ASTEROSEISMOLOGY OF THE NEARBY SN-II PROGENITOR: RIGEL. I. THE MOST HIGH-PRECISION PHOTOMETRY AND RADIAL VELOCITY MONITORING^*

Rigel was observed continuously with the MOST satellite (Walker et al. 2003) for 27.7 days from 2009 November 15 to December 13. The data set consists of 30,640 observations after correcting for the Southern Atlantic Anomaly. Figure 1(a) shows the light curve after removing an offset of ∼0.12 mag. The abscissa is in HJD−2,450,000.0. Because of the brightness of the target, the precision of the observations is of the order of ≈0.05–0.10 mmag. Starting with a sinusoidal variability pattern, oscillations die out and Rigel seems quiescent for several days, approximately from HJD 5164 to 5168 (enlarged in Figure 1(b)), followed by a gradual rise, and a steep decline in light flux; this resembles a possible beating pattern and we expect close, low-frequency modes to appear in the harmonic analysis (Section 4). There is a 20 hr gap around HJD 5175, arising from an interruption in communication between the satellite and the ground station. This is at the time when the star is decreasing in brightness almost monotonically.

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Rigel's optical spectrum has been monitored by the 2 m Tennessee State University Automatic Spectroscopic Telescope at Fairborn Observatory, AZ (TSU/AST; Eaton & Williamson 2007). A total of 2328 high-resolution (R ∼ 30,000 and 20,000) moderate signal-to-noise ratio (S/N ∼ 50–150) echelle spectra (4900–7100 Å) were obtained. The spectra were secured over 334 nights starting from 2003 December 11 to 2010 February 14. The density of time sampling was increased during the months centered on the MOST observations. Simultaneous with MOST, 442 spectra were collected over 20 nights.

The spectra were reduced at TSU by an automatic pipeline method without removing the telluric lines. The RV variations were derived by fitting a Gaussian to 29 metallic lines; Figure 1(c) shows the RV variations contemporaneous with MOST observations. An offset of +0.3 km s⁻¹ was applied to the AST instrumental RV measurements (Eaton & Williamson 2007). The standard deviation in the mean for each RV measurement was evaluated as the standard deviation of the 29 measured absorption lines in each spectrum. Averaged over all spectra, this value is 0.26 km s⁻¹. In the years prior to the MOST observations, Rigel was observed sporadically with AST. Also, the AST data set is interrupted by cloudy weather that occurred during the MOST observing run. Thus the sampling is not even, and the analysis of these data will suffer from daily and annual aliasing.

Extensive and highly variable velocity changes are clearly visible in Figure 1(c). During the plotted interval, the RVs vary from 11 to 25 km s⁻¹. There is evidence for a correlation between the RV and MOST brightness variations. During this interval, and other seasons, the RV variations appear to exhibit complex periodicity typical of a multi-periodic pulsating star.

In addition to the AST spectroscopy, Rigel was observed with the VLTI during the time of the MOST observations. A preliminary analysis of the spectro-interferometry by O. Chesneau et al. (in preparation) shows that during this interval Rigel was relatively quiet, with the Hα line appearing in absorption with sometimes a weak emission component appearing at times, and no outbursts being recorded. No major high velocity absorption event is identified during this interval. This is confirmed by seven raw Ritter echelle spectra (R = 26,000, S/N 100–200) taken during the same interval at Ritter Observatory (N. D. Morrison 2011, private communication). As a result, there is no evidence of a possible propagating atmospheric shock or chaotic activity during MOST photometry.

In conjunction with MOST photometry, 78 observations were collected with the high-resolution (R ∼ 65, 000) spectropolarimeter ESPaDOnS, installed at the Canada–France–Hawaii Telescope, and its clone Narval at Telescope Bernard Lyot (TBL, Pic du Midi). Analysis with the multi-line cross-correlation technique least-squares deconvolution (Donati et al. 1997) reveals no evidence of a magnetic field (Shultz et al. 2011), with error bars on the longitudinal field B_l of the order of 15 G. Matching synthetic disk-integrated Stokes V profiles to the observed Stokes V profiles has constrained the dipolar magnetic field to be B_d ≲ 25 G for high inclinations of the rotational axis and low obliquities of the magnetic axis, although B_d ∼ 50 G is possible at low-intermediate inclinations, in which case the rotational period is shorter and fewer observations can be binned together (M. Shultz et al., in preparation). If, as discussed above, Rigel is viewed nearly equator-on, then the first of the conditions necessary for a higher upper limit applies. However, these upper limits cannot rule out the possibility of significant magnetic effects. Thus, in the absence of any positive evidence for a magnetic field, or any pressing theoretical reason to suspect its existence, there is no reason at this time to complicate the pulsational analysis with its inclusion.

Kaufer et al. (1997) showed that different spectroscopic lines in BA supergiants emerge from different optical depths τ_λ, in a sense that lines from shallower photospheric depths have larger equivalent widths (hereafter EWs designated by W_λ). They classified the lines in the optical band accordingly to weak (50 mÅ ≲ W_λ ≲ 200 mÅ), medium (200 mÅ ≲ W_λ ≲ 500 mÅ), and strong (W_λ ≳ 500 mÅ). We chose C ii λ6583 as a weak line, Hα, C ii λ6578 and Si ii λ6371 as medium lines, and He i λ6678 and Si ii λ6347 as strong lines.

EWs of the above lines are calculated according to

$$\begin{equation} W_\lambda = \int (1-F_\lambda)\, d\lambda, \end{equation} \tag{ 1 }$$

where F_λ is the flux at each wavelength interval dλ, renormalized to the continuum on either side of the spectral line. To suppress the contribution from cosmic rays the spectra were filtered using a low-bandpass median filter; contamination from telluric lines was removed by identifying those regions so contaminated and using a higher-bandpass median filter. The corresponding errors for each EW measurement are evaluated with the per-pixel standard deviation of the flux from the mean value of a nearby section of the continuum and propagated through the measurement. Uncertainty in the location of the continuum is accounted for in the calculation of the per-pixel flux error from the S/N. As an example, Figure 1(d) shows the variations in W_λ for the Si ii λ6347 line during the MOST photometry. There is a weak correlation between the RV and EW variations during this season, but it is not always present.

Comparison of ESPaDOnS/Narval spectra with contemporaneous AST spectra reveal that, while the instruments generally yield consistent measurements of W_λ, the comparatively lower-resolution AST spectra show a small but persistent bias in which W_λ systematically increases with the uncertainty σ(W_λ). However, W_λ is accurate to within ∼5% which is sufficient to reveal at times a weak correlation between RV and the W_λ of, for instance, the Si II 6347 Å line, one of the stronger photospheric lines in the AST spectral window (see Figure 1(d)). Figure 2 is a 50 day representative interval, occurring one year prior to the MOST photometry, that shows the RV changes as a function of time (panel (a)). For the same time interval, panels (c) to (e) plot the EWs of five different lines. While Hα (having high sensitivity to wind) demonstrates the largest variation amplitude, the rest of the lines, though they have different absolute amplitudes, change moderately—on the order of 12%–19%. The two Si ii lines follow a similar trend and have similar amplitudes.

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Kaufer et al. (1996, 1997) derived several harmonics to fit to the season-by-season photometry, RV, and EW variations of Rigel and five other late BA supergiants. As they demonstrated, the Hα line EWs show systematic variations. Figure 7 in Kaufer et al. (1997) shows that α Cyg, the prototype of this class of pulsating stars, and the other BSGs exhibit variability with periods exceeding a week. For the specific case of Rigel, the periods they find are in the range of 4 to more than 50 days. Richardson et al. (2011) found evidence for season-to-season changes in the oscillation frequencies of α Cyg (Deneb, A2 Ia) from five years of spectroscopic and photometric monitoring.

The need for multiple modes to fit to our RV and EW data set is clearly shown in Figure 2. Similar behavior is observed in other seasons. Because different spectral lines are formed at various optical depths of the star's atmosphere, the temporal changes in W_λ for strong lines (such as Si ii λ6347) are significantly different from those of weak lines (such as C ii λ6578). The EW, however, is sensitive to local changes in the effective temperature δT_eff/T_eff and gravity δg/g. In a series of studies in β Cep and SPB stars (which are different from BA supergiants, but still have similarities in the nature of their pulsation), Dupret et al. (2002) and De Ridder et al. (2002) showed that a sinusoidal behavior in EW and RV time series with a common frequency is observed in only few cases. Given that Rigel (similar to SPBs) pulsates in g-modes which are transversal motions at the stellar surface, EWs are strongly affected by δT_eff/T_eff (De Ridder et al. 2002). Based on the observed behavior of Rigel's EWs (panels (c) to (e) in Figure 2), we believe that a complex non-RV field exists across different optical depths (Aerts et al. 2009; Simón-Díaz et al. 2010). As Kaufer et al. (1997) showed (in their Figure 4), RV variations are negligibly affected by this depth dependence. Furthermore, the RV variations show a weak (not to say absent) degree of correlation with the EW of any of the lines.

Compared to the MOST photometry time series (Rayleigh limit of 0.036 d⁻¹), the RV data have a much longer time span (Rayleigh limit of 4.43 × 10⁻⁴ d⁻¹). As a result, we base our search for intrinsic signals primarily on this data set using two widely used state-of-the-art programs—SigSpec (Reegen 2007) and Period04 (Lenz & Breger 2005). However, the RV data suffer from strong daily (f_d = 1 d⁻¹) and annual (f_a = 0.002769 d⁻¹) aliasing (inner panel in Figure 3). As shown in Figure 3, there is a repeating pattern every 5 d⁻¹ in the discrete Fourier transform (DFT) spectra that arises from our sampling rate. Thus we selected the upper frequency scan range at f_nyq = 5 d⁻¹, and our analysis did not identify frequencies higher than 1 d⁻¹.

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SigSpec computes the spectral significance level (sig) for the DFT amplitude spectrum based on the analytical solution for the probability density function of an amplitude level of any peak during the iterative prewhitening process. By default the program scans for peaks with sig>5, which is approximately equivalent to S/N = 3.8 (Reegen 2007). However, we conservatively chose just to search for highly significant modes to avoid a forest of low-frequency peaks. To accomplish this, the prewhitening process is stopped if the significance (sig) of each peak or the cumulative significance (csig) of the whole solution is below 15. A similar threshold was used by Chapellier et al. (2011) in the frequency analysis of one of the CoRoT primary targets. Since each RV measurement i has an error σ_i, we associate a normalized weight w_i = σ⁻²_i/∑_iσ⁻²_i to each data point. The resulting significance spectrum is shown in Figure 4. This procedure leads to a prewhitening of 19 significant modes. The corresponding list of detected harmonics is presented in Table 2. It tabulates the multimode solution to the RV data set from SigSpec. Analytical 1σ uncertainties are evaluated according to Montgomery & Odonoghue (1999); those in Table 2 are 4σ uncertainties. After prewhitening, the rms of the residual is 0.94 km s⁻¹ and csig = 15.18. This indicates that the probability that the presented frequency solution could be generated by noise is 1 in 10^csig. Figure 2(b) plots the residuals after prewhitening. As shown, the residuals are small but not perfectly featureless. To avoid misidentification of true frequencies from aliases, we conservatively discontinued the prewhitening at this step.

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A straightforward prewhitening of the data with the same weights used as above with Period04 starts with the same results as in SigSpec for the first few harmonics, but then shows some differences. Instead, we imported the frequencies from the SigSpec list and prewhitened sequentially. The maximum difference of 0.01 km s⁻¹ between the calculated amplitudes occurs in f₁₉. The final rms of the residual is 0.92 km s⁻¹ in Perod04 and 0.94 km s⁻¹ in SigSpec. The S/N within a box of 1.0 d⁻¹ around f₁₉ is 4.70.

Independent prewhitening of the MOST data set results in tens of modes that have no corresponding counterparts in the frequency list of Table 2. This is quite expected since the baseline of MOST observations (∼28 days) is shorter than some of the periods found for the star indicated from the analysis of the RV data. To resolve this, we used the "fixed" frequencies derived from the RV analysis (Table 2) and employed Period04 to determine the possible corresponding light amplitudes in the MOST data. We searched for stable amplitudes and phases (an additional harmonic at the orbital frequency of the satellite f_orb = 14.19827 d⁻¹ was also subtracted). Unfortunately, due to the presence of long-period modes comparable to (and longer than) the MOST photometry baseline, no converging solution could be achieved.

According to the asymptotic theory of non-radial stellar oscillations (Tassoul 1980), high-order p and g modes present regularity in frequencies and periods, respectively (Aerts et al. 2010a). In the case of Rigel and similar classes of stars, which are believed to be unstable against g modes, this regularity in period can bring a wealth of information about how stellar material is mixing in the stars' deep interior (see Degroote et al. 2010 for an example). However, as a prerequisite for applying this technique, identification of polar and azimuthal degrees (ℓ, m) for each individual mode is necessary. Unfortunately, the number of secured frequencies is not large enough for a statistical study of any possible period spacings. Moreover, the mode identification is beyond the scope of this paper.

However, after extraction of the frequencies with good precision, it is worthwhile to demonstrate how the modes are distributed according to their corresponding frequencies and amplitudes. For this purpose, the left panel in Figure 5 shows the distribution of detected frequencies between [0, 1] d⁻¹ and their corresponding RV amplitudes. Except for f₁₆ and f₁₉, the rest of the modes have frequencies below 0.4 d⁻¹. The right panel of Figure 5 shows the corresponding period distribution.

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It is possible that the rotation period of Rigel could be detected in the frequency analyses of the RV data. Pápics et al. (2011) found the rotational modulation as the only explanation for the variability in the B0.5 IV CoRoT double-line spectroscopic binary HD 51756. Rigel has two modern, self-consistent spectroscopic vsin i measures (Przybilla et al. 2010; Simón-Díaz et al. 2010) from which we adopt vsin i = 25 km s⁻¹. This vsin i is typical for a BSG and indicates that an inclination of i > 60° is likely (the inclination i is defined in the usual way as the angle of star's rotation axis relative to our line of sight). Adopting both i = 60° and i = 90° with R = 78.9 R_☉, yield P_rot(i = 60°) = 137 d (≡0.0073 d⁻¹) and P_rot(i = 90°) = 158 d (≡0.0063 d⁻¹), respectively. The value of i > 60° is also in accord with the simple angular momentum conservation assumption from the expected v_rot = 240–300 km s⁻¹ estimated for Rigel when it was an MS O9/B0 V star. As shown in Table 2, there are no significant frequencies in the range of 0.006–0.007 d⁻¹. So, rotation effects can be neglected in our interpretation of frequencies. Yet, we are aware that multiple integers of the rotational frequency might show up in the frequency analysis, and long-period modes could have a non-pulsational origin. Given the weak magnetic field on Rigel, rotational modulations induced by stellar spots are unlikely.

As panels (c)–(e) in Figure 2 clearly show, Si ii λ6347, Si ii λ6371, He i 6678, and Hα exhibit the large amplitude changes in EW. However, the binned EW measures have a very low duty cycle and DFT analyses do not lead to any significant frequencies above the 4σ noise threshold. There is only one mode detected for the Si ii λ6347 line with frequency, amplitude, and S/N of 0.0336 d⁻¹, 14.71 mÅ, and 4.24, respectively. Similarly for Si ii λ6371 line, the only dominant peak in the Fourier power spectrum corresponds to the frequency, amplitude, and S/N of 0.0336 d⁻¹, 11.32 mÅ, and 3.46, respectively. Therefore, not only do the frequencies from EWs show no apparent correspondences with any of the frequency entries in Table 2, the first two smallest frequencies, i.e., f₈ and f₁, do not appear in the DFT power spectrum of EWs. Hence, with the current time series of spectra at our disposal, we cannot associate the low-frequency range of RV variations with EW variations.