Gravitational Lensing by Power‐Law Mass Distributions: A Fast and Exact Series Approach

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Abstract
We present an analytical formulation of gravitational lensing using familiar triaxial power-law mass distributions, where the 3-dimensional mass density is given by $ ho(X,Y,Z) = ho_0 [1 + (frac{X}{a})^2 + (frac{Y}{b})^2 + (frac{Z}{c})^2]^{- u/2}$. The deflection angle and magnification factor are obtained analytically as Fourier series. We give the exact expressions for the deflection angle and magnification factor. The formulae for the deflection angle and magnification factor given in this paper will be useful for numerical studies of observed lens systems. An application of our results to the Einstein Cross can be found in Chae, Turnshek, & Khersonsky (1998). Our series approach can be viewed as a user-friendly and efficient method to calculate lensing properties that is better than the more conventional approaches, e.g., numerical integrations, multipole expansions.
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