3.6 Lens mapping3 Basics of Gravitational Lensing3.4 Image positions and magnifications

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

equation276

where tex2html_wrap_inline2287 is the one-dimensional velocity dispersion. Projecting the matter on a plane, one obtains the circularly-symmetric surface mass distribution

equation281

With tex2html_wrap_inline2289 plugged into Equation (4Popup Equation), one obtains the deflection angle for an isothermal sphere, which is a constant (i.e. independent of the impact parameter tex2html_wrap_inline2215):

equation286

In ``practical units'' for the velocity dispersion this can be expressed as:

equation289

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 tex2html_wrap_inline2293):

equation296

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 [14, 23, 86, 90, 101, 169].



3.6 Lens mapping3 Basics of Gravitational Lensing3.4 Image positions and magnifications

image Gravitational Lensing in Astronomy
Joachim Wambsganss
http://www.livingreviews.org/lrr-1998-12
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de