Evolution of the gravity offset of mixed modes in RGB stars

Observations of mixed modes in evolved low-mass stars enable us to probe the properties of not only the outer envelope of these stars, but also their deep layers. Among the seismic parameters associated with mixed modes, the gravity offset, denoted with $\varepsilon_{\rm g}$, is expected to reveal information on the boundaries of the inner buoyancy resonant cavity. This parameter was recently measured for hundreds of stars observed by the Kepler satellite and its value was shown to change during the evolution on the red giant branch. In this article, we theoretically investigate the reasons for such a variation in terms of structure properties. Using available asymptotic analyses and a simple model of the Brunt-V\"ais\"al\"a and Lamb frequencies, we derived an analytical expression of $\varepsilon_{\rm g}$ for dipolar modes and compared its predictions to observations. First, we show that the asymptotic value of $\varepsilon_{\rm g}$ well agrees with the mean value observed at the beginning of the ascent of the red giant branch, which results from the high density contrast between the helium core and the base of the convective envelope. Second, we demonstrate that the predicted value also explains the sharp decrease in $\varepsilon_{\rm g}$ observed for the more luminous red giant stars of the sample. This rapid drop turns out to occur just before the luminosity bump and result from the kink of the Brunt-V\"ais\"al\"a frequency near the upper turning point associated with the buoyancy cavity as stars evolve and this latter becomes close to the base of the convective envelope.The observed variation in $\varepsilon_{\rm g}$ and its link with the internal properties on the red giant branch are now globally understood. This motivates further analyses of the potential of this parameter as a seismic diagnosis of the region located between the helium core and the convective envelope.