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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 Lane–Emden equation ： ウィキペディア英語版 Lane–Emden equation In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden. The equation reads : $\backslash frac\; \backslash frac\; \backslash left(\}\backslash right)\; +\; \backslash theta^n\; =\; 0$ where $\backslash xi$ is a dimensionless radius and $\backslash theta$ is related to the density (and thus the pressure) by $\backslash rho=\backslash rho\_c\backslash theta^n$ for central density $\backslash rho\_c$. The index $n$ is the polytropic index that appears in the polytropic equation of state, : $P\; =\; K\; \backslash rho^\}\backslash ,$ where $P$ and $\backslash rho$ are the pressure and density, respectively, and $K$ is a constant of proportionality. The standard boundary conditions are $\backslash theta(0)=1$ and $\backslash theta\text{'}(0)=0$. Solutions thus describe the run of pressure and density with radius and are known as polytropes of index $n$. == Applications == Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. Thus, if we have a further equation that dictates how the pressure and density vary with respect to one another, we can reach a solution. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption. 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「Lane–Emden 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.