3.7 Time delay and ``Fermat's'' 3 Basics of Gravitational Lensing3.5 (Non-)Singular isothermal sphere

3.6 Lens mapping

In the vicinity of an arbitrary point, the lens mapping as shown in Equation (7Popup Equation) can be described by its Jacobian matrix tex2html_wrap_inline2295 :

equation304

Here we made use of the fact (see [26Jump To The Next Citation Point In The Article, 162Jump To The Next Citation Point In The Article]), that the deflection angle can be expressed as the gradient of an effective two-dimensional scalar potential tex2html_wrap_inline2297 : tex2html_wrap_inline2299, where

equation314

and tex2html_wrap_inline2301 is the Newtonian potential of the lens.

The determinant of the Jacobian tex2html_wrap_inline2295 is the inverse of the magnification:

  equation320

Let us define

equation324

The Laplacian of the effective potential tex2html_wrap_inline2297 is twice the convergence:

  equation328

With the definitions of the components of the external shear tex2html_wrap_inline2307,

  equation335

and

  equation341

(where the angle tex2html_wrap_inline2309 reflects the direction of the shear-inducing tidal force relative to the coordinate system), the Jacobian matrix can be written

equation346

The magnification can now be expressed as a function of the local convergence tex2html_wrap_inline2311 and the local shear tex2html_wrap_inline2307 :

equation358

Locations at which tex2html_wrap_inline2315 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.

  

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Figure 4: The critical curves (upper panel) and caustics (lower panel) for an elliptical lens. The numbers in the right panels identify regions in the source plane that correspond to 1, 3 or 5 images, respectively. The smooth lines in the right hand panel are called fold caustics; the tips at which in the inner curve two fold caustics connect are called cusp caustics.


3.7 Time delay and ``Fermat's'' 3 Basics of Gravitational Lensing3.5 (Non-)Singular isothermal sphere

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