# Elliptically Symmetric Lenses and Violation of Burke's Theorem - Astrophysics > Earth and Planetary Astrophysics

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Abstract: We show that the outside equation of a bounded elliptically symmetric lensESL exhibits a pseudo-caustic that arises from a branch cut. A pseudo-causticis a curve in the source plane across which the number of images changes byone. The inside lens equation of a bounded ESL is free of a pseudo-caustic.Thus the total parity of the images of a point source lensed by a boundedelliptically symmetric mass is not an invariant in violation of the Burke-stheorem. A smooth mass density function does not guarantee the validity of theBurke-s theorem.Pseudo-caustics of various lens equations are discussed. In the Appendix,Bourassa and Kantowski-s deflection angle formula for an elliptically symmetriclens is reproduced using the Schwarz function of the ellipse for an easyaccess; the outside and inside lens equations of an arbitrary set of truncatedcircularly or elliptically symmetric lenses, represented as points, sticks, anddisks, are presented as a reasonable approximation of the realistic galaxy orcluster lenses.One may consider smooth density functions that are not bounded but fallsufficiently fast asymptotically to preserve the total parity invariance. Anybounded function may be sufficiently closely approximated by an unboundedsmooth function obtained by truncating its Fourier integral at a high frequencymode. Whether to use a bounded function or an unbounded smooth function for anESL lens mass density, whereby whether to observe the total parity invarianceor not, incurs philosophical questions. For example, is it sensible to insistthat the elliptical symmetry of an elliptic lens galaxy be valid in the entiresky? How a pseudo-caustic close to or intersecting with a caustic must bewithered away during a smoothing process and what it means will be investigatedin a separate work.

Author: ** Sun Hong Rhie**

Source: https://arxiv.org/