Ray-tracing and Microphysics

Our ray-tracing and microphysics routines are based largely on the methods in Lim (2003) and on the C$^2$-ray method developed by Mellema (2006). We use operator splitting to first update the dynamics by a full timestep, and then run the microphysics update over the full timestep. In this work we only consider explicitly the ionisation of atomic hydrogen. We first describe the ray-tracing algorithm and then the microphysics calculation.

The Short Characteristics tracer (e.g. Raga, 1999) is used to trace out rays from a source in a causal manner, calculating the optical depth to a cell by interpolating between (previously calculated) optical depths to neighbouring cells closer to the source. Given that we are ignoring diffuse radiation (the On-the-Spot approximation) the diffusion in the ray-tracer is not significant, and is minimized using the weighting scheme given by Mellema (2006).

When the photo-ionisation time is short compared to other time-scales (cooling, recombination, and collisional ionisation times) the microphysics equations become difficult to solve explicitly so we adopt a dual approach. In cases of weak photo-ionisation, we use an explicit 5th order Runge-Kutta technique with adaptive step-size to a given relative accuracy (Press, 1992). For strong photo-ionisation we integrate explicitly until the Hydrogen ion fraction, $x$, satisfies $x\geq0.95$, and then analytically integrate the equations assuming a constant electron density (as described in Mellema 2006), with bisection substepping to convergence (typically $2-4$ substeps). For both of these methods we use a relative error tolerance of 0.001.

This algorithm also calculates the time-averaged optical depth through the cell $\Delta\tau$, which is then used by subsequent cells in the ray-tracer. Mellema (2006) use a simple time average of $\Delta\tau$, however we use a time average of $\exp(-\Delta\tau)$ since this gives a time average of the fraction of photons passing through the cell. This can be easily seen in the (extreme) case of an optically thick cell which is photo-ionised ``rapidly'' half way through a unit timestep, so that

$$\Delta\tau(t) = \left\{ \begin{array}{cc}100&t<0.5\\ 0&0.5<t<1 \end{array} \right\} \,.$$ (1.1)

The mean optical depth over the timestep is 50, but clearly half of the incident photons will pass through the cell, and $\int_0^1\exp(-\Delta\tau)dt=0.5$ gives the desired result. We do this integration at the same time as the microphysics variables, to the same accuracy criterion.

We use monochromatic radiation with a hydrogen photo-ionisation cross-section of $6.3\times10^{-18}\,\mathrm{cm}^{2}$ and an energy gain of $5.0\,$eV per photo-ionisation. Collisional ionisation rates are calculated with fitting functions from Voronov (1997), and radiative recombination (Case B) rates using the tables calculated by Hummer (1994). The difference between planar radiation and radiation from a point source can be quite significant if the size of the computational domain is comparable to the distance to the source. The rocket effect is weaker further from a point source due to the inverse square law, which may extend the lifetime of any structures that form. This effect can, however, reduce the length of such structures since the intensity of the radiation is higher at their heads. In the case of M16, the heads of the pillars are about $2\,$pc from the brightest star, and they are about $1\,$pc long, so the flux dilution is more than a factor of 2 along their length. We therefore use a point source in this work.