Rotating neutron stars within the macroscopic effective-surface approximation

The macroscopic model for a neutron star (NS) as a perfect liquid drop at the equilibrium is extended to rotating systems by incorporating the linear perturbation expansion over a small frequency $\omega$ near the Schwarzschild 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 is calculated by using the Kerr metric approach in the Boyer-Lindquist and Hogan coordinates for the perfect liquid-drop model of NSs. The off-diagonal metric element is derived analytically from equations of the General Relativity Theory (GRT) and is compared with Boyer-Lindquist and Hogan expressions. The gradient surface terms of the macroscopic NS energy density $\mathcal{E}(\rho)$ [Equation of State] are taken into account along with the volume ones 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 macroscopic NS angular momentum $I$ at small frequencies $\omega$, up to quadratic terms, are specified for calculations of the adiabatic moments of inertia (MI), $\Theta= d I/ d \omega$. The analytical NS MI expressions, $\Theta=\tilde{\Theta}/(1-\mathcal{T}_{t\varphi})$, has 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 due mainly to the $t,\varphi$ correlations and surface contributions. The adiabaticity condition is carried out for many neutron stars with a strong gravity.