  Number of  Maximum age  

name  region  asteroids  [m/s]  Complex  interlopers  interlopers  confirmed members  estimate [Gyr] 
Eunomia  Central mb  1416  155  S  33  1  1101  3.8 
Maria  Central mb  1416  155  S  16  1  386  3.8 
Koronis  Pristine zone  1015  135  S  18  1  502  4.6 
Themis  Outer mb  5472  220  CX  0  10  642  4.6 
Meliboea  Outer mb  5472  220  CX  0  4  78  4.6 
Ursula  Outer mb  5472  220  CX  0  11  172  4.6 
Alauda  HI Outer mb  286  250  CX  1  23  122  4.6 
3 Chronology
Monte Carlo methods to obtain estimates of the family age and ejection velocity parameters were pioneered by Vokrouhlický et al. (2006a, b, c) for the Eos and other asteroid groups. They were recently modified to account for the “stochastic” version of the YORP effect (Bottke et al. 2015), and for changes in the past values of Solar luminosity (Vokrouhlický et al. 2006a) for a study of dynamical groups in the Cybele region (Carruba et al. 2015). Age estimates obtained including the stochastic YORP effect in Carruba et al. (2015) were i) of better quality with respect those obtained with the static version of YORP for old families, in terms of confidence level, and ii) tend to produce younger age estimates. We refer the reader to Bottke et al. (2015) and Carruba et al. (2015) for a more in depth description of the method. Essentially, the semimajor axis distribution of various fictitious families is evolved under the influence of the Yarkovsky, both diurnal and seasonal versions, and YORP effect (and occasionally other effects such as close encounters with massive asteroids (Carruba et al. 2014a) or changes in past solar luminosity values (Carruba et al. 2015). This method, however, ignores the effect of planetary perturbations. The newly obtained distributions of a target function computed with the relationship:
(3) 
are then compared to the current distribution of real family members using a like variable (Vokrouhlický et al. 2006a, b, c), whose minimum value is associated with the bestfitted solution.
We applied this method to six out of the seven “Brož” families. For the case of the Maria family, since this group was significantly depleted at lower semimajor axis by interaction with the 3J:1A meanmotion resonance, the number of remaining intervals in the target function was too small to allow for a precise determination of the family age. Fig. 3 displays values in the () plane for Alauda family. Values of the parameter, that describes the spread in the terminal ejection velocities (Vokrouhlický et al. 2006), tend to be lower than the estimated escape velocity from the parent body (Bottke et al. 2015), 156.5 m/s for the case of the Alauda family. To estimate nominal values of the uncertainties associated with our estimate of the age and the parameter, here we used the approach first described in Vokrouhlický et al. (2006a,b,c). First, we computed the number of degrees of freedom of the like variable, given by the number of intervals in the distribution with more than 10 asteroids (we require a minimum number of 10 asteroid per interval so as to avoid the problems associated with dividing by small number when computing ), minus 2, the number of parameters estimated from the distribution. Then, assuming that the probability distribution is given by a incomplete gamma function of arguments and the number of degrees of freedom, we computed the value of associated with a 1sigma probability (or 68.3%) that the simulated and real distribution were compatible (Press et al. 2001).^{1}^{1}1We should caution the reader that other methods to estimate the uncertainties for the estimated parameters are also used in the literature. For instance, one can compute the 1, 2 or 3sigma values and sum these to the minimum observed value of to obtain estimates of the errors at 1, 2, or 3sigma levels (Press et al. 2001). Since age estimates for very old families, such as those studied in this work, tend to produce shallow minima, here we prefer to use the Vokrouhlický et al. (2006a,b,c) approach, so as to provide a more limited range of estimated values. But larger values of the error estimates are possible.
For the case of the Alauda family, we had 32 intervals with more than 10 asteroids, and 30 degrees of freedom. The Alauda family should be Myr old, with m/s, and a secondary minimum at lower values of . Our results for the six families are summarized in Table 2, were we display the name of the family, the estimated age, the values of , and the limit used for .
Name  Age  

[Myr]  [m/s]  
Eunomia  18.24  
Koronis  12.59  
Themis  15.41  
Meliboea  7.02  
Ursula  4.30  
Alauda  27.77 
Overall, we found that no family is nominally older than 2.7 Gyr, but uncertainties are too large for the Koronis, Themis, and Ursula families for a positive conclusion to be reached. We compared our results with the maximum age estimates from Brož et al. (2013) and from the more recent work of Spoto et al. (2015), that found estimates for the family ages with a Vshape method in the domain. Table 3 shows the maximum estimated ages from this work (mean values plus errors), from Brož et al. (2013) and from Spoto et al. (2015)^{2}^{2}2In Spoto et al. (2015) the authors found estimates for the left and right semimajor axis distributions of family members with respect to the family barycenter. Results in Table 3 are for the maximum possible estimates, among the two, when available (for the Maria and Ursula there was data for just one of the family wings, there was no estimate available for the Meliboea and Alauda families. The Eunomia parent body may have experienced two or more impacts, according to these authors. Table 3 reports the estimated age of the oldest impact)..
Name  Current  Brož et  Spoto et 
estimates  al. (2013)  al. (2015)  
[Gyr]  [Gyr]  [Gyr]  
Eunomia  2.67  3.00  2.35 
Maria  4.00  2.35  
Koronis  4.60  3.50  2.24 
Themis  4.15  3.50  4.60 
Meliboea  0.65  3.00  
Ursula  4.60  3.50  4.52 
Alauda  0.69  3.50 
To within the uncertainties, our estimates tend to be in agreement with previous results, with the two possible exceptions of the Alauda and Meliboea families, that could potentially be younger than previously thought (but uncertainties for these two families may be larger if different approaches for the error on the like variable were used.) Masiero et al. (2012) investigated the effect that changes in the nominal values of the parameters affecting the Yarkovsky and YORP forces may have on the estimate of the age of the Baptistina family and found that the parameters whose values most affected the strength of the Yarkovsky force were the asteroid density and the thermal conductivity. Since the largest variations were observed for changes in the values of the family thermal conductivity, for the sake of brevity in this work we concentrate our analysis on this parameter. Maximal ages can be found if one consider a value of the thermal conductivity of W/m/K. We repeated our analysis for the six “Brož” families, and this larger value of , and our results are summarized in Table 4, where we display the estimated ages, ejection velocity parameter, and maximum possible age.
Name  Age  Max. Age  

[Myr]  [m/s]  [Gyr]  
Eunomia  4.6  
Koronis  4.6  
Themis  4.6  
Meliboea  1.3  
Ursula  4.6  
Alauda  1.1 
The ages of the Stype families Koronis and Eunomia in these simulations was larger then 4.5 Gyr, but that it is just an artifact caused by the improbably high value of (0.1 W/m/K) used for these simulations (typical values of for Stype families are of the order of 0.001 W/m/K). More interesting were the results for the Ccomplex groups: while the maximum possible ages for the Themis and Ursula families were beyond 3.0 Gyr, none of the groups, even in this very favorable scenario, has nominal ages old enough to reach the earliest estimates for the LHB (3.8 Gyr ago). The implications of this analysis will be further explored in the next section.
4 Dynamical evolution of old families
To study the possible survival of any of the largest members of the oldest main belt family over 4 Gyr, we performed simulations with the integrator (SwiftYarkovskyStochastic YORPClose encounters) of Carruba et al. (2015), modified to also account for past changes in the values of the solar luminosity. The numerical setup of our simulations was similar to what discussed in Carruba et al. (2015): we used the optimal values of the Yarkovsky parameters discussed in Brož et al. (2013) for C and Stype asteroids, the initial spin obliquity was random, and normal reorientation timescales due to possible collisions as described in Brož (1999) were considered for all runs. We integrated our test particles under the influence of all planets, and obtained synthetic proper elements with the approach described in Carruba (2010b).
We generated fictitious families with the ejection parameter found in Sect. 3 (for the Maria family we used the same value found for the Eunomia group, i.e., ), and integrated these groups over 4.0 Gyr. Also, since only bodies larger than 4 Km in diameter were shown to survive in the Cybele region over 4.0 Gyr, following the approach of Carruba et al. (2015) we generated families with sizefrequency distributions (SFD) with an exponent that bestfitted the cumulative distribution equal to 3.6, a fairly typical value, and with diameters in the range from 2.0 to 12.0 km. The number of simulated objects was equal to the currently observed number of family members with diameters between 2.0 and 12.0 km.
For each of the simulated families, we computed the fraction of objects that remained in a box defined by the maximum and minimum value of associated with current members of the seven families as a function of time, the fraction of objects with km, and the time evolution of the exponent. These parameters will help estimating the dynamical evolution of the simulated families: the lower the values of these numbers, the more evolved and diffused should be the family. Also, to quantify the dispersion of the family members as a function of time, we also computed the nominal distance velocity cutoff for which two nearby asteroids are considered to be related using the approach of Beaugé and Roig (2001), that defines this quantity as the average minimum distance between all possible asteroid pairs, as a function of time (typical values are of the order of 50 m/s, significantly larger values would indicate that the family was dynamically dispersed beyond recognition).
Our results for the Koronis family are shown in Fig. 4, where we display in panel A the time evolution of the fraction of all asteroids remaining in the Koronis family region (blue line) and of the objects with km (green line). Panel B shows the time evolution of the exponent, while panel C displays as a function of time. The Koronis family is in a dynamically less active region, so a larger fraction of its original population survive the simulation, but this is not the case for all investigated families. Our results at Gyr, the minimum estimated age for the end of the late heavy bombardment, are reported in Table 5.
Name  Fraction of surviving  Fraction of surviving  

asteroids (all sizes) [%]  asteroids ( km) [%]  [m/s]  
Eunomia  36.8  53.3  1.7  77.3 
Maria  72.7  92.0  1.7  63.5 
Koronis  76.2  88.3  1.8  42.2 
Themis  69.5  89.4  2.0  92.3 
Meliboea  9.0  16.6  2.4  375.2 
Ursula  65.1  92.1  2.0  137.4 
Alauda  22.1  8.5  1.9  171.4 
Overall, the Meliboea and Alauda synthetic families were dispersed beyond recognition. All families had values of at Gyr much shallower than the initial value (3.6), and compatible with typical values of background asteroids (). This does not necessarily mean that all paleofamilies should be characterized by a shallow SFD. The initial SFD could have been much steeper, and collisionary evolution could have replenished the population of asteroids at smaller sizes. There are indeed some indications that some potential paleofamilies, such as Itha, could be characterized by a relatively steep SFD (Brož et al. 2012). However, dynamical effects alone indeed tend to remove smaller size bodies and to produce families with shallower SFD. In regimes where dynamical effects are prodominant and the initial SFD was not too steep, we would expect paleofamilies to be characterized by a shallow SFD. Also, according to the values of found in this work, only the synthetic Koronis, Maria, and possibly Eunomia family would be recognizable, with some difficulties, with respect to the background (typical values of depends on the local density of asteroids, but are usually of the order of 5060 m/s). To help visualize the difference between a completely dispersed family, such Alauda, and a relatively well preserved one, such Maria, we show in Fig. 5 a projection in the of the outcome of our simulations at Gyr. While only a handful of the largest members of Alauda survived up to this time, the simulated Maria family, while dispersed, could still be recognizable in this domain.
Can any paleofamily still be observable today? As we discussed, paleofamilies would be difficult to recognize with traditional methods such as HCM, being characterized by a shallow SFD, a significant depletion in small family members (those less than 5 km in diameter), and a large spread among the surviving members. Paleofamilies belonging to fairly typical taxonomical classes, such as C and Stype, would be extremely hard to recognize. It was however proposed that Vtype asteroids in the Eunomia orbital region could have been fragments of a paleoEunomia family associated with the disruption of Eunomia parent body crust (Carruba et al. 2007, 2014b). We checked the value of the 16 Vtype photometric candidates SFD currently in the Eunomia orbital region (defined according to our box criteria), and we found a value of 1.95. While this result should be considered with caution, given the limited number of Vtype asteroids in the region and possible limitations caused by observational incompleteness, the very shallow SFD of these objects suggests, in our opinion, that an origin from a paleoEunomia family is not incompatible with the results of this work.
5 Conclusions
In this work we:

Identified members of the seven old “Brož” families in a new domain of proper elements, slope and colors. Once taxonomical and dynamical interlopers were removed, preliminary estimates of the maximum possible family ages were obtained.

Used a “YarkoYORP” Monte Carlo approach (Carruba et al. 2015) that includes the effects of the stochastic YORP effect of Bottke et al. (2015) and past changes in values of the solar luminosity to obtain refined estimates of the family ages, when possible. Our nominal age estimates are lower than results of other groups that did not consider the stochastic YORP effect, as expected, but compatible to within the uncertainties. Even allowing for the maximum possible value in the thermal conductivity of the simulated families, no CXcomplex group could have a nominal age dating from the latest phases of the LHB.

Simulated with the symplectic integrators (Carruba et al. 2015) that accounts for the Yarkovsky and stochastic YORP effects, and past changes in solar luminosity, the dynamical evolution of members of fictitious original seven “Brož” families. Under the assumptions of our model (no collisional evolution, and an initial SFD with a exponent for the population of objects with km equal to 3.6), any “paleofamily” that formed between 2.7 and 3.8 Gyr ago would be characterized by a very shallow sizefrequency distribution, a depletion in smallest ( km) members, and a significant spread among the surviving fragments. Only families in dynamically less active regions, such as the Koronis family in the pristine zone of the main belt, could have potentially partially survived 3.8 Gyr of dynamical evolution and not be completely dynamically eroded. The Vtype asteroids in the Eunomia orbital region are characterized by a very shallow SFD, and could potentially be compatible with a paleoEunomia family, as suggested in the past (Carruba et al. 2007, 2014b).
Overall, the main result of this work is that some paleofamilies, particularly the initially most numerous ones, or those in dynamically less active parts of the main belt, could still be visible today, but would be of rather difficult identification, especially for the case of families belonging to fairly typical taxonomical types, such as the C and Stypes. Other effects not considered in this work, such as collisional cascading or comminution (Brož et al. 2013), close encounters with massive asteroids (Carruba et al. 2003), secular dynamics involving massive asteroids (Novaković et al. 2015) could all have contributed to further disperse paleofamily members, perhaps beyond recognition. Yet the quest for the identification of a paleofamily remain, in our opinion, a very worthy subject of research in asteroid dynamics. If such family could be found, such as is possibly the case for the Vtype asteroids in the Eunomia orbital region, it could provide precious clues about a very early stage of our Solar System. Finding and identifying paleofamilies remains therefore a very value line of research in asteroid dynamics.
Acknowledgments
We thank the reviewer of this paper, Miroslav Brož, for comments and suggestions that improved the quality of this work. We would like to thank the São Paulo State Science Foundation (FAPESP) that supported this work via the grants 14/067622 and 14/240717), and the Brazilian National Research Council (CNPq, grant 305453/20114). DN acknowledges support from the NASA Solar System Working (SSW) program. The first author was a visiting scientist at the Southwest Research Institute in Boulder, CO, USA, when this article was written. This publication makes use of data products from the Widefield Infrared Survey Explorer WISE and NEOWISE, which are a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
References
 beauge (2001) Beaugé, C., Roig, F., 2001, Icarus, 153, 391.
 Bendjoya (2002) Bendjoya, P., Zappalà, V. 2002. Asteroids III, Univ. of Arizona Press, Tucson, 613.
 Bottke (2012) Bottke, W. F., and 7 coauthors, 2012, Nature, 485, 78.
 Bottke (2014) Bottke, W. F., and 10 coauthors, 2015, Icarus, 247, 191.
 Brazil (2015) Brazil, P. I., Nesvorný, D., Roig, F., Carruba, V., Aljbaae, S., Huaman, M. E. 2015, Icarus, 266, 142.
 Broz (1999) Brož, M., 1999, Thesis, Charles Univ., Prague, Czech Republic.
 Broz et al. (2012) Broz, M., Cibulkova, H., & Rehak, M. 2012, AAS/Division for Planetary Sciences Meeting Abstracts, 44, 415.05.
 Broz (2013) Brož, M., Morbidelli, A., Bottke, W. F., Rozenhal, J., Vokrouhlický, D, and Nesvorný, D. 2013, A&A, 551, A117.
 Carruba (2003) Carruba, V., Burns, J., A., Bottke, W., Nesvorný, D. 2003, Icarus, 162, 308.
 Carruba (2007) Carruba, V., Michtchenko, T. A., Lazzaro, D, 2007, A&A, 473, 967.
 Carruba (2010a) Carruba, V., MNRAS, 2010a, 403, 1834.
 Carruba (2010b) Carruba, V., MNRAS, 2010b, 408, 580.
 Carruba (2011) Carruba, V., Machuca, J. F., Gasparino, H. P., MNRAS, 2011, 412, 2052.
 Carruba (2013) Carruba, V., Domingos, R. C., Nesvorný, D., Roig, F., Huaman, M. E., Souami, D., 2013, MNRAS, 433, 2075.
 Carruba (2014a) Carruba, V., Domingos, R. C., Huaman, M. E., Dos Santos, C. R., Souami, D., 2014a, MNRAS, 437, 2279.
 Carruba (2014b) Carruba, V., Huaman, M. E., Domingos, R. C., Dos Santos, C. R., Souami, D., 2014b, MNRAS, 439, 3168.
 Carruba (2015) Carruba, V., Nesvorný, D., Aljbaae, S., Huaman, M. E., 2015, MNRAS, 451, 4763.
 DeMeo (2013) DeMeo, F., Carry, B. 2013, Icarus, 226, 723.
 Ivezic (2002) Ivezić, Ž, and 34 coauthors, 2002, AJ, 122, 2749.
 Knezevic (2003) Knežević, Z., Milani, A., 2003, A&A, 403, 1165.
 Masiero (2012) Masiero, J. R., Mainzer, A. K., Grav, T., Bauer, J. M., and Jedicke, R., 2012, APJ, 759, 14.
 Neese (2010) Neese, C., 2010, Asteroid Taxonomy, NASA Planetary Data System, eARA5DDRTAXONOMYV6.0.
 Nesvorny (2013) Nesvorný, D., Vokrouhlický, D., Morbidelli, A., 2013, APJ, 768, 45.
 Nesvorny (2015) Nesvorný, D., Brož, M., Carruba, V., 2015, In Asteroid IV, (P. Michel, F. E.De Meo, W. Bottke Eds.), Univ. Arizona Press and LPI, 297.
 Novakovic (2015) Novaković, B., Maurel, C., Tsirvoulis, G., & Knežević, Z. 2015, ApJl, 807, L5.
 Press (2001) Press, V.H., Teukolsky, S. A., Vetterlink, W. T., Flannery, B. P., 2001, Numerical Recipes in Fortran 77, Cambridge Univ. Press, Cambridge.
 Spoto (2015) Spoto, F., Milani, A. Knežević, Z. 2015, Icarus, 257, 275.
 ( (2006a)) Vokrouhlický, D., Brož, M., Morbidelli, A., et al. 2006a, Icarus, 182, 92.
 Vokrouhlicky (2006b) Vokrouhlický D., Brož, M., Bottke, W. F., Nesvorný, D., Morbidelli, A. 2006b, Icarus, 182, 118.
 Vokrouhlicky (2006c) Vokrouhlický D., Brož, M., Bottke, W. F., Nesvorný, D., Morbidelli, A. 2006c, Icarus, 183, 349.
 Wright (2010) Wright, E. L., and 38 coauthors, 2010, AJ, 140, 1868.
 Zappala (1990) Zappalà, V., Cellino, A., Farinella, P., Knežević. Z., 1990, AJ, 100, 2030.
 Zappala (1995) Zappalà, V., Bendjoya, Ph., Cellino, A., Farinella, P., Froeschlé, C., 1995, Icarus, 116, 291.