Determining the Age for the Red Giants KIC 9145955 and KIC 9970396 by Gravity-dominated Mixed Modes

Determining the ages and helium core sizes of red giants is a challenging problem. To estimate the age and helium core size precisely requires a good understanding of the internal structure of the red giant. The properties of the g-dominated mixed modes of red giants are closely related to their inner radiative cores, especially the central helium core. Thus, the g-dominated mixed modes are useful indicators for probing the properties of the helium core and constraining the age of red giants. In our previous work, we have estimated the helium core sizes of the red giants KIC 9145955 and KIC 9970396 by asteroseismic models. In this work, we take a further step to calibrate the ages and core overshooting parameters for these two red giants. We find that the ages of these two stars are 4.61 ± 0.23 and 6.13 ± 0.19 Gyr, respectively. From a comparative study, we find that, for a single red giant, the age estimated by the asteroseismology of g-dominated mixed modes is likely to be more precise than that estimated by the combination of the asteroseismic (Δν and ΔP obs) and spectroscopic (T eff and [Fe/H]) observations. In addition, we estimate the core overshooting parameters of these two stars. We find that the overshooting parameter f ov of KIC 9145955 and KIC 9970396 was probably overestimated in previous works.


Introduction
Red giants are luminous evolved stars of low or intermediate masses. The Milky Way has a huge number of red giants. The ages of red giants are important for studying the evolution of stars and understanding the Galactic archeology (e.g., Stello et al. 2015). However, determination of stellar ages remains a challenging problem for traditional methods (Soderblom 2010). For example, isochrone fitting, a widely used method for age determination, works well for stars in open clusters whose ages are almost the same, but it does not work very well for a single star. The age estimation of an individual star highly depends on the stellar model (Soderblom 2010). The accuracy of the stellar model is crucial to the age estimation of the star. Asteroseismology, on the other hand, allows us to accurately determine the interior structure of a single star through the analysis of its global oscillation modes. The fundamental parameters derived by asteroseismic modeling have a relatively high precision (Jiang et al. 2011;Silva Aguirre et al. 2013;Lebreton & Goupil 2014;Zhang et al. 2018Zhang et al. , 2020Buldgen et al. 2019) that is hard to achieve by other methods (Aerts 2021). Thus, it has great potential for improving our understanding of the structure and evolution of stars.
The oscillation frequencies of pulsating stars are very sensitive to stellar interiors; thus, they can be used to detect the internal structures of stars. Different oscillation modes propagate in different layers of the stellar interior. The pure pmode oscillations, whose restoring force is pressure, are standing acoustic waves and mainly trapped in the outer layers of the stars. On the other hand, the pure g-mode oscillations, whose restoring force is buoyancy, are standing gravity waves and mainly trapped in the internal radiative region of the stars. The oscillations in low-mass main-sequence, subgiant, and red giant stars are stochastically excited by turbulent motions in the near-surface convection zone, where the restoring force is pressure (Aerts et al. 2010). As such oscillations are present in the Sun, oscillations excited in this manner are referred to as solar-like oscillations (Hekker & Christensen-Dalsgaard 2017). All stars with turbulent outer layers are expected to exhibit solar-like oscillations (Hekker & Christensen-Dalsgaard 2017). In the past decade, an immense number of solar-like stars with high-quality oscillation data were detected by CoRoT (Baglin et al. 2006), Kepler (Gilliland et al. 2010;Koch et al. 2010), and TESS (Ricker et al. 2014), which has led to great progress in understanding their structure and evolution by asteroseismology, for example, distinguishing between the hydrogen shell-burning giants and the helium core-burning giants (Bedding et al. 2011).
The oscillation spectra of red giants are more complex than the main-sequence solar-like stars because of the existence of mixed modes. The mixed modes, which behave as p-mode signatures in the stellar envelope and g-mode signatures in the stellar core, are mostly observed in dipole (l = 1) modes in red giant stages. They can propagate deeper into the star than p-modes and are easier to observe than g-modes. By taking this advantage, the internal structure can be determined more precisely when the mixed mode is included in asteroseismic modeling. However, the spectrum of the mixed mode deviates from the pure g-and p-mode spectra, which causes difficulty in the individual frequency fitting. In our previous work (Zhang et al. 2018(Zhang et al. , 2020, we proposed a "mode identification" method to identify the frequencies of the mixed modes. By this method, the most p-dominated mixed modes are separated from the g-dominated modes. As a result, the frequencies of the mixed modes can be well identified. With the identified mixed modes, stellar parameters are then calibrated from the best fitting of frequencies between the model and observational results. We have applied the mode identification method to two low-mass red giant stars, KIC 9145955 and KIC 9970396, and obtained their precise helium core masses and radii (Zhang et al. 2018(Zhang et al. , 2020. The g-dominated modes are closely related to the stellar core structure (Hekker & Christensen-Dalsgaard 2017); thus, it could possibly provide useful information on core convection and overshooting. For low-mass red giant stars, the ages mainly depend on the time spent in the mainsequence phase (τ MS ; Davies & Miglio 2016). In this paper, we take a further step to estimate the accurate age and convective core overshooting parameters of the two low-mass red giants KIC 9145955 and KIC 9970396 by considering the g-dominated mixed modes.
We present our work in detail in the following sections. We present photometric and spectroscopic data of KIC 9145955 and KIC 9970396 in Section 2. We describe the input physics and model grids in Section 3.1. The model fitting is described in Section 3.2. The determination of the stellar age, the acoustic radius τ 0 , and the size of the helium core of these two red giants is presented in Section 4.1. The selection of the best-fitting model is discussed in Section 4.2. We discuss the estimation of the convection core overshooting parameters of these two red giant stars in Section 5.1. We discuss the stellar age obtained by using the combination of asteroseismic and spectroscopic observations as the observational constraints in Section 5.2. We compare our results with previous work in Section 5.3. Our work is summarized in Section 5.4.

Photometric and Spectroscopic Observations
Object KIC 9145955 was continuously observed by Kepler for 4 yr in two different observing modes: long-cadence mode (29.4 minute exposure times) and short-cadence mode (58.9 s exposure times; Murphy 2012). There are 18 quarters (Q0-Q17) of long-cadence mode and one quarter (Q4.3) of shortcadence mode photometric data available for KIC 9145955. Zhang et al. (2018) extracted the oscillation frequencies of KIC 9145955 from all 18 quarters of Kepler long-cadence photometric data. In this paper, we also adopt 58 individual frequencies, including seven modes of l = 0, 44 modes of l = 1, and seven modes of l = 2 extracted by Zhang et al. (2018; see Table 1 of Zhang et al. 2018). The asteroseismic parameters and spectroscopic observations of KIC 9145955 in previous works are listed in Table 1.
First identified by Gaulme et al. (2013Gaulme et al. ( , 2014, KIC 9970396 is a pulsating red giant in a detached eclipsing binary system. The photometric data of KIC 9970396 were provided by Kepler and available in 15 quarters(Q0-Q6, Q8-Q10, Q12-Q14, and Q16-Q17) of long-cadence modes. Li et al. (2018) first extracted the oscillation frequencies from the long-cadence photometric data of KIC 9970396. In this work, we also use the oscillation frequencies extracted by Li et al. (2018; see Table 3 of Li et al. 2018 or Table 2 of Zhang et al. 2020). The asteroseismic parameters and spectroscopic observations of KIC 9970396 are listed in Table 2.

Input Parameters and Model Grids
The theoretical models are generated by the one-dimensional stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011Paxton et al. , 2013Paxton et al. , 2015Paxton et al. , 2018, version 10398. MESA is open-source and has many physical modules for a wide range of applications in stellar astrophysics. The submodule "pulse_adipls" developed by Christensen-Dalsgaard (2008) is adopted to compute the evolutionary models and their corresponding adiabatic oscillation frequencies in this work. In addition, we do not consider semiconvection, rotation, and magnetic field in our calculations.
For the calculation of equation of state (EOS), we use the OPAL EOS tables published by Rogers & Nayfonov (2002). For the calculation of opacity, we use the opacity tables of Iglesias & Rogers (1996) and Ferguson et al. (2005) for highand low-temperature regions, respectively. The surface temperature and pressure are calculated by the Eddington gray atmospheric model. The solar metal composition GS98 (Grevesse & Sauval 1998) is adopted as the initial composition in metal. Stellar convection is dealt with the mixing-length theory (MLT) of Böhm-Vitense (1958). The convective overshooting in the core is explained with the theory of Herwig (2000). The overshoot mixing diffusion coefficient D ov is defined as where f ov is an adjustable parameter, D conv,0 is the MLTderived diffusion coefficient at a user-defined location near the Schwarzschild boundary, H p,0 is the pressure scale height at that location, and z is the distance in the radiative layer away from that location (for more details, see Herwig 2000;Paxton et al. 2011). Each theoretical model is evolved from the pre-mainsequence stage to the red giant stage. The input parameters of KIC 9145955 and KIC 9970396 are listed in Table 3. For the mass fraction of helium Y in these two giants, we adopt the function Y = 0.245 + 1.54Z (Dotter et al. 2008;Thompson et al. 2014) as the initial value. In our previous work (Zhang et al. 2020), we found that the mixing-length parameter α can be well determined by the g-dominated mixed modes for these two red giants. The value of α is estimated to be about 2.0. Since the mixing-length parameter α has been well calibrated, we fix its value to be 2.0 to reduce computational cost.

Model Fittings
Before the model fitting process, we first need to classify the mixed modes into the most p-and g-dominated mixed modes. For the model frequencies, we define a mixed mode of the lowest mode inertia per radial order as the most p-dominated mixed mode and other modes in the same radial order as the gdominated mixed modes (Zhang et al. 2018(Zhang et al. , 2020 where ξ r (r) and ξ h (r) are the radial and horizontal displacement functions; M and R are the total mass and radius of the star, respectively; ρ 0 is the local density; and R phot is the photospheric radius. For the observational frequencies, the most p-dominated mixed modes of KIC 9145955 and KIC 9970396 are identified by different methods. For KIC 9145955, we use the characteristics of g-dominated mixed modes; their period spacings ΔP obs are close to those of pure g-modes, and the values of the period spacings ΔP obs are almost equal. Therefore, the period echelle diagram can help us to identify the most p-dominated mixed modes (for more details, see Zhang et al. 2018). For KIC 9970396, the red giant oscillation spectrum is regular (Mosser et al. 2011b), and the pure p-mode eigenfrequency can be calculated by the formula proposed by Mosser et al. (2012). We adopt the observational frequencies of the l = 1 modes, which are close to the values of the pure pmode frequency of the l = 1 modes, as the observational pdominated mixed modes. Here we just give a brief description of how to identify the most p-dominated mixed modes for KIC 9145955 and KIC 9970396. For more specific details, refer to our previous work (Zhang et al. 2018(Zhang et al. , 2020. As in our previous work (Zhang et al. 2018(Zhang et al. , 2020, we use the goodness-of-fit functions that have the χ 2 -type function of general form to select the model that best fits the oscillation frequencies in these two stars. These functions are defined as    The superscripts "obs" and "mod" in the above three equations denote the observational and model frequencies, respectively. The subscripts "all" and "l = 1g" mean that all frequencies and only l = 1 frequencies of the g-dominated modes are used in Equations (3) and (4), respectively. The subscript "l = 0, 1p, 2" means that l = 0 frequencies, l = 1 frequencies of the most p-dominated mixed modes, and l = 2 frequencies are used in Equation (5). Here σ obs is the observational error of the individual frequency, K all is the total number of observational oscillation frequencies, K ℓ=1g is the number of frequencies of l = 1g modes, and K ℓ=0,1p,2 is the number of observational frequencies of l = 0, l = 1p, and l = 2 modes. A small value of χ 2 means that the theoretical modes match the observational modes well. For each evolution track, we find the best-fitting model to be the model with the lowest value, e.g., m l g , 1 2 c = of l g 1 2 c = , among all theoretical models.

Asteroseismic Age
Based on the above input physics of Table 3, we calculated a total of 1430 evolutionary tracks for KIC 9145955 and 880 evolutionary tracks for KIC 9970396. Figure  2 c = on one evolutionary track. Stellar ages can be well estimated if the data distribution peaks at a small range. Figures 1(e) and (f) clearly show that the stellar age distributions shrink to a narrow range when the g-dominated mixed modes are used as the observed constraint. However, the stellar age distributions are more dispersed when all modes or p-modes are used as the observed constraints (see Figures 1(a)-(d)). The difference is more significant for KIC 9970396, where the age distributions are almost flat when all modes and pmodes are used. Therefore, compared with other modes, the stellar ages constrained by the g-dominated mixed modes are more precise. We can now estimate the stellar ages by constrained models of the g-dominated mixed modes. Candidate models are selected based on an arbitrary large threshold for m l g is also listed for reference. The best-fitting models for these two red giants are models A14 and B3, respectively. To further illustrate the goodness of fit, we present the frequency echelle diagrams of models A14 and B3 in Figures 2(a) and (b), respectively. As shown in Figure 2, the theoretical frequencies provided by the best-fitting model match the observational values very well, especially for the g-dominated mixed modes. Systematic offsets between the observational and model frequencies of the l = 0 and 2 modes are noted in Figure 2. This difference is caused by the surface effect due to the imperfect modeling of the near-surface convection (Kjeldsen et al. 2008). However, the g-dominated mixed modes are less affected by the surface effect due to their high mode inertia (Kjeldsen et al. 2008). As discussed in our previous works (Zhang et al. 2018(Zhang et al. , 2020, the frequency offsets of g-dominated mixed modes are so small that the influence of the surface effect can be ignored. For this reason, we did not correct the surface effect on the frequencies in our model. The good performance of model fitting by the g-dominated mixed modes ensures the accuracy of the estimated stellar ages. The mean values of the stellar age of the candidate models listed in Tables 4 and 5 are calculated by the following equations: x and σ x are the mean values and standard deviations of the output parameters of the models, respectively, and N is the number of models listed in Table 4 or 5. Finally, according to the asteroseismic models, the stellar ages of KIC 9145955 and KIC 9970396 are determined to be 4.61 ± 0.23 and 6.13 ± 0.19 Gyr, respectively. Figure 3 shows the mode inertia ratio of g-mode ζ for each calculated l = 1 mode for models A14 (KIC 9145955) and B3 (KIC 9970396). The ratio ζ, which represents the mode inertia in the g-mode cavity over the total mode inertia, is defined as (Goupil et al. 2013) Here E g has a similar definition as E in Equation (2), but its integration area is limited to the area where the g-mode propagates. The propagation of the p-mode requires |ω| > |N| and |ω| > S l , while the propagation of the g-mode requires |ω| < |N| and |ω| < S l . Here S l and N are the characteristic acoustic frequency and the Brunt-Väisälä frequency, which are defined as ( In Equations (9) and (10), c s is the adiabatic sound speed, p and ρ are the local pressure and density, and Γ 1 is the adiabatic is the local frequency of internal gravity waves of short horizontal wavelength and provide information on the conditions in the stellar core (Hekker & Christensen-Dalsgaard 2017). The g-mode inertia ratio measures the relative importance of the mode inertia in the g-mode cavity. For a pure g-mode, the g-mode inertia ratio is 1, and for a pure p-mode, the g-mode inertia ratio is zero. The model frequencies corresponding to the observed frequencies are shown as filled circles. Figure 3 shows that all of the l = 1 frequencies (filled circles) are mixed modes with both p-and g-mode characteristics. From the figure, we see that the g-mode inertia ratios are generally greater than 0.5 for these mixed modes (filled circles), indicating that the contribution from the g-mode cavity is significant.
To demonstrate that these mixed modes have both p-and gmode characteristics, we show the scaled displacement eigenfunctions of two selected mixed modes in Figures 4(c) and (d) for models A14 and B3, respectively. The displacement eigenfunction of a pure p-mode for each model is shown in Figures 4(a) and (b) for comparison. For the pure p-mode, we see that the displacement is mainly concentrated in the envelope (Figures 4(a) and (b)). For the mixed mode, on the other hand, the displacement function has a p-mode signature in the envelope and a g-mode signature in the core (Figures 4(c) and (d)). As a result, the mixed modes provide useful information for both stellar core and envelope. The acoustic radius can be used to calibrate the envelope and core sizes and is defined by (Aerts et al. 2010) Here c s is the adiabatic sound speed, and R is the stellar radius. Our previous work (Zhang et al. 2020) showed that τ 0 and the helium core mass M He can be precisely determined when using the g-dominated mixed modes as the observation constraint. It has been demonstrated that the helium core mass and the size of the stellar envelope can be determined precisely by the gdominated mixed modes. Following the procedure of Zhang et al. (2020), we calibrate the specific value of τ 0 and the helium core mass M He for KIC 9145955 and KIC 9970396 by the candidate models selected by the g-dominated mixed modes ( Figure 5). The candidate models are marked with filled circles in Figure 5, and the detailed parameters of these models are given in Table 4 for KIC 9145955 and Table 5 for KIC 9970396. Equations (6) and (7) are adopted to calculate the mean values and standard deviations for the acoustic radius and helium core size for KIC 9145955 and KIC 9970396. The acoustic radius and helium core size of the candidate models with different fundamental parameters in Table 4 are determined to be 0.479 ± 0.001 days, 0.2089 ± 0.0008 M e , and 0.03072 ± 0.00008 R e . Similarly, from Table 5, the where c s is the adiabatic sound speed, and R is the stellar radius. It should be noted that the τ 0 defined in Equation (11) is closely related to the mod n D defined in Equation (12). This relation is defined as The theoretical values of the large frequency separation mod n D are also listed in Tables 4 and 5. It is obvious that the   Note. The last two columns are the mass and radius of the helium core, respectively. The last two lines are the mean values and standard deviations of the output parameters for the candidate models, respectively.
systematic offsets between the observational and model frequencies of l = 0 caused by the surface effect lead to the deviations between the observed large frequency separation Δν and the calculated large frequency separation mod n D . If the observed large frequency separation Δν is taken as the stellar models' observational constraints, then the surface effect should be carefully treated. In other words, the performance of the fitting model relies heavily on the goodness of the treatment on the surface effect. However, the surface effect is less important when g-dominated mixed models are used as observational constraints (Zhang et al. 2018(Zhang et al. , 2020. By taking advantage of this, the uncertainty induced from the surface effect can be significantly reduced. As a result, the models obtained by using the g-dominated mixed modes as the observation limits can more accurately reflect the real situation of the stars: KIC 9145955 and KIC 9970396.

The Best-fitting Model
In this section, we examine whether the best-fitting models based solely on frequencies are consistent with the observational values of T eff and [Fe/H].  The errors in spectral observations will affect the selection of the best-fitting model. Asteroseismology shows its superiority, and its oscillation frequency observation value has higher precision than spectroscopic observations. Hence, the oscillation frequency observations can more accurately limit the theoretical model. Wu & Li (2017) independently and precisely determined the fundamental parameters of KIC 6225718 from seismic observations with no other nonseismic constraints. The fundamental parameters of KIC 9145955 and KIC 9970396 are independently determined by the oscillation frequencies in the work of Zhang et al. (2018Zhang et al. ( , 2020. In this work, the candidate models in Tables 4 and 5 are all within the error ranges of the observed frequencies. However, the [Fe/H] value of model A14 in our work is substantially higher than the observed value by several standard deviations. This discrepancy will require further investigation. The [Fe/H] value of model B3 is consistent with the observation errors of ACER and LAMOST. Based on the observational constraints of high-precision oscillation frequencies, it is appropriate that models A14 and B3 can be selected as the best-fitting models for KIC 9145955 and KIC 9970396.

Convective Core Overshooting Parameters
The Kippenhahn diagrams of models A14 and B3 are displayed in Figure 6 to describe the convective zone and hydrogen-burning region as a function of time. Figure 6 shows that the convection zone arises in the main-sequence stage for these two red giants.
The   . In this paper, the masses of KIC 9145955 and KIC 9970396 determined by the best-fitting models are 1.23 (model A14) and 1.14 (model B3) M e , respectively. The overshooting parameters f ov of Pérez Hernández et al. (2016) and Li et al. (2018) are greater than our results (KIC 9145955: f ov = 0.009; KIC 9970396: f ov = 0.008). Claret & Torres (2017 explored the influence on f ov of star masses using stellar evolution simulations for 50 well-studied double-lined eclipsing binaries. They discovered that f ov rises sharply with stellar masses in the range of 1.2-2.0 M e , and there was no measurable dependence on metallicity (Claret & Torres 2017). Despite the fact that the samples in Claret & Torres (2017 had masses larger than 1.2 M e , Claret & Torres (2018) also stated that stars with masses greater than 1.1-1.2 M e can generate convective cores on the main sequence and experience convective core overshooting. Panel (b) of Figure 6 shows that the convective zone of model B3 (M = 1.14 M e ) appears in the core during the main-sequence stage. The masses and overshooting parameters f ov of 50 eclipsing binaries listed in Claret & Torres (2017 showed that 1.6 M e is a boundary; when the mass of the star is greater than 1.6 M e , the convective overshooting parameter is greater than 0.01, and when the mass of the star is less than 1.6 M e , the convective overshooting parameter is less than 0.01. We list the samples from Claret & Torres (2017 with a convective core overshooting parameter f ov less than 0.01 in Table 6. In Table 6, except for one star whose mass is about 1.8 M e , the masses of the other stars are all below 1.6 M e . In addition, Guo & Li (2019) calibrated the value of the convective core overshooting parameter f ov by using the k-ω model proposed by Li (2012Li ( , 2017 and found that a suitable value of f ov is about 0.008 for the mass range of 1.0-1.8 M e stars. The overshooting parameters f ov of KIC 9145955 and KIC 9970396 in this work are in agreement with Guo & Li (2019). Based on the results of three independent methods, Claret & Torres (2017, Guo & Li (2019), and our work, the overshooting parameters f ov of KIC 9145955 and KIC 9970396 were overestimated by Pérez Hernández et al. (2016) and Li et al. (2018), respectively.

Period Spacing as Constraint
In this paper, we use ΔΠ 1 and ΔP obs to represent theoretical and observational period spacing, respectively. The theoretical period spacing ΔΠ 1 is related to the helium core of the star and defined as (Tassoul 1980;Aerts et al. 2010)  where r 1 and r 2 are the inner and outer boundaries of the region where gravity waves propagate, respectively. The mean values and standard deviations of the theoretical period spacing ΔΠ 1 of candidate models listed in Tables 4 and 5 are calculated by Equations (6) and (7). According to the asteroseismic models, the value of the theoretical period spacing ΔΠ 1 for KIC 9145955 and KIC 9970396 are determined to be 76.97 ± 0.02 and 68.98 ± 0.02 s. Davies & Miglio (2016) showed that the combination of observational constraints on the large frequency separation Δν, the frequency of maximum oscillation power max n , and the observational period spacing ΔP obs is, in principle, able to provide a tight constraint on the age of the star. Hjørringgaard et al. (2017) determined the physical parameters of HD 185351, which are in agreement with all observational constraints. Both the asteroseismic and spectroscopic observations, including observed period spacing ΔP obs , Δν, max n , T eff , and [Fe/H], are adopted as the observational constraints for the stellar models in their work (Hjørringgaard et al. 2017). In previous sections, we have demonstrated that the g-dominated mixed modes have a good performance in the determination of the age of a star. In the following, we compare the predictive Figure 6. Kippenhahn diagrams of models A14 and B3. Regions where convection takes place are hatched. Regions where nuclear burning produces more than 1 erg/ (g × s) are shown in gray for hydrogen burning. The vertical blue dashed lines in both panels indicate the age at the end of the main-sequence stage, that is, when the core hydrogen mass fraction is about 0.01. The vertical red dashed lines in both panels indicate the current age of models A14 and B3, respectively. performances between these two methods. Since the observed period spacing of KIC 9145955 has been well determined by several independent methods (Datta et al. 2015;Takeda et al. 2016;Vrard et al. 2016;Zhang et al. 2018), we take this star as an example for comparison. Similarly, we define the following χ 2 -type function to select the model that best fits the observational period spacing,  Datta et al. (2015). The observational period spacing ΔP obs of KIC 9145955 (76.98 ± 0.03 s) adopted in this paper was determined by Datta et al. (2015), who used an automated technique to determine the gmode period spacing by recognizing the vertical stacking of the gdominated modes and the roughly symmetrical deviation of the pdominated modes from this vertical ridge in the period echelle diagram. The value of the theoretical period spacing ΔΠ 1 for KIC 9145955 is determined to be 76.97 ± 0.02 s. The difference between observational and theoretical period spacing is smaller than the uncertainty in ΔP obs that would be introduced by the random uncertainties in the measurement of the stellar frequencies (Datta et al. 2015). Therefore, the observational period spacing ΔP obs of KIC 9145955 from observed frequencies can be considered as a reliable approximation to its asymptotic form as given by Equation (14). for reference in Table 7. First, we notice that both m l g , 1 2 c = and m,all 2 c in Table 7 are several orders of magnitude higher than those in Table 4. This means that the best-fitting model selected from the period spacing ΔP obs in combination with the large frequency separation Δν and the spectroscopic observables (T eff and [Fe/H]) has worse performances in the prediction of frequencies of individual modes for this star. The age of the best-fitting model is 5.48 Gyr. The age of this star determined by the g-dominated mixed modes, however, is more robust, with a value in the range of 4.61 ± 0.23 Gyr. Our results indicate that the g-dominated modes probably have more advantages than other methods in the estimation of the ages of red giants. Therefore, it Table 6 The Binaries with the Convective Core Overshooting Parameter f ov < 0.01 in the Sample of Claret & Torres (2017   is more advisable to employ the g-dominated mixed modes as the observed constraint in order to determine the precise ages of the red giants. Here we only considered one red giant, KIC 9145955, whose measured period spacing ΔP obs has been precisely determined by many different approaches. It will be interesting to conduct similar comparisons for more red giants in the future. For KIC 9970396, the value of ΔP obs has not been reported in previous works. Here we adopt the Kolmogorov-Smirnov (K-S) test method (Kawaler 1988) to search for the probable equal observational period spacing ΔP obs between 10 and 200 s for observational frequencies of l = 1 modes. The K-S test provides a measure of the probability that a set of numbers is drawn from a chosen distribution (Kawaler 1988). In the K-S test, any uniform (or at least systematically nonrandom) observational period spacings ΔP obs that are present will appear as minima in Q, where Q is the probability that the ΔP obs are randomly distributed (Kawaler 1988). The K-S test results for the range of 10-200 s are shown in Figure 8. An obvious minimum of period spacing ΔP obs at 56.73 s is presented in Figure 8. However, the theoretical value of ΔΠ 1 estimated by our g-dominated mixed modes method is about 68.98 ± 0.02 s. There is a significant discrepancy between the observational and theoretical values of the period spacing of KIC 9970396. This discrepancy has also been reported in previous works. Bedding et al. (2011) mentioned that the value of observational period spacing ΔP obs is less than the value of theoretical period spacing ΔΠ 1 by up to a factor of 2 due to the bumping of mixed modes. Mosser et al. (2011a) derived the relation between the observational and theoretical period spacing ΔP obs ; ΔΠ 1 /1.15 from the 5 month long CoRoT time series. In our work, the relation between the observational and theoretical period spacing of KIC 9970396 is about ΔP obs ; ΔΠ 1 /1.22, which is consistent with the results of Bedding et al. (2011) and Mosser et al. (2011a). The discrepancy between the observational and theoretical period spacings is probably caused by the structural glitches in the cores of red giants (Cunha et al. 2015). As discussed in Cunha et al. (2015), the buoyancy glitches left by the first dredge-up event affected the g-dominated mixed modes and the period spacing. As a consequence, the observed period spacing can significantly deviate from the asymptotic period spacing, depending on the glitch strength and position. In our previous work (Zhang et al. 2020), we found that KIC 9970396 is in an evolutionary stage between the first dredge-up event and the red giant bump phase. Thus, we suspect that the glitch effect of this red giant is important and responsible for the discrepancy between ΔP obs and ΔΠ 1 . If ΔP obs is used as an observational constraint for this red giant directly, uncertainty will arise because of the glitch effect.
In addition, the period spacings are sometimes difficult to obtain from observations. Grosjean et al. (2014) proposed that when the value of Δν is very low (that is, the luminosity of the red giant is very high), the mode inertia of the g-dominated mixed mode is very high, which makes it difficult to extract the period spacing ΔP obs in the observational oscillation spectrum. Grosjean et al. (2014) analyzed the red giant model with a mass of 1.5 M e from the bottom of the red giant branch to the helium burning in the core. The model concluded that when the frequency of maximum oscillation power max n 50 μHz and the large frequency separation Δν 4.9 μHz, l = 1 modes cannot be extracted from the power spectrum of the red giant star. The Δν value of KIC 9970396 is 6.3 μHz. In this case, the l = 1 modes can be marginally extracted from the photometric data. It is fortunate that the l = 1 modes of this red giant can be extracted from observations. However, as mentioned before, the observed period spacing deviates from the theoretical value because of the glitch effect. The accuracy of the calibration will be seriously affected if ΔP obs is used as the constraint. Instead, the asteroseismology method of g-dominated mixed modes provides us an alternative solution to precisely calibrate the age and inner structure of red giants.

Comparison with Previous Work
Here we compare our results with previous work on these two giants. Pérez Hernández et al. (2016) estimated the fundamental parameters of KIC 9145955 by the asteroseismology of l = 0 modes, l = 2 modes, and period spacing of l = 1 modes, and they obtained M = 1.196 M e , age = 3.912 Gyr, Z = 0.009, Y = 0.294, α = 1.941, f ov = 0.021, R = 5.543 R e , and L = 18.496 L e . The age of KIC 9145955 estimated by the g-dominated mixed modes in this work is 4.61 ± 0.23 Gyr, which is higher than the value indicated by the work of Pérez Hernández et al. (2016). They considered only the period spacing of l = 1 frequencies and l = 0 and 2 modes, while other l = 1 frequencies are not included. We have stated that there are systematic offsets between observational and model frequencies of l = 0 and 2 modes caused by the surface effect. The gdominated mixed modes are less affected by the surface effect compared to the l = 0 and 2 modes. In Section 5.2, we also estimate the age of KIC 9145955 by using the observed period spacing of l = 1 modes in combination with the large frequency separation Δν and the spectroscopic observables (T eff and [Fe/ H]). The result is 5.48 Gyr, which is also higher than the age determined by Pérez Hernández et al. (2016).
As we have discussed above, the g-dominated mixed modes are more informative than the observed period spacing ΔP obs when estimating the age of a red giant. Li et al. (2018) estimated the fundamental parameters of KIC 9970396 by the asteroseismology of individual modes, but they used a two-step method to identify the individual modes. They first fitted the l = 0 and 2 modes by theoretical models and then used these models as a guide to identify the l = 1 modes. The fundamental parameters of KIC 9970396 derived by Li et al. (2018) are M = 1.13 M e , age = 6.34 Gyr, R = 7.81 R e , L = 28.8 L e , Figure 8. The K-S test applied to the 28 eigenperiods of KIC 9970396 (oscillation frequencies listed in Table 2 of Zhang et al. 2020). Here Q is the probability that the ΔP obs are randomly distributed. α = 2.017, and f ov = 0.018. There are two major differences between Li et al. (2018) and our methods. First, the identification method of the l = 1 modes is different. Unlike the two-step method used by Li et al. (2018), we identified the l = 1 modes directly from the theoretical model of Mosser et al. (2011bMosser et al. ( , 2012. Second, the model selection criterion is different. Li et al. (2018) selected the best model from a combination of likelihood functions of both p-modes and gdominated mixed modes. When we use all modes, p-modes, and the g-dominated mixed mode as observational constraints in our work, we find that the g-dominated mixed mode has the strongest signal in predicting the internal structure of red giants. Thus, we use the g-dominated mixed modes to select the best model. The age of KIC 9970396 determined by our method is 6.13 ± 0.19 Gyr, which is in fact consistent with the estimation of Li et al. (2018) when the standard deviations are considered. In addition, significant differences are obtained in the estimations of overshooting parameters, and these differences are described in detail in Section 5.1.

Conclusion
In this work, we adopt observations of g-dominated mixed modes to constrain our asteroseismic models. According to our mode identification method, we are able to fit the observed and calculated values of mixed modes quite accurately. We determine the age and convective core overshooting parameter f ov of the red giants KIC 9145955 and KIC 9970396 according to our asteroseismic models. Our results are summarized as follows.
(1) The ages of KIC 9145955 and KIC 9970396 are determined by the g-dominated mixed modes to be 4.61 ± 0.23 and 6.13 ± 0.19 Gyr.
(2) The values of f ov of KIC 9145955 and KIC 9970396 were overestimated by the previous work (Pérez Hernández et al. 2016;Li et al. 2018). In our work, the convective core overshooting parameters f ov of KIC 9145955 and KIC 9970396 are determined to be 0.009 and 0.008, respectively.
(3) The g-dominated mixed modes have a good performance on the determination of the ages of red giants.