Scale height

In various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e (approximately 2.72, 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}{Mg}}

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:

{\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 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]

Uranus: 27.7 km^[11]
Neptune: 19.1–20.3 km^[12]
Pluto: ~60 km^[13]

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).^{[permanent dead link]}
^ "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. Retrieved 28 September 2013.
^ "Saturn Fact Sheet". NASA. 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.

[AMS-1] "Glossary of Meteorology - scale height". American Meteorological Society (AMS).

[wolfram-2] "Pressure Scale Height". Wolfram Research.

[Jacob1999-3] "Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999".

[iapetus_1-4] "Example: The scale height of the Earth's atmosphere" (PDF).^{[permanent dead link]}

[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.

[13] "Pluto Fact Sheet". NASA. Retrieved 2016-10-20.

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Scale height used in a simple atmospheric pressure model

Planetary examples

See also

References