In various scientific contexts, a scale height, usually denoted by the capital letter H, is a distance over which a quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718).
For planetary atmospheres, scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by
The pressure (force per unit area) at a given altitude is a result of the weight of the overlying atmosphere. If at a height of z the atmosphere has density ρ and pressure P, then moving upwards an infinitesimally small height dz will decrease the pressure by amount dP, equal to the weight of a layer of atmosphere of thickness dz.
where g is 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 an ideal gas of mean molecular mass M at temperature T, the density can be expressed as
Combining these equations gives
which can then be incorporated with the equation for H given above to give:
which will not change unless the temperature does. Integrating the above and assuming P0 is the pressure at height z = 0 (pressure at sea level) the pressure at height z can be written as:
This translates as the pressure decreasing exponentially with height.
In Earth's atmosphere, the pressure at sea level P0 averages about 1.01×105 Pa, 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 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.
These figures should be compared with the temperature and density of 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.
Approximate atmospheric scale heights for selected Solar System bodies follow.