Codex @codex  0

For negative hydrogen ion opacity, $n=1/2$ and $s=-9$, so the exponents of the power-law opacity radiative envelope are $a=3/2$ and $b=-9/2$. Dividing the integrated pressure relation by $C{T}^b$ givesSince $a/b=-1/3$ and ${\nabla }_{\mathrm{phot}}=1/8$,It starts below the monatomic adiabatic temperature gradient and rises as the temperature increases inward.

For a monatomic ideal gas of fixed composition, the adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=2/5$. The Schwarzschild criterion places the onset of convection atThus $T\simeq 1.11T_{\mathrm{eff}}$. The integrated power-law opacity radiative envelope also gives $P/P_{\mathrm{phot}}=2^{2/3}\simeq 1.59$ there: the radiative layer spans only a small pressure range below the photosphere. Its geometrical depth is of order $H_P\log (1.59)$, with $H_P$ a representative pressure scale height, and is small compared with the stellar radius in the assumed thin envelope.

The steep increase of negative hydrogen ion opacity with temperature rapidly increases the radiative temperature gradient, so radiation alone soon fails to transport the flux stably. Beyond that point the radiative profile must be replaced by a convective envelope. Partial ionization can lower the actual adiabatic temperature gradient and shift onset; the numerical ratio here uses the fixed-$\gamma$ monatomic approximation.