Доповідач
Опис
Macroscopic approach to rotating neutron stars
A.G. Magner$^{1}$, S.P. Maydanyuk$^{1,2}$, A. Bonasera$^{3}$, H. Zheng$^{4}$ , S.N. Fedotkin$^{1}$,
A.I. Levon$^{1}$, U.V. Grygoriev$^{1}$, T. Depastas$^{3}$, A.A. Uleiev$^{1}$
$^{1}$Institute for Nuclear Research, Kyiv, Ukraine
$^{2}$Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China
$^{3}$Cyclotron Institute, Texas A&M University, College Station, Texas, USA
$^{4}$School of Physics and Information Technology, Shaanxi Normal University, Xi'an, China
In this report we present the macroscopic model for a neutron star (NS), as a perfect liquid drop in equilibrium, in line of Tolman--Oppenheimer--Volkoff (TOV) equations derivations [1], for slow azimuthal rotation frequency $\omega$ around the symmetry axis. Within the Friedman--Ipser--Stergioulas [2] formalism for the Kerr metric in Boyer--Lindquist coordinates in the external region [3] and the Hogan metric inside of the star [4] we apply the effective-surface approximation [5] assuming a small crust thicknes $a$ with respect to the radius $R$ NS, ${a}/{R}\ll 1$. The surface gradient terms are taken into account in the energy density $E(\rho )$ for the macroscopic equation of state (EoS) within the Extended Thomas Fermi (ETF) approach but with a strong gravitational field. The angular momentum $I$ and the moment of inertia (MI), $\mathrm{\Theta }\mathrm{=}{dI}/{d\omega }$, are macroscopically calculated in the adiabatic approach. The adiabatic MI can be expressed as $\mathrm{\Theta }={\mathrm{\Theta }}_{av}/(1+G_{t\varphi })$, where ${\mathrm{\Theta }}_{av}$ is the statistically averaged MI, and $G_{t\varphi }$ is a correlation term due to the time--azimuthal coupling,
$$\begin{array}{c}
{\mathrm{\Theta }}_{av}=\int{dV}E\left(\rho \right)e^{-\nu }r^2{{\mathrm{sin}}^{\mathrm{2}} \theta \ },\ G_{t\varphi }=\frac{1}{M}\int{dV}E\left(\rho \right)e^{-\nu }\tau {{\mathrm{sin}}^{\mathrm{2}} \theta \ }.\#\left(1\right) \end{array}$$
Here, the Schwarzschild metric length element modified by a small rotation frequency $\mathrm{\Omega }\approx {I}/{M}\approx \omega \mathrm{\Theta }\mathrm{/}M$, asymtotically far from the NS, is given by ($c=G=1$):
$$\begin{array}{c}
ds^2=-e^{\nu }dt^2+2\tau \mathrm{\Omega }{{\mathrm{sin}}^{\mathrm{2}} \theta \ }dtd\varphi +e^{\lambda }dr^2+r^2d{\theta }^2+r^2{{\mathrm{sin}}^2 \theta \ }d{\varphi }^2,\#\left(2\right) \end{array}$$
where $\nu$ and $\lambda$ are its well-known parameters, $\tau = 1 - \left(\sqrt{1 - r^2/R_S^2} - 3\sqrt{1 - R^2/R_S^2}\right)^2/4$, $R_S = \sqrt{3/(8\pi E_0)}$ is the Schwarzschild radius [3], $R$ is the NS radius, and $E_0$ is an internal value of the NS energy density.
(Fig.1 and Fig.2 can be seen in the attachment)
Fig. 1 shows the moment of inertia $\Theta$ as a function of the radius $R$. As seen from Fig. 1, the correlation term $G_{t\varphi}$ [see Eq. (1)] in a strong gravitational NS field leads to a significant shift of the Schwarzschild $R_S$ to the Kerr $R_K$ asymptote. The moment of inertia contributions $\Theta_{av}$ and $G_{t\varphi}$ are the sums of the volume and surface components in terms of the ETF energy density $E(\rho)$, $\Theta_i = \Theta_{iV} + \Theta_{iS}$ ($i = av, t\varphi$), and $E = E_S + E_V$ is the total NS ETF energy. The subscript $V$ denotes the NS volume $V$, and $S$ represents the NS surface (proportional to the surface tension coefficient) terms. The leptodermic and incompressibility parameters are given by a small $a/R$ and a large $\kappa = 10$ for strong gravitation, respectively.
We also verify the adiabatic condition, $(1/2)\Theta \omega^2 \ll E$, for applications to several NS rotation periods, $P \gg P_0 = 2\pi \sqrt{\Theta/(2E)}$, where $E$ is the total ETF energy, and $P_0$ is the asymptotic boundary period limit; see Fig. 2. For a range of well-known pulsars with spin periods between about 5 and 3000 ms, the adiabatic approach is applicable to the description of these rotating NS systems, in good agreement with their observational data. The internal densities used in these calculations are typically two to three times larger than that of nuclear matter. For all these calculations, the NS radii $R$ are on the order of 10 km, and masses are 1.2–2.1 $M_\odot$ in NS stable equilibrium.
Fig. 2. Periods $P$ (in units of ms) as functions of $R/R_S$, where $R_S$ is the Schwarzschild radius. Black (blue) circles and red (green) squares represent the minimal and maximal values of $R/R_S$ for the first and second sets of experimental data for the J0030+0451 (J0740+6620) neutron star, respectively. Solid black (dotted magenta) and dashed red lines are the characteristic periods $P_0$ with (volume evaluations) and without accounting for the $t-\varphi$ correlations. Other parameters are the same as in Fig. 1.
With increasing incompressibility $\kappa$, the boundary periods $P_0$ become greater. To improve accuracy, we should account for non-adiabatic effects, especially for the J0740+6620 star with larger mass. As a perspective, one can apply our analytical macroscopic approach for calculations of NS rotation frequency corrections to the TOV equations.
- R. C. Tolman. Relativity, Thermodynamics, and Cosmology. (N.Y.: Dover Publications, 1987) 528 p.
- J. L. Friedman, J. R. Ipser, L. Parker, AJ 304 (1986) 115.
- R. H. Boyer, R. W. Lindquist. J. Math. Phys. 8 (1967) 265.
- P. A. Hogan, Lettere Al Nuovo Cimento 16 (1976) 33.
- A. G. Magner, S. P. Maydanyuk, A. Bonasera, H. Zheng, T. Depastas, A. I. Levon, U. V. Grygoriev. 2024. arXiv:2409.04745 [nucl-th, astro-ph.HE, astro-ph.SR]. Int. J. Mod. Phys. E 33 (2024) 2450043.
