Lensing properties of scale-free galaxies

The multiple images of lensed quasars provide evidence on the mass distribu tion of the lensing galaxy. The lensing invariants are constructed from the positions of the images, their parities, and their fluxes. They depend onl y on the structure of the lensing potential. The simplest is the magnificat ion invariant, which is the sum of the signed magnifications of the images. Higher order configuration invariants are the sums of products of the sign ed magnifications with positive or negative powers of the position coordina tes of the images. We consider the case of the four- and five-image systems produced by ellipt ical power-law galaxies with psi proportional to (x(2) + y(2)q(-2))(beta /2 ). This paper provides simple contour integrals for evaluating all their le nsing invariants. For practical evaluation, this offers considerable advant ages over the algebraic methods used previously. The magnification invarian t is exactly B = 2/(2 - beta) for the special cases beta = 0, 1, and 4/3; f or other values of beta, this remains an approximation, but an excellent on e at small source o+set. Similarly, the sums of the first and second powers of the image positions (or their reciprocals), when weighted with the sign ed magnifications, are just proportional to the same powers of the source o +set, with a constant of proportionality B. To illustrate the power of the contour integral method, we calculate full expansions in the source o+set f or all lensing invariants in the presence of arbitrary external shear. As a n example, we use the elliptical power-law galaxies to Dt to the data on th e four images of the Einstein Cross (G2237+030). The lensing invariants pla y a role by reducing the dimensionality of the parameter space in which the chi (2) minimization proceeds with consequent gains in accuracy and speed.