A Summing Up — Question 7

Question 7

Photon scattering determines when radiation can travel freely

At a trial epoch use electron density $n_e=3.0\times10^8$ m$^{-3}$, Thomson cross section $6.65\times10^{-29}$ m$^2$, c=$3.00\times10^8$ m/s and H=$4.0\times10^{-14}$ s$^{-1}$.

1. Find photon scattering rate and Gamma/H.

2. If recombination reduces the free-electron fraction to .001 with other quantities fixed, find the new rate ratio and mean free path.

Question 7 - Solution

Photon scattering determines when radiation can travel freely

(a) Method and calculation

$$\boxed{\Gamma=n_e\sigma_Tc=5.985\times10^{-12}\ {\mathrm{s^{-1}}}}.$$$$\boxed{\Gamma/H=149.625}.$$

(b) Interpretation and further calculation

$$\boxed{(\Gamma/H)_{\mathrm{after}}=.001(149.625)=.149625}.$$$$\boxed{\lambda_{\mathrm{after}}=(.001n_e\sigma_T)^{-1}\simeq5.01\times10^{22}\ {\mathrm{m}}}.$$

Physical check

The rate comparison is local. Last-scattering probabilities require optical depth integrated along the photon trajectory through the changing ionization history.