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Astron. Astrophys. 348, 831-842 (1999)

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5. Comparison with models for CQE's

The above description of the observations does not by necessity lead to the conclusion, that all 6 stars owe CQE's to the same mechanism. It even leaves open the possibility, that the CQE's observed in one and the same star, but different lines, have different origins. Especially the CQE's observed in lines with or without significant photospheric contributions may form differently. However, the homogeneity of the parameters shown in Table 2 is rather indicative of a common origin. For the sake of simplicity, such mixed stellar and circumstellar explanations are not considered here. Because of the unquestionable circumstellar origin of at least some of the CQE's, the main focus will in Sect. 5.2 be on such an explanation. However, since only stellar models have been considered in the past, the next subsection first briefly re-visits some purely photospheric models.

5.1. Photospheric origin of CQE's

Jeffery's (1991) Table 2 provides specific criteria for observational tests of his model. CQE's should hardly be seen in HeI [FORMULA] 4026 for any model, whereas they should be most prominent in SiIII [FORMULA] 4553, NII [FORMULA] 3995, and CII [FORMULA] 4267 for early B-type stars. However Fig. 1, left panel, shows that HeI [FORMULA] 4026 is particularly likely to exhibit CQE's. By contrast, in none of the other candidate lines identified by Jeffery CQE's could be detected in any of the program stars.

In the absence of rotationally induced gradients in effective gravity and temperature (von Zeipel's theorem), Zorec's (1994) model predicts the same profile for all lines with the same intrinsic width. This is not observed. By combining differential rotation and latitudinally varying atmospheric properties, a much larger range of sets of line profiles can be generated for an otherwise fixed set of parameters. However, as was pointed out by Baade (1990) and Jeffery (1991), this would still only work for stars with intermediate inclination angles. More specifically, Jeffery states that only sin i = 0.2 to 0.8 is a possible range for the formation of CQE's, the best being 0.4 to 0.6 for most models. The fact, that six out of six CQE stars are also shell stars, effectively excludes both Zorec's and again Jeffery's model since there is ample evidence that the shells around Be stars are equatorial, disk-like structures (cf. Sect. 6).

A final possibility to produce CQE's in the photosphere is a reduced polar chemical abundance of all elements showing CQE's (cf. Baade 1990). Over the broad wavelength range of the HEROS and FEROS spectra, there is no evidence of major chemical peculiarities. A still more severe problem is that, of a given ion, some lines may and others may not show a CQE. Finally, the restriction of CQE's to lines much narrower than those from the photosphere at large cannot be construed as evidence of a circumpolar origin. Fig. 3 shows that the width of a given line may vary as a consequence of the shell contribution to this line. Moreover, such an interpretation would be at variance with all else that is known about the formation of narrow lines in Be shell stars (cf. Sect. 6)

In summary, the present observations of CQE's do not invalidate the principles of any of the photospheric models proposed so far. But these models simply cannot account for the observations, which evidently require a circumstellar explanation.

5.2. Circumstellar origin of CQE's

Hanuschik (1995) has computed the iso-radial velocity contours of a gaseous Keplerian disk viewed edge-on and the associated fraction of the stellar disk that is occulted by gas having a given line-of-sight velocity. The scattering and absorption of stellar photons in an opaque spectral line formed in the circumstellar gas is roughly proportional to the obscured fraction of the stellar disk. This area reaches a maximum at a radial velocity [FORMULA] = [FORMULA], that corresponds to the orbital Keplerian velocity, [FORMULA], at the outer disk radius, [FORMULA], projected on to the line of sight towards the stellar limb. Since geometry and velocity field are symmetric with respect to the orbital axis, two such maxima exist, namely one each at [FORMULA] and [FORMULA]. Accordingly, a local minimum in absorption develops at zero velocity, which corresponds to a local maximum in flux, very much alike the CQE's. The principle is demonstrated in Hanuschik's Figs. 3 and 9; examples of the resulting line profiles are shown in his Fig. 8.

The local minimum in geometrical occultation by gas with zero radial velocity always exists in a Keplerian disk seen edge-on. Whether it leads to an observable CQE depends mainly on two basic circumstances:

  • Outer disk radius: Because [FORMULA] decreases with increasing outer disk radius as [FORMULA], the two line profile minima are the more separated the smaller the disk is. That is, a CQE can still be resolved even in the presence of some line broadening if the disk is small. However, if the size of the disk increases beyond some critical dimension, the separation of the two absorption maxima becomes smaller than the intrinsic line width. Then, the CQE disappears and a pure absorption core with a width corresponding to the intrinsic line width will appear instead. Since the ionization structure of the disk may change radially, "outer disk radius" actually denotes the outer radius at which a given line is formed. To some extent, this covers also the case of disks of finite thickness but not viewed exactly edge-on.

  • Turbulence and thermal broadening: Obviously, these two quantities need to be small in order not to reduce the contrast between a CQE and its adjacent minima beyond detectability. Lines of heavier elements are, therefore, more likely to display CQE's.

Finally, the zero-velocity condition with respect to the stellar photosphere also implies that CQE's occur at the stellar systemic velocity. Accordingly, they would supply a very reliable means of measuring variations of the latter even in the presence of other variations.

These theoretical conditions form an almost exact match of the empirical criteria for the occurrence of CQE's derived in Sect. 4.1. Because the inclination angle of all six stars considered will be somewhat different from 90 degrees and their vertical disk structure may not be the same as assumed by Hanuschik, the conclusion that the explanation for circumstellar CQE's is given by Hanuschik's model becomes even more robust.

However, although Hanuschik's model is very successful in reproducing CQE's (and shell line profiles in general), it is not physical in that it only assumes Keplerian rotation but does not predict it. This point, therefore, requires further scrutiny, which is the subject of the following section. Before that, a very abbreviated example of the practical usage of CQE's for the diagnosis of the structure of Be star disks, and especially their variability, is given in the following sub-section.

5.3. CQE's as a means to probe the disk structure

Hanuschik's (1995) Eq. 23 offers rather straightforward access to the radial disk structure, if the positions of the local minimum cusps blue- and redwards of the CQE are measured. This can be demonstrated well for the example of [FORMULA] Cen in 1996, when a similar outburst-relaxation sequence as in µ Cen was observed (Rivinius et al. 1998a).

Hanuschik (1995) normalized the equation to [FORMULA]. Even assuming [FORMULA], the rotation velocity of the disk at [FORMULA] is unknown. For a Keplarian disk this is the critical velocity, rather than the stellar rotational velocity as for a disk conserving angular momentum. As estimate [FORMULA] km/s is taken, being a plausible value for such an early B-type main sequence star. The cusp separation was measured for HeI [FORMULA] 6678. These numbers were derived from data secured just after an outburst and six weeks later, just before the following outburst event. The separation of the minimum cusps [FORMULA] is related to the distance from the star [FORMULA] up to which the lower levels of the respective transitions are populated. For HeI [FORMULA] 6678, this is the [FORMULA] singlet transition from 21.13 to 22.97 eV. HeI [FORMULA] 6678 moreover shows weak emission peaks, that can be used to probe the radius [FORMULA] up to which the upper level is populated sufficiently to produce net emission. Similarly, the H[FORMULA] peak separation can be used to derive an estimate of the dimensions of the H[FORMULA]-emitting disk. In the case of Keplerian rotation the respective equations are

[EQUATION]

and

[EQUATION]

The disk radius derived from the CQE as well as from the emission became larger during our observations 1996. It grew from 6.2 to [FORMULA] for HeI [FORMULA] 6678 absorption (CQE's) and from 7.3 to [FORMULA] for the H[FORMULA] emission. On the other hand, the emission radius derived from HeI [FORMULA] 6678 emission started at 2.8 and grew to [FORMULA]. The higher outbound velocity at smaller radii is consistent with the conclusions drawn already from µ Cen (Rivinius et al. 1998a), that part of the ejected material after a burst migrates slowly outwards as a ring of enhanced density, visible as migrating emission peaks. There it merges with an already present, relatively stable disk at larger radii. It should be noted that these values are derived on the basis of simplifying assumptions, as [FORMULA], and only estimated critical velocity, but the relative changes of the numbers should be reliable.

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Online publication: August 13, 199
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