  Families of asteroids generated by the collisional fragmentation of a common
parent body have been identified using clustering methods of asteroids in their
proper orbital element space. An alternative method has been developed in order
to identify collisional families from the correlation between the asteroid
fragment sizes and their proper semi-major axis distance from the family centre
(V-shape). This method has been shown to be effective in the cases of the very
diffuse families that formed billions of years ago. We obtained photometric
observations of asteroids in order to construct their rotational light curves;
we combine them with the literature light curves and sparse-in-time photometry;
we input these data in the light curve inversion methods to determine the shape
and the spin pole of the asteroids in order to assess whether an object is
prograde or retrograde. The ultimate goal is to assess whether we find an
excess of retrograde asteroids on the inward side of the V-shape of a 4 Gyr
asteroid family identified via the V-shape method. This excess of retrograde
rotators is predicted by the theory of asteroid family evolution. We obtained
the spin poles for 55 asteroids claimed to belong to a 4 Gyr collisional family
of the inner main belt that consists of low-albedo asteroids. After
re-evaluating the albedo and spectroscopic information, we found that nine of
these asteroids are interlopers in the 4 Gyr family. Of the 46 remaining
asteroids, 31 are found to be retrograde and 15 prograde. We also found that
these retrograde rotators have a very low probability (1.29%) of being due to
random sampling from an underlying uniform distribution of spin poles. Our
results constitute corroborating evidence that the asteroids identified as
members of a 4 Gyr collisional family have a common origin, thus strengthening
their family membership.
