5 The effect of external forces at large distances

As both gravity and the line driving force fall off as 1/r²at large distances, where the star appears as a point source, we can expect them to become unimportant far away from the star. In the reference model, the time-averaged wind has reached 92% of its terminal velocity at 10 R_*, suggesting a minor effect of external forces beyond that distance. It is not clear, however, to what extent this estimate pertains to the formation of structure. To examine the rôle of external forces on the outer wind variability, we now compare our full results with simulations that are assumed to be "force-free", i.e. no gravity or radiative force, beyond a certain distance.

To do this, we stored the time-series of the density, velocity and pressure from the reference model at some intermediate radius $R_{\rm in}$ and used this time-series as a fixed inflow inner boundary condition for the force-free model that extends from $R_{\rm in}$ to the usual outer boundary $R_{\rm max} = 101\ R_*$. We use two different values for $R_{\rm in}$: 11 and 31 R_*. Again, care was taken to run the model for a sufficiently long time, to allow the response to the initial condition to die away.

Figure 6 shows the statistical properties of the reference model (solid line), and two force-free models with $R_{\rm in} =11\ R_*$ (dotted line), $R_{\rm in} = 31\ R_*$ (dashed). The statistical properties of the $R_{\rm in} =11\ R_*$ model differ substantially from the reference model, indicating that, even though the wind has nearly reached its terminal velocity, the driving force still plays a rôle in the formation of structure. Beyond $31\ R_*$, this is no longer the case and the statistical properties of the force-free model are essentially the same as for the reference model.

This implies that the outer wind evolution beyond $r \approx 30\ R_*$ is a pure gasdynamical problem, for which radiation driving merely sets the inner boundary condition. In related work, we have utilised this principle to develop "pseudo-periodic" models that can be calculated out to very large distances (i.e. 1000 R_*) at relatively low cost, by repeating a fixed time-series at an inner boundary set at an intermediate radius $R_{\rm in} \approx 30\ R_*$. (For an initial discussion, see Owocki et al. 2000.)