The Homologousness of Polytropic Gaseous Spheres in the General-Relativistic Case

Abstract

TOV equations in the polytropic case $(P=K\rho^{1+{1/N}})$ are represented by the homologous invariants of $U, V$, and an additional one of $w=P/(\rho c^2)$, where $P$ and $\rho c^2$ are the pressure and the static energy density. The homologous core solutions form a curved surface in the space of $(U,V,w)$, and they are distinguished by the asymptotic surface values of $E(=UV^N)$ and $D(=wV)$. $U, V$, and $w$ lead the invariant variables of $x$ and $\mu$, expressing the radius and the mass function. The solution of $x$ and $\mu$ with a central value of $w_\mathrm{c}$, called the core bundle solution (CB), well describes the extreme general-relativistic state. Core solutions are represented by the usual Emden variables, defined by $\rho=\lambda\theta^N$ and $P=K\lambda^{1+{1/N}}\theta^{N+1}$, as the general-relativistic E-solution (gE), which are determined by the two parameters $\theta_\mathrm{c}$ and $\omega(=K\lambda^{{1/N}})$. However, these two parameters change into each other by a homologous transformation, under the condition of $w_\mathrm{c}=\theta_\mathrm{c}\omega$. Hence, the gE solutions form a continuous group of one-parameter families, one of which is a CB solution corresponding to the gE solution with $\omega=1$, and another of which the general-relativisitic Lane-Emden solutions (gLE), defined by gE solutions with $\theta_\mathrm{c}=1$. A gLE solution changes into a CB solution by homologous transformation between each other. In gLE, three ways of $\lambda=(K^{-1}\omega)^N, \rho_\mathrm{c}$, and $p_\mathrm{c}\omega^{-1}$ render the normalization by $K^{N/2}, \rho_\mathrm{c}^{-1/2}$, and $p_\mathrm{c}^{-1/2}$, respectively, so that three kinds of mass-radius relations, derived from each normalization, weave the mass–radius textile in the $(M,R,\omega)$ space, where it stands up besides the Schwarzschild-radius wall.