Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 3 / iv / d / Solution

The tangential condition $\bar{\kappa }=1$ gives $\sqrt{{\theta }_t^2+{\theta }_c^2}={\theta }_E$, henceThis is a nonzero gravitational-lensing critical curve precisely when ${\theta }_E>{\theta }_c$. It is also the radius of the Einstein ring for an aligned source in this cored model, so the strength parameter ${\theta }_E$ itself is not the cored model's ring radius.

For completeness the radial condition is equally explicit. Subtracting the mean lensing convergence from twice the local one givesThus the radial critical curve of an axisymmetric lens has radiusagain nonzero exactly when ${\theta }_E>{\theta }_c$.