Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 317 / 2 / i

Table of contents
- Solution i

Solution

For a homologous stellar sequence, mass conservation, hydrostatic equilibrium, and the ideal gas equation give

ρ_{c} \propto \frac{M}{R ^{3}}, P_{c} \propto \frac{G M ^{2}}{R ^{4}}, T_{c} \propto \frac{μ M}{R} .

Integrating the given stellar energy-generation rate over a fixed dimensionless profile gives

L \propto ϵ_{c} M \propto X ρ_{c} T_{c}^{13} M .

Therefore

L \propto X μ^{13} \frac{M ^{15}}{R ^{16}} .

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