Measuring the extent of convective cores in low-mass stars using Kepler data: toward a calibration of core overshooting

Context. Our poor understanding of the boundaries of convective cores generates large uncertainties on the extent of these cores and thus on stellar ages. The detection and precise characterization of solar-like oscillations in hundreds of main-sequence stars by CoRoT and Kepler has given the opportunity to revisit this problem. Aims. Our aim is to use asteroseismology to consistently measure the extent of convective cores in a sample of main-sequence stars whose masses lie around the mass limit for having a convective core. Methods. We first tested and validated a seismic diagnostic that was proposed to probe the extent of convective cores in a model-dependent way using the so-called r(010) ratios, which are built with l = 0 and l = 1 modes. We applied this procedure to 24 low-mass stars chosen among Kepler targets to optimize the efficiency of this diagnostic. For this purpose, we computed grids of stellar models with both the CESAM2K and MESA evolution codes, where the extensions of convective cores were modeled either by an instantaneous mixing or as a diffusion process. Results. We found that 10 stars in our sample are in fact subgiants. Among the other targets, were able to unambiguously detect convective cores in eight stars, and we obtained seismic measurements of the extent of the mixed core in these targets with a good agreement between the CESAM2K and MESA codes. By performing optimizations using the Levenberg-Marquardt algorithm, we then obtained estimates of the amount of extra mixing beyond the core that is required in CESAM2K to reproduce seismic observations for these eight stars, and we showed that this can be used to propose a calibration of this quantity. This calibration depends on the prescription chosen for the extra mixing, but we found that it should also be valid for the code MESA, provided the same prescription is used. Conclusions. This study constitutes a first step toward calibrating the extension of convective cores in low-mass stars, which will help reduce the uncertainties on the ages of these stars.