Asymptotic power spectra and visibilities of damped mixed modes

Recent observational studies of red giant stars have estimated the visibility of their mixed oscillation modes, which is a proxy of the average energy of these modes. Among other things, they demonstrated that although the damping rate of the oscillations in the core of many red giants appears to be negligible, other red giants exhibit high core damping rates that are sometimes consistent with the infinite value limit. Up until now, it has not been possible to link the mixed mode visibilities to core damping rates in a quantitative way. In this study, we use the progressive wave picture to derive an analytical function expressing the approximate resonance pattern of red giants up to a proportionality factor. This function can model the influence of the damping on the oscillations, as well as take into account other effects such as mode asymmetries. In particular, this expression can be used to obtain a quantitative estimate for the visibility of mixed modes and to predict the detectability of mixed mode and multiplet signatures under different core damping rates. Here, we conduct a parameter study to investigate how the damping processes affect these aspects. We find that the visibility approaches the value expected for an infinite core damping rate already at finite values. Furthermore, we find that both the mixed mode and the multiplet signatures disappear at finite core damping rates. This implies that the observational characteristics of red giants with finite core damping rates can appear as if their core damping rate were infinite, providing an explanation for the observed populations. Moreover, we have used our method to quantitatively estimate the core damping rates of red giants with unusually low mixed mode amplitudes from their observed visibilities. The analytical function describing the power spectrum developed in the present work is a flexible tool with many possible applications, which will be further explored in future studies.