The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude. The pressure drops approximately by 11.3 pascals per meter in first 1000 meters above sea level.
There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). The first equation is used when the value of standard temperature lapse rate is not equal to zero:
The second equation is used when standard temperature lapse rate equals zero:
where:
Or converted to imperial units:^{[1]}
where
The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In these equations, g_{0}, M and R^{*} are each single-valued constants, while P, L, T, and h are multivalued constants in accordance with the table below. The values used for M, g_{0}, and R^{*} are in accordance with the U.S. Standard Atmosphere, 1976, and the value for R^{*} in particular does not agree with standard values for this constant.^{[2]} The reference value for P_{b} for b = 0 is the defined sea level value, P_{0} = 101 325 Pa or 29.92126 inHg. Values of P_{b} of b = 1 through b = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when h = h_{b+1}.^{[2]}
Subscript b | Height above sea level | Static pressure | Standard temperature (K) |
Temperature lapse rate | |||
---|---|---|---|---|---|---|---|
(m) | (ft) | (Pa) | (inHg) | (K/m) | (K/ft) | ||
0 | 0 | 0 | 101 325.00 | 29.92126 | 288.15 | -0.0065 | -0.0019812 |
1 | 11 000 | 36,089 | 22 632.10 | 6.683245 | 216.65 | 0.0 | 0.0 |
2 | 20 000 | 65,617 | 5474.89 | 1.616734 | 216.65 | 0.001 | 0.0003048 |
3 | 32 000 | 104,987 | 868.02 | 0.2563258 | 228.65 | 0.0028 | 0.00085344 |
4 | 47 000 | 154,199 | 110.91 | 0.0327506 | 270.65 | 0.0 | 0.0 |
5 | 51 000 | 167,323 | 66.94 | 0.01976704 | 270.65 | -0.0028 | -0.00085344 |
6 | 71 000 | 232,940 | 3.96 | 0.00116833 | 214.65 | -0.002 | -0.0006096 |
The expressions for calculating density are nearly identical to calculating pressure. The only difference is the exponent in Equation 1.
There are two different equations for computing density at various height regimes below 86 geometric km (84 852 geopotential meters or 278 385.8 geopotential feet). The first equation is used when the value of standard temperature lapse rate is not equal to zero; the second equation is used when standard temperature lapse rate equals zero.
Equation 1:
Equation 2:
where
or, converted to English gravitational foot-pound-second units:^{[1]}
The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. The reference value for ?_{b} for b = 0 is the defined sea level value, ?_{0} = 1.2250 kg/m^{3} or 0.0023768908 slug/ft^{3}. Values of ?_{b} of b = 1 through b = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when h = h_{b+1}.^{[2]}
In these equations, g_{0}, M and R^{*} are each single-valued constants, while ?, L, T and h are multi-valued constants in accordance with the table below. The values used for M, g_{0} and R^{*} are in accordance with the U.S. Standard Atmosphere, 1976, and that the value for R^{*} in particular does not agree with standard values for this constant.^{[2]}
Subscript b | Height Above Sea Level (h) | Mass Density () | Standard Temperature (T') (K) |
Temperature Lapse Rate (L) | |||
---|---|---|---|---|---|---|---|
(m) | (ft) | (kg/m^{3}) | (slug/ft^{3}) | (K/m) | (K/ft) | ||
0 | 0 | 0 | 1.2250 | 2.3768908 x 10^{-3} | 288.15 | -0.0065 | -0.0019812 |
1 | 11 000 | 36,089.24 | 0.36391 | 7.0611703 x 10^{-4} | 216.65 | 0.0 | 0.0 |
2 | 20 000 | 65,616.79 | 0.08803 | 1.7081572 x 10^{-4} | 216.65 | 0.001 | 0.0003048 |
3 | 32 000 | 104,986.87 | 0.01322 | 2.5660735 x 10^{-5} | 228.65 | 0.0028 | 0.00085344 |
4 | 47 000 | 154,199.48 | 0.00143 | 2.7698702 x 10^{-6} | 270.65 | 0.0 | 0.0 |
5 | 51 000 | 167,322.83 | 0.00086 | 1.6717895 x 10^{-6} | 270.65 | -0.0028 | -0.00085344 |
6 | 71 000 | 232,939.63 | 0.000064 | 1.2458989 x 10^{-7} | 214.65 | -0.002 | -0.0006096 |
The barometric formula can be derived using the ideal gas law:
Assuming that all pressure is hydrostatic:
and dividing the by the expression we get:
Integrating this expression from the surface to the altitude z we get:
Assuming linear temperature change and constant molar mass and gravitational acceleration, we get the first barometric formula:
Instead, assuming constant temperature, integrating gives the second barometric formula:
In this formulation, R^{*} is the gas constant, and the term R^{*}T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere).
(For exact results, it should be remembered that atmospheres containing water do not behave as an ideal gas. See real gas or perfect gas or gas for further understanding.)