Definitions

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

$H = frac\left\{kT\right\}\left\{Mg\right\}$

where:

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:

$frac\left\{dP\right\}\left\{dz\right\} = -grho$

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:

$rho = frac\left\{mP\right\}\left\{kT\right\}$

Therefore combining the equations gives

$frac\left\{dP\right\}\left\{P\right\} = frac\left\{-dz\right\}\left\{frac\left\{kT\right\}\left\{mg\right\}\right\}$

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

$frac\left\{dP\right\}\left\{P\right\} = - frac\left\{dz\right\}\left\{H\right\}$

which will not change unless the temperature does. Integrating the above and assuming where P0 is the pressure at height z = 0 (pressure at sea level) the pressure at height z can be written as:

$P = P_0e^\left\{\left(-frac\left\{z\right\}\left\{H\right\}\right)\right\}$

This translates as the pressure decreasing exponentially with height.

In the Earth's atmosphere, the pressure at sea level P0 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:

1. 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 ρ0 roughly equal to 1.2 kg m−3
2. At heights over 100 km, molecular diffusion means that each molecular atomic species has its own scale height.

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