L16 Analytical Estimate

L16 Analytical Estimate#

Week 9, Thursday

Material covered and references#

We started making our first approximation that the opacity doesn’t depend on wavelength (called the “grey” case). From this we found some simple relations between \(\tilde{J}\), \(\tilde{F}\), \(\tilde{S}\), and \(\tilde{K}\), but we are still short one equation.

In the textbooks:

  • Gray chap 7

  • Leblanc 4.2 (note that Leblanc uses \(H = F/4\pi\))

  • We considered the case were the optical depth is very large (called the Eddington approximation).

  • Mathematically, we decomposed the source function at a Taylor expansion, kepping only the first term because tau is very large. The other mathematical approximation we did was that instead of integrating from \(\tau\) to \(\infty\), we only go from \(\tau\) to \(\tau+\tau\), which we found a good approximation for large \(\tau\). By doing a change of variable for \(x\) being the “distance from” \(\tau\) (\(x = |\tau' -\tau|\)), it enables us to combine both integral together, greatly simplifying the math.

In the textbooks:

  • Leblanc 3.8