MNRAS 457, 1332–1338 (2016) doi:10.1093/mnras/stw043

Constraints on the original ejection velocity fields of asteroid families

V. Carruba1,2‹ and D. Nesvorný1‹
1Department of Space Studies, Southwest Research Institute, Boulder, CO 80302, USA
2UNESP, Univ. Estadual Paulista, Grupo de dinâmica Orbital e Planetologia, Guaratinguetá, SP 12516-410, Brazil

Accepted 2016 January 5. Received 2016 January 5; in original form 2015 November 27

ABSTRACT
Asteroid families form as a result of large-scale collisions among main belt asteroids. The
orbital distribution of fragments after a family-forming impact could inform us about their
ejection velocities. Unfortunately, however, orbits dynamically evolve by a number of effects,
including the Yarkovsky drift, chaotic diffusion, and gravitational encounters with massive
asteroids, such that it is difficult to infer the ejection velocities eons after each family’s
formation. Here, we analyse the inclination distribution of asteroid families, because proper
inclination can remain constant over long time intervals, and could help us to understand the
distribution of the component of the ejection velocity that is perpendicular to the orbital plane
(vW). From modelling the initial break up, we find that the distribution of vW of the fragments,
which manage to escape the parent body’s gravity, should be more peaked than a Gaussian
distribution (i.e. be leptokurtic) even if the initial distribution was Gaussian. We surveyed
known asteroid families for signs of a peaked distribution of vW using a statistical measure of
the distribution peakedness or flatness known as kurtosis. We identified eight families whose
vW distribution is significantly leptokurtic. These cases (e.g. the Koronis family) are located in
dynamically quiet regions of the main belt, where, presumably, the initial distribution of vW was
not modified by subsequent orbital evolution. We suggest that, in these cases, the inclination
distribution can be used to obtain interesting information about the original ejection velocity
field.

Key words: celestial mechanics – Minor planets, asteroids: general.

1 IN T RO D U C T I O N

Asteroid families form as a result of collisions between asteroids.
These events can either lead to a formation of a large crater on the
parent body, from which fragments are ejected, or catastrophically
disrupt it. More than 120 families are currently known in the main
belt (Nesvorný, Brož & Carruba 2015) and the number of their
members ranges from several thousands to just a few dozens, for
the smaller and compact families. A lot of progress has been made
in the last decades in developing sophisticated impact hydrocodes
able to reproduce the main properties of families, mainly account-
ing for their size distribution, and, in a few cases (the Karin and
Veritas clusters) the ejection velocities of their members (Michel
et al. 2015). However, while the sizes of asteroids can be either mea-
sured directly through radar observations or occultations of stars, or
inferred if the geometric albedo of the asteroid is known, correctly
assessing ejection velocities is a more demanding task. The orbital
element distribution of family members can, at least in principle, be
converted into ejection velocities from Gauss’ equations (Zappalà

�E-mail: vcarruba@feg.unesp.br (VC); davidn@boulder.swri.edu (DN)

et al. 1996), provided that both the true anomaly and the argument
of perihelion of the family parent body are known (or assumed).

Orbital elements of family members, however, are not fixed in
time, but can be changed by gravitational and non-gravitational ef-
fects, such as resonant dynamics (Morbidelli & Nesvorný 1999),
close encounters with massive asteroids (Carruba et al. 2003), and
Yarkovsky (Bottke et al. 2001) effects, etc. Separating which part of
the current distribution in proper elements may be caused by the ini-
tial velocity field and which is the consequence of later evolution is a
quite complex problem. Interested readers are referred to Vokrouh-
lický et al. (2006a,b,c) for a discussion of Monte Carlo methods
applied to the distribution of asteroid families proper semimajor
axis. Yet, insights into the distribution of the ejection velocities are
valuable for better understanding of the physics of large-scale col-
lisions (Nesvorný et al. 2006; Michel et al. 2015). They may help
to calibrate impact hydrocodes, and improve models of the internal
structure of asteroids.

Here, we analyse the inclination distribution of asteroid fami-
lies. The proper inclination is the proper element least affected by
dynamical evolution, and it could still bear signs of the original
ejection velocity field. We find that a family formed in an event in
which the ejection velocities were not much larger than the escape

C© 2016 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society

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mailto:vcarruba@feg.unesp.br
mailto:davidn@boulder.swri.edu


Ejection velocity fields of asteroid families 1333

velocity from the parent body should be characterized by a peaked
(leptokurtic) initial distribution (relative to a Gaussian), while fam-
ilies formed in hyper-velocity impacts, such as the case of the Eos
family (Vokrouhlický et al. 2006a), should have either a normal
or less peaked (platykurtic) distribution. The subsequent dynam-
ical evolution should then act to bring this initial distribution to
appear more Gaussian (or mesokurtic). The relative importance of
the subsequent evolution depends on which specific proper element
is considered, and on how active the local dynamics are. Using
the proper inclination we attempt to identify cases where the local
dynamics either did not have time or was not effective in erasing
the initial, presumably leptokurtic, distributions. These cases can be
used to better understand the conditions immediately after a parent
body disruption.

This paper is divided as follows. In Section 2, we model the
distribution of ejection velocities after a family-forming event. We
explain how a peakedness of an expected distribution can be mea-
sured by the Pearson kurtosis. Section 3 shows how dynamics can
modify the initial distribution by creating a new distribution that
is more Gaussian in shape. In Section 4, we survey the known
asteroid families, to understand in which cases the traces of the ini-
tially leptokurtic distribution can be found. Section 5 presents our
conclusions.

2 MO D E L F O R TH E I N I T I A L vW

D I S T R I BU T I O N

Proper orbital elements can be related to the components of the
velocity in infinity, vinf, along the direction of the orbital motion
(vt), in the radial direction (vr), and perpendicular to the orbital
plane (vW) through the Gauss equations (Murray & Dermott 1999):

δa

a
= 2

na(1 − e2)1/2
[(1 + e cos (f )δvt + (e sin (f ))δvr ], (1)

δe = (1 − e2)1/2

na

[
e + cos(f ) + e cos2(f )

1 + e cos(f )
δvt + sin(f )δvr

]
, (2)

δi = (1 − e2)1/2

na

cos(ω + f )

1 + e cos(f )
δvW , (3)

where δa = a − aref, δe = e − eref, δi = i − iref, and aref, eref, iref define
a reference orbit (usually the centre of an asteroid family, defined in
our work as the centre of mass in a 3D proper element orbital space,1

and f and ω are the (generally unknown) true anomaly and perihe-
lion argument of the disrupted body at the time of impact. From the
initial distribution (a, e, i), it should therefore be possible to estimate
the three velocity components, assuming one knows the values of
f and ω. This can be accomplished by inverting equations (1), (2)
and (3). With the exception of extremely young asteroid families
(e.g. the Karin cluster, Nesvorný et al. 2002, 2006), this approach
is not viable in most cases. Apart from the obvious limitation that
we do not generally know the values of f and ω, a more funda-
mental difficulty is that several gravitational (e.g. mean-motion and
secular resonances, close encounters with massive asteroids) and

1 For families formed in cratering events, the centre of mass of the asteroid
family usually coincides with the orbital position of the parent body. One
notable exception to this rule is the Hygiea family, for reasons discussed
in Carruba et al. (2014), such as possible dynamical evolution caused by
close encounters with massive bodies with 10 Hygiea itself after the family
formation.

non-gravitational (e.g. Yarkovsky and YORP) effects act to change
the proper elements on long time-scales. The Gauss equations thus
cannot be directly applied in most cases to obtain information about
the original ejection velocities.

In this work, we propose a new method to circumvent this dif-
ficulty. Of the three proper elements, the proper inclination is the
one that is the least affected by dynamics. For example, unlike the
proper semimajor axis, the proper inclination is not directly ef-
fected by the diurnal version of the Yarkovsky effect [it is affected
by the seasonal version, but this is usually a much weaker effect,
whose strength is of the order of 10 per cent of that of the diurnal
version, for typical values of asteroid spin obliquities and rotation
periods (Vokrouhlický & Farinella 1999)]. Also, unlike the proper
eccentricity, which is affected by chaotic diffusion in mean-motion
resonances, the proper inclination is more stable. In addition, the
inclination is related to a single component of the ejection veloci-
ties, vW, via equation (3). This equation can be, at least in principle,
inverted to provide information about δvW.

What kind of a probability distribution function (pdf) is to be
expected for the original values of vw? Velocities at infinity Vinf are
obtained from the ejection velocities Vej through the relationship:

Vinf =
√

V 2
ej − V 2

esc, (4)

where Vesc is the escape velocity from the parent body. Following
Vokrouhlický et al. (2006a,b,c), we assume that the ejection velocity
field follows a Gaussian distribution of zero mean and standard
deviation given by:

σVej = VEJ

(
5km

D

)
, (5)

where D is the body diameter in km, and VEJ is a parameter de-
scribing the width of the velocity field. Only objects with Vej > Vesc

succeed in escaping from the parent body. As a result, the initial
distribution of Vinf should generally be more peaked than a Gaussian
one. Fig. 1, panel A, displays the frequency distribution function fdf
for 1031 objects with 2.5 < D < 3.5 km computed for a parent body
with an escape velocity of 130 m s−1 and VEJ = 0.5Vesc. We assumed
that f = 30◦ and (ω + f) = 50◦. Since f and (ω + f) appear only
as multiplying factors in the expression for vW, different choices of
these parameters do not affect the shape of the distribution. One can
notice how the vW distribution is indeed more peaked than that of
the Gaussian distribution with the same standard deviation (shown
in red in Fig. 1).

A useful parameter to understand if a given distribution is (or
is not) normally distributed is the kurtosis (Carruba et al. 2013b).
Pearson kurtosis (Pearson 1929), defined as γ2 = μ4

σ 4 − 3, where
µ4 is the fourth moment about the mean and σ is the standard
deviation, gives a measure of the ‘peakedness’ of the distribution.
The Gaussian distribution has γ 2 = 0 and is the most commonly
known example of a mesokurtic distribution. Larger values of γ 2

are associated with leptokurtic distributions, which have longer tails
and more acute central peaks. The opposite case, with γ 2 < 0, is
known as platykurtic.

We generated values of vW for various values of VEJ for ten
thousand 5 km bodies originating from a parent body with Vesc =
130 m s−1 (results can be re-scaled for any other body size using
equation (5), and a different value of D). Fig. 2 shows how the kur-
tosis value of the vW distribution changes as a function of VEJ/Vesc.
As expected, for smaller values of VEJ/Vesc the vW distribution is
more peaked, and the kurtosis values are larger. For VEJ/Vesc > 1,
the kurtosis value approaches 0, because the influence of parent

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1334 V. Carruba and D. Nesvorný

Figure 1. The vW distribution for a simulated family. Panel A shows the distribution for objects with diameters 2.5 < D < 3.5 km. Panel B shows the results
for 2.0 < D < 8.0 km. The size distribution was set to be N dN = CD−α dN with α = 3.6. We also set VEJ/Vesc = 0.5. A Gaussian distribution with the same
standard deviation of the vW distribution is shown in both panels for reference.

Figure 2. Dependence of the kurtosis γ 2 of the vW distribution on VEJ/Vesc

(solid blue line). The scaling of the tick marks on the x-axis are proportional
to the log10VEJ/Vesc. The blue dashed line is the second order polynomial
that best-fitted the data. The red line is γ 2 = 0, corresponding to a mesokurtic
distribution.

body’s gravity diminishes. Most families for which an estimate of
VEJ is available (see Nesvorný et al. 2015, Section 5 and references
within) have estimated values of VEJ/Vesc in the range from 0.4 to
1.2, and should therefore have been significantly leptokurtic when
they formed.

What typical values of γ 2 one would expect for a real asteroid
family, where different sizes of fragments are considered together?
As the large objects typically have lower values of vW, this implies
that they would contribute to the peak of the distribution, and when
considered with smaller fragments, the whole distribution should
correspond to a larger value of γ 2. To illustrate this effect, we
simulated a family using a size distribution N dN = CD−α dN with
α = 3.6, and VEJ/Vesc = 0.5. The computed value of γ 2 for bodies
in the size range from 2 to 8 km was found to be 0.96 (see Fig. 1,
panels B), while the one for a restricted range 2.5 < D < 3.5 km
was 0.21 (Fig. 1, panel A). This shows that one must be careful
when interpreting the real families, where the escape velocity and
size distribution effects can combine together to produce larger

values of γ 2. To isolate the escape velocity effect, it is best to use a
restricted range of sizes.

Finally, we compare our simple model for the vW distribution with
the results of impact simulations. We have taken these results from
Nesvorný et al. (2006), where a Smooth Particle Hydrodynamic
(SPH) code was used to model the formation of the Karin family. In
this case, the parent body was assumed to have D = 33 km, which
gives VESC � 35 m s−1. Different impact conditions (impact speed,
impact angle, projectile size, etc.) were studied in this work. The
dynamical evolution of fragments and their re-accretion following
the initial impact was followed by the PKDGRAV code.

Using these simulations, we extracted the values of vW for the
escaping fragments and estimated the value of VEJ from equation (4).
In a specific case that produced the best fit to the observed size
distribution of the Karin family, we obtained VEJ/Vesc = 0.2 for
fragments with 1 < D < 5 km. Finally, we found that the vW

distribution has γ 2 = 0.5, and is therefore significantly peaked. This
is in very good agreement with results from our simple method to
generate fragments (see above), which for the same size range and
VEJ/Vesc gives γ 2 = 0.53. This suggests, again, that the initial vW

distribution of asteroid families should be leptokurtic.

3 E F F E C T S O F L O N G - T E R M DY NA M I C S

Now that we have some expectations for the γ 2 values just after
a family formation, we turn our attention to γ 2 for real asteroid
families, where the inferred vW values may have been affected by
the long-term dynamics. We use the Koronis family to illustrate this
effect. The Koronis family is located at low eccentricities and low
inclinations, in a region that is relatively dynamically quiet (at least
in what concerns the proper inclination; Bottke et al. 2001).

Bottke et al. (2001) simulated the long-term dynamical evolution
of the Koronis family. Their simulation included the gravitational
effects of four outer planets and the Yarkovsky effect. Different
sizes of simulated family members were considered. We opted to
illustrate here the case with D = 2 km, for which Bottke et al.
(2001) numerically integrated the orbits of 210 bodies over 450 Myr.
The initial distribution of family members in Bottke et al. (2001)
was obtained using the same method as described in Section 2.
We computed the values of vw by inverting equation (3). Since
Pearson’s kurtosis is dependent on the presence of outliers, we
eliminated from our distributions objects with values more than 4σ

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Ejection velocity fields of asteroid families 1335

Figure 3. The time evolution of the kurtosis for a simulated Koronis family
(Bottke et al. 2001). As described in the main text, we computed γ 2 of the
vW component and D = 2 km bodies (black line). The blue line shows the
mean value over the 450 Myr integration span, and the red lines are one
standard deviation from the mean.

away from the mean. This allowed us to avoid possible distortions
in the computed γ 2 values caused by a few distant objects.

Fig. 3 shows that the initially leptokurtic distribution tends to
become more mesokurtic with time. This can be understood as fol-
lows. In statistics, the central limit theorem states that the averages
of random variables drawn from uncorrelated distributions are nor-
mally distributed. If the dynamical effects produce changes of iP

compatible with the central limit theorem, one would expect that,
with time, the distributions of vw should indeed become more and
more mesokurtic, as the contribution from dynamics increases.

To illustrate things in the case of the (real) Koronis family, we
computed the value of γ 2 for all its 5118 members taken from
Nesvorný et al. (2015), and for members with 4.5 < D < 5.5 km.
We obtained γ 2 = 0.993 and 0.712, respectively. Since both these
values are considerably leptokurtic, we believe that the dynamical
evolution of the Koronis family did not had enough time to alter
the initial distribution of inclinations. Therefore, in the specific
case of the Koronis family, we may still be observing traces of the
primordial distribution of vW.

4 K U RTO S I S O F TH E R E A L A S T E RO I D
FA MILIES

We selected the families identified in Nesvorný et al. (2015) that:
(i) have at least 100 members with 2 < D < 4 km, and (ii) have a
reasonably well defined age estimate.2 The first selection criterion is
required to have a reasonable statistics for the γ 2(vW) computation.
If we assume that the error is proportional to the square root of
the number of objects, a sample of 100 objects would give us an
uncertainty in γ 2(vW) of 10 per cent. We use a restricted size range,

2 Families were determined in Nesvorný et al. 2015 using the Hierarchi-
cal Clustering Method (HCM) in the (a, e, sin(i)) proper element domain.
This method may not identify peripheral regions of some large families, the
so-called halo of Brož & Morbidelli (2013), as belonging to the dynami-
cal group. For families such as the Eos and Koronis groups, results from
Nesvorný et al. (2015) should be considered as conservative estimates.

2 < D < 4 km, to avoid the influence of the size-dependent velocity
effects on the kurtosis. The age estimate is useful to have some
input for a physical interpretation of our results. Also, since γ 2(vW)
is sensitive to the presence of outliers in the distribution, we use
the Jarque–Bera statistical test (jbtest hereafter) to confirm that the
vW distribution differs (or not) from a Gaussian one. We choose to
work with this specific test, instead of others, because the jbtest is
particularly sensitive to whether sample data have the skewness and
kurtosis matching a normal distribution (Jarque & Bera 1987). The
test, implemented on MATrix LABoratory (MATLAB), provides a p
parameter stating the probability that a given vW distribution follows
(or not) the Gaussian pdf. The null hypothesis level is usually set at
5 per cent. Finally, we also checked that the vW distribution was not
too asymmetrical, and verified that its skewness, the parameter that
measures a possible asymmetry, was in the range between −0.25
and 0.25. A skewness value outside this range would indicate an
asymmetric ejection velocity field, or some dynamical effects that
are beyond the scope of this study.

Of the 122 families listed in Nesvorný et al. (2015), 48 cases
satisfied these requirements. At this stage of our analysis we do
not eliminate interlopers and do not consider in detail the local
dynamics. Also, families with haloes may have a further spread out
distribution in inclination than what accounted for by HCM, and
therefore larger values of γ 2(vW). For these families, our results
should be considered as conservative. Values of γ 2(vW) may be
affected by these factors, but first we would like to see what families
may be more interesting in terms of a simple first-order criteria.
Table 1 reports the values of γ 2(vW) for the whole family (3rd
column), for 2.0 < D < 4.0 km (D3) members (4th column), and
the pjbtest coefficient of the jbtest for the D3 population. Note that
the maximum value of pjbtest is 50.0 per cent.

Following the notation of Nesvorný et al. (2015), the first two
columns in Table 1 report the Family Identification Number (FIN)
and the family name. Finally, the sixth column displays the family
age estimate, with its error. The age estimate was obtained using
equation (1) in Nesvorný et al. (2015), and values of the C0 pa-
rameter, its error [also estimated from equation (1) and the value
of δC0 from Nesvorný et al. 20153], and geometric albedos from
table 2 of that paper. Densities were taken from Carry (2012) for the
namesake body of each family, when available. If not, a standard
value of 1.2 g cm−3 for C-complex families and 2.5 g cm−3 for the
S-complex families was used. In a few cases, for which a better age
estimate was available in the literature, we used these latter values.
This includes the following cases (labeled by † in Table 1): Karin
(Nesvorný et al. 2002), Eos (Vokrouhlický et al. 2006b), Hygiea
(Carruba et al. 2013a), and Koronis, Themis, Meliboea, and Ursula
(Carruba et al. 2015). Moreover, the families with values of skew-
ness not in the range from –0.25 to 0.25 are identified by a ‘(s)’
after family’s name in Table 1.

To select families whose vW component may have not changed
significantly with time, we adopted the following criteria: γ 2(vW)
for the D3 population has to be larger than 0.25, the estimated error
on values of γ 2(vW), skewness in the range from –0.25 and 0.25,
and pjbtest has to be lower than 5 per cent. After this pre-selection
was carried out, we were left with a sample of nine candidates This
is much higher than the two families one would expect to randomly

3 The error on family ages should be considered as nominal, since they do
not account for the uncertainties on values of fundamental parameters such
as the asteroids bulk densities and thermal conductivities (Masiero et al.
2012).

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1336 V. Carruba and D. Nesvorný

Table 1. Values of γ 2(vW) of the whole family (3rd column), the 2.0 <

D < 4.0 km (D3) members (4th column), the p coefficient of the jbtest
(5th column), and estimated family age with its error (6th column) for the
families in Nesvorný et al. (2015) that satisfy our selection criteria. Families
with values of skewness not in the range from −0.25 to 0.25 are identified
by a ‘(s)’ after the family’s name. Daggers identify the cases for which a
more refined age estimate was available in the literature.

FIN Family γ 2(vW) γ 2(vW) pjbtest Age
name All D3 (per cent) [Myr]

003 434 Hungaria (s) 0.42 0.21 0.5 135 ± 13
401 4 Vesta −0.42 −0.27 0.1 1440 ± 720
402 8 Flora −0.59 −0.60 0.1 4360 ± 2180
403 298 Baptistina −0.03 −0.16 0.1 110 ± 10
404 20 Massalia 0.26 1.07 0.1 320 ± 20
405 44 Nysa (s) 0.38 0.74 0.1 1660 ± 80
406 163 Erigone −0.32 −0.42 0.8 190 ± 25
408 752 Sulamitis (s) 1.28 1.56 0.1 320 ± 30
413 84 Klio (s) 0.12 0.19 2.5 960 ± 250
701 25 Phocaea −0.41 −0.40 1.4 1460 ± 1460
501 3 Juno (s) 1.05 1.05 0.1 740 ± 150
502 15 Eunomia −0.42 −0.44 0.8 3200 ± 2240
504 128 Nemesis 0.19 0.20 3.5 440 ± 20
505 145 Adeona 0.20 0.05 19.2 620 ± 190
506 170 Maria (s) −0.17 −0.38 0.1 1950 ± 1950
507 363 Padua −0.28 −0.18 4.8 410 ± 80
510 569 Misa 0.14 0.02 20.3 700 ± 140
511 606 Brangane (s) 0.48 −0.48 3.2 60 ± 5
512 668 Dora 0.16 0.28 1.0 1190 ± 600
513 808 Merxia 0.24 0.34 9.9 340 ± 30
514 847 Agnia (s) 0.47 0.24 0.3 200 ± 10
515 1128 Astrid (s) 3.69 4.83 0.1 140 ± 10
516 1272 Gefion 0.36 0.16 11.9 980 ± 290
517 3815 Konig 0.94 0.48 27.0 70 ± 10
518 1644 Rafita 0.63 0.72 0.3 480 ± 100
519 1726 Hoffmeister 2.21 1.82 0.1 270 ± 10
533 1668 Hanna 0.15 0.15 19.3 240 ± 20
535 2732 Witt −0.22 −0.21 50.0 790 ± 200
536 2344 Xizang (s) 2.47 1.87 0.1 220 ± 20
539 369 Aeria (s) −0.56 −0.21 1.6 180 ± 20
802 148 Gallia 2.02 3.39 0.1 650 ± 60
803 480 Hansa 0.81 1.17 0.1 2430 ± 600
804 686 Gersuind −0.12 −0.64 5.3 800 ± 80
805 945 Barcelona 1.48 1.32 0.5 250 ± 10
807 4203 Brucato 0.29 0.01 50.0 480 ± 100
601 10 Hygiea 0.58 0.29 0.1 2420 ± 580(†)
602 24 Themis 0.03 0.08 0.1 1500 ± 320(†)
605 158 Koronis 0.99 1.06 0.1 2360 ± 490(†)
606 221 Eos −0.60 −0.60 0.1 1300 ± 200(†)
607 283 Emma (s) 0.17 −0.69 0.8 400 ± 40
608 293 Brasilia −0.14 −0.45 17.7 160 ± 10
609 490 Veritas −0.16 −0.32 5.4 1820 ± 180
610 832 Karin (s) −0.29 0.89 1.0 5.8 ± 0.2(†)
611 845 Naema 0.10 −0.65 19.2 210 ± 10
612 1400 Tirela −0.14 −0.35 21.9 1980 ± 500
613 3556 Lixiaohua 1.30 −0.38 50.0 430 ± 20
631 375 Ursula (s) 0.06 0.70 4.3 1060 ± 60(†)
901 31 Euphrosyne (s) 1.97 1.02 0.4 1380 ± 70

fulfill this criteria, based on the number of families in our sam-
ple, and this suggests that a real effect might actually be observed
in our data. Ordered by a decreasing value of γ 2(vW(D3)), these
eight candidates are the Gallia, Hoffmeister, Barcelona, Hansa,
Massalia, Koronis, Rafita, Hygiea and Dora families. We discarded
the case of the Hoffmeister family, since this family is significantly

affected by the ν1C = g − gC resonance with Ceres (Novaković
et al. 2015).

Fig. 4 shows the distributions (blue line) of the vW values for D3

members of the Gallia, Barcelona, Massalia, and Koronis families,
four of our selected families with leptokurtic distributions of vW.
As can be observed from this figure, these families have vW more
peaked than a Gaussian, and this is particularly evident for the case
of the Gallia family. Fig. 5 displays values of γ 2(vW(D3)) versus the
estimated ages of the eight families satisfying our selection criteria.
One can notice that there is not a clear correlation between family
age and γ 2(vW(D3)). While there are four families younger than
700 Myr, three families (Hansa, Hygiea, and Rafita) are estimated
to be older than 2 Gyr. What these eight families have in common is
that they are located in orbital regions not very affected by dynam-
ical mechanisms capable of changing proper i. Gallia, Barcelona,
and Hansa are families in stable islands at high inclinations, all in
regions with not many mean-motion resonances (Carruba 2010).
As previously discussed, Koronis is located in a region at low incli-
nation relatively quiet in terms of dynamics, and the same can be
said for Hygiea (Carruba et al. 2013a), and, possibly, for the Rafita
and Massalia families (the latter also a relatively young family,
Vokrouhlický et al. 2006b).

Finally, among the families with large values of both skewness
and kurtosis, the cases of the Karin and Astrid families are of
a particular interest to us. Karin is a very young family, already
identified as such in Nesvorný et al. (2002, 2006), while Astrid is a
family located in a dynamically quiet region. Nesvorný et al. (2006)
showed that the velocity field of Karin was probably asymmetrical,
so the relatively high value of the skewness that we found for
the Karin cluster (−0.60) should not be surprising. Astrid has the
highest observed value of γ 2(vW(D3)). This family, however, also
interacts with a secular resonance that produces a large dispersion
of the inclination distribution. If the resonant population of objects
is removed, the value of γ 2(vW(D3)) drops to 0.3, with a skewness
within the acceptable range. The still large value of γ 2(vW(D3)) of
this revised Astrid group makes it a very good candidate for a family
with a partially pristine vW velocity field.

5 C O N C L U S I O N S

The main results of this work can be summarized as follows.

(i) We developed a model for how the original field of velocities
at infinity should appear for newly created families. Families with a
low vEJ/vesc ratio, where vEJ is a parameter describing the standard
deviation of ejection velocity field, assumed normal (equation 5),
and vesc is the escape velocity from the parent body, should have
values of velocities at infinity following a peaked distribution. This
distribution is characterized by positive values of kurtosis γ 2 (lep-
tokurtic distribution). This is confirmed by SPH simulations for the
Karin cluster, that show a leptokurtic distribution of vW values, the
component of the velocity of infinity perpendicular to the orbital
plane. Since the proper inclination is less affected by dynamical
evolution than proper semimajor axis and eccentricity, and since vW

can be obtained from the distribution in proper i (equation 3), we
decided to concentrate our analysis on the distribution of vW values.
We studied the influence that dynamics has on the distribution of vW

values. An initially peaked vW distributions tend to become more
normal with time.

(ii) We computed values of γ 2(vW(D3)), the kurtosis of the distri-
bution of vW values for objects with 2 < D < 4 (D3), for all families
listed in Nesvorný et al. (2015) that (i) have an age estimate, (ii) have

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Ejection velocity fields of asteroid families 1337

Figure 4. The vW distributions (blue line) of D3 members of the Gallia (panel A), Barcelona (panel B), Massalia (panel C), and Koronis families (panel D).
The red line displays the Gaussian distribution with the same standard deviation of the vW distribution and zero mean, normalized to the maximum value of
the frequency distribution of each family.

Figure 5. γ 2(vW(D3)) versus the estimated age of eight families satisfying
our selection criteria. The horizontal blue lines display the estimated error
on the ages, while the vertical ones show the nominal errors on γ 2(vW(D3)),
assumed equal to 0.1.

a D3 population of at least 100 objects, and (iii) are not following
a Gaussian distribution, according to the Jarque–Bera test. We also
required the families to have values of skewness, the parameter iden-
tifying possible asymmetries in the vW distribution, between −0.25
and 0.25. Eight families, the Gallia, Barcelona, Hansa, Massalia,
Koronis, Rafita, Hygiea and Dora families, satisfied these criteria
and have values of γ 2(vW(D3)) larger than 0.25, which suggests that
they could still bear traces of the original values of vW. Four of these

families are younger than 700 Myr. All of these groups appear to be
located in regions not strongly affected by dynamical mechanisms
able to modify proper inclination. Among asymmetrical families,
the Karin and Astrid groups stand out as interesting cases.

Overall, while a more in depth study of the families selected in
this work, accounting for a better analysis of local dynamics and of
the role of possible interlopers, should be performed, our analysis
already allowed us to select several asteroid families that could still
bear traces of the original distribution of velocities at infinity. While
to select families that could still bear traces of the original values of
vW we concentrated on the shape of the vW distribution, obtaining
better estimates of the magnitude of vW for these selected families
could much improve our knowledge of the physical mechanisms at
work in the family-forming event, and serve as a useful constraints
for simulations describing cratering and catastrophic destruction
events, other than the observed Size Frequency Distribution (Durda
et al. 2007).

AC K N OW L E D G E M E N T S

We thank an anonymous reviewer for comments and suggestions
that significantly improved the quality of this paper. This paper was
written while the first author was a visiting scientist at the Southwest
Research Institute (SwRI) in Boulder, CO, USA. We would like to
thank the São Paulo State Science Foundation (FAPESP) that sup-
ported this work via the grant 14/24071-7, the Brazilian National
Research Council (CNPq, grant 305453/2011-4), and the National
Science Foundation (NSF). The authors are grateful to W. F. Bottke
for allowing them to use the results of the Koronis family simula-
tions, to S. Aljbaae for revising earlier version of this manuscript,

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1338 V. Carruba and D. Nesvorný

and to D. Durda for comments and suggestions that improved the
quality of this work.

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