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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Tolman–Oppenheimer–Volkoff equation ： ウィキペディア英語版 Tolman–Oppenheimer–Volkoff equation In astrophysics, the Tolman–Oppenheimer–Volkoff (TOV) equation constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modelled by general relativity. The equation〔 〕 is :$\backslash frac=-\backslash frac\backslash left()\backslash left(r^3\; \backslash frac\backslash right)\backslash left()^\; \backslash ;$ Here, ''r'' is a radial coordinate, and ''ρ''(''r''0) and ''P''(''r''0) are the density and pressure, respectively, of the material at ''r'' = ''r''0. The equation is derived by solving the Einstein equations for a general time-invariant, spherically symmetric metric. For a solution to the Tolman–Oppenheimer–Volkoff equation, this metric will take the form〔 :$ds^2=e^\; c^2\; dt^2\; -\; (1-2GM(r)/rc^2)^\; dr^2\; -\; r^2(d\backslash theta^2\; +\; \backslash sin^2\; \backslash theta\; d\backslash phi^2)\; \backslash ;$ where ''ν''(''r'') is determined by the constraint〔 :$\backslash frac=-\; \backslash left(\backslash frac\; \backslash right)\; \backslash frac\; \backslash ;$ When supplemented with an equation of state, ''F''(''ρ'', ''P'') = 0, which relates density to pressure, the Tolman–Oppenheimer–Volkoff equation completely determines the structure of a spherically symmetric body of isotropic material in equilibrium. If terms of order 1/''c''2 are neglected, the Tolman–Oppenheimer–Volkoff equation becomes the Newtonian hydrostatic equation, used to find the equilibrium structure of a spherically symmetric body of isotropic material when general-relativistic corrections are not important. If the equation is used to model a bounded sphere of material in a vacuum, the zero-pressure condition ''P''(''r'') = 0 and the condition exp() = 1 − 2''GM''(''r'')/''rc''2 should be imposed at the boundary. The second boundary condition is imposed so that the metric at the boundary is continuous with the unique static spherically symmetric solution to the vacuum field equations, the Schwarzschild metric: :$ds^2=(1-2GM\_0/rc^2)\; c^2\; dt^2\; -\; (1-2GM\_0/rc^2)^\; dr^2\; -\; r^2(d\backslash theta^2\; +\; \backslash sin^2\; \backslash theta\; d\backslash phi^2)\; \backslash ;$ ==Total Mass== ''M''(''r''0) is the total mass inside radius ''r'' = ''r''0, as measured by the gravitational field felt by a distant observer, it satisfies ''M''(0) = 0.〔 :$\backslash frac=4\; \backslash pi\; \backslash rho(r)\; r^2\; \backslash ;$ Here, ''M''0 is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at ''r'' = ''r''B, continuity of the metric and the definition of M(r) require that :$M\_0=M(r\_B)=\backslash int\_0^\; 4\backslash pi\; \backslash rho(r)\; r^2\; \backslash ;\; dr\; \backslash ;$ Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value :$M\_1=\backslash int\_0^\; \backslash frac\; 4\backslash pi\; \backslash rho(r)\; r^2((1-2GM(r)/rc^2)^-1)\; \backslash ;\; dr\; \backslash ;$ will be the gravitational binding energy of the object divided by ''c''2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tolman–Oppenheimer–Volkoff equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.