7 The importance of the line-strength cut-off

As first noted by OCR, the high-speed rarefactions that arise in simulations of line-driven instability tend to continue to grow until they become optically thin to the strongest driving line. To keep such structure from becoming too steep to resolve with the assumed spatial grid, OCR introduced an opacity cut-off, which for relatively dense O-supergiant wind models had to be set to an artificially low value $\kappa_{\rm max} \approx 10^{-3}\, \kappa_{0}$. (As discussed in Sect. 2.2, a realistic value is $\kappa_{\rm max} \approx \, \kappa_{0}$.) Such a low cut-off has since been used in essentially all instability simulations of dense winds, and so also provides the basis for the standard case presented above. But in experimentation we have done with relatively fine grid resolution, we have found it possible to increase this cut-off to less artificial levels. Here we present an initial comparison of models with different levels of this cut-off parameter.

Figure 8: Snapshot of the inner wind at 2.0 Msec after the start of the simulation, for the model with $\kappa _{\rm max}=10^{-2} \kappa _0$. The dashed line in the upper panels represents time-averaged values.

Figure 9: Same as Fig. 8, but for the outer wind.

Specifically, Figs. 8 and 9 show snapshots of a model where $\kappa _{\rm max}$ has been increased from $10^{-3}\, \kappa_{0}$to $10^{-2} \,\kappa_{0}$. The effect of this increase is to include ${\sim}10$ strong lines that were not present in the reference model. It takes steeper velocity gradients for these lines to become optically thin than for the strongest lines in the reference model. They thus allow for much stronger rarefactions and shocks. Some of the gas is heated to very high temperatures. Although this hot gas is still only a minute fraction of the wind mass (as it was for the reference model), increasing $\kappa _{\rm max}$ does increase the mass fraction of X-ray emitting material. (For the snapshot shown above, 0.12% of the wind mass is in excess of a million degrees, compared to 0.02% for the reference model.)

Figure 10 shows the statistical properties of the model with $\kappa _{\rm max}=10^{-2}\, \kappa _0$ (dashed line) and the reference model (solid line), together with a model where $\kappa _{\rm max}$ has been decreased by a factor of ten (dotted line). Reducing $\kappa _{\rm max}$ by this amount leaves only the weaker lines to drive the wind and very little structure is formed.

Figure 10: Clumping factor and velocity dispersion for three different values of $\kappa _{\rm max}$. The solid line is the reference model, with $\kappa _{\rm max}=10^{-3} \,\kappa _0$. The dashed line corresponds to $\kappa _{\rm max}=10^{-2}\, \kappa _0$, and the dotted line to $\kappa _{\rm max}=10^{-4} \,\kappa _0$.

The strongest clumping in the model with increased $\kappa _{\rm max}$ occurs very far from the star (at 60 R_*) and is due to collisions forming dense and narrow clumps (with density enhancements reaching values over 100 and widths below 0.1 R_*). Resolving these clumps requires a small grid spacing over almost the entire grid. Hardly any collisions happen beyond 60 R_* and at the outer boundary the clumping factor is almost the same as for the reference model.

The change in $\kappa _{\rm max}$ also has a modest effect (of the order of 10%) on the mass loss rate and terminal velocity. This is due to the fact that with the smaller cut-off the distribution contains about 10% fewer optically thick lines. Accordingly, using the CAK/Sobolev line driving force in the point-star approximation as an estimate, we find the cut-off reduces the force by about 10% (we find a value of 0.9 for the factor in square brackets in Eq. (12) of OCR).