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 and
; (b) no variation of *V* and in SPh-shell phases ().

Assuming an ellipsoidal CE with ellipticity (*h* is the polar
height of the CE and 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 , which is also proportional to
the ratio of the envelope source function to the underlying stellar flux, is:

In (22) we have for *h* > *R*_{*},
where is
the radial optical thickness of the shell. For cases where *h* < *R*_{*} and (polar regions of the star not shielded
by the CE) it is . In (23) is the normalized projected area of the CE;
is the Planck function for the CE excitation temperature
; is the stellar flux. Except for when *h* < *R*_{*} so that , we see that
and are scaled by the same constant factor .
For a qualitative discussion of (22) we may then use either a given value of
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*_{*}, , *D*_{*}) are not variable
(Zorec & Briot 1991). Thus, using (22) with *E* = 1, the magnitude excess
, the colour excess and the *BD* discrepancy produced by the
CE, can be represented as:

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

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

We note that the sign of is a function of and 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 seen at angles ,
as we cannot have . On the other hand, it may
happen that , even if and .

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

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

For , where , from
(24) we derive the following relations:

where , with . 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 it is const., changes
of as a function of represent
variations of the CE extent . It follows from (22) that for
a given value of when ,
(and so ) is an increasing function of that easily reach , which implies a rather extended CE.
To an order of magnitude estimate, if we assume a B2e star (
22500 K), mag and
for K, it follows from (22) that .

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

In (a) we have and:

where mostly .

In (b) it is and it always
happens that:

but the behaviour of 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 whose solution for and for a wide range of 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 is a decreasing function
of . On the other hand, with (24) and (30), it can be
shown numerically that the *BD* difference increases with . 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 , it is also not possible to produce at to have , as needed for SPh-shell phases in pole-on Be stars with flattened CE where .

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 mag and are characteristic parameters. Writing where for most shell stars (Zorec & Garcia 1991) we get K so that . 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.