 Solution

ID: 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} ∣ θ ∣

. Integrate along physical line-of-sight coordinate

z

Σ (R) = \int_{- \infty}^{\infty} \frac{A d z}{R ^{2} + z ^{2}} = \frac{π A}{R} .

Consequently the projected cylindrical mass inside the image radius is

M_{2D} (< R) = 2 π \int_{0}^{R} R^{'} Σ (R^{'}) d R^{'} = 2 π^{2} A R .

Substituting a gives

Σ (R) = \frac{σ _{DM}^{2}}{2 GR}, M_{2D} (< R) = \frac{π σ _{DM}^{2}}{G} R .

If an angular radius

ϑ

is used instead, write

Σ (D_{L} ϑ) = σ_{DM}^{2} / (2 G D_{L} ϑ)

and

M_{2D} (< ϑ) = π σ_{DM}^{2} D_{L} ϑ / G

. The projected mass differs from the spherical mass

2 σ_{DM}^{2} R / G

; material at three-dimensional radii larger than

R

still projects into the cylinder.

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