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In this paper: Planet temperatures with surface cooling parameterized 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?

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    $\begingroup$ The optical depth 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? $\endgroup$ – B--rian Apr 20 at 20:07
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    $\begingroup$ "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!) $\endgroup$ – Eric Jensen Apr 22 at 1:24
  • $\begingroup$ @eshaya I did find another source that indeed had 1/2 instead of 1. Do you have a source that derives the equation? $\endgroup$ – Astavie Apr 22 at 17:39
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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 (e.g. Guillot (2010)). The ME-approximation is a standard result for grey radiation transport at optical depths >1, when the radiation flux is diffusive, i.e. $\rm F_{rad} = c/(3\rho \kappa_R)\nabla E_{rad}$.

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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