FUNDAMENTAL PROPERTIES OF KEPLER PLANET-CANDIDATE HOST STARS USING ASTEROSEISMOLOGY

Solar-like oscillations are acoustic standing waves excited by near-surface convection (see, e.g., Houdek et al. 1999; Houdek 2006; Samadi et al. 2007). The oscillation modes are characterized by a spherical degree l (the total number of surface nodal lines), a radial order n (the number of nodes from the surface to the center of the star), and an azimuthal order m (the number of surface nodal lines that cross the equator). The azimuthal order m is generally only important if the (2l + 1) degeneracy of frequencies of degree l is lifted by rotation.

Solar-like oscillations involve modes of low spherical degree l and high radial order n and hence the frequencies can be asymptotically described by a series of characteristic separations (Vandakurov 1968; Tassoul 1980; Gough 1986). The large frequency separation Δν is the spacing between modes with the same spherical degree l and consecutive radial order n, and probes the sound travel time across the stellar diameter. This means that Δν is related to the mean stellar density and is expected to scale as follows (Ulrich 1986):

$$\begin{equation} \Delta \nu = \frac{({M/M_{\odot }})^{1/2}}{(R/R_{\odot })^{3/2}} \Delta \nu _{\odot } \:. \end{equation} \tag{ 1 }$$

Another fundamental observable is the frequency at which the oscillations have maximum power (ν_max). As first argued by Brown et al. (1991), ν_max for Sun-like stars is expected to scale with the acoustic cut-off frequency and can therefore be related to fundamental stellar properties, as follows (Kjeldsen & Bedding 1995):

$$\begin{equation} \nu _{\rm max} = \frac{ M/M_{\odot }}{(R/R_{\odot })^{2}\sqrt{T_{\rm eff}/T_{\rm eff,\odot }}} \nu _{\rm max,\odot } \:. \end{equation} \tag{ 2 }$$

Equation (2) shows that ν_max is mainly dependent on surface gravity, and hence is a good indicator of the evolutionary state. Typical oscillation frequencies range from ∼3000 μHz for main sequence stars like our Sun down to ∼300 μHz for low-luminosity red giants, and a few μHz for high-luminosity giants. We note that while individual oscillation frequencies provide more detailed constraints on properties such as stellar ages (see, e.g., Doğan et al. 2010; Metcalfe et al. 2010; di Mauro et al. 2011; Mathur et al. 2012; Metcalfe et al. 2012), the extraction of frequencies is generally only possible for high signal-to-noise (S/N) detections. To extend our study to a large ensemble of planet-candidate hosts, we therefore concentrate solely on determining the global oscillation properties Δν and ν_max in this paper.

It is important to note that Equations (1) and (2) are approximate relations which require calibration. For comprehensive reviews of theoretical and empirical tests of both relations we refer the reader to Belkacem (2012) and Miglio et al. (2013a), but we provide a brief summary here. It is well known that Equation (2) is on less firm ground than Equation (1) due to uncertainties in modeling convection which drives the oscillations, although recent progress on the theoretical understanding of Equation (2) has been made (Belkacem et al. 2011). Stello et al. (2009) showed that both relations agree with models to a few percent, which was supported by investigating the relation between Δν and ν_max for a large ensemble of Kepler and CoRoT stars and by comparing derived radii and masses to evolutionary tracks and synthetic stellar populations (Hekker et al. 2009; Miglio et al. 2009; Kallinger et al. 2010a; Huber et al. 2010; Mosser et al. 2010; Huber et al. 2011; Silva Aguirre et al. 2011; Miglio et al. 2012b). More recently, White et al. (2011) showed that that Δν calculated from individual model frequencies can systematically differ from Δν calculated using Equation (1) by up to 2% for giants and for dwarfs with M/M_☉ > 1.2. Similar results were found by Mosser et al. (2013) who investigated the influence of correcting the observed Δν to the value expected in the high-frequency asymptotic limit. In general, however, comparisons with individual frequency modeling have shown agreement within 2% and 5% in radius and mass, respectively, both for dwarfs (Mathur et al. 2012) and for giants (di Mauro et al. 2011; Jiang et al. 2011).

Empirical tests have been performed using independently determined fundamental properties from Hipparcos parallaxes, eclipsing binaries, cluster stars, and optical long-baseline interferometry (see, e.g., Stello et al. 2008; Bedding 2011; Brogaard et al. 2012; Miglio 2012; Miglio et al. 2012a; Huber et al. 2012; Silva Aguirre et al. 2012). For unevolved stars (logg ≳ 3.8), no evidence for systematic deviations has yet been determined within the observational uncertainties, with upper limits of ≲ 4% in radius (Huber et al. 2012) and ≲ 10% in mass (Miglio 2012). For giants and evolved subgiants (log g ≲ 3.8) similar results have been reported, although a systematic deviation of ∼3% in Δν has recently been noted for He-core-burning red giants (Miglio et al. 2012a).

In summary, for stars considered in this study, Equations (1) and (2) have been tested theoretically to ∼2% and ∼5%, as well as empirically to ≲ 4% and ≲ 10% in radius and mass, respectively. While it should be kept in mind that future revisions of these relations based on more precise empirical data are possible, it is clear that these uncertainties are significantly smaller than for classical methods of determining radii and masses of field stars.

Our analysis is based on Kepler short-cadence (Gilliland et al. 2010b) and long-cadence (Jenkins et al. 2010) data through Q11. We have used simple-aperture photometry (SAP) data for our analysis. We have analyzed all available data for the 1797 planet-candidate hosts listed in the cumulative catalog by Batalha et al. (2013). Before searching for oscillations, transits need to be removed or corrected since the sharp structure in the time domain would cause significant power leakage from low frequencies into the oscillation frequency domain. This was done using a median filter with a length chosen according to the measured duration of the transit. In an alternative approach, all transits were phase-clipped from the time series using the periods and epochs listed in Batalha et al. (2013). Note that for typical transit durations and periods, the induced gaps in the time series have little influence on the resulting power spectrum. Finally, all time series were high-pass filtered by applying a quadratic Savitzky–Golay filter (Savitzky & Golay 1964) to remove additional low-frequency power due to stellar activity and instrumental variability. For short-cadence data, the typical cut-off frequency was ∼100 μHz, while for long-cadence data a cut-off of ∼1 μHz was applied.

To detect oscillations and extract the global oscillation parameters Δν and ν_max, we have used the analysis pipelines described by Huber et al. (2009), Hekker et al. (2010), Karoff et al. (2010), Verner & Roxburgh (2011) and Lund et al. (2012). Note that these methods have been extensively tested on Kepler data and were shown to agree well with other methods (Hekker et al. 2011a; Verner et al. 2011b; Hekker et al. 2012). We successfully detect oscillations in a total of 77 planet-candidate hosts (including 11 stars for which asteroseismic solutions have been published in separate studies). For 69 host stars short-cadence data were used, while 8 of them showed oscillations with ν_max values low enough to allow a detection using long-cadence data. The final values for Δν and ν_max are listed in Table 1 and were adopted from the method of Huber et al. (2009), with uncertainties calculated by adding in quadrature the formal uncertainty and the scatter of the values over all other methods. Note that in some cases the S/N was too low to reliably estimate ν_max, and hence only Δν is listed. For one host (KOI-1054) Δν could not be reliably determined, and hence only ν_max is listed. The solar reference values, which were calculated using the same method, are Δν_☉ = 135.1 ± 0.1 μHz and ν_{max, ☉} = 3090 ± 30 μHz (Huber et al. 2011).

We note that in the highest S/N cases, the observational uncertainties on Δν are comparable to or lower than the accuracy to which Equation (1) has been tested (see previous section). To account for systematic errors in Equation (1), we adopt a conservative approach by adding to our uncertainties in quadrature the difference between the observed Δν and the corrected Δν using Equation (5) in White et al. (2011). To account for the fact that Δν can be measured more precisely than ν_max, the same fractional uncertainties were added in quadrature to the formal ν_max uncertainties. The final median uncertainties in Δν and ν_max are 2% and 4%, respectively.

Figure 1 shows examples of power spectra for three stars in the sample, illustrating a main-sequence star (top panel), a subgiant (middle panel), and a red giant (bottom panel). Note that the power spectra illustrate typical intermediate S/N detections. Broadly speaking the detectability of oscillations depends on the brightness of the host star and the evolutionary state, because oscillation amplitudes scale proportionally to stellar luminosity (see, e.g., Kjeldsen & Bedding 1995; Chaplin et al. 2011a). Among host stars that are close to the main-sequence (log g > 4.2), the faintest star with detected oscillations has a Kepler magnitude of 12.4 mag.

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In addition to asteroseismic constraints, effective temperatures and metallicities are required to derive a full set of fundamental properties. For all stars in our sample high-resolution optical spectra were obtained as part of the Kepler follow-up program (Gautier et al. 2010). Spectroscopic observations were taken using four different instruments: the HIRES spectrograph (Vogt et al. 1994) on the 10 m telescope at Keck Observatory (Mauna Kea, HI), the FIES spectrograph (Djupvik & Andersen 2010) on the 2.5 m Nordic Optical Telescope at the Roque de los Muchachos Observatory (La Palma, Spain), the TRES spectrograph (Fürész 2008) on the 1.5 m Tillinghast reflector at the F. L. Whipple Observatory (Mt. Hopkins, AA), and the Tull Coudé spectrograph on the 2.7 m Harlan J. Smith Telescope at McDonald Observatory (Fort Davis, TX). Typical resolutions of the spectra range from 40,000 to 70,000. Atmospheric parameters were derived using either the Stellar Parameter Classification (SPC; see Buchhave et al. 2012) or Spectroscopy Made Easy (SME; see Valenti & Piskunov 1996) pipelines. Both methods match the observed spectrum to synthetic model spectra in the optical wavelengths and yield estimates of T_eff, log g, metallicity and vsin (i). Note that in our analysis we have assumed that the metal abundance [m/H], as returned by SPC, is equivalent to the iron abundance [Fe/H]. For stars with multiple observations each spectrum was analyzed individually, and the final parameters were calculated as an average of the individual results weighted by the cross-correlation function (CCF), which gives a measure of the quality of the fit compared to the spectral template. To ensure a homogenous set of parameters, we adopt the spectroscopic values from SPC, which was used to analyze the entire sample of host stars. Table 1 lists for each planet-candidate host the details of the SPC analysis such as the number of spectra used, the average S/N of the observations, the average CCF, and the instrument used to obtain the spectra.

As discussed by Torres et al. (2012), spectroscopic methods such as SME and SPC suffer from degeneracies between T_eff, log g, and [Fe/H]. Given the weak dependency of ν_max on T_eff (see Equation (2)), asteroseismology can be used to remove such degeneracies by fixing log g in the spectroscopic analysis to the asteroseismic value (see, e.g., Bruntt et al. 2012; Morel & Miglio 2012; Thygesen et al. 2012). This is done by performing the asteroseismic analysis (see next section) using initial estimates of T_eff and [Fe/H] from spectroscopy, and iterating both analyses until convergence is reached (usually after one iteration). We have applied this method to all host stars in our sample to derive asteroseismically constrained values of T_eff and [Fe/H], which are listed in Table 2. Note that since SPC has been less tested for giants, we have adopted more conservative error bars of 80 K in T_eff and 0.15 dex in [Fe/H] for all evolved giants with log g < 3 (Thygesen et al. 2012). For all stars in our sample, we have added contributions of 59 K in T_eff and 0.062 dex in [Fe/H] in quadrature to the formal uncertainties to account for systematic differences between spectroscopic methods, as suggested by Torres et al. (2012).

Our sample allows us to investigate the effects of fixing log g on the determination of T_eff and [Fe/H]. This is analogous to the work of Torres et al. (2012), who used stellar densities derived from transits to independently determine log g for a sample of main-sequence stars. Figure 2 shows the differences in log g, T_eff, and [Fe/H] as a function of T_eff. As in the Torres et al. (2012) sample, the unconstrained analysis tends to slightly underestimate log g (and hence T_eff and [Fe/H]) for stars hotter than ∼6000 K. More serious systematics are found for stars with T_eff ≲ 5400 K, which in our sample corresponds to subgiant and giant stars, for which log g is systematically overestimated by up to 0.2 dex. The effect of these systematics on planet-candidate radii will be discussed in detail in Section 3.1.

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Figure 2 shows that changes in log g are correlated with changes in T_eff and [Fe/H]. We have investigated the partial derivatives ΔT_eff/Δlog g and Δ[Fe/H]/Δlog g and did not find a significant dependence on stellar properties such as effective temperature. The median derivatives for our sample are ΔT_eff/Δlog g = 475 ± 60 K dex⁻¹ and Δ[Fe/H]/Δlog g = 0.31 ± 0.03, respectively. Hence, a change of log g = 0.1 dex typically corresponds to a change of ∼50 K in T_eff and 0.03 dex in [Fe/H].

The results in this study can also be used to test temperatures based on broadband photometry. A comparison of 46 dwarfs that overlap with the sample of Pinsonneault et al. (2012) showed that the photometric temperatures (corrected for the spectroscopic metallicities in Table 2) are on average 190 K hotter than our spectroscopic estimates with a scatter of 130 K. This offset is larger than previous comparisons based on a brighter comparison sample (see Pinsonneault et al. 2012), pointing to a potential problem with interstellar reddening. The results of our study, combined with other samples for which both asteroseismology and spectroscopy are available (Molenda-Żakowicz et al. 2011; Bruntt et al. 2012; Thygesen et al. 2012), will be a valuable calibration sample to improve effective temperatures in the Kepler field based on photometric techniques such as the infrared flux method (Casagrande et al. 2010).

Given an estimate of the effective temperature, Equations (1) and (2) can be used to calculate the mass and radius of a star (see, e.g., Kallinger et al. 2010b). However, since both equations allow radius and temperature to vary freely for any given mass, a more refined method is to include knowledge from evolutionary theory to match the spectroscopic parameters with asteroseismic constraints. This so-called grid-based method has been used extensively both for unevolved and evolved stars (Stello et al. 2009; Kallinger et al. 2010a; Chaplin et al. 2011b; Creevey et al. 2012; Basu et al. 2010, 2012).

To apply the grid-based method, we have used different model tracks: the Aarhus Stellar Evolution Code (ASTEC; Christensen-Dalsgaard 2008), the Bag of STellar Isochrones (BaSTI; Pietrinferni et al. 2004), the Dartmouth Stellar Evolution Database (DSEP; Dotter et al. 2008), the Padova stellar evolution code (Marigo et al. 2008), the Yonsei–Yale isochrones (YY; Demarque et al. 2004), and the Yale Rotating Stellar Evolution Code (YREC; Demarque et al. 2008). To derive masses and radii we have employed several different methods (da Silva et al. 2006; Stello et al. 2009; Basu et al. 2011; Miglio et al. 2013b; Silva Aguirre et al. 2013). In brief, the methods calculate a likelihood function for a set of independent Gaussian observables X:

$$\begin{equation} \mathcal {L}_{X} = \frac{1}{\sqrt{2\pi }\sigma _{X}} \exp {\left(\frac{-(X_{\rm obs}-X_{\rm model})^2}{2\sigma _{X}^2}\right)} \end{equation} \tag{ 3 }$$

with X = {T_eff, [Fe/H], ν_max, Δν}. The combined likelihood is:

$$\begin{equation} \mathcal {L} = \mathcal {L}_{T_{\rm eff}} \mathcal {L}_{\rm {Fe/H}} \mathcal {L}_{{\nu _{\rm max}}} \mathcal {L}_{{\Delta \nu }} \:. \end{equation} \tag{ 4 }$$

Note that for cases where only Δν could be measured, the ν_max term in Equation (4) was omitted. The best-fitting model is then identified from the likelihood distribution for a given physical parameter, e.g., stellar mass and radius. Uncertainties are calculated, for example, by performing Monte Carlo simulations using randomly drawn values for the observed values of T_eff, [Fe/H], ν_max, and Δν. For an extensive comparison of these methods, including a discussion of potential systematics, we refer the reader to Gai et al. (2011).

As an example, Figure 3 shows a diagram of log g versus T_eff for KOI-244 (Kepler-25), with evolutionary tracks taken from the BaSTI grid. The blue and green areas show models within 1σ of the observationally measured values of Δν and ν_max, respectively. The insets show histograms of Monte Carlo simulations for mass and radius. The best-fitting values were calculated as the median and 84.1 and 15.9 percentile (corresponding to the 1σ confidence limits) of the distributions.

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To account for systematics due to different model grids, the final parameters for each star were calculated as the median over all methods, with uncertainties estimated by adding in quadrature the median formal uncertainty and the scatter over all methods. Table 2 lists the final parameters for all host stars in our sample. The median uncertainties in radius and mass are 3% and 7%, respectively, consistent with the limits discussed in Section 2.1. Note that for 11 hosts we have adopted the solutions published in separate papers. We also note that for some stars the mass and radius distributions are not symmetric due to the "hook" in evolutionary tracks just before hydrogen exhaustion in the core (see Figure 3). In general, however, variations in the results of different model grids are larger than these asymmetries and hence it is valid to assume symmetric (Gaussian) error bars, as reported in Table 2. Table 2 also reports the stellar density directly derived from Equation (1). Importantly, the stellar properties presented here are independent of the properties of the planets themselves, and hence can be directly used to re-derive planetary parameters. Table 3 lists revised radii and semi-major axes for the 107 planet candidates in our sample calculated using the transit parameters in Batalha et al. (2013).

To check the consistency of the stellar properties, we compared the final radius and mass estimates in Table 2 with those calculated by solving Equations (1) and (2) (the direct method) for stars which have both reliable ν_max and Δν measurements. We found excellent agreement between both determinations, with no systematic deviations and a scatter consistent with the uncertainties from the direct method.

The planet-candidate catalog by Batalha et al. (2013) included a revision of stellar properties based on matching available constraints to Yonsei–Yale evolutionary tracks. This revision (hereafter referred to as YY values) was justified since KIC surface gravities for some stars, in particular for cool M-dwarfs and for G-type dwarfs, seemed unphysical compared to predictions from stellar evolutionary theory. When available, the starting values for this revision were spectroscopic solutions, while for the remaining stars KIC parameters were used as initial guesses. Our derived stellar parameters allow us to test this revision based on our subset of planet-candidate hosts.

Figure 4 shows surface gravity versus effective temperature for all planet-candidate hosts in the Batalha et al. (2013) catalog. The nearly horizontal dashed line shows the long-cadence Nyquist limit, below which short-cadence data are needed to sufficiently sample the oscillations. Thick red symbols in the plot show all host stars for which we have detected oscillations using short-cadence data (diamonds) and long-cadence data (triangles). Note that the T_eff and log g values plotted for these stars were derived from the combination of asteroseismology and spectroscopy, as discussed in Section 2 (see also Table 2). For each detection, a thin line connects the position of the star determined in this work to the values published in Batalha et al. (2013).

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Figure 4 shows that our sample consists primarily of slightly evolved F- to G-type stars. This is due to the larger oscillation amplitudes in these stars compared to their unevolved counterparts. We observe no obvious systematic shift in log g for unevolved stars, while surface gravities for evolved giants and subgiants were generally overestimated compared to the asteroseismic values. To illustrate this further, Figure 5 shows the difference between fundamental properties (log g, radius, mass and temperature) from the asteroseismic analysis and the values given by Batalha et al. (2013) as a function of surface gravity. Red triangles mark stars for which KIC parameters were used as initial values, while black diamonds are stars for which spectroscopic solutions were used. In the following we distinguish between unevolved and evolved stars using a cut at log g = 3.85 (see dotted line in Figure 5), which roughly divides our sample between stars before and after they reached the red-giant branch.

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Table 4 summarizes the mean differences between values derived in this work and the values given by Batalha et al. (2013). If we only use unevolved stars (log g > 3.85), the differences for stars with spectroscopic follow-up (black diamonds in Figure 5) are small, with an average difference of −0.04 ± 0.02 dex (scatter of 0.12 dex) in log g and 6 ± 2% (scatter of 15%) in radius. For stars based on KIC parameters (red triangles in Figure 5), the mean differences are considerably larger with −0.17 ± 0.10 dex (scatter of 0.29 dex) in log g and 41 ± 17% (scatter of 52%) in radius. This confirms previous studies predicting overestimated KIC radii for Kepler targets (Verner et al. 2011a; Gaidos & Mann 2013) and shows that, as expected, spectroscopy yields a strong improvement (both in reduced scatter and offset) compared to the KIC. This emphasizes the need for a systematic spectroscopic follow-up of all planet-candidate hosts.

The unevolved planet-candidate host with the largest change in stellar parameters from this study is KOI-268 (see annotation in Figure 4), which has previously been classified as a late K-dwarf with T_eff ∼ 4800 K, hosting a 1.6 R_⊕ planet in a 110 d orbit. Our asteroseismic analysis yields log g = 4.26 ± 0.01, which is clearly incompatible with a late-type dwarf. Follow-up spectroscopy revised T_eff for this star to 6300 K, correctly identifying it as a F-type dwarf. The revised radius for the planet candidate based on the stellar parameters presented here is 3.00 ± 0.06 R_⊕ (compared to the previous estimate of 1.6 R_⊕).

We note that four unevolved hosts in our sample are confirmed planetary systems that until now had no available asteroseismic constraints: Kepler-4 (Borucki et al. 2010b), Kepler-14 (Buchhave et al. 2011), Kepler-23 (Ford et al. 2012), and Kepler-25 (Steffen et al. 2012a). The agreement with the host-star properties published in the discovery papers (based on spectroscopic constraints with evolutionary tracks) is good, and the more precise stellar parameters presented here will be valuable for future studies of these systems. We have also compared our results for Kepler-4 and Kepler-14 with stellar properties published by Southworth (2011, 2012) and found good agreement within 1σ for mass and radius.

Turning to evolved hosts (log g < 3.85), the differences between the asteroseismic and YY values for stars with spectroscopic follow-up (black diamonds in Figure 5) are on average −0.27 ± 0.03 dex (scatter of 0.11 dex) in log g and 44 ± 5% (scatter of 17%) in radius. For stars based on KIC parameters (red triangles in Figure 5), the mean differences are −0.1 ± 0.2 dex (scatter of 0.5 dex) in log g and 24 ± 37% (scatter of 83%) in radius. Unlike for unevolved stars, the parameters based on spectroscopy are systematically overestimated in log g, and hence yield planet-candidate radii that are systematically underestimated by up to a factor of 1.5. This bias is not present in the YY properties based on initial values in the KIC (although the scatter is high), and illustrates the importance of coupling asteroseismic constraints with spectroscopy, particularly for evolved stars.

Figure 6 shows planet-candidate radii versus orbital periods for the full planet-candidate catalog, highlighting all candidates in our sample with revised radii <50 R_⊕ in red. As in Figure 4, thin lines connect our rederived radii to the values published by Batalha et al. (2013). For a few evolved host stars, the revised host radii change the status of the candidates from planetary companions to objects that are more compatible with brown dwarfs or low-mass stars. The first planet-candidate host that was identified as a false positive using asteroseismology was KOI-145.01, as discussed by Gilliland et al. (2010a). KOI-2640.01 is another example for a potential asteroseismically determined false positive, with an increase of the companion radius from 9 R_⊕ to 17.0 ± 0.8 R_⊕. Additionally, the companions of KOI-1230 and KOI-2481 are now firmly placed in the stellar mass regime with revised radii of 64 ± 2 R_⊕ (0.58 ± 0.02 R_☉) and 31 ± 2 R_⊕ (0.29 ± 0.02 R_☉), respectively. On the other hand, for KOI-1894 the companion radius becomes sufficiently small to qualify the companion as a sub-Jupiter size planet candidate, with a decreased radius from 16.3 R_⊕ to 7.2 ± 0.4 R_⊕. KOI-1054 is a peculiar star in the sample with a very low metallicity ([Fe/H] = -0.9 dex). Despite the lack of a reliable Δν measurement our study confirms that this star is a evolved giant with log g = 2.47 ± 0.01 dex, indicating that the potential companion with an orbital period of only 3.3 days is likely a false positive.

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A recent study by Mann et al. (2012) showed that ∼96% of all bright (K_p < 14) and cool (T_eff < 4500 K) stars in the KIC are giants. This raises considerable worry about a giant contamination among the cool planet-candidate host sample and has implications for studies of planet detection completeness and the occurrence rates of planets with a given size. Asteroseismology provides an efficient tool to identify giants using Kepler photometry alone without the need for follow-up observations. Since oscillation amplitudes scale with stellar luminosity (see, e.g., the y-axis scale in Figure 1), giants show large amplitudes that are detectable in all typical Kepler targets. In addition, oscillation timescales scale with stellar luminosity, with frequencies for red giants falling well below the long-cadence Nyquist limit (see, e.g., Bedding et al. 2010; Hekker et al. 2011b; Mosser et al. 2012). Hence, any cool giant should show detectable oscillations using long-cadence data, which is readily available for all Kepler targets.

A few caveats to this method exist. First, there is a cut-off in temperature below which the variability is too slow to reliably measure oscillations. For data up to Q11 this limit is about 3700 K, corresponding to a ν_max ∼ 1 μHz at solar metallicity. Second, there is evidence that tidal interactions from close stellar companions can suppress oscillations in giants, as first observed in the hierarchical triple system HD 180891 (Derekas et al. 2011; Borkovits et al. 2013; Fuller et al. 2013). Hence, giant stars that are also in multiple systems with close companions could escape a detection with our method.

To test the success rate of asteroseismic giant identifications, we analyzed 132 stars for which Mann et al. (2012) presented a spectroscopic luminosity class and for which several quarters of Kepler data are available. As an example, Figure 7 compares the power spectra of the dwarf KIC6363233 and the giant KIC9635876. The power spectrum of the dwarf shows a strong peak at ∼0.8 μHz accompanied by several harmonics, a typical signature of non-sinusoidal variability due to rotational spot modulation. The giant, on the other hand, shows clear power excess with regularly spaced peaks that are typical for solar-like oscillations, indicating the evolved nature of the object. Out of the 132 stars in the sample, 96 were identified as giants based on detection of oscillations, compared to 104 stars that were identified as giants by Mann et al. (2012). All eight stars that were missed by the asteroseismic classification are cool (≲ 3700 K) giants for which the oscillation timescales are likely too long to be resolved with the available amount of Kepler data. Hence, this result implies a very high success rate when T_eff restrictions are taken into account and suggests that giants with suppressed oscillations by close-in stellar companions are rare.

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As shown in Figure 4, our analysis did not yield a detection of oscillations compatible with giant stars in any of the cool main-sequence hosts. This confirms that the majority of these stars are indeed dwarfs. Additionally, we did not detect oscillations in stars near the 14 Gyr isochrone of the YY models (which can be seen as a "finger" between T_eff = 4700–5000 K and log g = 3.8–4.0 in Figure 4). The non-detection of oscillations confirms that these stars must have log g ≳ 3.5, and hence are either subgiants or cool dwarfs.

The observation of transits allows a measurement of the semi-major axis as a function of the stellar radius (a/R_*), provided the eccentricity of the orbit is known. For the special case of circular orbits, a/R_* is directly related to the mean density of the star (see, e.g., Seager & Mallén-Ornelas 2003; Winn 2010):

$$\begin{equation} \langle \rho _{\star }\rangle \approx \frac{3\pi }{GP^{2}}\left(\frac{a}{R_{\star }}\right)^3 \:, \end{equation} \tag{ 5 }$$

where G is the gravitational constant and P is the orbital period. Equation (5) can be used to infer host-star properties in systems with transiting exoplanets, for example, to reduce degeneracies between spectroscopic parameters (see, e.g., Sozzetti et al. 2007; Torres et al. 2012).

Figure 8(a) compares the stellar densities derived from Equation (5) assuming d/R_* = a/R_* in the table of Batalha et al. (2013) with our independent estimates from asteroseismology. We emphasize that Batalha et al. (2013) explicitly report the quantity d/R_* to point out that it is only a valid measurement of stellar density in the case of zero eccentricity, for which d/R_* = a/R_*. We also note that the uncertainties reported by Batalha et al. (2013) do not account for correlations between transit parameters, and hence the density uncertainties are likely underestimated. For this first comparison, we have excluded all hosts with a transit density uncertainty >50%.

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The comparison shows differences greater than 50% for more than half of the sample, with mostly underestimated stellar densities from the transit model compared to the seismic densities. To investigate the cause of this discrepancy, Figure 8(b) shows the fractional difference of stellar densities as a function of the impact parameter (the sky-projected distance of the planet to center of the stellar disc, expressed in units of the stellar radius) for each planet candidate. We observe a clear correlation, with large disagreements corresponding preferentially to high impact parameters. We have tested whether this bias is due to insufficiently sampled ingress and egress times by repeating the transit fits for a fraction of the host stars using short-cadence data, and found that the agreement significantly improves if short-cadence data is used. Further investigation showed that the larger disagreements are found for the shallower transits, while better agreement is found for transits with the highest S/N. Hence, it appears that impact parameters tend to be overestimated for small planets, which is compensated by underestimating the density of the star to match the observed transit duration. The reason for this bias may to be due to anomalously long ingress times for small planets caused by smearing due to uncorrected transit timing variations or other effects.

Additional reasons for the discrepancies between the transit and seismic densities include planet candidates on eccentric orbits, and false positive planet candidates. For eccentric orbits the stellar density derived from the transit can be either over- or underestimated (depending on the orientation of the orbit to the observer). However, in this case no correlation with the impact parameter would be expected, and the distribution of impact parameters should be uniform. Additionally, if eccentric orbits were responsible for the majority of the outliers, we would expect to detect a correlation of the fractional difference in density with orbital period, with planet candidates on short orbital periods showing preferentially good agreement due to tidal circularization. However, no such correlation is apparent in our data. For false positive scenarios (e.g., a transit around a fainter background star) the large dilution would lead to underestimated densities, and a correlation with the impact parameter would be expected. However, recent results by Fressin et al. (2013) showed that the global false positive rate is ≲ 10%. Hence, eccentric orbits and false positives are likely not responsible for the majority of the outliers.

The first comparison shown here underlines the statement in Batalha et al. (2013) that stellar properties derived from the transit fits in the planet-candidate catalog should be viewed with caution, with further work being needed to quantify these differences. We emphasize that the comparison shown here does not imply that transits cannot be used to accurately infer stellar densities. To demonstrate this, Figure 8(a) also includes HD 17156, a system with an exoplanet in a highly eccentric orbit (e = 0.68) for which high S/N constraints from transits, radial velocities and asteroseismology are available (Gilliland et al. 2011; Nutzman et al. 2011). The seismic and transit density are in excellent agreement, demonstrating that both techniques yield consistent results when the eccentricity and impact parameter can be accurately determined. Similar tests can be expected in future studies, making use of Kepler exoplanet hosts for which asteroseismic and radial velocity constraints are available (Batalha et al. 2011; Gilliland et al. 2013; G. W. Marcy et al. 2013, in preparation). Additionally, the precise stellar properties presented here will enable improved determinations of eccentricities using high S/N transit light curves (Dawson & Johnson 2012) and yield improved constraints for the study of eccentricity distributions in the Kepler planet sample compared to planets detected with radial velocities (see, e.g., Wang & Ford 2011; Moorhead et al. 2011; Kane et al. 2012; Plavchan et al. 2012).

The favored theoretical scenario for the formation of terrestrial planets and the cores of gas giants is the core-accretion model, a slow process involving the collision of planetesimals (Safronov & Zvjagina 1969), with giant planets growing massive enough to gravitationally trap light gases (Mizuno 1980; Pollack et al. 1996; Lissauer et al. 2009; Movshovitz et al. 2010). The efficiency of this process is predicted to be correlated with the disk properties and therefore the characteristics of the host star, such as stellar mass (Thommes et al. 2008).

Observationally, Doppler velocity surveys have yielded two important correlations: gas giant planets occur more frequently around stars of high metallicity (Gonzalez 1997; Santos et al. 2004; Fischer & Valenti 2005) and around more massive stars (Laws et al. 2003; Johnson et al. 2007; Lovis & Mayor 2007; Johnson et al. 2010). Early results using Kepler planet candidates showed that single planets appear to be more common around hotter stars, while multiple planetary system are preferentially found around cooler stars (Latham et al. 2011). Both results are in line with Howard et al. (2012), who found that small planets are more common around cool stars. More recently, Steffen et al. (2012b) showed evidence that hot Jupiters indeed tend to be found in single planet systems, while Fressin et al. (2013) found that the occurrence of small planets appears to be independent of the host star spectral type. While most of these findings have been interpreted in favor of the core-accretion scenario, many results have so far relied on uncertain or indirect estimates of stellar mass (such as T_eff). The sample of host stars presented in this study allow us to test these results using accurate radii and masses from asteroseismology.

Following Howard et al. (2012), we attempted to account for detection biases by estimating the smallest detectable planet for a given planet-candidate host in our sample as:

$$\begin{equation} R_{\rm min} = R_{\star } (\rm {S/N} \, \sigma _{\rm CDPP})^{0.5} \left(\frac{{n}_{\rm tr} {t}_{\rm dur}}{6\; \rm {hr}}\right)^{0.25}. \end{equation} \tag{ 6 }$$

Here, R_⋆ is the host-star radius, S/N is the required signal-to-noise ratio, σ_CDPP is the 6 hr combined differential photometric precision (Christiansen et al. 2012), n_tr is the number of transits observed and t_dur is the duration of the transits which, for the simplified case of circular orbits and a central transit (impact parameter b = 0), is given by:

$$\begin{equation} t_{\rm dur} = \frac{R_{*} P}{\pi a} \:. \end{equation} \tag{ 7 }$$

Here, P is the orbital period and a is the semi-major axis of the orbit. For each candidate, we estimate R_min by calculating t_dur, adopting the median 6 hr σ_CDPP from quarters 1–6 for each star and setting an S/N threshold of 25 (Ciardi et al. 2013). To de-bias our sample, we calculated for a range of planet sizes R_x the number of planet candidates that are larger than R_x and for which R_min < R_x. The maximum number of planet candidates fulfilling these criteria was found for a value of R_x = 2.4 R_⊕. In the following, our debiased sample consists only of planet candidates with R > 2.4 R_⊕ and with R_min < 2.4 R_⊕. Figure 9(a) shows planet radii versus host-star mass for all planet candidates in our sample (gray symbols), with the de-biased sample shown as filled symbols. Planet candidates in multi systems are shown as diamonds, while single systems are shown as triangles. For all candidates in the debiased sample, symbols are additionally color-coded according to the incident flux as a multiple of the flux incident on Earth.

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While the size of our de-biased sample precludes definite conclusions, we see that our observations are consistent with previous studies that found gas-giant planets to be less common around low-mass stars (<1 M_☉) than around more massive stars. A notable exception is KOI-1 (TReS-2), a 0.9 M_☉ K-dwarf hosting a hot Jupiter in a 2.5 day orbit (O'Donovan et al. 2006; Holman et al. 2007; Kipping & Bakos 2011; Barclay et al. 2012). However, this observation is only marginally significant: considering the full sample of stars hosting planet candidates with R > 4 R_⊕, the probability of observing one host with M < 1 M_☉ by chance is ∼1%, corresponding to a ∼2.5σ significance that the mass distribution is different. Using our unbiased sample, no statistically significant difference is found.

For sub-Neptune-sized objects, planet candidates are detected around stars with masses ranging over the full span of our sample (∼0.8–1.6 M_☉). This is illustrated in Figure 9(b), showing a close-up of the region of sub-Neptune planets on a linear scale. To demonstrate the improvement of the uncertainties in our sample, the dashed lines in Figure 9(b) shows an error bar for a typical best-case scenario when host star properties are based on spectroscopy alone (10% in stellar mass and 15% in stellar radius, Basu et al. 2012), neglecting uncertainties arising from the measurement of the transit depth. We note that, strictly speaking, the uncertainties in stellar mass and planet radius are not independent since asteroseismology constrains mostly the mean stellar density. The dash-dotted line in Figure 9(b) illustrates this by showing an error ellipse calculated from Monte Carlo simulations for a typical host planet pair in our sample.

The data in Figure 9(b) show a tentative lack of planets with sizes close to 3R_⊕, which does not seem to be related to detection bias in the sample. While this gap is intriguing, it is not compatible with previous observations of planets with radii between 3 and 3.5 R_⊕, such as in the Kepler-11 system (Lissauer et al. 2011). Further observations with a larger sample size will be needed to determine whether the apparent gap in Figure 9(b) is real or simply a consequence of the small size of the available sample.

The second main observable that can be tested for correlations with the host star mass is the orbital period of the planet candidates. Here, we do not expect a detection bias to be correlated with the host star properties, and hence we consider the full sample. Figure 10 compares the orbital period of the planet candidates as a function of host star mass. There does not appear to be an overall trend, although there is a tendency for more single planet candidates (black triangles) in close orbits (<10 days) around higher mass (≳ 1.3 M_☉) stars. This observation would qualitatively be consistent with previous findings that hot Jupiters are rare in multiple planet systems and are more frequently found around higher mass stars. However, we note that roughly half of the planet candidates with periods less than 10 days around stars with M > 1.3 M_☉ have radii smaller than Neptune and two have radii smaller than 2 R_⊕(see also Figure 9). Additionally, a K-S test yields only a marginal statistical difference (∼2.4σ) between host-star masses of single and multiple planet systems for periods <10 days.

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