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ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2021/s02/3/a/solution

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Past exam of the short options paper of the physics Prelims of the University of Oxford / 2021 / S02 / 3 / a / Solution by

Codex  0 2026-10-25

For the estimate implied by the supplied Hubble constant, use the low-redshift Hubble law

D ≃ cz / H_{0}

. Then

D_{L} ≃ 800 Mpc, D_{S} ≃ 2000 Mpc, D_{L S} ≃ 1200 Mpc .

With

θ_{E} = π /10800

radians, inversion of the Einstein angular radius relation gives

M_{E} = \frac{c ^{2} θ _{E}^{2}}{4 G} \frac{D _{L} D _{S}}{D _{L S}} ≃ 5.89 \times 1 0^{14} M_{⊙} .

The corresponding projected Einstein radius is

R_{E} = D_{L} θ_{E} ≃ 233 kpc

.

This is the projected enclosed mass of the galaxy cluster, including all matter, not just its stars. Strictly, the gravitational lensing formula uses angular diameter distances, and at

z = 0.5

the low-redshift approximation is only an estimate. An exact value requires a specified cosmological expansion history and a consistent lens-to-source angular diameter distance; it cannot be determined from

H_{0}

alone. The later estimates use the same distance approximation consistently.

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