A study of 3-dimensional shapes of asteroid families with an application to Eos

In order to fully understand the shapes of asteroids families in the 3-dimensional space of the proper elements $(a_{\rm p}, e_{\rm p}, \sin I_{\rm p})$ it is necessary to compare observed asteroids with N-body simulations. To this point, we describe a rigorous yet simple method which allows for a selection of the observed asteroids, assures the same size-frequency distribution of synthetic asteroids, accounts for a background population, and computes a $\chi^2$ metric. We study the Eos family as an example, and we are able to fully explain its non-isotropic features, including the distribution of pole latitudes $\beta$. We confirm its age $t = (1.3\pm0.3)\,{\rm Gyr}$; while this value still scales with the bulk density, it is verified by a Monte-Carlo collisional model. The method can be applied to other populous families (Flora, Eunomia, Hygiea , Koronis, Themis, Vesta, etc.).