Issue 
A&A
Volume 647, March 2021



Article Number  A187  
Number of page(s)  14  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/202040055  
Published online  02 April 2021 
Probing core overshooting using subgiant asteroseismology: The case of KIC10273246
IRAP, Université de Toulouse, CNRS, CNES, UPS, Toulouse, France
email: anoll@irap.omp.eu
Received:
3
December
2020
Accepted:
26
January
2021
Context. The size of convective cores remains uncertain, despite their substantial influence on stellar evolution, and thus on stellar ages. The seismic modeling of young subgiants can be used to obtain indirect constraints on the core structure during main sequence, thanks to the high probing potential of mixed modes.
Aims. We selected the young subgiant KIC10273246, observed by Kepler, based on its mixedmode properties. We thoroughly modeled this star, with the aim of placing constraints on the size of its mainsequence convective core. A corollary goal of this study is to elaborate a modeling technique that is suitable for subgiants and can later be applied to a larger number of targets.
Methods. We first extracted the parameters of the oscillation modes of the star using the full Kepler data set. To overcome the challenges posed by the seismic modeling of subgiants, we propose a method that is specifically tailored to subgiants with mixed modes and uses nested optimization. We then applied this method to perform a detailed seismic modeling of KIC10273246.
Results. We obtain models that show good statistical agreements with the observations, both seismic and nonseismic. We show that including core overshooting in the models significantly improves the quality of the seismic fit, optimal models being found for α_{ov} = 0.15. Higher amounts of core overshooting strongly worsen the agreement with the observations and are thus firmly ruled out. We also find that having access to two gdominated mixed modes in young subgiants allows us to place stronger constraints on the gradient of molecular weight in the core and on the central density.
Conclusions. This study confirms the high potential of young subgiants with mixed modes to investigate the size of mainsequence convective cores. It paves the way for a more general study including the subgiants observed with Kepler, TESS, and eventually PLATO.
Key words: asteroseismology / convection / stars: evolution / stars: interiors / stars: individual: KIC10273246
© A. Noll et al. 2021
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
One of the most important current open questions in stellar physics is the extent of convective cores. Several physical processes are known to extend the convective core boundaries beyond the standard Schwarzschild limit. The most frequently quoted are overshooting of ascending blobs of fluids due to their inertia, rotational mixing or semiconvection. All these processes remain poorly described by theory, and the way they interact is understood even less. They are therefore generally modeled, in stellar evolution codes, as an extension of the mixed core over a distance d_{ov}, which is often referred to as the distance of overshoot, even though other processes can contribute as well. In practice, this can either be achieved by simply extending the fully mixed central region, or by treating overshooting as a diffusive process, following a behavior found in numerical simulations (Freytag et al. 1996). In both cases, a free parameter controlling the extent of the additional mixing is required. Observations are therefore necessary to better constrain those processes.
Initial constraints have been obtained thanks to the study of the HR diagram of clusters (see e.g., Maeder & Mermilliod 1981, VandenBerg et al. 2006), and the modeling of eclipsing binaries (Claret & Torres 2016). Most of those studies favor adding overshooting, to various extents. Typically, d_{ov} is around 0.2 H_{p}, where H_{p} is the pressure scale height. Claret & Torres (2016) found that α_{ov}, the ratio between d_{ov} and H_{p}, increases with mass for stars under 2 M_{⊙} before reaching a plateau. However, this result is still debated (Constantino & Baraffe 2018, Claret & Torres 2019).
Over the last decade, asteroseismology allowed us to probe the structure of stellar cores. Thanks to the data of CoRoT (Baglin et al. 2006), Kepler (Borucki et al. 2010) and now TESS (Ricker et al. 2014) missions, we have been able to precisely measure the oscillation frequencies of numerous pulsators. The study of pressure (p) modes in lowmass main sequence (MS) stars, showed the need for core overshooting to correctly reproduce the observed frequencies (Goupil et al. 2011, Deheuvels et al. 2010, Silva Aguirre et al. 2013). Deheuvels et al. (2016), modeling several MS stars, found that α_{ov} increases with the mass. Moreover, gravity (g) mode pulsators, like slowlypulsating B (SPB) stars, are interesting targets to constrain the additional mixing around convective cores. Indeed, gravity modes probe the inner chemical structure of the star and allow detailed investigation of the convective core extensions. Moravveji et al. (2015, 2016), when modeling SPB stars, found that overshoot was necessary, and they favored diffusive overshooting over a simple extension of the central mixed region.
Postmainsequence stars are another way to put constraints on the amount of overshooting. Once the central hydrogen is exhausted, nuclear energy production stops, leaving an inert radiative helium core. This core then contracts, heating the surrounding hydrogen layers of the star until shell burning starts. At that moment, the star begins its subgiant phase, and evolves on a nuclear timescale for masses below about 1.5 solar masses (M_{⊙}). For stars that are close to the terminalage main sequence (TAMS), the structure and the evolution remain highly influenced by the properties of the MS convective core. Interestingly, the star begins to exhibit mixed modes at that moment. These modes behave like gravity modes in the internal propagation cavity and pressure modes in the outer one. Thus, they allow us to finely probe the deepest layers of the star, all the while being observable. This and the proximity of the subgiant to the TAMS make the mixed modes of young subgiants valuable data in studying the extension of convective cores.
Another particularity of mixed modes is their very fast evolution, compared to the nuclear evolution timescale of the subgiant phase. Indeed, mixed mode frequencies change dramatically over the course of a few million years. This makes their seismic modeling challenging. Recently, increasing efforts have been made to model subgiants (Huber et al. 2019; Stokholm et al. 2019; Metcalfe et al. 2020; Deheuvels et al. 2020; Li et al. 2019, 2020), driven by both their great physical interest and the sudden increase of seismic data for these stars. Most of those works focused on finding the optimal stellar parameters for one or several subgiants. So far, few studies have used subgiants as tools to test stellar physics, mainly due to the challenges of their modeling, as mentioned above.
Deheuvels & Michel (2011) successfully constrained α_{ov} from a subgiant observed by CoRoT, HD 49385, which exhibits only one gdominated mode and is therefore very close to the TAMS. They found that either no overshooting, or a model with α_{ov} = 0.19 were giving equally good results. In this work, we modeled a young subgiant, KIC10273246, which was observed by Kepler over almost 1000 days. That star exhibits two gdominated modes, which allows us to better constrain its inner structure. We performed a thorough seismic modeling of the star, in order to precisely estimate its stellar parameters and to place constraints on the extension of its MS convective core.
In Sect. 2, we show the utility of having access to two gdominated mixed modes in young subgiants. In Sect. 3, we present the surface observables of KIC10273246 and perform a fresh analysis of its oscillation spectrum using the full Kepler data set. We then describe, in Sect. 4, the modeling technique that we adopted, which is an improved version of the method developed by Deheuvels & Michel (2011). Section 5 presents our optimal stellar models and the constraints that were obtained from the extent of the MS convective core for KIC10273246. We discuss these results in Sect. 6, and Sect. 7 is dedicated to our conclusions.
2. Probing potential of mixed modes
Just after the main sequence, the oscillation spectra of solarlike pulsators show the presence of mixed modes, which are due to the coupling between the observed pmodes and low radialorder gmodes (n_{g} = 1, 2, 3, n_{g} being the number of nodes in the gmode cavity). The frequency spacing between loworder gmodes is large (several times the large separation of p modes), so that only a few are in the observable frequency window during the subgiant phase. Moreover, with n_{g} being low, the pure g modes that couple to p modes do not follow an asymptotic behavior (as described in Shibahashi 1979, Tassoul 1980). The oscillation spectra of subgiants therefore constrast with those of more evolved stars, which typically have more gdominated modes than pdominated modes, and for which n_{g} is of the order of several tens (e.g., Mosser et al. 2012).
The frequencies of mixed modes are mostly determined by two structural features of the star. The first is the gmode (G) cavity, which is delimited by the BruntVäisälä frequency N. The second is the evanescent zone between the gmode and pmode (P) cavities, the latter being delimited by the Lamb frequency S_{l}.
The G cavity affects the frequency of the gmode that is involved in the mixed mode frequency. Gmode frequencies, in the asymptotic theory, can be approximated by
l being the degree of the mode, r_{1} and r_{2} the turning points in the G cavity, and r the local radius of the star. In our case, n_{g} is low for the observed modes, so the asymptotic expression given in Eq. (1) should not apply. However, it has been shown that it can provide qualitative information about the behavior of the mixed mode frequencies (Deheuvels et al. 2010). It tells us that gdominated modes should give strong constraints on the BruntVäisälä frequency in the G cavity. One can write it in the following form (e.g., Kippenhahn et al. 2012):
where g is the local gravity, δ = −(∂lnρ/∂lnT)_{P, μ}, ϕ = (∂lnρ/∂lnμ)_{P, T}, ∇_{ad} = (∂lnT/∂lnP)_{ad}, ∇ = ∂lnT/∂lnP, and ∇_{μ} = ∂lnμ/∂lnP. The BruntVäisälä frequency consequently carries information about both the thermal (two first terms in parentheses) and compositional structure (last term) of the star.
The evanescent zone affects the coupling between the two cavities, whose strength is linked to the size of this region and the value of N inside it (e.g., Unno et al. 1989). Using a toy model, Deheuvels & Michel (2011) showed that a strong coupling induces a shift of the l ≥ 1 pdominated frequencies that are close to a gmode. The pdominated frequencies therefore provide complementary information about the internal structure of the subgiant.
In this work, we investigated whether having several gdominated modes in the observed oscillation spectrum could offer more information regarding the extension of the MS core. From above, we know that the frequencies of the gdominated mixed modes are related to the N/r integral between the turning points of the G cavity. Figure 1 shows the propagation diagram of a subgiant close to the TAMS, highlighting the frequencies of the first two gdominated l = 1 mixed modes, that is, those that arise due to the coupling of p modes with g modes of radial orders n_{g} = 1 and 2. We denote their turning points in the G cavity as r_{i1}, r_{o1} for the mode with n_{g} = 1, and r_{i2}, r_{o2} for the mode with n_{g} = 2. The difference between the two frequencies is thus mainly related to the BruntVäisälä frequency value between r_{o1} and r_{o2} (as the one in the [r_{i1}, r_{i2}] region is negligible). This region, as it can be seen in Fig. 1, is dominated by the μgradient contribution. This gradient is related to the characteristics of the hydrogenburning shell, especially the nuclear energy generation, and thus its temperature and composition. It has been shown that a Hburning shell structure depends on the MS core features, and especially on the amount of core overshooting. One can see this in Fig. 5 of Deheuvels & Michel (2010), which exhibits two BruntVäisälä profiles of stars having the same evolutionary stage and position in the HR diagram, but computed with different α_{ov}. The two profiles differ mainly in the peak caused by the μgradient, and these structural differences are large enough to have a significant impact on the eigenfrequencies of the star.
Fig. 1. Typical propagation diagram of a lowmass subgiant star. The BruntVäisälä frequency is represented in blue, delimiting the gmode cavity (light blue). The Lamb frequency, in orange, delimits the pmode cavity (light orange). Two gdominated mixed mode angular frequencies, with n_{g} = 1, 2, are represented (solid lines in propagation zones, dotted lines in evanescent zones). The G cavity turning points are noted as r_{i1}, r_{i2} and r_{o1}, r_{o2}. Finally, the thermal and chemical contributions to the BruntVäisälä frequency are represented in green (dashed) and red (dotdashed), respectively. 
For all those reasons, stars with two visible gdominated modes are therefore expected to be interesting targets on which to place constraints on the efficiency of core overshooting. That criterion led us to the choice of KIC10273246, a subgiant with two gdominated modes and a mass of 1.26 ± 0.10 M_{⊙} (Creevey et al. 2012), which places it safely in the mass range where stars are expected to have a convective core during the MS.
3. Observational properties of KIC 10273246
3.1. Surface constraints
3.1.1. Constraints from spectroscopy
Surface constraints were obtained for KIC10273246 by Creevey et al. (2012). The authors used different algorithms on the same spectra obtained with the FIES spectrograph. For our target, they found effective temperatures (T_{eff}) ranging from 5933 to 6380 K. A weighted mean gives us a value of 6150 ± 100 K, which we have used to constrain our seismic modeling. The star was also found to have to have a subsolar metallicity, with [Fe/H] = −0.13 ± 0.1 dex.
3.1.2. Constraints from broadband photometry
To obtain a reliable value of the luminosity of the star, we performed a spectral energy distribution (SED) fit, following the procedure of Stassun & Torres (2016). We extracted photometric values using the VizieR photometry tool. Those data come from the NUV filter from GALEX (Martin et al. 2005), the B_{T} and V_{T} filters from Tycho2 (Høg et al. 2000), the J,H and K_{s} filters from 2MASS (Skrutskie et al. 2006), the gri filters from SDSS (Skrutskie et al. 2006), the W1W4 filters from WISE (Wright et al. 2010), and the G magnitude from Gaia (Evans et al. 2018). The atmosphere model comes from the Kurucz atmosphere grid (Kurucz 2005), with the surface gravity (log g) derived from the seismic relations (see Sect. 3.2.4), and the metallicity coming from spectroscopic measurements. We then fit the photometry points to the spectrum, with the T_{eff} and extinction A_{v} as free parameters. We also used the spectroscopic data from Creevey et al. (2012) and the extinction from Green et al. (2019) as priors. With a reduced χ^{2} of 0.7, we found T_{eff} = 6000 ± 33 K, and . The fit spectrum and the photometric data are represented in Fig. 2. Finally, we integrated the flux over all the wavelengths and used the distance from Gaia to obtain the luminosity of the star. According to Zinn et al. (2019), a parallax bias exists in the Kepler field, which depends on the Gband magnitude and the pseudocolor ν_{eff} (effective wavenumber of the photon flux distribution in the Gaia band) of the star. We found ϖ − ϖ_{Gaia} = 39.15 ± 9.46 μas, which gives L = 5.74 ± 0.17 L_{⊙}. This result is, as expected, lower than the Gaia archive value (5.92 ± 0.13 L_{⊙}) due to the parallax offset.
Fig. 2. Best fit of the SED using Kurucz atmosphere models. The orange points represent the observations with the corresponding error bars, and the blue curve represents the best fit SED model. 
3.2. Seismic constraints
3.2.1. Preparation of Kepler light curve
The subgiant KIC10273246 was observed with Kepler between quarters Q0 and Q11 (total duration of 978 days) in short cadence (58.85 s). An early seismic analysis of the target was performed by Campante et al. (2011) using the first four quarters of Kepler observations (325 days of data). They estimated the frequencies of oscillation modes of degrees l = 0, 1, and 2 over eight overtones. We revisited this analysis using the complete Kepler data set. The light curve of the star was processed using the Kepler pipeline developed by Jenkins et al. (2010). Corrections from outliers, occasional drifts and jumps were performed following the method of García et al. (2011). The power density spectrum (PSD) was then obtained by applying the LombScargle periodogram (Lomb 1976, Scargle 1982).
The PSD is shown in the shape of an échelle diagram in Fig. 3. We recall that the échelle diagram is built by dividing the PSD in consecutive chunks with a length corresponding to the large separation of acoustic modes Δν, and piling them up. Here, we used the estimate of Δν = 48.2 μHz obtained by Campante et al. (2011). The main interest of échelle diagrams is that acoustic modes of the same degree align in nearly straight ridges, which eases mode identification. The neighboring l = 0 and l = 2 ridges are readily identified on the left part of the échelle diagram (modes indicated by crosses and triangles, respectively, in Fig. 3).
Fig. 3. Échelle diagram of KIC10273246, folded with Δν = 48.2 μHz. For clarity, the spectrum was smoothed over a 0.2μHz boxcar. The white symbols indicate the frequencies that have been extracted for modes of degree l = 0 (crosses), l = 1 (diamonds), and l = 2 (triangles) in Sect. 3.2.2. The two l = 1 gdominated mixed modes are circled in red. 
The ridge of l = 1 modes (indicated by diamonds in Fig. 3) deviates from the nearlyvertical line that is expected for purely acoustic modes. This behavior is known to arise for dipolar modes in the presence of avoided crossings between loworder g modes and p modes in subgiants. Each avoided crossing is characterized by the presence of an additional mode, which lies away from the ridge of theoretical purely acoustic l = 1 modes (which would be a nearly vertical line at an abscissa of about 35 μHz in Fig. 3). This mode is most strongly trapped in the core and is thus gdominated. The neighboring l = 1 modes are pdominated, but their frequencies are nevertheless affected by the presence of the gdominated mode. The modes with frequencies larger than the gdominated mode are shifted to higher frequencies (to the right in the échelle diagram) and those with frequencies below the gdominated mode are shifted to lower frequencies (to the left in the échelle diagram). These features are clearly visible in Fig. 3, corresponding to two l = 1 avoided crossings. The l = 1 gdominated modes associated to these avoided crossings are circled in red in Fig. 3.
3.2.2. Extraction of oscillation mode parameters
To extract the parameters of the oscillation modes, we followed the method of Appourchaux et al. (2008). Here, we briefly recall the main steps of the procedure and refer the reader to that paper for more details.
Prior to fitting the individual oscillation modes, we modeled the background of the PSD. The contribution from granulation was modeled by two Harveylike profiles, following the prescription of Karoff et al. (2013), and we added a white noise component to account for photon noise. The overall contribution from the oscillations was modeled as a Gaussian function. We fit this model to the PSD using maximumlikelihood estimation (MLE). The central frequency of the Gaussian function gives an estimate of the frequency of maximum power of the oscillations ν_{max}. To determine the error on this quantity, we subdivided the Kepler light curve in ten chunks of equal duration, and fit the background model on the PSD calculated with these time series. The error on ν_{max} was taken as the standard deviation of the measurements of this quantity for each chunk. We thus obtained ν_{max} = 843 ± 20 μHz. The PSD was then divided by the optimal background model.
We then performed a fit of the oscillation modes, which were modeled as Lorentzian functions to account for their finite lifetimes. Each mode profile of degree l, radial order n, and azimuthal order m was characterized by its central frequency ν_{n, l, m}, its height H_{n, l, m} and its line width Γ_{n, l, m}. Since dipolar modes have a mixed character, it cannot be assumed that they share similar heights and line widths with the neighboring radial modes, as is often done for main sequence solarlike pulsators. Most quadrupolar modes are expected to be pdominated, owing to the weak coupling between the P and G cavities for these modes. We therefore assumed that the l = 2 modes have the same heights and widths as their closest l = 0 modes, with the exception of one gdominated l = 2 mode, which is discussed below and in Sect. 3.2.3. Nonradial modes are split into multiplets by rotation. Owing to the slow rotation of subgiants, the effects of rotation on the mode frequencies can be found by applying a firstorder perturbation analysis. The components of a rotational multiplet are thus expected to be equally spaced by the rotational splittings δν_{n, l}. We also assumed that they share similar line widths, and that their height ratios depend only on the inclination angle i of the star following the expressions given by Gizon & Solanki (2003). In principle, mixed modes can have different rotational splittings, because they probe the rotation at different depths in the star. This has been used to probe the internal rotation of subgiants (e.g., Deheuvels et al. 2014).
To test whether individual rotational splittings can be measured in KIC10273246, we first performed local fits of the nonradial modes. Following the method described by Deheuvels et al. (2015), we fit each mode using two different models: one under the H_{0} hypothesis (no rotation, so that each mode is modeled as a single Lorentzian), and one under the H_{1} hypothesis (rotation is considered and each mode is modeled as a set of 2l + 1 Lorentzians separated by the rotational splitting). It is clear that hypothesis H_{1} necessarily provides better fits to the data than hypothesis H_{0} since it involves two additional free parameters (inclination angle and rotational splitting). The significance of hypothesis H_{1} can be tested using the likelihoods ℓ_{0} and ℓ_{1} of the best fits obtained under the H_{0} and H_{1} hypotheses, respectively. As shown by Wilks (1938), the quantity ΔΛ ≡ 2(lnℓ_{1} − lnℓ_{0}) follows the distribution of a χ^{2} with Δn degrees of freedom, where Δn is the difference between the number of free parameters involved in hypotheses H_{1} and H_{0} (here, Δn = 2)^{1}. For each multiplet, we thus obtained a value of ΔΛ. The falsealarm probability was then given by the pvalue p = P(χ^{2}(2 d.o.f.)≥ΔΛ), which corresponds to the probability that a mode under the null hypothesis can produce such a high value of ΔΛ.
For dipolar modes, the lowest pvalue that we found is 0.08, which is too high to consider the measurement as significant. This means that we cannot reliably extract individual rotational splittings for dipolar modes in this star. The most likely explanation is that the modes have large line widths compared to the rotational splitting. For quadrupolar modes, only one mode (the one with a frequency around 779.4 μHz) was found to have a low pvalue, of about 4 × 10^{−5}, which shows a very high significance level. A rotational splitting of 0.53 ± 0.03 μHz was obtained for this mode (see Fig. 4). This mode is in fact a gdominated mixed mode, as we show in Sect. 3.2.3.
Fig. 4. Oscillation spectrum of KIC102773246 in the vicinity of a quadrupolar mode that was found to be significantly split by rotation (see Sect. 3.2.2). The thick red curve corresponds to our bestfit model of the spectrum. Two quadrupolar mixed modes are visible (around 779.4 μHz and 783.9 μHz) and one radial mode (around 785.6 μHz). 
We then performed a global fit of the modes (all the modes are fit simultaneously). Since individual splittings cannot be measured, we assumed a common rotational splitting for all l = 1 and l = 2 modes (except for the aforementioned l = 2 mode around 779.4 μHz). Since most nonradial modes are pdominated, we expect the common rotational splitting to essentially measure the rotation in the envelope. The best fit corresponds to a rotational splitting of δν = 0.45 ± 0.02 μHz for nonradial modes and an inclination angle of i = 55 ± 6°. As was done for local fits, we also performed an additional fit of the modes without including the effects of rotation (null hypothesis). We could therefore estimate the pvalue corresponding to the measurement of a mean rotational splitting. We found p ∼ 10^{−4}, which indicates a high level of significance. Our results are compatible with the estimates of Campante et al. (2011), who had found i ≳ 20° for this star, and optimal values of the rotational splitting slightly below 0.5 μHz.
The bestfit parameters for the oscillation modes (frequencies, heights, and line widths) are given in Table A.1. The uncertainties of the fit dipolar mode frequencies range from 0.08 to 0.50 μHz. The measured mode frequencies are in quite good agreement with the ones found by Campante et al. (2011). Discrepancies at the level of 3σ were found for only two modes (the dipole mode around 1055 μHz and the quadrupole mode around 880 μHz). Using the complete Kepler data set enabled us to detect l = 0 and l = 2 modes over three additional radial overtones compared to Campante et al. (2011). Our results are also in very good agreement with the recent measurements of mode frequencies for KIC10273246 by Li et al. (2020) using the complete Kepler data set (agreement at the level of 2σ or better for all oscillation modes).
3.2.3. Detection of an l = 2 mixed mode
We mentioned above that the l = 2 mode with a frequency of about 779.4 μHz is the only mode for which an individual rotational splitting could be measured. This mode also has other distinctive features. It is separated from the closest radial mode by 6.1 ± 0.2 μHz. By comparison, for the other radial orders, the average separation between the l = 2 mode and the neighboring l = 0 mode is 4.4 μHz, with a standard deviation of 0.4 μHz. This suggests that this mode might be an l = 2 mixed mode, the frequency of which is modified by the coupling between the p and gmode cavities. This hypothesis is strengthened by the fact that it has a short line width (0.26 ± 0.08 μHz) compared to the width of the neighboring l = 2 modes (between 1.7 and 2.4 μHz). Indeed, if the mode under study is a gdominated mixed mode, it should have a higher inertia than pdominated l = 2 modes, and therefore a shorter line width. Figure 4 shows the profile of the radial mode that is closest to the l = 2 mode under study. There appears to be an additional mode in the left wing of the radial mode, at a frequency of about 783.9 μHz. To determine the significance of this mode, we performed local fits assuming either its presence (H_{1} hypothesis) or absence (H_{0} hypothesis). We found a pvalue of 0.01, indicating a reasonable significance level. This also supports the identification of the l = 2 mode at 779.4 μHz as a mixed mode. In this case, the additional mode at 783.9 μHz would also be an l = 2 mixed mode undergoing an avoided crossing with its close neighbor. As is shown in Sect. 5, the bestfit models for KIC10273246 do show a pair of mixed modes in the vicinity of these two modes, which confirms our identification.
3.2.4. First estimates of stellar parameters using seismic scaling relations
To obtain first estimates of the global stellar parameters of the star, we used seismic scaling relations, which relate the global seismic parameters Δν and ν_{max} to stellar properties such as the mass, radius and surface gravity (Brown et al. 1991). These relations could be derived because ν_{max} scales to an equally good approximation as the acoustic cutoff frequency (Brown et al. 1991; Stello et al. 2008; Belkacem et al. 2011).
To estimate the asymptotic large separation of acoustic modes, we followed the prescription of Mosser et al. (2013). We fit an expression of the type
to the observed radial modes, where Δν_{obs} is the observed large separation around ν_{max}, α measures the curvature corresponding the to the secondorder term in the asymptotic development, ε_{p} is an offset, and n_{max} = ν_{max}/Δν_{obs}. We thus obtained Δν_{obs} = 48.47 ± 0.02 μHz, which translates into an asymptotic large separation of Δν_{as} = 50.63 ± 0.02 μHz, following Mosser et al. (2013).
Using our estimates of Δν_{as}, ν_{max} from Sect. 3.2.2, and T_{eff} from Sect. 3.1, we could apply seismic scaling relations to derive preliminary estimates of the star’s mass, radius, and surface gravity. We obtained M = 1.24 ± 0.12 M_{⊙}, R = 2.10 ± 0.07 R_{⊙}, and log g = 3.88 ± 0.03.
4. Seismic modeling method
4.1. Physics of the models
We used MESA v10108 (Paxton et al. 2015) evolution models, with OPAL equation of states and opacity tables (Rogers et al. 1996, Iglesias & Rogers 1996), with the solar mixture from Asplund et al. (2009). The models were computed with an Eddingtongray atmosphere. The convection regions were treated using the standard mixing length theory (MLT) as prescribed in Cox & Giuli (1968), with a free parameter α_{conv} corresponding to the ratio between the mixing length and the pressure scale height. Microscopic diffusion was taken into account, unless otherwise specified, using the Burgers formalism (Burgers 1969) and diffusion coefficients from Stanton & Murillo (2016). However, radiative accelerations have not been included in the computed models, as the increase in computational time could not be afforded in this study. The impact of those processes are discussed in Sect. 6.2.
As Gabriel et al. (2014) stated, for stars that have a growing convective core, it is necessary to use the Ledoux criterion to determine the radius of the convective core R_{cc}. This way, we avoid the creation of unphysical convective zones outside the core in strongly chemically stratified regions, which may have an impact on the composition profile of the star and thus on its evolution. Moreover, we used the predictive mixing scheme (Paxton et al. 2018).
Core overshooting was modeled as a step extension of the convective core, over a distance
where d_{ov} is the distance of instant mixing overshooting, H_{p} the pressure scale height, and α_{ov} a free parameter quantifying the phenomenon. Equation (4) replaces the traditional expression d_{ov} = α_{ov}H_{p} in order to prevent d_{ov} from becoming unphysically large when the core is small (H_{p} → ∞ when r → 0). It is important to note that this definition varies from one evolution code to another (see, e.g., Eq. (1) of Deheuvels & Michel 2011 for CESAM2K). Lowmass stars have small convective cores, therefore those differences must be kept in mind when comparing models coming from different codes. Additionally, the impact on our results of using a diffusive overshooting, as proposed by Freytag et al. (1996), is discussed in Sect. 6.1.
The adiabatic oscillations of the models were computed using ADIPLS (ChristensenDalsgaard 2008), and the surface effects were corrected for using the cubic term of the prescription of Ball & Gizon (2014).
4.2. Why modeling subgiants is challenging
The frequencies of gdominated mixed modes evolve over a very short time, with a nonlinear change of several μHz per million years, which is much larger than the usual uncertainties coming from Kepler data. As this timescale is several orders of magnitude shorter than the typical nuclear evolution time of lowmass subgiants, reproducing the mixed modes with a traditional grid technique requires extremely small steps in mass and age. This makes this method prohibitive when the number of free parameters is large, as is required to test the model physics. Interpolation in age is possible (Li et al. 2020), but somewhat difficult for l = 2 gdominated modes, which we observed in KIC10273246. Interpolation across tracks (as used e.g., in AIMS, Rendle et al. 2019) could mitigate the need for extremely fine steps in mass, but needs to be tested for subgiants, especially regarding the extreme sensitivity of the gdominated frequencies to the masses of the models. Additionally, an “onthefly” optimization technique may perform badly due to the highly nonlinear behavior of the mixed modes, especially during the computation of the derivatives in the case of a gradientdescent kind of algorithm.
To overcome those difficulties, a new approach is necessary. We thus developed a nested optimization, where we optimize the physical parameters of models (e.g., metallicity, initial helium abundance etc.) that have been optimized in mass and age beforehand. This way, we can handle those two sensitive parameters using a dedicated and specific procedure, separately from the other ones for which a more traditional technique is possible. This modeling method originates from Deheuvels & Michel (2011) and has been adapted to make it more robust. In the following, we recall the basic principles of this method and highlight the differences with the one used in the present study.
4.3. Optimization in mass and age
In that part of the optimization process, we compute models with only two free parameters, the mass and the age of the star, the rest being fixed. The optimization of those two parameters can be made easier thanks to the fact that, if all the other physical parameters (such as metallicity, mixinglength parameter...) are fixed, reproducing only Δν and the frequency ν_{g} of a gmode is enough to precisely constrain the mass and the age.
A physical justification of that approach can be found in Deheuvels & Michel (2011). We remind the reader of it here using a HRdiagram represented in Fig. 5. It shows the isoΔν line, as for models with the same large separation, and the isoν_{g} line, computed from stellar evolution models. The two lines meet at a unique point, that can be reached by tuning only the mass (i.e., choosing the “right” evolution path) and age (i.e., stopping at the right moment on that path). In concrete terms, our first step is, at a given mass, to find the age that correctly reproduces the ν_{g} frequency.
Fig. 5. HRdiagram representing the evolution tracks of stellar models with masses varying from 1.2 M_{⊙} (lightest gray) to 1.3 M_{⊙} (darkest gray) and otherwise identical physics. Each evolution is stopped when ν_{g} is correctly reproduced. 
As we only see mixed modes and not pure gmodes, we cannot directly measure ν_{g}. A possible solution would be to choose a gdominated mode (i.e., a nonradial mode far from its ridge) frequency. Unfortunately, such a frequency does not evolve monotonously with age, as can be seen in the upper panel of Fig. 6. We thus preferred to look at the distance between that gdominated mode and its closest radial mode, which we call δν.
Fig. 6. Evolution of δν (top panel) and ν_{1, 11} (bottom panel) with age for a 1.3 M_{⊙} star, after the main sequence. Here, like in our modeling of KIC10273246, δν is defined as ν_{1, 11} − ν_{0, 13}, and the observed value is represented by the dotted line. The plot has been strongly magnified in order to see the 1σ uncertainties from the data. 
As we can see in the top panel of Fig. 6, this value always decreases with age, but it also keeps the interesting properties of the mixed modes as it evolves very quickly during an avoided crossing, allowing us to tightly constrain the age. This step would be equivalent to following a unique evolution path in Fig. 5 and stopping it when it crosses the isoν_{g} line.
We then optimize on those “goodage” models in order to correctly reproduce the large separation. In practice, to do this we minimize the χ^{2} of only the radial modes, which we define as
We do not take into account the nonradial modes at this stage to eliminate the complexity of behavior of the mixed modes. This approach differs from the one followed by Deheuvels & Michel (2011), who at this stage searched for models that minimized the difference between the observed average large separation and the one coming from the models. By using all the radial modes instead here, we found that the optimization process is more robust regarding the correction of nearsurface effects. It may be observed that the behavior of Δν (and, in turn, of the radial frequencies) is close to linear when varying the mass. Then, a simple Newtontype algorithm (such as a LevenbergMarquard algorithm) is enough to quickly find the optimal mass. This step would then be equivalent to the right evolution path that leads to the meeting points of the two isolines on Fig. 5.
Figure 7 shows the échelle diagram of a model that we can obtain after that first step, with arbitrary physical parameters: metallicity [Fe/H] = −0.2 dex, mixinglength parameter α_{conv} = 1.5, initial helium abundance Y_{0} = 0.28. We can see that the radial modes and δν = ν_{1, 11} − ν_{0, 13} (the proxy for ν_{g}) are, by construction, correctly reproduced. However, the other frequencies are far from the observed ones. Especially, the gdominated mode ν_{1, 18} is about 10 μHz away from the observations. Thus, to find a better matching model, we adjust the other parameters.
Fig. 7. Échelle diagram of a model optimized in mass and age (open symbols) and of the observed frequencies (full, with their 3σ error bars). The radial and dipolar modes are represented by crosses and diamonds, respectively, with their radial order indicated. 
4.4. Optimizing the other parameters
Now that we have a method to correctly find the mass and the age of a star at a given physics, we must find the other parameters of the stars, this time taking into account all the observational constraints. Thus, we define a new χ^{2} as
where N_{obs} is the total number of observational constraints, both seismic and nonseismic, the values of those observed constraints or their computed equivalent, and σ_{i} their respective uncertainties. We also introduced the quantities , which indicate the contributions of every observable to the χ^{2}, to be used later.
As those parameters have a lower impact on the frequencies than the mass and age, it is now possible to use more traditional approaches. One possibility is to compute grids of models, where each model of the grid is optimized in mass and age. Another option is to perform an optimization using an iterative method, where again each iteration consists of an optimization of the mass and age. To model KIC10273246, we opted for a hybrid method, which is described in the following section.
4.5. Fitting procedure adopted for KIC10273246
For the modeling of KIC10273246, we left the initial metallicity [Z/X]_{0}, the mixinglength parameter α_{conv}, the initial helium abundance Y_{0}, and, of course, the overshoot parameter α_{ov} as free parameters. At first, to explore the global behavior of the χ^{2}, we computed a very loose grid ([Fe/H] between −0.2 and 0.2, step 0.1; Y_{0} between 0.24 and 0.28, step 0.02; α_{conv} between 1.5 and 2.1, step 0.2 and α_{ov} between 0.0 and 0.2, step 0.05). We recall that each model of this grid is optimized in mass and age as explained in Sect. 4.3. The purpose of this loose grid was to investigate whether double solutions or local minima exist. No such features have been found. Moreover, those grids allowed us to identify the region of the best parameters.
We thereafter refined those parameters. As mentioned in Sect. 4.4, the optimization of [Z/X], α_{conv}, Y_{0}, α_{ov} can be performed either through a grid approach or an iterative procedure. We therefore conducted several tests, using stellar models as mock observations, to determine which method is preferable. We found the best robustness when following a hybrid approach: for given values of Y_{0} (0.26 through 0.31, step 0.01) and α_{ov} (0 through 0.25, step 0.05), we conducted iterative optimizations with the LevenbergMarquardt algorithm to find optimal values of [Fe/H] and α_{conv}. This method differs from the one used in Deheuvels & Michel (2011) where a single grid was used for all the free parameters.
Among those models, we considered only those that were compatible with the observational estimates of the chemical enrichment law ΔY_{0}/ΔZ_{0}. Typical values quoted for ΔY_{0}/ΔZ_{0} range from 1.4 to 4 (e.g., Chiappini et al. 2002, Balser 2006, Casagrande et al. 2006). We consequently had a conservative approach and took into account all models with ΔY_{0}/ΔZ_{0} < 5.
5. Results
In this section, we describe the general characteristics of the best models, before commenting the constraints on α_{ov}. We finally investigate at the internal structures of the best models and the constraints brought by the mixed modes.
5.1. General characteristics of the best models
Following the method described in Sect. 4.5, we obtained optimal models for each value of α_{ov}, whose characteristics are in Table 1. The best model, with α_{ov} = 0.15, has a reduced χ^{2} of 3.2. The échelle diagram of this model is represented in Fig. 8. Also, surface observables are consistent with the ones found in the literature, or with the SED fitting previously described: we find a less than 1σ difference for the effective temperature T_{eff}, the metallicity [Fe/H], and the luminosity L. We found a radius and a mass that are consistent with the seismic scaling relations as well. We can note that, as expected from the massbased prediction, all good models had a convective core during the MS. This supports our choice of using this star to constrain α_{ov}.
Fig. 8. Échelle diagram of the best model, with α_{ov} = 0.15. Symbols and colors are identical to those in Fig. 7. 
Characteristics of the best models, for every value of α_{ov}.
We note that our bestfit models are significantly less massive and are older than those of Li et al. (2020), who also performed a seismic modeling of KIC20173246 and found M = 1.49 ± 0.08 M_{⊙} and an age of 2.84 ± 0.60 Gyr. These discrepancies could be partially explained by the different assumptions made on the input physics. For instance, Li et al. (2020) considered a solarcalibrated mixing length parameter, while we left this parameter free in our study. Also, contrary to us, Li et al. (2020) neglected element diffusion and adopted the mixture of Grevesse & Sauval (1998). Finally, we stress that the agreement with the observed dipole mode frequencies, in particular for the gdominated mode ν_{1, 18}, is significantly better in the present study than it is for the bestfit models of Li et al. (2020) (compare Fig. 10 of Li et al. 2020 to Fig. 8 of the present paper). These mismatches between models and observations for dipole modes are acknowledged by Li et al. (2020), and the authors attribute them to an imprecise modeling of the core structure.
For each combination of (α_{ov}, Y_{0}), our minimization using the LM algorithm can be used to derive uncertainties in the stellar parameters. The error bars in the free parameters of the fit ([Fe/H] and α_{conv}) are obtained as the diagonal coefficients of the inverse of the Hessian matrix. The uncertainties on the other parameters can then be obtained using Eq. (10) of Deheuvels et al. (2016). We thus obtain very small error bars, of the order of 0.007 for [Fe/H] and 0.002 for α_{conv}, which translates into uncertainties of approximately 0.004 M_{⊙} for the stellar mass and 0.04 Gyr for the age. This means that for a given combination of (α_{ov}, Y_{0}), the available observables provide very strong constraints on the stellar parameters. By comparison, we find that optimal models with different Y_{0} can yield similar agreement with the observations (statistically equivalent χ^{2}) but have quite different stellar parameters. This degeneracy of stellar models with respect to Y_{0} is addressed in more detail in Sect. 6.3. It thus seems that the uncertainties in the stellar parameters are dominated by the model degeneracy in Y_{0}. We thus used the optimal models with different Y_{0} to estimate uncertainties in the stellar parameters. For a given α_{ov}, we fit a second order polynomial to the χ^{2} curve and retrieved the interval of values corresponding to . This gave us the 1σ uncertainties, which are reported in Table 1.
5.2. Constraints on core overshooting
Figure 9 shows the variation in the χ^{2} of the optimal models with α_{ov}. We can see that adding overshooting allows us to reproduce the observed frequencies significantly better, with the χ^{2} of the models without overshoot and with α_{ov} = 0.15 being 315 and 127, respectively.
Fig. 9. χ^{2} of the best model for every value of overshooting. The colored regions indicate the contributions to the χ^{2} (i.e., the sum of Δ_{i}) of the surface observables (T_{eff}, L and [Fe/H]) and the frequencies according to their degrees. 
To better understand which frequencies allow us to favor models with overshoot, we investigated the contributions of the different observables to the χ^{2} (Δ_{i} in Eq. (6)). We denote the χ^{2} contributions of the observables coming from the optimal models without overshoot as and the equivalent from models with α_{ov} = 0.15 as . Figure 10 represents the differences , positive values meaning that the observable is better reproduced by the α_{ov} = 0.15 model. As expected, we can see that the main χ^{2} difference is due to the dipolar modes, which have a mixed behavior. However, we observe that the gdominated modes (indicated by the dotted vertical lines) hardly contribute to distinguishing the models with and without overshooting. Both types of model fit the gdominated frequencies well. The main contributors to the χ^{2} differences are in fact the l = 1 pdominated modes in the neighborhood of the gdominated modes. As explained in Sect. 2, the frequencies of these modes are mainly influenced by the coupling between the P and the G cavities. The intensity of that coupling thus accounts for the main part of the differences between the models. We note that all those models correctly reproduce ν_{1, 18}, as the high sensitivity of this gdominated mode strongly constrains the region of parameters of the models with the smallest χ^{2}. The major role played by the dipolar modes in the constraints on α_{ov} is also illustrated in Fig. 9, where the colored regions indicate the contributions to the χ^{2} of the surface observables and modes depending on their degree.
Fig. 10. Difference of the χ^{2} contributions of the different observables, between α_{ov} = 0.0 and 0.15 models. The two dipolar gdominated modes are represented by dotted vertical lines. (a), (b), and (c) are T_{eff}, L, and [Fe/H], respectively. 
Moreover, Fig. 9 indicates that the contribution to the χ^{2} of the l = 2 modes hardly changes with α_{ov}. This was partly expected, because their evanescent zone is larger than that of the dipole modes, making the coupling between the G and P cavities weaker. Most of the detectable modes are therefore very strongly pdominated and do not constrain the deep structure of the star, hence α_{ov}. Yet, one gdominated l = 2 mode was detected (ν_{2, 10}, see Sect. 3.2.3). It is interesting to see that, in a similar way to the previous paragraph, its frequency is equally well reproduced by models with and without overshooting. One can see this in Fig. 10, where of that mode is less than 1σ. On the other hand, the (2, 11) mode, whose frequency is close enough to ν_{2, 10} to be influenced by the coupling, varies substantially with α_{ov}. Figure 10 shows a 3σ difference, despite the high 0.65 μHz observational uncertainty, confirming the key role of the coupling in the constraint on α_{ov}. Interestingly, however, while the α_{ov} = 0.15 model better reproduces the dipolar modes, the (2, 11) mode is better fit in the model without overshooting. Nevertheless, its large observational uncertainty prevents it from being too constraining.
Finally, we notice that adding a larger amount of overshooting (α_{ov} > 0.15) strongly worsens the quality of the fit, placing a strong maximum limit on the value of α_{ov}. To better understand this behavior, we investigate the seismic constraints on the internal structure of the models in the next section.
5.3. Constraints on the internal structure from mixed modes
5.3.1. Constraints on central density
A tight link is expected between the gdominated frequencies and the central density ρ_{c}. This comes from the relation between the gmode frequencies and the BruntVäisälä frequency, which approximately scales as ρ_{c} (see Eq. (15) from Deheuvels & Michel 2011). To verify this, we investigated the constraints placed on ρ_{c} by the frequency of a gdominated dipolar mode. For this purpose, we considered the values of ρ_{c} in the models computed in the loose grids defined in Sect. 4.5, in which models are optimized in mass and age in order to reproduce Δν and the frequency of a gdominated mode (here ν_{1, 11}, as described in Sect. 4.3). We observed that, despite the wide range of parameters considered, the standard deviation of ρ_{c} among those models is as low as 32.4 g cm^{−3}, which represents around 1% of the mean central density . This standard deviation even decreases to 10 g cm^{−3} if we only keep the 200 best models, illustrating the tight relation between ρ_{c} and the frequency of a gdominated mixed mode.
This plays a role in the increase of χ^{2} for α_{ov} > 0.15. To illustrate this point, we computed models with the same values of [Z/X], Y_{0}, and α_{conv}, but with different values of α_{ov}. Each of these models was optimized in mass and age, as described above. Figure 11 shows the evolution of ρ_{c} with age for those models. One can see that they all reach approximately the same value of central density, , in accordance with the previous paragraph. Moreover, the intensity of the ρ_{c} jump that is due to the postMS core contraction increases with α_{ov}. We can explain this by the fact that for bigger cores, the layers where hydrogen remains are more distant from the center. They are colder and require a stronger core contraction, hence a stronger jump in ρ_{c}, to reach the fusion temperature. Therefore, when α_{ov} increases, models that reach are closer to the TAMS. When the model gets too close to the TAMS, this impacts the whole stellar structure. Internally, the μgradient in the core is very much affected, because the nuclear reactions in the Hburning shell did not have enough time to smooth its shape. In the outer layers, the convective envelope, which expands during the subgiant phase, has a different size. Those processes alter the frequencies (the g and pdominated ones, respectively), which are thereby not compatible with the observations.
Fig. 11. Evolution of ρ_{c} with age, for models with different α_{ov} that have been optimized in mass and age in order to reproduce Δν and ν_{g}. 
5.3.2. Constraints on the BruntVäisälä profile.
Based on Eq. (1), we expect the frequency of mixed modes to place constraints on the integral of the BruntVäisälä frequency in the G cavity. To investigate this, in Fig. 12 we plot the N^{2} profiles for the models of the loose grid defined in Sect. 4.5, which all reproduce the lowfrequency gdominated mode (ν_{1, 11}) and Δν. We observe that reproducing both already strongly constrains the part of the BruntVäisälä frequency dominated by the thermal term (∇_{ad} − ∇), which corresponds to the most central layers (r < 0.05 R_{⊙}). This was expected because of the 1/r factor in the Eq. (1) integral. On the contrary, the part of N^{2} that is dominated by the μgradient changes significantly within the grid.
Fig. 12. N^{2} profiles of the loose grid models. The left and right panel profiles are colored in relation to the difference between the models and the observations of ν_{1, 18} and δ, respectively. Those differences are normalized by the observational uncertainties. The blue horizontal lines represent the observed gdominated frequencies. 
We expect that part to be strongly determined by the dipolar modes. We therefore investigated the constraints brought by the two most determining seismic features (see Sect. 2): the coupling intensity and the frequency of pure gmodes. As those two are not directly measurable, we used observational proxies. The intensity of the coupling can be quantified by δ ≡ ν_{1, 12} − ν_{1, 11}, which is from Deheuvels & Michel (2011). That value is the difference between the lowfrequency gdominated mode (ν_{1, 11}) and the following dipolar pdominated mode (ν_{1, 12}). Thus, δ increases with the coupling. The frequency of a pure gmode is measured through ν_{1, 18}, which is the highfrequency gdominated mode. For those two values, we colorcoded, in Fig. 12, the profiles of the BruntVäisälä and Lamb frequencies based on their agreement with the observations (left panel for δ and right panel for ν_{1, 18}).
One can see on the right panel that models correctly reproducing the coupling (i.e., dark profiles) have very similar Hburning shell positions (N^{2} peak around r = 0.05 R_{⊙}). However, the BruntVäisälä profiles become more degenerate for higher r: several different profiles can reproduce δ within 1σ. This degeneracy is lifted thanks to the highfrequency gdominated mode: on the left panel, models closely reproducing ν_{1, 18} all have similar BruntVäisälä profiles. This corroborates our theoretical approach in Sect. 2: the highfrequency gdominated mode adding tight constraints on the shape of the μgradient. Important gains in structural constraints are therefore obtained from having a second detectable gdominated mode.
6. Discussion
6.1. Diffusive overshooting
During this study, overshooting was modeled as a step extension of the convective core. However, Freytag et al. (1996) proposed another prescription, based on results coming from 2D simulations of Astars and white dwarfs. Overshoot is then modeled as a diffusive process, with a coefficient D_{ov} exponentially decaying from the boundary of core, following the expression
with D_{conv} being the MLT derived coefficient taken just below R_{cc,} and f_{ov} a free parameter that tunes the length scale of the overshooting region.
In order to compare the results coming from the two types of overshooting, we first found the f_{ov} that is equivalent to a given value of α_{ov}. In order to do this, we searched for the value of f_{ov} that gives models reaching the TAMS at the same age as models computed with a step overshooting and α_{ov} = 0.15. We found f_{ov} = 0.01. After that, we modeled it using a method similar to the one used in Sect. 5 and compared the best model with the one computed with a step overshoot.
As we can see in Fig. 13, the differences between the frequencies and observables of the best models with step and diffusive overshoot are mainly less than 1σ. We note the exception of the gdominated ν_{2, 10} mode, which is better reproduced by the diffusive overshoot model. However, its impact on the global χ^{2} is counterbalanced by the generally better reproduced dipolar frequencies of the step overshoot model. Moreover, the difference between the characteristics of the two models are within the uncertainties of Table 1. Therefore, we cannot discriminate between the two kinds of modelings with the current set of data.
Fig. 13. Differences in the χ^{2} contributions of the observables coming from the best models with step and diffusive overshoot. 
6.2. Effect of microscopic diffusion
The models presented in Sect. 5 of our study include both gravitational settling and chemical diffusion. Such processes, which happen to be necessary in order to correctly reproduce the helioseismic observations (ChristensenDalsgaard et al. 1993), are expected to have an impact on our results for two main reasons. The first is the sinking during the main sequence of heavier elements because of the gravitational settling. This reduces the hydrogen abundance in the core and shortens the main sequence, which eventually decreases the age of models with same mean density. The second is the smoothing of the structure of the subgiant. High μgradient peaks, like the one produced by the withdrawal of the convective core, are strongly smoothed out by the chemical diffusion, which impacts the mixed mode frequencies. Thus, it is interesting to see how gravitational settling and chemical diffusion change the characteristics and the quality of the fit. We therefore modeled the star following a similar methodology, but without including those two processes. We found that the best model also has α_{ov} = 0.15, but provides a significantly worse agreement with the observations than the best diffusion model, with . It is more massive (1.29 M_{⊙}) and older (4.85 Gyr), as was expected from the fact that heavy elements do not sink during the MS. We note that the surface observables are less well reproduced, with a best model that is too cold (5874 K, which represents 1.84 times the observational uncertainty) and has too low a luminosity (5.0 L_{⊙}, which represents 4.3 σ).
Moreover, similarly to what we found in Sect. 5, the quality of the fit improves as α_{ov} increases for α_{ov} ≤ 0.15. However, this is less significant (). For higher values, the quality of the fit strongly worsens, in a comparable way to what has been found with gravitational settling and chemical diffusion.
Figure 14 illustrates the differences in the BruntVäisälä profiles between the two best models, with and without both diffusive processes. One can see the high ∇_{μ} peak (at r = 0.06 R_{⊙}) is, as expected, smoothed out by the chemical diffusion. Otherwise, the two profiles are remarkably similar, despite the different physics of the models, which highlights the robustness of the constraints coming from the mixed modes.
Fig. 14. BruntVäisälä profiles of the best models with gravitational settling and chemical diffusion (blue) and without those two processes (orange). 
Finally, the effects of gravitational settling are expected to be somewhat counterbalanced in the envelope by radiative accelerations, which can have a significant impact on stars with masses greater than 1.1 M_{⊙} (Deal et al. 2018). However, including the effects of radiative accelerations in the models increases the computational time of stellar evolution calculation by orders of magnitude, and it could not be afforded in the present study. To test the impact of this process on our modeling, we computed a model that takes into account radiative accelerations, with the same parameters as the best model for α_{ov} = 0.15. We obtained slightly different large separations (Δν_{rad} − Δν_{norad} = 0.12 μHz) but very similar frequencies, once normalized by the large separation. Radiative accelerations are therefore not expected to change the conclusions of this study.
6.3. Helium degeneracy
To model KIC10273246, we initially performed optimizations with fixed α_{ov} and considering Y_{0}, α_{conv} and [Fe/H] as free parameters. In this case, we observed an unwanted sensitivity of the Y_{0} parameter to the guess value of our optimization process. This led us to the hybrid approach described in Sect. 4.5, using optimizations with fixed values of Y_{0} and varying [Fe/H], α_{conv}. We found that optimal models with different values of Y_{0} indeed have surprisingly close frequencies, despite their wide range of mass. This is illustrated in Fig. 15, which shows the échelle diagrams of the best model with Y_{0} = 0.28 (blue) and the best model with Y_{0} = 0.24 (orange), both of which have α_{ov} = 0.15. Those models have quite different characteristics, as reported in Table 2. However, their frequencies are almost indistinguishable, despite the very small uncertainties on the mode frequencies from Kepler data. Only the gdominated l = 2 mode allows us to slightly favor the Y_{0} = 0.28 model. Such degeneracy is related to the anticorrelation between mass and Y_{0}, that has been observed in MS stars (see e.g., Lebreton & Goupil 2014) as well as subgiant stars (Li et al. 2020). Additionally, we note that no monotonic behavior has been found between the age and Y_{0}. We therefore conclude that the seismic modeling of subgiants, despite bringing strong constraints on the deep structure of the star, does not lift the degeneracy between Y_{0} and the mass.
Fig. 15. Échelle diagrams of the best model (blue, Y_{0} = 0.28) and the best model for Y_{0} = 0.24 (orange). 3σ uncertainties from the observations are represented by black bars. 
6.4. Internal rotation
We mentioned in Sect. 3.2.2 that a rotational splitting of 0.53 ± 0.03 μHz could be measured with a high level of significance for the l = 2 mode at 779.4 μHz. This is obviously not enough to probe the internal rotation in detail. However, since this mode is gdominated, it can be used to place approximate constraints on the rotation in the core of the star.
Using our bestfit model from Sect. 5, we were able to compute the rotational kernel K(r), which relates the splitting δν_{s} of this mode to the rotation profile Ω(r):
This can be rewritten as δν_{s} = K_{g}⟨Ω_{g}⟩+K_{p}⟨Ω_{p}⟩, where ⟨Ω_{g}⟩ and ⟨Ω_{p}⟩ are average rotation rates in the g and pmode cavities, respectively, and K_{g} (resp. K_{p}) corresponds to the integral of K(r) in the gmode (resp. pmode) cavities. For the l = 2 mode under study, using our bestfit stellar model we found that 84% of the kernel energy is enclosed in the gmode cavity, which confirms that the mode is indeed gdominated.
Campante et al. (2011) found a clear rotational modulation in the Kepler light curve of KIC10273246. They thus estimated the surface rotation rate of the star to about 0.5 μHz (rotation period of about 23 days). This value is comparable to the average rotational splitting of 0.45 ± 0.02 μHz that we obtained in this study. As mentioned in Sect. 3.2.2, this average splitting is dominated by the contribution of pdominated modes, to the extent that it essentially measures the envelope rotation rate. Taken together, these two measurements suggest a low rotation contrast within the pmode cavity, which is in line with the conclusions of Benomar et al. (2015) and Nielsen et al. (2015) for mainsequence solarlike pulsators.
The splitting measured for the l = 2 gdominated mode is close to the rotation rate inferred for the envelope, which suggests a low amount of radial differential rotation in the star. If we take ⟨Ω_{p}⟩/(2π)≈0.45 μHz, we obtain a core rotation rate of about 0.65 μHz (rotation period of about 18 days). Clearly, more rotational splittings would be required to precisely measure the core rotation rate. However, our results indicate that KIC10273246 could be rotating nearly as a solidbody, like the two Kepler subgiants whose internal rotation profiles were recently measured (Deheuvels et al. 2020).
7. Conclusion
In this study, we performed a seismic modeling of KIC10273246, a subgiant observed by Kepler, and obtained strong constraints on the size of its MS convective core. We chose this target because it exhibits two dipolar gdominated modes, which we expected to bring stronger constraints on the internal structure. We extracted the mode parameters from the oscillation spectrum of the star using the full Kepler data set and thus updated the mode frequencies that were previously obtained by Campante et al. (2011).
The seismic modeling of subgiants is notoriously complex. We here elaborated on the algorithm proposed by Deheuvels & Michel (2011). This method consists of a twostep approach. The purpose of the first step is to find the mass and age that match the large separation of p modes and the frequency of one gdominated mixed mode. The second step optimizes the other free parameters ([Fe/H], Y_{0}, α_{conv} and α_{ov}) to reproduce the other frequencies as closely as possible. In this work, we improved this algorithm to make it more robust. This enabled us to perform a detailed seismic modeling of KIC10273246, with a particular emphasis on the investigation of the size of the MS convective core.
We found models in good agreement with the observations, with a reduced χ^{2} of 3.2 for the best model, and with surface observables that are reproduced to within less than 1σ. One key result of this study is that models with core overshooting during the MS reproduce the observations significantly better, with an optimal value of α_{ov} = 0.15. For higher values of α_{ov}, the quality of the fit significantly worsens. We found that such models are very close to the TAMS. Their internal structure thus differs from that of the lowerα_{ov} solutions, and their seismic properties show strong mismatch with the observations. We tested the robustness of our conclusions by considering other choices for the input physics. No significant difference was found when modeling core overshooting as a diffusive process. Models computed without microscopic diffusion also favor models with α_{ov} = 0.15, albeit less significantly, and show a strong mismatch compared with the observations for higher values of α_{ov}. However, they yield poorer agreement with the seismic and surface observables compared to the models computed with microscopic diffusion. This study thus confirms the high potential of young subgiants with mixed modes to measure the extent of the MS convective cores.
We also investigated the information conveyed by the mixed modes about the core structure. We showed that the combined knowledge of the large separation Δν and the frequency of one gdominated mixed mode is enough to estimate the central density ρ_{c} to a precision of about 1%. This helps us understand why models with a greater amount of core overshooting (α_{ov} > 0.15) are not compatible with the observations. Because of their larger MS convective core, they have a higher ρ_{c} just after the end of the MS, and they thus reach the optimal central density closer to the TAMS. We then studied the roles of the different mixed mode frequencies in determining the profile of the BruntVäisälä frequency inside the star. While the first gdominated mixed mode strongly constrains the thermal part, the second one helps constrain the part of the BruntVäisälä frequency that is dominated by the μgradient. We therefore confirm that having access to two gdominated mixed modes helps better characterize the BruntVäisälä profile.
Also, despite the strong constraints that were obtained on the internal structure, we noted the existence of a degeneracy between the stellar mass and the initial helium abundance Y_{0}. This degeneracy, which is already well known for MS stars (e.g., Lebreton & Goupil 2014), is not lifted by the mixed modes. We find that it is in fact the main source of uncertainties in the determination of the stellar parameters. This should be kept in mind when modeling subgiants. Current modeling techniques, such as traditional gridbased methods, tend to miss a significant fraction of the bestfit models because of the size of the mesh. In such conditions, the degeneracy between Y_{0} and mass could be explored only partially, thus causing the uncertainties on the stellar parameters to be underestimated.
As a byproduct of this work, we obtained partial constraints on the internal rotation of KIC10273246. We were not able not measure individual rotational splittings for the dipolar mixed modes, but we obtained a splitting of 0.53 ± 0.03 μHz for the only gdominated l = 2 mixed mode in the spectrum of the star. Interestingly, this value is close to the surface rotation rate of 0.5 μHz that was found for this star by Campante et al. (2011) using photometric data from Kepler. This suggests that this star might be rotating nearly as a solidbody, similarly to the two young subgiants recently studied by Deheuvels et al. (2020).
This work highlights the large potential of the seismic modeling of young subgiants to indirectly obtain constraints on the core structure of the star during the MS. The next step will be to use this method on a larger sample of stars drawn from the targets observed with Kepler and TESS, and therefore place quantitative constraints on the overshooting process in lowmass stars. The data from the upcoming PLATO mission (Rauer et al. 2014) will add a large amount of potential targets for this type of analysis.
Moreover, we show in this study that detecting several gdominated dipole modes places stronger constraints on the shape of the BruntVäisälä profile, and therefore on the μgradient in the stellar core. It could thus be anticipated that more evolved subgiants, which show a larger number of gdominated mixed modes, would be more favorable targets for our purpose. However, these stars are also further from the end of the MS, and a worry is that the chemical composition in the core might be less dependent on the properties of the MS core. We plan, therefore, to study this effect in a subsequent study.
We note that the definition of ΔΛ in Sect. 3.1 of Deheuvels et al. (2015) contains an erroneous minus sign. This is just a typo and the results presented in the paper consider the correct expression for ΔΛ.
Acknowledgments
We thank Morgan Deal for enlightening discussions about microscopic diffusion. We also thank the anonymous referee for comments that improved the clarity of this paper. We acknowledge support from from the project BEAMING ANR18CE31 0001 of the French National Research Agency (ANR) and from the Centre National d’Etudes Spatiales (CNES).
References
 Appourchaux, T., Michel, E., Auvergne, M., et al. 2008, A&A, 488, 705 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481 [NASA ADS] [CrossRef] [Google Scholar]
 Baglin, A., Auvergne, M., Barge, P., et al. 2006, in The CoRoT Mission PreLaunch Status  Stellar Seismology and Planet Finding, eds. M. Fridlund, A. Baglin, J. Lochard, L. Conroy, et al., ESA Spec. Pub., 1306, 33 [NASA ADS] [Google Scholar]
 Ball, W. H., & Gizon, L. 2014, A&A, 568, A123 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Balser, D. S. 2006, AJ, 132, 2326 [NASA ADS] [CrossRef] [Google Scholar]
 Belkacem, K., Goupil, M. J., Dupret, M. A., et al. 2011, A&A, 530, A142 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Benomar, O., Takata, M., Shibahashi, H., Ceillier, T., & García, R. A. 2015, MNRAS, 452, 2654 [NASA ADS] [CrossRef] [Google Scholar]
 Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W. 1991, ApJ, 368, 599 [NASA ADS] [CrossRef] [Google Scholar]
 Burgers, J. M. 1969, Flow Equations for Composite Gases (New York: Academic Press) [Google Scholar]
 Campante, T. L., Handberg, R., Mathur, S., et al. 2011, A&A, 534, A6 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Casagrande, L., Portinari, L., & Flynn, C. 2006, MNRAS, 373, 13 [NASA ADS] [CrossRef] [Google Scholar]
 Chiappini, C., Renda, A., & Matteucci, F. 2002, A&A, 395, 789 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 ChristensenDalsgaard, J. 2008, Ap&SS, 316, 113 [Google Scholar]
 ChristensenDalsgaard, J., Proffitt, C. R., & Thompson, M. J. 1993, ApJ, 403, L75 [NASA ADS] [CrossRef] [Google Scholar]
 Claret, A., & Torres, G. 2016, A&A, 592, A15 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Claret, A., & Torres, G. 2019, ApJ, 876, 134 [Google Scholar]
 Constantino, T., & Baraffe, I. 2018, A&A, 618, A177 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Structure (New York: Gordon and Breach) [Google Scholar]
 Creevey, O. L., Doǧan, G., Frasca, A., et al. 2012, A&A, 537, A111 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deal, M., Alecian, G., Lebreton, Y., et al. 2018, A&A, 618, A10 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., & Michel, E. 2010, Astron. Nachr., 331, 929 [NASA ADS] [CrossRef] [Google Scholar]
 Deheuvels, S., & Michel, E. 2011, A&A, 535, A91 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., Michel, E., Goupil, M. J., et al. 2010, A&A, 514, A31 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., Doğan, G., Goupil, M. J., et al. 2014, A&A, 564, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., Ballot, J., Beck, P. G., et al. 2015, A&A, 580, A96 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., Brandão, I., Silva Aguirre, V., et al. 2016, A&A, 589, A93 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deheuvels, S., Ballot, J., Eggenberger, P., et al. 2020, A&A, 641, A117 [EDP Sciences] [Google Scholar]
 Evans, D. W., Riello, M., De Angeli, F., et al. 2018, A&A, 616, A4 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Freytag, B., Ludwig, H. G., & Steffen, M. 1996, A&A, 313, 497 [NASA ADS] [Google Scholar]
 Gabriel, M., Noels, A., Montalbán, J., & Miglio, A. 2014, A&A, 569, A63 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 García, R. A., Hekker, S., Stello, D., et al. 2011, MNRAS, 414, L6 [NASA ADS] [CrossRef] [Google Scholar]
 Gizon, L., & Solanki, S. K. 2003, ApJ, 589, 1009 [Google Scholar]
 Goupil, M. J., Lebreton, Y., Marques, J. P., et al. 2011, in GONGSoHO 24: A New Era of Seismology of the Sun and SolarLike Stars, J. Phys. Conf. Ser., 271, 012032 [Google Scholar]
 Green, G. M., Schlafly, E., Zucker, C., Speagle, J. S., & Finkbeiner, D. 2019, ApJ, 887, 93 [Google Scholar]
 Grevesse, N., & Sauval, A. J. 1998, Space Sci. Rev., 85, 161 [Google Scholar]
 Høg, E., Fabricius, C., Makarov, V. V., et al. 2000, A&A, 355, L27 [Google Scholar]
 Huber, D., Chaplin, W. J., Chontos, A., et al. 2019, AJ, 157, 245 [Google Scholar]
 Iglesias, C. A., & Rogers, F. J. 1996, ApJ, 464, 943 [NASA ADS] [CrossRef] [Google Scholar]
 Jenkins, J. M., Caldwell, D. A., Chandrasekaran, H., et al. 2010, ApJ, 713, L120 [NASA ADS] [CrossRef] [Google Scholar]
 Karoff, C., Campante, T. L., Ballot, J., et al. 2013, ApJ, 767, 34 [NASA ADS] [CrossRef] [Google Scholar]
 Kippenhahn, R., Weigert, A., & Weiss, A. 2012, Stellar Structure and Evolution (SpringerVerlag Berlin Heidelberg) [Google Scholar]
 Kurucz, R. L. 2005, Mem. Soc. Astron. It. Suppl., 8, 14 [Google Scholar]
 Lebreton, Y., & Goupil, M. J. 2014, A&A, 569, A21 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Li, T., Bedding, T. R., Kjeldsen, H., et al. 2019, MNRAS, 483, 780 [Google Scholar]
 Li, T., Bedding, T. R., ChristensenDalsgaard, J., et al. 2020, MNRAS, 495, 3431 [Google Scholar]
 Lomb, N. R. 1976, Ap&SS, 39, 447 [Google Scholar]
 Maeder, A., & Mermilliod, J. C. 1981, A&A, 93, 136 [NASA ADS] [Google Scholar]
 Martin, D. C., Fanson, J., Schiminovich, D., et al. 2005, ApJ, 619, L1 [Google Scholar]
 Metcalfe, T. S., van Saders, J. L., Basu, S., et al. 2020, ApJ, 900, 154 [Google Scholar]
 Moravveji, E., Aerts, C., Pápics, P. I., Triana, S. A., & Vandoren, B. 2015, A&A, 580, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Moravveji, E., Townsend, R. H. D., Aerts, C., & Mathis, S. 2016, ApJ, 823, 130 [Google Scholar]
 Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A, 540, A143 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mosser, B., Michel, E., Belkacem, K., et al. 2013, A&A, 550, A126 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Nielsen, M. B., Schunker, H., Gizon, L., & Ball, W. H. 2015, A&A, 582, A10 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15 [Google Scholar]
 Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34 [Google Scholar]
 Rauer, H., Catala, C., Aerts, C., et al. 2014, Exp. Astron., 38, 249 [Google Scholar]
 Rendle, B. M., Buldgen, G., Miglio, A., et al. 2019, MNRAS, 484, 771 [Google Scholar]
 Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2014, in Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, SPIE Conf. Ser., 9143, 914320 [Google Scholar]
 Rogers, F. J., Swenson, F. J., & Iglesias, C. A. 1996, ApJ, 456, 902 [NASA ADS] [CrossRef] [Google Scholar]
 Scargle, J. D. 1982, ApJ, 263, 835 [Google Scholar]
 Shibahashi, H. 1979, PASJ, 31, 87 [NASA ADS] [Google Scholar]
 Silva Aguirre, V., Basu, S., Brandão, I. M., et al. 2013, ApJ, 769, 141 [NASA ADS] [CrossRef] [Google Scholar]
 Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163 [Google Scholar]
 Stanton, L. G., & Murillo, M. S. 2016, Phys. Rev. E, 93 [CrossRef] [Google Scholar]
 Stassun, K. G., & Torres, G. 2016, AJ, 152, 180 [Google Scholar]
 Stello, D., Bruntt, H., Preston, H., & Buzasi, D. 2008, ApJ, 674, L53 [NASA ADS] [CrossRef] [Google Scholar]
 Stokholm, A., Nissen, P. E., Silva Aguirre, V., et al. 2019, MNRAS, 489, 928 [Google Scholar]
 Tassoul, M. 1980, ApJS, 43, 469 [Google Scholar]
 Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. 1989, Nonradial Oscillations of Stars (Tokyo: University of Tokyo Press) [Google Scholar]
 VandenBerg, D. A., Bergbusch, P. A., & Dowler, P. D. 2006, ApJS, 162, 375 [Google Scholar]
 Wilks, S. S. 1938, Ann. Math. Stat., 9, 60 [Google Scholar]
 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868 [Google Scholar]
 Zinn, J. C., Pinsonneault, M. H., Huber, D., & Stello, D. 2019, ApJ, 878, 136 [NASA ADS] [CrossRef] [Google Scholar]
Appendix A: Observed frequencies
Estimated mode parameters for KIC10273246.
All Tables
All Figures
Fig. 1. Typical propagation diagram of a lowmass subgiant star. The BruntVäisälä frequency is represented in blue, delimiting the gmode cavity (light blue). The Lamb frequency, in orange, delimits the pmode cavity (light orange). Two gdominated mixed mode angular frequencies, with n_{g} = 1, 2, are represented (solid lines in propagation zones, dotted lines in evanescent zones). The G cavity turning points are noted as r_{i1}, r_{i2} and r_{o1}, r_{o2}. Finally, the thermal and chemical contributions to the BruntVäisälä frequency are represented in green (dashed) and red (dotdashed), respectively. 

In the text 
Fig. 2. Best fit of the SED using Kurucz atmosphere models. The orange points represent the observations with the corresponding error bars, and the blue curve represents the best fit SED model. 

In the text 
Fig. 3. Échelle diagram of KIC10273246, folded with Δν = 48.2 μHz. For clarity, the spectrum was smoothed over a 0.2μHz boxcar. The white symbols indicate the frequencies that have been extracted for modes of degree l = 0 (crosses), l = 1 (diamonds), and l = 2 (triangles) in Sect. 3.2.2. The two l = 1 gdominated mixed modes are circled in red. 

In the text 
Fig. 4. Oscillation spectrum of KIC102773246 in the vicinity of a quadrupolar mode that was found to be significantly split by rotation (see Sect. 3.2.2). The thick red curve corresponds to our bestfit model of the spectrum. Two quadrupolar mixed modes are visible (around 779.4 μHz and 783.9 μHz) and one radial mode (around 785.6 μHz). 

In the text 
Fig. 5. HRdiagram representing the evolution tracks of stellar models with masses varying from 1.2 M_{⊙} (lightest gray) to 1.3 M_{⊙} (darkest gray) and otherwise identical physics. Each evolution is stopped when ν_{g} is correctly reproduced. 

In the text 
Fig. 6. Evolution of δν (top panel) and ν_{1, 11} (bottom panel) with age for a 1.3 M_{⊙} star, after the main sequence. Here, like in our modeling of KIC10273246, δν is defined as ν_{1, 11} − ν_{0, 13}, and the observed value is represented by the dotted line. The plot has been strongly magnified in order to see the 1σ uncertainties from the data. 

In the text 
Fig. 7. Échelle diagram of a model optimized in mass and age (open symbols) and of the observed frequencies (full, with their 3σ error bars). The radial and dipolar modes are represented by crosses and diamonds, respectively, with their radial order indicated. 

In the text 
Fig. 8. Échelle diagram of the best model, with α_{ov} = 0.15. Symbols and colors are identical to those in Fig. 7. 

In the text 
Fig. 9. χ^{2} of the best model for every value of overshooting. The colored regions indicate the contributions to the χ^{2} (i.e., the sum of Δ_{i}) of the surface observables (T_{eff}, L and [Fe/H]) and the frequencies according to their degrees. 

In the text 
Fig. 10. Difference of the χ^{2} contributions of the different observables, between α_{ov} = 0.0 and 0.15 models. The two dipolar gdominated modes are represented by dotted vertical lines. (a), (b), and (c) are T_{eff}, L, and [Fe/H], respectively. 

In the text 
Fig. 11. Evolution of ρ_{c} with age, for models with different α_{ov} that have been optimized in mass and age in order to reproduce Δν and ν_{g}. 

In the text 
Fig. 12. N^{2} profiles of the loose grid models. The left and right panel profiles are colored in relation to the difference between the models and the observations of ν_{1, 18} and δ, respectively. Those differences are normalized by the observational uncertainties. The blue horizontal lines represent the observed gdominated frequencies. 

In the text 
Fig. 13. Differences in the χ^{2} contributions of the observables coming from the best models with step and diffusive overshoot. 

In the text 
Fig. 14. BruntVäisälä profiles of the best models with gravitational settling and chemical diffusion (blue) and without those two processes (orange). 

In the text 
Fig. 15. Échelle diagrams of the best model (blue, Y_{0} = 0.28) and the best model for Y_{0} = 0.24 (orange). 3σ uncertainties from the observations are represented by black bars. 

In the text 
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