The grey atmosphere (or gray) is a useful set of approximations made for radiative transfer applications in studies of stellar atmospheres (atmospheres of stars) based on the simplified notion that the absorption coefficient
The grey atmosphere approximation is the primary method astronomers use to determine the temperature and basic radiative properties of astronomical objects, including planets with atmospheres, the Sun, other stars, and interstellar clouds of gas and dust.
Although the simplified model of grey atmosphere approximation demonstrates good correlation to observations, it deviates from observational results because real atmospheres are not grey, e.g. radiation absorption is frequency-dependent.
The primary approximation is based on the assumption that the absorption coefficient, typically represented by an
Typically a number of other assumptions are made simultaneously: 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.
The Eddington approximation (see next section) may also be used optionally, to solve for the source function.
Deriving various quantities from the grey atmosphere model involves solving an integro-differential equation, an exact solution of which is complex.
Therefore, this derivation takes advantage of a simplification known as the Eddington Approximation.
Starting with an application of a plane-parallel model, we can imagine an atmospheric model built up of plane-parallel layers stacked on top of each other, where properties such as temperature are constant within a plane.
This means that such parameters are function of physical depth
is the so-called total source function defined as the ratio between emission and absorption coefficients.
This differential equation can by solved by multiplying both sides by
We now define some important parameters such as energy density
We see immediately that by dividing the radiative transfer equation by 2 and integrating over
By substituting the average specific intensity J into the definition of energy density, we also have the following relationship
Now, it is important to note that total flux must remain constant through the atmosphere therefore
Taking advantage of the constancy of total flux, we now integrate
We know from thermodynamics that for an isotropic gas the following relationship holds
where we have substituted the relationship between energy density and average specific intensity derived earlier.
Although this may be true for lower depths within the stellar atmosphere, near the surface it almost certainly isn't.
However, the Eddington Approximation assumes this to hold at all levels within the atmosphere.
This means we have solved the source function except for a constant of integration.
Substituting this result into the solution to the radiation transfer equation and integrating gives
This would represent radiation coming out of, say, the surface of the Sun.
Finally, substituting this into the definition of total flux and integrating gives
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 in
Note that the Eddington approximation is a direct consequence of these assumptions.
and applying the Stefan–Boltzmann law, realize this relation between the externally observed effective temperature and the internal blackbody temperature
The results of the grey atmosphere solution: The observed temperature
This approximation makes the source function linear in optical depth.