A theoretical model of the quasistatic electric field, formed at the rear surface of a thin solid target irradiated by a ultraintense subpicosecond laser pulse, due to the appearance of a cloud of ultrarelativistic bound electrons, is developed. It allows one to correctly describe the spatial profile of the accelerating field and to predict the maximum energies and the energy spectra of the accelerated ions. The agreement of the theoretical expectations with the experimental data looks satisfactory in a wide range of conditions. Previsions of regimes achievable in the future are given.