Modelling stochastic signatures in classical pulsators

We consider the impact of stochastic perturbations on otherwise coherent oscillations of classical pulsators. The resulting dynamics are modelled by a driven damped harmonic oscillator subject to either an external or an internal forcing and white noise velocity perturbations. We characterize the phase and relative amplitude variations using analytical and numerical tools. When the forcing is internal the phase variation displays a random walk behaviour and a red noise power spectrum with a ragged erratic appearance. We determine the dependence of the root mean square phase and relative amplitude variations (σ_Δφ and σ_ΔA/A, respectively) on the amplitude of the stochastic perturbations, the damping constant η, and the total observation time t_obs for this case, under the assumption that the relative amplitude variations remain small, showing that σ_Δφ increases with t_obs^{1/2} becoming much larger than σ_ΔA/A for t_obs ≫ η^-1. In the case of an external forcing the phase and relative amplitude variations remain of the same order, independent of the observing time. In the case of an internal forcing, we find that σ_Δφ does not depend on η. Hence, the damping time cannot be inferred from fitting the power of the signal, as done for solar-like pulsators, but the amplitude of the stochastic perturbations may be constrained from the observations. Our results imply that, given sufficient time, the variation of the phase associated with the stochastic perturbations in internally driven classical pulsators will become sufficiently large to be probed observationally.