3.5 (Non-)Singular isothermal sphere

A handy and popular model for galaxy lenses is the singular isothermal sphere with a three-dimensional density distribution of
2 -σv--1 ρ (r ) = 2πG r2 , (24 )
where σ v is the one-dimensional velocity dispersion. Projecting the matter on a plane, one obtains the circularly-symmetric surface mass distribution
σ2 1 Σ(ξ) = --v -. (25 ) 2G ξ
With ∫ M (ξ) = 0ξΣ(ξ′)2πξ′dξ′ plugged into Equation (4View Equation), one obtains the deflection angle for an isothermal sphere, which is a constant (i.e. independent of the impact parameter ξ):
σ2 α&tidle;(ξ) = 4π--v2 . (26 ) c
In “practical units” for the velocity dispersion this can be expressed as:
( )2 &tidle;α(ξ) = 1.15 ----σv----- arcsec. (27 ) 200 km s−1
Two generalizations of this isothermal model are commonly used: Models with finite cores are more realistic for (spiral) galaxies. In this case the deflection angle is modified to (core radius ξc):
σ2 ξ &tidle;α(ξ) = 4π -v------------. (28 ) c2(ξ2c + ξ2)1∕2
Furthermore, a realistic galaxy lens usually is not perfectly symmetric but is slightly elliptical. Depending on whether one wants an elliptical mass distribution or an elliptical potential, various formalisms have been suggested. Detailed treatments of elliptical lenses can be found in [13238993104170].
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