For a homologous stellar sequence, <mass conservation>, <hydrostatic equilibrium>, and the <ideal gas> equation give
$$
\rho_c\propto\frac M{R^3},
\qquad
P_c\propto\frac{GM^2}{R^4},
\qquad
T_c\propto\frac{\mu M}{R}.
$$
Integrating the given <stellar energy-generation rate> over a fixed dimensionless profile gives
$$
L\propto\epsilon_cM
\propto X\rho_cT_c^{13}M.
$$
Therefore
$$
\boxed{L\propto X\mu^{13}\frac{M^{15}}{R^{16}}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-317/2/i/solution