{{Short description|Distance over which a quantity decreases by a factor of e}} [[File:Pressure air.svg|thumb|300px|The earth atmosphere's scale height is about 8.5 km, as can be confirmed from this diagram of air pressure ''p'' by altitude ''h'': At an altitude of 0, 8.5, and 17 km, the pressure is about 1000, 370, and 140 hPa, respectively.]]
In physics, a '''scale height''', usually denoted by the capital letter ''H'', is a distance (vertical or radial) over which a physical quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718).<ref>{{Cite book |title=Encyclopedia of Astrobiology |date=2023-07-01 |publisher=Springer |isbn=978-3662650929 |edition=3rd |pages=2720 |chapter=S |access-date=2026-04-29 |chapter-url=https://link.springer.com/content/pdf/10.1007/978-3-662-65093-6_1406.pdf}}</ref>
== 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<ref name=AMS> {{cite web |title= Glossary of Meteorology - scale height |url=http://glossary.ametsoc.org/wiki/Scale_height |publisher= American Meteorological Society (AMS) }}</ref><ref name=wolfram> {{cite web |title= Pressure Scale Height |url=http://scienceworld.wolfram.com/physics/PressureScaleHeight.html |publisher= Wolfram Research }}</ref> <math display="block"> H = \frac{k_\text{B}T}{mg}, </math> or equivalently, <math display="block"> H = \frac{RT}{Mg}, </math> where : ''k''<sub>B</sub> = Boltzmann constant = {{physconst|k|round=3}} : ''R'' = molar gas constant = 8.31446 J⋅K<sup>−1</sup>⋅mol<sup>−1</sup> : ''T'' = mean atmospheric temperature in kelvins = 250 K<ref name=Jacob1999>{{cite web |title= Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999 |url= http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap2.html |access-date= 2013-04-18 |archive-date= 2013-04-10 |archive-url= https://web.archive.org/web/20130410233524/http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap2.html |url-status= dead }}</ref> for Earth : ''m'' = mean mass of a molecule : ''M'' = mean molar mass of atmospheric particles = 0.029 kg/mol for Earth : ''g'' = acceleration due to gravity at the current location
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 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 display="block"> \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 display="block"> \rho = \frac{MP}{RT}.</math>
Combining these equations gives <math display="block"> \frac{dP}{P} = \frac{-dz}{{k_\text{B}T}/{mg}},</math> which can then be incorporated with the equation for ''H'' given above to give <math display="block"> \frac{dP}{P} = - \frac{dz}{H},</math> which will not change unless the temperature does. Integrating the above and assuming ''P''<sub>0</sub> is the pressure at height ''z'' = 0 (pressure at sea level), the pressure at height ''z'' can be written as <math display="block"> P = P_0\exp\left(-\frac{z}{H}\right).</math> This translates as the pressure decreasing exponentially with height.<ref name="iapetus_1">{{cite web | title = Example: The scale height of the Earth's atmosphere | url = http://iapetus.phy.umist.ac.uk/Teaching/SolarSystem/WorkedExample4.pdf | archive-url = https://web.archive.org/web/20110716205659/http://iapetus.phy.umist.ac.uk/Teaching/SolarSystem/WorkedExample4.pdf | url-status = dead | archive-date = 2011-07-16 }}</ref>
In Earth's atmosphere, the pressure at sea level ''P''<sub>0</sub> averages about {{val|1.01|e=5|u=Pa}}, the mean molecular mass of dry air is {{val|28.964|ul=Da}}, and hence ''m'' = {{val|28.964|u=Da}} × {{val|1.660|e=-27|u=kg/Da}} = {{val|4.808|e=-26|u=kg}}. As a function of temperature, the scale height of Earth's atmosphere is therefore ''H''/''T'' = ''k''<sub>B</sub>/''mg'' = {{val|1.381|e=-23|u=J.K-1}} / ({{val|4.808|e=-26|u=kg}} × {{val|9.81|u=m.s-2}}) = {{val|29.28|u=m/K}}. 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/m<sup>3</sup> at sea level to 0.125 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.
Note: * Density is related to pressure by the ideal gas laws. Therefore, density will also decrease exponentially with height from a sea-level value of ''ρ''<sub>0</sub> roughly equal to {{val|1.2|u=kg.m-3}}. * At an altitude over 100 km, the atmosphere is no longer well-mixed, and each chemical species has its own scale height. * Here temperature and gravitational acceleration were assumed to be constant, but both may vary over large distances.
== Planetary examples == Approximate atmospheric scale heights for selected Solar System bodies: {| class="wikitable sortable mw-collapsible" |+ !Solar System body !Atmospheric scale height (km) !Mean temperature (K)<ref name="w054">{{cite web |date=1976-07-20 |title=standard planetary information, formulae and constants |url=https://atmos.nmsu.edu/jsdap/encyclopediawork.html |access-date=2025-05-07 |website=PDS Atmospheres Node}}</ref> !Mean molecular weight (g/mol)<ref name="w054" /> !Surface gravity (g<sub>earth</sub>)<ref name="w054" /> |- |Venus |15.9<ref>{{cite web |title=Venus Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html |accessdate=28 September 2013 |publisher=NASA |archive-date=8 March 2016 |archive-url=https://web.archive.org/web/20160308174416/http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html |url-status=dead }}</ref> |229 |44.01 |0.91 |- |Earth |8.5<ref>{{cite web |title=Earth Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html |accessdate=28 September 2013 |publisher=NASA |archive-date=8 May 2013 |archive-url=https://web.archive.org/web/20130508021904/http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html |url-status=dead }}</ref> |225 |28.96 |1.00 |- |Mars |11.1<ref>{{cite web |title=Mars Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html |accessdate=28 September 2013 |publisher=NASA |archive-date=12 June 2010 |archive-url=https://web.archive.org/web/20100612092806/http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html%7C |url-status=dead }}</ref> |210 |44.01 |0.38 |- |Jupiter |27<ref>{{cite web |title=Jupiter Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html |url-status=dead |archive-url=https://web.archive.org/web/20111013042045/http://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html |archive-date=13 October 2011 |accessdate=28 September 2013 |publisher=NASA |df=dmy-all}}</ref> |124 |2.22 |2.48 |- |Saturn |59.5<ref>{{cite web |title=Saturn Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html |url-status=dead |archive-url=https://web.archive.org/web/20110818175309/http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html |archive-date=18 August 2011 |accessdate=28 September 2013 |publisher=NASA |df=dmy-all}}</ref> |95 |2.14 |1.02 |- |Titan |21<ref>{{cite web |last=Justus |first=C. G. |author2=Aleta Duvall |author3=Vernon W. Keller |date=1 August 2003 |title=Engineering-Level Model Atmospheres For Titan and Mars |url=http://www.mrc.uidaho.edu/entryws/presentations/Papers/Justus.doc |accessdate=28 September 2013 |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}}</ref> |85 |28.67 |0.13 |- |Uranus |27.7<ref>{{cite web |title=Uranus Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/uranusfact.html |accessdate=28 September 2013 |publisher=NASA |archive-date=4 August 2011 |archive-url=https://web.archive.org/web/20110804224710/http://nssdc.gsfc.nasa.gov/planetary/factsheet/uranusfact.html |url-status=dead }}</ref> |59 |2.30 |0.90 |- |Neptune |19.1–20.3<ref>{{cite web |title=Neptune Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html |accessdate=28 September 2013 |publisher=NASA}}</ref> |59 |2.30 |1.13 |- |Pluto |~50<ref>{{cite web |title=Pluto Fact Sheet |url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html |accessdate=2020-09-28 |publisher=NASA |archive-date=2015-11-19 |archive-url=https://archive.today/20151119095810/http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html |url-status=dead }}</ref> | | | |}
== Scale height for a thin disk == {{citations needed|date=April 2026}} thumb|A schematic depiction of the force balance in a gas disk around a central object, e.g., a star For a disk of gas around a condensed central object, such as, for example, a protostar, one can derive a disk scale height which is somewhat analogous to the planetary scale height. We start with a disc of gas that has a mass small relative to the central object. We assume that the disc is in hydrostatic equilibrium with the ''z'' component of gravity from the star, where the gravity component is pointing to the midplane of the disk:<ref name="mkm12">{{Cite journal |last1=Müller |first1=T. W. A. |last2=Kley |first2=W. |last3=Meru |first3=F. |title=Treating gravity in thin-disk simulations |journal=Astronomy & Astrophysics |date=2012 |volume=541 |pages=A123 |doi=10.1051/0004-6361/201118737|arxiv=1203.1413 }}</ref> <math display="block"> \frac{dP}{dz} = -\frac{GM_*\rho z}{(r^2 + z^2)^{3/2}}, </math> where : ''G'' = Newtonian constant of gravitation ≈ {{physconst|G|round=3}} : ''r'' = the radial cylindrical coordinate for the distance from the center of the star or centrally condensed object : ''z'' = the height/altitude cylindrical coordinate for the distance from the disk midplane (or center of the star) : ''M''<sub>*</sub> = the mass of the star/centrally condensed object : ''P'' = the pressure of the gas in the disk : <math>\rho</math> = the gas mass density in the disk
In the thin disk approximation, <math>z \ll r</math>, and the hydrostatic equilibrium, the equation is<ref name="mkm12" /> <math display="block"> \frac{dP}{dz} \approx -\frac{GM_*\rho z}{r^3}.</math>
To determine the gas pressure, one can use the ideal gas law: <math display="block"> P = \frac{\rho k_\text{B} T}{\bar{m}}</math> with : ''T'' = the gas temperature in the disk, where the temperature is a function of ''r'', but independent of ''z'' : <math>\bar{m}</math> = the mean molecular mass of the gas
Using the ideal gas law and the hydrostatic equilibrium equation, gives <math display="block"> \frac{d\rho}{dz} \approx -\frac{GM_* \bar{m}\rho z}{k_\text{B} Tr^3},</math> which has the solution <math display="block"> \rho = \rho_0 \exp\left(-\left(\frac{z}{h_\text{D}}\right)^2 \right),</math> where <math>\rho_0</math> is the gas mass density at the midplane of the disk at a distance ''r'' from the center of the star, and <math>h_\text{D}</math> is the disk scale height with <math display="block"> h_\text{D} = \sqrt{\frac{2k_\text{B}Tr^3}{GM_* \bar{m}}} \approx 0.0306 \sqrt{\frac{(T/100~\text{K})(r/1~\text{au})^3}{(M_* / M_\odot)(\bar{m}/2~\text{Da})}}~\text{au}, </math> with <math>M_\odot</math> the solar mass, <math>\text{au}</math> the astronomical unit, and <math>\text{Da}</math> the dalton.
As an illustrative approximation, if we ignore the radial variation in the temperature <math> T </math>, we see that <math>h_\text{D} \propto r^{3/2}</math> and that the disk increases in altitude as one moves radially away from the central object.
Due to the assumption that the gas temperature ''T'' in the disk is independent of ''z'', <math>h_\text{D}</math> is sometimes known<ref>{{Cite book |last=Frank |first=J. |url=http://archive.org/details/accretionpowerin0000fran |title=Accretion power in astrophysics |date=1992 |publisher=Cambridge [England] ; New York, NY, USA : Cambridge University Press |others=Internet Archive |isbn=978-0-521-40306-1}}</ref> as the isothermal disk scale height.
== Disk scale height in a magnetic field == {{unsourced section|date=December 2024}} A magnetic field in a thin gas disk around a central object can change the scale height of the disk.<ref name="Lovelace">{{cite journal |last1=Lovelace |first1=R. V. E. |last2=Mehanian |first2=C. |last3=Mobarry |first3=C. M. |last4=Sulkanen |first4=M. E. |title=Theory of Axisymmetric Magnetohydrodynamic Flows: Disks |journal=Astrophysical Journal Supplement |date=September 1986 |volume=62 |page=1 |doi=10.1086/191132 |bibcode=1986ApJS...62....1L |url=https://ui.adsabs.harvard.edu/link_gateway/1986ApJS...62....1L/ADS_PDF |access-date=26 January 2022 |doi-access=free }}</ref><ref name="Campbell">{{cite journal |last1=Campbell |first1=C. G. |last2=Heptinstall |first2=P. M. |title=Disc structure around strongly magnetic accretors: a full disc solution with turbulent diffusivity |journal=Monthly Notices of the Royal Astronomical Society |date=August 1998 |volume=299 |issue=1 |page=31 |doi=10.1046/j.1365-8711.1998.01576.x |bibcode=1998MNRAS.299...31C |doi-access=free }}</ref><ref name="Liffman">{{cite journal |last1=Liffman |first1=Kurt |last2=Bardou |first2=Anne |title=A magnetic scaleheight: the effect of toroidal magnetic fields on the thickness of accretion discs |journal=Monthly Notices of the Royal Astronomical Society |date=October 1999 |volume=309 |issue=2 |page=443 |doi=10.1046/j.1365-8711.1999.02852.x |bibcode=1999MNRAS.309..443L |doi-access=free }}</ref> For example, if a non-perfectly conducting disk is rotating through a poloidal magnetic field (i.e., the initial magnetic field is perpendicular to the plane of the disk), then a toroidal (i.e., parallel to the disk plane) magnetic field will be produced within the disk, which will ''pinch'' and compress the disk. In this case, the gas density of the disk is<ref name="Liffman" /> <math display="block"> \rho(r, z) = \rho_0(r) \exp\left(-\left(\frac{z}{h_\text{D}}\right)^2\right) - \rho_\text{cut}(r) \left[1 - \exp\left(-\left(\frac{z}{h_\text{D}}\right)^2\right)\right], </math> where the ''cut-off'' density <math>\rho_\text{cut}</math> has the form <math display="block"> \rho_\text{cut}(r) = (\mu_0 \sigma_\text{D} r)^2 \frac{B_z^2}{\mu_0} \left(\frac{\Omega_*}{\Omega_\text{K}} - 1\right)^2,</math> where : <math>\mu_0</math> is the permeability of free space : <math>\sigma_\text{D}</math> is the electrical conductivity of the disk : <math>B_z</math> is the magnetic flux density of the poloidal field in the <math>z</math> direction : <math>\Omega_*</math> is the rotational angular velocity of the central object (if the poloidal magnetic field is independent of the central object, then <math>\Omega_*</math> can be set to zero) : <math>\Omega_\text{K}</math> is the keplerian angular velocity of the disk at a distance <math>r</math> from the central object.
These formulae give the maximum height of the magnetized disk as <math display="block"> H_\text{B} = h_\text{D} \sqrt{\ln\left(1 + \rho_0/\rho_\text{cut}\right)}, </math> while the e-folding magnetic scale height is <math display="block"> h_\text{B} = h_\text{D} \sqrt{\ln\left(1 + \frac{1 - 1/e}{1/e + \rho_\text{cut}/\rho_0} \right)}. </math>
== See also == * Time constant
== References == {{reflist}}
Category:Atmospheric dynamics Category:Vertical position