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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.

Let $e_v$ be the parcel's water vapor partial pressure. Fixed mole fraction before condensation means $e_v/p$ is constant. Its dew point $T_D$ is defined by $E(T_D)=e_v$, where $E$ is the saturation pressure. The Clausius-Clapeyron relation givesThe last approximation uses hydrostatic balance, the ideal gas law, and an environmental temperature close to the parcel's. Rearrangement yieldsFor a modest initial dew-point depression, $T_D/T\simeq 1$ in absolute temperature, soWith $L_v\simeq 2.5\times 10^6\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, $R_d\simeq 287\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, and $R_v\simeq 461.5\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, $H\simeq 1.58\times 10^5\mathrm{m}$. This dew point scale is much longer than the atmospheric pressure scale height.

The exponential function is an approximation, not the exact consequence of a dry adiabatic process. Retaining $T=T_s-{\Gamma }_{d}z$ gives $p/p_s=(T/T_s{)}^{c_p/R_d}$ and the useful comparisonIt also shows explicitly that expansion lowers the dew point even before any water vapor condenses.

A scale height is a characteristic distance over which a vertically stratified physical quantity varies. For a decreasing exponential $q(z)=q_0{e}^{-z/H}$, it is $H=-(d\log q/dz{)}^{-1}$. The local pressure scale height applies this logarithmic measure to pressure; density and pressure scale heights need not coincide when temperature varies.