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Concept
Scale height
Summary
In atmospheric, earth, and planetary sciences, a scale height, usually denoted by the capital letter H, is a distance (vertical or radial) over which a physical 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
or equivalently
where:
kB = Boltzmann constant =
R = gas constant
T = mean atmospheric temperature in kelvins = 250 K for Earth
m = mean mass of a molecule (units kg)
M = mean mass of one mol of atmospheric particles = 0.029 kg/mol for Earth
g = acceleration due to gravity at the current location (m/s2)
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.
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 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 1.01e5Pa, the mean molecular mass of dry air is 28.964 u and hence m = 28.964 × 1.660e-27 = 4.808e-26kg. As a function of temperature, the scale height of Earth's atmosphere is therefore H/T = k/mg = (1.
Official source
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