R-mode epitaph?

John Friedman, UW-Milwaukee, Nils Andersson, Univ. of Southampton UK friedman@thales.phys.uwm.edu, na@maths.soton.ac.uk

For the past few years, because the growth time of the gravitational-wave driven instability (CFS instability) is short for r-modes, it has seemed likely that the instability limits the spin of rapidly rotating, newly formed neutron stars - and possible that it limits the spin of old, accreting neutron stars as well. Because most of the initial angular momentum of the star could be radiated, the gravitational waves emitted by sources as close as the Virgo Cluster might be observable by detectors with the expected sensitivity of advanced LIGO. In the past year, however, studies of two mechanisms - rapid cooling and high viscosity associated with a hyperons in neutron-star cores, and amplitude saturation by nonlinear coupling to stable modes have sharply decreased the likelihood that the waves from nascent neutron stars will be detectable. Whether the instability limits the spin of neutron stars remains an open question. And, surprisingly, the hyperon studies may have increased the chance of detecting gravitational waves from old accreting stars.

One of the major questions about the r-mode instability was whether nonlinear couplings would sharply limit its amplitude. The first studies of this problem were numerical evolutions of the nonlinear equations: First, Stergioulas and Font [1] carried out a numerical evolution of the exact relativistic Euler equations on a background spacetime, with the initial data of a large-amplitude r-mode. They found, surprisingly, that there was no apparent energy transfer to daughter modes until the perturbation's amplitude was substantially larger than unity. Next was a numerical time-evolution of the full Newtonian perturbation equations by Lindblom, Tohline and Vallisneri [2]. Because the actual growth time from radiation-reaction is impractically long, they looked at the maximum amplitude for a perturbation driven by a greatly enlarged radiation-reaction term. They again found saturation only at amplitude large compared to unity; the limit appeared to be set by a shock wave, a dramatic breaking on the star's surface.

A clear implication of these numerical evolutions was that coupling to low-order modes did not set a stringent limit on the r-mode amplitude - not, at least, within a time of about ten rotational periods, and with the small initial daughter-mode amplitudes and numerical viscosity of the simulations. These evolutions can examine the full nonlinear coupling, but they were limited in their resolution and evolution time. The complementary way to address the question is in 2nd-order perturbation theory - by omitting higher-order couplings, one reduces the nonlinear evolution to a set of coupled ordinary differential equations. But the second-order perturbation theory of a rotating star had not been developed, and it is was a major undertaking, completed this year by Arras, Flanagan, Morsink, Schenk, Teukolsky, and Wasserman [3]. The implication of their work is opposite to the indications of the fully nonlinear evolutions: Nonlinear couplings sharply limit the amplitude of an unstable r-mode.

Their work is consistent with the fully nonlinear numerical evolutions, because the coupling they find to low-order modes is consistent with a saturation amplitude of order unity. It is, instead, the coupling to many short-wavelength modes with frequencies comparable to that of the r-mode, that saturates the mode at an amplitude smaller than $10^{-2}$. (And the most recent fully nonlinear numerical evolution, by Gressman et al.[4], finds that increasing the resolution yields rapid nonlinear decay of the mode at decreased amplitude.) Nevertheless, r-modes may set the limit on rotation of young rapidly rotating neutron stars (this is less likely if the actual saturation amplitude is $10^{-4}$); and the r-mode instability may account for the maximum rotation period observed in old neutron stars spun-up by accretion (i.e., in x-ray binaries).

Exotic particles in the core of a neutron star lead to a significantly stronger viscous damping than assumed in the first studies of the unstable r-modes. Of particular relevance is the presence of hyperons and non-leptonic weak reactions. Hyperon-mediated damping of neutron-star oscillations was proposed by Langer and Cameron in 1969 and by Peter Jones in 1970, but the minimum density expected for hyperons was apparently $10^{15}$ g/cm$^3$ (see, for example, Zel'dovich and Novikov, v. 1, Sect. 11.5, in paragraphs written by Thorne). Spurred by Jones' recent reminder [5] that the r-mode instability would be completely suppressed in stars with a significant hyperon fraction, Lindblom and Owen [6], in a tour-de-force calculation using fully relativistic cross sections, found the bulk viscosity coefficient for non-leptonic reactions. Independently Haensel, Levenfish and Yakovlev [7] considered superfluid hyperons and obtained a set of approximate formulas for various bulk viscosity coefficients. These results show that, when hyperons are present, the dissipation due to direct URCA is overwhelming. In fact, the hyperon cooling is so rapid that, even without the enhanced hyperon bulk viscosity, no mode of a nascent neutron star would have time to grow before it was stabilized by viscous damping.

There is, however, significant uncertainty in the critical density at which hyperons appear and in the central density of neutron stars. Even for the equation of state (due to Glendenning) that Lindblom and Owen consider, the central density of a 1.4 M$_\odot$ star is near that needed for hyperons to appear when the rotation reaches its maximum value. The averaged measured mass of neutron stars is somewhat smaller than this; and the compressibility consistent with our knowledge of matter above nuclear density ranges over a factor of 5 or more for a 1.4 M$_\odot$ star, an error bar easily large enough to prevent our knowing whether or not neutron stars have hyperon cores.

The possibility that gravitational waves from unstable modes could balance the accretion torque, and hence halt the associated spin-up, was first discussed by Papaloizou and Pringle and Wagoner for the f-modes. The analogous scenario for the r-modes was analyzed in detail by Andersson, Kokkotas and Stergioulas and Bildsten a few years ago. Should this happen, neutron stars in LMXBs would be promising sources for detectable gravitational waves. In independent work, Spruit and Levin pointed out that this scenario may not work: The unstable mode will heat the star up (via the shear viscosity), and, if the viscosity gets weaker as the temperature increases, the mode-heating will trigger a rapid thermal runaway during which the star will spin down. This mechanism would seem to rule out r-modes in galactic LMXBs as a source of detectable gravitational waves, because they will only radiate for a tiny fraction of the system's lifetime.

Interestingly, the recent results indicating a small saturation amplitude and strong ``exotic'' bulk viscosity may imply that r-modes in these systems would be relevant gravitational-wave sources after all. Very recently, Wagoner [8] reported work showing that old, accreting neutron stars with a hyperon core will not undergo a thermal runaway, but will reach a the quasi-equilibrium state in which gravitational wave emission balances accretion spin-up and cooling balances shear viscosity heating. Andersson, Jones and Kokkotas [9] had previously found the same balance for strange stars, and it might also characterize accreting neutron stars with quark cores. Most importantly, in these scenarios the r-mode amplitude required to balance the accretion torque is orders of magnitude smaller than the saturation amplitude estimated by Arras et al. Finally, Heyl [10] has investigated the effect of a small saturation amplitude on the thermal runaway in a ``normal'' neutron star. His results show that the phase during which the r-modes radiate gravitationally is significantly extended if the modes cannot grow to a large amplitude (as long as the modes are not wiped out by the saturation mechanism).

One must place hyperon damping and nonlinear saturation at the top of the list of mechanisms that may kill or maim unstable r-modes [11], but reports of their demise may be premature.


1. N. Stergioulas and J.A. Font, Phys. Rev. Lett 86 1148 (2001)

2. L. Lindblom, J.E. Tohline and M. Vallisneri Phys. Rev. Lett 86 1152 (2001); Phys. Rev. D 65 084039 (2002)

3. P. Arras, E.E. Flanagan, S.M. Morsink, A.K. Schenk, S.A. Teukolsky and I. Wasserman, Saturation of the r-mode instability preprint astro-ph/0202345 ; S.M. Morsink, Ap.J. 571, 435 (2002); A.K. Schenk, P. Arras, E.E. Flanagan, S.A. Teukolsky and I. Wasserman, Phys. Rev. D 65, 024001 (2002).

4. P. Gressman, L.-M. Lin, W.-M. Suen, N. Stergioulas and J. L. Friedman, Phys. Rev. D 66, 041303 (2002).

5. P.B. Jones, Phys. Rev. Lett. 86 1384 (2001); Phys. Rev. D 64, 084003 (2001).

6. L. Lindblom and B.J. Owen, Phys. Rev. D 65 063006 (2002)

7. P. Haensel, K.P. Levenfish and D.G. Yakovlev, Astron Astrophys. 381 1080 (2002)

8. R.V. Wagoner, Conditions for Steady Gravitational Radiation from Accreting Neutron Stars astro-ph/0207589

9. N. Andersson, D.I. Jones and K.D. Kokkotas, Strange stars as persistent sources of gravitational waves preprint astro-ph/0111582

10. J. Heyl, LMXBs may be important LIGO sources after all astro-ph/0206174

11. For references to other potentially fatal mechanisms including magnetic field wind-up, and turbulent viscosity and enhanced shear viscosity in a boundary layer near the crust, see for example, the review Int.J.Mod.Phys. D10 381-442,(2001), by N. Andersson and K.D. Kokkotas.

Jorge Pullin 2002-09-23