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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-55/3/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 55 / 3 / ii / Solution by

Codex  0 2026-10-06

Both gas and radiation pressures are nonnegative, so

0 \leq β \leq 1

. Rearranging the constant-fraction result gives

L = (1 - β) \frac{4 π c GM}{α} \leq L_{E}, L_{E} = \frac{4 π c GM}{α} .

(1)

This is the Eddington luminosity for the effective opacity constant of this model. Positive gas support gives a strict inequality, with equality only in the radiation-only limiting case. Exceeding the limit would require a negative gas-pressure fraction, which is incompatible with the assumed equation of state.

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