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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 fluctuations. 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 $t_{\rm obs}$ for this case, under the assumption that the relative amplitude variations remain small, showing that $σ_{Δφ}$ increases with $t_{\rm obs}^{1/2}$ becoming much larger than $σ_{ΔA/A}$ for $t_{\rm obs} \gg η^{-1}$. 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 to the stochastic perturbations in internally driven classical pulsators will become sufficiently large to be probed observationally.