S. F. Dermott, D. Li, & A. A. Christou Deconvolving the complex structure of
the asteroid belt

17 2021 10.1017/xxxxx

Proceedings Kavli-IAU Symposium No. 382, 2024

# Deconvolving the complex structure of the asteroid belt

Stanley F. Dermott1 Dan Li2 and Apostolos A. Christou3<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Astronomy,
University of Florida, Gainesville, FL 32611, US
email:
2NSF’s National Optical-Infrared Astronomy Research Laboratory, Tucson, AZ
85719, US
email:
3Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DG
email:

###### Abstract

The asteroid belt is a unique source of information on some of the most
important questions facing solar system science. These questions include the
sizes, numbers, types and orbital distributions of the planetesimals that
formed the planets, and the identification of those asteroids that are the
sources of meteorites and near-Earth asteroids. Answering these questions
requires an understanding of the dynamical evolution of the asteroid belt, but
this evolution is governed by a complex interplay of mechanisms that include
catastrophic disruption, orbital evolution driven by Yarkovsky radiation
forces, and chaotic orbital evolution driven by gravitational forces. While
the timescales of these loss mechanisms have been calculated using estimates
of some critical parameters that include the thermal properties, strengths and
mean densities of the asteroids, we argue here that the uncertainties in these
parameters are so large that deconvolution of the structure of the asteroid
belt must be guided primarily by observational constraints. We argue that
observations of the inner asteroid belt indicate that the size-frequency
distribution is not close to the equilibrium distribution postulated by
Dohnanyi (1969). We also discuss the correlations observed between the sizes
and the orbital elements of the asteroids. While some of these correlations
are significant and informative, others are spurious and may arise from the
limitations of the Hierarchical Clustering Method that is currently used to
define family membership.

###### keywords:

Asteroids, Main belt asteroids, Near-Earth objects, Small solar system bodies,
Meteorites

††editors: A. Lemaitre & A. S. Libert, eds.

##  1 Introduction

We discuss three classes of family asteroids. (1) Family asteroids that were
formed either by catastrophic disruption or crater formation. (2) Halo
asteroids that are members of known families. These are family members that,
due to the limitations of the Hierarchical Clustering Method (HCM), have not
been attached, unambiguously, to a specific family. (3) Asteroids in ghost
families. These are older families that were formed in the same way as those
in class (1) but are difficult to detect because their orbital elements have
dispersed. If we removed all the asteroids in these three classes, then we
would be left with the primordial asteroids. We note, however, that each
family must have, or have had, a precursor asteroid that is the root source of
the other family members. These precursor asteroids are also members of the
primordial class. Answering many of the questions facing solar system science
requires a determination of the structure of the primordial asteroid belt,
that is, the structure of the belt after the transport of the asteroids from
their formation locations to the main belt, and after the migration of the
major planets that triggered that transport had ceased, but before the time of
formation of any of the families. Determining that primordial structure and
its transformation into the current structure are the challenges discussed in
this paper.

We confine our discussion to the inner main belt (IMB) defined here by 2.1
$au$ $<a<$ 2.5 $au$ and $I<18$ $deg$, where $a$ is the proper semimajor axis,
and $I$ is the proper inclination. Family membership is usually defined using
the Hierarchical Clustering Method (HCM) introduced by Zappala et. al (1990).
This uses the metric

$d=na\sqrt{d_{1}^{2}+d_{2}^{2}+d_{3}^{2}}$ (1)

where

$na=\sqrt{GM_{\odot}/a},\,d_{1}=\sqrt{5/4}(\Delta a/a),\,d_{2}=\sqrt{2}\Delta
e,\,d_{3}=\sqrt{2}\Delta(\textrm{sin}I)$ (2)

and $na$ is the average orbital velocity of an asteroid having semi-major axis
$a=(a_{1}+a_{2})/2$, $\Delta a=a_{1}-a_{2}$, $\Delta e=e_{1}-e_{2}$,
$\Delta\textrm{sin}I=\textrm{sin}I_{1}-\textrm{sin}I_{2}$, where the indices 1
and 2 denote the two bodies whose mutual distance, $d$ is calculated. Each
member of a given family is separated from its nearest neighbor by
$d<d_{\textrm{crit}}$, where $d_{\textrm{crit}}$ is a critical distance chosen
such that the members of one family are not linked to the members of another
family. Asteroids in the IMB with absolute magnitude $H<16.5$ are
observationally complete (Dermott et. al, 2018; Hendler & Malhotra, 2020).
However, while this set of asteroids is devoid of observational selection
effects, some correlations between the sizes and the orbital elements of the
family asteroids may arise because of the way in which the HCM defines family
membership.

##  2 Orbital element and size correlations in the IMB

Figure 1: Proper orbital elements of the asteroids in the IMB with $H<16.5$
(the observational completeness limit). Panels (a)-(c) show the major asteroid
families defined by Nesvorný (2015). The green shaded region in panel (b) is
the Mars-crossing zone, and the dashed curve in panel (a) is the $\nu_{6}$
secular resonance. Panels (d)-(f) show the non-family asteroids, color-coded
according to the maximum Lyapunov characteristic exponent, $mLCE$ calculated
by Knežević & Milani (2000), with red being the most chaotic ($mLCE>0.00012$
per year) and black the most stable ($mLCE<0.00012$ per year) (taken from
Dermott et. al 2021). Figure 2: Scatter plot in $a-I$ space of all the
asteroids in the IMB with absolute magnitude, $H<12$. Non-family asteroids are
colored black. Asteroids in the prominent Flora family are colored cyan. A CC
asteroid with $H=12$ and albedo, $A=0.05$ has a diameter, $D=24\,km$. An NC
asteroid with $H=12$ and albedo, $A=0.24$ has a diameter, $D=11\,km$ (taken
from Dermott et. al 2021)

Fig. 1 shows the major families in the IMB as defined by Nesvorný (2015) using
the HCM. For those asteroids with $H<16.5$, the IMB is dominated by family and
halo asteroids and only an estimated 24% of the asteroids are non-family and
possibly primordial (Dermott et. al, 2022). The largest asteroids in the major
families have diameters of 525 $km$ (Vesta), 254 $km$ (Flora), 135 $km$, 142
$km$, 495 $km$ (the Nysa, Polana, Eulalia complex), 135 $km$ (Massalia), and
72 $km$ (Erigone), consistent with the argument that the planet-forming
planetesimals were large with diameters $D\sim 100$ $km$ (Morbidelli et. al,
2009). However, the percentage of asteroids in the IMB that are non-family is
size-dependent. Fig. 2 shows that the asteroid population with $H<12$,
particularly in the range 2.3 $au$ $<a<$ 2.5 $au$, is dominated by non-family
asteroids that are unlikely to be halo asteroids. For reference, a CC asteroid
(defined here as an asteroid with $A<0.13$) with $H=12$ and albedo, $A=0.05$
has a diameter, $D=24\,km$. An NC asteroid (defined here as an asteroid with
$A>0.13$) with $H=12$ and albedo, $A=0.24$ has a diameter, $D=11\,km$. These
diameters are considerably smaller than those associated with the precursor
asteroids of the major families and an outstanding question is what fraction
of these non-family, non-halo asteroids are members of ghost families
originating from larger primordial asteroids?

The ages of the families have been estimated from the V-shaped spread of the
asteroids in $a-1/D$ space, yielding the result that most families have ages
less than half the age of the solar system (Spoto et. al, 2015). Given that
catastrophic disruption was probably greater in the early solar system, this
implies that a large fraction of the non-family asteroids could be remnants of
ghost families. The spreading rate of the families is determined by the
strength of the diurnal Yarkovsky forces and is given by

$\frac{da}{dt}=\pm\xi\frac{3}{4\pi}\frac{1}{\sqrt{a}(1-e^{2})}\frac{L_{\odot}}{c\sqrt{GM}_{\odot}}\frac{1}{D\rho}$
(3)

where $L_{\odot}$ and $M_{\odot}$ are the solar luminosity and the solar mass,
$c$ is the speed of light, and $D$ and $\rho$ are the diameter and the mean
density of the asteroid (Greenberg et. al, 2020). The Yarkovsky efficiency,
$\xi$, depends on the spin pole obliquity and the thermal properties of the
asteroid (Bottke et. al, 2002). Observations of near-Earth asteroids (NEAs)
show that

$\xi=0.12_{-0.06}^{+0.16}$ (4)

(Greenberg et. al, 2020). Including the uncertainties in the mean densities,
the total uncertainties in the evolution rate may be as large as a factor of
two and therefore it is possible that the family ages span the age of the
solar system. It would be useful to have estimates of the strengths of the
Yarkovsky forces for the separate CC and NC main belt asteroids that do not
depend on estimates of the thermal parameters and mean densities, etc. Direct
observations of the changes in the asteroid orbits are not yet available for
main belt asteroids and we must look to other observations. It has been shown
that the Martian 1:2 mean motion resonance has a well-defined excess of
asteroids (Gallardo et. al, 2011; Dermott et. al, 2022). This excess depends
on the rate of orbital evolution and modeling could yield an estimate of the
strength of the Yarkovsky forces. However, that modeling has yet to be
undertaken.

Figure 3: Panel (a): the variation with absolute magnitude, $H$ of the mean
proper inclination of the high inclination ($I>9$ $deg$), non-family
asteroids. The data is shown binned in $H$, but the slope has been determined
from the individual points in the range $16.5>H>13.5$. Panels (b) and (c): in
these plots we have divided the data into CC and NC groups as determined by
their WISE albedos (taken from Dermott et. al 2021).

The outer half of the IMB (Fig. 2), the half that is not dominated by the
Flora family, has given us one window into the non-family asteroid population.
A second significant window is given by the $a-I$ distribution shown in Fig.
1a. Here, we see that the $I>9$ $deg$ region is devoid of major families and
the halo asteroids associated with those families. Fig. 3a shows that for the
non-family asteroids in this window there is a strong (7$\sigma$) correlation
between the asteroid magnitude, $H$ and the proper inclination, $I$ (Dermott
et. al, 2021, 2022). This correlation is supported by the correlations of the
separate subsets of CC (albedo, $A<0.13$) and NC (albedo, $A<0.13$) asteroids
(Figs. 3b, c). This correlation arises because the length of the escape route
in the IMB, that is, the distance between the $\nu_{6}$ secular resonance and
the 3:1 Jovian mean motion resonance decreases as $I$ increases with the
result that small asteroids are preferentially lost from the high inclination
orbits (Dermott et. al, 2021, 2022). To estimate the effectiveness of
Yarkovsky forces in the removal of small asteroids, we can evaluate eqn. (3)
and write

$\frac{1}{a}\frac{da}{dt}=\pm
2.0\xi\left(\frac{1\,km}{D}\right)\left(\frac{2000\,kgm^{-3}}{\rho}\right)Gyr^{-1}.$
(5)

Given that

$D=\frac{1329\,km}{10^{H/5}\sqrt{A}}$ (6)

and assuming that $a=2.4$ $au$, we can write

$\frac{da}{dt}=0.00043\,10^{H/5}\sqrt{A}\left(\frac{2000\,kgm^{-3}}{\rho}\right)au\,Gyr^{-1}.$
(7)

For NC asteroids, using $A\approx 0.24$ and $\rho\approx 2000\,kgm^{-3}$ we
have

$\frac{da}{dt}=0.0002\,10^{H/5}\,au\,Gyr^{-1}.$ (8)

For CC asteroids, more appropriate values would be $A\approx 0.05$ and
$\rho\approx 1000\,kgm^{-3}$. However, since $\sqrt{0.24/0.05}/2\approx 1$,
either set of parameters results in the same estimate for $da/dt$ and we can
usefully apply eqn. (8) to all of the IMB asteroids. If we assume that
$da\approx 0.2\,au$ typically results in the loss of an asteroid through one
of the escape hatches, then asteroids with $H\gtrsim 15$ will be lost from the
IBM in $\lesssim 1$ $Gyr$. $H=15$ corresponds to $D\approx 6\,km$ (CC) or
$D\approx 3\,km$ (NC).

Unless the Yarkovsky efficiency, $\xi$ estimated from the NEA observations is
an overestimate we must expect a significant loss of small asteroids from the
IMB. This loss could be compensated for by the creation of small asteroids
through the catastrophic destruction of larger asteroids. However, we have
other observations that place constraints on the creation rate. One constraint
is provided by the $H-I$ correlation shown in Fig. 3a. Dermott et. al (2021,
2022) have argued that the loss of asteroids from the IMB is dominated by
Yarkovsky orbital evolution, and if the initial distribution of asteroids in
$a-I$ space was uniform, and the loss of asteroids occurred over the age of
the solar system, then $\xi$ must be a factor of three smaller than the NEA
estimate or the correlation would be three times larger. Alternatively, if the
orbital evolution rate is consistent with the NEA observations, then the
asteroids must be younger than the age of the solar system by a factor of
three, implying that the majority of asteroids in the IMB with high
inclination ($I>9$ $deg$) orbits are not primordial but members of families
and the products of the catastrophic destruction of a small number of larger
asteroids. A diffuse cluster of high-$I$ asteroids, centered on $I=13.5$ $deg$
and $e=0.18$ is evident in Fig. 1c.

Figure 4: Panel (a): quadratic fits to the SFDs for asteroids in the separate
major families, and for the non-family ($I>9$ $deg$) asteroids. Nominal
asteroid diameters have been calculated from $H$ assuming an albedo of 0.13.
Panel (b): the variation with absolute magnitude, $H$ of the slopes to the
quadratic fits shown in panel (a). Panel (c): the dependence of the mean
proper inclination on asteroid size for all family asteroids (treated as one
group), for all non-family asteroids (also treated as one group), and for the
separate major families (taken from Dermott et. al 2018, 2021).

Other evidence for the net loss of small asteroids from the IMB is obtained
from the size-frequency distributions (SFDs). The data in Fig. 4 are
observationally complete and show that all the major families and the non-
family asteroids in the IMB have a depletion of small asteroids. As $H$
increases the observed slopes of all the SFDs tend to zero. Modeling of the
variation of the SFD with $H$ has shown that for $H\gtrsim 16.5$ we must
expect a marked lack of small asteroids (Dermott et. al, 2021, 2022). Thus,
the expectation that for high $H$ the SFD tends to the equilibrium
distribution postulated by Dohnanyi (1969) is not supported by the
observations. This is because Dohnanyi (1969) considered a closed system and
did not allow for the loss of small asteroids from that system. This has major
consequences with respect to calculations of the expected rate of collisional
disruption, and the rate of change of asteroid spin directions due to impacts
with small asteroids. Given that we do not know the SFD of the small asteroids
that impact the larger asteroids, we must accept that these rates are unknown.

##  3 The Mars-crossing zone

Figure 5: Panel (a): scatter plot of the proper eccentricity $e$ and the
semimajor axis, $a$ of the asteroids in the IMB with absolute magnitude
$H<15$. The green shaded zone on the left is the Mars-crossing zone. Panel
(b): histogram of the semimajor axes of the asteroids in the Mars-crossing
zone. Panel (c): quadratic fits to the SFDs (with $dH=0.5$) for the asteroids
in the separate major families, the non- family ($I<9$ $deg$) asteroids, the
non-family ($I>9$ $deg$) asteroids, and the asteroids in the Mars-crossing
zone. Nominal asteroid diameters have been calculated from $H$ assuming an
albedo of 0.13 (taken from Dermott et. al 2021).

The orbital eccentricities of the main-belt asteroids are largely capped by
the Mars-crossing zone (Fig. 5a). Given that the lifetime of the asteroids in
this zone is $\lesssim 10^{8}$ $yrs$, this indicates that Mars is currently
scattering asteroids out of the main belt and into the inner solar system. In
Fig. 5b, we observe that most of the asteroids in or above the crossing-zone
are in the inner main belt (IMB), suggesting that the IMB is the major source
of near-Earth asteroids (NEAs) and meteorites (Dermott et. al, 2021, 2022).
This conclusion is supported by the results of numerical investigations of the
likely escape routes from the main belt (Gladman et. al, 1997; Granvik et. al,
2017, 2018). The latter studies used estimates of the total number of NEAs
after correcting for the incompleteness of the observed NEA population,
estimates of the SFD of the main belt asteroids, and estimates of the
strengths of the Yarkovsky forces. However, the lifetimes of the asteroids in
the Mars-crossing zone are determined solely by the rate of close encounters
with Mars and it is possible to determine those lifetimes without making
unnecessary assumptions. The calculations for the asteroid set shown in Fig.
5b have yet to be performed, but it is possible, given that corrections for
observational selection and any assumptions about SFDs or Yarkovsky evolution
rates are not needed, that, if the Mars-crossing population is an equilibrium
population, then these calculations could give a robust estimate of the rate
of transfer of asteroids from the main belt to near-Earth space.

Figure 6: Panel (a): the variation with absolute magnitude, $H$ of the mean
proper eccentricity of the high inclination ($I>9$ $deg$), non-family
asteroids. Panel (b): histograms of the inclination distributions of all the
asteroids in the IMB with $H<16.5$ and the asteroids in the Mars-crossing zone
shown in Fig. 1e. These distributions have been normalized (taken from Dermott
et. al 2021).

Asteroids are lost from the IMB through dynamical mechanisms that include
chaotic and Yarkovsky driven orbital evolution. These mechanisms result in
asteroids diffusing into various weak mean motion resonances (Fig. 1e), and
then into the Mars-crossing zone (Morbidelli & Nesvorný, 1999; Farinella &
Vokrouhlický, 1999). A third mechanism depends on Yarkovsky forces driving
small asteroids into the $\nu_{6}$ secular resonance (Farinella et. al, 1994;
Migliorini et. al, 1998; Farinella, Vokrouhlický & Hartmann, 1998). Numerical
integrations have shown that all these mechanisms are viable. The questions
that remain are which mechanism is dominant and what is the rate of loss.
Greenberg et. al (2020) have shown from observations of the current orbital
evolution of 247 NEAs that the ratio of retrograde to prograde spin of these
objects is as high as $2.7_{-0.6}^{+0.3}$. This implies that prior to their
escape these objects were evolving towards the Sun and thus that the $\nu_{6}$
secular resonance is the dominant escape route. There are other observations
that are devoid of observational selection effects, that have yet to be
analyzed, that could also place constraints on the loss mechanisms. In Fig. 6a
we see that the mean proper eccentricity of the high eccentricity asteroids is
size-dependent, increasing with decreasing asteroid size, suggesting the
action of Yarkovsky forces. Fig. 6b shows the distribution of the inclinations
of the asteroids in the Mars crossing zone. This distribution is somewhat
similar to the distribution of the major families. There is also an excess of
asteroids with high inclinations, but it has not been determined that this
excess is strong enough to support the dominant role of the $\nu_{6}$ secular
resonance.

##  4 Vesta and the origin of the HED meteorites

The action of Yarkovsky forces has resulted in a marked net loss of small
asteroids from the IMB, implying that the remnants of many ghost families
could now be sparsely distributed among other large, background asteroids.
Progress with the detection of these dispersed families has been made by Delbó
et. al (2017, 2019). Here, we show that further evidence for the loss of small
asteroids from old families is provided by an analysis of the SFD of the Vesta
family.

One third of the $\sim$29,000 asteroids in the IMB with $H<16.5$ originate
from Vesta. This asteroid is the source of the HED meteorites and this is the
only firm asteroid-meteorite link (McCord, Adams & Johnson, 1970; McSween &
Binzel, 2022). Vesta is the largest asteroid in the IMB and has remained
intact since its formation, thus the asteroids in this large family must
originate from one or more craters. Over 25 years ago, Hubble observations
revealed an enormous crater, Rheasilvia, and this crater is thought to be the
source of the meteorites (Thomas et. al, 1997). But later the Dawn spacecraft
revealed a second enormous crater, Veneneia, partly underneath Rheasilvia
(Schenk et. al, 2012). Given that Veneneia has diameter of $\sim 400$ $km$,
its formation would probably have resulted in the formation of a large family.
Is it possible that Vesta is currently associated with two families of
asteroids, of different ages, that have sampled different surface locations
and mantle depths?

Figure 7: Panels (a) and (b) show the variations with absolute magnitude, $H$
of the mean proper $e$ and the mean proper $I$ of the asteroids in the Vesta
family. Panel (c) defines two independent regions of the Vesta family in $e-I$
space. Panel (d) shows the SFDs of the two regions shown in panel (c) after
normalization. Panel (e) shows the variation with absolute magnitude, $H$ of
the slopes to the quadratic fits of the SFDs shown in panel (d).

We have previously argued (Dermott et. al, 2018) that the merging of two
families with different mean proper orbital elements and different SFDs will
give rise to correlations between the sizes and the mean proper orbital
elements of the family members (treated as a whole) – see Fig. 4c. In Fig. 7a
we observe a strong correlation (4.7$\sigma$) between the eccentricities and
sizes of the Vesta family members. This observation, plus the asymmetric
distribution of the asteroids in $e-I$ space (Fig. 7c), suggests the current
existence of two families. However, if we isolate two different regions of the
Vesta family in $e-I$ space, as shown in Fig. 7c, we observe, after
normalization, that while the two regions have different mean orbital
elements, the SFDs of the two regions are identical. This implies that the
Vesta family must originate from one large crater and, while we believe that
the Veneneia crater must have resulted in the formation of a large asteroid
family, we can only account for the similarity of the SFDs if a majority of
the small asteroids in the older Veneneia family have been lost from the
system due to the action of Yarkovsky forces. This argument allows us to place
bounds on the age of the Rheasilvia crater and on the strength of the
Yarkovsky forces. Full details of these arguments will be published elsewhere.

##  5 Defects of the HCM and their consequences

What is the origin of the strong correlation observed between the mean
eccentricities and sizes of the Vesta family members? Here, we suggest that
this correlation may not have a physical explanation, but may be due to a
fundamental flaw in the way that the HCM defines family membership. The total
distance of the separation, $d$ used in eqn. (1) to define family membership
gives approximately equal weights to the $\Delta a/a$, $\Delta e$ and $\Delta
I$ components represented by $d_{1}$, $d_{2}$ and $d_{3}$. On the initial
formation of a family, these three components are determined by the structure
of the debris cloud released close to the crater. It is possible that the
distribution of kinetic energy between the particles in this cloud is size-
dependent (Leinhardt et. al, 2015). However, the second step in family
formation requires these particles to reaccumulate into rubble-pile asteroids
and it is not obvious that any size-energy correlation will be carried over to
the resultant rubble-piles.

Figure 8: Variations of the mean proper $e$ and the mean proper $I$ with
semimajor axis for asteroids in the Vesta family defined by Nesvorný (2015)
using the HCM. These variations are shown for two independent size ranges.

The two components of $d$ that depend on $\Delta e$ and $\Delta I$ are
comparatively stable: the changes due to Yarkovsky forces are negligible, but
we must expect some small changes due to chaotic orbital evolution. If the
changes in $\Delta e$ and $\Delta I$ are dominated by the changes that occur
while the asteroids are trapped in weak resonances, as shown by Christou,
Dermott & Li (2022), then the changes in $\Delta e$ and $\Delta I$ may be
size-dependent. However, in contrast, we can be certain that the changes in
the semimajor axes, due to Yarkovsky forces, are large and definitely size-
dependent. Thus, the final separations of the asteroids as determined by
$\Delta a/a$ are decoupled from the family formation mechanism and it is
questionable that they should be used in any criterion of family membership.
The inclusion of $\Delta a/a$ in the definition of $d$ in eqn. (1) could lead
to the criterion that separates family from non-family asteroids being size-
dependent. Evidence that this is the case is shown in Fig. 8, where we see
that there are strong correlations between the mean proper $e$ and mean proper
$I$ and the semimajor axes of the Vesta family asteroids.

Figure 9: The variation with semimajor axis of the total and the three
components of the mean minimum separation of the Vesta family asteroids in
$a-e-I$ space determined using the HCM

Figure 10: Comparison of the structure of the Vesta family as revealed by two
different methods. Panel (a) shows the major families as defined by Nesvorný
(2015) using the HCM. The contours in panel (b) show the variation of number
density, while the small cells show the variation of the mean proper
eccentricity, $e$ in $a-I$ space.

In Fig. 9, we show how the mean separations of the asteroids originating from
Vesta at $a=2.36$ $au$ increase with increasing distance from Vesta. We
conclude from this that if the criterion for family membership,
$d<d_{\textrm{crit}}$, is fixed but the mean separation of the asteroids in
$a-e-I$ space increases with increasing separation from Vesta, then the number
of asteroids classified as family members will decrease with increasing
separation from Vesta, while the number of non-family members that are halo
asteroids will increase. This is shown to be the case in Fig. 10. Panel (a) of
that figure shows the structure of the major families in $a-I$ space as
defined by Nesvorný (2015) using the HCM. We observe that the width of the
family, $\Delta I$ decreases as the semimajor axis increases to 2.5 $au$.
Panel (b) shows the variation of the mean eccentricity of the IMB asteroids in
$a-I$ space. In this plot, we have not defined the families, but we show those
features that allow us to recognize the families. The contours in panel (b)
show the variation of the number density, and by comparing the two panels we
can see that the highest number densities are associated with the Vesta family
and the Nysa-Polana-Eulalia complex. Fig. 10b also shows the variation of the
mean proper eccentricity in $a-I$ space.

Inspection of Fig. 1 shows a distinct contrast between these two major
families. While the Vesta family has a high mean inclination and a low mean
eccentricity, the opposite is the case for the Nysa-Polana-Eulalia complex. In
Fig. 10b, the band of low eccentricity asteroids with mean inclination close
to the inclination of Vesta is clearly associated with the Vesta family.
However, while the width of the Vesta family, $\Delta I$ shown in panel (a),
as determined by Nesvorný (2015) using the HCM, decreases as the semimajor
axis increases to 2.5 $au$, the opposite is true for the width of the band of
low eccentricity asteroids (i.e., the cyan-blue band between $I\sim 4.5$ $deg$
and $I\sim 8$ $deg$) shown in panel (b). The width of this low eccentricity
band increases markedly as the 3:1 Jovian resonance is approached, suggesting
that some of the asteroids in panel (a) close to $a=2.5$ $au$ that have been
classified as non-family by Nesvorný (2015) should be classified as halo
asteroids of the Vesta family. This shows a major deficiency in the ability of
the HCM to separate family asteroids from the asteroids in the associated
halo, and it is this deficiency that leads to the size, orbital element
correlations shown in Figs. 7 and 8.

As an aside, the distribution of the mean proper eccentricity in $a-I$ space
shown in Fig. 10b allows us to recognize a yellow region of high eccentricity
that could be associated with the cluster centered on $I=13.5$ $deg$ and
$e=0.18$ evident in Fig. 1c. Because this cluster does not extend the full
extent of the distance between the $\nu_{6}$ secular resonance and the 3:1
Jovian mean motion resonance, this cluster may be a recent family rather than
a ghost family.

##  6 Conclusions

The SFDs of the asteroids in the IMB are inconsistent with the equilibrium
distribution postulated by Dohnanyi (1969). The separation of the family and
halo asteroids achieved using the HCM is somewhat inadequate and this could
lead to an overestimate of the depletion of small family asteroids. However,
this is not a large concern because Fig. 5c shows that both family and non-
family are depleted in small asteroids. If this depletion applies to the main
belt as a whole, then while the SFD of the target asteroids may be well
constrained, the SFD of the smaller bullets is uncertain, and therefore we
must accept that the collisional lifetimes of the asteroids are uncertain,
that the rate of delivery of small asteroids from the inner belt to the inner
solar system is also uncertain, and that the rate of production of small
asteroids and the associated NEAs and meteorites may be a stochastic process
and time variable. This conclusion is consistent with previous arguments on
the origin of the solar system dust bands. Immediately after the discovery of
those bands by IRAS, two theories of their origin were proposed. The first
theory postulated that the dust bands were an equilibrium feature associated
with the three most prominent asteroid families: Eos. Themis and Koronis
(Dermott et. al, 1984). The second theory postulated that they were random
features associated with single collisions and not necessarily related to the
major families (Sykes & Greenberg, 1986). Dynamical modeling of the structure
of the “ten-degree band” showed that the initial mean proper inclination of
the dust particle orbits needed to account for that band is 9.3 $deg$ and
equal to the mean proper inclination of the Veritas family asteroids rather
than that of the Eos family (Grogan, Dermott & Durda, 2001; Dermott et. al,
2002), thus favoring the stochastic model.

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