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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 Gravitational binding energy ： ウィキペディア英語版 Gravitational binding energy A gravitational binding energy is the minimum energy that must be added to a system for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower (''i.e.'', more negative) gravitational potential energy than the sum of its parts — this is what keeps the system aggregated in accordance with the minimum total potential energy principle. For a spherical mass of uniform density, the gravitational binding energy ''U'' is given by the formula〔Chandrasekhar, S. 1939, ''An Introduction to the Study of Stellar Structure'' (Chicago: U. of Chicago; reprinted in New York: Dover), section 9, eqs. 90-92, p. 51 (Dover edition)〕〔Lang, K. R. 1980, ''Astrophysical Formulae'' (Berlin: Springer Verlag), p. 272〕 :$U\; =\; \backslash frac$ where ''G'' is the gravitational constant, ''M'' is the mass of the sphere, and ''R'' is its radius. Assuming that the Earth is a uniform sphere (which is not correct, but is close enough to get an order-of-magnitude estimate) with ''M'' = 5.97 · 1024kg and ''r'' = 6.37 · 106m, ''U'' is 2.24 · 1032J. This is roughly equal to one week of the Sun's total energy output. It is 37.5 MJ/kg, 60% of the absolute value of the potential energy per kilogram at the surface. The actual depth-dependence of density, inferred from seismic travel times (see Adams–Williamson equation), is given in the Preliminary Reference Earth Model (PREM). Using this, the real gravitational binding energy of Earth can be calculated numerically to U = 2.487 · 1032 J According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.〔 ==Derivation for a uniform sphere== The gravitational binding energy of a sphere with Radius $R$ is found by imagining that it is pulled apart by successively moving spherical shells to infinity, the outermost first, and finding the total energy needed for that. Assuming a constant density $\backslash rho$, the masses of a shell and the sphere inside it are: :$m\_\backslash mathrm=4\backslash pi\; r^\backslash rho\backslash ,dr$      and      $m\_\backslash mathrm=\backslash frac\backslash pi\; r^3\; \backslash rho$ The required energy for a shell is the negative of the gravitational potential energy: :$=-G\backslash frac\}$ Integrating over all shells yields: :$U\; =\; -G\backslash int\_0^R\; \backslash pi\; r^\backslash rho)\}\}\; dr\; =\; -G\}\backslash pi^2\; \backslash rho^2\; \backslash int\_0^R\; dr\; =\; -G\}^2^2\; R^5$ Since $\backslash rho$ is simply equal to the mass of the whole divided by its volume for objects with uniform density, therefore :$\backslash rho=\backslash frac\backslash pi\; R^3\}$ And finally, plugging this into our result leads to :$U=-G\backslash frac\; \backslash pi^2\; R^5\; \backslash left(\backslash frac\backslash pi\; R^3\}\backslash right)^2=\; -\backslash frac$ 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「Gravitational binding energy」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.