Contrail Threshold Temperature (Schmidt-Appleman Criterion)

An aircraft's exhaust mixes with the cold ambient air behind it. If the mixture passes through a state saturated over liquid water, droplets form and freeze into a contrail. The Schmidt-Appleman criterion marks the boundary: $T_{LM}$ is the warmest ambient temperature at which the mixing line of the exhaust just touches the saturation curve over liquid water. Air colder than $T_{LM}$ forms a contrail if it is also humid enough, and the contrail then persists if the air is supersaturated with respect to ice.

The exact condition is that the slope of the saturation curve over liquid water at $T_{LM}$ equals the mixing-line slope $G$ (Schumann 2012, appendix A5). The formula on this page is not that condition but a fit to it, Schumann's closed form from his 1996 paper, printed as equation 10 in his contrail cirrus model paper of 2012. $G$ is in pascals per kelvin and $T_{LM}$ in °C. The paper gives its error as below 0.07 K for $G$ between 0.24 and 23 Pa/K, so the result is an approximation that is reliable inside that range. The page's cross-check solves the exact condition by bisection, using the liquid-water equation of Murphy and Koop (2005).

The slope $G$ comes from the pressure, the fuel and the efficiency of the engine (the mixing-line-slope page). A higher overall efficiency raises $G$, because less of the fuel energy is left as heat in the exhaust, and so raises the threshold: a modern, efficient engine can leave contrails in a larger part of the sky than an old one. For kerosene at 250 hPa with an overall efficiency of 0.3 the slope is 1.64 Pa/K and the threshold is $-41.9$ °C. At $G = 1$ the fit's own arithmetic is $-46.46 + 9.43\ln(0.947) + 0.72\,[\ln(0.947)]^{2} = -46.46 - 0.51353 + 0.00214 = -46.97139$ °C. The exact condition gives $-46.946$ °C there, and $-39.827$ °C at $G = 2$, so the fit stays within the 0.1 K allowance.

This page covers only the temperature side. Whether a contrail forms also depends on how humid the ambient air is compared with what the mixing line allows, which the dew-point page covers, and whether it lasts depends on supersaturation with respect to ice, which the saturation vapour pressure over ice page covers. The cruise pressure that sets the slope in a standard atmosphere has its own page.