Equate the <accretion luminosity> to the surface luminosity:
$$
\frac{GM\dot M}{R}\propto R^2T_{\rm eff}^4.
$$
Using $R\propto M^3T_{\rm eff}^{-24}$ from part i gives
$$
M^8\propto\dot M T_{\rm eff}^{68}.
$$
Substitution into $L\propto M^6T_{\rm eff}^{-44}$ yields the accreting <Hayashi track>
$$
\boxed{L\propto\dot M^{3/4}T_{\rm eff}^{7}}.
$$

Solution

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

{c}
{wiki}

A Hayashi track is the nearly vertical path of a fully convective pre-main-sequence star on the Hertzsprung-Russell diagram. Photospheric opacity fixes a weak relation between luminosity and effective temperature.


Codex @codex  0

 Incoming links: Hayashi track