= Solution
The <Eddington closure approximation> with the standard surface boundary condition gives the <grey atmosphere> profile
$$
T^4=\frac34T_{\rm eff}^4\left(\tau+\frac23\right).
$$
Together with the specified pressure law, logarithmic differentiation gives
$$
\frac{d\log T}{d\tau}=\frac1{4(\tau+2/3)},\qquad
\frac{d\log P}{d\tau}=\frac{3}{4(1+3\tau/2)\log(1+3\tau/2)}.
$$
Consequently the <radiative temperature gradient> is
$$
\nabla_{\rm rad}=\frac{d\log T}{d\log P}=\frac12\log(1+3\tau/2).
$$
For a monatomic <perfect gas>, $\gamma=5/3$ and the <adiabatic temperature gradient> is $(\gamma-1)/\gamma=2/5$. Uniform composition makes the <Schwarzschild criterion> applicable. The first marginal layer therefore satisfies $\log(1+3\tau_c/2)=4/5$, giving the <convection onset in a logarithmic-pressure grey atmosphere>
$$
\boxed{\tau_c=\frac23(e^{4/5}-1)\simeq0.817.}
$$
The assumed profile becomes convectively unstable for $\tau>\tau_c$; further inward, a self-consistent model must account for convective transport.
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