In Figure
3
the corresponding angles and angular diameter distances
,
,
are indicated
. In the thin-lens approximation, the hyperbolic paths are
approximated by their asymptotes. In the circular-symmetric case
the deflection angle is given as
where
is the mass inside a radius
. In this depiction the origin is chosen at the observer. From
the diagram it can be seen that the following relation holds:
(for
,
,
; this condition is fulfilled in practically all astrophysically
relevant situations). With the definition of the reduced
deflection angle as
, this can be expressed as:
This relation between the positions of images and source can easily be derived for a non-symmetric mass distribution as well. In that case, all angles are vector-valued. The two-dimensional lens equation then reads:
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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 |