Scale height

Scale height used in a simple atmospheric pressure model

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

H = k T M g

or equivalently

H = R T g

where:

k = Boltzmann constant = 1.38 x 10⁻²³ J·K⁻¹

R = Specific gas constant

T = mean atmospheric temperature in kelvins = 250 K for Earth

M = mean mass of a molecule (units kg)

g = acceleration due to gravity on planetary surface (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 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:

d P d z = − g ρ

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

ρ = M P k T

Combining these equations gives

d P P = − d z k T M g

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

d P P = − d z 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 0 exp ⁡ ( − z H )

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 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)×10³ = 29.26 m/deg. This yields the following scale heights for representative air temperatures.

T = 290 K, H = 8500 m T = 273 K, H = 8000 m T = 260 K, H = 7610 m T = 210 K, H = 6000 m

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 0.5³ = .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.

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.

Planetary examples

Approximate atmospheric scale heights for selected Solar System bodies follow.

Venus: 15.9 km

Earth: 8.5 km

Mars: 11.1 km

Jupiter: 27 km

Saturn: 59.5 km

Titan: 40 km

Uranus: 27.7 km

Neptune: 19.1–20.3 km

Pluto: ~60 km