Asteroid families
are groups of asteroids that are supposed to have a common origin in
the collisional event that shattered the parent body. They are usually
determined by identifying clusters of objects close in proper elements
domain (a,e,sin(i)). The Hierarchical Clustering Method (HCM
hereafter) as described by Bendjoya and Zappalŕ (2002) operates by
identifying all objects that are closer than a given distance (cutoff)
with respect to at least one other member of a family.
If an object is closer than this distance, it is associated to the
dynamical family, and the procedure is repeated until no new family
members are found. For small values of the cutoff only the objects
closest in proper element domain are identified as family members: the
family ``core''. At larger cutoff one is able to identify objects that,
while still belonging to the collisional group, may have dynamically
evolved since the family formation and driftedapart from the core: the
family ``halo''.
One problem in obtaining a good determination of a family halo is
however the presence of objects in the orbital region of the halo that
might not be, for taxonomical reasons, associated with the local
family: the interlopers. 

An analysis of spectral
properties of local asteroids may provide insights on the possible
presence of such interlopers in groups found in proper elements
domains, but such information is usually available only for 2% or less
of the main belt asteroids.

The figure shows an (a,sin(i)) projection of the Hygiea family core
(blue crosses) and halo (red circles), as obtained in Carruba (2013).
Vertical red lines displays the location of the main meanmotion resonances in the region. 
Recently the Sloan Digital Sky SurveyMoving Object Catalog data,
fourth release (Ivezic et al. 2002, SDSSMOC4), provided multiband
photometry on a sample two order of magnitude larger than any available
in any current spectroscopic catalogs (about 60000 numbered objects).
Parker et al. (2008) constructed a colorcode diagram in a (a', iz) plane where:
a^{*}= C_{1} (gr)+C_{2}(ri)+C_{3} (1)
and C
_{1}, C
_{2}, and C
_{3} are numerical
coefficients that depend on the color values, on the number of
observations in the given database (Roig and GilHutton 2006),
and SDSS colors, known with an accuracy of about 0.03 magnitudes.
Bus and Binzel (2002a,b) and Nesvorný et al. (2005) found asteroid
families in extended domains of proper elements and SDSSMOC4 principal
components data in order to minimize the number of possible
interlopers, but such an analysis was not extended to asteroids'
geometrical albedo. Asteroid taxonomies are however also
determined by the body geometric albedo: Ctype asteroids tend to have
lower values of geometric albedo than Stype ones.
Tholen asteroid taxonomy used values of
p_{V}
to distinguish classes of asteroids inside the Xcomplex, such as the
M, E, and Ptypes. Until recently, however, only about
two thousand asteroids had reliable values of geometric albedos (see
Tedesco et al. 2002). Widefield Infrared Survey Explorer (WISE)
(Wright et al. 2010), and the NEOWISE (Mainzer et al. 2011) enhancement
to the WISE mission allowed to obtain diameters and geometric albedo
values for more than 100,000 Main Belt asteroids (Masiero et al. 2011),
increasing the sample of objects for which albedo values were known by
a factor 50. In this work we propose a new method quite effective
in determining objects belonging to a family halo, with low percentages
of likely interlopers, and results that are quite consistent in term of
taxonomy and geometric albedo of the halo members. We take full
advantage of the newly available WISE data and we introduce a new
hierarchical clustering method in a multidomain space composed by
asteroids proper elements (a,e,sin(i)), SDSSMOC4 colors
(a*,iz), and WISE geometric albedo
p_{V}, to identify halos associated with main belt asteroid families.
To avoid including too many interlopers in the family halo we proposed
to work with a metric in a multidomain space composed by proper
elements (a,e,sin(i)), SDSSMOC4 colors
(a*,iz), and WISE geometric albedo
p_{V}, to identify halos associated with main belt asteroid families. We eliminated from our sample asteroids with errors in :
a* or
(iz) larger than 0.1 magnitudes, and with
p_{V} larger than 0.05 if
p_{V} < 0.2, and
p_{V} larger than 0.1 if
p_{V} > 0.2. We also eliminated objects in the inner main belt with
p_{V}
> 0.5, since these objects may have large errors in albedos
associated with the way absolute magnitude are calculated (Masiero et
al. 2011). We then introduced a distance metric in this
multidomain space defined as:
d_{md}={d^{2}+C_{SPV} [(δa*)^{2}+( δ (i−z ))^{2}+(δpV )^{2} ]}^{1/2 } (2)
where d is the standard distance metric in proper element domain
(Zappalá et al. 1995)
, δa* = a^{*2} – a^{*1}, and similar relationships hold for
δ(iz) and δ
p_{V}.
C_{SPV} is a weighting factor equal to 10
^{6}.
As first halo members, we
selected asteroids that belong to the asteroids family, whose spectral
type is compatible with that of the other members. We then obtained dynamical groups using Eq. (2), for a value of cutoff d_{md} a bit less than the value for which the family halo merges with the local background.
The figure displays the number of members (blue line) and new members
(green line) of the Themis group as a function of the velocity cutoff d_{md}.


We then eliminated possible
interlopers, defined as objects in regions of the (a',iz) not
appropriate for the family, and with albedo values outside the range
usually expected (< 0.1 for CXcomplex objects, > 0.1 for
Scomplex objects).
The figure shows an (a*,iz) projection of Themis family asteroids,
with the division of the plane in the CXcomplex, Scomplex, and
Vtype asteroids areas.


Our result are summarized in the next webpages. Click on the
links in the table below to access the data on the five orbital regions
that we studied. The tables in each of the five regions
have links to the files with the information on the asteroid families
that we studied, in the format # of family member, asteroid
identification,
a,e,sin(i),n,g,s, H, Ly, p_{V}, δp_{V}, a*, iz, wher
n is the proper meanmotion,
g and
s are
the precession frequencies of the argument of pericenter and longitude
of the node, H is the absolute magnitude, and Ly is the Lyapunov
exponent.
Questions? Comments? We would like to hear from you. Please send us an
email.
N.B. The links above will be made available after publication of the MNRAS article that describes our method.