Here
is a three-dimensional vector in space, and
is a two-dimensional vector in the lens plane. The
two-dimensional deflection angle
is then given as the sum over all mass elements in the lens
plane:
For a finite circle with constant surface mass density
the deflection angle can be written:
With
this simplifies to
With the definition of the
critical surface mass density
as
the deflection angle for a such a mass distribution can be expressed as
The critical surface mass density is given by the lens mass
M
``smeared out'' over the area of the Einstein ring:
, where
. The value of the critical surface mass density is roughly
for lens and source redshifts of
and
, respectively. For an arbitrary mass distribution, the condition
at any point is sufficient to produce multiple images.
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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 |