ABSTRACT
Accurately determining the properties of stars is of prime importance for characterizing stellar populations in our Galaxy. The field of asteroseismology has been thought to be particularly successful in such an endeavor for stars in different evolutionary stages. However, to fully exploit its potential, robust methods for estimating stellar parameters are required and independent verification of the results is mandatory. With this purpose, we present a new technique to obtain stellar properties by coupling asteroseismic analysis with the InfraRed Flux Method. By using two global seismic observables and multi-band photometry, the technique allows us to obtain masses, radii, effective temperatures, bolometric fluxes, and hence distances for field stars in a self-consistent manner. We apply our method to 22 solar-like oscillators in the Kepler short-cadence sample, that have accurate Hipparcos parallaxes. Our distance determinations agree to better than 5%, while measurements of spectroscopic effective temperatures and interferometric radii also validate our results. We briefly discuss the potential of our technique for stellar population analysis and models of Galactic Chemical Evolution.
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1. INTRODUCTION
Studying the structure and evolution of the Milky Way requires detailed knowledge of the properties of the stellar populations comprising it. In this respect, asteroseismology is a powerful tool to determine masses and radii of single stars to a high level of precision (e.g., Mosser et al. 2010; Kallinger et al. 2010; Metcalfe et al. 2010). The CoRoT (Baglin et al. 2006; Michel et al. 2008) and Kepler missions (Gilliland et al. 2010; Borucki et al. 2010; Koch et al. 2010) have provided data on stellar oscillations of exquisite quality for thousands of stars, encouraging us to carry out a complete stellar census of the observed populations.
From the thousands of light curves obtained by the space missions, two asteroseismic parameters can be readily extracted (e.g., Hekker et al. 2011a; Huber et al. 2011; Chaplin et al. 2011). First, the power spectrum of solar-like oscillators is modulated in frequency by a Gaussian-like envelope, where the frequency of maximum power νmax scales approximately with the surface gravity and effective temperature. Second, the near-regular pattern of high overtones presents a dominant frequency spacing called the large frequency separation, Δν, which scales approximately with the square root of the mean stellar density. Applying scaling relations from solar values, these two asteroseismic observables may be used to estimate stellar properties of large numbers of solar-like oscillators, where individual frequencies are not available for all targets (e.g., Stello et al. 2008; Basu et al. 2010; Hekker et al. 2011a; Huber et al. 2011; Silva Aguirre et al. 2011b).
To gain insight about the formation history and evolution of our Galaxy, characteristics of stellar populations distributed across it must be accurately known. The best-studied sample of stars in the Milky Way is the solar neighborhood, where observations and analysis over many years have determined some of its key properties, such as orbits, kinematics, and metallicities (e.g., Edvardsson et al. 1993; Reddy et al. 2003; Nordström et al. 2004; van Leeuwen 2007; Feltzing & Bensby 2008; Casagrande et al. 2011). These data comprise the basic set of constraints for any model of chemical evolution of the Galaxy.
Models of Galactic Chemical Evolution are constructed under certain assumptions regarding the physical processes involved in the evolution of our Galaxy, and then calibrated against available observations. Those that reproduce them successfully are also used to predict other properties of the Galaxy, such as abundance gradients across the disk, gas infall episodes, and star formation rates (e.g., Tinsley 1980; Chiappini et al. 1997; Portinari et al. 1998; Schönrich & Binney 2009). Thus, their predictive power for our galactic history and morphology critically depends on how well they can reproduce these observations, most of which come from the solar neighborhood sample. Of particular importance among these restrictions is the age–metallicity relation, constructed using stellar isochrones and determinations of element abundances (e.g., Edvardsson et al. 1993; Nordström et al. 2004). The existence of an age–metallicity relation in the solar neighborhood is still a subject of debate (see Feltzing et al. 2001; Nordström et al. 2004; Freeman 2012 and references therein), and accurate age determinations are of prime importance to shed some new light in this issue. However, the solar neighborhood sample used to constrain these models is only complete to distances of ∼50 pc (e.g., Nordström et al. 2004), and accurate properties of stars further than ∼100 pc are difficult to measure, yet they are of crucial importance (e.g., Freeman & Bland-Hawthorn 2002; Steinmetz et al. 2006; Ivezić et al. 2008). To extend the sample used as a testbed for comparison, we need stellar parameters measured with high accuracy in different regions of the Galaxy.
Asteroseismology can help bridge this gap by providing accurate stellar properties, including distances, for field stars out to several hundred parsecs. These parameters, together with effective temperatures, metallicities, and kinematics, should make possible to study spatial gradients of stellar properties across the Galactic disk and provide insight into the formation process of our Galaxy (see Miglio 2012; Cheng et al. 2012; Miglio et al. 2012a). Moreover, robust age determinations obtained combining this information with evolutionary models will allow construction of the age–metallicity relation of the stellar populations observed by CoRoT and Kepler, and so provide tests outside the solar neighborhood of Galactic Chemical Evolution models.
The first comparison between asteroseismically determined parameters and predictions from Galactic Chemical Evolution models was made by Miglio et al. (2009) for a sample of CoRoT red giants. Chaplin et al. (2011) used the same technique to obtain masses and radii of Kepler main-sequence and subgiant stars, and found a slight but statistically significant difference between the observed and synthetic mass distributions. Continuing this line of work, Miglio et al. (2012b) obtained distances to CoRoT and Kepler red giants using bolometric corrections retrieved from the literature, while Creevey et al. (2012) took a similar approach for five Kepler subgiant stars.
Considering the enormous potential of the results, it is important to verify the techniques applied in asteroseismic analysis. So far, empirical tests of the scaling relations have used heterogeneous samples and relied on evolutionary models (see Stello et al. 2009a; Miglio 2012; Bedding 2011; Morel & Miglio 2012 and references therein). In this paper, we present a new method to derive stellar parameters in a self-consistent manner, combining seismic determinations with the InfraRed Flux Method (IRFM). We compare our results with Hipparcos parallaxes, high-resolution spectroscopic temperature determinations, and interferometric measurements of angular diameters. We briefly discuss the implications for models of Galactic Chemical Evolution and for age determinations of main-sequence stars.
2. SAMPLE SELECTION AND DATA EXTRACTION
From the more than 500 main-sequence and subgiant stars in which Kepler detected oscillations in its short-cadence mode (Chaplin et al. 2011), we selected the 32 stars that have Hipparcos parallaxes (van Leeuwen 2007). We discarded targets with parallax uncertainties larger than 20% as well as multiple systems and incorrect identifications from the Kepler Input Catalogue (KIC; Brown et al. 2011). This left us with a final sample of 22 stars. We also retrieved multi-band BTVT and JHKS photometry from the Tycho2 (Høg et al. 2000) and the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) catalogs, respectively. We have not used the ugriz photometry from the KIC since it suffers from zero-point uncertainties (Pinsonneault et al. 2012), while Tycho2 data have been extensively tested and tied to stellar parameters (Casagrande et al. 2010).
Using Kepler data, corrected as described by García et al. (2011), the global oscillation observables νmax and Δν were obtained in two ways. Set A was generated for all 22 stars from the power spectra using the pipeline described by Huber et al. (2009), while set B was derived from the individual mode frequencies for the 19 stars for which these are published (Appourchaux et al. 2012). Set B was based on post-survey data only, meaning that at least three-month-long time series have been used in the analysis. The analysis of the three stars belonging exclusively to set A (namely, KIC 5774694, KIC 5939450, and KIC 10513837) was based on survey data, i.e., of one-month-long duration. Set A includes all of set B.
The way in which seismic parameters were extracted from the frequency lists (set B) deserves a brief explanation: νmax was first estimated by fitting a Gaussian function to the envelope of radial (l = 0) mode amplitudes as a function of frequency. This least-squares fit was weighted by the uncertainties on the observed amplitudes. Using this value of νmax, we computed a proxy of Δν, Δνproxy, using the scaling relation given by Stello et al. (2009a). This proxy was then used to construct a Gaussian centered at νmax with an FWHM of 4Δνproxy, which served as the statistical weight when performing a least-squares fit to the radial frequencies as a function of radial order n to determine Δν (see White et al. 2011a, 2011b). The choice of such a wide weighting envelope around νmax ensures that any frequency dependence of Δν due to acoustic glitches is averaged out (see Mosser et al. 2011; Kallinger et al. 2012).
The second step in the selection of a final set of asteroseismic input parameters consisted of performing an internal check on the quoted (uncalibrated) uncertainties. The adopted procedure makes use of the results obtained for the stars common to both sets. We started by computing the rms of the relative residuals in νmax and Δν, i.e., (set B−set A)/set A and (set B−set A)/set B, to be subsequently used in the calibration of the quoted uncertainties given in sets A and B, respectively. This value of the rms, which can be regarded as an additional fractional uncertainty, was then added in quadrature to the uncalibrated uncertainties in order to obtain the final (calibrated) uncertainties.
Since our goal is to perform a uniform analysis, we must use a set of seismic parameters obtained from a single method of extraction. Set A has been selected as the final set (after error calibration) because it covers all the stars. Fractional differences between results in sets A and B are less than 3% for νmax and below 0.5% for Δν, which are within the uncertainties. The procedure used to obtain set A has been extensively tested for consistency with other methods of extracting seismic parameters (see Verner et al. 2011 and references therein).
3. DETERMINING STELLAR PARAMETERS
Our technique for estimating stellar parameters relies on an iterative approach that couples asteroseismic analysis with the results of the IRFM. We do this in such a way that the final values of Teff, bolometric flux, mass, and radius are dependent on one another and internally consistent.
3.1. The Direct and Grid-based Methods
To very good approximation, Δν scales as the square root of the mean density (e.g., Ulrich 1986), while νmax is related to the acoustic cutoff frequency of the atmosphere (e.g., Brown et al. 1991; Kjeldsen & Bedding 1995; Belkacem et al. 2011). These two quantities follow scaling relations from the accurately known solar parameters (e.g., Hekker et al. 2009; Stello et al. 2009a), which can be written as
Here, Teff, ☉ = 5777 K, Δν☉ = 135.1 ± 0.1 μHz, and νmax, ☉ = 3090 ± 30 μHz are the observed values in the Sun, derived using the same method as set A (Huber et al. 2011). Provided a value of Teff is available, these scaling relations give a determination of stellar mass and radius for each star that is independent of evolutionary models (see, e.g., Miglio et al. 2009; Hekker et al. 2011b; Silva Aguirre et al. 2011b), in what has come to be known as the direct method.
Another approach is to include models of stellar evolution when estimating the masses and radii. This so-called grid-based method uses evolutionary tracks constructed with a range of metallicities and searches for a best-fitting model, using Δν, νmax, Teff, and [Fe/H] as input parameters (e.g., Stello et al. 2009b; Basu et al. 2010, 2012; Gai et al. 2011).
Our reference grid of stellar models was computed with the Garching Stellar Evolution Code (GARSTEC; Weiss & Schlattl 2008). We have used Irwin's equation of state (Cassisi et al. 2003), OPAL opacities (Iglesias & Rogers 1996) that were complemented at low temperatures by those of Ferguson et al. (2005), and nuclear reaction rates by Adelberger et al. (2011). Models for masses below 1.4 M☉ included microscopic diffusion of helium and metals, following Thoul et al. (1994). This effect was not considered for masses above 1.4 M☉ since its overall evolutionary impact is small and the code does not include other processes that might become relevant in these cases, such as radiative levitation (e.g., Turcotte et al. 1998). Core and envelope convective overshooting has been included in all models, using the exponential decay description of Freytag et al. (1996) with a scale factor f = 0.02 and a geometric restriction for small convective cores (Magic et al. 2010). A mixing length parameter of α = 1.811 from a calibrated solar model and an Eddington T − τ relation for stellar atmospheres have been adopted.
The grid spans from 0.8 to 2.0 M☉ in steps of 0.01 M☉. For the metallicity, we defined [Fe/H] = 0 at the adopted present-day solar photospheric value of Z/X = 0.0230 (Grevesse & Sauval 1998). The initial composition of models was, however, defined in terms of that of a calibrated solar model with Z0 = 0.01876 and Y0 = 0.26896. Because of gravitational settling, this implies that the change in [Fe/H] of a solar-metallicity track in 4.57 Gyr of evolution is actually ∼ + 0.06 dex. Initial [Fe/H] values in the grid range from −0.54 to +0.36 dex in steps of 0.1 dex; the adopted Δ Y/Δ Z = 1.4 relation (e.g., Casagrande et al. 2007) allows for the complete determination of initial composition of models. The grid was restricted to models with ages above 0.05 Gyr to avoid degeneracy with pre-main-sequence models. The largest time step in the main sequence was constrained to 10 Myr, resulting in a very dense grid that comprises, in total, about 4.5 × 106 models.
When applying the grid-based approach, we obtained the Δν value of each model using frequencies of individual radial modes calculated with ADIPLS (Christensen-Dalsgaard 2008), in the manner described by White et al. (2011b). On the other hand, νmax was always computed from the acoustic cutoff frequency relation. In order to find the stellar parameters best fitting the input data, we followed a procedure similar to that presented by Basu et al. (2010). Briefly, we produced 10,000 Monte Carlo realizations of sets of input parameters using random Gaussian noise around the central observed values, and calculated the likelihood for models within 3σ of all the observables. Considering only those results with a likelihood larger than 95% of the maximum likelihood, we formed the probability distribution and assigned the uncertainties to be 34% of either side of the median value.
As it has been extensively discussed in Gai et al. (2011) and Basu et al. (2012), it is important to consider the dispersion in the grid results arising from the use of different evolutionary codes and input physics. For instance, the change in Teff during the main-sequence phase due to microscopic diffusion for stars more massive than ∼1.3 M☉ can reach approximately 150 K (Turcotte et al. 1998). Since this effect is not taken into account in our GARSTEC reference grid for masses above ∼1.4 M☉, when applying the grid-based method we also obtained stellar parameters using the Yale-Yonsei isochrones presented in Basu et al. (2010), the evolutionary tracks from Dotter (Dotter et al. 2008) and Marigo (Girardi et al. 2000; Marigo et al. 2008), the YREC set of models presented in Gai et al. (2011), and evolutionary tracks constructed with non-solar values of the mixing length parameter of convection (the "MLT" set from Basu et al. 2012). These sets of evolutionary tracks were constructed using different microphysics (e.g., equation of state, nuclear reactions, and opacities), as well as assumptions in the metallicity scale and treatment of convection, overshooting, and gravitational settling. The final results obtained from the grid-based method encompass these uncertainties using an error calibration process described in Section 3.3.
3.2. The InfraRed Flux Method
The IRFM is arguably one of the most direct and least model-dependent techniques to determine effective temperatures in stars. It was originally devised to obtain stellar angular diameters with an accuracy of a few percent (Blackwell & Shallis 1977; Blackwell et al. 1979, 1980). Our analysis is based on the IRFM described by Casagrande et al. (2006, 2010).
The basic idea is to recover for each star its bolometric 
and infrared monochromatic flux 
, both measured at the top of Earth's atmosphere. One must then compare their ratio to that obtained from the same quantities defined on a surface element of the star, i.e., the bolometric flux σT4eff and the theoretical surface infrared monochromatic flux. For stars hotter than ∼4200 K the latter quantity is relatively easy to determine because the near infrared region is largely dominated by the continuum and depends linearly on Teff (Rayleigh–Jeans regime), thus minimizing any dependence on model atmospheres. The problem is therefore reduced to a proper derivation of stellar fluxes, which can then be rearranged to return the effective temperature. Once 
and Teff are both known, the limb-darkened angular diameter, θ, is trivially obtained.
In the adopted implementation, the bolometric flux was recovered using multi-band photometry (Tycho2 BTVT and 2MASS JHKS), and the flux outside of these bands (i.e., the bolometric correction) was estimated using a theoretical model flux at a given Teff, [Fe/H], and log g. The infrared monochromatic flux was derived from 2MASS JHKS magnitudes only. We used an iterative procedure in Teff to cope with the mildly model-dependent nature of the bolometric correction and surface infrared monochromatic flux. For each star, we used the Castelli & Kurucz (2004) grid of model fluxes, starting with an initial estimate of its effective temperature and working at a fixed [Fe/H] and log g until convergence in Teff within 1 K was reached.
The uncertainties stemming from the adopted [Fe/H] and log g were taken into account in the error estimate, but their importance is secondary at this stage since the IRFM has been shown to depend only loosely on those parameters (see Casagrande et al. 2006). This makes the technique superior to most spectroscopic methods for determining Teff—provided that reddening is known—since the effects of Teff, log g, and [Fe/H] on the latter are usually strongly coupled and the model dependence is much more important. The metallicity adopted for each star and the coupling of the IRFM with asteroseismic gravities will be discussed in the next section, together with reddening effects.
3.3. Iterations and Error Determination
As described in Section 3.1, the asteroseismic methods provide a mass and radius based on an input Teff value (and [Fe/H] for the grid-based case). On the other hand, the IRFM gives Teff and the bolometric flux at a given input log g and [Fe/H]. In order to determine a unique set of stellar parameters for each star, we iterated the two methods in a consistent way, using both the direct and grid-based approach. A simplified version of this technique was first introduced by Silva Aguirre et al. (2011b).
We started by calculating sets of IRFM effective temperatures for each star at fixed log g = 2.0–5.0 in steps of 0.5 dex; this translates into Teff changes of less than 1% for each log g step. The metallicity of the targets must be given as an input, and we have considered them in the following order of preference, according to availability: the latest revision of the Geneva-Copenhagen Survey (GCS; Casagrande et al. 2011), spectroscopic determinations from Bruntt et al. (2012), or the value given in the KIC increased by 0.18 dex. The latter is the offset found between GCS and the KIC for the 11 stars common in our sample, and is similar to the +0.21 dex offset found by Bruntt et al. (2012).
Reddening must also be specified, and our calculations were made assuming distance dependent extinction values from Drimmel et al. (2003). These were obtained after an iteration in distance as described by Miglio et al. (2012b). If E(B − V) < 0.01 or no estimate was available we assigned E(B − V) = 0.0. We list in Table 1 the input parameters used in our analysis.
Table 1. Input Parameters for the 22-star Sample
| KIC ID | HIP | νmax | Δν | [Fe/H] | E |
|---|---|---|---|---|---|
| (μHz) | (μHz) | (B − V) | |||
| 3632418 | 94112 | 1144 ± 31 | 60.8 ± 0.2 | −0.01 | 0.024 |
| 3733735 | 94071 | 2145 ± 61 | 92.3 ± 0.3 | −0.10 | 0.025 |
| 4914923 | 94734 | 1887 ± 181 | 88.7 ± 0.3 | 0.17 | 0.018 |
| 5371516 | 96528 | 1018 ± 33 | 55.4 ± 0.2 | 0.13 | 0.020 |
| 5774694 | 93657 | 3442 ± 274 | 140.2 ± 4.0 | 0.01 | 0.000 |
| 5939450 | 92771 | 605 ± 25 | 30.5 ± 2.4 | −0.01 | 0.020 |
| 6106415 | 93427 | 2219 ± 60 | 104.3 ± 0.3 | −0.06 | 0.000 |
| 6225718 | 97527 | 2338 ± 66 | 105.8 ± 0.3 | −0.15 | 0.010 |
| 7747078 | 94918 | 946 ± 26 | 54.0 ± 0.2 | −0.26 | 0.018 |
| 7940546 | 92615 | 1081 ± 34 | 58.9 ± 0.2 | −0.04 | 0.010 |
| 8006161 | 91949 | 3570 ± 96 | 149.3 ± 0.4 | 0.34 | 0.000 |
| 8228742 | 95098 | 1175 ± 34 | 62.1 ± 0.2 | −0.14 | 0.025 |
| 8751420 | 95362 | 571 ± 15 | 34.6 ± 0.1 | −0.20 | 0.010 |
| 9139151 | 92961 | 2695 ± 74 | 117.3 ± 0.3 | 0.15 | 0.012 |
| 9139163 | 92962 | 1685 ± 45 | 81.1 ± 0.2 | 0.15 | 0.012 |
| 9206432 | 93607 | 1859 ± 50 | 84.7 ± 0.3 | 0.23 | 0.013 |
| 10068307 | 94675 | 976 ± 35 | 54.0 ± 0.2 | −0.13 | 0.014 |
| 10162436 | 97992 | 1016 ± 28 | 55.8 ± 0.2 | −0.08 | 0.023 |
| 10454113 | 92983 | 2310 ± 68 | 105.1 ± 0.3 | −0.06 | 0.011 |
| 10513837 | 91841 | 191 ± 7 | 14.6 ± 0.2 | 0.15 | 0.026 |
| 11253226 | 97071 | 1669 ± 45 | 77.0 ± 0.2 | −0.03 | 0.011 |
| 12258514 | 95568 | 1499 ± 40 | 75.0 ± 0.2 | 0.13 | 0.015 |
Note. See the text for details.
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The procedure applying the direct method works as follows. Using log g determinations from the KIC as an initial guess, we interpolated in gravity and computed Teff from the IRFM results. This Teff value, together with νmax and Δν, was fed to the scaling relations to obtain a mass, radius, and thus log g. Interpolating again in gravity gave an updated value of Teff, and the procedure was repeated until convergence in log g and Teff was reached.
We obtained 1σ uncertainties of the parameters during the iterations. Uncertainties in the seismic observables were taken into account, as well as variations in the Teff determinations arising from different photometric filters and log g determinations. The results are affected by the assumed value of extinction, and are mildly dependent on the metallicity considered. To account for possible errors in reddening and composition, we have also computed sets of results at log g = 3.5, one increasing E(B − V) by +0.01 (the decreasing case is essentially symmetric), and another one changing the metallicity by ±0.1 dex. Moreover, a Monte Carlo simulation was run to estimate the uncertainties in Teff from random photometric errors. Finally, we added an extra 20 K to the error budget to account for the uncertainty in the zero-point of the temperature scale.
The analysis was repeated using the grids mentioned in Section 3.1 to determine mass, radius, and log g values at each iteration. In all cases we used as input values the seismic observables and metallicities described above, considering an uncertainty in composition of ±0.1 dex consistent with what was applied for the IRFM. The final set of stellar parameters from the grid-based method and their corresponding uncertainties were obtained in the same manner as the seismic input parameters (see Section 2): we adopted the GARSTEC grid as the reference and performed an error calibration by computing the rms of the relative residuals in mass, radius, and gravity, and adding them in quadrature to the original GARSTEC uncertainties.
4. RESULTS
The procedure outlined in Section 3.3 provided final values of Teff, mass, radius, log g, and, as mentioned in Section 3.2, the associated bolometric flux and θ. Their corresponding 1σ uncertainties were also obtained during the iterations. It is important to mention that the log g values determined via the direct and grid-based method agree better than 0.03 dex, implying that their Teff values are also in agreement within 3 K. Using the asteroseismic radius and θ, it is straightforward to estimate the distance:
where C is the conversion factor to parsecs. In this manner, we determined asteroseismic distances for our 22 sample targets.
In Figure 1, we compare our distances with those obtained from Hipparcos parallax measurements. Note that, as described in Section 3.1, seismic radii determinations can be obtained by either the direct or grid-based method. The agreement is excellent, particularly for the close-by targets, boosting our confidence on the asteroseismic parameters and the robustness of our technique.
Figure 1. Upper panels: comparison of Hipparcos distances with those obtained via the seismic method. Lower panels: fractional difference (Hipp. − Seis.) between both determinations. The gray solid line shows the one-to-one correspondence, while the black dashed and dotted lines represent the weighted average difference and standard deviation, respectively. The red diamond in each panel is the star with the largest fractional error in Δν. See the text for details.
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Standard image High-resolution imageThere is one target that clearly deviates from the one-to-one relation in the results obtained via the direct method, shown with a red diamond in Figure 1. Not surprisingly, this star has the largest fractional error in Δν (∼8%, compared to the less than ∼2% of the rest of the targets; see Equation (2)). The large uncertainty in this parameter dominates the radius error budget, and thus propagates to the estimated distance. For this particular case, the grid method determined the radius of the star with a much higher accuracy, a result that is confirmed by the better agreement of its distance to that obtained from parallaxes. The fact that the grid-based approach restricts combinations of Δν, νmax, and [Fe/H] to a narrow range of possible Teff values seems to be sufficient to deal with large uncertainties in one measurement, provided the others are accurately determined (Gai et al. 2011).
For stars with accurate seismic measurements, the direct method provides results as reliable as the grid-based one. The weighted mean difference (Hipp. − Seis.) is 2.1% ± 1.8% for the direct method, while for the grid-based case is 2.4% ± 1.5%. Removing the outlier from the sample changes the average differences to 2.3% ± 1.8% and 3.1% ± 1.6%, respectively. One important factor to take into consideration is extinction. From Figure 1 we see that the uncertainties in asteroseismic distances seem to increase with distance. This points out to reddening as the cause, since the error in the seismic distance determinations should be comparable for stars with similar uncertainties in the global seismic parameters. Analysis of the error budget shows that reddening becomes the major contributor as distance increases, most likely due to the use of distance-dependent integrated maps of extinction. We discuss this further in Section 5.
As described in Section 2, the seismic input parameters were determined using two different methods, and the results shown in Figure 1 are those from set A (pipeline processing of the power spectrum). We have tested the impact on the distance determinations of using instead the results from individual frequency lists (set B): for the 19 targets common to both sets, containing stars up to ∼170 pc away, the weighted mean difference between the distances of Set A and Set B is below 0.5%.
All but one of our targets were included in the spectroscopic analysis made by Bruntt et al. (2012). Those authors obtained effective temperatures via the excitation balance of Fe i lines, using a fixed log g value in their analysis as determined by asteroseismology. In Figure 2, we compare their Teff values with ours and find excellent agreement. Individual fractional differences are all below 2%, while the weighted mean difference (Spec. − Seis.) is −0.8% ± 0.4%. This level of agreement is particularly impressive considering that the uncertainties quoted by Bruntt et al. (2012) are of 70 K for all the targets. However, there is a possible systematic offset to lower values in the spectroscopic temperatures compared to ours. Although the reason for this behavior is far from evident and goes beyond the scope of this paper, we mention that the effective temperatures determined in this work are supported by the analysis of hydrogen line profiles (M. Bergemann 2012, private communication). The Balmer-line Teff scale is known to be warmer than that given by the excitation balance of Fe i lines (see Casagrande et al. 2010, Figure 12), making it more consistent with the IRFM (M. Bergemann et al. 2012, in preparation).
Figure 2. Upper panel: comparison of effective temperatures using the asteroseismic method with those obtained from spectroscopy by Bruntt et al. (2012). Lower panel: fractional difference (Spec. − Seis.) between both determinations. The solid lines shows the one-to-one correspondence, while the dashed and dotted lines represent the weighted average difference and standard deviation, respectively.
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Standard image High-resolution imageAnother verification of our technique comes from comparing our derived angular diameters with results from interferometry. Four stars in our sample have been observed with the PAVO/CHARA long-baseline interferometer, as described by Huber et al. (2012). The residual mean value between our angular diameters and the interferometric ones is below 1%, consistent with the value found by Huber et al. (2012) for a larger sample. These results are also compared to the ones derived using the surface brightness technique of Kervella et al. (2004), obtaining an equally good level of agreement and further confirming the robustness of our method (see Figure 4 in Huber et al. 2012).
The results outlined above clearly show that our method provides accurate Teff, radii, angular diameters, and therefore distances for the sample of stars considered. In order to assess the level of accuracy of these asteroseismic distances, we separated the sample into three bins according to the uncertainty in the parallax measurements. There is a natural correlation between distance and quality of the parallaxes, where stars with smaller uncertainties are usually closer and therefore less affected by interstellar extinction. For those stars with parallax determined better than 5%, the rms between our grid-based distances and the Hipparcos ones is below 4.7%. Moreover, when considering the bin of uncertainties between 5% and 10%, as well as the bin with parallax errors between 10% and 20%, the rms of the relative residuals is also smaller than the typical Hipparcos error. Thus, provided extinction is properly taken into account, our distance determinations can be considered accurate to 5%.
We present in Table 2 the parameters obtained with the grid-based method. Note that, as mentioned in Section 3.1, the grid-based approach returns a distribution probability function for each parameter, and so the uncertainties in the mass, radius, and gravity are asymmetric.
Table 2. Stellar Parameters Derived Using the Grid-based Method for the 22-star Sample
| KIC ID | HIP | Teff | M/M☉ | R/R☉ | log g | Fbol | θ | d |
|---|---|---|---|---|---|---|---|---|
| (K) | (10−8 erg s−1 cm−2) | (mas) | (pc) | |||||
| 3632418 | 94112 | 6286 ± 70 | 1.396+0.074− 0.075 | 1.911+0.025− 0.026 | 4.018+0.003− 0.004 | 1.402 ± 0.074 | 0.164 ± 0.006 | 108 ± 4 |
| 3733735 | 94071 | 6824 ± 131 | 1.454+0.028− 0.055 | 1.427+0.019− 0.020 | 4.284+0.008− 0.006 | 1.251 ± 0.070 | 0.132 ± 0.006 | 101 ± 5 |
| 4914923 | 94734 | 5828 ± 56 | 1.174+0.058− 0.062 | 1.408+0.022− 0.023 | 4.209+0.003− 0.004 | 0.456 ± 0.023 | 0.109 ± 0.003 | 120 ± 4 |
| 5371516 | 96528 | 6360 ± 115 | 1.468+0.063− 0.080 | 2.066+0.022− 0.021 | 3.973+0.007− 0.010 | 1.167 ± 0.062 | 0.146 ± 0.007 | 131 ± 6 |
| 5774694 | 93657 | 5911 ± 106 | 1.079+0.041− 0.045 | 1.000+0.015− 0.016 | 4.467+0.007− 0.007 | 1.213 ± 0.076 | 0.173 ± 0.008 | 54 ± 3 |
| 5939450 | 92771 | 6380 ± 70 | 1.625+0.134− 0.069 | 2.916+0.062− 0.060 | 3.726+0.008− 0.009 | 3.207 ± 0.169 | 0.241 ± 0.008 | 113 ± 4 |
| 6106415 | 93427 | 6061 ± 89 | 1.110+0.036− 0.033 | 1.240+0.018− 0.018 | 4.296+0.004− 0.004 | 3.491 ± 0.024 | 0.279 ± 0.008 | 41 ± 1 |
| 6225718 | 97527 | 6338 ± 88 | 1.209+0.037− 0.034 | 1.256+0.014− 0.014 | 4.322+0.004− 0.004 | 2.664 ± 0.291 | 0.223 ± 0.014 | 52 ± 3 |
| 7747078 | 94918 | 5856 ± 112 | 1.135+0.086− 0.088 | 1.952+0.039− 0.039 | 3.910+0.004− 0.004 | 0.452 ± 0.099 | 0.107 ± 0.012 | 169 ± 20 |
| 7940546 | 92615 | 6287 ± 74 | 1.380+0.065− 0.104 | 1.944+0.024− 0.031 | 3.996+0.006− 0.012 | 2.913 ± 0.039 | 0.236 ± 0.006 | 76 ± 2 |
| 8006161 | 91949 | 5355 ± 107 | 0.959+0.035− 0.037 | 0.927+0.014− 0.014 | 4.484+0.004− 0.004 | 2.879 ± 0.153 | 0.324 ± 0.016 | 27 ± 1 |
| 8228742 | 95098 | 6130 ± 107 | 1.308+0.062− 0.060 | 1.855+0.027− 0.027 | 4.017+0.005− 0.004 | 0.457 ± 0.156 | 0.099 ± 0.017 | 175 ± 31 |
| 8751420 | 95362 | 5243 ± 162 | 1.285+0.082− 0.096 | 2.722+0.048− 0.057 | 3.674+0.006− 0.005 | 4.898 ± 0.368 | 0.441 ± 0.032 | 57 ± 4 |
| 9139151 | 92961 | 6141 ± 114 | 1.218+0.046− 0.046 | 1.178+0.018− 0.018 | 4.380+0.004− 0.004 | 0.527 ± 0.017 | 0.106 ± 0.004 | 104 ± 4 |
| 9139163 | 92962 | 6525 ± 111 | 1.405+0.034− 0.027 | 1.571+0.010− 0.010 | 4.195+0.004− 0.004 | 1.225 ± 0.031 | 0.142 ± 0.005 | 103 ± 4 |
| 9206432 | 93607 | 6614 ± 135 | 1.482+0.044− 0.044 | 1.544+0.015− 0.015 | 4.231+0.005− 0.005 | 0.605 ± 0.064 | 0.097 ± 0.007 | 147 ± 10 |
| 10068307 | 94675 | 6197 ± 97 | 1.366+0.062− 0.071 | 2.060+0.028− 0.033 | 3.943+0.003− 0.004 | 1.401 ± 0.018 | 0.169 ± 0.005 | 114 ± 4 |
| 10162436 | 97992 | 6245 ± 110 | 1.365+0.061− 0.068 | 2.015+0.025− 0.027 | 3.961+0.006− 0.004 | 0.947 ± 0.075 | 0.137 ± 0.007 | 137 ± 7 |
| 10454113 | 92983 | 6134 ± 113 | 1.165+0.045− 0.045 | 1.251+0.017− 0.017 | 4.309+0.005− 0.004 | 0.924 ± 0.050 | 0.140 ± 0.006 | 83 ± 4 |
| 10513837 | 91841 | 4955 ± 95 | 1.290+0.072− 0.076 | 4.788+0.083− 0.103 | 3.186+0.006− 0.007 | 0.585 ± 0.043 | 0.171 ± 0.009 | 261 ± 15 |
| 11253226 | 97071 | 6715 ± 97 | 1.458+0.032− 0.034 | 1.628+0.017− 0.018 | 4.176+0.006− 0.004 | 1.095 ± 0.027 | 0.127 ± 0.004 | 119 ± 4 |
| 12258514 | 95568 | 6064 ± 121 | 1.302+0.078− 0.084 | 1.630+0.029− 0.031 | 4.127+0.004− 0.005 | 1.532 ± 0.063 | 0.184 ± 0.008 | 82 ± 4 |
Note. Errors on the parameters come from probability distribution functions (see Sections 3.1 and 3.3 for details)
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Our parameters can be compared to other studies where the same stellar properties were determined. Recently, Pinsonneault et al. (2012) provided color–temperature relations consistent with the Casagrande et al. (2010) scale using the available Sloan Digital Sky Survey (SDSS) griz photometry from the KIC. Fourteen stars from our sample are present in their catalog, and comparison of the obtained Teff values is shown in Figure 3. The weighted mean difference (SDSS − Seis.) is −0.1% ± 0.6%, indicating that the corrected temperatures provided by Pinsonneault et al. (2012) are indeed on a scale consistent with ours and can be used for studies of field stars when this photometry is available.
Figure 3. Upper panel: comparison of effective temperatures using the asteroseismic method with those derived by Pinsonneault et al. (2012). Lower panel: fractional difference (SDSS − Seis.) between both determinations. The solid lines shows the one-to-one correspondence, while the dashed and dotted lines represent the weighted average difference and standard deviation, respectively.
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Standard image High-resolution imageMathur et al. (2012) performed a detailed modeling of several Kepler targets, including six stars from our sample, using individual frequency lists obtained from one-month-long observations. The input metallicities and effective temperatures used in that study were in some cases different from ours (see Table 3 in Mathur et al. 2012). Comparing the results reveals good overall agreement in the obtained radii, with two targets showing what appears to be a slight radius underestimation in their determinations with respect to ours. A more thorough investigation of this issue will be made when detailed modeling using longer time series is performed (T. M. Metcalfe et al. 2012, in preparation).
Although our sample of stars only represents a small fraction of the total short-cadence Kepler sample, they cover a wide range in metallicity, Teff and log g. In Figure 4 we present a log g–Teff diagram, where the 22 targets have been placed using the parameters derived with the grid-based method. Also plotted are stellar evolution tracks from the GARSTEC grid, at masses and metallicities compatible with those given in Tables 1 and 2. Our targets are distributed in different evolutionary stages, from the early main sequence to the beginning of the red giant branch. Thus, we have tested the accuracy of our method in stars showing a wide variety of masses, ages, compositions, and energy transport mechanisms.
Figure 4. Position in the log g–Teff plane of the 22-star sample as derived via grid-based approach and the IRFM. Also plotted are stellar evolutionary tracks from the GARSTEC grids for masses between 0.9 and 1.4 M☉ at two different initial metallicities: [Fe/H] = +0.35 (dotted blue lines) and [Fe/H] = −0.24 (dashed red lines).
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Standard image High-resolution imageWe find good agreement in the radii and bolometric fluxes (and hence distances) estimated by the direct and grid-based methods to the parallax data, which again supports our previous statement that the method has only a modest sensitivity to the stellar composition (see Section 3.2). Most importantly, it shows that we can extend the implementation of our method to stars evolved beyond the subgiant branch. This will be addressed in an upcoming publication.
A particularly interesting case is that of KIC 10513837, the most evolved and most distant star in the sample. We estimated its composition to be [Fe/H] = 0.15 by adding 0.18 dex to the KIC value (see Section 3.3 and Table 1). However, its position in Figure 4 is instead compatible with sub-solar metallicity. Spectroscopic analysis of this target made by Molenda-Żakowicz et al. (2008) found a value of [Fe/H] = −0.07 and an effective temperature consistent with our determination, further confirming the mild sensitivity of our method to composition.
In a natural continuation of this work, we applied our procedure using the direct method to the complete short-cadence Kepler sample and derived consistent parameters, including distances, for all these stars. The 565 targets considered are predominantly main-sequence and subgiant stars, with a handful of red giants also present in the sample. All the targets have Tycho2 photometry available, and we have used metallicities from the KIC increased by 0.18 dex. To account for the uncertainties in composition, we have computed IRFM sets of Teff varying the metallicity by ±0.3 dex (see Section 3.3). The seismic input parameters were computed as for set A, described in Section 2. Extinction values were obtained from Drimmel et al. (2003) after an iteration in distance. These preliminary determinations will be compared with results from several pipelines in a forthcoming publication (W. J. Chaplin et al. 2012, in preparation). In Figure 5 we show a histogram with the obtained distance distribution, where we have also plotted the distribution of our 22 sample stars for comparison.
Figure 5. Derived distance distributions using IRFM + asteroseismology. The red shaded region shows the results for the 22-sample stars, while the unshaded one depicts the results for all main-sequence and subgiant stars with available Kepler asteroseismic data.
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The resulting distance distribution shows that we can use Kepler data to probe populations of main-sequence and subgiant field stars as far as 1 kpc from the Sun. However, our distance determinations for this sample can be slightly undermined by faulty metallicities from the KIC (see Molenda-Żakowicz et al. 2011; Bruntt et al. 2011, for a discussion). We have estimated [Fe/H] for these stars as consistently as possible with the available data. The latest revision of the GCS is built upon 1500 stars with spectroscopic measurements agreeing with our Teff scale (Casagrande et al. 2011). Therefore, on average the +0.18 dex correction we find makes the metallicity scale of the KIC consistent with the underlying Teff scale we derive here. On the other hand, the good agreement with the results of Bruntt et al. (2012) implies that also the latter metallicity scale is broadly on the same zero point that we adopt (and in fact, a similar +0.21 dex correction is found). Thus, although the most self-consistent approach would be to iterate our Teff also for deriving metallicities, on average this condition is fulfilled.
As a final test of the impact on distances of incorrect metallicities, we have done calculations considering different assumptions for the composition of the full sample: using the [Fe/H] values as provided in the KIC (i.e., not including the +0.18 dex offset), and also considering a fixed mean metallicity for all the targets of [Fe/H] = −0.2 dex. The distance distribution obtained remained practically unchanged, reinforcing the notion of the mild metallicity dependence of our method.
5. CONCLUSIONS
Determining accurate stellar parameters is crucially important for detailed studies of individual stars as well as for characterizing stellar populations in the Milky Way. The asteroseismic revolution produced by the CoRoT and Kepler missions requires robust techniques to fully exploit the potential of the data and provide the community with the building blocks for ensemble analysis.
Using oscillation data and multi-band photometry, we have presented a new method to derive stellar parameters, combining the IRFM with asteroseismic analysis. The novelty of our approach is that it allows us to obtain radius, mass, Teff, and bolometric flux for individual targets in a self-consistent manner. This naturally results in direct determinations of angular diameters and distances without resorting to parallax information, further enhancing the capabilities of our technique.
Two asteroseismic methods were applied to the available data, one based entirely on scaling relations and the other one using grids of pre-calculated models. When accurate seismic data are available, comparison of our distance results with those from Hipparcos parallaxes shows an overall agreement better than 4%, regardless of the asteroseismic method employed. Furthermore, the obtained Teff values show a mean difference below 1% when compared to results from high-resolution spectroscopy. We have also compared our calculated angular diameters with those measured by long-baseline interferometry and found agreement within 5%. This provides verification of our radii, Teff values, and bolometric fluxes to an excellent level of accuracy.
Despite the encouraging results, systematics can arise from faulty determinations of reddening values and metallicities. In Section 3.3 we described how the effects of metallicity and reddening, as well as their uncertainties, were taken into account in the calculations. For our determinations to be completely self-consistent, we must be able to determine those parameters from a single set of data using the IRFM. An observational campaign is currently under way to obtain Strömgren photometry of Kepler stars that will provide a homogeneous set of values for [Fe/H] and extinction, as described by Casagrande et al. (2011).
For most of the stellar parameters included in our verifications, both asteroseismic methods produce equally good results. However, it should be kept in mind that the direct method can be significantly biased when large uncertainties in the seismic input parameters exist. Moreover, scaling relations are likely to have a different dependence on effective temperature beyond the main-sequence phase, as suggested by comparison to evolutionary calculations (see White et al. 2011b). The restrictions imposed by metallicity and by the theory of stellar evolution help to cope better with large errors in seismic data, and the use of the grid-based analysis in these cases is therefore recommended.
However, to take full advantage of the available parameters, asteroseismology must provide masses with a comparable level of accuracy. It is important to note that results on masses from the direct method for values above ∼1.5 M☉ can deviate significantly from those obtained using the grid-based approach. In fact, differences of more than ∼30% are not unusual in these cases. Using different grids of models, Gai et al. (2011) found that the fact that the direct method does not explicitly take metallicity into account could undermine its mass determinations. A thorough comparison of different grid-based techniques with the direct method is beyond the scope of this paper and will be presented in an upcoming publication (W. J. Chaplin et al. 2012, in preparation). Another method to obtain asteroseismic masses is via detailed modeling of targets, aiming at fitting the list of individual frequencies (e.g., Metcalfe et al. 2010; Mathur et al. 2012). This approach provides mass estimates with a high level of precision and, in principle, also with high accuracy. Regardless of the considered technique, one must keep in mind that verification of asteroseismic mass determinations in general is still needed.
Studies of the stellar populations in the CoRoT and Kepler fields can greatly benefit from accurate masses, radii, Teff, and distances (Miglio et al. 2012b, 2012a). Combining this information with evolutionary models can lead to an age–metallicity relation, opening the possibility of testing models of Galactic Chemical Evolution in stars outside the solar neighborhood (e.g., Chiappini et al. 1997; Schönrich & Binney 2009; Freeman 2012). Applying our method to the complete short-cadence Kepler sample reveals that we can probe stars as far as 1 kpc from our Sun, making this set of main-sequence and subgiant stars extremely interesting for population studies. Although much greater distances can be probed by analyzing oscillations in giants, the ages of these stars are mostly determined by their main-sequence lifetime (e.g., Salaris et al. 2002; Basu et al. 2011). Thus, the short-cadence sample is of key importance for helping to calibrate mass–age relationships of red giants and correctly characterize their populations.
A substantial number of the Kepler main-sequence and subgiant targets have been observed long enough to obtain individual frequency determinations (Appourchaux et al. 2012). Detailed modeling of these stars, particularly using frequency combinations (e.g., Roxburgh & Vorontsov 2003; De Meulenaer et al. 2010) and modes of mixed character (e.g., Deheuvels & Michel 2011; Benomar et al. 2012), can put tighter constraints on their masses and ages, providing anchor points for ensemble studies. In fact, certain combinations of frequencies can be used to probe the remaining central hydrogen content in stars (e.g., Christensen-Dalsgaard 1988), the existence and size of a convective core (e.g., Silva Aguirre et al. 2011a and references therein), and the position of the convective envelope and helium surface abundance (e.g., Christensen-Dalsgaard et al. 1991; Antia & Basu 1994). These techniques are currently being applied to several stars in the sample (e.g., S. Deheuvels et al. 2012, in preparation; A. Mazumdar et al. 2012, in preparation; V. Silva Aguirre et al. 2012, in preparation) and should help us obtain masses with higher accuracy and determine more robust differential ages.
Funding for this Discovery mission is provided by NASA's Science Mission Directorate. The authors thank the entire Kepler team, without whom these results would not be possible. We also thank all of the funding councils and agencies that have supported the activities of KASC Working Group 1. We are also grateful for support from the International Space Science Institute (ISSI). The authors acknowledge the KITP staff of UCSB for their warm hospitality during the research program "Asteroseismology in the Space Age." This KITP program was supported in part by the National Science Foundation of the United States under grant No. NSF PHY0551164. V.S.A. received financial support from the Excellence cluster "Origin and Structure of the Universe" (Garching). Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation. The research is supported by the ASTERISK project (ASTERoseismic Investigations with SONG and Kepler) funded by the European Research Council (Grant agreement No. 267864). S.B. acknowledges the NSF grant AST-1105930. T.L.C. and M.J.P.F.G.M. acknowledge financial support from project PTDC/CTE-AST/098754/2008 funded by FCT/MCTES, Portugal. W.J.C. and Y.E. acknowledge the financial support of the UK Science Technology and Facilities Council (STFC). D.H. is supported by an appointment to the NASA Postdoctoral Program at Ames Research Center, administered by Oak Ridge Associated Universities through a contract with NASA. A.M.S. is partially supported by the European Union International Reintegration Grant PIRG-GA-2009-247732 and the MICINN grant AYA2011-24704. S.H. acknowledges financial support from the Netherlands Organisation for Scientific Research (NWO). D.S. acknowledges support from the Australian Research Council. NCAR is partially funded by the National Science Foundation.