The Eos Family Halo (Or at Least a Was Used in Parker Et Al

arXiv:1302.1447v1 [astro-ph.EP] 6 Feb 2013 rpitsbitdt Icarus to submitted Preprint ofimdta seod r saigfo h o aiydue J9 family the Eos with the interaction from the to escaping are whi asteroids (2000) ob al. that Zappalà et spectroscopic confirmed by Detailed performed also indices. were colour servations SDSS the of terms aoadt xli t nsal ag pedi eccentrici in spread large unusually its explain to and halo esnmru tlre eiao xs ii hr sa ex- an is the there along (iii) family axes; the ar semimajor sizes of larger larger tension with at asteroids numerous and AU less 3.03 at family the vides .6A;(i h 9 the (ii) AU; 2.96 olwn:()teei hr ne onaycicdn wi th coinciding is boundary family inner the sharp 7 of a the is structure al. 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Object et the Parker Moving in Survey, visible clearly Sky is ital which family Eos nominal eccentricity yaia oe losu oidpnetyetmt h age the estimate independently to us allows model dynamical n neatoswt rvttoa eoacs otywi mostly resonances, gravitational with interactions Such and Survey. Sky Digital Sloan the by measured colours their xs.W a locntantegoer ftedsuto e disruption the of geometry the constrain also can We bas (2006) axis). al. Vokrouhlický et by estimate age previous the Keywords: u anmtvto st nesadteoii ftewhole the of origin the understand to is motivation main Our h o aiyi n ftebs-tde aiisi h main the in families best-studied the of one is family Eos The nti ok efcso hl’o seod rudthe around asteroids of ’halo’ a on focus we work, this In / enmto eoac ihJptra approximately at Jupiter with resonance mean-motion 3 nttt fAtooy hre nvriy rge Hole V Prague, University, Charles Astronomy, of Institute seod,dnmc,Rsnne,obtl lnt,migra Planets, orbital, Resonances, dynamics, Asteroids, e p n nlnto sin inclination and ff / enmto eoac ihJptrdi- Jupiter with resonance mean-motion 4 c nsmmjrai n h gravitational the and axis semimajor in ect ff c cetiiisadinclinations. and eccentricities ect bevtied aCoedAu,B 29 60 ieCdx4, Cedex Nice 06304 4229, BP Côte d’Azur, la de Observatoire / resonance. 4 z 1 I p ≡ l hs atse obe to seem fact these All . g / -ope seod in asteroids X-complex − g 6 + s − h o aiyhalo family Eos The s 6 s vˇ kah2 80 rge8 zc eulcemi:mira e-mail: Republic Czech ˇsoviˇckách 8, Prague 18000 2, secular yand ty hte9 the th .Morbidelli A. do ipie oe wihacut nyfrcagso se of changes for only accounts (which model simplified a on ed aigfo h aiy‘oe u oteYrosysemimajor Yarkovsky the to due ‘core’ family the from caping a etwihhdt cu ttetu anomaly true the at occur to had which vent ch ue .BroˇzM. th c- p ftefml 1 family the of h e e - - , hls fatri aiishv enrrl sda constr as used rarely been have families asteroid of ‘halos’ aiybcuete emt eitmtl eae,accordi related, intimately be to seem they because family s / enmto eoac ihJptra .3A.Our AU. 3.03 at Jupiter with resonance mean-motion 4 v .Adseneto h aiycr n halo and core family the of discernment A 2. discuss 4. We Section in observations. model results SDSS dynamical our the our of with consequences of comparison description a a to to and devoted is 3 the Section match to order in migration. giant-planet Gyr of 3.9 model approach Nice to have would family Mriel ta. 05.O ore nsc aeteproce the case a such in course, is Of 2005). popul al., small-body et all (Morbidelli signi of caused perturbations have gravitational would which icant migration giant-planet the to oko orulc´ ta.(06,bthr eaeinterest are we which here bodies but in exte (2006), substantial al. Vokrouhlický a et i of is reasonable work this Essentially, any with field. reconcile velocity to tial hard is which inclination ..Cluso o-ieasteroids Eos-like of Colours 2.1. space. proper-element the halo in a ’boxes’ define simple we using (iv) core Eos- finally, and with catalogue; we asteroids SDSS and all the colours select from we SDSS colours (iii) with range; family colour the a of define members (ii family; the Eos at nominal look the extract to method clustering cal HM aplae l 95 ihasial o cut-o identify low thus suitably We meth a with clustering 1995) so. hierarchical al. a do Zappalà et (HCM, using to family criterion Eos a nominal choose the to have we first cuto tion size-independent nScin2 edfieteEshl n oepopulations. core and halo Eos the define we 2, Section In nti eto,w rce sflos i eueahierarchi a use we (i) follows: as proceed we Section, this In ewr locrosi uhhlsmyb oeo related somehow be may halos such if curious also were We ewn oslc seod iia oteEsfml,but family, Eos the to similar asteroids select to want We ff = . 0m 50 o1 to 5 rne -al [email protected] e-mail: France, / wihlast iia xeto h family the of extent similar a to leads (which s . y.Ti sapoiaeyi gemn with agreement in approximately is This Gyr. 9 escaped n oevrteaeo h corresponding the of age the moreover and rmtenmnlfamily. nominal the from @sirrah.troja.m f ≃ 120 ◦ o180 to ff etme ,2018 6, September .cuni.cz ff ai drift -axis ◦ mimajor it for aints . so of nsion velocity N ations -body gto ng we ) like ni- od ed ss f- - 0.4 Table 1: The definitions of the core, halo and background populations in terms of intervals of proper eccentricity ep and proper inclination sin Ip. The range of proper semimajor axis ap (2.95, 3.16) AU is the same in all cases. 0.2 ∈ population ep sin Ip note 0 / mag core 0.04–0.10 0.15–0.20 z - i halo 0.00–0.15 0.12–0.24 and not in the core -0.2 background 0.00–0.15 0.06–0.12 togetherwith... 0.00–0.15 0.24–0.30 -0.4

-0.4 -0.2 0 0.2 0.4 a* / mag while the range of proper semimajor axis is always the same, a (2.95, 3.16) AU. p ∈ 0.4 Our results do not depend strongly on the selection criterion. For example, we tested a stringent definition: core was identi- 0.2 fied by the HCM at vcutoff = 50m/s and all remaining bodies in the surroundings belong to the halo. This approach makes the 0 / mag

z core as small as possible and the halo correspondingly larger - i but our results below (based on halo/core ratios) would be es- -0.2 sentially the same. According to our tests, not even a different definition of the background/haloboundary changes our results. -0.4 We are now ready to construct size-frequency distributions -0.4 -0.2 0 0.2 0.4 of individual populations. In order to convert absolute mag- a* / mag nitudes H to diameters D we computed the median geometric albedo pV = 0.16 fromthe WISE data (Masiero et al., 2011) for Figure 1: Colour indices i z and a∗ (defined in Parker et al. (2008)) of all as- the nominal Eos family members. The size-frequency distribu- teroids from the Sloan Digital− Sky Survey, Moving Object Catalogue version 4 tion (Figure 4) of the halo has a cumulative slope N(> D) Dγ and the corresponding colour palette (top panel) which is used in the following ∝ figures to distinguish colours of asteroids. We also plot the Eos family mem- equal to γ = 3.9 0.2 in the size range D = 6 to 15km and bers observed by the SDSS (bottom panel) with small photometric uncertain- is significantly−steeper± than that of the core (γ = 2.2 0.1). ties (less than 0.03 mag). The inferred range of colour indices (denoted by the Even this difference of slopes (1.7 0.2) indicates that− if± there a dashed yellow rectangle) is then used as a criterion for the selection of the Eos- ± like asteroids in the broad surroundings of the nominal family. The rectangle process transporting asteroids from the core to the halo it must does not encompass the outliers. be indeed size-dependent. A frequency analysis similar as in Carruba and Michtchenko (2007) or Carruba (2009) shows that there is approximately 5 % as in Vokrouhlickýet al. (2006)), and extract colour data from of likely z1 resonators (with the frequency g g6 + s s6 < the SDSS catalogue (see Figure 1). The majority of Eos-family − − 0.3 ′′/yr) in the halo region. However, the concentration of ob- asteroids have colour indices in the following intervals jects inside and outside the resonance is roughly the same, so that this secular resonance does not seem to be the most impor- a∗ (0.0, 0.1)mag, (1) ∈ tant transport mechanism. i z ( 0.03, 0.08)mag (2) − ∈ − which then serves as a criterion for the selection of Eos-like 3. Yarkovsky-driven origin of the halo asteroids in the broad surroundings of the nominal family. We also used an independentmethod for the selection of Eos- Motivated by the differences of the observed SFD’s, we now like asteroids employing a 1-dimensional colour index (which want to test a hypothesis that the Eos family halo (or at least a was used in Parker et al. 2008 to construct their colour palette) part of it) was created by the Yarkovsky semimajor-axis drift, and we verified that our results are not sensitive to this proce- which pushes objects from the core into neighbouring mean- dure. motion resonances and consequently to the halo region.

2.2. Boundaries in the proper element space 3.1. Initial conditions Next, we have to distinguish the family ’core’and ’halo’pop- We prepared an N-body simulation of the long-term evolu- ulations on the basis of proper orbital elements (ap, ep, sin Ip) tion of the Eos core and halo with the following initial condi- which will be consistently used for both the SDSS observations: we included the Sun and the four giant planets on current tions and our dynamical models. We also need to deﬁne ’back- orbits. We applied a standard barycentric correction to both ground’ population which enables to estimate how many as- massive objects and test particles to prevent a substantial shift teroids might have Eos-like colours by chance. We decided of secular frequencies (Milani and Knezevic, 1992). The to- to use a simple box criterion (see Figures 2, 3 and Table 1), tal number of test particles was 6545, with sizes ranging from 2 J7/3 J9/4 sin Ip = 0.12 to 0.24 Eos-like core halo 0.2

0.15 p e 0.1

0.05

ap = 2.95 to 3.16 AU 0.3 0 2.9 2.95 3 3.05 3.1 3.15 3.2

0.25 ap / AU

J7/3 J9/4 ep = 0 to 0.15 0.3 0.2 Eos-like core halo p I 0.15 0.25 sin

0.1 0.2 p I

0.05 sin 0.15 0 0 0.05 0.1 0.15 0.2 0.25 0.3

ep 0.1 ap = 2.95 to 3.16 AU 0.3 background 2.9 2.95 3 3.05 3.1 3.15 3.2 0.25 ap / AU

0.2 Figure 3: The proper semimajor axis ap vs proper eccentricity ep (top panel) and ap vs proper inclination sin Ip (bottom panel) for the observed Eos core p I (black dots), halo (red dots) and remaining Eos-like asteroids (gray dots) in the 0.15 core

sin surroundings. The sizes of symbols are (inversely) proportional to the absolute halo magnitudes H of asteroids. The positions of important mean-motion resonances 0.1 with Jupiter are also indicated (dotted vertical lines). background 0.05 10000 Eos family (WISE) Eos core (SDSS) 0 Eos halo (SDSS) 0 0.05 0.1 0.15 0.2 0.25 0.3

ep )

D 1000 (> N Figure 2: The proper eccentricity ep vs proper inclination sin Ip plot for asteroids included in the SDSS MOC 4 catalogue. The proper semimajor axis is conﬁned to the interval 2.95 to 3.16 AU, i.e. the Eos family zone. Colour coding 100 −2.3 corresponds to the SDSS colour indices according to Figure 1. The top panel includes all asteroids (regardless of their colours). The bottom panel shows only a subset of ’Eos-like’ asteroids with colours similar to those of the Eos −2.2 10

members (see Fig. 1, bottom). Moreover, we denote a box used for the defini- number of asteroids tion of the family ’core’ (dashed yellow line) a larger box for the ’halo’ (dotted green line) and two boxes considered as ’background’ (thin black line). For comparison, we also plot positions of the nominal Eos family members (black 1 −3.9 dots), identified for the velocity vcutoff = 50 m/s. 10 100 diameter D / km

Figure 4: The cumulative size-frequency distributions N(>D) of the Eos core and halo. We show power-law fits and corresponding slopes γ which clearly indicate that the halo population is significantly steeper than the core population. For comparison, we also plot the SFD of the nominal Eos family (as inferred from the WISE data, Masiero et al. 2011). The SFD’s of the core and halo are biased because they include only asteroids observed by the SDSS. Conse- quently, the core seems to be much less populated than the nominal Eos family, even though these SFD’s should be very similar. Nevertheless, the slopes and the halo/core ratios which we use in our analysis is not much affected by this bias.

3 0.24 D = 104 to 1.5 km and the distribution resembling the observed J9/4 J11/5 and 3J−2S−1 SFD of the Eos family. J7/3 0.22 z and other Material properties were as follows: the bulk density ρ = 1 2500 kg/m3, the surface density ρ = 1500 kg/m3, the thermal s 0.2 conductivity K = 0.001W/m/K, the specific thermal capacity p C = 680J/kg/K, the Bond albedo A = 0.1, the infrared emis- I 0.18 sivity ǫ = 0.9, i.e. all typical values for regolith covered basaltic sin asteroids. 0.16 Initial rotation periods were distributed uniformly on the interval 2 to 10hours and we used random (isotropic) orientations 0.14 of the spin axes. The YORP model of the spin evolution was 0.12 described in detail in Broˇzet al. (2011), while the efficiency 2.95 3 3.05 3.1 3.15 3.2 = . parameter was cYORP 0 33 (i.e. a likely value according to ap / AU Hanuˇset al. 2011). YORP angular momenta affecting the spin rate and the obliquity were taken from Capekˇ and Vokrouhlický Figure 6: The proper semimajor axis ap vs proper inclination sin Ip of the syn- (2004). We also included spin axis reorientationscaused by col- thetic family members at the moment when they have entered the halo region. It lisions1 with a time scale estimated by Farinella et al. (1998): is easy to distinguish objects injected by mean-motion resonances and by secular resonances in this projection, because the former have a particular value β1 β2 τreor = B (ω/ω0) (D/D0) , where B = 84.5 kyr, β1 = 5/6, of the semimajor axis. The objects injected by the J9/4 resonance are denoted β = 4/3, D = 2m and ω corresponds to period P = 5 hours. by red colour, the J11/5 and 3J 2S 1 by orange, the J7/3 by yellow, the z1 2 0 0 − − The initial velocity field was size-dependent, v v D /D, secular resonance and other resonances by black. ∝ 0 0 with v0 = 93m/s and D0 = 5km (i.e. the best-fit values from Vokrouhlickýet al. 2006). In principle, this type of halo: J9/4 57%, J11/5 (together with a three-body resonance size–velocity relation was initially suggested by Cellino et al. 3J 2S 1 with Jupiter and Saturn) 10%, J7/36%, and z1 sec- (1999), but here, we attempt to interpret the structure of the ular− resonance− 23%. The remaining few percent of bodies may family as a complex interplay between the velocity field and enter the halo by different dynamical routes.2 However, if we the Yarkovsky drift which is also inversely proportional to size. account for the fact that bodies captured by the z1 resonance We assumed isotropic orientations of the velocity vectors. The usually encounter also the J9/4 resonance that scatters them fur- geometry of collisional disruption was determined by the true ther away in to the halo, we obtain a modified statistics: J9/4 anomaly f = 150 , and the argument of perihelion ω = 30 . ◦ ◦ 70%, J11/5 12%, J7/3 5%, and z1 10% that better reflects the We discuss different geometries in Section 4. importance of different mechanisms. We use a modified version of the SWIFT package A saturation of the halo occurs after approximately 1Gyr, (Levison and Duncan, 1994) for numerical integrations, with because the halo population is affected by the Yarkovsky/YORP a second-order symplectic scheme (Laskar and Robutel, 2001), drift too, so that the injection rate roughly matches the removal digital filters employing frequency-modified Fourier transform rate. Nevertheless, the halo/core ratio steadily increases, which ˇ (Sidlichovskýand Nesvorný, 1996) and an implementation of is caused by the ongoing decay of the core population. the Yarkovsky effect (Broˇz, 2006). The integration time step In order to compare our model and the SDSS observations was ∆t = 91days, the output time step after all filtering proce- we compute the ratio = dNhalo/dNcore between the number dures 10Myr and the total integration time span reached 4Gyr. of objects in the haloR and in the core for a given differential size bin. This can be computed straightforwardly from our sim- 3.2. Results of the N-body simulation ulation data. In case of the SDSS observations, however, we Initially, almost all asteroids are located in the core (see Fig- think that there is a real background of asteroids with Eos-like ure 5). Only a few outliers may have velocities large enough to colours (may be due to observational uncertainties or a natural belong to the halo. Within a few million years the halo/core spread of colours; see Figure 2). Obviously, such background ratio quickly increases due to objects located inside the 9/4 overlaps with the core and the halo, so we need to subtract this resonance and injected to the halo by these size-independent contamination gravitational perturbations. Further increase is caused by the dNhalo 0.833dNbackground Yarkovsky/YORP semimajor axis drift which pushes additional obs − . (3) R ≡ dN 0.167dN orbits into the J9/4 and also other resonances. core − background We checked the orbital elements of bodies at the moment The numerical coefficients then reflect different ’volumes’ of when they enter the halo region (Figure 6) and we computed the halo, core and background in the space of proper elements the statistics of dynamical routes that had injected bodies in the (ap, ep, sin Ip), as defined in Table 1. As we can see in Figure7, a reasonablematch to the observed halo/core ratios can be obtained for ages 1.5Gyr (for smaller 1We do not take into account collisional disruptions because we model only that subset of asteroids which survived subsequent collisional grinding (and compare it to the currently observed asteroids). Of course, if we would like 2Other secular resonances intersecting this region, s s 2g + 2g or − 6 − 5 6 to discuss e.g. the size of the parent body, it would be necessary to model g + 2g5 3g6, do not seem to be important with respect to the transport from disruptive collisions too. the core− to the halo. 4 J7/3 J9/4 J11/5 and 3J-2S-1 10000 observed Eos core and halo synthetic core (rescaled to observed) 0.2 synthetic family at t = 0 synthetic halo at t = 0 core observed halo halo )

D 1000 p e 0.15 (> N

100 0.1

proper eccentricity 10

0.05 number of asteroids

0 1 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 10 100 proper semimajor axis ap / AU diameter D / km J7/3 J9/4 J11/5 and 3J-2S-1 10000 observed Eos core and halo synthetic core (rescaled to observed) 0.2 synthetic family at t = 1.7 Gy synthetic halo at t = 1.7 Gy core observed halo halo )

D 1000 p e 0.15 (> N

100 0.1

proper eccentricity 10

0.05 number of asteroids

0 1 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 10 100 proper semimajor axis ap / AU diameter D / km

Figure 5: Left panels: the proper semimajor axis ap vs proper eccentricity ep plots showing a dynamical evolution of our synthetic family. We can distinguish the core (black dots), the halo (red dots) and objects beyond the halo box (gray dots). There is a comparison to the observed Eos core and halo too (yellow crosses), as inferred from the SDSS data (the same as in Figure 2, bottom). The positions of important resonances are indicated by vertical dotted lines. We plot the initial situation at t = 0 (top panel) and the evolved family at t = 1.7 Gyr (bottom panel). The core of the synthetic family exhibits a slightly diﬀerent structure than the observed core which may indicate that: (i) the initial true anomaly was closer to f = 180◦, or (ii) the initial velocity ﬁeld deviated from the assumed v 1/D dependence. Right panels: the corresponding size-frequency distributions of the synthetic core (black line), which was always scaled to the observed SFD∝ of the Eos core, and the synthetic halo (red line) which can be then directly compared to the observed halo (gray line). It is clear that the halo’s SFD becomes steeper in the course of time and at t 1.7 Gyr it matches the observed SFD. ≃

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Broˇz, M., References fo Carruba V. paper. and this Cellino of Educa- reviews A. careful referees of both Ministry thank also Czech We tion. the Research the of and MSM0021620860 13-01308S) no. Program (grant Republic Czech the of Acknowledgements migration). giant-planet ute eraeisucrany othat so uncertainty, its decrease further beaea noelpo nevl flow of intervals of overlap prob- most an the as infer can age we able reasonable, be to seem – histogram arb,V,Mcthno .A,Dc 07 rqec appr frequency A 2007. Dec. A., T. Michtchenko, V., Carruba, ial,ltu mhsz htgvntedi the given that emphasize us let Finally, et o.Nt .Ato.Sc 1,2716–2727. 414, h Soc. late Astron. the R. during Not. form Mon. family ment? collisional Hilda the Did 2011. akvk-rvnoii em ob aua xlnto o explanation natural a be to seems origin Yarkovsky-driven sa motn ypout h rcs nbe st in- to us enabled process the by-product, important an As hss hre Univ. 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Solar the of Dynamics the and ect and χ 2 ◦ ( t o180 to / e vlto rfrt iue7 red 7, Figure to (refer evolution ) 10 p olarge to H ie iia praha in as approach similar a (i.e. ◦ epromdtsswith tests performed We . f / . a oeratios core t p a , ≃ p 120 e and p 1 ln:teobjects the plane: ) χ . ◦ o1 to 5 ff 2 sexcluded. is e rne between erences ( ,Btk,W . Jul. F., W. Bottke, ., t p z n hsway this and ) N 1 c.Fgr 5). Figure (cf. R aht identify- to oach bd model -body eua reso- secular . aybombard- eavy Gyr. 9 n the and aiy Mon. family. e5), re we , otal C a m ly ts n p - f r - Cellino, A., Michel, P., Tanga, P., Zappalà, V., Paolicchi, P., dell’Oro, A., Sep. 1999. The Velocity-Size Relationship for Members of Asteroid Families and Implications for the Physics of Catastrophic Collisions. Icarus 141, 79–95. DeMeo, F. E., Binzel, R. P., Slivan, S. M., Bus, S. J., Jul. 2009. An extension of the Bus asteroid taxonomy into the near-infrared. Icarus 202, 160–180. 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