Scale_height

Scale height

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:

k = Boltzmann constant = 1.38 x 10⁻²³ J·K⁻¹
T = mean planetary surface temperature in kelvins
M = mean molecular mass of dry air (units kg·mol⁻¹)
g = acceleration due to gravity on planetary surface (m/s²)

The pressure in the atmosphere is caused by the weight on 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:

$\frac{dP}{dz} = -g\rho$

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 Earth's 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).

For example:

T = 290 K, H = 8500 m

T = 210 K, H = 6000 m

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.