4 Discussion

Our numerical simulations of time-series analysis indicates that CLEANEST is more efficient than CLEAN in identifying aliases due to the observational window (see Gies & Kullavanijaya 1988, for a discussion about CLEAN on this matter). For that reason, in the following discussion more weight will be given to results obtained with CLEANEST but keeping in mind that this algorithm does not always succeed in eliminating aliases (see, for instance, Figs. 2, 3). Data sets obtained in five different epochs are analysed in this paper. We thus expect that power spectra will be essentially contaminated by one-day aliases (relatively easy to identify) and that periodicities appearing simultaneously in several data sets can reasonably be trusted to be present in the star. In the following, we will interpret the multiperiodicity detected in $\eta$Centauri in the framework of NRP. Low-degree sectoral NRP modes (l3) are detectable photometrically and in RV, EW and FWHM variations. They may also appear in lpv. In contrast, high-degree modes will not be detected photometrically due to cancellation effects. So, frequencies associated with low-degree modes can be identified by considering together the frequency spectra of the various kind of observable quantities.

4.1 Stellar parameters

The first minimum of the Fourier transformed mean spectrum can be used to estimate the projected stellar rotational velocity (Gray 1976). In the case of a Be star, factors such as gravitational darkening, shell components and shape deformation by the rapid stellar rotation can hamper the method and results must be treated with caution. Fourier transform has been applied to the mean spectrum derived from the six observing nights and from each night separately. While individual $\lambda$667.8nm HeI profiles show a considerable degree of asymmetry varying in time (that is, lpv) the mean profile is globally symmetric, indicating that positive and negative bumps corresponding to higher order NRP modes are averaged in time to its zero mean value (see Fig. 6, upper panel). Fourier frequencies are reduced to velocity units assuming that the first zero of the Fourier transform of a rotational profile points to the projected rotational velocity of the star (Fig. 6, lower panel). A limb darkening coefficient $\varepsilon$=0.6 was adopted. This analysis indicates a projected rotational velocity $V\sin i$=350 km s^-1 while the first minima of individual spectra indicate rather 360$\pm$30 km s^-1, as can be seen for instance in Fig. 7. However, note that the second zero of the rotational mean profile is not at the expected position of about 200 km s^-1, being biased towards 240 km s^-1 (the third zero occurs at the expected position of 140 km s^-1). This is certainly a consequence of the significant velocity fields introduced by other sources (as e.g., non-radial pulsations) present in the stellar atmosphere. Variations of several tens kilometers per second are sometimes observed in the Fourier transform minima from night to night. They are strong enough to smear the rotation pattern or to cause the first minimum to vanish. In the frame of the NRP model this behavior can be considered as a first indication of the presence of low degree (l  3) NRP velocity fields that are superposed on the global rotation velocity field, deforming individual rotation profiles and producing the corresponding variations of the measured velocities. In what follows we will adopt $V\sin i$=350 km s^-1 as the stellar equatorial velocity.

  

Figure 6: Mean spectrum for the observing run (upper panel) and its Fourier transform (bottom panel)

  

Figure 7: Positions of the first and second minima of Fourier transformed line profiles for March 15

The spectral classification of Be stars is a difficult task. Gravitational reddening caused by rapid rotation and variable influence of the changing, cooler and less dense envelope will often simulate a later spectral type and a higher luminosity class. The IAU CDS data bank (Strasbourg) lists spectral classifications in the range B1V-B3II. SBHB propose B1III-V based on photometric data. Hipparcos measured for the star a parallax $\pi$=10.57$\pm$0.39 mas. Using 2.3$\le$V$\le$2.4, 0.04$\le$E(B-V)$\le$0.06 (quoted by SBHB) and R=3.1 we get -2.84$\le$M_V$\le$-2.56. This puts an end to the controversy concerning the luminosity class of $\eta$Cen: its absolute magnitude is typical of a B1-B2 main-sequence star, a giant class being formally excluded (e.g. Lang 1992). That classification range will be adopted hereafter.

The estimation of the rotational frequency and of the critical (break-up) depends on its mass and radius. Extreme values of these parameters for main-sequence B stars are given by Harmanec (1988) and Underhill (1982). The physical dimensions of a B1-2V star are thus: R=4.3-4.9$R_{\hbox{$\odot$}}$, M=8.6-11.0$M_{\hbox{$\odot$}}$ (Harmanec) and R=6.0-6.8$R_{\hbox{$\odot$}}$, M=10.3-10.4$M_{\hbox{$\odot$}}$ (Underhill). In the Roche approximation, a star rotating at the critical (break-up) speed have $R_{\rm e}$=1.5R_*. For the parameters quoted above, the extreme critical rotational frequencies (velocities) will be $\nu_{\rm crit}$=2.00 c/d ($V_{\rm crit}$= 744 km s^-1) and $\nu_{\rm crit}$=0.85 c/d ($V_{\rm crit}$=440 km s^-1), respectively. Based in a line-fitting model, Hutchings et al. (1979) used UV observations to estimate independently rotation velocities and inclination angles for a sample of O-B stars. For $\eta$Cen they obtain i=55$\hbox{$^\circ$}$ (+12$\hbox{$^\circ$}$, $-7\hbox{$^\circ$}$)and $V_{\rm rot}/V_{\rm crit}$=0.65 (+0.07, -0.12). Taking these values for granted, $V\sin i$=350 km s^-1, yields for $R_{\rm e}$=R_* an equatorial velocity $V_{\rm e}$=427 km s^-1 (+44, -47) and 1.10 c/d$\nu_{\rm rot}$2.16 c/d (0.46 d$P_{\rm rot}$ 0.91 d). These results suggest that the mass and radius values of Harmanec (1988) are preferable in the present case. The critical frequency (period) of $\eta$Cen is thus probably $\nu_{\rm crit}$2.28 c/d ($P_{\rm crit}$0.44 d). The definition of these parameters is important for the interpretation of the periodicities found in our time series analysis. For instance, frequencies greater than $\sim$2.3 c/d cannot be assigned to stellar rotation alone.

4.2 NRP frequency and mode identification

In the following, we will discuss the main frequencies detected in our temporal analysis (Table 2, Figs. 2, 3 and Sect. 3). Note that in the FDI analysis a first harmonic will have twice the l value of the corresponding fundamental mode (cf. KWM). It should be kept in mind in the subsequent discussion that the frequency resolution for our photometric and spectroscopic data is 0.08 c/d and 0.14 c/d, respectively. The reader should refer to Figs. 2 and 3 for lpv results up to 6 c/d.

1) The frequency 0.78 c/d is present in RV, EW and lpv and can be partly due to a 1-day alias of 1.78 c/d. It is mentioned in the Hipparcos catalogue, implying a double-wave light curve, but for a single-wave modulation the signal would correspond to a 1.56 c/d modulation (see Sect. 4.3).

2) The strongest modulation detected by CLEANEST in lpv of our 1995 spectra when the global mean spectrum is used is 1.29 c/d. It is also present in RV (CLEANEST) and FWHM (CLEAN) variations, and in our 1993 photometric data. The FDI analysis of all our spectroscopic data sets (hereafter FDIall for the sake of brevity) associates to it l=1-3. One and half-day aliases of this frequency (2.3 and 3.3 c/d) appear in lpv and other data sets. They never occur at the same time at any given wavelength.

3) A 1.56 c/d periodicity, stable and dominant in pratically all the data sets (photometric and spectroscopic, treated with both CLEAN and CLEANEST) obtained from 1987 through 1992 including Hipparcos data, if the variation is interpreted as a single-wave light curve. In this context, SBHB speculated that 1.56 c/d could be the stellar rotation frequency, the variability being due to the rotation modulation effect caused by one pattern of photospheric spots spanning the entire stellar circumference. This is in fact equivalent to a l= $\vert m \vert$=1 NRP pattern, whereas a double-wave variability corresponds to l=$\vert m \vert$=2 (cf. SBHB) (see below). The 1.56 c/d frequency is absent or very weak in LNA 1993 photometry, but it can be identified with a 1.48 c/d secondary peak appearing in our 1995 lpv and FWHM periodograms. The same could be said with respect to a 0.61 d modulation ($\equiv$1.62 c/d) which was detected in a multiwavelength spectroscopic campaign conducted in 1991 March-April (Gies 1994). Beating and/or NRP amplitude variations could explain the fact that 1.56 c/d is by far the strongest periodicity present in $\eta$Cen in 1987, 1988 and 1992, whereas it is fairly weak during 1993 and 1995. We will return to the matter at the end of this discussion.

4) A 1.78 c/d periodicity is distinctly present in lpv (CLEAN and CLEANEST) and is also weakly present in some photometric, EW and FWHM data sets. It could be 2-day alias of 1.29 c/d. This periodicity has been recently reported by Stefl et al. (1997) as a secondary signal appearing in spectroscopic material obtained during 1995-1997. The one-day alias of this frequency (2.78 c/d) is clearly seen in our lpv periodogram.

5) The frequency 3.82 c/d is quite conspicuous in the lpv analysis performed with the nightly mean spectrum (CLEANEST, Fig. 3b). This frequency appears in FDIall associated with l=1-3, and the phase difference across the line profile indicates l=1.9.

6) A peak at $\sim$4.34 c/d appears in the periodogram of lpv obtained with CLEANEST, using the nightly mean spectra. This frequency corresponds to l=2-4 in the FDIall results.

7) A variation with 4.51 c/d is the strongest in the lpv analysis with CLEANEST and appears also in CLEAN results. FDIall map associates it to l=3-5. The 5.5 c/d frequency present in the lpv periodogram (Fig. 3b) (presumably appearing also in Table 2, Col. c as 5.35 c/d) is probably a one-day alias of this strong modulation.

8) The frequency 7.8 c/d is clearly present in the nightly analysis with CLEANEST and the FDI method, where it appears associated with l=6 (cf. Fig. 4a). This periodicity (or twice it) was detected by Leister et al. (1994), and tentatively associated with a tesseral l$\sim$7, m$\sim$6 mode.

9) Finally, periodicities of $\sim$9.1, 10.5, 11.5 and 17.0 c/d are also clearly present with moderate to low amplitude in some of the nights analysed separately with CLEANEST and in FDI results (see Fig. 4). It must be remembered that higher frequencies show up more easily when the nightly mean spectrum is used for constructing the residuals. Further observations with higher S/N are needed to establish the nature of such rapid periodicities, which would be characteristic of p modes. This has also been observed in the B star $\iota$Her (Mathias & Waelkens 1995, see below). A summary of the main frequencies that seem to be present in the star is presented in Table 4. The associated NRP degree l and/or azimuthal quantum number m found by various methods are also given.

  

Table 4: NRP degree l and azimuthal order m for the main frequencies found in this paper deduced with different methods. The corresponding superperiods are given in the last column

*Leister et al. (1994).

We included in Table 4 the corresponding superperiod (travelling period of bumps through the stellar disk), namely $\vert m \vert$P for the cases in which a value of m could be estimated. It can be seen that with the exception of the 0.78 c/d frequency, the superperiods agree within $\sim$8%: $\vert m \vert$P=$0.73\pm0.06$ d (the agreement is better if one takes into account the correction due to the Coriolis force). The reality of this travelling period of bumps is confirmed by an independent estimation using the acceleration of the "moving bumps", that are the signature of high-order NRP modes. They are clearly seen in Fig. 4. In the case of sectoral modes, the superperiod can be written $\vert m \vert$ $P =\,2\,\pi V\sin i /a$,where a is the bump acceleration at the line center. Taking $V\sin i$=350 km s^-1, one obtains for the 6 nights $\vert m \vert$P=$0.73\pm 0.04$ d.

4.3 NRP and rotation in $\eta$Centauri: A tentative scenario

It seems clear that $\eta$Centauri changed its variability pattern somewhere between May 1992 (data secured by SBHB) and May 1993 (LNA photometric data, this paper). The earlier pattern was dominated by a 1.56 c/d periodicity present in spectroscopic (EW, FWHM, RV) and photometric measurements. In particular, photometric data are well reproduced by a single-wave light curve which corresponds to a m=1 NRP mode. After the beginning of 1993 and up to mid-March 1995, the star shows a richer frequency spectrum, with four rather intense signals: 1.29, 1.78, 3.82 and 4.51 c/d (see Figs. 2, 3 and Table 4). The changing temporal behavior of the frequency spectrum of multiperiodic Be stars is a subject of current investigation (e.g. Smith 1985; Ruzic et al. 1994; Zhiping & Aying 1997; Stefl et al. 1997). In the case of NRP, nonlinearity, beating of high amplitude modes and resonant coupling between modes may render rather difficult to establish a definitive picture of the stellar pulsation characteristics. Even if the influence of the observational window function and its associated aliases is properly taken into account, possible intrinsic variability of mode amplitudes may entangle the situation even further. These considerations suggest that many factors can explain the changing variability pattern of $\eta$Cen.

A remarkable behavior appearing in the nightly FDI analysis of our data can be seen in Fig. 4. For the night of March 15, a mode with l=2 is largely dominant. The time resolution is insufficient to allow the corresponding frequency to be determined, but from Table 4, it can be identified with 1.29 c/d. In the subsequent night (March 16, Fig. 4b) the pattern radically changed, being now dominated by a mode with l=4 that can equally be associated with the frequency 4.51 c/d (see Table 4). This sort of alternating condition is also clearly observed for the other March 1995 data, namely that nights 11, 17 show a strong l=2 mode, whereas the pattern is dominated by a l=4 in nights 12, 18. This $\sim$2-day mode alternation can be due to a beat modulation linked to the travelling period of bumps/or to the stellar rotation: the repetition of the two distinct NRP dominant surface patterns occurs each $\sim$2 days $\simeq$3 superperiods (see above). This is similar to the proposition by SBHB, namely that 1.56 c/d could be the stellar rotation frequency.

NRP frequencies measured in an inertial frame can be written $\nu$=$\nu_{0}$$\pm$m$(1-C_{\rm nl})$$\Omega_{\rm rot}$, where $\nu_{0}$ is the frequency in the stellar frame for a non-rotating star, and $\Omega_{\rm rot}$ is the stellar rotation frequency. Adopting $\Omega_{\rm rot}$=1.5 c/d as a working hypothesis and supposing that NRP in $\eta$Cen are high radial order g modes (in which case $\nu_{0}$$\approx$0 and $C_{\rm nl}$$\simeq$1/(l(l+1)), cf. Dziembowski & Pamyatnykh 1993; Gautschy & Saio 1993), some of the frequencies in Table 4 can be reproduced: the corresponding l, m values are reported in the third column of the table. Note that they do not differ significantly from the other results, given the uncertainties. The frequencies in Table 4 are in good agreement with theoretical calculations by Gautschy $\&$ Saio (1993) of unstable NRP g-modes in hot B stars (with the exception of the 7.8 c/d signal). Other high value frequencies (9.1, 10.5, 11.5, 17 c/d) seem also to be present in the CLEANEST results for the nights analysed separately. They are more compatible with p modes of hot stars, which could exist in the limit between the $\beta$Cephei and the typical SPB stars in the HR diagram. In these "hybrid" stars, p and g modes would be unstable (Mathias $\&$ Waelkens 1995). Indeed, $\eta$Centauri could be another example of such intermediate stars: it is a rather luminous B star, located near the lower border of the $\beta$Cephei region. This subject will be further developed in a forthcoming paper (Janot-Pacheco et al., in preparation).