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# A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

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 dc.contributor.author Petters, Arlie en_US dc.contributor.author Teguia, A. M. en_US dc.date.accessioned 2011-04-15T16:46:39Z dc.date.available 2011-04-15T16:46:39Z dc.date.issued 2009 en_US dc.identifier.citation Petters,A. O.;Rider,B.;Teguia,A. M.. 2009. A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula. Journal of Mathematical Physics 50(12): 122501-122501. en_US dc.identifier.issn 0022-2488 en_US dc.identifier.uri http://hdl.handle.net/10161/3370 dc.description.abstract Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. en_US dc.language.iso en_US en_US dc.publisher AMER INST PHYSICS en_US dc.relation.isversionof doi:10.1063/1.3267859 en_US dc.subject functional analysis en_US dc.subject gravitational lenses en_US dc.subject morse potential en_US dc.subject probability en_US dc.subject random processes en_US dc.subject star clusters en_US dc.subject stochastic processes en_US dc.subject random complex zeros en_US dc.subject random polynomials en_US dc.subject number en_US dc.subject caustics en_US dc.subject physics, mathematical en_US dc.title A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula en_US dc.type Article en_US dc.description.version Version of Record en_US duke.date.pubdate 2009-12-0 en_US duke.description.endpage 122501 en_US duke.description.issue 12 en_US duke.description.startpage 122501 en_US duke.description.volume 50 en_US dc.relation.journal Journal of Mathematical Physics en_US