Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 4 / i

Table of contents
- Solution i

Solution

The gas-to-radiation pressure ratio gives

\frac{β}{1 - β} = \frac{P _{g}}{P _{rad}} = \frac{3 ρ k _{B}}{a _{rad} μ H T ^{3}} .

Hence

T (β, ρ) = [\frac{3 k _{B}}{a _{rad} μ H} \frac{1 - β}{β} ρ]^{1/3},

and either pressure component gives

P (β, ρ) = \frac{k _{B}}{μ H β} [\frac{3 k _{B}}{a _{rad} μ H} \frac{1 - β}{β}]^{1/3} ρ^{4/3} .

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