Data Set Overview  :  This asteroid dynamical family analysis has been carried out by David  Nesvorny (Nesvorny et al. 2015) using his Hierarchical Clustering Method  (HCM) code, with synthetic proper elements for 384,337 numbered asteroids. This analysis includes both low and high-inclination orbits.   The HCM code applies the HCM method of Zappala et al. (1990, 1994). The  synthetic proper elements were computed by Zoran Knezevic following the  method described in Knezevic et al. (2002). The distance cutoffs have  been selected by Nesvorny individually for each family based on a trial  and error method using visualization software. In three cases, namely  Vesta/Flora, Massalia/Nysa-Polana, Eunomia/Adeona, artificial cuts in  proper element space were used to prevent HCM from hopping between two  different families. See Nesvorny et al. (2015) for a general discussion of the family identification methods and specific description of this data  set.    Overview of the Method  :   The asteroid belt has collisionally evolved since its formation (see,  e.g., Davis et al. 2002). Possibly its most striking feature is the  presence of asteroid families that represent remnants of large,  collisionally disrupted asteroids (Hirayama 1918). Asteroid families can  be identified as clusters of asteroid positions in the space of proper  elements: the proper semimajor axis (a_P), proper eccentricity (e_P), and proper inclination (i_P) (Milani and Knezevic 1994, Knezevic et al. 2002). These orbital elements describe the size, shape and tilt of orbits.  Proper orbital elements, being more constant over time than the osculating orbital elements, provide a dynamical criterion of whether or not a group  of bodies has a common ancestor.   To identify an asteroid family, we use a numerical code that automatically detects a cluster of asteroid positions in 3-dimensional (3D) space of  proper elements. We briefly describe the code below. See Nesvorny et a  (2015) for a more thorough description.   The code implements the so-called Hierarchical Clustering Method  (hereafter HCM) originally proposed and pioneered in studies of the  asteroid families by Zappala et al. 1990, 1994). The HCM requires that  members of the identified cluster of asteroid positions in the proper  elements space be separated by less than a selected distance (the  so-called 'cutoff').   The procedure starts with an individual asteroid position in the space of  proper elements and identifies bodies in its neighborhood with mutual  distances less than a threshold limit (d_cutoff). Following Zappala et al. (1990, 1994), we define the distance in a_P, e_P, i_P space by d : n a_P  sqrt{C_a (da_P/a_P)^2 + C_e (de_P)^2 + C_i (dsin i_P)^2}, where n a_P is  the heliocentric velocity of an asteroid on a circular orbit having the  semimajor axis a_P, da_P : abs[a_P^(1) - a_P^(2)], de_P : abs[e_P^(1) -  e_P^(2)], and d sin i_P : abs[sin i_P^(1) - sin i_P^(2)]. The upper  indexes (1) and (2) denote the two bodies in consideration. C_a, C_e, and C_i are weighting factors; we adopt C_a : 5/4, C_e : 2 and C_i : 2  (Zappala et al. 1994). Other choices of C_a, C_e, and C_i yield similar  results.   Once bodies with d < d_cutoff are identified, each of them is used as a  starting point of the algorithm and new bodies are searched in its  neighborhood. The procedure is then iterated until no new bodies can be  found. The final result of the HCM method is a cluster of asteroids that  can be connected by a chain in proper elements space with segments shorter than d_cutoff. In other words, any member of the cluster will have, by  definition, at least one neighbor with distance d < d_cutoff. Also,  asteroids that were not classified as members cannot be connected to any  member by a segment shorter than d_cutoff. Note that spectroscopic  interlopers are not removed from the resulting family lists.    Selection of the Cutoff  :   The cutoff distance d_cutoff is a free parameter. With small d_cutoff the algorithm identifies tight clusters in proper element space. With large  d_cutoff the algorithm detects larger and more loosely connected clusters. For the main belt, the appropriate values of d_cutoff are between 1 and  200 m/s. To avoid an a priori choice of d_cutoff, we developed software  that runs HCM starting with each individual asteroid and loops over 200  values of d_cutoff between 1 and 200 m/s with a 1 m/s step. The result of  this algorithm can be conveniently visualized in a 'stalactite diagram'  (see Nesvorny et al. 2005).   The stalactite diagram is useful when we want to systematically classify  the asteroid families identified by HCM. With modern data (proper  asteroid catalog 2005 or newer), more than fifty families can be found  (see below). We also developed software that allows us to visualize, in  3D, the overall distribution of asteroid proper elements and highlight  families found by HCM. The software can also 'subtract' a cluster (or any number of clusters) from the distribution and show background. This is  helpful because it allows us to check on the 3D distribution of each  family, see if it ends at or steps over specific resonances, and give us a general idea about the appropriate range of d_cutoff values in each case.  By subtracting all identified families from the overall distribution, we  can also verify that no meaningful concentrations were left behind.   To select appropriate d_cutoff for each cluster, insights into the  dynamics of the main-belt asteroids are required. Nesvorny et al. (2005)  illustrated this for the Koronis family by discussing the number of  members of the cluster linked to (158) Koronis changes with d_cutoff. With small d_cutoff values, the algorithm accumulates members of a very tightly clustered group -- the product of the collisional breakup of a Koronis  family member about 5.8 My ago (Karin cluster, Nesvorny et al. 2002).  With d_cutoff about 20 m/s, the HCM starts to agglomerate the central part of the Koronis family. With even larger d_cutoff, the algorithm steps over the secular resonance that separates central and large semimajor axis  parts of the Koronis family (this particular shape resulted from long-term dynamics driven by radiation forces, Bottke et al. 2001). Finally, with  very large d_cutoff, the algorithm starts to select other structures in  the outer main belt that have unrelated origins. Therefore, according to  these considerations, d_cutoff : 10 m/s is the best choice for the Karin  cluster and d_cutoff : 50 m/s is best for the Koronis family (note that  these specific values are based on the 2008 update of proper element  catalog and may change as the catalogs grow).   For other families, we choose d_cutoff using similar criteria that are not explained here in detail. See Nesvorny et al. (2015) for additional  information. In summary, each family is treated individually, taking into account local resonances, radiation forces, etc. and this requires  significant human effort.  got to here.   Why is 'distance' a velocity?  -----------------------------   The 'distance' referred to by the distance cutoff is the distance in  proper elements phase space, which is related to the relative velocity of  the fragments as they were ejected from the sphere of influence of the  parent body. Thus this 'distance' has units of velocity.    Robustness and Completeness of the Resulting Families  :   To determine the statistical significance of each family we generated mock distributions of proper elements and applied the HCM to them. For example, to demonstrate a greater than 99% statistical significance of the Karin  cluster, we generated 1000 mock orbital distributions corresponding to the Koronis family determined at d_cutoff : 50 m/s , and applied the HCM  algorithm to these data. With d_cutoff : 10 m/s, we were unable to find a cluster containing more than a few dozen members, yet the Karin family  contains 541 members with this d_cutoff. We are thus confident that the  Karin cluster and other families to which we applied the same technique  are statistically robust.   For a discussion of the statistical significance of asteroid families with only a few known members (e.g. Datura, Lucascavin, Emilkowalski), see  Nesvorny et al. (2006) and Nesvorny and Vokrouhlicky (2006) In these  cases, the true membership was established based on past orbital histories of asteroids. As a byproduct of these numerical integrations, it was  determined that these families formed less than 1 My ago.   To determine whether our algorithm produced a reasonably complete list of  asteroid families, we searched for residual clusters in the background  asteroid population using proper elements and Sloan Digital Sky Survey  (SDSS; Ivezic et al. 2001) colors simultaneously. This method is based on  an assumption that each individual family represents a reasonably  homogeneous distribution of colors that may be different from that of  asteroids in the local background. We define the distance in a_P, e_P,  i_P, PC_1, PC_2 space, where PC_1 and PC_2 are the principal SDSS-color  components (see Nesvorny et al. 2005 for a definition), by d_2 : sqrt{d^2  + C_PC [(dPC_1)^2 + (dPC_2)^2]} , where d is the distance in a_P, e_P, i_P sub-space defined in the above equation, dPC_1 : abs[PC_1^(1) - PC_1^(2)]  and dPC_2 : abs[PC_2^(1) - PC_2^(2)]. The indexes (1) and (2) denote the  two bodies in consideration. C_PC is a factor that weights the relative  importance of colors in our generalized HCM search. With d in m/s, we  typically used C_PC : 10^6 and varied this factor in the 10^4 - 10^8 range to test the dependence of results.   We found no statistically robust concentrations in the extended proper  element/color space that could help us to identify new families. This  result shows that our list of dynamical families based on the proper  element data is reasonably complete. The current catalog (as of 2015)  includes 122 families. The increase relative to the previous family lists  is due to the increase in number of cataloged proper elements.   Data Products  :   Files listing the members for each of 119 families from synthetic proper  elements in this analysis are in the directory data/families. Note that  three of the 122 families listed in the families list are not represented  by files listing their members for the following reasons:   - 007 James Bond: James Bond is not an asteroid. (Asteroid 9007 James  Bond is a member of the Vesta family.)  - 503: This family has no members in the current analysis.  - 640 P/2012 F5 (Gibbs): This family was derived by another researcher  and is not included here. See Nesvorny et al. (2015).   The filenames are the concatenation of the family number and name. The  file familieslist.tab gives a listing of the 122 families with their  family number (FIN), family name, the distance cutoff used, and the number of members. The family name is the name of the largest asteroid in the  family. The family number is assigned as follows:   - less than 100 include Hungarias, Hildas, and Jupiter  Trojans,  - in the 400s include inner main belt families with  2.0 > a > 2.5 AU and i < 17.5,  - in the 500's include central belt families with  2.5 < a < 2.82 AU and i < 17.5,  - in the 600s include outer main belt families with  2.82 < a < 3.7 AU and i < 17,  - in the 700s include inner main belt families with  2.0 < a < 2.5 AU and i > 17,  - in the 800s include central main belt families with  2.5 < a < 2.82 AU and i > 17.5,  - in the 900s include outer main belt families with  2.82 < a < 3.5 AU and i > 17.5.'   Ancillary Data  :   The document directory includes the Nesvorny HCM code and the input file  of proper elements which were used to generate this family analysis. The  code and input file have been renamed to conform to PDS document  filenaming requirements. In the list below, the original filename is  given in parentheses. These files are provided as documentation of the  algorithms and input data used for generating the family memberships.   hcluster_syn_c.asc (hcluster_syn.c) - The Nesvorny HCM code for synthetic  proper elements (compile with nrutil.c and nrutil.h).   nrutil_c.asc (nrutil.c) - memory allocation functions used by hcluster.c  and hcluster_syn.c.   nrutil_h.asc (nrutil.h) - header of nrutil.c.   numb_syn.asc (numb.syn) - The Knezevic and Milani synthetic proper  elements for numbered asteroids, used as input to hcluster_syn.c.   Running the code:  On a Linux platform, code hcluster_syn.c can be compiled by typing: 'gcc  hcluster_syn.c -o hcluster -lm'. This works on Opteron 2360 workstation  running Fedora core. On other platforms, known compilation issues may  arise with memory allocation routines. On input, hcluster_syn requests  the designation number of an asteroid whose proper elements serve as a  starting point of the HCM algorithm (e.g., type 158 for the Koronis  family, where 158 stands for asteroid (158) Koronis). The second and last input is the cutoff distance in m/s (e.g., enter 45 for the Koronis  family, for which 45 m/s is adequate).    Modification History  :   Version 1.0 of this data set, archived in 2010, contained 55 families from the analytic proper elements of Milani and Knezevic. Version 2.0,  archived in 2012, contains a new HCM analysis of an expanded set of  analytic proper elements of M&K resulting in 64 families, plus an HCM  analysis based on synthetic proper elements of Knezevic et al. resulting  in 79 families. Version 3.0, archived in 2015, contains 122 families  based on synthetic proper elements, including high-inclination families.  Families based on analytic elements are not provided in this version  because the catalogs of analytic and synthetic elements are now almost the same size, and the synthetic elements are more precise.   The HCM analysis software has not been modified from one version to the  next.   References  :   Bottke, W.F., D. Vokrouhlicky, M. Broz, D. Nesvorny, and A. Morbidelli  2001. Dynamical Spreading of Asteroid Families via the Yarkovsky Effect.  Science 294, 1693-1696.   Davis, D.R., D.D. Durda, F. Marzari, A. Campo Bagatin, and R. Gil-Hutton  2002. Collisional Evolution of Small-Body Populations. In Asteroids III  (W.F. Bottke, A. Cellino, P. Paolicchi, and R. Binzel, Eds.). Univ. of  Arizona Press, Tucson, pp. 545-558.   Hirayama, K. 1918. Groups of asteroids probably of common origin. Astron. J. 31, 185-188.   Ivezic, Z., and 32 colleagues 2001. Solar System Objects Observed in the  Sloan Digital Sky Survey Commissioning Data. Astron. J. 122, 2749-2784.   Knezevic, Z., A. Lemaitre, and A. Milani 2002. The Determination of  Asteroid Proper Elements. In Asteroids III (W.F. Bottke, A. Cellino, P.  Paolicchi, and R. Binzel, Eds.). Univ. of Arizona Press, Tucson, pp.  603-612.   Milani, A., and Z. Knezevic, Asteroid Proper Elements and the Dynamical  Structure of the Asteroid Belt, Icarus 107, 219-254, 1994.   Nesvorny, D., W.F. Bottke, L. Dones, and H.F. Levison 2002. The recent  breakup of an asteroid in the main-belt region. Nature 417, 720-771.   Nesvorny, D., R. Jedicke, R.J. Whiteley, Z. Ivezic, 2005. Evidence for  asteroid space weathering from the Sloan Digital Sky Survey. Icarus 173,  132-152.   Nesvorny, D. and D. Vokrouhlicky, 2006. New Candidates for Recent Asteroid Breakups. The Astronomical Journal 132, 1950-1958.   Nesvorny, D., Vokrouhlicky, D., Bottke, W. F., 2006b. The Breakup of a  Main-Belt Asteroid 450 Thousand Years Ago. Science 312, 1490.   Nesvorny, D., Broz, M., Carruba, V., 2015. Identification and Dynamical  Properties of Asteroid Families. In Asteroids IV (P. Michel, F. DeMeo,  W.F. Bottke, R. Binzel, Eds.). Univ. of Arizona Press, Tucson, also  available on astro-ph.   Zappala, V., A. Cellino, P. Farinella, Z. Knezevic, Asteroid Families, I.  Identification by hierarchical clustering and reliability assessment,  Astronomical Journal, 100, 2030-2046, 1990.   Zappala, V., A. Cellino, P. Farinella, and A. Milani, 1994. Asteroid  families. II: Extension to unnumbered multiopposition asteroids. The  Astronomical Journal 107, 772-801.

Confidence Level Overview  :  Individual distance cutoffs have been selected subjectively for each  family with the intent of separating family members from background  objects. Family membership will vary with selection of different distance cutoffs.   The original review of V1.0 of this data set on June 7, 2010 found that  the documentation of the method was insufficient. After extensive  improvements to the data set description, a follow-up external peer review with was held on November 8, 2010. The follow-up review found that the  data set documentation is now sufficient to enable a data user to  understand and use the data and to understand the method by which the  family memberships were determined.