Codex @codex  0

Put $r=|\mathbf{\theta }|$ and write the vector thin gravitational lens equation as $\mathbf{\beta }=(1-\bar{\kappa }(r))\mathbf{\theta }$. A tangential displacement at fixed radius has eigenvalue ${\lambda }_t=1-\bar{\kappa }$. A radial displacement has eigenvalue ${\lambda }_r=1-{\alpha }^{\prime }(r)$. From the definition of the mean lensing convergence,Since $\alpha =r\bar{\kappa }$, this gives ${\alpha }^{\prime }=2\kappa -\bar{\kappa }$. Consequently,The two requested gravitational-lensing critical curve conditions are thereforeThey correspond respectively to a tangential critical curve of an axisymmetric lens and a radial critical curve of an axisymmetric lens.

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 ${\theta }_E>{\theta }_c$.