<p>I don't have time to write up a full answer with derivation, but you might want to have a look into the standard radiation transport literature (e.g. the book by Mihalas &amp; Mihalas) or the radiation transport in stellar interiors (e.g. Kippenhahn) or literature for irradiated exoplanets, see e.g. <a href="https://ui.adsabs.harvard.edu/abs/2010A%26A...520A..27G/abstract" rel="nofollow noreferrer">Guillot (2010)</a>.</p>
<p><strong>The Milne-Eddingtion-approximation (short: ME-approximation) is a standard <em>result</em> for grey radiation transport in another approximation</strong>, i.e. at optical depths &gt;1, when the radiation flux is diffusive:</p>
<p><span class="math-container">$$F_{\rm rad} = \frac{c}{3\rho \kappa_R} \nabla E_{\rm rad}$$</span></p>
<p>Then you can couple the equations for <span class="math-container">$\rm E_{rad}$</span> and <span class="math-container">$\rm E_{gas}$</span>, integrate and solve for <span class="math-container">$\rm E_{gas}$</span>, which is <span class="math-container">$\rm \rho c_vT$</span>. This gives you the ME relation, which is valid for the diffusive radiative transport in gases, and precludes interesting things like the greenhouse-effect, temperature inversions, etc.</p>

CC BY-SA 4.0
2021-04-23T17:49:59.403Z
2021-06-23T12:00:15.623Z
2021-06-23T12:00:15.623Z
AtmosphericPrisonEscape
<p>In this paper: <a href="http://saspcsus.pbworks.com/w/file/fetch/64696386/planet%20temperatures%20with%20surface%20cooling%20parameterized.pdf" rel="nofollow noreferrer">Planet temperatures with surface cooling parameterized</a> it states in the &quot;radiation model&quot; the following:</p>
<blockquote>
<p>The Eddington–Milne approximation relates <span class="math-container">$T_0$</span> and <span class="math-container">$T_e$</span> through the
gray IR optical depth <span class="math-container">$\tau$</span>:</p>
<p><span class="math-container">$T_0=T_e(1+\frac{3}{4}\tau)^{0.25}$</span></p>
</blockquote>
<p>I looked this approximation up on google, and found it was more commonly known as the Milne-Eddington Approximation. However, this approximation seems to (mainly) be applied to stars, and I can't exactly see the connection between it and the temperature of a planet. (I also don't completely understand what an &quot;optical depth&quot; is.)</p>
<p>What is this approximation and why can it be applied to approximate the temperature of a planet with an atmosphere?</p>

CC BY-SA 4.0
2021-04-20T20:07:19.110Z
The [optical depth](https://scienceworld.wolfram.com/physics/OpticalDepth.html) is "a measure of how opaque a medium is to radiation passing through it" which intuitively means relates to how far you can see into the atmosphere from outside for a given wavelength. I am stumbling upon the "gray" in your quote though - is "gray IR" a commonly accepted term?
B--rian
CC BY-SA 4.0
2021-04-22T01:24:06.330Z
"Gray" opacity or optical depth generally means "wavelength-independent", i.e. it is assumed to be the same at all wavelengths.  I haven't seen it used to modify IR before, but from context I take it to mean the assumption of a constant optical depth across some limited range of wavelengths, here the infrared (or part of it - the IR is huge as a part of the EM spectrum!)
Eric Jensen
CC BY-SA 4.0
2021-04-22T17:39:03.893Z
@eshaya I did find another source that indeed had 1/2 instead of 1. Do you have a source that derives the equation?
Astavie
<planet><solar-system><planetary-atmosphere><temperature><radiation>
What is the Milne-Eddington Approximation?
Astavie
# What is the Milne-Eddington Approximation?

(Asked by: username_0 on 2021-04-20)

In this paper: [Planet temperatures with surface cooling parameterized](http://saspcsus.pbworks.com/w/file/fetch/64696386/planet%20temperatures%20with%20surface%20cooling%20parameterized.pdf) it states in the "radiation model" the following:

> The Eddington–Milne approximation relates $T\_0$ and $T\_e$ through the
> gray IR optical depth $\tau$:
>
> $T\_0=T\_e(1+\frac{3}{4}\tau)^{0.25}$

I looked this approximation up on google, and found it was more commonly known as the Milne-Eddington Approximation. However, this approximation seems to (mainly) be applied to stars, and I can't exactly see the connection between it and the temperature of a planet. (I also don't completely understand what an "optical depth" is.)

What is this approximation and why can it be applied to approximate the temperature of a planet with an atmosphere?

**Tags:** planet, solar-system, planetary-atmosphere, temperature, radiation
> **Comments:**
> **username_1**: The [optical depth](https://scienceworld.wolfram.com/physics/OpticalDepth.html) is "a measure of how opaque a medium is to radiation passing through it" which intuitively means relates to how far you can see into the atmosphere from outside for a given wavelength. I am stumbling upon the "gray" in your quote though - is "gray IR" a commonly accepted term?  
> **username_2**: "Gray" opacity or optical depth generally means "wavelength-independent", i.e. it is assumed to be the same at all wavelengths.  I haven't seen it used to modify IR before, but from context I take it to mean the assumption of a constant optical depth across some limited range of wavelengths, here the infrared (or part of it - the IR is huge as a part of the EM spectrum!)  
> **username_0**: @eshaya I did find another source that indeed had 1/2 instead of 1. Do you have a source that derives the equation?  

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## 1 answers

### Answer by username_3 on 2021-04-23. Score: 2

I don't have time to write up a full answer with derivation, but you might want to have a look into the standard radiation transport literature (e.g. the book by Mihalas & Mihalas) or the radiation transport in stellar interiors (e.g. Kippenhahn) or literature for irradiated exoplanets, see e.g. [Guillot (2010)](https://ui.adsabs.harvard.edu/abs/2010A%26A...520A..27G/abstract).

**The Milne-Eddingtion-approximation (short: ME-approximation) is a standard *result* for grey radiation transport in another approximation**, i.e. at optical depths >1, when the radiation flux is diffusive:

$$F\_{\rm rad} = \frac{c}{3\rho \kappa\_R} \nabla E\_{\rm rad}$$

Then you can couple the equations for $\rm E\_{rad}$ and $\rm E\_{gas}$, integrate and solve for $\rm E\_{gas}$, which is $\rm \rho c\_vT$. This gives you the ME relation, which is valid for the diffusive radiative transport in gases, and precludes interesting things like the greenhouse-effect, temperature inversions, etc.


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