ar X iv : a st ro - p h / 01 02 08 1 v 1 6 F eb 2 00 1 Theory of Lensing Invariants I : The Power - Law Models

The multiple images of lensed quasars provide evidence on the mass distribution of the lensing galaxy. The lensing invariants are constructed from the positions of the images, their parities and their fluxes. They depend only on the structure of the lensing potential. The simplest is the magnification invariant, which is the sum of the signed magnifications of the images. Higher order configuration invariants are the sums of products of the signed magnifications with positive or negative powers of the position coordinates of the images. We consider the case of the four and five image systems produced by elliptical power-law galaxies with ψ ∝ (x 2 + y 2 q −2) β/2. This paper provides simple contour integrals for evaluating all their lensing invariants. For practical evaluation, this offers considerable advantages over the algebraic methods used previously. The magnification invariant is exactly B = 2/(2 − β) for the special cases β = 0, 1 and 4/3; for other values of β, this remains an approximation, but an excellent one at small source offset. Similarly, the sums of the first and second powers of the image positions (or their reciprocals), when weighted with the signed magnifications, are just proportional to the same powers of the source offset, with a constant of proportionality B. To illustrate the power of the contour integral method, we calculate full expansions in the source offset for all lensing invariants in the presence of arbitrary external shear. As an example, we use the elliptical power-law galaxies to fit to the data on the four images of the Einstein Cross (G2237+030). The lensing invariants play a role by reducing the dimensionality of the parameter space in which the χ 2 minimisation proceeds with consequent gains in accuracy and speed. Subject headings: gravitational lensing – galaxies: structure – quasars individual: G2237+030 – 2 –