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

The macroscopic model for a neutron star (NS) as a liquid drop for the equilibrium at zero temperature, $T=0$, is formulated for derivations of the equation of state (EoS) for the pressure P=P(ρ), the particle number density ρ, and the NS mass M expressions in the leptodermic approximation a/R 1, where a is the diffused NS effective surface (ES), and R is the effective NS radius. The macroscopic gravitation is taken into account in terms of the potential Φ(ρ) as a second order expansion in powers of ρ-ρ̄, where ρ̄ is the saturation density. The density distribution ρ across the NS ES in the normal direction to the ES was obtained analytically for a general form of the energy density E(ρ), in the leading order of the approximation a/R 1 for both spherical and deformed ES shapes. For a spherically symmetric NS and its typical parameters such as a crust thickness, a ~ 1 km, effective radii, R~ 10 km, and central particle-number density (saturation) value, ρ̄ ~ 2· 0.16 fm⁻³, we find the leading expression for the particle number density ρ. The surface NS mass correction MS to the volume part, MV, is found as MS ≈ - (2a/R) MV. The analytical solution to the Tolman-Oppenhamer-Volkoff (TOV) equations for the pressure P at different typical experimentally known N masses, M=(1.4-2.0) M_, is in good agreement with the corresponding numerical results. As shown from their comparison with our EoS pressure, P=P(ρ), we emphasize importance of its surface corrections PS near the NS edge.