Seismology of Altair with MOST

Altair is the fastest rotating star at less than 10 parsecs from the Sun. Its precise modelling is a landmark for our understanding of stellar evolution with fast rotation, and all observational constraints are most welcome to better determine the fundamental parameters of this star. We wish to improve the seismic spectrum of Altair and confirm the $\delta$-Scuti nature of this star. We used the photometric data collected by the Microvariability and Oscillations of STars (MOST) satellite in the form of a series of Fabry images to derive Altair light curves at four epochs, namely in 2007, 2011, 2012, and 2013. We first confirm the presence of $\delta$-Scuti oscillations in the light curves of Altair. We extend the precision of some eigenfrequencies and add new ones to the spectrum of Altair, which now has 15 detected eigenmodes. The rotation period, which is expected at $\sim$7h46min from models reproducing interferometric data, seems to appear in the 2012 data set, but it still needs confirmation. Finally, Altair modal oscillations show noticeable amplitude variations on a timescale of 10 to 15 days, which may be the signature of a coupling between oscillations and thermal convection in the layer where the kappa-mechanism is operating.The Altair oscillation spectrum does not contain a large number of excited eigenmodes, which is similar to the fast rotating star HD220811. This supports the idea that fast rotation hinders the excitation of eigenmodes as already pointed out by theoretical investigations.


Introduction
Altair (α Aquilae, HD187642) is one of the few fast rotating early-type stars in the solar neighbourhood, its distance being 5.13 pc according to HIPPARCOS data (van Leeuwen 2007). As such, it is an ideal target of interferometric observations, which have regularly measured its shape and surface brightness (van Belle et al. 2001;Domiciano de Souza et al. 2005;Monnier et al. 2007). These observations have led to the determination of the centrifugal flattening of Altair which turns out to be close to 22% while its angular velocity at the equator is 74% of the Keplerian one. Such a centrifugal distortion makes the use of two-dimensional models mandatory for a correct interpretation of observational data. Bouchaud et al. (2020) actually performed the first 2D modelling of Altair using ESTER 2D models which include, self-consistently, the 2D structure and the large-scale flows, namely differential rotation and meridional circulation, driven by baroclinicity (Espinosa Lara & Rieutord 2013;Rieutord et al. 2016). Besides demonstrating the young age of Altair (∼ 100 Myrs instead of 1 Gyrs as previously estimated with 1D models e.g. Domiciano de Souza et al. 2005), the work of Bouchaud et al. (2020) shows This work is based on data from the MOST satellite, a Canadian Space Agency mission, jointly operated by Dynacon Inc., the University of Toronto Institute for Aerospace Studies and the University of British Columbia, with the assistance of the University of Vienna.
us that the seismic spectrum of Altair is a key ingredient to further constrain the mass of this star. Indeed, acoustic modes are sensitive to the mean density (Reese et al. 2012;García Hernández et al. 2015), while interferometry, through the measurement of radii, determines the volume of the star.
Hence, the detection of (presumably) acoustic oscillations by Buzasi et al. (2005), in a photometric monitoring of Altair in 1999 by the star tracker of the WISE mission, was a good surprise. This detection meant that Altair is a δ-Scuti star, and actually the brightest one (Buzasi et al. 2005). However, these oscillations were never confirmed. Regarding their importance in the modelling of this star, we looked for new data that would confirm and improve the result of Buzasi et al. (2005). Altair has been quite intensively observed with the Microvariability and Oscillations of STars (MOST) satellite (Walker et al. 2003(Walker et al. ) in 2007(Walker et al. , 2011(Walker et al. , 2012(Walker et al. , and 2013, but no analysis of these data has been published so far. Since such data have the potential to confirm the previous results and possibly show new frequencies or variations of modes amplitudes, we embarked on a project to process them in order to once again exhibit the seismic spectrum of this fascinating star. This paper is organised as follows: We first describe the data and the processing we applied to extract the light curves (sect. 2). We then analyse the light curves and retrieve the oscillation spectra at the various epoch of the   data (sect. 3). A discussion and preliminary conclusions end the paper.

Observations and data reduction
Altair was observed in four sequences by the MOST satellite, which run around the Earth at a mean altitude of 830 km with an orbital period of 6084.9 s, corresponding to a frequency of 14.199 c/d (Walker et al. 2003). The characteristics of the time series are summarised in Tab. A.1.
Data were downloaded from the Canadian Astronomy Data Centre. They are available as a series of Fabry images, from which we derived light curves. In Appendix A, we provide some details on the first steps of this processing, but the main challenge of this data reduction is the clean suppression of stray light, which comes from the illuminated side of the Earth or from moonshine.
The technique we used is inspired by the one developed by Huber & Reegen (2008) and is based on calculating correlations of mean target and background flux. Since stray light is variable over timescales of approximately a day, the correlation was calculated using the mean background and target intensity in a moving window of 1 day rather than using the whole light curve. An example of a linear correlation between target and background pixels is shown for the 2011 data set in Fig. 1.
This method assumes that the influence of stray light is the same for target and background pixels, and that the background pixel intensity is not influenced by the star light.
The correction of the correlation was made by measuring the variations δm between the real target magnitude and the one given by the linear trend each time. Hence, where I tar,real (t) is the mean intensity of the target pixels at time t, and I tar,correl (t) = aI bck,real + b, where I bck,real (t) is the mean intensity of the background pixels at time t, and a, b are the coefficients given by the linear regression. As discussed in Huber & Reegen (2008), our procedure is a simplified version of the algorithm of Reegen et al. (2006), which searches for the background pixels that best correlate with each target pixel. In the procedure of Reegen et al. (2006), the background flux is removed from the target pixel intensity and the procedure is repeated on the next target pixel until all stray light contributions have been reduced to an acceptable level.
Such a procedure is able to remove a non-uniform stray light pattern, but it is quite demanding in computation time. We did not implement this type of processing since, in the end, the simple algorithm described previously, with some additional processing detailed below, satisfactorily removed the periodic signal coming from stray light.
Hence, to further remove the effects of the orbital period at a frequency of 14.199 c/d, we used a moving window of 28 periods (∼ 2 d) to create a mean light curve folded at the MOST orbital period and removed this mean light curve from the data. This step effectively suppressed the harmonics of the orbital frequency from the final power spectra.  To illustrate this processing, we show in Fig. 2 the evolution of the light curve at each step using a subset of the 2011 data. The final light curve, a glimpse of which is shown in Fig. 3, typically gives amplitude fluctuations of ±2 mmag. This is similar to those observed by Buzasi et al. (2005) and also quite similar to those of Rasalhague (α Ophiuchi), another fast rotating A-type star with δ-Scuti oscillations (Monnier et al. 2010). Fig. 4 further shows the role of the processing steps on the power spectra of the light curves using Period04 (Lenz & Breger 2005). The orbital frequency at 14.199 c/d clearly shows up in the unprocessed data, as well as in the decorrelated data (at a lower level of course). We also notice secondary peaks separated from the orbital frequency by δν = 1 c/d or multiples of it, namely associated with Earth's rotation. Fig. 4 clearly shows that the removal of the mean orbit signal and global decorrelation quite nicely suppresses most of the systematics coming from the orbital motion of MOST.

Algorithm -Validation
After the foregoing extraction of the light curves from the raw data, we proceeded with the derivation of the frequencies and amplitude of the modes. For that purpose, we used several types of software that are all based on the same classical method, involving computing the Lomb-Scargle periodogram, pre-whitening the signal, and adjusting a combination of sinusoidal signal as to the data through non-linear least-square optimisation. We used Period04 (Lenz & Breger 2005), FELIX (Charpinet et al. 2010;Zong et al. 2016), and PYPE, which is a python software specially built for this work from python libraries (Astropy 4.2 for the Lomb-Scargle periodogram and Scipy for the optimisation). These types of software differ from one another basically from their evaluation of uncertainties. Period04 uses Monte-Carlo simulations, FELIX uses formulae from Montgomery & O'Donoghue (1999), while PYPE is based on the bootstrap method. Comparison of the results from these three types of software is useful to appreciate the influence of the numerical procedure on the results. Basically, the programmes usually agree, especially when the quality of the data increases, not surprisingly. We show in Tab. 1 the results for two modes using the three programmes. This illustrates that amplitudes of the main mode are badly determined with the 2011 data due to their short length (4 days), mostly because of the mixture of this mode with its neighbouring one at 15.98 c/d.

Error bars, noise
Before presenting the results, we first discuss their sources of uncertainties. As written above, we assume that the flux variations of Altair are due to excited eigenmodes in the small amplitude regime and they can be represented by the series (2). The amplitudes A n , the frequencies ν n , or the phase φ n are the parameters of the model whose uncertainties are to be evaluated. These uncertainties basically have three sources: (i) residual stray light (ii), random noise (iii), and time sampling of the signal. As far as residual stray light is concerned, it is characterized by its frequencies that are typically of the form F nm = nf orb ± mf day , which we detected for 0 ≤ n ≤ 7 and 0 ≤ m ≤ 4. These spectral lines prevented us from detecting or correctly measuring any signal in frequency bands such as [F nm − ∆f, F nm + ∆f ], where ∆f is the frequency resolution of the window data set. As a consequence, Altair's frequency oscillation at 28.40 c/d 2f orb detected by Buzasi et al. (2005) was just screened by the stray light signal (and the time window which includes this harmonic). The same pollution affects the oscillation at 16.18 c/d close to the frequency f orb + 2f day .
Next, the influence of intrinsic noise was evaluated by either the bootstrap method (PYPE), Montgomery & The 2007 data set is obviously the noisiest due to large pointing errors, but its length (∼20 days) somehow compensates for this. The best data set is the one from 2012 because of its length and moderate noise. This is also illustrated in Fig. 5 where we plotted the noise spectrum for the four data sets. Clearly 2007 appears very much polluted by stray light, while 2012 appears to be the best series. We note that 2011 and 2013 have most of their noise in the [0, 50]c/d range, which is also the range of interest, unfortunately. Fig. 4 has already shown a view of the Altair oscillation spectrum with 2012 data. However, as discussed below, the modes amplitude varies with time. We therefore extracted, from each data set, the most significant frequencies along with their amplitude and they are represented in a periodogram-like plot in Fig. 6. From this spectrum, we extracted a list of modes, whose detection we consider as the most reliable. Their frequencies and amplitudes are gathered in Tab. 2. Most of them appear in several data sets, making their detection quite robust.

Altair's oscillation spectrum
The first impression one gets looking at Altair's spectrum is that it is rather sparse if we compare it to the well-studied (Chen et al. 2016  HD50844 observed by CoRoT (Poretti et al. 2009). Of course, HD50844 is near the TAMS and Altair is close to the ZAMS (Bouchaud et al. 2020). In addition, Altair's rotation period is eight times shorter than that of HD50844. Altair's oscillation spectrum better compares with that of stars picked out by Bedding et al. (2020). These stars are young with intermediate mass showing δ-Scuti oscillations just as Altair. But unlike them, Altair does not show modes with frequencies above 40 c/d, nor a clear regular spacing of frequencies. Actually we might note an approximately recurrent spacing of ∼2.5d −1 with frequencies f 6 , f 10 , f 11 , f 12 , f 13 , and f 15 , as already pointed out in Bouchaud et al. (2020).
Another regular spacing, ∼ 0.907c/d, may also be noticed within the low frequencies f 2 , f 3 , f 4 , f 5 . Monnier et al. (2010) observed a similar pattern for two sets of modes in the oscillation spectrum of α Ophiuchi, with a spacing of ∼ 1.71c/d and ∼ 1.74c/d, respectively. These authors interpreted these sets of modes as the possible signature of equatorial Kelvin waves. This is an appealing interpretation, but in our case we shall wait for the spectroscopic data that are expected shortly before deciding on the meaning of these quasi-regular patterns at low frequency (Rieutord et al. 2021 in prep.).
We also observed something similar to a triplet of frequencies f 6 , f 7 , f 8 around ν = 16c/d with the most powerful peak of Altair's spectrum at 15.7679 c/d. The frequency spacing of 0.2 c/d of course does not correspond to a rotational splitting which would be much larger. A search in the theoretical predictions based on the 2D model of Altair computed in Bouchaud et al. (2020) does not show conspicuous evidence of modes that could be identified with the detected frequencies. However, the theoretical modelling is not ripe yet, lacking the selection criterion of excited modes.

HD220811: Potential sister of Altair
Interestingly, Altair's spectrum shows some similarities with that of HD220811 taken from the list of Bedding et al. (2020). As shown in Fig.7, the two stars show their main excited modes in the range of 15-40 c/d and coincidentally both show a strongly excited mode at f ∼ 15.9 c/d. Not much is known about HD220811: It is a double star with a tiny separation of 0.4 . The Transiting Exoplanet Survey  Satellite (TESS) input catalogue says its effective temperature is 7527K, which is close to the average Altair's effective temperature of 7550K (Erspamer & North 2003). Its V sin i ∼ 270 km/s, according to Bedding et al. (2020), is compatible with the 313 km/s equatorial velocity of Altair (Bouchaud et al. 2020). The TESS input catalogue also mentions a Hipparcos parallax of 8.57 mas and a visual magnitude of 6.91. Using Altair's parallax 194.44 mas and visual magnitude of 0.76, we deduce that the apparent luminosity of HD220811 is slightly larger than that of Altair, namely L HD220811 1.8L Altair . Since this luminosity is only the apparent one, which depends on the inclination of the rotation axis, the two stars may well be very similar.
The probable similarity of Altair and HD220811 as far as their oscillation spectrum and spectral type are concerned, together with their very fast rotation (V eq ∼ 300 km/s), may illustrate the result of Reese et al. (2017) that fast rotation reduces the number of excited modes in a star. The parallel study of these two stars, and others with the same features, will be very useful to better appreciate all the effects of rotation.

Rotation
From interferometric and spectroscopic data, Bouchaud et al. (2020) derived the angular velocity of Altair which corresponds to ν rot = 3.08 ± 0.03 c/d. In fact, from the Bouchaud et al. (2020) model of Altair, we have an idea of the surface differential rotation of this star. It is shown in Fig. 8 along with its uncertainty. A flux pattern at the surface of Altair may therefore show up at a frequency in between 2.9 and 3.16 c/d. A close-up of the spectrum in this range (see Fig. 9) shows a signal at 2.99 c/d in the 2012 data set, which may be a signature of a such a flux pattern. Since Altair is known to be magnetically active (Robrade & Schmitt 2009), flux modulation by some magnetic feature at its surface is not impossible. However, this possible detection of rotation needs to be confirmed.

Time variability
The data sets collected by MOST over the years offer the possibility to investigate the time variation of the modes amplitude. However, the determination of the amplitude of a mode depends on the determination of its frequency. The length of the light curves and the time windows are unfortunately not constant from one set to another. Mode frequencies are therefore more or less well determined according to the data set at hand. However, for the modes listed in Tab. 2, frequencies can be considered as constant within their error bar. Thus, we shall assume their constancy in time. We therefore determined the best frequency of the main modes using all data and projected the desired subset of the light curve on the simple Fourier basis (e.g. Eq. 2) using least-squares to obtain the time evolution of amplitudes over the years.
A first view of these variations of the main modes over the years is shown in Fig. 10. The most prominent mode at f=15.7679 c/d shows a rather important growth (a factor of 1.5) between 1999 and 2007. With the origin of these variations being numerous (see Guzik et al. 2016 for a rather exhaustive list), a hint may be given by the timescales shown by these fluctuations. Fig. 10 shows that years should be considered, but phenomena such as convection may impose much shorter timescales and suggest for one to inspect variations over days. To this end, we concentrated on the 2012 data set, which is certainly the most appropriate with its 33 days length and its rather low noise. We first carried out a time-frequency analysis with a one-day time step and a 3.3 days interval. The result is shown in Fig. 11.
We clearly see amplitude oscillations near 15.8 c/d due to the two beating modes at 15.7679 and 15.983 c/d, but an evolving trend is also visible along the time interval. This evolution is better seen if we split the time interval into disjoint windows of 3.3 days. Shorter windows do not provide more information and they are influenced by noise or secondary window peaks too much. This is why we do not show the amplitude variation of the f 7 -mode, which is too close to the main f 6 -mode at f=15.7679 c/d. Hence, the amplitude variations of f 6 also include those of f 7 . Figure 12 shows the amplitude variations of the five most prominent modes. This figure suggests an amplitude variation on a timescale of 15 days for the main mode at f 6 and for the low amplitude mode at f 11 , whose variations are anti-correlated with those of f 6 . The amplitude of the beating effect between f 6 and f 7 is also shown in Fig. 12 (black dashed line). For that, we generated a light curve where the two modes f 6 and f 7 have a constant amplitude, and we processed this artificial light curve in a similar way as the real one. We can clearly see that the beating effect produces amplitude variations that are much smaller than those observed on f 6 . Mode f 10 also seems to show, mildly, an amplitude evolution on a 15-day timescale. The low-frequency mode at f 5 = 3.527 c/d seems to show an amplitude modulation on a 10-day timescale (e.g. Fig. 11), while the mode at f 12 also shows amplitude variations, but on a timescale similar to the window sampling, which hinders any conclusion being made. The driving of δ-Scuti oscillations is classically attributed to the κ-mechanism associated with the opacity bump generated by the partial second ionisation of helium around 50,000K (Baglin et al. 1973;Balona et al. 2015). However, such an opacity bump also destabilises a whole layer of the star where thermal convection arises. This convection occurs at high Reynolds numbers and thus excites a wide range of time and length scales. Using Bouchaud et al. (2020) Altair's model, we found that timescales of 10 or 15 days are well within the possible timescales of convection in the κ-exciting layer. This coincidence may not be just by chance and further work is needed to explain the way oscillations may be modulated on this timescale. Buzasi et al. (2005) discovered that Altair is the brightest δ-Scuti of the sky, and we confirm this. The oscillation spectrum of Altair now includes six new frequencies and we extend the range of frequencies from 0.57 c/d to 35.82 c/d. The MOST observations of Altair, distributed over several years, allowed us to bring the variations of the mode amplitudes to light. The time frequency analysis of the 2012 data set, which spans 33 days, showed characteristic timescales of ∼10 to 15 days, which can be imposed by the convective layer associated with the opacity bump of helium second ionisation. Since this opacity bump is also known to drive oscillations in δ-Scuti stars, a good modelling of the coupling of thermal convection with oscillations, in the spirit