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
)
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)