Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 2 / ii / Solution

At the core-envelope interface, continuity at temperature

T_{c}

requires the envelope pressure

P = C_{0} (\frac{M}{L})^{1/2} T_{c}^{17/4}, C_{0}^{2} = \frac{64 π a _{rad} c G R}{51 κ _{0}},

to equal the nonrelativistic electron degeneracy pressure

P = K (\frac{ρ}{μ _{e}})^{5/3}, ρ = \frac{P}{R T _{c}} .

Eliminating

ρ

gives

P = (R μ_{e} T_{c})^{5/2} K^{- 3/2} .

Equating the two pressures and solving for luminosity gives Mestel's cooling law

\frac{L}{M} = \frac{64 π a _{rad} c G}{51 κ _{0}} \frac{K ^{3}}{R ^{4} μ _{e}^{5}} T_{c}^{7/2} .

In particular,

L / M \propto T_{c}^{7/2}

.

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