Super-Nyquist asteroseismology of solar-like oscillators with Kepler and K2 – expanding the asteroseismic cohort at the base of the red giant branch

Abstract

INTRODUCTION

The NASA Kepler mission has provided photometric observations of exquisite quality for asteroseismic studies of a diverse range of pulsating stars (Gilliland et al. 2010). Particularly noteworthy has been the large volume of data collected on cool main-sequence, subgiant and red giant stars showing solar-like oscillations, pulsations that are excited and intrinsically damped by near-surface convection (Chaplin & Miglio 2013). The successful asteroseismology programme looks set to continue in the re-purposed Kepler mission, K2 (Howell et al. 2014). The quality of the K2 photometry for asteroseismic studies of coherent pulsators has already been demonstrated by results from engineering test data (Hermes et al. 2014; Jeffery & Ramsay 2014). The photometric performance has also confirmed the predicted potential for studies of solar-like oscillators (Chaplin et al. 2013), studies that will be made possible by the longer, full-campaign science data.

Kepler data were collected in two cadences (Koch et al. 2010) during the 4-yr-long nominal mission. Data on up to 170 000 targets were available in the 29.4-min long cadence (LC), whilst a much smaller total of up to 512 targets could be observed simultaneously in the 58.85-s short cadence (SC). These cadences establish Nyquist frequencies for the two modes of operation of ≃283 (in LC) and ≃ 8496 μHz (in SC). The LC Nyquist frequency coincides with the dominant periods of oscillation shown by cool stars that lie just above the base of the red giant branch. Both cadences have been maintained for the K2 mission, but the total number of targets for each observing campaign has been reduced – to approximately 10 000 and 50 targets, respectively – because of the need for larger pixel target masks.

The detection of solar-like oscillations in cool main-sequence and subgiant stars, and stars at the very bottom of the red giant branch, demands that the target-limited SC data be utilized since the dominant periods of oscillation are shorter than one hour. Observations made in the numerous LC slots are sufficient to detect oscillations in more evolved red giants, because the relevant pulsation periods are longer.

Oscillations have already been detected in around 16 000 red giants in the nominal mission LC data (Hekker et al. 2011; Stello et al. 2013; Huber et al. 2014). Whilst Kepler increased the number of cool main-sequence and subgiant stars with asteroseismic data by over an order of magnitude, to around 700 targets (Chaplin et al. 2011a, 2014; Huber et al. 2013), totals were nevertheless limited by the available number of SC target slots. The possibility of detecting solar-like oscillations in LC data above the LC Nyquist frequency – i.e. in the ‘super-Nyquist’ regime – offers the potential to increase significantly the number of evolved subgiant and low-luminosity red giant targets amenable to asteroseismic study.

The use of LC data for super-Nyquist studies of coherent pulsators has been considered in some detail by Murphy, Shibahashi & Kurtz (2013). Baran et al. (2012) also employed a super-Nyquist analysis of a compact pulsator observed in SC. While Gaulme et al. (2013) and Beck et al. (2014) identified a few red giants with oscillation frequencies above the LC Nyquist frequency, the prospects for studying solar-like oscillators in the LC super-Nyquist regime have not yet been explored in any detail. Our aim in this paper is to consider the utility of the archival Kepler and future K2 data for such studies.

The layout of the rest of the paper is as follows. We begin in Section 2 with a summary of some of the basic principles from Fourier analysis that are relevant to our study. We then consider in Section 3 the specific case of Kepler LC observations; and in Section 4, we go on to consider the characteristics of the super-Nyquist spectrum shown by solar-like oscillators, considering the impact of the finite mode lifetimes (Section 4.1), the use of a priori information to discriminate true from aliased peaks in the frequency spectrum (Section 4.2), and the reduced S/N above the Nyquist frequency (Section 4.3). We include model predictions for subgiant and low-luminosity red giant stars, and real super-Nyquist examples from the Kepler archive (Section 4.4). We finish the paper in Section 5 with concluding remarks.

BASIC PRINCIPLES

Kepler LC OBSERVATIONS

The Kepler LC data are comprised of Δt = 29.4-min cadences (Jenkins et al. 2010) that are exactly regular in the spacecraft frame of reference. This establishes a notional LC Nyquist frequency of ν_Nyq ≃ 283 μHz. Each 29.4-min cadence is in turn a summation of 270 individual ≃6-s readouts (e.g. Gilliland et al. 2011). Most of each cadence is therefore given over to the collection of photons. The very high fractional duty cycle for the Kepler integrations means the signal attenuation follows closely that described by equation (3).

The left-hand panel of Fig. 1 shows the frequency-power spectrum of idealized observations, made on a regular Kepler-like cadence, of several undersampled sinusoids. The sinusoids are all of unit amplitude and have frequencies lying in the super-Nyquist regime between 310 and 460 μHz. Peaks due to the true frequencies are rendered as black lines. The vertical dashed lines mark multiples of ν_Nyq. Peaks in grey are therefore aliases of the true frequencies. The spectrum is repeated every 2ν_Nyq because of the discrete nature of the calculation.

Figure 1.

Left-hand panel: frequency-power spectrum for a series of sinusoids of unit amplitude having frequencies between 310 and 460 μHz, sampled on a regular cadence. Peaks due to the true frequencies are rendered as black lines. The vertical dashed lines mark multiples of ν_Nyq. Peaks in grey are aliases of the true frequencies. The dotted line marks the sinc-squared attenuation envelope (i.e. η²). Right-hand panel: spectrum now given by irregular sampling of the undersampled sinusoids. The true peak and the aliases of one of the frequencies are rendered in thick linestyles (see text).

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This power spectrum has been calibrated¹ so that a sinusoid of unit amplitude with an infinite sample rate would show a maximum power per bin of unity. The observed powers at the true frequencies are much lower than unity, the true power. This is due to the sinc-function attenuation described by equation (3). The dotted line marks the sinc-squared suppression envelope in power, i.e. η².

The above is not quite the whole story as far as timing issues for the Kepler observations are concerned. Kepler lies in a 372.5-d heliocentric, Earth-trailing orbit. Observations of the pulsations of a Kepler target – be it one in the original field, or in the K2 fields in the ecliptic plane – will be phase modulated in the spacecraft frame of reference. This is because there is a component of the orbital motion of Kepler along the line-of-sight (target) direction, which delays or advances the arrival time of light from the star. The annual size of the effect is approximately ±190 s for targets in the original field, and approximately ±500 s for targets in the ecliptic (where the K2 fields lie).

To compensate for this effect² the Kepler time stamps are corrected to Barycentric arrival times (actually Barycentric Dynamical Time; see e.g. García et al. 2014). An important consequence is that the intervals between time stamps are no longer regular, but are modulated periodically on an ∼1-yr time-scale. This periodic modulation splits the aliased peaks in the frequency spectrum into many components, as discussed in detail by Murphy et al. (2013).

The right-hand panel of Fig. 1 shows the result of simulating LC observations of an idealized star in the original Kepler field undergoing coherent pulsations. The artificial data span four simulated years and include the aforementioned timing effects. The simulated pulsations are the same, undersampled sinusoids from the left-hand panel. Note how the aliases are no longer exact, reflected copies about multiples of ν_Nyq. Some copies have different maximum power spectral densities (i.e. peak heights in the spectrum) than others. The reason for this is that the splitting of the power into several components reduces the maximum heights in Fig. 1, relative to those expected for simple reflected aliases (the total integrated power in the aliases is conserved).

The top-left panel of Fig. 2 shows a zoom of the peak due to the true frequency marked ‘a’ in the right-hand panel of Fig. 1, whilst the other panels show zooms of the aliases marked ‘b’, ‘c’ and ‘d’ on Fig. 1. As explained by Murphy et al. (2013), the exact manner in which the alias peaks are affected by the timing modulation depends on the relation of their frequencies to the sampling frequency. If the true frequency is ν, then aliases at nν_s ± ν, or equivalently 2nν_Nyq ± ν, will share the same structure, e.g. what is predominantly a triplet structure when n = 1, or a quintuplet structure when n = 2, with the frequency splitting between adjacent components being ∼1 yr⁻¹.

Figure 2.

Top-left panel: zoom of the peak due to the true frequency marked ‘a’ on the right-hand panel Fig. 1. Other panels: zooms of the aliases marked ‘b’, ‘c’ and ‘d’ on Fig. 1.

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Murphy et al. (2013) pointed out that for high-amplitude, coherent pulsations the introduction of sideband structure, which is well resolved in the nominal mission Kepler data, allows one to discriminate the real and aliased peaks. What about solar-like oscillations, which are not coherent and typically have much lower amplitudes (and hence lower S/N levels in the frequency spectrum) than coherent pulsations?

SUPER-NYQUIST SPECTRUM OF SOLAR-LIKE OSCILLATIONS

Impact of finite mode lifetimes

The first issue we confront is that solar-like oscillations have finite lifetimes. In most cases, the lifetimes τ of detectable oscillations are significantly shorter than the duration T of the observations (e.g. Dupret et al. 2009). Only for some gravity-dominated mixed modes of red giants do the lifetimes of detected oscillations approach or exceed lengths commensurate with multi-month-long observations. One might therefore not usually expect to be able to resolve the aforementioned sideband structure in the frequency domain.

The underlying noise-free or ‘limit-spectrum’ profiles due to solar-like oscillations are to good approximation Lorentzian in shape, with the full width at half-maximum of the peaks given by Γ = (πτ)⁻¹ (e.g. see Chaplin et al. 2002). Fig. 3 shows how a finite mode lifetime would affect the appearance of the aliased quintuplet from the bottom-left panel of Fig. 2. The different linestyles show the composite, aliased limit-spectrum profiles expected when the real mode has a lifetime (linewidth) of 73 d (0.1 μHz; dashed line), 123 d (0.05 μHz; dotted line), 184 d (0.02 μHz; dark solid line) and 1 yr (0.01 μHz; grey solid line).

Figure 3.

Impact of finite mode lifetimes on the aliased quintuplet from the bottom-left panel of Fig. 2. The different linestyles show the composite, aliased profiles expected when the real mode has a lifetime (linewidth) of 73 d (0.1 μHz; dashed line), 123 d (0.05 μHz; dotted line), 184 d (0.02 μHz; dark solid line) and 1 yr (0.01 μHz; grey solid line).

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Lifetimes of solar-like oscillations in main-sequence and subgiant stars are typically of the order of several days in length. The lifetimes of detected radial mode oscillations in red giants can be as long as the 73-d lifetime considered above (e.g. Corsaro et al. 2012). The corresponding dashed line in Fig. 3 would be barely indistinguishable from a Lorentzian profile in the presence of real noise. The same would of course be true for shorter mode lifetimes. The longer lifetimes modelled in Fig. 3 show much more significant departures from a Lorentzian appearance. Provided S/N levels in the observed modes are sufficiently high, it might therefore be possible to discriminate visually the sidebands of some long-lived modes. It is worth noting once more that amplitudes of solar-like oscillations are usually much lower than for coherent pulsators, making it harder to distinguish the sideband structure.

We conclude that an important consequence of the finite mode lifetimes is that in most cases one would no longer be able to distinguish aliased from true oscillation peaks on account of the sideband structure. However, for solar-like oscillators we are fortunate in that we do not need to rely on such discrimination alone.

Use of a priori information for solar-like oscillators

The rich spectra of overtones shown by solar-like oscillators provides the necessary, a priori information to allow us to select the true spectrum from the aliased spectra. We illustrate how with a real example, the low-luminosity red giant KIC 4351319. This target was observed in both long and short cadence during the nominal Kepler mission. Here, we have used data prepared using the PDC-MAP pipeline (Smith et al. 2012; Stumpe et al. 2012). The top panel of Fig. 4 shows the frequency-power spectra of simultaneous LC (upper plot) and SC (lower plot) PDC-MAP data from Kepler Quarters Q8 through Q17. The nominal LC Nyquist frequency, ν_Nyq, is marked by the vertical dashed line. The luxury afforded by having contemporaneous data available also in SC – where the oscillations are oversampled – of course allows us to tag the peaks in the LC spectrum above ν_Nyq as the true oscillation frequencies (see the figure). However, two checks based on use of the LC spectrum alone allow us to make the same call.

Figure 4.

Top panel: frequency-power spectra of simultaneous LC (upper plot) and SC (lower plot) data collected by Kepler on the low-luminosity red giant KIC 4351319. Bottom panel: zoom of the LC spectrum above and below ν_Nyq. The lengths of the horizontal arrows in the bottom panel correspond to expected average separations, assuming a ν_max of ≃190 (peaks below ν_Nyq) and ≃ 380 μHz (peaks above ν_Nyq). The shaded regions mark the locations of pairs of adjacent l = 2 and l = 0 modes.

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The bottom panel of Fig. 4 shows a zoom of the LC spectrum above and below ν_Nyq. Our first check relates to the large-frequency separation, Δν, the spacing shown by consecutive overtones of the same angular (spherical) degree, l. An extensive body of literature has now established a fairly tight correlation between the average large separation and the observed frequency of maximum oscillation power, ν_max (e.g. see Hekker et al. 2009; Stello et al. 2009; Huber et al. 2011; Mosser et al. 2012). This relation may be expressed in the form |$\Delta \nu \propto \nu _{\rm max}^{\,\beta }$|⁠, with β ≃ 0.76 for the cool subgiants and low-luminosity red giants of interest here. While there is evidently spread in the relation due to, for example, dependences on mass and metallicity, the strong correlation nevertheless allows us to discriminate the true from the aliased spectrum. The lengths of the horizontal arrows in the bottom panel correspond to expected average separations, assuming a ν_max of ≃ 190 μHz (peaks below ν_Nyq) and ≃ 380 μHz (peaks above ν_Nyq). The shaded regions mark the locations of pairs of adjacent l = 2 and l = 0 modes. The frequency intervals between these regions provide a visual estimate of the observed large separation. Evidently, it is the spectrum above ν_Nyq that conforms with the expected average separation.

The second check relates to the relative power shown by adjacent l = 2 and l = 0 modes. Within each pair, the l = 0 mode lies at higher frequency. We expect to see more power in the l = 0 mode than in its l = 2 counterpart. Whilst predictions of the exact power ratios are rendered uncertain by the complexities of non-adiabatic calculations, observed ratios are usually not too far from the ≃2-to-1 ratio expected from the assumption of energy equipartition and geometric cancellation of the perturbations on the visible stellar disc (when projected on to spherical harmonic functions, with an appropriate limb-darkening law; e.g. see Aerts, Christensen-Dalsgaard & Kurtz 2009). Unless the star is observed with the rotation axis along the line-of-sight, power shown by the l = 2 modes will be spread across several components (e.g. Gizon & Solanki 2003). The ratio in total power then tends to be exaggerated in the frequency-power spectrum because it is the maximum power spectral densities, i.e. the heights of the mode peaks, that are immediately apparent from a visual inspection.

Inspection of the mode pairs in the bottom panel of Fig. 3 again implies that the spectrum above ν_nyq is the true one. The l = 0 modes, the higher frequency modes in each marked pair, have the higher observed powers.

Impact of signal attenuation and background aliasing

How far above ν_Nyq might we hope to detect solar-like oscillations in the LC Kepler data? A combination of several factors means that pushing the limit well above ν_Nyq is challenging. First, the sinc-function attenuation due to the finite integration time leads to significant reduction of the observed oscillation amplitudes at frequencies above ν_Nyq. Secondly, the impact of the sinc attenuation is exacerbated by the fact that the intrinsic maximum amplitudes of solar-like oscillations decrease with increasing ν_max. And thirdly, we must also contend with background power aliased from the region below ν_Nyq, which has contributions from stellar granulation, stellar activity, shot and instrumental noise. Note that any non-white or ‘red’ noise component, such as the granulation, will also have a contribution above ν_Nyq that is sinc-function attenuated, just like the oscillations (e.g. see Kallinger et al. 2014).

We come back to the example of KIC 4351319 to help illustrate these points. The top panel of Fig. 4 shows clearly the attenuated power of the oscillations in the LC spectrum of KIC 4351319, relative to those observed in the SC spectrum. The impact of the sinc-function attenuation on the observed S/N is compounded in the LC spectrum because the background in the region of ν_max is the aliased background from lower frequencies, which is dominated by granulation and is much higher – by more than an order of magnitude – than the shot-noise background. The spectrum of the oscillations is in reality coincident in frequency with the lower power, higher frequency part of the granulation spectrum, as shown by the SC spectrum of KIC 4351319. A much higher S/N is therefore observed in the oscillations when using SC data, because we do not have to contend with aliasing of power from the lower frequency, higher power part of the granulation spectrum. Our tests indicate that it is only when K_p is fainter than approximately 11–12 mag that the photon shot noise becomes an important factor in determining the detectability of the modes in the LC super-Nyquist regime.

From the observed SC spectrum of KIC 4351319, we may construct an (almost) accurate prediction of the LC spectrum. We begin by multiplying the observed SC spectrum³ by the sinc-squared attenuation filter, η². (Note we have not bothered to attempt to discriminate the white contribution due to shot noise, since it is very small compared to the other contributions.) The predicted LC power below ν_Nyq is given by adding the sub-Nyquist SC attenuated power to the aliased, super-Nyquist attenuated spectrum from above ν_Nyq (after reflecting the latter about ν_Nyq). The predicted LC power above ν_Nyq is then just the reflected prediction from below ν_Nyq. Fig. 5 shows the resulting prediction (lower plot), together with the LC spectrum from Fig. 4 (upper plot).

Figure 5.

Frequency-power spectrum of KIC 4351319: upper plot shows LC spectrum, lower plot predicted LC spectrum made from the available SC spectrum.

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Figure 6.

Idealized guide to the changing appearance of undersampled and oversampled oscillations spectra with varying ν_max, based on predictions using scaling relations applied to the fundamental properties of several 1 M_⊙ stellar models. The top-left panel shows the locations of the models in the log g–T_eff plane. The other panels show schematic representations of the limit (noise-free) power density spectra expected for LC (undersampled, upper plots) and SC (oversampled, lower plots) data, assuming each model was observed as a bright Kepler target having a Kepler apparent magnitude of K_p = 9. Spectra are plotted up to a frequency of just over 3ν_Nyq – with dashed lines marking ν_Nyq boundaries – to fully include the SC oscillation spectrum of the highest ν_max (least evolved) stellar model (see plot titles).

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After the top-left panel of Fig. 6, the other panels show schematic representations of the limit (noise-free) power density spectra expected for LC (upper plots) and SC (lower plots) data, where we assumed each model was observed as a bright Kepler target having a Kepler apparent magnitude of K_p = 9. The composite SC spectra were constructed using the scaling relations and formulae in Chaplin et al. (2011b), with contributions from oscillations, granulation and shot noise included. We refer the reader to that paper for further details.

The predicted SC spectra in Fig. 6 show where the true frequencies lie. Regarding the general appearance of the SC spectra: the power in, and the characteristic time-scale of, the granulation both increase with decreasing ν_max; we also recall that the power in the oscillations increases with decreasing ν_max. The nature of these changes is such that all composite SC spectra are, in their gross properties at least, approximately homologously scaled versions of one another.

The SC spectra were converted to predicted LC spectra by applying the sinc-squared attenuation in power, followed by suitable aliasing about the ν_Nyq boundaries (see Section 4.2). The schematic spectra have been plotted up to a frequency of just over 3ν_Nyq, to fully encompass the SC oscillation spectrum of the highest ν_max (least evolved) stellar model. The vertical dashed lines mark the ν_Nyq boundaries. Frequency aliasing adds to the complexity of the LC spectra, more severely in some cases than others depending on the placement of the true frequencies with respect to the multiple ν_Nyq boundaries. These plots show very clearly how attenuation of the oscillations can have a drastic impact on the observed S/N when the true frequencies are undersampled and hence lie in the super-Nyquist regime. The oscillation peaks are barely observable in the highest ν_max case shown here.

Before summarizing results from our simulations, it is worth noting that we have not included any low-frequency contributions due to stellar activity or instrumental or data-reduction noise. The predicted S/N ratios implied by the modelled spectra should therefore be regarded as upper-limit estimates. Nevertheless, starting from the detection recipe in Chaplin et al. (2011b), we have estimated detection probabilities for each of these modelled cases. We folded in predicted power from the non-radial modes, and modified the detection algorithm to operate on the composite, aliased spectrum in a single range of ν_Nyq. Our predictions suggest that for the targets in the nominal-mission Kepler archive with 4 yr of data, the upper limit for detections will be around ν_max ≃ 500 μHz. Instrumental noise levels will be higher for K2 (Howell et al. 2014); however, with reference to the above, intrinsic stellar noise should remain the limiting factor for brighter targets. Taking T = 75 d, the length of each K2 campaign, we estimate an absolute upper detection limit between ν_max ≃ 400 and ≃ 450 μHz.

Some real examples

Fig. 7 shows some more super-Nyquist examples from the Kepler archive. Five targets with Kepler LC and SC data have been selected that have undersampled oscillations in LC. The ν_max values range from ≃320 (just above ν_Nyq) to ≃ 550 μHz (just below 2ν_Nyq). It is worth stressing that there are not many targets like this in the Kepler archive with SC data (in particular at the low end of the range). The top-left panel plots the locations of the stars in the log g–T_eff plane, with stellar properties taken from Chaplin et al. (2014). The main plots of the other panels show zooms of the LC power spectra in the undersampled, super-Nyquist regions where the real oscillation frequencies lie. Note that we chose a slightly different layout compared to the previous figures to show more clearly the lower S/N cases at higher ν_max. The insets show power spectra computed from available SC data. Note that, unlike KIC 4351319, the SC coverage was much more limited for some of these stars. Even though the SC and LC data are not necessarily contemporaneous, the insets show clearly where the real oscillations are.

Figure 7.

Demonstration, using real Kepler LC data on five stars, of undersampled solar-like oscillation spectra in the LC super-Nyquist regime. Each target also has some SC data available, in which the oscillations are oversampled. The top-left panel shows the locations of the stars in the log g–T_eff plane. Other panels: main plots show the LC power spectra (oscillations undersampled), insets show power spectra computed from available SC data (oscillations oversampled).

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From our model predictions above, we expect to have good sensitivity up to ν_max ≃ 400 μHz, a conclusion that is borne out by these real data. At frequencies above this, detecting the modes becomes more difficult. Signatures of the oscillations are clearly apparent in the LC spectrum of KIC 6531928 (ν_max ≃ 450 μHz) but at a much reduced S/N compared to the other targets. KIC 8038445 has a ν_max of 490 μHz. Signatures of a few modes are just detectable in its LC spectrum. This case marks the approximate upper ν_max limit for making detections in the archival Kepler data.

Things become much more challenging as we move to higher frequencies, to the region where ν_max straddles or lies close to the boundary at 2ν_Nyq. Imposing a high-pass filter to suppress the very low frequency background power in the sub-Nyquist regime does not help to improve the S/N in the super-Nyquist regime: the filtering response is the same in each domain of ν_Nyq and so any oscillation signal is also filtered.

CONCLUSION

We have considered the prospects for detecting solar-like oscillations in the undersampled ‘super-Nyquist’ regime of LC Kepler data, i.e. above the LC Nyquist frequency of ≃ 283 μHz. We conclude that the LC data may be utilized for asteroseismic studies of targets whose dominant oscillation frequencies lie as high as ≃ 500 μHz. The frequency detection threshold for the shorter duration K2 science campaigns is lower. We estimate that the robust threshold will lie somewhere between ≃400 and 450 μHz.

These stars would usually be studied with Kepler SC data. The oscillations are then oversampled, since the associated Nyquist frequency of ≃ 8496 μHz lies well above the frequencies of the detectable oscillations. However, the number of SC available target slots is very limited: indeed, there are around only 50 targets of interest in the Kepler archive that have detected oscillations in SC data and ν_max between the LC ν_Nyq and ≃ 500 μHz. In most of these cases, there are only 1-month of data from the initial asteroseismic survey phase. This precludes very detailed studies of these targets, which demands high-frequency-resolution data sets for analyses of individual modes (including rotational frequency splittings).

The exploitation of the archival Kepler and new K2 data for LC super-Nyquist studies offers the prospect of increasing significantly the number of evolved subgiant and low-luminosity red giant targets available for asteroseismic study. By making use of the revised catalogue of Kepler star properties in Huber et al. (2014), we estimate that up to 400 targets may be available for study in the Kepler archive. Moreover, many of these targets will have data spanning the entire nominal mission, giving the frequency resolution needed for the above-mentioned analyses of individual modes.

The potential number of targets amenable to study with K2 data could run into the thousands, since new target lists will be generated for each campaign.

Funding for this Discovery mission is provided by NASA's Science Mission Directorate. The authors wish to thank the entire Kepler team, without whom these results would not be possible. WJC, YE, GRD, TLC, RH and AM acknowledge the support of the UK Science and Technology Facilities Council (STFC). Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). The research leading to these results has received funding from the European Community's Seventh Framework Programme ([FP7/2007-2013]) under grant agreement no. 312844 (SPACEINN).

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