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Lensing near critical points

  • Peter Schneider
  • Jürgen Ehlers
  • Emilio E. Falco
Chapter
  • 509 Downloads
Part of the Astronomy and Astrophysics Library book series (AAL)

Abstract

The lens equation, derived in Chap. 4 and discussed in some detail in the previous chapter, defines a surjective mapping \( f:{^2} \to {^2},x \to y \) from the lens plane to the source plane. We assume here that it is differentiable as often as is needed. If \( {y^{(0)}} = f({x^{(0)}}) \) is the image of x(0) and if the Jacobian D = det A of the derivative \( A = \frac{{\partial y}}{{\partial x}} = \frac{{\partial ({y_1},{y_2})}}{{\partial ({x_1},{x_2})}} \) of f does not vanish at x(0), there exist neighborhoods of x(0) and y(0) on which f is bijective, i.e., the lens mapping is locally invertible. For an infinitesimal displacement dy of the source, the corresponding image position in the lens plane changes by \( dx = {A^{ - 1}}({x^{(0)}})dy \).

Keywords

Critical Curve Catastrophe Theory Source Plane Critical Image Lens Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Peter Schneider
    • 1
  • Jürgen Ehlers
    • 2
  • Emilio E. Falco
    • 3
  1. 1.Max-Planck-Institut für AstrophysikGarchingGermany
  2. 2.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutPotsdamGermany
  3. 3.Harvard-Smithsonian Center for AstrophysicsCambridgeUSA

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