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 Temperature solution Integrating the first and second moments of the radiative transfer equation, applying the above relation and the Two-Stream Limit approximation