Gravitational waves from bumpy neutron stars

Benjamin J. Owen, Pennsylvania State University

It has often been overlooked in the recent fuss about unstable oscillation modes, but there is another way of getting gravitational waves out of single neutron stars. You might even say a more solid way: A crystalline crust might not be symmetric about the rotation axis of a star, giving the star a time-varying quadrupole moment.

This lack of symmetry could come either in the form of localized bumps (mountains), or a ``bump'' covering most of the star if the crust is symmetric about an axis different from the rotation axis. One problem is that, due to the nuclei having such low charge-to-mass ratios (being neutron-rich isotopes) and being so tightly packed, the so-called solid crust has essentially the same mechanical properties as Jell-O: It doesn't support bumps very well. Worse than Jell-O, neutron stars are expected to support bumps no higher than $10^{-7}$ times the star's radius. (For the derivation of this number, and other details and background, see the recent review by Ian Jones [1].)

Numerous mechanisms have been proposed since the 1970s for producing bumps or misaligned axes including strong ($10^{16}$G) magnetic fields, Magnus forces due to superfluid vortex pinning, and just plain settling as an aging star spins down and loses its centrifugal bulge. Until recently, however, there wasn't much hope of detecting gravitational waves from any bumpy neutron stars with LIGO or a comparable interferometer [2].

The revival of interest in bumpy neutron stars as gravitational-wave sources started about the same time as for unstable modes, when in 1998 Lars Bildsten [3] suggested that electron capture could make large mountains on the accreting neutron stars in low-mass x-ray binaries. The density of a neutron star crust doesn't increase smoothly as you go down towards the core, but rather in discrete jumps. The result is layered like an onion: At the bottom of each layer, a proton in each nucleus captures an electron due to the intense pressure and turns into a neutron (inverse beta decay). This changes the chemical composition in the next layer and allows the now less positively charged nuclei to come closer together. Since the pressure of each electron capture falls strongly with rising temperature, a rain of hot accreting matter falling unevenly around the star can create buried mountains many layers deep.

Bildsten revived an old argument by Wagoner [4] to show that, if there's a bump, you can work out the gravitational wave strain from the observed x-ray flux--assuming that torque-up from accretion is balanced by torque-down from gravitational waves. Some rapidly-accreting x-ray binaries, particularly Sco X-1, would produce gravitational wave strains as high as a few times $10^{-26}$ provided the assumption of torque balance holds.

Brady and Creighton [5] examined the details of data analysis and detection. Sco X-1 would be quite detectable by advanced LIGO (or a comparable interferometer) if its orbit and the small fluctuations of its spin about the torque-balanced value were precisely known, e.g. by radio observations. Sco X-1 and most of its cousins are seen only in x-rays, where the timing and orbital data are less precise. The need to search a large parameter space of possible data demodulations increases the minimum observable strain by 2.5, meaning that a broadband advanced LIGO configuration could barely detect Sco X-1 above the noise. A narrow-band signal-recycling configuration such as prototyped by GEO600 could get a signal-to-noise ratio several times higher. A good detection above the noise would allow the extraction of interesting physics of the crust such as breaking strain and probable thickness, which in turn tells us something about the equation of state.

Is torque balance a reasonable assumption? Ushomirsky et al. [6] checked by combining nuclear physics with geology. The bad news is that mountains settle into the mantle as well as poke up into the sky, reducing the quadrupole. The good news is that on an accreting neutron star the mountains extend over many layers, which brings the quadrupole back up. If a star has a 5% temperature variation over the surface and if the breaking strain of the crust is $10^{-2}$, even the fastest-accreting binaries such as Sco X-1 can achieve torque balance. More good news is that the crust won't smooth out a 5% temperature variation by conduction if the accretion is that irregular (and constant). More bad news is that $10^{-2}$ is on the very high end of predicted breaking strains, much higher than for terrestrial materials.

Another way of getting gravitational waves out of even axisymmetric neutron stars is free precession. If the crust's symmetry axis is tilted away from the rotation axis, both will precess about the fixed angular momentum vector. This free precession will radiate gravitational waves and tend to get damped both by radiation and by internal dissipation--the latter fairly quickly--and so the topic has been pretty quiet since the 1970s.

Free precession got a lot more notice recently when Stairs et al. [7] discovered a pulsar that shows the expected modulation in its radio signal. The star spins far too slowly for gravitational radiation to be significant, but it shed some light on the physics of precession (if there are vortices in a superfluid core they can't be pinned to the crust). Jones and Andersson [8] then systematically inventoried all known pumping mechanisms that could encourage rapid free precession and strong gravitational waves. Unfortunately the resulting signals are too faint even for advanced interferometers. Cutler and Jones [9] got excited briefly by the prospect of precessing neutron stars being unstable to gravitational radiation like the $r$-modes, but on closer inspection it turns out that an old result was wrong and radiation reaction always damps precession.

Stairs et al. aren't the only ones with a detection paper. Middleditch et al. [10] also claim to have observed a freely precessing pulsar with a 2ms spin period in the remnant of supernova 1987A. They also claim that its period derivative is consistent with spindown due to gravitational radiation! Obviously this is really hot if true, but there are some problems. Like the electron-capture scenario for Sco X-1, Middleditch's SN1987A would need a breaking strain of $10^{-2}$--tough but conceivable. Furthermore, the extra moment of inertia due to the bump would have to have changed by a factor of 2 in a few years while somehow maintaining a constant precession angle, which is much harder to believe. Finally, the pulsar has been seen only by Middleditch's team, and then only sporadically in the 1990s despite close observation.

Very recently the old mechanism of bumps through strong magnetic fields got a new twist from Curt Cutler [11]. It has been rediscovered several times since the 1970s that a magnetic field of more than a few times $10^{12}$G--a reasonable value for some young neutron stars--is secularly unstable in a neutron star with an elastic crust. That is, if the magnetic field axis is misaligned with the star's rotation axis, flexing of the crust due to free precession will let the field move into its lowest energy configuration. This happens to be one where the field axis is perpendicular to the rotation axis. Cutler points out that an internal toroidal field can be made quite strong by differential rotation in a young neutron star while keeping the external dipole magnetic field low, consistent with observations. This also means that gravitational radiation braking dominates electromagnetic braking. As a result stars in various scenarios--x-ray binaries, recycled millisecond pulsars, even (briefly) newborn neutron stars--could then be detectable by advanced interferometers.

To sum it up: The goods may be odd, but the odds are pretty good that within a few years we'll be seeing gravitational-wave signals from some known pulsars with big bumps on them. The old bumping mechanisms from the 1970s have been pepped up considerably, and with the observation of at least one precessing pulsar hopes are high for more to come.

References [1] D. I. Jones, Class. Quantum Grav. 19, 1255 (2002). Proceedings of the 4th Amaldi Conference.

[2] P. R. Brady et al., Phys. Rev. D 57, 2101 (1998).

[3] L. Bildsten, Astrophys. J. 501, L89 (1998).

[4] R. V. Wagoner, Astrophys. J. 278, 345 (1984).

[5] P. R. Brady and T. Creighton, Phys. Rev. D 61, 082001 (2000).

[6] G. Ushomirsky, C. Cutler, and L. Bildsten, MNRAS 319, 902 (2000).

[7] I. Stairs, A. G. Lyne, and S. L. Shemar, Nature 406, 484 (2000).

[8] D. I. Jones and N. Andersson, MNRAS 331, 203 (2002).

[9] C. Cutler and D. I. Jones, Phys. Rev. D 63, 024002 (2000).

[10] J. Middleditch et al., New Astronomy 5, 243 (2000).

[11] C. Cutler, astro-ph/0206051.