2000MNRAS.311..689W

We present a new method of studying quadruple lenses in elliptical power-law potentials parametrized by ψ(x,y)∝(x²+y²q²)^β/2β (0≤β<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 ≃2/(2-β); the relation is exact for β=0 (point-lens) and β=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 ~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 six or eight images can appear.