# 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 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[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 molecular molar mass of one atmospheric particle = 0.029 kg/mol for Earth*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:

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—with some departures caused by varying temperature—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, 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.

## See also

## References

- "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 2016-10-20.