7.3 Neutron star astrophysics

7.3.1 Gravitational collapse and the formation of neutron stars

The event that forms most neutron stars is the gravitational collapse that results in a supernova. It is difficult to predict the waveform or amplitude expected from this event. Although detecting this radiation has been a goal of detector development for decades, little more is known about what to expect than 30 years ago. The burst might be at any frequency between 100 Hz and 1 kHz, and it might be a regular chirp (from a rotating deformed core) or a more chaotic signal (from convective motions in the core). Considerable energy is released by a collapse, and on simple energetic grounds this source could produce strong radiation: if the emitted energy is more than about 0.01M ⊙, then second-generation detectors would have no trouble seeing events that occur in the Virgo Cluster. This energetic consideration drove the early development of bar detectors.

But numerical simulations tell a different story, and it seems very likely that radiation amplitudes will be much smaller, as described in Section 3. Such signals might be detectable by second-generation detectors from a supernova in our galaxy, but not from much greater distances. When they are finally detected, the gravitational waves will be extremely interesting, providing our only information about the dynamics inside the collapse, and helping to determine the equation of state of hot nuclear matter.

If gravitational collapse forms a neutron star spinning very rapidly, then it may be followed by a relatively long period (perhaps a year) of emission of nearly monochromatic gravitational radiation, as the r-mode instability (Section 7.3.4) forces the star to spin down to speeds of about 100 – 200 Hz [279Jump To The Next Citation Point]. If as few as 10% of all the neutron stars formed since star formation began (at a redshift of perhaps four) went through such a spindown, then they may have produced a detectable random background of gravitational radiation at frequencies down to 20 Hz [327].

7.3.2 Neutron-star–binary mergers

When two neutron stars merge, they will almost certainly have too much mass to remain as a star, and will eventually collapse to a black hole, unless they can somehow expel a significant amount of mass. The collision heats up the nuclear matter to a point where, at least initially, thermal pressure becomes significant. Numerical simulations can use theoretical equations of state (such as that of Lattimer and Swesty [237]) to predict the merger radiation, and observations will then test the nuclear physics assumptions that go into the equation of state. Simulations show that the choice of equation of state makes a big difference to the emitted waveform, as do the masses of the stars: there is no mass scaling as there is for black holes [62Jump To The Next Citation Point].

When a neutron star encounters a black hole in a stellar compact binary merger, the star may not be heated very much by the tidal forces, and the dynamics may be governed by the cold nuclear-matter equation of state, about which there is great uncertainty. Again, comparing observed with predicted waveforms may provide some insight into this equation of state. Simulations suggest that these systems may give rise to many of the observed short, hard gamma-ray bursts [155339]. Simultaneous gravitational wave and gamma ray detections would settle the issue and open the way to more detailed modeling of these systems.

7.3.3 Neutron-star normal mode oscillations

Gravitational wave observations at high frequencies of neutron-star vibrations may also constrain the cold-matter equation of state. In Figure 2View Image there is a dot for the typical neutron star. The corresponding frequency is the fundamental vibrational frequency of such an object. In fact, neutron stars have a rich spectrum of nonradial normal modes, which fall into several families: f, g, p, w, and r-modes have all been studied. These have been reviewed by Andersson and Comer [38Jump To The Next Citation Point]. If their gravitational wave emissions can be detected, then the details of their spectra would be a sensitive probe of their structure and of the equation of state of neutron stars, in much the same way that helioseismology probes the interior of the sun. Even knowing accurately the frequency and decay time of just the fundamental ℓ = 2 f-mode would be enough to eliminate most current equations of state [39Jump To The Next Citation Point].

This is a challenge to ground-based interferometers, which have so far focussed their efforts on frequencies below 1 kHz. But Advanced LIGO and the upgraded GEO-HF detector (Section 4.3.1) may have the capability to perform narrow-banding and enhance their sensitivity considerably at frequencies up to perhaps 2 kHz, which could put the f-modes of neutron stars into range.

The f-modes of neutron stars, which could be excited by glitches or by the nuclear explosions on accreting neutron stars that are thought to produce X-ray flares and soft gamma-ray repeater events. The rise-time of X-ray emission can be as short as a few milliseconds [173], which might be impulsive enough to excite acoustic vibrations. If the rise time of the explosion matches the period of the mode well enough, then a substantial fraction of the energy released could go into mechanical vibration, and almost all of this fraction would be carried away by gravitational waves, since other mode-damping mechanisms inside neutron stars are much less efficient.

Radio-pulsar glitches seem to release energies of order 1035 J, and X-ray and gamma ray events can be much more energetic. Using Equation (20View Equation), we can estimate that the release of that much energy into gravitational waves at 2 kHz at a distance of 1 kpc would create a wave of effective amplitude around 3 × 10–22. (The effective amplitude assumes we can do matched filtering, which in this case is not very difficult.) This kind of amplitude should be within the reach of Advanced LIGO (Figure 5View Image) and perhaps GEO-HF, provided they implement narrowbanding. This will not be easy, either scientifically or operationally, but the payoff in terms of our understanding of neutron star physics could be very substantial.

Observations of these modes would immediately constrain the cold-matter nuclear equation of state in significant ways [3938Jump To The Next Citation Point].

In fact, modes of neutron stars may have already been observed in X-rays [386]. But these are likely to be crustal modes, whose restoring force is the shear strength of the crust. While the physics of the crust is interesting in itself, such observations provide only weak constraints on the interior physics of the neutron star.

7.3.4 Stellar instabilities The CFS instability.
In 1971 Chandrasekhar [112] applied the quadrupole formula to calculate the corrections to the eignefrequencies of the normal mode vibrations of rotating stars, and he found to his surprise that some modes were made unstable, i.e., that coupling to gravitational radiation could destabilize a rotating star. Subsequent work by Friedman and Schutz [166] showed that there was a key signature for the mode of a Newtonian star that would be unstable in general relativity. This was the pattern speed of the mode, i.e., the angular velocity at which the crests of the pattern rotated about the rotation axis of the star. If this speed was in the same sense as the rotation of the star, but slower than the star, then the mode would be unstable in a perfect-fluid star. This instability has come to be known as the CFS instability, after the three authors who explained it. The basic theory was developed for perfect-fluid stars. However, Lindblom and Detweiler [240] showed that the effect of viscosity ran counter to that of radiation reaction, so that the instability was strongest in modes with the longest wavelengths, i.e., in the quadrupolar modes. Full numerical calculations on Newtonian stellar models with realistic viscosity models showed [241] that the standard fundamental and acoustic modes of rotating neutron stars were not vulnerable to this instability. Subsequent work on fully relativistic models [350] has hinted that the instability may be stronger than the Newtonian models indicate, but it is still at the margins of astrophysical interest. The r-mode instability.
The situation changed in 1997 when Andersson [37] pointed out that there is another class of modes of Newtonian stars that should be unstable in the same way, but which had not been studied in this context before, the Rossby or r-modes. These are momentum-dominated modes, where the gravitational radiation comes from the current-quadrupole terms, rather than from the mass quadrupole. Investigations by a number of authors [24340279] have shown that this instability could be very strong in hot, rapidly-rotating stars. This is particularly relevant to young neutron stars, which may well be formed with rapid spin and which will certainly be hot. For their first year, stars spinning faster than about 100 Hz could spin down to about 100 Hz by losing angular momentum to gravitational radiation. The instability might also operate in old accreting neutron stars, such as those in LMXB X-ray binaries (see the next section). However, the instability is, like other CFS instabilities, sensitive to viscosity, and there is great uncertainty about the amount of viscosity inside neutron stars [24222838].

7.3.5 Low-mass X-ray binaries

Observations by the Rossi satellite (RXTE) have given evidence that the class of X-ray sources called Low-Mass X-ray Binaries (LMXB’s) contains neutron stars with a remarkably narrow range of spins, between perhaps 250 Hz and 320 Hz [376]. These are systems in which it is believed that neutron stars are spun up from the low angular velocities they have after their lifetime as normal pulsars to the high spins that millisecond pulsars have. One would expect, therefore, that the spins of neutron stars in such systems would be spread over a wide range. The fact that they are not requires an explanation.

The most viable explanation offered so far is the suggestion of Bildsten [79] that gravitational radiation limits the rotation rate. The proposed mechanism is that anisotropic accretion onto the star creates a temperature gradient in the crust of the neutron star, which in turn creates a gradient in the mass of the nucleus that is in local equilibrium, and this in turn creates a density gradient that leads, via the rotation of the star, to the emission of gravitational radiation. This radiation carries away angular momentum, balancing that which is accreted, so that the star remains at an approximately constant speed.

According to the model, the gravitational wave luminosity of the star is proportional to the measured flux of X-rays, since the X-ray flux is itself proportional to the accreted angular momentum that has to be carried away by the gravitational waves. If this model is correct, then the X-ray source Sco X-1 might be marginally detectable by advanced interferometers, and other similar systems could also be candidates [385Jump To The Next Citation Point].

7.3.6 Galactic population of neutron stars

Neutron stars are known to astronomy through the pulsar phenomenon. As radio surveys improve, the number of known pulsars is pushing up toward 2000. There is a public catalogue on the web [57]. But the galactic population of neutron stars is orders of magnitude larger, perhaps as many as 108. Most are much older than typical pulsars, which seem to stop emitting after a few million years. X-ray surveys reveal a number of unidentified point sources, which might be hot neutron stars, but older neutron stars are probably not even hot enough to show up in such surveys.

Gravitational wave observations have the potential to discover more neutron stars, but in the foreseeable future the numbers will not be large. Spinning neutron stars can be found in searches for continuous-wave signals, but there is no a priori reason to expect significant deformations that would lead to large gravitational wave amplitudes. One mechanism, proposed by Cutler [128], is that a large buried toroidal magnetic field could, by pulling in the waist of a spinning star, turn it into a prolate spheroid. This is classically unstable and would tip over and spin about a short axis, emitting gravitational waves. Millisecond pulsars could, in principle, be spinning down through the emission of gravitational waves in this way. Only deep observations by Advanced LIGO could begin to probe this possibility.

In fact, strong emission of gravitational waves is in some sense counterproductive, since it causes a neutron star to spin down and move out of the observing band quickly. This places important limits on the likely distribution of observable continuous-wave amplitudes from neutron stars [220]. This is important input into the blind searches for such signals being conducted by the LSC.

Radio observations of pulsars have, of course, revealed a fascinating population of binary systems containing neutron stars, including the original Hulse–Taylor pulsar [203] and the double pulsar PSR J0737-3039 [248]. But radio surveys only cover a small fraction of our galaxy, so there may be many more interesting and exotic systems waiting to be discovered, including neutron stars orbiting black holes. In fact, not all neutron stars are pulsars, so there are likely to be nearby binary systems containing neutron stars that are not known as pulsars at all.

LISA has enough sensitivity to detect all such binaries in the galaxy whose gravitational wave emission is above 1 mHz, i.e., with orbital periods shorter than half an hour. Below that frequency, systems may just blend into the confusion noise of the white-dwarf background, unless they are particularly close. The Hulse–Taylor system is a bit below the LISA band, and even its higher harmonics are likely to be masked by the dense confusion noise of white-dwarf galaxies at low frequencies. Double pulsars should be detectable by LISA with low SNR (around five in five years) above the confusion background at a frequency of 0.2 mHz [211]. In all, LISA might detect several tens or even hundreds of double neutron-star systems, and potentially even a handful of double black hole binaries.

Neutron stars are the fossils of massive stars, and so a population census of binaries can help normalize our galaxy’s star-formation rate in the past. The mass distribution of such systems will also be of interest: do all neutron-star binaries have stars whose masses are near 1.4M ⊙, or is this only true of systems that become pulsars? LISA observations are likely to illuminate many puzzles of stellar evolution.

Finally, it is possible to search for gravitational waves from individual spinning neutron stars in binary systems. Although more rare than isolated neutron stars, these systems might have a different history and a different distribution of amplitudes. Searches are planned by the LSC, but they are difficult to do, since the parameter space is even larger than for isolated pulsars.

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