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We describe how the nonlinear development of the R mode instability ofneutron stars influences spin up to millisecond periods via accretion. Ourarguments 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 forwhich parametric instability leads to significant excitation of a particularpair of daughters. The lower bound on this limiting amplitude is proportionalto 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 avery thin boundary layer at the crust-core boundary then spin up to frequencieslarger than about 300 Hz does not occur. Within this conventional scenario theR mode saturates at an amplitude that is too large for angular momentum gainfrom accretion to overcome gravitational loss to gravitational radiation. Weconclude that lower dissipation is required for spin up to frequencies muchhigher than 300 Hz. We conjecture that if the transition from the fluid core tothe crystalline crust occurs over a distance much longer than 1 cm then a sharpviscous boundary layer fails to form. In this case, damping is due to shearviscosity dissipation integrated over the entire star; the rate is slower thanif a viscous boundary layer forms. We use statistical arguments and scalingrelations to estimate the lowest parametric instability threshold from firstprinciples. The resulting saturation amplitudes are low enough to permit spinup to higher frequencies. Further, we show that the requirement that the lowestparametric instability amplitude be small enough to allow continued spin upimposes an upper bound to the frequencies that may be attained via accretionthat may plausibly be about 750 Hz. Within this framework, the R mode isunstable for all millisecond pulsars, whether accreting or not.

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