Second-order rotational effects on the r -modes of neutron stars

Recently the r-modes have been found to play an interesting and important role in the evolution of hot young rapidly rotating neutron stars. Andersson [1] first realized and Friedman and Morsink [2] confirmed more generally that gravitational radiation tends to drive the rmodes unstable in all rotating stars. Lindblom, Owen, and Morsink [3] then showed that the coupling of gravitational radiation to the r-modes is sufficiently strong to overcome internal fluid dissipation effects and so drive these modes unstable in hot young neutron stars. This result has been verified by Andersson, Kokkotas, and Schutz [4]. This result seemed somewhat surprising at first because the dominant coupling of gravitational radiation to the r-modes is through the current multipoles rather than the more familiar and usually dominant mass multipoles. But it is now generally accepted that gravitational radiation does drive unstable any hot young neutron star with angular velocity greater than about 5% of the maximum (the angular velocity where mass shedding occurs). This instability therefore provides a natural explanation for the lack of observed very fast pulsars associated with young supernovae remnants. The r-mode instability is also interesting as a possible source of gravitational radiation. In the first few minutes after the formation of a hot young rapidly rotating neutron star in a supernova, gravitational radiation will increase the amplitude of the r-mode (with spherical harmonic index m = 2) to levels where non-linear hydrodynamic effects become important in determining its subsequent evolution. While the non-linear evolution of these modes is not well understood as yet, Owen et al. [5] have developed a simple non-linear evolution model to describe it approximately. This model predicts that within about one year the neutron star spins down (and cools down) to an angular velocity (and temperature) low enough that the instability is again suppressed by internal fluid dissipation. All of the excess angular momentum of the neutron star is radiated away via gravitational radiation. Owen et al. [5] estimate the detectability of the gravitational waves emitted during this spindown, and find that neutron stars spinning down in this manner may be detectable by the second-generation (“enhanced”) LIGO interferometers out to the Virgo cluster. Bildsten [6] and Andersson, Kokkotas, and Stergioulas [7] have raised the possibility that the r-mode instability may also operate in older colder neutron stars spun up by accretion in low-mass x-ray binaries. The gravitational waves emitted by some of these systems (e.g. Sco X-1) may also be detectable by enhanced LIGO [8]. Thus, the r-modes of rapidly rotating neutron stars have become a topic of considerable interest in relativistic astrophysics. The purpose of this paper is to explore further the properties of the r-modes of rotating neutron stars. The initial analyses of the r-mode instability [1–3] were based on a small angular-velocity expansion for these modes developed originally by Papaloizou and Pringle [9]. This expansion in powers of the angular velocity kept only the lowest-order terms in the expressions for the various quantities associated with the mode: the frequency, velocity perturbation, etc. This lowest-order expansion is sufficient to explore many of the interesting physical properties of these modes, including the gravitational radiation instability. However, some important physical quantities vanish at lowest order and hence a second-order analysis is needed [10]. For example the coupling of the r-modes to bulk viscosity vanishes in the lowest-order expansion. Estimates of this important bulk-viscosity coupling to the r-modes have been given by Lindblom, Owen, and Morsink [3], Andersson, Kokkotas, and Schutz [4,11], and by Kokkotas, and Stergioulas [12]. But none of these is based on the fully self consistent second-order calculation needed to evaluate this coupling accurately. Since bulk viscosity is expected