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

TL;DR

This work develops an analytic framework that uses Witt's rectangular hyperbola and Wynne's ellipse to localize the center of a cluster's gravitational potential from quadruple-image configurations. A new, offset-based figure of merit QC to prune poorly fitted quads is introduced, enabling robust combination of multiple quartets to estimate the cluster center. Applied to Abell 1689, the method retains three clean quads and yields a center, from the hyperbola intersections, that agrees with gas-based tracers and independent lensing centers within about 11 arcseconds, validating the approach as a fast, self-consistent estimator for cluster centers. The Wynne–Witt construction provides a transparent diagnostic of ellipticity and isothermality in cluster potentials and offers a practical cross-check among multi-wavelength and lensing-based center determinations.

Abstract

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.

Figures (4)

• Figure 1: Grayscale-inverted HST/ACS F814W view of the core of Abell 1689 with "primary" branches of Witt's Hyperbolae over-plotted. Red lines indicate well-fitting quadruples, while blue lines indicate quadruples that are discarded due to high offsets. The knot where the red hyperbolae intersect is the Wynne–Witt estimate of the cluster’s center of gravitational potential; its small dispersion visually demonstrates the precision of the analytic construction discussed in Sections \ref{['sec:method']} and \ref{['sec:results']}.
• Figure 2: Grayscale-inverted HST/ACS F814W view of the core of Abell 1689 with zoomed insets marking representative image positions used in our analysis (see Section \ref{['sub:impl']} and Section \ref{['sec:results']}). Primary branches of Witt's Hyperbolae are plotted in the background image, with insets showing the intersection of Witt's Hyperbolae and Wynne's Ellipses. The conics intersect precisely at the image coordinate so any visual displacement is the result of the dataset. Note: Not all insets have the same magnification.
• Figure 3: Comparison of fit quality between Quad 4 (green) and Quad 42 (red). The green and red squares represent the centers of Wynne's Ellipses for Quad 4 and 42 respectively. Quad 4 can seen to have a very low $|\Delta\mathbf x_{WW}|$, given the extremely small offset between Witt's Hyperbola and the center of Wynne's Ellipse. While Quad 42 is shown to have a larger $|\Delta\mathbf x_{WW}|$, representing a worse model fit.
• Figure 4: $(\alpha,\delta)_{\rm WW}$ (black circle) plotted over x-ray emission and SZ effect data. X-ray center and SZ center are marked with a square and diamond respectively but overlap considerably. X-ray emission data comes from Fig. 2 in Chappuis. SZ effect data is a gaussian interpolation of NASA's ACT DR6 + Planck Compton-y Map.