Constraining the Cometary Flux Through the Asteroid Belt During The

Home , 107 Camilla, 121 Hermione, 363 Padua, 375 Ursula, 5 Astraea, 785 Zwetana

arXiv:1301.6221v1 [astro-ph.EP] 26 Jan 2013 ais fti iwi orc,w a s tde flnran lunar of studies use lunar can all we encompass correct, not is does view mean and this would extent This If in 2012). basins. limited al. al. et is et Morbidelli LHB Bottke 2012, 2009, the al. Malhotra et & Marchi Minton 2010, 2005, al. b et main Gomes primordial bas the lunar of e so the out by driven or asteroids 12–15 by b last created made were the mainly were while basins planetesimals, oldest the leftover solution: best the 200 be al. may et Ryder 1974, al. tha et impacts Tera argued of (e.g., spike Ga school a 3.9 second near during see place The made took were review). 2007; basins a 2000, lunar for al. most that 2007 et al. Hartmann et 2001, Chapman al. plan et terrestrial (Neukum from over mation left were planetesimals ones, argue young impacting the school by including first basins, The lunar all end-memb LHB. nearly two the under- that were describing been there thought has past, of however, the schools LHB, In the revisions. recent of going extent Orienta or and and km Imbrium sources like 300 Moon (a The the basins on impact process crater) young the diameter relatively larger as but defined often huge is the It made system. th solar in the period of important an history is (LHB) bombardment heavy late The Introduction 1. ue1,2021 15, June srnm Astrophysics & Astronomy eetsuis oee,sgetta opoiescenari compromise a that suggest however, studies, Recent e words. con Key not does families asteroid of distribution observed the suiga“tnad iefeunydsrbto fprim of distribution size-frequency “standard” a Assuming 2fmle hc a erltdt h H codn otheir to according LHB the to related be may which families 12 vacliinlcsae,i)dsutoso oesbelow comets of disruptions ii) cascade), collisional (via oee,mr hn10atri aiiswith families asteroid 100 than more However, iceac a envrhls xlie ytefollowing the by explained nevertheless be can discrepancy ie h reo ntecmtr-irpinlw ecannot we law, cometary-disruption the in freedom the Given nsmmjrai,catcdi chaotic axis, semimajor in olsosbtencmt,mi-etatris n famil and asteroids, ev main-belt dynamical comets, long-term between the collisions into insight gain c to dynamical simulations and i data as albedo families asteroid data, observed colour of method, list clustering updated an present We LHB. the during flux seod.I atclr ewn ocekwehrteobser the whether check to want we particular, In asteroids. iprino rnnpuincmtr ik esuye study We disk. cometary transneptunian a of dispersion nteNc oe,telt ev obrmn LB srelate is (LHB) bombardment heavy late the model, Nice the In ??? accepted ???; Received 3 2 1 D PB -al otebudrsr.d,[email protected]. Institute [email protected], Research e-mail: Southwest Studies, Space of Department bevtied aCoedAu,B 29 60 ieCdx4, Cedex Nice 06304 4229, BP Côte d’Azur, la de Observatoire nttt fAtooy hre nvriy rge Hole [email protected] V e-mail: Prague, University, Charles Astronomy, of Institute ff cso aegatpae irto Tiai ta.2005, al. et (Tsiganis migration giant-planet late of ects ≥ osriigtecmtr u hog h seodbelt asteroid the through flux cometary the Constraining 0 m–cetddrn h H n subsequent and LHB the during created – km 200 eeta ehnc io lnt,atris eea c – general asteroids: planets, minor – mechanics celestial .BroˇzM. 1 .Morbidelli A. , aucitn.fams no. manuscript uigtelt ev bombardment heavy late the during ff so neccentricity in usion ff cn.z oenlosraoyc,[email protected] [email protected], .cuni.cz, 2 ..Bottke W.F. , D / nlnto,o osbeprubtosb h giant-plane the by perturbations possible or inclination, PB ≥ ≃ oeciia eieindsac ( distance perihelion critical some 0 msol ecetda h aetm hc r o observe not are which time same the at created be should km 100 tfor- et ebr,tepyia irpin fcmt,teYarkovsk the comets, of disruptions physical the members, y rdc ihacmtr LHB. cometary a with tradict edu made y fcliinleouin–wihsescnitn with consistent seems which – evolution collisional of Gyr 4 e olsoa aiispoiealwro nuprlmtfo limit upper an or lower a provide families collisional ved ff rilcmt,w rdc h ubro aiiswt parent with families of number the predict we comets, ordial lto fsnhtcLBfmle vr4Gr eacutfrt for account We Gyr. 4 over families LHB synthetic of olution ABSTRACT 3 that rcse:i seodfmle r e are families asteroid i) processes: cspoue yteehpteia oeaypoetlso projectiles cometary hypothetical these by produced ects .Rozehnal J. , s vˇ kah2 80 rge8 zc Republic, Czech ˇsoviˇckách 8, Prague 18000 2, ins etfidi h pc fsnhtcpoe lmnsb h hie the by elements proper synthetic of space the in dentified 00Wlu t,Sie30 ole,C 00,USA, 80302, CO Boulder, 300, Suite St., Walnut 1050 , 0). nieain n eetmt hi hsclprmtr.W parameters. physical their estimate we and onsiderations elt rvd tign iiso h oeayflx u ecnconc can we but flux, cometary the on limits stringent provide le. oa ria ntblt fgatpaeswihcue fa a causes which planets giant of instability orbital an to d er d y o d e t yaia gs ete sdcliinlmdl n -oyo N-body and models collisional used then We ages. dynamical rne -al [email protected] e-mail: France, h on nfc,gvnteepce oa asi h origin the in mass total on expected traces the obvious given no fact, be In to Moon. seems the there which of bombardment a been only 2010). have al. would et boundaries (Morbidelli contributor ast belt minor the main while 2012), current al. the et recorde (Bottke from as record LHB, the crater towards for lunar source 2012). the belt a al. in of main et enough (Bottke been the have edge could inner of last current extension its 20 of putative al. inwards et a Mars, (Morbidelli from belt a those main et the and (Gomes from disk asteroids transneptunian the original the 2005), the to from contributed comets In populations the review). gravitatio projectile a for the three 2010 scenario, by Morbidelli transneptu (see massive driven planetesimals a turn of and disk in planets insta said planets, th orbital between giant “saw- model, dynamical interactions a the late this rather of a to or by bility According spike triggered impact 2012). was an al. bombardment as et LHB (Morbidelli the tooth” of origin the of 2012). Nemchin 2009, & al. Norman et 2012, Swindle al. 2011, et 1995, (Bogard Bottke bill Moon 3.7–3.8 the to port on 4.1–4.2 ago basin-forming approximately years the from lasted that LHB suggest the of to dyn work, with together modelling events, ical impact by heated samples asteroid mt:gnrl–mtos numerical methods: – general omets: h iemdl oee,peit eyitnecometary intense very a predicts however, model, Nice The explanation coherent a provides model’ ‘Nice so-called The 1 .Vokrouhlický D. , q . 1 ffi . U r common. are AU) 5 inl etoe ycomminution by destroyed ciently 1 .Nesvorný D. , migration. t 3 h cometary the r y observations. tdynamical st / main-belt n OPdrift YORP bd sizes -body selected e emutual he

c rarchical uethat lude .This d. S 2021 ESO rbital eroids LHB: nian The am- this 10) ion ion nal al l. d 1 e - M. Broˇzet al.: Constraining the cometary flux through the asteroid belt during the late heavy bombardment transneptunian disk (Gomes et al. 2005) and the size distribution Finally, in Section 10 we analyse a curious portion of the of objects in this disk (Morbidelli et al. 2009), the Nice model main belt, located in a narrow semi-major axis zone bounded by predicts that about 5 × 104 km-size comets should have hit the the 5:2 and 7:3 resonances with Jupiter. This zone is severely Moon during the LHB. This would have formed 20km craters deficient in small asteroids compared to the other zones of the with a surface density of 1.7 × 10−3 craters per km2. But the main belt. For the reasons explained in the section, we think that highest crater densities of 20km craters on the lunar highlands this zone best preserves the initial asteroid belt population, and is less than 2 × 10−4 (Strom et al. 2005). This discrepancy might therefore we call it the “pristine zone”. We checked the num- be explainedby a grossoverestimate of the numberof small bod- ber of families in the pristine zone, their sizes, and ages and we ies in the original transneptunian disk in Morbidelli et al. (2009). found that they are consistent with the number expected in our However, all impact clast analyses of samples associated to ma- model invoking a cometary bombardmentat the LHB time and a jor LHB basins (Kring and Cohen 2002, Tagle 2005) show that subsequent collisional comminutionand dispersion of the family also the major projectiles were not carbonaceous chondrites or members. similar primitive, comet-like objects. The conclusions follow in Section 11. The lack of evidence of a cometary bombardment of the Moon can be considered as a fatal flaw in the Nice model. Curiously, however, in the outer solar system we see evidence 2. A list of known families of the cometary flux predicted by the Nice model. Such a flux is Although several lists of families exist in the literature (Zappalá consistent with the number of impact basins on Iapetus (Charnoz et al. 1995, Nesvornýet al. 2005, Parker et al. 2008, Nesvorný et al. 2009), with the number and the size distribution of the 2010), we are going to identify the families once again. The rea- irregular satellites of the giant planets (Nesvornýet al. 2007, son is that we seek an upper limit for the number of old families Bottke et al. 2010) and of the Trojans of Jupiter (Morbidelli et that may be significantly dispersed and depleted, while the pre- al. 2005), as well as with the capture of D-type asteroids in the vious works often focussed on well-defined families. Moreover, outer asteroid belt (Levison et al., 2009). Moreover, the Nice we need to calculate several physical parameters of the fami- model cometary flux is required to explain the origin of the col- lies (such as the parent-body size, slopes of the size-frequency lisional break-up of the asteroid (153) Hilda in the 3/2 resonance distribution, a dynamical age estimate if not available in the liter- with Jupiter (located at ≃ 4 AU, i.e. beyond the nominal outer ature) which are crucial for further modelling. Last but not least, border of the asteroid belt at ≃ 3.2 AU; Broˇzet al. 2011). we use more precise synthetic proper elements from the AstDyS Missing signs of an intense cometary bombardment on the database (Kneˇzević& Milani 2003, version Aug 2010) instead Moon and the evidence for a large cometary flux in the outer of semi-analytic ones. solar system suggest that the Nice model may be correct in its We employed a hierarchical clustering method (HCM, basic features, but most comets disintegrated as they penetrated Zappaláet al. 1995) for the initial identification of families in deep into the inner solar system. the proper element space (ap, ep, sin Ip), but then we had to per- To support or reject this possibility, this paper focusses on the form a lot of manual operations, because i) we had to select a main asteroid belt, looking for constraints on the flux of comets reasonable cut-off velocity vcutoff, usually such that the number throughthis region at the time of the LHB. In particularwe focus of members N(vcutoff) increases relatively slowly with increas- on old asteroid families, produced by the collisional break-up ing vcutoff. ii) The resulting family should also have a “reason- of large asteroids, which may date back at the LHB time. We able” shape in the space of proper elements, which should some- provide a census of these families in Section 2. how correspondto the local dynamical features.1 iii) We checked In Section 3, we construct a collisional model of the main taxonomic types (colour indices from the Sloan DSS MOC cat- belt population. We show that, on average, this population alone alogue version 4, Parker et al. 2008), which should be consistent could not have produced the observed number of families with among family members. We can recognise interlopers or over- DPB = 200–400km. Instead, the required number of families lapping families this way. iv) Finally, the size-frequency distri- with large parent bodies is systematically produced if the aster- bution should exhibit one or two well-defined slopes, otherwise oid belt was crossed by a large number of comets during the the cluster is considered uncertain. LHB, as expected in the Nice model (see Section 4). However, Our results are summarised in online Tables 1–3 and the po- for any reasonable size distribution of the cometary population, sitions of families within the main belt are plotted in Figure 1. the same cometary flux that would producethe correct numberof Our list is “optimistic”, so that even not very prominent families 2 families with DPB = 200–400km would produce too many fam- are included here. ilies with DPB ≃ 100km relative to what is observed. Therefore, There are, however, several potential problems we are aware in the subsequent sections we look for mechanisms that might of: prevent detection of most of these families. More specifically, in Sec. 5 we discuss the possibility that 1. There may be inconsistencies among different lists of families. For example, sometimes a clump may be regarded as families with DPB ≃ 100km are so numerous that they cannot be identified because they overlap with each other. In Sec.6 a single family or as two separate families. This may be the we investigate their possible dispersal below detectability due case of: Padua and Lydia, Rafita and Cameron. to the Yarkovsky effect and chaotic diffusion. In Sec. 7 we dis- 2. To identify families we used synthetic proper elements, cuss the role of the physical lifetime of comets. In Sec. 8 we which are more precise than the semi-analytic ones. analyse the dispersal of families due to the changes in the or- Sometimes the families look more regular (e.g., Teutonia) bits of the giant planets expected in the Nice model. In Sec. 9 1 For example, the Eos family has a complicated but still reasonable we consider the subsequent collisional comminution of the fam- shape, since it is determined by several intersecting high-order mean- ilies. Of all investigated processes, the last one seems to be the motion or secular resonances, see Vokrouhlickýet al. (2006). most promising for reducing the number of visible families with 2 On the other hand, we do not include all of the small and less- DPB ≃ 100km while not affecting the detectability of old fami- certain clumps in a high-inclination region as listed by Novakovićet al. lies with DPB = 200–400km. (2011). We do not focus on small or high-I families in this paper.

2 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

families background

Fig. 1. Asteroids from the synthetic AstDyS catalogue plotted in the proper semimajor axis ap vs proper eccentricity ep (top panels) and ap vs proper inclination sin Ip planes (bottom panels). We show the identified asteroid families (left panels) with the positions of the largest members indicated by red symbols, and also remaining background objects (right panels). The labels correspond to designations of the asteroid families that we focus on in this paper. There are still some structures consisting of small objects in the background population, visible only in the inclinations (bottom right panel). These “halos” may arise for two reasons: i) a family has no sharp boundary and its transition to the background is smooth, or ii) there are bodies escaping from the families due to long-term dynamical evolution. Nevertheless, we checked that these halo objects do not significantly affect our estimates of parent-body sizes.

or more tightly clustered (Beagle) when we use the syn- members as a first guess of the parent-body size, is essen- thetic elements. This very choice may, however, affect re- tially similar to our approach. sults substantially! A clear example is the Teutonia family, which also contains the big asteroid (5) Astraea if the syn- A complete list of all families’ thetic proper elements are used, but not if the semi-analytic members is available at our web site proper elements are used. This is due to the large differences http://sirrah.troja.mff.cuni.cz/˜mira/mp/fams/, between the semi-analytic and synthetic proper elements of including supporting figures. (5) Astraea. Consequently, the physical properties of the two families differ considerably. We believe that the family de- 2.1. A definition of the production function fined from the synthetic elements is more reliable. 3. Durda et al. (2007) often claim a larger size for the parent To compare observed families to simulations, we define a “pro- body (e.g., Themis, Meliboea, Maria, Eos, Gefion), because duction function” as the cumulative number N(>D) of fami- they try to match the SFD of larger bodies and the results of lies with parent-body size DPB larger than a given D. The ob- SPH experiments. This way they also account for small bod- served production function is shown in Figure 2, and it is worth ies that existed at the time of the disruption,but which do not noting that it is very shallow. The number of families with exist today since they were lost due to collisional grinding DPB ≃ 100km is comparable to the number of families in the and the Yarkovsky effect. We prefer to use DDurda instead of DPB = 200–400km range. the value DPB estimated from the currently observed SFD. It is important to note that the observed production function The geometric method of Tanga et al. (1999), which uses is likely to be affected by biases (the family sample may not be the sum of the diameters of the first and third largest family complete, especially below DPB . 100km) and also by long- term collisional/dynamical evolution which may prevent a de-

3 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

103 all observed families Table 4. Nominal thermal parameters for S and C/X taxonomic catastrophic disruptions D types of asteroids: ρbulk denotes the bulk density, ρsurf the sur- 2 + qtarget + 5/3 qproject > face density, K the thermal conductivity, Cth the speciﬁc thermal

PB 100 D capacity, ABond the Bond albedo and ǫ the infrared emissivity. with 10 type ρbulk ρsurf K Cth ABond ǫ 3 3 families (kg/m ) (kg/m ) (W/m/K) (J/kg/K) N S 2500 1500 0.001 680 0.1 0.9 1 C/X 1300 1300 0.01 680 0.02 0.9 10 100 1000 D [km]

Fig. 2. A production function (i.e. the cumulative number N(>D) where ac denotes the centre of the family, and C is the parameter. of families with parent-body size DPB larger than D) for all Such relation can be naturally expected when the semimajor-axis observed families (black) and families corresponding to catas- drift rate is inversely proportional to the size, da/dt ∝ 1/D, and trophic disruptions (red), i.e. with largest remnant/parent body the size is related to the absolute magnitude via the Pogson equa- mass ratio lower than 0.5. We also plot a theoretical slope ac- 2 2 tion H = −2.5log10(pV D /D0), where D0 denotes the reference cording to Eq. (1), assuming qtarget = −3.2 and qproject = −1.2, diameter and pV the geometric albedo (see Vokrouhlickýet al. which correspondto the slopes of the main belt population in the 2006 for a detailed discussion). The limiting value, for which range D = 100–200km and D = 15–60km, respectively. all Eos family members (except interlopers) are above the corresponding curve, is C = 1.5to2.0 × 10−4 AU. Assuming reasonable thermal parameters (summarised in Table 4), we calcu- tection of old comminutioned/dispersed families today (Marzari late the expected Yarkovsky drift rates da/dt (using the theory et al. 1999). from Broˇz2006) and consequently can determine the age to be From the theoretical point of view, the slope q of the produc- t < 1.5to2.0 Gyr. tion function N(>D) ∝ Dq should correspond to the cumulative The second method uses a histogram N(C, C + ∆C) of the slopes of the size-frequency distributions of the target and pro- number of asteroids with respect to the C parameter defined jectile populations. It is easy to show3 that the relation is above, which is fitted by a dynamical model of the initial velocity field and the Yarkovsky/YORP evolution. This enables us 5 q = 2 + qtarget + qproject . (1) to determine the lower limit for the age too (so the resulting age 3 +0.15 estimate is t = 1.3−0.2 Gyr for the Eos family). Of course, real populations may have complicated SFDs, with In the third case, we start an N-body simulation using a mod- different slopes in different ranges. Nevertheless, any popula- ified SWIFT integrator (Levison and Duncan 1994), with the Yarkovsky/YORP acceleration included, and evolve a synthetic tions that have a steep SFD (e.g. qtarget = qproject = −2.5) would inevitably produce a steep production function (q −4.7). family up to 4Gyr. We try to match the shape of the observed In the following analysis, we drop cratering events and family in all three proper orbital elements (ap, ep, sin Ip). In prin- discuss catastrophic disruptions only, i.e. families which have ciple, this method may provide a somewhat independent esti- largest remnant/parent body mass ratio less than 0.5. The rea- mate of the age. For example, there is a ‘halo’ of asteroids in the son is that the same criterion LR/PB < 0.5 is used in colli- surroundings of the nominal Eos family, which are of the same sional models. Moreover,cratering events were not yet systemat- taxonomic type K, and we may fit the ratio Nhalo/Ncore of the ically explored by SPH simulations due to insufficient resolution number of objects in the ‘halo’ and in the family ‘core’ (Broˇz et (Durda et al. 2007). al., in preparation). The major source of uncertainty in all methods are unknown bulk densities of asteroids (although we use the most likely val- 2.2. Methods for family age determination ues for the S or C/X taxonomic classes, Carry 2012). The age scales approximately as t ∝ ρ . Nevertheless, we are still able If there is no previous estimate of the age of a family, we used bulk to distinguish families that are young from those that are old, be- one of the following three dynamical methods to determine it: cause the allowed ranges of densities for S-types (2 to 3g/cm3) i) a simple (a , H) analysis as in Nesvornýet al. (2005); ii) a C- p and C/X-types(1 to 2g/cm3) are limited (Carry 2012)and so are parameter distribution fitting as introduced by Vokrouhlickýet the allowed ages of families. al. (2006); iii) a full N-body simulation described e.g. in Broˇzet al. (2011). In the first approach, we assume zero initial velocities, 2.3. Which families can be of LHB origin? and the current extent of the family is explained by the size- dependent Yarkovsky semimajor axis drift. This way we can ob- The ages of the observed families and their parent-body sizes tain only an upper limit for the dynamical age, of course. We are shown in Figure 4. Because the ages are generally very un- show an example for the Eos family in Figure 3. The extent of certain, we consider that any family whose nominal age is older the family in the proper semimajor axis vs the absolute magni- than 2Gyr is potentially a family formed ∼4Gyr ago, i.e. at the LHB time. If we compare the number of “young” (<2Gyr) and tude (ap, H) plane can be described by the parametric relation old families (>2 Gyr) with DPB = 200–400km, we cannot see |ap − ac| a significant over-abundance of old family formation events. On 0.2H = log , (2) 10 C the other hand, we almost do not find any small old families. Only 12 families from the whole list may be possibly dated 3 ⋆ 2 Assuming that the strength is approximately QD ∝ D in the gravity back to the late heavy bombardment, because their dynamical ⋆ 1/3 ∼ regime, the necessary projectile size d ∝ (QD) D (Bottke et al. 2005), ages approach 3.8Gyr (including the relatively large uncer- and the number of disruptions n ∝ D2Dqtarget dqproject . tainties; see Table 5, which is an excerpt from Tables 1–3).

4 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

− ∆C = 0.1 × 10 4 AU Table 5. Old families with ages possibly approaching the LHB. 16 They are sorted according to the parent body size, where DDurda determined by the Durda et al. (2007) method is preferred to 14 the estimate DPB inferred from the observed SFD. An additional ‘c’ letter indicates that we extrapolated the SFD down to D = 12

/ mag 0 km to accountfor small (unobserved)asteroids, an exclamation H − mark denotes a signiﬁcant mismatch between DPB and DDurda. 10 C = (1.5 to 2.0) × 10 4 AU Eos family, v = 50 m/s 8 cutoff designation DPB DDurda note (221) likely interlopers (km) (km) 2.95 3 3.05 3.1 3.15 24 Themis 209c 380–430! 10 Hygiea 410 442 cratering ap / AU 15 Eunomia 259 292 cratering Fig. 3. An example of the Eos asteroid family, shown on the 702 Alauda 218c 290–330! high-I proper semimajor axis ap vs absolute magnitude H plot. We 87 Sylvia 261 272 cratering 137 Meliboea 174c 240–290! also plot curves deﬁned by equation (2) and parameters ac = 3.019 AU, C = 1.5to2.0 × 10−4 AU, which is related to the up- 375 Ursula 198 240–280 cratering 107 Camilla >226 - non-existent per limit of the dynamical age of the family. 121 Hermione >209 - non-existent 158 Koronis 122c 170–180 young old 500 709 Fringilla 99c 130–140 cratering 10 24 170 Maria 100c 120–130

400 221

300 15 value is a simpliﬁcation, but about 90 % collisions have mutual 87 [km] 137 375 velocities between 2 and 8km/s (Dahlgren 1998), which assures 3556 3 107 PB 200 121 D 158 a similar collisional regime. 709 170 Thescaling law is described by the polynomialrelation (r de- 100 notes radius in cm)

0 ⋆ 1 a b 0 0.5 1 1.5 2 2.5 3 3.5 4 Q (r) = Q r + Bρr (3) D q 0 age [Gyr] fact with the parameters corresponding to basaltic material at 5 km/s Fig. 4. The relation between dynamical ages of families and the (Benz & Asphaug 1999): sizes of their parent bodies. Red labels correspond to catastrophic disruptions, while cratering events are labelled in black. Some of the families are denoted by the designation of the ρ Q0 a B bqfact largest member. The uncertainties of both parameters are listed (g/cm3) (erg/g) (erg/g) in Tables 1–3 (we do not include overlapping error bars here for 3.0 7 × 107 −0.45 2.1 1.19 1.0 clarity). Even though not all asteroids are basaltic, we use the scaling law above as a mean one for the main-belt population. Below, we If we drop cratering events and the families of Camilla discuss also the case of significantly lower strengths (i.e. higher and Hermione, which do not exist any more today (their exis- q values). tence was inferred from the satellite systems, Vokrouhlickýet fact We selected the time span of the simulation 4 Gyr (not al. 2010), we end up with only five families created by catas- 4.5Gyr) since we are interested in this last evolutionary phase trophic disruptions that may potentially date from the LHB time of the main belt, when its population and collisional activity is (i.e. their nominal age is more than 2Gy). As we shall see in nearly same as today (Bottke et al. 2005). The outcome of a sin- Section 4, this is an unexpectedly low number. gle simulation also depends on the “seed” value of the random- Moreover, it is really intriguing that most “possibly-LHB” number generator that is used in the Boulder code to decide families are larger than D ≃ 200km. It seems that old fam- PB whether a collision with a fractional probability actually occurs ilies with D ≃ 100km are missing in the observed sample. PB or not in a given time step. We thus have to run multiple simula- This is an important aspect that we have to explain, because it tions (usually 100) to obtain information on this stochasticity of contradicts our expectation of a steep production function. the collisional evolution process. The initial SFD of the main belt population conditions was 3. Collisions in the main belt alone approximated by a three-segment power law (see also thin grey line in Figure 5, 1st row) with differential slopes qa = −4.3 (for Before we proceed to scenarios involving the LHB, we try to D > D1), qb = −2.2, qc = −3.5 (for D < D2) where the size explain the observed families with ages spanning 0–4Gyr as a ranges were delimited by D1 = 80km and D2 = 16 km. We also result of collisions only among main-belt bodies. To this pur- added a few big bodies to reflect the observed shape of the SFD pose, we used the collisional code called Boulder (Morbidelli at large sizes (D > 400km). The normalisation was Nnorm(D > et al. 2009) with the following setup: the intrinsic probabilities D1) = 350 bodies in this case. −18 −2 −1 Pi = 3.1 × 10 km yr , and the mutual velocities Vimp = We used the observed SFD of the main belt as the first con- 5.28km/s for the MB vs MB collisions (both were taken from straint for our collisional model. We verified that the outcome the work of Dahlgren 1998). The assumption of a single Vimp our model after 4Gyr is not sensitive to the value of qc. Namely,

5 M. Broˇzet al.: Constraining the cometary flux through the asteroid belt during the late heavy bombardment a change of qc by as much as ±1 does not affect the final SFD in 15

any significant way. On the other hand, the values of the remain- / km/s –1 10 yr ing parameters (qa, qb, D1, D2, Nnorm) are enforced by the ob- imp V –2 5 served SFD. To obtain a reasonable fit, they cannot differ much 0.006

km V imp 0 mean (by more than 5–10%) from the values presented above. –18 0.004 Pi / 10 We do not use only a single number to describe the num- i P 0.002 ber of observed families (e.g. N = 20 for DPB ≥ 100km), but

we discuss a complete production function instead. The results mean 0 0 20 40 60 80 100 in terms of the production function are shown in Figure 5 (left t / Myr column, 2nd row). On average, the synthetic production function is steeper and below the observed one, even though there is Fig. 6. The temporal evolution of the intrinsic collisional proba- approximately a 5% chance that a single realization of the com- bility Pi (bottom) and mean collisional velocity Vimp (top) com- puter model will resemble the observations quite well. This also puted for collisions between cometary-disk bodies and the main- holds for the distribution of DPB = 200–400km families in the belt asteroids. The time t = 0 is arbitrary here; the sudden in- course of time (age). crease in Pi values corresponds to the beginning of the LHB.

In this case, the synthetic production function of DPB & 100 km families is not significantly affected by comminution. 4.1. Simple stationary model According to Bottke et al. (2005), most of D > 10 km fragments In a stationary collisional model, we choose an SFD for the survive intact and a DPB & 100km family should be recognis- able today. This is also confirmed by calculations with Boulder cometary disk, we assume a current population of the main belt; estimate the projectile size needed to disrupt a given target ac- (see Figure 5, left column, 3rd row). cording to (Bottke et al. 2005) To improve the match between the synthetic and the ob- ⋆ 2 1/3 served production function, we can do the following: i) mod- ddisrupt = 2QD/Vimp Dtarget , (4) ify the scaling law, or ii) account for a dynamical decay of the where Q⋆ denotes the specific energy for disruption and disper- MB population. Using a substantially lower strength (qfact = 5 D in Eq. (3), which is not likely, though) one can obtain a synthetic sion of the target (Benz & Asphaug 1999); and finally calculate production function which is on average consistent with the ob- the number of events during the LHB as servations in the DPB = 200–400km range. 2 Dtarget n = n P (t) n (t)dt , (5) Regarding the dynamical decay, Minton & Malhotra (2010) events 4 target Z i project suggest that initially the MB was three times more populousthan today while the decay timescale was very short: after 100 Myr where ntarget and nproject are the number of targets (i.e. main belt of evolution the number of bodies is almost at the current level. asteroids) and the number of projectiles (comets), respectively. In this brief period of time, about 50% more families will be The actual number of bodies (27,000) in the dynamical simula- created, but all of them will be old, of course. For the remaining tion of Vokrouhlickýet al. (2008) changes in the course of time, ∼ 3.9Gyr, the above model (without any dynamical decay) is and it was scaled such that it was initially equal to the numberof valid. projectiles N(>ddisrupt) inferred from the SFD of the disk. This is clearly a lower limit for the number of families created, since To conclude, it is possible – though not very likely – that the main belt was definitely more populous in the past. the observed families were produced by the collisional activity The average impact velocity is Vimp ≃ 10km/s, so we need in the main belt alone. A dynamical decay of the MB population the following projectile sizes to disrupt given target sizes: would create more families that are old, but technically speaking, this cannot be distinguished from the LHB scenario, which is ⋆ ρtarget Dtarget Ntargets Q ddisrupt for = 3to6 discussed next. D ρproject (km) intheMB (J/kg) (km) 100 ∼192 1 × 105 12.6 to 23 200 ∼23 4 × 105 40.0 to 73 4. Collisions between a “classical” cometary disk We tried to use various SFDs for the cometary disk (i.e., with and the main belt various differential slopes q1 for D > D0 and q2 for D < D0, the elbow diameter D0 and total mass Mdisk), including rather In this section, we construct a collisional model and estimate an extreme cases (see Figure 7). The resulting numbers of LHB expected number of families created during the LHB due to col- families are summarised in Table 6. Usually, we obtain sev- lisions between cometary-disk bodies and main-belt asteroids. eral families with DPB ≃ 200km and about 100 families with We start with a simple stationary model and we confirm the re- DPB ≃ 100km. This result is robust with respect to the slope sults using a more sophisticated Boulder code (Morbidelli et al. q2, because even very shallow SFDs should produce a lot of 2009). these families.4 The only way to decrease the number of fam- Using the data from Vokrouhlickýet al. (2008) for a “clas- ilies significantly is to assume the elbow at a larger diame- sical” cometary disk, we can estimate the intrinsic collisional ter D0 ≃ 150 km. probability and the collisional velocity between comets and as- 4 The extreme case with q2 = 0 is not likely at all, e.g. because of teroids. A typical time-dependent evolution of Pi and Vimp is the continuous SFD of basins on Iapetus and Rhea, which only ex- shown in Figure 6. The probabilities increase at first, as the hibits a mild depletion of D ≃ 100 km size craters; see Kirchoff & transneptunian cometary disk starts to decay, reaching up to Schenk (2010). On the other hand, Sheppard & Trujillo (2010) report − 6 × 10−21 km 2 yr−1, and after 100Myr they decrease to zero. an extremely shallow cumulative SFD of Neptune Trojans that is akin These results do not differ significantly from run to run. to low q2.

6 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

MB alone MB vs comets MB vs disrupting comets 1010 1010 1010 evolved MB 109 initial MB 109 109 observed MB 108 108 108 N(>D) N(>D) N(>D) 107 107 107 106 106 106 t = 4000 My 105 105 105 104 104 104 103 103 103 evolved MB evolved MB 102 102 comets at 190 My 102 comets at 190 My initial MB initial MB size-frequency distribution 10 size-frequency distribution 10 initial comets size-frequency distribution 10 initial comets observed MB observed MB 1 1 1 0.1 1 10 100 1000 0.1 1 10 100 1000 0.1 1 10 100 1000 D / km D / km D / km

104 104 104

D synthetic families with DLF > 10 km and LF/PB < 0.5 D synthetic families with DLF > 10 km and LF/PB < 0.5 D synthetic families with DLF > 10 km and LF/PB < 0.5

> stationary model MB vs MB > stationary model MB vs comets > observed families without craterings observed families without craterings observed families without craterings PB PB PB D 103 D 103 D 103 with with with

100 100 100 families families families N N N

10 10 10

1 1 1

production function 10 100 1000 production function 10 100 1000 production function 10 100 1000 D [km] D [km] D [km]

104 104 104

D observed families without craterings D observed families without craterings D observed families without craterings synthetic families with D > 10 km and LF/PB < 0.5 synthetic families with D > 10 km and LF/PB < 0.5 synthetic families with D > 10 km and LF/PB < 0.5 > LF > LF > LF families after comminution with remaining families after comminution with remaining families after comminution with remaining PB ≥ PB ≥ PB ≥ D N (D > 2 km) 10, 20, 50, 100 or 200 D N (D > 2 km) 10, 20, 50, 100 or 200 D N (D > 2 km) 10, 20, 50, 100 or 200 103 members 103 members 103 members with with with

100 100 100 families families families N N N

10 10 10

1 1 1

production function 10 100 1000 production function 10 100 1000 production function 10 100 1000 D [km] D [km] D [km] 500 500 500

400 400 400

300 300 300 [km] [km] [km]

PB 200 PB 200 PB 200 D D D

100 100 100

0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 age [Gyr] age [Gyr] age [Gyr]

Fig. 5. Results of three different collisional models: main-belt alone which is discussed in Section 3 (left column), main-belt and comets from Section 4 (middle column), main-belt and disrupting comets from Section 7 (right column). 1st row: the initial and evolved size-frequency distributions of the main belt populations for 100 Boulder simulations; 2nd row: the resulting family production functions (in order to distinguish 100 lines we plot them using different colours ranging from black to yellow) and their comparison to the observations; 3rd row: the production function affected by comminution for a selected simulation; and 4th row: the distribution of synthetic families with DPB ≥ 50km in the (age, DPB) plot for a selected simulation, without comminution. The positions of synthetic families in the 4th-row figures may differ significantly for a different Boulder simulation due to stochasticity and low-number statistics. Moreover, in the middle and right columns, many families were created during the LHB, so there are many overlapping crosses close to 4Gyr.

It is thus no problem to explain the existence of approxi- 1987), which is significantly younger than 4Gyr (Marchi et al. mately five large families with DPB = 200–400km, which are 2012). It is highly unlikely that Vesta experienced a catastrophic indeed observed, since they can be readily produced during the disruption in the past, and even large cratering events were lim- LHB. On the other hand, the high number of DPB ≃ 100km ited. We thus have to check the number of collisions between families clearly contradicts the observations, since we observe one D = 530km target and D ≃ 35km projectiles, which are almost no LHB families of this size. capable of producing the basin and the Vesta family (Thomas et al. 1997). According to Table 6, the predicted number of such events does not exceed ∼ 2, so given the stochasticity of the re- 4.2. Constraints from (4) Vesta sults there is a significant chance that Vesta indeed experienced zero such impacts during the LHB. The asteroid (4) Vesta presents a significant constraint for collisional models, being a differentiated body with a preserved basaltic crust (Keil 2002) and a 500km large basin on its surface (a feature indicated by the photometric analysis of Cellino et al.

7 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

Table 6. Results of a stationary collisional model between the cometary disk and the main belt. The parameters characterise the SFD of the disk: q1, q2 are diﬀerential slopes for the diameters larger/smaller than the elbow diameter D0, Mdisk denotes the total mass of the disk, and nevents is the resulting number of families created during the LHB for a given parent body size DPB. The ranges of nevents are due to variable density ratios ρtarget/ρproject = 1to3/1.

q1 q2 D0 Mdisk nevents notes (km) (M⊕) for DPB ≥ 100 km DPB ≥ 200 km Vesta craterings 5.0 3.0 100 45 115–55 4.9–2.1 2.0 nominal case 5.0 2.0 100 45 35–23 4.0–2.2 1.1 shallow SFD 5.0 3.5 100 45 174–70 4.3–1.6 1.8 steep SFD 5.0 1.1 100 45 14–12 3.1–2.1 1.1 extremely shallow SFD 4.5 3.0 100 45 77–37 3.3–1.5 1.3 lower q1 5.0 3.0 50 45 225–104 7.2–1.7 3.2 smaller turn-oﬀ 5.0 3.0 100 25 64–40 2.7–1.5 1.1 lower Mdisk 3 5.0 3.0 100 17 34 1.2 1.9 ρcomets = 500 kg/m 5.0 3.0 150 45 77–23 3.4–0.95 0.74 larger turn-oﬀ 5.0 0.0 100 10 1.5–1.4 0.5–0.4 0.16 worst case (zero q2 and low Mdisk)

q1 = 5.0, q2 = 3.0, D0 = 100 km, Mdisk = 45 ME was similar to the one in Section 3, qa = −4.2, qb = −2.2, q = 2.0 12 2 qc = −3.5, D1 = 80 km, D2 = 14km, and only the normalisation 10 q2 = 3.5 was increased up to Nnorm(D > D1) = 560 in this case. q2 = 1.1 D0 = 50 km The resulting size-frequency distributions of 100 indepen- 11 D0 = 150 km ﬀ 0 10 dent simulations with di erent random seeds are shown in D q1 = 4.5

> Figure 5 (middle column). The number of LHB families (ap-

D M = 25 M disk E proximately 10 with D ≃ 200kmand200with D ≃ 100 km) q2 = 0.0, Mdisk = 10 ME PB PB 1010 is even larger compared to the stationary model, as expected, because we had to start with a larger main belt to get a good ﬁt of valid only for

) the currently observed MB after 4Gyr of collisional evolution. D 109 (> To conclude, the stationary model and the Boulder code give N results that are compatible with each other, but that clearly con- tradict the observed production function of families. In partic- 8 10 ular, they predict far too many families with D = 100km parent bodies. At ﬁrst sight, this may be interpreted as proof that there was no cometary LHB on the asteroids. Before jump- 107 1 10 100 ing to this conclusion, however, one has to investigate whether D / km there are biases against identifying of DPB = 100 km families. In Sections 5–9 we discuss several mechanisms that all con- Fig. 7. Cumulative size-frequency distributions of the cometary tribute, at some level, to reducing the number of observable disk tested in this work.All the parametersof our nominalchoice DPB = 100km families over time. They are addressed in order are given in the top label; the other labels just report the param- of relevance, from the least to the most eﬀective. eters that changed relative to our nominal choice.

5. Families overlap 4.3. Simulations with the Boulder code Because the number of expected DPB ≥ 100km LHB families To confirm results of the simple stationary model, we also per- is very high (of the order of 100) we now want to verify if these formed simulations with the Boulder code. We modified the code families can overlap in such a way that they cannot be distin- to include a time-dependent collisional probability Pi(t) and im- guished from each other and from the background. We thus took pact velocity Vimp(t) of the cometary-disk population. 192 main-belt bodies with D ≥ 100km and selected randomly We started a simulation with a setup for the cometary disk 100 of them that will break apart. For each one we created an ar- resembling the nominal case from Table 6. The scaling law is tificial family with 102 members, assume a size-dependent ejec- described by Eq. (3), with the following parameters (the first set tion velocity V ∝ 1/D (with V = 50m/s for D = 5km) and corresponds to basaltic material at 5km/s, the second one to wa- the size distribution resembling that of the Koronis family. The ter ice, Benz & Asphaug 1999): choice of the true anomaly and the argument of perihelion at the instant of the break-up event was random. We then calculated ρ Q0 a B bqfact , , 3 proper elements (ap ep sin Ip) for all bodies. This type of anal- (g/cm ) (erg/g) (erg/g) ysis is in some respects similar to the work of Bendjoya et al. asteroids 3.0 7 × 107 −0.45 2.1 1.19 1.0 (1993). comets 1.0 1.6 × 107 −0.39 1.2 1.26 3.0 According to the resulting Figure 8 the answer to the question is simple: the families do not overlap sufficiently, and they −18 −2 −1 The intrinsic probabilities Pi = 3.1 × 10 km yr and cannot be hiddenthat way. Moreover,if we take only biggerbod- velocities Vimp = 5.28km/s for the MB vs MB collisions were ies (D > 10 km), these would be clustered even more tightly. again taken from the work of Dahlgren (1998). We do not ac- The same is true for proper inclinations, which are usually more count for comet–comet collisions since their evolution is domi- clustered than eccentricities, so families could be more easily nated by the dynamical decay. The initial SFD of the main belt recognised.

8 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

ν ν 6 3:1 5:2 7:3 2:1 6 3:1 5:2 7:3 2:1

0.3 0.3 p I p e

0.2 0.2

0.1 0.1 proper eccentricity proper inclination sin

0 0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 2 2.2 2.4 2.6 2.8 3 3.2 3.4

proper semimajor axis ap / AU proper semimajor axis ap / AU

Fig. 8. The proper semimajor axis ap vs the proper eccentric- Fig. 9. The proper semimajor axis ap vs the proper inclination ity ep for 100 synthetic families created in the main belt. It is the sin Ip for 100 synthetic asteroid families (black dots), evolved initial state, shortly after disruption events. We assume the size- over 4Gyr using a Monte-Carlo model. The assumed SFDs cor- frequency distribution of bodies in each synthetic family similar respond to the Koronis family, but we show only D > 10km to that of the Koronis family (down to D ≃ 2 km). Break-ups bodies here. We also include D > 10 km background asteroids with the true anomaly f ≃ 0 to 30◦ and 150◦ to 180◦ are more (grey dots) for comparison. easily visible on this plot, even though the choice of both f and the argument of perihelion ̟ was random for all families. sumption. We considered that no comet disrupt beyond 1.5AU, whereas all comets disrupt the first time that they penetrate in- 6. Dispersion of families by the Yarkovsky drift side 1.5AU. Both conditions are clearly not true in reality: some In this section, we model long-term evolution of synthetic fam- comets are observed to blow up beyond 1.5AU, and others are ilies driven by the Yarkovsky effect and chaotic diffusion. For seen to survive on an Earth-crossing orbit. Thus we adopted our one synthetic family located in the outer belt, we have performed disruption law just as an example of a drastic reduction of the a full N-body integration with the SWIFT package (Levison & number of comets with small perihelion distance, as required to Duncan 1994), which includes also an implementation of the explain the absence of evidence for a cometary bombardmenton Yarkovsky/YORP effect (Broˇz2006) and second-order integra- the Moon. tor by Laskar & Robutel (2001). We included 4 giant planets in We then removed all those objects from output of comet evo- this simulation. To speed-up the integration, we used ten times lution during the LHB that had a passage within 1.5AU from smaller sizes of the test particles and thus a ten times shorter the Sun, from the time of their first passage below this thresh- time span (400Myr instead of 4Gyr). The selected time step is old. We then recomputed the mean intrinsic collision probabil- ∆t = 91d. We computed proper elements, namely their differ- ity of a comet with the asteroid belt. The result is a factor ∼3 ences ∆ap, ∆ep, ∆ sin Ip between the initial and final positions. smaller than when no physical disruption of comets is taken into Then we used a simple Monte-Carlo approach for the account as in Fig. 6. The mean impact velocity with asteroids whole set of 100 synthetic families — we assigned a suitable also decreases, from 12km/sto8km/s. drift ∆ap(D) in semimajor axis, and also drifts in eccentric- The resulting number of asteroid disruption events is thus de- ∼ ity ∆ep and inclination ∆ sin Ip to each member of 100 families, creased by a factor 4.5, which can be also seen in the produc- respecting asteroid sizes, of course. This way we account for the tion function shown in Figure 5 (right column). The production Yarkovsky semimajor axis drift and also for interactions with of families with DPB = 200–400km is consistent with observa- mean-motion and secular resonances. This Monte-Carlo method tions, while the number of DPB ≃ 100km families is reduced to tends to smear all structures, so we can regard our results as the 30–70, but is still too high, by a factor 2–3. More importantly, upper limits for dispersion of families. the slope of the production function remains steeper than that While the eccentricities of small asteroids (down to D ≃ of the observed function. Thus, our conclusion is that physical 2km) seem to be dispersed enough to hide the families, there disruptions of comets alone cannot explain the observation, but are still some persistent structures in inclinations, which would may be an important factor to keep in mind for reconciling the be observable today. Moreover, large asteroids (D ≥ 10 km) model with the data. seem to be clustered even after 4Gyr, so that more than 50% of families can be easily recognised against the background (see Figure 9). We thus can conclude that it is not possible to disperse 8. Perturbation of families by migrating planets the families by the Yarkovsky effect alone. (a jumping-Jupiter scenario) In principle, families created during the LHB may be perturbed 7. Reduced physical lifetime of comets in the MB by still-migrating planets. It is an open question what the ex- crossing zone act orbital evolution of planets was at that time. Nevertheless, a plausible scenario called a “jumping Jupiter” was presented by To illustrate the effects that the physical disruption of comets Morbidelli et al. (2010). It explains major features of the main (due to volatile pressure build-up, amorphous/crystalline phase belt (namely the paucity of high-inclination asteroids above the transitions, spin-up by jets, etc.) can have on the collisional ν6 secular resonance), and is consistent with amplitudes of the evolution of the asteroid belt, we adopted here a simplistic as- secular frequencies of both giant and terrestrial planets and also

9 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

ν 3:1 5:2 7:3 2:1 ν 3:1 5:2 7:3 2:1 0.4 6 0.4 6 MB with D > 100 km MB with D > 100 km inner-belt family inner-belt family middle middle

p pristine p pristine I 0.3 outer I 0.3 outer

0.2 0.2

0.1 0.1 proper inclination sin proper inclination sin

0 0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 2 2.2 2.4 2.6 2.8 3 3.2 3.4

proper semimajor axis ap / AU proper semimajor axis ap / AU Fig. 10. The proper semimajor axis vs the proper inclination for four synthetic families (distinguished by symbols) as perturbed by giant-planet migration. Left panel: the case when families were evolved over the “jump” due to the encounter between Jupiter and Neptune. Right panel: the families created just after the jump and perturbed only by later phases of migration. with other features of the solar system. In this work, we thus ilies. To estimate the amount of comminution, we performed investigated this particular migration scenario. the following calculations: i) for a selected collisional simula- We used the data from Morbidelli et al. (2010) for the orbital tion, whose production function is close to the average one, we evolution of giant planets. We then employed a modified SWIFT recorded the SFDs of all synthetic families created in the course integrator, which read orbital elements for planets from an in- of time; ii) for each synthetic family, we restarted the simula- put file and calculated only the evolution of test particles. Four tion from the time t0 when the family was crated until 4Gyr and synthetic families located in the inner/middle/outerbelt were in- saved the final SFD, i.e. after the comminution. The results are tegrated. We started the evolution of planets at various times, shown in Figure 11. ranging from t0 to (t0 + 4Myr) and stopped the integration at It is now important to discuss criteria, which enable us to (t0 + 4My), in order to test the perturbation on families created decide if the comminutioned synthetic family would indeed be in different phases of migration. Finally, we calculated proper observable or not. We use the following set of conditions: DPB ≥ elements of asteroids when the planets do not migrate anymore. 50km, DLF ≥ 10 km (largest fragment is the first or the second (We also had to move planets smoothly to their exact current largest body,where the SFD becomes steep), LR/PB < 0.5(i.e.a orbital positions.) catastrophic disruption). Furthermore, we define Nmembers as the The results are shown in Figure 10. While the proper eccen- number of the remaining family members larger than observa- tricities seem to be sufficiently perturbed and families are dis- tional limit Dlimit ≃ 2km and use a condition Nmembers ≥ 10. The persed even when created at late phases of migration, the proper latter number depends on the position of the family within the inclinations are not very dispersed, except for families in the main belt, though. In the favourable “almost-empty” zone (be- outer asteroid belt that formed at the very beginning of the gi- tween ap = 2.825 and 2.955 AU), Nmembers ≥ 10 may be valid, ant planet instability (which may be unlikely, as there must be a but in a populatedpartof theMB onewouldneed Nmembers & 100 delay between the onset of planet instability and the beginning to detect the family. The size distributions of synthetic families of the cometary flux through the asteroid belt). In most cases, the selected this way resemble the observed SFDs of the main-belt LHB families could still be identified as clumps in semi-major families. axis vs inclination space. We do not see any of such (ap, sin Ip)- According to Figure 5 (3rd row), where we can see the clumps, dispersed in eccentricity, in the asteroid belt.5 production functions after comminution for increasing values The conclusion is clear: it is not possible to destroy low- of Nmembers, families with DPB = 200–400km remain more e and low-I families by perturbations arising from giant-planet prominent than DPB ≃ 100 km families simply because they migration, at least in the case of the “jumping-Jupiter”scenario.6 contain much more members with D > 10km that survive intact. Our conclusion is thus that comminution may explain the paucity of the observed DPB ≃ 100 km families. 9. Collisional comminution of asteroid families We have already mentioned that the comminution is not suffi- 10. “Pristine zone” between the 5:2 and 7:3 cient to destroy a DPB = 100km family in the current environ- resonances ment of the main belt (Bottke et al. 2005). We now focus on the zone between the 5:2 and 7:3 mean-motion However, the situation in case of the LHB scenario is differ- resonances, with ap = 2.825to 2.955AU, which is not as popu- ent. Both the large population of comets and the several-times lated as the surrounding regions of the main belt (see Figure 1). larger main belt, which has to withstand the cometary bombard- This is a unique situation, because both bounding resonances are ment, contribute to the enhanced comminution of the LHB fam- strong enough to prevent any asteroids from outside to enter this zone owing the Yarkovsky semimajor axis drift. Any family for- 5 High-inclination families would be dispersed much more owing to the Kozai mechanism, because eccentricities that are sufficiently per- mation event in the surroundings has only a minor influence on turbed exhibit oscillations coupled with inclinations. this narrow region. It thus can be called “pristine zone” because 6 The currently non-existent families around (107) Camilla and it may resemblethe belt prior to creation of big asteroid families. (121) Hermione — inferred from the existence of their satellites — We identified nine previously unknown small families that cannot be destroyed in the jumping-Jupiter scenario, unless the fami- are visible on the (ep, sin Ip) plot (see Figure 12). They are lies were actually pre-LHB and had experienced the jump. confirmed by the SDSS colours and WISE albedos, too.

10 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

Dcomplete Dcomplete Dcomplete 108 108 108 observed MB families synthetic families after disruption synthetic families after comminution 107 107 107

6 6 6

N(>D) 10 N(>D) 10 N(>D) 10

105 105 105

104 104 104

103 103 103

102 102 102

size-frequency distribution 10 size-frequency distribution 10 size-frequency distribution 10

1 1 1 0.1 1 10 100 1000 0.1 1 10 100 1000 0.1 1 10 100 1000 D / km D / km D / km Fig. 11. Left panel: the size-frequency distributions of the observed asteroid families. Middle panel: SFDs of 378 distinct synthetic families created during one of the collisional simulations of the MB and comets. Initially, all synthetic SFDs are very steep, in agreement with SPH simulations (Durda et al. 2007). We plot only the SFDs that fulﬁl the following criteria: DPB ≥ 50km, DLF ≥ 10km, LR/PB < 0.5 (i.e. catastrophic disruptions). Right panel: the evolved SFDs after comminution. Only a minority of families are observable now, since the number of remaining members larger than the observational limit Dlimit ≃ 2km is often much smaller than 100. The SFD that we use for the simulation in Section 10 is denoted by red.

0.35 0.35 a = 2.82 - 2.97 AU a = 2.82 - 2.97 AU 0.3 0.3 709 709 5567 15454 5567 15454

293 293 0.25 0.25

12573 12573 918 918 845 845 0.2 0.2 1189 1189

p 36256 p 36256 I I 0.4 sin 18405 sin 18405 0.15 0.15 81 81 5614 5614 0.2

10811 10811 0.1 627 0.1 627 15477 15477 0 i-z [mag]

0.05 0.05 -0.2 158,832 832158

-0.4 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.4 -0.2 0 0.2 0.4 ep ep a* [mag]

Fig. 12. The “pristine zone” of the main belt (ap = 2.825 to 2.955AU) displayed on the proper eccentricity ep vs proper inclination sin Ip plot. Left panel: the sizes of symbols correspond to the sizes of asteroids, the families are denoted by designations. Right panel: a subset of bodies for which SDSS data are available; the colours of symbols correspond to the SDSS colour indices a∗ and i − z (Parker et al. 2008).

Nevertheless, there is only one big and old family in this zone may speculate that the families like (918) Itha, (5567) Durisen, (DPB ≥ 100 km), i.e. Koronis. (12573) 1999 NJ53, or (15454) 1998 YB3 (all from the pristine That at most one LHB family (Koronis) is observed in the zone) are actually remnants of larger and older families, even “pristine zone” can give us a simple probabilistic estimate for though they are denoted as young. It may be that the age esti- the maximum number of disruptions during the LHB. We take mate based on the (ap, H) analysis is incorrect since small bodies the 192 existing main-belt bodies which have D ≥ 100km and were destroyed by comminutionand spread by the Yarkovskyef- select randomly 100 of them that will break apart. We repeat this fect too far away from the largest remnant, so they can no longer selection 1000 times and always count the number of families in be identified with the family. the pristine zone. The resulting histogram is shown in Figure 13. Finally, we have to ask an important question: what does As we can see, there is very low (<0.001) probability that the an old/comminutioned family with DPB ≃ 100 km look like in numberof families in the pristinezoneis zero or one.On average proper-element space? To this aim, we created a synthetic fam- we get eight families there, i.e. about half of the 16 asteroids ily in the “pristine zone”, and assumed the family has Nmembers ≃ with D ≥ 100km present in this zone. It seems that either the 100 larger than Dlimit ≃ 2km and that the SFD is already flat in number of disruptions should be substantially lower than 100 or the D = 1 to 10km range. We evolved the asteroids up to 4Gyr we expect to find at least some “remnants” of the LHB families due to the Yarkovsky effect and gravitational perturbations, us- here. ing the N-body integrator as in Section 6. Most of the D ≃ 2km It is interesting that the SFD of an old comminutioned family bodies were lost in the course of the dynamical evolution, of is very flat in the range D = 1 to 10km (see Figure 11) — simi- course. The resulting family is shown in Figure 14. We can also lar to those of some of the “less certain” observed families! We imagine that this family is placed in the pristine zone among

11 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

250 this paper the “production function”). We show that the collisional activity of the asteroid belt as a closed system, i.e. without 200 any external cometary bombardment, in general does not produce such a shallow production function. Moreover, the number 150 of families with parent bodies larger than 200km in diameter N 100 is in general too small compared to the observations. However, there is a lot of stochasticity in the collisional evolution of the 50 asteroid belt, and about 5% of our simulations actually ﬁt the observational constraints (shallowness of the production func- 0 tion and number of large families) quite well. Thus, in principle, 0 2 4 6 8 10 12 14 16 there is no need for a bombardment due to external agents (i.e. Nfamilies between a = 2.825 - 2.955 AU the comets) to explain the asteroid family collection, provided Fig. 13. The histogram for the expected number of LHB families that the real collisional evolution of the main belt was a “lucky” located in the “pristine zone” of the main belt. one and not the “average” one. If one accounts for the bombardmentprovided by the comets 0.35 crossing the main belt at the LHB time, predicted by the Nice a = 2.82 - 2.97 AU model, one can easily justify the number of observed families 0.3 t = 4 Gy with parent bodies larger than 200km. However, the resulting production function is steep, and the number of families pro-

0.25 duced by parent bodies of 100km is almost an order of magnitude too large.

0.2 We have investigated several processes that may decimate p I the number of families identiﬁable today with 100km parent

sin ff 0.15 bodies, without considerably a ecting the survival of families formed from larger parent bodies. Of all these processes, the collisional comminution of the families and their dispersal by the 0.1 Yarkovsky effect are the most effective ones. Provided that the physical disruption of comets due to activity reduced the effec- 0.05 tive cometary flux through the belt by a factor of ≈ 5, the resulting distribution of families (and consequently the Nice model) is 0 consistent with observations. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

ep To better quantify the effects of various cometary-disruption laws, we computed the numbers of asteroid families for dif- Fig. 14. The proper eccentricity vs proper inclination of one ferent critical perihelion distances qcrit and for different disrup- synthetic old/comminutioned family evolved dynamically over tion probabilities pcrit of comets during a given time step (∆t = 4Gyr. Only a few family members (N ≃ 101) remained from 500yr in our case). The results are summarised in Figure 15. the original number of N(D ≥ 2km) ≃ 102. The scales are the Provided that comets are disrupted frequently enough, namely same as in Figure 12, so we can compare it easily to the “pris- the critical perihelion distance has to be at least qcrit & 1 AU, tine zone”. while the probability of disruption is pcrit = 1, the number of DPB ≥ 100km families drops by the aforementioned factor of other observed families, to get a feeling of whether it is easily ≈ 5. Alternatively, qcrit may be larger, but then comets have to observed or not (refer to Figure 12). survive multiple perihelion passages (i.e. pcrit have to be lower It is clear that such family is hardly observable even in the than 1). It would be very useful to test these conditions by in- almost empty zone of the main belt! Our conclusion is that the dependent models of the evolution and physical disruptions of comminution (as given by the Boulder code) can explain the comets. Such additional constraints on cometary-disruption laws would then enable study of the original size-frequency distribu- paucity of DPB ≃ 100km LHB families, since we can hardly distinguish old families from the background. tion of the cometary disk in more detail. We can also think of two “alternative” explanations: i) physical lifetime of comets was strongly size-dependent so that 11. Conclusions smaller bodies break up easily comparedto bigger ones; ii) high- In this paper we investigated the cometary bombardment of the velocity collisions between hard targets (asteroids) and very weak projectiles (comets) may result in different outcomes than asteroidbelt at the time of the LHB, in the frameworkof the Nice in low-velocity regimes explored so far. Our work thus may also model. There is much evidence of a high cometary flux through the giant planet region, but no strong evidence of a cometary serve as a motivation for further SPH simulations. bombardment on the Moon. This suggests that many comets We finally emphasize that any collisional/dynamical models broke up on their way to the inner solar system. By investigat- of the main asteroid belt would benefit from the following ad- ing the collisional evolution of the asteroid belt and comparing vances: the results to the collection of actual collisional families, our aim was to constrain whether the asteroid belt experienced an intense i) determination of reliable masses of asteroids of various cometary bombardment at the time of the LHB and, if possible, classes. This may be at least partly achieved by the Gaia constrain the intensity of this bombardment. mission in the near future. Using up-to-date sizes and shape Observations suggest that the number of collisional families models (volumes) of asteroids one can then derive their den- is a very shallow function of parent-body size (that we call in sities, which are directly related to ages of asteroid families.

12 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

200 ≥ Bottke, W.F., Durda, D.D., Nesvorný, D., et al. 2005, Icarus, 175, 111 DPB 100 km 200 km Bottke, W.F., Levison, H.F., Nesvorný, D., & Dones, L. 2007, Icarus, 190, 203 150 observed Bottke, W.F., Vokrouhlický, D., & Nesvorný, D. 2007, Nature, 449, 48 Bottke, W.F., Nesvorný, D., Vokrouhlický, D., & Morbidelli, A. 2010, AJ, 139, 100 994 families

N Bottke, W.F., VokrouhlickýD., Minton, D., et al. 2012, Nature, 485, 78 50 Broˇz, M. 2006, PhD thesis, Charles Univ. Broˇz, M., Vokrouhlický, D., Morbidelli, A., Nesvorný. D., & Bottke, W.F. 2011, 0 MNRAS, 414, 2716 0 0.5 1 1.5 2 2.5 3 Carruba, V. 2009, MNRAS, 398, 1512 qcrit / AU Carruba, V. 2010, MNRAS, 408, 580 150 ≥ Carry, B. 2012, arXiv: 1203.4336v1 DPB 100 km 200 km Cellino, A., Zappalà, V., di Martino, M., Farinella, P., Paolicchi, P. 1987, Icarus, observed 70, 546 100 Chapman, C.R., Cohen, B.A., Grinspoon, D.H. 2007, Icarus, 189, 233 Charnoz, S., Morbidelli, A., Dones, L., & Salmon, J. 2009, Icarus, 199, 413 families

N 50 Cohen, B.A., Swindle, T.D., & Kring, D.A. 2000, Science, 290, 1754 Dahlgren, M. 1998, A&A, 336, 1056 Durda, D.D., Bottke, W.F., Nesvorný, D., et al. 2007, 186, 498 0 Foglia, S., & Masi, G., 2004, Minor Planet Bull., 31, 100 0.001 0.01 0.1 1 Gil-Hutton, R. 2006, Icarus, 183, 83 pcrit Gomes, R., Levison, H.F., Tsiganis, K., & Morbidelli, A. 2005, Nature, 435, 466 Hartmann, W.K., Ryder, G., Dones, L., & Grinspoon, D. 2000, in Origin of the Fig. 15. The numbers of collisional families for different critical Earth and Moon, ed. R.M. Canup, K. Righter, Tucson: University of Arizona perihelion distances qcrit at which comets break up and disrup- Press, p. 493 tion probabilities p during one time step (∆t = 500yr). In the Hartmann, W.K., Quantin, C., & Mangold, N. 2007, Icarus, 186, 11 crit Keil, K. 2002, in Asteroids III, ed. W.F. Bottke Jr., A. Cellino, P. Paolicchi, R.P. top panel, we vary qcrit while keeping pcrit = 1 constant. In the Binzel, Tucson: University of Arizona Press, p. 573 bottom panel, qcrit = 1.5AU is constant and we vary pcrit. We al- Kirchoff, M.R., & Schenk, P. 2010, Icarus, 206, 485 ways show the number of catastrophic disruptions with parent- Kneˇzević, Z., & Milani, A. 2003, A&A, 403, 1165 Koeberl, C. 2004, Geochimica et Cosmochimica Acta, 68, 931 body sizes DPB ≥ 100km (red line) and 200km (black line). Kring, D.A., & Cohen, B.A. 2002, J. Geophys. Res., 107, 41 The error bars indicate typical (1-σ) spreads of Boulder simu- Laskar, J., & Robutel, P. 2001, Celest. Mech. Dyn. Astron., 80, 39 lations with different random seeds. The observed numbers of Levison, H.F., & Duncan, M. 1994, Icarus, 108, 18 corresponding families are indicated by thin dotted lines. Levison, H.F., Bottke, W.F., Gounelle, M., et al. 2009, Nature, 460, 364 Margot, J.-L., & Rojo, P. 2007, BAAS, 39, 1608 Marzari, F., Farinella, P., & Davis, D.R. 1999, Icarus, 142, 63 ii) Development of methods for identifying asteroid families Masiero, J.P., Mainzer A.K., Grav, T., et al. 2011, ApJ, 741, 68 and possibly targeted observations of larger asteroids ad- Michel, P., Tanga, P., Benz, & W., Richardson, D.C. 2002, Icarus, 160, 10 Michel, P., Jutzi, M., Richardson, D.C., & Benz, W. 2011, Icarus, 211, 535 dressing their membership, which is sometimes critical for Minton, D.A., & Malhotra, R. 2009, Nature, 457, 1109 constructing size-frequency distributions and for estimating Minton, D.A., & Malhotra, R. 2010, Icarus, 207, 744 parent-body sizes. Morbidelli, A. 2010, Comptes Rendus Physique, 11, 651 iii) An extension of the SHP simulations for both smaller and Morbidelli, A., Tsiganis, K., Crida, A., Levison, H.F., & Gomes, R. 2007, AJ, larger targets, to assure that the scaling we use now is 134, 1790 Morbidelli, A., Levison, H.F., Bottke, W.F., Dones, L., & Nesvorný, D. 2009, valid. Studies and laboratory measurements of equations of Icarus, 202, 310 states for different materials (e.g. cometary-like, porous) are Morbidelli, A., Brasser, R., Gomes, R., Levison, H.F., & Tsiganis, K. 2010, AJ, closely related to this issue. 140, 1391 Morbidelli, A., Marchi, S., & Bottke, W.F. 2012, LPI Cont. 1649, 53 The topics outlined above seem to be the most urgent develop- Molnar, L.A., & Haegert, M.J. 2009, BAAS, 41, 2705 ments to be pursued in the future. Nesvorný, D. 2010, EAR-A-VARGBDET-5-NESVORNYFAM-V1.0, NASA Planetary Data System Nesvorný, D., & Vokrouhlický, D. 2006, AJ, 132, 1950 Acknowledgements Nesvorný, D., Morbidelli, A., Vokrouhlický, D., Bottke, W.F., & Broˇz, M. 2002, Icarus, 157, 155 The work of MB and DV has been supported by the Grant Nesvorný, D., Jedicke, R., Whiteley, R.J., & Ivezić, Z.ˇ 2005, Icarus, 173, 132 / / Nesvorný, D., Vokrouhlický, D., & Morbidelli, A. 2007, AJ, 133, 1962 Agency of the Czech Republic (grants no. 205 08 0064, and 13- Nesvorný, D., Vokrouhlický, D., Morbidelli, A., & Bottke, W.F. 2009, Icarus, 01308S) and the Research Programme MSM0021620860 of the 200, 698 Czech Ministry of Education. The work of WB and DN was Neukum, G., Ivanov, B.A., & Hartmann, W.K. 2001, Space Sci. Rev., 96, 55 supported by NASA’s Lunar Science Institute (Center for Lunar Norman, M.D., & Nemchin, A.A. 2012, LPI Cont., 1659, 1368 Origin and Evolution, grant number NNA09DB32A). We also Novaković, B. 2010, MNRAS, 407, 1477 Novaković, B., Tsiganis, K., & Kneˇzević, Z. 2010, Celest. Mech. Dyn. Astron., acknowledge the usage of computers of the Observatory and 107, 35 Planetarium in Hradec Králové. We thank Alberto Cellino for Novaković, B., Cellino, A., & Kneˇzević, Z. 2011, Icarus, 216, 69 a careful review which helped to improve final version of the Parker, A., Ivezić, Z.,ˇ Jurić, M, et al. 2008, Icarus, 198, 138 paper. Ryder, G., Koeberl, C., & Mojzsis, S.J. 2000, in Origin of the Earth and Moon, ed. R.M. Canup, K. Righter, Tucson: University of Arizona Press, p. 475 Sheppard, S.S, & Trujillo, C.A. 2010, ApJL, 723, L233 References Strom, R.G., Malhotra, R., Ito, T., Yoshida, F., & Kring, D.A. 2005, Science, 309, 1847. Bendjoya, P., Cellino, A., Froeschlé, C., Zappalà, V. 1993, A&A, 272, 651 Swindle, T.D., Isachsen, C.E., Weirich, J.R., & Kring, D.A. 2009, Meteor. Planet. Benz, W., & Asphaug, E. 1999, Icarus, 142, 5 Sci., 44, 747 Bogard, D.D. 1995, Meteoritics, 30, 244 Tagle, R. 2005, LPI Cont., 36, 2008 Bogard, D.D. 2011, Chemie der Erde – Geochemistry, 71, 207 Tanga, P., Cellino, A., Michel, P., Zappalà, V., Paolicchi, P., & Dell’Oro, A. 1999, Bottke, W.F., Vokrouhlický, D., Broˇz, M., Nesvorný, D., & Morbidelli, A. 2001, Icarus, 141, 65 Science, 294, 1693 Tedesco, E.F., Noah, P.V., Noah, M., & Price, S.D. 2002, AJ, 123, 1056

13 M. Broˇzet al.: Constraining the cometary ﬂux through the asteroid belt during the late heavy bombardment

Tera, F., Papanastassiou, D.A., & Wasserburg, G.J. 1974, Earth & Planet. Sci. Let., 22, 1 Tsiganis K., Gomes, R., Morbidelli, A., & Levison, H.F. 2005, Nature, 435, 459 Thomas, P.C., Binzel, R.P., Gaffey, M.J., et al. 1997, Science, 277, 1492 Vokrouhlický, D., & Nesvorný, D. 2011, AJ, 142, 26 Vokrouhlický, D., Broˇz, M., Bottke, W.F., Nesvorný, D., & Morbidelli, A. 2006, Icarus, 182, 118 Vokrouhlický, D., Nesvorný, D., & Levison, H.F. 2008, AJ, 136, 1463 Vokrouhlický, D., Nesvorný, D., Bottke, W.F., & Morbidelli, A. 2010, AJ, 139, 2148 Warner, B.D., Harris, A.W., Vokrouhlický, D., Nesvorný, D., & Bottke, W.F. 2009, Icarus, 204, 172 Weidenschilling, S.J. 2000, Space Sci. Rev., 92, 295 Zappalà, V., Bendjoya, Ph., Cellino, A., Farinella, P., & Froeschlé, C. 1995, Icarus, 116, 291

14 M. Broˇzet al.: Constraining the cometary flux through the asteroid belt during the late heavy bombardment, Online Material p 1 , 2 D the PB N Durda D oan DSS MOC 4 1 resonance / 2 resonance 2 resonance resonance, LL chondrites / / 6 the slope of the SFD for larger ν 1 q (a typical uncertainty is 10%), D velocity for the hierarchical clustering, ff 0002 0005 (490) is likely interloper (Michel et al. 2011) . . 005 0 0 . 05 Nesvornýet al. (2009), L chondrites 04 05 Novakovićet al. (2010) 04 (293) is interloper 0 . . . . 05 1 05 2 2 4 2 2 2 2 0 LHB? 2 0 LHB? 5 cut by J5 0 LHB? 1 1 5 LHB? Michel et al. (2002) 0 LHB? cratering 5 cut by 25 cratering, Marchi et al. (2012) ± ± 0 0 0 0 ...... PB the ratio of the volumes of the largest remnant to ± / 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 48 1 0058 2 0083 005 05 1 3 5 15 7 5 3 05 3 3 0 3 5 7 5 2 3 5 0 0 0 the escape velocity, 0.2 cut by J5 0.5 1.0 0.1 (1639) Bower is interloper 0.1 cratering 1.0 satellite 0.7 (1644) Rafita is interloper 0.2 cratering, close to J3 1.5 overlaps with Nysa 1.5 overlaps with the Polana family 3.0 old? 0.7 cratering, Nesvornýet al. (2005) ...... , LR < < < < < < < < < < < < age notes, references PB esc v D 4 5 0 6 0 7 4 1 0 8 0 9 0 4 0 3 0 3 0 1 0 9 0 7 0 7 0 0 3 0 0 2 5 1 1 8 3 4 1.0-3.8 LHB? cratering, Vokrouhlickýet al. (2010) 6 0 3 2 8 0 0 6 6(0.5) 8 4 2 3 0 0 0 3 2 2 2 9 1 9 1 2 ...... which do not have measured diameters (from Tedesco et al. 200 2 2 2 3 3 3 3 2 3 2 3 2 1 2 3 3 4 3 3 3 2 2 2 3 2 2 2 3 2 1 2 3 3 2 3 2 2 3 2 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − q is the selected cut–o 5) 3) 3) ff . . . cuto are known), 4 3 7 9 2 8 2 7 9 1 2 4(0 5 8 9 5 5(0 2 6(0 2 3 0 9 9 7 4 0 6 2 4 5 9 ...... v 2 4 1 3 4 2 5 2 3 6 4 1 4 1 4 3 2 5 3 5 3 7 6 1 2 3 5 5 4 3 4 4 mean albedo), taxonomic classification (according to the Sl 1 / − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − q Durda D s Gyr / esc v prolonged the SFD slope down to zero and PB D otherwise), dynamical age including its uncertainty. PB / LR Durda here are the following columns: D PB clamation mark denotes a significant mismatch with D e, a range is given if both y to use the WISE data to obtain median X 15 - 0.23 15 X 62 220! 0.029-0.001 35 X 34 ? 0.020 20 X 35 39 0.966-0.70 20 ? X 76 106 0.045-0.017 45 X 79 114 0.79-0.26 46 ? X 261 272 0.994-0.88 154 / / / / / / / tax. the adopted value of the geometric albedo for family members V p V N p parent body size, an additional ’c’ letter indicates that we PB s km km m ff / D 50 44 0.171w C m cuto (a typical uncertainty of the slopes is 0.2, if not indicated v D 12 meister 40 822 0.035 C 93c 134 0.022-0.007 55 ff A list of asteroid families and their physical parameters. T designation 8 Flora 60 5284 0.304w S 150c 160 0.81-0.68 88 4 Vesta 60 11169 0.351w V 530 425! 0.995 314 3 Juno 50 449 0.250 S 233 ? 0.999 139 87 Sylvia 110 71 0.045 C 46 Hestia 65 95 0.053 S 124 153 0.992-0.53 74 44 Nysa (Polana) 60 9957 0.278w S 81c ? 0.65 48 24 Themis 70 3581 0.066 C 268c 380-430! 0.43-0.09 158 20 Massalia 40 2980 0.215 S 146 144 0.995 86 15 Eunomia 50 2867 0.187 S 259 292 0.958-0.66 153 10 Hygiea 70 3122 0.055 C,B 410 442 0.976-0.78 243 847 Agnia 40 1077 0.177 S 39 61 0.38-0.10 23 832Karin - -- S - 40 - -- - 0 606 Brangane 30 81 0.102 S 37 46 0.92-0.48 22 ? 490 Veritas - - - C,P,D - 100-177 - - - - 0 410 Chloris 90 259 0.057 C 126c 154 0.952-0.52 74 ? 845 Naema 30 173 0.081 C 77c 81 0.35-0.30 46 808 Merxia 50 549 0.227 S 37 121! 0.66-0.018 22 569 Misa 70 543 0.031 C 88c 117 0.58-0.25 52 668 Dora 50 837 0.054 C 85 165! 0.031-0.004 50 293 Brasilia 60 282 0.175w C 396 Aeolia 20 124 0.171 C 363 Padua (Lydia) 50 596 0.097 C 283 Emma 75 345 0.050 - 152 185 0.92-0.51 90 ? 221 Eos 50 5976 0.130 K 208c 381! 0.13-0.020 123 170 Maria 80 3094 0.249w S 107c 120-130 0.070-0.048 63 163 Erigone 60 1059 0.056 C 158 Koronis 50 4225 0.147 S 122c 170-180 0.024-0.009 68 145 Adeona 50 1161 0.065 C 171c 185 0.69-0.54 101 142 Polana (Nysa) 60 3443 0.055w C 75 ? 0.42 45 137 Meliboea 95 199 0.054 C 174c 240-290! 0.59-0.20 102 128 Nemesis 60 654 0.052 C 189 197 0.987-0.87 112 the slope for smaller 1272 Gefion 60 19477 0.20 S 74c 100-150! 0.001-0.004 60 1128 Astrid 50 265 0.079 C 43c ? 0.52 25 9506 Telramund 40 146 0.217w S 22 - 0.05 13 1400 Tirela 80 1001 0.070 S 86 - 0.12 86 4652 Iannini - - - S ------0 3556 Lixiaohua 60 439 0.044w C 1726 Ho 3815 Konig 60 177 0.044 C 33 ? 0.32 20 ? 1658 Innes 70 621 0.246w S 27 ? 0.14 16 18405 1993 FY 2 corresponding number of family members, colours, Parker et al. 2008), Table 1. or Masiero et al. 2011, a letter ’w’ indicates it was necessar the parent body (an uncertainty corresponds to the last figur q size inferred from SPH simulations (Durda et al. 2007), an ex M. Broˇzet al.: Constraining the cometary flux through the asteroid belt during the late heavy bombardment, Online Material p 2 al 2. Table 01 a 0 5023 1-07 ? 5 0.77 - 11 S 0.273w 15 100 Lau 10811 86 95SU 1995 18466 42 mloasi---C - - - YC 1992 Emilkowalski 16598 14627 10 uacvn---S------0.0003-0.0008 ------S - - - Lucascavin 21509 19Trni 01 .7 6-0903 ? 33 0.990 - 56 C 0.070 18 50 Terentia 1189 82Lcen 0 7023 4-07 ? 8 0.71 - 14 S 0.223w 57 100 Lucienne 1892 33Kzi 02 .0wS1 .78? 8 0.57 - 16 S 0.206w 23 50 Kazvia 7353 20Dtr .04-.06 dnie noscul in identified 0.00045-0.00060 ------S - - - Datura 1270 34Shlo .07000 orulc´ Ne Vokrouhlický & 0.0007-0.0009 ------S - - - Schulhof 2384 04Tuoi 015 .4 710-01-.816-71 0.17-0.98 - 27-120 S 0.343 1950 50 Teutonia 1044 26Ade 041020 77 .1-.51-3? 10-43 0.010-0.95 - 17-74 S 0.290w 401 60 Andree 1296 07MCse 42600 9-04 7? 17 0.41 - 29 C 0.06 236 34 McCuskey 2007 05Hnn5 4 .0wS2 .316 0.13 - 27 S 0.200w 946 54 Henan 2085 22Mtdk 340004 97c-00708 26-46 0.037-0.81 - 49-79c C 0.064w 410 83 Mitidika 2262 23Buao------Brucato 4203 22Tn 1 7038S2 .41 ? 12 0.94 - 21 S 0.338 37 110 Tina 1222 7 hoad 514000C9c-02 7? 57 0.29 - 97c C 0.060 154 85 Theobalda 778 5 Koronis 158 0 lrsa3 5004C3 .62 ? 23 0.96 - 39 C 0.054 75 30 Clarissa 302 9 atsia5 29010 C 0.160w 1249 50 Baptistina 298 5 ege2 3009C6 .638 0.56 - 64 C 0.089 63 24 Beagle 656 2 hrs8 3 .8 S 0.081 235 80 Charis 627 3 ugra2049 .5E2 .515 0.15 - 25 E 0.35 4598 200 Hungaria 434 5 uaii 011002C6 .339 0.83 - 65 C 0.042 191 60 Sulamitis 752 7 rua8 7 .5wC23 4-8 .104 120 0.71-0.43 240-280 203c C 0.057w 777 80 Ursula 375 4 ala105 .6 8-0085 ? 58 0.058 - 98 S 0.169 57 150 Gallia 148 0 ail .5 - 0.054 ? ? Camilla 107 0 lua1071000B28 9-3!005129 0.025 290-330! 218c B 0.070 791 120 Alauda 702 2 emoe??008- 0.058 ? ? Hermione 121 8 as 5 5 .5 0-08 35 0.83 - 60 S 0.256 651 150 Hansa 480 4 acln 1 2 .4 8-07 6? 16 0.77 - 28 S 0.248 129 110 Barcelona 945 8 esid1018016S5c-04 0? 40 0.48 - 52c S 0.146 178 130 Gersuind 686 7Etre7 6 .6wS18 .9 70 0.998 - 118c S 0.260w 268 70 Euterpe 27 5Poaa1017 .2S9 .455 0.54 - 92 S 0.22 1370 160 Phocaea 25 1Eprsn 0 5 .5 5 .7153 0.97 - 259 C 0.056 851 100 Euphrosyne 31 als206 .6 9c-099 9 ? 295 0.9996 - 498c B 0.163 64 200 Pallas 2 designation otnaino al 1. Table of Continuation (2) 37 2 v cuto m 07 .4wS1 .4 ? 7 0.045 - 14 S 0.241w 71 40 / ff mk m km km s 5-----0 - - - - - 35 S - - - 0.00005-0.00025 ------S - - - p N V tax. / / 5 .721 0.17 - 35c X .01-.02 evr´ Vokrouhlický Nesvorný (2006) & 0.00019-0.00025 ------X > > > D 0-05 5? 35 0.53 - 60 2 .?LB yeergo,nneitn today, non-existent region, Cybele LHB? 3.8? ? ? ? ? ? 226 0 .?LB orulc´ ta.(2010) al. Vokrouhlický et LHB? 3.8? ? ? ? ? ? 209 PB D Durda LR / PB v esc / Gyr s q − − − − − − − − − − − − − 1 3 1 5 6 2 4 3 4 4 3 3 4 4 ...... 6 3 9 5 9 1 5 2 5 9 1 9 9 q − − − − − − − − − − − − − − − − − − − − − − − − − − − − 2 3 2 3 2 1 3 2 2 2 4 5 1 2 2 3 2 5 3 2 2 3 2 2 3 3 3 2 4 ...... 4 0 9 1 4 4 0 1 3 6? 8 4 0 8 2 3 9 9(0 6 2 2 2 6 4 4 2 5 9 7 1 . 5) g oe,references notes, age < < < < < < < < < < < < < < < < < < < < < < < < < < < . . . 1.0 . rtrn,Nesvorný (2010) cratering, 0.1 . ato h lr family Flora the of part 0.3 0.2 0.4 . cratering 0.2 0.1 0.3 0.3 0.1 . rtrn,Pre ta.(2008) al. et Parker cratering, 1.0 . old? 3.5 . eed n()Ataamembership Astraea (5) on depends 0.5 . eed n(9 uyoemembership Eurynome (79) on depends 1.0 . vraswt Nysa with overlaps 0.5 1.0 . eed n(8)Zeaamembership, Zwetana (785) on depends 1.0 . high- 0.5 .5high- 0.45 . l?high- old? 3.5 . l?high- old? 2.2 . high- 1.6 . nfe.space freq. in 1.3 .5high- 0.35 . rtrn,high- cratering, 1.5 . high- 0.8 .5high- 0.15 007 015 5 ± 0 ± ± . anre l (2010) al. et Warner 2 0 0 . . 0 rtrn,Novaković (2010) cratering, 002 0 rtrn,Mla agr (2009) Haegert & Molnar cratering, 005 26)i nelpr vraswt Juno with overlaps interloper, is (2262) Mro oo2007) Rojo & (Margot I I I I I I arb (2010) Carruba , ola&Ms (2004) Masi & Foglia , i-utn(2006) Gil-Hutton , I I u yJ2 by cut , / e u by cut , I ola&Msi(2004) Massi & Foglia , tn-lmn space, ating-element / svorný (2011) Polana / ν eoac,satellite resonance, 1 6 eoac,Crua(2009) Carruba resonance, M. Broˇzet al.: Constraining the cometary flux through the asteroid belt during the late heavy bombardment, Online Material p 3 ” 099, above (363) Padua . 0 > I 0.3 shallow SFD 1.5 0.5 shallow SFD 0.6 incomplete SFD 0.2 0.5 shallow SFD 1.5 shallow SFD 2.5 old? 0.5 cratering, less-certain families in the “pristine zone 1.0 only part with sin 0.04 close to (3) Juno 0.5 above (375) Ursula < < < < < < < < < < < < age notes, references 5) 5) 3) 5) 3) . . . . . 4(0 6(0 6(0 0(0 2 7 5 7 4 8 8(0 7 ...... 1 4 1 2 3 1 1 1 4 3 2 2 2 − − − − − − − − − − − − q 3) . 7 2 0(0 9 . . . . 2 6 4 3 1 − − − − q s Gyr / esc v PB / LR Durda D PB D tax. V N p s km km m ff / 60 30 0.210w S 17 - 0.037 10 ? 50 14 0.054w C 21 - 0.41 13 ? 40 13 0.190w C 15 - 0.13 9 ? m 110 144 0.098w S 25 - 0.065 14 ? cuto v 1 3 17 53 Continuation of Table 1. designation 81 Terpsichore 120 70 0.052 C 119 - 0.993 71 ? 918 Itha 140 63 0.23 S 38 - 0.16 22 709 Fringilla 140 60 0.047 X 99c 130-140 0.93-0.41 59 5614 Yakovlev 100 34 0.05 C 22 - 0.28 13 ? 5567 Durisen 100 18 0.044w X 42 - 0.89 25 ? 2732 Witt 60 985 0.260w S 25 - 0.082 15 1547 Nele 20 57 0.311w X 19 - 0.85 11 ? 1303 Luthera 100 142 0.043 X 92 - 0.81 54 36256 1999 XT 15477 1999 CG 15454 1998 YB 12573 1999 NJ Table 3.