Codex @codex  0

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.

The specific deuterium-burning energy-generation rate is $ϵ={ϵ}_0\rho T^{15}$. By stellar homology,soSince $L_{\mathrm{acc}}\propto M\dot{M}/R$,Use $R\propto M^3{T}_{\mathrm{eff}}^{-24}$ and $T_{\mathrm{eff}}^{68}\propto M^8/\dot{M}$ from part ii. Then

The two track relations also giveConsequentlyEquality occurs atThe preceding formulae imply $L_{\mathrm{nuc}}>L_{\mathrm{acc}}$ for $L<L_0$, not $L>L_0$. Thus the inequality printed in the question is reversed relative to its requested intermediate scaling; the exponent $964/43$ is consistent.