A Large ConvectiveCore Overshoot in Kepler Target KIC 11081729
Abstract
The frequency ratios and of KIC 11081729 decrease firstly and then increase with the increase in frequency. For different spectroscopic constraints, all models with overshooting parameter less than 1.7 can not reproduce the distributions of the ratios. However, the distributions of the ratios can be directly reproduced by models with in the range of about . The estimations of mass and age of the star can be affected by spectroscopic results, but the determination of the is not dependent on the spectroscopic results. A large overshooting of convective core may exist in KIC 11081729. The characteristics of and of KIC 11081729 may result from the effects of the large overshooting of convective core. The distributions of and of different stars with a convective core can be reproduced by the function . If the value of the critical frequency is larger than the value of frequency of maximum oscillation power , a star may have a small convective core and . But if the value of is less than that of , the star may have a large convective core and . The function aids in determining the presence of convective core and the size of the convective core including overshooting region from observed frequencies. The determination is not dependent on the calculation of stellar models.
stars: evolution – stars: interiors – stars: oscillations.
1 Introduction
Asteroseismology has proved to be a powerful tool for determining the fundamental parameters of stars, diagnosing the internal structure of stars, and probing physical processes in stellar interiors by comparing observed oscillation characteristics with those calculated from theoretical models (Mazumdar et al., 2006; Cunha & Metcalfe, 2007; ChristensenDalsgaard & Houdek, 2010; Yang et al., 2012; Silva Aguirre et al., 2013; Liu et al., 2014; Guenther et al., 2014; Chaplin et al., 2014; Metcalfe et al., 2014; Tian et al., 2014).
However, the stellar model found solely by matching individual frequencies of oscillation does not always properly reproduce other characteristics of oscillations (Deheuvels et al., 2010; Silva Aguirre et al., 2013; Liu et al., 2014), such as the ratios of smalltolarge separations, and , which are sensitive to conditions in stellar core (Roxburgh & Vorontsov, 2003). The small separations are defined as (Roxburgh & Vorontsov, 2003)
(1) 
and
(2) 
In calculation, equations (1) and (2) are rewritten as the smoother fivepoint separations.
Stars with a mass larger than may have a convective core during their main sequence (MS). The overshooting of the convective core extends the region of material mixing by a distance above the top of the convective core that is determined by Schwarzchild criterion, where is the local pressure scaleheight and is a free parameter. The overshooting brings more Hrich material into the core, prolongs the lifetime of the burning of core hydrogen, and changes the internal structure of stars and the global characteristics of the following giant stages (Schröder et al., 1997; Yang et al., 2012). The determination of the global parameters of stars by asteroseismology or other studies based on stellar evolution can be directly affected by the overshooting (Mazumdar et al., 2006; Yang et al., 2012).
However, for stars with a mass of around , there may or may not exist a convective core in their interior, depending on the input physics used in the computation of their evolutions (ChristensenDalsgaard & Houdek, 2010). Up to now, there is no a direct method to determine the presence and the size of the convective core and the extension of overshooting from observed data. Therefore, finding a method of determining the presence and the size of the convective core is important for understanding the structure and evolution of stars.
Deheuvels et al. (2010) argued that the presence or absence of a convective core in stars can be indicated by the small separations and that the star HD 203608 with and Gyr in fact has a convective core. De Meulenaer et al. (2010) studied the oscillations of Centauri A ( ) and concluded that the allows them to set an upper limit to the amount of convectivecore overshooting and that the model of Centauri A with a radiative core reproduces the observed significantly better than the model with a convective core. Moreover, Silva Aguirre et al. (2013) tried to detect the convective core of KIC 6106415 and KIC 12009504. They obtained that the mass of KIC 6106415 is and that of KIC 12009504 is , and concluded that a convective core and core overshooting exist in KIC 12009504, but could not determine whether a convective core exists in KIC 6106415. Tian et al. (2014) studied the oscillations of KIC 6225718 and concluded that either a small convective core or no convective core exists in the star.
For MS stars, the ratios generally decrease with frequency. Liu et al. (2014) argued that the ratios affected by the overshooting of convection core could exhibit an increase behavior. By using this seismic tool, the value of overshooting parameter is restricted between 0.4 and 0.8 for HD 49933 (Liu et al., 2014) and between 1.2 and 1.6 for KIC 2837475 (Yang et al., 2015). Moreover, Guenther et al. (2014) found that the value of is between 0.9 and 1.5 for Procyon, which is supported by the work of Bond et al. (2015). The values of are larger than the value of estimated by comparing the theoretical and observational colormagnitude diagram of star clusters (Prather & Demarque, 1974; Demarque et al., 1994) and characteristics of eclipsing binary stars (Schröder et al., 1997).
However, the presence of the large overshooting is supported by the convective theory of stars and numerical simulation. For example, Xiong (1985) shows that if the overshooting distance is defined as the distance up to which mixing of matter extends, the overshooting distance of the convective core can reach in his convective theory, where the value of the is between and ; the numerical simulation of Tian et al. (2009) shows that the penetration distance of downward overshooting of convection of giant stars can reach . Moreover, Valle et al. (2016) show that the calibration of the convective core overshooting of doublelined eclipsing binaries with the mass in the range of is poorly reliable. They suggested that asteroseismic data may be required for the calibration of .
KIC 11081729 is an F5 star (Wright et al., 2003). The frequencies of pmodes of KIC 11081729 have been extracted by Appourchaux et al. (2012). Combing asteroseismical and nonasteroseismical data, several authors (Huber et al., 2014; Metcalfe et al., 2014) have investigated the fundamental parameters of KIC 11081729. The mass determined by Huber et al. (2014) is for KIC 11081729, but that determined by Metcalfe et al. (2014) is . The observed ratios and of KIC 11081729 decrease firstly and then increase with the increase in frequencies. The characteristics may derive from the effects of overshooting of convective core. If these characteristics can be directly reproduced by stellar models, which may aid in understanding the origin of the characteristics and confirming the presence of the large overshooting of convective core.
In this work, we focus mainly on examining whether the observed characteristics of KIC 11081729 can be directly reproduced by stellar models and can be explained by overshooting of convective core. In order to find the best model of KIC 11081729, the chi squared method was used. First, an approximate set of models was found out with the constraints of luminosity, , and [Fe/H]. Then around this set of solutions, we sought for best models that match both nonseismic constraints and the individual frequencies extracted by Appourchaux et al. (2012), and then we compared the observed and with those calculated from models. And finally, a function was deduced and tested. In Section 2, stellar models of KIC 11081729 are introduced. In Section 3, the function is deduced, and in Section 4, the results are summarized and discussed.
2 Stellar Models
2.1 Input Physics
In order to obtain the best model of KIC 11081729, we construct a grid of evolutionary models using the Yale Rotation Evolution Code (YREC) (Pinsonneault et al., 1989; Yang & Bi, 2007). For the microphysics, the OPAL equationofstate table EOS2005 (Rogers & Nayfonov, 2002), OPAL opacity table GN93 (Iglesias & Rogers, 1996), and the lowtemperature opacity tables of Alexander & Ferguson (1994) are used. The standard mixinglength theory is adopted to treat convection. The value of the mixinglength parameter calibrated to the Sun is 1.74 for the YREC. But it is a free parameter in this work. The full mixing of material is assumed in overshooting region of models. The diffusion and settling of both helium and heavy elements are taken into account in models with a mass less than 1.30 by using the diffusion coefficients of Thoul et al. (1994). In the model calculation, the initial helium mass fraction is fixed at and . The value of is the standard big bang nucleosynthesis value (Spergel et al., 2007). The values of input parameters, mass (), and , and heavyelement abundance () of ZeroAge MS models are listed in Table 1. The models are constructed from ZAMS to the end of MS or the subgiant stage. Some of evolutionary tracks of the models are shown in Figure 1 as an example.
Variable  Minimum  Maximum  

M/M  1.01  1.50  0.02 
1.65  2.15  0.1  
0.0  1.8  0.2  
0.010  0.040  0.002 
Supplemented by = 1.7.
The low pmode frequencies of each model are calculated by using the Guenther’s adiabatic oscillation code (Guenther, 1994). For the modes with a given degree , the frequencies corrected from the nearsurface effects of the model are computed by using equation (Kjeldsen et al., 2008)
(2) 
where is fixed to be 4.9, is a parameter, is the adiabatic oscillation frequencies of the model, is the frequency of maximum power. The value of is Hz. The value of is determined from the observed and theoretical frequencies of the modes with the degree by using equations (6) and (10) of Kjeldsen et al. (2008).
2.2 Spectroscopic Constraints
The effective temperature of KIC 11081729 may be 6440 K (Wright et al., 2003), K (Ammons et al., 2006), 6359 K (Pinsonneault et al., 2012), K (Bruntt et al., 2012), or K (MolendaZakowicz et al., 2013). The value of [Fe/H] for KIC 11081729 is (Ammons et al., 2006). Combining the value of of the Sun given by Grevesse & Sauval (1998), the ratio of surface metal abundance to hydrogen abundance, , is estimated to be approximately between 0.024 and 0.047. The value of [Fe/H] determined by Bruntt et al. (2012) is , which corresponds to a approximately between 0.015 and 0.020; while that estimated by MolendaZakowicz et al. (2013) is , i.e., the value of is approximately between 0.009 and 0.025. MolendaZakowicz et al. (2013) have shown that the estimated atmospheric parameters of stars hotter than 6,000 K are affected by the method of spectral analysis. Thus the atmospheric parameters determined by Ammons et al. (2006), Bruntt et al. (2012), and MolendaZakowicz et al. (2013) were considered in this work.
The visual magnitude of KIC 11081729 is (Droege et al., 2006; Ammons et al., 2006). The value of bolometric correction and extinction is estimated from the tables of Flower (1996) and Ammons et al. (2006), respectively. The distance of KIC 11081729 is pc (Ammons et al., 2006; Pickles & Depagne, 2010). Then the luminosity of KIC 11081729 is estimated to be . The large uncertainty of the luminosity derives from the uncertainty of the distance.
To find the models that can reproduce the observed characteristics of KIC 11081729, we calculated the value of of models. The function is defined as
(3) 
where the symbol , the presents the observed values of these parameters, while represents the theoretical values. The observational error is given by . In the first step, the models with are chosen as candidate models.
2.3 Asteroseismic Constraints
In order to find the models that can reproduce the oscillation characteristics of KIC 11081729, we computed the value of for each model. The function is defined as
(4) 
where the quantity corresponds to the individual frequencies of modes from observation and modelling, is the observational error of the , presents the number of observed modes. The value of of corrected frequencies is also computed. The models with and or are chosen as candidate models.
When a model evolves to the vicinity of the errorbox of and in the HR diagram, the evolutionary time step of the model is set as small as possible to ensure that consecutive models have an approximately equal . In some cases, there may be a difference of a few Myr between the age of the model with the minimum and that of the model with the minimum . Yang et al. (2015) showed that the difference between the model with the minimum and the model with the minimum is insignificant. For a given mass and , the model with the minimum is chosen as the best model.
2.3.1 The models with the effective temperature and [Fe/H] of Bruntt
Model  age  


()  (K)  ()  ()  (Gyr)  (Hz)  
Mb1  1.21  6524  3.19  1.400  2.475  0.018  0.018  2.05  0.2  0.410  5.5  5.6  0.8  2255 
Mb2  1.22  6524  3.20  1.403  2.488  0.018  0.017  2.05  0.2  0.417  5.9  5.5  0.8  2255 
Mb3  1.23  6534  3.22  1.403  2.222  0.018  0.016  1.95  0.2  0.450  5.9  5.6  0.9  2346 
Mb4  1.24  6556  3.27  1.403  2.073  0.018  0.015  1.95  0.2  0.461  6.6  5.5  0.7  2347 
Mb5  1.25  6596  3.35  1.403  1.856  0.018  0.014  1.95  0.2  0.479  7.1  5.3  0.6  2347 
Mb6  1.26  6641  3.47  1.409  1.835  0.018  0.015  2.05  0.2  0.476  6.1  5.4  0.4  2347 
Mb7  1.27  6555  3.33  1.416  1.907  0.020  0.018  1.95  0.2  0.478  5.4  5.2  0.4  2347 
Mb8  1.28  6583  3.39  1.416  1.750  0.020  0.017  1.95  0.2  0.492  7.1  5.3  0.2  2347 
Mb9  1.29  6599  3.46  1.426  1.697  0.020  0.019  2.05  0.0  0.389  4.9  5.4  0.1  3562 
Mb10  1.30  6553  3.40  1.432  2.452  0.016  0.022  1.95  0.0  0.327  5.5  5.5  0.9  4010 
With the constraints of the effective temperature and the [Fe/H] of Bruntt et al. (2012), we calculated the values of , , and of each model. Table 2 lists the models that minimize + for a given mass. The value of of the models is less than 0.4. The model Mb8 has the minimum + , which seems to hint that Mb8 is the best model.
Using the posterior probability distribution function, the mass estimated from the sample of the models with and is for KIC 11081729, where the error bar indicates the level confidence interval. With the same constraints, Metcalfe et al. (2014) also obtained the same mass, i.e. . The radius is estimated to be that is also close to of Metcalfe et al. (2014). However, the age of Gyr is larger than Gyr given by Metcalfe et al. (2014). This is due to the fact that the effects of the settling of heavy elements and overshooting of convective core were considered that were not included in the models of Metcalfe et al. (2014). This indicates that the estimated age of stars can be significantly affected by physical effects that are considered in models.
However, the calculations show that the models with can not reproduce the distributions of the observed and of KIC 11081729 (see Figure 2). This shows that the internal structures of these models do not match that of KIC 11081729. The observed ratios change dramatically at high frequencies, which may derive from the fact that the observed frequencies are affected by largeline widths at high frequencies in the same way as that of KIC 6106415 and KIC 12009504 (Silva Aguirre et al., 2013). Silva Aguirre et al. (2013) suggested that the ratios at high frequencies should be excluded from the set of constraints.
Liu et al. (2014) argued that effects of overshooting of convective core can lead to the fact that ratios and exhibit an increase behavior. In order to test whether the characteristics of and of KIC 11081729 can be reproduced by the effects of overshooting, we computed the evolutions of models with as large as . The calculation shows that models with can not reproduce the distributions of the observed and . Table 3 lists the models that minimize + for a given mass and . Model Mb19 has the minimum + . Moreover, Figure 3 shows that the observed ratios can be reproduced well by models with and mass in the range of .
Model  age  


()  (K)  ()  ()  (Gyr)  (Hz)  
Mb11  1.19  6560  3.19  1.384  4.415  0.020  0.015  1.95  1.7  0.566  20.8  8.9  0.8  818 
Mb12  1.21  6529  3.20  1.400  4.567  0.024  0.020  2.05  1.7  0.557  10.6  8.1  0.9  818 
Mb13  1.23  6605  3.36  1.400  3.793  0.022  0.017  2.05  1.7  0.574  13.7  8.0  0.1  817 
Mb14  1.25  6596  3.38  1.409  3.603  0.024  0.019  2.05  1.7  0.576  12.4  7.4  0.2  817 
Mb15  1.27  6585  3.40  1.419  3.426  0.026  0.022  2.05  1.7  0.579  9.8  7.0  0.7  817 
Mb16  1.29  6582  3.40  1.419  2.856  0.026  0.021  1.95  1.7  0.594  14.4  7.5  0.4  816 
Mb17  1.21  6542  3.22  1.400  4.631  0.024  0.020  2.05  1.8  0.559  10.8  7.8  0.7  818 
Mb18  1.23  6536  3.22  1.403  4.000  0.024  0.019  1.95  1.8  0.574  14.9  7.7  0.6  817 
Mb19  1.25  6607  3.40  1.409  3.654  0.024  0.019  2.05  1.8  0.578  12.2  7.3  0.1  817 
Mb20  1.27  6640  3.49  1.413  3.210  0.024  0.018  2.05  1.8  0.588  14.0  7.5  0.1  816 
Mb21  1.29  6637  3.49  1.416  2.655  0.024  0.017  1.95  1.8  0.602  22.1  8.8  0.1  815 
Model Mb19 not only has the minimum + when is in the range of , it also reproduces the observed ratios. This shows that the structure of model Mb19 is similar to that of KIC 11081729, and KIC 11081729 may have a large overshoot of convective core. The mass, radius, and age of KIC 11081729 estimated from models with the large and with and are , , and Gyr, respectively. The mass and radius are close to those estimated by Metcalfe et al. (2014). But the age is larger than that determined by Metcalfe et al. (2014), which is due to the fact that overshooting of convective core brings more Hrich material into the region of hydrogen burning.
The value of of Mb19 is larger than that of Mb8. Moreover, the effective temperature and [Fe/H] determined by MolendaZakowicz et al. (2013) and Ammons et al. (2006) are different from those estimated by Bruntt et al. (2012). The determination of atmospheric parameters can be affected by spectroscopic methods (MolendaZakowicz et al., 2013). Thus we calculated the models with the effective temperature and [Fe/H] determined by MolendaZakowicz et al. (2013) and Ammons et al. (2006).
2.3.2 The models with the effective temperature and [Fe/H] of MolendaZakowicz
Model  age  


()  (K)  ()  ()  (Gyr)  (Hz)  
Mm1  1.17  6291  2.74  1.393  6.738  0.030  0.029  1.95  1.7  0.524  8.8  8.3  1.0  819 
Mm2  1.19  6326  2.82  1.400  6.070  0.030  0.028  1.95  1.7  0.533  8.5  8.0  0.8  818 
Mm3  1.21  6403  2.99  1.406  5.564  0.030  0.029  2.05  1.7  0.537  8.5  7.7  0.7  818 
Mm4  1.23  6476  3.15  1.413  4.707  0.028  0.026  2.05  1.7  0.553  8.3  7.6  0.5  818 
Mm5  1.25  6509  3.24  1.416  4.180  0.028  0.025  2.05  1.7  0.563  8.4  7.3  0.6  817 
Mm6  1.27  6543  3.33  1.421  3.706  0.028  0.025  2.05  1.7  0.572  8.4  6.9  0.8  817 
Mm7  1.29  6535  3.35  1.429  3.527  0.030  0.027  2.05  1.7  0.575  7.6  6.7  0.9  817 
Mm8  1.17  6389  2.89  1.390  6.048  0.026  0.023  1.95  1.8  0.539  10.5  8.4  0.2  818 
Mm9  1.19  6423  2.98  1.396  5.421  0.026  0.023  1.95  1.8  0.549  10.6  8.2  0.2  818 
Mm10  1.21  6457  3.08  1.403  5.296  0.028  0.025  2.05  1.8  0.546  8.6  7.9  0.5  818 
Mm11  1.23  6489  3.17  1.409  4.773  0.028  0.025  2.05  1.8  0.556  8.5  7.8  0.5  818 
Mm12  1.25  6441  3.11  1.417  4.388  0.030  0.027  1.95  1.8  0.564  8.8  7.6  0.6  817 
Mm13  1.27  6515  3.26  1.419  3.607  0.028  0.024  1.95  1.8  0.579  11.0  7.1  0.5  816 
Mm14  1.29  6466  3.19  1.425  3.256  0.030  0.026  1.85  1.8  0.588  12.1  7.1  0.5  816 
Table 4 lists the models that minimize + for a given mass and . The calculations show that the models with can not reproduce the observed ratios. Thus we do not considere these models when we estimate the mass and age of KIC 11081729. When the value of increases to 1.8, the observed ratios can be reproduced by models (see Figure 4). This indicates that the distributions of ratios and are mainly dependent on the effects of overshooting of convective core rather than on other effects.
The mass, radius, and age estimated from the models with in the range of and with and are , , and Gyr, respectively. The age of these models are larger than that of models with the effective temperature and [Fe/H] of Bruntt et al. (2012). Thus spectroscopic results can affect the estimation of age of stars. The values of of these models are as large as those of the models with the effective temperature and [Fe/H] of Bruntt et al. (2012). Thus these models are not better than the models with the effective temperature and [Fe/H] of Bruntt et al. (2012).
2.3.3 The models with the effective temperature and [Fe/H] of Ammons
The effective temperature determined by Ammons et al. (2006) is between that estimated by MolendaZakowicz et al. (2013) and that determined by Bruntt et al. (2012). Table 5 lists the models that minimize + for a given mass and . The distributions of and of models with less than 1.7 are also not consistent with those of the observed ratios. When the value of is in the range of , the observed ratios are reproduced well by the models with the large (see Figure 5).
Model  age  

()  (K)  ()  ()  (Gyr)  (Hz)  
Ma1  1.21  6529  3.20  1.400  4.567  0.024  0.020  2.05  1.7  0.557  10.6  8.1  0.7  818 
Ma2  1.23  6476  3.15  1.413  4.707  0.028  0.026  2.05  1.7  0.553  8.3  7.6  0.9  818 
Ma3  1.25  6509  3.24  1.416  4.180  0.028  0.025  2.05  1.7  0.563  8.4  7.3  0.5  817 
Ma4  1.27  6543  3.33  1.421  3.706  0.028  0.025  2.05  1.7  0.572  8.4  6.9  0.4  817 
Ma5  1.29  6536  3.35  1.429  3.527  0.030  0.027  2.05  1.7  0.575  7.6  6.7  0.3  817 
Ma6  1.30  6597  3.52  1.439  4.209  0.034  0.051  2.15  1.7  0.554  5.5  4.8  0.7  814 
Ma7  1.32  6630  3.61  1.442  3.735  0.034  0.051  2.15  1.7  0.563  5.1  4.6  0.9  814 
Ma8  1.34  6592  3.54  1.444  2.939  0.034  0.051  1.95  1.7  0.583  10.9  6.4  0.7  814 
Ma9  1.21  6542  3.22  1.398  4.631  0.024  0.020  2.05  1.8  0.559  10.8  7.8  0.7  818 
Ma10  1.23  6490  3.15  1.406  4.296  0.026  0.022  1.95  1.8  0.568  11.6  7.7  0.9  817 
Ma11  1.25  6562  3.32  1.413  3.934  0.026  0.022  2.05  1.8  0.572  10.0  7.2  0.5  817 
Ma12  1.27  6515  3.26  1.419  3.607  0.028  0.024  1.95  1.8  0.579  11.0  7.1  0.6  816 
Ma13  1.29  6507  3.28  1.426  3.424  0.030  0.026  1.95  1.8  0.582  9.0  7.0  0.5  816 
Ma14  1.30  6504  3.31  1.435  5.298  0.026  0.036  1.95  1.8  0.599  7.4  5.7  0.3  815 
Ma15  1.32  6487  3.32  1.443  5.054  0.028  0.039  1.95  1.8  0.602  6.5  5.8  0.5  815 
Ma16  1.34  6476  3.33  1.451  4.724  0.030  0.042  1.95  1.8  0.604  6.0  5.8  0.7  814 
Ma11b  1.25  6522  3.26  1.415  4.240  0.028  0.025  2.05  1.8  0.565  8.5  7.4  0.5  817 
Tables 1 5 show that model Ma6 has the minimum + in the calculations. Moreover, the observed ratios are reproduced by Ma6. Thus Ma6 is the bestfit model in the calculations. When the value of is equal to 1.8, models Ma14 has the minimum + . Figure 5 shows that the ratios of model Ma14 is almost consistent with the observed ones. Therefore, Ma14 is chosen as the bestfit model for = 1.8.
The mass, radius, and age estimated from the models with in the range of and with and are , , and Gyr, respectively. The median age of Gyr is larger than the ages of models with listed in Table 5. This is related to the fact that the table only lists part of the sample. For example, models Ma11 and Ma11b have the same mass and , and almost the same , and (see Figure 6), however, they have different age.
The calculations show that the estimated mass and age rely on spectroscopic results but the distributions of and are mainly dependent on the effects of overshooting of convective core. The estimation of is not affected by the difference in the spectroscopic results. KIC 11081729 may have a large overshooting of the convective core.
3 Fitting equation for the observed and theoretical ratios
The distributions of and are dependent on , which may provide an opportunity to diagnose the size of convective core from observed frequencies.
By making use of the asymptotic formula of frequencies, one can get (Liu et al., 2014)
(4) 
where
(5) 
In this equation, is the adiabatic sound speed at radius and is the fiducial radius of the star; is the inner turning point of the mode with the frequency . Equation (4) indicates that ratio is dependent on the quantity that is sensitive to the changes in the adiabatic sound speed . Figure 7 shows that the changes can directly affect the quantity . These hint to us that ratios and of stars with a convective core could be affected by the changes in the sound speed.
The pmode oscillations of stars are considered to be acoustic standing waves. The variation with time of a standing wave is proportional to , where the is angular frequency. The variation of the modes that penetrate into overshooting region can be affected by convection. For modes, there should be a critical angular frequency . When the angular frequency of modes is larger than , the modes penetrate into overshooting region. We assume that the variation with time of the acoustic wave that is affected by convection is related to , where is a free parameter. Due to the fact that oscillation frequencies of stars can be observed, thus we want to know that the effects of convection core on the characteristics of oscillations vary with frequency rather than with time. This can be achieved by making use of Fourier transform. Thus one can obtain
(6) 
where is an integer. Taken as and , equation (6) can be rewritten as
(7) 
where is a constant. The effects of convection core on the characteristics of oscillations should be shown by function .
The value of reaches maxima at around and , but reaches minima at around and . Thus the value of decreases with frequency between about and , but increases with frequency between about and . The value of decreases firstly and then increases with frequency in the range of about . These characteristics are consistent with those of and calculated from models with a convective core. Thus the ratios and of stars with a convective core could be described by equation (7).
The observed ratios of KIC 11081729 decrease firstly and then increase with frequency and have a minimum between 1804 and 1846 Hz. Thus one can estimate the value of is between about 800 and 820 Hz. Due to the fact that the value of is at and , hence is not a free parameter. For a given , the value of takes the value of ratio or at or . Thus we obtained the value of is between 0.019 and 0.026 from the ratios of Ma6. In addition, using equation (7) and the observed and , the values of parameters , , and are estimated to be , , and Hz,respectively. Figure 8 shows that the ratios of KIC 11081729 are reproduced well by equation (7) with , , and Hz. The distributions of and of Ma6 and Ma14 are reproduced well by equation (7). The ratios and of models Ma6 and Ma14 reach the minimum at about Hz that are consistent with Hz estimated by using equation (7).
3.1 Test the equation by other stars
Tian et al. (2014) studied the oscillations of KIC 6225718, and determined that the mass of KIC 6225718 is , but could not determine whether a convective core exists in KIC 6225718. From observed frequencies, one can obtain , , and Hz for KIC 6225718. The panel (a) of Figure 9 shows that the observed ratios of KIC 6225718 are reproduced well by equation (7) with , , and Hz. Using equation (16) of Liu et al. (2014) with , the value of for the best model 14 of KIC 6225718 (Tian et al., 2014) is estimated to be Hz. According to equation (7), ratios and reach the minimum at around . The ratios and calculated from the model 14 reach the minimum at Hz. The two values are consistent with Hz. The ratios of KIC 6225718 mainly exhibit a decreasing behavior. But the observed ratios have a maximum at about Hz, which is consistent with that has a maximum at around . The model 14 has a small convective core and (Tian et al., 2014). These indicate that a small convective core may exist in KIC 6225718. The value of of KIC 6225718 is Hz that is much less than the value of ( Hz).
The values of , , and computed from the observed frequencies of HD 49933 (Benomar et al., 2009) are , , and Hz. Using equation (16) of Liu et al. (2014) with , the value of for model M52 of HD 49933 (Liu et al., 2014) is estimated to be about Hz. The ratios of model M52 reach the minimum at about Hz. The panel (b) of Figure 9 shows that the ratios of model M52 are consistent with those computed by equation (7) with , , and Hz. The model M52 has a large convective core and (Liu et al., 2014). The value of of HD 49933 is Hz (Appourchaux et al., 2008) that is close to Hz of . The values of extracted frequencies of HD 49933 are in the vicinity of . The ratios of HD 49933 decrease and then increase with frequency, which is consistent with that calculated from equation (7).
Moreover, the panel (c) of Figure 9 shows that the observed and theoretical ratios of KIC 2837475 are reproduced by equation (7) with , , and Hz in the range of about Hz. The values of computed from the observed frequencies of KIC 2837475 is Hz. The ratios and calculated from model Ma14 of KIC 2837475 (Yang et al., 2015) reach the minimum at about Hz. The model has a large convective core and (Yang et al., 2015). The value of of KIC 2837475 is Hz that is much larger than Hz of . The values of extracted frequencies of KIC 2837475 are in the range between about and . The ratios of KIC 2837475 increase and then decrease with frequency, which is also consistent with that calculated from equation (7).
The distributions of and of different stars with a convective core can be reproduced well by equation (7). The value of of KIC 6225718 is less than the value of . KIC 6225718 may have a small convective core and a small . The value of of HD 49933 is close to the value of . HD 49933 may have a medium . The value of of KIC 2837475 and KIC 11081729 is larger than that of , respectively. KIC 2837475 and KIC 11081729 may have a large . The equation (7) could be used to determine the presence of convective core and estimate the size of overshoot from observed frequencies.
4 Discussion and Summary
4.1 Discussion
Mazumdar et al. (2014) showed that ratios and can be affected by the glitches at the BCZ and the layers of the HeIIZ. The changes in frequencies caused by the glitch at the BCZ or the layers of the HeIIZ have a periodicity of twice the acoustic depth of the corresponding glitch (Mazumdar et al., 2014). The value of the acoustic depth and is about 720 and 2060 s for model Mb8, respectively. The value of and of model Mb19 is 728 and 2100 s, respectively. The values of and of model Mb8 are approximately equal to those of model Mb19. Thus the glitches should result in a similar effect on the ratios of models Mb8 and Mb19. However, the distributions of ratios and of model Mb8 are obviously different from those of model Mb19. Thus, the difference between the ratios of Mb8 and those of Mb19 (see Figures 2 and 3) does not derive from the effects of the glitch at the BCZ or the HeIIZ.
For different spectroscopic results, only the models with in the range of about can reproduce the observed ratios. This indicates that the uncertainties of the effective temperature and [Fe/H] can not significantly affect the estimation of . This is due to the fact that the estimation of is mainly dependent on the distributions of ratios and that are only determined by the interior structure (Roxburgh & Vorontsov, 2003; Oti Floranes et al., 2005). Tables 2 5 list the value of adiabatic oscillation frequency at which and computed from models arrive at the minimum. The values of of models with in the range of are consistent with that determined by equation (7). The calculations show that the value of is related to the radius of overshooting region of models. The larger the radius of the core including the overshooting region, the smaller the value of . For KIC 6225718, HD 49933, KIC 2837475, and KIC 11081729 the values of estimated from observed ratios by using equation (7) are consistent with those obtained from and of the best model of the stars. Thus equation (7) aids in understanding the size of the core from the observed ratios. If the value of of a star is less than the value of estimated by using equation (7), the star may have a relatively small convective core. However, if the value of is larger than that of , the star may have a large convective core.
The value of of the is much larger than the value adopted usually. However, Xiong (1985) shows that the overshooting of convective core could mix material in the distance of . Moreover, the value is consistent with the numerical result for the downward overshooting of giants (Tian et al., 2009). A large means that an efficient mixing takes place in stellar interior. Rotation can lead to an increase in convective core, which depends on the efficiency of rotational mixing and rotation rate (Maeder, 1987; Yang et al., 2013a, b). The effects of rotation explain the extended MS turnoffs of intermediateage star clusters in the Large Magellanic Cloud, where rotational mixing plays an important role (Yang et al., 2013b). The effects of rotational mixing and convection overshoot can reconcile the lowZ solar models with helioseismology (Yang, 2016). Efficient mixing and convection overshoot also may exist in the Sun (Yang, 2016). The effect of overshooting can mimic the effects of rotational mixing to a certain degree. The surface rotation period of KIC 11081729 is about 2.78 days (McQuillan et al., 2014), which is obviously lower than the approximately 27 days of the Sun. The large might be related to rotation.
4.2 Summary
The observed ratios and of KIC 11081729 decrease firstly and then increase with frequency. There are different spectroscopic results for KIC 11081729, which can affect the determinations of mass and age of KIC 11081729 but can not affect the estimation of . This is due to the fact that the distributions of and are mainly dependent on the interior structure. The structure can be directly changed by overshooting of convective core. Thus the distributions of and are sensitive to . For the different spectroscopic constraints, the models with less than 1.7 can not reproduce the observed ratios. However, the distributions of the observed ratios can be reproduced well by models with in the range of about . A large overshooting of convective core may exist in KIC 11081729. The behavior of the ratios of KIC 11081729 may derive from the effects of the large overshooting of convective core.
The ratios and calculated from models with a convective core reach a minimum at about , arrive at a maximum at around , and then reach a minimum at about . These characteristics can be completely reproduced by equation (7). The distributions of observed and theoretical ratios of stars with a convective core can be reproduced well by the equation. The value of decreases with the increase in the radius of convective core and can be estimated by formula (7) from observed frequencies. Thus the equation aids in determining the presence and the size of convective core including overshooting region from observed frequencies. If the value of of a star is less than the value of determined from observed frequencies using equation (7), the star may have a small convective core and a small . However, if the value of is larger than that of , the star may have a large overshooting of convective core.
The author acknowledges the support from the NSFC 11273012, 11273007, 11503039, and the HSCC of Beijing Normal University.
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