 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2018/5/c/ii/solution

The general thin gravitational lens equation is a two-component angular relation,

β = θ - α (θ), α = \frac{D _{L S}}{D _{S}} ξ .

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

α, θ, β

can be signed scalars and

β = θ - \frac{D _{L S}}{D _{S}} ξ sgn θ .

They 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} \neq = D_{L} + D_{L S}

. For example, in a spatially flat expanding cosmology, comoving radial distances

χ_{L}, χ_{S}

give

D_{L} = \frac{χ _{L}}{1 + z _{L}}, D_{S} = \frac{χ _{S}}{1 + z _{S}}, D_{L S} = \frac{χ _{S} - χ _{L}}{1 + z _{S}} .

Thus

D_{S} - D_{L S} = χ_{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.

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