On the Magnfication Relations in Quadruple Lenses: A Moment Approach

We present a new method of studying quadruple lenses in elliptical power-law potentials parameterized by $\psi(x,y) \propto (x^2+y^2/q^2)^{\beta/2}/\beta (0 \leq \beta < 2)$. 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 $\simeq 2\mathrm{/}(2-\beta )$; 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.