Scale height

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 $${\displaystyle H={\frac {k_{\text{B}}T}{mg}},}$$ or equivalently, $${\displaystyle H={\frac {RT}{Mg}},}$$ 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 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: $${\displaystyle {\frac {dP}{dz}}=-g\rho ,}$$ 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 $${\displaystyle \rho ={\frac {MP}{RT}}.}$$

Combining these equations gives $${\displaystyle {\frac {dP}{P}}={\frac {-dz}{{k_{\text{B}}T}/{mg}}},}$$ which can then be incorporated with the equation for H given above to give $${\displaystyle {\frac {dP}{P}}=-{\frac {dz}{H}},}$$ 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 $${\displaystyle P=P_{0}\exp \left(-{\frac {z}{H}}\right).}$$ 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 Da, and hence m = 28.964 Da × 1.660×10⁻²⁷ kg/Da = 4.808×10⁻²⁶ kg. As a function of temperature, the scale height of Earth's atmosphere is therefore H/T = k_B/mg = 1.381×10⁻²³ J⋅K⁻¹ / (4.808×10⁻²⁶ kg × 9.81 m⋅s⁻²) = 29.28 m/K. 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 0.125 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.

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