Mon. Not. R. Astron. Soc. 000, 1–14 (2008) Printed 28 April 2008 (MN LATEX style file v2.2)
Asteroids families in the first order resonances with Jupiter
M. Broˇz1⋆, D. Vokrouhlick´y1, ... 1Institute of Astronomy, Charles University, Prague, V Holeˇsoviˇck´ach 2, 18000 Prague 8, Czech Republic 2Department of Space Studies, Southwest Research Institute, 1050 Walnut St., Suite 300, Boulder, CO 80302, USA
Accepted ???. Received ???; in original form ???
ABSTRACT Asteroids residing in the first-order mean motion resonances with Jupiter hold impor- tant information about the processes that set the final architecture of giant planets. Here we revise current populations of objects in J2/1 (Hecuba-gap group), J3/2 (Hilda group) and J4/3 (Thule group) resonances. The number of multi-opposition asteroids found is 274 for J2/1, 1197 for J3/2 and 3 for J4/3. By discovering a second and third objects in the J4/3 resonance, 2001 QG207 and 2006 UB219, this population becomes a real group rather than a single object. Using both hierarchical clustering technique and colour identification we characterise a collisionally-born asteroid family around the largest object (1911) Schubart in the J3/2 resonance. There is also a looser cluster around the largest asteroid (153) Hilda. Using numerical simulations we determine chaotic diffusion in the location of both families and we show they can be of pri- mordial origin, reminiscent of the formation time of the Hildas. We also prove that primordial objects in the J3/2 resonance were efficiently removed from their orbits during Jupiter’s and Saturn’s crossing of their mutual 2/1 mean motion resonance. As a result, the current population of asteroids in these resonances was trapped partly from the planetesimal disk, originally exterior to the giant planets, and partly from the nearby residing main asteroid belt. Key words: celestial mechanics – minor planets, asteroids – methods: N-body sim- ulations.
1 INTRODUCTION onances occupies the J3/2 resonance, and is frequently called the Hilda group. It was carefully studied in a parallel series Populations of asteroids in the Jovian first order mean mo- of papers by Schubart and Dahlgren and collaborators dur- tion resonances –J2/1, J3/2 and J4/3– are closely linked to ing the past few decades. Schubart (1982a,b, 1991) analysed the orbital evolution of the giant planets. This is because of short-term dynamics of Hilda-type orbits and introduced their orbital proximity to Jupiter.1 Stability or instability of quasi-constant orbital parameters that allowed their first these asteroid populations directly derives from the orbital classification. While pioneering, the Schubart’s work had the configuration of the giant planets. As such it is also sensitive disadvantage of having much smaller (i) sample of known on the nature and amount of Jupiter’s migration and other asteroids, and (ii) computer power than today. Dahlgren & finer details of its dynamics. As a result, the currently ob- Lagerkvist (1995) and Dahlgren et al. (1997, 1998, 1999) served asteroids in the Jovian first order resonances contain conducted the first systematic spectroscopic and rotation- valuable information about the early evolution of planets rate investigation of Hildas. They found about equal abun- and, if correctly understood and properly modelled, they dance of D- and P-type asteroids2 and suggested spectral- may help constraining it. size correlation such that P-types dominate large Hildas and Apart from the Trojan clouds (not studied in this paper) D-type dominate smaller Hildas. They also suggested small the largest known population in the Jovian mean motion res- Hildas have large lightcurve amplitude, as an indication of elongated and/or irregular shape, and distribution of their rotation rates is non-Maxwellian. Further analysis using the ⋆ E-mail: [email protected]ff.cuni.cz (MB); Sloan Digital Sky Survey (SDSS) data however does not sup- [email protected] (DV); ... port significant dominance of either of the two spectral types 1 Interestingly, at their discovery (158) Hilda and (279) Thule, residing in the J3/2 and J4/3 resonances, immediately attracted attention of astronomers by vastly extending asteroid zone toward giant planets and by their ability to apparently approach Jupiter 2 Note the former P-type objects were reclassified to X-type in a near aphelia of their orbits (e.g., K¨uhnert 1876; Krueger 1889). newer taxonomy by Bus and Binzel (e.g., Bus et al. 2002).
c 2008 RAS 2 M. Broˇzet al. for small sizes and indicates about equal mix of them (Gil- onance. A vast reservoir of planetesimals residing beyond Hutton & Brunini 2008; see also below). Smaller population the giant planets, and to some extend nearby regions in the of asteroids in the J2/1 and J4/3 received comparatively less outer part of the asteroid belt, re-populate these Jovian res- observational effort. onances at the same time and form thus the currently ob- Since the late 1990s powerful-enough computers allowed served populations. This is similar to the Trojan clouds of a more systematic analysis of fine details of the longer-term Jupiter (Morbidelli et al. 2005). dynamics in the Jovian first order resonances. Ferraz-Mello & Michtchenko (1996) and Ferraz-Mello et al. (1998a,b) de- termined that asteroids in the J2/1 resonance can be very 2 CURRENT ASTEROID POPULATIONS IN long-lived, possibly primordial, yet their motion is compar- THE JOVIAN FIRST ORDER RESONANCES atively more chaotic than those in the J3/2 resonance. The latter paper showed that commensurability between the li- Dynamics of asteroid motion in the Jovian first order reso- bration period and the period of Jupiter’s and Saturn’s great nances has been extensively studied by both analytical and inequality might have played an important role in depletion numerical methods in the past few decades (e.g., Murray of the J2/1 resonance. This would have occurred when both 1986; Sessin & Bressane 1988; Ferraz-Mello 1988; Lemaitre giant planets were farther from their mutual 5/2 mean mo- & Henrard 1990; Morbidelli & Moons 1993; Moons et al. tion configuration in the past. A still more complete analysis 1997; Nesvorn´y & Ferraz-Mello 1997; Roig et al. 2002; was obtained by Nesvorn´y& Ferraz-Mello (1997) who also Schubart 2007). In what follows we review a minimum in- pointed out that J4/3 resonance stable zone is surprisingly formation needed to understand our paper referring an in- void of asteroids, containing only (279) Thule. Roig et al. terested reader to the literature mentioned above for more (2001) and Broˇzet al. (2005) recently revised population insights. of asteroids in the J2/1 resonance and classified them into In the simplest framework of a circular restricted planar several groups according to their long-term orbital stabil- three-body problem (Sun-Jupiter-asteroid) the fundamental ity. While the origin of unstable population was successfully effects of the resonant dynamics is reduced to a one-degree interpreted using a model of a steady-state flow of main of freedom problem defined by a pair of variables (Σ, σ). For belt objects into this resonance with a rate consistent with J(p + 1)/p resonance (p = 1, 2 and 3 in our cases) we have the predicted Yarkovsky semimajor axis drift, the origin of Σ = √a 1 1 e2 , (1) the stable asteroids in J2/1 remains elusive. Population of − − ′p Hildas and Thule was assumed primordial or captured by an σ = (p + 1) λ p λ ̟ , (2) adiabatic migration of Jupiter (e.g., Franklin et al. 2004). − − It has been known for some time that the current con- where a is the semimajor axis, e the eccentricity, ̟ the lon- gitude of pericentre and λ the mean longitude in orbit of the figuration of giant planets does not correspond to that at ′ their birth. However, a new momentum to that hypothesis asteroid, and λ is the mean longitude in orbit of Jupiter. was given by the so called Nice model (Tsiganis et al. 2005; If the asteroid motion is not confined into the orbital Morbidelli et al. 2005; Gomes et al. 2005). The Nice model plane of the planet, we have an additional pair of resonant postulates the initial configuration of the giant planets was variables (Σz, σz) such that such that Jupiter and Saturn were interior of their mutual 2 2 i Σz = 2 a (1 e )sin , (3) 2/1 mean motion resonance (see also Morbidelli et al. 2008). − 2 The event of crossing this resonance had a major influence p ′ σz = (p + 1) λ p λ Ω , (4) on the final architecture of giant planets and strongly influ- − − enced structure of small-bodies populations in the Solar sys- where i denotes the inclination of asteroids orbit and Ω the tem. Morbidelli et al. (2005) showed that the population of longitude of its node. Remaining still with the simple av- Jupiter Trojan asteroids was destabilised and re-populated eraged model, orbital effects with shorter periods are ne- during this phase. In what follows we show that the same glected, the motion admits an integral of motion N given occurs for populations of asteroids in the J3/2 and J4/3 by resonances. p + 1 The paper is organised as follows. In Sec. 2 we revise in- N = √a 1 e2 cos i . (5) p − − formation about the current populations of asteroids in the p Jovian first order resonances. We use an up-to-date AstOrb Because of this integral of motion, variations of Σ imply database of asteroid orbits from the Lowell Observatory oscillations of both a and e. (ftp.lowell.edu) as of September 2007 and eliminate only The two-degree of freedom character of the resonant single-opposition cases to assure accurate orbital informa- motion prevents integrability. However, as an approxima- tion. In Sec. 3 we apply clustering techniques to determine tion we may introduce a hierarchy by noting that pertur- two families of asteroids on similar orbits in the J3/2 reso- bation described by the (Σ, σ) variables is larger than that nance. We strengthen their case with an additional colour described by the (Σz, σz) terms. This is usually true for real analysis using the SDSS broadband data. We also show the resonant asteroids of interest. Only the angle σ librates and origin of these families probably dates back to terminal re- σz circulates with a very long period. The (Σz, σz) dynam- population of the Hilda zone about 4 Gy ago. In Sec. 4.2 we ics thus produces a long-period perturbation of the (Σ, σ) determine stability of the putative primordial populations of motion. planetesimals in the Jovian first-order resonances. We show Within this model the minimum value of Σ in one reso- that those in the J3/2 are very efficiently eliminated when nant cycle (typically several hundreds of years) implies a is Jupiter and Saturn cross their mutual 2/1 mean motion res- minimum and e is maximum. These values do not conserve
c 2008 RAS, MNRAS 000, 1–14 Asteroids families in the first order resonances 3
exactly from one cycle to another because the (Σz, σz) mo- tion produces small oscillations. Since Σ+Σz N = √a/p 25 122 Zhongguos − − N one needs to wait until Σz reaches maximum over its cycle unstable Griquas and Zhongguos to attain ‘real’ minimum of a values and ‘real’ maximum 20 of e values over a longer time interval. From (3) we note the maximum of Σz occurs for the maximum of i variations. 15
This situations occurs typically once in a few thousands of extremely unstable years. In an ideal situation, these extremal values of (a, e, i) 10 would be constant and may serve as a set of proper orbital elements. 5 The motion of real asteroids in the Solar system is fur- number of asteroids in bin ther complicated by Jupiter having nonzero and oscillat- 0 1 2 10 70 100 1000 ing value of eccentricity. This brings further perturbations median lifetime in J2/1 t (Myr) (e.g., Ferraz-Mello 1988; Sessin & Bressane 1988 for a sim- J2/1 ple analytic description) and sources of instability inside Figure 1. A distribution of the median dynamical lifetimes for the resonance. Despite the non-integrability, we follow Roig objects in the J2/1 resonance. A division to several groups is et al. (2001) and introduce pseudo-proper orbital elements denoted: extremely unstable objects (tJ2/1 6 2 Myr), short-lived (ap, ep, sin ip) as the osculating elements (a, e, sin i) at the objects (tJ2/1 6 70 Myr) and long-lived objects (Griquas and moment, when the orbit satisfies the condition: Zhongguos, tJ2/1 > 70 Myr).
dσ ′ ′ σ = 0 < 0 ̟ ̟ = 0 Ω Ω = 0 , (6) ∧ dt ∧ − ∧ − ′ ′ 2.1 Hecuba-gap group where ̟ and Ω denote the longitude of pericentre and the longitude of node of Jupiter. As above, when (6) holds the In order to determine which objects are located in the J2/1 osculating orbital elements are such that a attains minimum, mean motion resonance we first extracted orbits from the e attains maximum and i attains maximum. Numerical ex- AstOrb database with osculating orbital elements in a broad periments show that with a complete perturbation model box around this resonance (see, e.g., Roig et al. 2001 for a and a finite time-step it is difficult to satisfy all conditions similar procedure). We thus obtained 7139 orbits, which we of (6) simultaneously. Following Roig et al. (2001) we thus numerically integrated for 10 kyr. We recorded and anal- ′ relax (6) to a more practical condition ysed behaviour of the resonance angle σ = 2λ λ ̟ from − − ◦ Eq. (2). Pericentric librators, for which σ oscillates about 0 , ◦ ∆σ ′ ◦ σ < 5 < 0 ̟ ̟ < 5 . (7) were searched. We found 274 such cases. That extends the | | ∧ ∆t ∧ | − | previous catalogue of Broˇzet al. (2005) almost twice (these Because this condition is only approximate, we numerically are mainly asteroids discovered or recovered after 2005 with integrate orbits of resonant asteroids for 1 Myr, over which accurate enough orbits). We disregard from our analysis as- the pseudo-proper orbital elements are recorded. We then teroids at the border of the resonance, for which σ(t) ex- compute their mean value and standard deviation, which is hibits alternating periods of libration and circulation, and an expression of the orbital stability over that interval of also those asteroids for which σ oscillates but are not res- time. For the case of the J3/2 and J4/3 resonances, we use onant anyway (N 6 0.8 in Eq. (5); see, e.g., Morbidelli & ∆σ a condition with different sign ∆t > 0. Moons 1993). The latter reside on low-eccentricity orbits in Aside to this short-term integration we perform long- the main asteroid belt adjacent to the J2/1 resonance. term runs to determine the stability of a particular reso- We conducted short- and long-term integrations of the nant orbit. With this aim we conduct integrations spanning resonant asteroids as described above. They allowed us to 4 Gyr for all resonant asteroids. Because of the inherent un- divide the population into 182 long-lived asteroids (with the certainty in the initial conditions (orbital elements at the median dynamical lifetime longer than 70 Myr) and 92 short- current epoch), we perform such integration for the nomi- lived (unstable) asteroids (the lifetime shorter than 70 Myr), nal orbit and 10 clones that randomly span the uncertainty see Figure 1.3 Among the objects residing on the unsta- ellipsoid. We then define dynamical lifetime of the orbit as ble orbits we found 14 have dynamical lifetimes less than the median of time intervals, for which the individual clones 2 Myr and we call them extremely unstable orbits. Broˇzet al. stayed in the resonance. (2005) suggested the unstable orbits in the J2/1 resonance Swift All integrations are performed using the pack- are resupplied from the adjacent main belt region due to a age (Levison & Duncan 1994), slightly modified to include permanent flux driven by the thermal forces while the ex- necessary on-line digital filters and a second order symplec- tremely unstable objects are temporarily captured Jupiter- tic integrator (Laskar & Robutel 2001). We include 4 outer family comets. The origin of the long-lived population in this planets and modify their and asteroid’s initial conditions by resonance is not known and it is not studied in this paper. a barycentric correction to partially account for the influ- ence of the terrestrial planets. The absence of the terrestrial planets as perturbers is a reasonable approximation in the 3 We summarise our results for both J2/1 and J3/2 resonances outer part of the main belt and for orbits with e < 0.8 in in tables available through a website http://sirrah.troja.mff. general. We nevertheless checked the short-term computa- cuni.cz/yarko-site/ (those for J4/3 bodies are given in Table 1). tions (pseudo-proper resonant elements determination) us- They contain listing of all resonant asteroids, their pseudo-proper ing a complete planetary model and noticed no significant orbital elements with standard deviations, their dynamical resi- difference in results. We use a time step of 91 days. dence time and some additional information.
c 2008 RAS, MNRAS 000, 1–14 4 M. Broˇzet al.
Figure 2 shows the pseudo-proper orbital elements of 0.7 the J2/1 asteroids projected onto the (ap, ep) and (ap, sin ip) Kozai planes. Our data confirm that the unstable population of
p 0.6 J2/1 asteroids populates the resonance outskirts near its e J2/1 separatrix, where several secular resonances overlap and trigger chaotic motion (e.g., Morbidelli & Moons 1993; 0.5 ν Nesvorn´y& Ferraz-Mello 1997; Moons et al. 1998). At low- 5 eccentricities the chaos is also caused by an overlap on 0.4 numerous secondary resonances (e.g., Lemaitre & Henrard A 1990). Two ‘islands’ of stability –A and B– harbour the long- 0.3 lived population of bodies. The high-inclination branch A is J2/1 much less populated, containing only 9 asteroids. The origin ν of this asymmetry is not known. pseudo-proper eccentricity 0.2 B 16 The size-frequency distribution of objects of a popula- tion is an important property, complementing that of the 0.1 orbital distribution. Figure 3 shows cumulative distribution 3.16 3.18 3.2 3.22 3.24 3.26 3.28 N(
c 2008 RAS, MNRAS 000, 1–14 Asteroids families in the first order resonances 5
diameter D (km) for albedo pV = 0.05 200 100 50 20 10 5 2 J3/2 0.4 p
1000 e J2/1 Griqua
) J4/3
H A Kohman (< +0.34 N 0.3 100
+0.70
Zhongguo 10 0.2 B pseudo-proper eccentricity number of asteroids
3.21 3.22 3.23 3.24 3.25 3.26 1 pseudo-proper semimajor axis ap (AU) 8 10 12 14 16 0.3 absolute magnitude H (mag) Kohman ) p Figure 3. Cumulative distributions N( Zhongguo the Thule population should be known at 10% complete- 0 ∼ 3.21 3.22 3.23 3.24 3.25 3.26 ness leaving only about 90% undiscovered population. We pseudo-proper semimajor axis a (AU) thus conclude that the objects in the H = 9 13 mag range p − are very likely missing in this resonance. Where does the Figure 4. A zoom of the (ap, ep) and (ap, sinip) plots (Fig. 2) existing population of small Thule-type asteroids begin? of the J2/1 population with relative sized indicated by crosses. Our initial search in the broad box around J4/3 de- Note the large bodies are far from each other. tected only 13 objects. Six of them, including the well- known extinct comet (3552) Don Quixote (e.g., Weissman et al. 2002), are on typical orbits of Jupiter-family comets parable amplitudes. The last object, (52007) 2002 EQ47, that happen to reside near this resonance with very high appears to reside on an unstable orbit outside the J4/3 res- eccentricity and moderately high inclination. Two more onance. Our search thus lead to the detection of two new are single-opposition objects and one has only poorly con- asteroids in this resonance, increasing its population by a strained orbit, leaving us with (279) Thule and three addi- factor of three. tional candidate objects: (52007) 2002 EQ47, 2001 QG207 According to a long-term numerical integration of the and 2006 UB219. nominal orbits plus 10 close clones, placed within an orbital Figure 8 shows short-term tracks of (279) Thule, uncertainty, we can state, that the orbit of (279) Thule seems 2001 QG207 and 2006 UB219 in resonant variables to be stable over 4 Gyr, but the orbits of 2001 QG207 and ′ (e cos σ, e sin σ) of the J4/3 resonance (σ = 4λ 3λ ̟ 2006 UB219 are partially . They are not ‘short-lived’, but − − in this case). In all cases their orbits librate about the peri- 45 % and 60 % of clones escaped well before 4 Gyr. This is centric branch (σ = 0) of this resonance, although this is also documented on a pseudo-proper semimajor axis vs time troubled –mainly in the low-eccentricity case of (279) Thule– plot (Fig. 10). by the forced terms due to Jupiter’s eccentricity (see, e.g., We would like to point out that it took more than a cen- Ferraz-Mello 1988; Sessin & Bressane 1988; Tsuchida 1990). tury from the discovery of (279) Thule (Palisa 1888; Krueger To make the resonance librations clearly revealed, we re- 1889) until further objects in this resonance were finally dis- moved the Jupiter forced term from (e cos σ, e sin σ) and ob- covered. This is because there is an anomalously large gap tained effectively the proper resonant mode (K,H) intro- in size of these bodies: (279) Thule is 127 km in size with duced by Sessin; motion in these coordinates is shown in pV = 0.04 AU (e.g., Tedesco et al.2002), while the estimated ◦ Fig. 9. The leftmost panel recovers the 40 libration of sizes of 2001 QG207 and 2006 UB219 for the same value of ∼ (279) Thule, given previously by Tsuchida (1990; Fig. 3). albedo are 8.9 km and 11.8 km only. It will be interest- The other two smaller asteroids show librations with com- ing to learn as much as possible about the Thule popula- c 2008 RAS, MNRAS 000, 1–14 6 M. Broˇzet al. Table 1. Data on presently known population of asteroids residing in the J4/3 Jovian mean motion reso- nance (Thule group). Pseudo-proper orbital elements (ap, ep, sin ip) are given together with their standard deviations (δap, δep, δ sin ip) determined from a 1 Myr numerical integration. σp, max is the maximum libra- tion amplitude in the Sessin’s (K,H) variables (see the text and Fig. 9), H is the absolute magnitude from the AstOrb catalogue and D is the estimated size using pV = 0.04 geometric albedo (Tedesco et al. 2002). No. Name ap ep sin ip δap δep δ sin ip σp, max H D [AU] [AU] [deg] [mag] [km] 279 Thule 4.2855 0.119 0.024 0.0005 0.012 0.003 ∼50 8.57 126.6 2001 QG207 4.2965 0.244 0.042 0.0003 0.014 0.003 25 14.36 8.9 2006 UB219 4.2979 0.234 0.102 0.0003 0.014 0.004 25 13.75 11.8 0.1 0.1 0.1 σ σ σ 0 0 0 sin sin sin e e e 0.1 0.1 0.1 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 e cos σ e cos σ e cos σ Figure 8. The orbits of (279) Thule [left], 2001 QG207 [middle] and 2006 UB219 [right] in resonant variables (e cos σ, e sin σ) of the J4/3 resonance, where e is the eccentricity and σ = 4λ′ − 3λ − ̟, with λ and λ′ the mean longitude in orbit of the asteroid and Jupiter and ̟ is the longitude of asteroid’s pericentre (in all cases the osculating orbital elements are used). Each panel shows a short-term, 10 kyr numerical integration of orbits of In all three cases the orbits librate about the pericentric branch (σ = 0) of the resonance. Perturbations due to Jupiter’s eccentricity and its variations make the regular libration move in an epicyclic manner (e.g., Ferraz-Mello 1988; Sessin & Brassane 1988). tion in the H = 13 15 mag absolute magnitude range us- of Jupiter, are typically too chaotic to hold stable aster- − ing future-generation survey projects such as Pan-STARRS oid populations or the populations were too small to enable (e.g., Jedicke et al. 2007). Such a completed population may families’ search. The only remaining candidate populations present interesting constraint on the planetesimal size dis- are those in the Jovian first order resonances, with Hilda tribution 4 Gyr ago. asteroids the most promising group. However, low expecta- tions for existence of collisional families likely de-motivated systematic search. Note, the estimated intrinsic collisional 3 COLLISIONAL FAMILIES AMONG HILDA probability of Hilda asteroids is about a factor 3 smaller ASTEROIDS than in the main asteroid belt (e.g., Dahlgren 1998; Dell’Oro et al. 2001) and the population is more than two orders of Collisions and subsequent fragmentations are ubiquitous magnitude smaller. process since planets formed in the Solar system. Because In spite of the situation outlined above, Schubart (1982, the characteristic dispersal velocities of the ejecta (as a rule 1991) repeatedly noticed subgroups of Hilda-type asteroids of thumb equal to the escape velocity of the parent body) with very similar proper elements. For instance, in his 1991 is usually smaller than the orbital velocity, the resulting paper he lists 5 members of what we call Schubart fam- fragments initially reside on nearby orbits. If the orbital ily below and pointed out their nearly identical values of chaoticity is not prohibitively large in the formation zone the proper inclination. Already in his 1982 paper Schubart we can recognise the outcome of such past fragmentations mentions a similarity of such clusters to Hirayama families, as distinct clusters in the space of sufficiently stable orbital but later never got back to the topic to investigate this prob- elements. More than 30 collisional families is known and lem with sufficient amount of data provided by the growing studied in the main asteroid belt (e.g., Zappal`aet al. 2002) knowledge about Hilda population.5 Even a zero order in- with important additions in the recent years (e.g., Nesvorn´y spection of Figs. 5 hints existence of two large clusters in the et al. 2002; Nesvorn´y, Vokrouhlick´y& Bottke 2006). Simi- larly, collisional families have been found among the Trojan clouds of Jupiter (e.g., Milani 1993; Beaug´e& Roig 2001), 5 Schubart lists 11 additional asteroids in the subgroup irregular satellites of Jupiter (e.g., Nesvorn´yet al. 2003, on his website http://www.rzuser.uni-heidelberg.de/~s24/ 2004) and even trans-Neptunian objects (e.g., Brown et al. hilda.htm, but again does not go into details of their putative 2007). Mean motion resonances, other than Trojan librators collisional origin. c 2008 RAS, MNRAS 000, 1–14 Asteroids families in the first order resonances 7 0.1 0.1 0.1 σ σ σ 0 0 0 sin sin sin e e e 0.1 0.1 0.1 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 e cos σ e cos σ e cos σ Figure 9. Filtered resonant variables (e cos σ, e sin σ), with short-period oscillations forced by Jupiter removed by digital filtering (for the three known asteroids in the J4/3 resonance: (279) Thule [left], 2001 QG207 [middle] and 2006 UB219 [right]). They are similar to Sessin’s (K,H) coordinates in Sessin & Brassane (1988) or Tsuchida (1990), In each of the cases the pericentric libration is clearly revealed. The maximum libration amplitude in these coordinates is denoted by σp, max in Table 1. Hilda population. In what follows we pay a closer analysis and Griquas is presented at Figure 13. No grouping seem to to both of them. be so significant, to be considered an impact generated clus- For the purpose of the HCM analysis, we average the ter. This is consistent with the discussion of the (ap, ep, ip) pseudo-proper elements over 10 Myr. (We also checked that plots in Section 2.1. our results practically do not depend on the width of this interval.) At first, we compute the number of bodies Nmin, which constitute a statistically significant cluster at a given 3.1 Schubart family value of the cutoff velocity v . We use a similar approach cutoff The Schubart family can be distinguished from the rest of as Beaug´e& Roig (2001): for all asteroids in the J3/2 count Hildas on a large range of cutoff velocities: from 50 m/s the number N (v ) of asteroids, which are closer than i cutoff to more than 100 m/s; it merges with the Hilda family at v , and then compute the average value N = N¯ . Ac- cutoff 0 i 200 m/s. Let us select v = 60 m/s as the nominal value. cording to Zappall`aet al. (1995), the cluster may be con- cutoff The total number of Schubart members is not too sensi- sidered significant, if N >N = N + 2√N . The plots min 0 0 tive to this value, but we do not select a higher value also N (v ) and the corresponding N (v ) for Hildas 0 cutoff min cutoff because the cluster attains a rather peculiar shape in the are at Figure 11. We use a standard metric (d defined by 1 (a , e , i ) space. Zappal`aet al., 1994) for this purpose: p p p The Figure 14 shows the cumulative absolute magni- tude distribution of the Schubart family, compared to the 5 δa 2 δv = na + 2(δe2) + 2(δ sin I)2 rest of Hildas. An important observation is, that the slope r 4 a γ = 0.48 is steeper for the Schubart, which supports the Note that we substitute resonant pseudo-proper elements hypothesis of its collisional origin. here, so in essence, it is also a measure of the libration am- According to the SDSS MOC data (Ivezi´cet al. 2002) plitude as in Beaug´e& Roig (2001). the Schubart family seems to be spectrally homogeneous (see We construct a ‘stalactite diagram’ for Hildas in a usual the histogram of spectral slopes in Fig. 15), and belongs to manner: we start with (153) Hilda as the first central body the C/X taxonomy class. This is also in concert with the and we find all bodies associated with it at vcutoff = 300 m/s, collisional origin. Hilda group as a whole exhibits rather a using a hierarchical clustering method (HCM, Zappal`aet al., bi-modal distribution, spanning from C/X-types to D-types. 1994). Then we select the asteroid with the lowest number Tedesco et al. (2002) derive D = 80 km size of (1911) (catalogue designation) from remaining (not associated) as- Schubart corresponding to a very low albedo pV = 0.025. teroids and repeat the HCM again, until no asteroids are The same authors determine D = 38 km size of (4230) van left. Then we repeat the whole procedure recursively for den Bergh and exactly the same albedo; this asteroid is all clusters detected at vcutoff = 300 m/s, but now for a among the five largest in the family. Assuming the same lower value, e.g., vcutoff = 299 m/s. We may continue un- albedo for all other family members, we can construct a size- til vcutoff = 0 m/s, but of course, for too low values of the frequency distribution (Fig. 16). The slope α = 2.8 is not − cutoff velocity, no clusters can be detected and all asteroids far from 2.5, i.e., that of a collisionally relaxed Dohnanyi − are single. The resulting stalactite diagram at Figure 12 is population. simply the asteroid number vs vcutoff plot, a dot at a given If we sum the volumes, we end up with a lower limit place is plotted only if the asteroids belongs to a cluster of for the parent body size DPB = 110 km, provided there no at least max(5,Nmin(vcutoff )) bodies interlopers. If we further assume an observational complete- We see two prominent clusters among Hildas: the first ness down to Dcomplete = 10 km and prolong the SFD slope one around the asteroid (153) Hilda itself, and the second α = 2.6 down to Dmin = 0, we can estimate the total vol- − one around (1911) Schubart. We discuss them separately. ume, including the contribution of small unobserved bodies, . The corresponding stalactite diagram for Zhongguos and DPB = 130 km, some sort of an upper limit. The vol- c 2008 RAS, MNRAS 000, 1–14 8 M. Broˇzet al. 0.4 (1911) Schubart stable 0.35 escaped 0.35 p e 0.3 0.3 [] 0.25 e 0.25 0.2 eccentricity 0.2 0.15 pseudo-proper eccentricity 0.15 0.1 0.1 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 pseudo-proper semimajor axis ap (AU) semimajor axis a [AU] 0.35 Figure 6. Pseudo-proper semimajor axis vs eccentricity plot for (1911) Schubart asteroids in the J3/2 resonance. Bold crosses denote orbits, which ) 0.3 escaped after 4 Gyr of evolution. p I 0.25 (1911) Schubart (279) Thule (153) Hilda N Hildas in J3/2 0.2 15 0.15 10 0.1 5 number of asteroids pseudo-proper inclination sin( 0.05 0 0 0.05 0.1 0.15 albedo p 0 V 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 Figure 7. Values of the geometric albedo p measured by pseudo-proper semimajor axis ap (AU) V Tedesco et al. (2002) for asteroids located inside the J3/2 res- Figure 5. Pseudo-proper orbital elements for 1197 Hildas pro- onance. Three individual values for (153) Hilda, (1911) Schubart jected onto the planes of semimajor axis ap vs eccentricity ep and (279) Thule (a J4/3 asteroid) are indicated on top. (top) and semimajor axis ap vs sine of inclination sin ip (bottom). Larger size of the symbol indicates larger physical size of the as- teroid. Because of Hildas orbital stability, the uncertainty in the pseudo-proper element values is typically smaller than the sym- bol size. Note a tight cluster around the proper inclination value 4.3 sin ip ≃ 0.0505, lead by the largest asteroid (1911) Schubart, and 2006 UB219 a somewhat looser cluster around the proper inclination value 2001 QG207 sin ip ≃ 0.151, lead by the largest asteroid (153) Hilda. Both are discussed in more detail in Sec. 3. [AU] p 4.29 a umetric ratio between the largest fragment and the parent . (279) Thule body is then VLF/VPB = 0.25. 4.28 0 1000 2000 3000 4000 3.2 Hilda family t [Myr] We repeat the same analysis as in Sec. 3.1 for the Hilda fam- ily. The results can be summarised as follows: the interval Figure 10. Pseudo-proper semimajor axis ap vs time t plot for the 3 orbits of asteroids located inside the J4/3 resonance and 10 of vcutoff is (130, 170) m/s we choose vcutoff = 150 m/s as the nominal value. The slope of the magnitude distribution close clones orbits for each of them (placed randomly within the orbital uncertainty ellipsoids). Thin lines denote the stable or- γ = 0.50 (Fig. 14) is again steeper than for the rest of J3/2 bits and thick lines unstable, which escaped from the J4/3 before bodies. 4 Gyr. The spectral slopes are not so distinct from the rest of the J3/2 population, but there is more C/X-type bodies than D-types. Tedesco et al. (2002) give albedo values for six mem- c 2008 RAS, MNRAS 000, 1–14 Asteroids families in the first order resonances 9 50 300 Nmin N0 40 250 max(Ni) 200 0 30 N , / m/s 150 min N 20 cutoff v 100 10 50 0 0 50 100 150 200 250 300 0 0 20 40 60 80 100 120 140 160 vcutoff [m/s] N Figure 11. The dependence of the number of asteroids Nmin Figure 13. A stalactite diagram computed for the long-lived J2/1 to be considered a statistically significant cluster on the cutoff population. There are no prominent groupings; 60 asteroids are velocity vcutoff , and the average number N0 of asteroids which not associated with any others, even at vcutoff = 300 m/s. Every are closer than vcutoff . Both quantities are valid for the J3/2 group plotted here has at least 5 members. population. diameter D (km) for albedo pV = 0.044 300 200 100 50 20 10 5 J3/2 250 1000 Hilda Schubart 0.37 200 ) Hilda H cluster (< / m/s 150 N Schubart 100 cutoff cluster v 0.50 100 50 0.48 10 0 0 200 400 600 800 1000 number of asteroids N Figure 12. A stalactite diagram computed for the J3/2 popu- 1 lation. Two prominent groupings, the Schubart family and the Hilda family, are indicated. Every group plotted here has at least 8 10 12 14 16 5 or Nmin members, whichever is larger. (see Fig. 11). absolute magnitude H (mag) Figure 14. Cumulative distributions N( with the exponent α = 3.25 and vesc being the escape veloc- ity from the parent body. 4 SIMULATED IMPACT EVENTS To simulate an impact that might create the Schubart In order to asses some limits for the ages of the asteroid family, we generated velocities randomly for 139 fragments families, we perform several numerical tests: we simulate a with the following parameters: vesc = 56 m/s (for DPB = 3 disruption of a parent body inside the resonance and calcu- 94 km and the density ρ = 2, 500 kg/m ), vmin = 1 m/s, late the long-term orbital evolution of the swarm of ejecta. vmax = 200 m/s; the resulting mean velocity vmean = 58 m/s. We assume an isotropic velocity field, wrt. the parent body, The swarm is depicted in osculating elements at Figure 17, ◦ and the distribution of velocities ”at infinity” according to for the impact geometry f = 0 and ω + f = 180 . Its dis- c 2008 RAS, MNRAS 000, 1–14 10 M. Broˇzet al. 20 C/X D 1000 J3/2 15 − Hilda J3/2 1.9 Schubart 10 5 ) −2.8 D (> 100 0 N 15 Hilda −2.7 10 5 10 number of spectra 0 number of asteroids 15 Schubart 10 1 5 10 100 0 diameter D / km 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 PC spectral slope ( 1) Figure 16. A cumulative distribution N(>D) of diameters D for the whole J3/2 population (solid) and for members of the Hilda Figure 15. Distribution of spectral slopes (PC components of 1 family (dashed) and the Schubart family (dotted). We assumed the 5 broad-band colours) of 153 asteroids in the J3/2 resonance the geometric albedo pV = 0.044 (and 0.025 for the Schubart) for (top), 21 Hilda family members (middle) and 4 Schubart family the conversion of absolute magnitudes to diameters. Labels are members (bottom). Data are from the 3rd release of the SDSS the best-fitted values of α using N(>D) ∝ Dα (straight lines) in catalogue of moving objects (Ivezi´cet al. 2002). Sometimes, there the interval (11, 20) km and (9, 20) km for the J3/2 population. are several spectra available for one asteroid. The slopes of Hildas range from neutral to steep, with roughly two groups separated by the value PC = 0.3: (i) C/X-types with PC < 0.3, and 1 1 233 fragments is mostly preserved over 4 Gyr and there is (ii) D-types with PC > 0.3. The Schubart family members are 1 a problem with the dispersion of eccentricities. spectrally similar; the median value PC1 = 0.20 corresponds to a C- or X-type parent body. If we generate a smaller swarm inside the J2/1 reso- nance, e.g., around (3789) Zhongguo, we realize the diffu- sion is faster than in the J3/2 and consequently, signs of persion in semimajor axis is typically much larger than that an impact disruption may disappear within 1 Gyr. This is of the Schubart family, but all fragments still fall within the in contrast with the observed steep SFD of Zhongguos and J3/2 resonance. The dispersion of eccentricities is on the Griquas, discussed in Section 2.1. other hand substantially smaller. Inclinations are roughly We conclude, the Schubart and Hilda families might be consistent. This holds even for calculated pseudo-proper el- primordial, because impact clusters in the J3/2 disperse only ements and for different isotropic-impact geometries (see negligibly. On contrary, any impacts in the J2/1 older than Fig. 18). The family only obtains a ‘strange’ shape, because 1 Gyr are not easily detectable. the bodies that fall to the left from the libration centre are ‘mapped’ to the right. 4.1 The influence of the Yarkovsky effect Surprisingly, if we evolve the swarm up to 4 Gyr, it re- mains almost the same (Fig. 19). Only about 10 % of bodies The semimajor axis drift, caused by the emission of thermal is removed from the J3/2, namely those with large ampli- radiation (the Yarkovsky effect), would have not negligible tude librations. (This is in agreement with Fig. 6.) The best influence for regular orbits outside resonances. An expected match between the simulated impact and the Schubart fam- ∆a during 4 Gyr is roughly 0.04 AU for a 10 km asteroid ◦ ily seems to be for f 135 . (We can state this already for (covered by low-conductivity surface layer and located 4 AU ≃ the initial pseudo-proper elements.) from the Sun). The overall extent of the Schubart family is However, the slow diffusion poses a problem, because smaller than that. the spread in eccentricities remains the same, i.e., smaller However, the resonant gravitational interaction of the than the spread of the Schubart family. Possibly, there is an- asteroid with Jupiter effectively inhibits the drift and the other mechanism we did not account for, e.g., non isotropic motion is almost the same as for the model without the − ejection of fragments, or encounters with massive Hildas Yarkovsky, at least for the drift of the order 10 5 AU/Myr. (like (153) Hilda, (190) Ismene or (748) Simeisa, with masses (In an adiabatic approximation, the drift in a would cause a −9 of the order 10 MJupiter), which can enhance the diffusion secular increase of e for the orbit captured in a mean motion a bit. resonance.) The situation is very similar with fragments ejected Thought, we may ask a question, for which size range at the location of (153) Hilda. A putative disruption of a the Yarkovsky effect becomes important? We perform a nu- 180 km parent body was simulated. Again the swarm of merical test: we multiply the sizes of the long-lived J2/1 ob- c 2008 RAS, MNRAS 000, 1–14 Asteroids families in the first order resonances 11 0.3 0.3 0.28 0.28 0.26 0.26 0.24 0.24 p p e 0.22 e 0.22 J3/2 0.2 Schubart cluster 0.2 initial osculating elements 0.18 pseudo-proper elements 0.18 J3/2 0.16 0.16 impact at 0 Myr 4 Gyr 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 ap / AU ap / AU Figure 17. The initial osculating elements of an impact- Figure 19. A comparison of the initial swarm of ejecta and the generated swarm at the location of (1911) Schubart (bot- same swarm after 4 Gyr, i.e., evolved due to planetary pertur- tom crosses), the corresponding pseudo-proper elements (upper bations. Note the shape does not change much, only ∼10 % of crosses) and a comparison with the observed Schubart family (cir- fragments escape. Even though the dispersion of eccentricities is cles), displayed on a semimajor axis vs eccentricity plot. Dots in still smaller than for the Schubart family (see Fig. 17). the background denote the rest J3/2 asteroids projected on the plane. Note the simulated cluster is very tight in eccentricity and very spread in semimajor axis as compared to the Schubart. 100 sizes multiplied by: 2 90 0.2 0.02 80 0.3 N 0.002 70 0.28 60 50 0.26 40 0.24 30 p number of asteroids in bin e 0.22 20 10 0.2 1 10 100 1000 0.18 median lifetime in J2/1 tJ2/1 (Myr) J3/2 ° Figure 20. Logarithmic histograms of dynamical lifetimes for the 0.16 impact at f = 0 f = 135° originally long-lived asteroids in the J2/1 resonance, in case the 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 Yarkovsky effect perturbs the orbits. The sizes of the objects (5– a / AU 20 km) were multiplied by 2, 0.2, 0.02 and 0.002 for comparison. p Note a stronger instability starts to occur for the factor 0.02. Figure 18. The same as Fig. 17, but for a comparison of two impact geometries (f = 0 and f = 135◦) in the proper element space. ure 21 shows clearly that the retrograde rotation increases the probability of an escape. This is consistent with the structure of the J2/1 resonance, for which low-a separatrix jects by factors 0.2, 0.02 and 0.002, for which the Yarkovsky does not continue to e = 0. We conclude the remaining very 1 small (yet unobservable) asteroids inside the J2/1 may ex- effect is stronger, because it scales roughly as D , and com- pare respective dynamical lifetimes with the original long- hibit a preferential prograde rotation. lived objects. According to Figure 20, we can state that a significant 4.2 Stability during planetary migration destabilisation of the resonant population occurs for sizes 0.1–0.5 km and smaller (provided the original population has As the first ‘motivation’ step to an analysis of the planetary sizes 5–20 km). This is well below the current observational migration and its influence on the resonant populations we limit ( 5 km in the outer belt). perform a simplified ‘static’ approach: we move Saturn into ∼ We can also check, which orientation of the spin axis is an exact resonance with Jupiter and watch the correspond- favourable with respect to the escape. We take 0.1–0.5 km ing change of dynamical lifetimes of the asteroids in the bodies, clone them 5 times and assign them different val- J2/1, J3/2 and J4/3 resonances with Jupiter. Of course, in ◦ ◦ ◦ ◦ ues of the obliquity γ = 0, 45 , 90 , 135 and 180 . Fig- reality both Jupiter and Saturn are forced to migrate away c 2008 RAS, MNRAS 000, 1–14 12 M. Broˇzet al. 100 100 90 sizes multiplied by 0.02 N no MMR obliquities: 0° 1:2 80 80 N 45° 4:9 90° 3:7 70 ° 135 60 2:5 180° G.I. vs J2/1 60 40 50 40 20 number of asteroids in bin 30 0 number of asteroids in bin 0.1 1 10 100 20 lifetime in J2/1 (Myr) 10 100 N 1 10 100 1000 no MMR median lifetime in J2/1 tJ2/1 (Myr) 1:2 80 4:9 3:7 Figure 21. The same as Fig. 20, but now the Yarkovsky effect 60 2:5 is calculated only for small bodies (0.1–0.4 km) and varies due to ◦ ◦ ◦ ◦ different values of the obliquity γ = 0, 45 , 90 , 135 and 180 . 40 Note the retrograde bodies are significantly more unstable. 20 number of asteroids in bin 415 9.5289 AU 0 Great Inequality 0.1 1 10 100 mean J2/1 libration lifetime in J3/2 (Myr) 410 (1362) Griqua 8 406.1 yr 405 N 7 no MMR 6 1:2 400 4:9 5 3:7 period / yr 2:5 395 4 3 390 2 385 number of asteroids in bin 1 9.525 9.526 9.527 9.528 9.529 9.53 0 semimajor axis of Saturn aS / AU 0.1 1 10 100 lifetime in J4/3 (Myr) Figure 22. The dependence of the Great Inequality period 5λ − S Figure 23. Histograms of dynamical lifetimes for asteroids in the 2λ on the osculating semimajor axis a of Saturn (at the epoch J S J2/1 [top], J3/2 [middle] and J4/3 [bottom] resonances, in case 2453100.5) and the comparison with the average period P¯ of J2/1 Jupiter and Saturn are at their current orbits, or in a mutual 1:2, librations in the J2/1 resonance (and with one particular asteroid 4:9, 3:7, 2:5 resonance, or in a Great Inequality resonance (in case (1362) Griqua). For a = 9.5289 AU the G.I. period is equal to S of the J2/1 only). The histograms were computed for 106 long- P . The current position of Saturn is a = 9.576 AU and the J2/1 S lived asteroids in the J2/1, first 100 Hildas in the J3/2 and 8 in 2:5 Jupiter–Saturn resonance would occur at a = 9.584 AU. S the J4/3 (including short-lived). from the mutual resonance by the interaction with the plan- in the J3/2 are very unstable (on the timescale 1 Myr) ∼ etesimal disk, but we may use this simplified simulations as with respect to the 1:2 Jupiter–Saturn resonance6 ; on con- the first hint, which resonances may cause greatest instabil- trary J2/1 asteroids may survive several 10 Myr (but the ities. 1:2 crossing would take 1 Myr only). Second, the 4:9 res- ∼ In case of the Nice model (Tsiganis et al. 2005), Jupiter onance has a larger influence on the J2/1 population than and Saturn started the migration before 1:2 resonance and on Hildas. Third, in the case of 3:7 resonance it is the op- consequently they had to encounter also low-order reso- posite: the J3/2 is more unstable than J2/1. Fourth, the nances 4:9, 3:7 and ended just before 2:5 (which we take Great Inequality resonance does indeed destabilises J2/1 on also as an test case). As the migration subsequently ceased, a time scale 50 Myr. Provided the last phases of the migra- Jupiter and Saturn might stay longer time in the vicinity of tion proceed very slowly, it may cause significant depletion the ‘later’ resonances. of a primordial J2/1 population. In the exact 2:5 resonance, For the J2/1 bodies, there is another important reso- the J2/1 population would not be affected at all. nance between the Great Inequality period and the period A more detailed analysis is postponed to the next paper. of libration inside J2/1. This situation occurs before Jupiter and Saturn reach the current configuration (see Figure 22). The chaotic diffusion is increased in this case (Ferraz-Mello 6 Jupiter Trojans, which are already known to be strongly un- et al. 1998ab) stable (Morbidelli et al. 2005) would have the dynamical lifetime The results are summarised in Figure 23. First, Hildas of the order 0.1 Myr in this kind of simulation. c 2008 RAS, MNRAS 000, 1–14 Asteroids families in the first order resonances 13 5 CONCLUSIONS AND FURTHER WORK Gil-Hutton R., Brunini A., 2008, Icarus, in press Gomes R., Levison H. L., Tsiganis K., Morbidelli A., 2005, The main results of this paper can be summarised as follows: Nature, 435, 466 (i) we provide an update of the observed J2/1, J3/2 and Ivezi´cZ., Juri´cM., Lupton R.H., Tabachnik S., Quinn T., J4/3 resonant populations, so their properties are gradually 2002, Survey and other telescope technologies and discov- better justified; (ii) we studied two asteroid families located eries, (J.A. Tyson, S. Wolff, Eds.) Proceedings of SPIE Vol. inside the J3/2 resonance (Schubart and Hilda) and pro- 4836 vide an evidence, they are of a collisional origin; (iii) 20 % of Jedicke R., Magnier E.A., Kaiser N., Chambers K.C., 2007, Hildas may escape from the J3/2 resonance within 4 Gyr in in Milani A., Valsecchi G.B., Vokrouhlick´yD., eds., Near the current configuration of planets; (iv) Hildas are strongly Earth Asteroids, our Celestial Neighbors. Cambridge Uni- unstable with respect to the 1:2 Jupiter–Saturn resonance, versity Press, Cambridge, p. 341 which may indicate their during a planetary migration pe- Krueger A., 1889, AN, 120, 221 riod by a re-population of the J3/2 zone. K¨uhnert F., 1876, AN, 87, 173 We postpone for the future work a more detailed model Laskar J., Robutel P., 2001, Celest. Mech. Dyn. Astron., with analytic or N-body migration of planets and its influ- 80, 39 ence on the stability of resonant populations. Lemaitre A., Henrard J., 1990, Icarus, 83, 391 Levison H., Duncan M., 1994, Icarus, 108, 18 Milani A., 1993, Celest. Mech. Dyn. Astron., 57, 59 ACKNOWLEDGEMENTS Milani A., Farinella P., 1995, Icarus, 115, 209 We thank ... The work of MB and DV has been sup- Morbidelli A., Moons M., 1993, Icarus, 102, 316 ported by the Grant Agency of the Czech Republic (grants Morbidelli A., Levison H. L., Tsiganis K., Gomes R., 2005, 205/08/0064 and 205/08/P196) and the Research Program Nature, 435, 462 MSM0021620860 of the Czech Ministry of Education. We Morbidelli A., Tsiganis K., Crida A., Levison H.F., Gomes also acknowledge the usage of the Metacentrum computing R., 2008, AJ, in press facilities (http://meta.cesnet.cz/). Moons M., Morbidelli A., Migliorini F., 1998, Icarus, 135, 458 Murray C.A., 1986, Icarus, 65, 70 Nesvorn´yD., Ferraz-Mello S., 1997, Icarus, 130, 247 REFERENCES Nesvorn´yD., Beaug´eC., Dones L., 2004, AJ, 127, 1768 Beaug´eC., Roig F., 2001, Icarus, 153, 391 Nesvorn´yD., Vokrouhlick´yD., Bottke W.F., 2006, Science, Brown M.E., Barkume K.M., Ragozzine D., Schaller E.L., 312, 1490 2007, Nature, 446, 294 Nesvorn´yD., Bottke W.F., Dones L., Levison H.F., 2002, BroˇzM., Vokrouhlick´yD., Roig F., Nesvorn´yD., Bottke Nature, 417, 720 W.F., Morbidelli A., 2005, MNRAS, 359, 1437 Nesvorn´yD., Alvarellos J.L.A., Dones L., Levison H.F., Bus S.J., Vilas F., Barucci M.A., 2002, in Bottke W.F., 2003, AJ, 126, 398 Cellino A., Paolicchi P., Binzel R.P., eds., Asteroids III. Nesvorn´yD., Jedicke R., Whiteley R.J., Ivezi´cZ., 2005, University of Arizona Press, Tucson, p. 169 Icarus, 173, 132 Dahlgren M., 1998, A&A, 336, 1056 O’Brien D.P., Greenberg R., 2003, Icarus, 164, 334 Dahlgren M., Lagerkvist C.-I., 1995, A&A, 302, 907 Palisa J., 1888, AN, 120, 111 Dahlgren M., Lagerkvist C.-I., Fitzsimmons A., Williams Quinn T.R., Tremaine, S., Duncan, M. 1991, AJ, 101, 2287 I. P., Gordon M., 1997, A&A, 323, 606 Roig F., Gil-Hutton R., 2006, Icarus, 183, 411 Dahlgren M., Lahulla J.F., Lagerkvist C.-I., Lagerros J., Roig F., Nesvorn´yD., Ferraz-Mello S., 2002, MNRAS, 335, Mottola S., Erikson A., Gonano-Beurer M., di Martino M., 417 1998, Icarus, 133, 247 Schubart J., 1982a, A&A, 114, 200 Dahlgren M., Lahulla J.F., Lagerkvist C.-I., 1999, Icarus, Schubart J., 1982b, Cel. Mech. Dyn. Astron., 28, 189 138, 259 Schubart J., 1991, A&A, 241, 297 Davis D.R., Durda D.D., Marzari F., Campo Bagatin A., Schubart J., 2007, Icarus, 188, 189 Gil-Hutton R., 2002, in Bottke W.F., Cellino A., Paolicchi Sessin W., Bressane B.R., 1988, Icarus, 75, 97 P., Binzel R.P., eds., Asteroids III. University of Arizona Tedesco E.F., Noah P.V., Noah M., Price S.D., 2002, AJ, Press, Tucson, p. 545 123, 1056 Dell’Oro A., Marzari F., Paolicchi P., Vanzani A., 2001, Tsiganis K., Kneˇzevi´cZ., Varvoglis H., 2007, Icarus, 186, A&A, 366, 1053 484 Dohnanyi J.W., 1969, J. Geophys. Res., 74, 2531 Tsiganis K., Gomes R., Morbidelli A., Levison H.L., 2005, Farinella P., Froeschl´eC., Gonczi R., 1994, in Milani A., Nature, 435, 459 Di Martino M., Cellino A., eds., Asteroids, comets, meteors Tsuchida M., 1990, Rev. Mexicana Astron. Astrophys., 21, 1993. Kluwer Academic Publishers, Dordrecht, p. 205 585 Ferraz-Mello S., 1988, AJ, 96, 400 Weissman P.R., Bottke W.F., Levison H.L., 2002, in Bot- Ferraz-Mello S., Michtchenko T.A., 1996, A&A, 310, 1021 tke W.F., Cellino A., Paolicchi P., Binzel R.P., eds., Aster- Ferraz-Mello S., Michtchenko T.A., Nesvorn´yD., Roig F., oids III. University of Arizona Press, Tucson, p. 669 Simula, A., 1998a, Planet. Sp. Sci., 46, 1425 Zappal`aV., Cellino A., Farinella P., Milani A., 1994, AJ, Ferraz-Mello S., Michtchenko T.A., Roig F., 1998b, AJ, 107, 772 116, 1491 Zappal`aV., Cellino A., Dell’Oro A., Paolicchi P., 2002, in c 2008 RAS, MNRAS 000, 1–14 14 M. Broˇzet al. Bottke W.F., Cellino A., Paolicchi P., Binzel R.P., eds., Asteroids III. University of Arizona Press, Tucson, p. 619 This paper has been typeset from a TEX/ LATEX file prepared by the author. c 2008 RAS, MNRAS 000, 1–14