Barometric formula

The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude.

Pressure equations

There are two equations for computing pressure as a function of height. The first equation is applicable to the atmospheric layers in which the temperature is assumed to vary with altitude at a non null lapse rate of $L_{b}$ : $P=P_{b}\left[1-{\frac {L_{M,b}}{T_{M,b}}}(h-h_{b})\right]^{\frac {g_{0}'M_{0}}{R^{*}L_{M,b}}}$ The second equation is applicable to the atmospheric layers in which the temperature is assumed not to vary^{[citation needed]} with altitude (lapse rate is null): $P=P_{b}\exp \left[{\frac {-g_{0}M\left(h-h_{b}\right)}{R^{*}{T_{M,b}}}}\right]$ where:

$P_{b}$ = reference pressure
$T_{M,b}$ = reference temperature (K)
$L_{M,b}$ = temperature lapse rate (K/m) in ISA
$h$ = geopotential height at which pressure is calculated (m)
$h_{b}$ = geopotential height of reference level b (meters; e.g., h_b = 11 000 m)
$R^{*}$ = universal gas constant: 8.3144598 J/(mol·K)
$g_{0}$ = gravitational acceleration: 9.80665 m/s²
$M$ = molar mass of Earth's air: 0.0289644 kg/mol

Or converted to imperial units:^[1]

$P_{b}$ = reference pressure
$T_{M,b}$ = reference temperature (K)
$L_{M,b}$ = temperature lapse rate (K/ft) in ISA
$h$ = height at which pressure is calculated (ft)
$h_{b}$ = height of reference level b (feet; e.g., h_b = 36,089 ft)
$R^{*}$ = universal gas constant; using feet, kelvins, and (SI) moles: 8.9494596×10⁴ lb·ft²/(lb-mol·K·s²)
$g_{0}$ = gravitational acceleration: 32.17405 ft/s²
$M$ = molar mass of Earth's air: 28.9644 lb/lb-mol

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, g₀, 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, g₀, 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, P₀ = 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	Geopotential height above MSL (h)		Static pressure		Standard temperature (K)	Temperature lapse rate		Exponent g0 M / R L
Subscript b	(m)	(ft)	(Pa)	(inHg)	Standard temperature (K)	(K/m)	(K/ft)	Exponent g0 M / R L
0	0	0	101 325.00	29.92126	288.15	0.0065	0.0019812	5.25588
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	-34.1626
3	32 000	104,987	868.02	0.2563258	228.65	-0.0028	-0.00085344	-12.2009
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	12.2009
6	71 000	232,940	3.96	0.00116833	214.65	0.002	0.0006096	17.0813

Density equations

The expressions for calculating density are nearly identical to calculating pressure. The only difference is the exponent in Equation 1.

There are two equations for computing density as a function of height. The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of $L_{b}$ ; the second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude.

Equation 1: $\rho =\rho _{b}\left[{\frac {T_{b}-(h-h_{b})L_{b}}{T_{b}}}\right]^{\left({\frac {g_{0}M}{R^{*}L_{b}}}-1\right)}$

which is equivalent to the ratio of the relative pressure and temperature changes

$\rho =\rho _{b}{\frac {P}{T}}{\frac {T_{b}}{P_{b}}}$

Equation 2: $\rho =\rho _{b}\exp \left[{\frac {-g_{0}M\left(h-h_{b}\right)}{R^{*}T_{b}}}\right]$

where

${\rho }$ = mass density (kg/m³)
$T_{b}$ = standard temperature (K)
$L$ = standard temperature lapse rate (see table below) (K/m) in ISA
$h$ = height above sea level (geopotential meters)
$R^{*}$ = universal gas constant 8.3144598 N·m/(mol·K)
$g_{0}$ = gravitational acceleration: 9.80665 m/s²
$M$ = molar mass of Earth's air: 0.0289644 kg/mol

or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.):^[1]

${\rho }$ = mass density (slug/ft³)
${T_{b}}$ = standard temperature (K)
${L}$ = standard temperature lapse rate (K/ft)
${h}$ = height above sea level (geopotential feet)
${R^{*}}$ = universal gas constant: 8.9494596×10⁴ ft²/(s·K)
${g_{0}}$ = gravitational acceleration: 32.17405 ft/s²
${M}$ = molar mass of Earth's air: 0.0289644 kg/mol

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 ρ_b for b = 0 is the defined sea level value, ρ₀ = 1.2250 kg/m³ or 0.0023768908 slug/ft³. Values of ρ_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, g₀, M and R^* are each single-valued constants, while ρ, L, T and h are multi-valued constants in accordance with the table below. The values used for M, g₀ 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	Geopotential height above MSL (h)		Mass Density ( $\rho$ )		Standard Temperature (T') (K)	Temperature Lapse Rate (L)
Subscript b	(m)	(ft)	(kg/m³)	(slug/ft³)	Standard Temperature (T') (K)	(K/m)	(K/ft)
0	0	0	1.2250	2.3768908×10⁻³	288.15	0.0065	0.0019812
1	11 000	36,089.24	0.36391	7.0611703×10⁻⁴	216.65	0.0	0.0
2	20 000	65,616.79	0.08803	1.7081572×10⁻⁴	216.65	-0.001	-0.0003048
3	32 000	104,986.87	0.01322	2.5660735×10⁻⁵	228.65	-0.0028	-0.00085344
4	47 000	154,199.48	0.00143	2.7698702×10⁻⁶	270.65	0.0	0.0
5	51 000	167,322.83	0.00086	1.6717895×10⁻⁶	270.65	0.0028	0.00085344
6	71 000	232,939.63	0.000064	1.2458989×10⁻⁷	214.65	0.002	0.0006096

Derivation

The barometric formula can be derived using the ideal gas law: $P={\frac {\rho }{M}}{R^{*}}T$

Assuming that all pressure is hydrostatic: $dP=-\rho g\,dz$ and dividing this equation by $P$ we get: ${\frac {dP}{P}}=-{\frac {Mg\,dz}{R^{*}T}}$

Integrating this expression from the surface to the altitude z we get: $P=P_{0}e^{-\int _{0}^{z}{Mgdz/R^{*}T}}$

Assuming linear temperature change $T=T_{0}-Lz$ and constant molar mass and gravitational acceleration, we get the first barometric formula: $P=P_{0}\cdot \left[{\frac {T}{T_{0}}}\right]^{\textstyle {\frac {Mg}{R^{*}L}}}$

Instead, assuming constant temperature, integrating gives the second barometric formula: $P=P_{0}e^{-Mgz/R^{*}T}$

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.)

References

^ ^a ^b Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C.
^ ^a ^b ^c ^d U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976. (Linked file is 17 Mb)

[conversion-1] Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C.

[USSA1976-2] U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976. (Linked file is 17 Mb)

[1]

[2]

Pressure equations

Density equations

Derivation

See also

References