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 [14, 23, 86, 90, 101, 169].
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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 |