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A convective envelope is an outer stellar region in which convection carries a substantial fraction of the energy and mixes material. Its nearly adiabatic response to rapid mass loss is important for the dynamical stability of binary mass transfer.

A stable stratification gives a restoring buoyancy force and positive squared buoyancy frequency for an adiabatically displaced parcel. An adverse specific entropy gradient can instead drive convection; other forces such as rotation can modify the full stability criterion.

Delayed cooling preserves a young giant planet's large radius by reducing the escape of internal energy. Enhanced atmospheric opacity, an irradiation-maintained outer radiative blanket, and composition gradients that inhibit convection can slow Kelvin-Helmholtz contraction. These mechanisms retain existing heat rather than supplying a new deep heat source.

A fully convective star has convection throughout nearly its entire interior. Very low-mass red dwarfs are fully convective; the transition to this structure is used in the disrupted magnetic braking model.

For uniform composition, the Schwarzschild criterion gives stability against convection whenA parcel displaced upward expands approximately adiabatically. If the ambient logarithmic temperature gradient is shallower than the adiabatic temperature gradient, the parcel cools more than its surroundings, becomes denser, and sinks back. Equality is neutral stability; a steeper gradient drives buoyant growth.

For a diatomic ideal gas with active translational and rotational degrees of freedom, $c_p/c_v=7/5$ and ${\nabla }_{\mathrm{ad}}=2/7$. If the infrared opacity is constant, $P\propto \tau$, so the profile's radiative temperature gradient isAn atmospheric thermal inversion has negative ${\nabla }_{\mathrm{rad}}$ and is stable by this criterion. Composition gradients require the Ledoux criterion; molecular dissociation and changing heat capacities can also change the numerical adiabatic temperature gradient.

This is a semi-grey irradiated atmosphere in radiative equilibrium. The quantities have these meanings:

The model assumes a static plane-parallel atmosphere, separate grey stellar and thermal bands with constant opacities, local thermodynamic equilibrium for thermal emission, negligible scattering, and transport by radiative transfer rather than convection. The Eddington closure approximation sets the thermal angular-moment ratio to $1/3$, and the upper boundary supplies the usual $2/3$ term. Treating the incoming radiation with representative direction cosine $1/\sqrt{3}$ gives its attenuation $e^{-\sqrt{3}\gamma \tau }$. Intrinsic flux enters from below, while the imposed external stellar flux is absorbed from above.

With constant gravity and hydrostatic equilibrium, $\tau \simeq {\kappa }_{\mathrm{IR}}P/g$, allowing conversion to an atmospheric pressure-temperature profile. This last relation additionally assumes constant thermal opacity. The grey treatment describes the energy balance approximately; it does not resolve individual molecular absorption lines.

Use the single-layer greenhouse model with a blackbody surface and one isothermal atmospheric layer. The layer is transparent to incoming stellar radiation, absorbs a fraction $\alpha$ of the surface's thermal radiation, and has the same thermal emissivity $\alpha$ by Kirchhoff's law of thermal radiation. Here $0\le \alpha \le 1$, and $T_e$ is the planetary equilibrium temperature defined by the absorbed global stellar flux ${\sigma }_{\mathrm{SB}}{T}_e^4$. Neglect convection, latent heat transport and intrinsic heat.

The layer absorbs $\alpha {\sigma }_{\mathrm{SB}}{T}_s^4$ and emits $\alpha {\sigma }_{\mathrm{SB}}{T}_a^4$ upward and downward. For $\alpha >0$, its energy balance givesThe surface receives the absorbed stellar flux plus downward atmospheric emission, soEliminating $T_a$ yieldsThe outgoing top-of-atmosphere flux is $(1-\alpha ){\sigma }_{\mathrm{SB}}{T}_s^4+\alpha {\sigma }_{\mathrm{SB}}{T}_a^4={\sigma }_{\mathrm{SB}}{T}_e^4$, confirming overall energy conservation. At $\alpha =0$, $T_s=T_e$ and the decoupled layer's temperature is not determined; at $\alpha =1$, $T_s=2^{1/4}{T}_e$. This warming is the restriction of thermal escape by the absorbing and emitting layer.

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.

For slow cooling of an astrophysical disk, let $t_{\mathrm{cool}}\gg {\Omega }^{-1}$ and assume a vertically symmetric laminar column with no significant radial mass exchange. Its vertical velocity is of order $H/t_{\mathrm{cool}}$, so the vertical material derivative of that velocity is of order $H/t_{\mathrm{cool}}^2$. Relative to gravity or pressure acceleration $\sim {\Omega }^{2}H$, the inertial correction is $O[(\Omega t_{\mathrm{cool}}{)}^{-2}]$.

The hydrostatic approximation therefore holds to leading order while the column slowly cools and contracts. Non-turbulent evolution removes the prescribed turbulent heating, but not the compressional work in the ideal gas energy equation:The continuity equation must still determine the slow vertical flow; $D_t$ is not generally ${\partial }_t$ at fixed height. This is a quasi-static approximation for a well-prepared, mechanically stable column, with fast free oscillations or growing convection excluded from the assumed slow solution.

A thin radiative envelope with constant enclosed mass, luminosity and mean molecular weight, ideal gas pressure and opacity $\kappa ={\kappa }_0{\rho }^n{T}^{-s}$ has $a=n+1$, $b=n+s+4$, and $C=16\pi a_{\mathrm{rad}}cGM/[3{\kappa }_0(\mu m_u/k_B{)}^{n}L]$. Combining radiative diffusion in a star with hydrostatic equilibrium gives ${\nabla }_{\mathrm{rad}}=P^a/(C{T}^b)$, so $P^a-P_0^a=(aC/b)(T^b-T_0^b)$ for nonzero exponents. Zero exponents give logarithmic limits. The model must be stopped or modified if the Schwarzschild criterion predicts convection.