3.5 (Non-)Singular isothermal sphere3 Basics of Gravitational Lensing3.3 Critical surface mass density

3.4 Image positions and magnifications

The lens equation (6Popup Equation) can be re-formulated in the case of a single point lens:

  equation240

Solving this for the image positions tex2html_wrap_inline2217 one finds that an isolated point source always produces two images of a background source. The positions of the images are given by the two solutions:

  equation244

The magnification of an image is defined by the ratio between the solid angles of the image and the source, since the surface brightness is conserved. Hence the magnification tex2html_wrap_inline2277 is given as

  equation250

In the symmetric case above, the image magnification can be written as (by using the lens equation):

equation255

Here we defined u as the ``impact parameter'', the angular separation between lens and source in units of the Einstein radius: tex2html_wrap_inline2281 . The magnification of one image (the one inside the Einstein radius) is negative. This means it has negative parity: It is mirror-inverted. For tex2html_wrap_inline2283 the magnification diverges. In the limit of geometrical optics, the Einstein ring of a point source has infinite magnification Popup Footnote ! The sum of the absolute values of the two image magnifications is the measurable total magnification tex2html_wrap_inline2277 :

  equation265

Note that this value is (always) larger than one Popup Footnote ! The difference between the two image magnifications is unity:

equation272



3.5 (Non-)Singular isothermal sphere3 Basics of Gravitational Lensing3.3 Critical surface mass density

image Gravitational Lensing in Astronomy
Joachim Wambsganss
http://www.livingreviews.org/lrr-1998-12
© Max-Planck-Gesellschaft. ISSN 1433-8351
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