Effects of the core radius of an isothermal ellipsoidal gravitational lens on the caustics and the critical curves

We study the effect of the core radius of an isothermal ellipsoidal gravitational lens on the caustics and the critical curves. We derive an analytic expression of the caustics for an isothermal ellipsoidal gravitational lens via a sixth-order algebraic equation. Since the expression is too long, by using another method we obtain a parametric representation of the critical curves in order to show analytically that there exist three cases: There are two curves for a small core radius, one for a quite large one, and no curves appear for an extremely large one, though the latter two cases are not realistic. The caustics are represented also by the same parameter.