© ESO, 2013

1. Introduction

In the past ten years asteroseismology has gone through an unprecedented evolution. The launch of the MOST (Microvariablity and Oscillations of STars, Walker et al. 2003) mission in 2003 and the CoRoT (Convection Rotation and planetary Transits, Auvergne et al. 2009) satellite in 2006 meant the start and the culmination of the industrial revolution in asteroseismology. Shortly afterwards, the belle époque arrived with the launch of the Kepler (Koch et al. 2010) satellite in 2009. This immense growth in observational data provides ever better input and testbeds to compare with theoretical calculations of stellar structure and evolution.

Although the primary science goal of Kepler is to detect Earth-like exoplanets (Borucki et al. 2010), its virtually uninterrupted photometry also provides us with micromagnitude-precision light curves of tens of thousands of stars situated in the 105 square degree field of view (FOV) fixed between the constellations of Cygnus and Lyra – a real goldmine for asteroseismology, too (Gilliland et al. 2010). While the photometric precision is roughly the same as the one of CoRoT (despite the fainter apparent magnitude of the typical asteroseismology sample of the Kepler mission), the fixed FOV and target list provide a timebase of several years for continuous observations. Such a long timebase is needed to resolve rotationally split multiplets – which can hardly be achieved from CoRoT data with a time base of five months – and to measure frequencies with the required high precision. The maximum frequency resolution we can aim for is given by the Loumos & Deeming (1978) criterion of ~2.5/T (where T is the timespan of the observations). This yields a resolution of 0.002 d^-1 in the periodogram, assuming 3.5 years of observations (the originally planned lifespan of the now extended mission). This resolution of ~10^-3 d^-1 is necessary for securely detecting the rotational splitting of individual modes because of the merging of various multiplets in the typically dense frequency spectra.

OB stars – intermediate mass to massive stars – are important in the chemistry and evolution of the universe. They make a significant contribution to the enrichment of the interstellar matter with heavy elements, moreover, some of them in suitable binary systems can end their lives as Ia type supernovae, the standard candles upon which the study of the expansion of the universe is based.

The first overview on the variability of B-type stars observed by Kepler was given by Balona et al. (2011b). The authors analysed a selection of 48 main sequence B-type stars, and concluded that 15 of them are slowly pulsating B (SPB) stars (with 7 SPB/β Cep hybrids). Apart from the pulsating stars, they have identified stars with frequency groupings similar to what is seen in Be stars, and suggested that this might be connected to rotation. Some of the stars showed clear sign of modulation due to proximity effects in binary systems or rotation. Balona et al. (2011b) have found more non-pulsating stars within the β Cep instability strip, and no pulsating stars between the cool edge of the SPB and the hot edge of the δ Sct instability strip.

The richness of the observed variable behaviour of B-type stars must imply that details in the internal physics of the various stars, such as their internal rotation, mixing, and convection, etc., are different. Some excitation problems may be solved by increasing the opacity in the excitation layers. Also, binary interaction can complicate the observational data and its interpretation, but possibly provide better constraints than for single stars. In an attempt to increase the number of well studied early-type stars, a sample of B-type stars (having no overlap with the sample analysed by Balona et al. – thanks to the different selection criteria) was selected for in depth analysis from the Kepler field.

2. Target selection

We have selected targets to construct a new B-type pulsator sample via a very careful iterative process, incorporating several different techniques. As a first step, based on the publicly available Q1 light curves of the Kepler mission (release: 15 June 2010), we carried out an automated supervised classification (Debosscher et al. 2011). This method uses limited Fourier-decomposition of the light curves with a maximum of three dominant frequencies, then after a least-squares fitting, the resulting parameters (frequencies and amplitudes) are compared to the typical values of known groups of pulsating variable stars (for further details – on the same method applied to the CoRoT exoplanet sample – see Debosscher et al. 2009). Because the oscillation properties of different classes of pulsators can be very similar, 2MASS colours were used to distinguish between these groups (SPB and γ Dor stars in this case). The remaining nine SPB candidates provided the starting point of our study. None of these pulsators are included in the asteroseismology target list of the mission (Balona et al. 2011b). The γ Dor stars are analysed by Tkachenko et al. (2013).

To make sure that not only their oscillation patterns but also their stellar parameters confirm that these stars are indeed B-type main-sequence stars (thus confirming the reliability of the previously outlined classification process), we took high resolution (R = 85   000) spectra of the four brightest targets with the HERMES spectrograph (Raskin et al. 2011) installed on the 1.2 Mercator telescope on La Palma, and estimated the fundamental parameters. We carried out a full grid search with four free parameters (T_eff, log g, and vsini, plus the metallicity Z), using the BSTAR2006 (Lanz & Hubeny 2007) and ATLAS (Palacios et al. 2010) atmosphere grids. The derived parameters place all stars within the SPB instability strip, which already suggested that the rest of the sample – chosen using the same criteria – should also fall into that area of the Kiel-diagram. The full spectroscopic campaign (see Sect. 3) revealed that two stars are double-lined spectroscopic binaries (SB2), and that one of the fainter objects is not a B star. As an orbital solution can provide additional constraints on the physical parameters of the system, we decided to analyse the two binary targets with first priority. This study is presented in the current paper.

Main sequence B stars are unlikely to have a planetary system, as fierce stellar radiation and winds destroy their primordial circumstellar disk before planets can form, which means that these targets are not interesting for the exoplanet community. As our objects are also not under investigation by the Kepler Asteroseismic Science Consortium (KASC, Gilliland et al. 2010), we had to take special measures to make sure that none of the stars are getting dropped from the target list of Kepler during the later phases of the mission. We have successfully applied for Kepler Guest Observer (GO) time in Cycles 3 and 4 for additional observations of our targets.

2.1. A priori knowledge

KIC 4931738:

our first target is a faint star in the constellation of Cygnus. It was also identified – independent of our study – as a B-type pulsator by McNamara et al. (2012), but no in-depth analysis was carried out, nor was its SB2 nature discovered due to the lack of spectroscopy. The known astrometric and photometric parameters are listed in Table 1.

KIC 6352430:

our second target (also known as HD 179506) is – by Kepler standards – a bright star in the constellation of Lyra at a distance of pc (from a parallax of 2.02 ± 0.57   mas, van Leeuwen 2007). It was first described by Sir John Frederick William Herschel in the first half of the 19th century as a visual multiple system (HJ 2857). The visual components HJ 2857 B and HJ 2857 C (V_mag = 13.6 and 12.1) are at a position angle of θ = 178° and 211° at a distance of ρ = 14.4″ and 36.6″, respectively (Dommanget & Nys 2000). We show below that the main component itself is a spectroscopic binary (HJ 2857 A = KIC 6352430 AB). The star was first listed among the γ Dor pulsators by Balona et al. (2011a), then Balona (2011) points out the large discrepancy between the effective temperature of 9438 K in the Kepler Input Catalog (Kepler Mission Team 2009) and the spectral type of B8 V (Hill & Lynas-Gray 1977)¹, and – given that the spectral classification as the KIC temperatures are well known to be unreliable in the B star range (Balona et al. 2011b) – attributes the variation to SPB pulsations. No detailed analysis has been carried out so far for this target either. The known astrometric and photometric parameters are listed in Table 1.

3. Spectroscopy

3.1. The spectroscopic data

In order to unravel the binary nature of the two pulsating systems, we have gathered 40 high-resolution spectra of KIC 4931738 using the HERMES spectrograph (Raskin et al. 2011) installed on the 1.2 m Mercator telescope on La Palma (Spain), the FIES spectrograph installed on the 2.5 m Nordic Optical Telescope on La Palma (Spain), the HAMILTON spectrograph (Vogt 1987) mounted on the 3 m Shane telescope at the Lick Observatory (USA), the ARCES spectrograph (Wang et al. 2003) mounted on the 3.5 m ARC telescope at the Apache Point Observatory (USA), and the Isis spectrograph mounted on the 4.2 m William Herschel Telescope on La Palma (Spain), and 27 high-resolution spectra of KIC 6352430 using exclusively HERMES. We refer to Tables 2 and 3 for a summary of the observations.

3.2. Data reduction

We worked with data produced by the standard pipelines of each instrument, except for Isis, where the data were reduced using standard STARLINK routines. We merged the separate orders (taking into account the variance levels in the overlapping range) where it was not yet done or where we felt more comfortable using the separate orders and apply our own merging method rather than using the merged spectra from the pipelines (ARCES, FIES, and HAMILTON spectra). Subsequently, we performed a careful one-by-one rectification of all spectra – in the useful range between 4040 Å and 6800 Å. The rectification was done with cubic splines which were fitted through some tens of points at fixed wavelengths, where the continuum was known to be free of spectral lines. For this we utilised an interactive graphical user interface (GUI) which displays a synthetic spectrum (built using a predefined parameter set) in the background enabling the user to select line-free regions. We used these rectified data sets in our further analysis. We note that in case of spectra from the HAMILTON spectrograph the pipeline-reduced orders were not blaze-corrected, and the blaze function was not available, so we used a 2D median filter followed by a 2D Gaussian filter on the 2D array containing the separate orders to construct an artificial blaze function which we used to correct the spectra. When these filters are parameterised properly – using suitable widths and sigma values for both dimensions – the output is well rectified spectral orders. This is a fast and efficient method for the rectification of spectra of early-type stars (whose spectra are dominated by continuum) obtained with échelle spectrographs. Furthermore, for the ARCES spectra the wings of the Balmer-lines were badly affected while applying the standard blaze-functions, so we could not rectify the region of the Balmer lines properly, but as we did not use them later on, we did not pay any further attention to this issue.

We have placed all observations in the same time frame by converting UTC times of the mid-exposures into Barycentric Julian Dates in Barycentric Dynamical Time (BJD_TDB, but we simply use the BJD notation from here on), and also calculated the barycentric radial velocity corrections. We checked and corrected for the zero point offset of different wavelength-calibrations by fitting Gaussian functions to a series of telluric lines between 6881 Å and 6899 Å, using an automated normalisation and fitting routine, choosing the first HERMES exposure of KIC 4931738 as a reference point. The fitted Gaussians also provided the instrumental broadening values for each exposure which we used later in Sect. 3.3. For the Isis spectra this was not possible because of the wavelength coverage of the spectrograph – more specifically of the red arm. Here we used a less pronounced region between 6275 Å and 6312 Å and cross-correlated it with a template spectrum (a slightly broadened version – to compensate for the different instrumental broadenings – of the first HERMES exposure) to determine the offset. As the blue arm does not contain any telluric features at all, we had to use an estimation of the instrumental broadening, and to assume that the zero point shift for the two arms is the same. For this reason, we decided not to use radial velocity measurements from this part of the spectrum.

3.3. Radial velocities

Radial velocities were measured using several selected narrow lines or line regions by fitting synthetic composite binary spectra to the observed ones.

First an initial set of fundamental parameters were obtained from a multi-dimensional grid-search (T_eff, log g, vsini, and the metallicity Z, each for both components) using the best (having the highest signal-to-noise ratio (S/N) combined with a good merging and normalisation; for KIC 4931738 a FIES exposure with an S/N of 84, while for KIC 6352430 a HERMES exposure with an S/N of 134) available exposure (for a more detailed description of the method see, e.g., Pápics et al. 2011). During the χ² calculation we computed the χ² values for several line regions (the ones which we later used during the radial velocity measurements, plus H_β and H_γ), then summed up the individual χ² values and corrected the sum for the number of used line regions. This way narrower regions had the same weight in the final result as broader ones. We used only these selected regions and not the full available wavelength range because we wanted the final result to fit the lines needed for the radial velocity determination the best. We will come back to the derivation of more precise fundamental parameters and abundances in Sect. 3.6.

In the next step we used the derived estimated parameters (see the third sections of Tables 4 and 5) to create synthetic spectra of the binary components and fit all observed spectra leaving only the radial velocity values of the components as variables. A full 2D grid search with sufficient resolution was carried out for all line regions independently for all spectra (using specific instrumental broadening values for each exposure derived in Sect. 3.2). In the case of KIC 4931738 we used the Mg ii line at 4481 Å, the He i lines at 4922 Å and 5016 Å, and the Si ii lines at 5041 Å, 5056 Å, 6347 Å, and 6371 Å, while in case of KIC 6352430 we used the region around the Fe ii lines at 4550 Å, 5018 Å, and 5317 Å, around the He i lines at 4471 Å, 4922 Å, and 5876 Å (this only for the fundamental parameters, not for radial velocities), around the Mg ii line at 4481 Å, and around the Si ii lines at 5041 Å, 5056 Å, 6347 Å, and 6371 Å for the measurements, containing both the mentioned lines of the primary, and several additional narrow – mostly Fe lines – from the secondary. The best fit values were provided by a third order spline-fit of the χ²-values in both dimensions, while the error bars were empirically fixed to the radial velocities where . The process is visualised in Figs. 1 and 2. All resulting values were corrected with the barycentric velocities and the zero point offset before proceeding further.

3.4. Orbital solutions

The orbital parameters were calculated by fitting a Keplerian orbit through the radial velocity measurements while adjusting the orbital period (P), the time of the periastron passage (T₀), the eccentricity (e), the angle of periastron (ω), the systematic velocities (γ), and the semi-amplitudes (K). We averaged the radial velocity measurements of individual lines and approximated the error-bars of these values as the average error-bars coming from individual lines corrected with a factor of , where n is the number of individual lines per spectra used in the radial velocity measurements. We have weighted the average radial velocities according to these errors as w = 1/σ in the fit.

From the orbital parameters it is possible to derive – as a function of the inclination angle – the masses and the semi-major axes (see, e.g., Ramm 2004). We give further notes and the results for the two systems in the subsections below.

3.4.1. KIC 4931738

During the fit we left out four measurements (one HERMES and three FIES exposures) where the components had nearly the same radial velocities such as the components were mixed up during the automated radial velocity determination due to the overlapping lines, as well as one measurement coming from the blue arm of Isis, where it was not possible to determine the zero point offset. The first fit showed that there is an order of magnitude higher scatter in the radial velocity measurements of the primary. We interpret this as a result of the underlying pulsations. For this reason we have redone the fit by first fitting the orbit of the secondary component, then fixing these values and fitting the remaining K₁ for the final parameters. This way we determined a binary orbit with a period of 14.197 ± 0.002 days and an eccentricity of 0.191 ± 0.001. The orbital and physical parameters derived from spectroscopy are listed in Table 4, while the radial velocity curves and the best fit solutions are plotted in Fig. 3.

The Scargle periodogram (Scargle 1982) of the residuals of the radial velocities of the primary (O₁ − C₁) shows one clear peak (and its 1   d^-1 aliases – see Fig. 5) above the noise level at 1.1040 ± 0.0006   d^-1 with an amplitude of 3.4 ± 0.5   km   s^-1, and a signal-to-noise value of 3.2 (calculated in a 3   d^-1 window centred on the peak). Although this is below the generally accepted significance level of 4, we will show in Sect. 4.3 that this is one of the strongest frequencies (f₂ in Table 6) found in the Kepler photometry of the star. The phase diagram of the O₁ − C₁ values calculated using f₂ (see Fig. 6) supports the suggested pulsational origin of the observed scatter, and also connects the pulsations seen in the light curve to the primary star. We also checked the O₁ − C₁ diagram using f₁ (in Table 6) as well and found nothing. We note that heat driven oscillation modes can have very different amplitudes in photometry, which is determined by the temperature variation, and in spectroscopy, which diagnoses the velocity variation. This can be the reason why two modes of the same l and m with similar photometric amplitudes (such as f₁ and f₂ – see Sect. 4.3.2 for more information on these modes) appear with different radial velocity amplitudes. There is no sign of periodic variability in the residuals of the secondary.

3.4.2. KIC 6352430

Thanks to the difference in the spectral type of the components and their projected rotational velocities, there was no need to drop measurements where the two components have nearly the same radial velocity. Our method works well in these circumstances, because there is no risk of misidentification of components during the fitting of synthetic spectra, as there was in the case of KIC 4931738. The first fits using the full data set brought our attention to the slight discrepancy between measurements made in 2012 and the three measurements in the previous two seasons. To resolve this issue we made the final fit using only the data points from 2012, which delivered a very well constrained orbit with an orbital period of 26.551 ± 0.019 days and an eccentricity of 0.370 ± 0.003. The orbital and physical parameters derived from spectroscopy are listed in Table 5, while the radial velocity curves and the best fit solutions are plotted in Fig. 4.

As it can be seen in Fig. 4, measurements from 2010 and 2011 are systematically and significantly off this fit, which we explain by the presence of a third body in a much wider orbit around the centre of mass. To put constraints on that component, further observations would be necessary. There is no significant frequency in the Scargle periodogram of the residuals.

3.5. Spectral disentangling

As introduced by Simon & Sturm (1994), in the method of spectral disentangling, one simultaneously solves for the individual spectra of stellar components of a multiple system and set of the orbital elements. Hadrava (1995) suggested an application of the technique in Fourier space which is significantly faster compared to the disentangling in wavelength domain. In this work, we apply the spectral disentangling technique in Fourier space as implemented in the FDBinary code (Ilijic et al. 2004).

We have used four spectral intervals centred on sufficiently strong helium and metal lines (He i at 4471 Å and Mg ii at 4481 Å; He i at 4921 Å; He i at 5016 Å and Si ii at 5041 Å; Mg i at 5167 Å and Fe ii at 5169 Å) for measuring the orbital elements. The final mean orbital elements and 1-σ error bars are in perfect agreement with the values determined from fitting synthetic composite spectra in Sect. 3.3, and are listed in the bottom section of Tables 4 and 5. Because the uncertainty in the normalisation of the observed spectra raises towards the blue edge of the spectra, we restricted our spectral decomposition to the wavelength range with λ > 4050 Å. It is well-known that spectral disentangling in Fourier space suffers from undulations in the resulting decomposed spectra (see, e.g., Hadrava 1995; Ilijic et al. 2004). To overcome this problem, we divided our spectra into ~50 Å overlapping wavelength regions and corrected continua of the resulting decomposed spectra in these short regions using spline functions. An exception has been made for wide Balmer lines for which the decomposition was done for the entire profile. The final, corrected decomposed spectra for each component have been obtained by merging all the short and Balmer lines disentangled intervals.

3.6. Fundamental parameters and abundances

For the analysis of the decomposed spectra, we used the GSSP package (Tkachenko et al. 2012; Lehmann et al. 2011). The code relies on a comparison between observed and synthetic spectra computed in a grid of T_eff, log g, ξ_t, [M/H], and vsini, and finds the optimum values of these parameters from a minimum in χ². Besides that, individual abundances of different chemical species can be adjusted in the second step assuming a stellar atmosphere model of a certain global metallicity. The error bars are represented by 1-σ confidence levels computed from χ² statistics. The grid of atmosphere models has been computed using the most recent version of the LLmodels code (Shulyak et al. 2004). For the calculation of synthetic spectra, we use the LTE-based code SynthV (Tsymbal 1996) which allows to compute the spectra based on individual elemental abundances.

The bottom sections of Tables 4 and 5 list the atmospheric parameters and elemental abundances for the individual stellar components of KIC 4931738 and KIC 6352430. The spectral types and the luminosity classes have been derived by interpolating in the tables published by Schmidt-Kaler (1982). Figures 7 and 8 show the quality of the observed and disentangled spectra and their fit for both components of both binary systems in different wavelength regions.

4. Kepler photometry

4.1. Data reduction

Due to the quarterly rolls of the Kepler satellite which are performed to keep the solar panels facing the Sun along the course of the ~370 day orbit, the photometric data is delivered in quarters (Q). Most data are obtained in long-cadence (LC) mode with an integration time of 29.4 min. The characteristics of the LC mode are described by Jenkins et al. (2010). In this paper we use long-cadence data from Q0 to Q13 (March 2009 to June 2012), which gives a timebase of slightly over three calendar years, which is nearly an order of magnitude longer than the one typically provided by CoRoT observations. Q6 and Q10 are missing for KIC 4931738 due to the malfunctioning of Module 3 of the camera, while Q0 is not available for KIC 6352430.

Instead of using the standard pre-extracted light curves delivered through MAST, we have extracted the light curves from the pixel data information using custom masks. Our masks are determined for each quarter separately, and contain all pixels with significant flux. Compared to the standard masks, which contain only the pixels with a very high signal to noise ratio, including lower signal-to-noise pixels results in light curves with significantly less instrumental trends than the standard light curves. The automated pixel selection process and the benefits in terms of reducing instrumental effects are explained in Bloemen et al. (in prep.). Examples of the sets of pixels we used are shown in Figs. 9 and 10.

We have manually removed obvious outliers from the data. To correct for the different flux levels of the individual quarters (which is a result of the targets being observed by different CCDs – modules – of the CCD array after each quarterly roll, having slightly different response curves and pixel masks) we have divided them by a fitted second order polynomial. As after these initial steps there were no clear jumps or trends visible anymore, we merged the quarters and used the resulting datasets in our analysis.

4.2. Frequency analysis

To derive the frequencies present in the light variation of the targets we performed an iterative prewhitening procedure whose description is already given by Degroote et al. (2009) and thus omitted here. This provided a list of amplitudes (A_j), frequencies (f_j), and phases (θ_j), by which the light curve can be modelled via n_f frequencies in the well-known form of

The prewhitening procedure was stopped when a p value of p = 0.001 was reached in hypothesis testing.

As shown by Pápics et al. (2012) the way one defines the significance criteria can strongly affect the final number of accepted frequencies in the Fourier-analysis. Here we decided to use the classical approach (Breger et al. 1993) and measure the signal-to-noise value (S/N) as the ratio of a given peak to the average residual amplitude in a 3   d^-1 window around the peak, and accept a peak as significant when S/N ≥ 4. We used this set of frequencies to model the light curve.

From the numbers listed in Sects. 4.3 and 4.4 it is clear that using this criterion, in the lower frequency ranges, the periodogram is basically saturated in frequency, which means that the average separation of significant peaks is close, or even less than the theoretical frequency resolution (within this, two close peaks will appear in the periodogram with a frequency which is affected by the other peak and vice versa), which is given by the Loumos & Deeming (1978) criterion of ~2.5/T = 0.0022   d^-1. Even though these peaks meet the classical signal-to-noise criterion, and they are needed to properly model the light curve, we must restrict ourselves to a more significant subset of these frequencies, to avoid confusion in our tests later on. For this purpose, we define a selection criterion which requires a peak to have S/N ≥ 4 in a 1   d^-1 window to be selected. We will use these selected peaks in our tests later on.

In the following paragraphs we describe the analysis of the frequencies found in the Kepler photometry for the two binary systems.

4.3. KIC 4931738

The cleaned and detrended Kepler light curve (see Fig. 11) containing 43578 data points covers 1152.52 days starting from BJD 2 454 953.53821024, and has a duty cycle of 77.26% due to the two quarter-long gaps caused by the malfunction (and loss) of Module 3 during Q4.

The iterative prewhitening process returned 1784 frequencies of which 1082 met our significance criteria. The Scargle periodogram is shown in Fig. 12. The model constructed using this set of frequencies provides a variance reduction of 99.94% while bringing down the average signal levels from 213.4–135.3–4.5–3.8–3.7 ppm to 2.0–1.9–0.6–0.3–0.4 ppm, measured in 2   d^-1 windows centred around 1, 2, 5, 10, and 20   d^-1, respectively. Most of the power is distributed between 0.25 and 2.5   d^-1: 887 peaks contribute to a total power density (integral power of significant peaks normalised with interval width) of 4.07 × 10⁷   ppm²   d, while there are only 40 peaks between 2.5 and 5   d^-1 (6.22 × 10³   ppm²   d), and 25 peaks above 5   d^-1 (38.6   ppm²   d). The 130 peaks below 0.25   d^-1 provide a power density of 1.00 × 10⁶   ppm²   d. 262 of these frequencies met the selection criterion and are listed in Table A.1.

The low frequency signal (especially the lowest frequencies) partly comes from minor and unavoidable instrumental trends left in the data (not 100% perfect merging of the quarters, long term CCD stability issues, guiding issues, etc.), but the other part still has a physical origin, most probably granulation noise and/or low amplitude pulsations.

The binary orbital period of 14.197 ± 0.002 days (f_SB2 = 0.0704374 ± 0.0000099   d^-1) can also be found in the frequency analysis of the light curve at 0.070457 ± 0.000014   d^-1 with an amplitude of 264 ± 8   ppm, along with twice the binary orbital frequency at 0.140869 ± 0.000016   d^-1 (216 ± 7   ppm), which are the two strongest peaks below 0.25   d^-1. The appearance of the harmonics is a sign of a slight ellipsoidal modulation. There is no sign of eclipses in the light curve phased with the orbital period. The light variation due to oscillations is too complicated to be prewhitened and interpret the photometric variability due to the orbital motion alone.

4.3.1. Spacings in Fourier space

In general, the Fourier spectrum is remarkably rich and structured. There is a clear comb-like structure between 0.3 and 2.3   d^-1, containing alternating regions of high and low amplitude peak groups. Both these regions and the peak groups within them are equidistantly spaced. This can be clearly seen from the autocorrelation function of the Scargle periodogram in Fig. 13. The two highest peaks are situated at δf = 0.03835081   d^-1 and Δf = 0.31799938   d^-1, and we call these values the small spacing (the separation of the peak groups) and large spacing (the separation of the high amplitude regions) from here on. The latter also appears as a significant frequency in the Scargle periodogram at 0.318087 ± 0.000016   d^-1 (221 ± 7   ppm), while the former one is responsible for the overall global beating pattern visible in the light curve.

We detect further details from an analysis of the échelle diagram of the 262 peaks which met our selection criterion. The échelle diagram constructed with δf (see Fig. 14) shows clear groupings of peaks, while the échelle diagram constructed using Δf (see Fig. 15) shows well defined vertical ridges. The groupings in Fig. 14 represent the different high amplitude regions of the Scargle periodogram. Although the different regions overlap slightly in frequency (the y axis in Fig. 14), they are still well separated when we look at the corresponding modulo values (the x axis). This means that what we see is not one long series of equidistantly spaced peak groups from 0.3 to 2.3   d^-1, but six separate regions separated by Δf. Within these regions, peaks in dense groups which have an equal spacing of δf occur.

From Fig. 15 it is clear that the latter three regions are slightly shifted relative to the first three, as the modulo value for the highest peaks in the first three regions is ~0.113   d^-1, while it is ~0.226 d^-1 for the last three. Thus the first three regions are separated by Δf + 0.112833   d^-1 from the second three. Given that 2 × Δf + 0.112833   d^-1 = 0.748832   d^-1 equals the frequency of the dominant peak in the Scargle periodogram (the difference is 0.000026   d^-1, while the Rayleigh-limit of the data set is 1/T = 0.000868), we interpret the peaks in the three regions above 1.25   d^-1 as low order combinations of peaks which can be found in the three regions below 1.25   d^-1. This is confirmed by an automated search for linear combinations using a selected subset of the significant frequencies following Pápics (2012). We restricted the search to second or third order combinations formed by two individual frequencies. Using the same amplitude criterion and frequency precision as in Pápics (2012), looking at the 46 peaks which have S/N   >   10 in a 1   d^-1 window (see Table 6), we find none of the strong peaks below 1.2   d^-1 to be combinations, while almost all above that limit are low order combinations.

For clarity purposes, we have restricted ourselves to the frequency range from 0.3 to 2.3   d^-1 during the explanation of the observed pattern. However, further examination of Figs. 14 and 15 reveals that there are at least two more peak regions visible above 2.3   d^-1, fitting perfectly into the described structure, representing higher order combinations with even lower amplitudes.

These observations suggest that we only need to interpret the signal of the first three regions. We would like to point out that the observed structure of power in the Fourier domain, such as consecutive high amplitude modes with an equal spacing of δf, forming a long continuous series except for two phase shifts (in frequency) between the consecutive higher amplitude regions which have a spacing of Δf, is very similar to the structure – even of the amplitudes – of the interference pattern of two closely spaced frequencies.

4.3.2. Seismic interpretation

It is common practice to obtain the short-time Fourier transform (STFT) of the light curve to perform a time-frequency analysis of the data set (see, e.g., Degroote 2010). This is done by calculating the discrete Fourier transform (DFT) of consecutive windowed subsets of the data (we used rectangular windows to maximise the frequency resolution) over the full timebase of the observations, and comparing these consecutive DFTs to each other to see if frequencies, amplitudes, or phases show temporal evolution. The STFT in Fig. 16 shows that all frequencies are stable over time, which would be atypical for any kind of stochastic excitation mechanism. Also, although all large amplitude peak regions show a very dense fine structure (e.g., f₁ − f₆ ≈ 0.0019   d^-1, slightly above the Rayleigh limit, but not meeting the Loumos & Deeming criterion), comparing the frequency analysis of the first half of the data set to the one of the second half, we see that even these structures are stable over time. This leaves us with two options: either the oscillations are excited by tidal forces (which is suggested by the structure in the periodogram along with the relatively close orbit and significant eccentricity), or internally by the κ-mechanism, which is the basic mechanism in pulsating B-type stars on the main sequence (β Cep and SPB stars, see, e.g., Pamyatnykh 1999). Independently of the excitation mechanism, the observed frequencies are in the range of g modes.

In SPB stars one expects that heat-driven g modes of the same low degree and of consecutive high radial order are equidistant in period space. We made a compatibility check to see if models having similar parameters to the primary component would show spacings similar to the measured ones. For this test we used the 4 strongest peaks around 0.75   d^-1, and compared their period spacings to the mean spacing values of l = 1 and l = 2 modes (Dupret 2001) from models (Scuflaire et al. 2008). We found (see Fig. 17) that the mean period spacing of l = 1 modes is compatible with the measured spacing of the 4 peaks in question. However, we cannot explain the period spacing of the high amplitude peaks around 0.43   d^-1 and 1.07   d^-1, which have a value a factor of ~3 and ~0.5, while having the same equal spacing in frequency.

Depending on the orbital parameters and the eigenfrequencies of the components, pulsation modes can be excited by tidal forces. In such case the forcing frequencies of the tide-generating potential,

with k,m ∈Z, must come into resonance with the free oscillation frequencies of the components. Thus we expect to see structure in the échelle diagram calculated using f_SB2. Indeed, this is the case here. Figure 18 shows two clear (but not exactly sharp) ridges for the strongest peaks in the three large amplitude regions. The lowest possible k,m combinations are the most probable, so we tried to find an optimal rotation period, which provides a frequency pattern matching the observed one.

The two simplest choices to reproduce the frequency pattern occur for m = 1, − 2 combinations and f_rot = (1/6)f_SB2 or f_rot = (7/6)f_SB2. The first value of f_rot is incompatible with the measured projected rotational velocity vsini_rot = v_min = 10.5 ± 1.0   km   s^-1, as it would require a star with a radius of at least 17.7   R_⊙, which is significantly higher than what is possible for stars in this log g − T_eff range. The second value is compatible with the vsini_rot measurement, as such rotation frequency would set the radius at 2.53   R_⊙/sini_rot. As a rough comparison, models displayed in Fig. 17, have a median radius of 3.16   R_⊙, which would translate into an i_rot of ~53°. These models have a median mass of 3.7   ℳ_⊙, which, using the ℳsin³i_orb value from Table 4, gives an i_orb of ~62°. The fact that i_rot and i_orb are close estimates, illustrates that the suggested f_rot = (7/6)f_SB2 is compatible with both the observations and the model. Using f_rot = (7/6)f_SB2 and m = 1, − 2 we can reasonably match almost all peaks in Table 6 which were not identified as linear combinations before in Sect. 4.3.1. The observed peaks have an average absolute deviation of 0.002669   d^-1 from the exact tidal frequencies. This is ~3.3 times lower than what is expected ((1/8)f_SB2) for a random distribution of frequencies and the suggested model, and only 3 times higher than the Rayleigh frequency. Those very few peaks, which cannot be matched in Table 6 are interpreted as excited free oscillation modes of the primary. We note here that we see no sign of rotational modulation in the light curve.

There are a couple of weak peaks at higher frequencies (see Table A.1), which cannot be explained or matched using linear combinations, thus we interpret them as low amplitude p modes.

4.4. KIC 6352430

The cleaned and detrended Kepler light curve (see Fig. 19) containing 51 903 data points covers 1141.54 days starting from BJD 2 454 964.51190674 with a 92.91% duty cycle.

The iterative prewhitening procedure returned 1784 frequencies of which 1175 met our significance criteria. The Scargle periodogram is shown in Fig. 20. The model constructed using this set of frequencies provides a variance reduction of 99.97% while bringing down the average signal levels from 216.6–217.0–4.5–3.6–3.2 ppm to 1.5–1.4–0.6–0.5–0.2 ppm, measured in 2   d^-1 windows centred around 1, 2, 5, 10, and 20   d^-1, respectively. Most of the power is distributed between 0.25 and 2.5   d^-1: 785 peaks contribute to a total power density of 4.83 × 10⁷   ppm²   d, while there are 107 peaks between 2.5 and 5   d^-1 (2.55 × 10⁵   ppm²   d), and 149 peaks above 5   d^-1 (1.20 × 10³   ppm²   d). The 134 peaks below 0.25   d^-1 provide a power density of 1.25 × 10⁶   ppm²d, and the statement on the low frequencies in Sect. 4.3 is valid here as well, i.e., one part of this power comes from small and unavoidable instrumental trends left in the data, while the other part has a physical origin, and most probably related to granulation noise and/or low amplitude pulsations. 555 of the significant frequencies met the selection criterion and are listed in Table A.2.

The binary orbital period of 26.551 ± 0.019 days (f_SB2 = 0.037663 ± 0.000027   d^-1) can also be found in the frequency analysis of the light curve at 0.037698 ± 0.000010   d^-1 with an amplitude of 285 ± 6   ppm, along with four of its harmonics with amplitudes of 197 ± 5   ppm, 108 ± 4   ppm, 69 ± 3   ppm, and 38 ± 2   ppm, for 2f_SB2, 3f_SB2, 4f_SB2, and 5f_SB2, respectively. The appearance of the harmonics is a sign of a slight ellipsoidal modulation. These are some of the most significant frequencies below 0.25   d^-1. There is no sign of eclipses in the light curve phased with the orbital period. The light variation due to oscillations is too complicated to be prewhitened with the aim of interpreting the photometric variability due to the orbital motion alone.

4.4.1. Spacings in Fourier space

The Fourier spectrum is dominated by two very strong and four medium peaks between 1 and 1.6   d^-1. The frequency difference of the two dominant peaks creates the ~6.8 day beating pattern which dominates the light curve. There is another group of peaks clearly standing out from the rest in amplitude between 2.6 and 3.1   d^-1, these are mostly low order combinations of the strongest peaks between 1 and 1.6   d^-1. Negative combinations (where – in the simplest case – the frequency of the combination peak is not the sum, but the difference of the two parent peaks) of the three strongest modes occur below 0.25   d^-1.

The clear presence of both positive and negative combinations is also shown by an automatic combination frequency search, similar to the one we have used in Sect. 4.3.1, but now looking at the 29 peaks having an S/N > 20 (see Table 7), and using a slightly different set of parameters to, e.g., allow for negative combinations. The use of a higher signal-to-noise cutoff – which was required to reach a similar limited sample to the one in Sect. 4.3.1 – was needed because of the different distribution of the frequencies and amplitudes of the most significant peaks. This test not only confirms the presence of various combination frequencies, but also shows that there are several independent modes present in higher frequency regions, unlike in the frequency spectrum of KIC 4931738.

Although it is not as strikingly clear as for KIC 4931738, the periodogram of KIC 6352430 also shows spacings in frequency. As the autocorrelation function is heavily dependent on the amplitudes, the presence of two strong peaks requires a different approach than for KIC 4931738. We checked the distribution of frequency differences for all possible frequency pairs in both the low and high frequency region (see Figs. 21 and 22). This method returns two small spacing values, δ₁f = 0.157 ± 0.001   d^-1 (more dominant in the lower frequency region), and δ₂f = 0.102 ± 0.001   d^-1 (more dominant towards higher frequencies), and two large spacing values, Δ₁f = 1.361 ± 0.001   d^-1, and Δ₂f = 1.519 ± 0.001   d^-1, both visible in the full frequency range, but dominant in the high frequency region. These spacing values seem to be connected to frequency values and differences, as δ₁f = f₂ − f₁ = 0.157086   d^-1, δ₂f = f₃ − f₁ = 0.102088   d^-1, Δ₁f = f₁ = 1.361640   d^-1, and Δ₂f = f₂ = 1.518726   d^-1.

Looking at the échelle diagram constructed with δ₁f (see Fig. 23) or δ₂f (see Fig. 24) of the 555 peaks which met our selection criterion, we can see vertical ridges of not only two, but several equidistantly spaced frequencies: Δ₁f and Δ₂f dominate in the pure p mode regime, which looks similar to, but more pronounced than in the case of HD 50230 (Degroote et al. 2012). The frequency pairs in Fig. 25 show a clear pattern in the region above 6   d^-1, where the equally spaced frequencies seem to create a quasi continuous series of peaks.

4.4.2. Seismic interpretation

The échelle diagram calculated using f_SB2 shows no structure at all, which suggests that the oscillations have no connection to the binary nature of the star. Typical for heat-driven oscillations in B stars, all modes are stable over time. Not surprisingly, we see g mode oscillations in the low frequency regime (characteristic of SPB stars), but the high frequency region is remarkable. We observe pulsation signal with frequencies and frequency spacings typical for p modes, but which are not expected to be excited by theory in this relatively low temperature range near 13   000 K, and were not seen in any of the well analysed B stars. We can exclude that this signal is coming from the secondary, as – beyond the big difference in luminosity – the frequency spacings are connected to strong pulsation modes seen in the g mode region of the primary.

We interpret the signal in the p mode region as a result of nonlinear resonant mode excitation due to the high-amplitude g modes. This explanation is suggested by the fact that the frequency spacing values in this region match the two highest amplitude g modes, and also by the richness of the frequency spectrum in low order combination frequencies. This could explain the appearance of all four observed spacing values. A compatibility check similar to the one we have done in Sect. 4.3.2 showed that the excited peaks fall in the low frequency end of the region where eigenfrequencies of low order low degree p modes are situated, and that the frequency spacings are compatible with the observed ones.

Similar nonlinear effects are expected to be seen in large-amplitude nonradial oscillators (e.g., Buchler et al. 1997), such as white dwarfs (e.g., Fontaine & Brassard 2008), δ Sct stars (e.g., Breger et al. 2005), and β Cep variables (e.g., Degroote et al. 2009; Briquet et al. 2009), but excitation of modes in the p mode regime through resonant coupling with dominant g modes were, as far as we are aware, not yet observed before in SPBs.

5. Conclusions

Our observational study is the first detailed asteroseismic analysis of main sequence B-type stars by means of more than three years of Kepler space photometry and high-resolution ground based spectroscopy. It led to the classification and detailed description of two SB2 binary systems. The derived observational binary and seismic constraints provide suitable starting points for in-depth seismic modelling of the two massive primary stars.

The first system, KIC 4931738 is a binary consisting of a B6 V primary and a B8.5 V secondary, both slow rotators on the main sequence, with an orbital period of 14.197 ± 0.002 d and an eccentricity of 0.191 ± 0.001. The observed pulsation spectrum is consistent with tidally excited g modes in the primary component. Tidal excitation of pulsation modes was observed only in a few unevolved main-sequence binaries before, e.g., by De Cat et al. (2000), Willems & Aerts (2002), Handler et al. (2002), Maceroni et al. (2009), and Welsh et al. (2011).

The second system, KIC 6352430 is multiple system with two observed components, a moderate rotator main sequence B7 V primary and a slow rotator F2.5 V secondary contributing less than 10% of the total light, with an orbital period of 26.551 ± 0.019 d and an eccentricity of 0.370 ± 0.003. The primary star shows typical g mode SPB pulsations, but in addition we detect a rich spectrum of modes in the p mode region, atypical for such a low temperature SPB star. As these modes are not expected to be excited by the κ mechanism at this temperature, we explain their presence by nonlinear resonant excitation by the two dominant g modes, as supported by the observed high number of combination frequencies exhibiting four main spacing values, which are all connected to these g-mode frequencies. Such excitation was not observed before in any SPB star.

In our next paper we will analyse the 6 single B-type pulsators from the sample described in Sect. 2 and followed-up be Kepler through our Kepler GO programme.

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Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement n° 227224 (PROSPERITY), as well as from the Belgian Science Policy Office (Belspo, C90309: CoRoT Data Exploitation). K.U. acknowledges financial support by the Spanish National Plan of R&D for 2010, project AYA2010-17803. We thank the spanish Night-Time Allocation Committee (CAT) for awarding time to the proposals 82-NOT4/12A and 61-Mercator3/11B. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. This work is partly based on observations made with the William Herschel Telescope, which is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos (ORM) of the Instituto de Astrofísica de Canarias (IAC).

References

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