| 35 | A number of simplifying assumptions make this significantly easier, the most important being a constant temperature (no heating or cooling terms from line transfer, gamma ~ 1). I may return to this and add the heating and cooling terms, which affect the recombination rate, and include the dependence on temperature in the ionization fraction. In addition, hydrodynamics were neglected. |
| 36 | |
| 37 | Since the line transfer done by the code is inherently 1D, ionization equilibrium was [attachment:"ionization front.nb" solved for] by setting the recombination rate equal to the ionization rate in each successive cell: |
| 38 | |
| 39 | {{{#!latex |
| 40 | |
| 41 | $\tau(1) = 0$ |
| 42 | |
| 43 | $RecomRate(c) = (X(c) n_H)^2 \alpha_{rec}$ |
| 44 | |
| 45 | $IonRate(c) = (1-X(c))n_H J_{phot} \sigma_H e^{-\tau(c)}$ |
| 46 | |
| 47 | $\tau(c+1) = \tau(c) + \sigma_H (1-X(c)) n_H dx$ |
| 48 | |
| 49 | }}} |
| 50 | |
| 51 | where dx is the cell size in cm, and we are solving for X(c), the ionization fraction in the current cell. A modified logistic curve was then fit approximately (i.e. by hand) to the ionization fraction solution and used as the initial conditions for a simulation, which was allowed to relax to true equilibrium. A better fit for initial conditions could be obtained with a minimization routine, but I think it is more instructive to see the simulation have some significant relaxation. |
| 52 | |
| 53 | One thing to note is that the mean free path of a photon in the ionization front is not particularly comparable to the actual width of the ionization front in this case, since the ionization front cannot be considered thin on our scale. |