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).

Mon Sep 7 17:37:02 EDT 1998