Saturation of thef-mode instability in neutron stars: Theoretical framework

Abstract

The basic formulation describing quadratic mode coupling in rotating Newtonian stars is presented, focusing on polar modes. Due to the Chandrasekhar-Friedman-Schutz mechanism, the f -mode (fundamental oscillation) is driven unstable by the emission of gravitational waves. If the star falls inside the so-called instability window, the mode's amplitude grows exponentially, until it is halted by nonlinear effects. Quadratic perturbations form three-mode networks inside the star, which evolve as coupled oscillators, exchanging energy. Coupling of the unstable f -mode to other (stable) modes can lead to a parametric resonance and the subsequent saturation of its amplitude, thus suppressing the instability. The saturation point determines the amplitude of the gravitationalwave signal obtained from an individual source, as well as the evolutionary path of the latter inside the instability window.

Using gravitational-wave data to constrain dynamical tides in neutron star binaries

“…However, as we have highlighted, this argument involves a number of unknowns. In order to address these uncertainties the underlying mechanism must be modelled at a more detailed level, but this is difficult given the intricate nature of the nonlinear mode-coupling problem [54,55] and the need to model the neutron star interior at an adequate level of realism. As long as such results are outstanding, we may still consider to what extent the imprint of the p-g instability would be distinct from that of the other mechanisms we have considered.…”

Gravitational Waves from Single Neutron Stars: An Advanced Detector Era Survey

“…A second key recent development concerns saturation amplitude (or equivalently saturation energy) of the instability. This arduous calculation was undertaken in [60,61] by means of a nonlinear mode-coupling model similar to the one that had been employed before for the r-modes (see Section V) but with the added property of non-uniform, stratified matter. According to the aforementioned work the mode's energy is primarily drained by the non-linear coupling to g-modes; this leads to a saturation energy that may fluctuate considerably across the instability's Ω − T parameter space.…”

Saturation of the f-mode instability in neutron stars

“…(1)- (3) to quadratic order, mode coupling is introduced and modes can no longer be described independently. It can be shown that modes couple in triplets, if a resonance condition is satisfied between their frequencies, and their amplitudes are described by 15,16 …”

“…When this happens, a parametric resonance occurs: after the parametric threshold (PT) is crossed by the parent mode, the daughter modes start growing by draining energy from it. The system eventually saturates if two additional conditions are met, given by 15,16 |γ β + γ γ | γ α and Δω…”