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Masa-aki Kondo, The Homologousness of Polytropic Gaseous Spheres in the General-Relativistic Case, Publications of the Astronomical Society of Japan, Volume 60, Issue 4, 25 August 2008, Pages 899–909, https://doi.org/10.1093/pasj/60.4.899
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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.
