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Convection mixes material vertically where the Schwarzschild criterion or Ledoux criterion allows it. Its chemical effect is controlled by competition between transport and reaction, rather than by a direct alteration of the equilibrium constant. Parametrize vertical transport by a vertical eddy diffusion coefficient $K_{zz}$; the eddy mixing time over a scale height isLet ${\tau }_{\mathrm{chem}}$ be the chemical relaxation time of the relevant interconversion. If reactions are faster, the gas follows local thermochemical equilibrium. If transport becomes faster, composition can be carried upward faster than it re-equilibrates. A chemical quench level is approximately where the two times match:For example, hot deep gas may contain abundant carbon monoxide. Strong mixing can maintain it in cooler layers where equilibrium would favour methane; this is carbon monoxide–methane quenching. Similar reasoning applies to nitrogen–ammonia quenching. The cooler nightside commonly has slower reaction rates and can preserve such quenched composition more readily, whereas sufficiently rapid dayside chemistry can restore local equilibrium. Neither statement is universal without the actual reaction network and temperature.

The observable atmosphere of a strongly irradiated hot Jupiter may be radiative and stable even if the deeper interior is convective. Deep convection influences the photosphere only if some mixing crosses the intervening radiative region, for example through overshoot or other dynamical transport. Moreover, transport from day to night is horizontal circulation, not vertical convection; its relevant competition is atmospheric advection time versus chemical relaxation time, giving horizontal chemical quenching. Convection can affect either hemisphere indirectly through connected vertical transport; strong deep convection alone does not guarantee a quenched observable composition.

For a quasistatic spherical star with fixed total mass, the stellar structure equations in radius areHere $s$ is the specific entropy of a moving mass shell, and the entropy term accounts for release or storage of thermal and gravitational energy. In thermal equilibrium it vanishes. An equation of state, opacity, energy-generation law and composition evolution close the equations. The absence of mass loss fixes the outer mass coordinate; it does not require the star to be in thermal equilibrium.

The radiative temperature gradient needed to carry the full luminosity isHere the actual temperature gradient is $\nabla =d\log T/d\log P$. For uniform composition the Schwarzschild criterion says that a radiative layer is stable if ${\nabla }_{\mathrm{rad}}<{\nabla }_{\mathrm{ad}}$, with marginal stability at equality, where the adiabatic temperature gradient is $(\partial \log T/\partial \log P{)}_s$. Use $\nabla ={\nabla }_{\mathrm{rad}}$ there. If ${\nabla }_{\mathrm{rad}}>{\nabla }_{\mathrm{ad}}$, convection carries some of the flux. Efficient convection gives $\nabla \simeq {\nabla }_{\mathrm{ad}}$; inefficient surface convection needs a transport prescription and can be superadiabatic. For composition gradients, use the Ledoux criterion, with threshold ${\nabla }_{\mathrm{ad}}+\varphi {\nabla }_{{\mu }_{\mathrm{mol}}}/{\delta }_{\mathrm{th}}$ instead. Here ${\nabla }_{{\mu }_{\mathrm{mol}}}=d\log {\mu }_{\mathrm{mol}}/d\log P$, $\varphi =(\partial \log \rho /\partial \log {\mu }_{\mathrm{mol}}{)}_{P,T}$ and ${\delta }_{\mathrm{th}}=-(\partial \log \rho /\partial \log T{)}_{P,{\mu }_{\mathrm{mol}}}$, using the mean molecular weight ${\mu }_{\mathrm{mol}}$.

Above the thin burning shell, take constant luminosity $L$, constant mean molecular weight ${\mu }_{\mathrm{mol}}$, ideal-gas pressure $P=\mathcal{R}T\rho$ with $\mathcal{R}=k_B/({\mu }_{\mathrm{mol}}{m}_u)$, and the Kramers' opacity law $\kappa ={\kappa }_0\rho T^{-7/2}$. Then the temperature equation isTo implement the printed assumption that $P,\rho$ and $M_r$ are all power laws while retaining mass conservation, write $M_r\propto r^m$, $\rho \propto r^b$, $P\propto r^p$ and $T\propto r^d$. The mass equation gives $m=b+3$; the ideal gas law gives $d=p-b$; hydrostatic equilibrium gives $d=m-1$. The diffusion equation gives $d-1=2b-2-13d/2$. Solving this linear system gives the self-gravitating Kramers power-law envelope:The coefficients can also be matched consistently. If $r_c$ is the core boundary and $M_c$ its enclosed mass,Radiative transfer fixes the constant luminosity throughwhose right-hand side is independent of radius for these exponents. Thus this is a solution of all four envelope equations, not just a dimensional estimate.

Matching the boundary temperature to the isothermal core and extrapolating the idealized power law to the specified photosphere givesFor reference $M_R/M_c=(R/r_c{)}^{1/11}=2500^{1/10}\simeq 2.19$, so the envelope mass is not negligible in this particular solution. The given $M_c\simeq M_{\odot }$ fixes the core-radius normalization $r_c=(11/42)G{M}_c/(\mathcal{R}10^7\mathrm{K})$; a numerical absolute radius needs the unspecified molecular weight. The exponent ratio also gives $\nabla =d/p=5/21<2/5$, so the fully ionized monatomic version of this idealization is convectively stable.

A different common approximation neglects the envelope's self-gravity and sets $M_r\simeq M_c$ in the force equation. It cannot obey the exact mass equation with a nonzero density and exactly constant $M_r$. Under that additional approximation, the same calculation instead givesThis is the constant-core-mass power-law opacity radiative envelope limit, not the full power-law solution with changing enclosed mass. Stating which approximation is used resolves the otherwise different numerical radius ratios.

A compact nearly isothermal core, a thin luminosity-producing shell and a much larger cool envelope describe a shell-burning red giant. The full power-law profile is a deliberately simple model. The Kramers' opacity law and an ionized ideal gas are not reliable at a $4000\mathrm{K}$ photosphere; partial ionization, other opacity sources and a convective envelope usually modify real cool giants. The quoted radius ratios are extrapolations within the stipulated 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.

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 $\varphi ,\delta >0$.

Uniform composition and adiabatic convection give $T\propto {\rho }^{2/3}$ for a monatomic perfect gas. Hence near the centreThe luminosity equation isSince ${\rho }^2{T}^{\eta }\propto 1-(2+2\eta /3)\lambda r^2$, integration givesSimilarly,

The radiative temperature gradient isExpanding $\kappa ={\kappa }_0{\rho }^a{T}^{-b}$ and all the preceding factors giveswhereThe Schwarzschild criterion requiresfor the centre to be convectively unstable. If $A>0$, the radiative gradient decreases outward and can cross $2/5$, producing a finite convective core.

LetA displaced fluid element remains in pressure balance, changes temperature adiabatically, and retains its composition. Comparing its density with the environment after an upward displacement gives the Ledoux criterion for stability,The reverse inequality causes convection. For uniform composition, or if composition is ignored, ${\nabla }_{\mu }=0$ and this reduces to the Schwarzschild criterionfor stability. A molecular weight increasing inward has ${\nabla }_{\mu }>0$ and stabilizes the stratification.

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.

For $P=C(T^b+K{)}^{1/p}$ with $p=n+1$ and $b=4+n+m$, the logarithmic radiative temperature gradient is $(p/b)(1+K/T^b)$. At a marginal boundary with an efficient ideal-gas convective envelope, the Schwarzschild criterion equates it to $(\gamma -1)/\gamma$. Hence $K=T_b^b[(b/p)(\gamma -1)/\gamma -1]$. The notation $K=T_0^b$ with real positive $T_0$ requires a nonnegative bracket. Vanishing $p$ or $b$ needs the corresponding logarithmic integration instead of this power form.

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.