Images for an isothermal ellipsoidal gravitational lens from a single real algebraic equation

¹ Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan
² Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
³ National Astronomical Observatory, Mitaka 181-8588, Japan

Corresponding author: H. Asada, asada@phys.hirosaki-u.ac.jp

Received: 30 September 2002
Accepted: 4 November 2002

Abstract

We present explicit expressions for the lens equation for a cored isothermal ellipsoidal gravitational lens as a single real sixth-order algebraic equation in two approaches; 2-dimensional Cartesian coordinates and 3-dimensional polar ones. We find a condition for physical solutions which correspond to at most five images. For a singular isothermal ellipsoid, the sixth-order equation is reduced to fourth-order one for which analytic solutions are well-known. Furthermore, we derive analytic criteria for determining the number of images for the singular lens, which give us simple expressions for the caustics and critical curves. The present formulation offers a useful way for studying galaxy lenses frequently modeled as isothermal ellipsoids.