Other
Scientific paper
Aug 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008pasj...60..899k&link_type=abstract
Publications of the Astronomical Society of Japan, Vol.60, No.4, pp.899--909
Other
Scientific paper
TOV equations in the polytropic case (P = Kρ1+1/N) are represented by the homologous invariants of U, V, and an additional one of w = P / ( ρc2), where P and ρc2 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 μ, expressing the radius and the mass function. The solution of x and μ with a central value of wc, called the core bundle solution (CB), well describes the extreme general-relativistic state. Core solutions are represented by the usual Emden variables, defined by ρ = λ θ N and P = K λ1+1/N θ N+1, as the general-relativistic E-solution (gE), which are determined by the two parameters θc and ω ( = K λ 1/N ). However, these two parameters change into each other by a homologous transformation, under the condition of wc = θcω. 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 ω = 1, and another of which the general-relativisitic Lane-Emden solutions (gLE), defined by gE solutions with θc = 1. A gLE solution changes into a CB solution by homologous transformation between each other. In gLE, three ways of λ = ( K-1ω )N, ρc, and pcω-1 render the normalization by K N/2, ρc-1/2, and pc-1/2, respectively, so that three kinds of mass-radius relations, derived from each normalization, weave the mass-radius textile in the (M, R, ω) space, where it stands up besides the Schwarzschild-radius wall.
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