4 Fundamental parameters

Once the lpv frequencies are obtained, it is also important to determine the fundamental parameters of $\eta$ Cen. This will allow us, on the one hand, to distinguish the periodicities associated with rotation from those related to NRP and, on the other hand, to determine the evolutionary state of the star.

4.1 Parameters derived from the continuum spectrum

To determine the fundamental parameters of HD 127972 we used its BCD spectrophotometric data (Chalonge & Divan 1952). They were derived from 12 low resolution spectra observed at ESO (La Silla, Chile) in May 1978 with the "Chalonge spectrograph" (Baillet et al. 1973). The advantages of using this system for Be stars were widely discussed in Zorec & Briot (1991). It provides observational parameters that describe the photospheric Balmer discontinuity (BD) of the star and which are free from interstellar extinction and circumstellar emission/absorption perturbations. As they concern stellar layers where the continuum spectrum is formed, they originate in regions that can be assumed to be less perturbed by the stellar activities affecting the outermost atmospheric layers and commonly seen in spectral lines. The ( $\lambda_1,D_*$) parameters ( $\lambda_1 =$ mean spectral position of the BD; D_* = energy jump measured at $\lambda =$ 3700 Å) represent the photospheric flux emitted by the observed stellar hemisphere, which in fast rotators is neither spherical nor has uniform surface temperature and gravity (Tassoul 1978). The emitted spectrum is then aspect angle dependent and may depict on average an object cooler and more evolved than it really is (Moss & Smith 1981). Thus, the observed spectral and photometric parameters do not reflect the actual stellar mass and its evolutionary stage, if the observed quantities are interpreted simply using the current calibrations of fundamental parameters, as it the star were rotationless.

Figure 7: Radial velocity (RV), equivalent width (EW) and FWHMfor all LNA spectra from 1996 to 2000.

 

 

Table 3: Comparative results among detected signals ( c/d).

lpv's	EW	RV	FWHM	V/R
0.6	0.5	0.6	0.7	0.5
1.5	1.6	1.6	1.5
3.8		3.6
5.3			5.0
9.2
10.3

Since the rotational energy of a rigid rotator is less than 1% of its gravitational energy (Zorec et al. 1988), we can assume that the evolutionary state of the star on the main sequence is described, in a first approximation, by models of non-rotating stars. So, to estimate the stellar mass and age from the usual evolutionary tracks, we need to determine the stellar rotationless parameters that underlie the observed ones. We disregarded subtle mixing effects on the stellar evolution produced by rotationally induced hydrodynamical instabilities (Endal & Sofia 1979; Zahn 1983; Meynet & Maeder 2000). We write then:

 

$$\left. \begin{array}{l} L(\lambda_1,D_*) = L_{\rm o}(M,t)F_L(M,\o... ...R_{\rm e}(M,\omega,t)\over R_{\rm c}(M,t)}\omega\sin i \\ \end{array}\right\},$$ (1)

Figure 8: Photometric Hipparcos light curve (ESA 1997) of HD 127972 folded modulo with its main detected frequency (1.5 c/d).

where $L_{\rm o}$, $D_{\rm o}$ and $\lambda_1^{\rm o}$ are respectively the bolometric luminosity, the BD and the $\lambda_1$ parameter of the star if it were rotationless; F_L, F_D and $F_{\lambda_1}$ are functions calculated assuming the stars are rigid rotators. M is taken as the "actual'' stellar mass, $\omega = \Omega/\Omega_{\rm c}$ is the angular velocity ratio ( $\Omega_{\rm c}$ is the critical angular velocity), i is the stellar aspect angle, t is the stellar age, $V_{\rm c}$ is the critical linear equatorial velocity, $R_{\rm c}$ is the critical equatorial radius and $R_{\rm e}$ the "actual'' equatorial radius at its rotational rate $\omega$. The function F_L, F_D and $F_{\lambda_1}$were obtained assuming a Roche model for the stellar deformation and von Zeipel's law for the effective temperature distribution, as previously done by many authors (cf. Maeder & Peytremann 1970, 1972; Collins & Sonneborn 1977; Collins et al. 1991). They were already used in Zorec & Briot (1997), Floquet et al. (2000, 2002), Frémat et al. (2002), Zorec et al. (2002a,b).

Figure 9: IPS diagram for the detected lpv signals from ${\nu }_{2}$ to ${\nu }_{6}$. Upper panels show the phase diagram across the He  I 6678 Å line profile. Lower panels present their respective amplitudes. Vertical lines indicate $\pm V\sin i$.

 

 

Table 4: Results of $\ell$ and $\vert m \vert$ parameters.

Frequencies	c/d	$\ell$ ($\pm$1)	$\vert m \vert$ ($\pm$ 2)
${\nu }_{2}$	1.5	3
${\nu }_{3}$	3.8	5
${\nu }_{4}$; ${\nu }_{6}$	5.3; 10.3	5	4
${\nu}_{5}$	9.2	7

The observed ( $\lambda_1,D_*$) parameters used to solve Eqs. (1) are given in Table 5. The corresponding apparent ( $\log L/L_{\odot},T_{\rm eff},\log g)_{(\lambda_1,D_*)}$ parameters obtained with calibrations suited to normal, rotationless B stars (Divan & Zorec 1982) are also reproduced in Table 5. The observed bolometric luminosity $\log L/L_{\odot}$ introduced in (1) was estimated, however, using integrated fluxes over the entire spectrum and the Hipparcos parallax of $\eta$ Cen. The method used to obtain it also helps us to test the consistency of the $(\lambda_1,D_*)-$calibration dependent fundamental parameters. By definition the flux effective temperature of a star is:

 

$$T^4_{\rm eff} = \frac{4}{\sigma_{\rm SB}}\frac{f}{\theta^2}$$ (2)

where $\theta$ is the angular diameter of the star, in radians, fis the flux received on earth, corrected from ISM extinction and integrated over the full extent of the spectrum and $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant. The angular diameter of the star is given by:

 

$$\theta = 2(f_{\lambda}/F_{\lambda})^{1/2}$$ (3)

where $f_{\lambda}$ is the absolute monochromatic flux received on earth, corrected from ISM extinction, and $F_{\lambda}$ is the absolute monochromatic flux emitted in the star. f was calculated using: 1) the 13-color photometry of Johnson & Mitchell (1975) calibrated in absolute fluxes; 2) the UV flux observed by IUE in 1982 and TD-1 satellite. The 13-color photometry of $\eta$ Cen apparently corresponds to a stellar non-emission phase, as it can be controlled by its apparent magnitude $V (V_{\rm o} =2.24\pm0.03$ mag dereddened; $E(B-V) = 0.030\pm 0.01$ mag) and the BD $D_* =0.161\pm 0.013$ dex obtained from the 13-color absolute fluxes. The wavelength interval used to derive $\theta$ is $\lambda\lambda$ 0.5 to 0.7 $\mu$m. The fluxes $F_{\lambda}$ and those employed to complete f in the non-observed spectral region are from the non-rotating normal stellar atmosphere models of Kurucz (1994). Adopting the $\log g(\lambda_1,D_*)$ parameter, we iterated relations (2) and (3) until a difference $\Delta T_{\rm eff} =1$ K was attained between two consecutive steps. The values of $T_{\rm eff}$, f and $\theta$ thus obtained are given in Table 5. From the Hipparcos parallax of $\eta$ Cen we have its distance $d_{\rm HIPP} =94.6^{+8.1}_{-6.9}$ pc that from f leads to the estimate of the stellar bolometric luminosity $\log L^f/L_{\odot}$ used in this work. With $\theta$ we can also obtain the stellar radius $R^f/R_{\odot}$. These parameters are also given in Table 5. It is worth noting the close resemblance of the bolometric luminosities, temperatures and radii obtained using these two completely different methods.

For comparison, let us quote the fundamental parameters of $\eta$ Cen recently determined by some authors. Harmanec (2000) obtained $T_{\rm eff} =22~400$ K, $\log L/L_{\odot} = 3.876$, $R_*/R_{\odot} = 5.7^{+0.5}_{-0.4}$ and $M = 9~M_{\odot}$. As compared with our determination, the slightly higher value of $\log L/L_{\odot}$ obtained by Harmanec (2000) could be due to its $T_{\rm eff}$ determination, which carries a difference of 0.23 mag in the bolometric correction (BC) used (Code et al. 1976). On the other hand, Code's et al. (1976) were calculated from the OAO-2 satellite far-UV energy measurements, which are overestimated as compared to those obtained since then with TD1 and IUE satellites. They may endanger the reliability of BC estimates for hot stars. The difference noticed in $T_{\rm eff}$ might be due to the use of $ubvy\beta$ Strömgren's photometry and the calibrations by Moon & Dworetsky (1985), which can lead to systematic deviations in the estimation of fundamental parameters (Frémat & Zorec 2002). Stefl et al. (1995) obtained $T_{\rm eff} = 21~860 \pm 480$ K also using $ubvy\beta$ photometry, but with Napiwotzki's et al. (1993) calibration. The use of photometric data can easily lead to overestimated effective temperatures, mostly for hot Be stars, because even a slight CE emission, which is difficult to clear up, can affect the u magnitude. Stefl et al. (1995) obtained $T_{\rm eff} = 21100\pm1370$ K by fitting spectral lines with model atmospheres. This value is no far from the temperature we derived by a similar method (Sect. 4.2), but which we did not adopt for the present study. The stellar mass and radius determined by Harmanec (2000) are similar to those we inferred in this paper, but neither the starting $T_{\rm eff}$ and $\log L/L_{\odot}$ parameters nor the methods used to obtain them are the same in the two attempts.

The system (1) was solved using a Monte Carlo method for the trials of the input set of parameters ( $\log L/L_{\odot},D_*,\lambda_1,V\sin i$). Each parameter, X, was sampled in turn between two extreme values, $X\pm \Delta X$ where the amplitudes $\Delta X$ are quoted in Table 5. We discarded about 10% of the solutions obtained, those which lead to $\omega >1$ and/or $\sin i > 1$. The average values of reliable solutions of (1) derived from the remaining 90% trials are given in Table 5. The quoted dispersions do not represent errors, but the range of acceptable solutions. The relations between luminosities, masses and stellar ages used are from the evolutionary tracks of non-rotating stars calculated by Schaller et al. (1992) for Z = 0.02.

Noting that $V\sin i$ is a key parameter to solve relations (1) and that the difference between Chauville's et al. (2001) and Slettebak's (1982) determinations is of the same order as their uncertainties, we sought the higher $V\sin i$ that, together with the above quoted uncertainties, still leads to reliable solutions of (1). We thus obtained $V \sin i = 370$ km s^-1. Solutions presented in Table 5 encompass the input $V\sin i$ values centered once on $V \sin i =$ 310 km s^-1 that range from 266 to 354 km s^-1 and then values centered on $V \sin i = 370$ km s^-1 ranging from 326 to 414 km s^-1.

Finally, we calculated the rotational frequency $\nu_{\rm r} = 0.02[V \sin i/\sin i])/[R_{\rm e}(\omega)/R_{\odot}]$ c/d, also given in Table 5, which can be compared with those obtained from time series analysis of lpv. They are, however, average values of only 27 determinations from the possible combinations of ( $V \sin i\pm\Delta_{V \sin i},i\pm\Delta_i,R_{\rm e}\pm\Delta_{R_{\rm e}}$) parameters, where each parameter takes in turn three values $X-\Delta X$, X and $X+\Delta X$. The frequency derived from lpv time series analysis that more closely resembles $\nu_{\rm r}$ is $\nu = 1.3$ c/d. A frequency $\nu = 1.29$ c/d was actually observed by Janot-Pacheco et al. (1999). Finally, we note that not only the rotational frequency $\nu_{\rm r}$ is far from the "critical'' rotational frequency $\nu_{\rm c}$, which ranges from 1.34 c/d to 1.39 c/d, depending on the value of $V\sin i$ adopted, but that it only marginally approaches the frequency $\nu = 1.5$ c/d suited for some lpv and the photometric variations.

Since for a star with mass $M \sim 8.6~M_{\odot}$ the main sequence (MS) life time is $t_{\rm MS} \simeq 3.0\times10^7$ yr, the ages estimated imply that $\eta$ Cen displays the Be phenomenon at $t \sim 0.6t_{\rm MS}$, roughly the earliest epoch at which this phenomenon seems to appear in open clusters (Fabregat & Torrejón 2000). Only if we adopted the observed fundamental parameters ( $T_{\rm eff},L/L_{\odot}$) without correcting them from rotational effects would the star apparently be at the end of its MS life span ( $t_{\rm app} = 2.8\times 10^7$ yr), as expected from theoretical predictions (Maeder & Meynet 2000). In these models, the high stellar rotation and hence, the Be phenomenon, are meant to appear as the consequence of the evolution of the stellar internal rotational law.

4.2 Parameters derived from the line spectrum

Unfortunately the only spectra we have at our disposal of HD 127972 covering a large spectral range are for the years 2000 and 2001, when this star displayed well developed H$\alpha$ emission. In spite of the fact that the estimation of fundamental parameters from these spectra brings in the influence of the CE, it can in principle furnish an extreme limit for these values. We calculated a grid of models with $T_{\rm eff}$ and $\log g$ around the values displayed in Table 5 with steps $\Delta_{T_{\rm eff}} = 200$ K and $\Delta_{\log g} = 0.1$ dex using the codes of non LTE model atmospheres TLUSTY (Hubeny 1988) and SYNSPEC (Hubeny et al. 1994) to synthesize line profiles. Fundamental parameter determination was attempted employing Wolf's (1973) method with a $\chi^2$ controlled fit of H$\gamma$ and H$\delta$ absorption lines combined with reproduction of the Si  II $\lambda 4131$/Si  III $\lambda 4553$ and He  I $\lambda 4388$/He  II $\lambda 4686$ equivalent width ratios. The results are summed up in Fig. 10. From this figure it seems that from the spectral lines used, the stellar parameters are $T_{\rm eff} = 21~600\pm960$ K and $\log g = 3.34\pm0.16$  dex (marked "x'' in Fig. 10).

Figure 10: $T_{\rm eff}$ and $\log g$ solutions from fitting of the observed H$\gamma$ and H$\delta$ line with predicted profiles by non-LTE model atmospheres and model representation of Si  II $\lambda 4131$/Si  III $\lambda 4553$ and He  I $\lambda 4388$/He  II $\lambda 4686$equivalent width ratios.

The difference between the solutions attempted for the H$\gamma$ and H$\delta$lines is explained by a different amount of line filling in with circumstellar emission. It can also be noted that H$\delta$ is not clean from this emission. In both cases the residual emission leads to a lower gravity and to a hotter stellar temperature estimation.

Notice the difference between the solutions obtained for the Si and He equivalent width ratios. Both series of lines are currently assumed to be little affected by circumstellar emission/absorption. On the other hand, according to von Zeipel's theorem, the local effective temperature of $\eta$ Cen ranges from 22 700 K at the pole to 19 900 K at the equator. As equivalent widths of the studied lines are increasing functions of  $T_{\rm eff}$ in this range of temperatures, the difference between solutions for the Si and He line ratios means that the stellar polar region more strongly favors the radiation fluxes in the Si than in He lines. This selection effect on formation region is also seen in the $V\sin i$ estimates, since from the Si we obtain $(V \sin i)_{\rm Si} = 250$ km s^-1, while from He lines it assumes $(V \sin i)_{\rm He} = 310$ km s^-1. This fact warns against the use of photospheric lines for $V\sin i$ determinations in fast rotators whose intensity increases with higher temperatures, since this parameter may result in underestimation.