# Scale height

In various scientific contexts, a scale height, usually denoted by the capital letter H, is a distance over which a quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718). The earth atmosphere's scale height is about 8.5km, as can be confirmed from this diagram of air pressure p by altitude h: At an altitude of 0, 8.5, and 17 km, the pressure is about 1000, 370, and 140 hPa, respectively.

## 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

$H={\frac {kT}{mg}}$

or equivalently

$H={\frac {RT}{Mg}}$

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:

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

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

$\rho ={\frac {MP}{RT}}$

Combining these 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 P0 is the pressure at height z = 0 (pressure at sea level) the pressure at height z can be written as:

$P=P_{0}\exp \left(-{\frac {z}{H}}\right)$

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 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 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, an atmosphere may no longer be well mixed. Then each chemical species has its own scale height.
• Here temperature and gravitational acceleration were assumed to be constant but both may vary over large distances.

## Planetary examples

Approximate atmospheric scale heights for selected Solar System bodies follow.