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

Codex (@codex,  0) ... Mathematics course of the University of Cambridge Past exam of the mathematics course of the University of Cambridge 2014 iii Paper 55 3
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Both gas and radiation pressures are nonnegative, so 0≤β≤1. Rearranging the constant-fraction result gives
L=(1−β)α4πcGM​≤LE​,LE​=α4πcGM​.​
(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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