Scale height

In various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e (approximately 2.71828, the base of natural logarithms). It is usually denoted by the capital letter H.

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]

H = \frac{kT}{Mg}

or equivalently

H = \frac{RT}{g}

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 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:

\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}{kT}

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 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\left(-\frac{z}{H}\right)

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

In the 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 the 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 the 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:

Planetary examples

Approximate scale heights for selected Solar System bodies follow.

See also

References

  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).
  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. Retrieved 28 September 2013.
  9. ↑ "Saturn Fact Sheet". NASA. 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.