Yarkovsky-Driven Spreading of the Eureka Family of Mars Trojans
aa r X i v : . [ a s t r o - ph . E P ] D ec Yarkovsky-Driven Spreading of the Eureka Family of Mars Trojans
Matija ´Cuk , Apostolos A. Christou , Douglas P. Hamilton Carl Sagan Center, SETI Institute189 North Bernardo Avenue, Mountain View, CA 94043 Armagh ObservatoryCollege Hill, Armagh BT61 9DG, NI, UK Department of Astronomy, University of Maryland1113 Physical Sciences Complex, Bldg. 415, College Park, MD 20742E-mail: [email protected] to IcarusDecember 3 rd Introduction
A Trojan (or ”tadpole”) coorbital companion is a small body that has thesame mean orbital distance as a planet, and librates around the so-calledtriangular Lagrangian points, which are located 60 ◦ ahead and behind theplanet. Trojans’ orbits can in principle be stable for star-planet (or planet-satellite) mass ratios above about 25 (Murray and Dermott, 1999). In ourSolar System, only Jupiter, Neptune and Mars are known to have long-termstable Trojan companions (Dotto et al., 2008). Additionally, Saturn’s moonsTethys and Dione also have two Trojan coorbitals each (Murray and Dermott,1999; Murray et al., 2005). Giant planets are thought to have acquired theirTrojans during a violent early episode of planetary migration and/or scatter-ing (Morbidelli et al., 2005; Nesvorn´y and Vokrouhlick´y, 2009; Nesvorn´y et al.,2013). Any primordial Saturn and Uranus Trojans would have been sub-sequently lost through planetary perturbations (Nesvorn´y and Dones, 2002;Dvorak et al., 2010; Hou et al., 2014), with the known Uranus Trojans thoughtto be temporarily captured from among the Centaurs (Alexandersen et al.,2013; de la Fuente Marcos and de la Fuente Marcos, 2014). Some hypothet-ical Trojans of Earth would have been long-term stable, with the situationat Venus being less clear (Tabachnik and Evans, 2000; Scholl et al., 2005;´Cuk et al., 2012; Marzari and Scholl, 2013); however, so far only temporarycoorbitals of these planets are known (Christou, 2000; Christou and Asher,2011; Connors et al., 2011). 4he first Mars Trojan to be discovered was 5261 Eureka in 1990 (Bowell et al.,1990). Since then, a total of nine Mars Trojans have been discovered andwere found to be stable (Mikkola et al., 1994; Mikkola and Innanen, 1994;Connors et al., 2005; Scholl et al., 2005; de la Fuente Marcos and de la Fuente Marcos,2013, and references therein). The three largest Mars Trojans do not formany kind of cluster: Eureka and 1998 VF are both in L but have verydifferent orbits, and 1999 UJ is in L . Recently, Christou (2013) proposedthat some of the smaller L Trojans form an orbital cluster together withEureka. Subsequently, multiple teams of researchers recognized a likely 6-member orbital cluster (de la Fuente Marcos and de la Fuente Marcos, 2013,A. Christou, 2014, pers. comm.): Eureka, 2001 DH , 2007 NS , 2011 SC ,2011 SL and 2011 UN , to which we add 2011 UB (based on latestorbital elements listed on JPL Solar System Dynamics web-page) . In thispaper, we will consider these seven objects only, as orbits of more newly dis-covered objects are likely to have large errors. This is especially true of MarsTrojans’ libration amplitudes, which can vary a lot due to relatively smallchanges in the solutions for their semimajor axes.Separately from orbital clustering, spectroscopy can resolve relationshipsbetween potential family members. Eureka and 1998 VF both have highalbedos (Trilling et al., 2007), but are not of the same surface composi-tion (Rivkin et al., 2003). Rivkin et al. (2007) find Eureka to be closest The recovery of this and other potential Mars Trojans was the result of a tar-geted campaign by A. A. Christou, O. Vaduvescu and the EURONEAR collaboration(Vaduvescu et al., 2013; Christou et al., 2014).
5o angrite meteorites, while Lim et al. (2011) find it to be better matchedby olivine-rich R-chondrites. Non-cluster member 1998 VF is likely tobe a primitive achondrite (Rivkin et al., 2007), while the sole L Trojan1999 UJ has a much lower albedo, and presumably very different composi-tion (Mainzer et al., 2012).In this paper, we explore if the Eureka cluster’s orbital distribution couldresult from a initially compact collisional family spreading due to planetaryperturbations and the radiative Yarkovsky effect. Orbits of asteroid families born in collisional disruptions spread due to bothgravitational and non-gravitational (usually radiative) forces (Bottke et al.,2001). In the main belt, the Yarkovsky effect (Farinella et al., 1998; Farinella and Vokrouhlicky,1999; Bottke et al., 2006) is by far the most important radiative force on theobservable asteroids. The details of a collisional family dispersal are likely tobe different among Mars Trojans from those of main-belt asteroids (MBAs).The coorbital relationship with Mars prevents the Trojans from drifting insemimajor axis, making their libration amplitudes, eccentricities and inclina-tions the only relevant parameters in which we can identify potential families.Additionally, they are largely free of collisions, allowing for the YORP ra-diation torques (Rubincam, 2000; Bottke et al., 2006) to fully dominate theevolution of their spins, with implications for the long-term behavior of the6arkovsky drift (sub-km Mars Trojans are expected to have their spins com-pletely re-oriented by YORP in a less than a Myr). All these factors makeit hard to use the lessons from MBA families for studying a potential familyamong Mars Trojans, and independent numerical modeling is clearly needed.In order to separate the effects of (purely gravitational) planetary perturba-tions and the radiation forces, we decided to first model spreading of a familydue to gravity alone. Such a simulation is certainly unlikely to reflect a real-life Mars Trojan family consisting mostly of sub-km bodies, but is valuablein providing a control for our Yarkovsky simulations.We used the SWIFT-rmvs4 symplectic integrator which efficiently in-tegrates perturbed Keplerian orbits, and is able to resolve close encoun-ters between massless test particles and the planets (Levison and Duncan,1994). While in previous versions of SWIFT the timestep used for integratingplanets depended on the timing of particle-planet encounters, SWIFT-rmvs4propagates planets using constant timestep, not dependent on the fate of testparticles. This enabled all of the 100 test particles to experience the samehistory of the chaotic inner Solar System, despite the computation being di-vided between five different processors. The initial conditions for the eightplanets and Eureka are based on the vectors for January 1st, 2000, down-loaded from JPL’s horizons ephemeris service. Test particles had the sameinitial positions as Eureka, with their velocities differing slightly from that of Vectors used for test-particle simulations were obtained in 2013, while those used toproduce Table 1 are from August 2014, leading to slight differences between the orbit ofEureka and the centers of the simulated clusters. − of the relevant velocity component, amounting to1 m/s in the y-direction, 0.5 m/s in the z-direction (comparable to Eureka’slikely escape velocity). This was in excess of a realistic collisional fragmentdispersion, but enabled us to sample a larger phase space (for comparison,escape from the Trojan region would require about 30 m/s). The simulationswere run for 10 years with a 5-day timestep.At the end of the simulation, 98 of the hundred particles were still MarsTrojans (the remaining two were destabilized). This agrees with the resultsof Scholl et al. (2005), who find that Eureka is most likely long-term stable,with only ≃
20% chance of escape over 4.5 Gyr. Also, just like Scholl et al.(2005), we find that it is the eccentricity that disperses most rapidly dueto planetary perturbations (Fig. 1). The eccentricity dispersion of our syn-thetic cluster reaches that of the actual cluster in at most a few hundredMyr. This is contrasted by the much slower dispersion in inclination; theinclination dispersion of the synthetic family does not approach the size ofthe Eureka cluster by the end of the integration. This discrepancy betweenthe eccentricity and inclination dispersals is independent of which bodies weinclude in the Eureka cluster. If we exclude outlying 2001 SC from thecluster, inclinations could be explained by a gravity-only dispersal over theSolar System’s age. However, the age of the family according to the scatterof member eccentricities would be only 10 years or so, clearly indicating that8his cannot be a collisional cluster that has been spreading due to gravityalone (Fig. 2).Our choice of initial conditions produced a relatively large dispersal inlibration amplitudes (2 − ◦ ), which has not changed appreciably in thecourse of the simulation. As we noted, our grid was too large for a realisticcollisional family, as some test particles received initial kicks as large as 6 m/s,much in excess of Eureka’s escape velocity. Overall, it appears that long-term planetary perturbations do not affect libration amplitudes as much asthey do eccentricities, as found by Scholl et al. (2005). This relatively rapiddispersal of eccentricities is the most important lesson from the simulationsdescribed in this section, and is certainly relevant for analysis of more realisticsimulations including the Yarkovsky effect. When introducing radiation forces into our simulation, we decided to restrictourselves to the simplest case: a constant tangential force on the particle.This is a good model of the net Yarkovsky acceleration over short timescales,but may not be accurate in the long term if the spin axis or the spin rate arechanging. In the absence of collisions, spin evolution of a Trojan is expectedto be dominated by the YORP torque (Rubincam, 2000; Bottke et al., 2006).Depending on the model, YORP torques may evolve bodies toward asymp-totic states (Vokrouhlick´y and ˇCapek, 2002; ˇCapek and Vokrouhlick´y, 2004),9r could cause reshaping that would settle the body into a quasi-stable state(Statler, 2009; Cotto-Figueroa et al., 2013). Depending on the outcome ofYORP, a sub-km asteroid not subject to collisions could keep the same senseof rotation indefinitely, or would cycle through obliquities and spin rates on1-Myr timecales. This has clear implications for Yarkovsky effect over Gyrtimescales. Therefore our constant Yarkovsky force should be seen as rep-resenting a long-term average of the thermal drift. Regardless of the actualbehavior of the body, the average thermal drift rate over 1 Gyr is a singlenumber, and by giving our particles a range of Yarkovsky drift rates we areeffectively exploring this averaged rate.Ninety-six clones of Eureka were assigned drift rates in the range − . × − AU/Myr < ˙ a < . × − AU/Myr and integrated for one billion years.While Yarkovsky-capable standard integrators are available (Broˇz, 2006), wewanted to have full control over our numerical experiment and used thehome-made SIMPL code ( ´Cuk et al., 2013). SIMPL does not allow for closeencounters, but stable Mars Trojans never have encounters with Mars orother planets. We used the parametric migration option in SIMPL to assign aconstant tangential force on the particle (i.e., a force in the plane of the orbit,perpendicular to the heliocentric radius vector). The force was assigned r − . dependence on heliocentric distance, in order to produce ˙ a ∼ a − typical ofthe Yarkovsky effect (Bottke et al., 2006). We also modified SIMPL so thatour Mars Trojans are treated as massless particles. The timestep was 9.1days. 10esults of this first Yarkovsky simulation are shown in Figs. 3 and 4.In all figures, particles with inward-type Yarkovsky drift (which can be in-terpreted as retrograde rotators) are represented with red plusses, and thosewith outward-type drift (which have to be prograde rotators) with blue Xs.Note that the direction of migration discussed here is one bodies would followif they were not co-orbitals; in our simulation the mean semimajor axes of testparticles always stays equal to that of Mars as must be the case for coorbitalmotion (unless an escape occurs). Also, the connection between Yarkovskydrift direction and rotation holds only for diurnal Yarkovsky effect, while theseasonal variety makes all bodies migrate inward; this will be discussed inmore detail in Section 4. The top panel in Fig. 3 shows evolution of theparticles’ eccentricities over time. There is a systematic trend that outwardmigrators increase their eccentricities, while the inward ones decrease theireccentricities over time. These two groups are not well-separated and thereis lots of overlap, likely due to gravitationally-generated chaos.Comparison with Fig. 1 shows that dispersion of eccentricities, even foroutward migrators, is somewhat suppressed in the Yarkovsky simulation.This is easy to understand for would-be inward migrators, as their eccen-tricities are directly decreased by the Yarkovsky effect. However, even theparticles with positive or near-zero Yarkovsky drift have more stable eccen-tricities in the second simulation. The most likely explanation is that, due tothe well-known chaotic nature of planetary orbits, the amount of chaotic dif-fusion was smaller in the Yarkovsky simulation as those particles experienced11 different dynamical history. The top panel in Fig. 5 shows the eccentricityof Mars in the two numerical experiments, and it is clearly lower for much ofthe Yarkovsky simulation.In order to get a better grip on the effect of martian eccentricity, weran seven more planets-only 1000 Gyr simulations (i.e. without Trojans).Initial conditions were very similar to our Yarkovsky simulation (shown inFig. 3), only with Earth initially shifted by few hundred meters along thex-axis. The bottom panel of Fig. 5 shows the (averaged) eccentricity ofMars in these simulations (together with that from the original Yarkovskyrun). It is clear that the test particles plotted in Fig. 3 experienced below-average eccentricity of Mars’ orbit over 1 Gyr. We then chose one of thehigher-eccentricity simulations (plotted with thick dashed line), and re-ranit with the same Trojan test-particles as in the first Yarkovsky simulation.The results of this second Yarkovsky simulation are plotted in Figs. 6 and7. The Trojans’ final eccentricities in Fig. 7 are more spread out than inFig. 4, confirming that the eccentricity dispersion of the family depends onthe history of martian eccentricity. However, both simulations are equallyvalid, and we cannot say (except probabilistically) what the orbit of Marswas doing over 100 Myr or longer timescales (Laskar et al., 2011). Thereforethe eccentricity is not very useful for determining the history of age of thefamily, and we must rely on inclinations and libration amplitudes which areless chaotic.The middle panels in Figs. 3 and 6 show the evolution of inclinations over12ime. The inclinations vary much more than in gravity-only simulations, withall of the inward-type migrators decreasing their inclinations, while all of theoutward migrators have growing inclinations. This is consistent with resultsof Liou et al. (1995) for Poynting-Robertson drag, and happens because theYarkovsky effect removes (or adds) angular momentum in the plane of theorbit, while the restoring force comes from Mars, which is less inclined thanthe Eureka cluster Trojans. The drift in inclination appears much more or-derly and linear than that in eccentricity, as we already know that inclinationis quite stable in the absence of the Yarkovsky effect (Fig. 1). The range ofinclinations at the end of 1 Gyr matches the spread of inclinations within theEureka cluster, with the peculiarity that all of the cluster members appear tohave been migrating in one direction. The bottom panels show the evolutionof the test-particles’ average libration amplitudes. They also appear to be af-fected primarily by the Yarkovsky drift, with outward-type migrators havingshrinking libration amplitudes, while the inward-type migrators have theirlibration amplitudes grow. Just as for inclination, cluster members (otherthan Eureka) show signs of a force that would have caused inward migration.Configurations at the end of our Yarkovsky simulations are shown in Figs4 and 7. The distribution of the true cluster eccentricities does not followthe synthetic cluster in Fig. 4, with the two high-eccentricity bodies beingoutliers. The second Yarkovsky simulation fares somewhat better (Fig. 7),but 2011 SC is still an outlier in eccentricity (as well as being most distantfrom Eureka in inclination). On the other hand, both synthetic clusters13atch the real one in the inclination-libration amplitude plots (top rightpanels of Figs. 4 and 7). Those cluster members that are well-separated fromEureka in their orbital elements (especially 2011 SC ) appear to have beentrying to migrate inward over the age of the cluster. The simplest conclusionfrom these results is that Eureka cluster is indeed a genetic family, and thatat least some of the cluster members show the effects of sustained inward Yarkovsky drift over Gyr timescales.
Before discussing the age of the family, we should consider the issue of thefamily originating in a single event, rather than having fragments ejectedfrom Eureka (or each other) at different times. When all three orbital pa-rameters are very similar (as in the case of 2001 DH and 2011 UB ),a very recent breakup (either rotational or collisional) is usually indicated(Vokrouhlick´y and Nesvorn´y, 2008). If some of the family members are morerecent fragments, we cannot make any statements about their Yarkovsky be-havior (or more accurately, lack thereof), as they may not have had the timeto evolve. So the ”age of the family” that we discuss here is the maximumage of the family, based on the dispersal of the most distant members (whichare likely to have been the first to separate).From comparison with test particles, 2011 SC should have experienceda Yarkovsky force equivalent to inward migration at 7.7 × − AU/Myr, if14t were to migrate from Eureka in 1 Gyr. If we assume it has albedo of0.4 (Trilling et al., 2007), this would made its diameter D = 300 m. Wecan compare that with rates of Yarkovsky drift computed by Bottke et al.(2006). Since ˙ a ∼ a − , rate of 7.7 × − AU/Myr at 1.52 AU is equivalentto the rate 2.8 × − AU/Myr at 2.5 AU (the benchmark distance used inBottke et al., 2006, Figure 5). Comparing this estimate to the theoreticalcurves in the same figure, we find that the inferred drift rate for 2011 SC isfirmly within the range of predictions for a 300 m body. The same calculationfor somewhat larger (and less divergent) 2007 NS gives a similar result.Somewhat smaller 2011 SL and 2011 UN , which have evolved about thesame amount in inclination and libration amplitude as 2007 NS , may implya slower-than-predicted rate, but uncertainties in predictions themselves areto big to make any firm conclusions.All of the above comparisons assumed the age of the family to be 1 Gyr.The fact that that assumption produces reasonable values for the Yarkovskydrift may imply that the real age is comparable. However, estimates ofYarkovsky drift vary by a factor of a few depending on asteroid properties(Bottke et al., 2006), and we do not know enough about the family membersto further refine these estimates. Evolution of the spin axis through YORPcould lead to much slower effective Yarkovsky drift, meaning that the familycould be significantly older.In Fig. 4, the eccentricities of bodies with zero or inward-type Yarkovskydrifts do not spread enough over 1 Gyr to match the dispersal of the family.15herefore, an age of the family longer than 1 Gyr would be consistent withthis particular simulation. On the other hand, the difference between theeccentricity dispersions of test particles in Figs. 2, 4 and 7 is due to thelower eccentricity of Mars in the first Yarkovsky simulation. Eccentricitiesof particles in Fig. 3 disperse fast until 300-400 Myr, when the dispersionrate (especially for would-be inward migrators) slows down. This correlateswell with the evolution of Martian eccentricity, as we see the eccentricityin the first Yarkovsky simulation (black solid lines inn Fig. 5) droppingbelow that in the gravitational and second Yarkovsky simulations (dashedred and blue lines, respectively, in Fig. 5) at about 300 Myr. Of course, thisdivergence is not in any way related to the Yarkovsky effect, but to slightlydifferent numerical integrators, timesteps or initial conditions used for thetwo simulations. While all of these integrations are equally valid in principle,additional planetary simulations indicate that higher martian eccentricity ismore likely than that seen in the first Yarkovsky simulation (bottom panelin Fig. 5). This is in agreement with previous work, for example the largesets of simulations done by Laskar (2008).Therefore, the age of the family based on eccentricity spreading is onaverage consistent with about 1 Gyr, but with very large uncertainties. Notethat 2011 SC , apart from being most evolved through Yarkovsky drift,is also an outlier in eccentricity, even in the second Yarkovsky simulation(which used a more realistic martian eccentricity). Such a correlation is likelyif the family members have different ages, with 2011 SC being presumably16he oldest extant fragment. This would be an argument for the origin ofthe family in a series of YORP disruptions, similar to those that producedasteroid pairs (Pravec et al., 2010).The fact that the four family members that are well-separated from Eu-reka in the orbital-element space all show signs of inward-type Yarkovsky mi-gration is not necessarily significant and may be due to chance. If future fam-ily members are found to exhibit outward-type average drift, the observed dis-persion would be consistent with the diurnal Yarkovsky effect. However, lackof any outward-type migrators could also be the sign of the dominance of theseasonal Yarkovsky effect. Using the theory of Vokrouhlick´y and Farinella(1999), we find that the predicted drift rate for a D = 300 m body at 1.5 AUshould be − × − AU/Myr (assuming a 90 ◦ obliquity), very similar to − . × − AU/Myr we infer for 2011 SC when assuming the Eurekafamily age of 1 Gyr. Here we also assumed that the thermal parameterΘ = 1, but otherwise this estimate is not affected by thermal properties ofthe asteroid, as the Yarkovsky seasonal effect relies on the heat from ab-sorbed sunlight penetrating to depths much smaller than the size of thebody (Vokrouhlick´y and Farinella, 1999). The presence or absence of re-golith cannot therefore be inferred, unless the diurnal Yarkovsky effect canalso be observed. Discoveries of more Trojans, and establishing the popula-tion outward-type migrators would be the best way of determining relativeimportance of the two Yarkovsky effects.The slopes of the synthetic families in top right panels of Figs. 4 and17 match that of the real family rather well, indicating the type of radiationforce included. We tested functional forms of the acceleration other than r − . (Fig. 8) and they do not lead to the same slope (although a somewhat steeperradial dependence like r − cannot be excluded at this point). In particular,we find that a constant negative tangential acceleration (leftmost lines inFig. 8) produces a decrease in the libration amplitude (this is equivalent tothe results of Fleming and Hamilton, 2000, for constant outward migrationof the planet), rather than an increase which is apparent among Eurekafamily members. However, since both diurnal and seasonal Yarkovsky effectshave the same distance dependence , there is no way to use this slope todistinguish between these two variants of the Yarkovsky effect.The closest match to Mars Trojans in terms of sizes and albedos (ifnot the collisional environment) would be sub-km Hungaria family asteroids(Warner et al., 2009). Warner et al. (2009) show a distribution of knownHungaria family members, and it appears that at 18th magnitude, whereincompleteness sets in, more bodies populate the inner part of the family.Of course, Hungarias are subject to observational bias that strongly favorscloser-in bodies. However, if this trend of inward-type migration perseveresas more Eureka family members are known, it would be interesting to revisitthis issue for small Hungarias, which offer a much larger, and yet similar,sample. This radial dependence of the seasonal Yarkovsky is true for asteroids but not formuch cooler Trans-Neptunian Objects (Bottke et al., 2006). Summary and Conclusions
In this paper, we have reached the following conclusions:1. The Eureka cluster of Mars Trojans is a genetic family, most likelyoriginating on (5261) Eureka. While the family could have formed in a sin-gle event, separation of different members at different times due to YORPbreakups may be more likely.2. Inclinations and libration amplitudes of Eureka family have spreadthrough the Yarkovsky effect. The shape of the family in the orbital param-eter space (Fig. 8) indicates that the force spreading the family has radialdependence close to ≃ r − . expected for the Yarkovsky effect.3. The age of the family is likely on the order of 1 Gyr. This age isconsistent with the expected rates of Yarkovsky drift, but we do not knowenough about family members to constrain it further. Spreading of eccen-tricities is driven by planetary chaos, and the rate of this spreading can varysignificantly from one simulation to another. We think that the observedspread in eccentricities is consistent with a ≃ would likely be older than 1 Gyr, while the remaining members would besignificantly younger.4. The family spreading in inclination and libration amplitude may bedominated by the seasonal, rather than the better-known diurnal, Yarkovskyeffect. The seasonal effect always removed orbital energy, and no family19embers known so far show signs of outward-type migration (indicative of thediurnal Yarkovsky effect acting on prograde rotators). Having more knownfamily members would answer the question of diurnal vs. seasonal Yarkovskyeffect, and would help us understand how the Yarkovsky effect acts on sub-kmasteroids in general. ACKNOWLEDGMENTS
M ´C is supported by NASA’s Planetary Geology and Geophysics (PGG)program, award number NNX12AO41G. Astronomical research at the Ar-magh Observatory is funded by the Northern Ireland Department of Culture,Arts and Leisure. M ´C thanks Hal Levison for sharing his SWIFT-rmvs4 in-tegrator. 20 eferences
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4, absolute magnitudes of H = 16 and H = 19 correspondto diameters of D = 1 . D = 0 .
33 km, respectively.Trojan Amplitude ( ◦ ) Eccentricity Inclination ( ◦ ) H (mag)(5261) Eureka 5.63 0.0593 22.22 16.12001 DH L i b r a t i on a m p li t ude ( ° ) Time (Myr)
18 20 22 24 26 I n c li na t i on ( ° ) E cc en t r i c i t y Figure 1: Evolution of the 100 Mars Trojan test particles’ eccentricities (top),inclinations (middle) and libration amplitudes (bottom) in the 1 Gyr sim-ulation which included only gravitational forces. Each point plots a meanvalue calculated over 10 Myr for each particle. Horizontal bars show meanelements for known cluster members.29 L i b r a t i on a m p li t ude ( ° ) Eccentricity
19 19.5 20 20.5 21 21.5 22 22.5 23 I n c li na t i on ( ° )
5 10 15 20
Libration amplitude ( ° ) Figure 2: Correlations between the test particles’ eccentricities and inclina-tions (top left), eccentricities and libration amplitudes (bottom) and incli-nations and libration amplitudes (top right) at the end of the gravity-only1 Gyr simulation. All of the elements were averaged over the last 10 Myr.Known family members are plotted as open squares, with Eureka shown asa filled square. 30 L i b r a t i on a m p li t ude ( ° ) Time (Myr)
18 20 22 24 26 I n c li na t i on ( ° ) E cc en t r i c i t y Figure 3: Evolution of the 96 Mars Trojan test particles’ eccentricities (top),inclinations (middle) and libration amplitudes (bottom) in the 1 Gyr simu-lation which included the Yarkovsky effect. Each point plots a mean valuecalculated over 10 Myr for each particle. Test particles with inward-typeYarkovsky drift are plotted as red pluses, those with outward-type drift areplotted as blue x’s. Horizontal bars show mean elements for known clustermembers. 31 L i b r a t i on a m p li t ude ( ° ) Eccentricity
18 20 22 24 26 I n c li na t i on ( ° )
2 4 6 8 10 12 14 16
Libration amplitude ( ° ) Figure 4: Correlations between eccentricities and inclinations (top left), ec-centricities and libration amplitudes (bottom) and inclinations and librationamplitudes (top right) at the end of the 1 Gyr simulations incorporating theYarkovsky effect. The elements have been averaged over the last 10 Myr. Testparticles with inward-type Yarkovsky drift are plotted as red pluses, thosewith outward-type drift are plotted as blue x’s. Known family members areplotted as open squares, with Eureka shown as a filled square.32 E cc en t r i c i t y Time (Myr) E cc en t r i c i t y Figure 5: Top: Evolution of the Martian eccentricity over 1 Gyr gravity-onlysimulation (red dots) and the first 1 Gyr Yarkovsky simulation shown in Figs.3-4 (black dots). Values averaged over 50 Myr bins are shown as red dashedand black solid lines for the non-Yarkovsky and the Yarkovsky simulations,respectively. Bottom: Eccentricity of Mars in eight simulations with similarinitial conditions, averaged over 50 Myr bins. The thick solid black line plotsthe original Yarkovsky simulation (same as solid black line in the top panel),while the remaining lines plot its ”clones”, including the case used for theYarkovsky simulation shown in Figs. 6 and 7 (thick blue dashed line).33 L i b r a t i on a m p li t ude ( ° ) Time (Myr)
18 20 22 24 26 I n c li na t i on ( ° ) E cc en t r i c i t y Figure 6: Evolution of the 96 Mars Trojan test particles’ eccentricities (top),inclinations (middle) and libration amplitudes (bottom) in the second 1 Gyrsimulation including the Yarkovsky effect. Each point plots a mean valuecalculated over 10 Myr for each particle. Test particles with inward-typeYarkovsky drift are plotted as red pluses, those with outward-type drift areplotted as blue x’s. Horizontal bars show mean elements for known clustermembers. 34 L i b r a t i on a m p li t ude ( ° ) Eccentricity
18 20 22 24 26 I n c li na t i on ( ° )
2 4 6 8 10 12 14 16
Libration amplitude ( ° ) Figure 7: Correlations between eccentricities and inclinations (top left), ec-centricities and libration amplitudes (bottom) and inclinations and librationamplitudes (top right) at the end of the second 1 Gyr simulations incorpo-rating the Yarkovsky effect. The elements have been averaged over the last10 Myr. Test particles with inward-type Yarkovsky drift are plotted as redpluses, those with outward-type drift are plotted as blue x’s. Known familymembers are plotted as open squares, with Eureka shown as the solid square.35
19 19.5 20 20.5 21 21.5 22 22.5 23 4 5 6 7 8 9 10 11 12 I n c li na t i on ( ° ) Libration amplitude ( ° ) Figure 8: Correlations between inclinations and libration amplitudes during10 Myr simulations using different orbital migration forces (with a greatlyaccelerated drift rate of -0.15 AU/Myr). Clockwise from top right, the fourparticle tracks used negative tangential acceleration with radial dependenceof r − , r − . , r − and r0