3.2 Einstein radius3 Basics of Gravitational Lensing3 Basics of Gravitational Lensing

3.1 Lens equation

The basic setup for such a simplified gravitational lens scenario involving a point source and a point lens is displayed in Figure  2 . The three ingredients in such a lensing situation are the source S, the lens L, and the observer O. Light rays emitted from the source are deflected by the lens. For a point-like lens, there will always be (at least) two images S tex2html_wrap_inline2193 and S tex2html_wrap_inline2195 of the source. With external shear - due to the tidal field of objects outside but near the light bundles - there can be more images. The observer sees the images in directions corresponding to the tangents to the real incoming light paths.

  

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Figure 2: Setup of a gravitational lens situation: The lens L located between source S and observer O produces two images tex2html_wrap_inline2137 and tex2html_wrap_inline2139 of the background source.

In Figure  3 the corresponding angles and angular diameter distances tex2html_wrap_inline2207, tex2html_wrap_inline2209, tex2html_wrap_inline2211 are indicated Popup Footnote . In the thin-lens approximation, the hyperbolic paths are approximated by their asymptotes. In the circular-symmetric case the deflection angle is given as

  equation118

where tex2html_wrap_inline2213 is the mass inside a radius tex2html_wrap_inline2215 . In this depiction the origin is chosen at the observer. From the diagram it can be seen that the following relation holds:

equation124

(for tex2html_wrap_inline2217, tex2html_wrap_inline2219, tex2html_wrap_inline2221 ; this condition is fulfilled in practically all astrophysically relevant situations). With the definition of the reduced deflection angle as tex2html_wrap_inline2223, this can be expressed as:

  equation131

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:

  equation135

  

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Figure 3: The relation between the various angles and distances involved in the lensing setup can be derived for the case tex2html_wrap_inline2221 and formulated in the lens equation (6Popup Equation).


3.2 Einstein radius3 Basics of Gravitational Lensing3 Basics of Gravitational Lensing

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