Nonlinear Development of the R Mode Instability and the Maximum Rotation Rate of Neutron Stars

We describe how the nonlinear development of the R mode instability of neutron stars influences spin up to millisecond periods via accretion. Our arguments are based on nearly-resonant interactions of the R mode with pairs of "daughter modes". The amplitude of the R mode saturates at the lowest value for which parametric instability leads to significant excitation of a particular pair of daughters. The lower bound on this limiting amplitude is proportional to the damping rate of the daughter modes that are excited parametrically. Based on this picture, we show that if modes damp because of dissipation in a very thin boundary layer at the crust-core boundary then spin up to frequencies larger than about 300 Hz does not occur. Within this conventional scenario the R mode saturates at an amplitude that is too large for angular momentum gain from accretion to overcome gravitational loss to gravitational radiation. We conclude that lower dissipation is required for spin up to frequencies much higher than 300 Hz. We conjecture that if the transition from the fluid core to the crystalline crust occurs over a distance much longer than 1 cm then a sharp viscous boundary layer fails to form. In this case, damping is due to shear viscosity dissipation integrated over the entire star; the rate is slower than if a viscous boundary layer forms. We use statistical arguments and scaling relations to estimate the lowest parametric instability threshold from first principles. The resulting saturation amplitudes are low enough to permit spin up to higher frequencies. Further, we show that the requirement that the lowest parametric instability amplitude be small enough to allow continued spin up imposes an upper bound to the frequencies that may be attained via accretion that may plausibly be about 750 Hz. Within this framework, the R mode is unstable for all millisecond pulsars, whether accreting or not.