Asteroid Families: Statistical Identification of New Families and Determination of Size Distributions of the Families’ Members

Home , 1272 Gefion, 15 Eunomia, 24 Themis, 43 Ariadne, 44 Nysa, 5 Astraea

Asteroid Families: Statistical identification of new families and determination of size distributions of the families’ members

J. Leliwa-Kopystyński University of Warsaw, Institute of Geophysics, Pasteura 7, 02-093 Warszawa, and Space Research Centre of Polish Academy of Sciences, Bartycka 18A, 00-716 Warszawa, Poland [email protected]

I. Włodarczyk Chorzow Astronomical Observatory (MPC 553), Al. Planetarium 4, 41-500 Chorzów, Poland [email protected] SUMMARY

1. Statistical approach to identification of the asteroid families. 2. Verification of the method: Known families. 3. Size distributions of the families members. (i) as a whole; (ii) variability with distance from the main body 4. Eunomia family as an example. 5. Identifications of the new families.

DATASET

In the present work we use the dataset of Knezevic and Milani (July 2006). It contains the proper elements of 130037 numbered asteroids belonging to the main asteroids belt.

(However, Knezevic and Milani, continuously update their dataset: e.g. version of December 2006 contains as many as 140798 asteroids.) Let Locally Largest Asteroid (LLA) has the proper elements (a0, e0, i0). We consider an asteroid with the proper elements a, e, i. The absolute values of the differences between the proper elements of this asteroid and the proper elements of the LLA are

∆a = |a – a0|, ∆e = |e – e0|, ∆i = |i – i0| .(1)

The considered asteroid is shifted off the point (a0, e0, i0) being the site of the LLA by the distances ∆a, a0∆e, a0sin(∆i) measured in the units of length [AU] along the axis a,e,i in the proper elements space.

Let Ω1 in the proper elements space is the volume of the sphere with the radius (∆a)1 centered in (a0, e0, i0). (The consecutive concentric volumes are Ωk = k Ω1): 2 2 2 2 ∆a + (a0 ∆e) + [a0 sin(∆i)] = (∆a)1 . (2)

The sphere cuts on the axis a,e,i, the segments with the lengths equal to (∆a)1, a0(∆e)1, and a0sin(∆i)1, respectively. The maximal values of the differences (∆e)1 and (∆i)1 express by the major semiaxis a0 of the LLA and by the parameter (∆a)1 as follows:

(∆e)1 = (∆a)1/a0 , (∆i)1 = arc sin[(∆a)1/a0] . (3) Space distributions of asteroids families members in Ω [AU3] centred in LLA.

DJuno = 244 km DAstraea = 125 km DEunomia = 272 km 1000 Karin Family

DKarin = 19 km 800

600

400

200 observed number of asteroids in the layer

0.000 0.001 0.002 0.003 0.004 volume of the sphere Table 1. Some of the well known families of asteroids.

Asteroid D a0 e0 i0 Ω Number Family Family B&Z = Bendjoya, Zappalà, 2002. of all population, population, 3 (LLA) km AU deg AU asteroids this work B&Z In: Asteroids III: HCM = Hierarchical Clustering Method N in Ω N – Nb in Ω. HCM WAM WAM = Wavelet Analysis Method 4 Vesta 501 2.362 0.099 6.39 0.005 10206 6406 ±400 231 242 63 Ausonia 108 2.395 0.121 6.22 0.005 8664 3664 ± 400 8 Flora 141 2.201 0.145 5.57 0.006 8536 3256 ±240 604 LLA = Locally Largest Asteroid 43 Ariadne 65.3 2.203 0.141 4.05 0.006 9278 3998 ± 240 575 10 Hygiea 429 3.142 0.136 5.10 0.006 2011 1387 ± 96 103 175 100 Hekate 92.0 3.095 0.150 5.22 0.006 1472 512 ± 96 Dataset ratio: 15 Eunomia 272 2.644 0.149 13.1 0.007 7359 4727 ± 224 439 303 85 Io 157 2.654 0.151 12.6 0.007 6953 4153 ± 224 This work / B&Z = 130037 / (~12000) 24 Themis 142 c 3.135 0.153 1.08 0.003 2071 1207 ± 72 550 517 62 Erato 99.3 3.122 0.150 1.29 0.003 2039 1079 ± 96 90 Antiope 125 3.146 0.154 1.33 0.003 2222 1262 ± 96 Family population ratio: 20 Massalia 151 2.409 0.162 1.42 0.006 9023 6143 ± 480 44 Nysa 73.3 2.423 0.174 3.06 0.006 10372 6052 ± 480 381 This work / B&Z ≈ 10 / 1 135 Hertha 82.0 2.429 0.174 2.67 0.006 9948 6588 ± 480 374 110 Lydia 127 c 2.733 0.050 5.25 0.004 2242 962 ± 96 26 50 137 Meliboea 106 c 3.119 0.186 14.4 0.004 418 98 ± 96 13 16 Errors are mainly originated from 158 Koronis 39.8 2.869 0.045 2.15 0.003 2504 1904 ± 96 325 299 estimations of density of background asteroids. 221 Eos 110 3.012 0.073 9.86 0.006 5676 4236 ± 240 477 482 1 Ceres 950 2.737 0.116 9.65 0.004 1771 1195 ± 96 93 Minerva 146 2.755 0.143 8.89 0.004 2252 1612 ± 96 1272 Gefion 10 c 2.784 0.129 9.01 0.004 1796 1156 ± 96 766 (ND) 3 Juno 244 2.669 0.234 13.25 0.003 778 298 ± 48 99 Dike 48 c 2.664 0.229 12.72 0.003 872 392 ± 48 170 Maria 46.2 2.554 0.101 15.09 0.005 2175 575 ± 120 70 (Z) 832 Karin 19 c 2.864 0.044 2.11 0.003 2458 1738 ± 120 39 (N)

Table 2. Search for the new families. The first 200 numbered asteroids is considered as the Locally Largest Asteroids (LLA).

LLA D a0 e0 i0 Ω Number Family 3 of all population (potential LFM) km AU deg AU asteroids N – Nb N in Ω

5 Astraea 125 2.576 0.198 4.51 0.004 2369 929 ± 96 53 Calypso 119 2.619 0.197 4.53 0.004 2351 751 ± 96

25 Phocaea 78 2.400 0.228 23.41 0.005 567 167 ± 40 31 Euphrosyne 248 3.155 0.208 26.54 0.002 124 76 ± 16

46 Hestia 131 2.526 0.117 2.49 0.004 2023 423 ± 64 91 Regina 114 2.590 0.103 2.75 0.004 2523 603 ± 64

58 Concordia 98 2.700 0.085 4.72 0.004 2106 346 ± 64 125 Liberatrix 69 c 2.743 0.086 4.46 0.004 2286 686 ± 64 128 Nemesis 137 c 2.750 0.088 5.13 0.004 1934 334 ± 64

73 Klytia 64 c 2.665 0.040 2.56 0.008 2582 1302 ± 128

79 Eurynone 104 2.444 0.177 5.00 0.002 1166 366 ±78

84 Klio 57 c 2.362 0.193 9.42 0.003 892 392 ± 48 108 Hecuba 112 c 3.486 0.093 9.69 0.004 798 222 ± 64

115 Thyra 111 c 2.380 0.180 12.5 0.004 599 215 ± 32 143 Adria 63 c 2.761 0.095 12.1 0.004 652 172 ± 32

177 Irma 56 c 2.460 0.053 12.9 0.002 140 76 ± 16

Diameters D of the asteroids are estimated from the formula (see e.g. Fowler and Chillemi, 1992): D = 1329 × 10-H/5 A –1/2 [km] = 3431 × 10-H/5 [km] H is the absolute magnitude, A is the albedo (it is assumed A = 0.15 for all asteroids).

D is used in order to calculate the integrated volume of the fragments in a given interval of size. So, the ratio (volume of the fragments) / (volume of the LLA) can be found. To do that we are fitting for all asteroid families identified in this paper (the previously recognized as well as the new ones) the logarithmic regression for size distribution lgn = – A lgD + B 15 Asteroids in the vicinity of Eunomia (a=2.644 AU, e=0.149, i=13.1 deg)

13 i_proper [deg] i_proper

< 3 km (924) 10 - 25 km (198) 50 - 3 - 10 km (4736) 25 - 50 km (6) 263 km (5) 11

2.52 2.56 2.60 2.64 2.68 2.72 a_proper [AU] 4 Positions of Eunomia family members

0 y [AU]

-2

ecliptic plane

-4

-4 -2 0 2 4 x [AU] 1000 15 Eunomia 4727 asteroids without background in Omega=0.007 AU^3 686 all asteroids in Omega (0.007, 0.009) AU^3

2305 family without background, D = 5-12 km log(N) = -3.319 * log(D) + 4.703, q = 0.929

D = 272 km 100 Eunomia N

256 background asteroids, D = 5-12 km log(N) = -4.342 * log(D) + 4.497, q = 0.741 Eunomia 1 110100 D[km] 1000 15 Eunomia 4727 asteroids without background in Omega=0.007 AU^3 2526 all asteroids in Omega (0.000, 0.001) AU^3

2305 family without background, D = 5-12 km log(N) = -3.319 * log(D) + 4.703, q = 0.929

D = 100 Eunomia 272 km N

Central part of Ω:

1330 family without background, D = 5-12 km, Omega (0.000, 0.001) AU^3 log(N) = -2.638 * log(D) + 3.884, q = 0.847 1 110100 D[km] Table 3. Size distributions of the asteroids families members fam – for a whole family without background, cent – for the centre of family without background, back – for background asteroids in the area adjacent to area containing the family.

Family Size range Ω Population lg n = – A lg D + B for 3 N in size AU D in kilometers regression range calculations A B q km – km 15 Eunomia – fam 5 – 12 0.000 - 0.007 2324 3.319 4.703 0.929 cent 0.000 - 0.001 1328 2.706 3.870 0.847 back 0.007 - 0.009 256 4.342 4.497 0.741 20 Massalia – fam 4 – 10 0.000 - 0.006 1087 4.945 5.296 0.955 cent 0.000 - 0.001 434 5.353 5.143 0.909 back 0.006 - 0.008 211 3.887 3.603 0.821 4 Vesta – fam 5 – 10 0.000 - 0.005 845 4.539 5.229 0.859 cent 0.000 - 0.001 520 5.122 5.512 0.838 back 0.005 - 0.007 214 4.281 4.445 0.808 8 Flora – fam 4 – 9 0.000 - 0.006 864 2.805 3.668 0.951 cent 0.000 - 0.001 315 2.194 2.761 0.808 back 0.006 - 0.008 440 4.329 4.452 0.830

and many other families considered but not presented here 20 15 Eunomia family

12 D, km 8

0 2.5 2.6 2.7 2.8 a , a.u. 20 20 Massalia family

Can be compared with Vokhrouhlicky et al., Icarus 182 (2006), Fig. 6. 16

12 D, km 8

0 2.30 2.35 2.40 2.45 2.50 a , a.u. 0.6 Distribution of (a, e) of the main belts asteroids according Knezevic and Milani, 2006 Previously known families

New families identified in this work

0.4 e

0.2

0.0 2.02.42.83.23.6 a [AU]