# Asteroid family identification using the Hierarchical Clustering Method and
WISE/NEOWISE physical properties

Joseph R. Masiero11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak
Grove Dr., MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, A. K. Mainzer11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak
Grove Dr., MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, J. M. Bauer11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove
Dr., MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>22affiliation: Infrared Processing and Analysis Center, Caltech, Pasadena, CA
, T. Grav33affiliation: Planetary Science Institute, Tucson, AZ<EMAIL_ADDRESS>, C. R. Nugent44affiliation: Department of Earth and Space Sciences,
University of California, Los Angeles, CA<EMAIL_ADDRESS>, R.
Stevenson11affiliation: Jet Propulsion Laboratory/Caltech, 4800 Oak Grove Dr.,
MS 183-601, Pasadena, CA 91109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using albedos from WISE/NEOWISE to separate distinct albedo groups within the
Main Belt asteroids, we apply the Hierarchical Clustering Method to these
subpopulations and identify dynamically associated clusters of asteroids.
While this survey is limited to the $\sim 35\%$ of known Main Belt asteroids
that were detected by NEOWISE, we present the families linked from these
objects as higher confidence associations than can be obtained from dynamical
linking alone. We find that over one-third of the observed population of the
Main Belt is represented in the high-confidence cores of dynamical families.
The albedo distribution of family members differs significantly from the
albedo distribution of background objects in the same region of the Main Belt,
however interpretation of this effect is complicated by the incomplete
identification of lower-confidence family members. In total we link $38298$
asteroids into $76$ distinct families. This work represents a critical step
necessary to debias the albedo and size distributions of asteroids in the Main
Belt and understand the formation and history of small bodies in our Solar
system.

##  1 Introduction

In the first publication in this series (Masiero et al., 2011, hereafter:
Mas11) we presented the preliminary results for Main Belt asteroids (MBAs)
from the Wide-field Infrared Survey Explorer (WISE) thermal infrared all-sky
survey (Wright et al., 2010) and the NEOWISE Solar system enhancement to the
core WISE mission (Mainzer et al., 2011a). Mas11 also presented the observed
albedo and diameter distributions for asteroid families drawn from the overlap
between the set of objects detected by NEOWISE and the families identified by
Nesvorný (2012) using the Hierarchical Clustering Method (HCM, Zappalà et al.,
1990, 1994, 1995; Benjoya & Zappalà, 2002). In this paper we perform new
analysis of the Main Belt using HCM, taking into account dynamical
associations as well as asteroid albedo and diameter. This method allows us to
incorporate two unique characteristics of asteroid families that both result
from an origin in a catastrophic disruption of a single parent: compositional
consistency and minimal orbital velocity differences.

Asteroid families were first identified as groups of objects that clustered
tightly in orbital element-space by Hirayama (1918) nearly a century ago.
Subsequent work has confirmed that families originate from the catastrophic
breakup of a single parent asteroid after an impact (see Cellino et al., 2009,
for a recent review of the current state of the field). This single
mineralogical origin causes families to cluster tightly not only when
comparing orbital elements but also when investigating colors (Ivezić et al.,
2002; Parker et al., 2008), reflectance spectra (e.g. Binzel & Xu, 1993;
Cellino et al., 2001) and albedo (Mas11). We note that while the asteroid (4)
Vesta shows an albedo range of $0.10-0.67$ across its surface (Reddy et al.,
2012), and thus any other differentiated asteroid may have similar large
variation, the Vesta asteroid family has a albedo distribution comparable in
width to other asteroid families (Mas11).

Building further on the origin of families as a result of collisionally-driven
breakups, the size-frequency distribution (SFD) of asteroid family members
acts as a tracer of the physical properties of the original parent body and
can even be used to constrain the impact velocity and angle (Zappalà et al.,
2002; Durda et al., 2007). However, a major deficiency in the field to date
has been the lack of measured diameters for the family members, forcing these
values to be estimated based on the apparent visible magnitude of the object.
Furthermore, the strong selection effects imposed by visible light surveys
against the discovery of low albedo objects (particularly the smallest low
albedo objects) results in a skew in the size distributions of linked
families. Albedo measurements of the largest bodies in a family are often
available from the Infrared Astronomical Satellite (IRAS) data set (Tedesco et
al., 2002) and can be used to assume an albedo for all family members, but
this can add a significant and systematic error to the diameters, especially
in the cases where it is unclear if the largest body in a family is indeed
associated with the other members (e.g. Cellino et al., 2001; Masiero et al.,
2012b). From the NEOWISE survey we now have measurements of diameters for over
$130,000$ Main Belt asteroids with relative errors of $\sim 10\%$ (see Mas11
and Mainzer et al., 2011b). Using this new data set along with the associated
proper orbital elements for these objects, we identify the high-confidence
associations of asteroid families detected by NEOWISE.

##  2 Data

Proper orbital elements, which are key to the determination of asteroid family
membership, are the time averaged values of the semimajor axis, eccentricity,
and inclination after removing the short-period perturbations by Jupiter and
Saturn and averaging over long-period variations (Milani & Knez̀ević, 1998).
Proper elements are preferred to osculating elements for family identification
as they are stable over long time periods, and thus asteroid families cluster
more tightly in proper orbital element-space (Carpino et al., 1986). We use
the proper orbital elements computed following Milani & Knez̀ević (1994, 1998)
that are provided on the AstDys
website111http://hamilton.dm.unipi.it/astdys/index.php to identify asteroid
families in this work.

There are two primary methods for determining proper elements: analytic and
synthetic. Using the analytic method, a Fourier expansion of the Hamiltonian
is solved for directly. Conversely the synthetic method integrates the present
day osculating orbits using numerical simulations and determines the proper
elements from that evolution over time. While the analytic method produces
more accurate results, it has a fundamental limit of $\sim 18^{\circ}$
inclination above which the solutions degrade due to the truncation of the
Hamiltonian that is typically employed (Milani & Knez̀ević, 1994). As NEOWISE
observed the entire sky, it detected and discovered over $15000$ objects at
inclinations $>18^{\circ}$, over ten percent of our total sample. In order to
consider all asteroids detected by NEOWISE we use the synthetic proper
elements for this work. This allows us to have the maximum sample size however
synthetic proper elements have their own inherent limitations: oscillations
with periods much longer than the integration time (typically $\sim 10$
million years) will not be removed properly and the forest of weak secular
resonances in the Main Belt can result in chaotic cases that show up as
banding in the structure of the semimajor axis distribution (Knez̀ević &
Milani, 2000).

Following the methods of (Knez̀ević & Milani, 2000) we have integrated the
orbits of all Main Belt asteroids that were detected by NEOWISE and have
measured diameters and albedos, but not represented in the AstDys catalog.
This will include objects with very short arcs, in particular those discovered
by NEOWISE that have had minimal ground-based followup and thus do not have
measured orbits with sufficient quality to accurately integrate their
positions over millions of years. We include these for completeness, but with
the appropriate caveats for our results.

We draw our physical properties for Main Belt asteroids primarily from the
measurements made by the WISE spacecraft as part of the NEOWISE project. We
include in our analysis MBAs that were detected and discovered throughout the
entire mission, both the cryogenic and post-cryogenic surveys (Mainzer et al.,
2011a, 2012). WISE surveyed the sky simultaneously in four thermal infrared
bands ($3.4~{}\mu$m, $4.6~{}\mu$m, $12~{}\mu$m, and $22~{}\mu$m) from a polar
low-Earth orbit, progressing one degree per day. WISE imaged the entire static
sky over the course of $6$ months starting 14 January 2010, and began a second
pass survey until the exhaustion of the outer cryogen tank on 6 August 2010.
At this point, the longest wavelength channel was lost and WISE carried out a
3-Band Cryo survey until 29 September 2010 when the inner cryogen tank was
exhausted. The NEOWISE Post-Cryo Survey to complete coverage of the largest
MBAs and discover new near-Earth objects began 30 September 2010 and ended on
1 February 2011, using only the two shortest wavelength bands. All phases of
the mission employed the WISE Moving Object Processing System (WMOPS) to
detect moving objects in the Level 1 WISE images. WMOPS required a minimum of
5 detections to link a track (although a typical track had $10-12$
detections), which was then submitted to the Minor Planet Center (MPC) for
further verification. In total, WMOPS detected $>158,000$ solar system
objects, the majority of which were MBAs.

Diameters and albedos for MBAs seen during the Fully Cryogenic survey were
given in Mas11, while physical properties for objects detected in the 3-Band
Cryo and Post-Cryo Surveys were presented in Masiero et al. (2012a). We also
include diameters and albedo measured for $150$ objects by the IRAS mission
(Tedesco et al., 2002). In most cases these objects were so large that they
saturated in the WISE images, though a small number were missed due to the
exhaustion of cryogen, observing geometry effects, background contamination,
or other filtering in the WISE or NEOWISE data processing pipelines. When
these data are combined with the synthetic proper orbital elements our total
sample size is $112,286~{}$MBAs, which we use for the study presented here. We
note that this is smaller than the total number of objects detected by NEOWISE
due to the fact that $\sim 24,000$ objects do not have stable proper orbital
element solutions. Nearly three quarters of the unstable objects have
observational arcs shorter than one month implying that their orbits are not
well known and thus their proper elements cannot be reliably computed. The
remaining $\sim 6000$ unstable objects have orbital elements that are
indistinguishable from other MBAs, and orbital arcs typically of one year or
more. We note that only nine of these objects have received number
designations, and thus the provisional orbits may still be uncertain, which
could result in non-converging proper elements. Strangely, we also find that
low albedo objects dominate these long-arc unstable MBAs ($\sim 90\%$ have
albedos of $p_{V}<0.11$), which is much larger than the fraction of low albedo
objects in any of the regions of the Main Belt (Mas11). Albedo is not
considered when calculating asteroid proper elements and thus should not
affect these results, however low albedo objects are more likely to be fainter
than a high albedo object during a given observation epoch. Thus we may be
seeing low signal-to-noise uncertainties in a number of individual
observations propagating into the final orbits. Conversely, these objects may
indeed be unstable and thus transitional in their current orbits, potentially
representing low albedo objects from more distant regions of the Solar system
that have been implanted in the Main Belt. Distinguishing between these two
possibilities, however, is beyond the scope of the current work and will be
the subject of future investigation.

We show in Figure 1 the distribution of albedo as a function of diameter for
all objects used in this work. Two populations are apparent, a low-albedo
population at $p_{V}\sim 0.06$ and a high-albedo population at $p_{V}\sim
0.25$ (c.f. Mas11). The low-albedo population is shifted to larger diameters
compared to the high-albedo population due to selection biases in optical
catalogs. While the sensitivity of NEOWISE was effectively albedo-independent
(Mainzer et al., 2011d; Grav et al., 2011a, b), followup observations of
NEOWISE discoveries by ground-based optical surveys suffer from a decreased
sensitivity to smaller, lower albedo asteroids. This will result in the
apparent shift in the low albedo population to larger diameters. Though we
assume for purposes of this analysis that families have uniform albedos, this
bias means that the derived membership lists for lower albedo families will be
missing more small members than would a high albedo family in the same region
of the Main Belt.

Figure 1: Asteroid albedo ($p_{V}$) vs diameter for the $112,286$ MBAs used in
this work. The color of the points also indicates albedo, as given in the
color bar. The white contours indicate the density of points in the saturated
regions. Two albedo components are apparent, with the shift in the low albedo
distribution to larger diameters a result of the selection bias against small,
dark objects in optical observations. The horizontal picket fence effect at
low albedos is an artifact of the precision of the quoted albedos. The subtle
diagonal linear trends at small ($D<2~{}$km) diameters are an artifact of the
precision of the literature $H$ absolute magnitudes. Albedos larger than $\sim
0.5$ are likely artifacts of bad $H$ or $G$ values (c.f. Pravec et al., 2012)

Recently, Pravec et al. (2012) have presented evidence of possible biases in
the various catalogs of asteroid absolute magnitudes ($H$), and they highlight
the effect that these biases have on the albedo values derived for a small
sample NEOWISE-observed MBAs. While it is clear that in many cases the
published absolute magnitudes do not reflect the true values when measured
independently, these deviations show a non-linear relationship, with a maximal
deviation at $H\sim 14~{}$mag. Additionally, the scatter in magnitude
difference at $H>6~{}$mag is large, and almost always comparable to the mean
of the differences. As such, we chose not to implement a blanket offset
correction to the catalog $H$ values, which would tend to offset the NEOWISE-
derived albedos. We note that diameters derived from thermal infrared
measurements are largely unaffected by offsets in $H$, and quoted albedo
errors include estimated uncertainties on $H$ and $G$ of $0.3$ and $0.1$
respectively. Large surveys recently begun or soon to come online (e.g. Pan-
STARRS, LSST) should greatly improve the $H$ catalog values for most
asteroids. Additionally, implementation of the $H$-$G_{1}$-$G_{2}$ magnitude
phase function Muinonen et al. (2010) may also improve the determined albedos
for cases where there is sufficient photometric data. For this work, we use
the albedos as published that are derived from the literature $H$ magnitudes,
but with appropriate error bars. Future work will revise the catalog of
NEOWISE-measured diameters and albedos using the updated NEOWISE science data
processing system and the most current values of $H$.

##  3 Hierarchical Clustering Method

Asteroid families were originally identified as pairs or groups of objects
with orbital elements that clustered more tightly than would otherwise be
expected from a random distribution of objects. Zappalà et al. (1990) present
a method for association of asteroid families based on their proper orbital
elements called the Hierarchical Clustering Method (HCM) which uses a distance
function and a velocity cut to link objects together into clusters. Iterating
over each body in the population, the distance between it and all other
asteroids is calculated by converting the difference in orbital elements into
a pseudo-velocity. All objects within a given velocity threshold are added
into the family. Each family member is then similarly tested, accreting nearby
objects into the family, until no further objects are added. The selected
velocity cutoff will strongly dictate the size of the family and the
reliability of the associations: a cutoff velocity that is too low will only
identify the core regions of the densest families, while a cutoff velocity
that is too large will include a large number of interlopers in the family
lists or accept spurious groupings of objects that do not have a real
collisional origin.

Following Zappalà et al. (1994), we apply a distance metric of

$d=2\pi
F\sqrt{\frac{\frac{5}{4}\left(\frac{a_{i}-a_{c}}{a_{c}}\right)^{2}+2(ecc_{i}-ecc_{c})^{2}+2(\sin{inc_{i}}-\sin{inc_{c}})^{2}}{a_{c}}}$

where $d$ is the distance in m/s between the objects, $F$ is the conversion
factor changing AU/yr to m/s, $a$, $ecc$, and $inc$ are the proper semimajor
axis, eccentricity, and inclination of the body, and the subscripts $c$ and
$i$ indicate the center body and the body being tested, respectively. We
perform HCM analysis on every object in our MBA sample, at velocity limits
ranging from $5~{}$m/s to $200~{}$m/s in steps of $5~{}$m/s. This allows us to
build up a database of all associations for each object at a range of
velocities.

Following Nesvorný (2012) we divide the Main Belt into three regions separated
by strong Jupiter mean motion resonances: the inner Main Belt (IMB,
$1.8<a<2.5~{}$AU), the middle Main Belt (MMB, $2.5<a<2.82~{}$AU), and the
outer Main Belt (OMB, $2.82<a<3.6~{}$AU). We set a limit on perihelion
distance of $q>1.666~{}$AU to ensure Mars-crossing asteroids are not included
in our data set. We use physical properties as a discriminant in family
identification by dividing each region of the Belt into two groups by albedo.
While the majority of MBAs show a bimodal albedo distribution (Figure 1), a
small number of asteroid families have a mean albedo that falls in between
these two peaks (Mas11). In order to ensure that these families can be
properly identified and that the wings of the albedo distribution of each
family are not truncated, we have separated the regions by albedo allowing for
an overlap region in between. As such, objects with moderate albedo will
appear in both lists. Our high albedo group for each Main Belt region includes
all objects with $p_{V}>0.065$ while the low albedo group includes objects
with $p_{V}<0.155$, which represents a buffer of $\pm 40\%$ relative to the
central minimum reflectance of the MBA albedo distribution at $p_{V}=0.11$
(Mas11). The total number of objects in each group searched as well as the
bounding semimajor axes and albedo are shown in Table 1.

Table 1: Main Belt regions used for HCM analysis

Region | Semimajor Axis Range (AU) | Albedo Range |  Number† |  QRL‡ (m/s)
---|---|---|---|---
IMBhigh | $1.8<a<2.5$ | $p_{V}>0.065$ | 21013 | $110\pm 6$
IMBlow | $1.8<a<2.5$ | $p_{V}<0.155$ | 10622 | $132\pm 7$
MMBhigh | $2.5<a<2.82$ | $p_{V}>0.065$ | 26214 | $101\pm 8$
MMBlow | $2.5<a<2.82$ | $p_{V}<0.155$ | 22958 | $102\pm 7$
OMBhigh | $2.82<a<3.6$ | $p_{V}>0.065$ | 24204 | $102\pm 7$
OMBlow | $2.82<a<3.6$ | $p_{V}<0.155$ | 38691 | $107\pm 8$

†Summed number of objects in all regions is greater than total population due
to overlap between albedos; see text for details
‡Quasi-Random Level used for determining significant family linkages

To enable rapid searching of each region investigated we employ a
k-dimensional (KD)-tree query to perform an initial reduction in the possible
associations for each object. KD-trees are computational methods of dividing
up multidimensional data to increase the efficiency of searches for specific
data points within that space. All objects identified by the KD-tree test are
then compared to the velocity limit of that run to test for family membership.
This process provides a dramatic reduction in run time of the search
procedure.

After identifying the family associations at each velocity cut for each region
of the Belt, we need to determine which velocity cut represents the optimal
blend between completeness and accuracy. We follow Zappalà et al. (1994) and
use a Quasi-Random Level (QRL) test to determine at which point background
objects begin to become a significant contributor to family lists. The
methodology behind a QRL test is to construct a synthetic population based on
the orbital elements of the real MBAs and determine at which velocity limit
the synthetic objects begin to be linked by HCM. This velocity can then be
considered the level at which a quasi-random population of objects begin to
contribute to family lists; any linkages below this limit are unlikely to be
random associations. We note that as the overall population of known MBAs
grows, the average distance between any two asteroids will decrease and thus
the QRL will get smaller over time (although as more family members are
observed the velocity cut at which they link will also shrink). We use a lower
limit of ten objects as the minimum size group we consider both for family
membership and QRL determination.

In order to construct a representative synthetic background population, one
must know which objects are in the background as opposed to belonging to a
family and thus a priori have a list of known family members. This becomes
especially critical when a large, dense family dominates a small area of the
phase space. If these families are not removed, the QRL will be made
artificially smaller as real family members begin to link together in the
quasi-random population.

Removing a too many objects will decrease the number density and thus increase
the calculated QRL, while removing too few will have the opposite effect.
Similarly, replacing removed objects with randomly generated ones will
artificially increase the number of objects not associated with families when
compared to the real population, causing them to link at a lower level than
the observed background. The two parameters determining the number of objects
removed are the linking velocity level chosen and the size of the smallest
family removed. By varying these two parameters, we tested their effect on the
final QRL determination for all regions of the Main Belt. We find that the
size of the smallest family removed (from $50-200$ members) has only a minimal
effect on the determined QRL, as it is the few largest families that dominate
the artificially reduced QRL levels. On the other hand, the velocity level
chosen (from $75-150~{}$m/s) does have a strong effect on the final determined
QRL, which track each other closely. We use the lists of members of the
largest families identified by Nesvorný (2012) to find a velocity cut in our
determination that most closely reproduces those families, and use that
velocity level as the limit for large family removal.

In this implementation we use limits of 100 members and a velocity cut of
$125~{}$m/s to remove the largest families. Any asteroid in one of these
families was removed from the sample used to generate the quasi-random
population. We then divided each region into subregions by semimajor axis so
that each eccentricity-inclination slice contained $10\%$ of the total number
of objects in the entire region. The quasi-random population was built by
randomly drawing $a$, $ecc$, and $inc$ values from the members of the slice.
This ensured that the distribution of these three orbital parameters remained
identical for the original and quasi-random populations, minus the removed
large families. The slices were then reassembled and HCM was run on this
population for all velocity cuts from $5-200~{}$m/s is steps of $5~{}$m/s.
This process was performed ten times for each region of the Main Belt, and the
velocity levels of the first five pseudo-families of each trial were averaged
to determine the QRL for the region. This provides both a mean QRL level for
each region as well as an estimate of the uncertainty on that value, both of
which are shown in Table 1.

A common way of representing asteroid families is a ‘stalactite’ plot, which
shows the membership of each family as a function of cutoff velocity as well
as which families merge or fragment at various cutoff levels. We show
stalactite plots for each of the six regions considered here in Figures 2-7.
We also show the $1\sigma$ range of the QRL as the grey box overlaid on the
stalactites. At each velocity step, the families are labeled with the
numerical designation of the member with the largest measured diameter.
Designations will change with decreasing velocity as objects connected more
loosely fall out of the family list. We have left the smallest families on
each plot unlabeled to preserve the clarity of the figures.

Figure 2: Stalactite diagram showing family associations for the high albedo
objects in the inner Main Belt. Families are identified by the designation
number of the largest asteroid contained in the family list and length of each
bar indicates the number of objects contained in that family at that velocity
cut. The count at the bottom of the plot is an arbitrary sum of the bars at
the topmost level. Small families are left unlabeled for clarity but are
discussed in the text. The grey box shows the $1\sigma$ range around the QRL
level: family associations within this range are likely contaminated by
background objects. The three dashed red lines indicate the velocities used to
extract family membership lists.

Figure 3: The same as Figure 2 but for the low albedo inner Main Belt
asteroids.

Figure 4: The same as Figure 2 but for the high albedo middle Main Belt
asteroids.

Figure 5: The same as Figure 2 but for the low albedo middle Main Belt
asteroids.

Figure 6: The same as Figure 2 but for the high albedo outer Main Belt
asteroids.

Figure 7: The same as Figure 2 but for the low albedo outer Main Belt
asteroids.

Stalactite plots are read from top to bottom, showing the least- to most-
significant clusters of objects. For example, in Figure 2 we can follow the
evolution of the objects linked to (8) Flora. At large velocities
($v_{link}>100~{}$m/s) a large fraction of the asteroids in the IMBhigh region
link to (4) Vesta. Even below the QRL level at $v_{link}=100~{}$m/s all of the
large families remain grouped together. At $v_{link}=95~{}$m/s the Vesta and
Flora clumps separate, and at $v_{link}=85~{}$m/s the Flora clump breaks up
into the Flora, (20) Massalia, and (135) Hertha families. At velocities below
$v_{link}=80~{}$m/s Flora no longer links with the family, and nearly half of
the other family members are rejected at this cutoff as well. In contrast,
Vesta remains linked to its family down to $v_{link}=65~{}$m/s, though this
family also loses a substantial fraction of linked members as the cutoff
velocity decreases.

##  4 Results and Discussion

###  4.1 Identified Families

We use our stalactite plots as a first cut to guide our selection of the
optimal cutoff velocity at which to extract each family list. For each region
we have extracted families at three different velocity cuts: one at or near
the lower edge of the identified QRL zone and two others at $10~{}$m/s and
$20~{}$m/s below that level to separate overlapping families (e.g. Vesta and
Flora) and winnow out background objects that may be connecting below the QRL
level. This is similar to the method employed by Milani et al. (2010) to
identify the Hungaria family. The extraction levels for each region are
indicated in Figures 2-7 by the dashed red lines. We note that due to the
assumptions inherent in our QRL determination, some of the associations
presented here may be incomplete or even spurious. A more detailed analysis of
each family individually can refine these associations, and will be the
subject of future work.

By plotting the diameter of each family member against their proper semimajor
axis, and color-coding the points to represent albedo, we construct “petal”
plots which can be used to diagnose the reliability of the family association.
An ideal family will have a large parent body at the bottom-center of the plot
with the family spreading to smaller diameters and larger distance in
semimajor axis from the parent, as would be expected from a family evolved by
Yarkovsky drift (Bottke et al., 2006). For families in dense regions of the
Main Belt, large families will link together even at levels below the QRL. By
choosing a lower velocity cutoff we can disentangle these overlapping
families. Figure 8 shows an example of this, where the Massalia family blends
with the Flora and Vesta family as the velocity cutoff is increased. We can
also use this technique to filter out large background objects that may be
linked to a family and thus mis-identified as the parent body. Figure 9 shows
an example of this scenario, where (500) Selinur is linked to the family at
larger velocities, but at the lowest cutoff is rejected from the family. The
albedo of (500) Selinur and its placement on the plots indicate it is unlikely
to be the parent of the family, and that (3811) Karma is a much more likely
parent body. This results in a family with a very classic petal shape.

Figure 8: Diameter vs proper semimajor axis for the Massalia family at three
different cutoff velocities (from top to bottom: $100~{}$m/s, $90~{}$m/s,
$80~{}$m/s). The color of each point indicates its albedo as in Figure 1,
while $N$ lists the number of objects linked at that cutoff. Each subplot is
titled with the largest object included in the family, the presumed parent.
Here, the family of (20) Massalia (marked as a square) blends with the
families of (8) Flora and (4) Vesta (both marked as triangles) at velocity
cutoffs larger than $80~{}$m/s. Figure 9: Diameter vs proper semimajor axis
for the Karma family at three different cutoff velocities (from top to bottom:
$90~{}$m/s, $80~{}$m/s, $70~{}$m/s). The color of each point indicates its
albedo as in Figure 1, while $N$ lists the number of objects linked at that
cutoff. Each subplot is titled with the largest object included in the family,
the presumed parent. Here, (500) Selinur (marked as a triangle) links with the
family at larger velocities, but is rejected at $70~{}$m/s, making (3811)
Karma (marked as a square) the most likely parent.

Using the physical and orbital parameters of the objects that were linked
together at each velocity cutoff, we identify the highest reliability families
in the Main Belt that can be detected with this technique. To ensure a high
level of reliability and measure statistically significant family properties
we have only included families with $>40$ members. We find $76$ families
throughout the Main Belt that pass these cutoffs representing $38,298$ MBAs
(approximately $35\%$ of all objects considered), and another $\sim 60$
candidate groupings below of $40$ member limit that are too small to be
definite detections. We give a list of the identified families, with the
number (in MPC-packed format) and name of the largest member, the HCM velocity
the family was extracted at ($v_{link}$), the median proper semimajor axis,
eccentricity, and inclination for the family ($a_{med}$, $ecc_{med}$,
$inc_{med}$, respectively), and the number of linked family members (N) in
Table LABEL:tab.stats. We also include average physical properties for each
family in Table LABEL:tab.stats (see Section 4.2 for description of these
parameters). Names with a ‘*’ suffix indicate cases of ambiguous parent bodies
(see below). In Table 3 we present the full list of asteroids associated with
families, their orbital and physical parameters, and identify the family to
which they have been linked.

Table 2: Orbital elements, median and maximum diameters, average albedos, and raw SFD slopes ($\alpha$) for observed asteroid families Number | Name | $v_{link}$ | $a_{med}$ | $ecc_{med}$ | $inc_{med}$ | $D_{max}$ | $D_{med}$ | $\overline{p_{V}}$ | $\sigma_{p_{V}}$ | $\alpha_{SFD}$ | $\sigma_{\alpha}$ | N
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (m/s) | (AU) |  | (deg) | (km) | (km) |  |  |  |  |
00004 | Vesta | 80 | 2.3469 | 0.0972 | 6.6786 | 468.30 | 2.50 | 0.361 | 0.111 | -3.448 | 0.038 | 1331
00008 | Flora | 80 | 2.2543 | 0.1409 | 5.4513 | 155.74 | 2.70 | 0.288 | 0.088 | -2.589 | 0.032 | 929
00020 | Massalia | 80 | 2.4035 | 0.1636 | 1.4211 | 135.68 | 1.91 | 0.243 | 0.066 | -3.929 | 0.164 | 203
00135 | Hertha | 80 | 2.3990 | 0.1796 | 2.4243 | 82.15 | 2.42 | 0.284 | 0.091 | -3.375 | 0.050 | 1113
00254 | Augusta* | 90 | 2.1986 | 0.1213 | 4.1490 | 11.85 | 2.45 | 0.305 | 0.094 | -1.727 | 0.111 | 72
00434 | Hungaria | 100 | 1.9466 | 0.0771 | 21.0020 | 8.93 | 1.66 | 0.722 | 0.156 | -2.102 | 0.397 | 48
00587 | Hypsipyle | 100 | 2.3354 | 0.2203 | 24.0667 | 12.23 | 3.01 | 0.318 | 0.093 | -2.285 | 0.389 | 43
01646 | Rosseland* | 100 | 2.3501 | 0.0983 | 8.0594 | 12.47 | 3.23 | 0.194 | 0.047 | -2.393 | 0.293 | 46
02409 | Chapman | 80 | 2.2746 | 0.1342 | 3.2275 | 8.70 | 2.31 | 0.288 | 0.076 | -2.391 | 0.205 | 78
04689 | Donn | 90 | 2.2763 | 0.1166 | 4.7007 | 6.14 | 2.72 | 0.278 | 0.074 | -3.326 | 0.351 | 60
13698 | 13698* | 80 | 2.4368 | 0.1129 | 6.3787 | 5.97 | 2.75 | 0.359 | 0.107 | -4.185 | 0.309 | 87
00012 | Klio* | 120 | 2.3859 | 0.1922 | 9.3462 | 126.64 | 3.57 | 0.062 | 0.019 | -2.750 | 0.082 | 269
01715 | Salli | 120 | 2.4114 | 0.2289 | 10.9966 | 24.16 | 3.49 | 0.062 | 0.021 | -2.907 | 0.103 | 178
00163 | Erigone | 120 | 2.3718 | 0.2088 | 5.0486 | 81.58 | 2.98 | 0.051 | 0.012 | -3.229 | 0.040 | 1093
00298 | Baptistina* | 100 | 2.2737 | 0.1451 | 5.6067 | 21.14 | 2.34 | 0.158 | 0.029 | -2.692 | 0.048 | 549
00302 | Clarissa | 120 | 2.3967 | 0.1082 | 2.6910 | 38.53 | 3.02 | 0.056 | 0.017 | -3.242 | 0.110 | 228
00554 | Polana* | 120 | 2.3566 | 0.1485 | 2.8258 | 102.78 | 3.11 | 0.057 | 0.015 | -2.376 | 0.016 | 2438
00623 | Chimaera | 130 | 2.4456 | 0.1500 | 14.8210 | 44.09 | 4.16 | 0.059 | 0.011 | -2.507 | 0.357 | 46
00752 | Sulamitis | 120 | 2.4407 | 0.0894 | 5.0428 | 60.85 | 3.38 | 0.052 | 0.013 | -2.408 | 0.090 | 191
00003 | Juno | 80 | 2.6654 | 0.2351 | 13.3530 | 246.60 | 2.15 | 0.252 | 0.062 | -3.318 | 0.121 | 196
00005 | Astraea | 80 | 2.5823 | 0.1986 | 4.4909 | 113.00 | 2.50 | 0.279 | 0.072 | -2.962 | 0.178 | 94
00015 | Eunomia | 70 | 2.6214 | 0.1497 | 13.1676 | 299.21 | 4.07 | 0.268 | 0.073 | -2.958 | 0.025 | 2140
00472 | Roma* | 80 | 2.6022 | 0.0915 | 14.8032 | 47.04 | 3.63 | 0.257 | 0.078 | -2.147 | 0.024 | 712
00480 | Hansa | 80 | 2.6557 | 0.0111 | 21.9749 | 65.67 | 3.12 | 0.249 | 0.091 | -2.358 | 0.231 | 65
00606 | Brangane | 70 | 2.5817 | 0.1803 | 9.6134 | 39.53 | 2.87 | 0.112 | 0.034 | -3.322 | 0.354 | 57
00808 | Merxia | 80 | 2.7431 | 0.1341 | 5.0083 | 37.68 | 3.16 | 0.229 | 0.062 | -2.624 | 0.180 | 90
00847 | Agnia | 80 | 2.7902 | 0.0723 | 3.8130 | 30.08 | 3.84 | 0.227 | 0.070 | -2.850 | 0.107 | 180
01658 | Innes | 80 | 2.5799 | 0.1727 | 7.5910 | 13.81 | 3.05 | 0.256 | 0.071 | -3.284 | 0.159 | 155
02595 | Gudiachvili* | 80 | 2.7718 | 0.1311 | 9.1140 | 14.62 | 3.97 | 0.265 | 0.069 | -3.087 | 0.055 | 584
00539 | Pamina | 80 | 2.7428 | 0.1628 | 8.2446 | 56.04 | 4.57 | 0.057 | 0.019 | -2.588 | 0.177 | 88
01734 | Zhongolovich* | 80 | 2.7861 | 0.1958 | 7.8512 | 26.70 | 4.86 | 0.054 | 0.014 | -2.557 | 0.034 | 903
00145 | Adeona | 70 | 2.6513 | 0.1663 | 11.6481 | 132.59 | 4.46 | 0.059 | 0.014 | -2.704 | 0.027 | 1321
00128 | Nemesis | 70 | 2.7343 | 0.0896 | 4.8703 | 193.08 | 3.64 | 0.071 | 0.023 | -3.718 | 0.094 | 390
00363 | Padua | 55 | 2.7344 | 0.0411 | 5.3190 | 86.04 | 4.15 | 0.067 | 0.018 | -2.655 | 0.048 | 512
00272 | Antonia* | 55 | 2.7842 | 0.0476 | 4.3932 | 25.67 | 4.01 | 0.046 | 0.011 | -2.966 | 0.040 | 861
00144 | Vibilia | 90 | 2.6715 | 0.1888 | 3.8417 | 142.38 | 4.33 | 0.064 | 0.013 | -3.095 | 0.123 | 184
00322 | Phaeo | 90 | 2.7875 | 0.1935 | 9.4217 | 73.15 | 3.37 | 0.068 | 0.017 | -2.956 | 0.282 | 72
00342 | Endymion* | 70 | 2.5719 | 0.1401 | 8.7950 | 64.27 | 3.39 | 0.043 | 0.013 | -2.812 | 0.086 | 230
00396 | Aeolia | 80 | 2.7399 | 0.1678 | 3.4398 | 37.29 | 2.85 | 0.094 | 0.027 | -2.823 | 0.289 | 62
00404 | Arsinoe* | 80 | 2.6257 | 0.2304 | 13.1558 | 105.41 | 4.17 | 0.051 | 0.015 | -2.391 | 0.137 | 113
00410 | Chloris | 90 | 2.7458 | 0.2522 | 8.7834 | 118.93 | 5.33 | 0.084 | 0.029 | -2.571 | 0.144 | 116
03811 | Karma | 70 | 2.5690 | 0.1066 | 10.7836 | 28.75 | 3.96 | 0.054 | 0.010 | -2.603 | 0.329 | 61
00569 | Misa | 80 | 2.6479 | 0.1778 | 2.2867 | 78.93 | 3.46 | 0.052 | 0.015 | -2.447 | 0.054 | 357
01128 | Astrid | 80 | 2.7773 | 0.0484 | 0.6818 | 48.63 | 3.50 | 0.046 | 0.011 | -2.585 | 0.083 | 201
01668 | Hanna | 80 | 2.7932 | 0.1766 | 4.2265 | 25.83 | 3.86 | 0.051 | 0.013 | -3.608 | 0.185 | 122
02669 | Shostakovich* | 90 | 2.7687 | 0.1739 | 9.2127 | 16.47 | 5.14 | 0.051 | 0.016 | -3.001 | 0.203 | 98
03567 | Alvema* | 90 | 2.7702 | 0.2806 | 8.3054 | 14.53 | 4.24 | 0.056 | 0.016 | -3.302 | 0.296 | 62
05079 | Brubeck* | 90 | 2.5722 | 0.2493 | 12.4977 | 16.95 | 4.00 | 0.066 | 0.018 | -3.011 | 0.071 | 441
00208 | Lacrimosa* | 90 | 2.8913 | 0.0489 | 2.1090 | 49.99 | 4.85 | 0.238 | 0.062 | -2.392 | 0.024 | 1175
00179 | Klytaemnestra | 80 | 2.9852 | 0.0674 | 8.7805 | 74.59 | 4.17 | 0.217 | 0.075 | -5.168 | 0.606 | 90
00221 | Eos* | 90 | 3.0248 | 0.0744 | 10.1660 | 95.63 | 4.98 | 0.157 | 0.045 | -2.320 | 0.010 | 5718
01040 | Klumpkea | 90 | 3.1230 | 0.1976 | 16.8220 | 22.67 | 4.08 | 0.235 | 0.066 | -2.934 | 0.091 | 333
03985 | Raybatson* | 90 | 2.8516 | 0.1221 | 15.0523 | 22.11 | 3.54 | 0.167 | 0.056 | -3.572 | 0.234 | 126
00010 | Hygiea | 100 | 3.1573 | 0.1308 | 5.2384 | 453.24 | 5.75 | 0.068 | 0.022 | -2.484 | 0.014 | 2757
00024 | Themis | 100 | 3.1371 | 0.1505 | 1.3982 | 193.54 | 6.88 | 0.066 | 0.021 | -2.177 | 0.012 | 3052
00031 | Euphrosyne | 100 | 3.1679 | 0.1940 | 26.5578 | 281.98 | 6.11 | 0.056 | 0.016 | -4.404 | 0.053 | 1392
00081 | Terpsichore | 90 | 2.8866 | 0.1848 | 8.1925 | 123.96 | 4.68 | 0.051 | 0.014 | -3.842 | 0.792 | 49
00087 | Sylvia | 90 | 3.4981 | 0.0558 | 9.8460 | 288.38 | 7.66 | 0.056 | 0.016 | -2.624 | 0.365 | 60
00096 | Aegle | 90 | 3.0570 | 0.1844 | 16.5379 | 177.77 | 6.63 | 0.071 | 0.016 | -3.418 | 0.296 | 86
03330 | Gantrisch | 90 | 3.1466 | 0.1974 | 10.1892 | 37.64 | 5.51 | 0.043 | 0.011 | -3.171 | 0.054 | 734
00276 | Adelheid* | 100 | 3.1953 | 0.0665 | 21.8977 | 100.35 | 8.76 | 0.065 | 0.019 | -2.244 | 0.050 | 358
00283 | Emma | 80 | 3.0508 | 0.1142 | 9.0734 | 145.55 | 5.74 | 0.044 | 0.013 | -3.160 | 0.081 | 340
00490 | Veritas | 65 | 3.1700 | 0.0615 | 9.2533 | 118.80 | 5.79 | 0.066 | 0.020 | -2.767 | 0.044 | 686
24649 | Balaklava* | 90 | 3.1880 | 0.2108 | 14.1109 | 16.79 | 4.23 | 0.056 | 0.011 | -2.244 | 0.148 | 88
00511 | Davida | 90 | 3.1476 | 0.1903 | 14.4627 | 285.84 | 7.66 | 0.058 | 0.018 | -1.455 | 0.068 | 104
00618 | Elfriede | 80 | 3.1886 | 0.0582 | 15.8548 | 131.23 | 5.16 | 0.054 | 0.018 | -3.621 | 0.799 | 45
01306 | Scythia* | 80 | 3.1408 | 0.0896 | 16.4274 | 72.24 | 6.59 | 0.057 | 0.017 | -2.482 | 0.035 | 705
01303 | Luthera | 90 | 3.2169 | 0.1210 | 18.7869 | 102.43 | 6.81 | 0.047 | 0.013 | -3.794 | 0.176 | 176
00702 | Alauda | 100 | 3.2184 | 0.0183 | 21.5554 | 196.47 | 10.22 | 0.065 | 0.015 | -1.747 | 0.197 | 49
00778 | Theobalda | 90 | 3.1745 | 0.2541 | 14.3001 | 55.32 | 6.04 | 0.061 | 0.020 | -3.199 | 0.156 | 144
00780 | Armenia | 90 | 3.1057 | 0.0690 | 18.1677 | 114.26 | 5.46 | 0.053 | 0.014 | -4.563 | 1.482 | 40
00816 | Juliana | 90 | 2.9887 | 0.1460 | 13.2853 | 50.08 | 5.62 | 0.042 | 0.014 | -3.260 | 0.640 | 42
00845 | Naema | 90 | 2.9333 | 0.0355 | 11.9642 | 58.53 | 5.25 | 0.059 | 0.016 | -3.955 | 0.141 | 246
00928 | Hildrun | 90 | 3.1459 | 0.1940 | 16.4752 | 62.54 | 5.55 | 0.052 | 0.013 | -3.096 | 0.159 | 111
02621 | Goto* | 80 | 3.0922 | 0.1207 | 12.2630 | 47.92 | 5.64 | 0.080 | 0.034 | -3.708 | 0.384 | 71
01113 | Katja | 90 | 3.1136 | 0.1324 | 13.7567 | 48.37 | 7.19 | 0.067 | 0.030 | -2.027 | 0.229 | 51
Table 2: (continued)

Table 3: Orbital and physical parameters for Main Belt asteroids associated
with dynamical families. Table 3 is published in its entirety in the
electronic edition of ApJ; a portion is shown here for guidance regarding its
form and content.

Name |  $a$ (AU) | $ecc$ |  $inc$ (deg) |  $D$ (km) | $\sigma_{D}$ | $p_{V}$ | $\sigma_{p_{V}}$ | Family
---|---|---|---|---|---|---|---|---
00004 | 2.3615 | 0.0988 | 6.3903 | 468.30 | 26.70 | 0.423 | 0.053 | 00004
00063 | 2.3952 | 0.1206 | 6.2173 | 109.51 | 2.25 | 0.142 | 0.022 | 00004
01273 | 2.3938 | 0.1226 | 6.2289 | 6.77 | 0.15 | 0.299 | 0.035 | 00004
01906 | 2.3736 | 0.0994 | 6.4076 | 8.06 | 0.08 | 0.228 | 0.047 | 00004
01929 | 2.3627 | 0.1141 | 7.0768 | 7.24 | 0.24 | 0.389 | 0.081 | 00004
01933 | 2.3530 | 0.0940 | 6.8229 | 5.48 | 0.07 | 0.454 | 0.044 | 00004
01959 | 2.3161 | 0.0945 | 6.8517 | 7.31 | 0.14 | 0.230 | 0.055 | 00004
01979 | 2.3740 | 0.1015 | 6.5229 | 4.52 | 0.16 | 0.357 | 0.030 | 00004
02011 | 2.3870 | 0.1113 | 6.3730 | 5.19 | 0.65 | 0.463 | 0.100 | 00004
02024 | 2.3254 | 0.0948 | 6.5575 | 8.64 | 0.22 | 0.180 | 0.016 | 00004

Most of the families we identify here are analogous to those given by Nesvorný
(2012), though we identify $28$ new families above our significance threshold
that were not previously known. In addition, $24$ families from Nesvorný
(2012) are lost in our analysis: fifteen fell below our cutoff size while nine
no longer link at the QRL level when the albedo components are considered
separately. We have cross-referenced the lists of family members for
overlapping families in the high-confusion IMB region and removed duplicate
objects in the larger family if they appear in the membership list of a
smaller family to ensure that the small family does not drop below our cutoff.
This resulted in objects being rejected from the Flora family list that were
linked to Baptistina, and objects being rejected from the Hertha family list
that appeared in the Polana list.

Figures 10 and 11 compare the orbital elements and albedos of objects
identified as part of an asteroid family in this work to those parameters of
the objects that were not linked to any family and thus assumed to be part of
the background. In Figure 12 we separate each semimajor axis and albedo region
and plot the eccentricities and inclinations of the identified families, while
Figure 13 shows all background objects in the same fashion. We see in these
plots evidence of halos in the background distributions around the locations
of known large families, indicating an incomplete identification (and thus
removal) of these families. For example, the background objects from the
OMBhigh region show a halo near $0<ecc<0.1$ and $8^{\circ}<inc<13^{\circ}$
corresponding to unlinked Eos family members. These halos likely represent
family members that could not be linked to the core of the family at the QRL
for that region, possibly because they have diffused into the background
population due to high initial ejection velocities, Yarkovsky drift, and/or
gravitational perturbation. We note two features in the IMB: a number of low
albedo objects from the Polana family have been dragged into the IMBhigh by
the Hertha family from the region of albedo overlap included in the high
albedo cutoff; and the moderately-high albedo Baptistina family is plotted in
the IMBlow region as this family blends with Flora and cannot be distinguished
in the high albedo population.

As is clearly shown in these plots, careful fine-tuning of the cutoff velocity
and extraction of each individual family could increase the population of
known families significantly. This is particularly true for the high
inclination regions where spatial densities are low; using a single QRL for
the high and low inclination components of each region is likely resulting in
a failure to identify a large number of family members at high inclinations
(e.g. near the Hungaria family at $a\sim 1.8~{}$AU, $inc\sim 20^{\circ}$, and
$ecc\sim 0.08$). However, subdividing each region’s QRL and fine-tuning each
family’s cutoff velocity increases the subjectivity of the family
determination; future work will attempt to investigate this more
quantitatively. It is likely that this extension to the present work will
increase the fraction of the MBAs that are members of a family significantly,
possibly to $\sim 50\%$ or more which would mean that most asteroids are by-
products of a catastrophic collision or large cratering event (cf. Bottke et
al., 2005).

Figure 10: Proper inclination (inc, in degrees) vs. proper semimajor axis (a,
in AU) for all identified family members (left) and all non-family background
objects (right). The color of the points indicates the albedo of the asteroid
as in Figure 1 and shown by the color bar. Figure 11: Proper eccentricity
(ecc) vs. proper semimajor axis (a, in AU) for all identified family members
(left) and all non-family background objects (right). The color of the points
indicates the albedo of the asteroid as in Figure 1 and shown by the color
bar. Figure 12: Proper inclination (inc, in degrees) vs proper eccentricity
(ecc) for identified family members for each of the six semimajor axis and
albedo regions considered in this work (see Table 1 for the definition of each
region). Color indicates the albedo of the asteroid as in Figure 1. Figure 13:
The same as Figure 12 but for background objects not part of any identified
family.

Gil-Hutton (2006) identified 13 nominal families at high inclinations using
proper orbital elements and HCM. Seven of those correspond to families that we
identify here, one was linked by our routine but was too small to pass the
reliability cuts. The remaining five found by Gil-Hutton (2006) were not
linked in our method, however four of those had only a small number of
members. In particular, we can not link any of the three families those
authors identify in the IMB below our nominal QRL. We are also unable to
identify any of the new families presented in (Novaković et al., 2011). These
differences in family lists are a result of our application of a single QRL to
both high and low inclination IMB objects despite the large differences in
spatial density between the two populations. We anticipate that with further
refinement of the QRL for high inclination objects these families will be
recoverable.

Recently, Broz et al. (2013) published a revised list of asteroid family
members, rerunning the HCM linking routine over the latest list of asteroids.
The majority of the families those authors present are similar to the ones we
discuss here, and most of their families we do not find are because they are
too small, at high inclination, or do not link at our QRL when confusion with
background sources is reduced. There are two specific examples of differences
between this work and theirs that we comment on here. First, Broz et al.
(2013) identify a family around (1044) Teutonia with $1950$ members, however
in our data this family does not link significantly at a level below our QRL,
and at the QRL only links to $\sim 65$ objects. Second, they identify a family
around (2085) Henan with $946$ members, however as with Teutonia this family
does not link below our QRL level. We interpret both of these cases as
instances where confusion with background sources is eliminated when albedos
are separated and thus linkages no longer become significant.

We compare in Figure 14 the albedo distribution of asteroids linked to a
dynamical family to the distribution for background objects for each of the
three orbital regions considered. While the family members show roughly equal
contributions from high and low albedo objects in all three regions, the
background objects show the opposite trend. The IMB background is dominated by
the higher albedo component, while the OMB background has only a minimal
contribution from these objects. This division is likely to become even more
extreme once the halo objects are properly accounted for, as these make up a
large fraction of the IMBlow and OMBhigh objects remaining in the background
population.

Figure 14: Albedo distribution for the IMB (red solid), MMB (blue dashed), and
OMB (black dotted) for all identified family members (left) and all background
objects (right)

In Figures 15-20 we show the individual petal plots for each of the families
we identify, separated into regions. Each subplot is labeled with the name of
the largest body; names in blue font with a ‘*’ suffix indicate cases of
ambiguous parent bodies, either because there is no clear largest body (e.g.
Endymion) or there is a small group of objects of similar sizes that could be
the parent or represent a completely shattered parent (e.g. the Eos family,
where (221) Eos, (639) Latona, and (579) Sidonia are clustered at similar
semimajor axes with diameters of D$=96~{}$km, $89~{}$km, and $86~{}$km
respectively). In some cases it is clear that the listed family likely
represents two overlapping families (e.g. Emma). We also indicate Baptistina
as an ambiguous case given the results found in Masiero et al. (2012b).

We note that as we have chosen to identify families by the name of the largest
member, rather than the lowest numbered object, in some cases well-known
families have changed name. A prime example of this is the Lacrimosa family,
which encompasses the well-known Koronis family. As (208) Lacrimosa has a
larger diameter than (158) Koronis (D$=50~{}$km vs D$=39~{}$km) our naming
system assigns the family to Lacrimosa, but this case is another example of a
family with an ambiguous parent.

Figure 15: Diameter vs semimajor axis petal plots for all significant families
in the high albedo inner Main Belt region. The parent body is indicated as the
larger square point. Names with a ‘*’ suffix have ambiguous parent bodies.
Colors of the points indicate the albedo of the asteroid as in Figure 1.
Figure 16: The same as Figure 15 but for the low albedo inner Main Belt.
Figure 17: The same as Figure 15 but for the high albedo middle Main Belt.
Figure 18: The same as Figure 15 but for the low albedo middle Main Belt.
Figure 19: The same as Figure 15 but for the high albedo outer Main Belt.
Figure 20: The same as Figure 15 but for the low albedo outer Main Belt.

###  4.2 Family Physical Properties

Using the high-confidence family lists we investigate the physical parameters
of each family including the characteristic size, raw size-frequency
distribution (SFD) and mean albedo as they relate to number of family members
identified and size of the largest remnant. We note that as shown in Figure 1
our sample, limited by the biases imposed predominantly by visual band
detection, is not complete to the same sizes for high- and low-albedo objects.
This means that the minimum and median sizes of low albedo families will be
shifted to larger values than for high albedo families. Additionally, for a
given family the wing of the albedo distribution at higher values will be
better sampled than the wing at lower values. Also note that our imposed
division of albedos into two components will remove objects that have albedos
very different from the bulk of the family due to incorrect measurements or
large physical differences. These biases will affect the measurement of SFD as
well, so care in interpretation is required.

We present the observed cumulative size distribution for each family in Figure
21, and the differential albedo distribution for each family in Figure 22.
Approximately two-thirds of the families show a single parent body much larger
than the remaining members, while the remaining families do not show a
dominant remnant. This may be indicative of differences between cratering
events and super-catastrophic disruptions as discussed by Durda et al. (2007),
however in the latter families we cannot rule out cases where the largest
remnant is present but did not link to the family due to the various cutoffs
imposed. As opposed to what was seen in Mas11, where $\sim 25\%$ of the
previously published families had non-uniform albedo distributions, almost
none of the families presented here show any bimodality in their albedo
distributions. The few exceptions (e.g. Chapman, Astraea) are expected to be
artifacts that will be removed with future planned revisions of the family
selection cutoffs. Given our separation of the regions into high- and low-
albedo components, this result is not unexpected.

Figure 21: Cumulative raw size distribution of family members identified in
this study for the $76$ families presented in Table LABEL:tab.stats. The red
dashed line shows the distribution for the Vesta family, for comparison.
Family names in blue indicate cases where the parent body is uncertain. As
discussed in the text, selection biases from visible light surveys have not
been quantified which will affect interpretation of these distributions.
Figure 22: Differential raw distribution of albedos for the $76$ families
presented in Table LABEL:tab.stats. The red dashed line shows the distribution
for the Vesta family, for comparison. Family names in blue indicate cases
where the parent body is uncertain. As discussed in the text, selection biases
from visible light surveys have not been quantified which will affect
interpretation of these distributions.

Using these distributions, we can determine initial physical properties for
each family with the caveat that selection biases imposed by visible light
surveys have not yet been removed and so will bias our results as well. We fit
a Gaussian profile to the albedo distribution of each family to derive the
mean albedo ($p_{V}$) and Gaussian width ($\sigma_{p_{V}}$), which is
$20\%-30\%$ of the mean for nearly all of the families we observe. This width
is comparable to the uncertainty we expect in the albedo determination from
the NEOWISE data (Mas11, Mainzer et al., 2011b). Using the diameter
distribution, we can size-sort the linked family to find the diameter of the
largest body ($D_{max}$), the median diameter of the observed sample
($D_{med}$), as well an initial estimate of the observed, raw SFD.

To determine the observed SFD we fit a function of $N=D^{\alpha}$ to each
cumulative distribution, using only those bins with more than five objects (to
avoid errors induced by a few large fragments in a cratering scenario) and
less than half of the sample (to minimize errors due to the incomplete
catalogs). Using a least-squares minimizer we derive the best-fit slope
parameter ($\alpha_{SFD}$) and the $1\sigma$ error on that slope
($\sigma_{\alpha}$). We see in our sample a range of slope parameters, from
$\alpha\sim-1.5$ at the shallowest to $\alpha\sim-5$ at the steepest. SFD
slope can be used to study the conditions of the breakup event (e.g. the curve
of the SFD can trace cratering vs. catastrophic disruption, Durda et al.,
2007), however these slopes can be modified over time as collisional grinding
alters the initial post-breakup population, flattening the SFD over time
(Marzari et al., 1995). Additionally, if the parent body prior to impact had a
shattered interior structure, this will also change the expected SFD of the
family by increasing the frequency of reaccumulation of large family members
(Michel et al., 2004).

The specific bins chosen for fitting will alter the fitted slope depending on
the shape of the SFD. Tests on our data show that varying the lower limit over
a range from five to fifty objects typically causes the fitted slopes to
become shallower by $\Delta\alpha\sim 0.1-0.3$, with the effect becoming more
pronounced for smaller families as the sample size decreases. This indicates
that most families have steeper size distributions in the biggest objects than
in the majority of the population, however this is probably at least partially
a result of survey incompleteness at smaller diameters. Conversely, when we
test the effect of changing the upper limit for fitting from half the
population down to only the largest $25\%$, we see that the fitted slopes
generally move away from a value of $\alpha\sim-2.5$ as fewer objects are
included in the fit (cf. Tanga et al., 1999). This shift is typically
$\Delta\alpha<0.5$ however for the most extreme case (the Hertha family) it is
as large as $\Delta\alpha=1.3$. As most families have measured slopes of
$\alpha<-2.5$ this means that their SFDs become steeper as objects are
removed. This effect may also be a result of survey incompleteness at the
smallest sizes, or may be tracing either a collisional equilibrium at smaller
sizes with the background population which has $\alpha\sim-2.5$ (Mas11) or an
increasing fraction of non-family members being included in family lists at
these sizes.

This uncertainty in the fit to the observed SFD is compounded by the visible-
light survey biases that are incorporated into the optical measurements we use
to determine albedo. Objects without albedos do not pass our initial cuts for
inclusion into our data set, thus our sample will include some effect from
these biases, which will be particularly pronounced for smaller, lower albedo
objects. These biases will result in systematic and potentially large errors
in the fits to the observed SFD. Debiasing of the MBA background and family
populations is critical to measuring the true SFDs and physical parameters for
asteroid families, and as such values given here should be regarded as
preliminary.

We present in Table LABEL:tab.stats the determined albedo, diameter, and slope
parameters for each of the $76$ significant families we identify. These values
are subject to large biases due to incomplete sampling of various
subpopulations within the MBAs, so care in interpretation is required. We note
that as families indicated by a ‘*’ suffix on their name have ambiguous parent
bodies, the size of the largest linked object is not necessarily indicative of
the size of the parent or largest remnant.

We show in Figure 23 a comparison of the mean family albedo to the median
diameter. As expected from the biases against smaller, lower albedo objects
imposed by visible light surveys used to create the catalog our sample is
drawn from, high albedo families have a smaller median diameter than low
albedo families. Despite this bias, nearly two-thirds of the observed families
are low albedo. As catalogs of asteroids become more complete at small sizes
for low albedo objects we expect that the number of low albedo families
identified will increase further. We also see indications for four distinct
albedo classes in the families: low albedo ($p_{V}\sim 0.06$), moderate albedo
($p_{V}\sim 0.16$), high albedo ($p_{V}\sim 0.25$) and the very high albedo
Hungaria family ($p_{V}>0.40$). We note that while updates to the measured
magnitudes of the Hungaria family including better fits of $H$ and $G$ are
expected to reduce the $H$ magnitudes and thus the determined albedos (B.
Warner, 2012, private communication), these family members still typically
have $p_{V}>0.40$ and are expected to remain narrowly distributed.

If we take albedo as a coarse tracer of the parent body’s composition (cf.
Mainzer et al., 2011c) we can search for any influence this has on the
resultant breakup. We show in Figure 24 a comparison between the
characteristic family albedo and the fitted observed SFD slope. If different
compositions traced different cohesion strengths, we would expect to see a
correlation between these two parameters, however we see no such relationship.
This may indicate that albedo does not correlate with internal structure, or
that impact geometry and/or energy play the dominant role in shaping the SFD
of the family, as shown by Durda et al. (2007), however debiasing must be
performed before definite conclusions can be drawn.

Figure 23: Characteristic albedo for each family compared to the median
diameter of all objects linked to that family. The size of the points
indicates the number of objects in the family, while the color also traces
albedo, following Figure 1. As expected, higher albedo families have small
median diameters as selection effects against smaller, lower albedo asteroids
imposed by visible light surveys have not been removed. Figure 24: The same as
Figure 23, now showing albedo vs. SFD slope parameter. For the largest
families the error on the slope fit is smaller than the size of the point,
however systematic errors due to survey bias have not been included.

Figure 25 shows the comparison between the observed raw SFD slope and the size
of the linked family. Clearly, families with smaller numbers of objects will
have more uncertain SFD slopes. However, there does appear to be a trend
toward shallower slopes for larger families. We note that in a diameter-
limited survey such as NEOWISE fewer objects are expected to link to families
with steep SFDs than shallow SFDs, as a larger fraction of the family members
will be at or below the survey detection threshold. This may be a contributing
factor to the lack of large, steep SFD families. Conversely, biases due to
variable survey completeness can induce systematic errors. As steep families
are thought to be young, and a large family indicates an energetic collision,
these families are expected to be rare, confusing any interpretation of their
absence. A survey probing a smaller size range would preferentially fill in
the membership lists of steep SFD families allowing better measurements of
their SFDs to be made.

Tanga et al. (1999) and Durda et al. (2007) showed that steeper slopes tend to
indicate families that originated from a cratering event, while shallower
slopes tend to follow disruptive impacts. This suggests that in the diameter
range we are sensitive to, most of the numerically largest families in the
Main Belt formed from disruptive events. An exception to this trend is the
(31) Euphrosyne family, which has $\sim 1400$ members but one of the steepest
slopes measured with $\alpha_{SFD}=-4.4$. This unusually steep observed SFD
may be indicative of a glancing impact between two large bodies resulting in a
large cratering event. We note that this observed SFD is significantly steeper
than what is observed for (4) Vesta ($\alpha=-3.45$, the next steepest SFD in
this family size range), which is known to have undergone two massive
cratering events from the results of the Dawn mission (Russell et al., 2012).
Conversely, the observed SFD of the Euphrosyne family may trace a low-speed
collision where reaccumulation onto the parent was highly efficient. In this
scenario only those objects with the highest ejection velocities,
preferentially the smallest ejecta, would fail to reaccrete and would become
independent family members. As grazing impacts have a much lower frequency of
binary production than slower, head-on impacts (Durda et al., 2004), a search
for binaries in the Euphrosyne family could differentiate between these two
scenarios. However, until proper debiasing of the MBA population is complete
and debiased SFDs can be measured, these possible formation scenarios are
speculative.

Figure 25: The same as Figure 23, now showing size of the linked family vs.
observed raw SFD slope parameter. For the largest families the error on the
slope is smaller than the size of the point, however systematic errors due to
survey bias have not been included.

##  5 Conclusions

In order to identify new asteroid families and improve the lists of previously
known families, we have used the databases of asteroid diameters and albedos
provided by the WISE/NEOWISE and IRAS surveys as a method of incorporating
asteroid physical properties into family determination. After splitting the
MBAs into two separate but overlapping albedo populations and three orbital
regions, we have applied the Hierarchical Clustering Method to these
subgroups. This technique is particularly useful for rejecting interlopers
from family lists, including objects that would otherwise be misidentified as
the family’s largest remnant. We identify $76$ families with more than $40$
members in our high-confidence set, of which $28$ were previously unreported.
One third of these families do not have an obvious parent asteroid that links
to the family. Approximately $35\%$ of MBAs link into a family, however there
are clear limitations to our method apparent at high inclinations and in
crowded regions. Carefully tuned family extraction will likely increase this
percentage significantly.

The observed albedo distribution of the $38,298$ identified family members
differs significantly from the distribution of the background asteroids, where
the background shows a more extreme change from the inner- to outer-Main Belt.
We see no correlation between the slope of the observed, raw size frequency
distribution of a family and its mean albedo. However, we note that a full
accounting for the selection biases imposed by the visible light surveys that
discovered the objects has not yet been performed, and an unquantified number
of objects are thus missing from each family; these biases must be accounted
for before firm conclusions about the SFDs for each family can be made. Of the
observed raw SFDs, we note a trend that families with larger numbers of
objects tend to have more shallow slopes. One possible explanation is that
these families were created could by catastrophic disruption or are very old
and have been ground down by collisional processing over very long timescales.
An exception to this trend is the (31) Euphrosyne family which has one of the
steepest size frequency distributions measured, but is also one of the largest
families observed. This could be the result of a cratering impact between two
large bodies, a recent giant impact, or both. While this work only includes
approximately one quarter of the known MBA population (those observed by
NEOWISE and IRAS), it sets the stage for improving our understanding of the
creation and evolution of MBA families. Future work will improve our family
identification routines to better differentiate family members from background
objects and debias the input catalog, improving our family lists and setting
the stage for the determination of ages for the majority of families in the
Main Belt.

## Acknowledgments

JM was partially supported by an appointment to the NASA Postdoctoral Program
at JPL, administered by Oak Ridge Associated Universities through a contract
with NASA. The authors thank the referee Alberto Cellino for comments that
greatly improved the paper, and John Dailey for helpful discussions on
implementation of algorithms. This publication makes use of data products from
the Wide-field Infrared Survey Explorer, which is a joint project of the
University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the National
Aeronautics and Space Administration. This publication also makes use of data
products from NEOWISE, which is a project of the Jet Propulsion
Laboratory/California Institute of Technology, funded by the Planetary Science
Division of the National Aeronautics and Space Administration. This research
has made use of the NASA/IPAC Infrared Science Archive, which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration. We gratefully
acknowledge the extraordinary services specific to NEOWISE contributed by the
International Astronomical Union’s Minor Planet Center, operated by the
Harvard-Smithsonian Center for Astrophysics, and the Central Bureau for
Astronomical Telegrams, operated by Harvard University.

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