Barometric Formula - Density Equations

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 (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

= Mass density (kg/m3)

= Standard temperature (K)

= Standard temperature lapse rate (see table below) (K/m) in ISA

= Height above sea level (geopotential meters)

= Universal gas constant for air: 8.31432 N·m/(mol·K)

= Gravitational acceleration (9.80665 m/s2)

= Molar mass of Earth's air (0.0289644 kg/mol)

Or converted to English gravitational foot-pound-second units:

Where

= Mass density (slug/ft3)

= Standard temperature (kelvins)

= Standard temperature lapse rate (degrees Celsius per foot)

= Height above sea level (geopotential feet)

= Universal gas constant (8.9494596×104 ft2/(s·°C))

= Gravitational acceleration (32.17405 ft/s2)

= 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 for b = 0 is the defined sea level value, = 1.2250 kg/m3 or 0.0023768908 slug/ft3. Values of 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

In these equations, g₀, 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, 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.