RELATIONSHIP BETWEEN LOW AND HIGH FREQUENCIES IN δ SCUTI STARS: PHOTOMETRIC KEPLER AND SPECTROSCOPIC ANALYSES OF THE RAPID ROTATOR KIC 8054146^*

Kepler observations have completely changed our understanding of stellar pulsations in the δ Sct instability strip. Earlier ground-based data had shown that γ Dor stars, which pulsate in frequencies typically below 5 day⁻¹, lie in a fairly small region on, or just above, the main sequence that partly overlaps the cool edge of the δ Sct instability strip. A few stars were known in which both γ Dor and δ Sct pulsations were present (the so-called hybrids). Kepler photometry has shown that practically all A and F stars, whether they are δ Sct stars or not, have low frequencies (Balona 2011). Thus the term "hybrid" has lost its meaning since nearly all δ Sct stars are hybrids. The convective blocking mechanism, which is thought to drive γ Dor pulsations, ceases to be effective for stars hotter than about 7500 K, yet low frequencies are found in the hottest A-type stars. It is, therefore, not possible to fully understand the low frequencies in the A-type stars as pulsation.

Statistical analyses of the low frequencies in A-type stars show that the frequencies are compatible with the expected rotational frequencies (Balona 2011). In fact, for the δ Sct star KIC 9700322, a secure identification of low-frequency peaks with stellar rotation was established from a variety of techniques: stellar line width from spectroscopy, the spacing of an observed ℓ = 2 p-mode quintuplet, and the timescales of the small amplitude modulation of the dominant modes (Breger et al. 2011; Breger 2011). The low and high frequencies in hybrid pulsators are also related through additional physical mechanisms, which are presently not fully understood. This relationship is seen in a number of δ Sct stars measured by space telescopes. An excellent example is the CoRoT star ID 105733033 studied by Chapellier et al. (2012), which shows a number of couplings between the low and high frequencies. Another star showing a number of numerical relationships between low and high frequencies is KIC 8054146, the star examined in the present paper. This star is interesting from another point of view as well: its very high rotational velocity (see below).

In general, asteroseismic studies of rapidly rotating stars of A and early F spectral type have been less successful than those of slow rotators. The reason is both theoretical and observational: rapid rotation cannot yet be reliably treated in theoretical modeling, while spectroscopically the extreme width of the spectral lines of the rapidly rotating stars makes detailed abundance analyses very difficult. Furthermore, observational studies (e.g., Breger 2000) have shown that in δ Sct stars low rotation is a prerequisite for high-amplitude radial pulsation, while the rapidly rotating δ Sct stars have small photometric amplitudes with mainly nonradial pulsation.

In this paper we shall study the asteroseismic properties of a very rapidly rotating A/F star, KIC 8054146, which is also a hybrid pulsator with both δ Sct and γ Dor properties (Uytterhoeven et al. 2011), i.e., a star with reported gravity and pressure modes.

The Kepler mission is designed to detect Earth-like planets around solar-type stars (Koch et al. 2010). To achieve that goal, Kepler is continuously monitoring the brightness of over 150,000 stars for at least six years in a 115 deg² fixed field of view. Photometric results show that after one year of almost continuous observations, pulsation amplitudes of 2 ppm (parts-per-million) are easily detected in the periodogram for stars brighter than V = 11 mag. Two modes of observation are available: long cadence (LC, 29.4 minute exposures) and short cadence (SC, 1 minute exposures). With SC exposures it is possible to observe the whole frequency range from 0 to 200 cycles day⁻¹ seen in δ Sct, γ Dor, and hybrid stars pulsating with both the δ Sct and γ Dor frequencies.

KIC 8054146 (K_p = 11.33) was extensively observed with the Kepler satellite in SC during quarters Q2.3 (BJD 245 5064–245 5091) as well as Q5 to Q10 (BJD 245 5276–245 5833), covering a time span of 27 days and 557 days, respectively. The large amount of accurate data covering more than a year leads to an unprecedented frequency resolution, permitting us to precisely examine the different frequency patterns (separations). The coverage is ideal to examine the relation between the low-frequency variability (possibly caused by gravity modes and stellar rotation) and the higher-frequency δ Sct pressure modes.

The Kepler SC photometry with a measurement per minute is available as calibrated and uncalibrated data. To avoid the known systematic errors present in the calibrated data, we only used the uncalibrated data. We performed our own reductions eliminating the zero-point trends, outliers, jumps, and known spurious effects such as those caused by the reaction wheels (spurious frequencies near 0.3 cycles day⁻¹). Our independently developed reduction process is similar to that published by García et al. (2011) for solar-type stars and the description will not be repeated here. Despite the careful reductions, any detections of very low frequencies below 0.3 cycles day⁻¹ should be treated with caution, but this does not affect the present conclusions. For Q8, a known spurious frequency, drifting from 31.5 to 32.2 cycles day⁻¹ with amplitudes around 60 ppm was taken out on a daily basis. A sample light curve is shown in Figure 1.

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The SC data were supplemented by LC data from quarters Q1 to Q4 (BJD 245 4964–245 5275). These measurements of two points per hour were reduced by us in a similar manner to the SC data. A comparison of the resulting LC and SC frequency spectra for the important Q2.3 time period gave an excellent average agreement to ±3 ppm for the low-frequency peaks, corresponding to the size of the formal uncertainties. Visual comparisons for subsequent quarters with both types of data confirmed that LC data can indeed be used for studies of low-frequency amplitude variability (only). This result is in agreement with a discussion of the characteristics of the Kepler data by Murphy (2012). Nevertheless, we emphasize that in this paper the LC data are used only to examine the low-frequency amplitude variability from Q1 to Q4.

The Kepler SC data of KIC 8054146 were analyzed with the statistical package PERIOD04 (Lenz & Breger 2005). This package carries out multifrequency analyses with Fourier as well as least-squares algorithms and the determination of significance of a peak does not rely on the assumption of white noise.

Following the standard procedures for examining the peaks with PERIOD04, we have adopted the amplitude signal/noise criterion of 4.0 (Breger et al. 1993). This standard technique is modified for all our analyses of accurate satellite photometry: the noise is calculated from prewhitened data because of the huge range in amplitudes of three orders of magnitudes. The extensive data led to very low rms noise levels in the Fourier spectrum: 1 ppm (0.5–7 cycles day⁻¹ range), 0.75 ppm (7–20 cycles day⁻¹), 0.6 ppm (20–30 cycles day⁻¹), 0.55 ppm (30–60 cycles day⁻¹), and 0.5 ppm (60–200 cycles day⁻¹).

We undertook extensive computations to eliminate the effects of amplitude and (small) frequency variations, while preserving the extremely high-frequency resolution. A large number of tests involving various amounts of prewhitening of the dominant frequencies were also undertaken with thousands of hours of computations.

We detected 349 statistically significant frequencies between 0.54 and 191.36 cycles day⁻¹. There is a clear separation in the density of frequency peaks between low and high frequencies: 116 frequencies have values smaller than 6 cycles day⁻¹, there is a single peak at 6.71 cycles day⁻¹, while 232 peaks are found between 7 and 192 cycles day⁻¹ (the high-frequency group). The results are shown in Figure 2, which presents the Fourier spectra in two different ways. The top panel ignores the small period and large amplitude variability of all frequencies and presents the average solution. The lower panel represents the average of separate Fourier analyses for each quarter and depicts the result with logarithmic amplitudes in order to emphasize the many low-amplitude peaks.

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The frequencies with average amplitudes of 5.0 ppm or larger are listed in Table 1. Due to the strong amplitude variability, these average amplitudes were not computed from a single solution, but were obtained from averaging the amplitudes of the three Q2, Q5–7, and Q8–10 time periods. The Q2 data cover only 27 days, but were given equal weight for determining the averages because of the dominance of f₁ and all its related sequences in the early quarters only. On the other hand, the lower frequency resolution of the Q2 data forced us to omit some close frequency pairs from the Q2 solution, explaining a small number of blank spaces in the table. The 120 frequencies in the table can only be a subset of the statistically significant frequencies and by themselves cannot reveal all the rich regularities present in the star. These will be covered below.

Table 1 also lists a number of identifications such as combination frequencies, harmonics, and other sequences, which will be covered below. We have restricted the listed identifications to only the main patterns, mostly involving the dominant frequencies f₁ to f₁₆. Multiple listings for each frequency component in a combination were also avoided in the interest of clarity. We also note the excellent frequency accuracy: the predicted and observed frequencies of the combinations fit within a few 0.0001 cycles day⁻¹.

Are there hundreds or thousands of additional frequencies that we have not detected? If we exclude the low-frequency region, in which the noise is definitely not white, we consider our amplitude limit of 2–3 ppm quite restrictive and can rule out a large number of additional modes with photometric amplitudes of this size or larger. In addition, no statistically significant peaks at higher frequencies up to the Nyquist frequency at 368 cycles day⁻¹ were detected.

The 117 low frequencies detected by us are not randomly distributed. They are mostly clustered in specific frequency bands: 1.6–1.8, 2.6–3.0, 3.5–3.7, and 5.2–6.0 cycles day⁻¹. We note that the last two bands are located at approximately twice the frequencies of the first two bands, although only some individual peaks obey a strict f, 2f relation. The clustering of the peaks is demonstrated in Figure 3. Since gravity modes of adjacent radial order, n, are expected to show equal spacing in period in the asymptotic limit, Figure 3 shows the distribution in period (inverse frequency). A cursory examination already reveals that equidistant period (or frequency) spacing is not dominant.

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For the hybrid pulsator, CoRoT ID 105733033, Chapellier et al. (2012) have discovered a remarkable sequence of asymptotically spaced frequencies with an equidistant period spacing of 0.0307 days. They interpret this to be caused by ℓ = 2 gravity modes of different radial order, n. Since for KIC 8054146 we also see a large number of closely spaced peaks at low frequencies, the question arises whether such gravity modes are also present. In KIC 8054146 the amplitudes of the low-frequency peaks are a factor of 10–100 smaller, but the extensive data set makes such a search possible.

However, a more cautious approach might be in order. Some aliasing exists in our Kepler data due to small gaps in coverage. Furthermore, the slow amplitude variability in this star can also lead to close frequencies. These effects need to be minimized for the low-frequency region with many close frequency peaks.

Consequently, we have repeated our previous analysis with some refinements. To reduce any effects of aliasing caused by the large gap between Q2 and Q5 as well as the small gaps in the Q5 to Q10 data, we omitted the 27 days of the Q2 data. This reduced the aliasing to four small peaks separated from the main frequencies by less than 0.01 cycles day⁻¹. To reduce the effects of amplitude variability of the dominant peaks, we calculated multifrequency solutions for the 10 dominant low frequencies with PERIOD04 allowing for quarterly amplitude and phase changes. The prewhitened data were then used to find additional, statistically significant peaks with amplitudes of 5 ppm or larger.

We then examined the four regions for regular frequency as well as period spacings of the individual peaks by constructing histograms of the observed frequency and period differences. Again, regular patterns could not be detected, in particular, the spacings observed by Chapellier et al. (2012) in another hybrid star were not found. We conclude that for KIC 8054146, gravity modes with equidistant period spacings were not detected.

The four dominant low frequencies in the 2.8–3.0 cycles day⁻¹ (32–35 μHz) show strong amplitude variability with timescales of months and years (Figure 4). An example is the peak at 2.814 cycles day⁻¹ (f₁), which reaches its maximum value during Q3, slowly decreases its amplitude during Q5 and is almost absent from Q6 to Q9. The 2f term follows this behavior. The 2.98 cycles day⁻¹ peak, however, appears in Q3, slowly grows and starts to disappear during Q9 and Q10. The behavior of the 2.93/5.86 cycles day⁻¹ group of peaks is more complex, and will be discussed below together with its related family of high-frequency modes.

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We have detected 232 statistically significant frequency peaks between 7 and 191 cycles day⁻¹. This large range in frequency is unusual for δ Sct stars. Not surprisingly, some of the frequency peaks are expected linear combinations of high frequencies and harmonic terms, but cannot explain most of the high frequencies.

The dominant high frequencies are three groups of triplets (T₁ to T₃), the difference between the T₁ and T₂ triplets (=25.952 cycles day⁻¹), and a "lonely" peak at 17.349 cycles day⁻¹. The triplets show very strong amplitude variability with a timescale of a year or longer. The most striking property of the detected frequency peaks is their separation in frequency, which often is identical to the previously detected low-frequency peaks.

A powerful method to look for regularities in frequency spacing consists of forming all possible frequency differences between all significant frequency peaks and to examine the histograms. The huge number of computed differences has to lead to some accidental agreements. In the present case, the high accuracy of the detected frequencies permits us to choose extremely narrow histogram boxes. Furthermore, we restrict the analysis to spacings smaller than 7 cycles day⁻¹. We have calculated all the frequency spacings between the 232 high-frequency peaks between 7 and 200 cycles day⁻¹. A histogram was computed with a very small box size of 0.0033 cycles day⁻¹. The resulting histogram with 2100 boxes is shown in Figure 5. We regard the result as remarkable.

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The four dominant low frequencies, f₁ to f₄, account for 136 frequency spacings among the 232 high-frequency peaks. A few of these spacings (within the narrow adopted range of only 0.0033 cycles day⁻¹) may be accidental. However, we believe the real number of exact spacings corresponding to observed low frequencies to be even higher because of our restriction to only simple agreements, e.g., spacings involving the (f₁+f₄) sum are ignored by our technique.

The preferred spacings involving f₁ to f₄ can be easily understood in terms of three families of pulsation modes. Each family behaves differently and may require different astrophysical explanations.

5.1. The f₂, f₃ Family and Triplets

Figure 6 shows the two triplets, T₁ and T₂, and their relationship to three low frequencies and the 25.9517 cycles day⁻¹ peak. All shown peaks have high amplitudes with a corresponding high accuracy of the frequency values (±0.0002 cycles day⁻¹). While data covering only one quarter would suggest equal separations, the two-year data made it possible to detect f₂ and f₃, which are clearly separated by 0.0040 ± 0.0003 cycles day⁻¹. This small difference has strong implications. We note:

• 1.  
  The two triplets have nearly identical properties, including amplitude variability. Their frequency separation of 25.9517 cycles day⁻¹ is also seen as a strong separate peak.
• 2.  
  The frequency differences between the central and right peaks correspond to an observed, dominant low-frequency peak at 2.9300 cycles day⁻¹ (f₂).
• 3.  
  The frequency differences between the central and left peaks correspond to an observed, dominant low-frequency peak at 2.9340 cycles day⁻¹ (f₃).
• 4.  
  The frequency differences between the left and right peaks correspond to an observed, dominant low-frequency peak at 5.8641 cycles day⁻¹ (f₂+f₃). This peak is not a harmonic of either f₂ or f₃.
• 5.  
  The amplitude changes of the f₂, f₃, and (f₂+f₃) peaks mirror the amplitude changes of the individual components of the triplets. The decrease of the f₂ and f₃ strengths after Q7 are closely mirrored by the decrease of the amplitudes of the central triplets components. On the other hand, the increase of the (f₁+f₂) amplitude after Q7 is mirrored by the increase in the non-central components of the triplets. (The lengthy detailed analysis of this phenomenon, which includes analyses of weekly amplitude and phase changes, will be the subject of a later report.)

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The observed measurements are incompatible with an interpretation that the 5.86 cycles day⁻¹ peak is a 2f harmonic of a mode near 2.93 cycles day⁻¹ and may challenge such interpretations in other hybrid stars. In Section 7 we will explore the possibility that the triplets are the counterparts of the ℓ = 1 triplets commonly seen in slowly rotating stars and the possibility of combination modes.

5.2. Equidistant S Sequences Involving f₁

5.3. Triplet 3 and Combinations with f₄

Three spectra of KIC 8054146 were obtained on 2011 June 29, July 16, and September 6 with the High Resolution Spectrograph (HRS) attached to the 9.2 m Hobby–Eberly Telescope at the McDonald observatory. This is a cross-dispersed echelle spectrograph yielding a resolving power (R) of 60,000. The signal-to-noise ratio (S/N) per pixel at λ ∼ 5000 Å is 70, 82, and 75, respectively.

The basic data reduction was carried out using the IRAF¹⁰ software suite. Bias subtraction was performed by removing an averaged zero frame for each night, and spectral images of a flat-field lamp taken with the same setup as used for our target observations were used to correct uneven pixel response across the two CCDs. Wavelength calibration (accurate to ∼6 × 10⁻⁵ Å) was derived from a ThAr emission lamp observed shortly before or after our stellar spectra were taken. Cosmic rays were identified as outliers to two-dimensional fits to the spectral orders and removed. In rare cases where a cosmic ray fell across the echelle order, defeating our normal rejection routine, we have removed it with IRAF's COSMICRAYS utility. The finalized spectrum for each night was produced by coadding the three cosmic ray splits, resulting in a total effective exposure time of 60 minutes.

After the application of the heliocentric correction, we checked that no radial velocity variation was present among the three acquired spectra. This excludes the possibility that the star is a member of a close binary system. This allowed us to average the three acquired spectra to increase the S/N, reaching an S/N of 131 for the final spectrum.

The average spectrum, normalized by fitting a low-order polynomial to carefully selected continuum points, covers the wavelength range 4100–7900 Å, with a gap between 5927 Å and 6024 Å, because one echelle order is lost in the gap between the two chips of the CCD mosaic detector. Given the very high projected rotational velocity (υ sin i) of the star, we adopted the hydrogen lines as primary indicators for the effective temperature T_eff determination. The normalization of hydrogen lines, observed with echelle spectrographs, gives the largest contribution to the total error budget in the T_eff determination by hydrogen line fitting. To reduce this uncertainty we normalized the Hα and Hβ lines individually for each of the three spectra, obtaining in this way six quasi-independent measurements of T_eff. We normalized the hydrogen lines making use of the artificial flat-fielding technique described in Barklem et al. (2002).

To compute model atmospheres, we employed the LLmodels stellar model atmosphere code (Shulyak et al. 2004). For all the calculations, local thermodynamical equilibrium and plane-parallel geometry were assumed. We used the VALD database (Piskunov et al. 1995; Kupka et al. 1999; Ryabchikova et al. 1999) as a source of atomic line parameters for opacity calculations with the LLmodels code. Finally, convection was implemented according to the Canuto & Mazzitelli (1991a, 1991b) model of convection. For more details see Heiter et al. (2002).

We performed the T_eff determination by fitting synthetic line profiles, calculated with SYNTH3 (Kochukhov 2007), to the observed Hα and Hβ line profiles. For each spectrum we derived that the best-fitting T_eff is 7600 K for the Hα line and 7700 K for the Hβ line, with a typical uncertainty of 150 K.

To decrease the uncertainty even more on the effective temperature due to the normalization, on 2011 September 2 we obtained a spectrum of KIC 8054146 centered on the Hα line with the Cassegrain low-resolution spectrograph attached to the 1.8 m telescope of the Dominion Astrophysical Observatory (DAO), Canada. The adopted configuration of the spectrograph provided a resolving power of about 15,000 and the obtained spectrum, covering the 6480–6780 Å wavelength region, has an S/N per pixel of 60, calculated at about 6600 Å. Since the spectrum covers the entire Hα line profile within a single order, the normalization line is a simple first-order polynomial that can be safely determined by using the available continuum regions on either side of the hydrogen line. This decreases greatly the uncertainty on the measurement of T_eff due to the continuum normalization. By fitting synthetic line profiles to this low-resolution spectrum, we derived a T_eff of 7500 ± 200 K, where the uncertainty is mostly due to the quality of the spectrum.

By averaging all these T_eff measurements, we finally adopted T_eff = 7600 K with a formal uncertainty of 80 K, which is not realistic, as described later. Figure 7 shows a comparison between the observed HRS and DAO spectra with synthetic spectra, calculated with the adopted T_eff of 7600 K and a T_eff of 7400 K, for comparison. In the procedure of fitting synthetic hydrogen line profiles to the observed spectra, we adopted a fixed value of log g = 3.9 (cgs) for the surface gravity and of 2.5 km s⁻¹ for the microturbulence velocity (υ_mic). All model atmospheres were produced assuming the solar chemical composition (Asplund et al. 2009). With this set of fundamental parameters, by fitting the most prominent blends we produced synthetic spectra of the available spectral region and measured a radial velocity (υ_r) of 31 ± 5 km s⁻¹ and a υ sin i of 300 ± 20 km s⁻¹.

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Following the procedure described in Fossati et al. (2007), we attempted a measurement of the iron abundance from a few Fe i and Fe ii lines, but the limited S/N of the spectrum, the extreme blending, and the uncertainty on the continuum normalization prevented us from obtaining any useful result. As a consequence, we were not able to determine any reliable log g value by means of the ionization equilibrium. At the temperature of KIC 8054146, the two analyzed hydrogen lines display a little reaction to gravity variations, which allowed us to determine at least an upper and lower log g value of 4.2 and 3.4, respectively. The best-fitting log g value is 3.9, indicating that the star is likely to be still on the main sequence. A more precise log g value could be obtained either with narrowband photometry,¹¹ or with a distance value, or with calibrated photometry spread over the region of the Balmer jump, but none of this is currently available. A more realistic value of can then be obtained by taking into account the limits given for log g, leading to  = 200 K.

The large uncertainties on the abundance values derived from the most prominent blends also made it impossible to determine a reliable value, which we therefore kept fixed at 2.5 km s⁻¹, typical of late A-type stars (see, e.g., Fossati et al. 2008). Figure 8 shows a comparison between the LLmodels theoretical fluxes, calculated with T_eff = 7600 K and log g = 3.9, and Johnson BV and Two Micron All Sky Survey (2MASS) JHK photometry. This comparison confirms that the derived T_eff fits well the available broadband photometry and the absence of infrared excess excludes the presence of warm dust around the star.

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In conclusion, the spectroscopic analysis allowed us to determine that KIC 8054146 is a very fast rotating late A-type star, which is likely to have a metallicity close to that of the Sun.

We have shown from the Kepler data that the low frequencies (up to 7 cycles day⁻¹) and the high frequencies (up to 200 cycles day⁻¹) are related in several families through preferred frequency spacings, frequency patterns, and similar amplitude variations in the individual families. Consequently, an explanation in terms of independent gravity and pressure modes appears too simple. The analyses of stars of spectral types A and F with the Kepler spacecraft have shown that the light output is modulated by the rotational frequency (Balona 2011). Even a very "simple" A star with radial pulsation modes and a very low rotation rate of ∼20 km s⁻¹ already shows complex rotational variations including small amplitude modulations of the radial modes (Breger et al. 2011). Consequently, a very rapidly rotating star such as KIC 8054146, presumably with differential rotation, might be very interesting. What is the rotation frequency of this star?

Based on the mean values of T_eff (7600 K) and log g (3.9) measured in the previous section we obtain a radius of about 1.9 R_☉ for the star (assuming a mass of 1.8 M_☉). Together with the mean measured value of the projected rotational velocity, υ sin i = 300 km s⁻¹ and assuming an equator-on view (i = 90^o) we derive a rotational frequency of 2.87 cycles day⁻¹ (corresponding to 0.75 of the Keplerian breakup rate). Taking into account the given uncertainty of 20 km s⁻¹ in υ sin i, we obtain a rotational frequency of 2.73 cycles day⁻¹ (i.e., 0.69 of Keplerian breakup rate) for the lower limit of υ sin i = 280 km s⁻¹. The rotational frequency is, therefore, very similar to that of the dominant low frequencies between 2.8 and 3.0 cycles day⁻¹, but the uncertainties exclude the possibility of identifying any one of the peaks as the rotational frequency.

Due to the high value of the surface rotation velocity near breakup velocity, we can conclude that it is more likely that the star is viewed nearly equator-on. This fact helps us to reduce the uncertainties in the fundamental parameters, since, due to gravity darkening, parameters such as the effective temperature are a function of aspect in rapidly rotating stars. Consequently, the measured temperature and gravity values are essentially the equatorial values. At the poles, we expect gravity brightening, since both the T_eff and log g values are higher there.

7.1. Rotationally Split ℓ = 1 Triplets?

7.2. Eclipses?

7.3. Spots and Surface Inhomogeneities

7.4. High-degree Nonradial Gravity Modes

7.5. Asymptotic Pulsation

7.6. Conclusion

M.B. is grateful to E. L. Robinson, K. Zwintz, and M. Montgomery for helpful discussions. This investigation has been supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung through project P 21830-N16. The authors wish to thank the Kepler team for their generosity in allowing the data to be released to the Kepler Asteroseismic Science Consortium (KASC) ahead of public release and for their outstanding efforts which have made these results possible. Funding for the Kepler mission is provided by NASA's Science Mission Directorate.