Mode Identification in the Pulsating Subdwarf B Star KIC 2697388 A.S

ACTA ASTRONOMICA Vol. 62 (2012) pp. 179–200

Mode Identiﬁcation in the Pulsating Subdwarf B Star KIC2697388 A.S. Baran

Mt. Suhora Observatory of the Pedagogical University, ul. Podchor ˛azych˙ 2, 30-084 Cracow, Poland email: [email protected] Missouri State University, Department of Physics, Astronomy, and Materials Science, 901 S. National Av., Springﬁeld, MO 65897, USA

Received June 28, 2012

ABSTRACT

We present our results on mode identification in the pulsating subdwarf B star KIC2697388 observed with the Kepler spacecraft. We detected 148 frequencies of which five were attributed to p-modes while 122, to g-modes. The remaining 21 frequencies are also likely g-modes. We used multiplets and asymptotic period spacing to constrain degrees of 89 peaks to either l = 1 or 2. Using splittings in multiplets we derived the rotation period of this star to be nearly 45days. The average period spacing between l = 1 and 2 overtones are nearly 240 s and 139 s, respectively. Our results show that combining tentative identification based on the presence of multiplets and asymptotic period spacing is indeed useful in mode identification. Mode degree consistently inferred from independent methods make the results reliable and will help to construct accurate models of subdwarf B stars. Key words: subdwarfs – Stars: oscillations – Asteroseismology

1. Introduction

Subdwarf B stars (sdB) are located on the blue extension of the horizontal branch. They are compact objects of half a solar mass, with typical radii of 0.2 R and typical effective temperatures of 30000K. We infer that an sdB progenitor on⊙ the main sequence has a mass similar to our Sun. After the hydrogen in its core is exhausted it moves from the main sequence heading toward the red giant branch. Initial masses of sdB progenitors indicate that they should pass through a helium ﬂash and according to our best knowledge of sdB evolution progenitors have to lose almost all of their hydrogen envelopes prior to, or during, that ﬂash. Only if this happened, can an object settle down on the blue edge of the horizontal branch, fusing helium into carbon in its core. After the helium is depleted, the stars head to the white dwarf cooling stage. 180 A. A.

The first sdB stars were discovered by Humason and Zwicky (1947) who searched for blue objects around the North Galactic Pole. Half century later Kilkenny et al. (1997) discovered that sdB stars can pulsate. This opened an asteroseismic way to study sdB stars and attracted others to search for pulsating subdwarf B stars (sdBV). The first observations of sdBV stars were conducted from the ground. In total more than 70 sdBV stars have been found including both p- and g-mode dominated pulsators. However, most sdBV stars discovered from the ground are p-mode dominated. We consider it to be a selection effect since p-modes oscillate on shorter time scales and typically have bigger amplitudes than g-modes. Space telescopes allowed us to monitor pulsating stars on longer time scales and detect g-modes with low amplitudes. CoRoT and Kepler are recent examples. Non-radial pulsations can be described by means of spherical harmonics identified by three natural numbers: n-radial order, l-degree and m-azimuthal order. It is not trivial to derive these three numbers for distant stars but they have to be known when making stellar models to compare with observations. Some efforts to identify modes were undertaken with ground-base data but with limited success. It includes multicolor photometry (Baran et al. 2008), spectroscopy (Telting and Østensen 2004) or multiplets (Baran et al. 2009). Typically, only a few modes have been identified. In the best case, three were identified in Balloon 090100001 (Baran et al. 2009), with two of them independently confirmed using another method of mode identification (Baran et al. 2008). Since only a few modes in sdB stars could be identified, an application of asteroseismology has been limited to straightfor- ward method of comparing model frequencies with the observed ones, assuming trial values of natural numbers. This method can easily produce a degenerate solution (Charpinet et al. 2011a). Space-age asteroseismology provides us with an opportunity to make better mode identifications, sometimes using two independent methods allowing for cross- correlation. In such cases we may always go through assignment of quantized numbers reducing the number of free parameters significantly. Resulting models should be more accurate. In this paper we present a mode identification using two independent methods which combined together make our conclusions even stronger. We used rotational multiplets and asymptotic period spacings between g-mode overtones.

2. Two Methods of Mode Identiﬁcation

2.1. Rotational Multiplets Preliminary work on KIC2697388 based on one month of data was published by Reed et al. (2010). In the current paper we present results obtained from nine months of data. This results in nine times better frequency resolution and three times lower average noise level. A longer coverage works in favor of detecting Vol. 62 181 more peaks and resolving those split by rotation. Since our goal is to make mode identifications we hoped to detect enough peaks to easily recognize sequences of consecutive overtones and detect multiplets to cross-correlate mode degrees. Non-radial modes of degree l have 2l + 1 components differing in m number. However, in the absence of rotation, they all have the same frequency becoming indistinguishable in an amplitude spectrum. Stellar rotation can lift this degeneracy, shifting frequencies of m = 0 components according to the following relation: 6 1 Cn l ν = ν + ∆ν = ν + m − , n,l,m n,l,0 n,l,m n,l,0 P where: ∆νn,l,m is a frequency splitting, P is a rotation period of a star and Cn,l expresses the Ledoux constant. This expression is valid for slow rotators and oscillations with periods shorter than the stellar rotation. According to this equation, components differing in m will be shifted in frequency proportionally to a rotation frequency. This will produce an equidistant (in frequency) pattern of peaks with a number of components which depends on the l value, i.e.,three peaks for l = 1, five peaks for l = 2 and so on. Detecting evenly spaced peaks in frequency may help to identify l values without using advanced methods for mode identifications. Unfortunately, the structure of multiplets depends on the inclination of the pulsation axis to the observer’s line of sight. A good example is provided in Fig.8.5 in the Supplementary material of Charpinet et al. (2011b). An unconstrained angle can result in two different explanations of a detected multiplet. It may be that we, for example, see three peaks creating a triplet at moderate angle or a quintuplet with central and two outermost peaks. In such a case independent information is needed to correctly identify l values. Even though we expect that stars rotate, it was a challenge to use multiplets for mode identification in ground based data. Balloon090100001 (Baran et al. 2009) is an exception with two multiplets detected. The rotation period estimated from frequency splitting is close to 7days. If a period of days or weeks is representative for sdB stars it can explain why multiplets are rarely seen in amplitude spectra. It requires long (weeks or months) monitoring to resolve close peaks in multiplets. We expected that nine months monitoring of KIC2697388 should help to resolve and detect multiplets, which then could help with mode identification. Identifying multiplets allows us to estimate a rotation period. From the above equation we derive: 1 Cn l P = − , . ∆νn,l,m If we know the splitting, m and the Ledoux constant we can calculate the stellar rotation period. A splitting ∆νn,l,m is our measurement for given m. The Ledoux constant is the only quantity we have to derive from theoretical models. From a theoretical work by Charpinet et al. (2002) we know that this constant has values close to zero for p-modes. The exceptions are the fundamental and first overtone 182 A. A. l = 2, 3 modes which may have values up to 0.3. For g-modes the Ledoux constant depends on the l parameter and may be calculated from the expression: Cn,l . 2 1 (l +l)− . It is close to 0.5 and 0.15 for l = 1 and 2 modes, respectively. The ratio between l = 2 and 1 splittings will be close to 1.7. 2.2. Asymptotic Period Spacing Another way to identify mode degree is a period spacing of consecutive overtones of the same degree. In the asymptotic limit n l , consecutive overtones of oscillation modes should be evenly spaced in period.≫ In sdB stars high order g-modes are usually detected, which makes period spacing useful for identifying g-modes. For given n and l values the period of oscillation modes is:

P0 Pl,n = n + ε pl(l + 1) where P0 is the period of the fundamental radial mode and ε represents a small value (Unno et al. 1979). The period spacings between two consecutive overtones are P0 ∆Pl = Pl,n+1 Pl,n = . − pl(l + 1) We can use the above equation to estimate the ratio between spacings of overtones of different degree. The ratio between l = 1 and 2 is equal to 1/√3 and between l = 2 and 3 equals to 1/√2. Since the surface cancellation effect is decreasing with smaller l , we expect that modes of low degree (l 3) will be detected easier ≤ with l = 1 modes to be detected the easiest. The above asymptotic relation is valid only for a homogeneous star. However, we may expect that compact objects, as sdB stars, might be compositionally strati- fied. This can produce zones with different mean molecular weight which will act as reflective walls. Oscillating modes will be confined to specific stellar regions. This so-called “mode trapping” will influence even spacing between overtones. The deviation from spacing may potentially be used to infer structures of those transition zones of compositional changes. This method of mode identification has never been successfully applied to sdB stars observed from the ground. The g-mode spectra were too sparse to find any signatures of evenly spaced overtones. Now, with continuous monitoring performed by space satellites we detect tens, if not hundreds, of g-modes which bear clear signs of evenly spaced peaks in period. Recently, Reed et al. (2011), applied period spacing to identify mode degrees in 14 sdBV stars observed from space. They obtained a common spacing of nearly 250 s for l = 1 overtones and nearly 145 s for for l = 2 overtones. No signatures of l 3 modes have been detected, however they draw a conclusion that some peaks ≥ did not fit into either l = 1 or 2 sequences, which indicates the possibility of modes of higher degrees. Vol. 62 183

In this work we intend to apply two methods of mode identiﬁcation. This cross- correlation should provide better constraints on mode degrees. In case of complete multiplets they may serve as a starting point to look for period sequences, while a few or several consecutive overtones evenly spaced by 250 s(l = 1) or 145 s ≈ ≈ (l = 2) may provide a clue on a numberof expected multipletcomponents. We have found this cross-correlation to be a very useful tool in the case of KIC2697388.

3. KIC2697388

KIC 2697388 (SDSS J190907.14+375614.2, Kp = 15.39) had been found in the Kepler field of view during the first year of monitoring. It has been observed during Q2.3 and the analysis along with the results have been already published by Reed et al. (2010). The authors cite this object to be cool, with Teff = 23900 ± 300 K and low gravity logg = 5.32 0.03 dex sdB star. One month of data uncov- ered nearly 40 frequencies in the g-mode± region, several in the intermediate region between g- and p-modes and only one peak at high frequency. The latter peak had the amplitude slightly above the detection threshold and needed to be confirmed. If true, this star would be another hybrid star but with g-modes dominating over the p-modes; opposite to hybrids detected from the ground. Reed et al. (2011) applied a period spacing to identify mode degree. This spacing has never been observed in any sdB star using ground-based data so it was the first application of the asymptotic relation in sdB stars for mode identification. They concluded that most of the detected modes are of low degree assigning 29 of 36 modes with either l = 1 or 2. This result proved the usefulness of the period spacing method for mode identification and should be even more efficient when more data help to detect more modes, particularly those filling incomplete period sequences. Charpinet et al. (2011a) undertook a theoretical study of KIC2697388. They derived new spectra and estimated Teff to be 25395 227 K and gravity of logg = ± 5.5 0.031 dex which is close to the parameters cited by Reed et al. (2010). They worked± with Q2.3 data and independently (from Reed et al. 2011) detected peaks. They found 43 independent frequencies that could be associated with oscillation modes. They have not constrained natural numbers prior to modeling but applied their forward-modeling approach. As it was mentioned in Section 1, this can lead to more than one possible solution and in fact, Charpinet et al. (2011a), obtained two indistinguishable models. This leads to some inconsistency in mode identification, though, the physical parameters of the star they obtained from those two models were not much different.

4. Kepler Photometry

The Kepler spacecraft was launched in 2009 with the primary goal to find ter- restrial planets in habitable zones of cool stars (Borucki et al. 2010). It monitors 184 A. A. nearly 150000 stars in the fixed field of view of about 115deg 2 . The spacecraft works in two time-resolution modes. A short cadence (SC) is built up of 9 exposures of 6.02 s each followed by 0.52 s overhead. This results in 58.85 s time resolution. A long cadence consists of 270 exposures resulting in almost 30min resolution. Limited on-board storage means that data have to be downloaded to Earth every month. The downlink is made with the high-gain antenna when point- ing toward Earth. This requires the spacecraft to be reoriented, resulting in, typically one-day gap of data every month. Moreover, every three months the spacecraft rolls 90 degrees to maximize electric power production and thermal control. Beside scheduled gaps, a safety mode can occur suspending data collection and making additional gaps.

25 20 15 10 5 0 -5 -10 -15 -20 -25 55307 55307.5 55308 55308.5 55309 55309.5 55310

flux [ppt] 25 20 15 10 5 0 -5 -10 -15 -20 -25 55307 55307.2 55307.4 55307.6 55307.8 55308

BJD-2400000 [days]

Fig. 1. Close-ups at the region of a downlink gap and a part of data to see ﬂux variation.

We downloaded data in FITS format from “Mikulski Archive for Space Tele- scopes”1 (MAST). These data are collected by the Flight System as described in the “Kepler Instrument Handbook”. We used PyKE software2 to extract ﬂuxes from target pixel data and account for time-series trends systematic to the spacecraft and its environment rather to our targets. Then, we extracted ﬂuxes and removed any additional trends on a time scale of weeks which could still exist in the data. This

1http://archive.stsci.edu/kepler/data_search/search.php 2http://keplergo.arc.nasa.gov/PyKE.shtml Vol. 62 185 additional de-trending was achieved with cubic spline curve fitted to 3-day sub- sets and subtracted from the data. Finally, we calculated amplitudes in per parts notation and used a Fourier technique to detect periodicities in the data. Pulsations in sdBV stars are rapid flux variations and cannot be reasonably sampled with long cadence data. SC sampling allows us to probe the frequency spectrum up to 8495 µHz. In this paper we present analysis of photometric SC data obtained during three quarters Q5-7 (276days) already available to the pub- lic. This coverage assures that the multiplets are resolved and is not too long to suffer from strong amplitude/phase changes. These changes are visible in the amplitude spectrum and will certainly become more severe with other quarters added. No safe-mode events occurred during Q5-7. In Fig. 1 we present close-ups of a downlink gap and a part of the data to visualize flux variations.

5. Fourier Analysis

We used a standard prewhitening procedure. We fitted peaks with Ai sin(2π fit + ϕi) by means of a non-linear least square method. We used a commonly used def- inition of the detection threshold as S/N = 4 (signal-to-noise). It equals 0.037ppt when calculated from the residual spectrum in the range from 0 up to the Nyquist frequency while 0.036 ppt when calculated between 1100 µHz and 8495 µHz. We chose the more conservative way and adopted the detection threshold of 0.037ppt. We were not able to evaluate accurate average noise level at low frequencies (up to 400 µHz) since the residual spectrum still contains substantial signal which clearly affects our estimation. We derived 0.068ppt at S/N = 4 in the region of 1/T to 400 µHz which clearly shows influence of the residual signal. We know, however, based on the analysis of other stars with no flux change, that noise in Kepler data does not depend on frequency. Therefore, we can interpolate a noise level between different regions of frequency. We fitted all signal above a detection threshold it- eratively derived from the last residual data unless the signal was unresolved or contained a very complex window pattern indicating an amplitude/phase variation over days or weeks. In Fig. 2 we present the window response to the distribution and data coverage. We marked recognizable aliases originating from two sources. The consecutive aliases starting from the second one are caused by the finite coverage (0.042 µHz = 1/276.07days). Every 0.386 µHz corresponding to 30-d coverage a slightly bigger alias shows up. It is due to monthly downlink gaps. It is worth noting that the first alias appears at 1.42 times the coverage (1/T) alias and we adopted that distance as the frequency resolution equals 0.06 µHz. We present the amplitude spectrum in three panels of Fig. 3. The top panel shows the lowest frequencies with the strongest and most abundant signals. The middle panel shows a transition region or a “g-mode tail”. It contains lower amplitude and fewer signals. The bottom panel include the high frequency region mostly 186 A. A.

0.9

0.8

0.7

monthly alias 0.6

0.5

0.4 Normalized amplitude

0.3 1.42 / T

0.2 1 / T 0.1

0 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency [µHz] Fig. 2. Window function calculated for Q5-7 data. We marked small aliases caused by the ﬁnite coverage and monthly downlink gaps.

dominated by long cadence artifacts and only two groups of signiﬁcant signals which we could not assign with any known artifacts.

In total we fitted 148 significant frequencies. Some of them are close to our detection threshold of 0.037 ppt and those should be considered tentative. We list all fitted frequencies in Table 1. We found five frequencies in the p-mode region above 2000 µHz. Other significant signals at high frequencies are already known and well documented instrumental artifacts. In the intermediate region between p- and g-modes regions we found 21 frequencies. They are all relatively low amplitude peaks with typical S/N < 10. In the low frequency region we found 122 peaks and associated them with g-modes. These modes are the most dominant in amplitude and the most abundant.

We noted that shapes of peaks are very distorted indicating amplitude/phase change. This leads to bigger uncertainties of frequencies and rotational splitting estimations (see Baran et al. 2012 for comparison). In many cases our multiplet candidates are not evenly spaced in frequencies and the splittings are not consistent. However, it may be that the centroids of those peaks, instead of a frequency of the tip of peaks, truly represent the correct frequencies. Upon this uncertainty, we assume that the candidates we picked represent multiplets. Vol. 62 187

Table1

The list of ﬁtted and signiﬁcant frequencies detected in KIC 2697388

ID Frequency [µHz] Period[s] Amplitude[ppt] S/N off1[s] off2[s] l m f1 68.5531(41) 14587.2(9) 0.051(9) 5.5 16.4 59.0 f2 68.6441(39) 14567.9(8) 0.056(9) 6.0 2.9 39.7 1 f3 68.7566(41) 14544.1(9) 0.050(9) 5.4 26.7 15.9 2 f4 69.5412(41) 14380.0(8) 0.049(8) 5.2 49.3 9.4 2 f5 70.0334(23) 14278.91(47) 0.090(9) 9.7 51.8 28.3 f6 70.1159(23) 14262.10(47) 0.089(9) 9.5 68.6 11.5 f7 70.2702(23) 14230.78(47) 0.086(9) 9.3 99.9 19.8 f8 78.9611(41) 12664.5(7) 0.048(8) 5.1 14.2 59.3 1 f9 82.0747(29) 12184.03(44) 0.067(9) 7.2 13.9 15.4 f10 82.2866(43) 12152.6(6) 0.046(9) 4.9 17.6 16.0 f11 83.3622(24) 11995.84(35) 0.082(8) 8.8 65.7 34.0 2* -2 f12 83.8019(16) 11932.90(23) 0.125(9) 13.4 2.8 41.9 2* 0 f13 84.0213(15) 11901.75(21) 0.132(9) 14.1 28.4 10.7 2* +1 f14 84.2730(15) 11866.19(21) 0.131(8) 14.1 63.9 24.8 2* +2 f15 86.7767(47) 11523.8(6) 0.042(8) 4.5 73.8 49.2 f16 87.3407(47) 11449.4(6) 0.041(8) 4.4 0.6 25.2 1 f17 89.0804(31) 11225.82(40) 0.062(8) 6.7 15.9 28.8 f18 93.4652(29) 10699.17(34) 0.067(8) 7.2 30.6 57.4 f19 94.6434(24) 10565.98(27) 0.081(8) 8.7 76.2 63.0 f20 97.2642(17) 10281.28(18) 0.118(8) 12.7 31.6 55.9 2 -2 f21 97.4363(20) 10263.11(21) 0.102(9) 10.9 13.4 37.7 2 -1 f22 97.7276(11) 10232.52(12) 0.174(9) 18.6 17.2 7.1 2 0 f23 97.9811(15) 10206.05(16) 0.129(9) 13.8 43.6 19.4 2 +1 f24 99.6970(26) 10030.39(26) 0.076(8) 8.2 20.8 56.2 f25 100.1386(27) 9986.16(27) 0.073(8) 7.8 23.5 38.4 f26 101.6982(11) 9833.02(11) 0.177(9) 19.0 63.4 24.0 2 -1 f27 101.8268(14) 9820.60(14) 0.143(9) 15.4 51.0 11.6 f28 101.9309(7) 9810.57(7) 0.268(9) 28.8 41.0 1.6 2 0 f29 105.0259(15) 9521.46(14) 0.128(8) 13.7 8.1 9.9 1or2 f30 111.06405(37) 9003.813(30) 0.534(8) 57.3 45.6 27.6 f31 111.3270(10) 8982.54(8) 0.200(8) 21.5 66.9 6.3 2 f32 113.67377(10) 8797.104(8) 1.887(8) 202.6 12.2 40.3 1 f33 114.15074(47) 8760.346(36) 0.420(8) 45.1 49.0 61.7 f34 114.4731(8) 8735.68(6) 0.250(8) 26.9 73.7 37.1 f35 116.6167(9) 8575.10(7) 0.217(8) 23.3 5.8 15.3 1 -1 f36 116.8749(12) 8556.16(9) 0.160(8) 17.2 13.1 3.6 1 +1 f37 119.8638(8) 8342.81(6) 0.235(8) 25.3 13.6 60.6 1 f38 123.8727(28) 8072.80(18) 0.071(8) 7.6 16.4 68.2 1 f39 124.9760(17) 8001.54(11) 0.118(8) 12.6 87.6 3.1 2 f40 127.9628(12) 7814.77(7) 0.162(8) 17.4 34.3 51.0 f41 128.1815(5) 7801.438(32) 0.383(9) 41.1 47.7 64.4 f42 128.4168(6) 7787.142(35) 0.349(9) 37.4 62.0 60.1 f43 128.68081(30) 7771.167(18) 0.662(8) 71.1 77.9 44.2 f44 130.9527(23) 7636.34(13) 0.086(8) 9.3 27.3 48.1 f45 131.3429(15) 7613.66(9) 0.130(8) 14.0 4.6 25.5 1# f46 135.8005(15) 7363.74(8) 0.130(8) 14.0 5.2 53.1 1 f47 135.9913(18) 7353.41(9) 0.114(8) 12.2 15.6 42.8 f48 136.9189(14) 7303.59(8) 0.139(8) 15.0 65.4 7.0 2 j1 and j2 denote period offsets from the average period spacings. The horizontal lines in columns 8 and 9 separate modes. 188 A. A.

Table1

Continued

ID Frequency [µHz] Period[s] Amplitude[ppt] S/N off1[s] off2[s] l m f49 140.1628(7) 7134.560(37) 0.278(9) 29.9 5.6 37.2 1 -1 f50 140.2752(17) 7128.84(9) 0.118(9) 12.6 0.1 43.0 1# 0 f51 140.4518(15) 7119.88(8) 0.129(9) 13.8 9.0 51.9 1 +1 f52 141.7780(22) 7053.28(11) 0.089(8) 9.5 75.6 20.3 f53 144.99752(9) 6896.6699(41) 2.277(8) 244.5 7.8 2.5 1# f54 146.6633(32) 6818.34(15) 0.062(8) 6.6 70.5 62.9 f55 148.2202(22) 6746.72(10) 0.089(9) 9.6 97.9 8.7 f56 148.3827(22) 6739.33(10) 0.091(9) 9.8 90.5 16.1 f57 150.62348(31) 6639.071(14) 0.640(9) 68.7 9.7 22.5 1* -1 f58 150.74992(25) 6633.503(11) 0.838(9) 89.9 15.3 16.9 1*# 0 f59 150.89398(34) 6627.170(15) 0.602(9) 64.7 21.6 10.6 1* +1 f60 156.12826(16) 6404.990(7) 1.194(8) 128.2 3.8 66.0 1*# -1 f61 156.382897(49) 6394.5612(20) 4.054(8) 435.4 14.2 55.6 1*# +1 f62 162.4337(18) 6156.36(7) 0.110(8) 11.8 12.3 43.8 1# f63 168.79569(30) 5924.322(10) 0.673(9) 72.3 4.3 1.7 1*# 0 f64 168.92403(8) 5919.8209(27) 2.590(9) 278.1 8.8 2.8 1* +1 f65 175.8014(12) 5688.237(39) 0.163(9) 17.5 0.3 43.2 1 -1 f66 175.96856(41) 5682.833(13) 0.481(9) 51.7 5.7 37.8 1# 0 f67 181.15077(48) 5520.264(15) 0.410(8) 44.0 71.8 14.1 2# -2 f68 181.9265(10) 5496.726(30) 0.200(8) 21.5 48.2 9.5 2# +2 f69 183.5494(34) 5448.13(10) 0.061(9) 6.6 0.4 58.1 1 -1 f70 183.6348(20) 5445.59(6) 0.104(9) 11.2 2.9 60.6 1# 0 f71 185.9323(10) 5378.300(29) 0.193(8) 20.8 70.2 10.9 2 -2 f72 186.1325(16) 5372.517(46) 0.124(8) 13.3 76.0 5.1 2# -1 f73 186.7933(30) 5353.51(9) 0.065(8) 7.0 95.0 13.9 2# +2 f74 200.44070(11) 4989.0067(27) 2.040(9) 219.1 20.6 38.0 1 -1 f75 200.51351(14) 4987.1951(35) 1.650(9) 177.2 18.8 36.2 1# 0 f76 200.58183(5) 4985.4966(14) 4.064(9) 436.5 17.1 34.5 1 +1 f77 210.5010(5) 4750.570(12) 0.412(9) 44.2 22.3 61.6 1*# 0 f78 210.6006(10) 4748.324(22) 0.217(9) 23.3 20.0 63.9 1* +1 f79 213.8210(31) 4676.81(7) 0.062(8) 6.7 51.5 3.4 2 -2 f80 214.2523(23) 4667.394(50) 0.086(8) 9.2 60.9 6.0 2# 0 f81 214.5326(37) 4661.30(8) 0.053(8) 5.7 67.0 12.1 2 +1 f82 220.2910(12) 4539.450(25) 0.162(8) 17.4 51.2 4.9 2# f83 222.2107(32) 4500.23(7) 0.061(8) 6.5 12.0 34.4 1# f84 227.0518(8) 4404.281(15) 0.258(8) 27.7 84.0 8.5 2# -2 f85 227.3010(5) 4399.452(10) 0.377(8) 40.5 88.8 3.7 2# -1 f86 234.37834(48) 4266.606(9) 0.413(8) 44.4 18.4 9.6 2* -2 f87 234.6179(5) 4262.249(10) 0.367(8) 39.4 14.0 5.2 2* -1 f88 234.8392(27) 4258.233(49) 0.073(9) 7.8 10.0 1.2 2*# 0 f89 235.0521(11) 4254.377(20) 0.180(9) 19.3 6.2 2.6 2* +1 f90 235.3346(8) 4249.270(14) 0.249(8) 26.8 1.1 7.7 2* +2 f91 239.2746(20) 4179.298(35) 0.097(8) 10.4 68.9 61.1 f92 240.3537(23) 4160.535(40) 0.084(8) 9.0 87.7 42.3 f93 242.2510(10) 4127.950(16) 0.204(8) 21.9 119.8 9.8 2* -2 f94 242.4530(8) 4124.510(13) 0.260(8) 27.9 116.4 6.3 2*# -1 f95 242.9345(17) 4116.336(29) 0.116(9) 12.4 108.2 1.9 2*# +1 f96 243.1402(12) 4112.854(20) 0.163(9) 17.5 104.7 5.3 2* +2 f97 250.5955(8) 3990.495(12) 0.255(8) 27.4 17.6 11.1 2* -2 f98 250.7803(8) 3987.553(12) 0.266(8) 28.5 20.6 8.2 2*# -1 Vol. 62 189

Table1

Concluded

ID Frequency [µHz] Period[s] Amplitude[ppt] S/N off1[s] off2[s] l m f99 251.3745(7) 3978.129(11) 0.273(8) 29.3 30.0 1.3 2*# +2 f100 260.5868(26) 3837.493(38) 0.075(8) 8.1 69.4 3.1 2*# -1 f101 261.0282(32) 3831.004(47) 0.061(8) 6.6 62.9 9.6 2*# +1 f102 261.2278(26) 3828.076(38) 0.076(8) 8.1 60.0 12.5 2* +2 f103 266.16250(38) 3757.103(5) 0.510(8) 54.8 11.0 55.3 1# f104 269.97694(19) 3704.0201(26) 1.055(8) 113.3 64.1 2.2 2* -2 f105 270.19128(32) 3701.0817(44) 0.617(8) 66.3 67.0 0.7 2*# -1 f106 270.62023(14) 3695.2152(19) 1.415(8) 152.0 72.9 6.6 2*# +1 f107 270.85675(21) 3691.9885(28) 0.958(8) 102.9 76.1 9.8 2* +2 f108 280.2972(22) 3567.642(28) 0.091(8) 9.7 39.6 4.6 2# -2 f109 281.1792(17) 3556.451(21) 0.117(8) 12.6 28.4 6.5 2# +2 f110 281.412(5) 3553.51(7) 0.036(8) 3.9 25.5 9.5 f111 284.27110(45) 3517.769(6) 0.455(9) 48.9 10.3 45.2 1 -1 f112 284.34549(46) 3516.848(6) 0.448(9) 48.1 11.2 46.2 1# 0 f113 292.3639(32) 3420.395(37) 0.061(8) 6.6 107.6 3.8 2# f114 303.9082(37) 3290.467(40) 0.053(8) 5.7 2.5 5.1 2 -1 f115 304.1143(45) 3288.238(49) 0.044(8) 4.7 0.3 2.8 2# 0 f116 304.3102(42) 3286.121(45) 0.047(8) 5.0 1.8 0.7 2 +1 f117 317.5301(21) 3149.308(20) 0.095(8) 10.2 101.4 2.7 2# f118 332.35562(22) 3008.8253(19) 0.906(8) 97.3 39.1 1.0 2# f119 354.8686(43) 2817.944(35) 0.045(8) 4.8 10.1 51.1 f120 356.964(5) 2801.401(40) 0.038(8) 4.1 6.4 67.6 f121 362.6480(12) 2757.495(9) 0.178(9) 19.1 50.3 27.3 f122 362.7110(9) 2757.016(7) 0.239(9) 25.7 50.8 26.8 f123 509.44876(46) 1962.9060(18) 0.426(8) 45.7 f124 509.9092(7) 1961.1334(28) 0.271(8) 29.1 f125 511.2037(7) 1956.1675(27) 0.281(8) 30.2 f126 540.8133(27) 1849.067(9) 0.072(8) 7.7 f127 651.8897(48) 1534.002(11) 0.041(8) 4.4 f128 652.0983(44) 1533.511(10) 0.045(8) 4.8 f129 708.9017(38) 1410.633(8) 0.052(8) 5.5 f130 736.1462(26) 1358.4259(48) 0.075(8) 8.0 f131 740.8301(30) 1349.837(6) 0.067(9) 7.2 1* 0 f132 740.9633(25) 1349.5944(46) 0.080(9) 8.6 1* +1 f133 780.0744(31) 1281.929(5) 0.064(8) 6.9 f134 781.2329(21) 1280.0280(35) 0.091(8) 9.8 f135 781.5183(29) 1279.5606(47) 0.068(8) 7.3 f136 864.3245(30) 1156.9728(40) 0.066(8) 7.0 f137 864.5638(46) 1156.653(6) 0.043(8) 4.6 f138 865.0258(36) 1156.0348(48) 0.055(8) 5.9 f139 891.6371(28) 1121.5326(35) 0.069(8) 7.5 f140 892.1343(34) 1120.9075(42) 0.061(9) 6.5 f141 892.2141(31) 1120.8072(39) 0.066(9) 7.1 f142 1020.2917(33) 980.1119(32) 0.059(9) 6.4 f143 1020.4480(32) 979.9618(31) 0.062(9) 6.6 f144 2868.7674(32) 348.58177(39) 0.061(8) 6.6 f145 2908.149(5) 343.8614(6) 0.037(8) 4.0 f146 3805.8885(46) 262.75074(31) 0.043(8) 4.6 f147 3806.356(5) 262.71844(36) 0.038(8) 4.1 f148 3810.262(5) 262.44911(36) 0.038(8) 4.0 190 A. A.

4.5 4.0 3.5 3.0 2.5

2.0 1.5 1.0 0.5 0.0 0 50 100 150 200 250 300 350 400

0.45 0.40 0.35 0.30 0.25 0.20 0.15 Amplitude [ppt] 0.10 0.05 0.00 400 500 600 700 800 900 1000 1100

0.14 0.12 LC 0.10 LC 0.08

s LC 0.06 s LC LC LC 0.04 0.02 0.00 2000 3000 4000 5000 6000 7000 8000 9000 Frequency [µHz] Fig. 3. Amplitude spectrum of KIC2697388. Top panel shows the low frequency region up to 400 µHz focusing on g-modes. Middle panel includes the intermediate frequency signal which we denoted as a “g-mode tail”. Peaks in this region are smaller in amplitudes with typical S/N < 10. Bottom panel presents frequencies above 1100 µHz up to the Nyquist frequency focusing on the p- mode region. Only ﬁve peaks organized in two groups (denoted as s) were ﬁtted in this region. Long cadence artifacts are marked with ’LC’.

6. Discussion

6.1. The High Frequency Region The signal at high frequencies is unexpected for such a cool sdB star. It was also reported in the previous work by Reed et al. (2010). They noticed a single small amplitude peak at 3805.906 µHz and concluded that if this peak is real, this object is a hybrid star with amplitudes of g-modes larger than p-modes. Charpinet et al. (2011a) considered this signal to be essential in their modeling approach and they adjusted their models to include that frequency in their solution. Vol. 62 191

In our analysis we found ﬁve signiﬁcant peaks. The signal is grouped at two frequencies, approximately 2900 µHz and 3800 µHz. The highest peak f144 has S/N = 6.5 while other peaks are closer to the S/N = 4 limit. Those low amplitude peaks should be considered tentative.

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00 2850 2860 2870 2880 2890 2900 2910 2920

0.045 0.040 0.035 0.030 0.025 0.020 0.015 Amplitude [ppt] 0.010 0.005 0.000 3790 3795 3800 3805 3810 3815

0.045 0.040 0.035 0.030 0.025

0.020 0.015 0.010 0.005 0.000 3804 3804.5 3805 3805.5 3806 3806.5 3807 3807.5 3808 Frequency [µHz]

Fig. 4. Signal at high frequency detected in KIC2697388. Top panel shows two frequencies f144 and f145 . Middle panels contain another three peaks f146 , f147 and f148 while bottom panel is a close-up of f147 and f148 showing other insigniﬁcant peaks with an average spacing of 0.022 µHz expected in rotationally split p-modes. This estimation is based on the splitting we detected for l = 1 g-modes. Horizontal dashed lines (here and in other ﬁgures) represent the detection threshold.

We searched for multiplets. In Fig. 4 we present three panels with signals detected at high frequencies. The top panel shows the ﬁrst region with the highest peak. This region is sparse and no signatures of multiplets can be seen. The middle panel shows the second region around other three peaks. We noted some signal excess above the average noise level detected at 3794 µHz, though, it is below the detection threshold. We present a close-up around two peaks f147 and f148 in the 192 A. A. bottom panel. A clear pattern of equidistant peaks is noticeable, however only two peaks cross our threshold. The average spacing is 0.22 µHz and as we will see in Section 6.3 it is twice the spacing among l = 1 g-modes so ﬁts the expected rotational splitting of p-modes. Another feature which is worth noting is that these two regions of peaks at approximately 2900 µHz and 3800 µHz are spaced by about 900 µHz. Such sep- arations, as derived from theoretical models, are typical for consecutive overtones among p-modes and have been already seen in other sdBV stars observed from the ground, i.e., Balloon090100001 (Baran et al. 2009) and RATJ0455+1305 (Baran et al. 2011).

6.2. The Intermediate Frequency Region

We found 21 frequencies in this region. It is dominated by signal at approximately 510 µHz with only three frequencies ﬁtted f123 , f124 and f125 . We show this region in the top panel of Fig. 5. The signal contains a very complex pattern which may indicate more independent frequencies and/or amplitude/phase change.

0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 508.5 509 509.5 510 510.5 511 511.5 512 740.4 740.6 740.8 741 741.2 0.09 Amplitude [ppt] 0.08 0.07 0.06 0.1332 0.05

0.04 0.03 0.02 0.01 0.00 -0.4 -0.2 0 0.2 0.4 Frequency [µHz] Fig. 5. Example of frequencies detected in the intermediate frequency region. Top panel presents frequencies with the highest amplitudes. The complex pattern is evident. Bottom panel shows the candidates for a triplet. Only two components were detected. We made our judgment based solely on frequency spacing as compared to evident multiplets found in the low frequency region. Numbers in panels (here and in the next two ﬁgures) represent the frequency difference between indicated components. Vol. 62 193

The entire intermediate region contains 21 peaks spread over 800 µHz. We could not ﬁnd peaks evenly spaced in period. We found one candidate for triplet and we show it in the bottom panel of Fig. 5. We only detected two components with splitting that is similar to other triplets found in the low-frequency region (Section 6.3). It is difﬁcult to convincingly conclude whether this candidate is truly multiplet, however based solely on a frequency spacing we assigned that candidate with l and m values according to our best estimate.

6.3. The Low Frequency Region

Since the frequency region of 50–400 µHz is the most abundant in peaks, including those with high amplitudes, it is the best place to look for asymptotic period spacings and multiplets. We fitted 122 frequencies in this region and based on their frequencies, Teff and logg we associated them all with g-modes. We looked for non-resonant combination frequencies only, since their frequencies are exactly combinations of the parent frequencies and their amplitudes are lower than those of parent peaks. We found only one indication of a combination peak. It is f120 which overlaps with a potential combination frequency of two highest peaks in the amplitude spectrum, f61 and f76 . Its amplitude barely reaches the detection threshold and it may explain why no combinations with smaller peaks are detected. We have not looked for resonant combination frequencies since they are not necessarily lower in amplitudes as compared to the parent peaks and their frequencies are only similar to the combinations of frequencies of parent peaks. It is a challenge in such a crowded spectrum to distinguish between independent peaks and resonant combinations. Based solely on even frequency splittings we have found four candidates for triplets. We plotted them in Fig. 6. Not all of the components of triplets could be fitted and the amplitude distribution among triplets is inconsistent. It makes it impossible to infer an inclination of the pulsation axis. It may also affect our identification of m parameters since when only a doublet is seen it is not clear which one is the central frequency (two bottom panels of Fig. 6). In such cases we made our decisions arbitrarily. The splittings inside triplets are not the same within the errors, however we keep in mind that the peaks show complex structures and the true frequency may be affected by amplitude/phase changes. The Ledoux constant also slightly depends on l and n parameters. The average splitting between consecutive components is approximately 0.125 0.016 µHz. ± In Fig. 7 we present our quintuplet candidates. We made our assignment based on even frequency splitting between peaks. We have found seven candidates with an average splitting of 0.219 0.023 µHz. It agrees with the expected values of ± splittings in quintuplets as constrained by approximately 1.7 ratio between l = 1 and 2 modes. In this case the ratio equals 1.75. Assuming that the multiplets are caused by stellar rotation we derived the spin period of KIC2697388. For l = 1 modes we assumed Cn,l = 0.5 and derived the 194 A. A.

150.4 150.6 150.8 151 151.2 0.9 0.8 0.12642 0.14415 0.7 0.6 0.5

0.4 0.3 0.2 0.1 0.0 -0.4 -0.2 0 0.2 0.4

155.8 156 156.2 156.4 156.6 4.5 4.0 3.5 3.0 2.5 2.0 0.25461 1.5 1.0 0.5 0.0 -0.4 -0.2 0 0.2 0.4

168.4 168.6 168.8 169 169.2 3.0 Amplitude [ppt] 2.5

2.0 0.12835

1.5

1.0

0.5

0.0 -0.4 -0.2 0 0.2 0.4

210 210.2 210.4 210.6 210.8 211 0.45 0.40 0.35 0.0997 0.30 0.25

0.20 0.15 0.10 0.05 0.00 -0.4 -0.2 0 0.2 0.4 Frequency [µHz]

Fig. 6. Triplet candidates we found in the low frequency region. We made our selection based solely on spacing. Vol. 62 195

83 83.5 84 84.5 0.14

0.12 0.4399 0.2192 0.2514 0.10 0.08

0.06 0.04 0.02 0.00 -1 -0.5 0 0.5 1

234 234.5 235 235.5 0.45 0.40 0.35 0.30 0.2396 0.2215 0.2126 0.2825 0.25

0.20 0.15 0.10 0.05 0.00 -1 -0.5 0 0.5 1

242 242.5 243 243.5 244 0.30

0.25 0.2021 0.4812 0.2059 0.20

0.15

0.10

0.05

0.00 -1 -0.5 0 0.5 1

250 250.5 251 251.5 252 0.40 0.35 0.1853 0.5938 0.30 0.25 0.20 0.15

Amplitude [ppt] 0.10 0.05 0.00 -1 -0.5 0 0.5 1

260 260.5 261 261.5 0.08 0.4414 0.1990 0.07 0.06 0.05

0.04 0.03 0.02 0.01 0.00 -1 -0.5 0 0.5 1

269.5 270 270.5 271 1.6 1.4 0.21442 0.42889 0.23645 1.2 1.0

0.8 0.6 0.4 0.2 0.0 -1 -0.5 0 0.5 1

303.5 304 304.5 305 0.06 0.2057 0.1954 0.05

0.04

0.03

0.02

0.01

0.00 -1 -0.5 0 0.5 1 Frequency [µHz] Fig. 7. Quintuplet candidates we found in the low frequency region. As in case of triplets we made our selection based solely on spacing, however, the spacing was constrained by triplets to follow 1.7 ratio. 196 A. A. average spin period of 46.2 5.2 days, while for l = 2 we assumed Cn l = 0.15 ± , and estimated the spin period to be 44.9 4.3 days. Our estimations derived from ± triplets and quintuplets agree within the errors. In Fig. 8 we show the low frequency region. Since we look for evenly spaced modes in periods, in all panels, we plotted the amplitude as a function of period instead of frequency. The top panel shows the entire region while in the bottom three panels we plotted close-ups to better visualize frequencies with small amplitudes and spacings between them. We started our search for sequences of even period spacings with already identified multiplets. We marked them in Fig. 8 with asterisks. We treated them as modes with fixed degree and the base for searching for other overtones in given sequences. Five quintuplets located between 3500 µHz and 4500 µHz create the complete, nearly-evenly-spaced sequence with an average spacing of 137 s. This evaluation helped us to extend that sequence over another five l = 2 modes toward shorter period and three such modes toward longer period. This sequence consists of 13 consecutive overtones and a few other shown in the third and fourth panel of Fig. 8. Since we know the relation between spacings of l = 1 and 2 modes we could estimate an expected spacing for l = 1 overtones. We derived 237 s and, in fact, it agrees with a value inferred from spacings between l = 1 modes already identified as triplets. We found the longest complete l = 1 sequence between 5500 µHz and 7200 µHz.

Table2 The period spacings between central components of identiﬁed multiplets in complete period sequences

ID period spacing [s] l = 1 f f 233.73 58− 60.5 f60.5 f63 2*237.72 f −f 4*234.28 63− 75 f f77 236.63 75− l = 2 f88 f 137.81 − 94.5 f94.5 f98.5 137.58 f − f 136.10 100.5− 105.5

We calculated spacings between central components of identiﬁed multiplets of the same l value. For l = 1 modes we used f58 , the average between f60 and f61 , f63 , f75 and f77 modes. For l = 2 we used f88 , the averages between f94 and f95 , f98 and f99 , f100 and f101 as well as f105 and f106 . We present the differences between consecutive central components in Table 2, where we denoted the missing Vol. 62 197

4.5 1 1* 4.0 3.5 3.0 1* 2.5 1

2.0 1 1.5 2* 1.0 2 1* 1 1 2* * 2* 2 1* 2 1 1,2 0.5 * 2 2 21 12,1 2 1 1 2 2 2* 2222 2 1 2 1 1 1 1 or 2 1 0.0 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

1.6 2* 1.4 1.2 1.0 2 0.8 0.6 1 1 2* 2* 2 1* 0.4 2* 2 0.2 2 2 2 2 2* 1 2 0.0 3000 3500 4000 4500

4.5 1 1* Amplitude [ppt] 4.0 3.5 3.0 1* 2.5 1

2.0 1.5 1.0 1* 1 0.5 1* 2 2 1 1 1 2 1 1 0.0 5000 5500 6000 6500 7000 7500

1.0 0.9 1 0.8 0.7 0.6 1,2

0.5 0.4 2 0.3 1 1 2 0.2 2 1 or 2 2* 1 0.1 1 0.0 8000 8500 9000 9500 10000 10500 11000 11500 12000 Period [sec] Fig. 8. Amplitude spectrum as a function of period. We show the entire region of period of our interest in the top panel. Next three panels present continuous close-ups with some overlapping. Numbers in panels represent l values according to our identification. Asterisks denote signal with mode degree identified based on multiplet pattern. 198 A. A. central components calculated from average values of adjacent components with half numbers. Our result shows no significant mode trapping as predicted by theoretical models. We used preliminary assigned l = 1 and 2 modes to estimate an average period spacings between consecutive overtones. We employed a linear regression to fit period spacing in function of arbitrarily assigned radial orders. The modes we used in these fits are marked with # (Table 1) and are limited to the detected central frequencies of multiplets. If the central frequency was not detected we calculated it from the nearest components assuming even splitting. In such cases we marked these nearest components with # (Table 1). In case of single peaks we assumed that they are m = 0 components. This assumption may not be always true, however it should not generate big deviations in linear regressions. The period difference between multiplet components is small (seconds) at short periods where most of the multiplets were found. We derived the average spacings of 240.06 0.46 s for l = 1 and 138.80 ± ± 0.19 s for l = 2 sequence. The empirical ratio between the average spacings of both sequences agrees with the theoretical one. We calculated offsets in period of all modes in this region from the empirical sequences. The offsets are included in Table 1. These quantities easily illustrate how individual frequencies deviate from the average spacings. For modes we included in the fit (those marked with #) the offsets are close to zero. However, there are few exceptions where the offsets from both sequences are small and comparable. This is the effect of sequence-crossing. In such cases we made our final assignment based on the structure of multiplets, amplitudes or continuity of sequences. We refer to frequencies f35 and f36 , f64 and f64 or f114 , f115 and f116 , as the examples. The peaks at the period of nearly 12000 s need more explanation. They resemble a quintuplet (with splittings closer to 0.22 µHz) rather than a triplet, therefore it was assigned with l = 2. Even though, it is marked with asterisks, the central frequency was not included in the fit. This quintuplet is located too far from other multiplets and an estimation of its relative order is very imprecise. An opposite assignment of l was achieved by means of a period spacing fit. The offsets from the fit are smaller for l = 1 sequence. This mode surely needs special attention while modeling. For frequencies which were not included in the fits we used those two offsets, to judge how likely the peaks belong to either l = 1 or 2 sequences. This assessment might not be accurate, particularly where both offsets are larger than 10 s. In total we identified 22 l = 1 and 23 l = 2 modes. Identification of one mode is only constrained to either l = 1 or 2. Not all of our identifications are confident. Those modes with asterisks were cross-correlated with the multiplet and period spacing methods. A period spacing does vary between overtones and the average spacing can only be used to hint where overtones may be expected. When mode trapping occurs, a spacing deviates from an average value by unpredictable amount. Another problem comes with the fact that the splittings in multiplets become com- Vol. 62 199 parable to the average period spacings. As an example we refer to the quintuplet at nearly 12000 s. The splitting between the two outermost components equals 128 s compared to 139 s average period spacing. There are still peaks that we did not identify. The offsets from the average spacings of both l = 1 and 2 sequences are comparable and relatively large (typically a few tens of seconds). In some cases the same radial order would be required for adjacent peaks, even though the frequency splitting refutes those peaks to be components of the same multiplet. A clear example are peaks f1 and f2 . We assigned the peak f2 having smaller off11 offset. It should be remembered that we limited our mode identification to two mode degrees. The l = 1 and 2 suffer from the surface cancellation effect the least and they should be the richest in our data. However, some of the modes we detected might be of l > 2 but too few detecteddo not allow us to find a sequence of e.g., l = 3 with an expected spacing of 98 s with high confidence. On the other hand, some of unidentified frequencies may be l = 1 or 2 “trapped” modes.

7. Summary

We presented our results of mode identification in the pulsating subdwarf B star KIC2697388 by means of rotational multiplets and asymptotic period spacing. Whenever possible, we used both methods making our identification more reliable. The amplitude spectrum shows that signal is mostly concentrated in the low frequency region (50–400 µHz), however a substantial number of peaks appeared in the intermediate region (400–1100 µHz) with just five peaks detected in the high frequency region above 1100 µHz. Signal detected above 277.45 µHz indicates that long cadence data cannot be used to fully analyze pulsations in KIC2697388. There are five high frequency peaks with low amplitudes which do not show multiplet patterns. We noted, however, that the peaks we detected at nearly 3800 µHz may be a part of a multiplet. High frequency peaks are located in two groups separated by nearly 900 µHz. We suspect that, similar to other sdBV stars, these two groups represent consecutive overtones. The intermediate region contains at least 21 peaks. We have not found any clear evidence of asymptotic period spacing. We did find one candidate for triplet. Most signal is located in the region between 50 µHz and 400 µHz. We detected many triplet and quintuplet candidates. The rotational splittings between those two kinds of multiplets are consistent as predicted by Ledoux theory. The ratio between l = 2 and 1 modes equals 1.75. We used frequencies of central components of identified multiplets to calculate the period spacings between overtones. We preferred complete sequences or those with nearby components. When central components of multiplets were not detected, we calculated them using the nearest components of those specific multiplets. We derived an average period spacing of 240.1 s for l = 1 modes and of 138.8 s for l = 2 modes. 200 A. A.

We used both, multiplets and period spacings to constrain a mode geometry. The best example of our cross-correlation are five quintuplets identified by their multiplet pattern, while later on, confirmed by being in a complete sequence, almost evenly spaced in period. Those peaks served as a starting point in our mode identification which allowed us to evaluate spacings between l = 2 modes. Then we could calculate l = 1 spacings and confront it with actual spacing between triplets. In total we identified45 modes or 89 peaks, which is nearly60% of all peaks detected in the amplitude spectrum of KIC2697388. Our mode identification mostly agrees with previous study presented by Reed et al. (2011). The agreement is better toward shorter periods. Comparing our identifications with the results of Charpinet et al. (2011a) we found that the mode identification associated with the optimal model solution 2 better agrees with our identification than with the optimal model solution 1. We found that the cross-correlated method of mode identification is very help- ful, particularly when incomplete sequences or multiplet patterns are found. It should provide a strong constraint in stellar models of pulsating sdBV stars.

Acknowledgements. This work was supported by the Polish Ministry of Sci- ence and Higher Education under project No.554/MOB/2009/0, National Science Centre under project No.UMO-2011/03/D/ST9/01914 and US NSF grant AST- 1009436. Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reﬂect the views of the National Science Foundation.

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