On the highly inclined v W leptokurtic asteroid families
aa r X i v : . [ a s t r o - ph . E P ] A ug MNRAS , 1–8 (2016) Preprint 15 October 2018 Compiled using MNRAS L A TEX style file v3.0
On the highly inclined v W leptokurtic asteroid families V. Carruba ⋆ , R. C. Domingos , , S. Aljbaae , M. Huaman UNESP, Univ. Estadual Paulista, Grupo de dinˆamica Orbital e Planetologia, Guaratinguet´a, SP, 12516-410, Brazil. UNESP, Univ. Estadual Paulista, S˜ao Jo˜ao da Boa Vista, SP, 13874-149, Brazil.
Accepted 2016 August 11. Received 2016 August 11; in original form 2016 June 29.
ABSTRACT v W leptokurtic asteroid families are families for which the distribution of thenormal component of the terminal ejection velocity field v W is characterized by apositive value of the γ Pearson kurtosis, i.e., they have a distribution with a moreconcentrated peak and larger tails than the Gaussian one. Currently, eight families areknown to have γ ( v W ) > .
25. Among these, three are highly inclined asteroid families,the Hansa, Barcelona, and Gallia families. As observed for the case of the Astrid family,the leptokurtic inclination distribution seems to be caused by the interaction of thesefamilies with node secular resonances. In particular, the Hansa and Gallia family arecrossed by the s − s V resonance with Vesta, that significantly alters the inclination ofsome of their members.In this work we use the time evolution of γ ( v W ) for simulated families under thegravitational influence of all planets and the three most massive bodies in the main beltto assess the dynamical importance (or lack of) node secular resonances with Ceres,Vesta, and Pallas for the considered families, and to obtain independent constraints onthe family ages. While secular resonances with massive bodies in the main belt do notsignificantly affect the dynamical evolution of the Barcelona family, they significantlyincrease the γ ( v W ) values of the simulated Hansa and Gallia families. Current valuesof the γ ( v W ) for the Gallia family are reached over the estimated family age only ifsecular resonances with Vesta are accounted for. Key words:
Minor planets, asteroids: general – celestial mechanics.
Of the three proper elements most commonly used to iden-tify an asteroid family, the inclination is the one usually lessaffected by dynamical evolution. Secular resonances involv-ing the precession frequency of the longitude of the nodeof an asteroid, like the linear secular resonance with Ceres s − s C , can, however, change the inclination distribution offamilies crossed by this kind of resonances (Novakovi´c et al.2015). This is, for instance, the case of the Astrid family, thatis characterized by a dispersion in inclination of its membersat a ≃ a, sin ( i )) plane (Novakovic et al.2016). Recently, Carruba (2016) used the time evolution ofthe Pearson kurtosis of the v W component of terminal ejec-tion velocities to set independent constraints on the Astridfamily age and ejection velocity parameter V EJ . Since v W can be obtained by inverting the third Gauss’equation and ⋆ E-mail: [email protected] is mostly dependent on δi = i − i ref , with i ref the inclina-tion of the barycenter of the family, families whose distri-bution in proper inclination is characterized by larger tailsand more concentrated peaks than that of a Gaussian distri-bution would have values of Pearson kurtosis γ ( v W ) largerthan zero. By simulating fictitious families for different val-ues of ejection velocities parameter V EJ under the influenceof the Yarkovsky non-gravitational force, and by observingwhen current values of γ ( v W ) were reached for the Astridfamilies it was possible to set independent constraints onthe family age, V EJ , and on key parameter determining thestrength of the Yarkovsky force such as the mean densityand surface thermal conductivity of family members.Of the eight v W leptokurtic families with γ ( v W ) > .
25, three are highly inclined families (sin ( i ) > .
3) in thecentral main belt: the Hansa, Barcelona, and Gallia families(Carruba & Nesvorn´y 2016). It has recently been suggestedthat these families could be interacting with node secularresonances with Vesta (Tsirvoulis and Novakovi´c 2016). Inthis work we attempt to use the numerical tools developedfor the Astrid family to i) assess the importance (or lack of) c (cid:13) V. Carruba, R. C. Domingos, S. Aljbaae, M. Huaman of the s − s V secular resonance and of possible analogousresonances with Pallas, and ii) set independent constraintson the three families ages. Overall, we found that the useof the γ ( v W ) could indeed provide valuable hints on theimportance of secular resonances with massive bodies, and,more generally, on the whole dynamical evolution of γ ( v W )leptokurtic asteroid families. As a first step in our analysis of the v W leptokurtic highly in-clined families, we start by identifying the Hansa, Barcelonaand Gallia family in the space of proper elements, and bystudying the local dynamics. For the first purpose, we usethe data from Nesvorn´y et al. (2015), where these familieswere identified in the domain of proper ( a, e, sin ( i )) usingthe hierarchical clustering method and cutoff velocities of200 m/s for the Hansa and Gallia families and of 150 m/sfor the Barcelona one. The groups so identified have 1094members for the Hansa family, 182 for the Gallia group, and306 for the Barcelona cluster. As also discussed in Carruba(2010), these three families have S-type taxonomies. Valuesof the mean geometric albedos for these three groups were0.26 for Hansa, 0.17 for Gallia, and 0.25 for Barcelona, re-spectively (Nesvorn´y et al. 2015).Carruba (2010) investigated in depth the local dynam-ical environment for these families. The author obtaineddynamical maps in the domain of synthetic proper ( a, e ),( a, sin ( i )), and ( e, sin ( i )) domains. Highly inclined asteroidfamilies in the central main belt are separated by the effectof the local secular dynamics. The strong ν = g − g sec-ular resonance acts as a dynamical barrier between highlyinclined and low inclined asteroids. Of importance are alsothe other two main linear secular resonances, the ν = g − g and the ν = s − s , that, together with the local mean-motion resonances 3J:-1A, 8J:-3A, and 5J:-2A separate theregions into eight different stable islands. The region is alsocrossed by several interesting non-linear secular resonance,whose detailed identification and description can be foundin Carruba (2010). Repeating the detailed dynamical anal-ysis of Carruba (2010) is of course redundant and beyondthe purposes of this paper. To allow the reader to have avisual understanding of the complex local dynamics, in thiswork we obtained dynamical maps of 7000 particles in thedomains of synthetic proper ( a, e ) and ( a, sin ( i )) with thesame approach described in Carruba (2010). We refer thereader to that paper for a discussion of the methods andinitial conditions used for obtaining these maps.Fig. 1 displays our results in the ( a, e ) (panel A) and( a, sin ( i )) (panel B) planes. Vertical lines display the lo-cation of local mean-motion resonances. Objects with ec-centricities larger than 0.35 are Mars-crossers in this re-gion of the main belt, and are lost on time-scales of ≃ ν secular resonance causes asteroids in librat-ing states to increase their eccentricity to Mars-crossing lev-els, and to become unstable. The region associated withthis resonance appears as a strip at sin ( i ) ≃ .
35 de-pleted of proper elements. Other secular resonances ap-pear as inclined alignments of test particles. Apart from the ν , important for its interaction with the Barcelona fam-ily (Froeschl´e and Scholl 1989), and the ν linear secular resonances, important non-linear secular resonances in theregion are the ν − ν , ν − ν , and 2 ν − ν + ν secu-lar resonances. Hansa, Barcelona, and Gallia members areshown as red, green, and yellow full dots, respectively.Recently, Tsirvoulis and Novakovi´c (2016) suggestedthat the linear secular resonances ν V = s − s V with Vestacould have played a role in the dynamical evolution of theBarcelona and Hansa family. To further investigate this hy-pothesis, we obtained a dynamic map in the ( a, sin ( i )) do-main, (node secular resonances change values of asteroidsproper inclinations) with the same initial conditions usedbefore, but adding Ceres, Vesta, and Pallas as massive per-turbers. Fig. 2 displays our results. Black circles identify ob-jects whose proper frequency s is to within ± . s is ± . g frequency of Pallas is outside the range of values coveredin this dynamical map. For these reasons, we did not fur-ther investigate the role of pericenter secular resonances withmassive asteroids in this work. The ν V could in principle beaffecting the dynamical evolution of the Hansa, Gallia, and,marginally, the Barcelona families. The ν P resonance, how-ever, does not seem to interact with any of these families,and could play a minor role just for the case of the Hansafamily. We will further investigate the role played by thoseresonances in the next sections.Having briefly revised the local dynamics, we then turnour attention to the taxonomic properties of the three stud-ied families. Taxonomic properties and geometric albedo val-ues of all highly inclined families in the central main beltwere studied in Carruba (2010) in some detail. The threefamilies all belong to the S-type taxonomic class. There are94 objects with photometric data compatible with a S-typecomposition in the Sloan Digital Sky Survey-Moving Ob-ject Catalog data, fourth release (SDSS-MOC4 hereafter,Ivezi´c et al. (2001)) in this region. 470 objects have geomet-ric albedo and absolute magnitude information in the WISEand NEOWISE database (Masiero et al. 2012).Fig. 3 display asteroids whose SDSS-MOC4 photomet-ric data is compatible with a S-type composition, accord-ing to the classifications method of DeMeo & Carry (2013)(panel A). Panel B displays objects whose WISE geomet-ric albedo p V has values compatible with a S-complex tax-onomy, (i.e., 0 . < p V < .
30, Masiero et al. (2012)). Ascan be seen from the figure, most of the objects with photo-metric and albedo S-type compatible data are indeed associ-ated with the three studied families. Using the SDSS-MOC4data, we tried to obtain halos for the three families with themethod discussed in Carruba et al. (2016) for the Koronisfamily. In this method, asteroids with SDSS-MOC4 data areconsidered to be members of the halo of the family if theirvalues of proper eccentricity and inclination are in a rangefrom the center of dynamical family to within four standarddeviations of e and sin ( i ) of the distribution observed for theHCM family. We applied this method for the three studiedfamilies, but, unfortunately, we are limited by small numberstatistics for the cases of the Gallia and Barcelona families, MNRAS , 1–8 (2016) nclined v W leptokurtic families q = Q Mars q = q
Mars
GalliaBarcelonaHansa
Semi−major axis [AU] E cc en t r i c i t y A GalliaBarcelonaHansa
Semi−major axis [AU] S i ne o f i n c li na t i on B Figure 1.
Dynamical maps in the proper ( a, e ) (panel A) and ( a, sin ( i )) (panel B) domains. Vertical red lines display the location ofthe local mean-motion resonances. The magenta line identifies the width of the unstable chaotic layer near the 3J:-1A resonance, asidentified in Morbidelli & Vokrouhlick´y (2003). The blue and red lines in the ( a, e ) plane identify the orbital location of asteroids whosepericenter q is equal to the apocenter and pericenter of Mars, respectively. The region depleted of test particles at sin ( i ) ≃ .
35 in the( a, sin ( i )) plane is associated with orbits in librating states of the ν secular resonance. Other secular resonances appear as inclinedbands of proper elements in the figure. Red, green and yellow full dots display the orbital location of members of the Hansa, Barcelona,and Gallia dynamical families, respectively. GalliaBarcelonaHansa
Semi−major axis [AU] S i ne o f i n c li na t i on Figure 2.
A dynamical map in the ( a, sin ( i )) domain for thesame region displayed in Fig. 1, but obtained also considering theeffect of Ceres, Vesta and Pallas as massive perturbers. Black cir-cles display the location of likely resonators in the ν V secularresonance, while magenta circles are associated with likely res-onators of the ν P resonance. Other symbols are the same as inFig. 1, panel B. that have halos of less than 10 members. The Hansa familyhas a halo of 63 members, but its distribution in proper in-clination is comparable to that of the HCM family. For thepurpose of studying the Kurtosis of the v W component ofterminal ejection velocities, we are therefore left using stan-dard HCM data alone. The apparent lack of significant halosfor these three families may be caused by the fact that thesegroups are contained in stable islands surrounded by unsta-ble regions, which limits the number of long-term survivingoutlying asteroids.Carruba & Nesvorn´y (2016) studied the shape of the v W component of the ejection velocity field of these three Table 1.
Values of γ ( v W ) of the whole family (3rd column), the2 . < D < . D ) members (4rd column), the p coefficientof the jbtest (5th column), and estimated family age with itserror (6th column) from Nesvorn´y et al. (2015) for the Hansa,Barcelona,and Gallia families.FIN Family γ ( v W ) γ ( v W ) p jbtest AgeName All D (%) [Myr]803 480 Hansa 0.81 1.17 0.1 2430 ± ± ± families, that are among the most leptokurtic among thestudied group. For the sake of the reader not familiar withthat work, we summarize in Table 1 the result of thatstudy. The first two column report the Family IdentificationNumber (FIN), as defined in Nesvorn´y et al. (2015), andthe family identification and name. The third and fourthcolumns report the values of γ ( v W ) for the whole familyand for the D population with 2 . < D < . v W component of simulated families will bediscussed later on in this paper. MNRAS , 1–8 (2016)
V. Carruba, R. C. Domingos, S. Aljbaae, M. Huaman
GalliaBarcelonaHansa
Semi−major axis [AU] S i ne o f i n c li na t i on A GalliaBarcelonaHansa
Semi−major axis [AU] S i ne o f i n c li na t i on B Figure 3.
Panel A: yellow full dots identify objects whose SDSS-MOC4 photometric data is compatible with an S-type composition.Panel B: objects with WISE albedo p V in the range from 0.15 to 0.30. Other symbols are the same as in Fig. 1, panel B. There is a considerable range of possible values for theage of the Hansa family in the literature. Carruba (2010),using the method of Yarkovsky isolines, provided an up-per limit for the family age of 1600 Myr old (see alsoBroˇz et al. (2013). The estimate from Nesvorn´y et al. (2015)was of 2430 ±
60 Myr, while Spoto et al. (2015), using a V-shape criteria, assessed the family age to be in the range420-1170 Myr. The age of the Barcelona family was esti-mated by Carruba (2010) (see also Broˇz et al. (2013)) tohave an upper limit of 350 Myr and to be in the rangeof 250 ±
10 Myr by Nesvorn´y et al. (2015). The Gallia fam-ily had an upper limit of 450 Myr in Carruba (2010) (seealso Broˇz et al. (2013)) and an age estimate of 650 ±
60 Myrin Nesvorn´y et al. (2015). No estimates for the ages of theBarcelona and Gallia family was provided in Spoto et al.(2015).Here we try to obtain a new estimate of the families agesusing the approach described in Carruba et al. (2015), thatuses a Monte Carlo method (Vokrouhlick´y et al. 2006a,b,c;Novakovi´c et al. 2010). The method has been described inseveral previous papers, so here we just shortly summarizedthe approach. Interested readers can found more details inCarruba et al. (2015). Basically, fictitious families with dif-ferent values of V EJ , a parameter describing the shape ofthe family ejection velocity field, are generated and thenevolved under the influence of the Yarkovsky and YORPeffects, and taking into account that solar luminosity wasless intense in the past. The obtained distribution of a C parameter, that depends on the asteroids semi-major axisand absolute magnitude, is then compared to the one ob-served for the real asteroid family, and a χ -like variable ψ ∆ C is then used to evaluate which fictitious family bestapproximate the C distribution of the real asteroid group.We applied this method to the Hansa, Barcelona, and Gal-lia families, and Fig. 4 displays our results in the ( Age, V EJ )plane. The radius of the family parent body, as estimatedfrom Nesvorn´y et al. (2015), and the escape velocity V ESC are reported in Table 2. Since Carruba & Nesvorn´y (2016)showed that most asteroid families have values of V EJ notgreater than 1.5 V ESC , we considered values of V EJ goingfrom 0 up to 90 m/s, i.e., equal to 1.5 the estimated es- cape velocity ( ≃
60 m/s) from the Hansa parent body, thebody with the largest escape velocity among the familieshere studied.At a 1-sigma level of probability of the simulated familydistribution being compatible with the real one (red curvein Fig. 4, associated ψ ∆ C = 10 .
73 and 12 degree of freedomsfor our distribution) we found that T = 460 +280 − Myr, and V EJ = 80 +10 − m/s for the Hansa family, T = 265 +45 − Myr,and V EJ = 15 +20 − m/s for the Barcelona family, and T =630 +30 − Myr, and V EJ = 5 +15 − m/s for the Gallia one. Again,our results are summarized in Table 2. V W LEPTOKURTIC FAMILIES
The time evolution of the kurtosis of the v W componentof the ejection velocity field was recently used by Carruba(2016) to set constraints on the age and acceptable valuesof key parameters describing the Yarkovsky force, such asthe surface thermal conductivity and asteroid density ofthe Astrid asteroid family. Here we use the same approachto investigate the dynamics of the three v W leptokurtichighly inclined families. The set-up of the simulations wasdiscussed in Carruba (2016), interested readers could findmore details in that paper. Basically, we simulated fictitiousfamilies with their currently observed size-frequency distri-bution, values of the parameters affecting the strength ofthe Yarkovsky force typical of S-type asteroids according toBroˇz et al. (2013), i.e., bulk and surface density, ρ bulk and ρ surf , equal to1500 and 2500 kg/m , respectively, thermalconductivity K = 0 .
001 W/m/K, thermal capacity equal to C th = 680 J/kg/K, Bond albedo A Bond = 0 . , and infraredemissivity ǫ = 0 .
9. We also generated fictitious families with Since the mean geometric albedo value of the Gallia family islower than that of the other two families, and since this couldimply a lower value of the Bond albedo, we also performed twoadditional sets of simulations for this family with a value of Bondalbedo A Bond = 0 .
07. The overall trend of the results of thissimulations was compatible with that of the standard simulationswith A Bond = 0 .
1. MNRAS , 1–8 (2016) nclined v W leptokurtic families
100 200 300 400 500 600 700 800 900 1000102030405060708090
Age [Myr] V E J [ m / s ] V EJ =V esc A −15−14−13−12−11−10−9−8−7
100 200 300 400 500 600 700 800 900 10005101520253035404550
Age [Myr] V E J [ m / s ] V EJ =V esc B −15−14.5−14−13.5−13−12.5−12−11.5−11
100 200 300 400 500 600 700 800 900 1000102030405060
Age [Myr] V E J [ m / s ] V EJ =V ESC C −30−28−26−24−22−20−18−16−14−12 Figure 4.
Target function ψ ∆ C values in ( Age, V EJ ) plane for the Hansa (panel A), Barcelona (panel B), and Gallia families (panelC). The horizontal green lines display the value of the estimated escape velocities V ESC from the parent body. The red lines display thecontour level of ψ ∆ C associated with a 1-sigma probability that the simulated and real distribution were compatible. Table 2.
Number of family members ( N mem ), mean geometric albedo ( p V ), estimated radius of the parent body ( R PB ), escape velocities( V ESC ), estimated V EJ , and ages T for the Hansa, Barcelona, and Gallia families.Family N mem p V R PB V ESC V EJ T [km] [m/s] [m/s] [Myr]Hansa 1094 0.26 28.0 30.0 80 +10 − +280 − Barcelona 306 0.25 13.5 14.5 15 +20 − +45 − Gallia 182 0.17 40.5 43.4 5 +17 − +30 − the optimal values of the ejection parameter V EJ found inSect. 3, for the three families Particles were integrated with SW IF T RMV SY , the symplectic integrator developed byBroˇz (1999) that simulates the diurnal and seasonal versionsof the Yarkovsky effect, over 1000 Myr for the Hansa fam-ily, 600 Myr for the Barcelona group, and 800 Myr for theGallia cluster, a time long enough to cover the putative es-timated ages of these families. Two sets of simulations were performed. In the first we accounted for all planets fromMercury to Neptune, while in the second we also includeCeres, Vesta, and Pallas as massive bodies. Once properelements were obtained, then values of v W were computedby inverting the third Gauss equation (Murray and Dermott1999): MNRAS , 1–8 (2016)
V. Carruba, R. C. Domingos, S. Aljbaae, M. Huaman δi = (1 − e ) / na cos ( ω + f )1 + ecos ( f ) δv W . (1)where δi = i − i ref , with i ref the inclination of thebarycenter of the family, and f and ω + f assumed equal to30 ◦ and 50.5 ◦ , respectively. As discussed in Carruba (2016),the shape of the v W distribution (and therefore its kurtosis),are not strongly dependent on the values of f and ω + f .Fig. 5 displays our results for the set of six simulations.Since the value of γ ( v W ) increased significantly when iso-lated objects drift beyond 4 sigma values in sin ( i ) from thecenter of the family, as in (Carruba & Nesvorn´y 2016) weeliminated from our computations of this parameters ob-jects with inclination beyond that range. In particular, thismeant considering asteroids with inclination between 21 . ◦ and 22 . ◦ for the Hansa family, between 27 . ◦ and 29 . ◦ for the Barcelona family, and between 24 . ◦ and 26 . ◦ forthe Gallia family. Results for the Hansa family without theeffect of massive asteroids show that values of γ ( v W ) arecompatible with the current one for times larger than 200Myr, which sets a lower limit on the family age. Includ-ing the massive asteroids only slightly alters this scenario,which suggest that the effect of resonances with Vesta for theHansa family should be minor, when compared with otherlocal resonances able to affect the inclination in the region.Spikes in the time behavior of γ ( v W ) are associated withisolated asteroids whose inclination temporarily approachedvalues of sin ( i ) close to the limits considered in our analysis.Of course, changing the allowed limits of sin ( i ) could mod-ify the length and the shape of the isolated spikes observedin Fig. 5. Since our goal in this paper is, however, to assessthe importance of different dynamical models and since weare using the same limits for the model with and withoutmassive asteroids, we believe that our approach should bereasonable.Concerning the Barcelona family, resonances with mas-sive asteroids are not important for this family, as shownin Fig. 5, panels C and D. Results are essentially identicalwith and without massive asteroids, as expected from the re-sults of Sect. 2, that showed that the Barcelona family is notactually crossed by the ν V resonance, or other resonanceswith massive bodies. This negative result, however, confirmthe usefulness of the γ ( v W ) as a tool to investigate thelong-term behavior of secular dynamics. As observed for theAstrid family (Carruba 2016), commonly used values of thekey parameters density and thermal conductivity for S-typefamilies are not able to produce the currently observed valueof γ ( v W ) over the estimated age of the family. This couldbe either caused by i) the fact that the mean values of den-sity and thermal conductivity for members of the Barcelonafamily could be higher, or ii) that the actual age of thisfamily could be younger than what obtained from estimatesfrom the Monte Carlo method of section 3. An analysis ofthe full dependence of the γ ( v W ) time behavior on values ofdensity and thermal conductivity for the Barcelona familyperformed in the same way as recently done for the Astridcluster seems to be outside the goals of this paper, that fo-cused on studying the effectiveness of the use of γ ( v W ) asa tool to investigate the long-term effect of secular dynam-ics. But it certainly remains an interesting topic for futureresearch. Finally, the case of the Gallia family is of particularinterest. If we do not consider the effect of secular resonanceswith Vesta, the simulated γ ( v W ) does not reach currentvalues over the estimated age of the family. But there isan excellent agreement if we include secular perturbationsfrom massive asteroids, as shown in Fig. 5, panel F. Thelarger values of γ ( v W ) obtained when massive asteroids areconsidered and the agreement with estimates of the Galliafamily age obtained with independent methods represent, inour opinion, some of the newest and most interesting resultsof this work. Our results could be summarized as it follows:(i) We identified the Hansa, Barcelona, and Gallia fami-lies in the domain of proper elements, obtained dynamicalmaps in the domains of proper ( a, e ) and ( a, sin ( i )), andmapped the location in the ( a, sin ( i )) of the three node res-onance with Ceres, Vesta, and Pallas. The Hansa and Galliafamilies are crossed by the ν V = s − s V secular resonances.All families are S-complex group, all characterized by rela-tively large values of γ ( v W ).(ii) We obtained age and terminal ejection velocities esti-mates for the three families using a Monte Carlo method tosimulate the Yarkovsky and stochastic YORP evolution inproper a of family members. At one sigma confidence levelwe found that T = 460 +280 − Myr, and V EJ = 80 +10 − m/sfor the Hansa family, T = 265 +45 − Myr, and V EJ = 15 +20 − m/s for the Barcelona family, and T = 630 +30 − Myr, and V EJ = 5 +15 − m/s for the Gallia one. Our results are summa-rized in Table 2.(iii) Simulated the dynamical evolution of fictitious fam-ilies with values of the ejection velocity parameters V EJ ob-tained from our previous analysis under the gravitationalinfluence of all planets, the Yarkovsky force, and the effectof Ceres, Vesta, and Pallas. The γ ( v W ) parameter was com-puted as a function of time for all simulated family members,and we monitored when its value was comparable to the cur-rently observed ones. The Gallia and, in a less measure, theHansa families were significantly affected by secular reso-nances with Vesta. Current values of γ ( v W ) for the Galliafamily could only be reached over the estimated family ageif secular resonances with Vesta were accounted for. Con-versely, secular resonances with main belt massive bodiesplay no significant role in the evolution of the Barcelonafamily. Independent constraints on the family age can beset by the time behavior of the γ ( v W ) parameter.Overall, we found that the γ ( v W ) parameter could bean invaluable tool for providing hints about the relative im-portance of secular dynamics, and to set constraints on theages, ejection velocity fields, and key parameters influenc-ing the Yarkovsky force, such as the mean density and sur-face conductivity of v W leptokurtic families, and could bein principle applied to other similar families identified inCarruba & Nesvorn´y (2016). MNRAS , 1–8 (2016) nclined v W leptokurtic families Age max
Age min
Time [My] γ Hansa familyA
Age max
Age min
Time [My] γ Hansa familyB
Age min
Age max
Age min
Age max
Time [My] γ Barcelona familyC
Age min
Age max
Time [My] γ Barcelona familyD
Age min
Age max
Time [My] γ Gallia familyE
Age min
Age max
Time [My] γ Gallia familyF
Figure 5.
Time dependence of the kurtosis of the v W component of the ejection velocity field ( γ ( v W )) for the simulated Hansa,Barcelona, and Gallia families (panels A, C, and E, respectively). Panels B, D, and E display the same, but for the simulation whereCeres, Pallas, and Vesta were considered as massive perturbers. The horizontal black line display the current values of the families γ ( v W ), while the vertical red lines show the estimated range of possible ages. ACKNOWLEDGMENTS
We are grateful to the reviewer of this paper, Dr. BojanNovakovi´c, for comments and suggestions that greatly im-proved the quality of this paper. We would like to thank theS˜ao Paulo State Science Foundation (FAPESP) that sup- ported this work via the grant 2016/04476-8, and the Brazil-ian National Research Council (CNPq, grant 305453/2011-4). This publication makes use of data products fromthe Wide-field Infrared Survey Explorer (WISE) and NE-OWISE, which are a joint project of the University of
MNRAS , 1–8 (2016)
V. Carruba, R. C. Domingos, S. Aljbaae, M. Huaman
California, Los Angeles, and the Jet Propulsion Labora-tory/California Institute of Technology, funded by the Na-tional Aeronautics and Space Administration.
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