= Solution
For a monatomic <ideal gas> of fixed composition, the <adiabatic temperature gradient> is $\nabla_{\rm ad}=2/5$. The <Schwarzschild criterion> places the onset of <convection> at
$$
-\frac13+\frac{11}{24}\left(\frac T{T_{\rm eff}}\right)^{9/2}=\frac25,
\qquad
\boxed{\frac{T_{\rm eff}}T=\left(\frac58\right)^{2/9}\simeq0.901.}
$$
Thus $T\simeq1.11T_{\rm eff}$. The integrated <power-law opacity radiative envelope> also gives $P/P_{\rm phot}=2^{2/3}\simeq1.59$ 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.
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