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:

• k = Boltzmann constant = 1.38 x 10^−23 J·K^−1
• R = Specific gas constant
• T = mean atmospheric temperature in kelvins = 250 K^[3] for Earth
• M = mean mass of a molecule (units kg)
• 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:

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

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 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—with some departures caused by varying temperature—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, molecular diffusion means that each molecular atomic species has its own scale height.

Planetary examples

Approximate 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: 40 km^[10]

• Uranus: 27.7 km^[11]
• Neptune: 19.1–20.3 km^[12]

See also

• Time constant

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.