SOURCES FOR CONTENT:
- Chandrasekhar, S., 1960. “Radiative Transfer”. Dover. 1.
- Choudhuri, A.R., 2010. “Astrophysics for Physicists”. Cambridge University Press. 2.
- Boyce, W.E., and DiPrima, R.C., 2005. “Elementary Differential Equations”. John Wiley & Sons. 2.1.
Recall from last time , the radiative transfer equation
where and are the emission and absorption coefficients, respectively. We can further define the absorption coefficient to be equivalent to . Hence,
which upon rearrangement and substitution in Eq. (1) gives
We may solve this equation by using the method of integrating factors, by which we multiply Eq.(3) by some unknown function (the integrating factor) yielding
Upon examining Eq.(4), we see that the left hand side is the product rule. It follows that
This only works if . To show that this is valid, consider the equation for only:
This is a separable ordinary differential equation so we can rearrange and integrate to get
where is some constant of integration. Let us assume that the constant of integration is , and let us also take the exponential of (6.2). This gives us
This is our integrating factor. Just as a check, let us take the derivative of our integrating factor with respect to ,
Thus this requirement is satisfied. If we now return to Eq.(4) and substitute in our integrating factor we get
We can treat this as a separable differential equation so we can integrate immediately. However, we are integrating from an optical depth to some optical depth , hence we have that
We find that
where if we add and divide by we arrive at the general solution of the radiative transfer equation
This is the mathematically formal solution to the radiative transfer equation. While mathematically sound, much of the more interesting physical phenomena require more complicated equations and therefore more sophisticated methods of solving them (an example would be the use of quadrature formulae or -th approximation for isotropic scattering).
Recall also that in general we can write the phase function via the following
Let us consider the case for which in the sum given by (11). This then would mean that the phase function is constant
Such a phase function is consistent with isotropic scattering. The term isotropic means, in this context, that radiation scattered is the same in all directions. Such a case yields a source function of the form
where upon use in the radiative transfer equation we get the integro-differential equation
Solution of this equation is beyond the scope of the project. In the next post I will discuss Rayleigh scattering and the corresponding phase function.