CHARACTERIZING TWO SOLAR-TYPE KEPLER SUBGIANTS WITH ASTEROSEISMOLOGY: KIC 10920273 AND KIC 11395018

Atmospheric properties of KIC 10920273 and KIC 11395018 were obtained from observations with the FIES spectrograph (Frandsen & Lindberg 1999) at the Nordic Optical Telescope (NOT on La Palma, Spain) at medium resolution (R  ∼  46, 000) in 2010 July and August. The reduced spectra were analyzed by several teams and the results were presented by Creevey et al. (2012). The constraints we used for our analysis are shown in Table 1 (see also the 1σ and 2σ error boxes in Figure 1). We adopted a set of atmospheric constraints for each star that were closest to the mean of the results from several methods described by Creevey et al. (2012). This approach was preferred for the sake of reproducibility, rather than using the mean values. However, we did not restrict our model-searching space to less than 3σ uncertainty around these constraints; therefore, the selected values represent well the overall results of the spectroscopic analysis.

Zoom In Zoom Out Reset image size

The spectroscopic υ sin i values for KIC 10920273 and KIC 11395018 are 1.5 ± 2.2 km s⁻¹ and 1.1 ± 0.8 km s⁻¹, respectively (Creevey et al. 2012). These low values indicate either slow rotation or low inclination angle i, although the latter is statistically unlikely. Rotational periods from modulation of the Kepler data due to spots on the surfaces of the two stars were measured to be ∼27 days for KIC 10920273 (Campante et al. 2011) and ∼36 days for KIC 11395018 (Mathur et al. 2011). The signal-to-noise ratios (S/Ns) of the peaks used for these measurements, particularly for KIC 10920273, were low, so the results should be used with caution. Mathur et al. (2011) inferred i ⩾ 45° for KIC 11395018 based on combining the rotational frequency splittings with the measured rotation period, which implies this star to be a slow rotator. A wider range of inclinations was possible for KIC 10920273 (Campante et al. 2011). However, the modulation in the light curve has been detected with less uncertainty in the new analysis performed using longer time series including Kepler Q9 and Q10 data of KIC 10920273 (R. A. García et al. 2013, private communication). If confirmed, this would be consistent with a relatively high inclination angle and slow rotation. The effects of the centrifugal force are negligible for slowly rotating stars.²² However, rotational mixing may lead to changes in the properties of the models even for slowly rotating stars because the efficiency of this mixing is more directly related to differential rotation in stellar interiors rather than to surface rotational velocities (Pinsonneault et al. 1990; Eggenberger et al. 2010). Studying the impact of rotation on post-MS stars would require a detailed discussion of the effects of rotational mixing on the chemical gradients in the central parts of the star. These influence the asteroseismic properties of the models, particularly the mixed modes. In the specific case of the evolved post-MS stars modeled here, however, we expect that these effects on the chemical gradients would already be erased, as found by Miglio et al. (2007) for models of 12 Bootis A in the thick-shell-H-burning phase (see also the discussion of the effects of microscopic diffusion in the subgiant HD 49385 by Deheuvels & Michel 2011). We therefore have not included the rotational effects for most of the analyses (see Section 3).

When the atmospheric properties alone are considered, these two stars are very similar. Due to the degeneracy inherent in the H-R diagram analysis (see, e.g., Fernandes & Monteiro 2003), it is not possible to determine the global stellar properties with sufficiently high precision to study their detailed characteristics without the help of seismic data, which we now discuss.

We used Kepler data from observations made in the period from 2009 May to 2010 March, i.e., from the commissioning run (Q0) through Quarter 4 (Q4). The formal frequency resolution is ∼0.05 μHz. From the power spectra, Campante et al. (2011) and Mathur et al. (2011) reported individual frequencies for KIC 10920273 and KIC 11395018, based on analyses performed by several teams. The final sets of results included a minimal and a maximal list of frequencies, where the former were those agreed upon by more than half of the fitters and the latter were those agreed upon by at least two fitters. Therefore, the frequencies that are in the maximal list but not the minimal list are less certain. For details of the frequency-extraction techniques and the selection methods, we refer the reader to Campante et al. (2011) and Mathur et al. (2011). The analysis of each star resulted in the extraction of up to a total of 25 individual oscillation frequencies for radial (l = 0), dipole (l = 1), and quadrupole (l = 2) modes, including several mixed modes. These mixed modes carry information from the core and hence provide stronger constraints on the evolutionary stage of the stars, as discussed in Section 1. We started by searching for models using the minimal-list of frequencies and then extended our analysis to include additional frequencies from the maximal lists.

To select the best models, we defined the two normalized χ² measures shown in Equations (3) and (4), which allowed us to evaluate the qualities of the fits for the atmospheric parameters and the seismic parameters separately. The seismic measure was

$$\begin{equation} \chi ^2_{\rm seis}=\frac{1}{N}\sum _{n,l}\left(\frac{\nu _{\mathrm{obs}}(n,l)-\nu _{\mathrm{corr}}(n,l)}{\sigma (\nu _{\mathrm{obs}}(n,l))}\right)^2, \end{equation} \tag{ 3 }$$

where N is the number of observed frequencies, ν_corr(n, l) represents the near-surface-corrected model frequencies with spherical degree l and radial order n, ν_obs(n, l) are the observed frequencies, and σ(ν_obs(n, l)) are the uncertainties on the observed frequencies. The measure for the atmospheric properties was

$$\begin{equation} \chi ^2_{\rm atm}=\frac{1}{3}\sum \left(\frac{{P}_{\mathrm{obs}}-{P}_{\mathrm{mod}}}{\sigma ({P}_{\mathrm{obs}})}\right)^2, \end{equation} \tag{ 4 }$$

where P = {T_eff, log g, [Fe/H]} and the subscripts "obs" and "mod" represent the observed and model properties, respectively, with σ(P_obs) denoting the observational uncertainties. The values of [Fe/H] for the models were calculated using the formula [Fe/H] = log (Z/X)_mod − log (Z/X)_☉, where the solar value was adopted from Grevesse & Noels (1993) as (Z/X)_☉ = 0.0245.

Each modeling team returned the model that best matched the observational constraints. We present the properties of these models in Tables 4 and 5, along with the normalized χ² values.²⁴ We also present models fitted using more or fewer frequencies than those in the minimal lists in order to see whether the model-fitting results change considerably. In each case, we calculated the χ²_seis in the tables using only the frequencies that were common constraints for all of the models, in order to achieve a consistent evaluation of the models.

There is a good agreement between the observed frequencies and the model frequencies. The quality of the fits can be seen in the échelle diagrams for a sample of models (Figure 2). Overall frequency patterns, including the dipolar mixed modes, are matched quite well. The fact that χ²_seis > 1.0 implies that either the observational uncertainties are underestimated or the models are incomplete representations of the observational data. The models with relatively high χ²_seis (≳ 20.0) are those that either could be improved with further refinement or that do not reproduce all of the modes simultaneously, in particular the mixed modes, which are more difficult to fit. Due to their strong sensitivity to stellar evolution, mixed modes tend to dominate the model-fitting results. The timescale on which the signatures of these modes evolve is very short, so it becomes difficult to find good fits unless the grid of models used is very fine.

Zoom In Zoom Out Reset image size

We present our results in Tables 4 and 5. Properties of all the models are listed, along with the weighted mean values and the standard deviations. Both stars have left the MS (central hydrogen mass fraction X_c = 0.0) but have quite different characteristics, as seen in Tables 4 and 5. The fact that KIC 10920273 is an old solar analog (with one solar mass and near-solar metallicity) makes it an interesting target for further studies. A typical solar model would turn off from the MS at around 9–10 Gyr. However, the metallicity and particularly the helium abundance alter this age estimate. In this case it is the high helium abundance that affects the MS turnoff age more than the low metallicity. The models with higher helium abundance behave similar to those with higher mass (higher luminosity), following an evolutionary track similar to that of a higher-mass star, and hence have shorter MS lifetimes.

The results in Tables 4 and 5 were weighted by the goodness of the seismic fit, i.e., the inverse of χ²_seis. This way, any misleading contribution due to coarse grids is eliminated. Although this does not provide a direct measurement of the systematic uncertainties, it still allows us to estimate the order of magnitude of the external errors expected from using different inputs, codes, and fitting methods. A similar determination of the systematic errors for the case of bright Kepler stars 16 Cyg A and B was carried out by Metcalfe et al. (2012) using different evolutionary codes and fitting methods. The uncertainties we determined are mostly greater due to the lower S/N in the data of our faint stars. Systematic errors in determination of stellar properties using grid-based pipelines caused by different observational constraints and different input physics were discussed more generally for a few Kepler stars by Creevey et al. (2012). We discuss the uncertainties further in the next section.

To further evaluate the typical uncertainties caused by different input physics, we calculated some additional models and small grids using KIC 10920273 as a test case. We selected SB1 as our base model. Keeping the input parameters (mass, Z/X_i, Y_i, and α) fixed, we first explored the effects of changing one single ingredient of input physics at a time. We also changed the value of α while keeping everything else fixed. On every new evolutionary sequence, we calculated the frequencies of the models that had all atmospheric properties (log g, T_eff, and [Fe/H]) within 3σ of the observed values. We then selected the model that best matched the observed frequencies. In the cases of including core overshoot, changing the convection treatment (to CGM formulation), and using several different nuclear rates (Bahcall et al. 1995; Parker 1986; Adelberger et al. 1998), the new models reproduced the observed frequencies as well as the base model with an age difference of only 0.7% at most (which corresponds to changes in radius of <0.2% and in log g of <0.1%). This means that the uncertainties on the final parameters caused by the corresponding inputs were negligible. We note that the effect of core overshoot would be more significant for an MS star.

For the cases where we did not obtain a model having a seismic χ² comparable to that of the base model, we carried on with the analysis. These cases resulted from including diffusion and gravitational settling of helium, using two different versions of the low-temperature opacities given in Table 2, and varying the value of mixing length parameter (in the range of 1.6–2.0). For each of these cases, we computed additional small grids around the base model by varying all the input parameters, in order to see how much the output properties were different for two models with different input physics but similar frequencies. We then calculated the weighted mean and standard deviation in the same way as in the original analysis. The standard deviation in this case represents the typical uncertainties caused by using a fixed set of input physics, hence decreasing the level of model dependence in the results substantially. The mean values for age, luminosity, radius, T_eff, and log g calculated from the additional analysis agreed with the original results within 1σ, while their standard deviations were of the same order as those given in Table 4. This confirms that the uncertainties presented here are realistic. Moreover, the resulting values of radius and log g from the additional analysis are essentially the same as the original results. This is reassuring given the importance of asteroseismology in determining the radius in a robust way.

The internal uncertainties were different for each method. However, the dominant source of uncertainty is the non-uniqueness of the solution rather than the statistical errors. Parameter correlations allow a tradeoff between parameters, leading to different families of solutions that are almost equally good seismic fits. The effective range of these correlations is narrowed substantially, but not eliminated, by the use of seismic data. We used a single method (AMP) to evaluate the uniqueness of the best models for both stars, since the uniqueness depends on the specific constraints adopted in each case. AMP finds the lowest value of χ² in the entire search space. Along the way it also identifies the secondary minima—which can be far away from, but not much worse than, the formally best solution. For both stars, these secondary minima (SA2 and BA2 in Tables 4 and 5) are marginally worse than the best solutions found using the same set of inputs (SA1 and BA1), but the values of the mass and helium abundance are quite different. This reflects the well-known mass–He degeneracy (see, e.g., Lebreton et al. 1993; Fernandes & Monteiro 2003; Metcalfe et al. 2009), which is not entirely lifted, even with the help of asteroseismology. The mass is constrained more strongly than it would be without asteroseismic data but in the absence of external constraints on the helium abundance, we cannot choose one particular model. Consequently, we used both the primary- and secondary-minimum solutions from AMP in the calculation of the weighted means. The range of the AMP results leads to relatively large "uniqueness uncertainties" in the fitted stellar properties. These are determined as follows for KIC 10920273 and KIC 11395018, respectively: 3% and 7% in stellar mass, 8% and 21% in the initial metallicity (Z/X)_i, 10% and 25% in Y_i, 7% and 8% in age, and 1% and 2% in radius. Note that the larger uncertainties in the mass and chemical composition for KIC 11395018 reflect the fact that it has two observed avoided crossings, which provide more stringent constraints such that neighboring models are significantly worse. Only a relatively large jump along the parameter correlations yielded a secondary minimum with a comparable seismic fit (cf. Metcalfe et al. 2010). When we evaluate the uncorrelated statistical uncertainties for each of these minima with a local analysis using singular value decomposition (SVD), we found very small errors (e.g., as low as 5 × 10⁻³% for mass), reflecting the steep χ² surfaces corresponding to these results. Although these "local" uncertainties are real, i.e., changing the values of the parameters by the correlated "tiny" uncertainties causes a large increase in the χ² (>50), the minima are not unique. Hence, a local analysis does not reflect the true uncertainties of the overall results.

Inclusion of the maximal list of frequencies as constraints in the analysis (five additional frequencies for KIC 10920273 and two for KIC 11395018) made the agreement between the model and observations better for both KIC 10920273 (SA3 in Table 4) and KIC 11395018 (see BA3 in Table 5). We also used a third list for KIC 10920273 to ensure that the less-certain frequencies did not bias our model fitting. For that purpose, we left out two frequencies in the analysis. One of these was identified as a dipole-mode frequency belonging to the minimal list, but tagged as being close to the second harmonic of the long-cadence frequency (with ν = 1135.36 ± 0.31 μHz), while the other one was identified as a quadrupole mode (with ν = 873.10 ± 0.32 μHz) that was tagged as a possible mixed mode introduced a posteriori (Campante et al. 2011). Both SA4 and SB2 were selected using this alternative frequency set for KIC 10920273, and the agreement between the model and observations was not affected substantially. Therefore, we cannot ascertain whether these two peaks are stellar in origin. Nevertheless, we do not completely rule out the possibility of these peaks being stellar as some of the models that result from using alternative frequency sets do contribute to the weighted mean values significantly. Furthermore, we note that the peak at ν = 873.10 μHz, which is tagged as a possible quadrupole mixed mode, is in the middle of the frequencies of a dipole mixed mode and a quadrupole mode in most of our models. So the observed peak may correspond to a dipole mixed mode, which, according to the models, has a relatively low normalized mode inertia indicating an observable amplitude.

Comparing our results with those from the pipeline analyses of Creevey et al. (also given here in Tables 4 and 5), we see that the mass determinations from the pipeline analyses were higher, which led to lower age estimates. We emphasize that the previous pipeline analyses used only the average seismic quantities, hence lacking additional information from the individual frequencies and being affected by the uncertainties of the scaling relations. Therefore, some deviation from their values was expected. Nonetheless, it is reassuring that the mass, radius, and age determinations of Creevey et al. (2012) are within 2σ uncertainty limits of our results for KIC 10920273, and within 1σ for KIC 11395018. We also confirm the robust determination of log g using scaling relations and grid-based analyses (see Table 7 in Creevey et al. 2012), with which our results are in agreement within 1σ. Additionally, we note that our results confirm that the mass and radius determined using only the scaling relations (e.g., Mathur et al. 2012) provide good initial estimates for these properties (1.06 ± 0.20 M_☉ and 1.80 ± 0.11 R_☉ for KIC 10920273; 1.31 ± 0.25 M_☉ and 2.21 ± 0.14 R_☉ for KIC 11395018).

There is excellent agreement, for both stars, between our results and the mass estimates of Benomar et al. (2012), who used the coupling strength of the observed mixed modes to determine the masses of several subgiants, including KIC 10920273 (1.04 ± 0.04(± 0.04) M_☉) and KIC 11395018 (1.21 ± 0.06(± 0.04) M_☉).

The most substantial improvement in this work comes from the use of individual frequencies which yield increased precision, with age being affected the most. The presence of the mixed modes in the data allowed us to determine the age with 5%–7% precision, although with some model dependency. This result is a major improvement on the 35%–40% precision in age achieved using atmospheric and mean seismic parameters. Both stars are determined to be post-MS subgiants with no hydrogen left in their cores. Evolutionary tracks of the selected models are shown in Figure 1.

Although we did not restrict the parameter search to be within 1σ uncertainty around the spectroscopic constraints, the weighted mean values of T_eff from the models are within 1σ limit for both stars, while log g results are in agreement with the spectroscopic values within 2σ (see Figure 1), and [Fe/H] within 1.5σ. Our log g results are in excellent agreement with asteroseismic log g values obtained from scaling relations (given by Creevey et al. 2012 and also in Tables 4 and 5). We also note that our temperature results are in good agreement with the revised photometric values for the Kepler Input Catalog (KIC) from Pinsonneault et al. (2012), who derived T_eff = 5872 ± 70 K for KIC 10920273, and T_eff = 5650 ± 59 K for KIC 11395018.

We discussed the asteroseismic constraints on the stellar age in Section 1. Here we discuss the implications of rotation and stellar activity on the age, as well as those of the surface lithium abundance.

Rotation and activity are potentially valuable diagnostics of stellar age. It was shown that the Ca⁺ emission luminosity, an indicator of stellar activity, decays roughly as t^−0.5 for some cluster stars and the Sun (Skumanich 1972); furthermore, rotational decay was shown to follow the same law. Large samples of stellar rotation periods have been collected, and the Kepler mission promises many more. There is therefore substantial interest in stellar rotation–mass–age, or gyrochronology, relations (see Barnes 2003, 2007; Mamajek & Hillenbrand 2008; Meibom et al. 2011; Epstein & Pinsonneault 2012). We discussed in Section 2.1 that relatively slow rotation is inferred for both stars. Slow rotation rates imply relatively old stars, which is consistent with our asteroseismic determinations. However, one would not expect MS spin-down relationships to apply directly to the evolved stars. Therefore, we cannot use the rotation rates for these stars to infer their ages with the age–rotation relations established for MS stars; these relations need to be calibrated for more evolved stars using larger samples.

We have analyzed the chromospheric activity in the Ca ii HK lines and found the levels of activity in both stars to be very low. Figure 3 shows the Ca ii K and H lines of KIC 10920273 and KIC 11395018 compared to the Sun. The solar spectrum was obtained from the solar light reflected by Ganymede, which was observed with HARPS in 2007 April,²⁵ when the Sun was close to the minimum of its activity cycle. We accounted for the different resolving power of HARPS (R ≃ 120, 000) compared to FIES spectrograph (R ≃ 46, 000) by convolving the solar spectrum with a Gaussian kernel of the appropriate width. It is clear that these two Kepler stars have chromospheric activity levels comparable to, or lower than, the quiet Sun. These low activity levels are consistent with the old ages we infer from asteroseismology; however, the rough nature of the empirical age–activity relations for post-MS stars does not allow us to make a quantitative analysis to infer ages.

Zoom In Zoom Out Reset image size

Another independent determination of stellar age may be obtained by measuring the Li content at the stellar surface. Lithium is easily destroyed in stellar interiors and is only produced under unusual circumstances; it has therefore been employed as an age indicator for low-mass stars. Lithium can be directly depleted if the surface convection zone is deep enough. It can also be mixed into the radiative interior, or it can be stored below the surface convection zone by microscopic diffusion processes. In standard stellar models, pre-MS depletion occurs for most low-mass stars when they have deep convection zones, and it is most severe in lower mass stars (Bodenheimer 1965). In a qualitative sense, a detection of Li in very cool stars is a strong indicator of youth. However, standard models also predict that stars of the order of 0.9 solar masses and higher would not experience MS Li depletion, and there is strong evidence from open clusters for a steady decrease in Li as a function of time, even for stars more massive than the Sun (see Zappala 1972; Pinsonneault 1997; and Sestito & Randich 2005 for reviews.)

It was also shown by Randich (2010) that for a fraction of solar-like stars with effective temperatures between 5750 K and 6050 K, Li is not further depleted after the age of ∼1 Gyr, unlike the Sun and many other stars. Due to this bimodal pattern, Li abundance alone cannot be used to determine the age for all stars, and a high Li abundance can only help define a lower limit for the age, since it may correspond either to a young star that has not yet depleted much Li or to an older star that has stopped depleting Li a long time ago.

Creevey et al. (2012) showed that the two stars considered here have strong Li absorption lines, which implies a high Li content at the surface (log N(Li) =2.4 ± 0.1 for KIC 10920273 and log N(Li) =2.6  ±  0.1 for KIC 11395018; where log N(Li) = log  [n(Li)/n(H)] + 12, with n being the number density of atoms and log N(H) = 12 by definition). Considering the empirical Li–age relation established by Sestito & Randich (2005), Creevey et al. then determined that the given Li abundances would indicate low ages (1–3 Gyr for KIC 10920273 and 0.1–0.4 Gyr for KIC 11395018), which are incompatible with the asteroseismic ages they determined through the pipeline modeling (see Tables 4 and 5) performed using the average asteroseismic quantities.

However, in addition to the bi-modality mentioned above, the age–Li relation has been shown to be valid for MS stars and does not necessarily extend to more evolved stars. This makes the age determination using the Li abundance ambiguous. Thus, despite the high Li abundance, we are confident that these are indeed evolved stars that have left the MS, due to the presence of the mixed modes in the observed oscillation spectra and as confirmed by our asteroseismic analysis.