CoRoT light curves of RR Lyrae stars The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA’s RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain.{}^{,} The CoRoT timeseries, Tables 1 and 2 are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/vol/page

CoRoT light curves of RR Lyrae stars thanks: The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA’s RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain. thanks: The CoRoT timeseries, Tables 1 and 2 are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/vol/page

CoRoT 101128793: long–term changes in the Blazhko effect and excitation of additional modes
E. Poretti INAF – Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy
ennio.poretti@brera.inaf.it
   M. Paparó Konkoly Observatory of the Hungarian Academy of Sciences, PO Box 67, H-1525 Budapest, Hungary    M. Deleuil LAM, UMR 6110, CNRS/Univ. de Provence, 38 rue F. Joliot-Curie, 13388 Marseille, France    M. Chadid Observatoire de la Côte d’Azur, Université Nice Sophia-Antipolis, UMR 6525, Parc Valrose, 06108 Nice Cedex 02, France    K. Kolenberg Institute of Astronomy, University of Vienna, Türkenschanzstrasse 17, A-1180 Vienna, Austria    R. Szabó Konkoly Observatory of the Hungarian Academy of Sciences, PO Box 67, H-1525 Budapest, Hungary    J.M. Benkő Konkoly Observatory of the Hungarian Academy of Sciences, PO Box 67, H-1525 Budapest, Hungary    E. Chapellier Observatoire de la Côte d’Azur, Université Nice Sophia-Antipolis, UMR 6525, Parc Valrose, 06108 Nice Cedex 02, France    E. Guggenberger Institute of Astronomy, University of Vienna, Türkenschanzstrasse 17, A-1180 Vienna, Austria    J.F. Le Borgne Laboratoire d’Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS, 14 Av. Edouard Belin, 31400 Toulouse, France    F. Rostagni Observatoire de la Côte d’Azur, Université Nice Sophia-Antipolis, UMR 6525, Parc Valrose, 06108 Nice Cedex 02, France    H.  Trinquet Observatoire de la Côte d’Azur, Université Nice Sophia-Antipolis, UMR 6525, Parc Valrose, 06108 Nice Cedex 02, France    M. Auvergne LESIA, Université Pierre et Marie Curie, Université Denis Diderot, Observatoire de Paris, 92195 Meudon Cedex, France    A. Baglin LESIA, Université Pierre et Marie Curie, Université Denis Diderot, Observatoire de Paris, 92195 Meudon Cedex, France    L.M. Sarro Dpt. de Inteligencia Artificial, UNED, Juan del Rosal 16, 28040 Madrid, Spain    W.W. Weiss Institute of Astronomy, University of Vienna, Türkenschanzstrasse 17, A-1180 Vienna, Austria
Received, accepted
Key Words.:
Stars: variables: RR Lyrae - stars: oscillations - stars: interiors - stars: individual: CoRoT 101128793  - techniques: photometric
Abstract

Context: The CoRoT (Convection, Rotation and planetary Transits) space mission provides a valuable opportunity to monitor stars with uninterrupted time sampling for up to 150 days at a time. The study of RR Lyrae stars, performed in the framework of the Additional Programmes belonging to the exoplanetary field, will particularly benefit from such dense, long-duration monitoring.

Aims: The Blazhko effect in RR Lyrae stars is a long-standing, unsolved problem of stellar astrophysics. We used the CoRoT data of the new RR Lyrae variable CoRoT 101128793  (=2.119 d, =0.4719296 d) to provide us with more detailed observational facts to understand the physical process behind the phenomenon.

Methods:The CoRoT data were corrected for one jump and the long-term drift. We applied different period-finding techniques to the corrected timeseries to investigate amplitude and phase modulation. We detected 79 frequencies in the light curve of CoRoT 101128793. They have been identified as the main frequency   and its harmonics, two independent terms, the terms related to the Blazhko frequency , and several combination terms.

Results:A Blazhko frequency =0.056 d and a triplet structure around the fundamental radial mode and harmonics were detected, as well as a long-term variability of the Blazhko modulation. Indeed, the amplitude of the main oscillation is decreasing along the CoRoT survey. The Blazhko modulation is one of the smallest observed in RR Lyrae stars. Moreover, the additional modes =3.630  and =3.159 d are detected. Taking its ratio with the fundamental radial mode into account, the term   could be the identified as the second radial overtone. Detecting of these modes in horizontal branch stars is a new result obtained by CoRoT.

Conclusions:

1 Introduction

The pulsation of RR Lyrae stars is paramount for advancing in several fields of stellar physics. Marconi (2009) emphasizes how we can reproduce all the relevant observables of the radial pulsation including only non-local, time-dependent treatment of the convection in nonlinear models. In particular, pulsational models are able to reproduce the correlation between the periods and the absolute magnitudes in the near infrared bands (Bono et al. 2003). The model-fitting technique (Marconi 2009) applied to a sample of RR Lyrae stars in the Large Magellanic Cloud was very useful to fix the problem of the distance scale (Marconi & Clementini 2005). Because they have been observed since the end of the XIXth century, RR Lyrae stars are also promising targets for studying stellar evolution in real time (Le Borgne et al. 2007).

What has not yet been understood in RR Lyrae stars is the Blazhko effect, a periodic modulation of both the amplitudes and phases of the main pulsational mode. Different mechanisms have been proposed to explain the phenomenon: the resonance model between nonradial modes of low degree and the main radial mode (Dziembowski & Mizerski 2004), the oblique pulsator model in which the rotational axis does not coincide with the magnetic axis (Kurtz 1982; Shibahashi 2000), and the action of a turbulent convective dynamo in the lower envelope of the star (Stothers 2006). Kovács (2009) reviews these models and points out why we cannot definitely accept any of these explanations. It seems well–established that Blazhko RR Lyrae stars do not show any strong magnetic field (Chadid et al. 2004; Kolenberg & Bagnulo 2009). The observation of Blazhko RR Lyrae stars was performed with remarkable success by means of extensive ground–based surveys. Well–defined findings (e.g., changes in the Blazhko period, modulation features, systematic changes in the global mean physical parameters, high–order multiplets, long–term changes) have recently been obtained on RR Lyr itself (Kolenberg et al. 2006), MW Lyr (Jurcsik et al. 2008), XZ Cyg (LaCluyzé et al. 2004), RR Gem (Jurcsik et al. 2005; Sódor et al. 2007), and DM Cyg (Jurcsik et al. 2009a).

The Additional Programmes in the exoplanetary science case of the CoRoT (COnvection, ROtation and planetary Transits; Baglin et al. 2006) space mission were focused on specific classes of stars with the aim of supplying a new and powerful tool for deciphering the physical reasons for their variability (Weiss 2006). RR Lyrae stars are being studied in the framework of the international RR Lyrae-CoRoTeam111The dedicated website is http://fizeau.unice.fr/corot.. Preliminary results were presented by Chadid et al. (2009), and the potential of the 150–day continuous monitoring of an RR Lyrae star has been demonstrated by the case of V1127 Aql (Chadid et al. 2010), not previously known as a Blazhko variable. Very high–order modulation sidepeaks were detected, up to the sepdecaplet structure. Additional modes have also been detected and interpreted as nonradial modes or secondary modulation. As the Blazhko effect remains misunderstood in its physical nature, we can look at the CoRoT data as a new opportunity for providing the observational facts we need to shed new light on it.

Figure 1: Top panel: the light curve obtained by removing the main frequency and its harmonics from the original data showing a long-term drift and a jump. Middle panel: the light curve of the residuals is corrected from the long-term drift and the jump. Bottom panel: the final light curve of CoRoT 101128793  is an example of the continuous, excellent quality monitoring of stars in the CoRoT exoplanetary field.

2 The CoRoT data

CoRoT 101128793USNOA2 0900-15089357 ( = , = +01° 13′ 3505, J2000) is a 16–mag star (=15.93, =+0.89, Deleuil et al. 2009) in the constellation of Aquila. Its variability was discovered during the first Long Run in the centre direction (LRc01), carried out continuously from May 15 to October 14, 2007, i.e., for 142 d. There is no relevant contamination from nearby stars, since the brightest star included in the CoRoT mask is 3.0 mag fainter than CoRoT 101128793  in light (Deleuil et al. 2009). The exposure time in the CoRoT exoplanetary channel was 512 sec and this time remained constant all over the LRc01. Thanks to its very high duty cycle, CoRoT collected 23922 data points, and the spectral window is free from any relevant alias structure. The star was classified as an RR Lyrae variable by the “CoRoT Variability Classifier” automated supervised method (Debosscher et al. 2009) and then confirmed by human inspection of the light curve. CoRoT 101128793, located close to the direction of the galactic centre, is therefore heavily reddened.

The absolute CoRoT photometry is affected by jumps, outliers, and a long-term drift. It is very hard to detect jumps in the original data of an RR Lyrae variable, since they have a small amplitude (few 0.01-mag) and are not discernible in a light curve having an amplitude of several tenths of a magnitude. As a matter of fact, we could detect a jump of 0.032 mag at JD 2454369.7 only a posteriori, after having performed the preliminary frequency analysis of the original data. Indeed, only the residuals obtained by subtracting the main frequency   and its harmonics from the original CoRoT data clearly show the jump. We re-aligned the whole dataset after removing the few corrupted measurements on the jump (Fig. 1, top and middle panels).

In addition to the jump, some oscillations and a continuous drift are clearly visible in the top panel of Fig. 1. The oscillations have a stellar origin (see Sect. 3.2), but the drift is an instrumental effect (Auvergne et al. 2009), so it should be removed before performing the frequency analysis. Different detrending algorithms can be used, based on moving means or polynomial fits. After several trials, we removed the drift by calculating the mean magnitudes of the least–squares fits of four consecutive cycles (i.e., 1.88 d). The main frequency and its harmonics were used, as in the previous step. At that point, the value of the mean magnitude was interpolated at the time of each observation and then subtracted from the original data. During this analysis we also removed the most obvious outliers. The final CoRoT timeseries is available in electronic form at the CDS. The re-aligned, de-jumped light curve disclosed the multiperiodic behaviour of CoRoT 101128793: continuous oscillations are clearly visible in the light curve prewhitened with   and harmonics (Fig. 1, middle panel) and in the light curve of the original data (a portion is shown in Fig. 1, bottom panel).

The subsequent frequency analysis was performed by using different packages such as Period04 (Lenz & Breger 2005), MuFrAn (Kolláth 1990), and the iterative sine–wave fitting (Vaniĉek 1971). The different algorithms led to the same results with only marginal differences at higher orders. We present here the results of the iterative–sine wave fitting, with a complementary frequency refinement obtained by means of the MTRAP algorithm (Carpino et al. 1987).

The realigned dataset was first analysed to search for the effects of the orbital frequency. Several frequencies were found at the orbital frequency =13.97 d and harmonics. Moreover, the term =1.0027 d was found. This perturbation comes from the passage of the satellite over the South Atlantic Anomaly (SAA). Since it occurs twice a day, the harmonic 2 is much stronger than , which corresponds to the passage of the satellite over the SAA on the same side of the Earth with respect to the Sun. The effects of these passages on the onboard instrumentation are described by Auvergne et al. (2009). They originate frequencies at

(1)

with and . The strongest terms are 27.94 and 41.91 d, i.e., and (3,0), respectively. The usually adopted technique of prewhitening the input data with the frequencies did not correct for the instrumental effects in a satisfactory way. The orientation of the CoRoT orbital plane with respect to the Earth–Sun line continuously changed over the course of the long run. Therefore, the environmental conditions (e.g., the eclipse effects on the electronics units, the eclipse durations, the difference in the Earth’s albedo of the overflown regions; see Sect. 3 in Auvergne et al. 2009) are affecting the CoRoT photometry in a complicated way.

Figure 2: Subsequent steps in the detection of frequencies in the amplitude spectra of CoRoT 101128793.

3 The frequency content

By using the packages previously mentioned, we identified 79 components of stellar origin, in addition to the frequencies and to the spurious peaks at very low frequencies, i.e., residuals of the long–term drift of the sensitivity drift of the CCDs. They can be divided into four categories:

  1. the main frequency   and its harmonics;

  2. the terms related to the Blazhko frequency ;

  3. other independent terms;

  4. the combination terms.

Figure 2 describes the different steps in the frequency detection. The spectrum in the top panel brings out the main frequency =2.119 d and its harmonics. The spectral window (inserted box) is almost free of aliases, and the peaks located at and 2 are too low to produce any significant effect. When   and harmonics are removed, the couple of sidepeaks (, with =0.056 d) due to the Blazhko effect becomes the most prominent structure (second panel, the zoom around   and 2  is shown in the inserted box).

The most intriguing peaks stand out in the region  d after subtracting the Blazhko sidepeaks (third panel and inserted box). The highest peaks in the third panel of Fig. 2 are at =3.157 d and =3.630 d. They show linear combination with   and harmonics and are therefore intrinsic to the RR Lyrae star. They provide evidence of excited modes other than the fundamental radial mode .

The residual spectrum does not show any other structure, except an excess of signal still centred on the largest amplitude modes and on the orbital frequencies of the satellite (Fig. 2, bottom panel). After removing the 79 frequencies, the average noise level resulted in  mag in the 0-100 d region of the residual spectrum (inserted box in Fig. 2, bottom panel). The lowest detected amplitude among the 79 frequencies led to 0.36 mmag, i.e., 5 times the level of the overall final noise. We note that at each step of the process in frequency detection we calculated the local noise centred on the detected peak, and we always got SNR3.5. This threshold was retained to accept a combination term, while independent terms have much higher SNR (17.45 and 9.25 for   and , respectively).

The final solution of the CoRoT light curve was calculated by means of a cosine series (T=2454308.2168) and their least–squares parameters, together with the local SNRs, are listed in Table 1. The listed values of the frequencies are corresponding to the highest peaks in the amplitude spectrum. The values calculated from the four independent frequencies and the identification listed in the last column of Table 1 (the so-called locked solution, obtained by using the MTRAP algorithm, Carpino et al. 1987) are generally in excellent agreement (=2.118977, =3.630499, =3.156776, and =0.00550 d). The observed discrepancies are probably due to the non–equidistance of the triplet structures and to other terms hidden in the residual noise. As an example, a third independent frequency is probably present close to 3.00 d, but we cannot identify it unambiguously. If this term were real, then some combination terms should be changed by substituting, e.g., 2 with 2. The solution with all independent frequencies gives the same residual rms of the solution with the locked frequencies (0.01006 and 0.01100 mag, respectively). These values are mostly affected by the residual peaks described above.

3.1 The main   term and its harmonics

Figure 3: Upper panel: The CoRoT data folded with , original (top) and after subtracting the 79 frequencies (bottom). Left panel, lower row: The amplitudes of the   terms. Right panel, lower row: The observed triplet structure around   terms. The filled circles indicate the component with the greatest amplitude in the triplet, the empty circles the second in amplitude, the star the third.

The light curve on the   term is very asymmetric (Fig. 3, upper curve in the top panel) and harmonics up to 13  are significant. Their amplitudes are not monotonically decreasing: the amplitude of 6  is larger than for 5, and that of 11  is larger than for 10, just before the final decline (Fig. 3, left panel in the bottom row). Indeed, the light curve of CoRoT 101128793  shows a couple of particularities, i.e., the bump near the minimum often observed in RRab stars and a change in slope on the rising branch. They are not very pronounced, but still discernible in the light curve (Fig. 1, bottom panel). The fit of these small particularities requires a more relevant contribution from the highest harmonics than in the case of smooth light curves. Moreover, the change in slope does not repeat in a regular way, since the plot of the residuals (Fig. 3, lower curve in the top panel) shows a wide spread in this phase interval. The non–white distribution of the photometric residuals is the cause of the small bunches of frequencies observed in the residual spectrum. The Blazhko variables RR Gem and DM Cyg show the same light curve shape and the same residual distribution as CoRoT 101128793  (Jurcsik et al. 2005, 2009a).

The measurements around the maximum and minimum brightnesses were fitted by means of a least-squares polynomial. We obtained the ephemeris


when fitting the times of maxima (Table 2) by means of a least–squares line. The O-C values (differences between the observed and calculated times of maxima) were determined by using this ephemeris.

3.2 The Blazhko frequency

As suggested by the residuals after subtracting the main oscillation (Fig. 1, middle panel), there is a periodic change in the shape of the light curve, and this change defines the Blazhko effect. The Blazhko effect translates into symmetric sidepeaks of   and its harmonics in the frequency domain (second panel in Fig. 2). In the case of CoRoT 101128793, the sidepeaks are    triplets (Fig. 3, right panel in the bottom row).

We obtained an independent confirmation of the Blazhko frequency from the magnitudes at the maximum brightness (see above) and from the application of the analytic signal method (Kolláth et al. 2002). The magnitudes at maximum oscillate in a peak-to-peak interval of 0.06 mag (Fig. 4, top panel): the power spectrum unambiguously identifies =0.056 d (Fig. 4, bottom panel). The instantaneous amplitudes and frequencies also vary with   (Fig. 4, middle panel). The period variations, and consequently the O-C range, are very small. As a matter of fact, CoRoT 101128793  shows the smallest period variation among the CoRoT RRab stars (see Fig. 2 in Szabó et al. 2009). Since the Blazhko effect is more evident in amplitude than in phase, the cycle–to–cycle variations in the light curve are undetectable when folding the data over , also considering the perfect coverage in phase ensured by the CoRoT observations. The Blazhko effect just causes a wider spread of the points, while the curve apparently remains very regular (Fig. 3, top panel). The Blazhko effect of CoRoT 101128793  seems to be particular since the harmonic 2  has an amplitude greater than   (Table 1 and middle panel of Fig. 1). This particularity could reflect the different forms in which the Blazhko effect can occur (see Table 1 in Szabó et al. 2009). However, we should also consider that the frequency and amplitude values of   and 2  could be affected by the slightly different separations between the sidepeaks of the triplets (Table 1) and by the correction of the low–frequency drift.

We observe a peak close to zero (Fig. 4, bottom panel) in the power spectrum of the magnitudes at maximum. This suggests that there is a very long–term variation, on a timescale longer than covered by the CoRoT data. The complicated behaviour of the light curve is made clear by the comparison between maxima and minima (Fig. 4, upper panel). The range in the magnitudes at minimum is about half that at the maximum. Also the amplitudes of the   component (calculated both as instantaneous values and by subdividing the timeseries in pulsational cycles) show the decreasing trend underlying the Blazhko periodicity (Fig. 4, middle panel).

The long–term change is a further complication of the frequency analysis (Benkő et al. 2009; Szeidl & Jurcsik 2009). Together with the change in slope on the rising branch, it causes the peaks around   and its harmonics in the residual spectrum (Fig. 2, bottom panel).

Figure 4: Evidence of the long–term variations in the CoRoT 101128793  light curve. Top panel: The magnitudes at maximum (upper curve) and minimum (lower curve) brightness in the different cycles. Middle panel: The amplitudes of the   component (upper curve) and the period values (lower curve). Reference line is displayed to show the long term variation in amplitude. Bottom panel: The power spectrum of the magnitudes at maximum brightness.

3.3 The independent terms

3.3.1 =3.630 d

The first peak not related to   and   is found at =3.630 d. The light curve related to this periodicity is slightly asymmetrical, since we found a small-amplitude first harmonic 2. It also shows several combination terms with    and . However,   is not affected by the Blazhko effect, since we did not detect terms of the form .

The ratio /=0.584 is very close to what is expected between the fundamental radial mode and the second overtone. To verify this possibility from a theoretical point of view, we computed linear RR Lyrae model grids on an extremely large parameter space (=40, 50, 60 and 70 L, =0.50–0.80 M with M=0.05 M, =5000–8000 K, =100 K, 0.001, 0.003, 0.01, 0.02 and 0.04). The other adopted parameters were standard RR Lyrae parameters (see Szabó et al. 2004). Nonlinearity introduces a negligible difference in the periods and period ratios. The Petersen diagram for different metallicities is shown in Fig. 5. The period ratio is fully compatible with an identification of   as the second radial overtone. In such a case, the models suggest a -metallicity of 0.002-0.004. Assuming a ratio of 0.74 between fundamental and first overtone radial modes, the latter should be around 2.863 d, but the frequency spectrum does not show any significant peak at this value.

Figure 5: Petersen diagram based on linear convective RR Lyrae models. The symbols denote different metallicities. The black square shows the position of CoRoT 101128793, assuming that the frequency is the second overtone radial mode.

3.3.2 =3.159 d

The amplitude of =3.159 d  is only a bit smaller than that of   (0.0021 and 0.0028 mag, respectively) and at the same level of that of the 8  harmonic. The ratio /=1.49083/2 could be the signature of the period doubling bifurcation (Moskalik & Buchler 1990) first noticed in some RR Lyrae stars observed with Kepler (Kolenberg et al. 2010).

Another characteristic of   is to be flanked by a Blazhko frequency at . This occurrence can have different explanations : i) the Blazhko variability also affects ; ii) it is a coincidence, and   and   are actually two independent modes; iii)   is a mere combination term, such as the difference between   and . We can try to disentangle the matter by examining the three possibilities. If   shows the Blazhko effect, it is strange that we do not detect   since we expect the sum term to have an amplitude greater than the difference term (see Fig. 3, right panel in the bottom row). The coincidence is also improbable, since the frequency spectrum is not very rich. The resonance mechanism is possible, but that it involves the Blazhko term makes it a very particular case. Finally, combination terms with  , , and   are detected. In particular, we found unusual combinations, such as ++  and +++, and they display a good SNR (5.4 and 4.6, respectively). Therefore, the hypothesis of a particular combination term seems the most plausible.

4 Discussion

The continuous, long monitoring offered by space photometry is a new observational tool to understand the pulsational behaviour of RR Lyrae stars. Through such data, cycle–to–cycle variations can be clearly pointed out. Indeed, the Blazhko modulation of CoRoT 101128793  is one of the smallest ever observed in RR Lyrae stars (Jurcsik et al. 2009b).

4.1 The Blazhko effect

CoRoT data already demonstrated that the Blazhko cycle of V1127 Aql is changing on a timescale of 143 d: the shift is much more evident in time than in magnitude (see Fig. 14 in Chadid et al. 2010). The Blazhko effect of CoRoT 101128793  is much smaller than that of V1127 Aql (0.06 mag vs. 0.35 mag in the full range of magnitudes at maximum, 0.02 p vs. 0.17 p in the phases of maximum).

Notwithstanding this small effect, the trend observed in the magnitudes at maximum and at minimum (Fig. 4, upper panel) supports a long–term change. The best observational evidence for a long–term change in the Blazhko period is given by RR Lyr itself. Ground–based photometry collected on several decades shows a decrease from 40.8 d to 38.8 d (Kolenberg et al. 2006). The modulation amplitude of RR Gem was also subjected to strong variations from the undetectable level (less than 0.04 mag in maximum brightness) to about 0.20 mag on a time baseline of 70 years (Sódor et al. 2007). In the case of the Blazhko effect of MW Lyrae (=0.060 d), Jurcsik et al. (2008) put in evidence secondary peaks around the main pulsation terms separated by a periodicity comparable with the time baseline, tentatively 500 d. Therefore, it seems that long–term changes are occurring in Blazhko RR Lyrae variables, and they can be detected when monitored in an intensive and/or continuous way.

4.2 The excitation of additional modes

The case of CoRoT 101128793  supplies new evidence of the excitation of additional modes in RR Lyrae stars. The two frequencies   and   are not related with the Blazhko or another modulation, as it could be for V1127 Aql (Chadid et al. 2010). The frequency   could be typified as the second overtone radial mode, while the nature of   is still unclear. We immediately note that also in the case of V1127 Aql we found frequency ratios compatible with that between fundamental and second overtone radial modes, i.e., 2.8090/4.8254=0.582, and with the possible period doubling bifurcation, i.e., 4.1916/2.8090=1.492. Moreover, the frequency values   and   are in the same interval of the nine additional modes detected in the frequency spectrum of V1127 Aql (3.64–4.82 d).

It is interesting to revisit the results obtained by Jurcsik et al. (2008) on MW Lyr. Those authors identified four frequencies in the 3.27–6.78 d interval (3.2701, 4.2738, 5.7847, and 6.7885 d)222We note that there is a 1 d spacing between the terms of the first couple and between those of the second one. This spacing is always a bit suspicious for observations collected mostly from one site, but we consider the frequency detection performed by Jurcsik et al. (2008) as a well-established one. as combination terms having the form (where =2.5146 d is the main pulsation mode and =0.0604 d the Blazhko frequency). Since the 12.5  spacing remains unexplained, we propose an alternative solution based on the additional modes =3.2701 d and =4.2738 d and the combination terms =5.7847 dand =6.7884 d. We find for the third time a frequency ratio (=0.588) that could be explained with the ratio between the fundamental and the second overtone radial modes.

These mode identifications are a new contribution to the debate on the excitation of the second overtone in RR Lyr stars (e.g., Alcock et al. 1996; Walker & Nemec 1996; Kaluzny et al. 2000; Soszyński et al. 2003; Olech & Moskalik 2009). We also note that the excitation of non-consecutive radial modes would be a new result for RR Lyr stars, so far sporadically observed in Cepheids (Soszyński et al. 2008).

5 Conclusions

The second detailed analysis of the CoRoT data on RR Lyrae variables allowed us to advance in the definition of their pulsational characteristics. It is confirmed that the Blazhko effect can span different ranges in the variations, both absolute and relative, of the amplitude and phase modulations. Moreover, there is new evidence that the Blazhko period is subjected to long–term variations, as can be directly detected from the consecutive cycles observed in the CoRoT LRc01. The mechanisms invoked to explain the Blazhko effect should reproduce “the strictly regular behavior of the modulation observed in many Blazhko stars” (Jurcsik 2009). This requirement should now be reconsidered in a slightly different way. The real mechanism must be able to reproduce both the regular structure of the sidepeaks in the frequency spectra and the observed variability on a long–term scale. In this context, it can be useful to stress that CoRoT 101128793, similar to DM Cyg (Jurcsik et al. 2009a) and RR Gem (Jurcsik et al. 2005), shows a bump on the rising branch of the light curve. These bumps are probably connected with hypersonic shock waves (Chadid et al. 2008), and the spreading of the residuals suggests a link between pulsation and atmosphere dynamics. More precisely, this could be the clue to an interaction between the Blazhko phenomenon and the atmosphere’s dynamics (Guggenberger & Kolenberg 2006), since monoperiodic RR Lyrae stars also have very regular light curves in the presence of this bump (Poretti 2001).

The other relevant result disclosed by the CoRoT data is the excitation of additional modes. A reanalysis of the V1127 Aql and MW Lyr cases seems to indicate that there is a narrow frequency interval where a few modes are excited. The recurrence of the ratio 0.58–0.59 between one of these modes and the fundamental radial mode suggests the possibility of the (preferred) excitation of the second overtone. The possible interplay between this type of double–mode pulsation and the Blazhko effect deserves further theoretical investigation. The Blazhko effect does not modulate the   term and this is particularly relevant in this scenario. Moreover, CoRoT detected other very significant peaks in the oscillation spectra of V1127 Aql and CoRoT 101128793, thus disclosing the evidence that nonradial modes are excited in horizontal branch stars. The theoretical prediction of these modes is the new challenge to the pulsation models of RR Lyrae stars launched by CoRoT.

Acknowledgements.
This research has made use of the Exo-Dat database, operated at LAM-OAMP, Marseille, France, on behalf of the CoRoT/Exoplanet programme. MC thanks F. Baudin and J. Debosscher for their help on the data reduction. JMB, MP, and RSz acknowledge the support of the ESA PECS projects No. 98022 & 98114. KK and EG acknowledge the projects FWF T359 and FWF P19962, and EP the Italian ESS project (contract ASI/INAF/I/015/07/0, WP 03170) for financial support.

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Frequency Amplitude Phase SNR ID
[mag] [0,2]
Main frequency and harmonics
2.1189511 0.235160 0.2021 83.170
4.2379699 0.083192 4.4638 81.173 2
6.3567948 0.025942 2.5909 69.957 3
8.4750872 0.003859 0.9586 25.783 4
10.5955076 0.004677 2.0603 26.972 5
12.7142172 0.005179 0.1356 32.102 6
14.8329821 0.003394 4.4305 24.368 7
16.9521809 0.002134 2.1695 16.597 8
19.0711250 0.000924 0.0034 9.907 9
21.1895828 0.000362 2.6349 5.089 10
23.3090458 0.000650 0.1582 8.902 11
25.4280815 0.000657 4.1858 9.371 12
27.5470715 0.000427 2.1154 5.925 13
Blazhko modulation
0.0557860 0.002250 3.5242 11.027
0.1130970 0.005032 2.0600 21.865 2
Modulation triplet frequencies
2.0706120 0.001003 4.3047 6.254
2.1719539 0.004580 5.3412 17.976
4.1836190 0.000864 4.5330 5.714 2
4.2913609 0.004452 2.8623 22.069 2
6.3063798 0.000914 1.7760 7.603 3
6.4105959 0.003799 1.2182 19.970 3
8.4165525 0.000603 2.4485 5.293 4
8.5297117 0.002889 6.1076 18.665 4
10.5445929 0.000403 4.8041 3.957 5
10.6492252 0.001437 4.1638 10.704 5
12.7694759 0.000618 2.3372 6.028 6
17.0050011 0.000589 2.2322 7.826 8
19.1245899 0.000586 0.3344 6.950 9
21.2440968 0.000399 4.5191 5.034 10
Additional terms identified as independent modes
3.1588659 0.002119 2.9665 9.251
3.6308789 0.002844 1.0416 17.450
Linear combinations between additional terms and ,
with or without
0.5557140 0.000579 1.4162 3.590
0.6137300 0.000967 4.3577 5.458
0.9815810 0.000673 5.9134 4.446
1.0310810 0.000907 4.4587 5.680
1.0892100 0.000625 0.2036 4.373
1.1364660 0.000529 0.0723 4.307
1.5060490 0.001430 3.0864 9.914
2.7380049 0.000817 0.5867 4.017
3.0997059 0.001693 2.0588 7.816
3.2565880 0.000691 5.0736 4.663
4.8528800 0.000808 4.6690 5.310
5.1331830 0.000699 1.0265 4.088
5.2227440 0.000559 0.6052 4.755
5.2794271 0.000596 6.1065 4.413
5.3320541 0.000404 0.1849 3.706
5.3743072 0.000564 2.6932 5.144
5.6918921 0.000502 1.0524 3.988
5.7496319 0.002907 5.7320 18.811
6.9680071 0.000622 3.9101 4.517
7.2510352 0.000450 3.3737 3.567
7.3422809 0.000408 6.1252 3.540
7.3839550 0.000509 2.9947 4.455
7.3946028 0.001058 5.1992 8.021
7.8696060 0.001498 3.8452 10.900
8.9037333 0.000488 1.7403 5.396
8.9577160 0.000431 3.1197 4.556
9.0911274 0.000523 1.7513 4.075
9.3717070 0.000840 2.8583 5.426
9.4592342 0.000536 4.5554 4.219
9.5042696 0.000520 1.3052 4.413
9.5122747 0.001021 3.4523 7.300
9.9870539 0.002025 2.1231 14.298
10.0394773 0.000660 4.7210 5.748
11.4905624 0.000896 0.9388 6.940
11.5774755 0.000494 2.9245 4.427
11.6231632 0.000689 6.1547 6.080
11.6310463 0.000901 1.6964 7.590
12.1040316 0.001536 0.2982 12.757
12.1594105 0.000564 3.3520 5.916
13.6098433 0.000621 5.4812 5.738
13.7422590 0.000518 4.7605 5.055
13.7495470 0.000543 5.8919 5.436
14.2228832 0.001424 4.3996 12.419
14.2800894 0.000424 1.1744 5.263
15.8649635 0.000590 3.1909 6.673
16.3416233 0.001096 2.4098 11.503
18.4694328 0.000364 2.2220 4.735
20.5785885 0.000401 5.4055 5.082
Table 1: continued.
Cycle Times Magnitude O–C Cycle Times Magnitude O–C
[E] of maximum [d] [E] of maximum [d]
[HJD-2450000] [HJD-2450000]
1 4237.1460 –0.3481 152 4308.4114 –0.3176
2 4237.6211 –0.3223 153 4308.8767 –0.3193
3 4238.0891 –0.3212 154 4309.3486 –0.3275
4 4238.5593 –0.3433 155 4309.8227 –0.3256
5 4239.0325 –0.3235 156 4310.2949 –0.3154
6 4239.5027 –0.3198 157 4310.7649 –0.3407
7 4239.9727 –0.3320 158 4311.2380 –0.3426
8 4240.4509 –0.3178 159 4311.7131 –0.3095
9 4240.9238 –0.2999 160 4312.1831 –0.3143
10 4241.3940 –0.3041 161 4312.6543 –0.3347
11 4241.8623 –0.3200 162 4313.1284 –0.3271
12 4242.3445 –0.2937 163 4313.5996 –0.3015
13 4242.8106 –0.2937 164 4314.0696 –0.3206
14 4243.2808 –0.2988 165 4314.5408 –0.3260
15 4243.7578 –0.2943 166 4315.0198 –0.3010
16 4244.2319 –0.2883 167 4315.4849 –0.3081
17 4244.6968 –0.2926 168 4315.9570 –0.3149
18 4245.1719 –0.3093 169 4316.4329 –0.3228
19 4245.6462 –0.2897 170 4316.9050 –0.3029
20 4246.1162 –0.2878 171 4317.3694 –0.3174
21 4246.5864 –0.3196 172 4317.8484 –0.3234
22 4247.0586 –0.3165 173 4318.3193 –0.3075
23 4247.5308 –0.2958 174 4318.7915 –0.2968
24 4248.0059 –0.3138 175 4319.2605 –0.3319
25 4248.4749 –0.3327 176 4319.7356 –0.3056
26 4248.9490 –0.3107 177 4320.2078 –0.2931
27 4249.4204 –0.3115 178 4320.6777 –0.3057
28 4249.8914 –0.3254 179 4321.1499 –0.3289
29 4250.3613 –0.3306 180 4321.6282 –0.2922
30 4250.8367 –0.3198 181 4322.0920 –0.3153
31 4251.3047 –0.3351 182 4322.5652 –0.3069
32 4251.7788 –0.3297 183 4323.0403 –0.3153
33 4252.2522 –0.3181 184 4323.5112 –0.3005
34 4252.7212 –0.3187 185 4323.9815 –0.3187
35 4253.1924 –0.3417 186 4324.4536 –0.3198
36 4253.6655 –0.3186 187 4324.9316 –0.3000
37 4254.1375 –0.3269 188 4325.3948 –0.3226
38 4254.6047 –0.3417 189 4325.8718 –0.3295
39 4255.0808 –0.3360 190 4326.3428 –0.3055
40 4255.5510 –0.3204 191 4326.8140 –0.3040
41 4256.0230 –0.3243 192 4327.2830 –0.3466
42 4256.4919 –0.3385 193 4327.7571 –0.3118
43 4256.9724 –0.3264 194 4328.2302 –0.3117
44 4257.4365 –0.3199 195 4328.6973 –0.3270
45 4257.9136 –0.3298 196 4329.1724 –0.3345
46 4258.3826 –0.3174 197 4329.6475 –0.3096
47 4258.8530 –0.3175 198 4330.1125 –0.3209
48 4259.3240 –0.3164 199 4330.5867 –0.3393
49 4259.8010 –0.3153 200 4331.0588 –0.3096
50 4260.2703 –0.3012 201 4331.5337 –0.2995
51 4260.7444 –0.3005 202 4332.0019 –0.3159
52 4261.2146 –0.3120 203 4332.4749 –0.3266
53 4261.6899 –0.2990 204 4332.9490 –0.2930
54 4262.1587 –0.2893 205 4333.4202 –0.3034
55 4262.6318 –0.3043 206 4333.8892 –0.3261
56 4263.1050 –0.3056 207 4334.3662 –0.3104
57 4263.5791 –0.3029 208 4334.8352 –0.2882
58 4264.0471 –0.3028 209 4335.3103 –0.3210
59 4264.5215 –0.3199 210 4335.7776 –0.3119
60 4264.9954 –0.3087 211 4336.2585 –0.2975
61 4265.4658 –0.3106 212 4336.7236 –0.3023
62 4265.9358 –0.3246 213 4337.1985 –0.2960
63 4266.4131 –0.3258 214 4337.6697 –0.3065
64 4266.8801 –0.3175 215 4338.1409 –0.3007
65 4267.3501 –0.3299 216 4338.6108 –0.3005
66 4267.8262 –0.3299 217 4339.0862 –0.2863
67 4268.2986 –0.3249 218 4339.5559 –0.3066
68 4268.7656 –0.3077 219 4340.0310 –0.2875
69 4269.2427 –0.3302 220 4340.5002 –0.3037
70 4269.7126 –0.3390 221 4340.9712 –0.2917
71 4270.1858 –0.3101 222 4341.4473 –0.2889
72 4270.6531 –0.3371 223 4341.9175 –0.3070
73 4271.1282 –0.3407 224 4342.3852 –0.3147
74 4271.6013 –0.3368 225 4342.8628 –0.2995
75 4272.0684 –0.3376 226 4343.3315 –0.3275
76 4272.5435 –0.3503 227 4343.8008 –0.3282
77 4273.0137 –0.3427 228 4344.2766 –0.3247
78 4273.4856 –0.3354 229 4344.7488 –0.3029
79 4273.9539 –0.3491 230 4345.2161 –0.3364
80 4274.4290 –0.3353 231 4345.6919 –0.3239
81 4274.9001 –0.3355 232 4346.1650 –0.3162
82 4275.3665 –0.3333 233 4346.6331 –0.3149
83 4275.8435 –0.3415 234 4347.1062 –0.3253
84 4276.3203 –0.3182 235 4347.5784 –0.3026
85 4276.7856 –0.3190 236 4348.0503 –0.3075
86 4277.2598 –0.3150 237 4348.5215 –0.3087
87 4277.7339 –0.3124 238 4348.9944 –0.3202
88 4278.2031 –0.3145 239 4349.4675 –0.2917
89 4278.6721 –0.3176 240 4349.9358 –0.3186
90 4279.1523 –0.3063 241 4350.4087 –0.2909
91 4279.6213 –0.3028 242 4350.8818 –0.3119
92 4280.0874 –0.3091 243 4351.3518 –0.2884
93 4280.5644 –0.3216 244 4351.8259 –0.3056
94 4281.0405 –0.3044 245 4352.2949 –0.3014
95 4281.5068 –0.3080 246 4352.7739 –0.3039
96 4281.9761 –0.3297 247 4353.2380 –0.2996
97 4282.4570 –0.3057 248 4353.7141 –0.3066
98 4282.9250 –0.3211 249 4354.1863 –0.2979
99 4283.3914 –0.3128 250 4354.6611 –0.2951
100 4283.8684 –0.3320 251 4355.1282 –0.2945
101 4284.3433 –0.3093 252 4355.6055 –0.2942
102 4284.8106 –0.3158 253 4356.0745 –0.2756
103 4285.2837 –0.3251 254 4356.5464 –0.2858
104 4285.7590 –0.3222 255 4357.0156 –0.2945
105 4286.2270 –0.3157 256 4357.4951 –0.2857
106 4286.6951 –0.3195 257 4357.9607 –0.2790
107 4287.1699 –0.3237 258 4358.4358 –0.2837
108 4287.6453 –0.3123 259 4358.9048 –0.2916
109 4288.1155 –0.3056 260 4359.3784 –0.3010
110 4288.5845 –0.3387 261 4359.8520 –0.2877
111 4289.0606 –0.3156 262 4360.3210 –0.3030
112 4289.5298 –0.3262 263 4360.7952 –0.3060
113 4290.0010 –0.3242 264 4361.2688 –0.3061
114 4290.4749 –0.3425 265 4361.7363 –0.3104
115 4290.9470 –0.3333 266 4362.2102 –0.3195
116 4291.4163 –0.3356 267 4362.6824 –0.3210
117 4291.8882 –0.3453 268 4363.1555 –0.3195
118 4292.3633 –0.3340 269 4363.6216 –0.3224
119 4292.8335 –0.3353 270 4364.0984 –0.3280
120 4293.3005 –0.3438 271 4364.5676 –0.3353
121 4293.7817 –0.3115 272 4365.0418 –0.3224
122 4294.2498 –0.3288 273 4365.5088 –0.3331
123 4294.7207 –0.3229 274 4365.9849 –0.3106
124 4295.1931 –0.3282 275 4366.4539 –0.3248
125 4295.6692 –0.3057 276 4366.9280 –0.3071
126 4296.1331 –0.3142 277 4367.3980 –0.3096
127 4296.6074 –0.3194 278 4367.8711 –0.3047
128 4297.0815 –0.3070 279 4368.3411 –0.3370
129 4297.5554 –0.3018 280 4368.8142 –0.3108
130 4298.0207 –0.3223 281 4369.2864 –0.3170
131 4298.4968 –0.3127 282 4369.7617 –0.3135
132 4298.9736 –0.2974 283 4370.2273 –0.3188
133 4299.4380 –0.3152 284 4370.7043 –0.2918
134 4299.9111 –0.3286 285 4371.1753 –0.3181
135 4300.3899 –0.2974 286 4371.6414 –0.2981
136 4300.8528 –0.2922 287 4372.1194 –0.3154
137 4301.3232 –0.3135 288 4372.5916 –0.2918
138 4301.8042 –0.3152 289 4373.0647 –0.2853
139 4302.2751 –0.3037 290 4373.5308 –0.3080
140 4302.7415 –0.3143 291 4374.0107 –0.2919
141 4303.2144 –0.3106 292 4374.4810 –0.2765
142 4303.6917 –0.2957 293 4374.9490 –0.2911
143 4304.1616 –0.3063 294 4375.4228 –0.2970
144 4304.6328 –0.3113 295 4375.8992 –0.2899
145 4305.1047 –0.3109 296 4376.3660 –0.2824
146 4305.5779 –0.2979 297 4376.8420 –0.3036
147 4306.0440 –0.3199 298 4377.3123 –0.2884
148 4306.5190 –0.3092 299 4377.7832 –0.2861
149 4306.9951 –0.3014 300 4378.2534 –0.2970
150 4307.4644 –0.3171 301 4378.7258 –0.3097
151 4307.9333 –0.3326
Table 2: continued.
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