Three essential ingredients of line driving are resonant amplification, self-shadowing, and the Doppler effect. Resonant amplification causes the cross-section of bound electrons to be many orders of magnitude larger than the Thomson cross-section of free electrons (Heitler 1954; Gayley 1995). Even accounting for their relative scarcity with respect to free electrons, bound electrons are thus capable of providing a much larger acceleration than what is needed to overcome the effective gravity. Self-shadowing of the scattering ions, however, blocks most of the flux available within the width of the spectral line and hence greatly reduces the magnitude of the line-driving force, bringing it back below the gravitational force.
The enormous potential of line driving would therefore remain untapped, were it not for the third ingredient, the Doppler effect. The motion of an ion away from the star allows unattenuated stellar flux to be red-shifted back into the line profile and increases the radiative force to which the ions are exposed. Due to Coulomb interactions the accelerated minor ions drag the bulk mass of hydrogen and helium ions with them. This brings the line driving force up to a level that is sufficient to overcome the inward pull of gravity and maintain a stellar wind. Line-scattering as the driving force of hot-star winds was first suggested by Lucy & Solomon (1970). A formalism to quantitatively model line-driven hot-star winds was introduced by Castor et al. (1975, hereafter CAK), and developed further (e.g. to account for the finite angle of the stellar core) by Pauldrach et al. (1986) and Friend & Abbott (1986). Gayley (1995) introduced an improved formalism that greatly helps clarify the physical interpretation of the line opacity and associated driving.
Long before being actually applied to hot-star winds, it was already realised (Milne 1926) that line driving is highly unstable. The reason for the instability is again the Doppler effect: a small increase in velocity of a fluid element exposes the element to fiercer radiation from the star and causes it to be further accelerated (MacGregor et al. 1979; Owocki & Rybicki 1984, 1985). Time-dependent hydrodynamical simulations of unstable line-driven stellar winds were first performed by Owocki et al. (1988, hereafter OCR), and later by Feldmeier (1995), Feldmeier et al. (1997b) and Owocki & Puls (1996, 1999, hereafter OP96 and OP99, respectively). All these simulations agree on the main features of instability-generated structure: the wind is pervaded by strong shocks (reverse and forward) that can heat the gas to temperatures in excess of a million degrees and compress it into clumps that are typically an order of magnitude denser than the ambient wind.
The instability-generated structure is stochastic in nature and should not be confused with large-scale, localised structures such as co-rotating interaction regions (CIRs). These CIRs are thought to cause the rotational modulation of discrete absorption components seen in ultraviolet spectral lines of hot stars (Prinja 1998). The observational evidence for instability-generated structure is more indirect. Soft X-rays from hot stars, as well as the presence of high ionisation stages such as N V, point to shocks that could be instability-generated (Lucy 1982a). Furthermore, the extended black troughs of saturated ultraviolet lines are indicative of increased back-scattering associated with a non-monotonic velocity (Lucy 1982b). Finally, the non-thermal radio emission detected for some of the brightest O-stars (Bieging et al. 1989) is thought to be synchrotron emission from shock-accelerated particles (Chen & White 1994).
Most previous papers on instability-generated wind structure focus on the inner or intermediate wind (below 30 stellar radii). There are however numerous reasons to study structure in the outer wind. One of the most reliable derivations of the mass-loss rates of hot stars is from the thermal radio continuum due to free-free emission (Bremsstrahlung). The conversion of a radio flux into a mass loss rate (Wright & Barlow 1975) depends on the degree of clumping of the gas (Abbott et al. 1981). Underestimating the degree of clumping results in an overestimate of the mass-loss rate.
The synchrotron radiation that is thought to be responsible for
non-thermal radio emission has to originate beyond
,
as photons
originating closer to the stellar surface would be shielded by the
large free-free opacity of the gas. Furthermore, the shock-acceleration itself
has to happen in situ, as Compton cooling prevents
relativistic particles from travelling large distances
(Chen & White 1994). This
points to the existence of shocks at large distances from the star.
Inverse Compton emission by relativistic particles produces non-thermal
X-rays, and from the comparison of the non-thermal X-ray
and radio emission
one can in principle derive the magnetic field strength
(Pollock 1987).
The formation region of thermal X-rays depends sensitively on the recombination of fully ionised helium to He+(Hillier et al. 1993; Feldmeier et al. 1997a). In winds where He recombines, observable X-rays must be emitted more than a hundred stellar radii above the stellar surface.
Finally, many Wolf-Rayet stars are surrounded by ring-nebulae. These nebulae are formed when a fast wind overtakes a slower wind from a previous evolutionary stage. High-resolution images have shown these nebulae to be highly structured. This structure can be caused by instabilities associated with the wind-interaction or could also be an imprint of structure in the wind itself (Grosdidier et al. 1998). This would require structure to persist up to very large distances from the star.
This paper presents radiation hydrodynamical simulations of wind structure out to distances of 100 stellar radii from the stellar surface. In Sect. 2 we discuss the general principles of our models. In Sect. 3 we give the statistical quantities which describe the wind structure. In Sect. 4 we present the reference model, with which all the other models are compared. In the next sections we discuss the effect of various processes and assumptions on the outer-wind structure: external forces such as line driving (Sect. 5), the spacing of the radial mesh (Sect. 6), the line-strength cut-off parameter (Sect. 7) and heating and cooling by radiation (Sect. 8). Conclusions are presented in Sect. 9.
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