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 of matter within a star's atmosphere is constant—that is, unchanging—for all frequencies of the star's incident radiation. Application 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. Approximations The primary approximation is based on the 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. 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 , 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