Ancient and primordial collisional families as the main sources of X-type asteroids of the inner main belt

Aims. The near-Earth asteroid population suggests the existence of an inner main belt source of asteroids that belongs to the spectroscopic X complex and has moderate albedos. The identification of such a source has been lacking so far. We argue that the most probable source is one or more collisional asteroid families that have escaped discovery up to now. Methods. We apply a novel method to search for asteroid families in the inner main-belt population of asteroids belonging to the X complex with moderate albedo. Instead of searching for asteroid clusters in orbital element space, which could be severely dispersed when older than some billions of years, our method looks for correlations between the orbital semimajor axis and the inverse size of asteroids. This correlation is the signature of members of collisional families that have drifted from a common centre under the effect of the Yarkovsky thermal effect. Results. We identify two previously unknown families in the inner main belt among the moderate-albedo X-complex asteroids. One of them, whose lowest numbered asteroid is (161) Athor, is ~3 Gyr old, whereas the second one, whose lowest numbered object is (689) Zita, could be as old as the solar system. Members of this latter family have orbital eccentricities and inclinations that spread them over the entire inner main belt, which is an indication that this family could be primordial, that is, it formed before the giant planet orbital instability. Conclusions. The vast majority of moderate-albedo X-complex asteroids of the inner main belt are genetically related, as they can be included into a few asteroid families. Only nine X-complex asteroids with moderate albedo of the inner main belt cannot be included in asteroid families. We suggest that these bodies formed by direct accretion of the solids in the protoplanetary disc, and are thus surviving planetesimals