PREDICTING THE DETECTABILITY OF OSCILLATIONS IN SOLAR-TYPE STARS OBSERVED BY KEPLER

The primary objective of the NASA Kepler Mission is to detect, by the transit method, Earth-sized planets in the habitable zones of solar-type stars (Borucki et al. 2010; Koch et al. 2010). Photometry of most of the stars is conducted at a long cadence (LC) of 29.4 minutes, but a subset of up to 512 stars can be observed at a short cadence (SC) of 58.85 s. The exquisite precision and accuracy of the Kepler data mean that they are also well suited to asteroseismic studies of stars (e.g., see Gilliland et al. 2010a; Jenkins et al. 2010). The cadence of the SC data is rapid enough to allow investigations of solar-like oscillations in solar-type stars, where dominant periods are of the order of several minutes (Chaplin et al. 2010, 2011a; Metcalfe et al. 2010).

Asteroseismology has an important role to play in the exoplanet program, in that the precise and accurate inferences on the stellar properties—i.e., radius, mass, and age—that are made possible by the seismic data allow very tight constraints to be placed on the exoplanetary systems, e.g., tight estimation of the stellar radius constrains the size of the planet (Kjeldsen et al. 2009; Stello et al. 2009a; Christensen-Dalsgaard et al. 2010). Solar-type stars flagged as potential planet hosts are therefore prime candidates for asteroseismic study.

The LC light curves of solar-type stars that are flagged by the Transiting Planet Search pipeline as having significant transit-like features are modeled, and those stars having companion radii less than two Jupiter radii are labeled as a Kepler Object of Interest (KOI; Batalha et al. 2010a). One of the factors that then determines whether or not to assign the star to one of the limited number of SC slots—allowing more rapid time-cadence observations—is whether the intrinsic properties and the apparent magnitude of the star would allow the detection of solar-like oscillations. This paper describes the straightforward procedure that has been implemented to answer this question.

We use simple scaling relations to predict the oscillations and granulation properties of any given KOI. The predictions use as input basic information available on each star from the Kepler Input Catalog (or KIC; see, e.g., Batalha et al. 2010b). Together with the known apparent magnitude of a given target, the scaling relation predictions are used to construct an appropriate measure of the signal-to-noise ratio (S/N) in the frequency-power spectrum of the solar-like oscillations, and, from that, we use some straightforward statistics to estimate the probability of detection for the star.

To calibrate and verify the recipe we use asteroseismic results on solar-type stars that were observed by Kepler during the first 7 months of science operations. About 2000 stars, down to Kepler apparent magnitude Kp ≃ 12.5, were selected as potential solar-type targets based upon their KIC parameters. Each was observed for 1 month at a time in SC. Time series were prepared for asteroseismic analysis in the manner described by García et al. (2011), and different teams applied their codes to attempt to detect, and then extract the basic properties of, the solar-like oscillations. Several modeling teams also then applied codes to estimate the stellar properties—using the basic oscillations parameters as inputs—which we then also made use of in validation of the recipe.

The layout of the rest of the paper is as follows. We introduce in Section 2 the basic principles underlying the detection test. We then detail the scaling relation predictions that are required for the test: Section 3 gives the scaling relations for the solar-like oscillations, while Section 4 gives scaling relations for the granulation and shot-noise contributions to the background power spectral density. The various ingredients are brought together in Section 5, in which the test is described in detail. The predictions of the test are compared in Section 6 to real results returned on the solar-type ensemble that was observed for asteroseismology. We conclude by presenting some general predictions for detectability of Kepler KOIs.

The frequency-power spectra of solar-like oscillations in solar-type stars present a pattern of peaks with near regular frequency separations. The mode powers are modulated by an envelope. The envelope has a bell-shaped appearance in many stars for which solar-like oscillations have been observed (including the Sun). Different techniques have been devised and applied to the Kepler SC data to detect signatures of solar-like oscillations (e.g., see Chaplin et al. 2010; Campante et al. 2010; Hekker et al. 2010; Huber et al. 2009; Karoff et al. 2010; Mosser & Appourchaux 2009; Mathur et al. 2010; Roxburgh 2009). Some techniques rely on extracting signatures of the near-regular frequency separations of the oscillations; others search for signatures of the Gaussian-like power excess due to the oscillations.

Here, we devise and apply a simple statistical detection test that is based upon the second family of techniques. Our choice is predicated on the requirement that the test should be straightforward to implement on a large number of stars. Making an absolute prediction of the detectability of the frequency separations is more complicated than the test we adopt. Strictly speaking, it demands that the damping rates of the solar-like oscillations, and the rotation and angle of inclination of the star, are known. We therefore opt for a simpler approach, which does not require assumptions to be made about these properties. We do so in the knowledge that once the oscillations' power excess has been detected, the large frequency separation in solar-type stars will also be readily extractable.

The detection test is based upon the total observed power due to p-mode oscillations in the star. Let P_tot be the total expected underlying mean power. As we shall show below, this is very easily estimated from scaling relations. In the frequency-power spectrum, the approximately Gaussian-shaped power envelope due to the oscillations sits on top of slowly varying background power, which we assume is dominated by contributions from shot/instrumental noise and stellar granulation. When the observed power in the oscillations relative to the background is high (i.e., high S/N), the power excess due to the oscillations will be clearly visible. However, at low S/N, statistical fluctuations in the background power may swamp the oscillations signal, so that the excess is much harder to see.

If B_tot measures the total underlying background power across the range in frequency occupied by the oscillations, then a "global" measure of S/N in the oscillations spectrum is given by

$$\begin{equation} {\rm {\rm S/N}_{tot}} = P_{\rm tot} / B_{\rm tot}. \end{equation} \tag{ 1 }$$

Provided ${\rm {\rm S/N}_{tot}}$ is high enough, we may be able to claim an unambiguous detection of the power excess due to the oscillations. If N independent frequency bins have contributed to the estimation of the S/N, we may then test it against χ² 2N-dof statistics to determine if the S/N is sufficiently high that it would be hard to explain by chance alone. This is a false-alarm approach.

Let us suppose that we demand that a false-alarm test be passed at, say, the 1% level to merit a likely detection. We first estimate a false-alarm threshold, ${\rm {\rm S/N}_{thresh}}$, i.e., the S/N value above which there is less than a 1 in 100 chance of a χ² 2N-dof distribution giving a normalized value greater than or equal to this purely at random. We then use the scaling relations to estimate the expected underlying ${\rm {\rm S/N}_{tot}}$ of the target, and, with that number in hand, we may then estimate the probability that an observed value of the S/N for the star would be greater than, or equal to, ${\rm {\rm S/N}_{thresh}}$, i.e., that it would pass the false-alarm test.³¹ This final probability serves as an estimate of the detection probability of the star.

3.1. ν_max and Δν

The frequency at the peak of the power envelope of the oscillations, where the observed modes present their strongest amplitudes, is commonly referred to as ν_max. Our ability to predict ν_max from the fundamental stellar properties allows us to estimate where in frequency the oscillations would be most prominent.

We assume that ν_max scales with the acoustic cutoff frequency (Brown et al. 1991; Kjeldsen & Bedding 1995) determined for an isothermal stellar atmosphere (as verified observationally by Mosser et al. 2010). This scaling has been shown to provide good predictions for observed solar-like oscillation spectra. This gives, in terms of the fundamental stellar properties, the following scaling relation:

$$\begin{equation} \nu _{\max } = \nu _{\max,\odot }\, \left(\frac{M}{M_{\odot }} \right) \left(\frac{R}{R_{\odot }} \right)^{-2} \left(\frac{T_{\rm eff}}{T_{\rm eff,\odot }}\right)^{-0.5}, \end{equation} \tag{ 2 }$$

where we have scaled against the solar values of ν_{max, ☉} = 3150 μHz and T_{eff, ☉} = 5777 K.

The most striking characteristic frequency separations in the oscillation spectra are the so-called large frequency separations, Δν. These are the separations between consecutive overtones, n, having the same spherical angular degree, l. The average value of the large frequency separation scales very well with the square root of the mean density of the star (e.g., see Christensen-Dalsgaard 1993). Here, we use this scaling and calibrate estimated values against the observed average large separation of the Sun, i.e.,

$$\begin{equation} \Delta \nu = \Delta \nu _{\odot } \left(\frac{M}{M_{\odot }} \right)^{0.5} \left(\frac{R}{R_{\odot }} \right)^{-1.5}, \end{equation} \tag{ 3 }$$

with Δν_☉ = 134.9 μHz. It has also recently been shown (Stello et al. 2009b) that solar-like oscillations in main-sequence stars follow to good approximation the following relation:

$$\begin{equation} \Delta \nu = \Delta \nu _{\odot } \left(\frac{\nu _{\rm max}}{\nu _{\rm max,\odot }} \right)^{0.75}. \end{equation} \tag{ 4 }$$

Use of Equation (4) allows us to eliminate M from Equations (2) and (3), to yield scaling relations that depend only on T_eff and R. The process of elimination yields the scaling relation M∝T^1.5_eff, which is close to the relation M∝T^1.7_eff in Noyes et al. (1984). The scaling relations for Δν and ν_max in R and T_eff are

$$\begin{equation} \nu _{\rm max} = \nu _{\rm max,\odot }\, \left(\frac{R}{R_{\odot }} \right)^{-2} \left(\frac{T_{\rm eff}}{{T_{\rm eff,\odot }}}\right), \end{equation} \tag{ 5 }$$

$$\begin{equation} \quad \;\Delta \nu = \Delta \nu _{\odot }\, \left(\frac{R}{R_{\odot }} \right)^{-1.5} \left(\frac{T_{\rm eff}}{{T_{\rm eff,\odot }}}\right)^{0.75}. \end{equation} \tag{ 6 }$$

3.2. Mode Amplitudes

In order to comment on mode detectability, we must be able to predict the mode amplitudes. We base our predictions on Kjeldsen & Bedding (1995) and Samadi et al. (2007). We make predictions for observations made in intensity (relevant to Kepler photometry). The prediction is that the maximum oscillation intensity amplitudes A_max of the radial (l = 0) modes scale like

$$\begin{equation} A_{\rm max} = A_{\rm max,\odot } \beta \, \left(\frac{L/{L_{\odot }}}{M/{M_{\odot }}} \right)^{s} \left(\frac{T_{\rm eff}}{{T_{\rm eff,\odot }}} \right)^{-2}, \end{equation} \tag{ 7 }$$

with the coefficient s often quoted between 0.7 and 1.0. We follow Kjeldsen & Bedding (1995), and assume that the temperature dependence is quadratic. We have also included another coefficient, β, because without any further correction the above relation is known to overestimate the amplitudes in the hottest solar-type stars (e.g., see Houdek 2006 and references therein). To estimate the solar rms value A_{max, ☉} we used the Kepler response function in wavelength and known measurements of the solar amplitudes made in narrow-band observations by the three wavelength channels of the VIRGO/SPM instrument on board the Solar and Heliospheric Observatory (see J. Ballot 2011, in preparation). From the SPM data we defined the observed solar amplitude as a function of wavelength, and then weighted this curve by the Kepler response function to give an estimated rms value of A_{max, ☉} ∼ 2.5 ppm.

We have used results of Kepler observations of solar-type stars to calibrate the size of the correction β. Results on Δν and ν_max were used by modeling teams, together with T_eff, to provide estimates of the stellar radii and masses (e.g., see Stello et al. 2009a; Basu et al. 2010; Quirion et al. 2010; Gai et al. 2011). These estimated stellar properties were used to calculate predicted maximum amplitudes using Equation (7), with the unknown (at this stage) β set to unity. We then calculated the ratios of the observed amplitudes, A_max (extracted by the Hekker et al. 2010 pipeline), to the predicted amplitudes, (A_max)_pred, for different assumed values of s. We now go on to discuss the results obtained for s = 1.0.

The ratios for s = 1.0 are plotted in Figure 1 against T_eff (left-hand panels) and T_red − T_eff (right-hand panels). We define T_red to be the temperature on the red edge of the radial-mode δ-Scuti instability strip, given in Houdek et al. (1999), at the luminosity, L, of a given star:

$$\begin{equation} T_{\rm red} = T_{\rm red,\odot } (L/{L_{\odot }})^{-0.093} \end{equation} \tag{ 8 }$$

with T_{red, ☉} = 8907 K. Both panels of Figure 1 show a decrease in the ratio of the observed to predicted amplitudes as T_eff increases. A more clearly defined trend is present in the T_red − T_eff plot, and we therefore chose to use fits performed on these data to fix the correction β. We fitted the following functions: first

$$\begin{equation} \beta = 1 - \exp \left(- \frac{T_{\rm red} - T_{\rm eff}}{\Delta T} \right), \end{equation} \tag{ 9 }$$

and then the less restrictive

$$\begin{equation} \beta = c_0 - c_1 \exp \left(- \frac{T_{\rm red} - T_{\rm eff}}{\Delta T} \right), \end{equation} \tag{ 10 }$$

where c₀, c₁, and ΔT were the free parameters to be optimized, and weights for the fits were fixed by the uncertainties on the amplitude ratios. The correction seeks to allow for the attenuation of the amplitudes at high T_eff (which maps to low T_red − T_eff), and since at progressively lower T_eff one expects the amplitude predictions to be much more robust, we require a function that turns over at high T_red − T_eff (hence the choice of an exponential). The solid blue and dashed red lines in the right-hand panel of Figure 1 show the best-fitting equations (9) and (10), respectively. The best-fitting χ² for the two models were almost indistinguishable (about 2.3), and as such we chose to adopt the simpler equation (9) as our model for β, with its best-fitting ΔT = 1550 K. A fit to a simple straight-line model gave a marginally better χ² than either of the two exponential models. The straight-line model does of course fail to capture the requirement that the amplitude ratio curve turns over at higher T_red − T_eff, and its use would lead to the prediction of unphysically high amplitudes in the coolest solar-type stars.

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We also performed the above analyses assuming the predicted amplitudes scaled like (L/M)^0.7 (i.e., with s = 0.7 in Equation (7)). However, we found that we had to increase arbitrarily our assumed value of A_{max, ☉}—from 2.5 to 3.1 ppm—otherwise most of the amplitude ratios lay significantly above unity. We therefore opted to fix s = 1.0 (as per the results outlined above).

To summarize, we have, using the scaling relations in R and T_eff, and taking s = 1.0,

$$\begin{equation} A_{\rm max} = 2.5\,\beta \, \left(\frac{R}{R_{\odot }} \right)^{2} \left(\frac{T_{\rm eff}}{{T_{\rm eff,\odot }}}\right)^{0.5}, \end{equation} \tag{ 11 }$$

with β calibrated by Equation (9) using ΔT = 1550 K.

We assume that there are two dominant contributions to the background power spectral density in the vicinity of ν_max in the frequency-power spectrum. The largest contribution comes from instrumental/shot noise, which we define in terms of a simple offset b_instr. When observations of the oscillations are made in intensity, stellar granulation also contributes significant power which must be accounted for in estimation of the total background. We model the granulation power as a Lorentzian function centered on zero frequency (e.g., Harvey 1985), with a characteristic power spectral density, b_gran, and a characteristic timescale, τ_gran. The power spectral density P_gran at frequency ν_max is then given by

$$\begin{equation} P_{\rm gran} = \frac{b_{\rm gran}}{1+(2\pi \tau _{\rm gran} \nu _{\rm max})^2}\;\rm ppm^2\;\mu Hz^{-1}. \end{equation} \tag{ 12 }$$

We now go on to describe scaling relations for each of the free parameters needed to model the total background, beginning with those describing the granulation term.

4.1. Scaling Relations for Stellar Granulation

We follow Huber et al. (2009) in assuming that τ_gran scales inversely with ν_max, as proposed by H. Kjeldsen & T. R. Bedding (2011, in preparation). This means that the denominator of Equation (12) will be constant for all stars. Taking solar values of τ_{gran, ☉} = 210 s and ν_{max, ☉} = 3150 μHz, we have that

$$\begin{equation} P_{\rm gran} \approx 0.055\, b_{\rm gran}\;\rm ppm^2\;\mu Hz^{-1}. \end{equation} \tag{ 13 }$$

To describe b_gran, we assume that the typical size of a convective granule is proportional to the scale height for an isothermal atmosphere, and that all granules behave in a statistically independent manner so that the total rms fluctuation scales inversely as the square root of the number of observed granules. This leads to (e.g., Schwarzschild 1975; H. Kjeldsen & T. R. Bedding 2011, in preparation; see also Baudin et al. 2006):

$$\begin{equation} b_{\rm gran} \propto \left(\frac{L}{L_{\odot }} \right)^{2} \left(\frac{M}{M_{\odot }} \right)^{-3} \left(\frac{T_{\rm eff}}{T_{\rm eff,\odot }} \right)^{-5.5}. \end{equation} \tag{ 14 }$$

With M∝T^1.5_eff (see Section 3.1), the above implies that

$$P_{\rm gran} \propto b_{\rm gran} \propto \left(\frac{R}{R_{\odot }} \right)^{4} \left(\frac{T_{\rm eff}}{{T_{\rm eff,\odot }}}\right)^{-2}$$

$$\begin{equation} \!\qquad \propto \left(\frac{\nu _{\rm max}}{\nu _{\rm max,\odot }} \right)^{-2}. \end{equation} \tag{ 15 }$$

We have tested Equation (15) using results from the Kepler survey on solar-type stars. The left-hand panel of Figure 2 plots the best-fitting P_gran of solar-type stars with detected oscillations (given by a fit of a Harvey-like model, plus a flat offset, to the background power spectral density). The best-fitting power law is found to be P_gran∝ν^−1.8_max, which is close to the prediction above. Using appropriate normalization from the real data, we therefore adopt

$$\begin{equation} P_{\rm gran} \approx 0.1 \left(\frac{\nu _{\rm max}}{\nu _{\rm max,\odot }} \right)^{-2}\;\rm ppm^2\;\mu Hz^{-1}. \end{equation} \tag{ 16 }$$

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The red line in Figure 2 shows the prediction from Equation (16).

4.2. Scaling Relations for Instrumental/Shot Noise

We use the "minimal term" noise model for Kepler presented in Gilliland et al. (2010b) in order to predict the rms noise, σ, per 58.85 s integration in the time domain:

$$\begin{equation} \sigma = \frac{10^6}{c} \left(c+9.5 \times 10^5(14/{\it Kp})^5 \right)^{1/2}\;\rm ppm, \end{equation} \tag{ 17 }$$

where c = 1.28 × 10^{0.4(12 − Kp) + 7} detections per cadence. The expected noise power spectral density in the frequency domain due to the instrumental noise is then given approximately by

$$\begin{equation} b_{\rm instr} = 2 \times 10^{-6}\;\sigma ^2\;\Delta t \;\rm ppm^2\;\mu Hz^{-1}, \end{equation} \tag{ 18 }$$

where Δt is the mean 58.85 s cadence.

The right-hand panel of Figure 2 plots the best-fitting b_instr of solar-type stars showing detected oscillations. The solid line shows the prediction from Equation (17). We see acceptable agreement between the observations and predictions.

5.1. Estimation of ${\rm {\rm S/N}_{tot}}$

First, we must compute the expected underlying global S/N of the target of interest, ${\rm {\rm S/N}_{tot}}$.

As noted previously, we assume that the power in the oscillations is modulated by a Gaussian-shaped envelope in frequency. We wish to calculate the total expected power underneath the envelope. The FWHM of the power envelope is assumed to be equal to ν_max/2 (e.g., see Stello et al. 2007, and references therein; also Mosser et al. 2010), implying that most of the total mode power is contained within a range ±ν_max/2 around ν_max.

Each segment Δν of the spectrum will contain power due to the visible l = 0, 1, 2, and 3 modes (although note that the relative contribution from the l = 3 modes is extremely weak). The l = 0 mode at the center of the Gaussian-shaped envelope will have a power approximately equal to A²_max. The neighboring l = 1, 2, and 3 modes in the central Δν will contribute a further ≈2.1 A²_max of power (e.g., see J. Ballot 2011, in preparation; Kjeldsen et al. 2008). This relative contribution is fixed by the relative visibilities of the different l. The total oscillations' power in the central segment is therefore equal to ≈3.1 A²_max.

To obtain the total mean power, P_tot, underneath the Gaussian-shaped envelope we must then add the contributions from all other segments of Δν that fall in the range ±ν_max/2 around ν_max. In all, ν_max/Δν segments will contribute, with the average power of the contributing segments being ∼0.5 times that of the central segment (the factor of 0.5 following from the fall-off in frequency of the Gaussian function). This implies that

$$\begin{equation} P_{\rm tot} \approx 1.55\; A_{\rm max}^2 \frac{\nu _{\rm max}}{\Delta \nu }. \end{equation} \tag{ 19 }$$

We must also make a correction for the fact that the high fractional duty cycle of the Kepler integrations, coupled with a cadence of nearly one minute, leads to apodization of the oscillations' signal the closer ν_max is to the Nyquist frequency (∼8496 μHz) of the SC sampling. The amplitudes are suppressed by the factor

$$\begin{equation} \eta = {\rm sinc} \left[ \pi /2 \left(\frac{\nu _{\rm max}}{8496\;\rm \mu Hz}\right) \right], \end{equation} \tag{ 20 }$$

so that Equation (19) should actually be written as

$$\begin{equation} P_{\rm tot} \approx 1.55\; A_{\rm max}^2\;\eta ^2\;\frac{\nu _{\rm max}}{\Delta \nu }. \end{equation} \tag{ 21 }$$

Figure 3 offers confirmation that Equation (21) does indeed provide a reasonably good description of P_tot, thereby implying that the Gaussian-envelope description provides a good model for the total observed oscillations power. It shows P_tot, as constructed with Equation (21) from the observed A_max, ν_max, and Δν of stars showing detected oscillations, against their actual measured P_tot, as given by the sum of the power excess above the fitted background in each frequency-power spectrum. The red line shows the one-to-one relation, which the data are seen to follow to good approximation.

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We take one additional step for the detection recipe, and use Equations (5), (6), and (11) to re-express ν_max, Δν, and A_max in terms of R and T_eff, to give the following relation:

$$\begin{equation} P_{\rm tot} \approx 225\;{\rm ppm^2}\;\beta ^2\;\eta ^2\, \left(\frac{R}{R_{\odot }} \right)^{3.5} \left(\frac{T_{\rm eff}}{{T_{\rm eff,\odot }}}\right)^{1.25}. \end{equation} \tag{ 22 }$$

Turning next to the background, the total underlying background power in the frequency-power spectrum across the range ν_max is given approximately by

$$\begin{equation} B_{\rm tot} \approx b_{\rm max} \nu _{\rm max}, \end{equation} \tag{ 23 }$$

where b_max is the background power spectral density at the frequency ν_max, i.e.,

$$\begin{equation} b_{\rm max} = b_{\rm instr} + P_{\rm gran}. \end{equation} \tag{ 24 }$$

For simplicity, we have ignored the frequency dependence of the background, assuming instead that the value b_max captures the average value of the background to sufficient accuracy. We may then calculate the global S/N in the total oscillations power as ${\rm {\rm S/N}_{tot}} = P_{\rm tot} / B_{\rm tot}$ (Equation (1)).

5.2. Estimation of Detection Probability

A total of N independent frequency bins will contribute to estimation of the underlying P_tot and B_tot, and hence S/N_tot. We may therefore test S/N_tot against χ² 2N-dof statistics to determine if the observed S/N would be large enough that it would be hard to explain by chance alone. If T is the length of the time series, then N will be given by

$$\begin{equation} N = \left(\frac{\nu _{\rm max}/{\rm \mu Hz} \times T/{\rm s}}{10^6} \right). \end{equation} \tag{ 25 }$$

The probability of obtaining any value ${\rm {\rm S/N}}^{\prime }$ above a given level ${\rm {\rm S/N}}$ is given by

$$\begin{equation} P({\rm {\rm S/N}}^{\prime } \ge {\rm {\rm S/N}}, N) = \int _{x}^{\infty } \frac{\exp {(-x^{\prime })}}{\Gamma (N)} x^{\prime (N-1)}\,dx^{\prime }, \end{equation} \tag{ 26 }$$

where x = 1 + S/N and Γ is the Gamma function. Note the normalization of x: as ${\rm {\rm S/N}}$ tends to zero (i.e., no oscillation signal present), we demand that x tends to unity.

To flag a possible detection, we would demand that the observed S/N of the star exceed some S/N threshold, ${\rm {\rm S/N}_{thresh}}$, corresponding to a fractional false-alarm probability, p (e.g., p = 0.01, or 1 %). The required false-alarm threshold follows by solving:

$$\begin{equation} P({\rm {\rm S/N}}^{\prime } \ge {\rm {\rm S/N}_{thresh}}, N) = p. \end{equation} \tag{ 27 }$$

The probability that the observed S/N would exceed ${\rm {\rm S/N}_{thresh}}$ is then given by

$$\begin{equation} P_{\rm final} = \int _{y}^{\infty } \frac{\exp {(-y^{\prime })}}{\Gamma (N)} y^{\prime (N-1)}\,dy^{\prime }, \end{equation} \tag{ 28 }$$

with

$$\begin{equation} y = (1+{\rm {\rm S/N}_{thresh}})/(1+{\rm {\rm S/N}_{tot}}). \end{equation} \tag{ 29 }$$

We may regard P_final as providing an approximate estimate of the probability of detecting solar-like oscillations in the star. The equation for N captures the sensitivity of the detection probability to the length of the observations, T. As T, and hence N, increases, so the relative statistical fluctuations in B_tot will decrease in magnitude; hence, for a given underlying ${\rm {\rm S/N}_{tot}}$, the excess power due to the oscillations will be more clearly visible against the background (as reflected in a reduction in the size of ${\rm {\rm S/N}_{thresh}}$).

We now apply our detection test to the solar-type stars observed for 1 month each by Kepler during the first 7 months of science operations. We use the KIC R and T_eff and apparent magnitude of each target to predict whether oscillations would be detectable in 1 month. We then compare our predictions, in a statistical sense, with the actual results from Kepler. We only consider stars having an observed, or predicted, ν_max ⩾ 350 μHz, which takes us approximately to the base of the red-giant branch.

The bottom panel of Figure 4 plots stars that showed detections of solar-like oscillations with ν_max ⩾ 350 μHz, with luminosities estimated from the seismically determined radii. We demanded that at least two of the asteroseismic data analysis pipelines returned consistent results on a star, giving a total of 499 stars with flagged detections. This is out of a total of 1750 observed stars having predicted ν_max ⩾ 350 μHz. The sizes of the symbols are proportional to the observed S/N_tot. The dotted lines in Figure 4 are evolutionary tracks (Padova models; Giradi et al. 2002, 2004; Marigo et al. 2008) computed for solar composition and masses ranging from 0.7 to 1.5 M_☉ in steps of 0.1 M_☉. The dashed red line marks the location of the red edge of the instability strip (for fundamental radial-mode δ-Scuti pulsations; see Houdek et al. 1999). The Sun is plotted with its usual symbol. The lack of detections close to the red edge, along with the amplitude ratio data plotted in Figure 1, will provide important inputs for modeling the excitation and damping of solar-like oscillations (i.e., the interplay between those oscillations and the convection).

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The top panels in Figure 4 show results from the predictions, which were made using the KIC parameters of the stars, and only stars for which P_final ⩾ 0.9 were plotted. (Note that here the luminosities were estimated from the KIC parameters—using the radius and T_eff—whereas the luminosities in the lower panel are the observed luminosities, from the seismically determined radii and T_eff.) Symbol sizes are proportional to the predicted S/N_tot. The top left-hand panel shows predictions without the β correction having been applied. A total of 1163 stars are flagged with predicted detections. This emphasizes the importance of applying the β correction, since without it the predicted detections run all the way to the red edge of the instability strip, and the predicted S/N_tot reach noticeably higher values across the ensemble than do the real observations. When the β correction is applied, we obtain the results plotted in the top right-hand panel. There is now much better agreement with the observations, and 765 stars are flagged with predicted detections.

Figure 5 shows histograms of the observed detections (solid red lines), the β-corrected predictions (dashed blue lines), and the total number of observed stars having predicted ν_max ⩾ 350 μHz (dotted lines), as functions of KIC apparent magnitude, Kp (left-hand panel), and KIC T_eff (right-hand panel). We see good agreement between the distributions of observed detections and β-corrected predictions.

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The predicted number of detections exceeds the observed number by about 50%. Several factors contribute to this overestimation. First, it turns out that there are several classical oscillators in the ensemble. The KIC parameters imply that these stars should show detectable solar-like oscillations; however, owing to uncertainties in the categorization it appears that they are actually too hot. Second, there are several eclipsing binaries in the sample, which can make automated extraction of the oscillation parameters more challenging (due to the presence of fairly sharp features in the light curves). Neither of the above categories of stars is sufficiently numerous or troubling to account for most of the overestimation. It appears instead that elevated levels of stellar activity might be responsible for suppression of the mode amplitudes, and hence levels of detectability, in a sufficient number of stars to possibly explain the mismatch (as reported in Chaplin et al. 2011b). This may also relate to an interesting difference between the predictions and observations in the set of stars covering 6 ≲ L/L_☉ ≲ 9 and 5300 K ≲ T_eff ≲ 5700 K in the upper right-hand panel of Figure 4, which are largely missing from the lower panel. Gilliland (1985) has noted that stars in this region may show interesting evolutionary effects in their stellar dynamos, which would manifest themselves in the surface magnetic activity. We also add a note of caution regarding stars in this part of the H-R diagram. They have ν_max ∼ 350–500 μHz. This range in frequency coincides with an instrumental artifact in the Kepler data (see García et al. 2011), which may hamper detection of the solar-like oscillations.

We finish by using the detection test to predict the detection probabilities for four canonical main-sequence stars: early-type and late-type K stars, the Sun (late G), and a mid-type F star. Figure 6 plots the detection probabilities, P_final, for different assumed lengths of observation, T (see figure annotation), as a function of Kepler apparent magnitude, Kp. The detection rates fall off quite sharply with Kp, and long timescale observations are required in order to detect oscillations in stars fainter than Kp ∼ 13.

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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. We also thank all funding councils and agencies that have supported the activities of KASC Working Group 1. W.J.C., Y.E., S.J.H., and G.A.V. acknowledge the support of the UK Science and Technology Facilities Council (STFC). S.H. acknowledges support from the Netherlands Organisation for Scientific Research (NWO). NCAR is supported by the National Science Foundation. We are also grateful for support from the International Space Science Institute (ISSI).

Facility: Kepler - The Kepler Mission