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4. Discussion

Using a simple phenomenological model for a star-CE system, we give in this section an outline for a discussion which aims at obtaining the first indications on the continuum opacity regimes of CE in Be stars, responsible for their most characteristic spectrophotometric changes, namely: (a) single relations in definite SPh-Be phases with slopes tex2html_wrap_inline6443 and tex2html_wrap_inline5915; (b) no variation of V and tex2html_wrap_inline4779 in SPh-shell phases (tex2html_wrap_inline6451).

4.1. Model of the CE

Assuming an ellipsoidal CE with ellipticity tex2html_wrap_inline6453 (h is the polar height of the CE and tex2html_wrap_inline6457 its equatorial radius), the radiation flux emitted by a Be star can, in a first approximation, be described with an ellipsoidal slab-like model as:


where the geometrical factor tex2html_wrap_inline6459, which is also proportional to the ratio of the envelope source function to the underlying stellar flux, is:
In (22) we have tex2html_wrap_inline6461 for h > R*, where tex2html_wrap_inline6465 is the radial optical thickness of the shell. For cases where h < R* and tex2html_wrap_inline6469 (polar regions of the star not shielded by the CE) it is tex2html_wrap_inline6471. In (23) tex2html_wrap_inline6473 is the normalized projected area of the CE; tex2html_wrap_inline6475 is the Planck function for the CE excitation temperature tex2html_wrap_inline6477; tex2html_wrap_inline6479 is the stellar flux. Except for tex2html_wrap_inline6481 when h < R* so that tex2html_wrap_inline6471, we see that tex2html_wrap_inline6487 and tex2html_wrap_inline6459 are scaled by the same constant factor tex2html_wrap_inline6491. For a qualitative discussion of (22) we may then use either a given value of tex2html_wrap_inline6491 or simply, for the sake of brevity, consider E = 1, which corresponds to a spherical CE. This approximation was widely used in the literature to discuss visible energy distributions of Be stars. We also assume that the stellar components (V*, tex2html_wrap_inline6499, D*) are not variable (Zorec & Briot 1991). Thus, using (22) with E = 1, the magnitude excess tex2html_wrap_inline4869, the colour excess tex2html_wrap_inline6507 and the BD discrepancy tex2html_wrap_inline6511 produced by the CE, can be represented as:

where tex2html_wrap_inline6513m; tex2html_wrap_inline6515 stand for the Paschen and Balmer sides of the continuum energy distribution at tex2html_wrap_inline6517m. The gradients are obtained using the classic definition (Allen 1973): tex2html_wrap_inline6519. The gradient tex2html_wrap_inline6521 is derived from tex2html_wrap_inline6523, so that tex2html_wrap_inline6525.

Before going into details of (24), let us comment on the curves tex2html_wrap_inline6527 shown in Fig. 7 which describe the main characteristics of (22) regarding the photometric variations of Be stars. Depending on the sign of tex2html_wrap_inline4835, two behaviours can be distinguished: (a) that corresponding to tex2html_wrap_inline4973, which is typical for a SPh-Be phase and where we always have tex2html_wrap_inline6533; (b) that of tex2html_wrap_inline4979, which corresponds to a SPh-shell phase and where it is always tex2html_wrap_inline6537. In (a) two distinct regimes may clearly be identified: (i) a low opacity regime characterized by tex2html_wrap_inline6541 (vertical side of tex2html_wrap_inline6543 curves); (ii) a high opacity regime, where tex2html_wrap_inline6547 (horizontal side of tex2html_wrap_inline6543 curves). Both regimes are schematically separated in Fig. 7 by dashed lines at tex2html_wrap_inline6551 (to sketch out the separation of opacity regimes tex2html_wrap_inline6551 was choosen so that tex2html_wrap_inline6555). In Fig. 7, it can also be seen that in the low opacity regime: tex2html_wrap_inline6557, changes in the magnitude V are mainly an opacity effect, because even if tex2html_wrap_inline6459 do not remain constant, small variations of tex2html_wrap_inline6563 will introduce marked changes of tex2html_wrap_inline4835. On the contrary, when tex2html_wrap_inline6547, variations in V mostly reflect changes of tex2html_wrap_inline6459. In (b), that is, for a SPh-shell behaviour, tex2html_wrap_inline6573, which corresponds to a high opacity regime. Curves tex2html_wrap_inline6575 and tex2html_wrap_inline6577 against tex2html_wrap_inline6563 also have the same kind of patterns as tex2html_wrap_inline6581, but their dependences on tex2html_wrap_inline6521 and tex2html_wrap_inline6585 reveal some more subtle behaviours.

We note that the sign of tex2html_wrap_inline4835 is a function of tex2html_wrap_inline6459 and tex2html_wrap_inline6563 only. Hence, the magnitude changes corresponding to both spectrophotometric phase variations in the same star can be described with a simple spherical CE. In the next subsection we shall also see that for the same star this model is enough to explain the main spectrophotometric characteristics of both SPh-Be and SPh-shell phases. We note again that for flattened CE with tex2html_wrap_inline6593 seen at angles tex2html_wrap_inline6595, as tex2html_wrap_inline6471 we cannot have tex2html_wrap_inline4979. On the other hand, it may happen that tex2html_wrap_inline6601, even if tex2html_wrap_inline6603 and tex2html_wrap_inline6605.

Figure 7: Ratio tex2html_wrap_inline6459 (for E = 1) as a function of optical depth tex2html_wrap_inline6563 for several values of tex2html_wrap_inline4835 corresponding to Be phases (tex2html_wrap_inline4973) and Be-shell phases (tex2html_wrap_inline4979). The vertical dashed lines schematically divide zones of low and high opacity for the respective tex2html_wrap_inline4973. The dotted curve corresponds to tex2html_wrap_inline6601

4.2. Spectrophotometric variations

In patterns of Fig. 6, two spectrophotometric behaviours seem to be the most relevant: (a) in SPh-Be phases, where tex2html_wrap_inline6627, there are linear correlations between (V,D) and tex2html_wrap_inline5803 so that tex2html_wrap_inline6443 and tex2html_wrap_inline5915; (b) in shell phases, where tex2html_wrap_inline6415, there is a very small variation, if any, of tex2html_wrap_inline4779 and V as a function of D. Let us then see in what opacity regime these variations can take place.

4.2.1. Low opacity regime

For tex2html_wrap_inline6645, where tex2html_wrap_inline6647, from (24) we derive the following relations:
where tex2html_wrap_inline6649, with tex2html_wrap_inline6651. Using (25) we readily realize that:
which are characteristic for SPh-Be phases. This suggests that SPh-Be variations are likely to be produced by CE in low opacity regimes. As for tex2html_wrap_inline6653 it is tex2html_wrap_inline6655 const., changes of tex2html_wrap_inline6459 as a function of tex2html_wrap_inline6653 represent variations of the CE extent tex2html_wrap_inline6661. It follows from (22) that for a given value of tex2html_wrap_inline4877 when tex2html_wrap_inline6653, tex2html_wrap_inline6459 (and so tex2html_wrap_inline6661) is an increasing function of tex2html_wrap_inline6671 that easily reach tex2html_wrap_inline6673, which implies a rather extended CE. To an order of magnitude estimate, if we assume a B2e star (tex2html_wrap_inline6675 22500 K), tex2html_wrap_inline6677 mag and tex2html_wrap_inline6679 for tex2html_wrap_inline6681 K, it follows from (22) that tex2html_wrap_inline6683.

4.2.2. High opacity regime

Two situations can be distinguished: (a) tex2html_wrap_inline4973, which implies emission excess in the Paschen continuum; (b) tex2html_wrap_inline4979, which indicates flux deficiency.

In (a) we have tex2html_wrap_inline6547 and:
where mostly tex2html_wrap_inline6699.

In (b) it is tex2html_wrap_inline6701 and it always happens that:
but the behaviour of tex2html_wrap_inline6585 deserves some additional explanations. Following the comments given in Sect. 3, the typical spectrophotometric behaviour in SPh-shell phases seems to be summarized by:
Using (29) in (24), we derive a differential equation for tex2html_wrap_inline6705 whose solution for tex2html_wrap_inline6707 and for a wide range of tex2html_wrap_inline4979 can roughly be represented by:


Hence, knowing that:


conditions (29) imply that SPh-shell phases reveal sensitivity of CE to temperature changes of the CE and that tex2html_wrap_inline6477 is a decreasing function of tex2html_wrap_inline6563. On the other hand, with (24) and (30), it can be shown numerically that the BD difference tex2html_wrap_inline6585 increases with tex2html_wrap_inline6563. So, the increase of D in SPh-shell phases is probably due to CE in high opacity regimes with a decreasing temperature as the opacity becomes higher.

As noted before for tex2html_wrap_inline4835, it is also not possible to produce tex2html_wrap_inline6725 at tex2html_wrap_inline6727 to have tex2html_wrap_inline6415, as needed for SPh-shell phases in pole-on Be stars with flattened CE where tex2html_wrap_inline6593.

Relation (28) implies that in SPh-shell phases CE are rather shrunk. As in the preceding section, we assume that for a shell phase of a B2e star tex2html_wrap_inline6733 mag and tex2html_wrap_inline6735 are characteristic parameters. Writing tex2html_wrap_inline6737 where for most shell stars tex2html_wrap_inline6739 (Zorec & Garcia 1991) we get tex2html_wrap_inline6741 K so that tex2html_wrap_inline6743. Using spectroscopic data, Kogure (1989) also concluded that regions of the CE responsible for spectroscopic shell spectra are smaller and denser than those producing spectroscopic Be phases. A more detailed discussion of these phenomena will be presented in a following paper.

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