Scale height

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A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of e. It is usually denoted by the capital letter H.

For planetary atmospheres, it is the vertical distance upwards, over which the pressure of the atmosphere decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by

H = \frac{kT}{Mg}

where:

The pressure in the atmosphere is caused by the weight of the atmosphere of the overlying atmosphere [force per unit area]. 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:

dP = − gρdz

where g is used to denote 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 a perfect gas of mean molecular mass m at temperature T, the density can be expressed as such:

\rho = \frac{mP}{kT}

Therefore combining the equations gives

\frac{dP}{P} = \frac{-dz}{\frac{kT}{mg}}

which can then be incorporated with the equation for H given above to give:

\frac{dP}{P} = - \frac{dz}{H}

which will not change unless the temperature does. Integrating the above and assuming where P0 is the pressure at height z = 0 (pressure at sea level) the pressure at height z can be written as:

P = P_0e^{(-\frac{z}{H})}

This translates as the pressure decreasing exponentially with height.

In the earths atmosphere the pressure at sea level P0 roughly equals 1.01×105Pa and the mean molecular mass of dry air is 28.964 u (1 u = 1.660×10−27 kg).

examples:

T = 290 K, H = 8500 m
T = 210 K, H = 6000 m

Note:

  1. 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
  2. At heights over 100 km, molecular diffusion means that each molecular atomic species has its own scale height.
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