Asteroseismic analysis of solar-mass subgiants KIC 6442183 and KIC 11137075 observed by Kepler
Key Words.:
Stars: oscillations – Stars: fundamental parameters – Stars: individual(KIC 6442183, KIC 11137075)Abstract
Context:Asteroseismology provides a powerful way to constrain stellar parameters. Solar-like oscillations have been observed on subgiant stars with the Kepler mission. The continuous and high-precision time series enables us to carry out a detailed asteroseismic study for these stars.
Aims:We carry out data processing of two subgiants of spectral type G: KIC 6442183 and KIC 11137075 observed with the Kepler mission, and perform seismic analysis for the two evolved stars.
Methods:We estimate the values of global asteroseismic parameters: Hz and Hz for KIC 6442183, Hz and Hz for KIC 11137075, respectively. In addition, we extract the individual mode frequencies of the two stars. We compare stellar models and observations, including mode frequencies and mode inertias. The mode inertias of mixed modes, which are sensitive to the stellar interior, are used to constrain stellar models. We define a quantity that measures the difference between the mixed modes and the expected pure pressure modes, which is related to the inertia ratio of mixed modes to radial modes.
Results:Asteroseismic together with spectroscopic constraints provide the estimations of the stellar parameters: , and Gyr for KIC 6442183, and , and Gyr for KIC 11137075. Either mode inertias or could be used to constrain stellar models.
Conclusions:
1 Introduction
KIC 6442183 | KIC 11137075 | |
(K) | ^{}^{} Bruntt et al. (2012) ^{}^{} Molenda-Żakowicz et al. (2013) | ^{}^{} Bruntt et al. (2012) ^{}^{} Molenda-Żakowicz et al. (2013) |
(dex) | ^{}^{} Bruntt et al. (2012) ^{}^{} Molenda-Żakowicz et al. (2013) | ^{}^{} Bruntt et al. (2012) ^{}^{} Molenda-Żakowicz et al. (2013) |
^{}^{} Bruntt et al. (2012) ^{}^{} Molenda-Żakowicz et al. (2013) | ^{}^{} Bruntt et al. (2012) ^{}^{} Molenda-Żakowicz et al. (2013) | |
KIC magnitude | 8.52^{}^{}Kepler Asteroseismic Science Operations Center: http://kasoc.phys.au.dk/. | 10.86^{}^{}Kepler Asteroseismic Science Operations Center: http://kasoc.phys.au.dk/. |
(Hz) | ^{}^{} Benomar et al. (2013), who stated that typical uncertainties on mass and radius were a few percent ^{}^{}This work | ^{}^{}This work |
(Hz) | ^{}^{} Benomar et al. (2013), who stated that typical uncertainties on mass and radius were a few percent ^{}^{}This work | ^{}^{}This work |
() | 0.94^{}^{} Benomar et al. (2013), who stated that typical uncertainties on mass and radius were a few percent ^{}^{}This work | ^{}^{}This work |
() | 1.60^{}^{} Benomar et al. (2013), who stated that typical uncertainties on mass and radius were a few percent ^{}^{}This work | ^{}^{}This work |
Asteroseismology provides a useful tool to probe stellar interiors, test internal physical processes and obtain stellar parameters accurately (e.g. Eggenberger et al. 2004; Bi et al. 2008; Kallinger et al. 2010; Montalbán et al. 2013). Many solar-like stars have been observed continuously and precisely with space missions such as CoRoT and Kepler (Appourchaux et al. 2008; Borucki et al. 2007), ushering in a golden age for the asteroseismic analysis of oscillating stars.
The oscillations of evolved stars include mixed modes, which behave as pressure modes (p-modes) in the envelope and gravity modes (g-modes) in the core (Osaki 1975; Aizenman et al. 1977). Mixed modes have been used to distinguish between red clump stars and RGB stars (Bedding et al. 2011; Mosser et al. 2011a), and to monitor stellar evolution status from the main sequence to the asymptotic giant branch (Mosser et al. 2014). The signature of avoided crossings raises the exciting possibility that detailed modeling of the star will provide a very precise determination of its age (Gilliland et al. 2010a; Chaplin et al. 2010; Metcalfe et al. 2010; Benomar et al. 2012; Bedding 2014).
As discussed by Dziembowski et al. (2001); Christensen-Dalsgaard (2004); Dupret et al. (2009); Benomar et al. (2014); Datta et al. (2015), the g-dominated non-radial modes have much higher inertia than the radial modes, whereas the p-dominated modes do not. Because the mixed modes possess the properties of partial g-modes, the inertia of mixed modes is usually higher than that of p-modes. The inertia of mixed modes could provide powerful constraints on stellar models.
Subgiants are the bridge between the main sequence (MS) and red giants on the H-R diagram. The study of subgiants could provide valuable insight into stellar structure and evolution. When stars leave the main sequence stage, the g-mode and p-mode frequencies overlap, which results in the appearances of mixed modes in post-main-sequence stars. KIC 6442183 (aka. ‘Dougal’) and KIC 11137075 (aka. ‘Zebedee’) are at the beginning of the subgiant stage. The magnitudes of the stars are 8.52 and 10.86 (KIC magnitude), which are bright enough to allow seismic observations. They have been observed by the Kepler mission continuously with a short cadence of 58.84 seconds (Gilliland et al. 2010b) for quarters 6.1–17.2 and 7.1–11.3, respectively. Benomar et al. (2013, 2014) extracted the mode frequencies and measured the mode inertia ratio for KIC 6442183. They estimated the mass and the radius for KIC 6442183 using scaling relations (Brown et al. 1991; Kjeldsen & Bedding 1995). With oscillation observations from Kepler and spectroscopic observations from Bruntt et al. (2012) and Molenda-Żakowicz et al. (2013), we can now conduct asteroseismic analyses and constrain stellar parameters for the two stars accurately.
In Section 2, we review the recent observations and extract the mode frequencies for the two stars. In Section 3, we construct stellar models and correct the near-surface term for the theoretical frequencies of the models. Seismic analyses and stellar parameters determinations are shown in Section 4. Finally, the discussions and conclusions are presented in Section 5.
2 Observations and Data Processing
Spectroscopic observations from Bruntt et al. (2012) and Molenda-Żakowicz et al. (2013) provided the values of metallicity (), effective temperature () and gravity () for these two stars. Results from the two groups show good consistency, and yield preliminary constraints on the stars. In this paper we adopt the parameters obtained by Bruntt et al. (2012) because they took the asteroseismic into account during the spectroscopic analysis.
We obtained the power spectra by applying the Lomb-Scargle Periodogram (Lomb 1976; Scargle 1982) to the short cadence time series corrected by KASC Working Group 1 (WG#1; ‘solar-like oscillating stars’) following García et al. (2011), which is available to the Kepler Asteroseismic Science Consortium (KASC; Kjeldsen et al. 2010) through the KASOC database ^{2}^{2}2Kepler Asteroseismic Science Operations Center: http://kasoc.phys.au.dk/..
The autocorrelation of the power spectrum can provide the estimate of periodic information, such as mean large frequency separation (Barban et al. 2009). In this work, we estimated the large frequency separations and the frequency of maximum power with the autocorrelation function (e.g. Roxburgh & Vorontsov 2006; Roxburgh 2009; Mosser & Appourchaux 2009) and collapsed autocorrelation function (e.g. Huber et al. 2009; Tian et al. 2014). These global oscillation parameters were evaluated as: Hz and Hz for KIC 6442183, Hz and Hz for KIC 11137075. The global oscillation parameters ( and ) and atmospheric constraints for the two stars are listed in Table1.
As discussed by Mathur et al. (2011), several methods have been developed to extract individual frequencies, and the key point is to fit a sum of Lorentzian profiles describing all oscillation modes. We followed the method described by Basu et al. (2000) and Régulo & Roca Cortés (2007), where each oscillation mode in the power spectrum was fitted to a Lorentzian profile using robust non-linear least squares method (Markwardt 2009, 2012). As discussed by Appourchaux et al. (1998a, b), the noise statistic of each bin of the power spectrum has a distribution with 2 degrees of freedom. However the non-linear least squares method assumes that this statistics is Gaussian. The mode extraction method adopted in this work might introduce a certain bias of amplitudes and linewidths, and fortunately does not affect the frequencies. To estimate the errors, we measured the individual frequencies quarter by quarter, and estimated the frequency and error of each mode by combining each subset. Asymptotic formulae show that the oscillation mode linewidthds are related to of stars with a power law (Chaplin et al. 2009; Appourchaux et al. 2012). The smallest linewidth of the detected modes for KIC 6442183 is of approximately 0.25 Hz (Benomar et al. 2013). Because the difference of between the two subgiants is about 150 K, they will obtain the linewidths of the same magnitude. Therefore, the frequency resolution of a Kepler quarter (0.1286 Hz) is sufficient to resolve the oscillation modes for the two stars.
This procedure provided us with independently obtained frequency sets with symmetric errors. We obtained 37 oscillation modes of degree for KIC 6442183 and 26 oscillation modes of degree for KIC 11137075. The individual frequencies for the two stars are listed in Tables 2 and 3. The oscillation frequencies for KIC 6442183 in this work and those given by Benomar et al. (2013) are consistent at the level. The mode frequencies of the power spectra are over-plotted on the échelle diagram in Figs. 3 and 4. The radial modes follow a vertical ridge in the figures, and the non-radial modes show avoided crossings, especially for the modes. In addition to the modes, there are five modes identified for KIC 6442183, which will provide extra constraints on theoretical models.
l | Frequencies (Hz) | error (Hz) |
---|---|---|
0 | 809.63 | 0.11 |
0 | 874.08 | 0.25 |
0 | 937.21 | 0.10 |
0 | 1001.04 | 0.12 |
0 | 1065.65 | 0.14 |
0 | 1130.50 | 0.09 |
0 | 1195.40 | 0.03 |
0 | 1260.32 | 0.12 |
0 | 1325.73 | 0.19 |
0 | 1391.69 | 0.55 |
1 | 783.35 | 0.31 |
1 | 840.62 | 0.11 |
1 | 901.27 | 0.14 |
1 | 960.26 | 0.09 |
1 | 1002.91 | 0.21 |
1 | 1037.60 | 0.09 |
1 | 1096.83 | 0.25 |
1 | 1159.96 | 0.06 |
1 | 1224.09 | 0.13 |
1 | 1288.75 | 0.22 |
1 | 1354.10 | 0.26 |
1 | 1419.52 | 0.37 |
2 | 801.63 | 0.35 |
2 | 868.11 | 0.17 |
2 | 931.16 | 0.30 |
2 | 996.92 | 0.15 |
2 | 1059.95 | 0.17 |
2 | 1124.73 | 0.23 |
2 | 1189.88 | 0.15 |
2 | 1254.61 | 0.14 |
2 | 1321.26 | 0.31 |
2 | 1386.81 | 0.41 |
3 | 1150.13 | 0.25 |
3 | 1215.93 | 0.31 |
3 | 1280.49 | 0.31 |
3 | 1347.26 | 0.37 |
3 | 1412.36 | 0.59 |
l | Frequencies (Hz) | error (Hz) |
---|---|---|
0 | 949.02 | 0.22 |
0 | 1013.37 | 0.22 |
0 | 1078.56 | 0.09 |
0 | 1144.34 | 0.10 |
0 | 1209.66 | 0.13 |
0 | 1275.11 | 0.08 |
0 | 1341.11 | 0.25 |
0 | 1407.67 | 0.45 |
1 | 916.94 | 0.33 |
1 | 978.01 | 0.11 |
1 | 1040.74 | 0.10 |
1 | 1101.50 | 0.10 |
1 | 1142.65 | 0.12 |
1 | 1180.26 | 0.09 |
1 | 1240.80 | 0.07 |
1 | 1304.94 | 0.19 |
1 | 1370.07 | 0.62 |
1 | 1436.91 | 1.05 |
2 | 943.76 | 0.47 |
2 | 1007.47 | 0.21 |
2 | 1072.59 | 0.21 |
2 | 1132.52 | 0.17 |
2 | 1141.38 | 0.23 |
2 | 1204.19 | 0.07 |
2 | 1269.64 | 0.20 |
2 | 1335.61 | 0.73 |
3 Stellar Models
To estimate the parameters of the two stars, a grid of stellar evolutionary models was constructed with the Yale stellar evolution code (YREC7; Demarque et al. 2008). We used the OPAL opacity table GN93 (Iglesias & Rogers 1996), the low-temperature table AGS05 (Ferguson et al. 2005), the OPAL equation-of-state tables EOS2005 (Rogers & Nayfonov 2002) and the Bahcall nuclear rates (Bahcall et al. 1995) for microphysics. We chose the Eddington grey atmosphere relation. We adopted the standard mixing-length theory (Böhm-Vitense 1958) and overshooting to treat convection. The coefficient of helium and heavy elements diffusion is from Thoul et al. (1994). We did not take rotation or magnetic field into consideration in our calculations.
The ratio of heavy-elements to hydrogen as a mass fraction was estimated through the formula:
(1) |
where we adopt the value of (Iglesias & Rogers 1996) and the values of [Fe/H] from Bruntt et al. (2012) for the two stars. The values of the ratio of the heavy element abundance to hydrogen abundance are in the range of 0.0165–0.0218 and 0.0186–0.0245 for KIC 6442183 and KIC 11137075, respectively. In the model calculation, we chose the initial helium abundance as (e.g., Dotter et al. 2008; Thompson et al. 2014), in terms of the initial metal abundance. For a given mass, stellar evolutionary models depend on three free parameters: initial chemical compositions, the mixing-length parameter and the overshooting parameter . The input parameters for model calculations are listed in Table4.
Variable | Range | |
---|---|---|
Mass | 0.90 – 1.08 | 0.01 |
0.012 – 0.018 | 0.001 | |
1.70 – 1.90 | 0.2 | |
0.0 – 0.2 | 0.2 |
We calculated three groups of tracks for these two stars: (i) models without diffusion and without overshooting, (ii) models with diffusion and without overshooting and (iii) models with diffusion and overshooting (). Because the observed luminosities of the two stars were not available, we used the large frequency separation to constrain stellar models in the H-R diagram. Considering the difference between of the models from the scaling relation and pulsation code, we assigned the large separation an error of approximately 2 Hz in the H-R diagram. There are 256 and 265 tracks falling into their error boxes for KIC 6442183 and KIC 11137075, respectively, as shown in Fig. 5.
4 Asteroseismic Diagnostics
4.1 Model calibration
For stellar models along the tracks falling inside the error box, we computed smaller and more finely sampled grids around these models, and calculated theoretical mode frequencies with Guenther’s pulsation code (Guenther 1994). It is well known that there is a systematic offset between observed and computed frequencies that arises from improper modeling of the near-surface layers for both the Sun and solar-type stars (Kjeldsen et al. 2008, and references therein). To correct the near-surface term, we followed the method described by Brandão et al. (2011), who corrected the near-surface term for radial modes with the method proposed by Kjeldsen et al. (2008) by fitting a power law:
(2) |
For the mixed modes, the corrected frequencies could be calculated through:
(3) |
where and are the order and degree of the modes, respectively, is the exponent calibrated from the Solar models, and is a constant for the stellar model. The quantity presents the ratio between the inertia of the non-radial mode and the inertia of a radial mode of the same frequency (Aerts et al. 2010). The inertia of mode is defined as:
(4) |
where and are the radial and horizontal displacements of the modes at the radius , respectively. The inertia ratio between non-radial and radial modes can be expressed as
(5) |
As discussed by Aerts et al. (2010), the typical values for are 0.01–0.1 for low-order p-modes, and 10–100 for high-order g-modes. This indicates that for pure acoustic modes and for p-g mixed modes.
To restrict the stellar parameters, we performed a minimization by a comparison of models with observations:
(6) |
where and the denote the observational errors. We chose models with as candidates for further pulsation analysis. In addition to the constraints from the atmospheric parameters and , we performed another minimization by a comparison of the near-surface-corrected model frequencies with the observed frequencies:
(7) |
where the superscripts ‘obs’ and ‘theo’ correspond to individual frequencies from observations and modeling, respectively, and denotes the observational errors. We list the candidate models with for KIC 6442183 in Table 5. Since the modes of KIC 11137075 were not available, we chose the candidate models with for this star, as listed in Table 5. Note that the Brunt-Väisälä frequency affects the distributions of oscillation modes undergoing avoided crossings. Models with the similar global properties can have different mixed mode frequencies due to the diverse positions of avoided crossings. This results in the large for some models listed in Table 5.
We note that the mass of these stars is not high enough to produce a convective core in the stellar interior, and the core overshooting will not have any impact on their structure and evolution. Therefore, we do not discuss the effects of overshooting in the following part for these low mass subgiants.
4.2 Optimal models
Model | Mass | t | ||||||||||||
() | (Gyr) | (K) | ) | () | (dex) | (Hz) | ||||||||
KIC 6442183 | ||||||||||||||
1 | 0.98 | 1.70 | No | 0.012 | 10.06 | 0.017 | 5703 | 2.51 | 1.63 | 4.01 | 64.86 | 0.44 | 29.85 | 2514.27 |
2 | 0.99 | 1.70 | No | 0.013 | 9.95 | 0.018 | 5685 | 2.50 | 1.63 | 4.01 | 64.85 | 0.27 | 28.32 | 599.80 |
3 | 1.00 | 1.70 | No | 0.014 | 9.80 | 0.019 | 5672 | 2.49 | 1.64 | 4.01 | 64.85 | 0.34 | 27.24 | 77.20 |
4 | 1.04 | 1.90 | No | 0.015 | 8.62 | 0.021 | 5807 | 2.82 | 1.66 | 4.01 | 64.85 | 0.49 | 26.46 | 70.98 |
5 | 1.03 | 1.90 | Yes | 0.016 | 8.85 | 0.019 | 5709 | 2.62 | 1.66 | 4.01 | 64.84 | 0.09 | 22.40 | 1429.14 |
6 | 1.04 | 1.90 | Yes | 0.015 | 8.30 | 0.017 | 5783 | 2.77 | 1.66 | 4.01 | 64.84 | 0.31 | 27.61 | 2208.02 |
7 | 1.04 | 1.90 | Yes | 0.016 | 8.50 | 0.019 | 5743 | 2.69 | 1.66 | 4.01 | 64.84 | 0.03 | 23.46 | 532.24 |
8 | 1.04 | 1.90 | Yes | 0.017 | 8.68 | 0.020 | 5707 | 2.63 | 1.66 | 4.01 | 64.84 | 0.15 | 22.04 | 46.23 |
9 | 1.04 | 1.90 | Yes | 0.018 | 8.85 | 0.021 | 5673 | 2.57 | 1.66 | 4.01 | 64.84 | 0.57 | 20.54 | 571.86 |
10 | 1.05 | 1.90 | Yes | 0.018 | 8.50 | 0.021 | 5707 | 2.65 | 1.67 | 4.02 | 64.85 | 0.32 | 23.59 | 1480.05 |
KIC 11137075 | ||||||||||||||
1 | 0.96 | 1.70 | No | 0.014 | 11.60 | 0.019 | 5549 | 2.19 | 1.60 | 4.01 | 65.50 | 0.26 | 81.36 | 1292.34 |
2 | 1.00 | 1.90 | No | 0.015 | 10.14 | 0.021 | 5686 | 2.49 | 1.63 | 4.02 | 65.51 | 0.64 | 64.19 | 470.62 |
3 | 1.00 | 1.90 | No | 0.016 | 10.39 | 0.022 | 5645 | 2.42 | 1.63 | 4.01 | 65.52 | 0.26 | 43.86 | 39.74 |
4 | 1.01 | 1.90 | No | 0.017 | 10.17 | 0.024 | 5645 | 2.43 | 1.63 | 4.02 | 65.53 | 0.44 | 19.19 | 839.92 |
5 | 0.99 | 1.90 | Yes | 0.016 | 10.37 | 0.019 | 5587 | 2.30 | 1.62 | 4.01 | 65.53 | 0.25 | 19.16 | 111.69 |
6 | 1.00 | 1.90 | Yes | 0.016 | 9.96 | 0.019 | 5622 | 2.38 | 1.63 | 4.01 | 65.48 | 0.32 | 135.53 | 1712.46 |
7 | 1.00 | 1.90 | Yes | 0.017 | 10.16 | 0.020 | 5586 | 2.32 | 1.63 | 4.01 | 65.50 | 0.05 | 65.69 | 431.52 |
8 | 1.00 | 1.90 | Yes | 0.018 | 10.36 | 0.022 | 5553 | 2.26 | 1.63 | 4.01 | 65.51 | 0.10 | 44.50 | 32.11 |
The frequency separation between consecutive modes varies rapidly during an avoided crossing and so the mixed modes together with pure p-modes provide an important constraint to the stellar evolutionary state (Christensen-Dalsgaard et al. 1995; Pamyatnykh et al. 2004). We compared the frequencies and inertia ratios from observations and theoretical models listed in Table 5. The theoretical inertias for models were calculated using equation 4, and the observed inertia ratio through the following function (e.g. Benomar et al. 2014):
(8) |
where and denote the amplitude and linewidth of modes, respectively, while the visibility is the square root of the ratio between theoretical heights for the dipole and radial modes. We adopted the observed amplitude, linewidth and the value of (=1.52) from Benomar et al. (2013) to deduce the observed inertia ratios.
In solar-like stars, p-mode oscillations are expected to follow the approximate relation (Tassoul 1980):
(9) |
where is related to the interior structure of the star and the offset is sensitive to the surface layers. The frequencies of the p-modes with the same degree show vertical ridges in the échelle diagram. The mean distance between dipole modes and radial modes with the same order would be , which is deduced from the approximate relation formulated in equation 9. For red giants, Mosser et al. (2011b) introduced the expression for the pure p-mode eigenfrequency pattern:
(10) |
where is the p-mode radial order, is the order of the mode with highest power and is a constant representing the mean curvature of the p-mode oscillation pattern. Although the dipole modes are p-g mixed modes in subgiants, we could estimate the expected pure acoustic modes through equations 9 or 10.
We define a quantity to measure the frequency difference between the mixed mode frequency and the pure acoustic modes . The quantity for dipole modes could be calculated using the following formula:
(11) |
The is the frequency difference between the mixed modes and the nearest p-mode. For g-dominated mixed modes, the bigger is the , the stronger is the coupling between p- and g-modes. We used inertia ratios and as criteria to constrain the stellar models of KIC 6442183 and KIC 11137075.
Through the comparisons of theoretical and observed inertia ratios and for KIC 6442183, we found that Models 3, 4 without diffusion and Model 8 with diffusion match the observations better than the other models. We estimated the following parameters for KIC 6442183: , and Gyr. By comparing Models 4 and 6, we found that the and luminosity estimated from the model without diffusion are higher than that estimated from diffusion models, which suggests that helium and heavy metal diffusions are not negligible. This is because the diffusion changes the opacity and equation of state for the models. The mass we find is approximately 11% higher than that deduced by Benomar et al. (2014) with the scaling relation.
In Fig. 8, we show the results for KIC 11137075. As shown in the figure, the maximum of theoretical dipole is lower than the observations. For the models presented in the third column of Fig. 8, Models 3, 5 without diffusion and Model 8 with diffusion match the observations much better than the other four models, which means that the three models reproduce the avoided crossing better than the other four models. Thus, Models 3, 5 and 8 were selected as the best models for KIC 11137075. Finally, we obtained the following parameters for KIC 11137075: , and Gyr. The models with the mixing length reproduce the observations better than the models with .
5 Discussions and Conclusions
We carried out data processing and performed seismic analysis for subgiants KIC 6442183 and KIC 11137075, which were observed by the Kepler Mission. The main results and discussions are summarized as follows.
We applied the Lomb-Scargle Periodogram to the corrected short cadence time series observed by Kepler to obtain the power spectra, and estimated the large frequency separation and frequency of maximum power for the two stars: Hz and Hz for KIC 6442183, Hz and Hz for KIC 11137075, respectively. Individual mode frequencies were also extracted in this work. After carrying out asteroseismic analysis, we estimated the stellar parameters: , and Gyr for KIC 6442183, , and Gyr for KIC 11137075. Both the subgiants are shown to be solar-mass stars.
For stars whose mass is similar to the Sun’s, the helium and heavy elements diffusion will affect the opacity and equation of state, which will change the thermal structure and affect the evolution of stars. Therefore, we have to consider the diffusion process in stellar models and seismic analysis.
Pure p modes frequencies decrease and g modes frequencies increase with age for subgiants (Christensen-Dalsgaard et al. 1995), which results in the change of the frequency at which the avoided crossing occurs (Bedding 2014). The frequencies of mixed modes sensitive to stellar interiors are useful to constrain stellar structures, especially for stellar interiors. The maximum of is coincident with the large inertia ratio, allowing us to locate the mixed modes with the most g-mode property through these quantities. We first constrain stellar models with inertia ratios and , which provides an impactful way to determine the parameters of evolved stars accurately.
In future work, we plan to select some evolved stars with mixed modes such as subgiants, red giants and red clumps which constitute an evolution sequence. Then we compare the differences of inertia ratios and between these stars and use these quantities to constrain the stellar interiors with the quantitis.
Acknowledgements.
We are grateful to the entire Kepler team. This work was supported by grants 10933002, 11273007 and 11273012 from the National Natural Science Foundation of China, and the Fundamental Research Funds for the Central UniversitiesReferences
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