In various scientific contexts, a scale height, usually denoted by the capital letterH, is a distance over which a quantity decreases by a factor of e.
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 or equivalently where:
The pressure at a given altitude is a result of the weight of the overlying atmosphere. If at a height of zthe atmosphere has densityρ 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: 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 massM 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 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 levelP0 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/×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/m3at sea level to 0.53 =.125 g/m3 at 70 km, a factor of 9600, indicating an average scale height of 70/ln = 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 ρ0 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.
Planetary examples
Approximate atmospheric scale heights for selected Solar System bodies follow.
