Chaotic Diffusion of the Vesta Family Induced by Close Encounters With

Home , 10 Hygiea, 15 Eunomia, 19 Fortuna, 324 Bamberga

Astronomy & Astrophysics manuscript no. article September 6, 2018 (DOI: will be inserted by hand later)

Chaotic diﬀusion of the Vesta family induced by close encounters with massive asteroids

J.-B. Delisle and J. Laskar

ASD, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau, 75014 Paris, France

September 6, 2018

Abstract. We numerically estimate the semi-major axis chaotic diffusion of the Vesta family asteroids induced by close encounters with 11 massive main belt asteroids : (1) Ceres, (2) Pallas, (3) Juno, (4) Vesta, (7) Iris, (10) Hygiea, (15) Eunomia, (19) Fortuna, (324) Bamberga, (532) Herculina, (704) Interamnia. We find that most of the diffusion is due to Ceres and Vesta. By extrapolating our results, we are able to constrain the global effect of close encounters with all the main belt asteroids. A comparison of this drift estimate with the one expected for the Yarkovsky effect shows that for asteroids whose diameter is larger than about 40 km, close encounters dominate the Yarkovsky effect. Overall, our findings confirm the standard scenario for the history of the Vesta family.

1. Introduction showed that close encounters among massive asteroids induce strong chaos which appears to be the main limit- It is now admitted that most of V-type near-Earth aster- ing factor for planetary ephemeris on tens of Myr. In the oids (NEAs) and howardite, eucrite and diogenite (HED) continuation of this work we concentrated our study on meteorites are fragments of a collision between Vesta and the diffusion induced by close encounters of Vesta family another object which also produced the Vesta family (see members with massive asteroids. Sears 1997). V-type NEAs and HED meteorites are former members of the Vesta family that have been carried from This effect has already been studied for different as- the main asteroid belt to Earth-crossing orbits by the two teroid families (e.g. the Flora and Lixiaohua families Nesvornýet al. 2002; Novakovićet al. 2010). The most main resonances in the neighborhood of Vesta : the ν6 and 3 : 1 resonances with Jupiter. The major issue in this sce- exhaustive study (in terms of number of asteroids taken nario is the fact that the average life time of fragments into account) concerned the Gefion and Adeona families in these two resonances is too short to explain the mean (Carruba et al. 2003). The authors considered all 682 as- age of HED meteorites and V-type NEAs (Migliorini et al. teroids of radius larger than 50 km for encounters with 1997). members of the families. They concluded that the four The commonly accepted explanation for this is that largest asteroids ((1) Ceres, (2) Pallas, (4) Vesta and (10) NEAs spent most of their life time in the main asteroid Hygiea) had a much larger influence than all the 678 re- belt before entering in resonance and being carried to their maining asteroids so the latters can be considered as negli- current orbits. This supposes that a mechanism can induce gible. The Vesta family has also been the object of such an a diffusion process that brings fragments to one (or both) analysis (Carruba et al. 2007) but only close encounters of the two main resonances. The Yarkovsky effect is now with (4) Vesta were taken into account. These different admitted to be the main mechanism explaining the diffu- studies show that the effect of close encounters with mas- sion of Vesta family members (e.g. Carruba et al. 2003). sive asteroids depends a lot on the considered family and its position in phase space with respect to those of massive However other processes also contribute to the diffu- arXiv:1110.5820v1 [astro-ph.EP] 26 Oct 2011 asteroids. sion. It is in particular the case for the chaos induced by overlaps of mean motion resonances or close encounters The purpose of this work is to explore further the between asteroids in the main belt. In the case of the res- effect of close encounters for the Vesta family by con- onances, Morbidelli & Gladman (1998) showed that the sidering the effect of 11 large asteroids : (1) Ceres, (2) chaotic zone around the 3 : 1 resonance is too narrow to Pallas, (3) Juno, (4) Vesta, (7) Iris, (10) Hygiea, (15) explain the needed diffusion. Recently, Laskar et al. (2011) Eunomia, (19) Fortuna, (324) Bamberga, (532) Herculina, (704) Interamnia. Moreover, we use the results from these Send offprint requests to: J.-B. Delisle, e-mail: interactions to extrapolate and obtain an estimate of the [email protected] effect which would be raised by the whole main asteroid 2 Delisle and Laskar: Chaotic diffusion induced by massive asteroids belt. Finally we compare the semi-major axis diffusion in- Simulation initial distribution duced by the Yarkovsky effect and the diffusion due to catalog objects close encounters (in the case of the Vesta family). 0.14 0.13 2. Numerical simulations 0.12 0.11 We ran two sets of numerical simulations. Both of them 0.10 e comprise the 8 planets of the Solar System, Pluto, the 0.09 Moon and the 11 asteroids enumerated above. In both sim- 0.08 ulations, we generate a synthetic Vesta family produced by 0.07 an initial collision of another object with Vesta. We do not 0.06 model a realistic collision process but instead we consider 0.05 a set of test particles initially at the same position as Vesta 8.0 but with different relative velocities. This is not a problem 7.5 because we are mainly interested in this work in exploring 7.0 the phase space in the region of the Vesta family. The rel- 6.5 ative velocities of the fragments of collision are sampled both in norm and direction. For the norm, we take 8 dif- I (deg) 6.0 1 ferent values (every 50 ms− ) between the escape velocity 5.5 1 at Vesta’s surface (around 350 ms− ) and twice this value 5.0 1 (700 ms− ). The inclination is sampled between 50◦ and − 4.5 50◦ every 10◦ (11 different values). Finally, the direction 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.5 in the invariant plane is taken every 10◦ (36 values). This a (AU) way we create 3168 test particles that represent the Vesta family. We plotted the positions of these particles in the Fig. 1. Initial distribution of the synthetic Vesta family (a, e) and (a, I) planes (where a is the semi-major axis, e in the (a, e) plane (top) and the (a, I) plane (bottom). the eccentricity and I the inclination) superimposed with Real members proper elements (from Nesvorný2010) are the catalog published by Nesvorný(2010) of current Vesta superimposed for comparison. family members (Fig.1). Note that despite we did not compute the proper elements whereas the catalog uses them 3. Global overview of the diffusion the two sets of points superimpose quite well. Moreover we constructed an initial set of particles just after the origi- The goal of this study is to understand whether the cur- nal collision whereas the catalog exposes the remnants of rent flux of V-type asteroids coming from the main belt the family after approximately 1.2 Gyr of evolution (see and carried to near-Earth orbits through strong planetary Carruba et al. 2007). resonances can be explained by the chaos induced by close In the first simulation (the reference simulation, S), encounters in the main asteroid belt (or at least that these the 11 asteroids are considered as test particles as well encounters contribute significantly). The simplest way to and thus do not perturb the fragments of collision. In the check this assumption is to look in both S and SE simula- second simulation (SE) the 11 asteroids are considered as tions to the number of NEAs as a function of time and to the planets and thereby interact with members of the syn- see if the consideration of asteroidal interactions implies thetic Vesta family. Both solutions are integrated over 30 a greater number of NEAs. We use the usual criterion to Myr using the symplectic integrator described in (Laskar decide whether an asteroid is a NEA or not : if the perias- et al. 2011) and references therein. The effects of all plan- tron of its orbit become smaller than 1.3 AU the asteroid ets and Pluto are taken into account, as well as general is considered as a NEA (see for example Morbidelli et al. relativity. 2002). In Fig.2 we plot the evolution of the number of In order to be able to analyze the evolution of the test NEAs in both simulations as a function of time. We can particles and in particular the impact of close encounters see on this graph that the close encounters have not a with the 11 asteroids we set up different logs. We record major effect on the population of NEAs on the duration for each particle its instantaneous orbital elements every of the simulation. However, we did not ran our simula- 1 kyr. We do not compute proper elements but instead we tions on the total age of the Vesta family (around 1.2 Gyr) use the minimum and maximum values (see Laskar 1994) and we can think that just after the collision that origi- taken by a, e and I on time spans of 10 kyr. Actually, we nated the Vesta family, the population of V-type NEAs is use the average of the minimum and maximum values as dominated by fragments of the collision that were directly proper elements. We also set up a log of close encounters injected in the two main resonances. The diffusion process for each particle. Each time that a particle passes within is supposed to provide these resonances on a much larger 0.01 AU from an asteroid, the asteroid number, the mini- time scale. mum distance of approach and the time of the encounter Another way to look for the effect of close encounters are recorded in the logs. is to compare the proper elements of the initial and final Delisle and Laskar: Chaotic diffusion induced by massive asteroids 3

70 0.010 S 60 0.009 SE 0.008 50 0.007 40 0.006

NEA 0.005 30 (AU) a

σ 0.004 20 0.003 10 S 0.002 SE 0 0.001 0 5 10 15 20 25 30 0.000 t (Myr) 0.10 S Fig. 2. Number of near-Earth asteroids (NEA) among the 0.09 SE synthetic Vesta family as a function of time in S and SE. 0.08 The criterion for considering a fragment as a NEA is based 0.07 on the minimum value taken by the periastron. When this 0.06 e value is less than 1.3 AU the fragment is considered as a σ 0.05 NEA. 0.04 0.03 distribution of fragments in both simulations. A usual way 0.02 to do that is to compute the standard deviation of diﬀerent 0.01 proper elements (and in particular the semi-major axis) on 0.00 the whole population of fragments at a time t with respect 2.0 to the initial values of these elements (see for example S SE Carruba et al. 2003) : 1.5 2 (xj(t) xj(0)) σ (t) = j − , (1) x 1.0

s N 1 (deg) P I

− σ where x can be the semi-major axis a, the eccentricity e or the inclination I of Vesta family fragments. As we men- 0.5 tioned it we use the averages of the minima and maxima as proper elements and in particular for the initial values we 0.0 compute them on the first 10 kyr of the simulation. We do 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 not use the initial conditions given in Fig.1 because they a (AU) are only instantaneous values. Fig. 3. Semi-major axis dependency of the diffusion in We measure the diffusion in semi-major axis only due semi-major axis (top), eccentricity (middle) and inclina- to close encounters, thereby we do not take into account tion (bottom) in S and SE. For each band of 0.01 AU, resonances and in particular strong ones. For this purpose we plot the standard deviation of the average of the mini- we plotted the value of the standard deviation between the mum and the maximum values of a (respectively e and I) beginning and the end (after 30 Myr of evolution) for both during the last 10 kyr of the simulations with respect to simulations (S, SE) as a function of the initial (proper) the initial values (see Eq.1). semi-major axis. We divided the interval of semi major axis in bands of 0.01 AU and for each band we computed the standard deviation (of the semi-major axis, the eccen- not representative of their instability. For instance the dif- tricity and the inclination) of the set of asteroids that are fusion is zero between 2.5 and 2.51 AU only because none initially in this band. The results of this calculation are of the fragments that were initially in this zone finished given in Fig.3. We can see that the strong resonance 3 : 1 the simulation. around 2.5 AU induces an important diffusion in both S Between 2.26 and 2.48 AU, S shows a very limited and SE and that for semi-major axis lower than 2.25 AU semi major-axis diffusion even if we can see small peaks both simulations are strongly affected by resonances (see (e.g. around 2.42 AU corresponding to the 1 : 2 resonance Nesvornýet al. 2008, for a list of resonances acting on the with Mars). SE undergoes a more important diffusion in Vesta family). Note that in strong resonances most frag- this band. We run our calculations on this band because it ments are highly unstable and for a great part of them the fulfills at the same time two important conditions : the dif- calculation stops before the end of the simulation (colli- fusion due to resonances is limited and the band contains sions with a planet or the Sun or escapes from the Solar (at the beginning of the simulation) a sufficient amount System). Thereby calculations of the diffusion between the of fragments (2617 of the 3168 fragments) which is impor- beginning and the end of the simulation in such zones is tant for the statistics. Note that for the eccentricity and 4 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

0.0009 S 2.398 0.0008 SE 0.0007 2.397 0.0006 2.396 2.395 0.0005 2.394 (AU)

a 0.0004 a (AU) σ 2.393 0.0003 2.392 0.0002 2.391 0.0001 2.39 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t (Myr) t (Myr)

Fig. 4. Evolution in time of the semi-major axis diffusion Fig. 5. Semi-major axis evolution of a fragment which un- in S and SE. We plot the standard deviation of the av- derwent various close encounters. We plot the maximum, erage of the minimum and the maximum values of a for minimum and instantaneous values of the semi-major axis 3 each fragment over 10 kyr steps with respect to the ini- of the fragment. All times of encounters closer than 10− tial values (see Eq.1) for asteroids whose semi-major axes AU are highlighted with dotted vertical lines. are initially between 2.26 and 2.48 AU (in order to avoid strong resonances). 2.33 2.325 2.32 2.315 the inclination, the diffusion is hidden in the remaining oscillations due to the secular evolution of these elements a (AU) 2.31 induced by the planets and that we cannot see any dif- 2.305 ference between both simulations in Fig.3. However, this 2.3 is not really a problem since in this work we are mainly interested in the semi-major axis diffusion. 0 5 10 15 20 25 30 The evolution (in time) of the standard deviation of t (Myr) the semi-major axis computed on the 2.26 - 2.48 AU band Fig. 6. Semi-major axis evolution of a fragment which for both simulations is plotted in Fig.4. The standard de- underwent a very close encounter (minimum distance of viation is approximately constant for S which reinforces 5 3.4 10− AU) with Vesta (at t 8.39 Myr). We plot our choice of this band. On the contrary, SE clearly under- the× maximum, minimum and instantaneous≈ values of the goes diffusion. Analyzing Fig.4 gives us an approximation semi-major axis of the fragment. All times of encounters of the drift in semi-major axis due to asteroidal interac- 3 closer than 10− AU are highlighted with dotted vertical tions during the simulation. After 30 Myr of evolution, we 4 lines. have a standard deviation around 9 10− AU for SE 4 × and 2 10− AU for S. In order to evaluate the diffusion × due to asteroidal interactions, we have to remove the dif- 4. Diffusion induced by asteroids close encounters fusion obtained in the reference simulation (which is probably due to resonances and output sampling of the secular The global analysis that we made does not allow us to evolution of the semi-major axis) from the total standard separate the contributions of the 11 asteroids in the diffu- deviation. Note that the total variance (the squared of the sion process and to extrapolate to the whole main asteroid standard deviation) is the sum of the variances of the dif- belt. In order to do that we have to examine logs of close ferent contributions. Thus we obtain a standard deviation encounters and estimate the diffusion resulting from each due to asteroidal interactions after 30 Myr of evolution of of these events and then make statistics. All this analysis 4 about 8.8 10− AU. We also observe several jumps in is still restricted to the 2.26 - 2.48 AU band in order to the semi-major× axis diffusion, which are due to very close avoid the strong resonances as previously. encounters of a single fragment with one of the 11 consid- For each encounter recorded in the logs we compare ered asteroids. Actually it has been shown (Carruba et al. the value of the semi-major axis before and after the en- 2007) that the distribution of jump sizes is not a Gaussian counter. Fig.5 gives an example of the evolution of the but has thicker wings. Thus the curve is not regular on semi-major axis of a fragment that underwent several close the time scale of this simulation because of the presence encounters which resulted in jumps of different sizes. Fig.6 of rare very important events. We can find the same kind gives a more unusual example of the evolution of a frag- of effects on Fig.3, the curve is not regular in the 2.26 - ment which underwent a very close encounter with Vesta 2.48 AU band and shows peaks due to a few very close resulting in a very important jump in semi-major axis. encounters. Then for each of the 11 asteroids we calculate the stan- Delisle and Laskar: Chaotic diffusion induced by massive asteroids 5 dard deviation of the distribution of jump sizes induced by With this method we compute the standard deviation of close encounters with this asteroid. This gives a measure the noise for different time intervals for the calculation of 5 of the average diffusion resulting from a single encounter extrema and we find that the minimal value (4.4 10− with the asteroid. As we already noticed, it is more con- AU) is obtained for an interval length of 200 kyr.× This is venient to manipulate variances than standard deviations the value we employ for all the following calculations. since the latter are not additive. If we assume that close When we compute the variance of jump sizes in SE encounters are independent events, every fragment under- (for real close encounters) we are affected by the same goes a random walk and after N encounters, the total noise. The value we obtain for the variance is the sum of variance is multiplied by a factor N : the variance of the real diffusion and the variance of the noise. Thus the real diffusion resulting from close encoun- 2 2 σa[N] = Nσa[1]. (2) ters is given by :

This gives a measure of the average diffusion resulting σ¯2[1] = σ2[1] σ2 . (5) from N encounters with the considered asteroid. We just a a − noise have then to replace N by the mean number of close en- Another problem we experienced is that the logs of counters per fragment with the selected asteroid during close encounters record all encounters within 0.01 AU but the whole simulation and we obtain the diffusion (vari- it appears that at this distance the effect is too small and ance) due to this asteroid on the duration of the simula- is completely hidden in the noise. Thus statistics are con- tion. The diffusion due to close encounters with all the 11 taminated by false events (with no real jumps). In or- asteroids on the duration of the simulation is the sum of der to avoid this contamination we take into account only the variances of the 11 asteroids. With the same reasoning encounters within 0.001 AU which are more significant as for Eq.2 it is possible to extrapolate the value of the events. variance on longer time scales : Finally, when a fragment undergoes several close en- t counters spaced by less than 200 kyr, we are not able to σ2(t) = σ2(T ). (3) separate the effect of each encounter. We thus ignore all a T a sim sim such multiple encounters and we keep only single encoun- Note that it as been shown (e.g. Nesvornýet al. 2002) ters in the calculation of the standard deviation of the size that close encounters do not exactly result in a random of the jumps. However, when we compute the total diffu- walk (there are some correlations between successive en- sion on the duration of the simulation we use the total counters) and that the variance is not exactly linear with number of encounters that occur during the simulation as time (or number of encounters). It can, indeed, be written multiple encounters should not be ignored. as a power law of the form : Table 1 sums up the results we obtain for each asteroid and for the 11 asteroids together. We give the numbers of B σa(t) = Ct . (4) encounter during the simulation (including multiple encounters), the standard deviation obtained from statistics For a random walk : B = 0.5. Numerical estimations of on single encounters (σ [1]), the corrected values from the this coefficient B for different families (see Nesvornýet al. a noise (¯σ [1]) and the standard deviations on the duration 2002; Carruba et al. 2003; Novakovićet al. 2010) show a of the simulation (¯σ (T )). Finally, the percentages of that it depends on the family considered and can even a sim contribution of the different asteroids are given in term of evolve with time. However the values found in the litera- variance. We obtain a total diffusion due to close encoun- ture range between 0.5 and 0.65 and we did not find any ters of 8.63 10 4 AU during the 30 Myr of the simulation estimations in the case of the Vesta family so we decided − (see table 1).× This number is to be compared with the drift to stick to the random walk hypothesis. rate obtained with the global approach (8.8 10 4 AU). Note that in this reasoning we do not take into account − The results given by both methods are in good× agreement. the influence of the noise in the calculation of jumps sizes. By using Eq.3 we can extrapolate our results to longer Indeed, we observe on Fig.5 and Fig.6 that the minimum times : and maximum values of the semi-major axis are not constant outside the close encounters (jumps) but undergo os- 4 σ¯a(t) = 1.57 10− t (Myr) AU. (6) cillations which are due to remaining secular evolution. It × is possible to eliminate an important part of this noise by Of course it is possiblep to do the same for the contribution taking the minimum of minima and the maximum of max- of each asteroid. This allows us to compare our results with ima over greater time scales before and after the encoun- the previous study of Carruba et al. (2007). In this article, +2.5 3 ters. In order to choose the best time interval to compute the authors find a drift rate of 2.0 2.0 10− AU/(100 − × these extrema and to evaluate the remaining noise after Myr) by considering only close encounters with Vesta. If this treatment we use the reference simulation S and we we extrapolate to 100 Myr and consider only Vesta we find 3 calculate the standard deviation of the difference of semi- a drift of 1.3 10− AU which is coherent with Carruba × major axis before and after random times. Since there et al. (2007) findings. are no close encounters thus no jumps in this simulation, If we look at the different contributions in table 1 we this standard deviation measures only the remaining noise. see that close encounters with Ceres and Vesta together 6 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

Table 1. Comparison of the contributions of the encoun- -6 regression ters with the 11 considered asteroids to the semi-major -6.5 slope 1 axis diﬀusion. All standard deviations are given in AU. -7 The contributions are given as percentages of the total

) -7.5 2 variance (which is additive unlike the standard deviation). a σ -8

log( -8.5 Ast. Nevents σa[1]σ ¯a[1]σ ¯a(Tsim) Contrib. 105 105 105 (%) -9 × × × 1 1175 76.9 76.8 51.4 35.55 -9.5 2 401 14.6 13.9 5.4 0.40 -10 3 868 5.9 3.8 2.2 0.07 0 0.5 1 1.5 2 2.5 3 4 6521 43.8 43.6 68.8 63.63 log(m) 7 2999 5.2 2.6 2.8 0.11 10 162 5.7 3.6 0.9 0.01 Fig. 7. Contribution to the semi major axis diffusion due 15 1064 5.6 3.4 2.2 0.06 to each considered asteroid as a function of its mass. 19 3210 4.8 1.9 2.1 0.06 324 603 6.5 4.8 2.3 0.07 7000 532 688 5.8 3.7 1.9 0.05 704 4 6.7 5.0 0.2 0.00 6000 All 17695 86.3 100 5000 − − 4000 N 3000 represent about 99% of the total diffusion (in variance). 2000 Contributions from the 9 other asteroids seem negligible. 1000 This is coherent with findings of previous studies on other 0 asteroid families (see in particular Carruba et al. 2003). 0 5 10 15 20 25 30 On the contrary, since Ceres represents 36% of the diffu- d (AU/yr) sion this asteroid should not be neglected (like in Carruba 3 et al. (2007)). We believe that two main reason can ex- Fig. 8. Number of close encounters (< 10− AU) between plain these proportions of contribution : the mass of the Vesta family fragments and each of the 11 considered as- asteroids and their proximity with the Vesta family mem- teroids as a function of the distance in phase space be- bers in phase space. The mass must influence the impor- tween these asteroids and Vesta. We use the mean values tance of the effect of a single close encounter whereas the of the semi-major axes, the eccentricities and the inclina- proximity in phase space influences the number of such tions of the 11 asteroids during the 30 Myr of the simula- events. In order to check for these correlations we plotted tion in order to compute the distances (see Eq.7). the variance due to close encounters as a function of the mass of the considered asteroid (Fig.7) and the number of events as a function of the distance in phase space of the tion we took into account the most massive objects of the considered asteroid with respect to Vesta (Fig.8). We use main asteroid belt. Actually, if we add the masses of the 11 asteroids of our simulation we obtain a total mass of the definition of the distance in phase space introduced by 10 8.36 10− M . The estimated total mass contained in Zappala et al. (1994) : × 10 the main asteroid belt is about 15 10− M (obtained × 2 from INPOP10a fits, Fienga et al. 2011) while Krasinsky 5 δa 2 2 10 d = na0 + 2(δe) + 2(δsini) . (7) et al. (2002) gave 18 10− M . The most massive as- s4 a0 × 10 teroid in the main belt is Ceres with 4.76 10− M , × This distance has the dimensions of a velocity since it is an thus the remaining mass (not considered in our simula- estimation of the ejection velocity that would be needed tion) represents about 1.5 to 2 times the mass of Ceres. to carry two objects initially at the same position to their The total mass contained in the main asteroid belt is not current positions. In Eq.7, we used the mean values of the well constrained but we can assume the upper limit of this semi-major axes, the eccentricities and the inclinations of remaining mass to about twice the mass of Ceres. the 11 asteroids during the whole simulation (30 Myr) in In Fig.7, the slope of the regression curve is about 1.2. order to compute the distances. We deduce from these We plotted a straight line of slope 1 for comparison. A plots that we indeed have an effect of the mass and the slope of 1 means that two objects of masses 0.5 have the proximity in phase space but we cannot obtain a simple same effect than one of mass 1. If the slope is higher than and precise law in order to evaluate the diffusion that 1 it is more efficient to have a single object whereas if would result from close encounters with other asteroids the slope is lower than 1 it is more efficient to have two than the ones considered in our simulation. objects. Even if this slope is not well constrained in Fig.7 However, we can constrain the global effect of the re- it is reasonable to consider that it is greater than 1 so maining objects of the main asteroid belt. In our simula- it is more efficient to have one big object than to split Delisle and Laskar: Chaotic diffusion induced by massive asteroids 7 it into smaller objects. Thereby, we can assume that the events is given by (see Farinella et al. 1998; Morbidelli & total contribution of the remaining objects of the main Vokrouhlický2003) : belt will be less than twice the contribution of Ceres. We still have the problem of the influence of the dis- 1/2 1/2 R D tance in phase space but it seems realistic to consider that τ = 15 Myr = 335 Myr. (9) coll 1 m 1 km the case of Vesta is singular since it is the parent body of the Vesta family and that the remaining objects must have a probability of close encounter closer to Ceres’s one than For kilometer-sized asteroids, the time between two reori- Vesta’s one (see Fig.8). Thus, the total variance due to enting collisions is about 300 Myr, it means that there are encounters with all the asteroids of the main belt should a few (3 or 4) reorientations on the time scale of the age not exceed (and is probably a lot smaller than) 1.7 times of the Vesta family. Thus it is likely that these reorienta- the value of the variance obtained in our simulation. In tions will reduce the diffusion due to the Yarkovsky effect terms of standard deviations it corresponds to a factor of but not by a large factor. Moreover, we can still consider about 1.3. that the YORP effect acts almost instantaneously between two reorientations and that the seasonal Yarkovsky effect is negligible and the diurnal effect is always maximal. 5. Comparison with the Yarkovsky-YORP effect Taking into consideration these different results we es- The Yarkovsky effect is a non-gravitational force which timate the diffusion of Vesta family members under the results from the interaction between asteroids and the effect of Yarkovsky, YORP and collisions by a simple ran- solar radiation. There are actually two versions of the dom walk process and the maximal diurnal Yarkovsky ef- Yarkovsky effect : a diurnal and a seasonal effect. The fect between each reorientations. Fig.9 shows a compari- diurnal effect can result in an increase or a decrease of son between the diffusion due to close encounters and the the semi-major axis depending on the obliquity () of the one due to the Yarkovsky effect as functions of the as- asteroid ( cos ), whereas the seasonal effect systemati- teroid diameter at different times (250 Myr, 1 Gyr and 3 cally decreases∝ the semi-major axis ( sin2 ). The diurnal Gyr). For the close encounters, we plotted the diffusion ∝ effect is maximal when = 0◦ or 180◦ and is zero when obtained for the 11 asteroids and the extrapolation we de- = 90◦. On the contrary, the seasonal effect is maximal duced for the whole main asteroid belt. For the Yarkovsky when = 90◦ and zero when = 0◦, 180◦. Both effects are effect we plotted both the maximal diurnal effect (without size dependent, for instance the diurnal effect is given by any reorientation) and the result of the random walk pro- (see in particular Nesvornýet al. 2008) : cess. We can see that reorienting collisions do not affect much the magnitude of the Yarkovsky effect for the range da 4 1 km = 2.5 10− cos AU/Myr. (8) of diameters we are interested in, even at 3 Gyr. We re- dt × D call that the estimated age of the Vesta family is about 1 In addition to the Yarkovsky effect, we have to consider Gyr (Carruba et al. 2007). For this duration, close encoun- the YORP (Yarkovsky - O’Keefe - Radzievskii - Paddack) ters are dominant for the diffusion of asteroids larger than effect which also comes from the solar radiation but affects 40 5 km (Fig. 9), but below this value the Yarkovsky ± the rotational velocity and the obliquity of asteroids. This effect is prevailing. For a diameter of 1 km the Yarkovsky effect depends a lot on the shape and the surface compo- effect is about 25 times greater than the effect of close sition of the asteroid but it is possible to do some statis- encounters. tics to understand what is the most probable scenario Regarding the time evolution of these effects, we as- (see Vokrouhlický& Capekˇ 2002; Capekˇ & Vokrouhlický sume in this work that close encounters generate a ran- 2004). It has been shown (Capekˇ & Vokrouhlický2004) dom walk process that evolves as √t whereas the maximal that for most of basaltic asteroids (more than 95%), the Yarkovsky effect is linear with time. When we take into obliquity is asymptotically driven to 0◦ or 180◦ by the account reorientations for the Yarkovsky effect we also YORP effect. The time scale for the YORP effect is about have a random walk process so the evolution law should 10-50 Myr (see Vokrouhlický& Capekˇ 2002; Morbidelli & be proportional to √t. However this is an asymptotic law Vokrouhlický2003). This means that on a 1 Gyr time scale which is valid when a great number of random walk steps (which is the order of magnitude for the age of the Vesta is reached. In the case of reorienting collisions the char- family (Carruba et al. 2007)), we can consider that the acteristic time is about 300 Myr so there are only a few obliquity is either 0◦ or 180◦ during the whole evolution steps per Gyr and the asymptotic law is invalid on this and that the seasonal effect is negligible and the diurnal time scale. It means that for the time scale we are inter- effect is maximal (in one or the other direction). ested in, the Yarkovsky effect evolves faster than √t so Nevertheless, the collisions must also be taken into ac- faster than close encounters. This is exactly what we ob- count. We have to distinguish whether a collision is de- serve on Fig.9 : for 250 Myr the close encounters take a structive or not. The collisional life time of a 1-10 km as- more important part in the diffusion process (equivalence teroid in the main belt is about 1 Gyr. But non-destructive of the two effects for D=19 km) whereas for 3 Gyr the collisions are more frequent and can reorient the spin Yarkovsky effect is even more prevailing (equivalence for axis of the asteroid. The characteristic time scale of such D=54 km). 8 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

1 ings are compatible with previous work on the Vesta fam- EN A TEN ily (Carruba et al. 2007). We use the results we obtained YMX YRW for the 11 asteroids to extrapolate and constrain the diﬀu- 0.1 sion that would result from close encounters with all the

(AU) main belt asteroids. We show that this diﬀusion should not exceed 1.3 times the diﬀusion due to the 11 asteroids.

250Myr Finally, we compare the diﬀusion due to close encounters a

∆ 0.01 with the Yarkovsky-driven drift of the semi-major axis of asteroids in the 1-100 km range (in term of asteroids diameters). We find that both effects are equivalent over 1 0.001 Gyr for a diameter of 40 5 km. For smaller asteroids the 1 Yarkovsky effect dominates± the semi-major axis diffusion B and for larger asteroids close encounters become more important. Thus we confirm that although asteroids close en- 0.1 counters have a significant influence, the main mechanism of semi-major axis diffusion that transports main belt as- (AU) teroids (and especially V-type asteroids) to Earth-crossing 1Gyr

a orbits via strong resonances is the Yarkovsky eﬀect. ∆ 0.01 Acknowledgements. This work was supported by ANR- ASTCM, INSU-CNRS, PNP-CNRS, and CS, Paris Observatory. Part of the computations were made at 0.001 1 CINES/GENCI. C References 0.1 Carruba, V., Burns, J. A., Bottke, W., & Nesvorn´y,D. (AU) 2003, Icarus, 162, 308 3Gyr

a Carruba, V., Roig, F., Michtchenko, T. A., Ferraz-Mello, ∆ 0.01 S., & Nesvorný,D. 2007, A&A, 465, 315 Farinella, P., Vokrouhlicky, D., & Hartmann, W. K. 1998, Icarus, 132, 378 0.001 Fienga, A., Laskar, J., Kuchynka, P., et al. 2011, Celestial 1 10 100 Mechanics and Dynamical Astronomy, 101 D (km) Krasinsky, G. A., Pitjeva, E. V., Vasilyev, M. V., & Fig. 9. Comparison of the semi-major axis diffusion due to Yagudina, E. I. 2002, Icarus, 158, 98 the close encounters and to the Yarkovsky effect after 250 Laskar, J. 1994, A&A, 287, L9 Myr (A), 1 Gyr (B) and 3 Gyr (C). EN is the diffusion due Laskar, J., Gastineau, M., Delisle, J.-B., Farrés,A., & to close encounters with the 11 asteroids considered in our Fienga, A. 2011, A&A, 532, L4+ simulations. TEN is the extrapolation we performed for Migliorini, F., Morbidelli, A., Zappala, V., et al. 1997, the entire main belt ( 1.3). YMX stands for the maximal Meteoritics and Planetary Science, 32, 903 diurnal Yarkovsky effect.× YRW stands for the Yarkovsky Morbidelli, A., Bottke, Jr., W. F., Froeschlé,C., & Michel, effect with reorientations (random walk). P. 2002, Asteroids III, 409 Morbidelli, A. & Gladman, B. 1998, Meteoritics and Planetary Science, 33, 999 6. Conclusion Morbidelli, A. & Vokrouhlický,D. 2003, Icarus, 163, 120 Nesvorný, D. 2010, Nesvorny HCM Asteroid Families In this work we evaluate the effect of close encounters be- V1.0. EAR-A-VARGBDET-5-NESVORNYFAM-V1.0. tween Vesta family members and massive asteroids on the NASA Planetary Data System semi-major axis diffusion of the family. We calculate this Nesvorný,D., Morbidelli, A., Vokrouhlický,D., Bottke, effect by two methods. We first look at the collective ef- W. F., & Broˇz,M. 2002, Icarus, 157, 155 fect of close encounters and deduce an estimation of the Nesvorný,D., Roig, F., Gladman, B., et al. 2008, Icarus, semi-major axis drift due to the 11 massive asteroids taken 193, 85 into account here. Then we study individually the close en- Novaković,B., Tsiganis, K., & Kneˇzević,Z. 2010, Celestial counters and make statistics over these events. With this Mechanics and Dynamical Astronomy, 107, 35 second method we are able to separate the contribution Sears, D. 1997, Meteoritics and Planetary Science, 32, 3 of each of the 11 asteroids to the diffusion. We show that Capek,ˇ D. & Vokrouhlický,D. 2004, Icarus, 172, 526 both methods give comparable results and that our find- Vokrouhlický,D. & Capek,ˇ D. 2002, Icarus, 159, 449 Delisle and Laskar: Chaotic diffusion induced by massive asteroids 9

Zappala, V., Cellino, A., Farinella, P., & Milani, A. 1994, AJ, 107, 772 Astronomy & Astrophysics manuscript no. article˙v1 September 6, 2018 (DOI: will be inserted by hand later)

Chaotic diﬀusion of the Vesta family induced by close encounters with massive asteroids

J.-B. Delisle and J. Laskar

ASD, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau, 75014 Paris, France

September 6, 2018

Abstract. We estimate numerically the semi-major axis chaotic diffusion of Vesta family asteroids induced by close encounters with 11 massive main belt asteroids : (1) Ceres, (2) Pallas, (3) Juno, (4) Vesta, (7) Iris, (10) Hygiea, (15) Eunomia, (19) Fortuna, (324) Bamberga, (532) Herculina, (704) Interamnia. Most of the diffusion is due to Ceres and Vesta. By extrapolating our results, we are able to constrain the global effect of close encounters with all the main belt asteroids. A comparison of this drift estimate with the one expected for the Yarkovsky effect shows that for asteroids whose radius is larger than about 35 km, close encounters dominate the Yarkovsky effect. Overall, our findings confirm the standard scenario for the history of the Vesta family.

1. Introduction of Myr. In the continuation of this work we concentrated our study on the diffusion induced by close encounters of It is now admitted that most of V-type near-Earth aster- Vesta family members with the massive asteroids. oids (NEAs) and howardite, eucrite and diogenite (HED) meteorites are fragments of a collision between Vesta and another object which also produced the Vesta family (see This effect has already been studied for different as- ?). V-type NEAs and HED meteorites are former members teroid families (e.g. the Flora and Lixiaohua families ??). of the Vesta family that have been carried from the main The most exhaustive study (in terms of number of aster- asteroid belt to Earth-crossing orbits by the two main res- oids taken into account) concerned the Gefion and Adeona families (?). The author considered all 682 asteroids of ra- onances in the neighborhood of Vesta : the ν6 and 3 : 1 resonances with Jupiter. The major issue in this scenario dius larger than 50 km for encounters with members of is the fact that the average life time of fragments in these the families. They concluded that the four largest aster- two resonances is too short to explain the the mean age oids ((1) Ceres, (2) Pallas, (4) Vesta and (10) Hygiea) had of HED meteorites and V-type NEAs (?). a much higher influence than all the 678 remaining aster- The commonly accepted explanation for this is that oids so the latters can be considered as negligible. The NEAs spent most of their life time in the main asteroid Vesta family has also been the object of such an analysis belt before entering in resonance and being carried to their (?) but only close encounters with (4) Vesta were taken current orbits. This supposes that a mechanism can in- into account. These different studies show that the effect duce a diffusion process that brings fragments to one (or of close encounters with massive asteroids depends a lot both) of the two main resonances. The Yarkovsky effect is on the considered family and its position in phase space now admitted to be the main mechanism explaining the with respect to those of massive asteroids. diffusion of Vesta family members (e.g. ?). However other processes also contribute to the diffu- The purpose of this work is to explore further the sion. It is in particular the case for the chaos induced effect of close encounters in the case of the Vesta family by considering more asteroids. We take into account

arXiv:1110.5820v1 [astro-ph.EP] 26 Oct 2011 by overlaps of mean motion resonances or close encounters between asteroids in the main belt. In the case of the effect of the 11 following asteroids : (1) Ceres, (2) the resonances, ? showed that the chaotic zone around Pallas, (3) Juno, (4) Vesta, (7) Iris, (10) Hygiea, (15) the 3 : 1 resonance is too narrow to explain the needed Eunomia, (19) Fortuna, (324) Bamberga, (532) Herculina, diffusion. Recently, ? showed that close encounters among (704) Interamnia. Moreover, we use the results from these massive asteroids induce strong chaos which appears to be interactions to extrapolate and obtain an estimate of the the main limiting factor for planetary ephemeris on tens effect which would be raised by the whole main asteroid belt. Finally we compare the semi-major axis diffusion in- Send offprint requests to: J.-B. Delisle, e-mail: duced by the Yarkovsky effect and the diffusion due to [email protected] close encounters (in the case of the Vesta family). 2 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

2. Numerical simulations Simulation initial distribution catalog objects We ran two sets of numerical simulations. Both of them comprise the 8 planets of the Solar System, Pluto, the 0.14 Moon and the 11 asteroids enumerated above. In both sim- 0.13 ulations, we generate a synthetic Vesta family produced by 0.12 an initial collision of another object with Vesta. We do not 0.11 0.10 model a realistic collision process but instead we consider e 0.09 a set of test particles initially at the same position as Vesta 0.08 but with different relative velocities. This is not a problem 0.07 because we are mainly interested in this work in exploring 0.06 the phase space in the region of the Vesta family. The rel- 0.05 ative velocities of the fragments of collision are sampled 8.0 both in norm and direction. For the norm, we take 8 dif- 7.5 1 ferent values (every 50 ms− ) between the escape velocity 1 7.0 at Vesta’s surface (around 350 ms− ) and twice this value 1 (700 ms− ). The inclination is sampled between 50◦ and 6.5 50 every 10 (11 different values). Finally, the− direction ◦ ◦ I (deg) 6.0 in the invariant plane is taken every 10 (36 values). This ◦ 5.5 way we create 3168 test particles that represent the Vesta family. We plotted the positions of these particles in the 5.0 (a, e) and (a, I) plans (where a is the semi-major axis, e 4.5 the eccentricity and I the inclination) superimposed with 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.5 a (AU) the catalog published by ? of current Vesta family members (Fig.1). Note that despite we did not compute the Fig. 1. Initial distribution of the synthetic Vesta family proper elements whereas the catalog uses them the two in the semi-major axis - eccentricity plan (top) and the sets of points superimpose quite well. Moreover we con- semi-major axis - inclination plan (bottom). Real mem- structed an initial set of particles just after the original bers proper elements (from ?) are superimposed for com- collision whereas the catalog exposes the remnants of the parison. family after approximately 1.2 Gyr of evolution (see ?). In the first simulation (the reference simulation, S), the 11 asteroids are considered as test particles as well and thus do not perturb the fragments of collision. In the encounters in the main asteroid belt (or at least that these second simulation (SE) the 11 asteroids are considered encounters contribute significantly). The simplest way to as the planets and thereby interact with members of the check this assumption is to look in both S and SE simula- synthetic Vesta family. Both solutions are integrated over tions to the number of NEAs as a function of time and to 30 Myr. see if the consideration of asteroidal interactions implies !!!!!!!!!!!!!!!!!!!!!! Description integrateur !!!!!!!!!!!!!! a greater number of NEAs. We use the usual criterion to In order to be able to analyze the evolution of the test decide whether an asteroid is a NEA or not : if the perias- particles and in particular the impact of close encoun- tron of its orbit become smaller than 1.3 AU the asteroid is ters with the 11 asteroids we set up different logs. We considered as a NEA (see for example ?). In Fig.2 we plot record for each particle its instantaneous orbital elements the evolution of the number of NEAs in both simulations every 1 Kyr. We do not compute proper elements but in- as a function of time. We can see on this graph that the stead we use the minimum and maximum values (see ?) close encounters have not a major effect on the population taken by a, e and I on time spans of 10 Kyr. Actually, we of NEAs on the duration of the simulation. However, we use the average of the minimum and maximum values as did not ran our simulations on the total age of the Vesta proper elements. We also set up a log of close encounters family (around 1.2 Gyr) and we can think that just after for each particle. Each time that a particle passes within the collision that originated the Vesta family, the popu- 0.01 AU from an asteroid, the asteroid number, the mini- lation of V-type NEAs is dominated by fragments of the mum distance of approach and the time of the encounter collision that were directly injected in the two main reso- are recorded in the logs. nances. The diffusion process is supposed to provide these resonances on a much larger time scale. 3. Global overview of the diffusion Another way to look for the effect of close encounters is to compare the proper elements of the initial and final The goal of this study is to understand whether the cur- distribution of fragments in both simulations. A usual way rent flux of V-type asteroids coming from the main belt to do that is to compute the standard deviation of different and carried to near-Earth orbits through strong planetary proper elements (and in particular the semi-major axis) on resonances can be explained by the chaos induced by close the whole population of fragments at a time t with respect Delisle and Laskar: Chaotic diffusion induced by massive asteroids 3

70 0.010 S 60 0.009 SE 0.008 50 0.007 40 0.006

NEA 0.005 30 (AU) a

σ 0.004 20 0.003 10 S 0.002 SE 0 0.001 0 5 10 15 20 25 30 0.000 t (Myr) 0.10 S Fig. 2. Number of near-Earth asteroids (NEA) among the 0.09 SE synthetic Vesta family as a function of time in S and SE. 0.08 The criterion for considering a fragment as a NEA is based 0.07 on the minimum value taken by the periastron. When this 0.06 e value is less than 1.3 AU the fragment is considered as a σ 0.05 NEA. 0.04 0.03 to the initial values of these elements (see for example ?) 0.02 : 0.01 0.00 (x (t) x (0))2 j j j 2.0 σx(t) = − , (1) s N 1 S P − SE where x can be the semi-major axis a, the eccentricity e 1.5 or the inclination I of Vesta family fragments. As we men- tioned it we use the averages of the minima and maxima 1.0 (deg) I

as proper elements and in particular for the initial values σ we compute them on the first 10 Kyr of the simulation. We do not use the initial conditions given in Fig.1 because 0.5 they are only instantaneous values. We want to measure the diffusion in semi-major axis 0.0 only due to close encounters, thereby we do not want to 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 take into account resonances and in particular strong ones. a (AU) For this purpose we plotted the value of the standard de- Fig. 3. Semi-major axis dependency of the diffusion in viation between the beginning and the end (after 30 Myr semi-major axis (top), eccentricity (middle) and inclina- of evolution) of both simulations (S, SE) as a function of tion (bottom) in S and SE. For each band of 0.01 AU, the initial (proper) semi-major axis. We divided the inter- we plot the standard deviation of the average of the mini- val of semi major axis in bands of 0.01 AU and for each mum and the maximum values of a (respectively e and I) band we computed the standard deviation (of the semi- during the last 10 Kyr of the simulations with respect to major axis, the eccentricity and the inclination) of the set the initial values (see Eq.1). of asteroids that are initially in this band. The results of this calculation are given in Fig.3. We can see that the strong resonance 3 : 1 around 2.5 AU induces an impor- Between 2.26 and 2.48 AU, S shows a very limited tant diffusion in both S and SE and that for semi-major semi major-axis diffusion even if we can see small peaks axis lower than 2.25 AU both simulations are strongly af- (e.g. around 2.42 AU corresponding to the 1 : 2 resonance fected by resonances (see ?, for a list of resonances acting with Mars). SE undergoes a more important diffusion in on the Vesta family). Note that in strong resonances most this band. We chose to run our calculations on this band fragments are highly unstable and for a great part of them because it fulfills at the same time two important condi- the calculation stops before the end of the simulation (col- tions : the diffusion due to resonances is limited and the lisions with a planet or the Sun or escapes from the Solar band contains (at the beginning of the simulation) a suf- System). Thereby calculations of the diffusion between the ficient amount of fragments (2617 of the 3168 fragments) beginning and the end of the simulation in such zones is which is important for the statistics. Note that for the ec- not representative of their instability. For instance the dif- centricity and the inclination, the diffusion is hidden in fusion is zero between 2.5 and 2.51 AU only because none the remaining oscillations due to the secular evolution of of the fragments that were initially in this zone finished these elements induced by the planets and that we can- the simulation. not see any difference between both simulations in Fig.3. 4 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

0.0009 S 2.398 0.0008 SE 0.0007 2.397 0.0006 2.396 2.395 0.0005 2.394 (AU)

a 0.0004 a (AU) σ 2.393 0.0003 2.392 0.0002 2.391 0.0001 2.39 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t (Myr) t (Myr)

a (AU) 2.31 2.305 However, this is not really a problem since in this work we 2.3 are mainly interested in the semi-major axis diffusion. The evolution (in time) of the standard deviation of 0 5 10 15 20 25 30 the semi-major axis computed on the 2.26 - 2.48 AU band t (Myr) for both simulations is plotted in Fig.4. The standard de- Fig. 6. Semi-major axis evolution of a fragment which viation is approximately constant for the reference sim- underwent a very close encounter (minimum distance of ulation which reinforces our choice of this band. On the 5 3.4 10− AU) with Vesta (at t 8.39 Myr). We plot contrary, the simulation which takes into account aster- the× maximum, minimum and instantaneous≈ values of the oidal interactions clearly undergoes diffusion. Analyzing semi-major axis of the fragment. All times of encounters Fig.4 gives us an approximation of the drift in semi-major 3 closer than 10− AU are highlighted with black vertical axis due to asteroidal interactions during the simulation. lines. After 30 Myr of evolution, we have a standard deviation 4 4 around 9 10− AU for SE and 2 10− AU for S. In or- × × der to evaluate the diffusion due to asteroidal interactions, 4. Diffusion induced by close encounters among we have to remove the diffusion obtained in the reference asteroids simulation (which is probably due to resonances and output sampling of the secular evolution of the semi-major The global analysis that we made does not allow us to axis) from the total standard deviation. Note that the to- separate the contributions of the 11 asteroids in the diffu- tal variance (the squared of the standard deviation) is the sion process and to extrapolate to the whole main asteroid sum of the variances of the different contributions. Thus belt. In order to do that we have to examine logs of close we obtain a standard deviation due to asteroidal interac- encounters and estimate the diffusion resulting from each 4 tions after 30 Myr of evolution of about 8.8 10− AU. of these events and then make statistics. All this analysis We also observe several jumps in the semi-major× axis dif- is still restricted to the 2.26 - 2.48 AU band in order to fusion, which are due to very close encounters of a single avoid the strong resonances as previously. fragment with one of the 11 considered asteroids. Actually For each encounter recorded in the logs we compare it has been shown (?) that the distribution of jump sizes the value of the semi-major axis before and after the en- is not a Gaussian but has thicker wings. Thus the curve counter. Fig.5 gives an example of the evolution of the is not regular on the time scale of this simulation because semi-major axis of a fragment that underwent several close of the presence of rare very important events. We can find encounters which resulted in jumps of different sizes. Fig.6 the same kind of effects on Fig.3, the curve is not regular gives a more unusual example of the evolution of a frag- in the 2.26 - 2.48 AU band and shows peaks due to a few ment which underwent a very close encounter with Vesta very close encounters. resulting in a very important jump in semi-major axis. Delisle and Laskar: Chaotic diffusion induced by massive asteroids 5

Then for each of the 11 asteroids we calculate the stan- With this method we compute the standard deviation of dard deviation of the distribution of jump sizes induced by the noise for different time intervals for the calculation of 5 close encounters with this asteroid. This gives a measure extrema and we find that the minimal value (4.4 10− of the average diffusion resulting from a single encounter AU) is obtained for an interval length of 200 Kyr.× This is with the asteroid. As we already noticed, it is more con- the value we employ for all the following calculations. venient to manipulate variances than standard deviations When we compute the variance of jump sizes in SE since the latter are not additive. If we assume that close (for real close encounters) we are affected by the same encounters are independent events, every fragment under- noise. The value we obtain for the variance is the sum of goes a random walk and after N encounters, the total the variance of the real diffusion and the variance of the variance is multiplied by a factor N : noise. Thus the real diffusion resulting from close encounters is given by : 2 2 σa[N] = Nσa[1]. (2) σ¯2[1] = σ2[1] σ2 . (5) This gives a measure of the average diffusion resulting a a − noise from N encounters with the considered asteroid. We just Another problem we experienced is that the logs of have then to replace N by the mean number of close en- close encounters record all encounters within 0.01 AU but counters per fragment with the selected asteroid during it appears that at this distance the effect is too small and the whole simulation and we obtain the diffusion (vari- is completely hidden in the noise. Thus statistics are con- ance) due to this asteroid on the duration of the simula- taminated by false events (with no real jumps). In order tion. The diffusion due to close encounters with all the 11 to avoid this contamination we chose to take into account asteroids on the duration of the simulation is the sum of only encounters within 0.001 AU which are much more the variances of the 11 asteroids. With the same reasoning credible events. than for Eq.2 it is possible to extrapolate the value of the Finally, when a fragment undergoes several close en- variance on longer time scales : counters spaced by less than 200 Kyr, we are not able to t separate the effect of each encounter. We thus chose to ig- σ2(t) = σ2(T ). (3) nore all such multiple encounters and we keep only single a T a sim sim encounters in the calculation of the standard deviation of Note that it as been shown (e.g. ?) that close encounters the size of the jumps. However, when we compute the total do not exactly result in a random walk (there are some diffusion on the duration of the simulation we use the to- correlations between successive encounters) and that the tal number of encounters that occur during the simulation variance is not exactly linear with time (or number of as multiple encounters should not be ignored. encounters). It can, indeed, be written as a power law of Table 1 sums up the results we obtained for each aster- the form : oid and for the 11 asteroids together. We give the numbers of encounter during the simulation (including multiple en- B σa(t) = Ct . (4) counters), the standard deviation obtained from statistics on single encounters (σ [1]), the corrected values from For a random walk : B = 0.5. Numerical estimations a noise (¯σ [1]) and the standard deviations on the duration of this coefficient B for different families (see ???) show a of the simulation (¯σ (T )). Finally, the percentages of that it depends on the family considered and can even a sim contribution of the different asteroids are given in term of evolve with time. However the values found in the litera- variance. We obtain a total diffusion due to close encoun- ture range between 0.5 and 0.65 and we did not find any ters of 8.63 10 4 AU during the 30 Myr of the simulation estimations in the case of the Vesta family so we decided − (see table 1).× This number is to be compared with the drift to stick to the random walk hypothesis. rate obtained with the global approach (8.8 10 4 AU). Note that in this reasoning we do not take into account − The results given by both methods are in good× agreement. the influence of the noise in the calculation of jumps sizes. By using Eq.3 we can extrapolate our results to longer Indeed, we observe on Fig.5 and Fig.6 that the minimum times : and maximum values of the semi-major axis are not constant outside the close encounters (jumps) but undergo os- 4 σ¯a(t) = 1.57 10− t (Myr) AU. (6) cillations which are due to remaining secular evolution. It × is possible to eliminate an important part of this noise by Of course it is possiblep to do the same for the contribution taking the minimum of minima and the maximum of max- of each asteroid. This allow us to compare our results with ima over greater time scales before and after the encoun- the previous study of ?. In this article, the authors find a +2.5 3 ters. In order to choose the best time interval to compute drift rate of 2.0 2.0 10− AU/(100 Myr) by considering − × these extrema and to evaluate the remaining noise after only close encounters with Vesta. If we extrapolate to 100 3 this treatment we use the reference simulation S and we Myr and consider only Vesta we find a drift of 1.3 10− × calculate the standard deviation of the difference of semi- AU which is coherent with ? findings. major axis before and after random times. Since there If we look at the different contributions in table 1 we are no close encounters thus no jumps in this simulation see that close encounters with Ceres and Vesta together this standard deviation measures only the remaining noise. represent about 99% of the total diffusion (in variance). 6 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

) -7.5 2 variance (which is additive unlike the standard deviation). a σ -8

log( -8.5 Ast. Nevents σa[1]σ ¯a[1]σ ¯a(Tsim) Contrib. 105 105 105 (%) -9 × × × 1 1175 76.9 76.8 51.4 35.55 -9.5 2 401 14.6 13.9 5.4 0.40 -10 3 868 5.9 3.8 2.2 0.07 0 0.5 1 1.5 2 2.5 3 4 6521 43.8 43.6 68.8 63.63 log(m) 7 2999 5.2 2.6 2.8 0.11 10 162 5.7 3.6 0.9 0.01 Fig. 7. Contribution to the semi major axis diffusion due 15 1064 5.6 3.4 2.2 0.06 to each considered asteroid as a function of the mass of 19 3210 4.8 1.9 2.1 0.06 this asteroid. 324 603 6.5 4.8 2.3 0.07 532 688 5.8 3.7 1.9 0.05 7000 704 4 6.7 5.0 0.2 0.00 6000 All 17695 86.3 100 − − 5000 4000 N Contributions from the 9 other asteroids seem negligible. 3000 This is coherent with findings of previous studies on other 2000 asteroid families (see in particular ?). On the contrary, 1000 since Ceres represents 36% of the diffusion this asteroid 0 should not be neglected (like in ?). We believe that two 0 5 10 15 20 25 30 main reason can explain these proportions of contribu- d (AU/yr) tion : the mass of the asteroids and their proximity with 3 the Vesta family members in phase space. The mass must Fig. 8. Number of close encounters (< 10− AU) between influence the importance of the effect of a single close en- Vesta family fragments and each of the 11 considered as- counter whereas the proximity in phase space influences teroids as a function of the distance in phase space be- the number of such events. In order to check for these tween these asteroids and Vesta. We use the mean values correlations we plotted the variance due to close encoun- of the semi-major axes, the eccentricities and the inclina- ters as a function of the mass of the considered aster- tions of the 11 asteroids during the 30 Myr of the simula- oid (Fig.7) and the number of events as a function of the tion in order to compute the distances (see Eq.7). distance in phase space of the considered asteroid with respect to Vesta (Fig.8). We use the definition of the dis- the 11 asteroids of our simulation we obtain a total mass tance in phase space introduced by ? : 10 of 8.36 10− M whereas the estimated total mass con- × 10 2 tained in the main asteroid belt is about 15 10− M 5 δa 2 2 × d = na0 + 2(δe) + 2(δsini) . (7) (obtained from INPOP10a fits, ?). The most massive as- s4 a0 10 teroid in the main belt is Ceres with 4.76 10− M , × This distance has the dimensions of a velocity since it is an thus the remaining mass (not considered in our simula- estimation of the ejection velocity that would be needed tion) represents about 1.5 times the mass of Ceres. The to carry two objects initially at the same position to their total mass contained in the main asteroid belt is not well current positions. In Eq.7, we used the mean values of the constrained and a previous determination of this mass by ? 10 semi-major axes, the eccentricities and the inclinations of gave 18 10− M which implies a remaining mass closer × the 11 asteroids during the whole simulation (30 Myr) in to twice the mass of Ceres. Thus, it seems reasonable to order to compute the distances. We deduce from these fix the upper limit of this remaining mass to about twice graphs that we indeed have an effect of the mass and the the mass of Ceres. proximity in phase space but we cannot obtain a simple In Fig.7, the slope of the regression curve is about 1.2. and precise law in order to evaluate the diffusion that We plotted a straight line of slope 1 for comparison. A would result from close encounters with other asteroids slope of 1 means that two objects of masses 0.5 have the than the ones considered in our simulation. same effect than one of mass 1. If the slope is higher than However, we can constrain the global effect of the re- 1 it is more efficient to have a single object whereas if maining objects of the main asteroid belt. In our simu- the slope is lower than 1 it is more efficient to have two lation we took into account the most massive objects of objects. Even if this slope is not well constrained in Fig.7 the main asteroid belt. Actually, if we add the masses of it is reasonable to consider that it is greater than 1 so Delisle and Laskar: Chaotic diffusion induced by massive asteroids 7 it is more efficient to have one big object than to split The characteristic time scale of such events is given by it into smaller objects. Thereby, we can assume that the (see ??): total contribution of the remaining objects of the main belt will be less than twice the contribution of Ceres. 1/2 1/2 We still have the problem of the influence of the dis- R D τcoll = 15 Myr = 335 Myr. (9) tance in phase space but it seems realistic to consider that 1 m 1 km the case of Vesta is singular since it is the parent body of the Vesta family and that the remaining objects must have For kilometer-sized asteroids, the time between two reori- a probability of close encounter closer to Ceres’s one than enting collisions is about 300 Myr, it means that there are Vesta’s one (see Fig.8). Thus, the total variance due to a few (3 or 4) reorientations on the time scale of the age of encounters with all the asteroids of the main belt should the Vesta family. Thus it is likely that these reorientations not exceed (and is probably a lot smaller than) 1.7 times will reduce the diffusion due to the Yarkovsky effect but the value of the variance obtained in our simulation. In not by large factor. Moreover, we can still consider that terms of standard deviations it corresponds to a factor of the YORP effect acts almost instantaneously between two about 1.3. reorientations and that the seasonal Yarkovsky effect is negligible and the diurnal effect is always maximal. 5. Comparison with the Yarkovsky-YORP effect Taking into consideration these different results we estimate the diffusion of Vesta family members under the The Yarkovsky effect is a non-gravitational force which re- effect of Yarkovsky, YORP and collisions by a simple ran- sults from interaction between asteroids and the solar radi- dom walk process and the maximal diurnal Yarkovsky ef- ation. There are actually two versions of the Yarkovsky effect between each reorientation. Fig.9 shows a comparison fect : a diurnal and a seasonal effect. The diurnal effect can between the diffusion due to close encounters and the one result in an increase or a decrease of the semi-major axis due to the Yarkovsky effect as functions of the asteroid depending on the obliquity () of the asteroid ( cos ), diameter at different times of the evolution (250 Myr, 1 whereas the seasonal effect systematically decreases∝ the Gyr and 3 Gyr). For the close encounters, we plotted the semi-major axis ( sin2 ). The diurnal effect is maximal diffusion obtained for the 11 asteroids and the extrapo- ∝ when = 0◦ or 180◦ and is zero when = 90◦. On the lation we deduced for the whole main asteroid belt. For contrary, the seasonal effect is maximal when = 90◦ and the Yarkovsky effect we plotted both the maximal diurnal zero when = 0◦, 180◦. Both effects are size dependent, effect (without any reorientation) and the result of the for instance the diurnal effect is given by (see in particular random walk process. We can see that reorienting colli- ?): sions do not affect much the magnitude of the Yarkovsky effect for the range of diameters we are interested in, even at 3 Gyr. We recall that the estimated age of the Vesta da 4 1 km = 2.5 10− cos AU/Myr. (8) dt × D family is about 1 Gyr (?). For this duration, close encounters are dominant for the diffusion of asteroids larger than In addition to the Yarkovsky effect, we have to consider 35 km but below this value the Yarkovsky effect is prevail- the YORP (Yarkovsky - O’Keefe - Radzievskii - Paddack) ing. For a diameter of 1 km the Yarkovsky effect is about effect which also comes from the solar radiation but affects 25 times greater than the effect of close encounters. the rotational velocity and the obliquity of asteroids. This Regarding the time evolution of these effects, we as- effect depends a lot on the shape and the surface composi- sume in this work that close encounters generate a ran- tion of the asteroid but it is possible to do some statistics dom walk process that evolves as √t whereas the maximal to understand what is the most probable scenario (see ??). Yarkovsky effect is linear with time. When we take into It has been shown (?) that for most of basaltic asteroids account reorientations for the Yarkovsky effect we also (more than 95%), the obliquity is asymptotically driven have a random walk process so the evolution law should to 0◦ or 180◦ by the YORP effect. The time scale for the be proportional to √t. However this is an asymptotic law YORP effect is about 10-50 Myr (see ??). This means that which is valid when a great number of random walk steps on a 1 Gyr time scale (which is the order of magnitude for is reached. In the case of reorienting collisions the char- the age of the Vesta family), we can consider that the acteristic time is about 300 Myr so there are only a few obliquity is either 0◦ or 180◦ during the whole evolution steps per Gyr and the asymptotic law is invalid on this and that the seasonal effect is negligible and the diurnal time scale. It means that for the time scale we are inter- effect is maximal (in one or the other direction). ested in, the Yarkovsky effect evolves faster than √t so Nevertheless, there is still another phenomenon which faster than close encounters. This is exactly what we ob- must be taken into account : the collisions. We have to serve on Fig.9 : for 250 Myr the close encounters take a distinguish whether a collision is destructive or not. The more important part in the diffusion process (equivalence collisional life time of a 1-10 km asteroid in the main belt of the two effects for D=19 km) whereas for 3 Gyr the is about 1 Gyr. But non-destructive collisions are more Yarkovsky effect is even more prevailing (equivalence for frequent and can reorient the spin axis of the asteroid. D=54 km). 8 Delisle and Laskar: Chaotic diffusion induced by massive asteroids

1 and that our ﬁndings are compatible with previous work EN A TEN on the Vesta family (?). We use the results we obtained YMX YRW for the 11 asteroids to extrapolate and constrain the diﬀu- 0.1 sion that would result from close encounters with all the

(AU) main belt asteroids. We show that this diﬀusion should not exceed 1.3 times the diﬀusion due to the 11 asteroids.

250Myr Finally, we compare the diﬀusion due to close encounters a

∆ 0.01 with the Yarkovsky-driven drift of the semi-major axis of asteroids in the 1-100 km range (in term of asteroids diameters). We find that both effects are equivalent over 1 Gyr 0.001 for a diameter of about 35 km. For smaller asteroids the 1 Yarkovsky effect dominates the semi-major axis diffusion B and for larger asteroids close encounters become more important. Thus we confirm that although asteroids close en- 0.1 counters have a significant influence, the main mechanism of semi-major axis diffusion that transports main belt as- (AU) teroids (and especially V-type asteroids) to Earth-crossing 1Gyr

a orbits via strong resonances is the Yarkovsky eﬀect. ∆ 0.01 Acknowledgements. This work was supported by ANR- ASTCM, INSU-CNRS, PNP-CNRS, and CS, Paris Observatory. 0.001 1 C

0.1 (AU) 3Gyr a

∆ 0.01

0.001 1 10 100 D (km) Fig. 9. Comparison of the semi-major axis diffusion due to the close encounters and to the Yarkovsky effect after 250 Myr (top), 1 Gyr (middle) and 3 Gyr (bottom). EN is the diffusion due to close encounters with the 11 asteroids considered in our simulations. TEN is the extrapolation we performed for the entire main belt ( 1.3). YMX stands for the maximal diurnal Yarkovsky effect.× YRW stands for the Yarkovsky effect with reorientations (random walk).

6. Conclusion In this work we evaluate the effect of close encounters between Vesta family members and massive asteroids on the semi-major axis diffusion of the family. We calculate this effect by two methods. We first look at the collective effect of these close encounters and deduce an estimation of the semi-major axis drift due to the 11 massive asteroids taken into account here. Then we study individually the close encounters and make statistics over these events. With this second method we are able to separate the contribution of each of the 11 asteroids to the diffusion. We show that both methods give comparable results