Rotating neutron stars within the macroscopic effective-surface approximation

The macroscopic model for a neutron star (NS) as a finite perfect fluid at the equilibrium is extended to rotating systems by incorporating the linear perturbation expansion over a small frequency $\omega$ near Schwarzschild outer-inner gravitational metric within the effective-surface (ES) approach. The NS angular momentum $I$ and moment of inertia (MI) for a slow stationary azimuthal rotation around the symmetry axis are calculated by using the Kerr metric approach in spherical coordinates, and compared with Boyer-Lindquist (outer) and Hogan (inner) metric results. The volume and gradient-surface terms of the macroscopic NS energy density $\mathcal{E}(\rho)$ (Equation of State) are taken into account at the leading order of the leptodermic parameter $a/R \ll 1$, where $a$ is the ES crust thickness and $R$ is the NS effective radius. The analytical macroscopic NS MI expressions, $\Theta = \mathrm{d}I/\mathrm{d}\omega = \tilde{\Theta}/(1-\mathcal{T}_{t\varphi})$, have been obtained in terms of the statistically averaged MI, $\tilde{\Theta}$, and its time and azimuthal-angle $t,\varphi$ correlation, $\mathcal{T}_{t\varphi}$, as sums of the volume and surface components. The MI $\Theta$ is changed significantly as function of the effective radius $R$ because of a strong gravity. We found the additional constraint for the NS radius to smaller accessible ranges which is due mainly to the $t,\varphi$ correlations and surface contributions. The adiabaticity conditions for applicability of the linear perturbation theory is carried out for several neutron stars with a strong gravity and relatively large rotation periods.