In various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e (approximately 2.71828, the base of natural logarithms). It is usually denoted by the capital letter H.
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^{[1]}^{[2]}
or equivalently
where:
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 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 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 where P_{0} 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.^{[4]}
In Earth's atmosphere, the pressure at sea level P_{0} averages about 1.01×10^{5} 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)×10^{3} = 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/m^{3} at sea level to 0.5^{3} = .125 g/m^{3} 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:
Approximate atmospheric scale heights for selected Solar System bodies follow.
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