Stellar atmospheres represents an ongoing area of investigation in astrophysics, and it's nice to get a handle on some basic concepts. To be sure, modeling stellar atmospheres is a very complex undertaking that often requires we make basic assumptions. The “gray atmosphere” is one such simplifying assumption. Let’s see how it works.
First, some preliminaries on essential technical terms, etc.
The Planck function describes the distribution of radiation for a black body, and can be expressed:
B(L) = {(2 hc^2)/ L^5} * [1/ (exp^hc/LkT - 1)]
where h is Planck’s constant, c is the speed of light, T is the absolute temperature, k is the Boltzmann constant, and L defines the wavelength.
In the plane-parallel treatment, we take layers of the gases in a stellar atmosphere to be like layers of a “sandwich”, where ds is an element of length or path perpendicular to the layers
----------------------
---------------------!-ds
----------------------
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This is opposed to employing curved layers (as would technically be the case, but for which the math is many times more complex!)
Now as a beam of radiation passes through stellar gases, there will be emission and absorption along the way
The “source function” specifies the ratio of one to the other and can be expressed:
S(L) = e(L)/ k(L)
where L again denotes wavelength and e(L) is the emission coefficent, and k(L) the absorption coefficient.
In the case of simple radiation transfer in a model stellar atmosphere (e.g. nothing changes with time), we have the relation of radiation intensity I(L) to source function S(L):
dI(L)/ds = -k(L) I(L) + k(L) S(L) = k(L)[S(L) – I(L)] - 0
or I(L) = S(L)
Now, for a black body (perfect absorber and perfect radiator, after which most stars are modelled), I(L) equals the Planck function B(L):
So, in effect, we have:
S(L) = I(L) = B(L)
And this is a condition – which for any stellar atmosphere – implies
LOCAL THERMODYNAMIC EQUILIBRIUM!
or LTE
LTE does NOT mean complete thermodynamic equilibrium!
(E.g. since in the outer layers of a star there is always large energy loss from the stellar surface)
Thus, one only assumes the emission of the radiation is the same as for a gas in thermodynamic equilibrium at a temperature (T) corresponding to the temperature of the layer under consideration.
Another way to say this is that if LTE holds, the photons always emerge at all wavelengths.
Now, in the above treatment, note that the absorption coefficient was always written as:
k(L) to emphasize its wavelength (L) dependence.
However, there are certain specific treatments for which we may eliminate the wavelength dependence on absorption, and simply write:
k
e.g. k has the same value at ALL wavelengths!
This is what is meant by the “gray atmosphere” approximation.
Specific application:
I’ll now give a specific application of the gray atmosphere approximation.
In a particular integral, the surface flux (π F(O)) =
2 π (I(cos (Θ)) = [a(L) + 2(b(L)/3 ] π
and F_L(0) = S(L) (t(L) = 2/3)
which states that the flux coming out of the stellar surface is equal to the source function at the optical depth t = 2/3. This is the important ‘Eddington-Barbier’ relation that paves the way for the understanding of how stellar spectra are formed.
Once one then assumes LTE, one can further assume k(L) is independent of L ("gray atmosphere") so that k(L) = k; t(L) = t and
F_L(0) = B_L(T(t = 2/3)
Thus, the energy distribution of F_L is that of a black body corresponding to the temperature at an optical depth t = 2/3.
From this, with some simple substitutions and integrations (hint: look at the Stefan-Boltzmann law!) energized readers should be able to easily determine:
π F(O) = oT_eff^4 and T_eff = T(t = 2/3)
where o is the Stefan-Boltzmann constant.
Thus, the temp. at optical depth 2/3 must equal the effective temperature!
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