 Softened isothermal lensing potential

This circular lensing potential has reduced deflection $\alpha (r)={\theta }_{E}r/\sqrt{r^2+{\theta }_c^2}$, 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.