Chaotic Transport and Chronology of Complex Asteroid Families

arXiv:0910.5857v1 [astro-ph.EP] 30 Oct 2009 o.Nt .Ato.Soc. Astron. R. Not. Mon.

⋆ particularly procedure, accurate most the Probably far. so science. asteroid the as in of issue dating important ag accurate an the Thus, is that unknown. families fact general, in the is, from family not arises a complications however, main is, the information of (Farle One relevant Earth the the on Obtaining “showers” dust 2006). related e of Cellino effects (e.g. th the bodies understand and parent to their 2007), of al. et structure disr Durda eralogical 2003; of al. exp laboratory et outcomes to Michel the inaccessible (e.g. range 2005), history size a collisional al. over et events the tion (Bottke understand Belt to Main e.g. science asteroid used, planetary Studie be for 2007). can important al. lies very et (Brown ex are region families to (Milan Transneptunian asteroid proposed recently, Trojans the most in the and ist among 2001), identified Roig Beaugé & been 1993; aster 20 have whole al. Nesvorný families et the Zappalà 2002; Also, & across se Bendjoya discovered (e.g. date, been Belt To Main have speeds. families orbital of their tens than wit lower orbits, t much heliocentric lead velocities nearby which into fragments asteroids, of among ejection collisions catastrophic from 7Spebr2018 September 17 [email protected](ZK) [email protected] .Novakovi B. Famil Asteroid Complex of Chronology and Transport Chaotic ruig ntesaeo ria lmns hs rus no groups, These elements. orbital promine as of well-known form days space can the asteroids (1918), in Hirayama groupings by noted first As INTRODUCTION 1 2 1 c eto fAtohsc,Atooy&Mcais Departmen Mechanics, & Astronomy Astrophysics, of Section srnmclOsraoy ogn ,106 egae3,Se 38, Belgrade 60 110 7, Volgina Observatory, Astronomical -al nvkvcabb.cy B) [email protected] (BN); [email protected] E-mail: 00RAS 0000 ubro g-eemnto ehd aebe proposed been have methods age-determination of number A c ´ seodfamilies asteroid 1 ⋆ 000 .Tsiganis K. , 0–0 00)Pitd1 etme 08(NL (MN 2018 September 17 Printed (0000) 000–000 , hsproe efis opt oa ifso coefficients, ( eccentricity diffusion proper local in compute sion simple first a we dynamic use for purpose, to estimates is this age goal accurate Our derive and elements. evolution proper the of space diff 3-D orbital the the describes of that model transport a present We ABSTRACT ot-al-yecd scntutdadue otaktee ( the in diffusion track coupling to by used asteroids), and constructed is code Monte-Carlo-type efidteaeo h iiou aiyt be to family mor Lixiaohua the much of thus age and Veritas the than find We older ou much apply is we which Finally, Lixiaohua, prope simulations. in randomk-walk members our family in chaotic duced of spreading the that show we f(9)Vrts o hc ercvrpeiu siae of estimates previous recover we which mode for Veritas, our validate (490) We of effects. thermal Yarkovsky/YORP the by e words: Key r eivdt aeresulted have to believed are , 2 ⋆ n .Kne Z. and eeta ehnc,mnrpaes seod,mtos n methods: asteroids, planets, minor mechanics, celestial t.r(KT); uth.gr l 2002) al. t fPyis rsol nvriyo hsaoii R5 12 54 GR Thessaloniki, of University Aristotle Physics, of t rbia utdfor suited relative h eriments Fami- . tal. et y min- e fthe of teroid the o veral easy. of s 06). of e wa- up- oid nt - i zevi e ˇ p n nlnto ( inclination and ) rprsm-ao xsand axis semi-major proper ino aiymmesi the di in the bodies, members large family than of faster drift tion Vokrouh bodies d & small axis (Farinella As semi-major forces 1999). thermal in Yarkovsky spread of slowly action asteroids the that fact the of ruso bet eiigo eua orbits. regular on residing objects of groups (5.8 cluster (8.3 Karin family the Veritas 2 of (2002, ages al. the Nesvorný et estimate by to applied ti successfully formation the was indicates method conjunction of time the of body, disruption ent the o after the immediately of only conjunction occur a can such elements As value. some orienta around orbital cluster the gles until time, in backwards members family ftefml antb rae hntetm eddfrismo its for needed time the than by greater dated be cannot approximately family be the of can zones, chronology chaotic chaotic in reside which on aiis(..age (i.e. families young owr oapyti ehdt aiislctdi h chaot the in belt. located asteroid families the of to st gions method not this is apply it to Again, forward (2006b). al. Vokrouhlický et and Vokrouhli by (2006a) families several torque to thermal applied YORP successfully of - in been action 1.5 the the for as and accounts much field which velocity as ejection method, of this factor of me a version this by improved age that An real so the neglected, overestimate were can families ogy the estimat (2005 these of al. In Nesvorný et sizes families. by initial asteroid used many of was ages method estimate That to clock. a as used c ´ 1 e ⋆ p o le aiis(..age (i.e. families older For iai&Frnla(94 ugse htatri families asteroid that suggested (1994) Farinella & Milani , I p A ihaditi rprsm-ao xs( axis semi-major proper in drift a with ) T E tl l v2.2) file style X 155 I p ,i eetdpaesaergo.Te,a Then, region. phase-space selected a in ), ± hsmto sbsdo h atta h age the that fact the on based is method This . ± . y) hsmto shwvrlmtdto limited however is method This Myr). 0.5 36 < Myr. 10 so fatrisi hoi regions chaotic in asteroids of usion H lycmlxatri aiis To families. asteroid complex ally hc hrceiecatcdiffu- chaotic characterize which y) st nert h riso the of orbits the integrate to is Myr), ouino admwles(i.e. walkers random of volution steaslt antd a be can – magnitude absolute the is t g ( age its oe otefml f(3556) of family the to model r ( lmnssaei elrepro- well is space elements r > a yapyn tt h family the to it applying by l admwl oe ostudy to model random-walk fetdb hra forces. thermal by affected e p H , hsaoii Greece Thessaloniki, 4 100 ) ∼ umerical y) n a aeuse make can one Myr), ln where – plane 8 ± . 3 . y)ado the of and Myr) 0.2 y) Moreover, Myr). a p ies induced ) a kye al. cký et e The me. p osthe ions inan- tion h par- the thodol- raight- stribu- sthe is ,has s, cre- ic rbital eto ue lický 003) itial 2. st ) , 2 B. Novakovic´ et al. chaotic members to escape from the family region. In its original of a few tens to a few hundreds of asteroids. Although this may be form, this method provides only an upper bound for the age. Re- adequate to compute the average values of the diffusion coefficients cently, Tsiganis et al. (2007) introduced an improved version of this (as in Tsiganis et al. 2007), a detailed investigation of the local dif- method, based on a statistical description of transport in the phase fusion characteristics requires a much larger sample of bodies. The space. Tsiganis et al. (2007) successfully applied it to the family latter can be obtained by adding in the fictitious bodies, selected in of (490) Veritas, finding an age of 8.7 1.7 Myr, which is statisti- such a way that they occupy the same region of the proper elements cally the same as that of 8.3 0.5 Myr,± obtained by Nesvornýet al. space as the real family members. Since we wish to study just these (2003)1. Despite these improvements,± the chaotic chronology still local diffusion properties and the effect of the use of variable coef- suffered from two important limitations. It did not account for the ficients in our chronology method, we are going to follow here this variations in diffusion in different parts of a chaotic zone, which can strategy. significantly alter the distribution of family members (i.e. the shape The 3-D space of proper elements, occupied by the selected of the family). Moreover, it did not account for Yarkovsky/YORP family, is divided into a number of cells. Then, in each cell, the dif- effects, thus being inadequate for the study of older families. fusion coefficients are calculated for both relevant action variables, 1 ap 2 1 ap 2 In this paper we extend the chaotic chronology method, by namely J1 ep and J2 sin Ip (aJ denotes 2 q aJ 2 q aJ constructing a more advanced transport model, which alleviates ∼ ∼ Jupiter’s semi-major axis, ep the proper eccentricity and Ip the the above limitations. We first use the Veritas family as bench- proper inclination of the asteroid). This is done by calculating the mark, since its age can be considered well-defined. Local diffu- 2 time evolution of the mean squared displacement (∆Ji) (i=1,2) sion coefficients are numerically computed, throughout the region in each action, the average taken over the set of bodiesh (reali or fic- of proper elements occupied by the family. These local coeffi- titious) that reside in this cell. The diffusion coefficient is then de- cients characterize the efficiency of chaotic transport at different 2 fined as the least-squares-fit slope of the (∆Ji) (t) curve, while locations within the considered zone. A Monte-Carlo-type model the formal error is computed as in Tsiganish et al. (2007).i is then constructed, in analogy to the one used by Tsiganis et al. The simulation of the spreading of family members in the (2007). The novelty of the present model is that it assumes vari- space of proper actions and the determination of the age of the fam- able transport coefficients, as well as a drift in semi-major axis due ily is done using a Markov Chain Monte Carlo (MCMC) technique to Yarkovsky/YORP effects, although the latter is ignored when (e.g. Gentle 2003; Berg 2004). At each step in the simulation the studying the Veritas family. Applying our model to Veritas, we find random walkers can change their position in all three directions, i.e. that both (a) the shape of its chaotic component and (b) its age the proper semi-major axis ap and the two actions J1 and J2. Al- are correctly recovered. We then apply our model to the family of though no macroscopic diffusion occurs in proper semi-major axis, (3556) Lixiaohua, another outer-belt family but much older than the random walker can change its ap value due to the Yarkovsky Veritas and hence much more affected by the Yarkovsky/YORP effect, while the changes in J1 and J2 are controlled by the local thermal effects. We find the age of the Lixiaohua family to be values of the diffusion coefficients. In the case of normal diffusion 155 Myr. ∼ the transport properties in action space are determined by the solu- We note that, depending on the variability of diffusion coeffi- tion of the Fokker-Planck equation (see Lichtenberg & Lieberman cients in the considered region of proper elements, this new trans- (1983)). The MCMC method is in fact equivalent to solving a port model can be computationally much more expensive than the discretized 2-D Fokker-Planck equation with variable coefficients, one applied in Tsiganis et al. (2007). This is because, if the values combined here with a 1-D equation for the Yarkovsky-induced dis- of the diffusion coefficients vary a lot across the considered region, placement in ap. The latter acts as a drift term, contributing to the one would have to calculate them in many different points. How- variability of diffusion in J1 and J2. ever, even so, this computation needs to be performed only once. The rate of change of ap due to the Yarkovsky thermal Then, the random-walk model can be used to perform multiple runs force, is given by the following equation (e.g. Vokrouhlický1999; at very low cost, e.g. to test different hypotheses about the original Farinella & Vokrouhlický1999): ejection velocities field or about the physical properties of the asteroids. On the other hand, for “smooth” diffusion regions in which da 2 = k1 cos γ + k2 sin γ (1) the coefficients only change by a factor of 2-3 across the consid- dt ered domain, the model can be simplified. In such regions, the age where the coefficients k1 and k2 depend on parameters that de- of a family can be accurately determined even by assuming an av- scribe physical and thermal characteristics of the asteroid and γ de- erage (i.e. constant over the entire region) diffusion coefficient, as notes the obliquity of the body’s spin axis. For km-sized asteroids, we show in Section 3. the drift rate is inversely proportional to their radius. This simplified Yarkovsky model assumes that the asteroid follows a circular orbit, and thus linear analysis can be used to describe heat diffusion across the asteroid’s surface. The obliquity of the spin axis and the 2 THEMODEL angular velocity of rotation (ω) of the asteroid are subject to ther- Our study begins by selecting the target phase-space region. This is mal torques (YORP) that change their values with time, according done by identifying the members of an asteroid family crossed by to the following equations: resonances, from a catalog of numbered asteroids. Apart from the dω dγ g(γ) largest Hirayama families, for the other smaller and more compact = f(γ) , = (2) dt dt ω ones, in the current catalog one typically finds up to several hun- dred members. Thus the chaotic component of the family consists (e.g. Vokrouhlický& Capekˇ 2002; Capekˇ & Vokrouhlický2004), where the functions f and g describe the mean strength of the YORP torque and depend on the asteroid’s surface thermal con- 1 Of course, Tsiganis et al. (2007) and Nevornýet al. (2003) used a chaotic ductivity (Capekˇ & Vokrouhlický2004). and a regular subsets of the family, respectively. The length of the jump in ap that a random-walker under-

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Chaotic Transport and Chronology of Complex Asteroid Families 3 takes at each time-step dt in the MCMC simulation, is determined 3 THERESULTS by equations (1)-(2), in their discretized form. Of course, a set In this section we use our model to study the evolution of the of values of the physical parameters must be assigned to each chaotic component of two outer-belt asteroid families: (490) Ver- body. As the majority of the Veritas family members are of C- itas and (3556) Lixiaohua. In both cases, a number of mean motion type (Di Martino et al. 1997; Mothé-Diniz et al. 2005), while the resonances (MMR) cut-through the family, such that a significant Lixiaohua family members seem to be C/X-type (Lazzaro et al. fraction of members follow chaotic trajectories. On the other hand, 2004; Nesvornýet al. 2005), the following values for these pa- their ages differ significantly, according to previous estimates. In rameters (Broˇzet al. 2005; Broˇz2006) are adopted: thermal con- − this respect, the Yarkovsky effect can be neglected in the study of ductivity K = 0.01 0.5 [W(m K) 1], specific heat capacity −− Veritas, but not in the study of Lixiaohua. C = 1000 [J(Kkg) 1], and the same value for surface and bulk − We begin by performing an extensive study of the local dif- density ̺ = 1500 [kg m 3]. In Tedesco et al. (2002), the geomet- fusion properties in the chaotic region of the Veritas family. Then, ric albedos (pv) of several Veritas and Lixiaohua family members the MCMC model is used to simulate the evolution of the chaotic are listed, yielding a mean pv = 0.068 0.018 for Veritas and ± members and to derive an estimate of the age of the family. The re- pv = 0.049 0.027 for Lixiaohua. The rotation period, P , is cho- ± sults are compared to the ones given by the model of Tsiganis et al. sen randomly from a Gaussian distribution peaked at P = 8 h, (2007). Finally, we apply the MCMC model to the Lixiaohua fam- while the distribution of initial obliquities, γ, is assumed to be uni- ily and derive estimates of its age, for different values of the form2. To assign the appropriate values of absolute magnitude H Yarkovsky-related physical parameters. to each body, we need to have an estimate of the cumulative distribution N(< H) of family members. A power-law approximation is used (e.g. Vokrouhlickýet al. 2006b) 3.1 The Veritas family N(

4 B. Novakovic´ et al.

0.075 3 3 -2 5 -2 -2 7 -7 -2

3.177 0.07 3.176

0.065 3.175 p e 3.174 [AU] 0.06 p a 3.173

0.055 3.172 3.171 50 100 150 200 250 300 0.05 t [kyr] 3.15 3.155 3.16 3.165 3.17 3.175 3.18 3.185 3.19

ap [AU]

0.168 3 3 -2 5 -2 -2 7 -7 -2 0.166 350 0.164 300 0.162 250 p

0.16 [deg] 200 sin I 0.158 150 (5,-2,-2) σ 100 0.156 50 0.154 0 0.152 50 100 150 200 250 300 3.15 3.155 3.16 3.165 3.17 3.175 3.18 3.185 3.19 t [kyr]

ap [AU]

Figure 1. Distribution of the real Veritas family members in the (ap,ep) Figure 2. The time evolution of the mean semi-major axis a (top) and the and (ap,sinIp) planes. The superimposed ellipses represent equivelocity resonant angle (bottom), for a ﬁctitious body inside the (5, -2, -2) three-body curves, computed according to the equations of Gauss (e.g. Morbidelli et al. MMR. Note the correlation between the two quantities. Temporary capture 1995), for a velocity of 40m s−1, true anomaly f = 30◦ and argument at the resonance border occurs when a is at maximum (resp. minimum) and ◦ of pericentre ω = 150 . The vertical dashed lines represent approximate σ(5,−2,−2) circulates in a positive (resp. negative) sense. borders of the main three-body MMRs, as indicated by the corresponding labels.

number of cells in ap and a small number of cells in ep and sin Ip, outer-belt asteroids. Note that the integration time used here is in except in the (5,-2,-2) region. The efficiency of the computation is fact longer than the known age of the Veritas family. This is done in improved if we use a moving-average technique (i.e. overlapping order to study the convergence of the computation of the diffusion cells), instead of a large number of static cells, because in the latter coefficients, with respect to the integration time-span. case we would need a significantly larger number of fictitious bod- For each body mean elements are computed on-line, by ap- ies. We selected the size of a cell in each dimension as well as an applying digital filtering, and proper elements are subsequently com- propriate step-size, by which we shift the cell through the family, as −4 puted according to the analytical theory of Milani & Kneˇzević follows: for ap, the cell-size was ∆ap = 5 10 AU and the step- −4 × −4 (1990, 1994). Synthetic proper elements (Kneˇzević& Milani 2000) size 10 AU; for J1 the cell-size was ∆J1 = 10 and the step- −5 −4 are also calculated, for comparison and control. Since the mapping size 4 10 ; finally, for J2, the cell-size was ∆J2 = 2 10 × − × from osculating to proper elements is not linear, the distribution of and the step-size 10 4. Thus, the total number of (overlapping) the fictitious bodies in the space of proper elements is not uniform, cells used in our computations was 2, 196. which can complicate the statistics. A smaller sample of 10, 000 The time evolution of the mean squared displacement in J1 ∼ bodies, with practically uniform distribution in proper elements, is and J2 is shown in Fig. 3, for a representative cell in the (5,-2,-2) therefore chosen. Thus, our statistical sample, on which all com- resonance. The evolution is basically linear in time, as it should be putations are based, is in fact 25 times larger than the actual for normal diffusion. The slope of the fitted line defines the value population of the family. ∼ of the local diffusion coefficient. When performing such computa- As explained in the previous section, we computed local dif- tions, one needs to know (i) what is the shortest possible integration fusion coefficients, by dividing the space occupied by the Veritas time-span, and (ii) what is the smallest possible number of fictitious family in a number of cells. Our preliminary experiments suggested bodies per cell, for which reliable values of the coefficients can be that, while a large number of cells is needed to accurately represent obtained. For several different groups of fictitious bodies (i.e. dif- the dependence of the coefficients on ap, the same is not true for ferent cells), we calculated the diffusion coefficients using differ- ep and sin IP , except for the wide chaotic zone of the (5, 2, 2) ent values of the integration time-span, between 1 and 10 Myr. Our − − MMR. Thus, we decided to follow the strategy of using a large results suggest, that an integration time of 4 Myr is sufficient ∼ c 0000 RAS, MNRAS 000, 000–000

Chaotic Transport and Chronology of Complex Asteroid Families 5

1.6e-007

5 -2 -2 ap = 3.174 AU 1.4e-007 2.5e-014 1.2e-007 2e-014 1e-007 1.5e-014 ) > 1 2 ) 1

J 8e-008 D(J ∆ 1e-014 <( 6e-008 5e-015 4e-008 0 0 1 2 3 4 5 6 7 8 9 10 11 2e-008 tint [Myr] 0 0 2e+006 4e+006 6e+006 8e+006 1e+007 t [yr] 1.6e-007

5 -2 -2 ap = 3.174 AU 1.4e-007 2.5e-014 1.2e-007 2e-014 1e-007 1.5e-014 ) > 2 2 ) 2

J 8e-008 D(J ∆ 1e-014 <( 6e-008 5e-015 4e-008 0 0 1 2 3 4 5 6 7 8 9 10 11 2e-008 tint [Myr] 0 0 2e+006 4e+006 6e+006 8e+006 1e+007 t [yr]

Figure 3. The mean squared displacement in J1 and J2 as a function of Figure 4. The values of the diffusion coefficients D(J1) and D(J2) as a time, for a cell inside the (5,-2,-2) MMR. The evolution is basically linear function of the integration time span. Each curve on these graphs corre- in time, as can be seen from the respective fit given on each graph, with a sponds to a different cell. superimposed small amplitude oscillation of a period of 5 Myr. ∼ −14 −1 local minima are D(J1) = (0.73 0.01) 10 yr , and D(J2) −14 −1 ± × to obtain reliable values, as shown in Fig. 4. This saturation time = (1.16 0.02) 10 yr . This form of dependence in ap is is about half the known age of the Veritas family. Hence, for this in agreement± with× the results of Nesvorný& Morbidelli (1999), case, computing diffusion coefficients is practically as expensive where it was shown that the dynamics in this resonance are simi- as studying the evolution of the family by long-term integrations. lar to those of a modulated pendulum (see also Morbidelli (2002)), However, the saturation time is related to the resonance in ques- with an island of regular motion persisting at the center of the res- tion and not to the age of the family, which could be much longer. onance. However, the size of the island decreases with decreasing Thus, as a matter of principle, the computational gain can become ep, so that the diffusion coefficients may decrease as the resonance important when dealing with much older families. Resonances of center is approached, but do not go to zero. similar order are characterized by similar Lyapunov and diffusion As also seen in Fig. 5, the (5,-2,-2) is by far the most important times (see (Murray & Holman 1997)), and thus similar computa- resonance in the Veritas region, associated with the widest chaotic tion time-spans (i.e. a few Myr) should be used, for various reso- zone. Bodies inside this resonance exhibit a complex behaviour, nances throughout the belt. as already noted by Kneˇzević& Pavlović(2002). Most resonant The dependence of the diffusion coefficients on the number bodies show oscillations in proper semi-major axis around ap = of bodies considered in each cell was also tested. For a number 3.174 AU, but some are temporarily trapped near the resonance’s of different cells, we calculated the coefficients, using from 10 to borders (see also Fig. 2), at ap = 3.172 AU or ap = 3.1755 AU. This 100 bodies for the computation of the corresponding averages. Our “stickiness” can be important, as it can affect the diffusion rate. In results suggest that at least 50 bodies per cell are needed, for an fact, we find that the ap values of these bodies shift towards the accurate computation. resonance borders, where slower diffusion rates are also measured. The values of the diffusion coefficients in J1 and J2, along The diffusion properties are quite different at the (3,3,-2) (lo- with their formal errors, are given as functions of ap in Fig. 5. The cated at ap 3.168 AU) and (7,-7,-2) (at ap 3.18 AU) MMRs, largest diffusion rate is measured in the (5,-2,-2) MMR, which cuts which are of≈ higher order in eccentricity with≈ respect to the (5,-2,- through the family at ap 3.174 AU. Both coefficients increase as 2) MMR (see Nesvorný& Morbidelli 1999). The values of D(J1) the center of the resonance∼ is approached, but show local minima for the (3,3,-2) resonance (see Fig. 5a) are also increasing as the at ap 3.174 AU, which is approximately the location of the cen- center of the resonance is approached, but no local minimum is ≈ −14 −1 ter. The maximum values are D(J1) = (1.28 0.02) 10 yr , seen near the center of the resonance, at least in this resolution. −14 −1 ± × −16 −1 and D(J2) = (1.36 0.02) 10 yr , while the values at the The maximum values are only D(J1) = (8.30 0.09) 10 yr ± × ± × c 0000 RAS, MNRAS 000, 000–000

6 B. Novakovic´ et al.

1.4e-014

3 3 -2 7 -7 -2 1.2e-014 1e-015 1e-015 1.8e-014 1e-014 5e-016 5e-016 0.165 ]

-1 8e-015 0 0 1.6e-014 3.167 3.168 3.169 3.17 3.1795 3.18 3.1805 ) [yr 1 6e-015 D(J 0.16 1.4e-014

4e-015 p 1.2e-014 sin I 2e-015 0.155 1e-014 0 3.165 3.17 3.175 3.18 3.185 ap [AU] 0.15 8e-015

1.4e-014 6e-015 3 3 -2 7 -7 -2 1.2e-014 0.055 0.06 0.065 0.07 4e-017 4e-017 e 1e-014 p 2e-017 2e-017 ]

-1 8e-015 0 0 3.167 3.168 3.169 3.17 3.1795 3.18 3.1805 ) [yr 2 6e-015 D(J

4e-015 2e-014 0.165 2e-015 1.8e-014

0 3.165 3.17 3.175 3.18 3.185 0.16 1.6e-014 ap [AU] p 1.4e-014 sin I Figure 5. The local diffusion coefﬁcients D(J1) and D(J2) (with their cor- 0.155 responding error-bars) in the Veritas family region, shown here as functions 1.2e-014 of the proper semi-major axis ap. The embedded rectangles are magniﬁca- tions of the graph, in the vicinity of the (3, 3, -2) and (7, -7, 2) MMRs. Note 0.15 1e-014 that D(J2) is practically zero everywhere, except in the (5,-2,-2) region. 8e-015

−16 −1 0.055 0.06 0.065 0.07 and D(J2) = (0.22 0.03) 10 yr , clearly much smaller than ± × e in the (5,-2,-2) MMR. Note that D(J2) has practically zero value. p The region of the (7,-7,-2) resonance is even less exciting. D(J1) is almost constant across the resonance, with a very small value −16 −1 (4.1 0.06 10 yr ), and D(J2) is practically zero. Figure 6. The local diffusion coefficients D(J1) (top) and D(J2) (bottom) ± × The above results suggest that the (5,-2,-2) MMR is essentially inside the (5,-2,-2) region, shown here in grey-scale as functions of ep and the only resonance in the Veritas region characterized by apprecia- sin Ip. The values are averaged over the resonant range in semi-major axis. ble macroscopic diffusion. We now focus on the variation of the diffusion coefficients with respect to (ep, sin Ip) along this resonance. A set of six values ( , , , , , ) is assigned to each random As shown in Fig. 6, the dependence of the diffusion rate on the ini- ap J1 J2 P γ H 4 walker in the simulation. All bodies are initially distributed uni- tial values of the actions is very complex. The values of D(J1) −14 −1 −14 −1 formly inside a region of predefined size in δJi(0) and semi-major vary from (0.60 0.01) 10 yr to (1.66 0.02) 10 yr ± × ± ×−14 −1 axes in the range [3.172, 3.176]. The age of the family, (τ), is de- while, for D(J2), they vary from (0.63 0.02) 10 yr to −14 −1 ± × fined as the time needed for 0.3% of the random walkers to leave (2.31 0.03) 10 yr . Consequently, chaotic diffusion along an ellipse in the ( ) plane, corresponding to a 3- confidence this resonance± × can in principle produce asymmetric “tails” in the J1, J2 σ interval of a 2-D Gaussian distribution. We note that, for Veritas, distribution of group-A members. Note, however, that the coeffi- the mobility in semi-major axis due to Yarkovsky is very small and cients only vary by a factor of 2 3 and that their average values practically insignificant for what concerns the estimation of its age, are essentially the same as in Tsiganis− et al. (2007). since the family is young and distant from the Sun. For older families one should also define appropriate borders in ap. 3.1.2 The MCMC Simulations - Chronology of Veritas Using the values of the coefficients obtained above, and the − − values of σ(J )=(2.31 0.22) 10 4, σ(J )=(3.97 0.30) 10 4 , We can now use the MCMC method to simulate the evolution and 1 2 calculated from the distribution± × of the real group A± members× , we determine the age of the Veritas family, assuming that all the dy- simulate the spreading of group A and estimate its age. Of course, namically distinct groups originated from a single brake-up event. the model depends on some free parameters: the initial spread of the group in (δJ1(0), δJ2(0)), the time-step, dt, and the number 4 The values of the diffusion coefficients are calculated for a regular grid of random-walkers, n. Therefore, the dependence of the age, τ, on in J1 and J2 and then translated into proper elements space. these parameters was checked. Uncertainties in the values of the

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Chaotic Transport and Chronology of Complex Asteroid Families 7 current borders of the group (i.e. the conﬁdence ellipse) and the values of the diffusion coefﬁcients were taken into account, when δ -4 δ -4 calculating the formal error in τ. n=2000, J1(0)=1.25 x 10 , J2(0)=5.6 x 10 Different sets of simulations were performed, the results of 11 10 which are given in Fig. 7(a)-(d). Each “simulation” (i.e. each point 9 8

in a plot) actually consists of 100 different realizations (runs) of [Myr] τ 7 the MCMC code. In each run, the values of D(Ji) and σ(Ji) were 6 varied, according to the previously computed distributions of their 5 values. The values of the free parameters were the same for all runs 0 1000 2000 3000 4000 5000 6000 dt [yr] in a given simulation. The first set of simulations was performed in order to check how the results depend on the time step, dt. Five simulations were made, with dt ranging from 1000 to 5000 yr (Fig. 7a). The stan- dt=2000, δJ (0)=1.25 x 10-4, δJ (0)=5.6 x 10-4 dard deviation of τ is relatively small, suggesting that τ is roughly 1 2 11 independent of dt. According to this set of simulations, the age of 10 the family is τ = τ ∆τ = 8.5 1.3 Myr, where τ is the 9 h i± ± h i 8 mean value and ∆τ the standard error of the mean. This estimate [Myr] τ 7 is in excellent agreement with those of Nesvornýet al. (2003) and 6 Tsiganis et al. (2007). 5 The second set of simulations was performed in order to check 0 1000 2000 3000 4000 5000 6000 n how the results depend on the number of random walkers, n. As shown in Fig. 7b), τ is weakly dependent of n, the variation of the mean value of τ is slightly larger than in the previous case. The age of the family, according to this set, is τ = 8.6 1.3 Myr. δ -4 ± n=2000, dt=2000 yr, J1(0)=2.30 x 10 The third group of simulations was performed in order to 11 check how the results depend on the assumed initial spread of group 10 9 A in δJ2(0), a parameter which is poorly constrained from the re- 8 [Myr] spective equivelocity ellipse in Fig. 1. We fixed the value of δJ1(0) τ 7 − to 2.3 10 4, which can be considered an upper limit, according 6 × 5 5 to Fig. (1) . Six simulations were performed, with δJ2(0) ranging −4 −4 2 4 6 8 10 12 from 3.5 10 to 11.0 10 , and the results are shown in Fig. δ 4 × × J2(0) x 10 7c. The values of τ tend to decrease as δJ2(0) increases. This is to be expected, since increasing the initial spread of the family, while targeting for the same final spread, should take a shorter time for a given diffusion rate. The results yield τ = 8.8 1.1 Myr. n=2000, δJ (0)=2.3 x 10-4, δJ (0)=11.3 x 10-4 As a final check, we performed a set± of simulations with 1 2 −4 −4 11 δJ1(0) = 2.3 10 and δJ2(0) = 11.0 10 . These values cor- × × 10 respond to equivelocity ellipses that contain almost all regular and 9 8 [Myr]

(3,3,-2)-resonant family members, except for the very low inclina- τ 7 tion bodies (sin Ip < 0.156). Five sets of runs, for five different 6 values of dt, were performed (see Fig. 7d). As in our first set of 5 0 1000 2000 3000 4000 5000 6000 simulations, τ is practically independent of dt. On the other hand, dt [yr] τ turns out to be smaller than in the previous simulations, since the assumed values for δJ1(0) and δJ2(0) are quite large. Even so, we find τ = 7.6 1.1 Myr, which is still an acceptable value. Combining± the results of the first three sets of simulations and Figure 7. Dependence of τ and σ(τ) on three free parameters: (a) time step dt; (b) number of random walkers n; (c) initial size of the family δJ (0). taking into account all uncertainties, we find an age estimate of τ = 2 The bottom panel, 7(d), shows the dependence of τ on dt for a run in which 8.7 1.2 Myr for the Veritas family. This result is very close to the ± the maximum values of δJ1(0) and δJ2(0) were assumed. The values of one found by Tsiganis et al. (2007), the error though being smaller the free parameters that are constant in each group of simulations are indi- by 30%. cated on top of each plot. The horizontal lines and dashed areas denote the ∼ In addition to the determination of the family’s age, we would age of the Veritas family (and the corresponding errorbar), as obtained by like to know how well the MCMC model reproduces the evolu- Nesvornýet al. (2003). tion of the spread of group-A bodies, in the (J1,J2) space. For this purpose we compared the evolution of group A for 10 Myr in the future, as given (i) by direct numerical integration of the orbits, and Fig. 8(a), the random-walkers of the MCMC simulation (triangles) (ii) by an MCMC simulation with variable diffusion coefficients. practically cover the same region in (J1,J2) as the real group-A Figures 8(a)-(b) show the outcome of this comparison. As shown in members (circles). Moreover, the time evolution of the ratio of the standard deviations σ(J2)/σ(J1), which characterizes the shape of the distribution, is reproduced quite well, as shown in Fig. 8(b). 5 Assuming that an equivelocity ellipse in (ap,ep), large enough to encom- An additional MCMC simulation with constant (i.e. average pass both the regular part of the family and the (3,3,-2) bodies, is a better with respect to ep and sin Ip) coefficients of diffusion was also per- constraint. formed. As shown in Fig. 8(b), the value of σ(J2)/σ(J1) in this c 0000 RAS, MNRAS 000, 000–000

8 B. Novakovic´ et al.

this way estimated the age of this family to τ = 300 200 Myr6. 0.0115 ± +10 Myr Thus, we choose to study the Lixiaohua family because, on one 0.011 hand, it is relatively old, so Yarkovsky/YORP effects are important, but, on the other hand, it has the feature we need (i.e. a signiﬁcant 0.0105 chaotic zone) to test the behaviour of our model on longer time scales. Here, we use our MCMC method to derive a more accurate 2

J 0.01 estimate of its age, taking into account also the Yarkovsky/YORP effects. 0.0095 The distribution of the family members in (ap,ep) and (ap, sin Ip) is shown in Fig. 9. Using velocity cut-off vc = 0.009 50 m s−1 we find 263 bodies (database as of February 2009) linked to the family. The shape of this family is, as in the Veritas case, 0.0085 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 intriguing. For a> 3.15 AU, the family appears to better fit inside the equivelocity ellipse shown in the figure, with only a few bodies J1 showing a significant excursion in ep and sin Ip. On the other hand, 1.8 for a < 3.145 AU, the family members occupy a wider area in ep and sin Ip. Throughout the family region we find thin, “vertical”, 1.7 7 4 strips of chaotic bodies, with Lyapunov times TL < 2 10 y. These strips are associated to different mean motion resona× nces. 1.6 The most important chaotic domain (hereafter Main Chaotic Zone, ) 1 1.5 MCZ) is the one centered around ap 3.146 AU; indicated by (J ≈ σ the grey-shaded area in both plots of Fig. 9. A number of two- and )/ 2

(J 1.4 three-body MMRs can be associated to the formation of the MCZ, σ such as the 17:8 MMR with Jupiter and the (7, 9, -5) three-body 1.3 MMR. Note that the two largest members of this family, (3330) 1.2 Gantrisch and (5900) Jensen, are located just outside the MCZ, as 4 indicated by their larger values of TL (> 2 10 y). In fact, a 1.1 × 0 2e+06 4e+06 6e+06 8e+06 1e+07 significant group of bodies just outside the MCZ (see Fig. 9a) has time [yr] higher values of TL but similar spread in ep as the MCZ bodies. This suggests that bodies around the chaotic zone could have once Figure 8. Comparison of the time evolution of group A Veritas members resided therein, evolving towards high/low values of ep by chaotic in (i) a numerical integration of their orbits for 10 Myr, and (ii) an MCMC diffusion. Numerical integrations of the orbits of selected Lixiao- realization, corresponding to the same time interval. Top: the final distribution, as taken from the integration (circles) and the MCMC run (triangles). hua members for 100 Myr indeed confirmed that bodies could enter (or leave) the MCZ. We believe that the distribution of family mem- Bottom: time evolution of the ratio σ(J2)/σ(J1). The thick solid (resp. dashed) curve corresponds to the MCMC run with variable (resp. constant) bers on either side of the MCZ is strongly indicative of an interplay diffusion coefficients, while the thin, “noisy” one corresponds to the numer- between Yarkovsky drift in semi-major axis and chaotic diffusion ical integration of the equations of motion. in ep and sin Ip, induced by the overlapping resonances; bodies can be forced to cross the MCZ, thus receiving a “kick” in ep and sin Ip, before exiting on the other side of the zone. A population of 5, 000 fictitious bodies was selected and simulation appears to slowly deviate from the one measured in the used for calculation of≈ the local diffusion coefficients. Here, we re- previous MCMC simulation, as time progresses. However, this de- strict ourselves in calculating coefficients as functions of ap only viation is not very large, also compared to the result of the numeri- (i.e. averaged in ep and sin Ip). As shown in Fig. 10, there are sev- cal integration. Thus, we conclude that an MCMC model with con- eral diffusive zones, corresponding to the low-TL strips of Fig. 9. stant coefficients is adequate for deriving a reasonably accurate es- However, as in the Veritas case, only one zone appears diffusive timate of the age of a family, provided that the variations of the local in both actions; D(J2) is practically zero everywhere outside the diffusion coefficients are not very large. Given this result, we de- MCZ (a 3.146 AU), and significant dispersion in proper ele- cided to use average coefficients for the Lixiaohua case. Note also ments is observed∼ only in this zone. that, in the Veritas case, the observed deviation in σ(J2)/σ(J1) Given the above results, we conclude that the MCZ family between the two MCMC models is reflected in the error of τ (i.e. members can be used to estimate the age of the family, much like 1.7 Myr vs. 1.2 Myr). the Veritas (5,-2,-2) resonant bodies. Given the fact that random- walkers can drift in ap, the way of computing the age is accordingly modified. A large number of random walkers, uniformly distributed across the whole family region (i.e. the equivelocity ellipses) is 3.2 The age of the Lixiaohua family used. The simulation again stops when 0.3% of MCZ-bodies are The Lixiaohua family is another typical outer-belt family, crossed found to be outside the observed (J1, J2) borders of the family. by several MMRs. This results into a significant component of family members that follow chaotic trajectories. At the same time, a clear ‘V’-shaped distribution is observed in the (ap,H) plane (see 6 This age was estimated neglecting the initial size of the family. Fig. 9), suggesting that the family is old-enough for Yarkovsky to 7 For details on the computation of Lyapunov times see e.g. Tsiganis et al. have significantly altered its size in ap. Nesvornýet al. (2005) in (2003)

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Chaotic Transport and Chronology of Complex Asteroid Families 9

0.215 9e-15 8e-15 0.21 7e-15 0.205 6e-15 ] 0.2 -1 5e-15 p e ) [yr 0.195 1 4e-15 D(J 3e-15 0.19 2e-15 (5900) Jensen 0.185 (3330) Gantrisch 1e-15 0.18 0 3.13 3.135 3.14 3.145 3.15 3.155 3.16 3.165 3.17 3.13 3.135 3.14 3.145 3.15 3.155 3.16 3.165 3.17

ap [AU] ap [AU]

0.184 4e-15 (3330) Gantrisch 0.182 3.5e-15

0.18 3e-15

] 2.5e-15

0.178 -1 p 2e-15 ) [yr 2

sin I 0.176

D(J 1.5e-15 0.174 1e-15 0.172 (5900) Jensen 5e-16 0.17 0 3.13 3.135 3.14 3.145 3.15 3.155 3.16 3.165 3.17 3.13 3.135 3.14 3.145 3.15 3.155 3.16 3.165 3.17

ap [AU] ap [AU]

Figure 10. The same as Fig. 5, but for the Lixiaohua family. Note that while 16 several diffusive peaks are visible in D(J1), only the region of the MCZ (ap 3.146 AU) shows signiﬁcant diffusion in J2. 15 ∼

H [mag] 13 In order to compute the age of the family we performed 2400 12 MCMC runs. This was repeated three times, for three different val- (5900) Jensen ues of thermal conductivity (see above) and once more, neglecting 11 (3330) Gantrisch the Yarkovsky effect. Given the uncertainty in determining the ini- 3.13 3.135 3.14 3.145 3.15 3.155 3.16 3.165 3.17 tial size of the family, we repeated the computations for two more

ap [AU] sets of δJ1(0), δJ2(0). The results of these computations are presented in Fig. 11. We find an upper limit of 230 Myr for the age ∼ of the family and a lower limit of 100 Myr. Taking into account Figure 9. Top and middle panels: The same as Fig. 1, but for the Lixi- ∼ aohua family. The different symbols correspond to different values of the only the runs performed for our “nominal” initial size and includ- 5 ing the Yarkovsky effect, we find the age of Lixiaohua family to be Lyapunov time, TL. Triangles correspond to TL > 10 y, empty circles to 4 5 4 2 10 TL 10 y and solid circles to TL < 2 10 y. The equiv- 155 36 Myr. This value lies towards the lower end but comfort- × ≤ ≤ − × ◦ ◦ ± elocity curves were obtained for v = 40 ms 1, f = 80 and ω = 300 . ably within the range (300 200 Myr) given by Nesvornýet al. ± Bottom: The distribution of Lixiaohua members in the (ap,H) plane. The (2005). grey-shaded region denotes the extent of the main chaotic zone (MCZ) in We note that, when the Yarkovsky effect is taken into account, ap. The two largest members of the family are also denoted. the age of the family turns out to be longer by 30 Myr (see also Fig. 11). This is a purely dynamical effect, related∼ to the fact that bodies can drift towards the MCZ from the adjacent non-diffusive However, the number of MCZ bodies is not constant during the regions. However, as more bodies enter the MCZ near its center simulation, because bodies initially outside (resp. inside) the MCZ in ep and sin Ip, it takes longer for 0.3% of random walkers to can enter (resp. leave) that region. Thus, the aforementioned per- diffuse outside the confidence ellipse in (ep, sin Ip). At the same centage is calculated with respect to the corresponding number of time, bodies that are initially inside the MCZ can also drift outside, MCZ bodies at each time-step. to lower/higher values of ap, thus slowing down or even stop diffus- The size of the MCZ in the space of proper actions is given by ing in ep and Ip. This also explains the large spread in (ep, sin Ip) −4 −4 σ(J1)= (8.49 0.51) 10 and σ(J2)=(3.17 0.28) 10 . observed for family members located just outside the MCZ. ± × ± × c 0000 RAS, MNRAS 000, 000–000

10 B. Novakovic´ et al.

chronology. Both methods are basically statistical and make use of the quasi-linear time evolution of certain statistical quantities 220 (either the spread in ∆a or the dispersion in ep and sin Ip), de- 200 scribing a family. There are, however, important differences. The Yarkovsky/YORP chronology method works better for older fami- 180 lies and the age estimates are more accurate for this class of asteroid families (provided there are no other important effects on that time

[Myr] 160

τ scale). On the other hand, for our method to be efficient, we need that diffusion is fast enough to cause measurable effects, but slow 140 enough so that most of the family members are still forming a ro- 120 bust family structure (i.e. there is no dynamical “sink” that would lead to a severe depletion of the chaotic zone). Thus, our model 100 can be applied to a limited number of families that reside in com- 3e-08 5e-08 7e-08 9e-08 1.1e-07 plex phase-space regions, but, in the same time, this is the only Size of the initial ellipse model that takes into account the chaotic dispersion of these families. There are at least a few families for which both chronology Figure 11. The age of Lixiaohua family, as measured by our MCMC runs. methods can be applied, thus leading to more reliable age estimates, The mean value and standard deviation (error-bar) of the age are given, as as well as to a direct comparison of the two different chronologies. functions of the assumed size of the initial equivelocity ellipse. Four sets of For example, the families of (20) Massalia and (778) Theobalda points are shown, corresponding to our estimate (i) including Yarkovsky and assuming different values of the thermal conductivity (K = 0.01 = dotted would be good test cases. We, however, reserve this for future work. line, K = 0.1 = dashed line and K = 0.5 = solid thin line), and (ii) An important advantage of the model is that it can be used to neglecting Yarkovsky (thick solid line). estimate the physical properties of a dynamically complex asteroid family, provided that its age is known by independent means (e.g. by applying the method of Nesvornýet al. (2003) to the regular members of the family). A large number of MCMC runs can 4 CONCLUSIONS be performed at low computational cost, thus allowing a thorough We have presented here a refined statistical model for asteroid analysis of the physical parameters of family members or the prop- transport, which accounts for the local structure of the phase-space, erties of the original ejection velocities field that better reproduce by using variable diffusion coefficients. Also, the model takes into the currently observed shape of the family. account the long-term drift in semi-major axis of asteroids, induced by the Yarkovsky/YORP effects. This model can be applied to simulate the evolution of asteroid families, also giving rise to an ad- ACKNOWLEDGEMENTS vanced version of the ”chaotic chronology” method for the determination of the age of asteroid families. The work of B.N. and Z.K. has been supported by the Ministry of We applied our model to the Veritas family, whose age is well Science and Technological Development of the Republic of Serbia constrained from previous works. This allowed us to assess the (Project No 146004 ”Dynamics of Celestial Bodies, Systems and quality and to calibrate our model. We first analyzed the local dif- Populations”). fusion characteristics in the region of Veritas. Our results showed that local diffusion coefficients vary by about a factor of 2 3 − across the (ep, sin Ip) region covered by the (5,-2,-2) MMR. Thus, REFERENCES although local coefficients are needed to accurately model (by the MCMC method) the evolution of the distribution of group-A mem- Beaugé, C., Roig, F., 2001, Icarus, 153, 391 bers, average coefficients are enough for a reasonably accurate es- Bendjoya, Ph., Zappalà, V., 2002, in: W. F. Bottke Jr., A. Cellino, timation of the family’s age. We note though that the variable co- P. Paolicchi, and R. P. Binzel (eds.), Asteroids III, University of efficients model reduces the error in τ by 30%, but requires Arizona Press, Tucson, 613 a computationally expensive procedure. Using∼ the variable coeffi- Berg, A.B., 2004, Markov Chain Monte Carlo Simulations and cients MCMC model, we found the age of the Veritas family to be Their Statistical Analysis, World Scientific, Singapore τ = (8.7 1.2) Myr; a result in very good agreement with that of Bottke, W.F., Durda, D.D., Nesvorný, D., Jedicke, R., Morbidelli, Nesvornýet± al. (2003) and Tsiganis et al. (2007). A., Vokrouhlick`y, D., Levison, H.F., 2005, Icarus 179, 63 We used our model to estimate also the age of the Lixiao- Broˇz, M., Vokrouhlický, D., Roig, F., Nesvorný, D., Bottke, W.F., hua family. This family is similar to the Veritas family in many re- Morbidelli, A., 2005, MNRAS, 359, 1437 spects; it is a typical outer-belt family of C-type asteroids, crossed Broˇz, M., 2006, Yarkovsky effect and the dynamics of Solar sys- by several MMRs. Like the Veritas family, only the main chaotic tem. Ph.D. Thesis, Faculty of Mathematics and Physics, Charles zone (MCZ) shows appreciable diffusion in both eccentricity and University, Prague. inclination. On the other hand, this is a much older family and the Brown, M.E., Barkume, K.M., Ragozzine, D., Schaller, E.L., Yarkovsky effect can no longer be ignored. This is evident from the 2007, Nature, 446, 294 distribution of family members, adjacent to the MCZ. Our model Capek,ˇ D., Vokrouhlický, D., 2004, Icarus, 172, 526 suggests that the age of this family is between 100 and 230 Myr, Carruba, V., Burns, J.A., Bottke, W.F., Nesvorný, D., 2003, Icarus the best estimate being 155 36 Myr. 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