MNRAS 000, 1–10 (2016) Preprint 8 June 2021 Compiled using MNRAS LATEX style file v3.0 On the Astrid asteroid family V. Carruba1⋆ 1UNESP, Univ. Estadual Paulista, Grupo de dinˆamica Orbital e Planetologia, Guaratinguet´a, SP, 12516-410, Brazil. Accepted ... Received 2016 ...; in original form 2016 May 02 ABSTRACT Among asteroid families, the Astrid family is peculiar because of its unusual inclina- tion distribution. Objects at a ≃ 2.764 au are quite dispersed in this orbital element, giving the family a “crab-like” appearance. Recent works showed that this feature is caused by the interaction of the family with the s − sC nodal secular resonance with Ceres, that spreads the inclination of asteroids near its separatrix. As a consequence, the currently observed distribution of the vW component of terminal ejection veloci- ties obtained from inverting Gauss equation is quite leptokurtic, since this parameter mostly depends on the asteroids inclination. The peculiar orbital configuration of the Astrid family can be used to set constraints on key parameters describing the strength of the Yarkovsky force, such as the bulk and surface density and the thermal conduc- tivity of surface material. By simulating various fictitious families with different values of these parameters, and by demanding that the current value of the kurtosis of the distribution in vW be reached over the estimated lifetime of the family, we obtained that the thermal conductivity of Astrid family members should be ≃ 0.001 W/m/K, and that the surface and bulk density should be higher than 1000 kg/m3. Monte Carlo methods simulating Yarkovsky and stochastic YORP evolution of the Astrid family show its age to be T = 140±30 Myr old, in good agreement with estimates from other +17 groups. Its terminal ejection velocity parameter is in the range VEJ =5−5 m/s. Values of VEJ larger than 25 m/s are excluded from constraints from the current inclination distribution. Key words: Minor planets, asteroids: general – minor planets, asteroids: individual: Astrid– celestial mechanics. 1 INTRODUCTION of its standard deviation, that for a distribution of n random variable x is given by: The Astrid asteroid family is characterized by an unusual i distribution in the (a, sin (i)) plane, with a dispersion in in- 1 n 4 clination of its members at a ≃ 2.764 au much larger than (xi− <x>) k = n Pi=1 , (1) that of members at other semi-major axis. Novakovic et al. ( 1 n (x − <x>)2)2 n Pi=1 i arXiv:1606.05189v1 [astro-ph.EP] 16 Jun 2016 (2016) recently showed that this feature of the Astrid fam- 1 n ily is caused by its interaction with the s − sC nodal sec- where <x>= x is the mean value of the distri- n Pi=1 i ular resonance with Ceres. Asteroid crossing this resonance bution, then Pearson γ2 kurtosis is equal to k-3. Gaussian are significantly dispersed in inclination, causing the crab- distributions are characterized by values of γ2 equal to 0. like appearance of the family. The unusual distribution in The value of the Pearson γ2 parameter for the whole Astrid inclination of the Astrid family also produces other conse- family is quite high, but is closer to mesokurtic values if quences. Carruba & Nesvorn´y (2016) observed that the cur- asteroids in the resonant region are excluded. rent distribution of the vW component of terminal ejection In this work we investigate what information on key pa- velocities field computed from inverting Gauss equation for rameters describing the Yarkovsky effect, such as the ther- this family is characterized by a leptokurtic distribution, mal conductivity of material on the surface and the mass i.e., a distribution with larger tails and more peaked than density, can be obtained by studying the orbital diffusion of a Gaussian. If we define as kurtosis the ratio of the fourth fictitious members of several simulated Astrid families. By momenta of a distribution with respect to the fourth power checking on what time-scales the current value of γ2(vW ) can be reached, and for what values of the parameters describing the Yarkovsky force, constraints on the allowed range of val- ⋆ E-mail: [email protected] ues of these parameters can be, in principle, obtained. The c 2016 The Authors 2 V. Carruba independent constraints provided by secular dynamics (and massive body1 with SWIFT MVSF , the symplectic inte- from the current inclination distribution of Astrid members) grator based on SWIFT MVS from the Swift package of could then be used to estimate the age of the Astrid family Levison and Duncan (1994), and modified by Broˇz (1999) with a higher precision than that available for other families. to include on line filtering of osculating elements. The ini- tial osculating elements of the particles went from 2.730 to 2.828 au in a and from 1.00◦ to 2.45◦ in i. We used 50 inter- vals in a and 31 in i. The other orbital elements of the test particles were set equal to those of Ceres at the modified 2 FAMILY IDENTIFICATION AND LOCAL Julian date of 57200. DYNAMICS Fig. 3 displays our results for the two maps. For the case without Ceres (panel A) the orbital region of the Astrid fam- As a first step in our analysis we selected the Astrid family, ily is quite stable and regular, with most of the perturbations as identified in Nesvorn´yet al. (2015) using the Hierarchi- caused the 3J:-1S:-1A and 5J:-2A mean-motion resonances. cal Clustering Method (HCM, (Bendjoya & Zappal`a2002)) More interesting is the case where Ceres was treated like and a cutoff of 60 m/s. 489 members of the Astrid dynami- a massive body (panel B). As observed by Novakovic et al. cal group were identified in that work. We also selected as- (2016), the linear nodal secular resonance s − sC now ap- teroids in the background of the family, defined as a box in pears in the region. Objects whose pericenter frequency is the (a,e, sin (i)) domain. We selected asteroids to within the within ±0.3 arc-sec/yr from sC = −59.17 arc-sec/yr, likely minimum and maximum values of Astrid proper elements, resonators in the terminology of Carruba (2009), are shown plus or minus 0.02 au, 0.02, and 0.02 in proper a,e, and as black full dots in this figure. Two other secular resonances sin (i), respectively, with the exception of the maximum val- involving the nodal frequency sC of Ceres are also observed. ues in a that was given by the semi-major axis of the center Since the difference for the values of the g5 and g7 preces- of the 5J:-2A mean-motion resonance. 588 asteroids, 99 of sion frequency of the pericenter of Jupiter and Uranus is which not members of the Astrid group, were identified in small (4.257 and 3.093 arcsec/yr, respectively, which yield the background of the family so defined. a difference of 1.164 arcsec/yr (Kneˇzevi´cand Milani 2000)), − Fig. 1 displays the orbital location of family members resonances of resonant argument involving s sC and combi- (black full dots) and local background asteroids (black open nations of these two frequencies that satisfy the D’Alembert dots) in the (a,e) (top panel) and (a, sin (i)) (bottom panel) rules of permissible arguments are close in proper element − domains. The Astrid family numerically dominates the pop- space with respect to the main resonance s sC . In this ulation in the local background: 83.1% of the asteroids in the work we called such resonances “harmonics” of the main res- − − − − region are members of the HCM family. One can also notice onance. We identified the s sC g5+g7 and s sC 2(g5+g7) the spread in sin (i) of Astrid members at a ≃ 2.765 au, harmonics, whose likely resonators are shown in Fig. 3 as full caused by the nodal linear secular resonance with Ceres squares and full hexagons, respectively. s − sC , as shown in Novakovic et al. (2016). To study the resonant dynamics of the Astrid family We then turned our attention to the physical prop- members, we integrated the 489 HCM Astrid asteroids with erties of objects in the Astrid region. We checked which the same scheme used to obtain the dynamical map in Fig. 3, asteroids have information in the three major photo- panel B. We then i) identify the likely resonators in the − metric/spectroscopic surveys (ECAS (Eight-Color Aster- s sC resonance, and studied the time evolution of the res- − oid Analysis, Tholen et al. (1989)), SMASS (Small Main onant argument Ω ΩC . We identified 96 likely resonators, Belt Spectroscopic Survey, Bus & Binzel (2002a,b)), and and 19 objects (19.8% of the total) whose resonant argu- ◦ S3OS2 (Small Solar System Objects Spectroscopic Survey, ment librated around ±90 for 20 Myr, the length of the Lazzaro et al. (2004)), in the Sloan Digital Sky Survey- integration. Unfortunately, the limited number of objects − Moving Object Catalog data, fourth release (SDSS-MOC4 in librating states of the s sC resonance does not allow hereafter, Ivezi´cet al. (2001)), and in the WISE sur- to use conserved quantities of this resonance to obtain in- vey (Masiero et al. 2012). Taxonomic information was de- formation on the initial ejection velocity field, as done by duced for the SDSS-MOC4 objects using the method of Vokrouhlick´yet al. (2006b) for the Agnia family and the DeMeo & Carry (2013). We obtained taxonomic informa- z1 secular resonance, or, more recently, by Carruba et al.