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{{Technical|date=December 2009}} | |||
'''Grey atmosphere''' (or gray) is a useful set of approximations made for radiative transfer applications. | |||
== Application == | |||
The application of the grey atmosphere approximation is the primary method astronomers use to determine the temperature and basic radiative properties of astronomical objects including the sun, planets with atmospheres, other stars, and interstellar clouds of gas and dust. | |||
== Approximations == | |||
The primary approximation is assumption that the [[absorption coefficient]], typically represented by an α, has no wavelength or frequency dependence for the frequency range being worked in. Typically a number of other assumptions are made simultaneously: | |||
# The atmosphere has a [[plane-parallel|plane-parallel atmosphere]] geometry. | |||
# The atmosphere is in a thermal radiative equilibrium. | |||
This set of assumptions leads directly to the mean [[radiance|intensity]] and [[source function]] being directly equivalent to a [[blackbody]] [[Planck function]] of the temperature at that [[optical depth]]. | |||
== Temperature solution == | |||
Integrating the first and second moments of the radiative transfer equation, applying the above relation and the [[Two-Stream Limit]] approximation leads to information about each of the higher moments. The first moment of the mean intensity <math>H</math> is constant regardless of [[optical depth]]: | |||
<math> H(\tau) = H </math> | |||
The second moment of the mean intensity <math>K</math> is then given by: | |||
<math> K(\tau) = \tau H + 2/3 H = 1/3 J(\tau) </math> | |||
Note that the [[Radiative transfer#The Eddington approximation|Eddington approximation]] is a direct consequence of these assumptions. | |||
Defining an effective temperature <math>T_{eff} </math> for the Eddington flux <math>H</math> and applying the [[Stefan-Boltzmann law]], realized this relation between the externally observed effective temperature and the internal blackbody temperature <math>T</math> of the medium. | |||
<math> T^4 = T_{eff}^4 \frac{3}{4} \left( \tau + \frac{2}{3} \right) </math> | |||
The results of the grey atmosphere solution: The observed temperature <math> T_{eff} </math> is a good measure of the true temperature <math> T </math> at an optical depth <math>\tau = 2/3 </math> and the surface temperature is <math> 0.841T_{eff} </math>. | |||
This approximation makes the [[source function]] linear in optical depth. | |||
== References == | |||
<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically --> | |||
{{Reflist}} | |||
{{Cite book | |||
|first1= George | |||
|last1= Rybicki | |||
|first2= Alan | |||
|last2= Lightman | |||
|title=Radiative Processes in Astrophysics | |||
|year = 2004 | |||
|publisher=Wiley-VCH | |||
|isbn=978-0-471-82759-7 | |||
}} | |||
== External links == | |||
* http://www.astro.uu.se/~ke/summ9.pdf | |||
* http://xweb.geos.ed.ac.uk/~stephan/pdf/lect_PhysClim2003_07_Notes.pdf | |||
{{DEFAULTSORT:Grey Atmosphere}} | |||
[[Category:Observational astronomy]] | |||
[[Category:Astrophysics]] | |||
Revision as of 20:41, 6 September 2013
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Grey atmosphere (or gray) is a useful set of approximations made for radiative transfer applications.
Application
The application of the grey atmosphere approximation is the primary method astronomers use to determine the temperature and basic radiative properties of astronomical objects including the sun, planets with atmospheres, other stars, and interstellar clouds of gas and dust.
Approximations
The primary approximation is assumption that the absorption coefficient, typically represented by an α, has no wavelength or frequency dependence for the frequency range being worked in. Typically a number of other assumptions are made simultaneously:
- The atmosphere has a plane-parallel atmosphere geometry.
- The atmosphere is in a thermal radiative equilibrium.
This set of assumptions leads directly to the mean intensity and source function being directly equivalent to a blackbody Planck function of the temperature at that optical depth.
Temperature solution
Integrating the first and second moments of the radiative transfer equation, applying the above relation and the Two-Stream Limit approximation leads to information about each of the higher moments. The first moment of the mean intensity is constant regardless of optical depth:
The second moment of the mean intensity is then given by:
Note that the Eddington approximation is a direct consequence of these assumptions.
Defining an effective temperature for the Eddington flux and applying the Stefan-Boltzmann law, realized this relation between the externally observed effective temperature and the internal blackbody temperature of the medium.
The results of the grey atmosphere solution: The observed temperature is a good measure of the true temperature at an optical depth and the surface temperature is .
This approximation makes the source function linear in optical depth.
References
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