## ON THE AGES OF ASTEROID FAMILIES

In our works Paper I (Milani et al. 2014)
and Paper II (Knežević et
al. 2014) we have introduced a new methods to classify
asteroids into families, applicable to an extremely large
dataset of proper elements, to update continuously this
classification, and to estimate the collisional ages of
large families. In later
works, Paper III (Spoto et al. 2015)
and Paper IV (Milani et al. 2016), we
have systematically applied an uniform method (an
improvement of that proposed in Paper I) to estimate
asteroid family collisional ages, and solved a number of
problems of collisional models, including cases of complex
relationship between **dynamical families** (identified
by clustering in the proper elements space)
and **collisional families** (formed at a single time
of collision). In
our latest paper (arXiv preprint), we
solve several difficult cases of families for which either
a collisional model had not been obtained, e.g., because
it was not clear how many separate collisions were needed
to form a given dynamical family, or because our method
based on V-shapes (in the plane with coordinates proper
semimajor axis a and inverse of diameter 1/D) did not
appear to work properly.

The figure shows the **chronology of the asteroid
families**; the grouping on the horizontal axis corresponds
to fragmentation families, cratering families, young
families and families with one-sided V-shape (be they
cratering of fragmentation).

## THE METHOD

In Paper III, we have computed the ages of **37**
collisional families. The members of these collisional
families belong to **34** dynamical families, including
30 of those with more than 250 members. Moreover, we have
computed uncertainties based on a well defined error
model.

The computation of the family ages can be performed by
using the V-shape plots in the proper *a*
- *1/D* plane. The key idea is to compute the
diameter D from the absolute magnitude, assuming a common
geometric albedo. The geometric albedo is the average value
of the known WISE albedos for the asteroids in the
family. Then we use the least squares method to fit the
data with two straight lines, one for the low proper a (IN
side) and the other for the high proper a (OUT side), with
an outlier rejection procedure.

The method we use to convert the inverse slopes from
the V-shape fit into family ages consists in finding a
Yarkovsky calibration, which is the value of the Yarkovsky
driven secular drift *da/dt* for an hypothetical
family member of size *D* = *1 km* and with
spin axis obliquity 0° for the OUT side and 180°
for the IN side. Since the inverse slope is the change
accumulated over the family age by a family member with
unit *1/D*, the age is this change divided by the Yarkovsky
calibration.

It is a fact that the V-shape method, when applied without the prejudice that each dynamical family (found as density contrast in the space of proper elements) must correspond to one and only one collisional family (a single originating collision, with a single age), results in many case with two ages.

In some cases a **W-shape** is actually visible ,
and all 4 sides are used in the fit for slopes (see the
V-shape of
569 Misa, or
the case of
847 Agnia with the 3395 Jitka subfamily). In other
cases some of the sides are either partially or totally
obliterated by the superposition of the substructure.

The OUT side of the family of 15 Eunomia at higher a corresponds to a much younger age that the IN side at lower a, thus the difference in age is statistically very significant (see the table of the slopes for the ratio of the inverse slopes). Nevertheless, only 2 slopes have been fit, although the OUT one is based essentially only on data points with proper a>2.67 au. For 2.62 < a < 2.67 it would be possible to fit a third slope which can be interpreted as the OUT slope for the family with older age, and would be consistent with the age from the IN slope. The fourth side of the W, the IN side for the younger family, is obliterated by the superposition of the two V-shapes.

## MAIN RESULTS

Among the first 34 dynamical families for which we have computed the ages in Paper III, we find:

**3**cases in which a dynamical family corresponds to at least 2 collisional ones:**4 Vesta**,**(15) Eunomia**, and**1521 Seinajoki**.**2**examples of two separate dynamical families together forming a single V-shape:**101955 Harig**and**19466 Darcydiegel**,**163 Erigone**and**5026 Martes**. In these cases we use the definition of**family joint**.**2**cases of families containing a conspicuous**subfamily**, with a sharp number density contrast, such that it is possible to measure the slope of a distinct V-shape for the subfamily, thus the age of the secondary collision: the subfamily**3395 Jitka**of**847 Agnia**and**15124 2000 EZ39**of**569 Misa**.**2**cases in which the parent body is an interloper in his dynamical families: we are going to speak of the family**1272 Gefion**instead of**93 Minerva**, and of the family**1521 Seinajoki**instead of**293 Brasilia**. Both are obtained by removing the interlopers selected because of the albedo data, and the namesake is the lowest numbered after removing the interlopers

### LIST OF DYNAMICAL FAMILIES INCLUDING MULTIOPPOSITION ASTEROIDS

In Paper IV we present a new and larger classification, upgraded by using a proper elements catalog with more than 500.000 asteroids, numbered and multi-oppositions.

- We simplify the classification, by decreasing the number of families: indeed, 7 small/tiny families (with less than 100 members) and 2 medium families (173 and 19466) have been merged with larger ones. 1 tiny family (less than 30 members) has been removed because its lack of growth suggests that it should be a statical fluke.
- The increase in family members is an important improvement.
- We have been able to compute
**6 new ages.****3**fragmentations:**(221) Eos**,**1040 Klumpkea**, and**1303 Luthera**,**1**young:**302 Clarissa**, and**2**one-sided:**650 Amalasuntha**and**752 Sulamitis**.

### RESONANT, ERODED AND FOSSIL ASTEROID FAMILIES

We attempt to give a collisional model to a number of families for which the same attempt had previously failed. Most of these families were either locked in resonances or anyway significantly affected by resonances, both mean motion and secular. To estimate an age for the family required in each of these resonant cases we apply a specific calibration for the Yarkovsky effect, which in principle could be different in each case.

- In the Hilda region
the Yarkovsky effect results in a secular change in
eccentricity, thus the V-shape technique had to be
applied in the
*(e,1/D)*plane. We find family**(1911) Schubart**with a good age determination, and family**(153) Hilda**of the eroded type. - For the Trojans we
presents a new
classification which identifies a number of
families by using synthetic proper elements and a full
HCM method. Numerical calibration have shown that
the
**Yarkovsky perturbations are ineffective in determining secular changes in all proper elements**. Thus all Trojan families are fossil families, frozen with the original field of relative velocities. We find no way to estimate the ages of the families. - Of the 25 families
with more than 1000 members, we have computed at
least
**one age**for all but 490, which has a too recent age, already known and unsuitable for out method. Of the 19 families with 300 < N < 1000 members, excluding 778 which has a too recent age, already known,**there is only one case left without age**, namely 179. On the 24 families with 100 < N < 300 we have computed**6**ages, the others we believe could only give low reliability results.

## TABLES, V-SHAPES AND HISTOGRAMS

The numerical data with the computation of ages are collected in the tables. Tables and Figures are partitioned into sections for families of type fragmentation, of type cratering, of type young and with one side only.### TABLES

**Fit region**: family
number and name, explanation of the choice, minimum value
of proper *a*, minimum value of the diameter
selected for the inner and the outer
side. xls
and pdf format.
**Fit region**: family
number and name, explanation of the choice, minimum value
of proper *e*, minimum value of the diameter
selected for the inner and the outer
side. xls
and pdf format.

**Family albedos**:
family number and name, albedo of the parent body with
standard deviation and code of reference, maximum and
minimum value for computing mean, mean and standard
deviation of the
albedo. xls
and pdf format.

**Slopes of the
V-shapes**: family number and name, side, slope
(S), inverse slope (1/S), standard deviation of the
inverse slope, ration OUT/IN of 1/S, and standard
deviation of the
ratio. xls
and pdf format.
**Slopes of the
V-shapes** for the families in the 3/2 resonance:
family number and name, side, slope (S) in
the *(e,1/D)* plane, inverse slope (1/S), standard
deviation of the inverse slope, ration OUT/IN of 1/S, and
standard deviation of the
ratio. xls
and pdf format.

**Data for the Yarkovsky
calibration**: family number and name, proper
semimajor axis *a* and eccentricity *e* for
the inner and outer side, 1-A, density value at *1*
km, taxonomic type, a flag with values m (measured), a
(assumed) and g (guessed), and the relative standard
deviation of the
calibration. xls
and pdf format.

**Age estimation**:
family number and name, *da/dt*, age estimation,
uncertainty of the age due to the fit, uncertainty of the
age due to the calibration, and total uncertainty of the
age estimation. xls
and pdf format.
**Age estimation** for
the families in the 3/2 resonance: family number and
name, *de/dt*, age estimation, uncertainty of the
age due to the fit, uncertainty of the age due to the
calibration, and total uncertainty of the age
estimation. xls
and pdf format.