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

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

Codex  0 2026-10-06

Divide the radiation-pressure gradient by the total-pressure gradient. Radiative diffusion in a star and hydrostatic equilibrium give

\frac{d P _{r}}{d r} = - \frac{κ ρ L _{r}}{4 π c r ^{2}}, \frac{d P}{d r} = - \frac{G m _{r} ρ}{r ^{2}}, \frac{d P _{r}}{d P} = \frac{κ L _{r}}{4 π c G m _{r}} = \frac{αL}{4 π c GM} =: f .

(1)

This last quantity is independent of radius under the specified opacity assumption. Integration gives

P_{r} = f P + C

. With the standard idealized zero-pressure outer boundary, where both components vanish,

C = 0

. The stellar gas-pressure fraction is therefore

β = \frac{P _{g}}{P} = 1 - f = 1 - \frac{αL}{4 π c GM},

(2)

constant throughout the model. This is the Eddington standard model idealization. A finite photospheric pressure offset would need its boundary treatment; the differential relation alone does not set the integration constant to zero.

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