= Solution
The radiative family is a stylized approximation to gas-pressure-supported <main sequence> stars with mostly radiative interiors and moderately temperature-sensitive hydrogen burning, such as the roughly solar-mass regime. Its opacity exponents are local approximations, not a universal composition-independent opacity law. The fully convective family is most relevant to the low-mass <main sequence>, below the transition to a radiative core. Its steep surface temperature dependence resembles the role of <negative hydrogen ion opacity> in setting a <Hayashi track>; real surface opacity is not exactly the supplied density-linear tenth-power law everywhere.
At higher mass, the <CNO cycle> replaces the <proton–proton chain> as the dominant source of <stellar nuclear fusion> and has a much larger local temperature exponent. This promotes a <convective core>. Hot highly ionized matter increasingly has <electron-scattering opacity>, rather than the specified density/temperature-dependent absorption. <Radiation pressure> becomes important in the <equation of state>, changing $T_c\propto M/R$ and eventually making the <luminosity> relation approach an <Eddington luminosity> scaling. Composition gradients and strong <mass loss (astrophysics)> also break strict homology.
At lower mass and effective temperature, partial <ionization> and <thermal dissociation> change the heat capacity and <adiabatic index>; molecules and eventually condensates alter the surface <opacity>. Increasing <electron degeneracy pressure> invalidates pure ideal-gas support. Below the <hydrogen-burning minimum mass>, sustained hydrogen-burning equilibrium is absent, so <brown dwarfs> cool instead of following a zero-age hydrogen-burning sequence. These changes explain why neither derived slope should be extrapolated indefinitely.
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