= Solution
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 $\epsilon=R_A/R_B$. For a dilute <ideal gas> mixture,
$$
r=\frac{\epsilon e_s}{p-e_s}\simeq\epsilon\frac{e_s}p,\qquad
\frac{d\ln e_s}{dT}=\frac{L}{R_BT^2}.
$$
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 order
$$
c_{pA}\,dT-R_AT\,d\ln p=-L\,dr.
$$
Since $dr=r[L\,dT/(R_BT^2)-d\ln p]$, the resulting <moist adiabat> has logarithmic slope
$$
\frac{d\ln T}{d\ln p}=\frac{R_A}{c_{pA}}\frac{1+Lr/(R_AT)}{1+L^2r/(c_{pA}R_BT^2)}.
$$
Consequently a sufficient <dry adiabat> criterion is
$$
\boxed{\frac{e_s}p\ll1,\quad \frac{r c_{pB}}{c_{pA}}\ll1,\quad\frac{Lr}{R_AT}\ll1,\quad\frac{L^2r}{c_{pA}R_BT^2}\ll1.}
$$
The 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 $\beta=L/(R_BT)$, the latent conditions are $\beta e_s/p\ll1$ and $x\beta^2e_s/p\ll1$. Usually $x\beta\gg1$, 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 and
$$
\boxed{T(p)\simeq T_s(p/p_s)^x.}
$$
These 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.
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