Lensing near critical points

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


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 \( dx = {A^{ - 1}}({x^{(0)}})dy \).


Critical Curve Catastrophe Theory Source Plane Critical Image Lens Model 
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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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