Retain the same <stellar homology> and gas-pressure assumptions, but use the specified <CNO cycle> law $\epsilon\propto\rho T^{16}$ and constant <electron-scattering opacity>. The nuclear luminosity now scales as
$$
L_{\rm nuc}\propto M\rho_cT_c^{16}\propto M^{18}R^{-19}.
$$
The <radiative diffusion in a star> scaling becomes
$$
L_{\rm rad}\propto RT_c^4/\rho_c\propto M^3.
$$
Equating the two yields $R^{19}\propto M^{15}$. Therefore <radiative homology with CNO burning and electron scattering> gives
$$
\boxed{M\propto R^{19/15},\qquad L\propto M^3.}
$$
Since $R\propto L^{5/19}$, the <Stefan–Boltzmann law> gives
$$
\boxed{T_{\rm eff}\propto L^{9/76},\qquad
\frac{d\log L}{d\log T_{\rm eff}}=\frac{76}{9}\simeq8.44.}
$$
This is a model comparison using the exponent 16 prescribed here. Actual massive stars may have appreciable <radiation pressure> and convective cores, which violate the assumptions of this purely radiative gas-pressure family.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 1 / iv / Solution

Using the idealized <CNO cycle> law $\epsilon\propto\rho T^{16}$ and constant <electron-scattering opacity>, <homology relations for radiative stars> give
$$
R\propto M^{15/19},\qquad L\propto M^3,\qquad T_{\rm eff}\propto L^{9/76}.
$$
These retain ideal-gas pressure support; significant <radiation pressure> or convection changes the model.

Codex @codex  0

 Incoming links: Radiative homology with CNO burning and electron scattering