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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.

For typical well-mixed structures, the dominant processes are:

Representative temperatures must specify the level: Earth has about $288\mathrm{K}$ at the surface (about $255\mathrm{K}$ effective emission temperature); Jupiter has about $165\mathrm{K}$ near one bar (about $125\mathrm{K}$ effective temperature); hot Jupiters commonly have photospheric temperatures of order $1000$–$2500\mathrm{K}$; and the Sun's photosphere is about $5800\mathrm{K}$. Upper layers, nightsides, deep interiors and the solar corona have different temperatures. These are characteristic values, not constant temperatures throughout each atmosphere.

The transition above a planetary troposphere, usually near a temperature minimum or the change to a weakly varying or outward-increasing temperature profile. The overlying planetary stratosphere is stable against ordinary thermal convection. Its pressure is planet- and location-dependent.