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For the single-line greenhouse deficit, use hydrostatic balance, fixed surface pressure $p_s$, constant gravity $g$, and the dry adiabatThe gas constant and specific heat refer to the background gas. Let ${\mu }_d=\cos \tilde{\theta }>0$ be the supplied effective ray cosine. The vertical optical depth and slant transmittance from level $p$ to space areThis permits pressure-dependent line widths. With local thermodynamic equilibrium, negligible infrared scattering, a black thermal surface, and no incident thermal radiation from space, the formal radiative transfer solution isThe weights add to one. Subtract the transparent-surface radiance and put $B(p)=B_{{\nu }_0}(T(p))$, using the narrow-line approximation only in the line deficit. Define its pressure-dependent equivalent widthApproximating each hemispheric flux by $\pi$ times the effective-ray radiance preserves the black-surface flux. The outgoing longwave flux is thereforeIntegration by parts gives the particularly transparent formBoth boundary terms vanish: $E(0,q)=0$ and $B(p_s)-B(p_s)=0$. The dry adiabat and monotonic Planck function give $dB/dp>0$. Atmospheric emission replaces part of the warm surface radiation by emission from cooler levels, reducing emission to space for a specified $T_s$. The inequality is strict when opacity occurs over a nonzero range with temperature contrast; an isothermal absorbing layer at the surface temperature would not produce such a contrast.

For a spherical planet the mean absorbed instellation is $F_{\ast }=(1-\alpha )L_{⊛}/4$. The required global energy balance isConsequently every equilibrium of this model has $T_s\ge T_{\mathrm{bare}}=[(1-\alpha )L_{⊛}/(4{\sigma }_{\mathrm{SB}}){]}^{1/4}$, with strict warming when $D>0$. At fixed temperature profile, increasing $q$ increases $E$ and hence $D$. This argument does not assume that the greenhouse gas changes the stellar-band albedo, surface pressure, or background thermodynamics; those are held fixed. For sufficiently broad or overlapping saturated wings the full frequency-dependent Planck function must replace the narrow-line approximation.

For a unit mass of ideal gas, the first law of thermodynamics in enthalpy form is $\delta Q=c_{p}dT-vdp$. A reversible adiabatic displacement has $\delta Q=0$, and the ideal gas law gives $v=RT/p$. ThusThe atmospheric profile approaches this dry adiabat when convection efficiently mixes potential temperature and there is no condensation or other strong diabatic source. The first law by itself determines a parcel path, not every possible environmental temperature profile; the convective mixing assumption is needed. Hydrostatic balance gives the corresponding height lapse rate $dT/dz=-g/c_p$.

For rigid linear molecules with translation and rotation active but vibration frozen, the molecular heat capacities are $C_V=(5/2)R_u$ and $C_P=(7/2)R_u$, hence $R/c_p=2/7$. Earth is dominated by the diatomic gases nitrogen and oxygen. Mars is dominated by carbon dioxide, which is triatomic but linear and has the same two rotational degrees of freedom. In the rigid-molecule approximation it therefore has the same ratio. The value is not exact for carbon dioxide when its bending vibrations contribute appreciably to heat capacity, and variable heat capacities give a nonconstant dry-adiabatic exponent. Here $\gamma$ denotes $R/c_p$, not the distinct adiabatic index $c_p/c_v$.

A hydrostatic pressure interval contains $dp/(gM)$ moles of background gas per area. Its absorbing column is $x(p)dp/(gM)$, so for a nadir rayThe assumed Planck-temperature power and dry adiabat give $B(p)/B_0=[T(p)/T_0{]}^{3.5}=p/p_0$. To first order in optical depth, $-d\mathfrak{T}\simeq d\tau$, and the radiance solution becomesFor a uniform mixing ratio this also evaluates to $L\simeq B_0[1-k_{0}x{p}_0/(6gM)]$. Its total optical depth is ${\tau }_0=k_{0}x{p}_0/(2gM)$, so the deficit is $B_0{\tau }_0/3$.

A pressure-resolved perturbation defines the infrared absorber sensitivity kernel by $\delta L=\int K_p(p)\delta x(p)dp$, withIt is zero at the top, negative in the interior, and zero again at the surface. The magnitude is largest where $d{K}_p/dp=0$, namelySign and coordinate qualification. Literally the maximum of the signed kernel is zero at the endpoints; $p_0/2$ is its minimum and the maximum of its absolute value. The derivative with respect to a single vertically uniform $x$ is the integrated constant $-k_0{B}_0{p}_0/(6gM)$ and has no pressure location. For equal-width logarithmic pressure perturbations the kernel is instead $K_{\log p}=p{K}_p$ and its magnitude peaks at $2p_0/3$. The stated pressure-interval convention is therefore essential.

The low top-of-atmosphere sensitivity reflects the small pressure-broadened absorption coefficient and small molecular column per $dp$ there; the low surface sensitivity reflects small thermal contrast. The printed phrase that temperature decreases with pressure reverses the actual stipulated relation: with positive exponent, temperature decreases as pressure decreases upward. The derivation uses the explicit adiabatic formula.

Let $R_A,R_B$ be the specific specific gas constants and $c_{pA},c_{pB}$ the specific heat capacities of the noncondensible gas and vapour. Write $e_s(T)$ for the saturation vapor pressure, $r$ for vapour mass per unit dry-gas mass, and $ϵ=R_A/R_B$. For a dilute ideal gas mixture,The second equation is the Clausius-Clapeyron equation, neglecting condensed-phase volume and taking constant latent heat $L$. The first law of thermodynamics for saturated ascent, with condensate removed and negligible condensate heat capacity, is to leading orderSince $dr=r[LdT/(R_B{T}^2)-d\ln p]$, the resulting moist adiabat has logarithmic slopeConsequently a sufficient dry adiabat criterion isThe first two conditions justify neglecting the vapour's effects on the gas law and sensible heat capacity; the last two suppress latent heating. With $x=R_A/c_{pA}$ and $\beta =L/(R_{B}T)$, the latent conditions are $\beta e_s/p\ll 1$ and $x{\beta }^2{e}_s/p\ll 1$. Usually $x\beta \gg 1$, so the squared-latent term gives the stronger condition. At sufficiently low surface temperature, the exponential smallness of $e_s(T_s)$ makes these conditions hold andThese are local conditions along the column. They can be checked at the surface when $x{\beta }_s>1+2x$: on this dry profile the strongest latent parameter is proportional to $\exp [(1+2x)\zeta -{\beta }_s(e^{x\zeta }-1)]$ and decreases upwards. Without that restriction a surface-only test need not bound every altitude. If suspended condensate is retained, its own heat capacity and loading must also be negligible; saturation alone does not specify those quantities.