= Solution
For fixed composition, <electron-scattering opacity> is effectively independent of <mass density> and <temperature>, so $\lambda=\nu=0$. With the specified <CNO cycle> exponent $\eta=16$, the <homology relations for radiative stars> give
$$
X=\frac{15}{19},\qquad Y=18-19X=3.
$$
The <radiative homology with CNO burning and electron scattering> is therefore
$$
\boxed{R\propto M^{15/19},\qquad L\propto M^3.}
$$
The <Stefan–Boltzmann law> now gives
$$
T_{\rm eff}\propto M^{27/76},\qquad
\boxed{\frac{d\log L}{d\log T_{\rm eff}}=\frac{4\cdot3}{3-30/19}
=\frac{76}{9}\simeq8.444.}
$$
Equivalently $T_{\rm eff}\propto L^{9/76}$. These slopes are illustrated with arbitrary separate reference-model normalizations:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-317-radiative-homology.png]
{title=Logarithmic Hertzsprung-Russell slopes for the two idealized radiative homology sequences, with temperature increasing to the left}
\b[The proton–proton/Kramers model applies to the lower-mass regime; the CNO/electron-scattering model applies to the higher-mass regime.] Higher-mass cores are hotter, making the more temperature-sensitive <CNO cycle> efficient, and highly ionized material favors <electron-scattering opacity>. These are the stated idealized radiative, gas-pressure-supported sequences: a fully convective low-mass star or a radiation-pressure-dominated massive star needs a different structural model. The intersection of independently normalized lines in the figure does not specify a physical crossover mass.
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