A fully convective monatomic <ideal gas> is an <adiabatic stellar polytrope> with index $n=3/2$. Its <polytropic mass-radius relation> gives
$$
K=\frac{P}{\rho^{5/3}}\propto GM^{1/3}R.
$$
At the base of a thin <grey atmosphere>, the <stellar surface boundary condition> and electron-scattering opacity give
$$
P_{\rm ph}\propto\frac{g}{\kappa}
\propto\frac{M}{R^2(1+X)}.
$$
The ideal-gas equation evaluated on the convective adiabat gives
$$
K\propto\left(\frac{T_{\rm eff}}{\mu}\right)^{5/3}
P_{\rm ph}^{-2/3}.
$$
Combining these relations,
$$
T_{\rm eff}\propto
\frac{\mu M^{3/5}}{(1+X)^{2/5}R^{1/5}}.
$$
The <Stefan–Boltzmann law> then gives
$$
\boxed{L\propto
\frac{\mu^4}{(1+X)^{8/5}}M^{12/5}R^{6/5}}.
$$

Solution

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

A fully convective monatomic perfect-gas star follows an $n=3/2$ <adiabatic stellar polytrope>. At the photosphere, $\tau=2/3$ and <hydrostatic equilibrium in optical depth> gives
$$
P_{\rm ph}\sim\frac{g}{\kappa_{\rm ph}}
\propto\frac{M}{R^2}T_{\rm eff}^{-19/2}.
$$
For an ideal gas on an adiabat,
$$
K=\frac{P}{\rho^{5/3}}
\propto T^{5/3}P^{-2/3}.
$$
Evaluating this at the photosphere gives
$$
K\propto M^{-2/3}R^{4/3}T_{\rm eff}^{8}.
$$
The $n=3/2$ <polytropic mass-radius relation> is $R\propto KM^{-1/3}$, so
$$
R\propto M^3T_{\rm eff}^{-24}.
$$
Finally the <Stefan–Boltzmann law> gives
$$
\boxed{
L=4\pi R^2\sigma T_{\rm eff}^4
\propto M^6T_{\rm eff}^{-44}}.
$$
This is the specified <Hayashi track>.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 3 / i / Solution

An adiabatic stellar polytrope has $P=K\rho^\gamma$ with constant specific entropy, so $n=1/(\gamma-1)$. For a monatomic perfect gas, $\gamma=5/3$ and $n=3/2$.


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 Incoming links: Adiabatic stellar polytrope