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Gravitational Waves from Neutron Star Oscillations: New Theoretical Results

Gravitational Waves from Neutron Star Oscillations: New Theoretical Results

- Contributed by Gregory Mendell

Neutron stars are born in supernova explosions when approximately one solar mass of burnt nuclear matter in the core collapses from about the size of the Earth to the size of an average city. This collapse accomplishes three things. First, it compresses the matter to several times nuclear density, squeezing together electrons and protons, which interact to form neutrons at a density approaching a billion tons per teaspoon. Second, it spins up the star like an ice skater (while conserving angular momentum). The result is that neutron stars are the most rapidly rotating objects in the universe, with observed spin rates of up to 642 times per second. (The fastest ones have actually been respun in old age by accretion from a companion star. "Accretion" in this sense means the neutron star is draining matter off the companion onto itself.) And third, the collapse increases the magnetic field to a strength that is a trillion times the Earth's field (while conserving magnetic flux). (You wouldn't want to get your credit cards too close to a neutron star!) The combination of spin and magnetic field switches the neutron star on like a lighthouse, and it emits beams of electromagnetic waves that sweep through space. On Earth, radio telescopes detect these beams, which make a flash each time they sweep by (one flash per rotation of the star). These observed rapidly-flashing radio sources are called pulsars, and there are over 1000 known pulsars in the sky, all thought to correspond to rapidly rotating neutron stars.

Now, with the construction of LIGO and other gravitational wave observatories around the world, it may soon become possible to directly observe gravitational waves from neutron stars. (Gravitational waves have already been observed indirectly in binary pulsar systems.) The direct detection of such waves should help clarify many uncertainties about neutron star structure. (For example, the interior neutrons might form a superfluid, or they might condense into a soup of quarks and other strange stuff.) One way neutron stars might generate gravitational waves depends on an amazing prediction of Einstein's General Theory of Relativity, first worked out by S. Chandrasekhar in 1970: gravitational waves tend to drive all rotating stars unstable. The instability occurs when a vibration of the star travels opposite the rotation direction when viewed by observers rotating with the star, but travels in the same direction as the rotation when viewed by observers that remain motionless with respect to the distant stars. The situation is illustrated in Figure 1 below. In this figure, the blue circle represents the equatorial cross-section of a nonvibrating star. The central arrow shows the direction of rotation, which is taken to be clockwise. The dashed red and green lines represent a quadrupole vibration of the star at two instances in time (one quarter period apart). Such a vibration, according to General Relativity, will emit gravitational waves that carry energy and angular momentum away from the star. If the vibrational pattern rotates counterclockwise when viewed by observers rotating with the star, then the vibrational mode reduces the star's angular momentum. However, if the star spins fast enough, the vibrational pattern will be dragged clockwise when viewed by observers that remain motionless with respect to the distant stars. In this case the emitted gravitational waves carry angular momentum away from the star. The star must lose angular moment and it does so by increasing the amplitude of the angular-momentum-reducing vibrational mode. Thus an instability has set in! In more dynamical terms, the gravitational waves emanating from the vibrations push back on the vibrating matter in the same direction that the vibrations are trying to travel around the star. It is just like blowing on your cup of coffee. The waves build up.

Figure 1.
    Figure 1.   A vibrating star emits gravitational waves.

A few years ago N. Andersson along with J. Friedman and S. Morsink showed that a certain mode of oscillation, known as the r-mode, is always subjected to the gravitational-wave driven instability. R-modes are vibrational modes that exist in rotating stars. The mode corresponds to oscillating flows of material (currents) in the star that arise due to the Coriolis effect. When the instability is triggered (for example by perturbations left over from the star's violent birth) the amplitude of the mode grows exponentially. However, internal friction within the star (such as that due to viscosity) is expected to completely suppress the instability of the r-modes (and other modes) in all stars except neutron stars. In fact, a flurry of work by researchers showed that the r-mode instability was not only not suppressed in neutron stars, but that it was very strong in these stars (see Lindblom, Owen, & Morsink, Andersson, Kokkotas, and Schutz, Owen et al.). They found it could cause newborn neutron stars to emit most of their rotational energy as gravitational waves within the first year after their birth. This was a far larger effect than predicted for any other mode of neutron star vibration, making it possible that gravitational waves from the r-modes could be detected by the proposed enhanced version of LIGO.

One crucial factor that will determine whether gravitational waves from the r-mode instability can be detected by LIGO is the critical angular velocity (the spin rate) for the onset of the instability. Copious amounts of gravitational radiation are emitted from the r-modes only while the star spins faster than this critical rate. However Bildsten and Ushomirsky showed that viscous dissipation that occurs at the crust-core interface of a neutron star with a solid crust and fluid core will completely suppress the instability except for very rapid spin rates. (Their results were later extended by Andersson et al., Rieutord, Levin and Ushomirsky, Lindblom, Owen, and Ushomirsky.) To understand why this is so, consider the situation illustrated in Figure 2 below. Here, the red arrows represent the flow of an oscillating unbound fluid at one moment in time. In this example this flow is spatially uniform. However, if the fluid is bounded by a solid wall then an additional flow must develop (represented by the blue arrows) that cancels the original flow at the boundary. (A viscous fluid cannot slip at a perfectly rigid solid boundary.) The additional flow is large only near the boundary and falls off exponentially (in the case the fluid is oscillating) as one moves away from the boundary. Thus, elements of the fluid experience a large shear near the boundary, forming what is called a viscous boundary layer (VBL). The large shear in the VBL leads to a large amount of viscous dissipation which damps the oscillation of the fluid. (For example, consider the fluid element shown. The difference in the gradient of the flow on either side of the fluid element in the x-direction produces a net shear force on the fluid in the y-direction. This force damps the oscillation of the fluid, and the energy of the oscillation is converted to heat by the rubbing that occurs between the sheared layers of the fluid.)

Figure 2.
        Figure 2.   A viscous fluid cannot slip past a perfectly rigid boundary. This causes a large
                       shear flow to develop (blue arrows) in an oscillating fluid near a boundary.

The effect of viscous dissipation in a VBL on the r-modes seemed bad news for the detection of the r-modes by LIGO, since a newborn neutron star with a hot fluid crust would only spin briefly above the critical value needed to make the mode unstable before the star cooled and a solid crust formed. (This happens within minutes after its birth.) To make matters seemingly worse, it has recently been shown that a neutron star's magnetic field enhances the viscous boundary layer suppression of the r-mode instability (Mendell). (A neutron star, despite its name, has charged particles to produce and interact with the magnetic field: electrons and heavy ions exist in the crust, and a small fraction of protons and electrons exist in the core.) When magnetic fields exist perpendicular to the boundary, magnetic forces cause the viscous boundary layer to spread out, which would seem to decrease the shear. However, short wavelength vibrations of the magnetic field lines produces a large shear again which acts over the increased volume of the VBL, increasing the viscous dissipation rate. Mendell shows that the presence of typical neutron star magnetic field of a trillion Gauss will cause dissipation in the VBL to completely suppress the r-mode instability in any neutron star with a solid crust. The magnitude of the magnetic field oscillations is shown in Figure. 3 below, for the example shown in Fig. 2 generalized to include a magnetic field in the x-direction. (Note that the two figures are not drawn to scale. For example, the length-scale over which the shear flow, indicated by the blue arrows, exponentially decays is actually much larger for the case shown in Fig. 3 than that shown in Fig. 2.)

Figure 3.
         Figure 3.   Adding a magnetic field perpendicular to the boundary creates short wavelength oscillations
                        of the field lines that enhances the shear dissipation of the flow.

As just explained, the suppression of the r-mode instability by a VBL (magnetically enhanced or not) in a neutron star with a solid crust seems disheartening news for those interested in the detection of gravitational waves from the r-modes by LIGO. But all is not as it seems. The reason is that the previous results of Lindblom, Owen, and Ushomirsky showed that if the r-mode amplitude grows sufficiently large during a newborn neutron star's brief, hot, fluid-crust phase then a solid crust does not immediately form. Instead one gets an "ice-pack" of lumps of crust mixed with fluid while the r-mode spins down the star. The reason an "ice-pack" can form in the presence of an r-mode is that if a solid crust forms, heat from viscous dissipation of the r-mode in the VBL melts the solid crust, but once melted the star quickly cools below the melting temperature due to neutrino emission and the crust begins to reform. Thus, this competition between the heating and cooling forces the system into a mixed state of solid lumps and fluid, until "ice-pack" equilibrium at the melting temperature is reached. In this state, the r-modes can continue to spin down the star to very low spin rates, with it thus emitting large amount of gravitational waves in the process. Ironically, the increased viscous dissipation rate caused by the presence of a magnetic field may reduce the amplitude of the r-modes needed to make "ice-pack" formation favorable. However, more calculations are needed to decide whether this is so.

A second crucial factor that will determine whether the r-mode instability can be detected by LIGO is the saturation amplitude of the mode, i.e., the amplitude for which some effect causes it to stop growing. A new 3D hydrodynamic simulation carried out by Caltech's Lee Lindblom and Michele Vallisneri, and Louisiana State University's Joel E. Tohline, shows that the breaking of waves at the star's surface and shock formation causes saturation. Furthermore, they find that the saturation amplitude can be quite large (e.g., the perturbed velocity at the surface of the star can grow to 3.4 times the equilibrium value). This is a very important result, and good news for LIGO! But rather than going into the details in this brief article, an explanation, including a movie, can be found at:


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