The barometric formula is a formula used to model how the air pressure (or air density) changes with altitude.
Model equations
The U.S. Standard Atmosphere gives two equations for computing pressure as a function of height, valid from sea level to 86 km altitude. The first equation is applicable to the atmospheric layers in which the temperature is assumed to vary with altitude at a non-zero temperature gradient of L M , b {\displaystyle L_{M,b}} :
P = P b ⋅ [ T M , b T M , b + L M , b ⋅ ( H − H b ) ] g 0 ′ ⋅ M 0 R ∗ ⋅ L M , b . {\displaystyle P=P_{b}\cdot \left[{\frac {T_{M,b}}{T_{M,b}+L_{M,b}\cdot (H-H_{b})}}\right]^{\frac {g_{0}'\cdot M_{0}}{R^{*}\cdot L_{M,b}}}.}
The second equation is applicable to the atmospheric layers in which the temperature is assumed not to vary with altitude (zero temperature gradient):
P = P b ⋅ exp [ − g 0 ′ ⋅ M 0 ( H − H b ) R ∗ ⋅ T M , b ] . {\displaystyle P=P_{b}\cdot \exp \left[{\frac {-g_{0}'\cdot M_{0}(H-H_{b})}{R^{*}\cdot T_{M,b}}}\right].}
In both equations:
P b {\displaystyle P_{b}} = reference pressure;
T M , b {\displaystyle T_{M,b}} = reference temperature (K);
L M , b {\displaystyle L_{M,b}} = temperature gradient (K/m), e.g. -6.5 K/km at sea level (this is the lapse rate with the opposite sign convention);
H {\displaystyle H} = geopotential height at which pressure is calculated (m);
H b {\displaystyle H_{b}} = geopotential height of reference level b (meters, e.g., Hb = 11000 m);
R ∗ {\displaystyle R^{*}} = universal gas constant, taken to be 8.31432×103 J/(kmol·K), although the actual constant's value in those units rounds to 8.31446;
M 0 {\displaystyle M_{0}} = mean molar mass of air at sea level = 28.9644 kg/kmol as of 1976;
g 0 ′ {\displaystyle g_{0}'} = the gravitational acceleration in units of geopotential height = 9.80665 m/s2. Or converted to imperial units:
P b {\displaystyle P_{b}} = reference pressure;
T M , b {\displaystyle T_{M,b}} = reference temperature (K);
L M , b {\displaystyle L_{M,b}} = temperature gradient (K/ft);
H {\displaystyle H} = height at which pressure is calculated (ft);
H b {\displaystyle H_{b}} = height of reference level b (feet, e.g., Hb = 36089 ft);
R ∗ {\displaystyle R^{*}} = universal gas constant, using feet, kelvins, and (SI) moles, taken to be roughly 8.9494596×104 lbm·ft2/(lbm-mol·K·s2) by correctly converting the (incorrectly) taken constant from metric to imperial;
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