The Crusts of Neutron Stars Revisited: Approximations within a Polytropic Equation of State Approach
In this work, we revisit several thin-crust approximations presented in the literature and compare them with the exact solutions of the Tolman--Oppenheimer--Volkoff (TOV) equations. In addition, we employ three different equations of state (EoSs), including one with a pasta phase, each based on a distinct theoretical framework: the variational method, relativistic Brueckner--Hartree--Fock theory, and relativistic mean-field theory. We emphasize that these approximations require only the TOV solutions for the core and the EoS properties at the core--crust interface; in our approach, only the energy density is needed. Finally, the relativistic approximation, as well as the Newtonian approximation with corrections, shows good agreement with the exact solutions. This indicates that a simple treatment of the crust is sufficient for structural purposes, independently of the uncertainties in the sub-nuclear equation of state, which are not very large. The unified EoS SINPA (relativistic mean-field theory), including the pasta phase, was used to study the thin-crust approximation, while degeneracy in the $M$--$R$ relation is demonstrated through: (i) anisotropic pressure in the modified TOV equations, (ii) the $f(R, L_m, T)$ gravity model, and (iii) dark matter admixture. As demonstrated, modifications to the description of gravitation introduce degeneracies in the mass--radius relation that are challenging to disentangle or quantify precisely.