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Though generally agreed that the symmetry energy plays a dramatic role indetermining the structure of neutron stars and the evolution of core-collapsingsupernovae, little is known in what concerns its value away from normal nuclearmatter density and, even more important, the correct definition of thisquantity in the case of unhomogeneous matter. Indeed, nuclear mattertraditionally addressed by mean-field models is uniform while clusters areknown to exist in the dilute baryonic matter which constitutes the maincomponent of compact objects outer shells. In the present work we investigatethe meaning of symmetry energy in the case of clusterized systems and thesensitivity of the proto-neutron star composition and equation of state to theeffective interaction. To this aim an improved Nuclear Statistical Equilibrium(NSE) model is developed, where the same effective interaction is consistentlyused to determine the clusters and unbound particles energy functionals in theself-consistent mean-field approximation. In the same framework, in-mediummodifications to the cluster energies due to the presence of the nuclear gasare evaluated. We show that the excluded volume effect does not exhaust thein-medium effects and an extra isospin and density dependent energy shift hasto be considered to consistently determine the composition of subsaturationstellar matter. The symmetry energy of diluted matter is seen to depend on theisovector properties of the effective interaction, but its behavior withdensity and its quantitative value are strongly modified by clusterization.

Clusterized nuclear matter in the (proto-)neutron star crust and the symmetry energy