arXiv:1505.04329v1 [astro-ph.EP] 16 May 2015 tefis eoatatri 16)Giu a recognized century ⋆ was 20th Griqua the (1362) in observations asteroid by resonant realised first bod was small-sized (the it several as harbours asteroids, ies of depleted be 1964). (Schubart exhibi commensurability not is 2:1 does which technically exact Hecuba but Cy- the (108) asteroid the from main-belt and derived outer is belt an name main Its outer region. be the bele to separating considered borderline commonly is the (Kirkwood which gap 1969), Hecuba associat Schweizer the res- as is 1867; known Jovian It gap first-order Kirkwood belt. the asteroid major with main the the of hereinafter intersecting one Jupiter, onances is with J2/1, as resonance denoted mean-motion 2:1 The INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon.

h rgno oglvdatrisi h : mean-motion 2:1 the Jupiter in with asteroids resonance long-lived of origin The cetd21 a 2 eevd21 a 2 noiia om2 form original in 12; May 2015 Received 12. May 2015 Accepted c 2 1 3 .Chrenko O. eateto pc tde,SuhetRsac Institute Research Hol Southwest V Prague, Studies, Space in of University Department Charles Astronomy, of Institute eateto hsc,AitteUiest fThessaloni of University Aristotle Physics, of Department -al [email protected]ff.cuni.cz E-mail: 05RAS 2015 h : eoac,wihwsoiial eivdto believed originally was which resonance, 2:1 The 1 ⋆ .Broˇz M. , 000 –0(05 rne 1Spebr22 M L (MN 2021 September 11 Printed (2015) 1–20 , uzigadteeoeteoii fteln-ie seod a not has asteroids constra dis long-lived observational the orbital These of the origin far. cluster. the but collisional therefore steep and a inclined, puzzling is highly of distribution predominantly evidence size-frequency are any The asteroids A-island B. of island orbits The 10. ueoseog osbtnilycnrbt oteosre pop observed words: the Key to we contribute migration substantially grou m the to outer long-lived surviving enough hypothetical numerous asteroids the of Primordial that part migration. narrow is a planetary conclusion from of Our capture evolution resonant Gyr. collisional 4 by the past model over fift sur also escaping population the We an du instability. (i) with population jumping-Jupiter investigate scenario the a we described of recently Then a capture B. following (ii) gration, island and with asteroids comparison resonant dynamica primordial in the faster, that is demonstrate we A included, effect Yarkovsky the inadrvstispyia rpris sn an Using properties. physical its revisit and tion ABSTRACT sad ftepaesaedntdAadB iha nvnpopulatio uneven an with near B, reside and A They denoted Gyr. 4 space phase to gro the comparable transient of long-lived lifetimes islands a The dynamical drift. be axis exhibit to semimajor can Yarkovsky known the by is state population steady short-lived The asteroids. 1 ihteamt rvd ibeepaain efis paetereson the update first we explanation, viable a provide to aim the With h : enmto eoac ihJptrhror w itntg distinct two harbours Jupiter with resonance mean-motion 2:1 The .Nesvorn´y D. , io lnt,atris eea ehd:numerical. methods: – general asteroids: planets, minor i R514Teslnk,Greece Thessaloniki, GR–54124 ki, 00Wlu tet ut 0,Budr 0832 USA C0–80302, Boulder, 300, Suite Street, Walnut 1050 , ed ˇ oic ´ c ,C–80 rge8 zc Republic Czech eˇsoviˇck´ach 8, Prague CZ–18000 2, t - 2 .Tsiganis K. , 1 pi 04 April 015 hs ein eedsgae ssal sad n B and A 1). islands Fig. stable e.g. see as 1997, 1997). designated Ferraz-Mello (Nesvorn´y & Ferraz-Mello were & regions Michtchenko phase the quasi- These the 1994; of of of part (Franklin the regions resonant behaviour the space of separate chaotic inside two located regions strongly motion are regular the a there while orbits, to asteroidal that rise discovered give was overlap par- resonances In 1995). it secondary secu- Moons ticular, & and overlapping Watanabe 1993) Henrard, and 1987; Moons (Wisdom compli- J2/1 & a (Morbidelli the by the lar between affected investigate interplay orbits to resonant cated and various resonance of the character the of to insight structure get internal to (e.g analysis helped frequency 1997) and Ferraz-Mello Nesvorn´y 1999) 1986; mapping & Ferraz-Mello symplectic & Murray Roig 1998), an- (e.g. (e.g. Migliorini on & based methods Morbidelli studies Moons, semi-analytical follow-up and The 1959)). alytical (Rabe 1943 in 3 A .K Skoulidou K. D. , T E tl l v2.2) file style X N bd oe ihsvnpaesand planets seven with model -body i-etfml during family ain-belt ulation. elto fisland of depletion l seod,however, asteroids, nsaesomewhat are ints igpaeaymi- planetary ring in lntand planet giant h enepandso explained been 3 epoal not probably re pssandin sustained up rbto lacks tribution ratio n a created was p oprdto compared h resonant the w isolated two iaiiyof vivability n popula- ant op of roups B/A ≃ . 2 O. Chrenko et al.

However, it was also suggested by account the paucity of bodies in island A, the differences in Ferraz-Mello, Michtchenko & Roig (1998) that the sta- the inclination and the non-equilibrium size distribution. bility of the islands could have been weakened in the past. The long dynamical lifetimes of the dynamically sta- The mechanism responsible for this increase of chaos was ble asteroids, together with the failure of the aforemen- introduced as 1:1 resonance of the J2/1 libration period tioned hypotheses, strongly indicates that their origin may with the period of Jupiter-Saturn great inequality (GI). be traced back to the epoch of planetary migration. Re- Although the present value of the GI period is 880 yr, cent advances in migration theories suggest that a primor- it was probably smaller when configuration of planetary dial compact configuration of planetary orbits and subse- orbits was more compact (Morbidelli & Crida 2007) and quent violent planetary migration might have led to a desta- it slowly increased during Jupiter’s and Saturn’s divergent bilization of several regions that are otherwise stable un- migration. If the value of the GI period was temporarily der the current planetary configuration. The populations of comparable with the libration period of the J2/1 asteroids Trojans and Hildas (Nesvorn´y, Vokrouhlick´y& Morbidelli (which is typically 420 yr), the aforementioned resonance 2013; Roig & Nesvorn´y2014) may serve as examples of such ≃ would have occurred. a process: the primordial populations in the 1:1 and 3:2 reso- With ongoing improvements of the observation tech- nances with Jupiter were totally dispersed and the observed niques and sky surveys, there is now a population of aster- populations were formed by resonant capture of bodies, orig- oids inside the J2/1 which is large enough to be analyzed sta- inating either in the outer main belt or in the transneptunian tistically with numerical methods. Long-term integrations in disc of comets. It seems inevitable that the 2:1 resonance a simplified model with four giant planets were performed by with Jupiter also undergoes significant changes of its loca- Roig, Nesvorn´y& Ferraz-Mello (2002) who identified both tion, inner secular structure and asteroid population, during short-lived and long-lived asteroids within the observed pop- the planetary migration. ulation, which consisted of 53 bodies by then. Investigat- The structure of the paper is as follows. We first use the ing resident lifetimes of the resonant asteroids, Roig et al. latest observational data from the databases AstOrb, Ast- (2002) demonstrated that part of the population escapes DyS, WISE and SDSS to update the observed population from the resonance on time-scales 10 Myr, while other as- in Section 2. We also briefly review the characterization of ∼ teroids may have dynamical lifetimes comparable to the age resonant orbits and describe a method of dynamical map- of the Solar System. The orbits of these long-lived aster- ping. In Section 3, we study whether the planetary migra- oids were embodied in island B and part of them was also tion may cause depletion or repopulation of the long-lived present in its boundaries, being affected by chaotic diffusion. J2/1 asteroids. The dynamical models we create are based Surprisingly, the stable island A was empty. on simulations with prescribed evolution of planets in con- text of the modern migration scenario with five giant plan- In a series of papers focused on the 2:1 mean-motion res- ets and jumping-Jupiter instability (Nesvorn´y& Morbidelli onance, Broˇzet al. (2005) and Broˇz& Vokrouhlick´y(2008) 2012). Section 4 is focused on dynamical simulations cover- identified significantly more bodies as resonant. The latter ing the late stage of planetary migration during which the catalogue contained 92 short-lived and 182 long-lived as- GI period evolution might have influenced the stability of teroids. They also realised that nine of the long-lived as- the long-lived asteroids. In Section 5, we examine the effects teroids were located in island A, while the rest was resid- of collisions on the long-lived population. Finally, Section 6 ing in the island B and its vicinity. They successfully ex- is devoted to conclusions. plained the origin of the short-lived asteroids by a numer- ical steady-state model, in which the resonant population was replenished by an inflow of outer main-belt asteroids driven by the Yarkovsky semimajor axis drift. However, this model fails in the case of the long-lived population which was thought to be created by Yarkovsky induced injection of the nearby Themis family asteroids that exhibit similar incli- nations as the B-island asteroids. Broˇzet al. (2005) showed 2 OBSERVED RESONANT POPULATION that the objects transported from Themis are usually per- turbed shortly after entering the resonance and rarely reach In this section, we first describe methods we use for identi- fication and description of resonant orbits as well as for dy- the islands. namical mapping of the resonance. Our approach is similar Roig et al. (2002) argued that the steep size-frequency to the one used by Roig et al. (2002). We identify resonant distribution (SFD) of the long-lived asteroids, which was orbits in a recent catalogue of osculating orbital elements inconsistent with a steady state (Dohnanyi 1969), could im- and we study their dynamical lifetimes on the basis of long- ply recent collisional origin. On the contrary, there was no term numerical integrations in a simplified model with four collisional cluster identified in the orbital distribution by giant planets only. Our goal is also to survey available data Broˇz& Vokrouhlick´y(2008), thus a collisional family would for physical properties of the population, namely the abso- have to be older than 1 Gyr. Moreover, the presence of the lute magnitudes, sizes and albedos. In the last subsection, highly inclined A-island asteroids would require unrealistic we revisit results of Skoulidou et al. (in preparation) in or- ejection velocities, assuming that the disruption occurred in der to compare the simplified four-giant-planet model with the more populated island B. a more sophisticated framework including terrestrial planets As there is no self-consistent explanation, the main goal and the Yarkovsky effect. Especially, we look for the differ- of this paper is to provide a reasonable model for the forma- ences in the dynamical decay rates in the stable islands A tion and evolution of the long-lived population, taking into and B.

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 3

2.1 Characterization of resonant orbits of motion. The situation for the unstable orbits is just the opposite and the intersections slowly disperse in time. This The 2:1 resonance critical angle is defined as: fact propagates numerically to the resulting resonant ele-

σ 2λJ λ ̟ , (1) ments, if we compute their standard deviation as an error ≡ − − of the mean value obtained over a significant period of time where λJ and λ are the mean longitudes of Jupiter and of ( 100 kyr). ∼ an asteroid, respectively, ̟ is the asteroid’s longitude of Nevertheless, due to higher order perturbations and sec- perihelion. The critical argument of any body trapped inside ular effects, the above conditions are seldom satisfied exactly the J2/1 resonance librates (quasi-periodically changes on and one has to use less confined criteria in the following form an interval < 2π) with a typical period of about 420 yr. On when numerically integrating the orbits: the other hand, the critical argument of the asteroids outside ◦ ∆σ ◦ the resonance circulates. This provides us a useful tool for σ < 5 > 0 ̟ ̟J < 5 . (4) the identification of the resonant asteroids. | | ∧ ∆t ∧ | − | The libration of σ is linked to periodic changes of the We use the difference between successive numerical time osculating semimajor axis a, the eccentricity e and the incli- steps (denoted as ∆σ) rather then the time derivative. Note nation I. These changes are coupled together as described that we completely omit the condition for Ω. This simplifi- by the adiabatic invariant N of the asteroid’s motion in the cation can be compensated by verifying that the maximal circular and planar restricted Sun–Jupiter–asteroid system value of the inclination is reached when recording the reso- nant elements. N = √a 2 1 e2 cos I . (2) − −   The presencep of other planets and variable eccentricity of 2.2 Dynamical maps Jupiter’s orbit give rise to multiple perturbations in the J2/1 In this section, we summarize our approach to dynamical and prevent integrability of the orbits. mapping of the 2:1 resonance and its secular structure. Our None the less, the a,e,I coupling is preserved to a cer- method is verified by comparing the map computed for the tain degree. Semimajor axis oscillates around the libration present configuration of planets with the separatrices and lo- centre, which is positioned approximately at the exact res- cations of secular resonances derived by Moons et al. (1998). onance ares 3.27 AU. The eccentricity and inclination at- ≃ An important outcome is our ability to record global changes tain their maximal values when the oscillation of a is at its inside the resonance when the planetary orbits are recon- minimum and vice versa. figured and also to examine dynamical stability in various An inconvenient consequence of this behaviour is that regions of the resonant phase space. the standard averaging methods for the computation of Our method employs the definition of the resonant ele- proper elements (Kneˇzevi´c& Milani 2003) do not retain any ments. The respective values should change systematically in information about the libration amplitude (i.e. the proper the case of unstable resonant orbits, whereas stable orbits semimajor axes of all resonant asteroids approach the value should only oscillate with small variations. The same be- of ares). Therefore the proper elements are not the appro- haviour is exhibited by actions of a dynamical system; time priate choice when studying resonant orbits. series of their extremal or mean values are therefore often In order to properly characterize the libration ampli- used for dynamical mapping (e.g Laskar 1994; Morbidelli tude, we use an alternative set of resonant (or pseudo- 1997; Tsiganis, Kneˇzevi´c& Varvoglis 2007). proper) elements (Roig et al. 2002). The idea is to record We proceed as follows. First, we divide an investigated the osculating orbital elements at the moment when they part of the phase space into the boxes of the same size. Let reach their extremal values during the libration cycle. These the central coordinates (a,e,I) of each box represent a set of values can be found as the intersections with a suitably de- initial resonant elements (this is accomplished by setting the fined reference plane in the osculating elements space. A set osculating angular elements so that condition (3) is fulfilled). of conditions determining this plane can be written as We integrate the initial orbits over several millions of years dσ and track the evolution of the resonant elements. Finally, σ = 0 > 0 ̟ ̟J = 0 Ω ΩJ = 0 , (3) ∧ dt ∧ − ∧ − we determine the differences δar, δer and δ sin Ir between the initial and final (or the last recorded) resonant elements. where Ω denotes the longitude of node and subscript J is These values represent the dynamical stability of the initial used for Jupiter. The purpose of the conditions for ̟ and Ω orbit and we use them as a characteristics of the entire box. is to eliminate secular variations of the resonant elements. For the purpose of expressing the total displacement in The whole set enables the resonant elements to be recorded the phase space, we define the distance when the osculating semimajor axis a reaches its minimum, the eccentricity e and the inclination I attain maxima. Note 2 δar 2 2 that as a consequence, the J2/1 asteroids are always depicted d +(δer) +(δ sin Ir) , (5) ≡ a¯r on the left-hand side (closer to the Sun) of the libration s  centre in the resonant elements space (see e.g. Fig. 2). wherea ¯r denotes the arithmetic mean of the initial and Moreover, temporal evolution of resonant elements may final resonant semimajor axis. The distance d is similar serve as the first indicator of the stability. The reason is that to the metric used in the Hierarchical Clustering Method stable orbits exhibit stable libration with only a small vari- (Zappal`aet al. 1995) for family identification. ation around the mean value. Therefore, successive intersec- In order to verify our method, we constructed a dynam- tions with the reference plane do not move significantly and ical map of the 2:1 resonance in the present configuration the recorded resonant elements are nearly exact constants of planets. We investigated the phase space in the intervals

c 2015 RAS, MNRAS 000, 1–20 4 O. Chrenko et al.

[!h] putation of the resonant elements based on criterion (4) 0.5 1 (Broˇzet al. 2005). This symplectic integrator enables us to ν use a time-step ∆t = 91.3125 d. The simulations we perform 5 are simplified: as we study the outer main-belt we take into

r 0.8 e 0.4 Kozai account only the gravitational interactions with the Sun and the four giant planets. The terrestrial planets are neglected,

island A d 0.6 except for a barycentric correction applied to the initial con- 0.3 island B ditions. We use the Laplace plane to define our initial frame ν

16 0.4 distance of reference. We do not consider any non-gravitational ac- celeration at this point. 0.2 resonant eccentricity separatrix 0.2 The final step was to calculate the distance d for each ¯ secondary test particle. The average value d for the particles initially resonances placed in the same box was taken as a measure of the dy- 0.1 0 3.2 3.22 3.24 3.26 namical instability in the box. Note that there is no temporal measure in this method, i.e. the map reflects the maximal resonant semimajor axis ar [AU] displacement in each box during the integration timespan tspan but it does not say how fast did this displacement oc- 1 cur. 0.5 We plot the mean values d¯ in the top panel of Figure 1. r I 0.8 The map is an average projection of all sections in sin Ir to 0.4 the (ar,er) plane. It can be compared with several separa- island A 0.6 d trices derived by Moons et al. (1998). It is clear that the 0.3 map reflects the inner secular structure of the 2:1 resonance very well: we can locate both the stable islands A and B,

0.4 distance 0.2 island B separated by the ν16 secular resonance. The borderline at higher values of er is formed by the Kozai and ν5 separa- resonant inclination sin 0.1 0.2 trices. The low-eccentricity region near the libration cen- tre is affected by presence of multiple secondary resonances 0 0 (Roig & Ferraz-Mello 1999). 3.2 3.22 3.24 3.26 The bottom panel of Fig. 1 displays the projection of resonant semimajor axis a [AU] r our map to the (ar, sin Ir) plane but this time for a small interval of er because the shape of the separatrices and stable Figure 1. A dynamical map of the 2:1 mean-motion resonance islands strongly depends on er. The map demonstrates how with Jupiter computed in the present configuration of giant plan- the shape of the stable islands changes with the resonant ets. Top panel: the map is plotted in the resonant semimajor axis inclination Ir. ar vs resonant eccentricity er plane and it is averaged over five sec- Let us conclude that the suitability of the resonant ele- tions in the resonant inclination sin Ir. The color coding of boxes ments for the dynamical mapping serves as an independent represents the average distance in the phase space travelled by a confirmation that they indeed reflect the regularity of res- test particle with initial orbital elements placed within the box. onant orbits and they retain important properties of the Additionally, separatrices and borders of the secular resonances, which were computed by Moons et al. (1998), are plotted over the proper elements at the same time. map. The left solid line corresponds to the separatrix of the J2/1. The near-vertical solid line is the libration centre of the J2/1. The 2.3 Lifetimes of asteroids in the 2:1 resonance dotted lines indicate the ν16 and ν5 resonance, and the dashed line represents the Kozai resonance. We mark the stable islands Let us now discuss our approach to the identification and denoted A and B and also the region of overlapping secondary classification of the resonant asteroids. We numerically prop- resonances (Roig & Ferraz-Mello 1999). Bottom panel: the map agated the orbits of known numbered and multi-opposition plotted in the resonant semimajor axis a vs sine of resonant in- r asteroids in the broad surroundings of the J2/1 to identify clination sin I plane. It was computed for test particles with the r those trapped inside, using a time series of the critical argu- resonant eccentricity er = 0.25. ment σ. We extracted the osculating elements of the main- belt objects from the AstOrb database (Bowell 2012) as of ◦ ◦ a (3.195, 3.275) AU, e (0.1, 0.5), I (0 , 25 ) and split 2012 November 15 to set-up the initial conditions for the first ∈ ∈ ∈ them into a grid of 40 40 5 boxes. With the aim to im- short-term integration. Our choice of the borders in the os- × × prove our statistics, we randomly generated two more test culating (a,e) plane was e1 = 0.45 (a 3.24) / (3.1 3.24) − − particles in the vicinity of each central particle (the disper- and e2 = 0.5(a 3.24) / (3.46 3.24). We ended up with − − sion of particles is not exceeding 20 per cent of the box size). 11, 469 orbits. The corresponding planetary ephemeris were In this way we created the initial conditions for 24, 000 test taken from JPL DE405 (Standish 2004) for the given Julian particles. date. We numerically integrated the orbits for 10 kyr and In the next step, we integrated the orbits for tspan = we recorded the critical argument σ for each asteroid during 10 Myr. For all these simulations, we use the symplectic the simulation. We have found 374 librating asteroids. integrator swift (Levison & Duncan 1994) with a built- The following long-term integration requires a different in second-order symplectic scheme (Laskar & Robutel 2001) approach. Our goal is to determine the future orbital evolu- and with an implementation of digital filters for the com- tion of the resonant asteroids and estimate their dynamical

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 5 lifetime. However, we are limited by strong chaotic diffusion 0.5 in the J2/1 resonance. In other words, even a slight change ν 5 of the initial conditions can significantly alter the orbital r evolution. Because the orbital elements of each observed as- e 0.4 Kozai island A teroid are only known with a finite accuracy, one should consider all values within the observational uncertainty as possible initial conditions to cover all alternatives of future 0.3 orbital evolution. ν To account for this chaotic behaviour, we first matched 16 2/1 centre 0.2 2/1 separatrix island B

the AstOrb data for librating asteroids with correspond- resonant eccentricity ing AstDyS uncertainties (Kneˇzevi´c& Milani 2003). The matching failed in four cases1 only and hence we discarded these bodies from the J2/1 population. We then used a 0.1 3.2 3.22 3.24 3.26 pseudo-random generator to create a bundle of 10 synthetic resonant semimajor axis a [AU] orbits for each asteroid which are close to its nominal orbit. r The generated orbital elements fall within the Gaussian dis- tribution (over 3σ interval) in the nonsingular osculating 0.6 ± element space. These synthetic test particles are called close 0.5 r clones. In this manner, we obtained 4, 070 orbits (1 nom- I inal and 10 synthetic for each body) which we integrated 0.4 over 1 Gyr. By this procedure, we can study several possible realisations of future orbital motion. 0.3 Following again Roig et al. (2002), we define the dy- namical lifetime τ as the asteroid’s timespan of residence 0.2 inside the resonance. Test particles leaving the resonance are usually discarded due to highly eccentric or inclined or- resonant inclination sin 0.1 bits, which lead to planetary crossing or fall into the Sun. We calculated the median dynamical lifetimeτ ¯ as the median 0 value of the residence lifetimes of close clones. 3.2 3.22 3.24 3.26 We divided the J2/1 asteroids into three groups by resonant semimajor axis ar [AU] virtue of their median dynamical lifetimeτ ¯ as follows: τ¯ 6 70 Myr: short-lived/unstable/Zulus. Number of Figure 2. Orbital distribution of the J2/1 asteroids in the reso- • identified bodies: 140. nant semimajor axis ar vs the resonant eccentricity er plane (top) and in a vs sine of the resonant inclination sin I plane (bottom). τ¯ (70, 1000) Myr: long-lived/marginally sta- r r • ∈ The symbols correspond to the dynamical lifetime of each body: ble/Griquas. Number of identified bodies: 106. filled circles denote dynamically stable Zhongguos, empty circles τ¯ > 1 Gyr: long-lived/stable/Zhongguos. Number of • marginally stable Griquas, crosses unstable Zulus. The error bars identified bodies: 124. indicate standard deviations of the computed orbits. We show the same borders and structures as in Fig. 1. Only a relatively small For reference, the numbers of resonant asteroids identified part of the unstable population is depicted. in the previous paper Broˇz& Vokrouhlick´y(2008) were 92 short-lived and 182 long-lived asteroids. We used the same long-term integration of the resonant orbits to construct orbital distribution of asteroids in the jections to the (ar,er) and (ar, sin Ir) planes in Fig. 2. The J2/1. We calculated average values of the resonant elements orbital distribution exhibits similar properties as in the pre- for each asteroid and each close clone over the time interval vious studies. Zhongguos reside at the very centre of the of 1 Myr; then we computed the arithmetic mean for the stable islands, 11 Zhongguos in island A and 113 in island asteroid and its close clones together. We define the standard B. The majority of B-island Zhongguos have low resonant in- deviation of the resonant elements as the uncertainty of the clinations while A-island Zhongguos are more inclined. Gri- arithmetic mean2. quas partially overlap with the Zhongguo group, but they We depict the resulting mean resonant elements as pro- either reside on orbits with lower semimajor axis or higher inclination than Zhongguos. There is no clear separation be- tween the orbits of Griquas and Zhongguos. It is therefore 1 These particular bodies have the following designations: questionable whether the division of the long-lived popula- 2003 EO31, 2003 HH4, 2006 SE403 and 2006 UF152. The rea- tion into these two groups has a solid physical foundation, son for the databases mismatch might be in different methods apart from the dynamical lifetime. The orbital position of used for orbit identification. Griquas with respect to Zhongguos indicates that Griquas 2 Let us note that the standard deviations of the mean resonant might be chaotically diffusing part of Zhongguos which have elements will be slightly overestimated. The reason for this arises from our approach to the generation of initial conditions because haphazardly entered the periphery regions of the stable is- the initial elements of the close clones are generated randomly lands. thus resulting in uncorrelated sets of quantities even though they Since we focus on the long-lived population, a great part should be correlated, as described by the correlation matrices of the short-lived bodies is not displayed in Fig. 2 (they are available in orbital elements databases. located outside the range of the plot). The depicted short-

c 2015 RAS, MNRAS 000, 1–20 6 O. Chrenko et al.

12 group nominal γ variation of γ

10 short-lived −3.2 (−2.5, −3.7) N long-lived −4.3 (−3.7, −5.1) 8 Griquas −3.0 (−3.0, −3.3) Zhongguos −5.1 (−3.9, −5.1) 6

4 Table 1. The slopes γ resulting from the method of least squares which we applied to fit the SFDs of dynamical groups with a

number of asteroids γ 2 power law function N (>D) ∝ D . The middle column shows 2001 RN2 the result of the fit in the nominal interval of diameters D ∈ 0 (7.5, 18) km, the third column reflects the variation of γ if the 0 0.05 0.1 0.15 0.2 0.25 limits of the interval are shifted. geometric albedo pV

600 Figure 3. A histogram of the albedo distribution among the J2/1 asteroids: the visual geometric albedo pV vs the number of aster- oids N. The data for 44 asteroids were available in Masiero et al. 100 (2011). γ = -4.3 γ = -3.2 10 γ = -2.5 lived asteroids are chaotically drifting towards the Kozai separatrix. N(>D) short-lived long-lived 1 2.4 Albedo, colour and size-frequency 600 distributions In order to study the basic physical properties of the J2/1 100 population, we first searched the WISE database. We were number of asteroids γ able to extract the visual geometric albedos pV and the effec- = -5.1 γ tive diameters D, inferred from NEATM thermal models by 10 = -3.0 Masiero et al. (2011), for 44 asteroids inside the J2/1 (out of 370). Griquas Zhongguos The data obtained allow us to examine the albedo dis- 1 tribution in the resonance (Fig. 3). The lowest albedo is 1 5 10 20 1 5 10 20 (0.034 0.002) while the largest value is (0.24 0.03). This ± ± diameter D [km] relatively large value corresponds to asteroid 2001 RN2 and indicates that this object likely belongs to the S taxonomic Figure 4. Size-frequency distributions (SFDs) of individual dy- type. The majority of the asteroids (98 per cent) have albedo namical groups residing inside the J2/1: the diameter D vs the lower or equal to (0.136 0.004). The shape of the albedo cumulative number N (>D) of asteroids larger than D. A loga- ± distribution is typical for the outer main-belt region where rithmic scaling is used in the plot. The distributions correspond C-types dominate (e.g. DeMeo & Carry 2013). to the short-lived population (top left) and long-lived popula- As the next step, we investigated the diameters of as- tion (top right), which is then divided to marginally stable Gri- quas (bottom left) and stable Zhongguos (bottom right). The teroids. For the 44 cases we simply use the value D and its steep part of each distribution is approximated with a power-law uncertainty. For the rest of the asteroids we calculate their N (>D) ∝ Dγ . The nominal borders of the fitted regions (rang- approximate diameter using the following relation (Harris ing from 7.5 km to 18 km) are shown by vertical lines. The value 1998): of the slope γ is also given. We also plot stationary Dohnanyi-like slope (γ = −2.5) for comparison. 1329 − H D = 10 5 , (6) √pV where we insert the absolute magnitude H from the AstOrb interval except for SFD of Griquas which remains steep up catalogue and we use the mean albedop ¯V = (0.08 0.03). to D 4 km. We also record the variation of the slope γ by ± ≃ The standard deviation of D is then computed as the prop- slightly changing the fitted range of diameters. The results agated uncertainty of both H andp ¯V. are summarized in Table 1. Unlike the results of Broˇzet al. We construct the cumulative size-frequency distribu- (2005), the SFD of updated Griquas is not shallower than tions (SFDs) of the resonant population and individual a Dohnanyi-like SFD with γ = 2.5 (Dohnanyi 1969). In − groups, as shown in Fig. 4. We fit the steep part of the Section 5, we shall use the observed SFDs for comparison distribution with a power-law function N (> D) Dγ to with the outcomes of our collisional models. ∝ estimate the slope parameter γ for further comparison. The Finally, we also searched the SDSS MOC catalogue value of γ clearly depends on the chosen interval of diam- (Parker et al. 2008). We found astrometric and photomet- eters over which we approximate the SFDs with the power ric data for 81 resonant bodies for which we constructed law. We set the nominal fitted interval as D (7.5, 18) km. a colour-colour diagram (see Fig. 5). The principal compo- ∈ ⋆ The SFDs start to bend at the lower limit of this nominal nent denoted as a is defined on the basis of measurements

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 7

1 0 0.4

N(t)/N 0.8

29524 0.2 340646 358448 146951 261696 314325 303830 0.6 157354

[mag] 0 z

island B −

i 0.4 -0.2 island A 0.2

-0.4 272090

188795 normalized number of asteroids 0 -0.4 -0.2 0 0.2 0.4 0 1 2 3 a* [mag] time t [Gyr]

Figure 5. The optimized colour a⋆ vs the i − z colour diagram Figure 6. Time evolution of the number of resonant objects, of the J2/1 resonant asteroids found in the SDSS MOC cata- (N (t) /N0), normalized to the initial value. This is the result of logue. Error bars represent the standard deviations of the dis- a 3 Gyr-long simulation, within the framework of the 7-planets played quantities. The outliers with a⋆ > 0 are labeled with cor- model and including Yarkovsky-induced drift in a. The two responding catalogue numbers. curves refer to the two islands (A=open triangles and B=solid circles). Superimposed are the least-square fitted exponentials, yielding e-folding times of τA = (0.57 ± 0.02) Gyr and τB = in filters r, i and g as a⋆ = 0.89 (g r) + 0.45(r i) 0.57 (0.94 ± 0.02) Gyr, respectively. − ⋆− − and it can be used to distinguish C-complex (a < 0) and ⋆ S-complex (a > 0) asteroids. The distribution in the dia- In Fig. 6 we present the results of a similar simula- gram is typical for outer main-belt C-type asteroids, but one ⋆ tion that spanned 3 Gyr, in a model that contained the can also see several outliers with a > 0. We did not found seven major planets (Venus to Neptune; ‘7pl’ model). This any significant relation between the colours and the orbital simulation treated the 2:1 population, as was defined by distribution. Broˇz& Vokrouhlick´y(2008). Also, an approximate treat- ment of the Yarkovsky acceleration was included in the equa- tions of motion, assuming a constant drift rate in semimajor − − − 2.5 The long-term stability of the islands A and B axis equal to 2.7 10 4D 1 AU Myr 1. For each object its × The observed resonant population has likely evolved diameter D was estimated, using its catalogued H value and assuming an albedo of 0.06–0.08, and a value of either 0◦ or over Gyr-long time-scales. Hence, it is important ◦ to assess its dynamical stability over similarly long 180 obliquity was randomly assigned. The population was time-spans. This problem was recently revisited by found to decay exponentially in time, according to what was Skoulidou, Tsiganis & Varvoglis (2014). Here we will use described in the previous paragraph. the main results of this study that are relevant for our However, what is more interesting here is that the two work; a complete dynamical study will be presented in a sub-populations that are contained in the two quasi-stable different paper (Skoulidou et al. in preparation). resonant islands are not diffusing out at the same rate. Fig. 6 When studying the outer asteroid belt, we typically shows two sets of curves (one for each island): clearly island ignore the weak perturbative effects of the terrestrial A is depleted faster than island B. Fitting exponentials on planets, as this speeds up the numerical propagation. the data, we get the corresponding e-folding times: τA = (0.57 0.02) Gyr for island A and τB = (0.94 0.02) Gyr Skoulidou et al. (2014) showed that this is not a safe op- ± ± tion, when studying the 2:1 resonant population for time- for island B. This uneven depletion is an important dynami- spans longer than 1 Gyr, as numerous weak encounters with cal property of the resonance and is likely one of the reasons Mars actually induce small-scale chaotic variations. These behind the observed A/B asymmetry. The importance of variations build-up slowly with time and eventually assist in this observation will become more evident in the following breaking the phase-protection mechanism that would other- sections. wise prevent asteroids on moderately eccentric orbits (e ∼ 0.4) from encountering Jupiter on this time-span. Moreover, when the Yarkovsky effect is added, asteroids with D < 20 km can escape from the resonance more efficiently. This is because, as also shown in Broˇz& Vokrouhlick´y(2008), the combined action of resonance and slow Yarkovsky drift forces the asteroid orbits to slowly develop higher eccentric- ities as a consequence of adiabatic invariance. The compu- tations presented in Skoulidou et al. (2014) show that, in a physical model that takes into account the aforementioned phenomena, the 2:1 population would decay roughly expo- nentially in time, with an e-folding time of order 1 Gyr.

c 2015 RAS, MNRAS 000, 1–20 8 O. Chrenko et al.

3 EFFECTS OF THE JUMPING-JUPITER INSTABILITY 30 We investigate the role of the jumping-Jupiter instability on the depletion and eventual repopulation of the J2/1 in this 25 Neptune section. We study survival of a hypothetical primordial pop- ulation, and resonant capture from the outer main belt, both 20 [AU] in the fifth giant planet scenario (Nesvorn´y& Morbidelli Q Uranus , q

2012). Our simulations cover only the instability phase itself, , 15 a demonstrating at first which process is plausible. 5th planet 10 Saturn 3.1 Simulations with prescribed migration Jupiter 5

We adopt the technique suggested by Nesvorn´yet al. (2013) 0 2 4 6 8 10 for simulating the jumping-Jupiter instability. We modified time t [Myr] the swift rmvs3 integrator (Levison & Duncan 1994), so that the evolution of massive bodies is given by a prescribed Figure 7. Orbital evolution of giant planets in the fifth gi- input file and the evolution of test particles is computed by ant planet scenario, adopted from Nesvorn´y& Morbidelli (2012), the standard symplectic algorithm. This method allows us during the jumping-Jupiter instability, as it was reproduced by to exclude the transneptunian disc of planetesimals from our our modified integrator. We plot the time t vs the semimajor axis integrations, because its damping and scattering effects are a, the pericentre q and the apocentre Q. Each evolutionary track no longer needed; planets evolve ‘the way they are told’ by is labeled with the name of the corresponding giant planet. the input file. Moreover, the exact progress of the migration scenario is then exactly reproducible. ◦ ◦ e (0, 0.35), I (0 , 15 ). The angular elements were ran- The evolution of migrating planets is prescribed ∈ ∈ ◦ ◦ domly distributed over (0 , 360 ) interval. by the fifth giant planet scenario, developed in Nesvorn´y& Morbidelli (2012) (see Fig. 7). This scenario has already been proved reliable in terms of reproducing the Evolution of test particles. We recorded the time series orbital architecture of planetary orbits, the period ratios of resonant elements of the test particles during the integra- and secular frequencies, and several distributions of minor tion and processed them off-line using the Savitzky–Golay solar-system bodies. The prescribed input file is the result smoothing filter (Press et al. 2007) with 0.1 Myr range of the of a complete self-consistent simulation with five giant running window and a second order smoothing polynomial. planets and a massive transneptunian disc of planetesimals The result of our simulation is shown in Fig. 8. Note that for the sake of simplicity, we use the resonant elements (Mdisc = 20 MEarth). We use only a 10 Myr portion of 3 the simulation, containing the jumping-Jupiter phase. The for all particles, even for those not trapped inside the J2/1 . Shortly after the onset of migration, the test particles still input data sampling is ∆tinput = 1 yr which is sufficient for our purpose. retain the uniform character of the initial distribution. A Our integrator transforms the input data to the Carte- few minor mean-motion resonances can be identified. The sian coordinates (i.e. orbital elements to positions and ve- apparent gap between the resonance centre and the synthetic locities) and interpolates them according to the integration population is intentional, because we do not want any of the time step ∆t = 0.25 yr. The interpolation is done by a for- test particles to be initially placed inside the resonance. ward/backward drift along Keplerian ellipses, starting at After 2 Myr, the relaxation processes start to take place. input data point preceding/forthcoming to required time The 2:1 resonance changes its position with respect to the value. The results of both drifts are averaged using the initial one and starts to perturb several eccentric orbits at weighted mean, where the weight depends on temporal dis- the outskirts of the synthetic population. All of the per- tance between the data points and the required time value. turbed asteroids fall into the short-lived unstable zone (dis- cussed in Broˇzet al. 2005). The minor mean-motion reso- nances begin to pump the eccentricity of the test particles 3.2 Capture from the outer main belt residing inside them. At 6.26 Myr, an instantaneous discontinuity, known as We investigated a possibility that the resonant asteroids Jupiter’s jump, occurs. The 2:1 resonance, co-moving with were captured from outer main-belt orbits near the J2/1 jumping Jupiter, changes its location suddenly and a large during Jupiter’s jump. We present a simulation of the cap- number of test particles flow through the libration zone, ture in this section. while others are excited and ejected, or left behind in the emerging inner Cybele region. Initial conditions. Initially, we distributed 5, 000 test par- Finally at 10 Myr, a population between the left-hand ticles uniformly over the region which is about to be covered separatrix and the libration centre is stabilised, as the result by the moving 2:1 resonance. This can be estimated easily by Kepler’s third law since we know the evolution of Jupiter’s semimajor axis a priori. Our choice of the semimajor axes 3 The resonant elements of the non-resonant asteroids do not distribution was a (3.06, 3.32) AU. The distribution of ec- ∈ have any special physical meaning, but they correspond to the centricities and inclinations was also chosen as uniform and extremal secular variations of osculating elements as given by corresponds to the extent of a moderately excited main belt: equation (4).

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 9

0.5 t = 2 Myr migrating 0.4 outer main belt 2:1 centre

0.3

0.2 r e 0.1

0.0 0.5 t = 6.26 Myr t = 10 Myr Jump! 0.4

resonant eccentricity Cybele region 0.3

0.2

0.1

0.0 3.1 3.2 3.3 3.4 3.1 3.2 3.3 3.4

resonant semimajor axis ar [AU]

Figure 8. A simulation of the resonant capture from the outer main belt in the fifth giant planet scenario (Nesvorn´y& Morbidelli 2012). We plot the evolution of test particles in the resonant semimajor axis ar vs resonant eccentricity er plane. The vertical line indicates an approximate location of the 2:1 resonance centre which migrates inwards together with Jupiter. The integration time is shown in three of panels (t = 0, 2, 6.26 and 10 Myr) and corresponds to the timeline in Fig. 7. The dashed arrows and lines indicate an approximate extent of the libration zone in case of the migrating and stable resonance, respectively. From top to bottom and left to right, the plots show: the initial conditions, the relaxed population of test particles prior to jump, the state during the jumping-Jupiter instability and the final state. Note that all test particles are depicted in terms of resonant elements for simplicity (even the non-resonant orbits). of the resonant capture. The bodies outside the left-hand present in the depicted part of the phase space, which ef- separatrix are simply remnants of the initial population of fectively enlarges the A island with respect to the present test particles. A few bodies between the libration centre and state. The selected candidates for long-lived orbits are also the right-hand separatrix are in fact outside the 2:1 reso- shown in Fig. 9. Note that there are very few A-island bodies nance, as well as the asteroids in the Cybele region. captured on highly eccentric orbits, therefore the aforemen- tioned difference in shape of island A should not strongly affect the results. Using the dynamical mapping, we identi- Captured population. Instead of another long-term inte- synth synth fied NA = 69 candidates in island A and NB = 254 gration, we use the rather efficient dynamical mapping intro- candidates in island B. duced in Section 2.2 to check the stability of the captured population in the post-migration configuration of planets. The next step is a rescaling of the initial population so This allows us to identify the counterparts of the stable is- its particle density would reach realistic values. We used lands A and B in the phase space for this planetary configu- the observed particle density in the main belt for bod- ies with diameters D > 5 km within the following region: ration. Hence, we can trace the long-lived candidates of our ◦ ◦ a (2.95, 3.21) AU, e (0, 0.35), I (0 , 15 ). The intervals captured bodies simply by selecting particles with resonant ∈ ∈ ∈ elements falling within the range of the islands. of e and I correspond to those of our synthetic initial pop- ulation. On the other hand, the interval of semimajor axes The dynamical map for the 2:1 resonance is shown in the Fig. 9. It covers the phase space in the intervals a was shifted into the region which is currently not depleted ◦ ◦ ∈ (3.115, 3.195) AU, e (0.1, 0.5) and I (0 , 25 ), which by the presence of the J2/1. We further increased this par- ∈ ∈ ticle density by a factor of three (Minton & Malhotra 2010) were divided into a grid of 40 40 5 boxes (and we finally × × took an average over all sections in the inclination). We set to account for the dynamical depletion of the main belt after the reconfiguration of planets during the last 3.85 Gyr. If up and followed three test particles per box for up to 10 Myr. ≃ we increase the particle density throughout the initial popu- Comparing the map with Fig. 1, one can see very similar lation, the number of captured long-lived bodies larger than structures. The stable islands A and B are there, separated scaled scaled 5 km would be NA = 1552 and NB = 5857 in island by the ν16 secular resonance. The Kozai resonance separa- A and B, respectively. trix can also be easily identified. A major difference is the shape and size of the A island – the ν5 resonance is not Finally, we assume that the dynamical depletion rate

c 2015 RAS, MNRAS 000, 1–20 10 O. Chrenko et al.

[!h] τA = (0.57 0.05) Gyr and τB = (0.94 0.05) Gyr. We ± ± 0.5 1 use t = (3.9 0.1) Gyr as a value of the decay time pe- ± riod. Combining the results together and identifying the limit values, our model predicts that we should observe

r 0.8

e model 0.4 NA = 1–3 long-lived bodies larger than 5 km in island model A and NB = 62–121 asteroids in island B. The observed 0.6 d obs obs values are NA = 2and NB = 71. The results of our model 0.3 and the observed values are in good agreement, which sup-

0.4 distance ports the resonant capture scenario. Turning our attention to the bottom panel in Fig. 9, we 0.2 resonant eccentricity 0.2 can discuss whether the inclination distribution of captured synthetic bodies can evolve towards the observed one. It can be clearly seen that the number of low-I asteroids in island 0.1 0 3.12 3.14 3.16 3.18 A is lower than the number of high-I asteroids. Assuming strong depletion of the whole island and considering that resonant semimajor axis ar [AU] only 1–3 objects larger than 5 km may survive up to the present according to our model, the low-I asteroids are more 0.8 likely to be depleted completely. That is in agreement with B island A island the observations because the observed asteroids in island A

r high-I I reside exclusively on highly inclined orbits. 0.6 On the other hand, the population captured in island B borderline

approximate stretches almost uniformly over a relatively large interval of 0.4 inclinations. Considering the shape of the real island shown in Fig. 1, which shrinks with increasing inclination, the long- term diffusion in the high-I region will partially deplete this 0.2 part of the captured population. But in order to reproduce resonant inclination sin the observed state in island B, it is necessary for the source low-I population to contain low-I asteroids predominantly. 0 3.12 3.14 3.16 3.18

resonant semimajor axis ar [AU] 3.3 Survival of the primordial population Let us study hypothetical long-lived primordial orbits and Figure 9. A final result of resonant capture from the outer main their survival during the planetary migration. The proce- belt in the fifth giant planet scenario. Top: the resonant semi- dure is similar to that in the previous section, the major major axis ar vs the resonant eccentricity er. The background of the plot is the dynamical map of the 2:1 mean-motion resonance difference is of course in the initial conditions setup. Here with Jupiter computed for the post-migration configuration of the we aim to study only the bodies which reside inside the J2/1 giant planets. All white symbols indicate orbits of test particles when the instability simulation begins. Another important captured during the simulation (they correspond to the final state property that should be satisfied is the long-term stability in Fig. 8, but only a part of the phase space containing the stable of the initial orbits – otherwise one could unintentionally islands is plotted). Test particles embedded in the dark regions simulate different effects such as survival of the short-lived of the dynamical map are candidates for a long-term stability population, etc. and we distinguish them by circles. Other particles are marked by crosses. Bottom: the resonant semimajor axis a vs sine of the r Initial conditions. How to create a set of test particles resonant inclination sin Ir. We plot only the particles captured inside the stable islands. Black circles indicate test particles cap- on the long-lived orbits at the beginning of the migration tured in island B and gray open squares indicate those captured scenario? The only viable solution is to use the dynamical in A island. The horizontal dashed line is plotted for reference, mapping once again. To this effect, we slightly modify the because vast majority of the observed B-island Zhongguos reside standard procedure described in Section 2.2. First, the in- on low inclinations sin Ir 6 0.1 and all of the observed A-island tegration is not done in a stable unvarying configuration of Zhongguos have high inclinations sin Ir > 0.1. planets. Is is instead carried out during the first 5 Myr of the prescribed migration scenario, starting in the pre-migration configuration of planets and evolving into a configuration due to chaotic diffusion is the same for the post-migration closely preceding the instability. islands as for the islands in the observed configuration. Second, we use only the recorded changes of the res- Knowing the approximate number of captured long-lived as- onant inclination δ sin Ir when constructing the map (i.e. teroids, we employ an exponential decay law and the ex- we omit δar and δer when evaluating the metric given by 4 tremal values of the e-folding times in island A and B equation (5)). This is necessary because as Jupiter migrates

4 These values were derived in Section 2.5 but we use a slightly oids of all sizes. In this section, however, we focus on asteroids higher standard deviation (0.05 instead of 0.02). The reason is larger than 5 km and consequently, the e-folding times are rather that the e-folding times were obtained from a model with seven upper limits for us. We thus artificially increase the original de- planets and the Yarkovsky drift included, with long-lived aster- viation to account for this difference.

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 11

0.6 model model Ninit(D > 5 km) NA NB 0.4 0.5 2000 0 0 – 1 r

e 5000 0 1 – 2 I 0.4 0.3 10000 0 2 – 4 sin

δ 100000 1 – 3 18 – 35 = 0.3 d 0.2 model model Table 2. The numbers NA and NB of the asteroids sur- 0.2 viving in the islands A and B up to the present as predicted distance by our model of the primordial population survival. The table resonant eccentricity 0.1 represents how the resulting population changes with increas- 0.1 ing number Ninit of initial primordial asteroids. The bodies with D > 5 km are considered. First line roughly corresponds to the 0 0 3.34 3.36 3.38 3.4 3.42 3.44 primordial population with particle density of the present outer main belt, the next two cases approximately consider the particle resonant semimajor axis ar [AU] density proposed in Minton & Malhotra (2010). In the last line, we assume particle density of primordial main belt shortly after Figure 10. A dynamical map of the 2:1 mean-motion resonance its creation, as suggested by Morbidelli et al. (2009); we note that with Jupiter computed during the pre-instability evolution of gi- this case is not very probable. ant planets. The map is displayed in the resonant (ar,er) plane ◦ corresponding to Ir = 2.5 . Note that only the displacement in the resonant inclination δIr is used to represent the dynamical tially destabilizes the islands and enables significant number stability. of bodies to leak out and also to populate other regions of the 2:1 resonance. At the end of migration, 85 per cent of inwards, the resonance follows and the resonant semima- the initial population is lost and the rest is dispersed all over jor axis ar of bodies residing inside decreases. This in turn the resonant region, except for the low eccentricity region. changes the resonant eccentricity er because of the adiabatic This indicates that the only possible change in resonant ec- invariance of resonant orbits. These changes are systematic centricity during the migration of Jupiter is to increase. and have different amplitudes for different orbits. Conse- quently, they should not be incorporated when calculating Surviving population. Following our procedure from Sec- the distance d. On the other hand, our numerical runs sug- tion 3.2, we used the same dynamical map to identify the gest that the resonant inclination Ir of test particles does long-lived asteroids (see Fig. 12) and we applied the rescal- not undergo substantial systematic changes under the influ- ing and the long-term dynamical decay. Because the parti- ence of migrating giant planets. If a large change in inclina- cle density in a hypothetical primordial population is not tion is registered, it usually means that a secular resonance known, we varied the number of initial synthetic long-lived affected the orbit. The resulting dynamical map thus rep- particles and calculated the expected number of asteroids in resents what we need – the location of regions crossed by islands A and B surviving up to the present. secular resonances and the stable islands lying in between The selected results are listed in Table 2 and the com- for the time period before Jupiter’s jump. parison is made for asteroids with D > 5 km (we remind the obs obs Our choice of boundaries in the phase space to construct reader that there are NA = 2 and NB = 71 larger than the dynamical map was a (3.32, 3.45) AU and e (0, 0.6). 5 km in the observed population). If we assume the parti- ∈ ∈ We created 40 40 grid in the (a,e) plane and covered it cle density from Minton & Malhotra (2010), which we also × uniformly with six particles per each cell, assigning a fixed used in Section 3.2, the contribution of the primordial aster- ◦ value of I = 2.5 to all of them. As already mentioned, oids to the observed population is negligible after long-term the planetary migration is not able to strongly change the dynamical evolution. inclinations, that is why we map the region of low inclina- In order to observe a substantial contribution, the tions where the observed long-lived population dominates. particle density in the J2/1 would have to be consid- The resulting pre-instability dynamical map is displayed in erably larger than in the neighbouring main belt (at Fig. 10. least ten times larger). Such particle density gradient We randomly distributed a group of 2, 000 test parti- is not very probable, because the 2:1 resonance would cles over the identified stable islands (see Fig. 11) with low have to be dynamically protected against depleting mech- ◦ inclinations I < 5 to set up a synthetic primordial popu- anisms and perturbations, arising e.g. from planetary lation for our simulation. The angular osculating elements embryos (O’Brien, Morbidelli & Bottke 2007) or due to were chosen in such a way that condition (3) holds and the the compact primordial configuration of planetary orbits osculating elements are therefore identical to the resonant (Masset & Snellgrove 2001; Roig & Nesvorn´y2014). elements at t = 0. We also investigated if A-island orbits can become B- island and vice versa. We found that < 1 per cent of the Evolution of test particles. 2 Myr after the onset of the asteroids initially placed inside island A survive there and migration, the orbits are not significantly dispersed which 1 per cent drift into B island during the migration. On the is in agreement with our aim to study only the dynamically other hand, the primordial population of island B preserves stable resonant bodies. After 6.1 Myr, the close encounters 2 per cent of the original bodies and 2 per cent of them of giant planets begin to occur and start to perturb the populate island A. Because the primordial B island is larger islands inside the J2/1. Progression of Jupiter’s jump par- than the A island, it harbours more primordial asteroids

c 2015 RAS, MNRAS 000, 1–20 12 O. Chrenko et al.

0.5 migrating 0.4 2:1 centre

0.3 primordial islands

0.2 r e

0.1 t = 2 Myr t = 6.1 Myr

0.5 Jump!

0.4 resonant eccentricity

0.3

0.2

0.1 t = 6.2 Myr t = 10 Myr

3.1 3.2 3.3 3.4 3.1 3.2 3.3 3.4

resonant semimajor axis ar [AU]

Figure 11. A simulation of the primordial population survival in the fifth giant planet scenario (Nesvorn´y& Morbidelli 2012). The evolution of primordial long-lived orbits in the (ar,er) plane is displayed. The vertical line indicates the migrating 2:1 resonance. The initial population was placed inside the islands identified in Fig. 10, top left panel represents its orbital distribution 2 Myr after the onset of the migration. The following panels show the situation during Jupiter’s jump and also the final state. under the assumption of homogeneous particle density. The describe the long-lived resonant population of test particles contribution of island B to the surviving long-lived popula- because we employed fast and efficient method of dynamical tion therefore dominates. Also note that after the migration mapping. The natural question then arises, whether the late the ratio NA/NB 1, thus the migration itself is not able to migration stages can invoke perturbations that would affect ≃ create an asymmetric population out of surviving primordial the overall stability of the 2:1 resonance significantly. asteroids. Further orbital evolution is needed in this case. In particular, we want to check whether the period The orbital distribution in the (ar,Ir) plane at the end Pσ of libration of the resonant asteroids can be compa- of the planetary migration indeed resembles the initial in- rable with the Jupiter–Saturn great inequality period PGI terval of inclinations, as both islands are populated by low-I (which is the period of circulation of the 2λJ 5λS angle) − orbits. To obtain high-I orbits, which are observed in island and what effect would it have on the resonant population A, we would have to assume the presence of their analogues (Ferraz-Mello et al. 1998, see also Section 1). in the pre-migration islands. But this problem is redundant In order to mimic the smooth late planetary migration since the primordial population probably do not survive up and lead giant planets towards their current orbits, we used to the present, as we argued above. a modified version of the swift rmvs3 integrator developed in Broˇzet al. (2011). The integrator introduces an ad hoc dissipation term which modifies planetary velocity vectors v in each time step ∆t according to the following relation: 4 EFFECTS OF JUPITER–SATURN GREAT ∆v ∆t t t0 INEQUALITY v (t +∆t)= v (t) 1+ exp − , (7) v τmig − τmig In Section 3, we investigated evolution of the resonant pop-    ulation during a major instability of the planetary system where ∆v = GM/ainit GM/afin is the total dissipation − known as Jupiter’s jump. In the original migration experi- determined as the difference of the initial and final mean ve- p p ments performed in Nesvorn´y& Morbidelli (2012), the vio- locity, τmig is the migration time-scale, t is the time variable lent evolution of planetary orbits was usually followed by a and t0 is an arbitrary initial time. The eccentricity damp- residual smooth migration during which giant planets slowly ing is also included and can be set independently for each approached their current orbits. We are in a similar situa- planet by choosing the damping parameter denoted as edamp tion, the planetary configuration at the end of our insta- (Morbidelli et al. 2010). bility simulations slightly differs from the observed one. In To set up the model, we used the final configuration principle, this fact does not affect our ability to locate and of giant planets and test particles from our simulations of

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 13

0.5 1 680 r 0.8 N e 0.4 620

0.6 d 560 0.3 500

0.4 distance

0.2 number of asteroids 440 resonant eccentricity 0.2 30 Myr time-scale 60 Myr time-scale 380 0.1 0 3.12 3.14 3.16 3.18 800 representative Pσ evolution resonant semimajor axis ar [AU]

[yr] 600 σ P ,

GI 400 P 0.6 B island A island r I high-I 200 PGI evolution 0 10 20 30 40 50 borderline

0.4 approximate time t [Myr]

Figure 13. Results of two simulations of smooth late migra- tion and its effects on the resonant population. Two runs with 0.2 tfin ≃ 27 Myr (red and black curves and arrows) and tfin ≃ 58 Myr resonant inclination sin (blue and gray curves and arrows) are presented (migration time- scales τ = 30 Myr and τ = 60 Myr were initially chosen low-I mig mig but the integration is evaluated only until the GI period reaches 0 3.12 3.14 3.16 3.18 880 yr). Top: temporal evolution of the number N of test parti- cles. Bottom: temporal evolution of the GI period P and the resonant semimajor axis ar [AU] GI libration period Pσ. The latter is plotted for two representative cases. The arrows in the top panel approximately mark the time Figure 12. Top: The same post-migration dynamical map as period during which PGI ≃ Pσ. We note that ‘real’ PGI evolves in Fig. 9. The primordial test particles surviving the jumping- in an oscillatory manner but only with a small amplitude; a poly- Jupiter instability are plotted over the map. Circles represent nomial fit of the real PGI evolution is plotted here for clarity. candidates for long-lived orbits and crosses are the surrounding orbits. Bottom: the resonant semimajor axis ar vs sine of the resonant inclination sin Ir. We plot only the particles identified cal resonant asteroids. While Pσ oscillates around a nearly above as long-lived. Black circles indicate those residing in the constant value 420 yr, PGI initially lies below this value island B and gray open squares indicate those residing in the A ≃ island. and rises smoothly as Jupiter and Saturn undergo divergent migration, and their eccentricities are damped. The evolution of the N(t) curves does not follow any resonant capture5, but we only selected test particles lo- simple decay law. This may serve as a proof that the level of cated in broader surroundings of the stable islands to dis- chaotic diffusion inside the islands indeed evolves during the card major part of short-lived asteroids. After this proce- smooth late migration. The steepest slope of the N(t) depen- dure, we launched a set of integrations with different val- dency is reached during the time interval when PGI Pσ. ≃ ues of edamp and different migration time-scales, bearing But one can also notice that the slope changes are smooth in mind that the time-scale of the original experiments in rather than sharp; the destabilization is not strictly bounded Nesvorn´y& Morbidelli (2012) was τmig 30 Myr. When the by the PGI Pσ condition. This is in agreement with the ≃ ≃ integrations finished, we selected only the runs in which gi- results of Ferraz-Mello et al. (1998) who argued that even a ant planets ended up with orbital parameters similar to the near-resonant state of PGI and Pσ can enhance the chaotic observed ones and we investigated the results at the time diffusion. However, it is clear that the number of asteroids tfin when PGI = 880 yr and the ratio of the Saturn’s and surviving the smooth late migration depends mainly on the Jupiter’s orbital periods is approximately PS/PJ 2.49. length of the time interval during which PGI Pσ; the pop- ≃ ≃ In the following, we will discuss results of two runs with ulation is more depleted in the run with tfin 58 Myr. ≃ tfin 27 Myr and tfin 58 Myr. Fig. 13 shows the temporal With the aim to evaluate the result of the models with ≃ ≃ evolution of the number N of test particles in these two runs, residual migration included, we first identified test particles the GI period PGI and the libration period Pσ of two typi- which were located inside the stable islands at the time tfin. At this time, the orbital architecture of giant planets ap- proximately corresponds to the observed one and therefore 5 The results of simulations with primordial asteroids are not the identification of long-lived asteroids can be achieved us- considered here, as these asteroids probably do not significantly ing the current dynamical map. Finally, we derived expected model model contribute to the observed population. numbers NA and NB of asteroids that should be ob-

c 2015 RAS, MNRAS 000, 1–20 14 O. Chrenko et al. servable in the stable islands after 4 Gyr of orbital evolution. colliding Pi Vi − For this purpose, we rescaled the initial population of test populations [10−18 km 2yr−1] [kms−1] particles and we applied simple exponential dynamical decay law, exactly as in Section 3.2. Zhongguos vs MB 3.82 5.36 Griquas (I 6 8◦) vs MB 3.65 5.52 We calculated following values for asteroids larger ◦ model model Griquas (I > 8 ) vs MB 1.81 7.57 than 5 km: NA = 0–1 and NB = 40–78 in case of model model tfin 27 Myr and NA = 0–1 and NB = 22–42 in case average 3.09 6.15 ≃ obs of tfin 58 Myr (while the observed values are NA = 2 and obs ≃ NB = 71). We can conclude that the hypothesis of resonant Table 3. The intrinsic probability Pi and the mean impact ve- 6 capture is still valid and explains the existence of the long- locity Vi for collisions of main-belt and long-lived J2/1 asteroids. lived population, if the giant planets finished their late stage migration on a time-scale comparable to τmig 30 Myr. ≃ Larger time-scales lead to a slower passage of the GI period 15 / km/s over the values of the J2/1 libration period and this in turn ±1 10 yr causes an enhanced depletion of the stable islands. Even in imp V ±2 5 such a case, a significant part of the observed population 0.006 km V imp 0 mean should originate from the resonant capture. ±18 0.004 Pi / 10 i

P 0.002

5 COLLISIONAL MODELS mean 0 0 20 40 60 80 100 Our results of Section 3 and 4 imply that the long-lived t / Myr J2/1 population was created by resonant capture during Figure 14. Temporal evolution of the intrinsic probability Pi(t) planetary migration. In this section, we further develop the (bottom curve) and the mean impact velocity Vi(t) (top curve) framework of this hypothesis. We study collisional evolution for collisions between transneptunian comets and main-belt as- of the J2/1 in order to confirm whether the size-frequency teroids, as it was calculated in Broˇzet al. (2013). We emphasize distribution of the long-lived resonant asteroids can survive that we modify these dependences by numerical factors to allow a time period of 4 Gyr in a non-stationary state. We thus for realistic lifetimes of comets. The values Pi/3 and Vi/1.5 serve have to also account for the epoch of the late heavy bom- as an input for our models. bardment (LHB) (Gomes et al. 2005; Levison et al. 2009) during which the transneptunian disc was destabilized and enough, we input all long-lived orbits and the first 50, 000 the flux of cometary projectiles through the solar system main-belt objects from the AstOrb catalogue. increased substantially. We split the long-lived population into Zhongguos, Gri- In the following collisional models, three populations ◦ quas with inclination I 6 8 and Griquas with inclina- of minor solar-system bodies are included: main-belt aster- ◦ tion I > 8 and calculate the intrinsic probability Pi and oids, transneptunian comets and Zhongguos, i.e. we consider weighted mean impact velocity Vi for them, separately. Us- only a subset of the long-lived population. The reason for ing this separation, we want to check if the intrinsic proba- this simplification is that the observed SFDs of long-lived bilities for Griquas may differ from Zhongguos significantly. population and its dynamical subgroups share similar fea- The results are summarized in the Table 3. For refer- tures (see Fig. 4), but Zhongguos have the steepest slope −18 −2 −1 ence, Dahlgren (1998) computed Pi = 3.1 10 km yr γ = 5.1. Solving an inverse problem, we aim to explain −1 × − and Vi = 5.28 km s for collisions between main-belt as- the formation of an initial SFD with an even steeper slope, teroids. We can see that both Zhongguos and low-inclined which would evolve towards the observed one due to colli- Griquas have Pi and Vi only slightly higher than these refer- sions over the 4 Gyr timespan. Because of certain level of ence values. This is caused by the moderate values of eccen- parametric freedom in collisional models, the same process tricities in the J2/1 population. The orbits then more likely of formation should also apply to Griquas with shallower intersect with those in the main belt. The higher collisional observed SFD. velocity is also plausible because we are comparing an outer main-belt population with the rest of the main belt. 5.1 Intrinsic probabilities and impact velocities On the other hand, Griquas with high inclinations can avoid intersecting some of main-belt orbits, thus their in- As the first step in the construction of a collisional model, we trinsic probability is lower by about a factor of two. Even compute the intrinsic probability Pi and the mean impact at this point, we can conclude that the observed difference velocity Vi of colliding main-belt and long-lived resonant as- in slopes of SFDs (when Griquas and Zhongguos are sepa- teroids. We adopt the method of Bottke et al. (1994) based rated) cannot be explained by the differences in Pi and Vi on a geometrical formalism of orbital encounters introduced because the shallower SFD of Griquas would require higher by Greenberg (1982). To make our samples of orbits large Pi in order to collide more often.

6 Although the values derived here do not overlap the observa- tions in case of island A, the difference is only one asteroid. We think that this discrepancy is not significant as we are comparing small numbers and the evolution is definitely stochastic.

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 15

109 5.2 Capture from the main-belt population t = 10 Myr 108 Let us ask a question whether the J2/1 SFD can originate 7 from the main-belt SFD. We assume that the capture of ) 10 D the resonant population is not size dependent and thus the (> 6 N 10 captured SFD resembles that of the main belt at the time 105 of planetary migration, scaled down by a numerical factor. Our arbitrary choice of this factor is such that the number 104 of the largest captured bodies is approximately the same as 3 10 initial comets we observe in the J2/1 population. This allows us to imme- evolved comets

number of asteroids 100 initial MB diately compare the slope of captured and observed SFD of evolved MB Zhongguos. 10 observed MB synthetic Zhongguos The main belt itself was likely more populous 4 Gyr observed Zhongguos 1 ago (by a factor of three according to Minton & Malhotra 0.1 1 10 100 1000 (2010)) but has remained in a near-stationary collisional diameter D [km] regime ever since (Bottke et al. 2005a). The only possible 109 stage of evolution, which could have temporarily increased t = 4 Gyr the slope of the main belt SFD, was the late heavy bombard- 108 ment. We therefore investigate this early period of collisions 7 ) 10

D between the main belt and transneptunian comets. We fo-

(> 6 N 10 cus on diameters D < 25 km to see whether the steepness

5 in this interval can be instantaneously increased – if so, a 10 population captured from such SFD would adopt that slope. 104 We use the Boulder code (Morbidelli et al. 2009) to 3 construct a suitable collisional model. We setup initial SFDs 10 initial comets evolved MB by a piecewise power law function which is characterized by

number of asteroids 100 initial MB observed MB three differential slope indices qa, qb and qc in the intervals 10 synthetic Zhongguos of diameters D > D1, (D2, D1) and D < D2, respectively. initial Zhongguos observed Zhongguos The SFD is normalized by setting the number Nnorm of bod- 1 0.1 1 10 100 1000 ies larger than D1. The following values are used in case of diameter D [km] the main-belt SFD: D1 = 100 km, D2 = 14 km, qa = 5.0, − qb = 2.3, qc = 3.5, Nnorm = 1110. A few D 1000 km as- 109 − − ∼ t = 4 Gyr teroids are added in order to properly reproduce the present 108 state. Using the same assumptions as Broˇzet al. (2013), the 7 ) 10 cometary disc is characterized as: D1 = 100 km, qa = 5.0,

D 7 −

(> qb = qc = 3.0, Nnorm = 5 10 . 6 N − × 10 Because the evolution of the transneptunian disc is 105 dominated by its fast dynamical dispersion, the intrinsic probability Pi(t) and mean impact velocity Vi(t) of collisions 104 with comets are time-dependent quantities. The temporal 3 10 initial comets evolution of Pi(t) and Vi(t), which was derived in Broˇzet al. evolved MB

number of asteroids 100 initial MB (2013), is shown in Fig. 14. Broˇzet al. (2013) also argued observed MB that it is necessary to modify this dependence in order to 10 synthetic Zhongguos initial Zhongguos mimic the effects of spontaneous cometary breakups due to observed Zhongguos 1 various physical processes. Regarding these effects, they de- 0.1 1 10 100 1000 rived quantities P˜i(t) = Pi(t)/3 and V˜i(t) = Vi(t)/1.5 as a diameter D [km] feasible modification. ⋆ Figure 15. Results of collisional models created with the Boul- The specific energy QD required to disperse 50 per cent der code, plotted as temporal evolution of the SFDs. Description of the shattered target is described by the polynomial scaling of individual curves is given in the legend of each plot. The evolv- law (Benz & Asphaug 1999) ing curves are displayed in the time of the simulation t, which is ⋆ 1 a b also shown. Top: synthetic Zhongguos are created by capture from QD (r)= Q0r + Bρr , (8) qfact the main-belt background population affected by collisions with   transneptunian comets (see also Section 5.2). Middle: the situa- where r denotes the target’s radius in cm. Material param- tion from top left panel is reproduced, but a single D ≃ 100 km eters ρ, Q0, a, B, b and qfact for basaltic rock (used for asteroid is added to Zhongguos and we study the possibility of its asteroids) and water ice (used for comets) are listed in Ta- catastrophic breakup over 4 Gyr (see also Section 5.3). Bottom: ble 4. we demonstrate what parameters of the initial SFD of Zhongguos We simulated the collisional evolution for a time period are needed in order to evolve it towards the observed one due to of 10 Myr in which Pi (t) reaches its maximum (see Fig. 14 collisions (see Section 5.4 for discussion). and top panel of Fig. 15). The flux of cometary projectiles in- duced by ongoing planetary migration reaches its climax and strongly affects the main-belt SFD. We performed 100 nu- merical realisations of the model with different seeds of the

c 2015 RAS, MNRAS 000, 1–20 16 O. Chrenko et al.

ρ Q0 a B bqfact DPB ddisrupt Nproject Ncol [gcm−3] [erg g−1] [erg g−1] [km] [km]

Basalt: 50 6 25821 0.18 3.0 7 × 107 −0.45 2.1 1.19 1.0 70 9 7837 0.10 Water ice: 100 15 2589 0.07 1.0 1.6 × 107 −0.39 1.2 1.26 3.0 120 19 1777 0.07

Table 4. Material parameters ρ, Q0, a, B, b and qfact of Table 5. The results of a stationary collisional model: the number the polynomial scaling law (see equation (8)) adopted from Ncol of catastrophic breakups of a parent body with the diameter Benz & Asphaug (1999). The first line is used in case of main- DPB over 4 Gyr timespan. We also list the minimal size ddisrupt belt or resonant asteroids and the second line is used for comets. and the number Nproject of suitable projectiles. built-in random-number generator to account for stochastic- are given in Table 5. It turns out that the probability of a ity of the evolution and possible low-probability breakups. catastrophic disruption is 6 18 per cent when considering a At the end of the simulation, we assume the resonant cap- −4 single target only. ture takes place with the efficiency factor 10 . The steep Note that a more realistic case should take into consid- part of the captured SFD can be characterized by the slope eration that there is no observable evidence of a collisional γ = 3.0. Although the region of the km-sized bodies is − cluster, thus the hypothetical breakup event must have oc- steeper with respect to the initial state of the main-belt curred more than 1 Gyr ago (Broˇzet al. 2005). Of course, SFD, it is not steep enough to reach the slope of the SFD of one should also account for the influence of cometary flux observed Zhongguos. We thus consider this scenario unlikely. during LHB which can temporarily increase the rate of col- lisions but also tends to speed up the evolution of SFDs. We therefore test the possibility of a catastrophic 5.3 Catastrophic disruption of a captured parent breakup once again in the more sophisticated framework of body the Boulder code and we also check the influence of comets. From the result of the previous section, it is obvious that The setup for main-belt asteroids and comets is the same as a different explanation of the steep initial SFD is needed. in Section 5.2, as well as the SFD of synthetic Zhongguos, Here we test the possibility that a large parent body was which is only modified by adding a single D 100 km aster- ≃ captured inside the 2:1 resonance by chance, it was subse- oid. The intrinsic probability and the mean impact velocity quently disrupted and its fragments formed a family. for MB vs. Zhongguos collisions are taken as the average We first use a stationary model to investigate the prob- of values from Table 3. We simulated 4 Gyr of collisional ability of such a catastrophic breakup event. As a prereq- evolution. uisite, we estimated a lower limit of the parent body size The middle panel of Fig. 15 shows a set of 100 realisa- DPB. To achieve that, we searched the outcomes of hy- tions of our model. At the beginning of the simulations, the drodynamic disruption models in Durda et al. (2007) and synthetic population of Zhongguos is strongly affected by the Benavidez et al. (2012) for SFDs which have approximately cometary flux, which mostly causes cratering of the large the same slope as the observed SFD of Zhongguos. We body in the resonance. Moreover, a family-forming event rescaled the sizes of synthetic asteroids in these datasets so takes place in a few runs. Simultaneously, the comets speed that the diameter of the largest remnant would correspond up the collisional evolution in and below the region of mid approximately to the size of (3789) Zhongguo. We selected sized asteroids. As a result, families inside the J2/1 are com- all reasonable fits and derived range of admissible values minutioned too fast and the steepness of their SFDs drops DPB (50, 120) km with the median value D˜ PB = 70 km. below the observed one. ∈ We then use the following relation expressing the num- When studying the family-forming events after the ber Ncol of catastrophic disruptions of parent bodies with LHB, we focused only on cases with the size of the largest the diameter DPB due to collisions with a population of fragment (or remnant) DLF > 10km to obtain a body simi- projectiles: lar to (3789) Zhongguo. In three such cases, which occurred 2 at the time of simulation t = 2.1, 2.8 and 3.1 Gyr, a parent DPB Ncol = PiNPBNproject ∆t , (9) body with the diameter DPB 85 km was shattered into a 4 ≃ family with the largest fragment having DLF 18km and ≃ where Pi denotes the intrinsic probability, NPB is the num- the largest remnant having DLR 60 km. Corresponding 7 ≃ ber of parent bodies, Nproject is the number of projectiles evolved SFDs match the observed one, except for the pres- capable of disrupting the parent body and ∆t is the consid- ence of the parent-body remnant. ered timespan. We can conclude that a family-forming event inside the Since there are no large bodies observed in the cur- 2:1 resonance is a very unlikely process because a complex rent J2/1 population, we assume NPB = 1. To provide the set of constraints has to be satisfied: A major collision would lower limit on the diameter of the projectiles ddisrupt, we use have to occur despite its low probability, which is typically the model of Bottke et al. (2005b) in combination with the scaling law given by equation (8) (Benz & Asphaug 1999). We then take Nproject for different values of ddisrupt as the 7 In fact, the SFDs resulting from our simulations lie slightly be- number of main-belt asteroids with diameters D > ddisrupt. low the observed case but this discrepancy can be easily removed Several resulting values of Ncol over ∆t = 4 Gyr timespan assuming a bit larger parent body.

c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 17

16 Then we discuss possible process which could have led to formation of such an SFD. 14 Using trial and error, we established the following ini-

[mag] tial SFD as an appropriate one: D1 = 10 km, D2 = 5km, H 12 qa = 6.6, qb = 6.6, qc = 3.5, Nnorm = 12. The change of − − − slope at D2 is necessary to reasonably satisfy the mass con- 10 servation law. The main-belt and cometary SFDs are the synthetic family same as in the previous models. 8 selected part

= 2.54 to 2.56 AU The result of 100 runs of the collisional code is displayed p a absolute magnitude in bottom panel of Fig. 15. The evolved SFD of the resonant 6 population corresponds to the observed one very well. The steep parts have the same slope, the only difference can be 2.52 2.56 2.6 2.64 2.68 proper semimajor axis a [AU] seen in the region of small bodies where the synthetic SFD p is abundant. But one has to realize that the observed SFD is probably biased in this region. 10000 The mechanism responsible for creation of the high ini- tial slope may be the following (and it is related to the effect ) D 1000 described in Carruba, Aljbaae & Souami (2014)). Let us as- (>

N sume resonant capture from a hypothetical outer main-belt synthetic family instead of capture from the background main-belt family 100 selected part population. The stable islands are able to capture only a ap = 2.54 to 2.56 AU fraction of this family because jump of the resonance is thought to be fast (it does not sweep through the resonance), 10 the capture efficiency is limited (see Section 3.2), and also

number of asteroids the width of the islands in semimajor axis is relatively small − 3.9 (see e.g. Fig. 1). If the captured part of the family is located −11.0 1 farther away from the original position of the parent body, 1 10 100 its SFD will be very steep because smaller fragments have diameter D [km] higher ejection velocities and therefore land at orbits that are more distant from the parent body. Figure 16. Dependence of the absolute magnitude H on the proper semi-major axis ap for a synthetic family (top panel). The We employed a simple Monte-Carlo test to investigate family is used in a simple Monte-Carlo test, in which we study the described possibility. We generated a uniform distribu- how the shape of the SFD changes if we select only a part of the tion of 10, 000 test particles satisfying the bounds 2.52 6 −4 family. An example, for the part of the family bordered by the a 6 2.70 AU and log a ac /2 10 /0.2 6 H 6 16 in vertical lines, is given in the bottom panel. We compare cumu- | − | × the (a,H) plane where ac 2.64 AU is value of the central ≃
lative size-frequency distributions of the entire synthetic family semimajor axis and H is the absolute magnitude. Finally, (dashed curve) and of its part (solid curve). The steep segments we assigned diameters D to the test particles on the basis are approximated with a power law function and obtained slope indices are shown for reference. of the H distribution, assuming single value of the geomet- ric albedo pV = 0.05 throughout our sample. This way we created a synthetic family (similar e.g. to Eunomia family). 10 per cent, as given by the stationary model, or 3 per cent, Then we randomly moved a ∆a = 0.02 AU window in the as derived from the more sophisticated model. The created semimajor axis over the collisional cluster and we monitored SFD would have to be very steep, preferably without pres- the SFDs in the successively selected regions. ence of a parent-body remnant. If the remnant was present, one would have to rely on its subsequent elimination due to An example is given in Fig. 16. Obviously, extremely dynamical depletion because there is no large asteroid ob- high slope can be reached by this process. The resulting served in the J2/1. Finally, the breakup would have to take slope is well above the estimated lower limit which is needed place after the LHB, but early enough to allow for dispersion for the initial SFD of Zhongguos. Note that this mechanism of the collisional cluster. does not require the family itself to have steep SFD. The steepness is achieved afterwards by selective resonant cap- ture from an appropriate region. 5.4 Capture from a family Finally, we remark that an interesting link can be found Previous collisional models imply that it is difficult to ex- between the hypothesis of capture from a family and our dy- plain the steep SFD of Zhongguos by standard processes namical simulations. In Section 3.2, we argued that a source (such as direct capture from the background main-belt pop- population for resonant capture should contain greater num- ulation or family-forming event) that could have occurred ber of asteroids with low inclination in order to explain the during resonant capture or later on. We thus employ a sort observed high concentration of low-I B-island bodies. Thus of ‘reversed’ method in this section. We first try to discover we can conclude that if the hypothetical outer main-belt what parameters of the initial SFD are needed in order to family was indeed captured, it probably must have been lo- obtain the observed one after 4 Gyr of collisional evolution. cated on low inclinations.

c 2015 RAS, MNRAS 000, 1–20 18 O. Chrenko et al.

6 CONCLUSIONS orbits. However, we argued that the primordial asteroids probably do not contribute to the observed population at Let us briefly review the content of this paper. We updated all. The reason is that the primordial population would have the population of asteroids residing in the 2:1 mean-motion to exhibit particle density larger than the one proposed by resonance with Jupiter using recent observational data. The Minton & Malhotra (2010) by a factor of ten, in order to new list of resonant objects now contains 370 bodies, which survive 4 Gyr of post-migration dynamical evolution. Such can be divided on the basis of their mean dynamical lifetime particle density would create a gradient with respect to the to 140 short-lived and 230 long-lived asteroids. Our revision neighbouring main belt which we assume dubious. of physical properties of the resonant population is generally Finally, by creating several collisional models, we in agreement with the conclusions of previous studies; new demonstrated that the observed steep SFD of the long-lived result is our estimate of the mean albedop ¯V = (0.08 0.03), ± asteroids cannot be explained by capture from the back- based on the data from the WISE database. ground main-belt SFD, affected by the late heavy bombard- The long-term dynamics of the two quasi-stable islands ment. We also proved that a family-forming catastrophic A and B was studied, using the results of Skoulidou et al. (in disruption inside the J2/1 is very unlikely. preparation) who took into account also the perturbations Our main conclusion is that the long-lived J2/1 popu- induced by the terrestrial planets and the semimajor axis lation was probably formed by capture from a hypothetical drift caused by the Yarkovsky effect. The population in the outer main-belt family during Jupiter’s jump. If this is the islands decays exponentially, but the escape rate is signifi- case, then the long-lived asteroids in the 2:1 resonance with cantly faster in island A, the e-folding times being 0.57 Gyr Jupiter represent the oldest identifiable remnants of a main- for island A and 0.94 Gyr for island B. Hence, the dynami- belt asteroidal family. cal evolution on Gyr-long time-scales results into differential There are several improvements that are needed to con- depletion of the islands, which is certainly one of the reasons clude the debate on the long-lived population in the 2:1 res- for the observed asymmetric A/B population ratio. onance and its origin. For example, it is appropriate to as- The primary goal of this paper was to explain the for- sess the possibility of the Themis family formation event as mation of the long-lived population, satisfying the observa- a contributor to the resonant population. Although we did tional constraints. We tested two hypotheses: not study this hypothesis in detail, we summarized several (i) capture from the outer main belt, results of our preliminary tests in Appendix A. Our simu- (ii) survival of primordial long-lived resonant asteroids, lations suggest that this possibility is not viable and this in turn supports the hypothesis of resonant capture. both in the framework of the five giant planets migration sce- Other improvements of our work might include a devel- nario (Nesvorn´y& Morbidelli 2012) with a jumping-Jupiter opment of a self-consistent model of both dynamical and col- instability. lisional evolution of long-lived asteroids. Finally, we suggest Our simulations of the instability imply that both pro- to study the hypothetical primordial population in the epoch cesses could have been at work. The capture itself, to- before the jumping-Jupiter instability in order to properly gether with subsequent differential depletion due to long- estimate primordial particle density inside the stable islands. term dynamical evolution, can explain the observed pop- ulation in both islands, if we assume that the number of asteroids in the main belt 4 Gyr ago was three times larger ACKNOWLEDGEMENTS than the current one, as suggested by Minton & Malhotra (2010). Namely, the numbers of asteroids with D > 5 km, The work of OC and MB has been supported by the Grant model which should survive to this day, are NA = 1–3 and Agency of the Czech Republic (grant no. 13-01308S) and by model NB = 62–121, as predicted by our model. For compari- Charles University in Prague (project GA UK no. 1062214; obs obs son, the observed numbers are NA = 2 and NB = 71. project SVV-260089). The work of DN was supported by We also modeled the late residual planetary migration NASA’s Solar System Workings program. We thank an to check on the effect of possible resonance between the li- anonymous reviewer for valuable comments, which improved bration period in the 2:1 commensurability and the raising the final version of the paper. period of Jupiter–Saturn great inequality. The impact of this effect on the long-lived population depends on the time-scale of the late migration. If the time-scale was 30 Myr (which ≃ REFERENCES corresponds with the scenarios in Nesvorn´y& Morbidelli (2012)), the destabilization of the islands would be weak Benavidez P. G., Durda D. D., Enke B. L., Bottke W. F., enough for the captured population to survive in a state Nesvorn´yD., Richardson D. C., Asphaug E., Merline similar to the observations. More specifically, if we account W. J., 2012, Icarus, 219, 57 for the effect of great inequality in our model, the resulting Benz W., Asphaug E., 1999, Icarus, 142, 5 model model values are NA = 0–1 and NB = 40–78. If the time- Bottke W. F., Durda D. 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c 2015 RAS, MNRAS 000, 1–20 Long-lived asteroids in the 2:1 resonance 19

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IAU Symp. 310, Long-term evolution of aster- Carruba V., Aljbaae S., Souami D., 2014, ApJ, 792, 46 oids in the 2:1 mean-motion resonance. Cambridge Univ. Dahlgren M., 1998, A&A, 336, 1056 Press, Cambridge, p. 178 DeMeo F. E., Carry B., 2013, Icarus, 226, 723 Standish E. M., 2004, A&A, 417, 1165 Dohnanyi J. S., 1969, JGR, 74, 2531 Tsiganis K., Kneˇzevi´cZ., Varvoglis H., 2007, Icarus, 186, Durda D. D., Bottke W. F., Nesvorn´yD., Enke B. L., Mer- 484 line W. J., Asphaug E., Richardson D. C., 2007, Icarus, Wisdom J., 1987, Icarus, 72, 241 186, 498 Zappal`aV., Bendjoya P., Cellino A., Farinella P., Froeschl´e Farinella P., Gonczi R., Froeschle C., 1993, Icarus, 101, 174 C., 1995, Icarus, 116, 291 Ferraz-Mello S., Michtchenko T. A., Roig F., 1998, AJ, 116, 1491 Franklin F. A., 1994, AJ, 107, 1890 APPENDIX A: CONTRIBUTION OF THE Gomes R., Levison H. 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F., Gounelle M., Morbidelli A., may fall directly in the J2/1 resonance and thus contribute Nesvorn´yD., Tsiganis K., 2009, Nature, 460, 364 somehow to the stable population. There are several caveats, Levison H. F., Duncan M. J., 1994, Icarus, 108, 18 however. Masiero J. R., et al., 2011, ApJ, 741, 68 (i) One has to assume that a substantial part of Masset F., Snellgrove M., 2001, MNRAS, 320, L55 fragments ( 10 per cent) have very large ejection ve- ≃ Michtchenko T., Ferraz-Mello S., 1997, P&SS, 45, 1587 locities with respect to (24) Themis, or the respective Minton D. A., Malhotra R., 2010, Icarus, 207, 744 parent body with DPB 400 km. We tried to use ≃ Moons M., Morbidelli A., Migliorini F., 1998, Icarus, 135, an ‘extreme’ size-independent velocity field prescribed by 458 Farinella, Gonczi & Froeschle (1993) relation, with param- −1 Morbidelli A., 1997, Icarus, 127, 1 eters vesc = 170 m s and α = 3.25. Then up to N(D > Morbidelli A., Bottke W. F., Nesvorn´yD., Levison H. F., 5 km) = 10–30 fragments land within island B, according 2009, Icarus, 204, 558 to our tests. This number should be further decreased by a Morbidelli A., Brasser R., Gomes R., Levison H. F., Tsiga- subsequent long-term orbital evolution, as Themis family is nis K., 2010, AJ, 140, 1391 (2.5 1.0) Gyr old (Broˇzet al. 2013). Let us also note that ± Morbidelli A., Crida A., 2007, Icarus, 191, 158 the collisional evolution of resonant bodies was accounted for Morbidelli A., Moons M., 1993, Icarus, 102, 316 automatically, as we did this simulation with the currently Murray C. D., 1986, Icarus, 65, 70 observed size-frequency distribution of Themis family. Nesvorn´yD., Ferraz-Mello S., 1997, A&A, 320, 672 (ii) At the same time, it is required that the breakup ◦ Nesvorn´yD., Morbidelli A., 2012, AJ, 144, 117 takes place when the true anomaly fimp 0 for this ejection ≃ Nesvorn´yD., Vokrouhlick´yD., Morbidelli A., 2013, ApJ, scenario to work; otherwise, the number of objects landing 768, 45 in the islands decreases as well. But this particular impact O’Brien D. P., Morbidelli A., Bottke W. F., 2007, Icarus, geometry does not seem to be compatible with the observed 191, 434 shape of Themis family, namely with a large eccentricity Parker A., Ivezi´c Z.,ˇ Juri´cM., Lupton R., Sekora M. D., dispersion of the family below the J11/5 resonance, at ap = Kowalski A., 2008, Icarus, 198, 138 3.03 AU (see Fig. A1). Press W. R., Teukolsky S. A., Vetterling W., Flannery (iii) Starting with a more reasonable size-dependent ve- B. P., 2007, Numerical Recipes: The Art of Scientific Com- locity field – as used in Broˇz& Morbidelli (2013) for Eos puting. Cambridge Univ. Press, Cambridge family, which has a similar parent body size – makes the Rabe E., 1959, AJ, 64, 53 contribution to the long-lived population as low as N(D > Roig F., Ferraz-Mello S., 1999, P&SS, 47, 653 5 km) = 2.

c 2015 RAS, MNRAS 000, 1–20 ν 5

20 O. Chrenko et al.

0.3

t = 100 My J2/1

0.25 r e

ν 16 0.2

0.15 resonant eccentricity

0.1 observed Themis family synthetic family 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35

resonant semimajor axis ar [AU]

Figure A1. A simulation of a synthetic Themis-like family for- mation event. The figure represents the state 100 Myr after the breakup. The orbital distribution of observed Themis family is displayed in the proper elements (black squares), while the orbital distribution of the synthetic family is displayed in the resonant el- ements (blue circles). This causes the mutual shift in eccentricity. Note the differences in extent of both families and the eccentricity dispersion beyond the 11:5 mean-motion resonance with Jupiter.

(iv) There is no way to explain the existence of aster- oids in island A by the ejection, both the eccentricities and inclinations of the observed A-island objects are too large. Therefore, the ejection from Themis is not a viable hypoth- esis in case of island A.

This paper has been typeset from a TEX/ LATEX file prepared by the author.

c 2015 RAS, MNRAS 000, 1–20