Rotating neutron stars within the macroscopic effective-surface approximation

Abstract

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 $ω$ 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 $\mathcal{E}(ρ)$ [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 $ω$, up to quadratic terms, are specified for calculations of the adiabatic moments of inertia (MI), $Θ= d I/ d ω$. The analytical NS MI expressions, $Θ=\tildeΘ/(1-\mathcal{T}_{t\varphi})$, has been obtained in terms of the statistically averaged MI, $\tildeΘ$, and its time and azimuthal-angle $t,\varphi$ correlation, $\mathcal{T}_{t\varphi}$, as sums of the volume and surface components. The MI $Θ$ 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.

Figures (12)

• Figure 1: Numerical inner solutions of the off-diagonal GRT equation (\ref{['14tauin']}) for the key quantity $\tau$ of Kerr metric in the linear perturbation approach are shown as functions of the radial variable $r$ in units of the Schwarzschild radius $R_{\rm S}$, Eq. (\ref{['rgRS']}), $x=r/R_{\rm S}$, $r\leq R$, for several values of parameter $R/R_{\rm S}$ by different lines (logarithmic scales for both axes). Arrows show the end values of $R/R_{\rm S}=0.1$ and $0.5$. For the same arbitrary initial conditions, $\tau(0)=0$ and $\tau^\prime(0)=0$, all $\tau(x)$ are calculated in terms of the "Mathematica InterpolatingFunction" solution of Eq. (\ref{['14tauin']}) for $\tau(x)$.
• Figure 2: Volume contributions to the moment of inertia $W_1$, Eq. (\ref{['I1tV']}) (green dotted), and $\mathcal{T}_{V}\equiv W_2(\xi)$, Eq. (\ref{['I2tV']}) for $W_2$ (red dashed), and full volume MI $W_1/(1-W_2)$, Eq. (\ref{['MI']}) and its several approximations: the numerical red dotted "1a", Eq. (\ref{['IntI2']}), its analytical approximation "1" in terms of the Gauss hypergeometric functions [see black solid, Eq. (\ref{['q2expJp']})], and two asymptotic approximations, "1b" blue dash dotted, Eq. (\ref{['q2exp']}); and "1c" thin solid magenta, Eq. (\ref{['q2as6exp2']}), as functions of the dimensionless variable, $R/R_{\rm S}$, and $c^{}_1$ is given by Eq. (\ref{['c1']}). Heavy dashed cyan line displays the Hogan approach for MI $W_1/(1-W_2)$ from Ref. AM25npae. The asymptotes are shown by the vertical lines.
• Figure 3: Masses $M$, Eq. (\ref{['MNStot']}), in units of the volume component, $M_V$ [Eq. (\ref{['MNSV']})], and their surface part, $M_S$ [Eq. (\ref{['MNSSfin']})], as function of radius $R$ in units of the Schwarzschild radius $R_{\rm S}$, $\xi=R/R_{\rm S}$ [Eq. (\ref{['rgRS']})]. Black solid and dashed "1" and "2" versus red dashed "3" and dotted "4" curves show the results for $a/R=0.08$ and $0.04$ for the vdW&Skyrme interaction, respectively.
• Figure 4: Adiabatic NS moments of inertia, $\Theta$, Eq. (\ref{['MI']}), in units of the uniform sphere MI, $\Theta_{\rm sph}=2 M R^2/5$ with the same radius $R$ and mass $M$, as functions of the effective radius $R$ are shown for several values of the inner densities $\overline{\rho}/\rho^{}_0$=1 (solid), 2 (dashed), 3 (dotted), and 4 (dash-dotted lines) where $\rho^{}_0=m^{}_{N}n^{}_0$=2.68$\cdot 10^{14} \hbox{g}/\hbox{cm}^3$ ($m^{}_{N}$ is the nucleon mass, $n^{}_0=0.16$ fm$^{-3}$ is the nuclear-matter particle-number density) for the incompressibility $\kappa=10$, Eq. (\ref{['kappa']}), and leptodermic parameter $a/R=0.08$ for Hogan's metric, Eq. (\ref{['tauin']}). Vertical lines and arrows 1 - 4 show the asymptotes, which are related to the rotational critical radius $R_{\rm K}$ corresponding to a root of Eq. (\ref{['RKpole']}).
• Figure 5: The same as in Fig. \ref{['fig4']} but for our expression (\ref{['14tauinexpsol']}) for $\tau$.