Locating Centers of Clusters of Galaxies with Quadruple Images: Witt's Hyperbola and a New Figure of Merit

For any elliptical potential with an external parallel shear, Witt has proven that the gravitational center lies on a rectangular hyperbola derived from the image positions of a single quadruply lensed object. Moreover, it is predicted that for an isothermal elliptical potential the source position both lies on Witt's Hyperbola and coincides with the center of Wynne's Ellipse (fitted through the four images). Thus, by fitting Witt's Hyperbolae to several quartets of images - ten are known in Abell 1689 - the points of intersection provide an estimate for the center for the assumed isothermal elliptical potential. We introduce a new figure of merit defined by the offset of the center of Wynne's Ellipse from Witt's Hyperbola. This offset quantifies deviations from an ideal elliptical isothermal potential and serves as a discriminant to exclude poorly fitted quadruples and assign greater weight to intersections of hyperbolae of better fitting systems. Applying the method to 10 quads (after excluding 7 poorly fitted quads) in Abell 1689, we find the potential is centered within 11" of the BCG, X-ray center, flexion-based center and the center found from a total strong lensing analysis. The Wynne-Witt framework thus delivers a fast, analytic, and self-consistency-checked estimator for centers in clusters with multiple quads.