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For PP chain burning and the Kramers' opacity law, insert $\eta =4$, $\lambda =1$, and $\nu =-7/2$ into the homology relations for radiative stars:Thus the radiative homology with proton-proton burning and Kramers opacity isTo obtain the Hertzsprung-Russell diagram slope, apply the Stefan–Boltzmann law $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$:Equivalently $T_{\mathrm{eff}}\propto L^{69/284}$. The Hertzsprung-Russell slope of a stellar homology sequence refers here to logarithmic luminosity versus logarithmic effective temperature; a conventional diagram reverses its horizontal axis, so the line descends toward the right.

Using $R\propto M^{1/13}$ in the radiative diffusion in a star scaling from part (i),Therefore the mass-luminosity relation isThe nuclear scaling independently gives $L\propto M^6{R}^{-7}\propto M^{6-7/13}=M^{71/13}$, consistent with radiative homology with proton-proton burning and Kramers opacity.

Assume stellar homology: fixed dimensionless profiles, uniform fixed composition, negligible radiation pressure, ideal gas support, and radiative transport throughout the model. Mass conservation gives ${\rho }_c\propto M/R^3$. The hydrostatic pressure support equation gives $P_c\propto G{M}^2/R^4$; combining with $P\propto \rho T$ gives $T_c\propto M/R$.

Integrating specific proton–proton chain energy generation over mass givesFor radiative diffusion in a star, $T_c/R\propto {\kappa }_c{\rho }_{c}L/(R^2{T}_c^3)$, hence $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. With the Kramers' opacity law ${\kappa }_c\propto {\rho }_c{T}_c^{-7/2}$,Equating generated and transported luminosity gives $M^{1/2}\propto R^{13/2}$, so the radiative homology with proton-proton burning and Kramers opacity relation isThe proportionality constants are fixed only within the adopted homologous family, not for arbitrary evolving stars.