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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.

The stellar adiabatic exponent ${\Gamma }_2$ is defined at fixed specific entropy and composition byCombining parts (iii) and (iv), the radiative temperature gradient of the constant-opacity grey atmosphere isFor $1<{\Gamma }_2<4/3$, the adiabatic temperature gradient is strictly below $1/4$, and the Schwarzschild criterion is eventually violated. If ${\Gamma }_2$ is constant, solving for the crossing givesAt ${\Gamma }_2=4/3$ the radiative gradient approaches marginality from below, so there is no finite crossing in this model. If ${\Gamma }_2$ varies, the conclusion requires its subcritical value to persist in the deep layers. Uniform composition is needed for this use of the Schwarzschild criterion; a composition gradient requires the Ledoux criterion.

The Eddington closure approximation with the standard surface boundary condition gives the grey atmosphere profileTogether with the specified pressure law, logarithmic differentiation givesConsequently the radiative temperature gradient isFor 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 atmosphereThe assumed profile becomes convectively unstable for $\tau >{\tau }_c$; further inward, a self-consistent model must account for convective transport.

Consider an element displaced upward by $\xi$ in hydrostatic equilibrium. Assume it quickly reaches the ambient pressure, but moves adiabatically and retains its composition. Put $\nabla =d\log T/d\log P$, ${\nabla }_{\mu }=d\log \mu /d\log P$, and let ${\nabla }_{\mathrm{ad}}$ be the adiabatic temperature gradient. The equation of state has differentialwhere $\delta =-(\partial \log \rho /\partial \log T{)}_{P,\mu }$ and $\varphi =(\partial \log \rho /\partial \log \mu {)}_{P,T}$.

At the new pressure, the parcel-to-environment density difference isWith positive pressure scale height $H_P=-(d\log P/dr{)}^{-1}$, $\Delta \log P=-\xi /H_P$. The buoyancy acceleration is therefore $\ddot{\xi }=-N^2\xi$, with stellar buoyancy frequencyA negative $N^2$ amplifies the displacement. For a radiative background, $\nabla ={\nabla }_{\mathrm{rad}}$, so the Ledoux criterion for instability isFor uniform composition, ${\nabla }_{\mu }=0$, giving the Schwarzschild criterionEquality is marginal stability. A mean molecular weight increasing inward has ${\nabla }_{\mu }>0$ and supplies a stabilizing composition term when $\varphi ,\delta >0$.

Hydrostatic balance and the ideal-gas adiabatic temperature gradient implyThe radiative region is stable while $K\rho /T^3<g/c_p$. Equality at the radiative-convective boundary, together with $\rho =\mu P/(\mathcal{R}T)$, givesFor radiative diffusion carrying intrinsic flux $F_{\mathrm{int}}=\sigma T_{\mathrm{int}}^4$, $K=3\kappa F_{\mathrm{int}}/(16\sigma )$, soFor an irradiated hot Jupiter with $g\simeq 10\mathrm{m}{\mathrm{s}}^{-2}$, ${\nabla }_{\mathrm{ad}}\simeq 2/7$, $\kappa \simeq 10^{-2}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$, $T_{\mathrm{rc}}\simeq 1500\mathrm{K}$, and $T_{\mathrm{int}}\simeq 150$--$200\mathrm{K}$, this gives roughly $P_{\mathrm{rc}}\sim 50$--$200\mathrm{bar}$.

Convection begins when the radiative temperature gradient equals the adiabatic temperature gradient ${\nabla }_{\mathrm{ad}}$. To convert optical depth into pressure, assume a pressure-law opacity $\kappa ={\kappa }_0{P}^n$ and constant gravity. Hydrostatic balance givesHenceThe exact radiative-convective boundary is the positive solution ofDeep enough that the exponential term is negligible,which requires $n+1>4{\nabla }_{\mathrm{ad}}$. This exposes why constant opacity is inadequate for a molecular atmosphere: its limiting radiative gradient is $1/4$, below ${\nabla }_{\mathrm{ad}}\simeq 2/7$.

In a grey scaling, $A\sim T_{\mathrm{eq}}^4$ measures irradiation and $B\sim T_{\mathrm{int}}^4$ measures intrinsic flux. ThusFor a hot Jupiter with $T_{\mathrm{eq}}\sim 1500\mathrm{K}$ and $T_{\mathrm{int}}\sim 100\mathrm{K}$, irradiation pushes the boundary to hundreds of bars for typical increasing opacity. Jupiter has $T_{\mathrm{eq}}$ and $T_{\mathrm{int}}$ both of order $10^2\mathrm{K}$ and becomes convective near the bar scale. The estimate is order-of-magnitude because real opacities depend on both pressure and temperature.

This model solves a vertical atmospheric pressure-temperature profile consistent with heating, radiative transfer and a convective adjustment where the radiative gradient exceeds the adiabatic temperature gradient. Irradiation, intrinsic flux, gravity and opacity determine the radiative-convective boundary. A one-dimensional model describes a column or average rather than resolving all atmospheric winds.

Stellar convective instability occurs when an adiabatic process displaces a fluid element and buoyancy amplifies the displacement. For uniform composition the Schwarzschild criterion gives instability when the actual temperature gradient exceeds the adiabatic temperature gradient; composition gradients instead require the Ledoux criterion.

The radiative temperature gradient needed to carry a spherical stellar luminosity follows by dividing radiative diffusion in a star by the hydrostatic pressure support equation. It is a logarithmic pressure gradient, not $dT/dr$ itself. Comparing it with the adiabatic temperature gradient by the Schwarzschild criterion determines whether purely radiative transport is locally unstable at uniform composition.