Neutron star crust properties: comparison between the compressible liquid-drop model and the extended Thomas-Fermi approach

We present a detailed analysis of three models predicting the properties of non-uniform matter in the crust of neutron stars: the compressible liquid-drop model, the fourth order Extended Thomas Fermi (ETF) method, and ETF plus Strutinsky integral (ETFSI) correction. The former treats the nuclear clusters as uniform hard spheres, the second takes into account the density distribution which can be different for neutrons and protons, and the last one includes the proton shell effects within the Strutinsky approach. The purpose of this work is to understand the importance of the improvements in the nuclear modeling and to analyze the quantities which are the most sensitive to them. We find that thermodynamic quantities such as pressure, energy and chemical potential, as well as the electron fraction, are in very good agreement among the three models. This confirms previous results where we have shown that the improvement in the finite-size description of the nuclear clusters has a small impact on these quantities, since they are mainly constrained by the bulk properties. The refinements in the finite-size modeling are shown to impact mostly the composition of the nuclear clusters ($Z_{cl}$, $N_{cl}$) in an ordering which ranks according to the leptodermous expansion. This analysis is performed considering both the r-cluster and the e-cluster representations. The proton shell effects are shown to stabilize $Z_{cl}$, which consequently impacts the neutron number $N_{cl}$ as well.