On the relevance of the r-mode instability for accreting neutron stars and white dwarfs

We present a case study for the relevance of the r-mode instability for accreting compact stars. Our estimates are based on rough approximations that facilitate back-of-the-envelope calculations. We discuss two different cases: 1) For recycled millisecond pulsars we find that the r-mode instability can limit the rotation provided that the stars are hotter than about 2 × 10 K. The limiting rotation period that we infer is, in fact, close to the observed 1.56 ms of PSR B1937+21, and agrees well also with the recently discovered 2.49 ms X-ray pulsar SAX J1808.4-3658. Furthermore, our data suggests that the instability should be active in the rapidly spinning neutron stars in low mass X-ray binaries. This provides a new explanation for the remarkably similar rotation periods inferred from the kHz quasi-periodic oscillations that are seen in these systems. The possibility that the rotation of recycled pulsars may be gravitational-radiation limited is interesting because the gravitational waves from a neutron star rotating at the instability limit and accreting at the Eddington limit should be detectable with the new generation of interferometric detectors. 2) We also consider white dwarfs, and find that the r-mode instability may possibly be active in short period white dwarfs. Our order of magnitude estimates (for a white dwarf of M = M⊙ and R = 0.01R⊙ composed of C ) yield a critical period P ≈ 27 s. This number is in interesting agreement with the observed periods (> 28 s) of the rapidly spinning DQ Herculis stars. However, we argue that the instability grows too slowly to affect the rotation of these stars.