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
where 
is the one-dimensional velocity dispersion. Projecting the matter on a plane, one obtains the
circularly-symmetric surface mass distribution
With 
plugged into Equation (4), one obtains the deflection angle
for an isothermal sphere, which is a constant (i.e. independent of the impact parameter 
):
In “practical units” for the velocity dispersion this can be expressed as:
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 
):
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 [13, 23, 89, 93, 104, 170].
), one obtains the deflection angle
for an isothermal sphere, which is a constant (i.e. independent of the impact parameter 
