On the Magnfication Relations in Quadruple Lenses: A Moment Approach

Abstract

We present a new method of studying quadruple lenses in elliptical power-law potentials parameterized by $\psi (x,y)\propto (x^2+y^2/q^2{)}^{\beta /2}/\beta (0\le \beta <2)$. For this potential, the moments of the four image positions weighted by signed magnifications (magnification times parity) have very simple properties. In particular, we find that the zeroth moment -- the sum of four signed magnifications satisfies $\simeq 2/(2-\beta )$; the relation is exact for $\beta =0$ (point-lens) and $\beta =1$ (isothermal potential), independent of the axial ratio. Similar relations can be derived when a shear is present along the major or minor axes. These relations, however, do not hold well for the closely-related elliptical density distributions. For a singular isothermal elliptical density distribution without shear, the sum of signed magnifications for quadruple lenses is $\approx 2.8$, again nearly independent of the ellipticity. For the same distribution with shear, the total signed magnification is around 2-3 for most cases, but can be significantly different for some combinations of the axial ratio and shear where more than four images can appear.

16 pages, 4 figures, submitted to MNRAS

References in corpus (6)

• Evidence for substructure in lens galaxies?
• Gravitational Lensing By Spiral Galaxies
• Probing the Structure of Lensing Galaxies with Quadruple Lenses: The Effect of the ``External'' Shear
• The magnification invariant of simple galaxy lens models
• Gravitational Lensing by Power-Law Mass Distributions: A Fast and Exact Series Approach
• Ultraviolet images of the gravitationally lensed quadruple quasar Q2237+0305 with the HST

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