Gravitational waves from accreting neutron stars

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Abstract

The observational evidence for gravitational wave emission from accreting millisecond pulsars, specifically the spin frequency distribution of the population, is reviewed. These continuous wave sources are promising candidates for detection by LIGO. Four theoretical mechanisms are discussed for producing the mass quadrupole moments inferred from the spin frequency data: thermal mountains (formed by gradients in the electron capture rate), r-modes in the core, magnetic mountains (formed by polar magnetic burial), and superfluid circulation in the core. For magnetic mountains, it is shown that the gravitational wave strain is inversely proportional to the magnetic moment, and that the gravitational wave spectrum displays distinctive sidebands as the magnetically confined mountain wobbles around. (The compression of the magnetic field towards the equator also modifies the physics of thermonuclear type I X-ray bursts in these objects, e.g., by thermally insulating the two hemispheres and leading to burst pairs.) For superfluid circulation in the core, it is shown that the high-Reynolds-number flow is nonaxisymmetric and emits a distinctive gravitational wave spectrum with two broad peaks.

Introduction

Progress in experimental gravitational wave physics has accelerated recently, following steady technological advances in the design and operation of long-baseline gravitational wave interferometers. Most excitingly, the Laser Interferometer Gravitational Wave Observatory (LIGO) has achieved its design sensitivity in the frequency range 0.1 kHz  f  1 kHz, the astrophysical ‘sweet spot’, with a spectral density minimum of 3 × 10−23  Hz−1/2 at f  0.15 kHz (Lazzarini, 2006). This represents a 300-fold sensitivity improvement in the strain noise amplitude from the first science run (S1, 17 days) to the fifth (S5, two calendar years when complete, corresponding to one year of triple coincidence data). The Advanced LIGO upgrade, equipped with more powerful lasers, better test-mass suspensions, and better thermal noise mitigation, and following a seismic retrofit, is expected to attain its optimum design sensitivity (2 × 10−24 Hz−1/2) by 2013.

LIGO was initially optimised to detect coalescing neutron-star binaries, whose Galactic archetype is the Hulse–Taylor binary pulsar. Following the discovery of the double pulsar PSR J0737–3039, which is predicted to coalesce in an unexpectedly short 85 Myr, Kalogera et al. (2004) performed a Bayesian analysis (including survey selection effects, e.g., small number and faint object biases) and concluded that the modal event rate is boosted six-fold relative to previous estimates, to 0.1 (360) yr−1 at 20 (350) Mpc for Initial (Advanced) LIGO.

Recently, another promising class of gravitational wave emitters has attracted attention: periodic (continuous wave) sources, an example of which is a nonaxisymmetric rotor. Continuous wave sources enjoy several advantages vis-à-vis coalescing neutron star binaries: (i) they are persistent, not transient; (ii) their frequency and sky position are often known a priori, from radio or X-ray timing, enabling the interferometer to integrate coherently over many cycles, and obviating the need for computationally expensive hierarchical Fourier searches; and (iii) their waveform is sinusoidal, not chirped (to a first approximation; the full reality is more complicated, as we explain below). Unlike coalescing neutron-star binaries, however, the amplitude of the signal from continuous wave sources is poorly known, as it depends on the messy internal physics of the emitting objects, e.g., the viscosity and magnetic field of neutron stars.

In this paper, we concentrate on one particular continuous wave source: accreting neutron stars. We review the observational evidence that these objects are gravitational wave emitters, namely the cutoff in their spin distribution above 0.7 kHz. We then describe four physical mechanisms for generating nonaxisymmetries in these objects: thermal mountains, r-modes in the core, magnetic mountains, and superfluid turbulence in the core.

Section snippets

Spin distribution of low-mass X-ray binaries

Low-mass X-ray binaries (LMXBs), in which a neutron star accretes gas from an evolved binary companion of mass <1M, are strong X-ray emitters. The spin frequency f of the neutron star in an LMXB can be measured in two different ways: (i) directly, from its X-ray pulsations; and (ii) indirectly, from millisecond oscillations in the tails of thermonuclear (type I) X-ray bursts. The accreting millisecond pulsars SAX J1808–3658 and XTE J1814–338, which simultaneously exhibit pulsations and burst

Thermal mountains

An accreting neutron star naturally develops lateral temperature gradients across its surface, e.g., between the accretion hot spot and the remainder of the star, or in the wake of a thermonuclear burst. Nuclear reactions occur at nonuniform rates as a result. For example, electron capture, which converts nuclei with (atomic weight, atomic number) = (A,Z) to (A,Z  1), occurs at lower densities in hot spots, creating ‘wavy’ capture layers whose mean composition and hence density vary across the

R-modes

R-modes (Rossby waves) are linear, nearly nonradial oscillations in a rotating fluid, in which the restoring force is provided by the Coriolis force. They are not axisymmetric, nor are they stationary in the corotating fluid frame. It has been postulated that r-modes are continuously excited in the core of an accreting neutron star (Andersson et al., 1999). Core r-modes of this sort are to be distinguished from surface r-modes excited in the neutron star ocean (Heyl, 2004).

The amplitude and

Magnetic mountains

Matter accreting onto a neutron star is funnelled onto the magnetic poles, where it accumulates to form a mountain. As the mountain grows, it spreads equatorward, dragging along the frozen-in polar magnetic field. The local (Alfvén) time-scale for hydromagnetic adjustment is much shorter than the accretion, Hall, and ohmic time-scales, so the mountain passes through a sequence of hydromagnetic (Grad-Shafranov) equilibria, in which the hydrostatic pressure at the base of the accreted column

Superfluid turbulence

Even when an accreting neutron star rotates steadily, without precession, subjected only to an accretion (spin-up or spin-down) torque acting on the crust, the flow in its superfluid interior is inevitably a source of gravitational waves. To understand why, we recall a well-known fact from laboratory experiments: a viscous fluid in a differentially rotating, spherical vessel cannot rotate uniformly. Ekman pumping drives meridional circulation within the vessel, and, at high Reynolds numbers Re  

Conclusion

The design sensitivity of Initial LIGO has recently been surpassed, and the technological upgrade path to Advanced LIGO is increasingly free of obstacles. Simultaneously, X-ray timing of LMXBs has revealed a cutoff in their spin frequency distribution above 0.7 kHz, which can be attributed to gravitational wave spin down if the neutron star has ellipticity ϵ  10−8. Taken together, these advances engender some confidence that LMXBs will be detected as continuous wave sources in the near future,

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