# 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).

## Scale height used in a simple atmospheric pressure modelEdit

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^{[1]}^{[2]}

or equivalently

where:

*k*= Boltzmann constant = 1.38 x 10^{−23}J·K^{−1}*R*= gas constant*T*= mean atmospheric temperature in kelvins = 250 K^{[3]}for Earth*m*= mean mass of a molecule (units kg)*M*= mean mass of one mol of atmospheric particles = 0.029 kg/mol for Earth*g*= acceleration due to gravity at the current location (m/s^{2})

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:

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

Combining these equations gives

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

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

This translates as the pressure decreasing exponentially with height.^{[4]}

In Earth's atmosphere, the pressure at sea level *P*_{0} averages about 1.01×10^{5} 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)×10^{3} = 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/m^{3} at sea level to 0.5^{3} = .125 g/m^{3} 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
*ρ*_{0}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 examplesEdit

Approximate atmospheric scale heights for selected Solar System bodies follow.

- Venus: 15.9 km
^{[5]} - Earth: 8.5 km
^{[6]} - Mars: 11.1 km
^{[7]} - Jupiter: 27 km
^{[8]} - Saturn: 59.5 km
^{[9]}

- Titan: 21 km
^{[10]}

- Titan: 21 km

## See alsoEdit

## ReferencesEdit

**^**"Glossary of Meteorology - scale height". American Meteorological Society (AMS).**^**"Pressure Scale Height". Wolfram Research.**^**"Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999".**^**"Example: The scale height of the Earth's atmosphere" (PDF). Archived from the original (PDF) on 2011-07-16.**^**"Venus Fact Sheet". NASA. Retrieved 28 September 2013.**^**"Earth Fact Sheet". NASA. Retrieved 28 September 2013.**^**"Mars Fact Sheet". NASA. Retrieved 28 September 2013.**^**"Jupiter Fact Sheet". NASA. Archived from the original on 5 October 2011. Retrieved 28 September 2013.**^**"Saturn Fact Sheet". NASA. Archived from the original on 21 August 2011. Retrieved 28 September 2013.**^**Justus, C. G.; Aleta Duvall; Vernon W. Keller (1 August 2003). "Engineering-Level Model Atmospheres For Titan and Mars".*International Workshop on Planetary Probe Atmospheric Entry and Descent Trajectory Analysis and Science, Lisbon, Portugal, October 6–9, 2003, Proceedings: ESA SP-544*. ESA. Retrieved 28 September 2013.**^**"Uranus Fact Sheet". NASA. Retrieved 28 September 2013.**^**"Neptune Fact Sheet". NASA. Retrieved 28 September 2013.**^**"Pluto Fact Sheet". NASA. Retrieved 2020-09-28.