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. When nearly resonant interactions of the l = m = 2 R-mode with pairs of ”daughter modes” are included, the R-mode saturates at the lowest amplitude which leads to significant excitation of a pair of modes. The lower bound for this threshold amplitude is proportional to the damping rate of the particular daughter modes that are excited parametrically. We show that if dissipation occurs in a very thin boundary layer at the crust-core boundary, the R-mode saturation amplitude is too large for angular momentum gain from accretion to overcome loss to gravitational radiation. We find that lower dissipation is required to explain 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. We estimate the lowest parametric instability threshold from first principles. The resulting saturation amplitude is low enough to permit spin up to higher frequencies. The requirement to allow continued spin up imposes an upper bound to the frequencies attained via accretion that plausibly may be about 750 Hz. Within this framework, the R-mode is unstable for all millisecond pulsars, whether accreting or not. DOI: https://doi.org/10.1088/0004-637X/778/1/9 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-90725 Accepted Version Originally published at: Bondarescu, Ruxandra; Wasserman, Ira (2013). Nonlinear development of the R-mode instability and the maximum rotation rate of neutron stars. Astrophysical Journal, 778(1):9. DOI: https://doi.org/10.1088/0004-637X/778/1/9 ar X iv :1 30 5. 23 35 v2 [ as tr oph .S R ] 5 S ep 2 01 3 Draft version September 6, 2013 Preprint typeset using LTEX style emulateapj v. 5/2/11 NONLINEAR DEVELOPMENT OF THE R MODE INSTABILITY AND THE MAXIMUM ROTATION RATE OF NEUTRON STARS Ruxandra Bondarescu Institute for Theoretical Physics, University of Zurich, CH-8057, Switzerland and Ira Wasserman Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853 Draft version September 6, 2013 ABSTRACT 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.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. 1. THE R MODE INSTABILITY VERSUS THE SPIN UP LINE The fastest spinning radio pulsar has a rotational frequency ν = 716 Hz (Hessels et al. 2006) and 39 have been detected with ν > 400 Hz (Manchester et al. 2005). See ATNF Pulsar Catalogue at http://www.atnf.csiro.au/research/pulsar/psrcat/. Moreover, there are 14 pulsars in X ray binaries with inferred ν > 400 Hz, but none demonstrated convincingly to be faster than 620 Hz (Watts 2012; Patruno & Watts 2012); Chakrabarty has argued that the population of neutron star spins cuts off sharply at around 730 Hz (Chakrabarty 2005, 2008, 2012). In the standard picture, millisecond pulsars are thought to be spun up via accretion (Alpar et al. 1982) and the P − Ṗ diagram for radiopulsars is consistent with the idea that accreting neutron stars reach spin equilibrium (e.g. Bildsten et al. 1997) in that there appear to be no neutron stars outside the boundary set by the “spinup line” (e.g. Arzoumanian et al. 1999)