Éanna Flanagan, Cornell
eef3@cornell.edu
The gravitational waves that bathe the Earth presumably do not vary wildly in strength from year to year. However, our very imperfect understanding of their strengths occasionally does change abruptly, prompted either by new astrophysical observations or by new theoretical predictions and discoveries. Such has been the case in the last year for our expectations for periodic gravitational waves from neutron stars, in two different scenarios: (i) accreting neutron stars in low mass X ray binaries (LMXBs), and (ii) hot, young neutron stars in the first year or so after their formation.
Accreting neutron stars in LMXBs:
As is well known, rotating neutron stars can radiate via three
mechanisms: (i) non-axisymmetry of the star, (ii) non-alignment
between the axis of rotation and a principle axis of the moment of
inertia tensor, and (iii) excitation of the stars' normal modes. For
mechanism (i), the amount of non-axisymmetry can be parameterized by
the equatorial eccentricity 
,
where 
is the moment of inertia tensor and the rotation axis
is the z axis. Advanced LIGO interferometers can see
non-axisymmetric neutron stars out to 
kpc for 
with 1/3 year integration time, where the
rotation frequency is 
Hz (Thorne 1998). The likely values of 
for various
neutron star populations are highly uncertain. Aside from the
millisecond pulsar population which is highly constrained, we know only that
(Thorne 1998).
Lars Bildsten has recently given fairly convincing theoretical and
observational arguments that many LMXBs like Scorpius X-1 should have
values of 
of order 
or larger and should thus
be fairly strong sources (Bildsten 1998).
First, recent observations by the Rossi X-Ray Timing Explorer
satellite indicate that many of the rapidly accreting stars have spin
frequencies clustered near 300 Hz. This is somewhat of a puzzle
since the accretion would be expected to spin up the stars to much
higher frequencies. Bildsten suggests an explanation for this puzzle:
that gravitational wave emission is preventing these sources from
being spun up any further, i.e., that all the angular momentum being
accreted is being radiated into gravitational waves. The limiting
angular velocity then scales as the 
th power of 
and is thus fairly insensitive to the amount of non-axisymmetry.
Second, Bildsten suggests a specific mechanism for generating the required
non-axisymmetry: that lateral temperature
gradients due to non-uniform accretion over the surface of the star lead (via
temperature-dependent electron capture reactions) to lateral density
variations in the crust. The resulting estimated values of
are of the order 
, consistent with what is
required.
The wave strengths for these sources can be predicted directly from
the the observed X-ray flux and the inferred accretion rate; the
amount of non-axisymmetry (quadrupole) is determined by demanding
equality of the spin-up and spin-down torques. The strongest source,
Sco X-1, is predicted to be detectable with 
years integration
with initial LIGO interferometers (Bildsten 1998). Thus, a priority
for the early data runs for LIGO and also VIRGO and GEO will be
directed searches for periodic signals from known accreting neutron
stars.
Hot young neutron stars -- the r-mode instability:
Just over a year ago, Andersson discovered that the r-modes of rotating neutron stars are unstable in the absence of viscosity, for all values of the star's angular velocity (Andersson 1998, see also Friedman and Morsink 1998). The instability is driven by gravitational radiation via the Chandrasekhar-Friedman-Schutz (CFS) mechanism (Chandrasekhar 1970, Friedman and Schutz 1978), and was reviewed by Sharon Morsink in Issue 10 of Matters of Gravity (Morsink 1997). Over the past year a flurry of papers have explored the dramatic astrophysical consequences of the r-mode instability. In this review I'll describe these predicted consequences, and summarize the uncertainties and implications. [For a detailed review of instabilities in rotating stars see Stergioulas 1998].
The picture that is emerging is the following. When a neutron star is
first formed it is likely spinning at a substantial fraction of its
maximum angular
velocity. While the star cools from 
to 
via neutrino emission over the first few years of its life, the
stars' r-modes are excited and radiate copious amounts of
gravitational radiation, carrying away as much as 
of
energy and most of the initial angular momentum of the star
(Lindblom et al. 1998, Andersson et al. 1998). When the
transition to a superfluid state occurs at 
, the star
is left with angular velocity of 
, the exact value being somewhat uncertain. Here 
is the maximum allowed angular velocity. The predicted wave
strengths are such that these sources could be seen out to the VIRGO
cluster (
) with enhanced LIGO interferometers
(Owen et al. 1998). This is quite an exciting prospect since the event
rate could be many per year.
This scenario is consistent with the inferred spin after formation of
the Crab pulsar of about 
, and also (within the
uncertainties of the predictions) of the initial spin period of 
of the recently discovered young pulsar PSR J0537-6910
(Owen et al. 1998). It also resolves the observational puzzle that
neutron stars seem to be formed with rather small spins despite one's
expectation of near maximal initial spins due to conservation of angular
momentum during stellar core collapse. It rules out
accretion-induced-collapse of white dwarfs as a mechanism for forming
millisecond pulsars; millisecond pulsars must form instead via accretion in which
the temperature never gets hot enough to trigger the r-mode
instability. Finally, since it now seems more likely than before that
typical stellar core collapses involve rapid rotation rates, it
improves the prospects of our detecting supernovae.
Turn now to the assumptions and calculations that underlie these
predictions. There
are two conditions for a mode in a realistic neutron star
to be CFS-unstable: (i) The mode must be retrograde with
respect to the star but prograde with respect to distant inertial
observers, the classical CFS condition. Not all r-modes will
satisfy this condition (Lindblom and Ipser 1998), but
the dominant l=m=2 r-mode will do so.
A crucial point is that this condition is satisfied for
all values of 
for unstable r-modes, whereas it is only
satisfied at large 
for the previously considered f-modes.
(ii) When one measures the mode's energy in the rotating frame, the
amount of energy
lost to viscous dissipation per cycle must be less than the amount of
energy per cycle that gravitational radiation reaction adds to the
mode. In other words, the viscous dissipation timescale must be longer
than the instability growth timescale. For the original CFS
instability, calculations in Newtonian and
post-Newtonian gravity (Lindblom 1995) had shown that these
conditions are satisfied for the l=m=2 f-mode only in a certain region
in the 
plane (where T is the stellar temperature) with
and 
,
where 
is the maximum angular velocity. The
dependence on temperature arises due to the strong dependence of the
coefficients of bulk and shear viscosity on temperature.
Since neutron stars are at temperatures 
only for the
first few years after their formation, and since it was not clear that
the initial value of 
would be 
,
the conventional view was that the CFS instability would probably not
be important in practice (see, eg, Thorne 1998).
This picture changed when the r-mode instability was discovered.
Two independent calculations in Newtonian gravity using a slow
rotation approximation have indicated that
the instability region in the 
plane is much larger for r-modes, extending down to 
(Lindblom et al. 1998, Andersson et al.\
1998).
For a newly born neutron star, the evolution of the stars angular velocity and of the r-mode amplitude was solved for by making the following assumptions (Owen et al. 1998): Assume that only the dominant, l=m=2, r-mode is relevant. Assume that the mode amplitude grows due to the instability until it saturates at a value of order unity due to nonlinear effects. [The predictions are not very sensitive to the assumed saturated value of the mode amplitude]. Then, use conservation of angular momentum to solve for the spin down of the star. While these assumptions seem reasonable, it will be important to verify the qualitative predictions by numerical calculations that allow for nonlinear mode-mode couplings, perhaps using post-Newtonian hydrodynamic codes. There are also uncertainties related to the values of the viscosity coefficients and the temperature of the transition to superfluidity; investigation into these issues is continuing. However, the overall picture of rapid spindown seems very robust.
To conclude, it is not often that elegant but somewhat arcane aspects of general relativity (like radiation reaction due to current multipoles) have such dramatic astrophysical and observational consequences. Let us hope for many more such discoveries.
References:
Andersson, N., 1998, Ap. J. 502, 708A (also gr-qc/9706075).
Andersson, N., Kokkotas, K., and Schutz, B.F., 1998, astro-ph/9805225. Bildsten, L., 1998, Ap. J. 501, L89 (also astro-ph/9804325).
Chandrasekhar, S., 1970, Phys. Rev. Lett., 24, 611.
Friedman, J.L., 1978, Commun. Math. Phys., 62 247. Friedman, J.L. and Morsink, S.M., 1998, Ap. J. 502, 714 (also gr-qc/9706073).
Friedman, J.L. and Schutz, B.F., 1978, Ap. J, 222, 281.
Lindblom, L., 1995, Ap. J. 438, 265. Lindblom, L. and Ipser, J.R., 1998, gr-qc/9807049. Lindblom, L., Owen, B.J., and Morsink, S.M., 1998, Phys. Rev. Lett.\ 80, 4843 (also gr-qc/9803053).
Morsink, S.M., 1997, Instability of rotating stars to axial perturbations, MOG No. 10, (also gr-qc/9709023.
Owen, B.J., Lindblom, L., Cutler, C., Schutz, B.F., Vecchio, A., and Andersson, N., 1998,
Stergioulas, N., 1998, Living Reviews in Relativity, Vol 1 (also gr-qc/9805012). Thorne, K.S., 1998, In R. M. Wald, editor, Black Holes and Relativistic Stars, pages 41-77, University of Chicago Press (also gr-qc/9706079).