Past exam of the astrophysics course of the University of Oxford / 2018 / 5 / d / Solution

The Einstein angular radius is the image-ring radius of an exactly aligned source. For the singular isothermal sphere lens, set $\beta =0$ in ii:Choose $\beta >0$ without loss of generality. The positive branch always gives ${\theta }_{+}=\beta +{\theta }_E$. A negative-side image would have ${\theta }_{-}=\beta -{\theta }_E$; this branch is admissible only when its result is negative. ThereforeAt $\beta =0$, symmetry produces an Einstein ring of angular radius ${\theta }_E$. At $\beta ={\theta }_E$, the negative branch reaches the singular origin and is not an ordinary additional finite image; the one finite image is at $2{\theta }_E$. Reversing the source sign reverses the signed image coordinates, so the multiplicity criterion uses $|\beta |$. These features rely on the singular isothermal profile; a smooth finite-density core changes the central-image behaviour.