The underlying physical meaning of the $\nu_{\rm max}-\nu_{\rm c}$ relation

TL;DR

This paper provides a theoretical explanation for the scaling law between the maximum oscillation frequency and the cut-off frequency in solar-like stars, linking it to damping rates and turbulence properties.

Contribution

It establishes a physical basis for the $ u_{ m max}- u_{ m c}$ relation, connecting it to damping mechanisms and turbulence characteristics in stellar atmospheres.

Findings

Confirmed the role of damping rate plateau in oscillation power spectrum.

Derived a linear relation between $ u_{ m max}$ and $ u_{ m c}$.

Linked the relation to turbulence Mach number and mixing-length parameter.

Abstract

Asteroseismology of stars that exhibit solar-like oscillations are enjoying a growing interest with the wealth of observational results obtained with the CoRoT and Kepler missions. In this framework, scaling laws between asteroseismic quantities and stellar parameters are becoming essential tools to study a rich variety of stars. However, the physical underlying mechanisms of those scaling laws are still poorly known. Our objective is to provide a theoretical basis for the scaling between the frequency of the maximum in the power spectrum ($\nu_{\rm max}$) of solar-like oscillations and the cut-off frequency ($\nu_{\rm c}$). Using the SoHO GOLF observations together with theoretical considerations, we first confirm that the maximum of the height in oscillation power spectrum is determined by the so-called \emph{plateau} of the damping rates. The physical origin of the plateau can be traced to the destabilizing effect of the Lagrangian perturbation of entropy in the upper-most layers which becomes important when the modal period and the local thermal relaxation time-scale are comparable. Based on this analysis, we then find a linear relation between $\nu_{\rm max}$ and $\nu_{\rm c}$, with a coefficient that depends on the ratio of the Mach number of the exciting turbulence to the third power to the mixing-length parameter.