Standard Atmosphere Density (Troposphere)

The standard atmosphere is not a weather forecast. It is an agreed fiction — a defined average that lets an altimeter be calibrated, an engine be rated, an aircraft's performance be published and two wind tunnels in different countries be compared. The troposphere layer of it, which is the part this equation covers, says the temperature starts at 15 °C at sea level and falls linearly at 6.5 K per kilometre, and that the air is in hydrostatic balance and behaves as an ideal gas.

Those assumptions are enough to derive everything else. Hydrostatic balance gives $dp = -\rho g\,dh$; the ideal gas law gives $\rho = p/RT$; the linear temperature profile gives $T$ as a function of $h$. Combining them and integrating produces the pressure as a power of the temperature ratio, and dividing by $RT$ once more gives the density with the exponent one lower: $\rho = \rho_0(1 - Lh/T_0)^{g/(RL)-1}$. The exponent works out to 4.2559, and the fact that it is not a round number is a small reminder that it is a derived quantity and not a fitted one.

The model stops at 11 km, and this page refuses to go past it. Above the tropopause the temperature stops falling and holds at −56.5 °C through the lower stratosphere, so the lapse rate $L$ becomes zero — and an equation with $L$ in the denominator of its exponent simply has nothing to say. The stratospheric layer uses an exponential decay with a different constant. The reason for a hard error rather than a quiet extrapolation is that the tropospheric formula does not blow up above 11 km; it returns a plausible, confidently wrong number, and a plausible wrong number is worse than no answer at all.

What a pilot actually needs from this is DENSITY ALTITUDE — the altitude at which the standard atmosphere has the density the aircraft is currently sitting in. It is what governs the takeoff roll, the climb rate and the power the engine makes, and on a hot day it can be thousands of feet above the field elevation. A 35 °C afternoon at a 1,500 m aerodrome gives the wing the air it would find above 2,500 m on a standard day, and the aircraft performs accordingly. Humidity works the same way and is nearly always ignored: water vapour is lighter than dry air, so humid air is thinner, though the effect is a small fraction of the temperature one.

One technical footnote for anyone comparing against a published table. This model's vertical coordinate is GEOPOTENTIAL altitude, which absorbs the weakening of gravity with height, and the familiar table values are indexed on GEOMETRIC altitude. At sea level the two coincide; at 11 km the geopotential figure sits about 19 m lower, which is why tables read 0.36480 kg/m³ at 11 km geometric and 0.36392 at 11 km geopotential. Below about 3 km the difference is not worth carrying.