ROTATIONALLY MODULATED g-MODES IN THE RAPIDLY ROTATING δ SCUTI STAR RASALHAGUE (α OPHIUCHI)^*

The photometry presented here was obtained by the MOST satellite. MOST houses a CCD photometer fed by a 15 cm Maksutov telescope through a custom broadband optical filter (350–750 nm). The satellite's Sun-synchronous 820 km polar orbit (period = 101.4 minutes = 14.20 cycles day⁻¹) enables uninterrupted observations of stars in its Continuous Viewing Zone (−18°< dec < +36°) for up to 8 weeks. A pre-launch summary of the mission is given by Walker et al. (2003) and on-orbit science operations are described by Matthews et al. (2004).

The MOST photometry of the α Ophiuchi binary system covers 30 days during 2009 May 27 to 2009 June 26, and was obtained in Fabry Imaging mode, where the telescope pupil, illuminated by the star, is projected onto the CCD by a Fabry microlens as an annulus covering about 1300 pixels. The data were reduced using the background decorrelation technique described by Reegen et al. (2006). The reduction includes filtering for cosmic particle impacts, correction of the stray light background modulated with satellite orbital parameters, filtering of long-term trends, and removal of obvious outliers.

The final heliocentric corrected time series provided by the MOST instrument team contained 81,018 measurements with an exposure time and sampling of 30 s (based on thirty 1 s exposures stacked onboard the spacecraft). The datastream was remarkably consistent and continuous, being on-target with data for about 94% of the α Oph observation. Instrument problems did cause six data gaps in excess of 30 minutes, with the longest being 5 hr.

The first step our team applied in our data analysis was to remove local 5σ outliers from the data. Because a similar procedure had already been applied by the MOST instrument team, this step had only a minor effect—reducing the data set to 80,002 independent data points, a data volume reduction of only 1.3%. These flux points are presented in Figure 2 in their entirety.

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The next step was to remove the harmonics of the 14.2 c/d orbital period. Similar to the method described by Walker et al. (2005), we used a moving window of 10 days to create a high signal-to-noise mean light curve folded at the orbital period. We then remove this component from the original light curve. The long averaging window ensures we only remove frequencies within a few (Nyquist) resolution elements of the harmonics. We have inspected data both with and without this correction and found the only significant effect was to remove the harmonics from our final power spectra.

Lastly, we "detrended" the light curve in order to filter out slow variations on long timescales. The "slow trend" light curve was constructed by fitting a line segment to the flux measurements (after removal of orbital modulation) within a moving 1-day window. The slow trends were subtracted from the light curve resulting in a final data set on which subsequent Fourier analysis was performed. The slow trend light curve can be seen in Figure 2. At our request, spacecraft telemetry was inspected to see if any camera or telescope diagnostic showed variations in sync with the slow variations seen here—none were found.

We have graphically summarized the steps in our data reduction in Figure 3. Here, we have zoomed up on a single day in our time series and show explicitly the original data, our estimate of the residual orbital modulation, and the slow trend. We also show the final corrected light curve with and without a 5 minutes averaging window to demonstrate the overall quality of our data.

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After more than 7 years in orbit, the MOST Team has a very robust understanding of the performance of the instrument, especially the point-to-point precision of the photometry as a function of target magnitude and the noise levels at various frequency ranges in Fourier space. Many of the pulsating variable stars detected in the MOST Guide Star sample (in the magnitude ranges 6 < V < 10) had a "discovery amplitude" (i.e., amplitude of the largest oscillation signal) of only about 1–2 mmag, with many other oscillation peaks 10 times smaller and a noise level in Fourier space 50 times lower (e.g., the richly multi-periodic δ Scuti star HD 209775 Matthews 2007). Some examples of stars which characterize the photometric precision of the MOST instrument are: HD 146490 (a guide star used as a calibration by Kallinger et al. 2008b; see their Figure 1), with V = 7.2 and point-to-point scatter of 0.12 mmag; tau Bootis (V = 4.5), an exoplanet-hosting star which shows clear modulation at the planet's orbital period with a peak-to-peak amplitude of 2 mmag (Walker et al. 2008) for which its mean instrumental magnitude is repeatable to within 1 mmag in observations spaced several years apart; and Procyon (V = 0.35), with point-to-point precision of 0.14 mmag. In these cases, the noise floor in the Fourier spectrum is well defined and at a level of 0.01 mmag or less.

Another MOST target, epsilon Oph (Barban et al. 2007; Kallinger et al. 2008a), a cool giant (V = 3.24), is stable to 0.35 mmag at frequencies below 0.1 c/d. At these low frequencies, non-periodic noise intrinsic to the star is usually associated with granulation. The MOST Team has never seen instrumental artifacts with the amplitudes and coherence of the variations evident at low frequencies in the MOST light curve of α Oph presented here. For the results in this paper, the low-frequency signals have no direct implications, so we do not address them in further detail. However, they are likely intrinsic to the star and hence do not raise any alarms of possible unrecognized photometric artifacts at other frequency ranges in the α Oph time series.

We have carried out an exhaustive Fourier analysis of the MOST photometry. Although the MOST data are remarkably complete and evenly spaced compared to ground-based data, we nonetheless employed a numerical Fourier Transform followed by a "CLEAN"ing step based on the algorithm of Roberts et al. (1987) in order to remove the small sidelobes from the strongest peaks in the power spectrum that arise due to the gaps in the temporal sampling.

Figure 4 shows the amplitude spectrum of our photometry in units of millimagnitude for the range 0–64 c/d. The power for frequencies below ∼ 1 cycle day⁻¹ was suppressed by the detrending step described in Section 3. In fact, without detrending, nearly all the frequencies below ∼ 0.75 c/d show significant power above background noise levels. We note that all frequencies up to 1440 c/d were inspected and frequencies above 64 c/d are not shown here because no statistically significant peaks were found.

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It was clear from the power spectrum that the background noise level is higher for frequencies below 30 c/d compared to higher frequencies. In order to properly estimate the statistical significance of a "peak," it is crucial to establish the level of the frequency-dependent noise background. We have used a median filter with a 2 cycles day⁻¹ window in order to crudely estimate the background noise power for a given frequency, and this background level is included in Figure 4. We will discuss the physical nature of this non-white noise in Section 5.1; for now we simply treat it as a noise source that can produce spurious spikes in the amplitude spectrum.

Armed with the frequency-dependent background noise spectrum estimated in Section 4.2, we can now place confidence limits on a given amplitude peak being a "real" distinct pulsation mode rather than having arisen by chance due to background fluctuations. We have chosen a statistical measure such that we only expect one of our identified modes in the range between 0.5 cycle day⁻¹ and 64 c/d to be a false positive. Since there are ∼1800 independent frequencies that can be probed in our data in this spectral range, we chose only peaks in the power spectrum above 3.5-σ confidence, a criteria that should produce roughly one false positive assuming pure Gaussian random noise. The 57 modes that survived this analysis are listed in Table 2 and are plotted in Figure 4. We also confirmed the statistical significance of our peaks through a bootstrap analysis, and we note that we would have erroneously identified >500 distinct modes if we had ignored the frequency-dependence of the noise background.

The strongest modes are 18.668 c/d (half-amplitude 0.66 mmag), the closely spaced pair at 16.124/16.174 c/d (0.35/0.41 mmag) and at 11.72 c/d (0.41 mmag). These frequencies are typical for the dominant p-mode oscillations in δ Scuti stars and bear some resemblance to the dominant frequencies observed for another rapid rotator Altair (Buzasi et al. 2005), which had dominant peaks at approximately 15.7, 20.8, and 26.0 c/d. The highest-frequency mode we have detected with high confidence is at 48.35 c/d (0.036 mmag).

Inspection of Figure 4 reveals many additional peaks that appear real—more than can be attributed to random fluctuations of the background. However, we have chosen not to report these peaks in Table 2 in order to maintain a highly pure set of frequencies with minimal false positives. In other words, we understand that many of the other peaks are real pulsational frequencies but are also sure that some of them are contaminated by background fluctuations. Future mode analysis might allow some of the 2- or 3-σ peaks to become bonafide pulsation frequencies, but we have been conservative in our mode identifications for this paper.

In addition, inspection of the amplitude spectrum at lower frequencies easily reveals increased power at harmonics of 1.7 c/d, close to the expected rotational frequency (1.65 c/d, see Section 2). We will discuss the physical origin of these frequencies in Section 5.

With such a long data set of high quality, we investigated the stability of the detected modes. Figure 4 contains also the power spectrum split into four 1 week chunks and is depicted as a two-dimensional background image in each panel. One can see the lower-frequency modes, <10 c/d, tend to be variable—strong in some weeks and not present in others. Degroote et al. (2009) found amplitude variability in modes in some higher-mass objects with lifetimes from one to a few days, substantially shorter than those here which tend to be approximately one week. Most of the higher-frequency modes are more stable, although not all. For instance, the 16.1 c/d dominant mode shows amplitude varying by a large amount, consistent with two equal modes with a frequency difference just barely resolved by our 30 day run. The lifetime of modes is another clue to their physical origin and will be discussed in Section 5.

Poretti et al. (2009) observed the δ Scuti star HD 50844 using COROT, identifying >1000 statistically significant pulsational modes using standard Fourier analysis. Kallinger & Matthews (2010) argued it was physically unreasonable to have so many distinct modes and interpreted the higher power for frequencies lower than ∼30 c/d as due to granulation noise. Indeed, these authors presented a compelling correlation of the granulation cutoff frequency with stellar mass and radius. We subscribe to the interpretation of Kallinger & Matthews, choosing to treat the higher average Fourier power at the lower frequencies as a kind of non-white background noise (see Section 4.2).

In order to better estimate the granulation cutoff frequency, we have plotted the median-filtered power (2 c/d window) in Figure 5. We identified an obvious artifact from the orbital harmonics at 42.6 c/d, and 56.8 c/d and likely present at 14.2 c/d and 28.4 c/d, probably due to imperfect background subtraction of scattered light. By modeling the harmonics beyond 35 c/d, we have crudely estimated the contamination at lower frequencies based on the isolated features at 42.6 and 56.8 c/d. Figure 5 shows our ad hoc model for the contributions from the orbital harmonics and also includes our corrected granulation noise spectrum. Here, the granulation cutoff frequency can be estimated to be 26 ± 2 c/d.

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As explained in Kallinger & Matthews (2010), the granulation cutoff frequency should be related to the inverse of the sound crossing time of one pressure scale height in the outer convective layer. This leads to a dependence of $\nu _{\rm granulation} \propto \frac{M}{R^2 T^\frac{1}{2}}$. It is beyond the scope of this paper to compare our result with their more elaborate power-law fitting method; however, we can compare our measured granulation cutoff with that predicted by the empirical relation discovered by Kallinger & Matthews (2010, see their Figure 3). Using a mass of 2.18 M_☉, effective radius of 2.7 R_☉, and T_eff = 8300 K for α Oph, we find that MR⁻² T^−0.5∼ 0.25 (in solar units), leading to an expected ν_granulation ∼ 300μHz = 26 c/d, consistent with our observations.

Following treatment of Buzasi et al. (2005), we can estimate the frequency of the fundamental radial (p-) mode using the relation

$$\begin{equation} P \sqrt{\rho /\rho _\odot } = Q. \end{equation} \tag{ 1 }$$

For Q = 0.033 days (Breger 1979) and using the stellar parameters from Table 1, we find f_fun∼ 10.1 c/d (compared to 15.6 c/d for the 1.79 M_☉ star Altair). Thus, we will begin by assuming p-modes all have frequencies above or near to this cutoff. A full analysis of the p-modes with tentative mode identifications will be the subject of a future paper. Here, we focus primarily on the unexpected discovery of rotationally modulated g-modes in α Oph.

We showed in Section 4 (see also Figure 2) that there was a ∼1 mmag slow (week-timescale) variation seen in the photometry. We also have identified 15 strong modes with frequencies smaller than the fundamental radial mode that cluster around harmonics of 1.7 c/d, close to the rotation frequency. This distinctive structuring suggests we are witnessing the rotational modulation of modes whose frequency in the co-rotating frame is small in comparison to the rotation frequency.

The tabulated frequencies were extracted directly from peaks in the CLEANed Fourier Transform, and we assigned frequency errors based simply on the total length of the data set. We realize that higher precision on the frequency localization of pure modes can sometimes be achieved through a multi-component least-squares fit (e.g., Walker et al. 2005); however, this procedure has convergence problems for very closed spaced frequencies as encountered here. The errors on amplitudes were assigned based on the local background noise as estimated in Section 4.2. We caution that the formal amplitude errors may be underestimated due to non-stationary statistics of the background or due to amplitude variations in the modes themselves. The full time series is available upon request to researchers interested in exploring alternative analysis schemes, but our straightforward approach adopted here leads to conservative errors in the frequencies and is sufficient for the analysis presented here.

Gravity (g-) modes are expected to have long periods compared to p-modes, although it has been difficult to estimate this theoretically for rapid rotators. Indeed, our amplitude spectrum is consistent with a complex and time-variable set of g-modes with co-rotating frequencies ≲0.1 c/d, corresponding to approximately 10 days periods. Normally, the properties of these modes are quite difficult to measure due to the long time frames needed to see the modes travel around the star. However, in our case, the ∼1.65 c/d rotational rate of this rapid rotator leads to strong rotational modulation. Note also that modes of north–south asymmetry are invisible to us while the symmetric l − |m| even-parity modes are strongly modulated due to our near equator-on viewing angle.

Specifically, we see small clusters of modes around frequencies ∼1.8, 3.5, 5.4, 7.0, 8.6, 10.4, 12.0 c/d. While some of the higher frequencies here might be p-modes, it seems that the strong periodicity at about 1.7 c/d suggests that each cluster represents a new non-radial family of modes with increasing m values. Stellar rotation causes light curve variations with frequencies proportional to mf_rot, since the m parameter represents the number of nodes around the equator. With this interpretation, we are seeing non-radial modes corresponding to m = 1, 2, 3, 4, 5 and possible m = 6, 7. We believe this is the first time such a rich set of g-modes has been detected and partially identified around a rapid rotator, although perhaps a related phenomenon may be at work around the late Be star HD 50209 (Diago et al. 2009).

The simultaneous appearance of g-modes and p-modes would make α Oph a "hybrid" γ Dor-δ Sct pulsator, a class recently explored by Grigahcène et al. (2010) using first Kepler data. Indeed, these authors present a frequency spectrum of KIC9775454 that bears some similarity to the data presented here for α Oph, although our frequency spectrum does not show the characteristic "gap" between 5 and 10 c/d. Further comparison with this work is not possible until more is known about the rotational properties of the Kepler sample, especially since many hybrid pulsators are known to be slowly rotating Am stars.

To further investigate the periodic spacing of the g-modes of α Oph, we searched for families of modes that follow a linear frequency relationship of the form

$$\begin{equation} f = f_{0} + m f_{1} \end{equation} \tag{ 2 }$$

for integer m. We found two families, each comprising more than three modes, that can be well represented by this formula. Family A contains five modes; assuming the lowest-frequency mode corresponds to m = 1, a least-squares fit gives f₀ = 0.073 ± 0.017 c/d and f₁ = 1.7095 ± 0.0002 c/d. Family B contains four modes, with a fit f₀ = 0.219 ± 0.017 c/d and f₁ = 1.7424 ± 0.0002 c/d. These modes are noted in Table 2 and included in Figure 6 along with the observed amplitude spectrum for f < 16 c/d. There are other solutions that fit three modes or fewer, but in this treatment, we consider just families A and B, since the larger number of modes mean the fit is more reliable and less likely due to chance superpositions.

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There are a number of possible interpretations for the linear frequency relationship exhibited by the two mode families, which we shall consider in turn. To first order in the rotation frequency f_rot, the observed frequency of a mode with radial order n, harmonic degree ℓ, and azimuthal order m is given by the well-known Ledoux (1951) formula

$$\begin{equation} f = f_{n,\ell } + m f_{\rm rot} (1 - C_{n,\ell }). \end{equation} \tag{ 3 }$$

Here, f_n,ℓ is the frequency the mode would have in the absence of rotation, and is independent of m. The parentheses combine together the effects of the Doppler shift (in transforming from co-rotating to inertial reference frame) and the Coriolis force–the latter represented by the C_n,ℓ term, which again is independent of m. For high-order g-modes, C_n,ℓ approaches [ℓ(ℓ + 1)]⁻¹ (e.g., Aerts et al. 2010).

If we make the identifications f₀ = f_n,ℓ and f₁ = f_rot(1 − C_n,ℓ), then the Ledoux formula is apparently able to explain the linear frequency relationship of the two families, assuming each comprises modes having the same n and ℓ but differing m. However, there are strong arguments against this conclusion. For both families, the mode frequencies in the co-rotating frame

$$\begin{equation} f_{\rm c} \equiv f - m f_{\rm rot} \end{equation} \tag{ 4 }$$

are small (on the order of 0.1–0.3 c/d) compared to the rotation frequency f_rot = 1.65 c/d (Table 1). Therefore, the spin parameter ν ≡ 2|f_rot/f_c| of the modes must be much larger than unity, implying that their dynamics are dominated by the Coriolis force (see, e.g., Townsend 2003). This represents a fundamental inconsistency with the use of the first-order Ledoux formula (3), which is valid only for ν ≲ 1.

A possible way around this inconsistency is to relax the assumption that lowest-frequency mode in each family corresponds to azumithal order m = 1. If we instead assume m ⩾ 3 for these modes, then by Equation (4)⁶ we can achieve ν ≲ 1. However, this "fix" itself runs into difficulty when we note that for both mode families f₁ > f_rot, implying that the Coriolis coefficients C_n,ℓ are negative. This is incompatible with the above-mentioned limit C_n,ℓ ≈ [ℓ(ℓ + 1)]⁻¹ > 0 (indeed, we are not aware of any physically plausible pulsation model that predicts negative Coriolis coefficients).

Accordingly, we are led to abandon the Ledoux formula as an explanation for the linear frequency relationship of the two mode families. In searching for alternative interpretations, we note that the small magnitude of f₀ suggests we may be observing seven dispersion-free low-frequency modes. In rotating stars, there are two classes of mode that are dispersion free: equatorial Kelvin modes (which are prograde) and Rossby modes (which are retrograde). Rossby modes tend not to generate light variations because they are incompressible, so let us put them aside for the moment (although not forget about them completely).

Focusing therefore on the Kelvin modes, these are prograde sectoral (ℓ = m) g-modes which are confined to an equatorial waveguide by the action of the Coriolis force (see Townsend 2003). A defining characteristic of these modes is that their frequencies are directly proportional to m. At high radial orders, these frequencies in the co-rotating frame can be approximated by

$$\begin{equation} f_{\rm c} \approx \frac{m I_{\rm g}}{2\pi ^{2}n}; \end{equation} \tag{ 5 }$$

here, following Mullan (1989), we define

$$\begin{equation} I_{\rm g} = \frac{1}{\sqrt{2} \pi ^{2}} \int _{0} \frac{N}{r} \,{\rm d}r, \end{equation} \tag{ 6 }$$

with N is the Brunt-Väisälä frequency. The corresponding observed frequencies are then given by

$$\begin{equation} f \approx m \left(\frac{I_{\rm g}}{2 \pi ^{2} n} + f_{\rm rot} \right). \end{equation} \tag{ 7 }$$

For a group of modes all having the same n, this expression predicts a linear frequency relationship with zero intercept. If n varies within the group, however, then there will be some scatter about a linear relationship, with a relative amplitude of ∼Δn/n.

To examine whether the two mode families could be Kelvin modes, we re-fit their frequencies assuming a zero intercept (again identifying the lowest-frequency mode as m = 1). This gives f_c/m = 0.07 ± 0.03 c/d for family A and f_c/m = 0.10 ± 0.03 c/d for family B. To interpret these values, we assume a relationship $I_{\rm g} \approx 10^{-3} \sqrt{(M/M_{\odot })/(R/R_{\odot })^{3}}\,{\rm Hz}$, derived from the n = 3 polytropic model considered by Mullan (1989). Using the stellar parameters from Table 1, this yields I_g ≈ 34 c/d. Hence, the approximate radial orders of the modes are derived as n ≈ 24 (family A) and n ≈ 17 (family B), with corresponding scatter (based on the quoted uncertainties in the f_c/m fits) of Δn ≈ 10 and Δn ≈ 5, respectively.

These values are consistent with the typical radial orders of unstable g-modes observed in the more-massive Slowly Pulsating B (SPB) stars (e.g., Cameron et al. 2008; Pamyatnykh 1999). This lends strong support to the identification of the two mode families as high-order (n ≈ 17 and n ≈ 24) g-modes transformed by the Coriolis force into equatorial Kelvin modes.

As a last comment, according to Figure 4, most of the g-modes seem to be time-variable within our one month time frame, in contrast to the p-modes whose amplitudes are more typically constant within the errors. The g-modes tend to change on a timescale similar to the intrinsic mode period (∼ 10 days) or a bit longer. This could be due to superposition of many closely spaced modes in what appears to be indeed a complex spectra. Alternatively, many excitable modes may be transferring energy between themselves, leading to a dynamic and time-variable amplitude spectrum. The ability to see so many independent l- and m-modes by using strong rotational modulation is invaluable to deriving strong constraints on the mode properties up to high order (here up to m = 7) and promises to reveal interior stellar structure with additional analysis.