SOURCE FOR CONTENT: Chandrasekhar, S., 1960. Radiative Transfer. 1.
In this post, I will be discussing the basics of radiative transfer theory necessary to understand the methods used in this project. I will start with some definitions, then I will look at the radiative transfer equation and consider two simple cases of scattering.
The first definition we require is the specific intensity, which is the amount of energy associated with a specific frequency passing through an area constrained to a solid angle in a time . We may write this mathematically as
We must also consider the net flux given by
where if we integrate over all solid angles we get
Let be an element of the surface in a volume through which radiation passes. Further let and denote the angles which form normals with respect to elements and . These surfaces are joined by these normals and hence we have the surface across which energy flows includes the elements and , given by the following:
where is the solid angle subtended by the surface element at a point and volume element is the volume that is intercepted in volume . If we take this further, and integrate over all and we arrive at
where if the radiation travels some distance in the volume, then we must multiply Eq.(5) by , where is the speed of light.
We now define the integrated energy density as being
while the average intensity is
and the relation between these two equations is
I will now introduce the radiative transfer equation. This equation is a balance between the amount of radiation absorbed and the radiation that is emitted. The equation is,
where if we divide by we get
where represents the source function given by
The source function is typically the ratio between the absorption and emission coefficients. One of the terms in the source function is the phase function which varies according to the specific scattering geometry. In its most general form, we can represent the phase function as an expansion of Legendre polynomials:
where we have let (in keeping with our notation in previous posts).
In Part II, we will discuss a few simple cases of scattering and their corresponding phase functions, as well as obtaining the formal solution of the radiative transfer equation. (DISCLAIMER: While this solution will be consistent in a mathematical sense, it is not exactly an insightful solution since much of the more interesting and complex cases involve the solution of either integro-differential equations or pure integral equations (a possible new topic).)