A PRECISE ASTEROSEISMIC AGE AND RADIUS FOR THE EVOLVED SUN-LIKE STAR KIC 11026764

The spectrum was acquired with the Fibre-fed Echelle Spectrograph (FIES) at the 2.56 m Nordic Optical Telescope (NOT) on 2009 November 9 (HJD 2455145.3428). The 1800 s exposure covers the wavelength range 3730–7360 Å at a resolution R ∼ 67,000 and signal-to-noise ratio (S/N) =80 at 4400 Å. The Th–Ar reference spectrum was acquired immediately after the stellar spectrum. The reduction was performed with the FIESTOOL software, which was developed specifically for the FIES instrument and performs all of the conventional steps of échelle data reduction³⁷. This includes the subtraction of bias frames, modeling and subtraction of scattered light, flat-field correction, extraction of the orders, normalization of the spectra (including fringe correction), and wavelength calibration.

We derived the atmospheric parameters of KIC 11026764 using several methods to provide an estimate of the external errors on T_eff, log g, and [Fe/H], which would be used in the asteroseismic modeling. The five independent reductions included: the VWA³⁸ software package (Bruntt et al. 2004, 2010a), the MOOG³⁹ code (Sneden 1973), the ARES⁴⁰ code (Sousa et al. 2007), the SYNSPEC method (Hubeny 1988; Hubeny & Lanz 1995), and the ROTFIT code (Frasca et al. 2003, 2006). The principal characteristics of the methods employed by each of these codes are described below, and the individual results are listed in Table 2.

For the VWA method, the T_eff, log g, and microturbulence of the adopted MARCS atmospheric models (Gustafsson et al. 2008) are adjusted to minimize the correlations of Fe i with line strength and excitation potential. The atmospheric parameters are then adjusted to ensure agreement between the mean abundances of Fe i and Fe ii. Additional constraints on the surface gravity come from the two wide Ca lines at 6122 and 6162 Å, and from the Mg-1b lines (Bruntt et al. 2010b). The final value of log g is the weighted mean of the results obtained from these methods. The mean metallicity is calculated only from those elements (Si, Ti, Fe, and Ni) exhibiting at least 10 lines in the observed spectrum. The uncertainties in the derived atmospheric parameters are determined by perturbing the computed models, as described in Bruntt et al. (2008). Having computed the mean atmospheric parameters for the star, VWA finally determines abundances for all of the elements contained in the spectrum (see Figure 2). No trace of Li i 6707.8 absorption is seen in the spectrum of KIC 11026764. We estimate an upper limit for the equivalent width (EW) ⩽5 mÅ.

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The 2002 version of the MOOG code determines the iron abundance under the assumption of local thermodynamic equilibrium (LTE), using a grid of one-dimensional model atmospheres by Kurucz (1993). The LTE iron abundance was derived from the EWs of 65 Fe i and 10 Fe ii lines in the 4830–6810 Å range, measured with a Gaussian fitting procedure in the IRAF⁴¹ task splot. For the analysis, we followed the prescription of Randich et al. (2006), using the same list of lines as K. Biazzo et al. (2010, in preparation). The effective temperature and microturbulent velocity were determined by requiring that the iron abundance be independent of the excitation potentials and the EWs of Fe i lines. The surface gravity was determined by requiring ionization equilibrium between Fe i and Fe ii. The initial values for the effective temperature, surface gravity, and microturbulence were chosen to be solar (T_eff = 5770 K, log g = 4.44 dex, and ξ = 1.1 km s⁻¹).

ARES provides an automated measurement of the EWs of absorption lines in stellar spectra: the LTE abundance is determined differentially relative to the Sun with the help of MOOG and a grid of ATLAS-9 plane-parallel model atmospheres (Kurucz 1993). We used ARES with two different lists of iron lines: a "short" one composed of isolated iron lines (Sousa et al. 2006) and a "long" one composed of iron lines suitable for automatic measurements (Sousa et al. 2008). Our computations resulted in two consistent sets of atmospheric parameters for KIC 11026764.

SYNSPEC provides synthetic spectra based on model atmospheres, either calculated by TLUSTY or taken from the literature. We used the new grid of ATLAS-9 models (Kurucz 1993; Castelli & Kurucz 2003) to calculate synthetic spectra, which were then compared to the observed spectrum. Based on the list of iron lines from Sousa et al. (2008), we derived the stellar parameters in two ways to estimate the uncertainty due to the normalization of the observed spectrum. For the first approach, we determined the minimum χ² of the deviation between the synthetic iron lines and the observed spectrum for a fixed set of stellar parameters. For the second approach, we determined the best-fitting effective temperature and surface gravity for each iron line from the list, and adopted stellar parameters from the mean. The primary uncertainty in the final parameters arises from the correlation between the effective temperature and surface gravity: a reduction of the effective temperature can be compensated by a reduction of the gravity. While the χ² method results in lower values for both parameters (T_eff = 5560 K, log g = 3.62 dex), the averaging approach yields higher values (T_eff = 5701 K, log g = 3.95 dex). The differences between the two results exceed the formal errors of each method. We therefore adopt the mean, and assume half of the difference for the uncertainty. The metallicity is determined by minimizing the scatter in the parameters derived from individual lines.

ROTFIT performs a simultaneous and fast determination of T_eff, log g, and [Fe/H] for a star—as well as its projected rotational velocity vsin i—by comparing the observed spectrum with a library of spectra for reference stars (see Katz et al. 1998; Soubiran et al. 1998). The adopted estimates for the stellar parameters come from a weighted mean of the parameters for the 10 reference stars that most closely resemble the target spectrum, which is quantified by a χ² measure. We applied the ROTFIT code to all échelle orders between 21 and 69, which cover the range 4320–6770 Å in the observed spectrum. We also derived a projected rotational velocity for KIC 11026764 of 2.8 ± 1.6 km s⁻¹.

The effective temperature of KIC 11026764 derived from the five methods outlined above generally have overlapping 1σ errors. They all point to a star that is hotter than the KIC estimate by 100–300 K. We find reasonable agreement between the derived values for surface gravity and the KIC estimate. Most of the applied methods result in log g above 4.0 dex, again slightly higher than in the KIC. Only VWA and SYNSPEC yield slightly lower values, which is not surprising considering the correlation between T_eff and log g. Finally, all of the methods agree that the star is slightly metal-rich compared to the Sun, in contrast to the photometric estimate of [Fe/H] =−0.255 dex from the KIC. The initial set of spectroscopic constraints provided to the modeling teams (see Section 4) came from a mean of the preliminary results from the analyses discussed above, with uncertainties large enough to cover the full range for each parameter: T_eff = 5635 ± 185 K, log g = 3.95 ± 0.25 dex, and [Fe/H] =−0.06 ± 0.25 dex.

For the final estimate of the atmospheric parameters (see Section 5), we adopted the results from the VWA method, since it has been carefully tested against direct methods for 10 nearby solar-type stars. Specifically, Bruntt et al. (2010a) used VWA to determine T_eff from high-quality spectra and these values were compared to a direct method (nearly independent of model atmospheres) using the measured angular diameters from interferometry and bolometric flux measurements. A comparison of the direct (interferometric) and indirect (VWA) methods showed only a slight offset of −40 ± 20 K, and this offset has been removed for KIC 11026764. Similarly, Bruntt et al. (2010a) determined the spectroscopic log g parameter, which agrees very well for the three binary stars in their sample where the absolute masses and radii (and hence log g) have been measured. Based on these comparisons of direct and indirect methods, Bruntt et al. (2010a) also discuss the issue of realistic uncertainties on spectroscopic parameters, and their estimates of the systematic uncertainties have been incorporated into the adopted values from VWA listed in Table 2.

We employed the Aarhus stellar evolution code (ASTEC; Christensen-Dalsgaard 2008a) for stellar evolution computations, and the adiabatic pulsation package (ADIPLS; Christensen-Dalsgaard 2008b) for frequency calculations. The input physics for the evolution calculations included the OPAL 2005 equation of state (Rogers & Nayfonov 2002), OPAL opacity tables (Iglesias & Rogers 1996) with low-temperature opacities from Ferguson et al. (2005), and the NACRE nuclear reaction rates (Angulo et al. 1999). Convection was treated according to the mixing-length theory of Böhm-Vitense (1958). We did not include diffusion or convective overshoot in the models.

We computed several grids of evolutionary tracks spanning the parameter space around the values given by Chaplin et al. (2010). We primarily adjusted the stellar mass and metallicity in our grids, while fixing the mixing-length parameter α to 1.7. We scanned the parameter space in mass M from 1.00 to 1.35 M_☉, initial heavy-element mass fraction Z_i from 0.009 to 0.025, and initial hydrogen mass fraction X_i from 0.68 to 0.76. These values of Z_i and X_i cover a range of (Z/X)_i = 0.012–0.037, or [Fe/H] =−0.317 to +0.177 dex using [Fe/H] =log(Z/X) − log(Z/X)_☉, where (Z/X) is the ratio at the stellar surface and (Z/X)_☉ = 0.0245 (Grevesse & Noels 1993). This range of [Fe/H] is compatible with the initial spectroscopic constraints. However, we later extended our grids to determine whether there was a better model with lower or higher metallicity. For all of the models on our tracks, we calculated the oscillation frequencies when the values of T_eff and log g were within 2σ of the derived values (see Section 3.3). We then assigned a goodness of fit to the frequency set of each model by calculating χ²:

$$\begin{equation} \chi ^{2}=\frac{1}{N}\sum _{n,l}\left(\frac{\nu _{l}^{\rm {obs}}(n)-\nu _{l}^{\rm {model}}(n)}{\sigma \big(\nu _{l}^{\rm {obs}}(n)\big)}\right)^{2}, \end{equation} \tag{ 2 }$$

where N is the total number of modes included, $\nu _{l}^{\rm {obs}}(n)$ and $\nu _{l}^{\rm {model}}(n)$ are the observed and model frequencies for a given spherical degree l and radial order n, while $\sigma (\nu _{l}^{\rm {obs}}(n))$ represents the uncertainties on the observed frequencies. We calculated χ² after correcting the frequencies for surface effects, following Kjeldsen et al. (2008):

$$\begin{equation} \nu _{\rm {obs}}(n)-\nu _{\rm {best}}(n)=a\left[\frac{\nu _{\rm {obs}}(n)}{\nu _{0}}\right]^b, \end{equation} \tag{ 3 }$$

where $\nu _{\rm {obs}}(n)$ and $\nu _{\rm {best}}(n)$ are the observed and best model frequencies with spherical degree l = 0 and radial order n, and ν₀ is the frequency of maximum power in the oscillation spectrum, which is 857 μHz for KIC 11026764. We fixed the exponent b to the value derived for the Sun (b = 4.90) by Kjeldsen et al. (2008), and a was calculated for each model. We computed smaller and more finely sampled grids around the models with the lowest χ² to refine the fit. The properties of the best model are listed in Table 3. Although we found models with higher or lower log g that had large separations quite close to the observed value, the individual frequencies were not close to the observations, and the larger range of metallicity did not yield improved results. We also found more massive models (around 1.3 M_☉) with a total χ² value comparable to our best fit, but they did not include the mixed modes. Model A is the best match from the family of solutions (also including Models F, H, J, and K) with masses near 1.1 M_☉.

We used the Geneva stellar evolution code including rotation (Eggenberger et al. 2008) for all computations. This code includes the OPAL equation of state (Rogers & Nayfonov 2002), the OPAL opacities (Iglesias & Rogers 1996) complemented at low temperatures with the molecular opacities of Alexander & Ferguson (1994), the NACRE nuclear reaction rates (Angulo et al. 1999), and the standard mixing-length formalism for convection (Böhm-Vitense 1958). Overshooting from the convective core into the surrounding radiatively stable layers by a distance d_ov ≡ α_ovmin [H_p, r_core] (Maeder & Meynet 1989) is included with an overshoot parameter α_ov = 0.1.

In the Geneva code, rotational effects are computed in the framework of shellular rotation. The transport of angular momentum then obeys an advection–diffusion equation (Zahn 1992; Maeder & Zahn 1998), while the vertical transport of chemicals through the combined action of vertical advection and strong horizontal diffusion can be described as a purely diffusive process (Chaboyer & Zahn 1992). Since the modeling of these rotational effects has been described in previous papers (e.g., Eggenberger et al. 2010), we simply note that the Geneva code includes a comprehensive treatment of shellular rotation and that meridional circulation is treated as a truly advective process. For a detailed analysis of the effect of centrifugal force on the oscillation frequencies, see Appendix A. In addition to rotation, atomic diffusion of helium and heavy elements is included with diffusion coefficients calculated according to the prescription of Paquette et al. (1986).

The properties of a stellar model including rotation depend on six parameters: the mass M, the age t, the mixing-length parameter α ≡ l/H_p for convection, the initial rotation velocity on zero-age main sequence (ZAMS), and two parameters describing the initial chemical composition of the star. For these two parameters, we chose the initial helium abundance Y_i and the initial ratio between the mass fraction of heavy elements and hydrogen (Z/X)_i. This ratio can be related to the metallicity [Fe/H] assuming that log(Z/X) ≅ [Fe/H] + log(Z/X)_☉; we adopt the solar value (Z/X)_☉ = 0.0245 given by Grevesse & Noels (1993). For these computations, the mixing-length parameter was fixed to a solar calibrated value (α_☉ = 1.7998) and the initial rotation velocity on ZAMS was 50 km s⁻¹. The braking law of Kawaler (1988) was used to reproduce the magnetic braking experienced by low-mass stars during main-sequence evolution.

With the above assumptions, the characteristics of a stellar model depend on only four parameters: M, t, Y_i, and (Z/X)_i. The determination of the parameters that best reproduce the observational constraints was then performed in two steps as described in Eggenberger & Carrier (2006). First, a grid of models with global properties in reasonable agreement (within 2σ) with the adopted spectroscopic constraints was constructed. Theoretical frequencies of l ⩽ 2 modes in the observed range of 590–1100 μHz were computed using ADIPLS (Christensen-Dalsgaard 2008b) along with the characteristic frequency separations. The mean large separation was determined by considering only radial modes. The effects of incomplete modeling of the external layers on computed frequencies were taken into account using the empirical power law given by Kjeldsen et al. (2008). This correction was applied to theoretical frequencies using the solar calibrated value of the exponent (b = 4.90) and calculating the coefficient a for each stellar model.

Using spectroscopic measurements of [Fe/H], T_eff, and log g together with the observed frequencies, a χ² minimization was performed to determine the set of model parameters that resulted in the best agreement with all observational constraints. The properties of the best model are listed in Table 3. This model correctly reproduces the spectroscopic measurements of the surface metallicity and log g, but exhibits a slightly higher effective temperature. It is in good agreement with the asteroseismic data and in particular with the observed deviation of the l = 1 modes from asymptotic behavior. Model B is the best match from the family of solutions (also including Models C, E, and G) with masses near 1.2 M_☉.

The Garching Stellar Evolution Code (GARSTEC; Weiss & Schlattl 2008) is a one-dimensional hydrostatic code which does not include the effects of rotation. For the model calculations, we used the OPAL equation of state (Rogers et al. 1996) complemented with the MHD equation of state at low temperatures (Hummer & Mihalas 1988), OPAL opacities for high temperatures (Iglesias & Rogers 1996) and Ferguson's opacities for low temperatures (Ferguson et al. 2005), the Grevesse & Sauval (1998) solar mixture, and the NACRE compilation of thermonuclear reaction rates (Angulo et al. 1999). Mixing is performed diffusively in convective regions using the mixing-length theory for convection in the formulation from Kippenhahn & Weigert (1990), and convective overshooting can optionally be implemented as a diffusive process with an exponential decay of the convective velocities in the radiative zone. The amount of mixing for overshooting depends on an efficiency parameter A calibrated with open clusters (typically A = 0.016). Atomic diffusion can be applied following the prescription of Thoul et al. (1994), and we use a plane-parallel Eddington gray atmosphere.

We started all of our calculations from the pre-main-sequence phase. The value of the mixing-length parameter for convection was fixed (α = 1.791 from our solar calibration), the Schwarzschild criterion for definition of convective boundaries was used, and we did not consider convective overshooting or atomic diffusion. We constructed a grid of models in the mass range between 1.0 M_☉ and 1.3 M_☉ (in steps of 0.01) for several [Fe/H] values from the spectroscopic analysis: 0.06, 0.09, 0.12, and 0.15. To convert the observed values into total metallicity, we applied a chemical enrichment law of ΔY/ΔZ = 2 and used the primordial abundances from our solar calibration. We did not explore variations in either the hydrogen or helium abundances.

Once all of the tracks were computed, we restricted our analysis to those models contained within the spectroscopic uncertainties. For these cases, we calculated the oscillation frequencies using ADIPLS (Christensen-Dalsgaard 2008b) and looked for the model which best reproduced the large frequency separation (no surface correction was applied to the calculated frequencies). Several models were found to fulfill these requirements for each metallicity grid, and among those best-fit models to the large frequency separation we then performed a χ² test to obtain the global best fit to the individual frequencies and the spectroscopic constraints. Our global best-fit model came from the grid with [Fe/H] =0.12, and the properties are listed in Table 3. This model reproduces well the observed mixed modes.

The Asteroseismic Modeling Portal (AMP) is a web-based tool tied to TeraGrid computing resources that uses ASTEC (Christensen-Dalsgaard 2008a) and ADIPLS (Christensen-Dalsgaard 2008b) in conjunction with a parallel genetic algorithm (GA; Metcalfe & Charbonneau 2003) to optimize the match to observational data (see Metcalfe et al. 2009). The models use the OPAL 2005 equation of state (see Rogers & Nayfonov 2002) and the most recent OPAL opacities (see Iglesias & Rogers 1996), supplemented by Kurucz opacities at low temperatures. The nuclear reaction rates come from Bahcall & Pinsonneault (1995), convection is described by the mixing-length theory of Böhm-Vitense (1958), and we can optionally include the effects of helium settling as described by Michaud & Proffitt (1993).

Each model evaluation involves the computation of a stellar evolution track from ZAMS through a mass-dependent number of internal time steps, terminating prior to the beginning of the red giant stage. Rather than calculating the pulsation frequencies for each of the 200–300 models along the track, we exploit the fact that the average frequency separation of consecutive radial orders <Δν₀> in most cases is a monotonically decreasing function of age (Christensen-Dalsgaard 1993). Once the evolution track is complete, we start with a pulsation analysis of the model at the middle time step and then use a binary decision tree—comparing the observed and calculated values of <Δν₀>—to select older or younger models along the track. This allows us to interpolate the age between the two nearest time steps by running the pulsation code on just eight models from each stellar evolution track. The frequencies of each model are then corrected for surface effects following the prescription of Kjeldsen et al. (2008).

The GA optimizes four adjustable model parameters, including the stellar mass (M) from 0.75 to 1.75 M_☉, the metallicity (Z) from 0.002 to 0.05 (equally spaced in log Z), the initial helium mass fraction (Y_i) from 0.22 to 0.32, and the mixing-length parameter (α) from 1 to 3. The stellar age (t) is optimized internally during each model evaluation by matching the observed value of <Δν₀> (see above). The GA uses two-digit decimal encoding, such that there are 100 possible values for each parameter within the specified ranges. Each run of the GA evolves a population of 128 models through 200 generations to find the optimal set of parameters, and we execute four independent runs with different random initialization to ensure that the best model identified is truly the global solution. The resulting properties of the optimal model are listed in Table 3.

The extreme values in this global fit arose from treating each spectroscopic constraint as equivalent to a single frequency. Since the adopted spectroscopic errors are large, they provide much more flexibility for the models compared to the individual frequencies with relatively small errors. Consequently, the fitting algorithm found it advantageous to shift the effective temperature and metallicity by several σ from their target values to achieve significantly better agreement with the 22 oscillation frequencies. The improvement in the frequency match outweighed the degradation in the spectroscopic fit for the calculation of χ². The solution to this problem may be to calculate a separate value of χ² for the asteroseismic and spectroscopic constraints, and then average them to provide more equal weight to the two types of constraints. This is an important lesson for future automated searches, and explains why Model D does not align with either of the two major families of solutions.

We used the Yale Stellar Evolution Code (YREC; Demarque et al. 2008) in its non-rotating configuration to model KIC 11026764. All models were constructed with the OPAL equation of state (Rogers & Nayfonov 2002). We used OPAL high-temperature opacities (Iglesias & Rogers 1996) supplemented with low-temperature opacities from Ferguson et al. (2005). The NACRE nuclear reaction rates (Angulo et al. 1999) were used. We assumed that the current solar metallicity is that given by Grevesse & Sauval (1998). We have not explored the consequences of using the lower metallicity measurements of Asplund et al. (2005) or the intermediate metallicity measurements of Ludwig et al. (2010). We searched for the best fit within a fixed grid of models. There were eight separate grids defined by different combinations of the mixing-length parameter (α = 1.83 or α = 2.14), the initial helium abundance (either Y_i = 0.27 or Y_i calculated assuming a ΔY/ΔZ = 2, with Y_i for [Fe/H] =0 being the current solar CZ helium abundance), and the amount of overshoot (0 H_p or 0.2 H_p). All models included gravitational settling of helium and heavy elements using the formulation of Thoul et al. (1994).

Our fitting method included two steps. In the first step, we calculated the average large frequency separation for the models and selected all of those that fit the observed separation within 3σ errors. We adopted the observed value of Δν = 50.8 ± 0.3 μHz from Chaplin et al. (2010). A second cut was made using the effective temperature: all models within ±2σ of the observed value were chosen. A third cut was made using the frequencies of the three lowest-frequency l = 0 modes. Given the small variation in mass, this process was effectively a radius cut. The selected models had radii around 2R_☉. Note that the Yale-Birmingham radius pipeline (Basu et al. 2010) finds a radius of 2.18^+0.04_−0.05 R_☉ for this star using the adopted values of Δν, T_eff, log g, and [Fe/H]. In the second step of the process, we made a finer grid in mass and age around the selected values and then compared the models with the observed set of frequencies. The properties of our best-fit model are listed in Table 3. This model was constructed with Y_i = 0.27, Z_i = 0.0147, and core overshoot of 0.2 H_p. We were unable to find a good model without core overshoot.

We modeled KIC 11026764 with the Catania Astrophysical Observatory version of the GARSTEC code (Bonanno et al. 2002) using a grid-based approach. The input physics of this stellar evolution code included the OPAL 2005 equation of state (Rogers & Nayfonov 2002) and the OPAL opacities (Iglesias & Rogers 1996) complemented in the low-temperature regime with the tables of Alexander & Ferguson (1994). The nuclear reaction rates were taken from the NACRE collaboration (Angulo et al. 1999) and the standard mixing-length formalism for convection was used (Kippenhahn & Weigert 1990). Microscopic diffusion of hydrogen, helium, and all of the major metals can optionally be taken into account. The outer boundary conditions were determined by assuming an Eddington gray atmosphere.

A grid of evolutionary models was computed to span the 1σ uncertainties in the spectroscopic constraints obtained from ground-based observations. When a given evolutionary track was in the error box, a maximum time step of 20 Myr was chosen and the frequencies were computed with the ADIPLS code. A non-uniform grid of mass in the range 1.0–1.24 M_☉, helium abundance in the range Y_i = 0.26–0.31, mixing-length parameter α = 1.6–1.9, and initial surface heavy-element abundances (Z/X)_i = 0.022–0.029 was scanned. A global optimization strategy was implemented by minimizing the χ² for all of the l = 0, l = 1, and l = 2 modes. The empirical surface effect, as discussed by Kjeldsen et al. (2008), was used to correct all theoretical frequencies. The properties of the best model with heavy-element diffusion and surface corrected frequencies are listed in Table 3.

We used a version of ASTEC (Christensen-Dalsgaard 2008a) which includes the OPAL 2001 equation of state (Rogers et al. 1996), OPAL opacities (Iglesias & Rogers 1996), Bahcall & Pinsonneault (1995) nuclear cross sections, and the mixing-length formalism (Böhm-Vitense 1958) for convection. We computed several grids of models by varying all of the input parameters within the range of the observed errors (Chaplin et al. 2010). In particular, we calculated evolutionary tracks by varying the input mass in the range M = 0.9–1.2 M_☉, the metallicity in the range Z = 0.009–0.03, and the hydrogen abundance in the range X = 0.67–0.7. We also adopted different values of the mixing-length parameter in the range α = 1.67–1.88. We calculated additional evolutionary models using the Canuto & Mazzitelli (1992) convection formulation. The mixing-length parameter α of the CM formulation was chosen in the range α = α_CM = 0.9–1.0. To obtain the deviation from asymptotic behavior observed in the l = 1 modes (the mixed modes), we did not include overshooting in the calculation, following the conclusion of Di Mauro et al. (2003). These models are distinct from the grid used to produce Model A, not only because they employ a slightly older equation of state (EOS) and nuclear reaction rates, but also because the grid search included α and calculated fewer models within the specified range of parameter values. We used the adiabatic oscillation code (ADIPLS; Christensen-Dalsgaard 2008b) to calculate the p-mode eigenfrequencies with harmonic degree l = 0–2. The characteristics of the model which best fits the observations are listed in Table 3.

For the evolution calculations, we used the publicly available Dartmouth stellar evolution code (DSEP; Chaboyer et al. 2001; Guenther et al. 1992; Dotter et al. 2007), which is based on the code developed by Pierre Demarque and his students (Larson & Demarque 1964; Demarque & Mengel 1971). The input physics includes high-temperature opacities from OPAL (Iglesias & Rogers 1996), low-temperature opacities from Ferguson et al. (2005), the nuclear reaction rates of Bahcall & Pinsonneault (1992), helium and heavy-element settling and diffusion (Michaud & Proffitt 1993), and Debye-Hückel corrections to the equation of state (Guenther et al. 1992). The models employ the standard mixing-length theory. Convective core overshoot is calculated assuming that the extent is proportional to the pressure scale height at the boundary (Demarque et al. 2004). The models used the standard conversion from [Fe/H] and ΔY/ΔZ to Z and Y (Chaboyer et al. 1999). The oscillation frequencies were computed using the adiabatic oscillation codes of Kosovichev (1999) and Christensen-Dalsgaard (2008b). No surface corrections were applied.

The strategy to find a model matching the observed spectroscopic constraints involved calculating a series of evolutionary tracks in the log g–T_eff plane for a mass range of 1.0–1.3M_☉, heavy-element abundance Z = 0.01–0.03, initial helium abundance Y_i = 0.25–0.30, and mixing-length parameter α = 1.70–1.75. We then selected the models closest to the target values within half of the specified uncertainties. All models for the search were calculated assuming an overshoot parameter α_ov = 0.2, and included element diffusion. For comparison, the corresponding models without diffusion and convective overshoot were also calculated. The oscillation spectra were matched to the observed frequencies, first by comparing the frequencies of radial (l = 0) modes with the corresponding observed frequencies, and then selecting a model with the closest frequency values for the l = 1 and l = 2 modes. The properties of the final model are listed in Table 3. This model matches the observed frequencies quite well except for the first two l = 1 modes, which deviate by about 12–13 μHz. Our search demonstrated that the behavior of the mixed mode frequencies is sensitive to element diffusion and convective overshoot. This requires further investigation.

To characterize KIC 11026764, we constructed a grid of stellar models with the CESAM code (Morel 1997), and computed their oscillation frequencies with the adiabatic oscillation code FILOU (Suárez 2002; Suárez & Goupil 2008). Opacity tables were taken from the OPAL package (Iglesias & Rogers 1996), complemented at low temperatures (T ⩽ 10⁴ K) with the tables provided by Alexander & Ferguson (1994). The atmosphere was constructed from an Eddington T–τ relation and was assumed to be gray. The stellar metallicity (Z/X) was derived from the [Fe/H] value assuming (Z/X)_☉ = 0.0245 (Grevesse & Noels 1993), Y_pr = 0.235, and Z_pr = 0 for the primordial helium and heavy-element abundances, and a value ΔY/ΔZ = 2 for the enrichment ratio. No microscopic diffusion of elements was considered.

The main strategy was to search for representative equilibrium models of the star in a database of 5 × 10⁵ equilibrium models, querying for those matching the global properties of the star, including the effective temperature, gravity, and metallicity. Using this set of models, we then applied the asteroseismic constraints, including the individual frequencies and large separations. The global fitting method involved a χ² minimization, taking into account all of the observational constraints simultaneously. No correction for surface effects was applied, and no a priori information on mode identification was assumed when fitting the individual frequencies. The properties of the best model we found are listed in Table 3, and includes overshooting with α_ov = 0.3. Note that the analysis did not adopt the identifications of spherical degree (l) from Section 2, and it included l = 3 modes to perform the match. As a consequence, the final result is much different than any of the others and it does not fall into either of the two major families of solutions.

The SEEK procedure makes use of a large grid of stellar models computed with ASTEC (Christensen-Dalsgaard 2008a). It compares the observations with every model in the grid and makes a probabilistic assessment, with the help of Bayesian statistics, about the global properties of the star. The model grid includes 7300 evolution tracks containing 5,842,619 individual models. Each track begins at ZAMS and continues to the red giant branch or a maximum age of t = 15 Gyr. The tracks are separated into 100 subsets with different combinations of metallicity Z, initial hydrogen content X_i, and mixing-length parameter α. These combinations are separated into two regularly spaced and interlaced subgrids. The first subgrid comprises tracks with Z = [0.005, 0.01, 0.015, 0.02, 0.025, 0.03], X_i = [0.68, 0.70, 0.72, 0.74], and α = [0.8, 1.8, 2.8] while the second subset has Z = [0.0075, 0.0125, 0.0175, 0.0225, 0.0275], X_i = [0.69, 0.71, 0.73], and α = [1.3, 2.3]. Every subset is composed of 73 tracks with masses between 0.6 and 3.0 M_☉. The spacing in mass between the tracks is 0.02 M_☉ from 0.6 to 1.8 M_☉ and 0.1 from 1.8 to 3.0 M_☉. A relatively high value of Y_☉ = 0.2713 and Z_☉ = 0.0196 for the Sun has been used for the standard definition of [Fe/H] in SEEK. This value is used to calibrate solar models from ASTEC to the correct luminosity (Christensen-Dalsgaard 1998). The input physics includes the OPAL equation of state, opacity tables from OPAL (Iglesias & Rogers 1996) and Alexander & Ferguson (1994), and the metallic mixture of Grevesse & Sauval (1998). Convection is treated according to the mixing-length theory of Böhm-Vitense (1958) with the convective efficiency characterized by the mixing-length to pressure height scale ratio α, which varies across the grid of models. Diffusion and overshooting were not included.

The grid allows us to map the physical input parameters of the model p ≡ {M, t, Z, X_i, α} into the grid of observable quantities q^g ≡ {Δν, δν, T_eff, log g, [Fe/H], ...}, defining the transformation

$$\begin{equation} \mathbf {q}^\mathbf {g} = \mathcal {K}(\mathbf {p}). \end{equation} \tag{ 4 }$$

We compare these quantities to the observed values q^obs with the help of a likelihood function $\mathcal {L}$,

$$\begin{equation} \mathcal {L} = \left({\displaystyle \prod _{i=0}^n}\frac{1}{\sqrt{2\pi }\sigma _i}\right)\exp (-\chi ^2/2), \end{equation} \tag{ 5 }$$

and the usual χ² definition

$$\begin{equation} \chi ^2 = \frac{1}{N}{\displaystyle \sum _{i=0}^{N}}\left(\frac{{q}_i^{\rm obs} - {q}_i^{\rm g}}{\sigma _i} \right)^2, \end{equation} \tag{ 6 }$$

where σ_i is the estimated error for each observation q^obs_i and N is the number of observables. The maximum likelihood is then combined with the prior probability of the grid f₀ to yield the posterior, or the resulting probability density

$$\begin{equation} f(\mathbf {p}) \propto f_0(\mathbf {p}) \mathcal {L}(\mathcal {K}(\mathbf {p})). \end{equation} \tag{ 7 }$$

This probability density can be integrated to obtain the value and uncertainty for each of the parameters, as listed in Tables 3 and 4. It can also be projected onto any plane to get the correlation between two parameters, as shown in Appendix B. The details of the SEEK procedure, including the choice of priors, and an introduction to Bayesian statistics can be found in P. O. Quirion et al. (2010, in preparation).

SEEK uses the large and small separations and the median frequency of the observed modes as asteroseismic inputs. Values of Δν₀ = 50.68 ± 1.30 (computed with l = 0 modes only) and δν_0,2 = 4.28 ± 0.73 (computed from l = 0, 2 modes) were derived from Kepler data around a central value of 900 μHz. Each separation is the mean of the individual observed separations, while the error is the standard deviation of the individual values from the mean. These values differ slightly from those given in Chaplin et al. (2010) because they are calculated from the individual frequencies rather than derived from the power spectrum. Using these asteroseismic inputs along with the initial spectroscopic constraints, we obtained the parameters listed in Table 4. For a SEEK analysis of the importance of the asteroseismic constraints, see Appendix B.

To investigate how well we can find an appropriate model without comparing individual oscillation frequencies, we used the RADIUS pipeline (Stello et al. 2009a), which takes T_eff, log g, [Fe/H], and Δν as the only inputs to find the best-fitting model. The value of Δν = 50.8 ± 0.3 μHz from Chaplin et al. (2010) was adopted. The pipeline is based on a large grid of ASTEC models (Christensen-Dalsgaard 2008a) using the EFF equation of state (Eggleton et al. 1973). We use the opacity tables of Rogers & Iglesias (1995) and Kurucz (1991) for T < 10⁴ K with the solar mixture of Grevesse & Noels (1993). Rotation, overshooting, and diffusion were not included. The grid was created with fixed values of the mixing-length parameter (α = 1.8) and the initial hydrogen abundance (X_i = 0.7). The resolution in log Z was 0.1 dex between 0.001 < Z < 0.055, and the resolution in mass was 0.01 M_☉ from 0.5 to 4.0 M_☉. The evolution begins at ZAMS and continues to the tip of the red giant branch. To convert between the model values of Z and the observed [Fe/H], we used Z_☉ = 0.0188 (Cox 2000).

We made slight modifications to the RADIUS approach described by Stello et al. (2009a). First, the large frequency separation was derived by scaling the solar value (Kjeldsen & Bedding 1995) instead of calculating it directly from the model frequencies. Although there is a known systematic difference between these two ways of deriving Δν, the effect is probably below the 1% level (Stello et al. 2009b; Basu et al. 2010). Second, we pinpointed a single best-fitting model based on a χ² formalism that was applied to all models within ±3σ of the observed properties. The properties of the best-fitting model are listed in Table 3. This model shows a frequency pattern in the échelle diagram that looks very similar to the observations if we allow a small tweaking of the adopted frequency separation (52.1 μHz) used to generate the échelle. This basically means that we found a model that homologously represents the observations quite well. In particular, we see relative positions of the l = 1 mode frequencies that are very similar to those observed.

Comparing the theoretical frequencies of Models A and B with the maximal frequency set from Section 2, we identified four of the seven additional oscillation modes that could be used for refined model fitting (see Table 1). Recall that the maximal frequency set comes from the individual analysis with the smallest rms deviation with respect to the maximal list, so the frequencies of the modes from the minimal set are slightly different in the maximal set. Without any additional fitting, these subtle frequency differences improve the χ² of Models A and B when comparing them to those modes from the maximal set that are also present in the minimal set. There is one additional l = 0 mode (n = 12) and three additional l = 2 modes (n = 11, 16, 19) in the maximal set that are within 3σ of frequencies in both Models A and B. Considering the very different input physics of these two models, we took this agreement as evidence of the reliability of these four additional frequencies and we incorporated them as constraints for our refined model fitting. Two of the remaining frequencies in the maximal set (n = 12,  l = 1 and 2) were not present in either Models A or B, while one (n = 10,  l = 2) had a close match in Model B but not in Model A. We excluded these three modes from the refined model fitting. Given that the l = 1 modes provide the strongest constraints on the models (see Section 5.2), the additional l = 0 and l = 2 modes are expected to perturb the final fit only slightly.

In addition to the 26 oscillation frequencies from the maximal set, we also included stronger spectroscopic constraints in the refined model fitting by adopting the results of the VWA analysis instead of using the mean atmospheric parameters from the preliminary analyses (see Section 3.3). Although the uncertainties on all three parameters are considerably smaller from the VWA analysis, the actual values only differ slightly from the initial spectroscopic constraints. These were just three of the 25 constraints used to calculate the χ² and rank the initial search results in Table 3. Since the 22 frequencies from the minimal set were orders of magnitude more precise, they dominated the χ² determination. Although the spectroscopic constraints from VWA are more precise than the initial atmospheric parameters, they are still much less precise than the frequencies and should only perturb the χ² ranking slightly. Consequently, we do not need to perform a new global search after adopting the additional and updated observational constraints.

Several modeling teams used the updated asteroseismic and spectroscopic constraints for refined model fitting with a variety of codes. Each team started with the parameters of Models A and B from Table 3, and then performed a local optimization to produce the best match to the observations within each family of solutions. The results of this analysis are shown in Table 4, where the refined Models A and B are ranked separately by their final χ² value. Each model is labeled with a letter from Table 3 to identify the modeling team, followed by either A or B to identify the family of solutions. The two pipeline approaches labeled J' and K' simply adopted the revised constraints to evaluate any shift in the optimal parameter estimates and errors. Note that both the SEEK and RADIUS pipelines identified parameters in the high-mass family of solutions when using the revised constraints, but the low-mass family was only marginally suboptimal. The apparent bifurcation of results in Table 4 into two values of α arises from the decision of most modelers to fix this parameter in each case to the original value from Table 3. For each family of solutions, the modeling teams adopted the appropriate input physics: neglecting overshoot and diffusion for the refined Models A, while including both ingredients for the refined Models B. One team produced two additional models (labeled AA' and AB') to isolate the effect of input physics on the final results. Model AA' started from the parameters of Model A but included overshoot and helium settling, while Model AB' searched in the region of Model B but neglected overshoot and diffusion.

An inspection of the results in Table 4 reveals that models in either family of solutions can provide a comparable match to the observational constraints. This ambiguity cannot be attributed to the input physics, since the models that sample all four combinations of the input physics and family of solutions (AA, AA', AB, and AB') have comparable χ² values. The individual frequencies of these models all provide a good fit to the data, including the l = 1 mixed modes. The observations are compared to two representative models in Figure 3 using an échelle diagram, where we divide the frequency spectrum into segments of length <Δν> and plot them against the oscillation frequency. This representation of the data aligns modes with the same spherical degree into roughly vertical columns, with l = 0 modes shown as circles, l = 1 modes shown as triangles, and l = 2 modes shown as squares. The 26 modes from the maximal frequency set that were included in the final fit are indicated with solid points. Open points indicate the model frequencies, with Model AA shown in blue and Model AB shown in red. A gray-scale map of the power spectrum (smoothed to 1  μHz resolution) is included in the background for reference. The different models appear to be sampling comparable local minima in a correlated parameter space. Without additional constraints, we have no way of selecting one of these models over the other.

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We can understand the two families of models by considering the general properties of subgiant stars, where a wide range of masses can have the same stellar luminosity with minor adjustments to the input physics and other model parameters—in particular the helium mass fraction. This degeneracy between mass and helium abundance has been discussed in the modeling of specific subgiant stars (e.g., Fernandes & Monteiro 2003; Pinheiro & Fernandes 2010; Yang & Meng 2010), and it adds a large uncertainty to the already difficult problem of determining the helium abundance in main-sequence stars (e.g., Vauclair et al. 2008; Soriano & Vauclair 2010). The luminosity of stellar models on the subgiant branch is mainly determined by the amount of energy produced at the edge of the helium core established during the main-sequence phase. The rate of energy production depends on the temperature and the hydrogen abundance in that layer. As a consequence, it is possible to find models at the same luminosity with quite different values of total mass, by adjusting the other parameters to yield the required temperature at the edge of the helium core. Fortunately, for subgiant stars the presence of mixed modes can provide additional constraints on the size of the helium core. For KIC 11026764, this mixed mode constraint significantly reduces the range of possible masses to two specific intervals, where the combination of stellar mass and core size is compatible with the atmospheric parameters and the frequencies of the mixed modes.

Regardless of which family of solutions is a better representation of KIC 11026764, the parameter values in Table 4 already yield precise determinations of some of the most interesting stellar properties—in particular, the asteroseismic age and radius. If we consider only the four models that were produced by the same modeling team with an identical fitting method (ASTEC models: AA, AA', AB, and AB'), we can calculate the mean value and an internal (statistical) uncertainty in isolation from external (systematic) errors arising from differences between the various codes and methods. The asteroseismic ages of the two families of solutions range from 5.87 to 5.99 Gyr, with a mean value of t = 5.94 ± 0.05 Gyr. The stellar radii of the two families range from 2.03 to 2.08 R_☉, with a mean value of R = 2.05 ± 0.03 R_☉. The luminosity ranges from 3.45 to 3.80 L_☉, with a mean value of L = 3.56 ± 0.14 L_☉. These are unprecedented levels of precision for an isolated star, despite the fact that the stellar mass is still ambiguous at the 10% level. Of course, precision does not necessarily translate into accuracy, but we can evaluate the possible systematic errors on these determinations by looking at the distribution of parameter values for the entire sample of modeling results, not just those from ASTEC.

Considering the full range of the best models (χ² < 10) in Table 4, there is broad agreement on the value of the stellar radius. The low value of 2.03 R_☉ is from one of the ASTEC models considered above, while the high value of 2.09 R_☉ is from the Catania-GARSTEC code, leading to a systematic offset of ^+0.04_−0.02 R_☉ compared to the mean ASTEC value. There is a slightly higher dispersion in the values of the asteroseismic age. Again considering only the best models, we find a full range for the age as low as 4.99 Gyr from Catania-GARSTEC up to 5.99 Gyr from ASTEC, for a systematic offset of ^+0.05_−0.95 Gyr relative to the ASTEC models. The best models exhibit the highest dispersion in the values of the luminosity, with a low estimate of 3.45 L_☉ from ASTEC and a high value of 4.44 L_☉ from Catania-GARSTEC. These models establish a systematic offset of ^+0.88_−0.11 L_☉ compared to the ASTEC results.

To evaluate the relative contribution of each observational constraint to the final parameter determinations, we can study the significance S_i using singular value decomposition (SVD; see Brown et al. 1994). An observable with a low value of S_i has little influence on the solution, while a high value of S_i indicates an observable with greater impact. The significance of each observable in determining the parameters of KIC 11026764 for Model AA is shown in Figure 4. From left to right, we show the spectroscopic constraints followed by the l = 0, 1, and 2 frequencies labeled with the reference radial order from Table 1, and each group of constraints is shown in a different shade. It is immediately clear that the l = 1 and l = 2 frequencies have more weight in determining the parameters. If we examine how the significance of each spectroscopic constraint changes when we combine them with modes of a given spherical degree, we can quantify the impact of each set of frequencies because the information content of the spectroscopic data does not change. The spectroscopic constraints contribute more than 25% of the total significance when combined with the l = 0 modes (13% from the effective temperature alone), indicating that these two sets of constraints contain redundant information. By contrast, the total significance of the spectroscopic constraints drops to 4% for the l = 2 modes, and 7% when combined with the l = 1 modes—confirming that these frequencies contain more independent information than the l = 0 modes, as expected from the short evolutionary timescale for mixed modes. Although the significance of the atmospheric parameters on the final solution appears to be small, we emphasize that accurate spectroscopic constraints are essential for narrowing down the initial parameter space.

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