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

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.

Let $\alpha$ be the reduced, observed angular displacement between source and image. In the small-angle lens geometry, the physical bending angle is weighted by the lens-to-source and observer-to-source angular diameter distances:In a Euclidean diagram this follows because the transverse displacement at the source plane is $D_{LS}\xi$, whose observed angle is that displacement divided by $D_S$. In cosmology the same thin-lens expression uses the corresponding angular diameter distances. For a signed one-dimensional coordinate, the direction is supplied by the image's side of the lens.

The general thin gravitational lens equation is a two-component angular relation,For a spherical lens, image, source and lens centre lie in one plane containing the line of sight. Choose its sky direction as a signed axis. Then $\alpha ,\theta ,\beta$ can be signed scalars andThey are not fundamentally scalar quantities for a general lens; the reduction uses this symmetry and axis choice. An exactly aligned source requires the full ring geometry, not a unique preferred sky axis.

In general, $D_S\ne D_L+D_{LS}$. For example, in a spatially flat expanding cosmology, comoving radial distances ${\chi }_L,{\chi }_S$ giveThus $D_S-D_{LS}={\chi }_L/(1+z_S)$ differs from $D_L$ when the redshifts differ. Ordinary Euclidean addition applies only when the same static physical-distance convention is appropriate.