First evidence of inertial modes in $\gamma$ Doradus stars: The core rotation revealed.

Gamma Doradus stars present an incredibly rich pulsation spectra, with gravito-inertial modes, in some cases supplemented with delta Scuti-like pressure modes and in numerous cases with Rossby modes. The present paper aims at showing that, in addition to these modes established in the radiative envelope, pure inertial modes, trapped in the convective core, can be detected in Kepler observations of gamma Doradus stars, thanks to their resonance with the gravito-inertial modes. We start by using a simplified model of perturbations in a full sphere of uniform density. Under these conditions, the spectrum of pure inertial modes is known from analytical solutions of the so-called Poincare equation. We then compute coupling factors which help select the pure inertial modes which interact best with the surrounding dipolar gravito-inertial modes. Using complete calculations of gravito-inertial modes in realistic models of gamma Doradus stars, we are able to show that the pure inertial/gravito-inertial resonances appear as dips in the gravito-inertial mode period spacing series at spin parameters close to those predicted by the simple model. We find the first evidence of such dips in the Kepler gamma Doradus star KIC5608334. Finally, using complete calculations in isolated convective cores, we find that the spin parameters of the pure inertial/gravito-inertial resonances are also sensitive to the density stratification of the convective core. In conclusion, we have discovered that certain dips in gravito-inertial mode period spacings observed in some Kepler stars are in fact the signatures of resonances with pure-inertial modes that are trapped in the convective core. This holds the promise to finally access the central conditions , i.e. rotation and density stratification, of intermediate-mass stars on the main sequence.