Scale height

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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.

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[edit]

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


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.


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 P0 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 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.


  • 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[edit]

Approximate atmospheric scale heights for selected Solar System bodies follow.

See also[edit]


  1. ^ "Glossary of Meteorology - scale height". American Meteorological Society (AMS).
  2. ^ "Pressure Scale Height". Wolfram Research.
  3. ^ "Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999".
  4. ^ "Example: The scale height of the Earth's atmosphere" (PDF). Archived from the original (PDF) on 2011-07-16.
  5. ^ "Venus Fact Sheet". NASA. Retrieved 28 September 2013.
  6. ^ "Earth Fact Sheet". NASA. Retrieved 28 September 2013.
  7. ^ "Mars Fact Sheet". NASA. Retrieved 28 September 2013.
  8. ^ "Jupiter Fact Sheet". NASA. Archived from the original on 5 October 2011. Retrieved 28 September 2013.
  9. ^ "Saturn Fact Sheet". NASA. Archived from the original on 21 August 2011. Retrieved 28 September 2013.
  10. ^ 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.
  11. ^ "Uranus Fact Sheet". NASA. Retrieved 28 September 2013.
  12. ^ "Neptune Fact Sheet". NASA. Retrieved 28 September 2013.
  13. ^ "Pluto Fact Sheet". NASA. Retrieved 2020-09-28.