L11 Source Function

L11 Source Function#

Week 6, Tuesday

Notebook: L11-SourceFunction-template.ipynb

Material covered and references#

The change in intensity in the case of pure emission (the solution is a quite simple, straight integral, for once!).

Micro-objectives

  • I can list the two different ways that intensity can be gained going through matter

  • I can list and describe the factors that contributes to an element of emission (\(dI\))

  • I can derive the equation for intensity as a function of position \(I(s)\), and I can calculate and sketch it if given a functional form for the characteristics of the matter (e.g. \(\rho(s)\), \(\kappa(s)\), etc).

In the textbooks:

  • Gray Chap 5

  • Leblanc 3.6

  • The formal solution of the radiative transfer equation. This solution is expressed in terms of the optical depth \(\tau\), and make use of a quantity called the source function \(S_\lambda = j_\lambda/\kappa_\lambda\), and has units of intensity.

  • The source function is what the intensity will become after going through a number of optical depth. Of course, the intensity might not be able to become equal to the source function, if the source function itself varies significantly over a few optical depth.

Micro-objectives:

  • I can state and explain all of the terms in the formal solution of radiative transfer.

  • I can derive the equation for the solution of radiative transfer for constant density, opacity, and source function, starting from the formal solution.

  • I can make a sketch of \(I(s)\) for constant density, opacity, and source function.

  • I can calculate the solution for \(I(s)\) if given a simple functional form for e.g. \(\rho(s)\) or \(S(\tau)\).

In the textbooks:

  • Gray Chap 7 (note that Gray do a flat atmosphere geometry straight from the start - this is where the \(\cos(\theta)\) are coming from)

  • Leblanc 3.4, 3.6 (same remark as Gray)

  • Rybicki 1.4 (in this textbook, \(\alpha = \kappa * \rho\), and \(j = \) our \(j * \rho\))

  • Mullan 2.5