| DATA_SET_DESCRIPTION |
Data Set Overview
=================
This asteroid dynamical family analysis has been carried out by David
Nesvorny using his Hierarchical Clustering Method (HCM) code, with proper
elements for 293,368 asteroids calculated by Milani and Knezevic. This
analysis includes only low-inclination asteroids (proper sine of
inclination less than 0.3).
The HCM code applies the HCM method of Zappala et al. (1990, 1994). The
input proper elements files were calculated by Milani and Knezevic using a
semianalytical method described in Milani and Knezevic (1994). 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.
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 al.
(2005, 2006) 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
150 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 150
values of d_cutoff between 1 and 150 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. (2005, 2006) for additional
information. In summary, each family is treated individually, taking into
account local resonances, radiation forces, etc. and this requires
significant human effort.
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 99.9% 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 493 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 2010)
includes 55 families while Nesvorny et al. (2005) list 41. This increase
is due to the increase in number of cataloged proper elements (100,000 in
2005, more than 300,000 in 2010), which allowed us to identify new cases.
Data Products
=============
Files listing the members for each of the 55 families in this analysis
along with their proper elements are in the directory data/families. The
filenames are the concatenation of the family number and name. The file
familieslist.tab gives a listing of the 55 families with their family
number, 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 numbers are 1XX for inner belt (2-2.5 AU), 2XX for central belt
(2.5-2.82 AU) and 3XX for outer belt (2.82-3.6 AU). Number XX increases
with the designation number of the asteroid after which the family is
named (e.g., 101 is the Vesta family named after (4) Vesta, listed on on
the 1st line of 101_vesta.tab; 102 is the Flora family named after (8)
Flora). Families with very few members, typically 3 to 6, are highlighted
by using XX > 50 (e.g., 151 is the Datura family with 6 members).
Ancillary Data
==============
The document directory includes the Nesvorny HCM code and the two input
files of proper elements which were used to generate this family analysis.
The code and input files 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_c.asc (hcluster.c) - The Nesvorny HCM code (compile with nrutil.c
and nrutil.h).
nrutil_c.asc (nrutil.c) - memory allocation functions used by hcluster.c
nrutil_h.asc (nrutil.h) - header of nrutil.c
allnum_pro.asc (allnum.pro) - The M&K proper elements for numbered
asteroids, used as input to hcluster.c.
ufitobs_pro.asc (ufitobs.pro) - The M&K proper elements for unnumbered
asteroids, used as input to hcluster.c.
Running the code:
On a Linux platform, code hcluster.c can be compiled by typing: 'gcc
hcluster.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 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 50 for the Koronis
family, for which 50 m/s is adequate).
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.
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.
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