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

Use $R$ for the physical projected radius, or impact parameter, in the lens plane. Although the source calls it angular, a physical length is required in the mass and deflection equations; its relation to an angle is $R=D_L|\theta |$. Integrate along physical line-of-sight coordinate $z$:Consequently the projected cylindrical mass inside the image radius isSubstituting a givesIf an angular radius $\vartheta$ is used instead, write $\Sigma (D_L\vartheta )={\sigma }_{\mathrm{DM}}^2/(2G{D}_L\vartheta )$ and $M_{2D}(<\vartheta )=\pi {\sigma }_{\mathrm{DM}}^2{D}_L\vartheta /G$. The projected mass differs from the spherical mass $2{\sigma }_{\mathrm{DM}}^{2}R/G$; material at three-dimensional radii larger than $R$ still projects into the cylinder.