= Solution
For fully ionized <hydrogen> and <helium> with mass fractions $X$ and $Y=1-X$, the inverse <mean molecular weight> counts nuclei plus free electrons:
$$
\mu^{-1}=2X+\frac34Y=\frac34+\frac54X.
$$
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),
$$
T_e\propto\mu^{1/2}M^{1/4}R^{1/8},\qquad L\propto\mu^2MR^{5/2}.
$$
Eliminating $R$ gives the <composition shift of a convective pre-main-sequence track>:
$$
\boxed{L\propto\mu^{-8}M^{-4}T_e^{20}}.
$$
Consequently the tracks in a <Hertzsprung-Russell diagram> are almost vertical, with $d\log L/d\log T_e=20$. At equal <luminosity>, $T_{e,1}/T_{e,2}=(14/13)^{2/5}\simeq1.030$: \b[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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-63-convective-tracks.png]
{title=Convective contraction tracks at equal mass, with evolution downward and rightward}
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).
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