Probing the mid-layer structure of red giants: I. Mixed-mode coupling factor as a seismic diagnosis

Context. The space-borne missions CoRoT and Kepler have already brought stringent constraints on the internal structure of low-mass evolved stars, a large part of which results from the detection of mixed modes. However, all the potential of these oscillation modes as a diagnosis of the stellar interior has not been fully exploited yet. In particular, the coupling factor or the gravity-offset of mixed modes, q and epsilon(g), are expected to provide additional constraints on the mid-layers of red giants, which are located between the hydrogen-burning shell and the neighborhood of the base of the convective zone. The link between these parameters and the properties of this region, nevertheless, still remains to be precisely established.Aims. In the present paper, we investigate the potential of the coupling factor in probing the mid-layer structure of evolved stars.Methods. Guided by typical stellar models and general physical considerations, we modeled the coupling region along with evolution. We subsequently obtained an analytical expression of q based on the asymptotic theory of mixed modes and compared it to observations.Results. We show that the value of q is degenerate with respect to the thickness of the coupling evanescent region and the local density scale height. On the subgiant branch and the beginning of the red giant branch (RGB), the model predicts that the peak in the observed value of q is necessarily associated with the important shrinking and the subsequent thickening of the coupling region, which is located in the radiative zone at these stages. The large spread in the measurement is interpreted as the result of the high sensitivity of q to the structure properties when the coupling region becomes very thin. Nevertheless, the important degeneracy of q in this regime prevents us from unambiguously concluding on the precise structural origin of the observed values. In later stages, the progressive migration of the coupling region toward the convective zone is expected to result in a slight and smooth decrease in q, which is in agreement with observations. At one point just before the end of the first-dredge up and the luminosity bump, the coupling region becomes entirely located in the convective region and its continuous thickening is shown to be responsible for the observed decrease in q. We demonstrate that q has the promising potential to probe the migration of the base of the convective region as well as convective extra-mixing during this stage. We also show that the frequency-dependence of q cannot be neglected in the oscillation spectra of such evolved RGB stars, which is in contrast with what is assumed in the current measurement methods. This fact can have an influence on the physical interpretation of the observed values. In red clump stars, in which the coupling regions are very thin and located in the radiative zone, the small variations and spread observed in q suggest that their mid-layer structure is very stable.Conclusions. A structural interpretation of the global observed variations in q was obtained and the potential of this parameter in probing the dynamics of the mid-layer properties of red giants is highlighted. This analytical study paves the way for a more quantitative exploration of the link of q with the internal properties of evolved stars using stellar models for a proper interpretation of the observations. This will be undertaken in the following papers of this series.