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INTERLOPERS WITHIN ASTEROID FAMILIES
F. Migliorini 1 , V. Zappal`a 2 , R. Vio 1 , A. Cellino 2
1 Dipartimento di Astronomia, Universit`a di Padova, vicolo
dell'Osservatorio 5, I­35131 Padova, Italy; e­mail: astrpd::migliorini,
astrpd::vio
2 Osservatorio Astronomico di Torino, I­10025 Pino Torinese (TO), Italy;
fax: (39)11 4619030; e­mail: zappala@to.astro.it, cellino@to.astro.it
Manuscript pages: 64
Figures: 11
Tables: 2
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Running title: Asteroid families interlopers
Send editorial communications to:
V. Zappal`a
Osservatorio Astronomico di Torino
I­10025 Pino Torinese (TO), Italy
fax: (39) 11 4619030
e­mail: zappala@to.astro.it
2

Abstract
A statistical method for the evaluation of the probable amount of in­
terlopers included in the most recently identified asteroid families is
developed. The method is based on the analysis of the local properties
of the asteroidal distribution in the zones of the proper element space
in which families are located. An application to the most important
families listed by Zappal`a et al. [1995, Icarus in press] allows to
derive the probable amount of interlopers at different size ranges. In
some cases, the results suggest that the criteria adopted to derive the
nominal memberships have been quite conservative. The procedure
developed in this paper allows to assess ``how many'' interlopers are
probably included in the nominal member lists, but not ``which ones''.
However, such a quantitative estimate of the amount of probable inter­
lopers is very useful and should be taken into account in any analysis of
the physical properties of the families, since the presence of interlopers
can explain some apparent discrepancies in several cases.
3

1 Introduction
Nature is incomparably better than mankind as a performer of experiments
of catastrophic break--up of solid bodies. Apart from the evidence of the
occurrence in the past of giant events involving large planetesimals (anoma­
lous tilt of Uranus, origin of our Moon, etc.), we know that in the asteroid
main belt there are many outstanding examples of very violent ``natural ex­
periments'', in which bodies having diameters ranging from a few tens to a
few hundreds kilometers have been collisionally disrupted, leaving swarms
of fragments orbiting around the Sun: these are the so--called dynamical
families of asteroids. Due to their catastrophic origin, asteroid families are
a very important source of information for the study of the minor planets.
The reason is that collisional evolution is generally recognized as the major
process which has affected the physical properties of the asteroids since the
end of the epoch of planetary accretion in our Solar System. As a conse­
quence, understanding the physics of catastrophic break--up of asteroidal
bodies is one of the major tasks of the modern planetary science.
From the theoretical point of view, this is a very complex and difficult
problem. Only in recent years, many improvements have been achieved in
4

the attempts to model the events of catastrophic break--up by means of
complex hydrocodes (Melosh et al., 1992; Benz et al., 1994) and by means
of semi--empirical models (Paolicchi et al., 1989, 1993, 1995; Verlicchi et
al., 1994). The major source of physical evidence constraining the present
day models comes from the results of extensive laboratory experiments per­
formed by several authors (see Fujiwara et al., 1989 for a review; more re­
cently, see Davis and Ryan, 1990; Nakamura and Fujiwara, 1991; Nakamura
et al., 1992; Giblin et al., 1994).
On the other hand, only very recently the possibility to directly compare
the models with the observational evidence coming from the properties of
the real asteroid families has been made possible by the improvements of the
techniques of family identification (Zappal`a et al., 1990, 1994, 1995; Bend­
joya et al., 1991; Bendjoya, 1993; Lindblad, 1992, 1994). These techniques
have been developed after the pioneering work of many authors active in
this field (see, in particular the extensive family searches by Williams, 1979,
1989, 1992; see also Valsecchi et al., 1989, for a review of this subject), and
have allowed to largely improve the statistical reliability of the identified
families. We recall here that families are found as clusterings in the space
of the orbital proper elements which, statistically speaking, cannot be due
5

to chance. In particular, it is now possible to extract a sub­sample of very
reliable groupings independently identified by means of different statistical
techniques, which give also nominal lists of members in a very good agree­
ment. These families should be analyzed in details, in order to extract all
the relevant physical information related to the collisional events from which
they have been originated.
This possibility constitutes a very big improvement of the situation from
the point of view of the theory of catastrophic impacts, since in this way
the available models can be compared directly with the events they should
reproduce. In this respect, laboratory experiments, though very important,
are far less satisfactory, since it is very difficult to derive reliable physical
information concerning the break--up of km--sized bodies starting from the
results of experiments involving cm--sized targets. This is the well known
scaling problem, which has been extensively investigated in recent years
(Fujiwara et al., 1989; Davis and Ryan, 1990; Holsapple, 1993, 1994). On
the other hand, several problems must be faced when we want to analyze
directly the physical properties of the asteroid families. Among them, there
is the problem of the so--called interlopers.
In this respect, there is here another major difference between our small--
6

scale laboratory experiments, and those carried out by Nature. In our lab­
oratories the experiments are mostly performed in closed chambers, which
are obviously cleaned before the disruption of each target, in order to allow
a precise recovery of all the fragments originated by the impact (or the ex­
plosion). Nature did not care about that. The parent bodies of the presently
recognizable families orbited the Sun in populous regions of the main belt.
As a consequence, the swarms of fragments originated by the impacts cannot
be easily distinguished from the nearby background of objects as far as the
pure orbital properties are analyzed.
The possibility to include interlopers, that is objects having nothing to
do with the real members of a given family, introduces undesirable compli­
cations in the job of studying the general physical properties of the families.
The main effect of interlopers is that of influencing the trend of the distri­
butions of the most important physical properties of the members. This can
affect noticeably the overall apparent behaviour when the number of inter­
lopers is not negligible, and especially when their sizes are relatively large.
In particular, the presence of big interlopers can have a strong effect on the
resulting size (mass) distribution, and, as a consequence, on the estimate of
the parent body mass. Moreover, the presence of large interlopers relatively
7

far from the center of mass of the family, introduces a spurious shift of its
barycenter and anomalies in the derived kinematical structure of the family.
In these respects, smaller interlopers are usually less important, since they
only modestly affect the derived total mass of the parent body.
For the above reasons, the problem of the presence of interlopers, mainly
at relatively large sizes, is a serious one for any physical study of the fam­
ilies, and some effort must be done in order to identify, or at least reliably
evaluate the amount of existing interlopers in each family, at different ranges
of diameters.
As for the possibility to directly identify interlopers, this can be done in
principle on the basis of the optical surface properties of the family members.
As an example, a large C--type asteroid in a family composed by S--type ob­
jects should be considered as a probable interloper. This criterion, however,
can hardly been applied in practice for several reasons. The first reason
is the scarcity of data about the taxonomic types and the albedos of most
family members. The second reason is that, simply, the albedo and/or tax­
onomic types do not work as interloper indicators in cases in which both
family members and interlopers share the same optical properties. In prin­
ciple, moreover, the break--up of differentiated parent bodies can produce
8

a swarm of fragments having different mineralogic compositions, and con­
sequently not uniform optical properties, leading to complications for any
method of interloper identification based on these properties.
This does not mean that extensive observational campaigns aimed at
deriving information on the spectroscopic properties of family members are
not useful. Instead, we are convinced that such campaigns should get a
very high priority in the present observational activity, as can be suggested
also by the spectacular spectroscopic results on the family of Vesta (Binzel
and Xu, 1993). On the other hand, in this paper we try to investigate the
possibility to evaluate the amount of possible interlopers in the families by
means of a statistical approach, starting from the local properties of the
asteroidal population in the zones of the proper element space in which the
main families are located.
It is reasonable to expect that the number of probable interlopers changes
greatly in different cases, as well as as a function of the diameters. In
particular, the number of interlopers should depend on the volume of the
family in the proper element space, as well as on the overall density of the
background objects in that particular region of the proper element space.
In other words, we expect only a few, or no interloper for a small compact
9

family located in a relatively empty region of the proper element space,
whereas this is not true for a big family covering a large volume in a densely
populated region of the space. In the present paper, this general idea is
developed and made quantitative by means of refined statistical techniques
that will be explained in the following Sections.
As for the families analyzed here, we have chosen the most ``robust'' fam­
ilies identified by Zappal`a et al. (1995) by means of two independent statisti­
cal procedures of identification, namely the so--called Hierarchical Clustering
method (hereinafter HCM) and the Wavelet Analysis method (hereinafter
WAM). The efficiency of these procedures have been thoroughly discussed in
the above mentioned paper, as well as in the previous family searches by the
same authors, quoted therein. Taking also into account the numerical sim­
ulations performed by Bendjoya et al. (1993) we can be confident about the
excellent overall reliability of the main families independently found by both
HCM and WAM, apart from some differences in the nominal memberships,
which are not exactly the same due to the different statistical approaches
used.
For what concerns the amount of possible interlopers that we could pre­
dict a priori on the basis of the results of the family identification procedures,
10

there is a big difference between the so--called clusters and the clans, fol­
lowing the nomenclature introduced by Farinella et al. (1992). Clusters are
generally dense and compact groupings, whose memberships do not depend
appreciably upon the statistical criteria adopted, in the sense that more or
less restrictive criteria give essentially the same members. Clans are more
complex groupings. They are generally very populous, extend over large vol­
umes in the proper element space, and their nominal members depend more
strictly upon the statistical criteria adopted. As shown by Bendjoya et al.
(1993), in the case of the clans adopting more liberal criteria of identification
may lead to include a significant fraction of interlopers. The memberships of
the clans considered in the present paper are the nominal ones as listed by
Zappal`a et al. (1995). As quoted in that paper, the criteria adopted both by
HCM and by WAM for the nominal memberships of the resulting families
are not particularly liberal, and may be instead quite restrictive in some
cases. However, the fact that some of the most populous families (especially
the clans) occupy relatively large volumes of the proper element space, en­
sures that a fraction of interlopers should exist, and their plausible amount
at different size ranges should be assessed as a necessary pre--requisite for
any physical study of these families.
11

In the following Sections, we develop a general statistical technique to
evaluate the amount of chance interlopers among families. The analysis is
carried out for different size ranges, as the number of possible interlopers
is a function of size (since in the main belt the general size distribution is
a Pareto power--law, see Cellino et al., 1991, and the number of existing
objects strongly increases at smaller sizes). At the same time, this analysis
can allow to infer precise indications about the possibility that the nominal
lists of family members are too restrictive. In other words, not only can
we assess the most probable amounts of background objects included in the
family lists, but we can also identify cases in which a number of real family
members are probably not included in the nominal families. We are aware
that this kind of result has apparently important implications from the point
of view of the techniques of family identification, since it seems to suggest
some improvements of the statistical criteria used to derive the nominal
family members. However, we want to stress that we are not dealing here
with a method of family identification. Instead, we have developed a method
to assess the amount of expected interlopers in the families, which is based
on the fact that it acts a posteriori on families that have been previously
identified on the basis of very objective statistical techniques. Zappal`a et al.
12

(1995) have clearly stated that the nominal lists of members coming from the
application of HCM and WAM should not be considered as too rigid. While
the reliability of the resulting families is not questionable (at least for the
most ``robust'' ones) the same is not true for the nominal memberships. In
this respect, the method for interlopers assessment presented in this paper
is mainly aimed at supporting the physical studies of the families. It can be
considered also as a useful complementary tool for the existing techniques
of family identification, but not as a method of identification itself.
For each family analyzed, the results are presented in terms of the poisso­
nian probability to have some amounts of interlopers at different size ranges,
on the basis of the local density of background objects in the same region of
the proper element space. This information should be carefully taken into
account in any future analysis of the physical properties of these families
(Zappal`a et al., in preparation). We can also recall that a very prelimi­
nary evaluation of the possible interlopers in some clans by means of a local
analysis of the background was sketched by Cellino and Zappal`a (1993) and
more in general by Zappal`a et al. (1994). However, the general idea was not
fully developed in these papers, and it was used only to give very qualitative
indications.
13

This paper is organized as follows: in Section 2 we illustrate the sta­
tistical method developed to derive the probable amount of interlopers at
different size ranges for any given family. In Section 3 we discuss the data--
set of family members and background objects used in the analysis. In
Section 4 the results are shown and briefly discussed for each family, while a
general discussion about the main conclusions of the present work and their
implications is given in Section 5.
2 The statistical method
The general idea of our model is very simple: we want to predict the number
of objects present in a certain volume (in this case, a volume in the space of
the orbital proper elements), on the basis of the knowledge of the distribution
of objects in the zone of the space surrounding this volume. The fundamental
hypothesis, therefore, is that the distribution of the objects in the volume is
not completely uncorrelated to the distribution of objects in that local region
of the space. Roughly speaking, the rules of the game are the following:
take a cube of side 3l. This can be subdivided in 27 smaller cubes of side l.
Fill the whole cube with a given number of objects, following any rule, (or
completely by chance, if we want to contradict the basic assumption). Now,
14

count the objects in all the small sub­cubes but the central one (the one
completely surrounded by the other 26 small cubes). The goal is to predict
the number of objects present in the central sub­cube, from the knowledge
of the numbers of objects present in the other 26 ones. In other words,
we take into account the overall properties of the local distribution of the
objects: in practice, we account for a possible linear variation of the number
of objects as a function of some direction. This appears sufficient for our
purposes. More sophisticated models, involving a fit of a higher degree of
the object distribution seem too artificially complicated, and have not been
considered.
This kind of approach is not arbitrary, but it is suggested by the actual
properties of the distribution of the asteroids in the proper element space.
It is well known that this distribution is far to be homogeneous. Apart
from the regions depleted by mean--motion and secular resonances, and from
the concentrations due to the major families, the background population
varies greatly as a function of the proper semi­major axis, eccentricity and
inclination. Such a variation reflects the overall dynamical and physical
history of the asteroid belt. In order to establish what is the expected
number of objects in a certain volume of the belt, it is therefore essential to
15

analyze the zone surrounding it. As a first approximation, the assumption
of a linear variation of the distribution of asteroids is fully reasonable locally
for limited regions of the proper element space.
In the practical application of the method sketched above, the volume
containing the family is not really a cube, but it is a parallelepiped in the
space of the proper elements a 0 , e 0 and sin i 0 . Its sides are determined by
the width of the family along the three coordinates a 0 , e 0 and sin i 0 . As a
consequence, also the whole surrounding volume and the other 26 subvol­
umes have parallelepipedal shapes, and the absolute value of the volume is
not fixed for all the families, but it depends on the extension of each group­
ing. Of course, the families have not in general a parallelepipedal structure
in the space of the proper elements: for this reason, they do not fill com­
pletely their circumscribing parallelepipeds. In general, only a fraction of
the parallelepiped is occupied by the real family. This fact must be taken
into account, since the number of interlopers expected in each parallelepiped
associated with a family must be portioned out between the ``real'' family
interlopers and the background objects expected in the family volume. This
is a quite serious complication, and the solution of the problem will be given
in Subsection 2.2.
16

For each family, having determined the size and shape of all the equi­
sized volumes of interest (the parallelepiped containing the family and the
26 surrounding ones) we have proceeded in the following way:
ffl it is assumed that the expected number N(x i ; y i ; z i ) i = 1; 2; ::::; 27, of
objects within each sub­volume i can be modeled by a linear relation­
ship
N(x i ; y i ; z i ) = a + bx i + cy i + dz i (1)
The reliability of this hypothesis has been checked by looking at the
gradient of N(x i ; y i ; z i ) along several directions passing through the
subvolume containing the considered family;
ffl the model (1) has been fitted to the data by means of a least--squares
technique. In the fit the subvolume containing the family has not been
considered;
ffl the quantity
N \Lambda = “a + “ bx \Lambda + “ cy \Lambda + “
dz \Lambda
has been assumed as the estimated contribution of the background
objects within the volume occupied by the family. Here x \Lambda ; y \Lambda ; z \Lambda are
17

the coordinates of the center of the volume containing the family, and
“a, “ b, “
c are the values of the estimated parameters in this volume.
In order to check the stability of the obtained results, we have also esti­
mated the parameters in the relation (1) by means of a maximum--likelihood
approach. Here, assuming that within each subvolume the probability dis­
tribution function of the observed background objects can be approximated
by means of a Poisson distribution, then the log--likelihood function of the
data can be written as
ln P (a; b; c; d) =
26
X
i=1
[N i ln N(x i ; y i ; z i )] \Gamma
26
X
i=1
[N(x i ; y i ; z i )] + constant (2)
Here N i represents the number of observed points in the i­th subvolume. By
maximizing the quantity (2) with respect to a, b, c and d, it is possible to
obtain an estimate of the parameters of the linear model (1). The results
obtained with this method do not significatively differ from those obtained
with the least--squares approach.
In practice, in the actual application of the method, we work in terms
of number density, that is the values of N(x i ; y i ; z i ) are taken as the num­
bers of objects per unit volume. This does not imply any practical change
of the method, but it allows to account more easily for the possible pres­
18

ence of empty zones associated with mean motion resonances in the zone
immediately surrounding some families. The proximity of some of the most
important Kirkwood gaps to several families is a well known fact. Recently
Morbidelli et al. (1995) have carried out an extensive analysis of the dynam­
ics of asteroids in the neighbouring of the main Kirkwood gaps, and have
shown that some families can have injected a fraction of their fragments into
some of these resonances, with very important implications for the origin of
meteorites. For the purposes of the present analysis, the presence of ``for­
bidden'' zones related to such resonances must be taken into account, since
it can affect significantly the results, in the sense that some of the paral­
lelepipeds surrounding a family can be crossed by some resonance, lowering
significantly the fraction of the volume within which can be really populated
by background objects. Since we work in terms of density, such situations
can be easily accounted for, by computing the fraction of ``forbidden'' volume
associated to resonances in each parallelepiped. This can be done without
great complications, from the knowledge of the location of the resonance
edges in the a 0 --e 0 space. In this way, the effect of nearby mean--motion reso­
nances has been fully taken into account in the present analysis. As for the
data about the precise location of the mean--motion resonances, they have
19

been provided by M. Moons and A. Morbidelli (private communication), on
the basis of their most recent dynamical analysis of the orbital motion close
to the major Kirkwood gaps. In particular, we have adopted the ``mean''
values of the resonances in the a, e plane, e ranging between 0 and 0:3. See
the quoted paper by Morbidelli et al. (1995) for an explanation.
Since we follow a statistical approach, it is clear that the value of N \Lambda
derived as explained above does not give a fixed amount of objects that
should be considered as the ``exact'' solution of the problem. Instead, this
value of N \Lambda should be considered as the expectation value – of a poissonian
distribution of expected interlopers:
f(k) = – k
k!
e \Gamma–
The results presented in Section 4 are given accordingly under the form
of tables and figures, in which we give the resulting poissonian parameter
–, and, when possible, the ranges of interloper amounts corresponding to
certain ranges of probability. This is done, in particular, when – turns
out to be sufficiently high to allow a good gaussian approximation of the
poissonian distribution of interlopers.
In particular, for values of – larger than 5, the poissonian distribution,
20

although it is intrinsically discrete, can be adequately fitted by a gaussian,
having mean – and oe =
p
–. This is no longer true for smaller values of –.
In order to give a visual aid in the interpretation of the results concerning
the families analyzed in this paper, we have plotted in Fig. 1a the probabil­
ity to have k interlopers for different values of – ranging from 0.25 to 6. In
particular, the size of the circular symbols used in this Figure is directly pro­
portional to the probability to have k interlopers for each value of –. In Fig.
1b the gaussian behaviour of the poissonian distribution for higher values
of – is used to represent the ranges of expected interlopers corresponding to
ranges of probability of 1oe and 2oe from the mean, respectively, as a function
of –. For the meaning of the crosses in these Figures, see later.
2.1 Numerical tests
Of course, the effectiveness of the approach described in this Section should
not be assessed only by means of a generic a priori discussion. Instead, such
an approach is suitable for an a posteriori check, based on an application
to a number of zones of the real asteroid belt, sufficiently free from the
problems related to the presence of families. In other words, we can check
whether such an approach is able to predict with a satisfactory accuracy the
21

number of ``normal'' background objects located in different zones of the
proper element space on the basis of the observed distribution of ``normal''
background asteroids in the surrounding region.
The results of this exercise are graphically shown in Fig. 1 by means
of the cross--like symbols. The location of each cross is determined by the
value of the poissonian – parameter determined in each test, and by the
corresponding amount k of objects really present within each zone analyzed.
As for the locations of the test--regions that we have chosen in the proper
element space, they are graphically shown in Fig. 2(a,b), together with the
overall set of HCM background objects adopted in this paper, as explained
in Section 3. As can be seen, we have considered a large variety of cases
(30) corresponding to different locations in the proper element space, and
to varying volume extensions, corresponding to the range of values that are
typical of the real families. Also the size ranges considered largely varied
according to the typical ranges exhibited by the real families.
The results shown in Fig. 1(a,b) support the statistical approach adopted
in this paper: as can be seen, when the resulting poissonian parameter is
low, the real amount of objects present in the considered volumes corre­
spond generally to the most probable values predicted by the poissonian
22

distribution (large circles in Fig. 1a). For higher values of –, the poissonian
can be approximated by a gaussian, and the real amount of objects in the
test zones are generally located within 1oe or 2oe from the mean of the dis­
tribution, apart from a few cases as could be expected on the basis of the
resulting probability distributions (Fig. 1b). On the basis of these tests, we
can conclude that our statistical method is effectively suited to analyze the
observed distribution of asteroids in the proper element space. Of course,
the statistical nature of the problem must always be reminded, but we can
be confident that the whole set of the results concerning the families ana­
lyzed in this paper should give a fairly good estimate of the most probable
amounts of chance interlopers present in these families.
As a final remark, we can notice that the tests show that the method
works fine when it is applied to the normal background population. This
means that the background population in the 26 equal--sized parallelepipeds
surrounding the one containing a given family should not include family
members of this or other close families. For this reason, as explained in
Section 3, a data file of background objects not included in any identified
family must be built at the beginning. We know that the nominal family
memberships are not too rigid, and we will see in what follows that in some
23

cases they have been probably underestimated. For this reason, there is
potentially the possibility that some family members could be located, in
the worst cases, in some of the 26 parallelepipeds surrounding the family
volume. Although in principle this can lead to a spurious increase of the
number N \Lambda of objects predicted in the family parallelepiped, such an effect is
generally negligible in practice, since the number of members located outside
the family--circumscribed parallelepiped is fairly small and not uniformly
distributed in the whole surrounding region.
2.2 Evaluation of the ``real'' family interlopers
As quoted above, the determination of the numbers N \Lambda of expected interlop­
ers in the parallalepipeds circumscribed to the families in the proper element
space does not solve fully the problem of the assessment of the amount of
interlopers within each family. The reason is that families have not a paral­
lelepipedal shape in the space of the proper elements a 0 , e 0 and sin i 0 , thus
they do not fill their circumscribed parallelepipedal volume, but only a frac­
tion of it. For this reason, we do have in general a number of non--family
asteroids inside the parallelepipeds circumscribing each family. Having de­
termined the value N \Lambda of random interlopers expected in the parallelepiped
24

associated to a given family, this value must be subdivided in a number N f
that must be associated to the volume really occupied by the family (the
really expected family interlopers) and a number N \Lambda \Gamma N f that should be
associated to the rest of the volume, and should be compared with the real
number N ext of non--family objects present in the volume. This is not qual­
itatively different from the local tests in family--free volumes, described in
Section 2.1.
Of course, the difficult problem here is to find a good method for the
determination of the ``real'' family volume. A simple method could be to
assume a fixed shape for all the families, like a triaxial ellipsoid. This would
be less exotic than a parallelepiped, and would allow to portion out N \Lambda in
a very easy way (the ratio between an ellipsoid and its circumscribed par­
allelepiped). However, such a simple approach seems not very satisfactory.
The real families have not in general an ellipsoidal shape in the space of
the proper elements. In some cases we have populous families showing quite
complicated structures, while in other cases some families can be very small
and composed by only a few members randomly filling the whole circum­
scribed parallelepiped. A numerical approach seems better, and has been
adopted here. For sake of simplicity, let us assume that the volume circum­
25

scribing the family is a simple cube. The idea is to subdivide this volume
in many sub­cubes, and to compute the ratio between the number of sub­
cubes occupied by family members, and the number of empty sub­cubes.
Of course, this technique must be carefully studied in the details, since it is
not free from difficult problems. In particular, we must take into account
that if we subdivide the family cube in an increasingly larger number of
sub­cubes, the volume fraction associated to the family tends to become 0.
This is evident if we imagine to subdivide the cube in a number of sub­cubes
tending to infinity. Some kind of cut--off for the minimum allowable size of
the sub­cubes must be introduced. In particular, we want to avoid creating
sub­cubes so small that empty zones (empty sub­cubes) are created within
the family ``body''.
This requirement suggests a possible criterion: one starts by subdividing
the family cube in 2 \Theta 2 \Theta 2 subcubes. Then the cube is subdivided in
3 \Theta 3 \Theta 3, 4 \Theta 4 \Theta 4 . . . , and so on until we find that, by subdividing the
cubes in n \Theta n \Theta n sub­cubes, one of the cubes occupied by at least one
family member is completely surrounded by empty subcubes. At this point
the ``right scale'' is the previous one, that is the one given by a subdivision
in (n \Gamma 1) \Theta (n \Gamma 1) \Theta (n \Gamma 1) parts. At that scale, the ratio between the empty
26

sub­cubes and those occupied by family members should give a (somehow
rough) evaluation of the relative volume of the family within the original
cube.
However, another fact must be taken into account in this procedure,
and is suggested by the actual mechanism of the statistical method adopted
for the family identification. This is true in particular in the framework of
the families identified by means of the HCM. We cannot repeat here the
details of the HCM procedure, which are extensively explained in several
papers (Zappal`a et al., 1990, 1994, 1995). However, one basic fact must be
taken into account here. This fact is that the families identified by HCM
are such that the minimum distance between one member and the rest of
the family in the space of the proper elements cannot be greater than some
fixed value. In particular, such critical distance value for the nominal family
members is called Quasi Random Level (QRL), and is computed according
to a metric function which is not the ordinary distance
p
x 2 + y 2 + z 2 of
the 3D Euclidean space, but in the space of the proper elements a 0 , e 0 , sin i 0
takes the following form:
d jk = na 0
s
5
4 ( ffi a 0
a 0
) 2 + 2(ffie 0 ) 2 + 2(ffi sin i 0 ) 2
27

where d jk is the distance between two generic objects j and k, having proper
element differences ffia 0 , ffi e 0 and ffi sin i 0 , a 0 being the average proper semi­
major axis, and na 0 the corresponding circular velocity. One property of
this choice of the metric function is that the distances computed according
to this definition have the dimensions of velocities, and are expressed in
m=s. The HCM allows to identify all the clusterings existing at different
levels of the distance, and the QRL is the critical level at which families are
defined. As a consequence of the HCM mechanism, it cannot happen that
the distance between a nominal family member and the closest member of
the same family is greater than the QRL value.
This fact is important, since it allows to derive another value of the
cut--off value for our cubic dissection of the family volume. Not only do
we stop the process when one sub--cube occupied by a family member is
completely surrounded by empty sub--cubes, but also we stop a priori the
cubic dissection of the whole volume when the side of the sub--cubes be­
comes smaller than the QRL of the family, according to the above distance
definition. The reason is that subdividing more finely the volume would
be senseless, since the HCM puts such an upper limit for the distance be­
tween any couple of closest members. We should notice that such a limit is
28

generally encountered before with respect to the other quoted criterion for
stopping the volume subdivision.
Such a result leads to a further complication: in particular, an intrinsic
lower limit for the cut--off does not exist for the families identified by means
of the WAM, because this method works in a completely different way with
respect to HCM. Since in the present paper we analyze families identified
both by HCM and by WAM, for the latter some compromise must be found,
if we want to avoid using the pure criterion of the family--occupied sub--
cube surrounded by empty ones, which leads in general to underestimate
the family volume. In practice, for the WAM families we have choosen to
consider as a lower limit for the cubic dissection a value of distance of 160
m=s, corresponding to the ``intermediate'' scale used in the framework of the
WAM analysis (see Zappal`a et al. 1995 and references within for a deeper
explanation of the WAM procedure).
As a final remark, we should remind now that the family volumes are
not generally cubes, but parallelepipeds. For this reason, on the basis of
the facts explained above, the family volume has been subdivided in smaller
parallelepipeds by subdividing separately in n parts each side, and the cut--
off for this subdivision was found in each case when the shortest side of the
29

resulting parallelepiped reached the critical value (the corresponding QRL
for the HCM families, or 160 m=s for the WAM families). In particular, the
value of n derived in this way turns out to be generally 3 or 4. Only in some
cases are larger values of n encountered, but never beyond 9 (Eunomia). We
have to notice that in some cases a number of non--family members are found
in the family volume. This fact is unavoidable, since the family volume is
given by a set of parallelepipeds fitting the structure of the family, and some
background objects can be located in some cases in the peripheral parts of
this jigsaw puzzle. These objects must be located in the very outer part
of the family volume, and even a little smoothing of this volume would let
them to remain outside. However, the presence of these objects is not a
problem, since they are not different in principle from the other background
objects located in the family--circumscribed parallelepiped. Anyway, their
amount is given in each case in Table II (see later).
The procedure described in this Section allows to derive an approxi­
mated estimate of the family volume, allowing to portion out the expected
number of random objects in the parallelepiped circumscribing each family,
in a number of ``real'' family interlopers, and a number of expected non--
family objects. In particular, since the results are given in terms of the –
30

parameter of the Poissonian distribution of probability, we have in general a
value – int corresponding to the expected family interlopers, and a value – ext
corresponding to the expected background objects within the fraction of the
family--circumscribed parallelepiped not included in the family volume.
As will be explained in Section 4, this procedure allows to point out
the existence of cases in which the nominal list of members severely un­
derestimates the real family membership. A graphic representation of the
overall procedure applied to the families analyzed in this paper is given in
Figures 3 to 10, referring to some of the considered size ranges. Figs. 3 and
5 show the region of the proper element space around the families of Dora
and Liberatrix. Both the parallelepiped circumscribing the family (dashed)
and the other 26 ``local'' parallelepipeds are shown in both cases, together
with the populations of background objects present in these regions. The
sizes of the symbols are directly proportional to the diameters, derived as
explained in the next Section. Figs. 4 and 6 show then the partition of the
shaded regions of Figs. 3 and 5 in a grid of sub--volumes, in which the ones
occupied by one or more family members are dashed. Again, the different
levels (``slices'') in the considered interval of sin i 0 are plotted in the a 0 --
e 0 plane. This allows in each case to give an estimate of the structure of
31

the family, and the relative fraction of volume occupied by it. Figs. 7 to
10 are analogous, but they refer to the more populous families of Eos and
Nysa, spanning over much bigger volumes of the proper element space. We
are aware that such estimates of the family shapes and sizes are somehow
rough, but we are confident that the procedure explained in this Section is
sufficiently accurate for the purposes of the present analysis.
3 The input data
The data--base used in the present paper is the same set of orbital proper el­
ements used by Zappal`a et al. (1995) in their most recent search for asteroid
families, using independently both the HCM and WAM methods. The sam­
ple is composed of 12,487 asteroids of the main belt, whose proper elements
a 0 , e 0 and i 0 have been computed by Milani and KneŸzevi'c (1994, version 6.8.5
of their fully analytical theory). It does not include objects having either
e 0 or sin i 0 larger than 0.3, nor does it include objects outside the interval
2.065--3.278 in a 0 (these limits corresponding to the 4:1 and 2:1 mean--motion
resonances with Jupiter). In addition to 4616 objects numbered up to the
beginning of 1993, the sample includes nearly 8000 non--numbered objects,
whose osculating orbital elements are known with a sufficient accuracy for
32

a reliable computation of the proper elements.
We have extracted from the whole data--set all the objects being nominal
members of the families found by Zappal`a et al. (1995). The remaining
objects constitute the background population, used to derive the number of
objects present in the local volumes of the proper element space associated
with the families. In particular, since we have at disposal both the lists
of families found by means of HCM and WAM, two different background
populations have been considered: one is obtained by removing from the
whole data--set the nominal members (that is, those found to belong to each
family at its critical distance level QRL) of the nominal HCM families. In
other words, we did not consider here the existence of the clumps (smaller
and less reliable groupings; for their definition, see Zappal`a et al., 1995). An
exception to the above criterion is given by the two populous clans of Flora
and Nysa. For both of them, the nominal QRL has been found at 120 m=s
by Zappal`a et al. (1995). However, for these two families the membership
adopted in the present paper is given by the objects found to belong to
them at the distance value of 130 m=s. This difference corresponds to the
nominal 1oe uncertainty of the derived QRL (see the above paper for an
explanation), and such a choice for both the above families is suggested by
33

important reasons: in the case of Nysa, using the nominal membership at
QRL would lead to neglect (and separately analyze) the important sub--clan
of Polana, which joins the Nysa family at 130 m=s, and appears to be a
statistically reliable grouping of objects genetically related to the Nysa clan.
In the case of Flora, the choice to adopt as family members the objects found
at 130 m=s was suggested a posteriori on the basis of the clear indications
that the nominal membership severely underestimated the real family (see
next Section). In particular, we should take into account that the Flora
clan is very populous, has a very complex and not very well understood
structure, and gives a very important contribution to the total population
of the asteroids in the inner part of the main belt. For this reason, we
tried to avoid that an excessive underestimate of the Flora clan could be
responsible of an important overestimate of the background population in
that region, leading to erroneous results also in the analysis of the nearby
families. The HCM background population derived as explained above is
shown in Fig. 2.
The other background population was obtained by removing from the
whole data set all the members of the WAM dynamical families (following
the nomenclature used in Zappal`a et al., 1995). In other words, we did
34

not consider here the ``tribes'' and the ``marginal groupings'' (see the above
paper for these definitions).
In general, the families considered are very reliable and show a good
agreement (actually excellent in some cases) between the nominal HCM and
WAM memberships. For this reason, for the cases in which a family is found
by both HCM and WAM, we considered in this paper only its HCM version,
that is we did not repeat the procedure of interlopers assessment also for
the WAM version of the same family. In this way we avoid also the problem
related to the fact that for the WAM families it is not easy to define a cut--off
value for the subdivision of the family circumscribed parallelepiped, needed
to compute the real family volume inside it (see Section 2.2.) Therefore,
the WAM background population was only used for the analysis of the few
important WAM families having not an HCM counterpart. Moreover, we
did not analyze the nominal families (both HCM and WAM) having less
than 10 nominal members. They were taken into account for what concerns
the background (see above) but we did not apply to them the procedure of
interloper assessment, due to their generally tiny volumes and scarcity of
known members.
Since in this paper we derive the number of family interlopers at different
35

size ranges, the sizes of the considered asteroids are obviously needed. They
have been taken, when available, from the last issue of the IRAS catalogue
of sizes and albedos (Tedesco et al., 1992). In most cases, however, there
are not IRAS size determinations for the asteroids of our sample. In these
cases, we adopted different methods to derive a size estimate. In the cases of
the family members, we computed the size according to the known absolute
magnitude H of the objects, and assuming a fixed albedo within each family,
equal to the average albedo of the IRAS observed family members. For
the background objects, we assigned to each non--IRAS--observed object a
size computed by means of its known absolute magnitude, and an albedo
value equal to the average albedo of the IRAS--observed asteroids in the
same region of heliocentric distance. This mean albedo value was computed
by averaging the IRAS data in bins of (proper) semi--major axis such to
contain at least 100 IRAS--observed objects, starting from the minimum
value a 0 = 2:065. In this way the width of the bins in a 0 is not fixed, but we
have an adequate number of IRAS--observed objects in each bin.
Two exceptions to the above rules are associated with the two important
Vesta and Nysa families. For them, we adopted as the average albedo the
values 0.3 and 0.05 respectively, assuming that the albedo of the majority of
36

the members should correspond to the typical V and F--type, respectively.
For the family of Rafita, Henan, Natascha and Yangel, finally, we adopted
as family albedo that of the corresponding semi­major axis bin, since no
member of these families was observed by IRAS.
The size ranges that we explored in each case are not constant, but they
were decided case by case on the basis of the observed size distribution of
each family. In particular, we took into account for each family its complete­
ness diameter, that is the limiting size value D c for which we can expect
that all the family members larger than D c are known. This is derived on
the basis of the mean family albedo and assuming that asteroids brighter
than V=15.5 at opposition have been fully discovered (Zappal`a and Cellino,
1994). We are aware that the choices of the size ranges for the single fami­
lies analyzed in this paper are somehow subjective in some cases, sometimes
including only a few family members. We tried to avoid as much as possi­
ble these situations, since the statistical approach adopted loses its meaning
when it is applied to empty size ranges. For this reason, in some cases in
which a family includes a largest member which is very large with respect
to the other members and also with respect to the nearby background pop­
ulation, we did not take into account this object in the analysis. This does
37

not mean that we do not believe it to be a real member, but simply that
we cannot apply our method to a size range populated by only one object.
In particular, this happened in the cases of the Gefion (Ceres), Eunomia,
Hygiea, Themis, Vesta, Juno. In general, we can say that the choices of the
size ranges are suggested by the observed properties of the families. Some
general criterion to be applied to all the families can hardly been found,
since the size ranges and the size distributions are very different in different
cases. The general statistical procedure we have developed in this paper can
be applied without problem to any choice of the size ranges, chosen in each
individual case on the basis od the scientific goal being pursued.
For each family, and for each chosen size range, we explore the distri­
bution of the background population in the local surrounding zone, follow­
ing the technique described in Section 2. In particular, the ``family par­
allelepiped'' is the one circumscribing the family, and having as sides the
width of a 0 , e 0 and sin i 0 over which the family extends.
Table I lists for each family analyzed in this paper the extrema in proper
semi--major axis, eccentricity and inclination; the corresponding adimen­
sional volume \Deltaa 0 =a 0 \Delta \Deltae 0 \Delta \Delta sin i 0 (we note that the corresponding volume
of the whole belt between 2.065 and 3.278 in a 0 , and between 0.0 and 0.3
38

in both e 0 and sin i 0 turns out to be 1:322 \Delta 10 \Gamma1 ); the last column gives the
fraction of the family--circumscribed parallelepiped occupied by the family,
computed as explained in Section 2.2. We can observe that even the largest
families extend over fractions not larger than 10 \Gamma4 of the adimensional vol­
ume of the whole belt. The general results of the analysis and a brief case
by case comment for each family are given in the next Section.
4 The results
Table II gives a summary of the results obtained for the families analyzed in
this paper. In particular, it lists for each family and for each considered size
range the number of nominal family members in the size range; the corre­
sponding amount of background objects present in the family--circumscribed
parallelepiped, given by the sum of the objects located in the parallelepiped
in the part external to the family volume, plus the (generally few) non--
family objects included in the family volume computed by means of the
parallelepipedal dissection explained in Section 2.2; the resulting expecta­
tion values – ext and – int of the resulting poissonian distributions of proba­
bility for the amount of objects located in the family parallelepiped outside
(– ext ) and within (– int ) the family volume.
39

As can be seen, in some cases the listed N ext is not defined. This cor­
responds to situations in which the family occupies all its circumscribed
parallelepiped. In other cases, no value is given for – int . These cases corre­
spond to situations in which the family is small and relatively isolated, thus
no objects are found in the 26 parallelepipeds surrounding the one circum­
scribing the family. In such cases, of course, the whole statistical procedure
based on the properties of the local background cannot be applied, although
we have good reasons to believe that no interlopers are probably present.
While the meaning of – int is obvious, since it represents the goal of the
present paper, aimed at deriving a quantitative estimate of the probable
amount of interlopers within asteroid families, the meaning of – ext should
be stressed here. In particular, we note that a comparison between – ext and
N ext can give a clear indication of a possible underestimate of the family
membership. This happens when the number of real objects N ext external
to the family volume turns out to be much larger than the plausible amount
predicted by – ext . In other words, N ext AE – ext means that there is an ex­
cessive amount of officially non--family members in the close neighbourings
of the family. This should be considered as a clear indication that most of
these objects are real family members not included in the nominal list of
40

members, due to an exaggerately conservative criterion of family member­
ship. Interestingly enough, such a situation is encountered more frequently
in the cases of the major clans identified by Zappal`a et al. (1995), as can be
seen by looking at Table II, and as has been graphically shown in Figure 11,
analogous to Figure 1b. As can be seen, in most cases N ext is fully compati­
ble with the corresponding value of – ext , the main exceptions corresponding
to the major clans, like Eunomia, Nysa, Vesta, and others. In these cases,
the results obtained for each family at the different size ranges considered
are connected by lines. The case of Flora is not shown in the Figure, because
for it the values of N ext turns out to be even more underestimated by the
resulting – ext .
The opposite case, that is N ext Ü – ext is not expected to be encountered
in practice, if the statistical procedure developed in this paper is not faulty.
Such cases could not be interpreted in terms of not reliable family mem­
berships, and should correspond only to very unlikely random fluctuations
of the asteroidal distribution. The fact that this does not happen in the
set of families analyzed, gives an independent confirmation of the overall
reliability of the adopted statistical procedure.
An interesting general result of the analysis is that, as can be seen in
41

Table II, the relative fraction of expected interlopers with respect to the
amount of known family members appears to increase for increasing diam­
eters. Such a behaviour seems quite frequent, and this is not surprising if
we take into account the results obtained by Cellino et al. (1991) about the
general size distribution of the asteroids. In that paper it was shown that
the size distribution of the families appears to be significantly steeper with
respect to the normal background population, in the sense that families are
characterized by a higher value of the exponent of the Pareto (power law)
distribution of sizes. If this is true, we can predict that the relative amount
of interlopers, which are typical background objects, should be increasingly
lower with respect to the amount of family members at smaller sizes. The
fact that the results shown in Table II confirm such a prediction should be
therefore considered as a fact of high interest.
Having outlined the most important general comments that can be done
by taking a simple glance at the results shown in Table II, we will briefly
analyze separately the individual families in the following Subsections.
4.1 HCM families
Adeona The family size seems well determined, that is no additional mem­
42

bers should be expected in the neighbouring region, since there is no
discrepancy between N ext and the resulting – ext at all the considered
diameter ranges. Below the diameter of completeness (17 km) a couple
of chance interlopers are predicted. One interloper cannot be excluded
even among the four objects having larger sizes, between 17 and 60
km.
Astrid, Bower, Hoffmeister, Meliboea, Veritas For all these families
the sizes appear well determined, and no interlopers should be present
among the nominal members.
Chloris A couple of interlopers are possible below the completeness diame­
ter (19 km). Family size well determined. The presence of two largest
members unusually bigger with respect to the rest of the family ap­
pears puzzling, but the expectation parameter – int turns out to be
only 0.04 in this size range.
Dora Family size well determined. A couple of possible interlopers below
the completeness diameter (22 km).
Eos The nominal membership of this important family seems severely un­
derestimated. At sizes smaller than 17 km, 80% of the objects sur­
43

rounding the nominal members could be real family members. At the
same time, some interlopers are probably present at all the considered
size ranges, even at D ? 40 km. As suggested by D.D. Durda (per­
sonal communication) the fact that the nominal Eos family is proba­
bly underestimated, might be useful to interpret the difficulties so far
encountered in associating this family with the 10 degree dust band
found by the IRAS satellite, since a larger inclination dispersion of the
family seems necessary to fit the observed band profile.
Erigone Family size well determined. From 2 to 4 expected interlopers
below the completeness diameter (15 km).
Eunomia This large clan appears severely underestimated (N ext = 71,
against a value of – ext = 17:9 below 11 km). A fairly large amount of
interlopers is predictable at the smallest size ranges, while two or three
could be present in the size range between 10 and 50 km, and possibly
one at D ? 50 km. These estimates open the possibility that most
of the largest members of the family could be chance interlopers, so
allowing to find a solution to the apparent geochemical discrepancies
(taxonomic anomalies) of this family. We should note that 15 Euno­
44

mia, being much bigger than any other family member, has not been
considered in the analysis.
Flora This is probably the most complicated and controversial grouping
presently identified in the belt. Due to the very dense population in
this zone of the proper element space, even the most modern methods
of family identification find a big difficulty in unambiguously deriv­
ing reliable results. Flora is an extreme case of a complex clan. Its
nominal membership is mostly tentative, and the number of members
greatly varies even for modest changes of the adopted critical distance
level. The present analysis indicates that the nominal list of members
is highly underestimated. Even considering the membership at the
distance level of 130 m=s instead of the nominal QRL, as explained in
Section 3, it turns out that 5=6 background objects around the family
should be real family members, up to a size of 50 km. In addition,
several interlopers should be expected, and it is not impossible that
even the only two members above 50 km could be chance interlopers.
We should also take into account, however, that underestimating the
real amount of members we can also underestimate the ``right'' fam­
45

ily volume, and some family members can actually raise spuriously
the background population in the parallelepipeds surrounding the one
circumscribed to the family, leading to a slight overestimate of both
– ext and – int . However, such an effect should not be very important,
considering also that the values of – int listed in Table II predict that
only a tiny fraction of the nominal members could be interlopers. As
a conclusion, we confirm the very complex nature of the Flora clan,
preventing from stating firmly the resulting membership of this family.
Gefion (Ceres) We have excluded a priori from the family the asteroid 1
Ceres, which is the nominal largest member of this family according
to the results of Zappal`a et al. (1995), since it is difficult to admit
that a family could have been originated from an impact on Ceres,
due to the very high amount of energy required for such an event. On
the other hand, this family did not include Ceres in previous family
searches (Zappal`a et al., 1990, 1994) and was identified as the family of
Gefion. This family appears to be slightly underestimated at sizes less
than 15 km (diameter of completeness). Some of the seven background
objects could be real family members. In the same size range a couple
46

of family interlopers are predicted, while at least one interloper could
be present even in the upper size range, between 15 and 65 km.
Henan From 1 to 4 interlopers in the size range 0 -- 15 km could be present
in this small family, whose size appears well determined (– ext of the
order of N ext ).
Hygiea We excluded from the analysis 10 Hygiea, one of the largest as­
teroids of the belt. The membership list of this family turns out to
be underestimated, with 60% of the neighbouring background objects
being potential family members at sizes less than the completeness
diameter (25 km). In the same size range a not negligible amount of
interlopers (3 -- 8) are predictable, while the percentage of possible
interlopers increases in the size range between 25 and 65 km.
Koronis This family size appears to be well determined, or very slightly
underestimated below the completeness diameter (13 km) and up to
25 km. A small fraction of interlopers is expected at low sizes, while
– int turns out to be quite low at larger sizes.
Liberatrix Family size well determined. Some interlopers below the com­
pleteness diameter, but one cannot be excluded also among the 6
47

known larger members.
Lydia Although this family appears to be close to others present in this
region of the proper element space (Zappal`a et al., 1995) the resulting
size appears well determined, since N ext is of the order of – ext . A
number of interlopers that should not exceed 2 or 3 is expected, mainly
below the completeness diameter.
Maria This high--inclination family seems to have a well determined size,
with some interlopers, that should be comprised between 1 and 4 in
the size range 9 -- 25 km.
Massalia This family, which is close to the Nysa clan in the proper ele­
ment space, appears to be only slightly underestimated, and includes
probably a couple (or a few more) interlopers. 20 Massalia was not
included in the analysis, since it is the only object with a size much
larger than 8 km.
Merxia While the family size appears to be well determined, a couple of
interlopers cannot be excluded in the whole size range of the family.
Misa A small family with a well determined size. A not negligible fraction
48

of interlopers could be present at sizes smaller than the completeness
diameter (27 km).
Nemesis Another case of family size well determined, with up to 3 possible
interlopers possibly present in the whole size range.
Nysa Even having considered the membership at the distancce level of 130
m/s instead of the nominal QRL of 120 m/s (see above), the Nysa clan
appears noticeably underestimated. Several interlopers should also be
present mainly at small sizes, while one or two cannot be excluded
between 16 and 90 km. This fact opens the possibility that either 44
Nysa or 135 Hertha, or even both of them could be interlopers. This
fact should be taken into account in any physical analysis of this clan
(Zappal`a et al., in preparation).
Rafita A small family well determined (– ext ' N ext = 0). One or two
interlopers cannot be excluded in the whole size range.
Themis This large family appears to be much underestimated for what
concerns the members smaller than 40 km. 80% of the neighbouring
background objects could be real family members. The expected frac­
tion of interlopers is relatively low, but one or two cannot be excluded
49

even at sizes larger than 40 km. 24 Themis was not included in the
analysis, due to its relatively very big size.
Vesta Also in this case the biggest member, 4 Vesta, was not included in
the analysis due to its exceedingly large size. The nominal membership
of this clan appears to be severely underestimated. At the same time
several interlopers can be present at sizes up to 15 km. The results
leave open the possibility for the only member between 15 and 50 km,
306 Unitas, to be an actual interloper.
4.2 WAM families
Brasilia, Juno These two families have probably well determined sizes,
without interlopers. In the case of Juno, we excluded from the analysis
3 Juno itself, having a size completely different from the rest of the
family.
Eurynome This is the only example of a large WAM family with no HCM
counterpart (Zappal`a et al., 1995). It is located in a densely populated
zone in the inner belt. For this family we find a very high value for
– int (36.1 at sizes below the completeness diameter of 11 km). This
fact can hardly be explained. Even taking into account a possible
50

contamination of the WAM background by a number of neglected Flora
members, this could affect only the background population within one
or two of the 26 parallelepipeds surrounding the one circumscribed to
the family. For this reason, we do not expect that the value of – int
(and – ext ) could be lowered very much even taking into account such
a background contamination. These fact leads to conclude that this
family is probably badly defined, and includes many interlopers.
Gabriella This family could have a slightly underestimated membership.
A few interlopers are present at sizes smaller than the completeness
diameter (9 km), while it cannot be ruled out the possibility that 355
Gabriella itself could be an interloper.
Hestia, Natascha, Yangel These families appear to have well determined
sizes, and could include at most a couple of interlopers in the whole
size ranges spanned by the family members. Due to its comparatively
big diameter, 46 Hestia itself was excluded from the analysis of its
family.
Nina Well determined family size, with 1 -- 3 interlopers below 13 km (com­
pleteness diameter).
51

5 Conclusions
The motivation that led the authors of this paper to undertake the anal­
ysis presented here comes from the problems encountered in the physical
analysis of the asteroid families. The fact to have at disposal a list of fami­
lies very reliable from the statistical point of view allows now to undertake
a systematic effort aimed at extracting from the families a lot of informa­
tion on the collisional events from which they originated, with important
implications about the physical properties of the parent bodies and on the
overall process of collisional evolution of the asteroid belt (see also Zappal`a
and Cellino, 1994).
It is evident that this paper allows to give an answer to the question
``how many interlopers are there'', and not to the question ``which ones''
However, this is already an important first step, since in many cases there
are reasons to believe that some family members are not compatible with
the rest of a family (anomalous taxonomic type, anomalous size distribution,
anomalous location within the family structure, etc.), but in the absence of
direct physical observations it is difficult to justify any exclusion from family
lists that have been obtained on the basis of careful and strict statistical
52

criteria. Now, the present paper allows to give a first quantitative estimate
of the amount of probable interlopers present at different size ranges, on
the basis of the local properties of the asteroid distribution in the proper
element space.
It is evident that the present analysis is not aimed at substituting an
extensive observational activity. Only observations, mainly spectroscopy,
can provide definitive indications about the precise identity of interlopers.
However, the present analysis gives some suggestions about the more prof­
itable strategy for the undergoing observational activity. In other words, we
can now predict which families and at which size ranges contain the larger
amounts of probable interlopers.
The results presented in this paper are statistical, and must be considered
as such. The values for – int listed in Table II must be understood for what
they are, that is expectation parameters of poissonian distributions. They
should represent a good approximation of the reality on the average, but we
can expect in a few cases the real amount of interlopers to be significantly
different, due to the statistical nature of the problem.
One of the main results of the present analysis is the evidence found for
a severe underestimate of the memberships of several families, mainly the
53

larger clans. In these cases, the results of the present paper (in particular
those related to the – ext =N ext ratio) gives a quantitative indication about
how fruitful could be to extend the physical observations also to the back­
ground objects surrounding the nominal family members. This could be a
very good approach for finding the actual family borders.
In the present paper we limited ourselves to a systematic analysis of the
nominal memberships of the families listed by Zappal`a et al. (1995). The
authors of that paper already stated clearly that the ``official'' memberships
of these families were not too rigid, and mainly tentative in some cases,
corresponding to the ones for which we find now the evidence of an underes­
timate. In this respect, a more complicated analysis could performed, taking
into account the memberships obtained at different levels of statistical ev­
idence: the ``stalactite diagrams'' of the HCM allow to do this very easily.
In other terms, we could present an analysis of the interlopers predictable
for any given clan as found at QRL, at QRL + 10 m/s, etc. We are aware
that this could be interesting, but obviously quite complicated too, taking
into account, for instance, that the general background population could be
obtained only by an iterative procedure, after having determined separately
the best cutting level for each family. We decided to postpone such an anal­
54

ysis to a series of forthcoming papers that will be devoted to the detailed
physical analysis of single families, where the local background population
will be taken into account case by case.
The scope of the present paper is to present a tool that should be useful
for the physical studies of the families. Apart from some ``technical'' com­
plications (Section 2.2) the basic idea is very simple, and the results shown
in Table II should be considered as a useful first step, to be improved by
means of a more detailed analysis of the individual cases.
6 Acknowledgements
M. Moons and A. Morbidelli are kindly acknowledged for providing us with
the values of the edges of the mean motion resonances with Jupiter. We also
thank the referees E.F. Tedesco and D.D. Durda for their useful comments.
This work has been partially supported by the Italian Space Agency (ASI)
and by the Italian Ministry for University and Scientific Research (MURST).
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Figure captions
Figure 1. The probability to have k events following a poissonian distribu­
tion is shown in (a) as a function of the expectation parameter – by
means of circles, having sizes proportional to the corresponding prob­
ability. In (b) higher values of – are considered, and for them we take
into account that the poissonian distribution can be approximated by
a gaussian. Therefore, the open and filled circles correspond to ranges
of probability of 1­oe and 2­oe from the mean, respectively. In both fig­
ures, the crosses correspond to the set of tests performed in different
regions of the proper element space (ses text). In particular, for them
– is the resulting poissonian parameter, while k is the true amount of
objects located within the test regions.
Figure 2. The locations of the test­regions in the a 0 --e 0 (a) and in the a 0 --
sin i 0 (b) plane, respectively.
Figure 3. The region of the proper element space around the family of
Dora. Each plot corresponds to an a 0 --e 0 projection of the family--
circumscribed parallelepiped (dashed) and the other 26 equipollent
volumes surrounding it (see text). The width of the inclination ranges
61

corresponding to each plot is listed in the bottom--right corner, the
first being the actual family range. The quasi--vertical lines crossing
the parallelepipeds correspond to the borders of the 5/2 mean­motion
resonance with Jupiter. The open circles show the location of the back­
ground objects in the size range 0--35 km, which includes all the known
members of the family. The sizes of the symbols are proportional to
the corresponding diameters.
Figure 4. This figure shows the a 0 --e 0 projections of the 3 \Theta 3 \Theta 3 subdivi­
sion of the family--circumscribed parallelepiped corresponding to the
Dora family as explained in Section 2.2. The subvolumes occupied by
(all size) family members (shown by crosses) are dashed. The range
of sin i 0 listed at the bottom corresponds to the whole family width.
Each plot (from top--left) corresponds to one ``slice'' of the sin i 0 range,
starting from the lowest edge of sin i 0 . The whole set of dashed regions
corresponds to the resulting ``family volume'' (see text).
Figure 5. The same as Figure 3 but for the family of 125 Liberatrix. The
location of the 5/2 mean motion resonance with Jupiter, and of the
7/3 mean motion resonance (not crossing the analyzed region) are also
62

shown. The size range considered here is 0--60 km.
Figure 6. The same as Figure 4 but for the family of 125 Liberatrix. Here
the subdivision turns out to be 4 \Theta 4 \Theta 4. The circles correspond
to the background objects in the listed size range located inside the
family--circumscribed parallelepiped.
Figure 7. The same as Figure 3 but for the clan of 44 Nysa. The location of
the important 3/1 mean motion resonance with Jupiter is also shown.
The size range here is 10--16 km.
Figure 8. The same as Figure 4 but for the 7 \Theta 7 \Theta 7 subdivision of the
clan of 44 Nysa. For the meaning of the circles, see Figure 6.
Figure 9. The same as Figure 3 but for the family of 221 Eos. From left to
right, the edges of the 5/2, 7/3, 9/4 and 2/1 mean motion resonances
with Jupiter are shown. Here, the size range is 0--17 km.
Figure 10. The same as Figure 4 but for the 6 \Theta 6 \Theta 6 subdivision of the
family of 221 Eos. For the meaning of the circles, see Figure 6.
Figure 11. The same as Fig. 1b. Here crosses correspond to the real num­
ber of background objects found in the ''external'' part of the family--
63

circumscribed parallelepipeds (see text) plotted against the resulting
values of the poissonian parameter –, for all the size ranges listed in
Table II. Values corresponding to different size­ranges of a few fami­
lies showing a well established under­estimation of their members are
connected with straight lines. The peculiar case of 79 Eurynome is
also evidenced for the lowest size range listed in Table II.
64