Here we made use of the fact (see [26,
162
]), that the deflection angle can be expressed as the gradient of
an effective two-dimensional scalar potential
:
, where
and
is the Newtonian potential of the lens.
The determinant of the Jacobian
is the inverse of the magnification:
Let us define
The Laplacian of the effective potential
is twice the convergence:
With the definitions of the components of the external shear
,
and
(where the angle
reflects the direction of the shear-inducing tidal force
relative to the coordinate system), the Jacobian matrix can be
written
The magnification can now be expressed as a function of the
local convergence
and the local shear
:
Locations at which
have formally infinite magnification. They are called
critical curves
in the lens plane. The corresponding locations in the source
plane are the
caustics
. For spherically symmetric mass distributions, the critical
curves are circles. For a point lens, the caustic degenerates
into a point. For elliptical lenses or spherically symmetric
lenses plus external shear, the caustics can consist of cusps and
folds. In Figure
4
the caustics and critical curves for an elliptical lens with a
finite core are displayed.
![]() |
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 |