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A fully convective monatomic ideal gas is an adiabatic stellar polytrope with index $n=3/2$. Its polytropic mass-radius relation givesAt the base of a thin grey atmosphere, the stellar surface boundary condition and electron-scattering opacity giveThe ideal-gas equation evaluated on the convective adiabat givesCombining these relations,The Stefan–Boltzmann law then gives

For a spherical stellar polytrope with $P=K{\rho }^{1+1/n}$, the Lane-Emden equation givesEliminating the central density at fixed composition and entropy gives the polytropic mass-radius relationAn incompressible rocky body has $n=0$ and $\beta =1/3$. A moderately massive gas giant is approximately an $n=1$ polytrope and has $\beta =0$, explaining its weak radius dependence on mass. In a more strongly degenerate nonrelativistic regime, $n=3/2$ gives $\beta =-1/3$.

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 givesFor an ideal gas on an adiabat,Evaluating this at the photosphere givesThe $n=3/2$ polytropic mass-radius relation is $R\propto K{M}^{-1/3}$, soFinally the Stefan–Boltzmann law givesThis is the specified Hayashi track.