© ESO, 2013

1. Introduction

Asteroseismology is a powerful tool for diagnostics of deep stellar interiors that are not accessible through observations otherwise. Detection of as many oscillation modes as possible that propagate from the inner layers of a star all the way up to the surface is a clue towards better understanding of stellar structure at different evolutionary stages via asteroseismic methods. The recent launches of the Microvariability and Oscillations of STars (MOST, Walker et al. 2003), Convection Rotation and Planetary Transits (CoRoT, Auvergne et al. 2009), and Kepler (Gilliland et al. 2010a) space missions delivering high-quality photometric data at a micro-magnitude precision led to a discovery of numerous oscillating stars with rich pulsation spectra and opened up a new era in asteroseismology.

Besides acoustic waves propagating throughout the stars and having the pressure force as their dominant restoring force (p-modes), some stars pulsate also in the g-modes for which gravity (or buoyancy) is the dominant restoring force. Given that gravity modes have large amplitudes deep inside stars, they allow us to study properties of the stellar core and nearby regions far better than acoustic modes.

In this paper, we focus on the class of main-sequence, non-radially pulsating stars named after the prototype star Gamma Doradus (γ Dor, hereafter), whose multiperiodic variable nature was first reported by Cousins (1992). γ Dor stars are assumed to pulsate in high-order, low-degree gravity modes driven by the flux blocking mechanism near the base of their convective zones (Guzik et al. 2000; Dupret et al. 2005). These stars have masses ranging from 1.5 to 1.8 M_⊙ (Aerts et al. 2010) and A7–F5 spectral types (Kaye et al. 1999). They are usually multiperiodic pulsators exhibiting both low-amplitude photometric and spectroscopic variability with periods between 0.5 and 3 days (Kaye et al. 1999). γ Dor stars are grouped in a region close to the red edge of the classical instability strip in the Hertzsprung-Russell (HR) diagram. The theoretical γ Dor instability strip overlaps with the region where the Delta Scuti (δ Sct, hereafter) stars pulsating in low-order p-modes are located. Pulsators in the overlapping region are expected to show both high-order g-modes probing the core and low-order p- and g-modes probing the outer layers. These hybrid pulsators are among the most interesting targets for asteroseismic diagnostics, as the co-existence of the two types of oscillation modes yields the potential of constraining the whole interior of the star, from the core to the outer atmosphere.

The first-order asymptotic approximation for non-radial pulsations developed by Tassoul (1980) shows that the periods of high-order, low-degree g-modes are equally spaced in the case of a model of a typical γ Dor pulsator. However, the steep chemical composition gradient left by the shrinking core in stars with masses above ~1.5 M_⊙ causes sharp features in the Brunt-Väisälä frequency to occur, which in turn determines how the gravity modes are trapped in the stellar interiors. The trapping also leads to the deviation from uniform period spacing of the g-modes of consecutive radial order n (Miglio et al. 2008). Moreover, the rotational splitting of the g-modes provides a unique opportunity to deduce the internal rotation law and to estimate the amount of rotational mixing that may be partly responsible for the chemically inhomogeneous layer exterior to the core. This type of diagnostic is not available for stars pulsating purely in p-modes. Consequently, a non-uniform period spacing together with the rotational splitting of gravity modes represent a powerful tool for the diagnostics of the properties of the inner core and surrounding layers.

This idea was worked out in full detail for white dwarfs by Brassard et al. (1992) and applied to these type of objects by, e.g., Winget et al. (1991) and Costa et al. (2008). Miglio et al. (2008) developed similar methodology for gravity modes of main-sequence pulsators. Recently, Bouabid et al. (2013) studied the effects of the Coriolis force and of diffusive mixing on high-order g-modes in γ Dor stars, adopting the traditional approximation. The authors conclude that rotation has no significant influence on instability strips but does allow to fill the gap between δ Sct-type p-modes and γ Dor-type g-modes in the frequency spectrum of a pulsating star by shifting the g-mode frequencies to higher values. Bouabid et al. (2013) also conclude that the deviations from a uniform period spacing is no longer periodically oscillating around a constant value; as far as we are aware, this theoretical description, however, was not yet applied to data.

A first concrete application of detected period spacings to main-sequence stars was done by Degroote et al. (2010), based on CoRoT data of the slowly pulsating B (hereafter called SPB) star HD 50230. Among some hundreds statistically significant frequencies, the authors identify eight high-amplitude peaks in the g-mode regime that show a clear periodic deviation from the mean period spacing of 9418 s. They link this deviation with the chemical gradient left by the shrinking core and constrain the location of the transition zone to be at about 10 percent of the radius. Additionally, the authors suggest an extra-mixing around the convective core to occur. Another indication of a non-uniform period spacing was reported by Pápics et al. (2012) for the SPB/β Cep hybrid pulsator HD 43317. The authors detect two series of ten and seven components in the g-mode regime that show mean period spacings of 6339 and 6380 s, respectively. Given that the two values of the period spacing are almost the same, the authors conclude that both series belong to the same value of degree l but have different azimuthal numbers m. However, they do not present any physical interpretation of the observed period spacings, claiming that the detected modes are in the gravito-inertial regime, and thus full computations with respect to the rotation of the star, rather than approximations based on a perturbative approach are required.

The class of γ Dor-type pulsators should contain both stars with a shrinking and expanding core during their main-sequence evolution, and seismic probing of the core holds the potential to distinguish between those two scenarios. Based on the analysis of almost 11 500 light curves gathered with the CoRoT space mission, Hareter et al. (2010) confidently detect 34 γ Dor stars and 25 γ Dor/δ Sct hybrid pulsators with about 50 stars still being candidates of either of the class. For five out of 34 γ Dor stars, the authors detect regular patterns of period spacings of high-order gravity modes ranging from 650 to 2400 s. Hareter (2012) extends the sample up to 95 and 54 γ Dor and hybrid pulsators, respectively, and attempts to interpret the period spacings from the previous study. The author finds that the observed spacings are too short to be due to high-order l = 1 modes and concludes that these spacings are either coincidental due to the poor Rayleigh limit of CoRoT data or need some other explanation. Recently, Maceroni et al. (2013) studied an eclipsing binary, CoRoT 102918586, with a γ Dor pulsating component based on space photometry and ground-based spectroscopy. The authors detect a nearly equidistant frequency spacing of about 0.05 d^-1 (0.58 μHz), which was found to be too small to be due to the stellar rotation. The corresponding mean period spacing of 3110 s was found to be consistent with the theoretical spacings of l = 1 high-order g-modes based on the models computed without convective overshooting. Neither of the two studies present a detailed modelling of the deviation from the constant period spacings, however, and both rely on the comparison to theoretically predicted mean values of the period spacings for an F-type star. This encourages us to start this work with the aim of studying the frequency content of γ Dor stars observed in the Kepler data, which have a much more suitable Rayleigh limit, in more detail to unravel the different structures observed in their amplitude spectra.

Our ultimate goal is to deduce the extent of the stellar core and mixing processes with high precision near the core for γ Dor-like stars that burn hydrogen in their cores. This requires the selection of a large enough sample of slowly to moderately rotating γ Dor stars for which this paper is devoted to this purpose. Both photometric and spectroscopic data have been acquired and are described together with the data reduction process in Sect. 2. The sample of stars is introduced in Sect. 3. Photometric characterization of the sample is given in Sect. 4; the results of the spectroscopic analysis of all stars for which spectra are available are outlined in Sect. 5. The conclusions and an outlook for future research are given in Sect. 6.

2. Observations and data reduction

We base our analysis on high-quality photometric and spectroscopic data. The light curves have been gathered by the Kepler space telescope and are of micro-magnitude precision. The data are acquired in two different modes, the so-called long (LC) and short (SC) cadences. In both cases, the exposure time is 6.54 s, of which 0.52 s is spent for the readout. The LC integrate over 270 single exposures, giving a time-resolution of 29.42 min, whereas one data point in the SC mode contains 9 exposures, corresponding to a 58.85-s time resolution. The data are released in quarters (~3 months of nearly continuous observations), which are the periods between spacecraft rolls needed to keep its solar panels pointing towards the Sun. More information on the characteristics of the SC and LC data can be found in Gilliland et al. (2010b) and Jenkins et al. (2010), respectively.

For the analysis, we use all publicly available data at the time of the analysis (Q0-Q8) and our data from the Kepler Guest Observer data (GO, Still et al. 2011). LC data are well-suited to our analysis, as the corresponding Nyquist frequency of 24.47 d^-1 (283.12 μHz) is well above the γ Dor-typical oscillation periods that lie between 0.5 and 3 days. Instead of using the standard pre-extracted light curves delivered through MAST (Mikulski Archive for Space Telescopes), we have extracted the light curves from the pixel data information using custom masks based on software developed by one of us (SB, Bloemen et al., in prep). Contrary to the standard masks, which only use pixels with the highest signal-to-noise ratios (S/Ns), our masks contain as many pixels as possible with significant flux. The light curves obtained by including these lower S/N pixels have significantly less instrumental trends than the standard light curves. Table 1 shows the influence of the proper mask choice on the frequency analysis of one of our γ Dor stars. The fifteen crossed out low frequencies present in the standard pre-extracted light curve are gone from the data when we extract the light curve from the pixel data using a custom mask. An appropriate removal of the instrumental effects is essential for γ Dor stars pulsating in high-order g-modes that show up in the low-frequency domain. One must thus cautiously interpret frequencies deduced from the standard light curves (as listed in, e.g. Grigahcène et al. 2010; Uytterhoeven et al. 2011; Balona et al. 2011).

We acquired high-resolution spectroscopy for the brightest targets with the HERMES (High Efficiency and Resolution Mercator Echelle Spectrograph, Raskin et al. 2011) spectrograph attached to the 1.2-m Mercator telescope (La Palma, Canary Islands). The spectra have a resolution of 85 000 and cover the wavelength range from 377 to 900 nm. Typical S/Ns measured at λλ 5500 Å are about 40 for each spectrum. An overview of spectroscopic observations is given in Table 2.

Fig. 1 The target sample representation. The plot illustrates all frequencies that we consider as independent ones for all 69 stars. Note the different y-scales for the two panels.

Fig. 2 Light curves (left) and amplitude spectra (right) of selected stars. From top to bottom: KIC 02710594, 03448365, 03744571, 04547348, 04749989, and 10080943. The y-scale of the left panel represents differential magnitude after subtraction of the corresponding mean value for each target. Note a mmag scale for KIC 04749989.

The spectra have been reduced using a dedicated pipeline. The data reduction procedure included bias and stray-light subtraction, cosmic rays filtering, flat fielding, wavelength calibration by ThArNe lamp, and order merging. The normalisation to the local continuum has been performed in two steps: first, by fitting a master function (cubic spline connected through some tens of carefully selected continuum points) that accounts for the artificial spectral signature induced by the flat field procedure. The spectral signature of the flat field lamps is attenuated by red blocking filters (Raskin et al. 2011), which induce a specific spectral signature in the reduced merged spectra. Second, a correction for small-scale effects is performed by selecting more knot points at wavelengths free of spectral lines and connecting them with linear functions. At this phase, all spectra are corrected for their individual radial velocities (RVs) as the position of the selected knot points is fixed in wavelength. For more information on the normalisation procedure, we refer to Pápics et al. (2012).

3. The target sample

Our selection of γ Dor targets is based on an automated supervised classification method, applied to the entire Kepler Q1 dataset of about 150 000 light curves. For detailed information on the method, we refer to Debosscher et al. (2011). Basically, we summarize the main characteristics of each light curve by means of a uniform set of parameters, obtained from a Fourier decomposition. These include, e.g., significant frequencies present in the light curve, their amplitudes, and their relative phases. Statistical class assignment is then performed by comparing these parameters with those derived from a training set of known class members for several types of variable stars. This classification procedure only uses information derived from the Kepler light curve, which is not sufficient to distinguish all variability types. Therefore, we refined the classification in a second stage by using 2MASS colour information. In this way, we could distinguish between pulsating stars that show the same type of variability in their light curves but reside at different locations in the HR diagram (e.g. γ Dor versus SPB pulsators).

In total, we selected 69 γ Dor candidate stars. Figure 1 represents the entire sample by showing all frequencies that we consider as independent ones (cf. Sect. 4) for all 69 targets. The total number of shown frequencies is 344, which results from about 5 individual peaks per object. These values are statistical and, in practice, turn out to be higher. In any sufficiently large set of frequencies, it is not expected to find more than ~10 (depending on the criteria used to search for combination frequencies) independent modes to occur as there is no way to distinguish for the high-order combination and lower amplitude modes. Most of the frequencies in Fig. 1 are clearly grouped in the low-frequency region below 3.5 d^-1 (40.50 μHz). There is also a clear contribution in the high-frequency domain. This points to the fact that γ Dor/δ Sct hybrid pulsators have been included into the sample as well.

Figure 2 illustrates Kepler light curves and amplitude spectra of six selected stars: KIC 02710594, 03448365, 03744571, 04547348, 04749989, and 10080943. The first four targets belong to the same class of γ Dor-type variable stars and exhibit similar variability in their light curves but with different beating patterns caused by the closely spaced individual peaks in the low frequency domain. A much higher frequency contribution is easily recognizable in the light curves of the two other stars, KIC 04749989 and 10080943, suggesting the hybrid nature of these objects. Indeed, lots of high and moderate amplitude peaks characteristic of p-mode pulsations show up in the high frequency domain in the corresponding amplitude spectra of these stars.

Table 3 presents photometric characterization and classification of all 69 stars in our sample. The first column gives the KIC-number of the star; the second and the third columns represent the total frequency range in d^-1 and μHz, accordingly. In the next three columns, we give the amplitude range in mmag and frequency of the highest amplitude peak both in d^-1 and μHz, accordingly. The total number of statistically significant frequencies and the photometric classification according to the type of variability are given in the last two columns. Despite our advanced data reduction procedure (cf. Sect. 2) for the majority of the stars, low-frequency (<0.1 d^-1 or 1.2 μHz) individual contributions have been detected. As stated by Balona (2011), the amplitudes of these low frequencies are affected by the instrumental effects, even in the case where the frequencies represent oscillation modes of the stars. We are aware of that and thus do not consider these low-frequency contributions when classifying the targets. Assignment of a star to a certain variability class (γ Dor or hybrid pulsator) has been made also by considering the information on combination frequencies into account (cf. Sect. 4). As such, the detection of individual peaks in the high frequency domain is not by itself a sufficient condition for the star to be classified as a γ Dor/δ Sct hybrid pulsator: the peaks in the p-mode regime must be independent modes and not (low-order) combinations of the low frequency g-modes. There are several objects marked with superscript asterisk (^∗) in Table 3 for which classification is uncertain in a sense that some low-amplitude, high-order (typically, higher than 6–8) combination frequencies have been detected in the high frequency domain as well. These high-order combinations, however, can be a plain mathematical coincidence and we thus put an uncertainty mark on these targets and are aware that they can also exhibit very low-amplitude p-modes.

4. Frequency analysis of the Kepler light curves

For the extraction of frequencies, amplitudes and phases of pulsation modes from the Kepler light curves, we used the Lomb-Scargle periodogram (Scargle 1982) and consecutive prewhitening procedure. A detailed description of the whole procedure can be found in Pápics et al. (2012).

In our case, we optimize the amplitudes and phases of the modes by means of least-squares fitting of the model to the observations at each step of the prewhitening procedure while the frequencies are fixed to those obtained from the Scargle periodogram. In this way, the frequency uncertainty is determined by the frequency resolution given by the Rayleigh limit 1/T; the errors in amplitudes and phases are the formal errors from the least-squares fitting. We keep iterating until the commonly used significance level of 4.0 in S/N is reached (Breger et al. 1993), where the noise level is computed from a 3 d^-1 (34.7 μHz) window before prewhitening the frequency peak of interest. However, there are several objects (typically those for which less than three quarters of data are available) for which only a couple of dozens of peaks can be detected following this standard criterion, resulting in a fairly bad fit of the observed light curve. In these few cases, we exceed the lower limit in S/N and iterate until the value of S/N = 3.0 is reached, making sure the model fits the observations well.

For every star in our sample, we check for evidence of non-linear effects in the light curve by looking for low-order combination frequencies. We assume that any combination frequency should have a lower amplitude than the parent frequencies, i.e. it should appear in a list, which is sorted according to amplitude, below the frequency(ies) from which it is formed. We allow up to three terms when computing the sum and/or difference combination frequencies, assuming that harmonics can enter the combination as well (e.g., f_i = 2f₁ + f₂ − f₃ or f_i = f₁ − 3f₂ + f₃). The frequency of interest is accepted to be a combination of the independent frequency if the difference between the combination and true values is less than or equal to the Rayleigh limit. For example, the arbitrary frequency f_i is assumed to be a combination of the independent frequencies, f₁ and f₂ when, e.g., |f_i − (2f₁ + f₂) | ≤ 1/T.

Table B.1 summarizes the results of the frequency analysis for all 69 stars. The total time span of the observations and thus Rayleigh limit can vary from star to star and is indicated in the parenthesis following the star designation. The table lists the frequencies (both in d^-1 and μHz) and corresponding amplitudes (in mmag), S/N, the corresponding lowest order combination, and the number of combinations that can be built of independent frequencies only. The latter number obviously needs more explanation. Each frequency in the list (except for the largest amplitude peak which is assumed to be independent) is first considered to be a possible combination term of the larger-amplitude parent frequencies. In the first step, we check for all possible combinations of the parent frequencies (all frequencies that have larger amplitudes than the one in question) that are capable of representing the frequency in question within the Rayleigh limit. Among the variety of all the obtained combinations, we select those that involve the independent frequencies only. The total number of these combinations is given in the last column of Table B.1, which is designated “N”. A value of zero in this column means that the frequency of interest can only be represented as a combination of the frequencies which are combinations of the independent peaks. In practice, this means that the frequencies with the number N = 0 can only be represented as very high-order (typically, higher than 10) combinations, which is basically just a mathematical coincidence and has no physical meaning. The independent frequencies are highlighted in Table B.1 in boldface, whereas their second- and third-order combinations are shown in italics and with the superscript asterisk (^∗), respectively.

Pápics (2012) investigated the problem of occurring combination frequencies in the light curves of Kepler B-type stars. The author simulated thousands of light curves having similar (but random) power spectrum as the main-sequence SPBs. Pápics (2012) concluded that one has to restrict to the low-order (up to the 3rd) combinations when interpreting the light curves, as the higher-order combination peaks are likely mere coincidence than having any physical meaning. This is the reason why we consider only low-order (2nd and 3rd) combination peaks as having physical meaning while we are skeptical with respect to all other, higher-order combinations.

5. Spectroscopic analysis

As mentioned in Sect. 2, we obtained high resolution (R = 85   000) spectroscopic data for half of the stars in our sample. At least two spectra have been obtained for each star to check for possible RV variations due to binarity. Besides the binarity check, a collection of N independent spectra allows us to increase S/N by roughly a factor of by combining the single measurements and building an average spectrum. The gain in S/N is very important given that we aim for estimation of the fundamental stellar parameters like effective temperature T_eff, surface gravity log g, projected rotational velocity vsini, and metallicity [M/H].

As for the second half of the sample stars, their observations are planned for the next summer when the Kepler field is best visible on La Palma, Canary Islands. Besides that, we also plan more extensive spectroscopic monitoring of the stars for which more than just a couple of measurements would be essential (e.g., spectroscopic binaries for which a precise orbital solution and possibly decomposed spectra in the case of double-lined binaries (SB2) can be derived given that good orbital phase coverage is provided). In the following, we describe the results of our spectrum analysis of 36 of the sample stars and compare the spectroscopic classification with the one based on the method by Debosscher et al. (2011).

5.1. Fundamental parameters and position in the HR diagram

For estimation of the fundamental parameters of the stars, we use the GSSP code (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, vsini, [M/H], and microturbulent velocity ξ 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 grid of atmosphere models has been computed using the most recent version of the LLmodels code (Shulyak et al. 2004) in [–0.8, +0.8] dex, [4500, 22 000] K, and [2.5, 5.0] dex range for metallicity, effective temperature, and surface gravity, respectively (Tkachenko et al. 2012). Synthetic spectra are computed using the SynthV code (Tsymbal 1996), which computes the spectra based on individual elemental abundances, considering vertical stratification of a chemical element and/or microturbulent velocity.

The errors of measurement (1σ confidence level) are calculated from the χ² statistics using the projections of the hypersurface of the χ² from all grid points of all parameters onto the parameter in question. In this way, the estimated error bars include any possible model-inherent correlations between the parameters. Possible imperfections of the model like incorrect atomic data, non-LTE effects, or continuum normalization are not considered. In a recent study by Molenda-Żakowicz et al. (2013), the use of different methods and codes to derive atmospheric parameters for F, G, K, and M-type stars is compared and led the authors to conclude that the realistic accuracy in the determination of atmospheric parameters for these types of stars is ±150 K in T_eff, ±0.15 dex in [Fe/H] and 0.3 dex in log g, even though error calculations for individual programs might result in smaller errors. Hence, we are aware of a possible underestimation of errors in Table 4.

Fig. 3 Distribution of effective temperatures (top left), surface gravities (top right), projected rotational velocities (bottom left), and metallicities (bottom right) of the sample targets.

Given the low S/N of our spectra (typically, below 80 for the average spectrum), we decided to fix the microturbulent velocity ξ to the standard value of 2 km s^-1 and optimise the global metallicity (which is a scaling of all chemical elements heavier than hydrogen and helium by the same factor), instead of evaluating individual abundances of chemical elements. Table 4 summarizes the results of the spectroscopic analyses of the 36 stars in our sample listing the KIC-number, T_eff, log g, vsini, and [M/H] values, as well as the number of individual spectra used for building the average spectrum.

Figure 3 shows distributions of the four fundamental parameters of the sample stars. The majority of the targets (about 66%) have effective temperatures in the range between 6900 and 7200 K, though a few objects show slightly higher/lower (±200 K) temperature values. The distribution of the surface gravities suggests that all but one stars are luminosity class IV-V sub-giant or main-sequence stars, which together with the above mentioned temperature regime, perfectly fits the classical definition of the γ Dor stars group (see, e.g. Kaye et al. 1999; Aerts et al. 2010). The largest majority of the targets within the error of measurements show solar metallicities though a couple of stars exhibit slightly depleted overall chemical composition compared to the Sun. Our sample is dominated by slowly (vsini< 30 km s^-1) and moderately (30 <vsini< 100 km s^-1) rotating stars, only seven objects have vsini values above 100 km s^-1.

The derived values of T_eff and log g allow us to place the stars into the log (L/L_⊙)–log T_eff diagram and classify them by evaluating their positions relative to the γ Dor and δ Sct theoretical instability strips. The luminosity of each star has been estimated from an interpolation in the tables by Schmidt-Kaler (1982) using our spectroscopically derived T_eff and log g. The errors in luminosity were evaluated by considering the errors of both T_eff and log g. However, we base our spectroscopic classification mainly on the position of the stars, according to the derived temperatures as the luminosity errors can still be underestimated due to the uncertainties in the empirical relations.

Figure 4 shows the position of the stars in the HR diagram together with the γ Dor and δ Sct theoretical instability strips. The latter are based on Dupret et al. (2005, Figs. 2 and 9). The edges of the δ Sct instability region have been computed for the fundamental mode and a mixing-length parameter of α = 1.8 (solid lines). The borders of the γ Dor instability regions computed with α = 2.0 and 1.5 are shown by dashed thin and thick lines, respectively. All stars but one are nicely clustering in the region of the HR diagram, where the γ Dor-type g-mode pulsations are expected to be excited in stellar interiors. KIC 08378079 is the only outlier located the top left (towards higher temperatures and more luminous objects) of the γ Dor instability region in the diagram (cf. Fig. 4). Given that our average spectrum is of very low S/N and thus the error bars in both temperature and luminosity are large for this faint (V = 12.5) star, it can safely be considered as a γ Dor-type variable from our spectroscopic values of T_eff and log g.

As an overall conclusion, our spectroscopic classification perfectly agrees and thus confirms the photometric one based on the information derived from the Kepler light curves only (cf. Sect. 3). A second important conclusion for future asteroseismic modelling is that most stars are slow to moderate rotators. Of course, the estimated value of the projected rotational velocity is not a measure of the true rotational period of a star but allows to put some constraints (lower limit) on the stellar rotation frequency.

Fig. 4 Location of 36 stars (see Table 4) and the γ Dor and δ Sct (solid lines) theoretical instability strips (Dupret et al. 2005) in the HR diagram. Dashed thin and thick lines represent the γ Dor instability regions computed with α = 2.0 and 1.5, respectively. Open circles refer to the hybrid pulsators.

Fig. 5 vsini versus independent frequencies for the sample stars. Open triangles refer to the dominant oscillation mode for each star; the two other symbols represent all other independent frequencies. Filled boxes and open circles refer to γ Dor and hybrid pulsators, respectively. The solid black line shows linear fit to all data points (there are three frequencies between 18 (208.26) and 21 (242.97) d-1 (μHz) in the δ Sct domain); the red dashed line represents a linear fit to the frequencies in the γ Dor range after removing the δ Sct frequencies.

Fig. 6 Same as Fig. 5 but for the amplitudes. The solid black line shows a linear fit to all data points; the red dashed line represents a linear fit after removing two stars with vsini> 150 km s-1 from the sample.

We also checked for a possible correlation between the spectroscopically derived value of vsini and photometric frequencies of the independent pulsation modes for each star. The distribution is illustrated in Fig. 5. There is a clear trend in the vsini-frequency diagram implying that the independent mode frequencies are higher for the larger vsini. The correlation becomes even stronger if we include the three frequencies from the δ Sct domain that we detected for two of the shown stars (solid, black line to be compared with the dashed, red line in Fig. 5). For completeness, in Fig. 6, we also present the vsini-amplitude distribution for the sample stars. Though a negative trend shows up when fitting all the data points linearly (solid line), it is almost gone when the two stars with vsini> 150 km s^-1 are removed (dashed line). Thus, we conclude that the vsini-amplitude correlation presented here is at best weak and is not a characteristic of γ Dor stars.

To check whether a similar vsini-frequency correlation is also characteristic of δ Sct stars, we used the data from the catalogues of Rodríguez et al. (2000, hereafter called R2000) and Uytterhoeven et al. (2011). All stars for which vsini measurements are available were selected, resulting in 189 and 16 objects extracted from R2000 and Uytterhoeven et al. (2011), respectively. The full list of the retrieved objects is given in Table A.1. We are also aware of the two recent extensive studies by Balona & Dziembowski (2011) and Chang et al. (2013) focusing on the δ Sct stars but these authors do not present vsini-frequency correlations in their papers. The vsini-frequency distributions obtained based on the data from R2000 and Uytterhoeven et al. (2011) are illustrated in Fig. 7. In both cases, linear fits (solid lines in both panels) show a negative trend, though with different slopes: it is steeper for the Uytterhoeven et al. (2011) data than for those from R2000. However, the reliability of the correlation shown in the bottom panel of Fig. 7 is not convincing because of too few objects that could be retrieved from the sample of Uytterhoeven et al. (2011). Comparison of the R2000 distribution (cf. Fig. 7, top panel) with the one we obtained for the γ Dor stars in this paper (cf. Fig. 5) reveals opposite behaviour: For the γ Dor stars, the frequencies of oscillation modes are found to increase as the vsini increases, whereas they decrease or, at most, remain constant for the δ Sct stars.

Fig. 7 Frequency of the dominant mode as a function of vsini for δ Sct stars from the catalogues of Rodríguez et al. (2000, top and Uytterhoeven et al. (2011, bottom. The solid line represents a linear function fit to the shown data points. Detailed information on individual stars is provided in Table A.1.

From Fig. 5, it is clear that not only the dominant mode frequencies (open triangles) but also all the other independent frequencies (filled boxes and open circles) show a correlation with vsini. If the correlation was due to rotational modulation, one would expect only the dominant, rotation frequency to show dependence on vsini. Since we observe several independent frequencies for each star and they all show similar behaviour with respect to vsini (cf. Fig. 5), we conclude that the observed correlation is due to stellar pulsations rather than rotational modulation. Besides the above mentioned correlation for the rotation frequency, one would also expect the rotational modulation to show up with a series of harmonics of that frequency (see e.g., Thoul et al. 2013). We thus checked how many harmonics in total (including those of combination frequencies) per star could be detected and present the distribution in Fig. 8 (top). Though there is a clear peak at three harmonics per star, the distribution is random, revealing stars that show up to a couple of dozens of harmonics in their light curves. The majority of these detections are harmonics of (very low-amplitude) combination frequencies, which are likely to be just a mathematical coincidence rather than having any physical meaning (see Pápics 2012). Indeed, none of the stars show a series of harmonics but single peaks for combination frequencies. We thus reconsidered our distribution to include only harmonics of the independent frequencies now (see Fig. 8, bottom). There is a clear peak at three harmonics per star, and the largest number of the detected harmonics per object decreased from the previous 29 to the present 9. A careful look at the independent frequencies and their harmonics showed that the latter are homogeneously distributed among all independent frequencies, and there is no star showing a long series of harmonics of a single independent frequency, as one would expect for rotational modulation.

Fig. 8 Distributions of the total number of the detected harmonics (top) and those formed of independent frequencies only (bottom).

6. Discussion and conclusions

Non-uniform period spacings of gravity pulsation modes is a powerful tool for the diagnostics of the properties of the inner core and its surrounding layers. We aim for the application of the methodology described by Miglio et al. (2008) and Bouabid et al. (2013) (for the first practical application, see Degroote et al. 2010) to γ Dor-type pulsating stars. This paper is the first step towards this goal.

Based on an automated supervised classification method (Debosscher et al. 2011) applied to the entire Kepler Q1 dataset of about 150 000 high-quality light curves, we compiled a sample of 69 γ Dor candidate stars. We presented the results of frequency analyses of the Kepler light curves for all stars in our sample. For each star, we checked the results for evidence of non-linear effects in the light curve by looking for low-order combination frequencies. All our stars show at least several second-order and third-order combination frequencies (including harmonics), suggesting that non-linear effects are common for γ Dor-type pulsating stars.

From the Kepler light curve analysis, we identified 45 γ Dor stars, 14 hybrid pulsators, and 10 stars that are marked as “possible γ Dor/δ Sct hybrids” and show low amplitude, high-order (typically, higher than 6–8) combinations in the p-mode regime of the Fourier spectrum. A more detailed study of the amplitude spectra would be needed for these ten stars to classify them correctly. According to the largest amplitude frequency distribution (cf. Fig. 9), the γ Dor-type g-mode pulsations dominate our sample, and there is only one star, KIC 06778063, where the peak in the p-mode regime is the dominant one in the data.

Additionally, high-resolution spectroscopic data have been obtained for half of our stars with the HERMES spectrograph (Raskin et al. 2011) that is attached to the 1.2-m Mercator telescope. All 36 stars for which spectra have been acquired fall into the γ Dor instability region confirming the photometric classification as either γ Dor- or of γ Dor/δ Sct hybrid-type pulsating stars. The effective temperatures are distributed within a narrow window of 700 K; the surface gravities distribution suggests luminosity class IV-V sub-giant or main-sequence stars. The applied method of photometric classification as described in Debosscher et al. (2011) hence proves to be very robust.

Fig. 9 Distribution of the highest amplitude frequency detected in the Fourier spectra of the sample stars.

Uytterhoeven et al. (2011) presented a characterization of a large sample of A- to F-type stars based on Kepler photometry and where available, ground-based spectroscopy. Among the stars showing γ Dor, δ Sct, or both types of pulsations in their light curves, the authors identified about 36% as hybrid pulsators. In our case, this fraction is about 35% if we consider 10 “possible γ Dor/δ Sct hybrids” as being indeed hybrid pulsators and about 20% if we exclude them. An extrapolation of the results obtained by Uytterhoeven et al. (2011) for a sub-sample of 41 targets to the entire sample led the authors to the conclusion that many of γ Dor and δ Sct pulsators are moderate (40 <vsini< 90 km s^-1) to fast (vsini> 90 km s^-1) rotators. Our sample consists of mainly slow to moderate rotators with the vsini distribution peaking at the value of 65 km s^-1 (cf. Fig. 3, bottom left). Major conclusion that Uytterhoeven et al. (2011) made concerning γ Dor and δ Sct theoretical instability strips was that they have to be refined; the stars with a characteristic type of pulsation seems to exist beyond the respective instability regions. The studies performed by Grigahcène et al. (2010) and Tkachenko et al. (2012, 2013) led to similar conclusions. Hareter (2012), who studied a large number of the CoRoT light curves, also found that γ Dor stars mainly cluster at the red edge of the δ Sct instability strip with a significant fraction of them having cooler temperatures, whereas δ Sct-γ Dor hybrid pulsators fill the whole δ Sct instability region with a few stars being beyond its blue edge. The author also concludes that there is no close relation between γ Dor stars and hybrid pulsators from the position of the studied objects in the HR diagram. In this paper, we found that all stars for which fundamental parameters could be measured from spectroscopic data reside in the expected range of the HR diagram and cluster inside the theoretical γ Dor instability region. Our results are based on different, much stricter, selection criteria than those applied in the above papers. That is, the criteria is such that the morphology of the light curves in combination with a spectroscopic T_eff determination are used rather than a poor T_eff estimate or colour index alone. This implies that our sample is much cleaner and not contaminated by stars with other causes of variability than pulsations.

Balona et al. (2011) suggested that the pulsation and rotation periods must be closely related. Given that vsini values were obtained for a very limited number of stars in Uytterhoeven et al. (2011), the authors did not look for any correlations between vsini and pulsation period/frequency. In our case, we have found a clear correlation between the spectroscopically derived vsini and the photometric frequencies of the independent pulsation modes (cf. Fig. 5) in the sense that the modes have higher frequencies when the rotation is faster. This correlation is not caused by rotational modulation of the analysed stars but is valid for the γ Dor g-mode stellar oscillations. These findings perfectly agree with the results of the recent theoretical work by Bouabid et al. (2013), who showed that the rotation shifts the g-mode frequencies to higher values in the frequency spectrum of a pulsating star, which fill the gap between the δ Sct-type p-modes and γ Dor-type g-modes. Comparison with the δ Sct stars reveals that they behave in the opposite way. Namely, they show decreasing frequency as the vsini increases. We also checked for a vsini-amplitude correlation for our γ Dor stars but did not get any convincing results.

Compared to the study by Uytterhoeven et al. (2011) where a large sample of 750 A- to F-type stars has been analysed and of which about 21% was found to belong to neither the classes of γ Dor and δ Sct nor to the hybrid pulsators, 12% were identified as binary or multiple star systems, and many objects were found to be constant stars. Our sample is much more homogeneous and does not include a bias, which is when all selected stars are multiperiodic pulsators. Compared to the sample of CoRoT stars compiled by Hareter (2012), nearly continuous Kepler observations spread over several years offer an order of magnitude higher frequency resolution that resolves the frequency spectrum and makes the detection of frequency and period spacings possible. Hence, our sample selection and frequency analysis offer a good starting point for seismic modelling of γ Dor stars.

In follow-up papers, we plan to present the detailed frequency analysis results for all individual stars, a spectroscopic analysis of the fainter stars in the sample, and seismic modelling based on the observational results.

Acknowledgments

The research leading to these results received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement N° 227224 (PROSPERITY). Funding for the Kepler mission is provided by NASA’s Science Mission Directorate. We thank the whole team for the development and operations of the mission. This research made use of the SIMBAD database, operated at CDS, Strasbourg, France, and the SAO/NASA Astrophysics Data System. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France.

References

All Tables

All Figures

	Fig. 1 The target sample representation. The plot illustrates all frequencies that we consider as independent ones for all 69 stars. Note the different y-scales for the two panels.
In the text

	Fig. 2 Light curves (left) and amplitude spectra (right) of selected stars. From top to bottom: KIC 02710594, 03448365, 03744571, 04547348, 04749989, and 10080943. The y-scale of the left panel represents differential magnitude after subtraction of the corresponding mean value for each target. Note a mmag scale for KIC 04749989.
In the text

	Fig. 3 Distribution of effective temperatures (top left), surface gravities (top right), projected rotational velocities (bottom left), and metallicities (bottom right) of the sample targets.
In the text

	Fig. 4 Location of 36 stars (see Table 4) and the γ Dor and δ Sct (solid lines) theoretical instability strips (Dupret et al. 2005) in the HR diagram. Dashed thin and thick lines represent the γ Dor instability regions computed with α = 2.0 and 1.5, respectively. Open circles refer to the hybrid pulsators.
In the text

Fig. 5 vsini versus independent frequencies for the sample stars. Open triangles refer to the dominant oscillation mode for each star; the two other symbols represent all other independent frequencies. Filled boxes and open circles refer to γ Dor and hybrid pulsators, respectively. The solid black line shows linear fit to all data points (there are three frequencies between 18 (208.26) and 21 (242.97) d-1 (μHz) in the δ Sct domain); the red dashed line represents a linear fit to the frequencies in the γ Dor range after removing the δ Sct frequencies.

In the text

	Fig. 6 Same as Fig. 5 but for the amplitudes. The solid black line shows a linear fit to all data points; the red dashed line represents a linear fit after removing two stars with vsini> 150 km s-1 from the sample.
In the text

	Fig. 7 Frequency of the dominant mode as a function of vsini for δ Sct stars from the catalogues of Rodríguez et al. (2000, top and Uytterhoeven et al. (2011, bottom. The solid line represents a linear function fit to the shown data points. Detailed information on individual stars is provided in Table A.1.
In the text

	Fig. 8 Distributions of the total number of the detected harmonics (top) and those formed of independent frequencies only (bottom).
In the text

	Fig. 9 Distribution of the highest amplitude frequency detected in the Fourier spectra of the sample stars.
In the text