An update on the r-mode instability

Nils Andersson, University of Southampton, UK
N.Andersson@maths.soton.ac.uk
In the last two years the r-modes in rotating neutron stars have attracted a lot of attention. The main reason for this is that they are unstable due to the emission of gravitational waves via a mechanism that was discovered by Chandrasekhar, Friedman and Schutz more than 20 years ago. Until recently the r-modes --which are essentially horizontal currents associated with very small density variations-- had not been considered in this context. Hence the discovery that they are unstable at all rates of rotation in a perfect fluid star [1] came as a slight surprise. And even more of a surprise was the subsequent realization [2] that the unstable r-modes (which radiate mainly through the current multipoles) provide a much more severe constraint on the rotation rate of viscous stars (viscosity tends to counteract mode-growth due to gravitational radiation) than the previously considered f-modes (which are dominated by the mass multipoles). A direct comparison shows that the f-mode becomes unstable when the star is spun up to roughly 95% of the mass-shedding limit, while the dominant r-mode becomes unstable already at 5% of the maximum spin rate (at some temperature). With these early estimates, the r-mode instability emerged as a potential agent for spinning nascent neutron stars down to rotation rates similar to the initial period inferred for the Crab pulsar ( $P\approx 19$ ms), in the process radiating an amount of gravitational waves that should be detectable with LIGO II for sources in the Virgo cluster [3]. It was also suggested [4] that the instability could operate in older, colder neutron stars and perhaps explain the clustering of spin periods in the range 260-590 Hz of accreting neutron stars in Low-Mass X-ray Binaries (LMXB) indicated by the kHz QPOs. Understandably, these possibilities created some excitement among workers in this field and some 50 papers discussing the r-mode instability have since appeared. My intention here is to provide an update on the current status of this discussion without going into too much detail. The interested reader is referred to an exhaustive review on the subject [5] (and the original papers, of course). My main aim is to describe how our understanding of the r-modes and the associated instability has changed since the first studies. To some extent this is a simple task, because all the original ideas remain relevant. No one has yet provided a demonstration that the mechanism cannot work or that our original thinking was seriously flawed. This is, of course, good news. Less comforting is the fact that the questions that need to be addressed to make further progress are very hard and involve a lot of essentially unknown physics. A natural point of departure for this survey is the case for r-modes in hot young neutron stars emerging from supernova explosions. A newly born neutron star should cool to the temperature at which the dominant r-mode goes unstable (a few times 1010 K) in a few seconds. Provided that the star spins fast enough the r-mode will then grow with an ``e-unfolding'' time of a few tens of seconds until it enters the nonlinear regime and... then what? In the first studies of the problem it was assumed that nonlinear effects (e.g. coupling to other modes) would lead to the mode saturating at some large amplitude [3]. The mode would continue to radiate away angular momentum and the star would spin down from the mass-shedding limit to a period of 15-20 ms in a year or so [2]. In these models a crucial parameter is the amplitude of saturation. In order for the instability to have a dramatic effect on the spin-evolution of a young neutron star, the r-mode must be allowed to grow to a reasonably large amplitude. Intuitively, one might expect non-linear effects to become relevant at much smaller mode-amplitudes than those considered in the early work. However, the indications are now that the mode will be able to grow surprisingly large. This is demonstrated by very recent 3D time-evolutions (using a fully nonlinear relativistic hydrodynamics code with the spacetime ``frozen'') of Stergioulas and Font [6]. The first results of investigations into the nonlinear coupling between r-modes and other modes seem to point in the same direction [7]. There are no signs of mode-saturation until at very large amplitudes. It should, of course, be noted that much work remains to be done on this problem before we can draw any firm conclusions. The original spin-down scenarios were based on the assumption that the star evolves along a sequence of uniformly rotation equilibrium models as it loses angular momentum. Recent work indicates that this is unlikely to be the case. One might expect that a large amplitude unstable mode will lead to differential rotation in the stellar fluid. It is well-known that this is the case for the bar-mode instability in the Maclaurin spheroids. Once spun up to the point where the bar-mode becomes unstable, the Maclaurin spheroids evolve through a sequence of differentially rotating Riemann S-ellipsoids. One might expect an analogous evolution for stars governed by the r-mode instability. Evidence in favor of this possibility have been presented by Rezzolla, Lamb and Shapiro [8]. They argue that the r-mode leads to a nonlinear differential drift of the various fluid elements. Their calculation is based on inferring higher order (in the mode-amplitude) results from established linear results, and may not be quantitatively reliable, but it provides an indication that nonlinear effects will severely alter the fluid motion. This result is supported both by the time-evolutions of Stergioulas and Font [6] and a shell toy model studied by Levin and Ushomirsky [9]. In the latter case the nonlinear effects can be determined exactly, and they lead to the anticipated differential drift. Furthermore, the shell toy-model shows that, once radiation reaction is implemented, another source of differential rotation comes into play. Thus it would seem almost certain that differential rotation will play a key role in any realistic r-mode scenario. Differential rotation immediately brings magnetic field effects into focus. While effects due to electromagnetic waves generated by an oscillation mode are typically small [10], differential rotation may lead to a twisting of the field lines and a dramatic increase in the field strength. In the case of the r-modes the instability scenario may lead to the generation of a very strong toroidal magnetic field [8]. Following an original suggestion by Bildsten and Ushomirsky [11], much work in the last eight months or so has been focussed on the interface between the fluid core and the solid crust in a slightly older neutron star. Given that the crust is likely form already at a temperature of the order of 1010 K this discussion is relevant for all but very young neutron stars. Bildsten and Ushomirsky showed that a viscous boundary layer at the crust-core interface would lead to a very strong dissipation mechanism that would prevent the instability from operating unless the rotation period was very short. The original estimates seemed to suggest that the r-mode instability would not be relevant in the LMXBs and that it would not be able to spin a newly born neutron star down to spin periods beyond a few milliseconds. With more detailed studies these suggestions have been revised [12], and it now seems as if the instability could well be relevant for the LMXBs (perhaps leading to a cyclical spin-evolution [13]). But the uncertainties are large and many issues remain to be explored in this context. The crust-core discussion has led to suggestions that the heat released in the viscous boundary layer may, in fact, melt the crust. An interesting possibility, suggested by Lindblom, Owen and Ushomirsky [14], is that the final outcome is a kind of mixed state, with ``chunks of crust'' immersed in the fluid. To estimate the mode-dissipation associated with such a situation is, of course, very difficult. Also worth mentioning in this context are the results of Wu, Matzner and Arras [15]. They argue that the crust-core boundary layer is likely to be turbulent which would provide a mechanism for saturation. However, one can infer that the resultant saturation amplitude is of order unity for rapidly rotating stars. This could well indicate that the modes saturate due to some alternative, as yet unspecified, mechanism. Progress on all these issues is somewhat hampered by the lack of detailed quantitative results. It may be appropriate to provide a contrast by concluding this discussion by emphasizing two particular cases where hard calculations have provided relevant results. The first of these concerns r-modes in superfluid stars. This is an important issue since the bulk of a neutron star is expected to become superfluid once it cools below a few times 109 K. At this point some rather exotic dissipation mechanisms come into play, and it turns out (somewhat paradoxically) that a superfluid star is more dissipative than a normal fluid one. The most important new mechanism is the so-called mutual friction which has been shown to completely suppress the instability associated with the f-modes. The initial expectations were that mutual friction would also have a strong effect on the r-modes [2,3]. Detailed calculations by Lindblom and Mendell [16] have shown that this is not necessarily the case. The outcome depends rather sensitively on the detailed superfluid model (the parameters of the so-called entrainment effect), and only in a small set of the models considered by Lindblom and Mendell do mutual friction affect the r-modes in a significant way. It would thus seem as if the r-mode instability may prevail also in superfluid stars. Another important issue regards r-modes in fully relativistic stars. After all, the instability is a truly relativistic effect (being driven by gravitational radiation) and a relativistic calculation is required if we want to understand radiation reaction ``beyond the quadrupole formula''. And it should be recalled that the quadrupole formula leads to a significant error (it deviates from the true result by 20-30% already for $M/R\approx 0.03$) in estimates of the gravitational wave damping of the f-mode [17]. Furthermore, it is known that relativistic effects tend to further destabilize the f-modes [18]. While the quadrupole f-mode does not become unstable below the mass-shedding limit in a Newtonian star it does so in the relativistic case. For all these reasons the modeling of relativistic r-modes is a crucial step towards improved estimates of the instability timescales. It turns out that relativistic modes whose dynamics is mainly determined by the Coriolis force generally have a ``hybrid'' nature. This makes the calculation rather complicated, but significant progress on determining the relativistic analogue of the Newtonian r-modes has been made recently. These results are detailed in Lockitch's PhD thesis [19], as well as a recent paper [20] where the post-Newtonian corrections to the l=m r-modes of uniform density stars are calculated. Estimates of the growth rate of the unstable modes in the fully relativistic case are currently being worked out as an extension of this work. At this point I hope it is clear that, despite some recent progress, the uncertainties regarding the astrophysical role of the r-mode instability remain considerable. This is obviously somewhat disconcerting since it means that our understanding of this mechanism is not detailed enough to provide reliable theoretical templates that can be used to search for the associated gravitational waves in data taken by LIGO, GEO600, VIRGO or TAMA. In fact, I think it is quite unlikely that theorists will be able to provide this kind of information any time soon. After all, a detailed understanding of the involved issues demands a successful modeling of a regime where many extremes of physics meet. In view of this, I believe the challenge is to invent a pragmatic detection strategy based on general principles rather than detailed theoretical information. After all, would it not be quite exciting if an actual detection would provide us with some of the missing pieces of this pulsar puzzle, and help improve our understanding of general relativity, supranuclear physics, magnetic fields, superfluidity etcetera?

References:

[1]  N. Andersson Astrophys. J., 502, 708 (1998); J.L. Friedman and S.M. Morsink Astrophys. J., 502, 714 (1998)

[2]   L. Lindblom, B.J. Owen and S.M. Morsink Phys. Rev. Lett. , 80, 4843 (1998); N.W Andersson, K.D. Kokkotas and B.F. Schutz Astrophys. J., 510, 846 (1999)

[3]  B.J. Owen, L. Lindblom, C. Cutler, B.F. Schutz, A. Vecchio and N. Andersson Phys. Rev. D, 58, 084020 (1998)

[4]  L. Bildsten Astrophys. J. Lett., 501, 89 (1998); N. Andersson, K.D. Kokkotas and N. Stergioulas Astrophys. J., 516, 307 (1999)

[5]  N. Andersson and K.D. Kokkotas to appear in Int. J. Mod. Phys. D (2000)

[6]  N. Stergioulas and J.A. Font Nonlinear r-modes in rapidly rotating relativistic stars preprint astro-ph/0007086

[7]  S. Morsink private communication

[8]  L. Rezzolla, F.K. Lamb and S.L. Shapiro Astrophys. J. Lett., 531, 139 (2000)

[9]  Y. Levin and G. Ushomirsky Nonlinear r-modes in a spherical shell: issues of principle preprint astro-ph/9911295

[10]  W.C.G. Ho and D. Lai R-mode oscillations and spindown of young rotating magnetic neutron stars preprint astro-ph/9912296

[11]  L. Bildsten and G. Ushomirsky Astrophys. J. Lett., 529, 33 (2000)

[12]  N. Andersson, D.I. Jones, K.D. Kokkotas and N. Stergioulas Astrophys. J. Lett., 534, 75 (2000); S. Yoshida and U. Lee R-modes of neutron stars with a solid crust preprint astro-ph/0006107; Y. Levin and G. Ushomirsky Crust-core coupling and r-mode damping in neutron stars: a toy model preprint astro-ph/0006028

[13]  Y. Levin Astrophys. J., 517, 328 (1999)

[14]   L. Lindblom, B.J. Owen and G. Ushomirsky Effect of a neutron-star crust on the r-mode instability preprint astro-ph/0006242

[15]  Y. Wu, C.D. Matzner and P. Arras R-modes in Neutron Stars with Crusts: Turbulent Saturation, Spin-down, and Crust Melting preprint astro-ph/0006123

[16]  L. Lindblom and G. Mendell Phys. Rev. D, 61, 104003 (2000)

[17]  E. Balbinski, B.F. Schutz, S. Detweiler and L. Lindblom Mon. Not. R. Astr. Soc., 213, 553 (1985)

[18]  N. Stergioulas, Living Reviews in Relativity, 1998-8 (1998)
http://www.livingreviews.org/Articles/Volume1/1998-8stergio/

[19]  K.H Lockitch Stability and rotational mixing of modes in Newtonian and relativistic stars Ph.D. Thesis, University of Wisconsin - Milwaukee (1999). Available as preprint gr-qc/9909029

[20]  K.L. Lockitch, N. Andersson and J.L. Friedman to appear in Phys. Rev. D (2000)