Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/preprints/2011/589.pdf Äàòà èçìåíåíèÿ: Mon Oct 24 15:35:09 2011 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 02:58:40 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: comet linear

A population of main belt asteroids co-orbiting with Ceres and Vesta

arXiv:1110.4810v1 [astro-ph.EP] 21 Oct 2011

Apostolos A. Christoua,, Paul Wiegert
a b

b

Armagh Observatory, Col lege Hil l, Armagh BT61 9DG, Northern Ireland, UK The University of Western Ontario, Department of Physics and Astronomy, London, Ontario N6A 3K7, Canada

Abstract We have carried out a search for Main Belt Asteroids (MBAs) co-orbiting with the large MBA Vesta and the dwarf planet Ceres. Through improving the search criteria used in Christou (2000b) and numerical integrations of candidate co orbitals, we have identified approximately 51 (44) ob jects currently in co-orbital libration with Ceres (Vesta). We show that these form part of a larger population of transient co orbitals; 129 (94) MBAs undergo episo des of co-orbital libration with Ceres (Vesta) within a 2 Myr interval centred on the present. The lifetime in the resonance is typically a few times 105 yr but can exceed 2 â 106 yr. The variational properties of the orbits of several co-orbitals were examined. It was found that their present states with respect to the secondary are well determined but knowledge of it is lost typically after 2 â 105 years. Ob jects initially deeper into the co orbital region maintain their co orbital state for longer. Using the mo del of Namouni et al. (1999) we show that their dynamics are similar to those of temporary co orbital NEAs of the Earth and Venus. As in that case, the lifetime of resonant libration is dictated by planetary secular perturbations, the inherent chaoticity of the orbits and close encounters with massive ob jects other than the secondary. In particular we present evidence that, while in the co orbital state, close encounters with the secondary are generally avoided and that Ceres affects the stability of tadpole librators of Vesta. Finally we demonstrate the existence of Quasi-satellite orbiters of both Ceres and Vesta
Corresponding author Email addresses: aac@arm.ac.uk (Apostolos A. Christou), Fax: (Apostolos A. Christou), pwiegert@uwo.ca (Paul Wiegert)


+44 2837 522928

Preprint submitted to Icarus

October 24, 2011


and conclude that decametre-sized ob jects detected in the vicinity of Vesta by the DAWN mission may, in fact, belong to this dynamical class rather than be bona-fide (i.e. keplerian) satellites of Vesta. Keywords: Asteroids, Dynamics, Asteroid Ceres, Asteroid Vesta 1. Intro duction The co orbital resonance, where the gravitational interaction between two bo dies with nearly the same orbital energy leads to stable and predictable motion, is ubiquitous in the solar system. Ob jects attended by known co-orbital companions include Jupiter, Mars as well as the saturnian satellites Tethys, Dione, Janus and Epimetheus (see Christou, 2000a, for a review). More recently, the planet Neptune was added to this list (Sheppard and Trujillo, 2006) while an additional co orbital of Dione was discovered by the Cassini mission (Murray et al., 2005). In all these cases, the motion has been shown to be stable against all but the most slow-acting perturbations (Lissauer et al., 1985; Levison et al., 1997; Brasser et al., 2004; Scholl et al., 2005). The discovery of a co orbital attendant of the Earth on a highly inclined and eccentric orbit (Wiegert et al., 1997, 1998) motivated new theoretical work in the field. Namouni (1999), using Hill's approximation to the Restricted Three Bo dy Problem (R3BP) showed analytically that the intro duction of large eccentricity and inclination mo difies considerably the topology of co orbital dynamics near the secondary mass. It results in the app earance of bounded eccentric orbits (Quasi-Satellites; Mikkola and Innanen, 1997) and, in three dimensions, "compound" orbits and stable transitions between the different mo des of libration. Further, he demonstrated numerically that these results hold when the full R3BP is considered. In this case, the appearance of new types of compound orbits, such as asymmetric mo des or compounds of tadpoles and retrograde satellites, was shown in Namouni et al. (1999) to be due to the secular evolution of the co orbital potential. Such types of co orbital libration were identified in the motion of the ob ject highlighted by Wiegert et al. (1997) as well as other near-Earth asteroids but the secular forcing of the potential in that case is provided by planetary secular perturbations (Namouni et al., 1999; Christou, 2000a). The expected characteristics of the population of co-orbitals of Earth and Venus were investigated by Morais and Morbidelli (2002) and Morais and Morbidelli (2006) respectively. Christou (2000b), motivated by Pluto's ability, as demonstrated by Yu and Tremaine 2

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

(1999) and Nesvorny et al. (2000), to trap other Edgeworth-Kuiper Belt ob´ jects in co-orbital motion with itself, demonstrated in turn that main belt asteroids can co-orbit with the dwarf planet 1 Ceres and the large Main Belt Asteroid (MBA) 4 Vesta. Four such asteroids were identified, two co-orbiting with Ceres and two with Vesta. Here we report the results of a search for additional co orbitals of these two massive asteroids. This was motivated partly by the large growth in the number of sufficiently well-known MBA orbits during the intervening decade, but also a refinement of the search criterion used in the work by Christou. As a result, we find over 200 new transient co-orbital MBAs of Ceres and Vesta. In this work we examine their ensemble properties, use existing dynamical mo dels to understand how they arise and identify similarities with co-orbital populations elsewhere in the solar system, in particular the transient co orbital NEAs of Earth and Venus. The paper is organised as follows: In the next Section we expose our Search metho dology, in particular those aspects which differ from the search carried out by Christou (2000b). In Section 3 we describe the statistics of co orbital lifetime and orbital element distribution found in our integrations. In addition, we examine the robustness of the dynamical structures we observe. In Section 4 we investigate the degree to which the mo del of Namouni et al. (1999) can repro duce the observed dynamics. Section 5 deals with the effects of additional massive asteroids in the three-bo dy dynamics while Section 6 fo cuses on the stability of so-called Quasi-Satellite orbits. Finally, Section 7 summarises our conclusions and identifies further avenues of investigation. 2. Search Metho dology Christou (2000b) searched for candidate Ceres co orbitals by employing the osculating semima jor axis ar of the asteroid relative to that of Ceres (equal to (a - aCeres ) /aCeres ) to highlight ob jects that merited further investigation. This metho d, although it led to the successful identification of two co orbitals, 1372 Haremari and 8877 Rentaro, ignores the existence of high frequency variations in a due to perturbations by the planets, especially Jupiter. This is illustrated in the upper panel of Fig. 1 where the evolution of ar for asteroid 1372 Haremari, one of the ob jects identified by Christou, is depicted. A perio dic term of amplitude 0.001 apparently causes ar to move in and out of Ceres' co orbital zone (bounded by the dashed horizontal lines) 3

57

58 59 60 61 62 63 64 65 66 67


68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

every 20 years. On the other hand, an ob ject with ar would have a conjunction with Ceres every 6 â 104 years. Here = (µ/3)1/3 denotes Hill's parameter in the Circular Restricted Three-Bo dy Problem for a secondary bo dy with a mass µ scaled to that of the central bo dy (in this case, the Sun). This suggests that co orbital motion should be insensitive to these variations and that a low-pass filtered semima jor axis would work better as an indicator of libration in the 1:1 resonance with Ceres. We have chosen this to be the synthetic proper semima jor axis (hereafter referred to as "proper semima jor axis") as defined by Kneevi´ and Milani (2000). Synthetic proper elements zc are numerically-derived constants of an asteroid's motion under planetary perturbations. The proper semima jor axis, in particular, is invariant with respect to secular perturbations up to the second order in the masses under Poisson's theorem and, hence, an appropriate metric to use in this work. In the bottom panel of Fig. 1, we plotted the proper relative semima jor axis ar, p = ap - ap, Ceres /ap, Ceres ; bold horizontal line) superimposed on the ar history of the asteroid but filtered with a 256-point running-box average. The slopes at the beginning and the end of this time series are the result of incomplete averaging. The dots in the bottom panel represent the unaveraged osculating semima jor axis sampled with a constant timestep of 1 yr. It is observed that the proper semima jor axis agrees with the average of its osculating counterpart, whereas the osculating value sampled at t = 0 ( Julian Date 2451545.0; dashed horizontal line) do es not. For our search we used the database of synthetic proper elements for 185546 numbered asteroids computed by Novakovi´, Kneevi´ and Milani c zc that was available as of 09/2008 at the AstDys online information service (http://hamilton.dm.unipi.it/astdys/propsynth/numb.syn). A total of 648 and 514 main-belt asteroids respectively with proper semima jor axes within ±a of those of Ceres and Vesta were identified. Table 1 shows the parameters relevant to this search for the massive asteroids. The second row provides the mass ratio µ of the asteroid relative to the mass of the Sun, the third row its Hill parameter , the fourth row the asteroids' synthetic proper semimaxor axis ap as given in the database and the fifth row the pro duct ap â which is the half-width of the co orbital region in AU. The state vectors of these asteroids at Julian Date 2451545.0 were retrieved from HORIZONS (Giorgini et al., 1996) and numerically integrated one million years in the past and in the future using a mo del of the solar system consisting of the 8 ma jor planets and the asteroids 1 Ceres, 4 Vesta, 2 Pallas and 10 Hygiea. Mass values adopted for these four asteroids were 4


106 107 108 109 110 111 112 113 114 115 116 117 118

taken from Konopliv et al. (2006). Their initial state vectors as well as planetary initial conditions and constants were also retrieved from HORIZONS. The integrations were carried out using the "hybrid" scheme which is part of the MERCURY package (Chambers, 1999). This scheme is based on a second-order mixed variable symplectic (MVS) algorithm; it switches to a Bulirsch-Sto er scheme within a certain distance from a massive ob ject. For all the integrations reported here, this distance was 5 Hill radii. A time step of 4 days or 1/20th of the orbital perio d of Mercury was chosen in order to mitigate the effects of stepsize resonances (Wisdom and Holman, 1992). Trials of this scheme vs an RA15 RADAU integrator (Everhart, 1985) included in the same package and with a tolerance parameter of 10-13 showed the results to be indistinguishable from each other while the hybrid scheme was significantly faster. 3. Results 3.1. Population Statistics In our runs, we observed a total of 129 and 94 asteroids enter in one of the known libration mo des of the 1:1 resonance with Ceres or Vesta respectively at some point during the integrations. A full list of these asteroids is available from the corresponding author upon request. Table 2 shows a statistical breakdown of the observed population according to different types of behaviour. The first row identifies the secondary (Ceres or Vesta). The top part of the Table shows the number of asteroids that were captured into co-orbital libration at some point during the 2 â 106 yr simulation (second column), the number of asteroids that were found to be co-orbitingat present ( "current" co orbitals; third column) and the number of current L4 and L5 tadpole librators (fourth and fifth rows respectively). The bottom part of the table shows the number of current horsesho e librators (second column), the number of ob jects currently in transition between two distinct libration mo des (third column) and the number of current Quasi-Satellite (QS) librators. The fifth column shows the number of asteroids that were librating for the full duration of the numerical integration, although not necessarily in the same mo de ("persistent" co orbitals). Numbers in brackets refer to MBAs that remained in the same libration mo de throughout the integration ("single mo de persistent" co orbitals). A total of 95 MBAs are currently in the 1:1 resonance with either Ceres or Vesta with the ratio of ob jects corresponding to the two asteroids (51/44) being roughly the same as for the 5

119

120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141


142 143 144 145 146 147 148

original sample (129/94). The number of MBAs currently in tadpole orbits is 19 and 12 for the two asteroids respectively while horsesho es are evenly matched at 20 and 21. 11 ob jects for each asteroid are in the pro cess of transitioning between different mo des of libration. One ob ject, MBA 76146, is currently in a Quasi-Satellite (QS) orbit around Ceres with a guiding centre amplitude of 10o in r (rightmost panels in Fig 2) while two other MBAs, 138245 and 166529, are currently transitioning into such a state. 3.2. Some Examples In most cases, the librations were transient i.e. changing to another mo de or to circulation of the critical argument r = - Ceres/Vesta . In that sense, there is a similarity to what is observed for co-orbitals of the Earth and Venus (Namouni et al., 1999; Christou, 2000a; Morais and Morbidelli, 2002, 2006). Compound mo des (unions of Quasi-Satellite and Horsesho e or Tadpole librations) also appear, although they are rare. Examples of different types of behaviour are shown in Figs 2 and 3 for Ceres and Vesta respectively. In the former case, asteroid 71210 (left column) is currently (t = 0 at Julian Date 2451545.0) in a horsesho e configuration with Ceres, transitioning to tadpole libration at times. The position of Ceres at r = 0 is avoided. Asteroid 81522 (centre column) librates around the L4 triangular equilibrium point of Ceres for the duration of the integration. In the right column we observe stable transitions of the orbit of asteroid 76146 between different libration mo des. The maximum excursion of the guiding centre of the asteroid from that of Ceres during the integration is ar = 10-4 or 0.2 , well within the co orbital region. This asteroid becomes a Quasi-Satellite of Ceres at t = -2 â 105 yr and transitions into a horsesho e mo de at t = +105 yr. In the case of Vesta, asteroid 22668 (left column) transitions from a passing to a horsesho e mo de and back again. It is currently a horsesho e of Vesta. Near the end of the integration it enters into Quasi-Satellite libration with Vesta, where it remains. At t = +5 â 105 yr it executes half a libration in a compound Horsesho e/Quasi Satellite mo de. Asteroid 98231 (centre column) is an L5 tadpole of Vesta at t = -1â106 yr. The libration amplitude increases gradually until transition into horsesho e libration o ccurs at t = -3 â 105 yr. During this perio d the libration amplitude begins to decrease until the reverse transition back into L4 libration takes place at t = +6 â 105 yr. This behaviour should be compared with the case of asteroid 71210 in Fig. 2. The evolution of asteroid 156810 (right column) is similar to that of 98231 except

149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177

6


178 179 180

that the rate of increase of the libration amplitude reverses sign before transition into horsesho e libration takes place. As a result, the asteroid librates around the L4 equilibrium point of Vesta for the duration of the integration. 3.3. Variational Properties Although these results can be regarded statistically, their value is increased if we establish their robustness against the ephemeris uncertainties of the asteroids. To investigate this, we have picked 6 asteroids in the sample, populated their 1-sigma uncerainty ellipsoids with 100 clones per object using states and state covariance matrices retrieved from the AstDys online orbital information service, and propagated them forward in time for 106 yr under the same mo del of the solar system as before but using an integrator developed by one of us (PW) with a 4-day time step. This co de utilises the same algorithm as the HYBRID co de within MERCURY. In the original integrations, 65313 and 129109 persist as L5 Tro jans of Ceres, 81522 and 185105 as L4 Tro jans of Ceres and 156810 persists as an L5 Tro jan of Vesta. Asteroid 76146 is a Quasi-Satellite of Ceres until +105 yr. Their Lyapounov times, as given by the proper element database of Novakovi´ et al, are: 3 â 106 , c 5 6 5 5 5 5 â 10 , 2 â 10 , 6 â 10 , 10 and 5 â 10 yr respectively. The results of this exercise are shown in Fig. 4. We find that all clones of 65313 and 81522 remain as Tro jans of Ceres for the full integration. 76146 persists as a Quasi-Satellite of Ceres until +6 â 105 yr. Three clones of 129109 escape from libration around L5 after 5 â 105 yr, but most remain in libration until the end. Those of 185105 begin to diverge at +3 â 105 yr until knowledge of the ob ject's state is lost near the middle of the integration timespan. The clones of the Vesta Tro jan 156810 suffer a similar fate, but divergence is more general in nature; most of the clones enter into other libration mo des by +6 â 105 yr. We conclude that the present state of these ob jects, as determined by the original integrations, is robust. The differences we observe in their evolution could be due to several causes. Firstly, although we use the same force mo del and the same type of integrator, the initial states of the asteroids in the new integrations were taken from AstDys, not HORIZONS, and correspond to a later epo ch of osculation. In addition, the volumes of space sampled by the ob ject's ephemeris uncertainties, being proportional to the eigenvalue pro duct of the respective covariance matrices, are smallest for 65313 and 81522. We therefore expect that, assuming similar Lyapounov times, clones of low-numbered asteroids generally those with longer observational arcs - would be slower to disperse 7

181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214


215 216 217 218 219

than those of high-numbered ones. Interestingly, the clones disp erse faster than one would expect from their Lyapounov times, however some differences would be expected since the mo del used to compute these did not include the gravitational attraction of large asteroids. In Section 5, the role of impulsive perturbations during close encounters is investigated in detail. 3.4. Proper element distribution In order to understand the ensemble properties of the population and how these might differ from those of other MBAs, we have compared the distribution of their proper elements to that of the broader population. Fig 5 shows the distribution of the relative proper semima jor axis (blue curve) ar, p = ap - ap, Ceres/Vesta /ap, Ceres/Vesta of these MBAs, scaled to , superposed on that of all MBAs in the interval [-2 , + 2 ] (red curve). The width of each bin for both panels is 0.06. The sharp peak at ar = -1.2 in the plot for Vesta reflects an increased concentration of asteroid proper elements probably asso ciated with the 36:11 mean motion resonance with Jupiter. Although interesting in its own right, we do not, at present, have reason to believe that its existence near the co-orbital region of Vesta is anything more than coincidental. Hence, we refrain from dicussing it further in this paper. In both cases, the distributions appear to be centred at ar, p = 0. Gaussian fits to the centre µ and the standard deviation of the distribution give a Full Width at Half Maximum (FWHM; 2 2 log 2 ) of 0.334 ± 0.033 and a centre at -0.053 ± 0.017 for the Ceres distribution. The slight offset to the left is probably due to slightly higher counts for the bins left of ar, p = 0. The Vesta distribution is slightly narrower (FWHM of 0.292 ± 0.028) but more symmetric around the origin (centre at -0.006 ± 0.014). No cases of co orbital libration were observed for asteroids with |ar, p | > 0.42 while only two cases (both with Vesta) had |ar, p | > 0.30. Seeking additional insight into the dependence of the co orbital state on the semima jor axis, we examined the distributions of different types of coorbitals - as observed in our simulations - normalised to the total number of MBAs in each bin. In Fig. 6 we show the distributions of all co orbitals (red curve), current co orbitals (blue curve) and persistent co orbitals (gray curve). Fitted values of the Gaussian parameters (µ, FWHM) for the three populations are given in Table 3. The distributions for Vesta co orbitals are consistently narrower than those of Ceres implying that this is a real difference between the two.

220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250

8


251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288

Interestingly, both cases exhibit a hierarchy in the three populations: the persistent population is embedded into the current one, which in turn is embedded into the distribution of all ob jects. One important consequence of this observation is that one can robustly define the boundaries of each population. This is, of course, partly due to the criteria used to define each population but the fact that there are clear differences between the three populations (ie no two populations coincide) is not a trivial one. Hence, persistent co orbitals are confined in the domain |ar, p | < 0.12, current co orbitals within |ar, p | < 0.24 and all co orbitals in the domain |ar, p | < 0.42. The shape of the distributions in Figs. 5 and 6 can be partly attributed to the co orbital dynamics (see Section 4). However, they must also be affected by chaotic `noise' in the determination of the proper elements which we used in our search (Milani and Kneevi´, 1994). In their 2 Myr integrazc tions, Kneevi´ and Milani (2000) regarded the derived proper elements of zc MBAs with ap < 3 â 10-4 , ep < 3 â 10-3 and Ip < 10-3 as `go o d'. All but four of the asteroids considered here belong to this catelogy. The bound for the proper semima jor axis corresponds to 20% of the width of the co orbital region of Ceres and 35% of that of Vesta (Table 1). It is also comparable to the fitted widths of the distributions of the co orbitals found here. Hence, the actual distributions are likely significantly altered by a convolution with an error function. On the other hand, this convolution do es not completely smear out the true distribution of ar, p since, in that case, the observed sorting of the populations according to residence time in the resonance would not o ccur (Fig. 6). The distribution of the proper eccentricities and inclinations of individual co orbitals in relation to those of other MBAs are shown in Fig. 7. Plus symbols denote MBAs that have tested negative for co-orbital motion within the perio d [ - 106 yr, 106 yr]. Asterisks and squares refer to the respective populations of current and persistent co orbitals while the filled circle marks the lo cation of the secondary (either Ceres or Vesta). In the interests of clarity, we have not plotted the distribution of all co orbitals. Instead, we show as triangles those persistent co orbitals that remained in either L4 or L5 tadpole libration for the full simulation. Their lo cation deep into the co orbital region are in agreement with the theoretical upper limit - (8/3)µCeres/Vesta - for near-planar, near-circular tadpole orbits which evaluates to 0.065 for Ceres and 0.053 for Vesta. The significant size of the sample of co orbitals under study prompted a search for trends in the distribution of r . Fig. 8 shows a histogram of this 9


289 290 291 292 293 294 295

quantity at t = 0 for all current co orbitals and with a bin size of 30 . On this we superimpose, as a dashed line, a histogram of all ob jects which are not currently co orbiting with either Ceres or Vesta. The vertical line segments indicate square-ro ot Poissonian uncertainties. There appear to b eno features that stand out above the uncertainties. Hence the co orbital resonance do es not measurably affect the phasing of the populations of current co orbitals with respect to their secondary in this case. 4. Analysis of the Dynamics The dynamical context presents some similarities with co orbitals of the Earth and Venus such as non-negligible eccentricities and inclinations. Here we attempt to mo del the evolution using the framework of the restricted three bo dy problem where a particle's state evolves under the gravity of the Sun and the secondary mass (Namouni et al., 1999). This is done through the expression 8µ a2 = C - S (r ,e,I , ) (1 ) r 3 where 1 aS r · rS S= - d . (2) 2 - | r - r S | a aS

296

297 298 299 300 301 302

303

304 305 306 307 308

Here, a, e, I , and denote the semima jor axis, eccentricity, inclination, argument of pericentre and mean longitude of the particle's orbit. The subscript "S" is used to denote the same quantities for the secondary. The helio centric position vectors of the particle and the secondary are denoted as r and rS respectively. The relative elements ar and r are defined as ar = (a - aS ) /aS , r = -
S

(3)

309 310 311 312 313 314 315 316 317

and µ is the mass of the secondary scaled to the total system mass. Eq. 1 can be seen as a conservation law where C is the constant "energy" of the particle, a2 its kinetic energy and the term containing S its potential energy. r Namouni et al. showed that, as the left-hand side of this expression cannot be negative, it restricts, in general, the evolution of (ar , r ). A collision (r = rS ) can only o ccur for specific combinations of values for e, I and . Hence, actual collisions are rare and the above formulation is generally valid. For computational purposes, Eq. 2 may be evaluated using standard two bo dy formulae (eg Murray and Dermott, 1999) as e, I , and r . One 10


318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355

other consideration that is specific to this paper is that the high frequency harmonics of a are external to the R3BP (hence external to the mo del) and must somehow be removed before the above expressions may be used. However, at t = 0, the relative proper semima jor axis ar, p can be considered to be this low-pass filtered value of ar in the Sun - Ceres/Vesta - MBA problem (see also Section 2). Hence, the constant C can be evaluated and the mo del can be readily applied. In Fig. 9 we show S profiles for each of the MBAs in Figs 2 and 3 compared to the lo cation of the point (r , E = 3C/(8µ)) at t = 0 (dotted circle). As motion is restricted to the domain above S , the mo del predicts that the Ceres co orbitals 71210, 81522, 76146 are currently in L5 tadpole, L4 tadpole and Quasi-Satellite libration respectively. Similarly the Vesta co orbitals 22668, 98231 and 156810 are predicted to be in horsesho e, horsesho e and L5 libration respectively. Referring to the Figure, the mo del apparently succeeds in 5 out of the 6 cases, but fails in the case of the Ceres co orbital 71210 where the observed mo de of libration is a horsesho e. This is probably due to the fact that Eq 1 is evaluated when ar = 0 i.e. at the turning points of the libration. In Namouni et al. (1999), r e, I , and the orbit can be considered "frozen" during a libration cycle. In our case, however, we observe that > r because of the small mass of the secondary. Incidentally, this parity in timescales may also account for the general lack of compound libration mo des for these co orbitals, as controls the relative height of the maxima of S on either side of r = 0. To quantify the effect that this has to our mo del, we evaluated E against S for the example MBAs shown in Figs 2 and 3 but at different values of r and . We found that determination of S is generally insensitive to , except near the lo cal maxima bracketing the origin on the r axis. Physically, these correspond to the closest possible cartesian distances between the particle and the secondary so it is not a surprise that they are sensitive to the orbital elements. Particularly for the case where the mo del failed, E - S 0.6 when the asteroid reaches the far end of the mo del tadpole (r -130o ) and the potential maximum at r = 180o i.e. the ob ject is classified as a horsesho e in agreement with the numerical integrations. Hence, this metho d for determining the resonant mo de is formally valid where the ob ject is currently near the turning point of the libration i.e. those of MBAs 81522, 76146 and 156810. In the other cases, the more involving pro cess of monitoring the quantity E - S in the integrations for a time perio d comparable to a libration cycle would be necessary to establish the libration mo de. Finally, we wish to understand the stability of the QS librator of Ceres, 11


356 357 358 359 360 361 362 363 364 365

76146, in the context of our findings. The sensitivity of S to for the lo cal maxima near r = 0 is important as these features of the potential are the "gatekeepers" for evolution in and out of the QS. This ob ject was captured as a QS from a passing orbit and completes 7 cycles in this mo de before becoming a horsesho e. It is currently at r 8o . Irrespective of the value of E , escape from the QS is certain when St=0 Sr =0 i.e. when the extremum at the origin becomes a lo cal maximum. According to the mo del this o ccurs for -340 < < +12o . Keeping in mind the perio dicity of S in , the asteroid will turn back 360/2/46 4 times before it escapes, in go o d agreement with what is observed in the numerical simulation. 5. The role of other massive asteroids Christou (2000a) showed that Venus and Mars play a key role in the evolution of Earth co-orbitals. These can force transitions between different libration mo des or escape from the resonance altogether. Here the only candidates available to play a similar role are other massive asteroids. In this paper we have fo cused on the effects of Ceres and Vesta - as well as Pallas - on Vesta or Ceres co-orbitals respectively. In a first experiment to determine their role (if any), we have integrated the same six asteroids as in Section 3 but, in the first instance leaving Pallas out of the mo del ("No Pallas" or NP) and, in the second, only under the gravity of the secondary (ie Ceres or Vesta as appropriate; "Secondary Only" or SO). We find that the evolution of the Ceres Tro jans 65313 and 81522 and of the Quasi-Satellite 76146 are the same in both of these runs as well as the original runs where all three massive asteroids were present ("All Masses" or AM). In contrast, we find significant differences in the evolution of the remaining three asteroids, 129