Saturation of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math>-mode instability in neutron stars: Theoretical framework

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 gravitational-wave signal obtained from an individual source, as well as the evolutionary path of the latter inside the instability window.

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