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Up: Self-similar evolution of wind-blown


   
1 Introduction

Numerous observations of wind-blown bubbles (WBBs) have led to the conclusion that their structures and evolution are significantly affected by mass entrainment from embedded clumps. As a case in point, bubbles blown by the winds of highly evolved stars can be mass-loaded from clumps of stellar material ejected during prior stages of massloss (Smith et al.1984; Hartquist et al.1986; Meaburn et al.1991; Dyson & Hartquist 1992; Hartquist & Dyson 1993). The properties of winds mass-loaded by material from clumps (or stellar sources) has also received theoretical attention in the context of ultracompact H II regions (Dyson et al. 1995), globular cluster winds (Durisen & Burns 1981), galactic winds and starburst galaxy superwinds (Strickland & Stevens 2000), and accretion flow structures at the centers of active galactic nuclei (David & Durisen 1989; Toniazzo et al.2001).

Investigations into the self-similar nature of supernova remnant (McKee & Ostriker 1977; Chièze & Lazareff 1981; Dyson & Hartquist 1987) and stellar wind-blown bubble (Pittard et al.2001) evolution in tenuous media with embedded clumps have been performed. Detailed one-dimensional, time-dependent hydrodynamic models of specific WBBs associated with evolved stars have also been constructed (Arthur et al. 199319941996). In at least some isothermal mass-loaded winds, Arthur et al.(1994) and Williams et al. (1995, 1999) showed that a global shock does not occur in the region where the Mach number is large. Instead, a global shock occurs only after the wind has travelled far enough for mass loading to decelerate it sufficiently that its Mach number is less than a few. In some of these cases, a global shock within the mass loading region does not exist at all, and the wind continues until it encounters a termination shock caused by the interaction of the wind with an external medium.

Observations supporting mass loading in WBBs include the detection of blue-shifted absorption features of species with a range of ionization potentials in the spectrum of the central star of the Wolf-Rayet nebula RCW 58. Smith et al.(1984) argued that the observed velocity spread for the detected features is much larger than would be expected for $T \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle ... K gas consisting only of stellar wind material decelerated through a terminal shock, and suggested that the velocity spread originates due to mass loading of the shocked wind through the entrainment of material from clumps. Crucially, one can obtain an estimate for the ratio of wind mass to entrained clump mass: the observed velocity spread implies a value of 40 or 50 to one. A detailed time-dependent, hydrodynamic model including nonequilibrium ionization structure is consistent with these conclusions (Arthur et al.1996). Spectroscopic data also provide evidence for transonic flows in the halo of core-halo planetary nebula (Meaburn et al.1991), again consistent with high mass loading rates. Finally, it is also suspected that the gradual deceleration of a wind by mass loading and the associated weakening of an inner shock may in fact contribute to the radio quietness of some WBBs, although this is a conjecture which currently cannot be proven (see Williams et al.1995, 1999). In Pittard et al.(2001, hereafter PDH), similarity solutions were derived for the structures and evolution of mass-loaded WBBs, under the assumption that conductive evaporation from embedded clumps was the dominant mass loading process, and that both evaporation from the cold swept-up shell and radiative losses were negligible. To obtain a similarity solution with these conditions, specific radial power-laws on the clump and interclump density distribution, and temporal power-laws on the wind mass-loss rate and terminal velocity, were required. Approximate similarity solutions for evaporatively mass-loaded WBBs with the assumptions of constant mass-loss rate and wind velocity, and an isobaric shocked wind region have previously been obtained by Weaver et al.(1977; evaporation from the cool swept-up shell) and Hanami & Sakashita (1987; mass-loading from clumps). A central assumption in both of these papers was that the shocked wind was approximately isobaric. However, PDH demonstrated that this was not necessarily a good assumption (e.g. see their Fig. 4). Indeed, imposing this condition is likely to set a limit to the amount of mass loading that can occur.

A central conclusion of PDH was that there exist maximum possible values for the ratio of evaporated mass to stellar wind mass, as a consequence of the evaporation rates dependence on temperature and the lowering of temperature by mass loading. In particular it was difficult to find ratios approaching what was observationally required. The work in this paper complements PDH by considering the case in which hydrodynamic ablation is the dominant mass addition process. As conductively driven evaporation has a very temperature sensitive rate, ablation is likely to regulate clump dispersal into lower temperature media.

Our solutions are again potentially relevant to such diverse objects as WBBs created by a faster wind interacting with a clumpy AGB superwind, by the wind of a young star interacting with surrounding molecular material, and the wind of an active galactic nucleus impacting its environment.


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