Convergence

Ionization cycles in Python are intended to establish the correct ion densities and temperature of the various cells in the wind. The degree to which this happens for a given number of ionization cycles depends on how far the initial guess of electrons temperatures in various portions of the wind and the number of photons generated during each photoionization cycles. Furthermore, the accuracy of the final model depends on the number of photons that pass through each cell. As a result, the accuracy with which ion abundances and temperature are determined will differ on a cell by cell basis. In a typical model with a biconical outflow, a small cells at the outer edge of the accretion disk will record fewer photon passages than one in the middle of the grid that is exposed to large numbers of photons from the disk.

To estimate whether a a model calculation has reached a steady state, Python carries out three tests, one comparing the difference in the radiation temperature between the current and the preceding ionization cycle, one comparing the electron temperature in the same manner and once comparing heating and cooling rates in the current cycle. More specifically, if a cell satisfies the following 3 tests,

\[\left | \frac{t_e^n-t_e^{n-1}}{t_e^n+t_e^{n-1}} \right | < \epsilon\]

\[\left | \frac{t_r^n-t_r^{n-1}}{t_r^n+t_r^{n-1}} \right | < \epsilon\]

\[\left | \frac{Heat_{tot}^n-Cooling_{tot}^{n}}{Heat_{tot}^n+Cooling_{tot}^{n}} \right | <\epsilon\]

where \(\epsilon = 0.05\), then cell is said to have converged.

The routine run_check.py produces two plots related to convergence, one showing the fraction of cells that have passed each of the tests above as a function of cycle, and the other showing the number of failed checks for each cell in the wind.