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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.

A cold self-gravitating core initially contracts on a dynamical scale $t_{\mathrm{ff}}\sim (G\rho {)}^{-1/2}$ if cooling prevents pressure from halting collapse. Increasing optical depth eventually traps radiation and forms a pressure-supported first hydrostatic core. Dissociation of molecular hydrogen absorbs compressional energy, softens the equation of state and triggers a second protostellar collapse to a compact protostar. Rotation produces a disk; angular-momentum transport permits continuing accretion, while outflows remove some mass and angular momentum.

An accreting object's luminosity includes a contribution of order $L_{\mathrm{acc}}\sim GM\dot{M}/R$, with a coefficient set by where accretion energy is radiated or retained. After accretion subsides, gravitational contraction supplies internal luminosity over the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(R{L}_{\mathrm{int}})$. Hydrostatic contraction heats the interior even while energy is radiated, as follows from the virial theorem and the negative heat capacity of a virialized gravitational system. accretion luminosity must not be substituted for $L_{\mathrm{int}}$ in the contraction time without an energy-budget justification.

Low-mass, nearly fully convective pre-main-sequence stars contract down a Hayashi track at roughly constant effective temperature and decreasing radius and luminosity. A developing radiative core can lead onto a Henyey track toward higher effective temperature; the route depends on mass and opacity. The deuterium-burning energy-generation rate can temporarily moderate contraction, but sustained core hydrogen burning marks arrival on the main sequence. Massive objects may begin hydrogen burning while still accreting, and substellar objects never establish long-lived hydrogen burning. An infrared spectral class reflects envelope and disk emission as well as inclination and interstellar extinction, so it is not a unique stellar age or interior evolutionary track.

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.