Parameterized neutron star and proto-neutron star equation of state at finite temperature

Abstract

Read et al. proposed a parameterized equation of state (EOS) giving the pressure $P$ as a function of the rest mass density $\rho$ in a piecewise polytropic form for the study of neutron stars. The parameterized piecewise polytropic EOS can accurately represent many realistic EOS tables at zero temperature. We showed that the treatment can be extended to finite temperature to cover the study of neutron star at non zero temperature and proto-neutron stars. At finite temperature, two types of thermodynamic relations are needed, one for the pressure $P$ and one for the internal energy density $u$. We show that for many finite temperature realistic EOS tables, both the $P$ and $u$ can be accurately represented in piecewise polytropic forms as functions of the rest mass density $\rho$ and temperature $T$. The polytropic parameterization provides a framework to compare and understand the properties of the different realistic EOSs.

Appendix A: The parameter (\(\varGamma_{i}\), \(K_{i}\), \(b_{i}\), \(a_{i}\)) of the piecewise polytropic EOSs

In this appendix we give the parameters ($K_{i}$, $\varGamma_{i}$, $a_{i}$, $b_{i}$) of the piecewise polytropic EOSs for the 4 different EOSs at different temperature. The parameters ($K_{i}$, $\varGamma_{i}$, $a_{i}$, $b_{i}$) of the four density range ($\rho< 2.7\times10^{-4}$, $2.7\times10^{-4} < \rho< 8.0\times10^{-4}$, $8.0\times10^{-4} < \rho< 1.9\times10^{-3}$ and $\rho>1.9\times10^{-3}$) are showed in the Tables 5–8 for the 4 EOSs. The temperature $T$ studied ranged from $2~\mbox{MeV}$ to $60~\mbox{MeV}$.

Table 5 The parameters ($\varGamma_{i}$, $K_{i}$, $b_{i}$, $a_{i}$) at different temperature, for the Lattimer-Swesty EOS with $Y_{p} =0.1$. The ($\varGamma_{1}$, $K_{1}$, $b_{1}$, $a_{1}$) are for the density range $\rho< 2.7\times10^{-4}$, the ($\varGamma_{2}$, $K_{2}$, $b_{2}$, $a_{2}$) are for the density range $2.7\times10^{-4} < \rho<8.0\times10^{-4}$, the ($\varGamma_{3}$, $K_{3}$, $b_{3}$, $a_{3}$) are for the density range $8.0\times10^{-4} < \rho< 1.9 \times10^{-3}$, and the ($\varGamma_{4}$, $K_{4}$, $b_{4}$, $a_{4}$) are for the density range $\rho>1.9\times10^{-3}$. ($T$ given in terms of MeV per baryon, and $\rho$ and $K$ are given in units of $G=c=1$ and $M_{\odot}=1$)

Table 6 The parameters ($\varGamma_{i}$, $K_{i}$, $b_{i}$, $a_{i}$) at different temperature, for the H. Shen EOS with $Y_{p} =0.1$. The ($\varGamma_{1}$, $K_{1}$, $b_{1}$, $a_{1}$) are for the density range $\rho< 2.7\times10^{-4}$, the ($\varGamma_{2}$, $K_{2}$, $b_{2}$, $a_{2}$) are for the density range $2.7\times10^{-4} < \rho<8.0 \times10^{-4}$, the ($\varGamma_{3}$, $K_{3}$, $b_{3}$, $a_{3}$) are for the density range $8.0\times10^{-4} < \rho< 1.9\times10^{-3}$, and the ($\varGamma_{4}$, $K_{4}$, $b_{4}$, $a_{4}$) are for the density range $\rho>1.9\times10^{-3}$. ($T$ given in terms of MeV per baryon, and $\rho$ and $K$ are given in units of $G=c=1$ and $M_{\odot}=1$)

Table 7 The parameters ($\varGamma_{i}$, $K_{i}$, $b_{i}$, $a_{i}$) at different temperature, for the G. Shen EOS with $Y_{p} =0.1$. The ($\varGamma_{1}$, $K_{1}$, $b_{1}$, $a_{1}$) are for the density range $\rho< 2.7\times10^{-4}$, the ($\varGamma_{2}$, $K_{2}$, $b_{2}$, $a_{2}$) are for the density range $2.7\times10^{-4} < \rho<8.0 \times10^{-4}$, the ($\varGamma_{3}$, $K_{3}$, $b_{3}$, $a_{3}$) are for the density range $8.0\times10^{-4} < \rho< 1.9\times10^{-3}$, and the ($\varGamma_{4}$, $K_{4}$, $b_{4}$, $a_{4}$) are for the density range $\rho>1.9\times10^{-3}$. ($T$ given in terms of MeV per baryon, and $\rho$ and $K$ are given in units of $G=c=1$ and $M_{\odot}=1$)

Table 8 The parameters ($\varGamma_{i}$, $K_{i}$, $b_{i}$, $a_{i}$) at different temperature, for the M. Hempel EOS with $Y_{p} =0.1$. The ($\varGamma_{1}$, $K_{1}$, $b_{1}$, $a_{1}$) are for the density range $\rho< 2.7\times10^{-4}$, the ($\varGamma_{2}$, $K_{2}$, $b_{2}$, $a_{2}$) are for the density range $2.7\times10^{-4} < \rho<8.0\times10^{-4}$, the ($\varGamma_{3}$, $K_{3}$, $b_{3}$, $a_{3}$) are for the density range $8.0\times10^{-4} < \rho< 1.9 \times10^{-3}$, and the ($\varGamma_{4}$, $K_{4}$, $b_{4}$, $a_{4}$) are for the density range $\rho>1.9\times10^{-3}$. ($T$ given in terms of MeV per baryon, and $\rho$ and $K$ are given in units of $G=c=1$ and $M_{\odot}=1$)