Abstract
We discuss gravitational lensing by elliptical galaxies with some particular mass distributions. Using simple techniques from the theory of quadrature domains and the Schwarz function (cf. [18]) we show that when the mass density is constant on confocal ellipses, the total number of lensed images of a point source cannot exceed 5 (4 bright images and 1 dim image). Also, using the Dive-Nikliborc converse of the celebrated Newton’s theorem concerning the potentials of ellipsoids, we show that “Einstein rings” must always be either circles (in the absence of a tidal shear), or ellipses.
The third author gratefully acknowledges partial support from the National Science Foundation under the grant DMS-0701873. The first and third authors are also grateful to Kavli Institute of Theoretical Physics for the partial support of their visit there in 10/2006 under the NSF grant PHY05-51164.
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References
J. An and N.W. Evans. The Chang-Refsdal lens revisited. Mon. Not. R. Astron. Soc., 369:317–334, 2006.
W.L. Burke. Multiple gravitational imaging by distributed masses. Astrophys. J., 244:L1, 1981.
Ph. Griffiths and J. Harris. Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley-Interscience, New York, 1978.
B. Gustafsson and H.S. Shapiro. What is a quadrature domain? In Quadrature domains and their applications, volume 156 of Oper. Theory Adv. Appl., pages 1–25. Birkhäuser, Basel, 2005.
Ch. Keeton, S. Mao, and H.J. Witt. Gravitational lenses with more than four images, I. classification of caustics. Astrophys. J., pages 697–707, 2000.
Ch. Keeton and J. Winn. The quintuple quasar: Mass models and interpretation. Astrophys. J., 590:39–51, 2003.
D. Khavinson. Holomorphic partial differential equations and classical potential theory. Universidad de La Laguna, Departamento de Análisis Matemático, La Laguna, 1996.
D. Khavinson and G. Neumann. On the number of zeros of certain rational harmonic functions. Proc. Amer. Math. Soc., 134(4):1077–1085 (electronic), 2006.
D. Khavinson and G. Neumann. From the fundamental theorem of algebra to astrophysics: a “harmonious” path, Notices Amer. Math. Soc., Vol. 55, Issue 6, 2008, 666–675.
D. Khavinson and G. Światek. On the number of zeros of certain harmonic polynomials. Proc. Amer. Math. Soc., 131(2):409–414 (electronic), 2003.
S. Mao, A.O. Petters, and H.J. Witt. Properties of point-mass lenses on a regular polygon and the problem of maximum number of images. In T. Piron, editor, Proc. of the eighth Marcell Grossman Meeting on General Relativity (Jerusalem, Israel, 1977), pages 1494–1496. World Scientific, Singapore, 1998.
R. Narayan and B. Bartelman. Lectures on gravitational lensing. In Proceedings of the 1995 Jerusalem Winter School, http://cfa-www.harvard.edu/~narayan/papers/JeruLect.ps, 1995.
A.O. Petters. Morse theory and gravitational microlensing. J. Math. Phys., 33:1915–1931, 1992.
A.O. Petters, H. Levine, and J. Wambsganss. Singularity Theory and Gravitational Lensing. Birkhäuser, Boston, MA, 2001.
S.H. Rhie. Can a gravitational quadruple lens produce 17 images? www.arxiv.org/pdf/astro-ph/0103463, 2001.
S.H. Rhie. n-point gravitational lenses with 5n-5 images, www.arxiv.org/pdf/astroph/0305166, 2003.
T. Sauer. Nova Geminorum of 1912 and the origin of the idea of gravitational lensing, lanl.arxiv.org/pdf/0704.0963, 2007.
H.S. Shapiro. The Schwarz function and its generalization to higher dimensions, volume 9 of University of Arkansas Lecture Notes in the Mathematical Sciences.
N. Straumann. Complex formulation of lensing theory and applications. Helvetica Phys. Acta, arXiv:astro-ph/9703103, 70:896–908, 1997.
Ch. Turner. The early history of gravitational lensing, www.nd.edu/~/turner.pdf, 2006.
J. Wambsganss. Gravitational lensing in astronomy. Living Rev. Relativity, www.livingreviews.org/lrr-1998-12, 1:74 pgs., 1998. Last amended: 31 Aug. 2001
A. Wilmhurst. The valence of harmonic polynomials. Proc. Amer. Math. Soc., 126:2077–2081, 1998.
J. Winn, Ch. Kochanek, Ch. Keeton, and J. Lovell. The quintuple quasar: Radio and optical observations. Astrophys. J., 590:26–38, 2003.
H. J. Witt. Investigations of high amplification events in light curves of gravitationally lensed quasars. Astron. Astrophys., 236:311–322, 1990.
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Fassnacht, C., Keeton, C., Khavinson, D. (2009). Gravitational Lensing by Elliptical Galaxies, and the Schwarz Function. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_6
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