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Astron. Astrophys. 349, 151-168 (1999)

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3. Interpretation of SPh variations

3.1. Constraints for CE models

In the preceding section we have shown that mean [FORMULA] and [FORMULA] give insights for mean aspect angle and temperature dependencies. However, from star to star the CE regions near the central star responsible for the observed SPh behaviours can have a variety of geometric and physical structures. At the moment, we would like to investigate only the most outstanding characteristics of these regions using simple but sufficiently general CE models: (a) to estimate the first order effects produced on the visible spectrum by different geometrical structures of CE (discs, ellipsoids); (b) to draw out easily global statistical conclusions. To do so, we shall first quote some relevant characteristics that must be respected by any CE model to ensure basic consistencies with observations.

(a ) - On the extent of CE regions producing the SPh changes

Interferometric measurements of [FORMULA] Cas (Stee et al. 1998) show that the diameter of the CE zone producing the continuum flux excess in [FORMULA]m at the observing epoch is 3.5[FORMULA] and 2.8[FORMULA] for [FORMULA]m. It is then expected that the radius of the region forming the continuum radiation at [FORMULA]m will not exceed 1.6[FORMULA]. The bulk of visible continuum radiation was predicted by Poeckert & Marlborough's disc CE model for [FORMULA] Cas to originate from within twice the observed extent: [FORMULA]. In general, as radiation in the visible continuum is dominated by H recombination and by stellar light scattered by free electrons, the studied CE region must be rather highly ionized. H ionization proceeds efficiently from the second level (Höflich 1988, Millar & Marlborough 1998). So, the volume of the visible continuum formation region should be proportional to the generation rate of stellar Balmer photons (Osterbrock 1989): [FORMULA] [FORMULA] [FORMULA] [FORMULA] ([FORMULA] = mean temperature of stellar layers supplying the Balmer photons). Thus, the expected range of CE zone radii [FORMULA], assuming [FORMULA] cm-3, is [FORMULA](B9V) [FORMULA] [FORMULA](B0V) in close agreeemnt with observations of [FORMULA] Cas. This suggests that CE zones contributing to the observed magnitude V are very near the central star: about or less than 2 stellar radii .

(b ) - On the density of CE regions implied by the SPh changes

Among the most noticeable characteristics of SPh variations is the appearance and variation of the second component of the BD. This component is either in emission (SPh-E phases), or in absorption (SPh-A phases) (Barbier & Chalonge 1939, 1941, Divan 1979, Zorec & Briot 1991). The emission rise (or the absorption fall) of the second BD begins closely at [FORMULA] Å in almost all Be stars. This emission (or absorption) reaches its highest value at [FORMULA] Å. Excepting some rare and short lasting cases, no coninuum emission rise (or absorption fall) is noticeable from [FORMULA] to [FORMULA]. Using the Vidal's (1966) relation, or simply the known Inglis-Teller formula, this wavelength limit implies that there can be an upper limit to the CE base density at about [FORMULA] cm-3. We note that emission intensities in H[FORMULA], H[FORMULA], H[FORMULA] lines and that in the visible continuum of some Be stars have been consistently explained by Höflich (1988) using spherical CE with base densities [FORMULA] cm-3.

(c ) - SPh phases vs. aspect angle of Be stars

We note that some program Be stars, with aspect angles probably [FORMULA], have shown widely recorded spectroscopic Be-shell phases which also unavoidably imply strong differences [FORMULA]. However, simultaneous photometric observations in both sides of the BD were curiously not carried out during their phase changes. Among these stars, the most significant are HD 5394 (Underhill & Doazan 1982) and HD 200120 (Hubert-Delplace & Hubert 1981, Underhill & Doazan 1982). Concerning HD 200120, a decrease of more than 1 mag in the far-UV energy distribution was observed with the TD-1 satellite (Beeckmans 1976) during one of its spectroscopic Be-shell phases (1972-mid 1974). Hummel (1998) explains the line variations during these rather short-lasting phase changes with a temporarily tilted circumstellar disc. The photometric behaviour in these phases can also be explained in terms of huge prominence-like ejections (Zorec et al. 1999). However, in our sample there are a number of Be stars with rather low [FORMULA], which are seen in long duration spectroscopic Be-shell and SPh-A phases: HD 56014, HD 184279, HD 187811 and HD 218674. On the other hand, Be stars not included in our sample have shown strong shell characteristics in their spectra, like HD 45542 ([FORMULA] [FORMULA] km s-1) (Hanuschik et al. 1996). According to the discussion in Sect. 2.3.1, the expected mean aspect angle of this group of stars is small: [FORMULA] (upper inclination for [FORMULA]; lower inclination for [FORMULA]; mean dispersion of means [FORMULA]).

Contrary to the preceding situation, where Be stars with low [FORMULA] seen at SPh-A phases, there are Be stars with [FORMULA], which are more frequently seen as Be-shell and SPh-A, but which have shown long lasting Be and SPh-E phases. The best known is Pleione (HD 23862), for which there is a vast literature about its phase variations (Hirata & Kogure 1976, Gulliver 1977). This star is seen nearly equator-on: [FORMULA]. On the other hand, stars like HD 4180, HD 35439, HD 83953, HD 91465, HD 162428, HD 191610 and HD 217543 which are more or less in genuine spectroscopic Be and SPh-E phases have a mean probable inclination [FORMULA] (lower limit of inclination for [FORMULA]; upper limit of inclination for average value of maximum attainable [FORMULA] of these stars [FORMULA]; mean dispersion of means [FORMULA]). In these stars spectroscopic Be-shell and SPh-A phases are rare and weak, or they have not yet been seen (Hubert-Delplace & Hubert 1979). In the frame of flattened CE models, such as those of Poeckert & Marlborough (1978a,b), these objects must all be seen at low inclinations [[FORMULA], also see (f )].

Though it would not be impossible to account for SPh-E [FORMULA] SPh-N [FORMULA] SPh-A changes when [FORMULA] ([FORMULA] half-opening angle of CE) with flattened CE, SPh-A phases cannot be produced with them if [FORMULA]. Hence, a great number of Be stars, far from being an exception, seem to be seen at inappropriate angles in regard to predictions of strongly flattened CE models.

(d ) - Interferometric data and the aspect angles

Interferometric measurements of Be stars (Quirrenbach et al. 1993, 1994, 1997; Stee et al. 1995, Stee et al. 1998) were done for the following Be stars: HD 5394, HD 10516, HD 22192, HD 23630, HD 25940, HD 37202. The apparent ellipticities [FORMULA] of the H[FORMULA] emission line formation regions in the respective CE, aspect angles [FORMULA] calculated as discussed in Sect. 2.3.1 and the lower limits predicted from [FORMULA] are given in Table 2. We also reproduce the adopted stellar parameters. HD 10516, an interacting binary whose shell appears twice per orbital cycle, was excluded from this table.


[TABLE]

Table 2. Measured CE ellipticities and probable aspect angles of Be stars.
Note:
We used BCD spectral classification; [FORMULA] was derived from [FORMULA]) calibrations in stellar parameters


From Table 2, taking into account the possible uncertainties in [FORMULA] and [FORMULA], we can see that the predicted aspect angles [FORMULA] and [FORMULA] do not contradict each other, though [FORMULA] is not a genuine measured parameter. HD 23630 and HD 25940 are sometimes considered as slightly hotter than the BCD classification. This leads to a lower estimate of their [FORMULA]. On the other hand, the H[FORMULA] emission line profiles of these stars are bottle-shaped, which means that their markedly pole-on aspect may be due to non-coherent scattering in the CE (Hummel & Dachs 1992). Hence, they do not necessarily correspond to stars seen at very low aspect angles [FORMULA]. Nevertheless, [FORMULA] is systematically higher and i is lower when stars are in a Be phase, which favors interpretion of observations in terms of flattened CE. We have however shown in (c ) that there is an important fraction of Be stars where spectroscopic and SPh phases do not obey the phase-aspect angle tendency which would be expected if only discs or strongly flattened CE existed.

(e ) - Linear polarization of visual continuum radiation

Assuming that Be CE envelopes are Thomson scattering dominated axial-symmetric ellipsoids with ellipticity E, a power-law [FORMULA] electron density distribution and an extended central source of unpolarized light, using a Monte Carlo method for a multi-scattering approach, it was possible to show that a given linear polarization in the V magnitude in Be stars may correspond to two different optical depths and that observations are consistent on average with [FORMULA] and [FORMULA] (Höflich & Zorec 1989, Hö et al. 1989, Höflich 1991). We shall see in Sect. 3.3.1 that [FORMULA] carry [FORMULA] which implies that moderate densities and flattenings of CE may explain the observed polarization in V.

Among the most detailed spectropolarimetric calculations for Be stars are those of Poeckert & Marlborough (1978a) done for [FORMULA] Cas (HD 5394), and of Wood et al. (1997) for [FORMULA] Tau (HD 37202). Shortcomings of Poeckert & Marlborough's attempts were discussed in the work of Wood's et al. It is thus worth commenting the model of [FORMULA] Tau, because of its connection with the SPh studied in the present paper. The model is characterized by a thin CE with a half-opening angle [FORMULA], constant temperature, density distribution [FORMULA] with [FORMULA] [FORMULA] [FORMULA] cm-3 and inclination [FORMULA]. The model accounts for the BD polarimetric jump, visible and IR spectropolarimetric distributions. It indeed fails to describe the polarimetric distribution in the Balmer continuum and the energy distribution around the BD. In fact, SPh BCD observations spanning about 50 years of [FORMULA] Tau have always shown two well defined BD components: (a) a constant photospheric component [FORMULA] dex; [FORMULA] Å, so that from Chalonge & Divan's (1973) [FORMULA] calibration into MK classification, the spectral type of [FORMULA] Tau is B2III; (b) a variable second component in absorption [FORMULA] dex. Wood's et al. thin CE predicts however a second BD in emission . On the other hand, due to the very sudden fall of the second BD at [FORMULA], the expected electron density at the base of the CE should be nearly an order of magnitude lower than that used in Wood's et al. model. This shows that the thin CE model (Bjorkman & Cassinelli 1993, Owocki et al. 1994) is probably not suitable to explain the visible and near UV energy distribution of [FORMULA] Tau. We still note that the wind-compressed disc formation can be inhibited by radiation pressure and rotation (Owocki et al. 1996).

(f ) - Visual emission from flattened CE

Waters & Marlborough (1994) and van Kerkwijk et al. (1995) have discussed difficulties in explaining with disc-like CE models (Waters 1986, Poeckert & Marlborough 1978a) the observed correlations in Be stars between H[FORMULA] emission and [FORMULA] colour excess. Regarding the SPh aspects, flattened CE models worked out in detail by Poeckert & Marlborough (1978a,b) show that continuum emission becomes detectable only when [FORMULA] and [FORMULA] cm-3, which conflicts with the density limits discussed in (b ). For other inclinations and whatever the density, these models always produce continuum absorption ([FORMULA]). Hence, the main difficulty of such models lies in the fact that for low inclinations ([FORMULA]) the SPh-E [FORMULA] SPh-N [FORMULA] SPh-A cannot be produced by them, and that for [FORMULA] the model fails to produce the SPh-E aspects.

3.2. The model

Taking into account the arguments developed in the preceding subsection, we must be able to explain the observed SPh variations with CE models: (a) where densities do not exceed [FORMULA] cm-3; (b) with regions producing the visible continuum which are not greater than some [FORMULA]; (c) which easily produce SPh-A and SPh-E phases and variations from one to another, whatever the aspect angle; (d) that explain the bivalued SPh relations [Sect. 1.1(2)]. As we have already noted, strongly flattened CE cannot fulfill requirement (c). Mass loss in Be stars is a strongly variable phenomenon. In addition, these stars show light outbursts which imply short lasting but strong mass ejections (Hubert et al. 1997, Hubert & Floquet 1998). SPh variations are then probably a consequence of dynamic phenomena taking place in the CE. Calculation of SPh variations as a function of these phenomena is outside the scope of this paper. We shall however take them into account in a simplified way using axi-symmetrical CE by varying their extent, base density and/or their flattening (ellipticity).

3.2.1. Electron density distribution in the CE emitting region

The electron density distribution in a disc-like CE is currently represented in the literature by a power-law [FORMULA] [FORMULA] [FORMULA] with [FORMULA] (tefl 1998). In the present work we generalize this law by assuming that the electron density is constant over ellipsoidal surfaces, which all have the same ellipticity E. Such a particle density distribution is not far from some theoretical predictions for CE in Be stars (Araújo & Freitas Pacheco 1989, Stee & Araújo 1994, Stee et al. 1995). Making the conservative assumption [FORMULA], the electron density over ellipsoidal CE isopicnic surfaces is then given by [FORMULA] [FORMULA], where [FORMULA] [FORMULA] is the polar radius of an ellipsoidal surface; x and y are coordinates in the equatorial plane and z is measured from the equatorial plane); [FORMULA] is the base density.

As in what follows an equivalent slab will represent the effect of radiation resulting from an ellipsoidal structure of the CE, its effective aspect angle dependent opacity is assumed to be represented by the one calculated in the radial direction of the viewing angle i. It is then given by (MZH):

[EQUATION]

where [FORMULA] is the total [scattering ([FORMULA] + absorption bound-free+free-free ([FORMULA])] radial optical depth of the CE in the polar direction. Although this representation looks oversimplified, we shall see that it is able to account for the main angle dependent characteristics of the radition field emerging from the star-CE system.

Some SPh variations studied in Sect. 3.3.2 are for CE which preserve their mass while they change their volume. In CE with constant mass and changing volume, where the index n is constant, the main variation of [FORMULA] is produced by the subsequent drop of the density at the base of the CE. The analytic expressions for [FORMULA] are cumbersome and depend on the configuration whether the CE covers or not the polar regions of the central star. For shortness, if we use power representations [FORMULA] for the R dependence, [FORMULA] can be approximated to a sufficient degree of accuracy by:

[EQUATION]

All values of s and [FORMULA] ("e" stands for electron scattering and "a" for bb+ff absorption) studied in this paper are between those of to two limiting cases: spherical CE and flat discs. Hence, the scattering component of the total opacity [FORMULA]has: [FORMULA] for spherical CE and [FORMULA] for discs; [FORMULA] for both cases. The absorption (bf+ff) component [FORMULA] has: [FORMULA] (spherical CE); [FORMULA] (disc) and [FORMULA] for both. [FORMULA] for spherical CE and [FORMULA] for discs.

3.2.2. The radiation field

For a given radial-dependent particle distribution in an ellipsoidal CE, in a good first order approximation, radiation transfer can be reduced to an addition of emitting/absorbing equivalent thin slabs. Neglecting second order effects due to tangential optical depths and assuming isothermal slabs, the emitted radiation flux of a star-CE system like the one depicted in Fig. 7 is given by:

[EQUATION]

[FIGURE] Fig. 7. Geometric elements of the star-CE system ([FORMULA]; [FORMULA]). Star-CE system observed equator-on (left); Star-CE system projected on the plane perpendicular to the line of sight (right; [FORMULA]).

The first term accounts for the energy emitted by the fraction of stellar surface which is not covered by the CE. The second term represents the fraction of stellar energy absorbed by the CE layers between the star and the observer and the energy emitted by the same layers. The third term gives the amount of energy emitted by the CE layers beyond the stellar surface. [FORMULA] is the fraction of the uncovered stellar surface seen by the observer and projected on the plane perpendicular to the line of sight. [FORMULA] is the fraction of stellar surface eclipsed by the CE. [FORMULA] is the area, projected on the background, of the ellipsoidal CE which does not eclipse the star. All quantities [FORMULA], [FORMULA] and [FORMULA] are given in units of the stellar disc area ([FORMULA]) and their explicit forms for several inclination intervals are given in Appendix A. The relevant parameters needed to represent the fractional surfaces are shown in Fig. 7. We still have in (11): [FORMULA], the stellar photospheric flux and [FORMULA], the equatorial radius of the equivalent ellipsoidal shell representing the CE.

Due to electron densities expected in hydrogenic CE of Be stars, opacity due to electron scattering can be significant (Hirata & Kogure 1984, Millar & Marlborough 1998). The scattered radiation field then has to be taken into account in the source function [FORMULA] of the emitting shells (here `shell' has the literal meaning) (Mihalas 1978):

[EQUATION]

where [FORMULA] is the ratio of scattering to the total absorption coefficient, [FORMULA] is the Planck function and [FORMULA] is the local mean radiation field, which based on the assumptions used for (11) is given by (Moujtahid 1998):

[EQUATION]

where [FORMULA]. The first member of (13) represents the diffuse component and the second is the scattered stellar radiation field. For extreme cases: [FORMULA] and [FORMULA], (13) reduces to the known simplified expression [FORMULA] [FORMULA] [FORMULA] (Mihalas 1978), where [FORMULA] is the geometrical dilution factor. Having introduced the opacity ratio [FORMULA] in the source function, the characteristics of the BD will be due to the absorption discontinuity not only through the absorption/emission terms [FORMULA] but also to the discontinuous source function [FORMULA] given by (12).

To test the validity of the approximations made to calculate the angle dependent radiation fluxes, we compare in Fig. 8 the magnitude differences [FORMULA] obtained from (11)(full lines) with those predicted by a Monte Carlo simulation (open dots) (Höflich & Zorec 1989, Höflich 1991) as a function of the aspect angle i. Both calculations refer to a system where the central star has [FORMULA] K and [FORMULA], and an ellipsoidal CE with [FORMULA] where the density distribution follows the [FORMULA] law. The CE has a constant temperature [FORMULA], equatorial extension [FORMULA], base electron densities (a): [FORMULA] and (b): [FORMULA] cm-3. We can see, that despite the roughness of our model, it closely reproduces the global variation of [FORMULA] against i predicted by the Monte Carlo method.

[FIGURE] Fig. 8. [FORMULA] magnitude differences for a CE with [FORMULA] against i calculated with (11) (full lines) and from a Monte Carlo simulation (open dots) for two base densitites. (a): [FORMULA]; (b): [FORMULA] cm-3

3.2.3. Temperature of the CE emitting region

Except for rare examples, models of CE in Be stars assume either a constant temperature [FORMULA] or [FORMULA] K whatever the photospheric effective temperature. In both cases the choice of [FORMULA] is independent of the CE density and/or its optical depth. Recently Millar & Marlborough (1998) discussed the temperature structure in Poeckert & Marlborough's (1978a) CE model for [FORMULA] Cas. Assuming that the diffuse radiation in the CE can be neglected, they obtain that the temperature in the CE is [FORMULA] [FORMULA] and fairly constant up to 100 [FORMULA]. It ranges from [FORMULA] K in the dense equatorial plane to about [FORMULA] K in the less dense regions.

This result can be qualitatively understood as follows. In dense layers ([FORMULA] cm-3) where collisional ionization rates of H dominate, the temperature should approach [FORMULA] K. In less dense CE regions, where photoionizations and radiative recombinations dominate, the ionization energy is provided by stellar irradiation. As in Be stars ionization of H atoms comes mostly from the Balmer level (Höflich 1988 and Millar, private comm.). Combining the statistical and radiative equilibrium conditions [as done for the Lyman level (Cayrel 1963, Cram 1978)] we obtain: [FORMULA] [FORMULA] [FORMULA], which implies that [FORMULA], the mean temperature of the photospheric Balmer continuum formation regions. Using the Kurucz' (1994) models of stellar atmospheres we get [FORMULA]. These values can be reproduced with the interpolation formula: [FORMULA]. For [FORMULA] K used for [FORMULA] Cas, it follows that [FORMULA] K, close to Millar & Marlborough's (1998) temperature estimate for layers where the density is [FORMULA] cm-3. For SPh-E phases we shall then assume [FORMULA].

In CE where densities are expected to be rather high, we should have [FORMULA] K and even lower. This may be suited for SPh-A phases, as discussed in MZH, where it was shown that the second component of the BD during a SPh-A phase deepens in absorption while [FORMULA] decreases according to an increase of the CE total optical depth.

3.3. Results of calculations

In Sect. 2.2 it was noted that SPh-A slopes do not show any detectable dependence on aspect-angle and on effective temperature, as we have [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA]. So, calculations here below concern SPh-E phases only. On the other hand, only a[FORMULA] and a[FORMULA] SPh slopes are worth studying with models because they seem to show the most noticeable [FORMULA] and [FORMULA] dependencies.

3.3.1. SPh energy distributions

Before going on to predictions of characteristics of SPh parameters, in the present section we shall answer very briefly goal (c) mentioned in Sect. 3.2. To do so, we show in Fig. 9 the energy distributions obtained by (11) for a same model-Be star, seen in a SPh-E and in a SPh-A phase. This calculation was done for a B2IV-V central star: [FORMULA] K, [FORMULA], log g = 3.9. In both phases [FORMULA]. Temperatures for the studied CE zones were adopted as discussed in Sect. 3.2.3: T [FORMULA] [FORMULA] K for SPh-E and T [FORMULA] K for SPh-A. According to Höflich & Zorec's (1989) results, we used [FORMULA] and [FORMULA]. The remaining free parameters [FORMULA] and [FORMULA] were chosen so as to reproduce the mean observed values of [FORMULA] and [FORMULA]. The CE model parameters hence used and the resulting SPh quantities are resumed in Table 3.

[FIGURE] Fig. 9. Model SPh-E ([FORMULA] mag; [FORMULA] dex) and SPh-A ([FORMULA] mag; [FORMULA] dex) energy distributions normalized to the photospheric flux at [FORMULA]m. The dashed line represents the photospheric energy distribution for [FORMULA] K and log g = 3.9


[TABLE]

Table 3. Model parameters characterizing the energy distributions shown in Fig. 9.


At once we can see that for the SPh-E phase the model reproduces an intrinsic reddening of the Paschen continuum that accompanies [FORMULA] and [FORMULA]. In the SPh-A phase we note that there is a [FORMULA] with a very small absorption [FORMULA] and that there is almost no reddening in the Paschen continuum, as shown by observations (Sect. 1.1). For both phases we have [FORMULA] cm-3, [FORMULA] [FORMULA] and the CE region producing the SPh-A aspect is more compact (denser and less extended) than that producing the SPh-E phase, as previously discussed in MZH. It follows then that rather small changes in density, extent and temperature of the CE can be sufficient to explain SPh phase variations in a given Be star.

3.3.2. Assumed CE variations

As it is outside the scope of this paper to calculate in detail SPh variations produced by changes of the CE structure due to dynamic phenomena driven by successive more or less continuous or discrete mass loss events, we shall only sketch them assuming three extreme cases: (a) Surrounding the star there is an elliptical expanding shell (literal meaning) whose mass and ellipticity are maintained. The mass of the shell is parametrized through an initial scattering optical depth [FORMULA] measured in the polar direction; (b) Expanding shell as in (a), but the shell evolves preserving the vertical semi-axis h, so that its equivalent ellipticity is variable as the shell expands: [FORMULA] [FORMULA] (for simplicity we used [FORMULA]); (c) The CE has constant ellipticity ([FORMULA] const) and equatorial extent ([FORMULA] const), but its density increases with time.

For all three cases we assumed that the particle density distribution is preserved with [FORMULA] even if the shape of the CE will change. The central star was characterized by [FORMULA] 22000 K and [FORMULA] 5.7. In all calculations we used [FORMULA]; [FORMULA] and we always used the same parametrization steps: [FORMULA]; [FORMULA] to better appreciate possible differences in rates of SPh variations. Each time the parametrization implied a total optical depth [FORMULA] calculations were stopped.

As an exemple for properties of models discussed in the next section for explaining (a) variations, the following model parameters are listed in Table 4 for five steps of the development of a CE having [FORMULA]: [FORMULA] = CE total equatorial extent; [FORMULA] = base electron density; [FORMULA] [FORMULA] [FORMULA] = total radial opacity for [FORMULA]m in the direction i; [FORMULA] and [FORMULA] that reproduce the SPh variations for case (a) with [FORMULA] and a layer with initial [FORMULA], which corresponds to [FORMULA] cm-3 and an ejected mass [FORMULA] roughly.


[TABLE]

Table 4. Parameters characterizing the case (a) of SPh variations with [FORMULA], [FORMULA] and initial [FORMULA]


3.3.3. Main characteristics of results obtained

Characteristics of the aspect angle dependency of the [FORMULA] variations against [FORMULA] are depicted in Fig. 10. Case (a) is illustrated for [FORMULA], three constant ellipticities: [FORMULA] 0.1, 0.3, 0.7 and for aspect angles: [FORMULA] = 0, 30, 50, 60, 70, 80 and 90. Case (b) is illustrated for initial densities represented by [FORMULA] 0.5, 0.7, 1 and for the same aspect angles as in case (a). Case (c) is illustrated for the same ellipticities as in (a) and for two CE extents: [FORMULA] 2.5 (points), 6 (crosses). In case (c) densities range from [FORMULA] (where [FORMULA]) to [FORMULA] cm-3. For [FORMULA] results are shown for [FORMULA] 0, 30, 50, and 90, while for [FORMULA] they are also given for [FORMULA] 60, 70, 80 and 90. Other results for different effective temperatures and [FORMULA] are shown numerically in Table 5. The SPh slopes were obtained using the least squares method in the BD interval [FORMULA]. In this variation interval of D, which corresponds to that most frequently observed, a number of combinations of model parameters lead to SPh curves that strongly deviate from linear behaviour or simply there is no SPh ([FORMULA]) relation, which justifies some missing SPh curves in Fig. 10 or missing slopes in Table 5. Other missing curves are discussed in Sect. 3.3.4.

[FIGURE] Fig. 10a-c.  [FORMULA] variations against [FORMULA] for several CE configurations. Case a : expanding envelope with E = constant. Case b : expanding envelope with E = variable but [FORMULA]. Case c : variable density in a CE with constant volume. In c points are for [FORMULA] 2.5 and crosses for [FORMULA] 6


[TABLE]

Table 5. SPh slopes calculated for cases (a) and (c). Values are presented by couples a[FORMULA]a[FORMULA] (mag dex[FORMULA]µm dex-1).
Note:
Letters "u" and "l" in column Sol. (solutions) in case (a) are respectively for "upper" and "lower" braches of ([FORMULA]) loops.


The obtained results can be summed up as follows:

1 - Observed amplitudes (typically less than or equal to 0.5 mag) of [FORMULA] variations can be accounted for with events of type (a) to (c) using densities [FORMULA] cm-3 and equivalent CE extents [FORMULA] [roughly the turnover point of the [FORMULA] loops in (a) and (b)]. Emissions are as a mean the strongest the more flattened the CE, but only for [FORMULA]. In strongly flattened CE emission in V is lost for [FORMULA] and when [FORMULA]. However, emission in V produced in CE with more moderate flattenings ([FORMULA]) is seen from whatever aspect angle, though [FORMULA] can be somewhat smaller. In case (b), which more closely corresponds to a disc-like configuration, the emission in the V magnitude is lost at inclinations [FORMULA] and/or when [FORMULA], quite similarly as in Poeckert & Marlborough's (1978a,b) models [Sect. 3.1(f)].

2 - For a given choice of E and i, amplitudes of [FORMULA] and [FORMULA] variations are a function of [FORMULA]. To produce by processes (a) or (b) BD discrepancies [FORMULA] [FORMULA] dex, which are similar to the mean observed ones (Zorec & Briot 1991, Ballereau et al. 1995), the amount of gathered mass in the CE is characterized by [FORMULA]. However, if the same amount of emission in V is expected to be produced by process (b), we would not only need [FORMULA] but also [FORMULA] which can easily conflict with the density limitations discussed in Sect. 3.1(b).

3 - Most of the calculated a[FORMULA] slopes in SPh-E phases are positive. However, negative ones, although rarely observed, are expected for very flattened CE seen nearly equator-on (ex. [FORMULA], [FORMULA]).

4 - Events of type (a) and (b) produce "double valued" a[FORMULA] and a[FORMULA] slopes so that two phases are distinguished, which are discussed in Sect. 3.3.4.

5 - Process (c) produces a single valued a[FORMULA] slope, which is a function of E, i and [FORMULA]. At given values of E and i the rate [FORMULA] strongly depends on [FORMULA]. When [FORMULA] at inclinations from [FORMULA] to [FORMULA], respectively for [FORMULA] 2.5 to 6, in (c) it is [FORMULA] even though [FORMULA] as long as opacity [FORMULA]. For compact CE ([FORMULA] 2.5) when [FORMULA], the SPh evolution is characterized by [FORMULA] constant and a strong emission rise in the magnitude V. In more extended CE ([FORMULA] 6) there is a single monotonic slope in the commonly observed variation interval of [FORMULA], whatever the inclination. The same [FORMULA] and [FORMULA] differences are obtained in the more extended CE using lower values of [FORMULA] than in the compact ones.

6 - Processes (a) and (b) produce a[FORMULA] as observed in most Be stars. Process (c) leads to a reddening effect of the visible energy distribution, but the reddening decreases as emission increases so that a[FORMULA]. This effect was also observed, though rarely.

Finally we note that for given aspect-angle and effective temperature, but depending on the value of [FORMULA] as well as on the degree to which one of the above processes is dominant, a wide variety of SPh slopes can be obtained. This accounts for significant dispersion of SPh slopes among stars with similar fundamental parameters, as seen in Figs. 2 to 5. On the other hand, an effective temperature dependency of SPh slopes can be derived from inspection of Table 5 which is in the direction depicted by the statistical results of Figs. 2 to 5. However, actual temperature dependency may still be different, depending on the model used and on the combination of CE parameters.

3.3.4. Discussion

Though probably cases (a), (b) and (c) do not represent individually an actual dynamic behaviour/evolution of CE, they can be thought of as extreme CE variation frames to explore the first order effects on SPh changes produced by CE as different as discs and spheroids. In particular, they give as clue to understanding the double valued SPh slopes, which is intrinsically important, because it leads us to a possible mechanism of CE formation in Be stars. The double valued SPh slopes can be understood in terms of well known optical depth variations of expanding shells (Ambartsumian 1966, Sobolev 1990). Two main phases are distinguished. In the first phase, the opacity of the shell (literal meaning) is rather high and the photometric evolution of the system is determined by the increase of [FORMULA]. It corresponds to the upper part of [FORMULA] loops in Fig. 10. Assuming the mass of the expanding shell is preserved, it can be easily shown that opacity is a decreasing function of [FORMULA], if [FORMULA] in the particle density distribution function. The emission in V and D reduces accordingly. This characterizes the second phase, the lower part of the [FORMULA] loops in Fig. 10. The gradient [FORMULA] is stronger in the first phase (rapid variation) than in the second phase (slow variation).

If behaviours (a) and (b) are to be driven by an isolated huge mass ejection, an initial value [FORMULA] implies that a shell of mass [FORMULA] would be ejected. The same effect can also be obtained if there is a low-density circumstellar environment where a snowplough like drag of mass is set up by less massive ejecta. The mentioned `second phase' will then exist whenever the rate of accumulating mass in the shell, due to the snowplough effect, does not prevent an opacity decrease produced by the expansion of the outermost layers. Phases of isolated massive ejections, a possible explanation of light outbursts described by Hubert et al. (1997) and Hubert & Floquet (1998), can then be followed by more or less variable continuous mass loss that fills up the environment emptied by the dragged out matter. The latter corresponds in a way to the phenomenon described by case (c). In such an interpretation, the building up of CE in Be stars will then be due to sequences of phases with massive `parcel' ejections, followed by continuous mass loss phenomena. This generalizes the mechanism of CE formation by sequences of variable mass loss events proposed by Zorec (1981). The massive isolated ejections might finally also initiate non-radial global oscillations in CE, which are thought today to produce the V/R line variations (Okazaki 1991, 1996, Hanuschik et al. 1995). We note however, that the density perturbations needed to explain the V/R variations are so tiny that they cannot be responsible for any noticeable SPh changes like those studied in the present paper.

We note that there is still an additional physical reason which explains some missing ([FORMULA]) relations in Fig. 10 and missing slopes in Table 5. They correspond to highly flattened CE configurations, where in some wavelength regions we obtain [FORMULA] as [FORMULA]. In such limits we frequently have [FORMULA], while [FORMULA] in the near-UV carries [FORMULA] implying a strong continuous emission. However, for those wavelengths where [FORMULA] the CE behaves as a pseudo-photosphere and an approximation like (11) is probably no longer suitable. As in classic Be stars, whatever the SPh phase, there is always a distinguishable underlying photospheric energy distribution in the Paschen continuum near the BD [but with rare exceptions such as [FORMULA] Cas in 1932-42 (Chalonge & Safir (1936)], the fact that for [FORMULA] in strongly flattened CE we obtain [FORMULA], a base density [FORMULA] cm-3 leading to [FORMULA] in cases (a) and (b) may be considered as too high, though density limits established in Sect. 3.1(b) are not violated.

It is worth noting that due to the number of CE parameter combinations which lead to similar SPh quantities and the variety of CE geometrical configurations implied by double valued SPh behaviours, it can hardly be maintained that the observed SPh phenomena in a given Be star can be explained in the framework of a unique and constant geometry of CE.

We conclude that the main characteristics of the observed SPh behaviours were schematically described using simple models with parameters: [FORMULA], [FORMULA] and [FORMULA] whose values are not beyond the limits imposed by observations. These models can account for: amplitudes of [FORMULA], [FORMULA] and [FORMULA] variations; single and bivalued SPh slopes; signs and absolute values of a[FORMULA] and a[FORMULA] SPh slopes; mean aspect-angle and temperature dependencies of SPh slopes that compare with those shown in Figs. 2 to 5. However, as the combinations of model parameters and physical frames producing a given SPh variation from star to star and even in the same star at different epochs are numerous, stars should be analyzed individually to determine the type of phenomenon that most likely produces an observed SPh variation. Only then should estimates of CE mean densities, temperatures, extents and ellipticities be used to produce averages of SPh slopes that can be compared with the statistical results shown in Figs. 2 to 5. We finally note that the SPh slopes we calculated in this work were obtained for electron distributions with [FORMULA]. However, as the discussion of characteristics of SPh variations dependent on the value of n is outside the scope of the present paper, it will be given elsewhere.

Let us finally note that the light outbursts found by Hubert & Floquet (1998), which the most likely correspond to type (a) or (b) variations, have characteristic time scales of the order of 1 year. To confirm the counter-clockwise loop-like SPh behaviours described in this paper, regular multicolour photometric and/or SPh observations should have been carried out during the whole time elapsed by these phenomena. On the other hand, light outbursts of short time scales (days), which can also imply discontinuous mass ejection events, were reported in the literature (tefl et al. 1994). However, successive discontinuous mass ejection events, will produce a series of colliding layers with backward anf forward directed shocks, that cannot be treated with the simple scenarios presented here. Though we have foreseen the possibility of loop-like SPh behaviours, those reported in MZH may actually correspond to physical frames that widely spill the simple ones studied in this paper.

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Online publication: August 25, 1999
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