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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 Einstein radius ： ウィキペディア英語版 Einstein radius The Einstein radius is the radius of an Einstein ring, and is a characteristic angle for gravitational lensing in general, as typical distances between images in gravitational lensing are of the order of the Einstein radius. == Derivation == In the following derivation of the Einstein radius, we will assume that all of mass ''M'' of the lensing galaxy ''L'' is concentrated in the center of the galaxy. For a point mass the deflection can be calculated and is one of the classical tests of general relativity. For small angles ''α''1 the total deflection by a point mass ''M'' is given (see Schwarzschild metric) by :$\backslash alpha\_1\; =\; \backslash frac\backslash frac$ where : ''b''1 is the impact parameter (the distance of nearest approach of the lightbeam to the center of mass) : ''G'' is the gravitational constant, : ''c'' is the speed of light. By noting that, for small angles and with the angle expressed in radians, the point of nearest approach ''b''1 at an angle ''θ''1 for the lens ''L'' on a distance ''D''L is given by , we can re-express the bending angle ''α''1 as :$\backslash alpha\_1(\backslash theta\_1)\; =\; \backslash frac\backslash frac\backslash frac\; =\; \backslash theta\_\backslash ;\; D\_\; +\; \backslash alpha\_1\; \backslash ;\; D\_$ which can be rearranged to give :$\backslash alpha\_1(\backslash theta\_1)\; =\; \backslash frac\}\; (\backslash theta\_1\; -\; \backslash theta\_)$ ..... (Eqn. 2) By setting (eq. 1) equal to (eq. 2), and rearranging, we get :$\backslash theta\_1-\backslash theta\_\; =\; \backslash frac\; \backslash ;\; \backslash frac\; \backslash ;\; \backslash frac\; D\_\}$ For a source right behind the lens, , and the lens equation for a point mass gives a characteristic value for ''θ''1 that is called the Einstein radius, denoted ''θ''E. Putting and solving for ''θ''1 gives :$\backslash theta\_E\; =\; \backslash left(\backslash frac\backslash ;\backslash frac\; D\_\}\backslash right)^$ The Einstein radius for a point mass provides a convenient linear scale to make dimensionless lensing variables. In terms of the Einstein radius, the lens equation for a point mass becomes :$\backslash theta\_1\; =\; \backslash theta\_\; +\; \backslash frac$ Substituting for the constants gives :$\backslash theta\_E\; =\; \backslash left(\backslash frac\}\backslash right)^\; \backslash left(\backslash frac/\; D\_\}\; \backslash rm$ In the latter form, the mass is expressed in solar masses ( and the distances in Gigaparsec (Gpc). The Einstein radius is most prominent for a lens typically halfway between the source and the observer. For a dense cluster with mass }} at a distance of 1 Gigaparsec (1 Gpc) this radius could be as large as 100 arcsec (called macrolensing). For a Gravitational microlensing event (with masses of order ) search for at galactic distances (say ), the typical Einstein radius would be of order milli-arcseconds. Consequently separate images in microlensing events are impossible to observe with current techniques. Likewise, for the ''lower'' ray of light reaching the observer from below the lens, we have :$\backslash theta\_2\; \backslash ;\; D\_\; =\; -\; \backslash ;\; \backslash theta\_\backslash ;\; D\_\; +\; \backslash alpha\_2\; \backslash ;\; D\_$ and :$\backslash theta\_2\; +\; \backslash theta\_\; =\; \backslash frac\; \backslash ;\; \backslash frac\; \backslash ;\; \backslash frac\; D\_\}$ and thus :$\backslash theta\_2\; =\; -\; \backslash ;\; \backslash theta\_\; +\; \backslash frac$ The argument above can be extended for lenses which have a distributed mass, rather than a point mass, by using a different expression for the bend angle α. The positions ''θ''I(''θ''S) of the images can then be calculated. For small deflections this mapping is one-to-one and consists of distortions of the observed positions which are invertible. This is called weak lensing. For large deflections one can have multiple images and a non-invertible mapping: this is called strong lensing. Note that in order for a distributed mass to result in an Einstein ring, it must be axially symmetric. 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「Einstein radius」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.