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
Contents
Application
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.
Approximations
The primary approximation is assumption that the absorption coefficient, typically represented by an
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.
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. 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
We now define optical depth as
where
where
where
where we have used the limits
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
Furthermore, by multiplying the same equation by
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
where
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
Therefore,
Temperature solution
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
The second moment of the mean intensity
Note that the Eddington approximation is a direct consequence of these assumptions.
Defining an effective temperature
The results of the grey atmosphere solution: The observed temperature
This approximation makes the source function linear in optical depth.