There are two different equations for computing pressure at various height regimes below 86 km. 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 temperaturelapse rate equals zero: where: Or converted to imperial units: 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, 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. The reference value for Pb for b = 0 is the defined sea level value, P0 = 101 325 Pa or 29.92126 inHg. Values of Pb 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 = hb+1.
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 different equations for computing density at various height regimes below 86 geometric km. 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: 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 ρb for b = 0 is the defined sea level value, ρ0 = 1.2250 kg/m3 or 0.0023768908 slug/ft3. Values of ρb 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 = hb+1. 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.
Derivation
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.
