Here we discuss potential effects of other geometries than the ‘patch geometry’ assumed in all our wind models in Chapters 4 and 5. (See these chapters for a description of the patch model.) Let us first point out, however, that most of the effects discussed here still are subjects to (sometimes quite lively) debates among practitioners in the field, and that a real consensus regarding their importance has not yet been reached.
As a first example, let us consider a wind consisting of spherical, isotropic clumps with characteristic length scales l, volumes≈l3(neglectingπfactors of order unity), and distancesΛbetween them that are equal in all spatial directions. AssumingΛ>>l, we have fv≈l3/Λ3and h=l/fv≈fcl2/3Λ. The
equation of continuity in a spherical expansion may be used as a constraint forΛ’s radial dependence, implying ncl∝(r2v)−1with clump number density ncl. Since ncl=1/Λ3, it follows thatΛ ∝(r2v)1/3.
(This is the model used by Oskinova et al. 2007, in their application of an isotropic porosity formalism, Sect. 3.4, to line opacity.) But consider now clumps that are spread out over the surface of the star and start propagating radially outwards, obeying some velocity law v(r). If clumps do not collide or merge, the physical distances between clumps in theΘ direction will be ΛΘ ∝r, but the radial distances between clumps will be controlled by the velocity law. So ifΛr=ΛΘ is to hold, we must
have v∝r. Thus, if we interpret the assumption ofΛ equal in all spatial directions strictly, we may not let the clumps flow according to aβ-type velocity law and simultaneously assume equal distances between clumps in all directions.
A similar problem is encountered in the combined assumptions of clumps of equal length scales l in all spatial directions and a constant fv; the radial distances between clumps no longer increase
130 APPENDIX A. MORE ON THE RADIATIVE TRANSFER CODES
directions still increase asΛΘ∝r. Then if we demand that fvmust be kept constant, l must increase in
all directions to preserve the equal length scales, and will eventually become larger thanΛr. To avoid
this somewhat strange picture of radially overlapping clumps (which in principle is the same as having clumps that are ‘infinite’ in the radial direction, at least as long as clumps are radially aligned, i.e. have no or very small lateral velocity components), one must make also the radial distances between clumps proportional to r (either by a homologous expansion, v(r)∝r, or by not letting clumps flow with the velocity field). Naturally, this would further imply that h∝r.
However, it may be that these apparent problems are only illusionary. The way around them is to not interpretΛ and l in the strict meanings of physical distances, but instead as quantities determining an average mean free path between the clumps,
m f p= 1 nclAcl ≈ Λ3 l2 = l fv =h. (A.45)
In this picture, the average mean free paths, and thereby the porosity lengths h, remain isotropic, independent of the assumed expansion of the medium, and despite the fact that the physical distances between the clumps are not equal in all directions. Then it is not necessary to invoke v∝r to obtain a consistent wind model with isotropic clumps. In terms of the above mentioned picture of radially elongated clumps that overlap, the essential point is that the spherical expansion opens up for holes in between these through which photons can escape, so that the mean free path still is preserved, also in the radial direction. However, at least within a modification of our present patch geometry, this kind of model might be problematic, as now discussed.
Modifying the patch geometry. Clumps in our stochastic models expand in the tangential direc- tions preserving their solid angles. Thus for clump length scales l, lΘ,lΦ ∝r. On the other hand, the clumps’ radial widths lr=vβδt fvare calculated by pre-describing fvandδt . This leads to very
anisotropic ‘pancake shaped’ clumps, as is easily seen from density contour plots of corresponding models (Figs. 2.1 and 4.1). Moreover, clumps are released from the stellar surface with a complete covering fraction.
It might be, however, that clumps do not preserve their solid angles, but experience ‘lateral break-up’ when traversing outward in the wind. As a first guess, let us assume that such lateral break-up scales as r−2 (which essentially means that clumps keep their initial lateral extensions). Then lr= fvΛrr2,
and we encounter the same problem as above with clumps that eventually all overlap each other in the radial direction (since the radial distances between them are determined by vβδt, i.e. is constant when the terminal speed is reached). Physically, it may be questionable that such a clump geometry, i.e. one in which clumps are extremely long in the radial direction but have large lateral holes between them, could exist in a hot star wind. To circumvent the radially overlapping clumps, one might assume that also fvdecreases with r−2(which recovers the original expression for lr), but this would produce much
higher clumping factors in the outer wind than in the inner, which is not consistent with observational constraints from radio emission (Puls et al., 2006).
Another way to modify the geometry of the patch model might be to assume that, in the tangential directions, only a fixed fraction of the total wind volume of one slice is covered by clumps. Denoting with Cc the fraction of the total wind slice that is covered by clumps, we obtain lr = fvΛrCc−1, i.e. a radial extension of the clumps for a given volume filling factor, which compensates for the lateral holes created. However, for a radial photon within a given wind slice that encounters precisely one clump within its resonance zone, the probability of actually hitting this is now Ccinstead of 1, as in the
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original patch model. This coverage fraction therefore becomes important when clumps are optically thick; for the resonance lines a lower than unity Cc would open up for additional escapes, which in turn would reduce the profile strengths. We may connect this to the ‘broken shell’ porosity models that were presented in Sect. 3.4; if clumps were optically thick for a specific continuum process, no radial photons could escape through our original patch models. However, if Cc were to be used and clumps were to be randomly positioned within the wind slice, holes would open up, and radial photons would have a chance to escape without ever encountering a clump1. In principle, this corresponds to the assumptions of fragmentation and lateral randomization that are inherent in the broken shell wind models for porosity.
As illustrated, details on clump geometry as well as on coverage fractions might be important for the radiation transport in clumped hot star winds. Unfortunately, however, little is known of either. Therefore we have adhered to the ‘patch geometry’ in Chapters 4 and 5 in this thesis, deferring to future studies, e.g., the inclusion of coverage fractions into our wind models.