4 Reference model

The stellar parameters and wind parameters used for the reference model are the same as those in OP99, and are given in Table 1. Although not specifically intended to model a particular star, the stellar parameters correspond to an early O supergiant such as $\zeta$ Pup. The chosen parameters result in a mass loss rate of $3.5 \times 10^{-6}\ M_{\odot}$/yr and a terminal speed of 2000 kms^-1. It is assumed that H and He are fully ionised.

We use a fixed spatial mesh of 10637 points. In order to adequately resolve the region of rapid acceleration, the initial 1025 points have a spacing that increases linearly from 0.001 to 0.01 R_* over the radius range r = 1-5 R_*. The remaining points use a constant spacing of 0.01 R_*spanning the range r= 5-101 R_*.

  
4.1 Wind structure

 

 

Table 1: Stellar parameters and wind parameters for the reference model.

quantity	symbol	value
stellar mass	M	40 $M_{\odot}$
photospheric radius	R*	19 $R_{\odot}$
effective temperature	$T_{\rm eff}$	37800 K
CAK exponent	$\alpha$	0.7
opacity constant	$\kappa_0 v_{\rm th}/c$	3500 cm2/g
line strength cut-off	$\kappa _{\rm max}$	0.001 $\kappa_0$
H abundance by mass	X	0.73
He abundance by mass	Y	1-X
thermal speed	$v_{\rm th}$	0.28 $a^{\dagger}$

$\dagger$ Isothermal sound speed $a=\sqrt{p/\rho}$.

Figure 1: Snapshot of the reference model in the inner wind, at 2.0 Msec after the start of the simulation. The dashed line in the upper panels represents time-averaged values.

Figure 2: Same as Fig. 1, but for a representative portion of the outer wind. Note that the range in radius is larger than on the previous figure.

Figure 3: Spatial and temporal evolution of the normalised density $\rho /<\rho >$. To highlight the dense clumps, the lower and upper cut-offs of the grey-scale have been set to 1 and 2, respectively; gas with a density below the mean appears as white, and all shells with a density larger than $2<\rho >$ as black.

Figure 4: Snapshot of the reference model at 2 Msec, now plotted versus the Lagrangian mass coordinate m defined in Eq. (10). The upper panel shows the Eulerian radius, while the remaining panels show the velocity, density, and temperature. The dashed lines in these lower panels show the corresponding time-averaged values.

Figures 1 and 2 show snapshots of the radial variation of velocity, density and temperature in the inner and outer wind. Many of the features have been extensively discussed in other papers (e.g. OCR, Feldmeier 1995; Feldmeier et al. 1997b, OP99), and so here we focus mainly on the radial evolution of the overall structure properties.

The wind can be divided into a number of geometric regions for which the boundaries are somewhat fuzzy, but which nevertheless have their own typical characteristics.

The outflow at the base of the wind is almost steady. But starting at $r \sim 1.3 \;R_*$irregular variations appear in the velocity and density. The variations rapidly steepen into shocks (for a detailed discussion of the onset of structure in SSF models, see OP99). These initial shocks are reverse type, which means that they propagate inward relative to the gas, although the overall outflow advects them outward relative to the star. They decelerate and compress rarefied gas that has been accelerated to high-speed by the instability. Most of the velocity peaks in the upper panel of Fig. 1 are steep rarefaction waves terminated by a reverse shock.

In this initial structure, most of the stellar wind material thus becomes collected into in a sequence of dense clumps bounded on the inside by a reverse shock that separates the clumps from the much more rarefied, high speed flow in between them. After a few stellar radii many clumps also become bounded on the outside by a weaker forward shock, whenever clumps flow faster than material ahead of them. For example, a weak forward shock is visible around 5.3 R_*, just beyond a strong reverse shock. The structure then finds its definite form: a sequence of dense clumps bounded on both sides by shocks that feed rarefied gas into the clumps. Most of the shocks in Fig. 2 occur in such reverse-forward pairs.

The clumps are typically an order of magnitude denser than the average wind. They move at approximately the terminal velocity but have finite relative velocities, causing them to collide and form denser clumps. This can be seen from Fig. 3, which shows the density (normalised to the mean density $<\rho>$) as a function of radius and time, with the dark streaks indicating the motion of a clump. The importance of clump collisions for X-ray production has been emphasised by Feldmeier et al. (1997b). In the simulations by these authors, the base of the wind was perturbed by a sound wave or turbulence. Our results show that clump collisions can also occur when the structure is self-excited, although generally with lower relative speeds than in the perturbed models. As they result in denser clumps, collisions also play an important rôle in maintaining structure. Note moreover that collisions can persist to quite large distances from the star.

Fig. 4 shows a wind snapshot at 2 Msec plotted versus a Lagrangian mass coordinate (OCR, OP99) defined by

$$m(r) \equiv \int_r^{R_{\rm max}} 4 \pi \rho (r') r'^2 \, {\rm d}r' .$$ (10)

Such a plot emphasises the relative amount of mass to be associated with various flow structures. In particular, note that the high-speed rarefactions that form from the initial instability actually contain very little mass, and so are not likely to have much direct effect on, e.g., observable line-profile variations. Likewise, the lowermost panel makes clear that there is really very little material that is shock-heated to temperatures substantially above the wind floor temperature. As such, these models of intrinsic variability generally yield far less X-ray emission than is commonly observed for hot-star winds. Moreover, this plot shows that the purely spatial average of the temperature given by the dashed curve is actually not very representative of the temperature for most of the wind mass.

On the other hand, the mass plots of velocity and density illustrate quite well the persistence of substantial velocity dispersion and clumping through an extended range of material in the outer wind.

Figure 5: Statistical properties of the reference model. The three panels, from top to bottom, show the clumping factor, the velocity dispersion, and the velocity-density correlation, all as a function of radius. The full line corresponds to averages taken between 2 and 2.5 Msec, the dashed line to averages taken between 2.5 and 3 Msec. The zero level for the correlation function is indicated by a dotted line.

4.2 Statistical properties

The statistical properties of the reference model (Fig. 5) also reflect the distinct characteristics of the different stellar wind regions. To judge the influence of the time interval over which the statistical quantities are calculated, we continued the reference model for another 500 ksec, i.e. over 2.5-3 Msec. Quantities for the standard model (from 2 to 2.5 Msec) are indicated by solid lines, and those for the 500 ksec extension by dashed lines. The good overall agreement between these indicates that the statistical quantities do not depend much on the integration interval and so are intrinsic properties of the model.

At the base of the wind (below $r \sim 1.5 \;R_*$), the mass distribution is smooth ( $f_{\rm cl} =1$) and the variations in the velocity are extremely small ( $v_{\rm disp} \ll a$). As is typical for SSF calculations (OP99), structure appears with almost perfectly anti-correlated variations of density and velocity ( $C_{\rm {v\rho}} \approx -1$). These variations grow dramatically and steepen into shocks. The subsequent nonlinear interaction from clump collisions quickly disrupts this flow anti-correlation, so that above $r \approx 2 \; R_*$there is little net velocity-density correlation, indicating there is roughly equal mixture of forward and reverse propagating structure.

The steep initial rise in velocity dispersion reflects the initial strong amplification of velocity variations by the line-driven instability. This initial rise is temporarily halted as the high-speed rarefactions are filled in and the anti-correlation vanishes. But then, quite surprisingly, there develops a second rise in dispersion, characterised now by little net correlation between velocity and density. Moreover, even after the dispersion reaches an absolute maximum rms amplitude of ${\sim}200$ kms^-1 at around $r \approx 3 \; R_*$, the subsequent decline is quite gradual, with a residual dispersion of ${\sim}50$ kms^-1persisting even at $r= 100 \; R_*$.

Perhaps even more surprising, the density clumping factor actually continues to rise out to nearly $r \approx 20 \; R_*$. The supersonic collisions among the clumps tend to compress them further, causing a steady rise in the clumping factor. But as these collisions become weaker and less frequent in the outer wind, the pressure-driven expansion of individual clumps into the rarefied regions between them eventually causes the overall clumping factor to slowly decline.

One simple way to maintain structure up to large distances would be to have cold, high density gas in pressure equilibrium with hot, rarefied gas. Shocks do heat some gas to very high temperatures, but due to the efficiency of radiative cooling, only the most rarefied gas can remain hot. It is, however, not hot enough to balance the pressure of the dense clumps. It appears therefore, that under the present assumptions, clump collisions are the key mechanism to maintain a structured wind.

Figure 6: Statistical properties of the reference model (solid line), compared to two models where the external forces have been set to zero beyond 11 R*(dotted line) and 31 R* (dashed line).

Figure 7: Influence of the spatial grid on the statistical properties of the model. The solid line corresponds to the reference grid, the dashed line to the grid with doubled spacing and the dotted line to a grid with spacing proportional to r.