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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 Faber–Jackson relation ： ウィキペディア英語版 Faber–Jackson relation The Faber–Jackson relation is an early empirical power-law relation between the luminosity $L$ and the central stellar velocity dispersion $\backslash sigma$ of elliptical galaxies, first noted by the astronomers Sandra M. Faber and Robert Earl Jackson in 1976. The original relation can be expressed mathematically as: :$$ L \propto \sigma^ \gamma where the index $\backslash gamma$ is observed to be approximately equal to 4, but depends on the range of galaxy luminosities that is fitted. The Faber–Jackson relation is now understood as a projection of the fundamental planes of elliptical galaxies. One of its main uses is as a tool for determining distances to external galaxies. == Theory == The gravitational potential of a mass distribution of radius $R$ and mass $M$ is given by the expression: :$$ U=-\alpha \frac Where α is a constant depending e.g. on the density profile of the system and G is the gravitational constant. For a constant density, $\backslash alpha\backslash \; =\; \backslash frac$ The kinetic energy is (recall $\backslash sigma$ is the 1-dimensional velocity dispersion. Therefore $3\backslash sigma^2\; =\; V^2$): :$$ K = \fracMV^2 :$$ K = \fracM \sigma^2 From the virial theorem ($2\; K\; +\; U\; =\; 0$ ) it follows :$$ \sigma^2 =\frac\frac. If we assume that the mass to light ratio, $M/L$, is constant, e.g. $M\; \backslash propto\; L$ we can use this and the above expression to obtain a relation between $R$ and $\backslash sigma^2$: :$$ R \propto\frac. Let us introduce the surface brightness, $B\; =\; L/(4\; \backslash pi\; R^2)$ and assume this is a constant (which from a fundamental theoretical point of view, is a totally unjustified assumption) to get :$$ L=4\pi R^2 B. Using this and combining it with the relation between $R$ and $L$, this results in :$$ L \propto 4\pi\left(\frac\right)^2B and by rewriting the above expression, we finally obtain the relation between luminosity and velocity dispersion: :$$ L \propto\frac, that is :$$ L \propto \sigma^4. When account is taken of the fact that massive galaxies originate from homologous merging, and the fainter ones from dissipation, the assumption of constant surface brightness can no longer be supported. Empirically, surface brightness exhibits a peak at about $M\_V=-23$. The revised relation then becomes :$$ L \propto \sigma^ for the less massive galaxies, and :$$ L \propto \sigma^ for the more massive ones. With these revised formulae, the fundamental plane splits into two planes inclined by about 11 degrees to each other. Even first-ranked cluster galaxies do not have constant surface brightness. A claim supporting constant surface brightness was presented by astronomer Allan R. Sandage in 1972 based on three logical arguments and his own empirical data. In 1975, Donald Gudehus showed that each of the logical arguments was incorrect and that first-ranked cluster galaxies exhibited a standard deviation of about half a magnitude. 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「Faber–Jackson relation」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.