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>.

Solution

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

In a plane-parallel atmosphere with $d\tau=-\kappa\rho\,dr$ and $dP/dr=-\rho g$, one has $dP/d\tau=g/\kappa$. If gravity and opacity vary slowly, the pressure at optical depth $\tau$ is approximately $P\simeq g\tau/\kappa$.


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 Incoming links: Hydrostatic equilibrium in optical depth