Neutron stars as a dense liquid drop at equilibrium within the effective surface approximation

The macroscopic model is formulated for a neutron star (NS) as a perfect liquid drop at the equilibrium. We use the leptodermic approximation $a/R\ll 1$, where $a$ is the crust thickness of the effective NS surface (ES), and $R$ is the mean radius of the ES curvature. Within the approximate Schwarzschild metric solution to the general relativity theory equations for the spherically symmetric systems, the macroscopic gravitation is taken into account in terms of the total separation particle energy and incompressibility. Density distribution $\rho$ across the ES in the normal direction to the ES was obtained analytically for a general form of the energy density $\mathcal{E}(\rho)$. For the typical crust thickness, and effective radius, one finds the leading expression for the density $\rho$. NS masses are analytically calculated as a sum of the volume and surface terms, taking into account the radial curvature of the metric space, in reasonable agreement with the recently measured masses for several neutron stars. We derive the simple macroscopic equation of state (EoS) with the surface correction. The analytical and numerical solutions to Tolman-Oppenheimer-Volkoff equations for the pressure are in good agreement with the volume part of our EoS.