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| In various scientific contexts, a '''scale height''' is a distance over which a quantity decreases by a factor of ''[[e (mathematical constant)|e]]'' (approximately 2.71828, the base of [[natural logarithms]]). It is usually denoted by the capital letter ''H''.
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| ==Scale height used in a simple atmospheric pressure model==
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| For planetary atmospheres, '''scale height''' is the vertical distance over which the [[pressure]] of the atmosphere changes by a factor of ''e'' (decreasing upward). The scale height remains constant for a particular temperature. It can be calculated by<ref name=AMS>
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| {{cite web
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| |title= Glossary of Meteorology - scale height
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| |url=http://amsglossary.allenpress.com/glossary/search?id=scale-height1
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| |publisher= [[American Meteorological Society]] (AMS)
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| }}</ref><ref name=wolfram>
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| {{cite web
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| |title= Pressure Scale Height
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| |url=http://scienceworld.wolfram.com/physics/PressureScaleHeight.html
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| |publisher= [[Wolfram Research]]
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| }}</ref>
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| :<math>H = \frac{kT}{Mg}</math>
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| where:
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| * ''k'' = [[Boltzmann constant]] = 1.38 x 10<sup>−23</sup> J·K<sup>−1</sup>
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| * ''T'' = mean atmospheric [[temperature]] in [[kelvin]]s = 250 K<ref name=Jacob1999>
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| {{cite web
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| |title= Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999
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| |url=http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap2.html
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| }}</ref>
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| * ''M'' = mean [[molecular mass]] of dry air (units kg)
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| * ''g'' = [[acceleration]] due to [[gravity]] on planetary surface (m/s²)
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| 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''.
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| Thus:
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| :<math>\frac{dP}{dz} = -g\rho</math>
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| 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
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| :<math>\rho = \frac{MP}{kT}</math>
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| Combining these equations gives
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| :<math>\frac{dP}{P} = \frac{-dz}{\frac{kT}{Mg}}</math>
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| which can then be incorporated with the equation for ''H'' given above to give:
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| :<math>\frac{dP}{P} = - \frac{dz}{H}</math>
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| which will not change unless the temperature does. Integrating the above and assuming where ''P''<sub>0</sub> is the pressure at height ''z'' = 0 (pressure at [[sea level]]) the pressure at height ''z'' can be written as:
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| :<math>P = P_0\exp(-\frac{z}{H})</math>
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| This translates as the pressure [[exponential decay|decreasing exponentially]] with height.<ref name=iapetus_1>
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| {{cite web
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| |title= Example: The scale height of the Earth's atmosphere
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| |url=http://iapetus.phy.umist.ac.uk/Teaching/SolarSystem/WorkedExample4.pdf
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| }}</ref>
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| In the [[Earth's atmosphere]], the pressure at sea level ''P''<sub>0</sub> averages about 1.01×10<sup>5</sup> Pa, the mean molecular mass of dry air is 28.964 [[unified atomic mass unit|u]] and hence 28.964 × 1.660×10<sup>−27</sup> = 4.808×10<sup>−26</sup> kg, and ''[[standard gravity|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<sup>3</sup> = 29.26 m/deg. This yields the following scale heights for representative air temperatures.
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| :''T'' = 290 K, ''H'' = 8500 m
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| :''T'' = 273 K, ''H'' = 8000 m
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| :''T'' = 260 K, ''H'' = 7610 m
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| :''T'' = 210 K, ''H'' = 6000 m
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| 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<sup>3</sup> at sea level to 0.5<sup>3</sup> = .125 g/m<sup>3</sup> 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.
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| Note:
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| * 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''<sub>0</sub> roughly equal to 1.2 kg m<sup>−3</sup>
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| * At heights over 100 km, molecular [[diffusion]] means that each molecular atomic species has its own scale height.
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| ==Planetary examples==
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| Approximate scale heights for selected Solar System bodies follow.
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| *[[Venus]] : 15.9 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html|title=Venus Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| *[[Earth]] : 8.5 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html|title=Earth Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| *[[Mars]] : 11.1 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html|title=Mars Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| *[[Jupiter]] : 27 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html|title=Jupiter Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| *[[Saturn]] : 59.5 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html|title=Saturn Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| :*[[Titan (moon)|Titan]] : 40 km<ref>{{cite web|url=http://www.mrc.uidaho.edu/entryws/presentations/Papers/Justus.doc|title=Engineering-Level Model Atmospheres For Titan and Mars|last=Justus|first=C. G.|coauthors=Aleta Duvall, Vernon W. Keller|date=November 2003 and February 2004|work=International Workshop on Planetary Probe Atmospheric Entry and Descent Trajectory Analysis and Science, Lisbon, Portugal, October 6-9, 2003, Proceedings: ESA SP-544|publisher=ESA|accessdate=28 September 2013}}</ref>
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| *[[Uranus]] : 27.7 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/uranusfact.html|title=Uranus Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| *[[Neptune]] : 19.1 - 20.3 km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html|title=Neptune Fact Sheet|publisher=NASA|accessdate=28 September 2013}}</ref>
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| ==See also==
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| *[[Time constant]]
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| ==References==
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| {{reflist|colwidth=30em}}
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| [[Category:Atmosphere]]
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| [[Category:Atmospheric dynamics]]
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