Scale height

A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of "e." It is usually denoted by the capital letter "H".

For planetary atmospheres, it is the vertical distance upwards, over which the pressure of the atmosphere decreases by a factor of "e." The scale height remains constant for a particular temperature. It can be calculated by

:$H\; =\; frac\{kT\}\{Mg\}$

where:

* "k" = Boltzmann constant = 1.38 x 10⁻²³ J·K⁻¹
* "T" = mean planetary surface temperature in kelvins
* "M" = mean molecular mass of dry air (units kg)
* "g" = acceleration due to gravity on planetary surface (m/s²)

The pressure in the atmosphere is caused by the weight on the atmosphere of the overlying atmosphere [force per unit area] . If at a height of "z" the atmosphere has density "ρ" and pressure "P", then moving upwards at an infinitesimally small height "dz" will decrease the pressure by amount "dP", equal to the weight of a layer of atmosphere of thickness "dz".

Thus:

:$frac\{dP\}\{dz\}\; =\; -g\; ho$

where "g" is used to denote the acceleration due to gravity. For small "dz" it is possible to assume "g" to be constant; the minus sign indicates that as the height increases the pressure decreases. Therefore using the equation of state for a perfect gas of mean molecular mass "m" at temperature "T," the density can be expressed as such:

:$ho\; =\; frac\{mP\}\{kT\}$

Therefore combining the equations gives

:$frac\{dP\}\{P\}\; =\; frac\{-dz\}\{frac\{kT\}\{mg$

which can then be incorporated with the equation for "H" given above to give:

:$frac\{dP\}\{P\}\; =\; -\; frac\{dz\}\{H\}$

which will not change unless the temperature does. Integrating the above and assuming where "P"₀ is the pressure at height "z" = 0 (pressure at sea level) the pressure at height "z" can be written as:

:$P\; =\; P\_0e^\{(-frac\{z\}\{H\})\}$

This translates as the pressure decreasing exponentially with height.

In the Earth's atmosphere, the pressure at sea level "P"₀ roughly equals 1.01×10⁵Pa, the mean molecular mass of dry air is 28.964 u and hence 28.964 × 1.660×10⁻²⁷ = 4.808×10⁻²⁶ kg, and "g" = 9.81 m/s². As a function of temperature the scale height of the Earth's atmosphere is therefore 1.38/(4.808×9.81)×10³ = 29.26 m/deg. This yields the following scale heights for representative air temperatures.

:"T" = 290 K, "H" = 8500 m:"T" = 273 K, "H" = 8000 m:"T" = 260 K, "H" = 7610 m:"T" = 210 K, "H" = 6000 m

These figures should be compared with the temperature and density of the Earth's atmosphere plotted at NRLMSISE-00, which shows the air density dropping from 1200 g/m³ at sea level to 0.5³ = .125 g/m³ at 70 km, a factor of 9600, indicating an average scale height of 70/ln(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K.

Note:
# Density is related to pressure by the ideal gas laws. Therefore with some departures caused by varying temperature—density will also decrease exponentially with height from a sea level value of "ρ"₀ roughly equal to 1.2 kg m⁻³
# At heights over 100 km, molecular diffusion means that each molecular atomic species has its own scale height.