Lensing Properties of Cored Galaxy Models

N.W. Evans, C. Hunter

A method is developed to evaluate the magnifications of the images of galaxies with lensing potentials stratified on similar concentric ellipses. In a quadruplet system, there are two even parity images, two odd parity images, together with a de-magnified and usually missing central image. A simple contour integral is provided which enables the sums of the magnifications of the even parity or the odd parity images or the central image to be separately calculated without explicit solution of the lens equation. We find that the sums for pairs of images generally vary considerably with the position of the source, while the signed sums of the two pairs can be remarkably uniform inside the tangential caustic in the absence of naked cusps. For a family of models in which the lensing potential is a power-law of the elliptic radius, $\psi \propto (a^2 + x^2 + y^2 q^{-2})^{\beta/2}$, the number of visible images is found as a function of flattening $q$, external shear $\gamma$ and core radius $a$. The magnification of the central image depends on the size of the core and the slope $\beta$ of the gravitational potential. It grows strongly with the source offset if $\beta>1$, but weakly if $\beta<1$. For typical source and lens redshifts, the missing central image leads to strong constraints; the slope $\beta$ must be $\lta 1$ and the core radius $a$ must be $\lta 300$ pc. The mass distribution in the lensing galaxy population must be nearly cusped, and the cusp must be isothermal or stronger. This is in good accord with the cuspy cores seen in high resolution photometry of nearby, massive, early-type galaxies, which typically have $\beta \approx 0.7$ (or surface density falling like distance$^{-1.3}$) outside a break radius of a few hundred parsecs. Cuspy cores by themselves can provide the explanation of the missing central images. Dark matter at large radii may alter the slope of the projected density; however, provided the slope remains isothermal or steeper and the break radius remains small, then the central image remains unobservable. The sensitivity of the radio maps must be increased fifty-fold to find the central images in abundance.