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.