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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.
Contents
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]}
- <math>H = \frac{kT}{Mg}</math>
or equivalently
- <math>H = \frac{RT}{g}</math>
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:
- <math>\frac{dP}{dz} = -g\rho</math>
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
- <math>\rho = \frac{MP}{kT}</math>
Combining these equations gives
- <math>\frac{dP}{P} = \frac{-dz}{\frac{kT}{Mg}}</math>
which can then be incorporated with the equation for H given above to give:
- <math>\frac{dP}{P} = - \frac{dz}{H}</math>
which will not change unless the temperature does. Integrating the above and assuming where P_{0} is the pressure at height z = 0 (pressure at sea level) the pressure at height z can be written as:
- <math>P = P_0\exp\left(-\frac{z}{H}\right)</math>
This translates as the pressure decreasing exponentially with height.^{[4]}
In the 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 the 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 the 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 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]}
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).
- ^ "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.