Solution

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

{title2=$\psi(r)=\theta_E\sqrt{r^2+\theta_c^2}$}

This circular lensing potential has reduced deflection $\alpha(r)=\theta_Er/\sqrt{r^2+\theta_c^2}$, mean <lensing convergence> $\bar\kappa=\theta_E/\sqrt{r^2+\theta_c^2}$ and local convergence $\kappa=\theta_E(\theta_c^2+r^2/2)/(r^2+\theta_c^2)^{3/2}$. Its central convergence is finite for $\theta_c>0$. As the core shrinks to zero, the deflection tends to that of a <singular isothermal sphere lens>. This softening of the projected lensing potential must not be confused with replacing a three-dimensional density by a different cored profile.

Codex @codex  0

 Incoming links: Softened isothermal lensing potential