View source for Scale height

[[File:Pressure air.svg|thumb|300px|The earth atmosphere's scale height is about 8.5km, 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:Atmospheric science|atmospheric]], [[Earth:Earth science|earth]], and [[Astronomy:Planetary science|planetary]] sciences, a '''scale height''', usually denoted by the capital letter ''H'', is a distance (vertical or [[Polar coordinate system|radial]]) over which a [[Physics:Physical quantity|physical quantity]] decreases by a factor of [[E (mathematical constant)|e]] (the base of natural logarithms, approximately 2.718).

==Scale height used in a simple atmospheric pressure model==

For planetary atmospheres, '''scale height''' is the increase in altitude for which the [[Physics:Atmospheric pressure|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= [[Company:Wolfram Research|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> = [[Physics:Boltzmann constant|Boltzmann constant]] = {{physconst|k|round=3}}
* ''R'' = [[Physics:Gas constant|gas constant]]
* ''T'' = mean atmospheric [[Physics:Temperature|temperature]] in [[Kelvin|kelvin]]s = 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
}}</ref> for Earth
* ''m'' = mean mass of a molecule (units kg)
* ''M'' = mean mass of one mol of atmospheric particles = 0.029&nbsp;kg/mol for Earth
* ''g'' = [[Physics:Acceleration|acceleration]] due to [[Physics:Gravity|gravity]] at the current location (m/s<sup>2</sup>)

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 [[Physics:Density|density]] ''&rho;'' 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&nbsp;''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 [[Physics:Equation of state|equation of state]] for an [[Physics:Ideal gas|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}{\frac{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|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 [[Exponential decay|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:Atmosphere of Earth|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 28.964 [[Physics:Unified atomic mass unit|u]] and hence m = 28.964 &times; {{val|1.660|e=-27}} = {{val|4.808|e=-26|u=kg}}. As a function of temperature, the scale height of Earth's atmosphere is therefore ''H''/''T'' = ''k''/''mg'' = (1.38/(4.808&times;9.81))×10<sup>3</sup> = {{val|29.26|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 [[Earth:NRLMSISE-00|NRLMSISE-00]], which shows the air density dropping from 1200 g/m<sup>3</sup> at sea level to 0.5<sup>3</sup> = 0.125 g/m<sup>3</sup> at 70&nbsp;km, a factor of 9600, indicating an average scale height of 70/ln(9600) = 7.64&nbsp;km, consistent with the indicated average air temperature over that range of close to 260 K.

Note:
* Density is related to pressure by the [[Physics:Ideal gas|ideal gas]] laws. Therefore, density will also decrease exponentially with height from a sea level value of ''&rho;''<sub>0</sub> roughly equal to 1.2&nbsp;kg&nbsp;m<sup>&minus;3</sup>
* At heights over 100&nbsp;km, an atmosphere may no longer be well mixed. Then 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 follow.
{{div col}}
*[[Astronomy:Venus|Venus]]: 15.9&nbsp;km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html|title=Venus Fact Sheet| publisher=[[Organization:NASA|NASA]]|accessdate=28 September 2013}}</ref>
*[[Astronomy:Earth|Earth]]: 8.5&nbsp;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>
*[[Astronomy:Mars|Mars]]: 11.1&nbsp;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>
*[[Astronomy:Jupiter|Jupiter]]: 27&nbsp;km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html|title=Jupiter Fact Sheet|publisher=NASA|accessdate=28 September 2013|archive-url=https://web.archive.org/web/20111013042045/http://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html|archive-date=13 October 2011|url-status=dead|df=dmy-all}}</ref>
*[[Astronomy:Saturn|Saturn]]: 59.5&nbsp;km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html|title=Saturn Fact Sheet|publisher=NASA|accessdate=28 September 2013|archive-url=https://web.archive.org/web/20110818175309/http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html|archive-date=18 August 2011|url-status=dead|df=dmy-all}}</ref>
**[[Astronomy:Titan (moon)|Titan]]: 21&nbsp;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.|author2=Aleta Duvall |author3=Vernon W. Keller |date=1 August 2003|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>
*[[Astronomy:Uranus|Uranus]]: 27.7&nbsp;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>
*[[Astronomy:Neptune|Neptune]]: 19.1–20.3&nbsp;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>
*[[Astronomy:Pluto|Pluto]]: ~50&nbsp;km<ref>{{cite web|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html|title=Pluto Fact Sheet| publisher=NASA|accessdate=2020-09-28}}</ref>
{{div col end}}

== Scale height for a thin disk ==
[[File:Scale Height Force Balance.jpg|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 which is 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:

<math display="block">\frac{dP}{dz} = - \frac{GM_*\rho z}{(r^2+z^2)^{3/2}}</math>

where:

* ''G'' = [[Physics:Gravitational constant|Gravitational constant]] ≈ {{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 equation is
<math display="block">\frac{dP}{dz} \approx - \frac{GM_*\rho z}{r^3}</math>

To determine the gas pressure, one can use the [[Physics:Ideal gas law|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 [[Physics:Ideal gas law|ideal gas law]] and the hydrostatic equilibrium equation, gives:
<math display="block">\frac{d\rho}{dz} \approx - \frac{GM_* \bar{m}\rho z}{k Tr^3}</math>
which has the solution
<math display="block"> \rho = \rho_0 \exp\left(-\left(\frac{z}{h_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_D </math> is the disk scale height with

<math display="block"> h_D = \sqrt{\frac{2kTr^3}{GM_* \bar{m}}} \approx 0.0306 \sqrt{\frac{\left(T/100 \ \text{K}\right)\left(r/1 \text{ au} \right)^3}{\left(M_* / M_\odot\right)\left(\bar{m}/2 \text{ amu} \right)}} \ \text{ au} </math>
with <math> M_\odot </math> the [[Astronomy:Solar mass|solar mass]], <math> \text{au} </math> the [[Astronomy:Astronomical unit|astronomical unit]] and <math> \text{amu} </math> the atomic mass unit.

As an illustrative approximation, if we ignore the radial variation in the temperature, <math> T </math>, we see that <math> h_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 in the disk, ''T'', is independent of ''z'', <math> h_D </math> is sometimes known as the isothermal disk scale height.

== Disk scale height in a magnetic field ==

A [[Physics:Magnetic field|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_D}\right)^2 \right) - \rho_\text{cut}(r)
\left[1 - \exp \left(- \left(\frac{z}{h_D}\right)^2 \right) \right] </math>
where the ''cut-off'' density <math> \rho_\text{cut} </math> has the form
<math display="block"> \rho_{\rm cut}(r) = (\mu_0 \sigma_D r)^2 \left(\frac{B_z^2}{\mu_0}\right)
\left(\frac{\Omega_*}{\Omega_K} - 1 \right)^2 </math>
where
* <math> \mu_0 </math> is the permeability of free space
* <math> \sigma_D </math> is the [[Physics:Electrical conductivity|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 [[Physics:Angular velocity|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_K </math> is the keplerian [[Physics:Angular velocity|angular velocity]] of the disk at a distance <math> r </math> from the central object.

These formulae give the maximum height, <math> H_B </math>, of the magnetized disk as

<math display="block"> H_B = h_D \sqrt{\ln\left(1 + \rho_0/\rho_{\rm cut} \right)} ,</math>
while the e-folding magnetic scale height, <math> h_B </math>, is
<math display="block"> h_B = h_D \sqrt{\ln\left(1 + \frac{1 - 1/e}{1/e + \rho_\text{cut}/\rho_0} \right)} \ . </math>

==See also==
*[[Physics:Time constant|Time constant]]

==References==
{{reflist|colwidth=30em}}

[[Category:Atmospheric dynamics]]
[[Category:Vertical position]]

{{Sourceattribution|Scale height}}