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

In the weak-field thin-lens limit, circular symmetry makes the physical deflection depend on the projected enclosed mass. With the mass calculated in b,For ${\sigma }_{\mathrm{DM}}=155\mathrm{km}{\mathrm{s}}^{-1}$, this is $3.36\times 10^{-6}\mathrm{rad}$, orIt is independent of impact radius in this ideal profile.

Enclosed-mass qualification. The source's appeal to Birkhoff's theorem must not be read as permission to use the mass of a three-dimensional sphere of radius $R$ here. Deflection accumulates along the light path. A concrete counterexample is a thin spherical shell of radius $r_s>R$: its spherical enclosed mass at $R$ is zero, but its projected mass is $M[1-\sqrt{1-R^2/r_s^2}]>0$, and it produces a nonzero deflection. The formula above is the axisymmetric projected-mass lens result; it is consistent with b and is not that incorrect spherical-mass interpretation.