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: m 106

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

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(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

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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


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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

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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

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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

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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.

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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


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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

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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

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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


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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


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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, 129109, 156810 and 185105 (Fig. 10). In the SO runs (top row), all clones of these Tro jan librators persist as such. This is not the case for the NP and AM runs (middle and bottom rows respectively). Most clones ultimately leave this libration mo de, with the exception of 129109 in the AM runs. Although it is difficult, on the basis of a sample of six, to attempt to decouple the effects of the individual massive asteroids, these results indicate that their presence do es have an effect on the lifetime of the co-orbital configurations we have found. To quantify these in a more statistically robust sense, we used MERCURY as before to re-integrate the original 129 and 94 co-orbital MBAs of Ceres and Vesta respectively under the SO, NP and AM mo dels for 106 yr in the 12

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367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391


392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429

future (ie half the timespan of the original integrations). We then check which MBAs survive, either as "persistent" or "single mo de" libration in the new runs. The results are shown in Table 4. As far as persistency is concerned, the fraction of those co-orbitals in the respective samples is between 20% and 25% for the three mo dels examined (SO, NP and AM) but also for both Ceres and Vesta co-orbitals. Upon closer inspection, we find that approximately the same number of persistent co-orbitals (16 for Ceres and 14 for Vesta) are common between the three mo dels. Thus, Vesta harbours a slightly larger fraction of these co-orbitals than do es Ceres. The remainder of the persistent population is composed of ob jects that gain or lose the p ersistency property from one mo del to the next. There is no apparent trend towards one direction or the other; ob jects that lose this property are replenished by the ob jects which gain it. Thus, it seems that persistency of co-orbital motion is independent of the adding or removing massive asteroids from the mo del. If this premise is correct, then a mechanism that can explain the observations is the intrincic chaoticity of the orbits. We saw in Section 3 that the amount of chaotic `noise' in the proper semima jor axis is comparable to the width of the co orbital zones of Ceres and Vesta. It is thus reasonable to expect that some ob jects are removed from the resonance while others are injected into it as a consequence of that element's random walk. The behaviour of single-mo de co orbital libration is altogether different. For Ceres, we find that the addition of other massive asteroids increases the number of persistent single-mo de librators, from 6 (SO) to 14 (NP, AM). For Vesta, we observe the opposite trend: adding Pallas and Ceres to the mo del decreases the number of such librators from 11 (SO) to 3 (NP) and 5 (AM). In addition, only one ob ject, 87955, persisted as a Ceres L4 Tro jan in all three mo dels. The statistics are marginally significant for these low counts. Nevertheless, they compelled us to explore possible causes. It is tempting at this point to try and correlate persistency with the ob jects' Lyapounov Characteristic Exponent (LCE), an established quantifier of chaoticity, available from the proper element database used in this paper. However, we point out that these LCEs were computed under a dynamical mo del that do es not involve any massive asteroids. As it is not clear how the presence of the co-orbital resonance will affect the determination of the LCE, we refrain from attempting such a correlation in this work. In Fig. 11 and 12 we present the distribution of the number of Ceres or Vesta co orbitals respectively that undergo a given number of encounters within 5 Hill radii (RH = a ) of the massive asteroids in the three mo dels. 13


430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467

A number of interesting features are evident. Firstly, persistent co orbitals (green and blue columns), as well as a certain fraction of non-persistent co orbitals (red columns; 25-30% for Ceres co orbitals, 50% of Vesta co orbitals), do not, in general, approach the secondary. This feature was also identified in Christou (2000a) for Earth co-orbitals (cf Fig. 8 of that paper). It seems to be a generic feature of co orbital dynamics at high eccentricity and inclination orbits although the cause-and-effect relationship is not yet clear. In addition, the distribution of encounters for non-persistent co orbitals (red) shows a tail; in other words, many of these ob jects undergo many encounters with the secondary. This is probably due to the slowness of the evolution of the relative longitude of the guiding centre compared to the epicyclic motion. Indeed, we find that these encounters are not randomly distributed in time but o ccur in groups typically spanning a few centuries. In contrast, those related to massive asteroids other than the secondary have an upper cutoff (5 encounters for Pallas and Vesta, 10 for Ceres). The shapes of the distributions for the three classes of co orbitals are similar. The apparently high frequencies observed in the case of Ceres encounters for single-mo de persistent co orbitals of Vesta are probably due to the small size of the population in those classes. The better populated distributions in the AM and SO mo dels mimic the distributions of the other two classes we investigated. A different way to lo ok at the data is to create histograms of minimum distances, since distance is one of the factors (velo city being the other) that determine the magnitude of the change in an MBA's orbit. Fig. 13 and 14 show the distribution of these distances in units of RH . Bin i contains all recorded encounter distances between (i - 1) RH and iRH and have been normalised with respect to the area of the corresponding annulus of width RH on the impact plane. For Ceres co orbitals we do not discern any statistically significant variations, in other words the counts within the different bins are the same given the uncertainties. For Vesta co orbitals, the situation is similar with one exception: we note that single mo de persistent co orbitals (blue) do not approach Ceres closer than one Hill radius. Due to the low counts, we cannot exclude the possibility that we are lo oking at statistical variation in the data. However, if it is the signature of a real trend, it would mean that encounters with Ceres may cause Vesta co orbitals to exit a particular libration mo de. Regardless of their significance at population level, there is clear evidence in the data that Ceres encounters can affect the orbital evolution of co orbitals of Vesta. Fig 15 shows two instances where Vesta co orbital MBAs 45364 and 14


468 469 470 471 472 473

164791 leave the resonance following close encounters with Ceres deep within the Hill sphere of that dwarf planet. We have searched for similar o ccurences in the evolution of Ceres co orbitals but without success. Neither Pallas nor Vesta appear capable of playing a similar role. This is probably due to their smaller masses (hence physically smaller Hill spheres) but also, specifically for the case of Pallas, the higher encounter velo cities. 6. Quasi-Satellites This Section is devoted to the existence, as well as stability, of so-called "Quasi-Satellite" (QS) or "bound" orbits. These appeared first in the literature as Retrograde Satellite (RS) orbits (as it turned out, a sp ecial case of the QS state; Jackson, 1913) and later studied in the context of dynamical systems analysis (H´non, 1969; H´non and Guyot, 1970). More recently, e e the survival of this libration mo de in the real solar system was examined by Mikkola and Innanen (1997) and Wiegert et al. (2000). Currently, two known Quasi-Satellites exist for the Earth (Wa jer, 2010) and one for Venus (Mikkola et al., 2004). Several instances of QS libration relative to Ceres and Vesta were found in our simulations. Before these are discussed in detail, we intro duce some elements of a convenient theoretical framework to study QS motion dynamics, namely that by Namouni (1999, see also Hen´n and Petit, 1986). It employs o the set of relative variables xr = er cos (nt - r )+ ar yr = -2er sin (nt - r )+ zr = Ir sin (nt - r ) (4) (5) (6 )

474

475 476 477 478 479 480 481 482 483 484 485 486 487 488

r

489 490 491 492 493 494 495 496 497 498

where n is the mean motion and er , Ir , r and r the relative eccentricity, relative inclination, relative longitude of pericentre and relative longitude of the ascending no de as defined in Namouni (1999) respectively. In this formulation, the motion is composed of the slow evolution of the guiding centre (ar , r ) to which a fast, three-dimensional epicyclic motion of frequency n and amplitude proportional to er and Ir is superposed. Examples of guiding centre libration while in QS mo de have been shown in Figures 2 (bottom right panel) and 3 (bottom middle panel). To illustrate the relationship between the two components of the motion, we show in Fig 16 two examples of QS motion recovered from our simulations. The left panel shows the cartesian 15


499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536

motion of MBA 50251 in a cartesian helio centric frame that rotates with the mean motion of Ceres (`C') and for a perio d of 5 â 104 yr. The guiding centre libration is indicated by the short arc straddling the secondary (actually a closed lo op of width 10-3 in ar ) while the cartesian motion along the epicycle path appears as a lo op. The inset shows the evolution of the relative longitude r for 106 yr including the perio d of libration around 0 . The example on the right panel shows the motion of MBA 121118 in a frame rotating with the mean motion of Vesta (`V') during a perio d of QS libration around Vesta lasting for 7 â 105 yr. As in the previous case, the guiding centre libration is indicated by the short arc straddling the secondary. Here, the higher amplitude of the fast harmonics in the evolution of ar act to smear out the epicycle to some extent. It is also evident in these plots that QS librators are physically lo cated well outside the secondary's Hill sphere and should not be confused with keplerian satellites. Statistically, we find that 39 (24) out of the 129 (94) Ceres (Vesta) coorbitals exhibited QS motion at some point during the 2 â 106 yr perio d covered by the simulations reported in Section 3, a fraction of 25-30% in both cases. In the case of Vesta we find three episo des of QS libration of unusually long (> 4 â 105 yr) duration. One is that of 121118 illustrated in Fig. 16, the others concern MBAs 22668 and 134633. In Fig. 17 we compare the distribution of the relative proper semima jor axes of all ob jects that became temporary QS librators with the members of the persistent co orbital population. The two are generally separate with the former population further away and on either side of the latter. We believe this is because MBAs capable of becoming QSs are highly energetic. In other words, and referring to the top left and top right panels of Fig. 9, the value of the energy integral 3C/8µ - represented by the horizontal line - is generally well separated from the potential S . Hence the amplitude of ar libration is only small near r = 0 i.e. if the ob ject is in QS mo de at t = 0. This is the case for the one current and the two imminent QS orbiters of Ceres: 76146, 138245 and 166529. The values of ar for these ob jects are -0.0068, -0.0155 and -0.0083 respectively. A particularly interesting subtype of QS librator is that with vanishing guiding centre amplitude. These are the "retrograde satellites" (RS orbits) of Jackson (1913). In that case, the motion can be studied through an energy integral analogous to Eq. 1 but valid only in Hill's approximation to motion near the secondary. It depends explictly on er , Ir , r and r (cf Eqs. 28 and 29 of N99). 16


537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572

Activating the QS mo de requires er while physical proximity of the ob ject to the secondary in an RS orbit is controlled by er , Ir and r . For the small values of considered here, the relative eccentricity can still be small in absolute terms (eg 10 or 10-2 ). If Ir is also small (typically < er for bound orbits; see Namouni, 1999), then such ob jects can remain, in principle, within a few times 106 km of Ceres or Vesta. Note that small er implies a small libration amplitude for r since r < er (see Section 3.3 of N99). The long-term stability of these configurations depends, through the energy integral, on the evolution of the relative orbital elements. If these vary slowly, the asteroid remains trapped in QS motion for many libration cycles. To mo del the secular evolution of the asteroid's orbit we need a theoretical mo del of co-orbital motion within an N-bo dy system. Such mo dels do exist (eg the theory of Message, 1966 valid near L4 and L5 and the more general mo del of Morais, 1999, 2001). However, the case in hand violates several assumptions for which those theories are strictly valid: low to mo derate e and I of the asteroid (here e, I ), a co orbital mo de (QS) that was not examined in those works and a clear separation of timescales between co orbital motion and the secular evolution of the orbit with , (in fact, here we observe that < , ). To check that the secular evolution of e, I , and of the asteroids do es not depend on the presence of absence of the secondary mass we integrated the orbit of Vesta co orbital MBA 139168 with the eight ma jor planets both with and without the massive asteroids (including Vesta) for 1 Myr. We found that the differences in the eccentricity and inclination vectors - e exp i and I exp i respectively - are < 5% between the two cases. In addition, the asteroids' elements do not exhibit any of the features that arise from the three-bo dy dynamics (Namouni, 1999). Hence, it is reasonable to assume that the secular forcing of er and Ir is fully decoupled from the co orbital dynamics and can be mo delled by N-bo dy secular theory. In the absence of mean motion resonances, the secular evolution of the eccentricity and inclination vectors within a system of N bo dies in nearcircular, near-planar orbits around a central mass can be approximated by the so-called Laplace-Lagrange system of 2N first order coupled differential equations which are linear in e and I (Murray & Dermott 1999). The corresponding secular solution for the eccentricity and inclination vectors of a particle intro duced into that system has the form of a sum of N+1-perio dic

17


573

complex functions:
N

e = ef exp i (gf t + f )+
i=1 N

Ei exp i (gi t + i ) Ii exp i (si t + i )
i=1

(7 ) (8 )

I = If exp i (sf t + f )+
574 575 576 577 578 579 580 581 582 583 584

where the first and second terms on the right-hand-side are referred to as the free and forced components respectively. The parameters of the forced component are derived directly from the Laplace-Lagrange solution. They, as well as the free eigenfrequencies gf and sf , are dependent only on the semima jor axes of the planets and the particle. As with the theory of Morais, the validity of these expressions is limitedto low-to-mo derate e, I and gf = gi , sf = si . Brouwer & van Wo erkom (1950) showed that the 5:2 near-resonance between Jupiter and Saturn mo difies the Laplace-Lagrange parameters of the real solar system. However, the independence of the forced component on the particle's e and I still holds; this allows us to write er = ef exp i (gf t + f ) - ef exp i gf t + f I
r

(9) (10)

= If exp i (sf t + f ) - If exp i sf t +

f

585 586

where the superscript ` ' refers to Ceres or Vesta as appropriate. Hence, er and Ir are given by er = e2 + ef 2 +2ef ef cos f I
r

= I +I

2 f

2 f

+2If If cos

sf - s

gf - g

f

f

t + f -

t + f - f
f

(11) · (12)

587 588 589 590

591 592 593

where f = f - f , f = f - f , gf = gf - gf and sf = sf - sf . All these criteria, except the last one, can be tested for by making use of proper elements. The last criterion involves the proper phases which are not 18

The requirement for slow-evolving er and Ir implies gf g f , sf sf . Further, a necessary but not sufficient condition for er and Ir to be small is ef ef , If If . For these to be concurrently small, the following condition must also hold - f gf = (13) - f sf


594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631

included in the database but can, in principle, be recovered through harmonic analysis of the orbital element time series. As an example, we specify an upper limit in ef and sin If of 0.01 and a limit in gf and sf of 1 arscec per year. For Ceres co-orbitals, these correspond to a maximum epicycle excursion of 0.05 AU and a perio d of > 1 Myr in the evolution of er and Ir . 15 and 7 of the 39 QS librators identified in this work satisfy either of these criteria respectively but none do both. On the other hand, 2 co orbitals of Vesta do satisfy both criteria (3 and 8 respectively satisfy either one). Although neither of these two ob jects (78994 and 139168) satisfies the condition for concurrently small er and Ir we find the dynamical evolution of 139168 particularly interesting and we discuss it in some detail below. This asteroid has ef = -0.0029379, sin If = -0.0006901, gf = 0.020303 arcsec/yr and sf = 0.075063 arcsec/yr. Inspection of our simulations of the nominal orbit of the asteroid shows that currently er 0.013, sin Ir 0.15. Both elements are slowly increasing in time from the present with a perio d significantly longer than the 2 â 106 yr spanned by our integrations. Towards the past, The relative inclination continues to decrease with sin Ir 0.10 at t = -106 yr while er reaches a minimum of < 0.008 at t = -8.5 â 105 yr. For most of the integrated timespan, the ob ject is a horsesho e with a very small opening angle, suggesting that it is sufficiently energetic to enter a QS mo de (top left panel of Fig. 9) with 3C/8µ 3.5. Indeed, we find two instances, indicated by the vertical arrows in the top panel of Fig. 18, when the asteroid is trapped into a QS mo de for 104 yr and an r amplitude of 1 ( 10-2 rad). Since this is comparable to er (middle panel), these are not, strictly speaking, RS orbits. Nevertheless, it implies that the planar component of the motion o ccurs within a few times r a of Vesta. In Fig. 19 we show the asteroid's motion for the phase of QS libration at t = -3.15 â 105 yr in a helio centric cartesian frame co-rotating with Vesta's mean motion. In the XY plane (bottom panel), the distance from Vesta varies between 10-2 and 8 â 10-2 AU. However, the excursion in Z (bottom panel) is significantly larger, 0.3 AU. Note that the centre of the planar motion is offset from Vesta's position. This is partly because one is lo oking at the superposition of two harmonic mo des (guiding centre and epicycle) of comparable amplitude. In addition, the potential minimum may not be exactly at r = 0 since the maxima that bracket it are generally not equal unless r = k . This becomes apparent if one expands the Hill potential to order higher than 2 or utilises the full potential of the averaged 19


632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657

motion i.e. Eq. 1. Since er < Ir the QS-enabling potential minimum at the origin exists only for values of r sufficiently far from k while the maxima on either side are highest for r = k + /2, becoming singularities if Ir = 0. These are the K > 2 orbits of N99. We can verify that this is the case here by overplotting r on the top panel of Fig. 18 (dashed curve). We find that, during QS libration, the value of r is near ±90 (horizontal dotted lines) as expected. Eventually, er and Ir will pass through their minima sufficiently closely in time to make long-term capture in a low amplitude QS mo de possible. For the given values of gf and sf this should o ccur every 6.4 â 107 yr. A crude estimate on the expected number of such ob jects may be made under the assumptions that (a) this is the only ob ject of its type, (b) the proper element catalog is complete for MBAs with a < 2.8 AU down to an absolute magnitude H 16 and (c) the duration of QS capture is 105 yr, similar to large amplitude QS phases observed for other asteroids. The resulting average frequency of such ob jects at any one time is 1.5 â 10-3 . Adopting an absolute magnitude distribution law of 100.3H (Gladman et al., 2009), we find that the frequency reaches unity for H 26 or D = 12 - 37 m ob jects. Unlike large amplitude Quasi-Satellites, the cartesian velo city of such ob jects with respect to Vesta is not high. For 139168 it ranges from 2.2 to 3.2 km sec-1 for the integration spanning the last 1 Myr and will be lower if Ir is smaller. Hence, and in view of possible in situ satellite searches by missions such as DAWN (Cellino et al., 2006), newly discovered ob jects in apparent proximity to Vesta in the sky would need to be carefully followed up to determine whether they are "true" (ie keplerian) satellites or small-amplitude Quasi-Satellites. 7. Conclusions and Discussion In this work we have demonstrated the existence of a population of Main Belt Asteroids (MBAs) in the co orbital resonance with the large asteroid (4) Vesta and the dwarf planet (1) Ceres. Libration within the resonance is transient in nature; our integrations show that these episo des can last for > 2 â 106 yr. Partly due to the significant eccentricities and inclinations of these asteroids, we find that their dynamics are similar to those that govern the evolution of near-Earth asteroids in the 1:1 resonance with the Earth and Venus. However, due to the high density of ob jects as a function of semima jor axis, a steady state population of 50 co-orbitals of Ceres and 45 20

658

659 660 661 662 663 664 665 666 667


668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701

of Vesta is maintained. Apart from the natural dynamics of eccentric and inclined 1:1 librators, we identify two other mechanisms which contribute to the temporary nature of these ob jects. One is the inherent chaoticity of the orbits; with some exceptions, particles started in neighbouring orbits evolve apart over 105 yr timescales. The other is close encounters with massive asteroids that do not participate in the Sun-Secondary-Particle three bo dy problem. We show individual cases of asteroids leaving the co orbital resonance with Vesta following a deep encounter with Ceres. It is not clear if the latter effect is significant at the population level. It may be adversely affecting the o ccurence of long-lived tadpole librators of Vesta. Finally, we show that bound or Quasi-Satellite orbits around both Ceres and Vesta can exist and identify 3 current Quasi-Satellite librators around Ceres as well as an ob ject which may experience episo des of long-lived bound motion within a few times 10-2 AU from Vesta. The demonstration of a resonant mechanism within the asteroid belt which acts independently of the ma jor planets raises some interesting questions to be addressed by future work. One of these is whether there exists a threshold below which a mass becomes a particle in real planetary systems. In partial response to this question - and as part of the integrations reported in Section 3 - we simulated the motion of several MBAs with proper semima jor axes within the co orbital region of 2 Pallas and 10 Hygiea. None of these was trapped in co-orbital libration which leads us to conclude that this threshold for the solar system's Main Asteroid Belt is 10-10 solar masses. However, this may not be true of other planetary systems with fewer and/or less massive planets. We speculate that co-orbital trapping of planetesimals by terrestrial planets or large asteroids may give rise to observationally verifiable dynamical structures. This is not a new idea (eg Moldovan et al., 2010) but our results show that even small bo dies (10-10 < µ < 10-6 ) can maintain transient populations of Tro jans in a steady state and that the dynamical excitation of planetesimals in a disk do es not necessarily imply that the co orbital resonance becomes ineffective. In this context, it may be relevant to the dynamical evolution of the larger planetesimals in protoplanetary disks and the planets or protoplanet cores embedded within them (Papaloizou et al., 2007).

21


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Acknowledgements The authors would like to thank Dr Fathi Namouni for kindly answering our numerous questions regarding eccentric and inclined co orbital motion. Part of this work was carried out during a visit of AAC at UWO, funded by NASA's Meteoroid Environment Office (MEO). Astronomical research at the Armagh Observatory is funded by the Northern Ireland Department of Culture, Arts and Leisure (DCAL).

703 704 705 706 707 708

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805

List of Figures 1 Comparison of proper, averaged and osculating semima jor axis for asteroid 1372 Haremari during a numerical 104 yr numerical integration of its orbit. See text for details. . . . . . . . . Examples of dynamical evolution of specific asteroids co orbiting with Ceres as determined by our numerical integrations over 2 â 106 yr. The upper row shows the evolution of the relative semima jor axis ar while the bottom row that of the relative longitude r . . . . . . . . . . . . . . . . . . . . . . As Fig. 2 but for co orbitals of Vesta. . . . . . . . . . . . . . Dynamical evolution of 600 clones of asteroids 65313 (Ceres co orbital; top left), 76146 (Ceres co orbital; top right), 81522 (Ceres co orbital; centre left), 129109 (Ceres co orbital; centre right), 156810 (Vesta co orbital; bottom left) and 185105 (Ceres co orbital; bottom right) for 106 yr as explained in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Histograms of the statistical distribution of co-orbital asteroids of Ceres (upper panel) and Vesta (lower panel) within the co orbital regions of these massive asteroids as quantified by the relative proper semima jor axis ar, p . . . . . . . . . . . Histograms of the statistical distribution of different types of Ceres (top) and Vesta (bottom) co-orbital asteroids divided over the total number of asteroids in each bin. The x-axis is in the same units as Fig. 5. . . . . . . . . . . . . . . . . . . . Proper element distribution of different types of Ceres (top) and Vesta (bottom) co-orbital asteroids. The two left panels show the proper eccentricity as a function of the relative proper semima jor axis (units of as in Fig. 5) while the right two panels show the proper inclination. . . . . . . . . . . . . Histogram of the relative longitudes r of asteroids currently co orbiting with Ceres (upper panel) or Vesta (lower panel). The position of the secondary (r = 0) is at bin 6. Counts have been normalised to the average value per bin. . . . . .

806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837

. 34

2

3 4

. 35 . 36

. 37

5

. 38

6

. 39

7

. 40

8

. 41

27


838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870

9

10

11

12 13

14 15

16

17

18

Mo del fits to the motion of Ceres (top row) and Vesta (bottom row) co orbital MBAs shown in Figs 2 and 3 respectively. The curve, horizonal line and dotted circle correspond to the profile of S (Eq. 2), the quantity 3C/8µ and the point (r , 3C/8µ) respectively at t = 0 as functions of the relative longitude r in degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical evolution of 300 clones of asteroids 129109, 156810 and 185105 (left, centre and middle column respectively) for the three mo dels discussed in the text. . . . . . . . . . . . . Histograms of numbers of encounters (< 5RH ) of Ceres coorbitals with massive asteroids in our three mo dels. Bins have been normalised by the total number of ob jects in each persistency class. . . . . . . . . . . . . . . . . . . . . . . . . . . As Fig. 11 but for Vesta co-orbitals. . . . . . . . . . . . . . . Histograms of numbers of encounters of Ceres co-orbitals with massive asteroids as a function of encounter distance in the three mo dels described in Section 5. . . . . . . . . . . . . . . As Fig. 13 but for Vesta co-orbitals. . . . . . . . . . . . . . . Two examples of co orbitals of Vesta ejected from the resonance after close encounters with Ceres. The vertical arrows indicate the moment of encounter with Ceres (`C') while the number in brackets is the closest approach distance in RH . . . . . . . Two examples of MBAs that exhibited Quasi-Satellite libration with Ceres and Vesta in our simulations. The position of the Sun is indicated by the character `S'. . . . . . . . . . . . Histogram of the relative proper semima jor axis (in units of ) of those MBAs that persisted in co orbital libration (blue line) and those that became temporary quasi-satellites (red line) of Ceres (left) and Vesta (right). . . . . . . . . . . . . . . . . . Detail of the past dynamical evolution of 139168 in our simulations. The arrows indicate instances when the asteroid became an eccentric retrograde satellite of Vesta with r almost stationary at 0 . . . . . . . . . . . . . . . . . . . . . . . . . .

. 42

. 43

. 44 . 45

. 46 . 47

. 48

. 49

. 50

. 51

28


871 872 873 874 875 876

19

Motion of 139168 with respect to Vesta in cartesian ecliptic co ordinates during an episo de of QS capture at t -3 â 105 yr indicated in Fig. 18. The dashed lines mark the lo cation x = ap, Vesta 2.3615 au, y = z = 0. Note the large amplitude of the vertical motion ( right panel) compared to the planar pro jection (left panel). . . . . . . . . . . . . . . . . . . . . . . 52

29


Table 1: Dynamical parameters of large asteroids considered in this work. See text for details.

µ â 10 â 104 aproper (AU) aproper (â104 AU)

10

Ceres Vesta Pallas Hygiea 4.699 1.358 1.026 0.454 5.42 3.56 3.25 2.47 2.7670962 2.3615126 2.7709176 3.1417827 15.0 8.4 9.0 7.8

30


Table 2: Statistical results of the numerical simulations reported in Section 3. See text for details.

Ceres Vesta Ceres Vesta

Full 129 94 Horsesho e 20 21

Now 51 44 Transition 11 11

L4 Tadpole 1 5 (2 ) 5 QS 1 0

L5 Tadpole 4 (2 ) 7 (1 ) Always 18 10

31


Table 3: Estimated mean and Full Width at Half Maximum (FWHM) of distributions presented in Fig. 6.

Ceres Vesta

All co orbitals µ FWHM ±0.011 -0.029 0.341±0.021 -0.006±0.009 0.301±0.019

Current co orbitals µ FWHM ±0.002 -0.019 0.169±0.005 -0.011±0.005 0.151±0.009

Persistent co orbitals µ FWHM ±0.001 -0.023 0.096±0.001 -0.010±0.001 0.052±0.002

32


Table 4: Results of the numerical simulations for the models described in Section 5. See text for details.

Secondary Bo dy Ceres Vesta

Ceres/Vesta Only Persistent Single Mo de 26/129 6/129 23/94 11/94

No Pallas Persistent Single Mo de 28/129 14/129 18/94 3/94

All Massive Asteroids Persistent Single Mo de 32/129 15/129 18/94 6/94

33


0.002
Ceres Ceres

)/a

0.001 0.000 -0.001 -0.002 0 0.0004 50 100 150 200

Ceres Ceres

(a1372 - a )/a (a1372 - a

0.0002 0.0000 -0.0002 -0.0004 0 2000 4000 6000 time (years) 8000 10000

Figure 1: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

34


71210
0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 -0.0001 -0.0002 -0.0003 -0.0004 -0.0005 180 120 r (degrees) 60 0 -60 -120 -180 -10 -8 -6 -4 -2 0 2 4 6

81522

76146

a

r

8 10 -10 -8 -6 -4 -2 0 2 4 time ( x1e5 yr)

6

8 10 -10 -8 -6 -4 -2 0

2

4

6

8 10

Figure 2: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

35


22668
0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 -0.0001 -0.0002 -0.0003 -0.0004 -0.0005 180 120 r (degrees) 60 0 -60 -120 -180 -10 -8 -6 -4 -2 0 2 4 6

98231

156810

a

r

8 10 -10 -8 -6 -4 -2 0 2 4 time (x 1e5 yr)

6

8 10 -10 -8 -6 -4 -2 0

2

4

6

8 10

Figure 3: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

36


37
Figure 4: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta


70 60 50 40 30 20 10 0 -2 70 60 50 40 30 20 10 0 -2 -1.5 -1.5

MBAs < 2 epsilon from Ceres Coorbiting within 1E6 yrs from t=0

-1

-0.5 0 0.5 a_r, p (units of epsilon)

1

1.5

2

MBAs < 2 epsilon from Vesta Coorbiting within 1E6 yrs from t=0

-1

-0.5 0 0.5 a_r, p (units of epsilon)

1

1.5

2

Figure 5: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

38


1.2 ALL 1.1 CURRENT (t=0) PERSISTENT 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.6 -0.4 -0.2 0 0.2 0.4 a_r, p (units of 1.2 ALL 1.1 CURRENT (t=0) PERSISTENT 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.6 -0.4 -0.2 0 0.2 0.4 a_r, p (units of

COORBITALS COORBITALS COORBITALS

0.6 0.8 epsilon)

1

1.2

1.4

COORBITALS COORBITALS COORBITALS

0.6 0.8 epsilon)

1

1.2

1.4

Figure 6: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

39


0.4 0.35 proper eccentricity 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.3 0.4 0.35 proper eccentricity 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.3 -0.2 -0.2

proper inclination (deg)

ALL MBAs CURRENT COORBITALS PERSISTENT COORBITALS CERES PERM. TADPOLES

20

15

ALL MBAs CURRENT COORBITALS PERSISTENT COORBITALS CERES PERM. TADPOLES

10

5

-0.1

0

0.1

0.2

0.3

0 -0.3 20

-0.2

-0.1

0

0.1

0.2

0.3

a_r, p (units of epsilon) ALL MBAs CURRENT COORBITALS PERSISTENT COORBITALS VESTA PERM. TADPOLES

a_r, p (units of epsilon) ALL MBAs CURRENT COORBITALS PERSISTENT COORBITALS VESTA PERM. TADPOLES

proper inclination (deg) 0.2 0.3

15

10

5

-0.1

0

0.1

0 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

a_r, p (units of epsilon)

a_r, p (units of epsilon)

Figure 7: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

40


3

2

1 0 0 2 4 6 8 10 12

3

2

1 0 0 2 4 6 8 10 12

Figure 8: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

41


4 3.5 3 2.5 2 1.5 1 0.5 -150 4 3.5 3 2.5 2 1.5 1 0.5 -150 -100 -50 0 50 100 150 -100 -50 0 50 100 150 71210

4 3.5 3 2.5 2 81522

2 1.8 1.6 1.4 1.2 1 76146

o

o

1.5 1 0.5 -150 4 3.5 3 2.5 2 1.5 1 0.5 -150 -100 -50 0 50 100 150 98231 -100 -50 0 50 100 150

o

0.8 0.6 -150 2.5 156810 2 1.5 -100 -50 0 50 100 150

22668

o

o
1

o
0.5

-150

-100

-50

0

50

100

150

Figure 9: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

42


Figure 10: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

43


1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 CERES COORBITALS CERES, PALLAS, VESTA CERES ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 CERES COORBITALS CERES, PALLAS, VESTA PALLAS ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 CERES COORBITALS CERES, PALLAS, VESTA VESTA ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 CERES COORBITALS CERES, VESTA CERES ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 CERES COORBITALS CERES, VESTA VESTA ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 CERES COORBITALS CERES CERES ENCOUNTERS

Figure 11: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

44


1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 VESTA COORBITALS CERES, PALLAS, VESTA VESTA ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 VESTA COORBITALS CERES, PALLAS, VESTA PALLAS ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 VESTA COORBITALS CERES, PALLAS, VESTA CERES ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 VESTA COORBITALS CERES, VESTA VESTA ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 VESTA COORBITALS CERES, VESTA CERES ENCOUNTERS

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 VESTA COORBITALS VESTA VESTA ENCOUNTERS

Figure 12: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

45


50 45 35 30 25 20 15 10 5 1 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 CERES COORBITALS CERES, VESTA CERES ENCOUNTERS 2 3 4 5 CERES COORBITALS CERES ENCOUNTERS 40 CERES, PALLAS, VESTA

50 45 35 30 25 20 15 10 5 1 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 CERES COORBITALS CERES, VESTA VESTA ENCOUNTERS 2 3 4 5 CERES COORBITALS PALLAS ENCOUNTERS 40 CERES, PALLAS, VESTA

50 45 35 30 25 20 15 10 5 1 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 CERES COORBITALS CERES CERES ENCOUNTERS 2 3 4 5 CERES COORBITALS VESTA ENCOUNTERS 40 CERES, PALLAS, VESTA

Figure 13: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

46


50 45 35 30 25 20 15 10 5 1 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 VESTA COORBITALS CERES, VESTA VESTA ENCOUNTERS 2 3 4 5 VESTA COORBITALS VESTA ENCOUNTERS 40 CERES, PALLAS, VESTA

50 45 35 30 25 20 15 10 5 1 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 VESTA COORBITALS CERES, VESTA CERES ENCOUNTERS 2 3 4 5 VESTA COORBITALS PALLAS ENCOUNTERS 40 CERES, PALLAS, VESTA

50 45 35 30 25 20 15 10 5 1 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 VESTA COORBITALS VESTA VESTA ENCOUNTERS 2 3 4 5 VESTA COORBITALS CERES ENCOUNTERS 40 CERES, PALLAS, VESTA

Figure 14: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

47


180 120 60
C(0.11 RH)

r (degrees)

0 -60 -120 -180 180 120 60
C(0.31 RH) 45364

r (degrees)

0 -60 -120 -180 0 2 4 6 time ( x1e5 yr) 8 10

Figure 15: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

48


2

2

0 Y (AU)
180 120 r (degrees)

0 Y (AU)
180 120 r (degrees)

-2

60 0 -60 -120 -180 0 2 4 6 8 time ( x1e5 yr) 10

-2

60 0 -60 -120 -180 0 2 4 6 8 time ( x1e5 yr) 10

-4 -4

-4 2 4 -4

-2

0 X (AU)

-2

0 X (AU)

2

4

Figure 16: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

49


16 14 12 10 8 6 4 2

PERSISTENT COORBITALS QUASI SATELLITES

16 14 12 10 8 6 4 2

PERSISTENT COORBITALS QUASI SATELLITES

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 a_r, p (units of epsilon)

0.3

0.4

0.5

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 a_r, p (units of epsilon)

0.3

0.4

0.5

Figure 17: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

50


180 120 60 0 -60 -120 -180 0.016 0.014 0.012 0.010 er 0.008 0.006 0.004 0.002 0.140 Ir (rad) 0.135 0.130 0.125 -6 -5 -4 time ( x1e5 yr) -3 -2

Figure 18: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

r (degrees)

51


0.4

0.4

0.2

0.2

Y (AU)

0.0

Z (AU) 2.2 2.3 2.4 X (AU) 2.5 2.6

0.0

-0.2

-0.2

-0.4 2.1

-0.4 2.1

2.2

2.3 2.4 X (AU)

2.5

2.6

Figure 19: Christou and Wiegert 2010, Coorbitals of Ceres and Vesta

52