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A pre-main-sequence star with a developing radiative core can contract along a relatively horizontal path toward higher effective temperature in the Hertzsprung-Russell diagram. This differs from the nearly vertical fully convective Hayashi track.

The radiative family is a stylized approximation to gas-pressure-supported main sequence stars with mostly radiative interiors and moderately temperature-sensitive hydrogen burning, such as the roughly solar-mass regime. Its opacity exponents are local approximations, not a universal composition-independent opacity law. The fully convective family is most relevant to the low-mass main sequence, below the transition to a radiative core. Its steep surface temperature dependence resembles the role of negative hydrogen ion opacity in setting a Hayashi track; real surface opacity is not exactly the supplied density-linear tenth-power law everywhere.

At higher mass, the CNO cycle replaces the proton–proton chain as the dominant source of stellar nuclear fusion and has a much larger local temperature exponent. This promotes a convective core. Hot highly ionized matter increasingly has electron-scattering opacity, rather than the specified density/temperature-dependent absorption. Radiation pressure becomes important in the equation of state, changing $T_c\propto M/R$ and eventually making the luminosity relation approach an Eddington luminosity scaling. Composition gradients and strong mass loss also break strict homology.

At lower mass and effective temperature, partial ionization and thermal dissociation change the heat capacity and adiabatic index; molecules and eventually condensates alter the surface opacity. Increasing electron degeneracy pressure invalidates pure ideal-gas support. Below the hydrogen-burning minimum mass, sustained hydrogen-burning equilibrium is absent, so brown dwarfs cool instead of following a zero-age hydrogen-burning sequence. These changes explain why neither derived slope should be extrapolated indefinitely.

For fully ionized hydrogen and helium with mass fractions $X$ and $Y=1-X$, the inverse mean molecular weight counts nuclei plus free electrons:Thus ${\mu }_1=8/13$ for $X_1=0.7$, ${\mu }_2=4/7$ for $X_2=0.8$, and ${\mu }_1/{\mu }_2=14/13$. Keep the same $M,{\kappa }_0$ and dimensionless structure constants when comparing these toy sequences. From part (a),Eliminating $R$ gives the composition shift of a convective pre-main-sequence track:Consequently the tracks in a Hertzsprung-Russell diagram are almost vertical, with $d\log L/d\log T_e=20$. At equal luminosity, $T_{e,1}/T_{e,2}=(14/13{)}^{2/5}\simeq 1.030$: the $X=0.7$ track lies to the hotter, left-hand side of the $X=0.8$ track. Since contraction decreases $R$, both $L$ and $T_e$ decrease in this particular opacity model. Both arrows therefore point downward and slightly rightward on the conventional hot-left diagram.

The plotted reference scales are arbitrary; the horizontal separation is the predicted $\Delta {\log }_{10}{T}_e=(2/5){\log }_{10}(14/13)$. They are schematic Hayashi tracks under the specified atmospheric opacity law, rather than calibrated stellar models.

A protostar leaves this contraction sequence when its assumptions fail. Increasing central stellar temperature makes stellar nuclear fusion significant: burning of deuterium can temporarily slow contraction, and sustained hydrogen burning brings the star to the main sequence. In a sufficiently massive pre-main-sequence star, a radiative core develops, the interior ceases to share one adiabat, and evolution can proceed along a hotter Henyey track. Changing atmospheric opacity, partial ionization, or substantial continuing accretion also changes the predicted path. For sufficiently low mass, electron degeneracy pressure instead limits the increase of central temperature, as in part (c).

The helium mass fraction is $Y=1-X-Z$. For the two compositions, $Y_1=0.28$ and $Y_2=0.18$. The mean molecular weight is thereforeAt fixed mass and with the same prescribed opacity coefficient, part (a) gives $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating radius givesThis is the composition shift of a convective pre-main-sequence track. The higher-helium mass fraction case $X_1=0.7$ is hotter at a given luminosity, lying to the left of the $X_2=0.8$ case in a Hertzsprung-Russell diagram. Its fixed-luminosity temperature ratio is $(349/324{)}^{2/5}\simeq 1.030$.

Both stars contract, so both $L$ and $T_e$ decrease in this particular opacity model. Their nearly vertical Hayashi tracks therefore run downward and slightly to the right, with arrows in that direction:

The plotted luminosity and effective temperature are normalized to arbitrary common references; no opacity normalization or absolute age was specified. The comparison holds the given opacity coefficient fixed.

These paths cease to apply when their assumptions fail. Deuterium-burning energy-generation rate can temporarily slow contraction; sustained hydrogen burning ends the purely gravitational phase. Formation of a radiative core breaks the fully convective adiabat and can lead to a Henyey track toward hotter effective temperature. Partial ionization, a changing atmospheric opacity, and electron degeneracy pressure alter the equation of state or boundary relation. Accretion luminosity, changing mass, and magnetic inhibition of convection can also modify a young star's path.

Equate the accretion luminosity to the surface luminosity:Using $R\propto M^3{T}_{\mathrm{eff}}^{-24}$ from part i givesSubstitution into $L\propto M^6{T}_{\mathrm{eff}}^{-44}$ yields the accreting Hayashi track

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.

A protostar is a forming star whose energy comes chiefly from gravitational contraction and accretion, before sustained core hydrogen burning supplies its luminosity. A fixed-mass contraction model approximates the stage after most accretion has ended; an actively accreting protostar need not have fixed mass or the same Hayashi track.

A stellar core in which radiative diffusion in a star carries most of the energy, rather than convection. Its formation in a pre-main-sequence star breaks the fully convective assumption behind an idealized Hayashi track.