# Scale height

In atmospheric, earth, and planetary sciences, 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).

## 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]\displaystyle{ H = \frac{k_\text{B}T}{mg} }[/math] or equivalently [math]\displaystyle{ H = \frac{RT}{Mg} }[/math] where:

*k*_{B}= Boltzmann constant = 1.381×10^{−23}J⋅K^{−1}^{[3]}*R*= gas constant*T*= mean atmospheric temperature in kelvins = 250 K^{[4]}for Earth*m*= mean mass of a molecule (units kg)*M*= mean mass of one mol of atmospheric particles = 0.029 kg/mol for Earth*g*= acceleration due to gravity at the current location (m/s^{2})

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]\displaystyle{ \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]\displaystyle{ \rho = \frac{MP}{RT} }[/math]

Combining these equations gives
[math]\displaystyle{ \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]\displaystyle{ \frac{dP}{P} = - \frac{dz}{H} }[/math]
which will not change unless the temperature does. Integrating the above and assuming *P*_{0} is the pressure at height *z* = 0 (pressure at sea level) the pressure at height *z* can be written as:
[math]\displaystyle{ P = P_0\exp\left(-\frac{z}{H}\right) }[/math]

This translates as the pressure decreasing exponentially with height.^{[5]}

In 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 m = 28.964 × 1.660×10^{−27} = 4.808×10^{−26} kg. As a function of temperature, the scale height of Earth's atmosphere is therefore *H*/*T* = *k*/*mg* = (1.38/(4.808×9.81))×10^{3} = 29.26 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^{3} at sea level to 0.5^{3} = 0.125 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, 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, 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.

- Venus: 15.9 km
^{[6]} - Earth: 8.5 km
^{[7]} - Mars: 11.1 km
^{[8]} - Jupiter: 27 km
^{[9]} - Saturn: 59.5 km
^{[10]}- Titan: 21 km
^{[11]}

- Titan: 21 km
- Uranus: 27.7 km
^{[12]} - Neptune: 19.1–20.3 km
^{[13]} - Pluto: ~50 km
^{[14]}

## Scale height for a thin disk

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]\displaystyle{ \frac{dP}{dz} = - \frac{GM_*\rho z}{(r^2+z^2)^{3/2}} }[/math]

where:

*G*= Gravitational constant ≈ 6.674×10^{−11}m^{3}⋅kg^{−1}⋅s^{−2}^{[15]}*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*_{*}= the mass of the star/centrally condensed object*P*= the pressure of the gas in the disk- [math]\displaystyle{ \rho }[/math] = the gas mass density in the disk

In the thin disk approximation, [math]\displaystyle{ z \ll r }[/math] and the hydrostatic equilibrium equation is [math]\displaystyle{ \frac{dP}{dz} \approx - \frac{GM_*\rho z}{r^3} }[/math]

To determine the gas pressure, one can use the ideal gas law: [math]\displaystyle{ 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]\displaystyle{ \bar{m} }[/math] = the mean molecular mass of the gas

Using the ideal gas law and the hydrostatic equilibrium equation, gives:
[math]\displaystyle{ \frac{d\rho}{dz} \approx - \frac{GM_* \bar{m}\rho z}{k Tr^3} }[/math]
which has the solution
[math]\displaystyle{ \rho = \rho_0 \exp\left(-\left(\frac{z}{h_D}\right)^2 \right) }[/math]
where [math]\displaystyle{ \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]\displaystyle{ h_D }[/math] is the disk scale height with

[math]\displaystyle{ 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]\displaystyle{ M_\odot }[/math] the solar mass, [math]\displaystyle{ \text{au} }[/math] the astronomical unit and [math]\displaystyle{ \text{amu} }[/math] the atomic mass unit.

As an illustrative approximation, if we ignore the radial variation in the temperature, [math]\displaystyle{ T }[/math], we see that [math]\displaystyle{ 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]\displaystyle{ h_D }[/math] is sometimes known as the isothermal disk scale height.

## Disk scale height in a magnetic field

A magnetic field in a thin gas disk around a central object can change the scale height of the disk.^{[16]}^{[17]}^{[18]} 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:^{[18]}

[math]\displaystyle{ \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]\displaystyle{ \rho_\text{cut} }[/math] has the form
[math]\displaystyle{ \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]\displaystyle{ \mu_0 }[/math] is the permeability of free space
- [math]\displaystyle{ \sigma_D }[/math] is the electrical conductivity of the disk
- [math]\displaystyle{ B_z }[/math] is the magnetic flux density of the poloidal field in the [math]\displaystyle{ z }[/math] direction
- [math]\displaystyle{ \Omega_* }[/math] is the rotational angular velocity of the central object (if the poloidal magnetic field is independent of the central object then [math]\displaystyle{ \Omega_* }[/math] can be set to zero)
- [math]\displaystyle{ \Omega_K }[/math] is the keplerian angular velocity of the disk at a distance [math]\displaystyle{ r }[/math] from the central object.

These formulae give the maximum height, [math]\displaystyle{ H_B }[/math], of the magnetized disk as

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

## See also

## References

- ↑ "Glossary of Meteorology - scale height". American Meteorological Society (AMS). http://glossary.ametsoc.org/wiki/Scale_height.
- ↑ "Pressure Scale Height". Wolfram Research. http://scienceworld.wolfram.com/physics/PressureScaleHeight.html.
- ↑ "2018 CODATA Value: Boltzmann constant".
*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?k. Retrieved 2019-05-20. - ↑ "Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999". http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap2.html.
- ↑ "Example: The scale height of the Earth's atmosphere". http://iapetus.phy.umist.ac.uk/Teaching/SolarSystem/WorkedExample4.pdf.
- ↑ "Venus Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html. Retrieved 28 September 2013.
- ↑ "Earth Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html. Retrieved 28 September 2013.
- ↑ "Mars Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html. Retrieved 28 September 2013.
- ↑ "Jupiter Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html. Retrieved 28 September 2013.
- ↑ "Saturn Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html. Retrieved 28 September 2013.
- ↑ Justus, C. G.; Aleta Duvall (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. http://www.mrc.uidaho.edu/entryws/presentations/Papers/Justus.doc. Retrieved 28 September 2013. - ↑ "Uranus Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/uranusfact.html. Retrieved 28 September 2013.
- ↑ "Neptune Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html. Retrieved 28 September 2013.
- ↑ "Pluto Fact Sheet". NASA. http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html. Retrieved 2020-09-28.
- ↑ "2018 CODATA Value: Newtonian constant of gravitation".
*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?bg. Retrieved 2019-05-20. - ↑ Lovelace, R.V.E.; Mehanian, C.; Mobarry, C. M.; Sulkanen, M. E. (September 1986). "Theory of Axisymmetric Magnetohydrodynamic Flows: Disks".
*Astrophysical Journal Supplement***62**: 1. doi:10.1086/191132. Bibcode: 1986ApJS...62....1L. https://ui.adsabs.harvard.edu/link_gateway/1986ApJS...62....1L/ADS_PDF. Retrieved 26 January 2022. - ↑ Campbell, C. G.; Heptinstall, P. M. (August 1998). "Disc structure around strongly magnetic accretors: a full disc solution with turbulent diffusivity".
*Monthly Notices of the Royal Astronomical Society***299**(1): 31. doi:10.1046/j.1365-8711.1998.01576.x. Bibcode: 1998MNRAS.299...31C. - ↑
^{18.0}^{18.1}Liffman, Kurt; Bardou, Anne (October 1999). "A magnetic scaleheight: the effect of toroidal magnetic fields on the thickness of accretion discs".*Monthly Notices of the Royal Astronomical Society***309**(2): 443. doi:10.1046/j.1365-8711.1999.02852.x. Bibcode: 1999MNRAS.309..443L.

Original source: https://en.wikipedia.org/wiki/Scale height.
Read more |