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A dry parcel in rapid pressure balance with an ambient ideal gas has $d{T}_g/dP=T_g\mathcal{R}/(P{C}_p)$. Ambient hydrostatic equilibrium gives $dP/dz=-Pg/(\mathcal{R}{T}_{\mathrm{env}})$, hence $d{T}_g/dz=-(g/C_p)T_g/T_{\mathrm{env}}$. At a temperature-matched launch point this equals the dry adiabatic lapse rate, supplying the local linear stability test. A substantially hotter parcel does not exactly follow the ambient density's hydrostatic adiabat.

Use a dry ideal gas of fixed composition with specific gas constant $\mathcal{R}=C_p-C_v$ and constant specific heat capacity at constant pressure $C_p$. For a fixed-mass parcel, $P{V}^{\gamma }=$ constant and $PV=m_{\mathrm{parcel}}\mathcal{R}{T}_g$ implyAlong a hydrostatic adiabat, $dP/dz=-{\rho }_{g}g$; hence the dry adiabatic lapse rate isFor an actual pressure-balanced parcel rising in an ambient atmosphere, $dP/dz=-{\rho }_{\mathrm{env}}g$ instead gives $d{T}_g/dz=-(g/C_p)(T_g/T_{\mathrm{env}})$. The usual lapse-rate expression is exact for a hydrostatic adiabatic column and is the local first-order result at the launch point where $T_g=T_{\mathrm{env}}$, as needed in a linear stability test. Treating an already much hotter parcel as an exact copy of the ambient hydrostatic column would be an extra approximation.

After a small upward displacement $\delta z$ from temperature equilibrium, its temperature excess isAt equal pressure, warmer gas is less dense and continues to rise. Thus the Schwarzschild criterion in altitude form isThe supplied non-strict inequality includes the marginal case; strict growth requires the strict inequality. A downward displacement gives the same stability conclusion. Efficient convection normally adjusts an initially superadiabatic gradient to a nearly adiabatic one.

Deep envelopes of gas giants and ice giants commonly transport intrinsic heat by convection, as do the planetary tropospheres of many weakly irradiated atmospheres. Earth's dry troposphere provides another approximate example, with moisture changing the lapse rate. Strongly irradiated hot Jupiters can still have deep convective interiors, while their upper radiative regions need not be convective. Composition gradients can modify the homogeneous-gas criterion and inhibit overturning even in an interior.

In local thermodynamic equilibrium, thermal intensity samples the Planck function near an optical depth of order unity. The Eddington-Barbier relation makes this explicit: $I_{\lambda }(0,\mu )\simeq B_{\lambda }[T({\tau }_{\lambda }=\mu )]$. A molecular band has greater opacity than its adjacent continuum and therefore samples a higher layer. A band in emission relative to the continuum implies that this higher layer is hotter: the line-forming region has an atmospheric thermal inversion under the assumed LTE, thermal interpretation.

The continuum is thermal radiation from an optically thick, deeper photosphere, with comparatively smooth opacity. In an H/He hot Jupiter, collision-induced absorption by H2-H2 and H2-He collisions supplies an important continuum; weak overlapping molecular lines and opaque exoplanet clouds can contribute too. It is not a separate blackbody emitter floating above the gas. A strongly isothermal layer would erase LTE molecular contrast rather than generate emission peaks.

In the emitting inversion, $dT/dz>0$, whereas the dry adiabatic lapse rate has $dT/dz=-g/C_p<0$. Thereforewhich lies on the stable side of the Schwarzschild criterion. An upward-displaced parcel cools and becomes denser than the ambient hot upper gas. The region can thus carry and redistribute thermal energy by radiative transfer, not by unstable thermal convection. Winds may transport energy horizontally; stability rules out the specified buoyant vertical convection, not every possible motion.