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The Grey atmosphere (or gray) is a useful set of approximations made for radiative transfer applications in studies of stellar atmospheres based on the simplification that the absorption coefficient of matter within the atmosphere is constant for all frequencies of incident radiation.
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. Although the model 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 assumption that the absorption coefficient, typically represented by an , has no dependence on frequency for the frequency range being worked in, e.g. .
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.
The Eddington approximation (see next section) may also be used optionally, to solve for the source function. This greatly simplifies the model without greatly distorting results.
Derivation of source function using the Eddington Approximation
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 , where the direction of positive points towards the upper layers of the atmosphere. From this it is easy to see that a ray path at angle to the vertical, is given by
We now define optical depth as
where is the absorption coefficient associated with the various constituents of the atmosphere. We now turn to the radiation transfer equation
where is the total specific intensity, is the emission coefficient. After substituting for and dividing by we have
where 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 , re-writing the lefthand side as and then integrating the whole equation with respect to . This gives the solution
where we have used the limits as we are integrating outward from some depth within the atmosphere; therefore . Even though we have neglected the frequency-dependence of parameters such as , we know that it is a function of optical depth therefore in order to integrate this we need to have a method for deriving the source function. We now define some important parameters such as energy density , total flux and radiation pressure as follows
We also define the average specific intensity (averaged over all frequencies) as
We see immediately that by dividing the radiative transfer equation by 2 and integrating over , we have
Furthermore, by multiplying the same equation by and integrating w.r.t. , we have
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
This condition is known as radiative equilibrium. Taking advantage of the constancy of total flux, we now integrate to obtain
where is a constant of integration. 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. Substituting this in the previous equation for pressure gives
and under the condition of radiative equilibrium
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
Here we have set the lower limit of to zero, which is the value of optical depth at the surface of the atmosphere. 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
Therefore, and the source function is given by
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 atmosphere top temperature is .
This approximation makes the source function linear in optical depth.