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A pulsation zoo in the hot subdwarf B star KIC 10139564 observed by Kepler

We present our analyses of 15 months of Kepler data on KIC 10139564. We detected 57 periodicities with a variety of properties not previously observed all together in one pulsating subdwarf B star. Ten of the periodicities were found in the low-frequency region, and we associate them with nonradial g-modes. The other periodicities were found in the high-frequency region, which are likely p-modes. We discovered that most of the periodicities are components of multiplets with a common spacing. Assuming that multiplets are caused by rotation, we derive a rotation period of 25.6(1.8) days. The multiplets also allow us to identify the pulsations to an unprecedented extent for this class of pulsator. We also detect l<=2 multiplets, which are sensitive to the pulsation inclination and can constrain limb darkening via geometric cancellation factors. While most periodicities are stable, we detected several regions that show complex patterns. Detailed analyses showed these regions are complicated by several factors. Two are combination frequencies that originate in the superNyquist region and were found to be reflected below the Nyquist frequency. The Fourier peaks are clear in the superNyquist region, but the orbital motion of Kepler smears the Nyquist frequency in the barycentric reference frame and this effect is passed on to the subNyquist reflections. Others are likely multiplets but unstable in amplitudes and/or frequencies. The density of periodicities also make KIC 10139564 challenging to explain using published models. This menagerie of properties should provide tight constraints on structural models, making this subdwarf B star the most promising for applying asteroseismology.


INTRODUCTION
Hot subdwarf B stars (hereafter: sdB) are horizontal branch stars that consist of helium-rich cores surrounded by thin hydrogen envelopes. The helium core in each sdB star sustains nuclear burning of helium into carbon and oxygen, though in some cases the core may have exhausted helium at the center. Their progenitors are lowmass, main sequence stars, and each is likely to have undergone a core helium flash (M 2M⊙). Once hydrogen in the core is exhausted, it evolves towards the tip of the red giant branch, where an sdB progenitor retains less than 0.01M⊙ of hydrogen during (or prior to) the helium flash. Those red giant stars lie on the high T eff (∼30 000K) end of the horizontal branch. This small fraction of hydrogen has an influence on the future evolution of sdB stars. Since hydrogen-shell burning cannot be sustained by a thin hydrogen envelope, the stars will move directly to the white dwarf cooling track instead of going through the asymptotic giant branch phase. Several mechanisms for producing this small residual hydrogen envelope involving single-star or binary evolution have been proposed; e.g. D'Cruz et al. (1996); Han et al. (2002Han et al. ( , 2003. The sdB stars have effective temperatures and surface gravities near 30 000 K and log g ∼ 5.5, respectively. It has been more than 14 years since the discovery of the first pulsating sdB star (hereafter sdBV, Kilkenny et al. 1997). This star and several other stars discovered later were dominated by short period oscillations, which were identified as pressure (p-modes) modes by Charpinet et al. (1997). They are driven in the envelopes and have allowed us to use asteroseismology to study the interiors of sdB stars. Several years later, Green et al. (2003) reported the discovery of another kind of variation in sdB stars. These brightness changes were smaller in amplitude and longer in period. According to the work done by Fontaine et al. (2003), the new, longer period variations were assigned to gravity modes (g-modes) which dominate cooler, lower-gravity sdB stars.
The g-modes in sdB stars probe deeper than p-modes and should allow us to study different parts of sdB stars. Recently, Reed et al. (2011) used g-mode period spacings for mode identifications in sdB stars. It was found that l = 1 period spacings were roughly 250 sec while l = 2 spacings were approximately 150 sec. There are sdB stars pulsating in both p-and g-modes at the same time. These stars are the most favorable, since they allow us to probe larger parts of those stars. At first, only p-mode dominated, hybrid sdBV stars were detected from the ground, but the Kepler spacecraft has discovered several hybrid stars where the g-modes dominate across the entire amplitude spectrum ; Baran et al. 2011a).
Nonradial pulsations can be described using spherical harmonics with quantized indices n, l, and m, where n corresponds to the radial order of the mode, l is the number of nodal lines on the surface, and m is the number of those lines passing through the poles. Following the formalism given in Ledoux (1951), σ n,l,m = σ n,l,0 + ∆σ n,l,m = σ n,l,0 + mΩ(1 − C n,l ) where σ n,l,m is the shifted frequency and σ n,l,0 is the (degenerate) frequency without rotation. If the Ledoux constant C n,l is estimated and the multiplet splittings ∆σ n,l,m are measured, we can derive the rotation frequency Ω. It should be remembered that this way, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. † E-mail:sfbaran@cyf-kr.edu.pl equation is valid for slow, solid-body rotators with pulsation periods much shorter than the rotation period. In the absence of rotation, all modes of the same degree but with differing m values are present in pulsating stars, but they are degenerate in frequency, though they differ in angular eigenfunctions. In this case, we only see the σ n,l,0 frequency. However, rotation lifts the degeneracy in m, resulting in (2l + 1) periodicities with different frequencies.
Periodicities with positive and negative m are shifted in the opposite direction of σ n,l,0 . The m = 0 frequency remains unaffected by rotation.
It is generally accepted that p-modes detected in sdB stars are of low order and of low degree. The Ledoux constant for such modes is typically close to zero. However, for the fundamental and first overtone l = 2, 3 modes, C n,l may have values up to 0.3 (Charpinet et al. 2002). This increase in the Ledoux constant creates a decrease in the frequency splittings. In the case of high-order g-modes, the Ledoux constant depends on the l parameter according to the following equation: C n,l (l 2 + l) −1 . It is around 0.5 for l = 1 and close to 0.16 for l = 2. If we assume a star rotates as a solid body, the ratio of the Ledoux constants between the p-and g-mode regions can be used to constrain the degree of the g-modes. A p/g splitting ratio of 0.55 indicates l = 1 g-modes and a ratio of 0.81 indicates l = 2 g-modes.
Observations of long period sdBV stars, including detection of new pulsators, are not easy from the ground. Relatively long period photometric variations (with only a few cycles per night) can be affected by variable sky transparency. To make a definitive determination of the pulsation frequencies requires extended photometric campaigns, preferably at several sites widely spaced in longitude to reduce day/night aliasing. However, Kepler provides long-duration, continuous, homogenous, evenly spaced timeseries photometry, making it an ideal instrument for asteroseismology. The Kepler science goals, mission design, and overall performance are reviewed by Borucki et al. (2010), Koch et al. (2010) and Jenkins et al. (2010). This is currently the state-of-the-art way to observe variable stars. With this instrument, we can reach down to 19 mag and still achieve better signal-to-noise (S/N) observations than from the ground.
In this paper, we present analyses on KIC 10139564 (2MASS 19245816+4707536), a p-mode dominated sdBV star detected in Kepler data. Our analyses are completed on combined Q5 through Q9 data spanning 15 months of continuous observations. Such long, continuous coverage -never achieved before for compact pulsators -significantly improves the S/N ratio in the amplitude spectra. This coverage allowed the detection of more frequencies and helped to improve mode identification, as compared to the preliminary results from the one-month survey data (Kawaler et al. 2010). In addition to Kepler data, we collected spectroscopic data, which we present in the next section.

SPECTROSCOPIC DATA
On the nights of 31 May (2 spectra), 19 June (2) and 23 July 2010 (6), we obtained 10 spectra in total of KIC 10139564 to monitor possible radial-velocity variations and, thus, check its binary status. We used ALFOSC at the Nordic Optical Telescope with grism #16 that yield spectra of the H β -H θ region with dispersion of 0.77Å/pix and a resolution of 2.2Å, with a median S/N = 48 per pixel. The exposure times were 900 seconds. The spectra of 23 July were not taken consecutively so cover 0.3 days.
The data were homogeneously reduced and analyzed. Standard reduction steps within IRAF were bias subtraction, removal of pixel-to-pixel sensitivity variations, optimal spectral extraction, and wavelength calibration based on arc lamp spectra. The target spectra and mid-exposure times were shifted to the barycentric frame of the solar system. The spectra were normalized to place the continuum at unity by comparing with a model spectrum for a star with similar physical parameters as we find for KIC 10139564.
Radial velocities were derived with the FXCOR package in IRAF, with resulting errors of about 10 km/s per spectrum. We used the lines of H β , Hγ , H δ and H ζ to determine the radial velocities.
We find an average radial velocity of 1.0 ±2.9 km/s and find all data points except the two from 31 May consistent within 1σ with average radial velocity. The two measurements of 31 May combine to a radial velocity of 15.5 ±7.1 km/s. We conclude that there is no clear evidence for orbital radial-velocity variations for periods shorter than half a day.
The 10 individual spectra were combined into an average spectrum with S/N∼150, which was used for an improved determination of the atmospheric parameters of KIC 10139564. We fitted the mean spectrum using the same method and grid of models as that used in Østensen et al. (2010a). (For a description of these models, see Heber, Reid & Werner 2000). The fit is shown in Figure 1. The physical parameters resulting from the fitting procedure are listed in Table 1, and places KIC 10139564 at a temperature some 600 K lower and at a surface gravity 0.14 dex lower than that obtained by Østensen et al. (2010a) whose data were based on a single low S/N spectrum. Figure 1. The fit of an LTE model spectrum to a mean of 10 spectra obtained with the NOT/ALFOSC. Note that the Hǫ line was kept out of the fit due to interstellar Ca II contamination.

PHOTOMETRIC DATA
All data included in this paper were obtained through the Kepler Asteroseismic Science Consortium (KASC) 1 . Since pixel data for Q6 and Q8 are still not available to us, we used flux measurements already calculated using default apertures defined by the in-house extraction pipeline (Jenkins et al. 2010). We have chosen to use the raw fluxes, which are calibrated for instrumental effects (bias, flatfield, cosmic rays) but not corrected for on-board systematics or flux excesses (often called contamination). Contamination does not influence our main goal, since we rely on periods and not amplitudes. We have corrected the data for on-board systematics to remove discontinuities in the data. Table 1 contains an observational report and the number of points used in our analysis. It is worth noting the unprecedented length of continuous coverage which cannot be obtained from the ground. The duty cycle is 92%. We decided not to plot the photometric data, since they do not differ from those presented in the previous paper by Kawaler et al. (2010) other than being substantially longer in time.
We analyzed short cadence (SC) data. Each data point is a sum of nine integrations of 6.02 s followed by a 0.52 s readout, resulting in an SC sampling of 58.85 s. This translates into a Nyquist frequency of 8496.356 µHz in the spacecraft reference frame. Time stamps given in that frame were corrected to the barycenter of the Solar System. This correction has smeared the Nyquist frequency because data are no longer evenly sampled.
In preparing data for Fourier analysis, we first removed all SC points flagged with errant times or fluxes. Then, all remaining points were subject to detrending which was done via a cubic spline fit calculated on 0.5 day subsets and subtracted from the original data. Detrending with this interval will strongly suppress all frequencies below 12 µHz. Then we removed all outliers using 3.5σ clipping before the data were subjected to Fourier analysis. We did not include the 30 days of data from the survey phase, since they were too temporally separated from the data presented here, which could cause a complex pattern of peaks in the amplitude spectra and hinder our prewhitening process.
During Q5, Q6, Q7 and Q9, there were no safe-mode events, and cadences were only lost while the spacecraft was not in fine pointing (which occurred during monthly data transmissions to Earth). This makes these four quarters failure free. Unfortunately during Q8, the spacecraft experienced three safe-mode events: one which delayed the start of Q8 by about two weeks, one which caused a three-day gap, and one which prematurely ended Q8 observations 6 days early.

AMPLITUDE SPECTRUM
We used Fourier analysis to identify candidate pulsation frequencies. Then we completed a nonlinear least-squares fit, including each periodicity in the form Ai sin(ωit + φi). This iterative process followed the standard prewhitening procedure and continued until all peaks with amplitudes above a threshold of S/N = 4 had been removed. While a threshold of S/N=4 is the commonly-used limit for sdB pulsations, it is not a strict limit and so frequencies near this limit (S/N 4.5) may be considered tentative, and are marked as such in Table 2. Continued observations by Kepler may be used to confirm or refute them. The frequency resolution is 0.0375 µHz, defined as 1.5/T, where T is the time baseline of the data. We have coverage fifteen times longer than the 30 days of survey data, and the noise level in the amplitude spectra are lower roughly by a factor of √ 15 = 3.87. When the spacecraft is rolled every 3 months, the stars move onto different CCDs. Moreover, any deviation in pointing accuracy and other on-board systematics may cause different flux levels between quarters, which in turn leads to a different amplitude of light variation. Although we detrended the data using 0.5 day subsets and 4σ clipping, it is possible that instrumental systematics remain in the data. Such instrumental flux variations would decrease the effectiveness of prewhitening and manifest themselves as increased residuals and residual peaks with fork-like structures (multiple, closely-spaced peaks). Since many of the residuals in KIC 10139564 do show fork-like structures we examined other options to further correct the data for instrumental flux changes. The best possible method would be to use a source external to the spacecraft, such as the star itself, for systematic flux corrections. Pulsation amplitudes can be used for this purpose, so long as the amplitudes themselves are stable with time. While many sdB pulsation amplitudes are variable, some do appear to be remarkably stable (see §4.2 of Reed et al. 2012). As such, we sought to correct these data using the amplitudes of the pulsations themselves. While this re-normalization could affect pulsation ampli-tudes, we are not interested in absolute amplitudes but rather the best possible frequency solution, one measure of which are residuals after prewhitening. The procedure we used was to split the data into month-long segments and fit the highest-amplitude pulsations. We then used those amplitudes to correct the original data on a month-by-month basis and re-prewhitened and analyzed the resultant residuals. We did this normalization one high amplitude pulsation frequency at a time. Only for the peak f18 (Table 2) did we achieve significant improvement. An example region is shown in Figure 2. In this figure, the top panels show the unnormalized data and residuals while the bottom panels show the same for the amplitude-normalized data. The frequency solution is the same in both cases but the normalized residuals are smaller with less forklike structure. The detection thresholds (horizontal dotted lines) for un-and normalized data are equal to 0.034 and 0.033 ppt, respectively. Since the residuals were reduced for all frequencies, any amplitude changes in f18 were most likely systematic and not intrinsic to that peak. For KIC10139564 this procedure did not change the frequency solution and so the sole benefit was in reducing several of the residual peaks below the detection threshold and removing some of the fork-like structures, which is a purely cosmetic result.
For this amplitude-normalized dataset we calculated the detection threshold as four times the average level of the residual spectrum in the range of 230 µHz to the Nyquist frequency. The detection threshold is 0.033 ppt (parts per thousand). but can deviate locally by 3% when using 500 µHz bins in signal-free regions. Please note that observed amplitudes are smeared by a finite sampling, leading to a decrease of 15-20% for most periods and up to 35% for the shortest periods.
KIC 10139564 is the only short period dominated (p-mode) pulsator among sdBV stars detected by Kepler. Based on 30 days of survey data, Kawaler et al. (2010) noted that the amplitude spectrum of this star is dominated by several peaks between 160 and 200 s (6200 and 5050 µHz) with the highest amplitude being about 9.9 ppt. One single peak in the low-frequency region was also mentioned, but it had yet to be confirmed. Our analysis confirms all signals reported in the previous analysis. However, our frequency resolution is fifteen times better than in the survey data, allowing us to better resolve crowded peaks that lead to small changes in frequencies. In total, we have identified and successfully fit 57 periodicities with amplitudes above our detection threshold of 0.033 ppt ( Table 2). Note that some signals prewhitened in the previous analysis were not prewhitened in ours. With our improved resolution, we found that the signal in some regions is very complex and difficult to fit. We discuss these regions in §5.5. Table 2. The frequency list as a result of our prewhitening process for KIC 10139564. The detection threshold is 0.033 ppt. The numbers in parentheses are the 1-σ fitting errors of the last digits. This list contains only peaks prewhitened from our data, which does not include some complex signal patterns described in §5.5 and LC artifacts. Observed amplitudes are smeared by a finite sampling, leading to a decrease of 15-20% in most periods and 35% for the shortest periods. The asterisks denote tentative peaks with S/N 4.5. The single horizontal line separates low and high-frequency regions. Columns six and seven provide l and m identification as a result of our analysis. This identification is based on full or partial multiplets. The last column, denoted K10, provides the notation of frequencies used in Kawaler et al. (2010).

Signal in the low-frequency region
In the low-frequency region, we have detected ten peaks (fA to fJ). All low-frequency peaks are listed in the upper part of Table 2 and are shown in Figure 3. We associated all of them with g-modes.
In each region, the signal is split into two or three significant peaks as seen in Figure 4. The frequency separations between peaks are similar and includes periodicities with amplitudes that have S/N<4, shown in Figure 4. In Figure 5, we plot the window function to show that the splitting is not a result of the aliasing pattern. Since rotation can remove degeneracies in pulsation frequencies of modes with the same l but different m value, we suggest that these are rotationally split multiplets. With two or three peaks observed, we can consider them to be l = 1 triplets; however with more data, it is possible they may turn into quintuplets. Two frequencies fD and fE are different. They show larger splittings of about 0.72 µHz. An insignificant frequency (S/N=3.8), marked with a vertical dash-dotted line, between them could create a multiplet with a splitting of 0.36 µHz on average. This value is expected for a quintuplet since the Ledoux constant is 67% smaller for l = 2 modes. These frequencies also overlap an artifact listed in Data Release Notes 12 Notes as "wide" and it is possible they are not intrinsic to the star. However, that artifact was discovered by one of us (ASB) in Q0 data. In addition, our two peaks do not resemble the artifact found in Q0, so we infer them to be produced by the star.
All peaks except fD and fE are organized into three groups with the separations between them of 610 and 630 sec, respectively. This is not surprising, since g-modes should be equally spaced in period in accordance with asymptotic theory. In KIC 10139564 we could have a sequence of l = 2 modes but only with every fourth multiple (620/4=155) detected. However we assigned them as l = 1 modes based on their multiplet structures, and that makes them a sequence with spacings of (620/2=)310 or (620/3=)207 sec, if only the second or third multiples were detected, respectively. While the l = 1 period spacings would not be consistent with other pulsating sdB stars, those stars are g-mode dominated while KIC 10139564 is p-mode dominated and the frequency splittings indicate these gmodes are likely l = 1.

Signal in the high-frequency region
The high-frequency region contains 47 significant peaks. We show this region in Figure 6. The highest peak has an amplitude of 7.950 ppt and four other peaks are between one and six ppt. Most of the peaks, however, are of low amplitudes, barely exceeding S/N = 10. Signals prewhitened from our data are denoted as p, LC means long cadence artifacts and s stands for complex signal regions, which we were not able to remove from the data using standard techniques. LC artifacts are peaks at the long cadence frequency (566.41 µHz) and its harmonics up to the Nyquist frequency. They were identified during the first year of the Kepler mission, and their frequencies are given in the Kepler Instrument Handbook (KIH). Besides LC artifacts, Kepler data may also contain other spurious peaks. Some of them have been identified in KIH, but there are others that are still not documented very well. Baran et al. (2011) reported another set of artifact signals detected in many stars analyzing Kepler public data. It is also possible that artifacts may come and go, and a frequency of a specific artifact    Figure 6. The high-frequency region of KIC 10139564. This plot is truncated at 0.5 ppt to show low-amplitude peaks. Labels help to see how this region is organized. We found at least one strong LC artifact, many peaks we could prewhiten (p) and complex signal (s) which we left untouched.
may also change. In the amplitude spectra we have detected all LC harmonics starting from the 6 th up to the 14 th , which is the Nyquist frequency.
The first few periodicities (f1 to f4) are well separated from the bulk of p-modes. Frequencies f1 and f2 appeared to be singlets. At 3540 µHz, we fit two significant peaks (f3 and f4) as seen in Figure 7. In addition, two other peaks seem likely to represent real signal (particularly the peak close to 3541 µHz with S/N = 3.6), but we need more data to confirm this. Both significant peaks have low amplitudes and the separation between them is 0.9755 µHz. Estimating frequencies for the two other peaks and calculating separations reveals that all four peaks create an almost equally spaced quadruplet. We found a similar case at 5049 µHz shown in Figure 8. We detected two low-amplitude, yet significant, peaks (f5 and f6) separated by 0.9682 µHz with another insignificant (S/N=3) peak 0.482 µHz away from f6. The separations between peaks in the group at 3540 and 5049 µHz are similar. The peak at f7 stands alone  Figure 7, we found two low-amplitude peaks f 5 and f 6 and another peak slightly below the significance limit. It is worth noting that separations between these peaks are comparable to the peaks in the Figure 7. and barely exceeds the detection threshold. If f7 is a part of a multiplet, other components are currently undetected. The region presented in Figure 9 contains a clearly resolved quintuplet (f8 to f12) with separations of ∼ 0.41 µHz, comparable to those in the previous p-mode regions. As may be seen in the residual spectrum (bottom panel), all peaks prewhitened with almost no residual signal left. Figure 10 shows the region near 5412 µHz that also contains five significant peaks (f13 to f17). However, these peaks do not create an evenly spaced structure, yet the separations look familiar. One separation is ∼ 0.4256 µHz, while the other three appear approximately twice that near 0.9 µHz. In addition, after all significant peaks were removed, we found another peak (with S/N=3.9) between f13 and f14 that may fit into this sequence. Two possible interpretations exist. Either all the peaks create one large l >3 multiplet with separations of ∼0.44 µHz or, as we prefer, f13 to f15 are part of a quintuplet, f16 and f17 are part of a triplet and the two multiplets just happen to be separated by ∼0.9 µHz.
The peak f18 shown in Figure 11 is different. It seems to be single with some low-amplitude complex signal at slightly lower frequencies. The amplitude of this peak was used to normalize data   Figure 13. The frequency region near 5748 µHz with five peaks (f 28 to f 32 ).
Peaks f 30 to f 32 might be part of a quintuplet, since the separation between them is twice the smallest splitting in other multiplets. between individual months, which significantly removed residuals. This is the only single high amplitude peak we have detected in our data, and its lack of multiplets, despite its high amplitude, leads us to conclude it is a radial mode. In the residual spectrum (bottom panel of Figure 11), we notice an obvious and very wide signal excess. We did not try to prewhiten any peaks from this "forest". It resembles other regions of peaks between 5900 and 6400 µHz. The signal at 5572 µHz, shown in Figure 12, might be of special interest. We detect five peaks, f19 to f23, with an average spacing of 0.45 µHz assuming that two peaks just below the detection threshold are intrinsic to KIC 10139564. The residual spectrum has a peak at the expected frequency to complete the sequence but it is not significant (S/N=3.41). This region could contain an l = 3 septuplet. Thus far, the detection of l>2 modes was generally regarded as unlikely, since these modes suffer from large surface cancellation effects, making their observed amplitudes greatly reduced, assuming the intrinsic amplitudes to be similar. With Kepler data we can reach down to the ppm level, making the detection of septuplets possible. More data are definitely needed to confirm the nature of the peaks in this region, but it is a good candidate for the first detection of l 3 modes in any sdB star.  Figure 15. The amplitude spectra of the highest amplitude periodicities (f 33 to f 35 ) calculated from data split into quarters. Notice the amplitude shift between the outer frequencies. In Q8 the peak on the right is slightly higher than the one on the left.
We do not show the next region at 5618 µHz, since it only contains two low amplitude peaks f24 and f25 separated by 2.482 µHz. It may be the case of another septuplet with the two peaks being the first and last in a sequence of seven modes separated approximately by 0.42 µHz. The residual spectrum shows one peak near the detection threshold and fits the potential sequence well. Similarly, only two low-amplitude peaks f26 and f27 were found at 5709 µHz. More data are necessary to determine the nature of these two regions.
Similar to the region shown in Figure 12, we have detected a triplet at 5748 µHz (f30 to f32) with separations of 0.98 µHz on average ( Figure 13). This could easily fit a quintuplet with intrinsic splittings of 0.49 µHz, similar to other multiplets. In addition, f28 and the peak (S/N∼4) at 5747.55 µHz could potentially fit this sequence. Another option could be that f28 to f29 are part of a different multiplet with shorter separations of 0.3442 µHz which is similar to the multiplets found at 5840 and 6235 µHz.
The next p-mode region is shown in Figure 14. It is centered  around 5760 µHz and contains three significant peaks (f33 to f35).
Since radial modes suffer no geometric cancellation, it is usually anticipated that the highest amplitude modes are, therefore, radial. This does not seem to be the case for KIC 10139564. All three peaks are spaced by the distance of roughly 0.42 µHz, forming a triplet. These spacings in the highest amplitude frequencies provide assurance that this frequency spacing, which is common to many regions of KIC 10139564, is intrinsic and related to rotationally split multiplets. The residual spectrum shown in the bottom panel of Figure 14 indicates that all those peaks are unstable in their amplitudes/phases. In Figure 15 we show how the peaks change their amplitudes between quarters; in Q8 the amplitude of the right peak was the highest. The region at 5840 µHz contains low amplitude peaks f36 and f37 with very complex shapes shown in Figure 16. They appear to be suffering from amplitude/phase variations or as unresolved frequencies, which makes our fitting process difficult and our estimation of their frequencies and amplitudes less precise. Regardless, we have fitted f36 and f37 with a separation of 2.1953 µHz. Other peaks in this region cannot be fitted. We will discuss this region in detail in §5.5.
The region at 6001 µHz is one with complex signal patterns and is described in Sect.5.5. Two frequencies f38 and f39 look relatively high and stable in amplitudes, so we fitted and prewhitened them (listed in Table 2). The next two regions around 6076 µHz (f40) and 6106 µHz (f41) are not shown since we fitted just one significant low amplitude peak each. The region at 6235 µHz shows three significant peaks (Figure 17), but their structure is very complex. We could easily prewhiten two of them (f42 and f43), leaving the third one untouched. The residual spectrum clearly shows amplitude and/or phase variation of all peaks. The smallest (lowest frequency) signal has a fork-like appearance, which may mean it is unresolved in the original spectrum, and we decided not to make an attempt to remove it. The spacing between the two prewhitened peaks is slightly smaller than the common value of 0.45 µHz, as derived in other regions. This deviation can be explained by a difference in the Ledoux constants ( §1). The remaining signal appears at exactly the right frequency to fit a triplet, leading us to conclude this region is another rotational multiplet. There is another significant signal in the amplitude spectrum around 7633 µHz with a single frequency f44 detected.
The region at 8118 µHz is the highest frequency region with three peaks f45 to f47 that we could easily prewhiten. In fact, it is   Figure 18. The frequency region around 8118 µHz with three peaks detected (f 45 to f 47 ). The rightmost peak is very wide and is likely unresolved, as the residual seems to confirm. Including the two near the detection limit, this region form a quintuplet.
very close to the Nyquist frequency, leaving some doubts whether this region is mirrored across that frequency. If so, the real signal could appear at approximately 8874 µHz instead. Similar to the first region at 3540 µHz, this region is also well separated from the bulk of signals we detected in the amplitude spectrum. We show this region in Figure 18. It seems obvious that five peaks are spaced by the common splitting (two just below the detection threshold), and we interpret them as another l = 2 quintuplet.

Multiplets
We expect all nonradial modes of a slowly rotating star to be split according to the Ledoux equation (Eq. 1). This is indeed what we see in the amplitude spectrum of KIC 10139564. Most peaks are split with roughly consistent splitting among modes of the same kind (p-or g-modes).
In Figure 19 we show spacings between p-modes with S/N>5.5. Most group at approximately 0.45 µHz, with a few having double that spacing at 0.9 µHz and two with three times that spacing near 1.4 µHz. We interpret the group with the smallest spacings to arise from modes of consecutive m values (∆m = 1),  Figure 19. A histogram of frequency spacings in the p-mode region. the next group due to ∆m = 2, and the final pair due to ∆m = 3.
Using the average splitting between p-modes of 0.452±0.030 µHz) and C n,l = 0 (splittings below 0.4 µHz were not taken into account since we suspect they are caused by a non-zero Ledoux constant), we derived an average rotation period of 25.6±1.8 days.
Multiplets at low frequencies have average splittings of 0.2504±0.0019 and 0.3646±0.0042 µHz. As mentioned in §1 the ratio of the Ledoux constant between the p-and g-mode regions can be used to constrain the degree of the g-modes. The smaller splitting has a ratio of 0.55 and the larger splitting has a ratio of 0.81 which matches the Ledoux ratio if the g-modes are l = 1 and l = 2, respectively. Baran et al. (2009) reported the discovery of a symmetrical but not evenly spaced quintuplet in the sdBV star Balloon 090100001 (Bal09) and explained this uneven spacing as differential rotation on the surface. In KIC 10139564, we detected the quintuplet f8 to f12, which is symmetric, but components with |m| = 1 are spread wider (0.41505±0.00042 µHz on average) compared to the |m| = 2 modes (0.41290±0.00044 µHz). This leads us to the conclusion that rotation in KIC 10139564 may be slowest on the equator which would be opposite to the result obtained by Baran et al. (2009).
Pulsation amplitudes of identified multiplet components can also be used to constrain the pulsation inclination (supplementary materials of Charpinet et al. 2011). If the components of individual multiplets are intrinsically driven to the same amplitude, then observed amplitude differences are caused by the inclination at which they are viewed. This is particularly true for l 3 modes which have a higher degree of geometric cancellation (Figure 8.5 of supplementary material of Charpinet et al. 2011). For KIC 10139564 we detected four possible triplets (some with components below the detection threshold) yet they do not have consistent amplitudes between the various components. This is particularly evident for the complete quintuplet where the m = +1 and +2 components have triple and double the amplitudes of their counterparts, respectively. Likewise, we observe a triplet with spacings which indicate we are seeing the m = −2, +1 and +2 components, which cannot be attributed to viewing inclination. As such, the amplitudes are not sufficiently consistent to be used for constraining the pulsation inclination. Silvotti et al. (2007) have detected the first planet around an sdB star. They showed that pulsations detected in sdBV stars can be used to search for periodic variations of pulsation phases, which could be caused by orbiting bodies. However, the pulsation phases have to be monitored for a minimum of 2 to 3 years and measured with low uncertainties. It means that resolved high-frequency and high-amplitude modes are preferred. With several high-amplitude modes, KIC 10139564 is one of the best candidates where we can look for phase variations in terms of planet hunting. KIC 10139564 has been observed for almost two years, but at the time of this analysis, we had access to only 15 months of data. There are a few highamplitude modes among the p-modes that we used in our phase analysis. Unfortunately, the highest amplitude periodicity is part of a triplet, that exhibits amplitude variation, and when the data are divided into subsets, a strong beating pattern in the phase analysis appears. There is a single periodicity f18 with a relatively high amplitude. We have calculated amplitudes and phases of that periodicity in 30-day bins. The phases are stable to within 1 sec except for three points deviating by 5σ. The uncertainties of the phases are of the order of 0.5 sec. A similar flat trend was found using components of the quintuplet at 5287 µHz but with correspondingly larger uncertainties. It is beneficial to continue monitoring these periodicities for the remainder of the Kepler mission which should determine whether a companion (planet) exists. Spectroscopically, KIC 10139564 appears as a single star.

Exotic species in the zoo
Several regions in the amplitude spectrum appear to be frequencycrowded, unresolved, or "forked" which could indicate amplitude and/or phase variability. Investigating these regions has yielded unanticipated results, adding to the diversity of the pulsation zoo within KIC 10139564.
The first exotic result is that of the regions around 5472 and 5645 µHz. While the region looks quite messy in the combined amplitude spectrum (top panels of Figure 20), the phases (shown in the bottom panels of Figure 20), calculated using five-day sliding bins, drifted in a smooth fashion at a period of approximately one year. As such, we suspected an artifact related to the spacecraft but detected something even more interesting. The periodicities actually occur above the Nyquist frequency so are reflected across it to the regions listed above. The true frequencies are centered around 11521 and 11047 µHz. Careful examination reveals these frequencies are, in fact, combinations from high-amplitude frequencies in the subNyquist region. For the periodicities near 11521 µHz, there is a quintuplet formed from combinations of the main triplet (f33 to f35) near 5760 µHz. Similarly, the quintuplet near 11047 µHz is caused by combinations of that same triplet (f33 to f35) with the quintuplet near 5287 µHz (f8 to f12). All combination frequencies we detected are listed in Table 3. While they are reflected across the Nyquist, they become smeared by the spacecraft's motion. The exposure cadence is fixed in the spacecraft reference frame. However, the spacecraft is moving relative to the observed field, so the Nyquist frequency varies in the barycentric reference frame; this movement leads to a smearing of all frequencies whenever they are reflected across the Nyquist frequency. Therefore, these two messy regions are, in fact, reflections of combination frequencies above the Nyquist frequency.
For other regions, this approach did not yield results, as shown in the right panels of Figure 20. Our best solutions produced another mixed set of exotic results. Amplitude variability has been previously observed in many sdBV stars (Reed et al. 2007), so it is not surprising also to encounter it here. One solution, therefore, is to investigate shorter sets of data over which the amplitudes and/or phases may be fairly stable. Our first approach was to divide the data into quarters, which is how the data are released to us. Figure 21 shows a region where this method resulted in a solution. The  region around 5840 µHz was known to have portions of multiplets from previous quarters of data. Using Q5 through Q8 data, the amplitude spectrum contained a complete septuplet above the detection threshold. The peaks were not symmetric, indicating phase or amplitude variation, and when Q9 was added, several peaks diminished below the detection limit. This sort of behavior is common for pulsations which are damped and re-excited. In Figure 21, the septuplet frequencies of Q5 through Q8 are marked with dashed lines. The quarterly amplitude spectra reveal that no frequencies are above the detection limit during all five quarters, yet this region is quite frequency-rich. Quarters 5, 7 and 9 all show several substantive peaks, which fit the previously established multiplet splittings, and dotted lines in Figure 21 indicate two previously unrevealed frequencies. The average spacing between all these peaks is 0.37 µHz. The splittings are not identical, and there is likely some frequency wandering between quarters. This may be caused by amplitude variations and/or phase interactions with other nearby periodicities which are also amplitude variable. We cannot claim all these periodicities are intrinsic to the star, but the evidence suggests this region contains an l 4 mode. We also examined the multiplets near 3540 and 5049 µHz by quarter, but no peaks were detected except those listed in Table 2. If quarterly data did not provide a solution for a given region, we examined them using shorter data sets. We calculated amplitude spectra for sets of data spanning 50, 60 and 70 days, avoiding the gap between days 270 and 290. The top panels of Figure 22 and Table 4 show our solution for the region near 6001 µHz. There appears to be some frequency wandering, but from these subsets of data, we estimate frequencies lie near 6001.10, 6001.47 and 6001.88 µHz, which give an overall separation of 0.39 µHz, roughly the established rotational multiplet splitting. Most likely, this region contains a triplet with a separation of the spin period.
The situation near 6173 µHz is more complex (Figure 22), largely because the amplitudes become very low between days 130 to 220 and after day 330. Additionally, there are more frequencies involved in this region which covers ∼ 3µHz. It appears there is a triplet near 6170.83, 6171.17 and 6171.66 µHz, which switches power from the right pair to the center frequency to the outside pair prior to day 150. Then another triplet, which is modest at the beginning, gets even weaker in the middle but comes back at the end of the run. These frequencies are near 6172.39, 6172.83 and 6173.32 µHz, which would be another triplet. However, these cannot both be l = 1 triplets, since they are too close together, so one of them is likely a part of an l = 2 multiplet. This region, which is shown in the bottom panels of Figure 22, is very complex, and we can only roughly estimate the frequencies of Table 4. Any interpretations are fairly speculative. We estimate that this region contains two distinct triplets with splittings of the rotation period, but we certainly cannot rule out a single multiplet.
The last region we discuss is near 6235 µHz (Figure 23). In this region, there is a high amplitude peak in the overall amplitude spectrum with two sidelobes, none of which prewhiten well. Breaking the data into 60-day subsets to resolve the frequencies clarifies what is occurring. In this case the main frequency f42 at 6234.7 is frequency and phase stable with an amplitude that diminishes after day ∼80. There are two other periodicities, both with frequencies and amplitudes that vary during the observations. The higher frequency (f43) drifts from 6235.07 to 6235.02 µHz and back again during the run, while the lower frequency drifts from 6234.4 to 6234.2 µHz prior to day 180, after which it is no longer detectable. This is shown in Figure 23 where the original amplitude spectra are shown in the top panels, the amplitude spectra prewhitened by f42  Table 4.

SUMMARY
KIC 10139564 is the only p-mode dominated sdBV star observed by Kepler. It was discovered during the survey phase of the mission and has been observed continuously in SC mode since the start of the second year. The SC mode allows us to study short period pulsations that would be unobservable with LC sampling. KIC 101539564 contains frequencies higher than the Nyquist frequency associated with the SC sampling interval, but thanks to the long duration of these observations the barycentric correction induces frequency modulations that resolves the ambiguity of modes reflected across the Nyquist frequency into the subNyquist region.
The amplitude spectrum calculated from 15 months of nearly continuous data shows more than 50 periodicities in a pulsation zoo, which includes both anticipated and exotic species. We detected the anticipated p-modes, found several g-modes and two periodicities between the p-and g-mode regions. These possibly mixed modes have been observed in some other sdBV stars Baran et al. 2011a), and are particularly abundant in the complex pulsator 2M1938+4603 (Østensen et al. 2010c). The highest amplitude is around 8 ppt, and only a few other frequencies are of comparable height. Most of the signal is below 0.5 ppt, which would be difficult to detect from the ground.
Almost all periodicities are split into frequency multiplets. We interpret this splitting as caused by rotation and derive an average rotation period of 25.6±1.8 days. The observed splittings vary by a small amount, leading to an uncertainty in the rotation period. This is most likely caused by small differences in the Ledoux constant as has been observed in models calculated by Charpinet et al. (2002). The scale of variation in the Ledoux constant matches the variation in frequency spacings. The exact rotation rate can be established once the order n and degree l are identified to better constrain the Ledoux constant. Most gravity modes appear as triplets, and their frequency differences are consistent with identifying them as l = 1 modes.
They also show nearly equal period spacings, which we interpret as asymptotic overtones of a single degree. Multiplets indicate they are l = 1 modes, but the period spacings are too large to be of consecutive overtones. They are more likely 207 or 310 s which is dissimilar to the 250 s observed for the other Kepler-observed pulsating sdB stars (Reed et al. 2011). However, those other stars are all g-mode dominated and so are cooler than KIC 10139564. The only other sdB star that is a p-mode dominated hybrid is Bal09. Ground-based observations detected 20 g-mode periodicities yet there is no indication of consistent period spacings between them. So at this time Bal09 is the only pulsating sdB star with rich gmode spectrum which does not show asymptotic g-mode period spacings. It is also both cooler and more evolved (lower log g) than KIC 10139564 and so may not be any more suitable for comparison than the other g-mode pulsators. Theoretical works in progress (by VVG) are seeking to explain this discrepancy.
Several of the multiplets show signal excesses at the frequencies of likely missed components and are indicated in the figures with vertical dash-dotted lines. We included them in our list as the S/N = 4 threshold is adopted to separate intrinsic peaks from random noise. When there is good reason to anticipate pulsation frequencies, in this case multiplet splitting, the significance threshold does not necessarily apply. We do not claim that any of these frequencies are intrinsic to the star, yet we include this speculation to compel future investigation with additional data.
Recent spectroscopic measurements indicate KIC 10139564 is not a radial velocity variable and allowed us to refine its temperature and gravity, placing it well away from the typical hybrid pulsator region (see Fig. 2 of Østensen 2010b). This raises the possibility that low-amplitude, g-mode pulsations are common place in V361 Hya class stars, but an instrument with Kepler's accuracy is required to observe them. A re-examination of pulsation driving models would seem necessary as the most recent g-mode driving publications (  . pulsations to the temperature of KIC 10139564. Improvements for diffusion of iron-group elements (in progress by HH) seems likely to solve this issue.
Among the p-modes at high frequencies, we have noticed several regions to be unresolved in frequencies, and our first guess led us towards stochastic oscillations as a possible explanation. Detailed analysis showed, however, that they are likely not stochastic in nature, but rather require multiple explanations. The regions all appear as multiplets and frequency splittings are consistent with our derived rotation period. Two regions are caused by reflections across the Nyquist frequency of combination frequencies, which are smeared by the difference in spacecraft and barycentric reference frames. Others are caused by highly variable amplitudes, and some appear to be changing their frequencies slightly. In the combined data set, this leads to multiple-peaked or even transient periodicities with amplitudes that are greatly reduced.
As a result of the abundant multiplets, we were able to associate most frequencies with modes, which is a very rare result for sdBV stars. Also of interest are the probable detections of l = 3 or greater modes. In many other sdBV stars, frequency density has strongly suggested that higher l modes are excited (Reed et al. 2007), yet because of geometric cancellation, there has been some debate whether we would most likely observe l = 3 or 4 modes. Mode density in this star also creates a new problem because current models (e.g. Charpinet et al. 2002) cannot produce as many modes as are observed. From a sample of representative models, the overtone spacings for l = 0 to 4 averaged from 904 to 939 µHz. Near the fundamental order, this could be reduced to about 700 µHz. For the better-established sequences, we detect overtone separations of 237, 124, 176, 241 and 234µHz. Such a dense spectrum is clearly challenging for current models yet could go a long way to explain the high-density observed in several sdBV stars. KIC 10139564 seems well placed to solve these issues, and the many multiplets (especially those of higher degree) should provide constraints on the rotation axis as well.
Only one high-amplitude p-mode is detected to be a single peak. Its amplitude is almost 4 ppt. We used this peak to match amplitudes between months to reduce residual signal as much as possible. This helped to diminish residuals for most of the peaks, which could mean either the change of intrinsic amplitudes of those periodicities is consistent or they are stable and any difference in amplitude between quarters is caused by systematics.
Fifteen months of continuous data coverage opened a way to study phase stability of detected periodicities to search for possible companions to KIC 101039564. Although we have detected more than 50 periodicities, we could not use most of them for checking phase stability. Close neighbouring peaks or residuals, which indicate amplitude/phase variation, cause a beating pattern in both amplitude and phase. We found only one well-suited frequency around 5472 µHz, which we used to look for phase stability. It is still affected by some nearby low-amplitude, complex signals. However, the beating between the peak and that complex signal is about 10 days, so the beat period is shorter than the period from rotational splitting. As a result of our phase analysis, we have not noticed any periodic variation in phase that could be interpreted as a companion to KIC 10139564. There seems to be a real deviation in phase between day 420 and 520, but we are not yet able to interpret this variation.
The results we presented in this paper clearly show KIC 10139564 to be very interesting and allowed us to detect features not accessible from ground-based data. As most frequency regions contained multiplets, mode identifications could be determined for most of the peaks we detected. The identified modes should pose strong constraints on stellar models and should, obviously, allow the exclusion of non-unique solutions, making the theoretical study of this object easier. In fact, this is the only sdBV star with so many modes identified. As a natural implication of the pulsation variety and all the remarkable results we obtained from only 15 months of data, we definitely should continue monitoring this star to confirm several features as well as to uncover more properties still hidden in the data.