3.4 Image positions and magnifications3 Basics of Gravitational Lensing3.2 Einstein radius

3.3 Critical surface mass density

In the more general case of a three-dimensional mass distribution of an extended lens, the density tex2html_wrap_inline2245 can be projected along the line of sight onto the lens plane to obtain the two-dimensional surface mass density distribution tex2html_wrap_inline2247, as

equation188

Here tex2html_wrap_inline2249 is a three-dimensional vector in space, and tex2html_wrap_inline2251 is a two-dimensional vector in the lens plane. The two-dimensional deflection angle tex2html_wrap_inline2253 is then given as the sum over all mass elements in the lens plane:

equation192

For a finite circle with constant surface mass density tex2html_wrap_inline2255 the deflection angle can be written:

equation200

With tex2html_wrap_inline2229 this simplifies to

equation207

With the definition of the critical surface mass density tex2html_wrap_inline2259 as

  equation215

the deflection angle for a such a mass distribution can be expressed as

equation223

The critical surface mass density is given by the lens mass M ``smeared out'' over the area of the Einstein ring: tex2html_wrap_inline2263, where tex2html_wrap_inline2265 . The value of the critical surface mass density is roughly tex2html_wrap_inline2267 for lens and source redshifts of tex2html_wrap_inline2237 and tex2html_wrap_inline2239, respectively. For an arbitrary mass distribution, the condition tex2html_wrap_inline2273 at any point is sufficient to produce multiple images.



3.4 Image positions and magnifications3 Basics of Gravitational Lensing3.2 Einstein radius

image 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