Properties of observable mixed inertial and gravito-inertial modes in Doradus stars

In Doradus stars, gravito-inertial modes in the radiative zone and inertial modes in the convective core can interact resonantly, which translates into the appearance of dip structures in the period spacing of modes. Those dips are information-rich, as they are related to the star core characteristics. Our aim is to characterise these dips according to stellar properties and thus to develop new seismic diagnostic tools to constrain the internal structure of Doradus stars, especially their cores. We used the two-dimensional oscillation code TOP to compute sectoral prograde and axisymmetric dipolar modes in Doradus stars at different rotation rates and evolutionary stages. We then characterised the dips we obtained by their width and location on the period spacing diagram. We found that the width and the location of the dips depend quasi-linearly on the ratio of the rotation rate and the Brunt-V\"ais\"al\"a frequency at the core interface. This allowed us to determine empirical relations between the width and location of dips as well as the resonant inertial mode frequency in the core and the Brunt-V\"ais\"al\"a frequency at the interface between the convective core and the radiative zone. We propose an approximate theoretical model to support and discuss these empirical relations. The empirical relations we established could be applied to dips observed in data, which would allow for the estimation of frequencies of resonant inertial modes in the core and of the Brunt-V\"ais\"al\"a jump at the interface between the core and the radiative zone. As those two parameters are both related to the evolutionary stage of the star, their determination could lead to more accurate estimations of stellar ages.