A&A 449, 305-311 (2006)
DOI: 10.1051/0004-6361:20054142

Multiperiodicity in the large-amplitude rapidly-rotating $\beta~$Cephei star HD 203664[*],[*]

C. Aerts1,2 - P. De Cat3 - J. De Ridder1,[*] - H. Van Winckel1 - G. Raskin1,4 - G. Davignon1,4 - K. Uytterhoeven1,4,[*]

1 - Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
2 - Department of Astrophysics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
3 - Koninklijke Sterrenwacht, Ringlaan 3, 1180 Brussel, Belgium
4 - Mercator Telescope, Calle Alvarez de Abreu 70, 38700 Santa Cruz de La Palma, Spain

Received 2 September 2005 / Accepted 28 October 2005

Aims. We made a seismic study of the young massive $\beta~$Cephei star HD 203664 with the goal of constraining its interior structure.
Methods. Our study is based on a time series of 328 new Geneva 7-colour photometric data of the star spread over 496.8 days.
Results. The data confirm the frequency of the dominant mode of the star, which we refined to f1=6.02885 c d-1. The mode has a large amplitude of 37 mmag in V and is unambiguously identified as a dipole mode ($\ell =2$) from its amplitude ratios and non-adiabatic computations. Besides f1, we discovered two additional new frequencies in the star with amplitudes above $4\sigma$: f2=6.82902 c d-1 and f3=4.81543 c d-1, or one of their daily aliases. The amplitudes of these two modes are only between 3 and 4 mmag, which explains why they were not detected before. Their amplitude ratios are too uncertain for mode identification.
Conclusions. We show that the observed oscillation spectrum of HD 203664 is compatible with standard stellar models but that we have insufficient information for asteroseismic inferences. Among the large-amplitude $\beta~$Cephei stars, HD 203664 stands out as the only one rotating at a significant fraction of its critical rotation velocity ($\sim$$ 40\%$).

Key words: stars: oscillations - stars: variables: general - stars: early-type - stars: individual: HD 203664 - stars: interiors

1 Introduction

The $\beta~$Cephei stars are a homogeneous group of oscillating B0-B3 stars that have been studied as a class for more than a century now. Stankov & Handler (2005) recently compiled an overview of the observational properties of this group of stars. The oscillations of $\beta~$Cephei stars are explained in terms of the $\kappa$ mechanism operating in the ionisation layer of the iron-peak elements (Cox et al. 1992; Kiriakidis et al. 1992; Moskalik & Dziembowski 1992; Dziembowksi & Pamyathnykh 1993). Given that mainly low-degree low-order pressure and gravity modes are excited, these stars are good potential targets for in-depth seismic studies of the interior structure of massive (i.e. pre-supernova) stars. Indeed, the luminosity classes of the known $\beta~$Cephei stars range from V up to I (Stankov & Handler 2005; see also Waelkens et al. 1998), and theory predicts the occurrence of oscillations for this whole area in the HR diagram (Pamyatnykh 1999).

Recent progress in the seismic interpretation of selected $\beta~$Cephei stars was remarkable in the sense that standard stellar structure models are unable to explain the oscillation data for the best-studied stars: HD 129929 (Aerts et al. 2003), $\nu~$Eridani (Pamyatnykh et al. 2004; Ausseloos et al. 2004), 12 Lacertae (Handler et al. 2005; Ausseloos 2005). Pamyatnykh et al. (2004) have suggested including radiative diffusion processes in a new generation of stellar models in an attempt to resolve the excitation problem in $\nu~$Eridani. This has not been achieved so far. These three well-studied $\beta~$Cephei stars are all slow rotators with vsini below 40 km s-1and a mass between 9 and 12 $M_\odot$.

In view of these recent achievements, and in an attempt to obtain similar results for a star with higher surface rotation velocity, we selected one of the most rapid rotators among the large-amplitude $\beta~$Cephei stars for a long-term photometric monitoring programme on which we report here.

The star HD 203664 (B0.5V, $m_{\rm V}=8.59$) was discovered to be a new $\beta~$Cephei star by Aerts (2000), who derived one frequency of 6.0289 c d-1 from the HIPPARCOS photometry. This frequency was confirmed by her in 49 Geneva measurements spread over about one year taken with the P7 photometer attached to the 0.7 m Swiss telescope at La Silla observatory. The scarce multicolour data set did not allow discrimination between $\ell =1$ or 2 for the spherical degree of this oscillation mode but did seem to exclude a radial mode. HD 203664 is among the top ten of the class as far as photometric amplitude is concerned, with a value of $\sim$ 30 mmag in the Geneva V filter (Aerts 2000). It is by far the most rapid rotator among the large-amplitude members (see Fig. 4 of Stankov & Handler 2005), with $v \sin i=200$ km s-1derived from high-resolution spectra by Little et al. (1994). It also happens to be one of the very few class members that is situated at a high galactic latitude ( $l=61\hbox{$.\!\!^\circ$ }93, b=-27\hbox{$.\!\!^\circ$ }46$) at a distance of 3200 pc, thanks to which high-precision spectroscopic data is available and carefully analysed (Little et al. 1994).

In this paper, we report the findings of our observational study of HD 203664 in an attempt to contribute to a better understanding of $\beta~$Cephei stellar structure models, considering the diversity of this type of stars.

2 Data description and stellar parameters

We included HD 203664 in the long-term photometric monitoring programme of pulsating stars performed with the 1.2 m Mercator telescope at Roque de los Muchachos in La Palma, Canary Islands. In this framework we obtained 328 Geneva 7-colour high-precision photometric measurements between HJD 2 452 085.6 and HJD 2 452 582.3. The time span of these new data is 496.8 days. The integration times were typically 4 min, resulting in a precision of about 7 mmag per measurement in U and 6 mmag in V. Aerts (2000) had already obtained 49 datapoints for the star between HJD 2 450 391 and HJD 2 450 790 with the same instrument but attached to the 0.7 m Swiss telescope at La Silla. The Southern and Northern Geneva standard star systems are carefully calibrated so that measurements in both hemispheres should be compatible, even over a long baseline. All reduced data are provided in Table 1.

The basic stellar parameters of HD 203664 were derived from different sources. Aerts (2000) used the old Geneva photometry and positioned the star in the HR diagram with respect to the $\beta~$Cephei star instability strip (see her Fig. 1). Using standard stellar models published by Schaller et al. (1992), she thus derived a mass of 13.8 $M_\odot$. We recomputed the estimates using the same method as in Aerts (2000) from the average value of the 6 Geneva colours for all new data and find refined values of $\log~T_{\rm eff}=
4.47\pm 0.01$, $\log g=3.9\pm0.3$.

It is well known, however, that fundamental parameter estimates for B stars from high-resolution spectroscopy often result in a lower effective temperature and gravity (see, e.g. De Ridder et al. 2004, for a discussion about this for the $\beta~$Cephei star $\nu~$Eridani). Moreover, in the case of HD 203664, we cannot rely on an accurate value of the parallax, as shown by the large discrepancy between the result of 0.32 mas by Little et al. (1994) and of $2.23\pm 1.08$ mas from HIPPARCOS (Perryman et al. 1997). Little et al. (1994) derived the stellar parameters from high-resolution spectra and used these to estimate the distance. They find $\log~T_{\rm eff}= 4.447$, $\log
g=3.7$, and a mass of 14 $M_\odot$ (based on evolutionary models by Maeder & Meynet 1988), as well as a normal (i.e. solar) abundance for B stars in our vicinity. Unfortunately, these authors did not provide error estimates. Finally, the few $\beta~$Cephei stars with accurate seismic modelling have always ended up outside their observationally determined error box in effective temperature and gravity, to the cooler and less massive part (Thoul et al. 2003, for 16 Lac; Aerts et al. 2003, for HD 129929; Pamyatnykh et al. 2004; and Ausseloos et al. 2004, for $\nu~$Eridani).

The final estimate of the parameters that we conservatively adopted for HD 203664, based on all these arguments, is $\log~T_{\rm eff}= 4.45\pm 0.02$and $\log g=3.8\pm 0.2$, while we did not use any constraint at all on its luminosity.

3 Frequency analysis

\end{figure} Figure 1: The periodograms for the Geneva U ( left) and B ( right) data of HD 203664 after subsequent stages of prewhitening with the frequencies provided in the text. The amplitudes are expressed in magnitudes. The dashed horizontal line indicates the 4$\sigma $ level determined as explained in the text.
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We searched for frequencies in the Geneva U, B, V filters with the method outlined in Scargle (1982). The results were similar in the B and V filters, so we only provide the detailed analysis for the U and B filters. We accepted frequencies as long as their amplitude is more than 4 times the noise level, which corresponds to a 99.9% confidence level of having found an intrinsic variation rather than a frequency due to noise (Breger et al. 1993; Kuschnig et al. 1997). This criterion is common practise among asteroseismologists these days (e.g. Handler et al. 2003, 2004, 2005). The noise level was computed by averaging the periodogram peaks in the range [0,10] c d-1 after final prewhitening.

The accuracy of the frequencies was calculated as $\sigma_f\sim\sigma/\sqrt{N} A
T$ (Horne & Baliunas 1986; Montgomery & O'Donoghue 1999) where the proportionality constant is of order 1 depending on the author (see Cuypers 1987, for a discussion), A is the amplitude of the frequency f, N the number of data points, T the total time base, and $\sigma $ the average error on each individual measurement. We estimated the last by computing the standard deviation of the noise after final prewhitening, and found it to be 7.1 mmag for the Geneva U filter, 6.5 mmag for B, and 7.1 mmag for V.

During a first step, we confirmed the dominant frequency already found by Aerts (2000) from the HIPPARCOS photometry. This frequency appeared clearly in all three filters considered: f1=6.02885(2) c d-1 (where the uncertainty of the last digit is given in parenthesis). There was strong aliasing due to the single-site nature of the data (see top panels in Fig. 1); but as the HIPPARCOS data gave us the same value without any daily alias, we were sure that we picked the correct frequency for the dominant mode.

After prewhitening, we encountered strong frequency peaks at n c d-1. These peaks were due to the slight difference in the average magnitude between the older 0.70 m data and the 1.2 m Mercator data, which introduces daily aliases of the yearly periods due to the observing seasons. This small difference in average magnitude probably results from the fact that the standard stars used for the Mercator measurements are fainter than those used for the 0.70 m Swiss telescope at La Silla. In order not to be disturbed by these daily aliases, we ignored the 49 older measurements in our subsequent frequency analyses.

Prewhitening with f1 then led to a clear second frequency f2=6.82902(13) c d-1 and its aliases in the U filter. This frequency and its aliases safely fulfilled the amplitude requirement (Fig. 1). In the B filter, we found the yearly alias of f2: f2'=7.83176 c d-1. The HIPPARCOS photometry was of no use discriminating among the aliases as of this stage, because there are no significant frequencies in that dataset after prewhitening with f1. We proceeded by considering f2 as well as f2' in a biperiodic fit together with f1, but this did not help to discriminate between the two options. Therefore, we concluded that either f2 or f2' is the true second frequency. Phase plots for f1 and f2 for the three Geneva UBV filters are provided in Fig. 2. The curves in the lower panel are indistinguishable in quality from those for f2'.

\end{figure} Figure 2: Phase diagrams of the U ( left), B ( central), V ( right) lightcurves of HD 203664 for f1 ( upper) and for f2 after prewhitening with f1( lower). The dots are the data and the full line is a sinusoidal fit with fixed indicated frequency. Note the different scale of the y-axes of the upper versus lower plots.
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After subsequent prewhitening with either f2 or f2', we continued the search for new frequencies. In this way, we found f3=4.81543(14) c d-1 or one of its aliases. This frequency fulfilled the amplitude requirement in B, but not in V and only barely in U. It must therefore be regarded as a candidate frequency only without further observational confirmation. Moreover, we could not distinguish the frequency from its aliases from multiperiodic fits to the lightcurves. Unlike for the second frequency, numerous aliases of f3 gave equally good fits.

The best overall fits to the U, B, V data was achieved by taking the values for $f_1,\ldots,f_3$ as listed above, but we cannot exclude the possibility that we missed a true frequency and have taken an alias for any one of f2, f3. We will not use the latter frequency to make seismic inferences further on and, in the case of the second frequency, we consider both f2 and f2' each time. Any other frequencies found after prewhitening do no longer reach four times the noise level (see bottom panels of Fig. 1) so we stop the frequency analysis at this point.

A firm conclusion is that HD 203664 is a multiperiodic $\beta~$Cephei star with one dominant mode having an amplitude about ten times larger than the ones of its other modes. The three frequencies f1, f2, f3 reach an amplitude of, respectively, 38.8, 4.5, 4.0 times the noise level in U, of 38.4, 4.7, 4.2 times the noise level in B, and of 31.8, 4.5, 3.2 times the noise level in V.

4 Amplitude ratios and mode identification

Table 2: Results of harmonic fits to the Geneva lightcurves of HD 203664. A stands for the amplitude, expressed in millimag, and $\phi $ for the phase, expressed in 2$\pi $ radians. The adopted reference epoch for $\phi =0.0$corresponds to HJD 2 450 000.0.

\end{figure} Figure 3: Observed amplitude ratios AX/AU (filled circles) and their uncertainties for the dominant mode with frequency f1 of HD 203664, where AX stands for any of the amplitudes in the seven Geneva filters U, B1, B, B2, V1, V, G. The lines represent theoretical predictions in the non-adiabatic treatment of the oscillations for stellar models within the mass range $[12,16]~M_\odot$ during the main-sequence phase and fulfilling the error box in $T_{\rm eff}$ and $\log g$ derived in Sect. 2.
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\end{figure} Figure 4: Same as Fig. 3, but for the mode with frequency f2.
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\end{figure} Figure 5: Theoretical frequency spectra for stellar models with $M=13~M_\odot$, X=0.70, Z=0.02 without core overshooting from the ZAMS until the TAMS. The star's position is to the left of the vertical dashed-dot line according to the $T_{\rm eff}$-range derived in Sect. 2. The symbol convention for the modes is as follows: squares for $\ell =0$, triangles for $\ell =1$, plus sign for $\ell =2$, circles for $\ell =3$. Higher $\ell $-values are not shown for the sake of clarity. The observed frequency values f1($\ell =2$) and f2 (likely $\ell =2$) are indicated by the horizontal lines. The vertical line indicates the model for which an exact match to f1and f2 is obtained for two zonal $\ell =2$ modes.
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Table 2 lists the results for the amplitudes and phases of least-squares fits with f1, f2, f3 fixed at their values mentioned above. The overall variance reduction is also listed. Using f2' rather than f2, or any alias of f3 rather than these frequencies, led to amplitude and phase values within the error bars listed in Table 2. As a comparison, we mention that a monoperiodic fit with f1 has a variance reduction that is typically 5 to 8% lower depending on the filter. Each time the amplitudes were largest in the U filter, as is expected for the oscillation modes in the $\beta~$Cephei stars.

We observed that the phases are equal to within their accuracy for the three modes in the seven filters, so we made use only of amplitude ratios as a mode identification diagnostic. This is according to the common procedure for the $\beta~$Cephei stars.

In order to attempt mode identification, we proceeded as follows. We computed stellar models using the Code Liégeois d'Évolution Stellaire (CLES) kindly provided by R. Scuflaire. For details on the input physics, opacity tables used and the equation of state, we refer to Ausseloos et al. (2004). We restricted ourselves to models with X=0.70, Z=0.02, in agreement with the abundances derived by Little et al. (1994), and without core convective overshooting. For each value of the mass, we computed evolutionary tracks from the ZAMS until the TAMS, and we selected those that are within the observed range of $T_{\rm eff}$ and $\log g$ derived in Sect. 2. In doing so we considered a range in mass from 12 to 16 $M_\odot$ in steps of 0.5 $M_\odot$. Subsequently, we computed eigenfrequencies and eigenfunctions for each of the models using the non-adiabatic oscillation code MAD (Dupret 2001) kindly made available by M.-A. Dupret. For each of the models, we selected the eigenfrequency that was closest to the measured $f_i, i=1,\ldots,3$ and considered its theoretical non-adiabatic amplitude ratios following the formalism by Dupret et al. (2003). Finally, we compared all these theoretically predicted amplitude ratios with the observed ones for each of the detected frequencies. In this way, we performed a mode identification that is independent of one particular stellar model but that considers a range of theoretical models safely covering the current mass estimate of HD 203664.

The result of this procedure for the dominant mode can be found in Fig. 3 for $\ell=0,\ldots,4$. Thanks to the small error bars of the observed ratios, we readily identified this mode as a dipole mode. Indeed, the $\ell =2$ solution was the only one compatible with the high-quality data. The strongest mode of HD 203664 is therefore clearly non-radial. A similar situation occurs in several other $\beta~$Cephei stars (see, e.g., Heynderickx et al. 1994).

The other two modes had errors on their amplitude ratios that were too high to be able to discriminate among the $\ell $-values. We illustrate this for the mode with frequency f2 in Fig. 4. While the agreement between the theoretical predictions and the observations is best for $\ell =2$, we cannot firmly exclude the other $\ell $-values of the non-radial modes. We do find that the mode with frequency f2 is unlikely to be radial. The same result is obtained if we take f2' rather than f2. The ratios for the modes with f3 are even more uncertain and do not provide any constraint at all so the corresponding mode identification plot is omitted here.

In an attempt to confront these seismic observational constraints with standard models, we used the ones mentioned above, i.e. with masses ranging from 12 to 16$~M_\odot$, X=0.70, Z=0.02 and without core overshooting ( $\alpha_{\rm
ov}=0.0$). We computed all their oscillation frequencies for zonal modes of $\ell=0,\ldots,3$ without taking the effects of rotation into account. We point out that this is a simplification of the true situation, because second-order rotational effects imply frequency shifts, even for zonal modes. Moreover, mode coupling occurs due to rotation for modes whose degree differs by 2 if their frequencies are close together (e.g. Soufi et al. 1998; Daszynska-Daszkiewicz et al. 2003). However, we postpone a sophisticated interpretation of HD 203664's observed frequencies for the time being and restrict ourselves to a confrontation with standard models here.

Typically three or four radial orders of an $\ell =2$ zonal mode fitting the frequency f1 resulted from the standard models from the ZAMS to the TAMS, for each of the considered masses between 12 and 16$~M_\odot$. Forcing a simultaneous fit to an $\ell =2$ zonal mode for f2 was achieved for several of the models that already fit f1. An example of the mode spectrum of evolutionary models with $M=13~M_\odot$ is provided in Fig. 5, in which all the $\ell=0,\ldots,3$ mode frequencies are compared with f1 and f2. The vertical dashed-dot line indicates the lower limit of the observed effective temperature interval, i.e. HD 203664 must occur to the left or close to this line, pointing out its young age as already found by Aerts (2000). The model indicated by the full vertical line provides a good fit to f1 and f2for zonal $\ell =2$ modes. This model has parameters X=0.70, Z=0.02, $M=13~M_\odot$, $\log~T_{\rm eff}=4.429$, $\log g=3.98$, $\log
L/L_\odot=4.24$, $R=6.12~R_\odot$, a central hydrogen abundance $X_{\rm c}=0.438$, an age of $7.5\times 10^6$ yr, and no core overshooting. These $T_{\rm eff}$ and $\log g$ values are compatible with the estimates from observations provided in Sect. 2. Moreover, non-adiabatic computations with MAD indicate that the two modes are excited for this model.

Fitting f1 and f2' as two zonal $\ell =2$ modes is also possible if their radial order differs by 2, but the models for which such a fit is obtained are somewhat more evolved, which is less likely due to the observational constraint on the effective temperature. Moreover, the frequency fit is not as good as for f2. Any of the models with $M\neq~13~M_\odot$ while keeping X, Z, $\alpha_{\rm ov}$ fixed do not provide as good a fit to both f1 and f2 (or f2') as the model indicated by the vertical line in Fig. 5.

We thus find that a good fit for f1 and f2 as $\ell =2$ modes can only be achieved close to an avoided crossing if we restrict to zonal modes. Modes near avoided crossings have very good potential to probe the interior of a star. However, we have no conclusive observational evidence to restrict ourselves to m=0, so other evolutionary models may also lead to an equally good fit of the two considered frequencies for non-zonal modes.

At present we have observational constraints that are too restrictive to perform a precise fitting procedure (e.g. as outlined in Ausseloos et al. 2004), i.e. to scan in detail the parameter space of any possible stellar model as a function of (X, Z, $\alpha_{\rm ov}$, M). Indeed, besides the fact that we cannot exclude a different $\ell $-value for the mode with frequency f2 (or f2'), we neither have information on the azimuthal number of the modes nor do we know the surface rotation frequency of the star. Assuming that f1 and f2represent a good approximation of the central peaks of frequency multiplets is too restrictive because the rotational frequency must be of order 0.5 c d-1 or larger given that $v \sin i=200$ km s-1.

We can only conclude that the currently observed oscillation spectrum of HD 203664 can be explained easily by standard stellar models and excitation computations and that we have insufficient information to further constrain the stellar parameters from asteroseismology at present.

5 Discussion

We have shown the large-amplitude $\beta~$Cephei star HD 203664 to be a multiperiodic non-radial oscillator from single-site long-term multicolour Geneva photometry. The dominant mode was unambiguously identified as a dipole $\ell =2$ mode from its amplitude ratios. The frequencies of the star are compatible with standard stellar models of massive stars during the main sequence phase.

There are at present nine other $\beta~$Cephei stars known to have a peak-to-peak V amplitude that is larger than the one of HD 203664 (Stankov & Handler 2005). Only one of these ten stars is monoperiodic (BW Vul) as multiperiodicity was found in HD 180642 (Aerts et al., in preparation). Three of them have a confirmed dominant radial mode ($\nu~$Eri - Handler et al. 2004; BW Vul - Aerts et al. 1995; HD 180642 - unpublished), three have an $\ell =1$ mode (IL Vel - Handler et al. 2003; 12 Lac - Handler et al. 2005; KP Per - Saesen et al., in preparation), and two have an $\ell =2$ mode (KZ Mus - Handler et al. 2003; HD 203664 - this paper). We do not find any obvious relation between these results and the projected rotation velocity of the stars (Stankov & Handler 2005). HD 203664 is by far the most rapid rotator among them, with $v \sin i=200$ km s-1, which is more than twice as high as for any of the other large-amplitude $\beta~$Cephei stars. From the mass and $\log g$ estimates given in Sect. 2, we find a critical velocity of 480 km s-1 and hence $v \sin i / v_{\rm crit} = 42$%.

The stellar parameters and internal structure of HD 203664 can only be constrained further by means of elimination of the alias problems presented here for the low-amplitude modes (i.e. from a multisite campaign such as those performed by Handler et al. 2003, 2004, 2005), by subsequent unambiguous identification of their degree from amplitude ratios and by further identification of their azimuthal number through high-resolution spectroscopy. Given its visual magnitude of 8.6, its oscillation periods near 4 hours, and its low-amplitude modes, together with the data requirements for such type of spectroscopic analysis (e.g. Aerts & Eyer 2000), one needs, besides a photometric multisite campaign, at least one week of 8-m class telescope time to achieve mode identification and subsequent interpretation of the oscillations of this massive hot rapid rotator among the $\beta~$Cephei stars.

We are grateful to the staff of the Geneva observatory for the generous awarding of telescope time at La Silla and for the data reduction. We thank our colleagues from the Institute of Astronomy at Leuven University who contributed to the gathering of the photometric data at La Silla and at La Palma. Moreover, we are much indebted to R. Scuflaire, M.-A. Dupret, and M. Ausseloos for making their computer source codes CLES, MAD, and SCAN available.



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