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Astron. Astrophys. 335, 1003-1008 (1998)

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3. Clumping in first approximation

3.1. Formulation

For a first-order approach to clumped stellar winds, we make the following simplifying assumptions:

  • The wind consists of clumps with density [FORMULA], where [FORMULA] is the density stratification of the homogeneous model with the same mass-loss rate. The factor [FORMULA] thus gives the density enhancement, and is assumed to be constant all over the atmosphere.

  • The interclump space is void. Thus the volume filling factor of the clumps is [FORMULA].

  • The clumps are assumed to have small size, compared to the photon free path. Thus the radiative transfer can be calculated with "effective" emissivities and opacities averaging between the clump and interclump medium.

Under these assumptions, clumping can be accounted for by limited modifications in existing model codes. The statistical equations have to be solved for the enhanced density [FORMULA]. The population numbers, which add up to the enhanced particle density of the clump medium, are applied as usual for the calculation of the non-LTE emissivity and opacity. However, the latter two are scaled down by the volume filling factor [FORMULA] before they enter the radiative transfer calculation.

In our code we obtain the consistent solution of the co-moving frame radiative transfer equation and the equations of statistical equilibrium by an iterative scheme using approximate lambda operators. The latter are set up with the same, modified opacities which are used in the "exact" radiative transfer. The described modifications of emissivity and opacity also apply in the Formal Integral which is finally performed in the observer's frame once the population numbers are established.

3.2. Scaling properties

In the described formulation, the density enhancement D would cancel out in the emissivities and opacities, if they were linear in the density [FORMULA]. However, Wolf-Rayet spectra are known to be dominated by processes which scale with the square of the density. This can be concluded from the scaling property of Wolf-Rayet models, which has been discovered first by Schmutz et al. (1989). They defined a so-called transformed radius [FORMULA] as

[EQUATION]

and found that models with same [FORMULA] exhibit the same emission line equivalent widths, irrespective of different combinations of [FORMULA], [FORMULA] and [FORMULA] (while, of course, [FORMULA], composition etc. are fixed). This invariance was validated by various numerical experiments with reasonable accuracy. In a stricter sense, one might compare only models with same terminal velocity [FORMULA]. Then even the line profiles and the total shape of the emergent spectra are invariant for models with same [FORMULA], except of a scaling of the absolute flux with [FORMULA]. This property greatly facilitates any spectral analyses, as only two essential parameters ([FORMULA], [FORMULA]) must be adapted.

The described invariance can be understood by adopting that the relevant emission processes scale with the square of the density. This can be seen as follows. If the emissivity (at any considered wavelength) scales with [FORMULA], then the flux emitted from a volume V of that density scales with [FORMULA]. Keeping [FORMULA] constant implies [FORMULA] (Eq. 1), while we have [FORMULA] from the continuity equation. Combining both proportionalities yields [FORMULA]. Any emitting volume scales with [FORMULA], and thus the emitted flux scales like [FORMULA], just as empirically established. The relative spectral shape remains unchanged.

The likely physical explanation is that most continuum photons are created by free-free emission or photo-recombination, while the line photons are emitted in recombination cascades. The number of created line photons thus depends only on the corresponding recombination rate, even when they are trapped for some time in optically thick scattering lines before they escape. Both, free-free emission and radiative recombination, are [FORMULA]-processes.

The dominance of [FORMULA] emission processes explains the invariance of the radiation field, when comparing models with same [FORMULA] but different combinations of [FORMULA] and [FORMULA], as long as the model structure (stratification of population numbers) is the same. This, however, is not to be expected, as the density structure is different (namely scaling with [FORMULA], see above) and the density never can cancel out in the ratio between any ionization and recombination process. Indeed, a detailed comparison between models with same [FORMULA] reveals the expected differences in the ionization stratification. However, these differences do not, in most cases, lead to significantly different spectra. We suggest that this is due to the relatively weak dependence of [FORMULA] on [FORMULA], and on the fact that the main emission in a specific line comes from those radial zones where the relevant ion (i.e. the next higher stage) is the dominating one.

The concept of the "transformed radius" now can be generalized for clumped models. In order to cancel out in quadratic processes, the clump density enhancement D must be compensated by diminishing the mass-loss rate by a factor [FORMULA]. Thus we define

[EQUATION]

and expect that models with same [FORMULA] exhibit the same line equivalent widths, irrespective of different combinations of D, [FORMULA], [FORMULA] and [FORMULA]. For constant [FORMULA], the absolute spectra should only differ by a scaling with [FORMULA].

Note that this scaling invariance holds only for the [FORMULA] processes, i.e. for the main spectral features. The electron scattering opacity, however, scales linearly with density. Thus, in inhomogeneous models the enhanced clump density is already fully compensated by the volume filling factor; scaling down the mass-loss rate in order to keep the same [FORMULA] decreases the effective Thomson opacity. Hence, for a series of models with same [FORMULA] the extended line wings caused by frequency redistribution of electron-scattered line photons should become weaker with increasing D, while the main spectral feature remain unchanged. This is the reason why the too strong electron scattering wings predicted by the standard, non-clumped models are considered as evidence for inhomogeneities.

Among the manifold combinations of [FORMULA], D and [FORMULA] which give the samed transformed radius [FORMULA], one can restrict the subspace for which the processes linear in [FORMULA] are simultaneously invariant in the above sense, i.e. for which the electron scattering wings are expected to have the same strength. The condition that [FORMULA] also give fluxes that scale with [FORMULA] yields [FORMULA]. In other words, if going to bigger stars with same transformed radius, one might increase the clump density enhancement [FORMULA], and (despite of the constant factor [FORMULA]) the whole spectrum should remain invariant, the main line features as well as the electron scattering wings. Vice versa, this connection has an interesting consequence. If the degree of clumping (i.e. the parameter D) would turn out to be a universal or otherwise predictable property of WR type stellar winds, one could, in principal, estimate the stellar luminosity (and thus the distance etc.) from the spectrum alone. At least, one expects that less luminous stars have weaker electron-scattering wings than more luminous counterparts with otherwise similar spectra. By this one might, e.g., distinguish post-AGB stars with WR type spectra from Pop. I Wolf-Rayet stars.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
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