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12.5 Gravitational wave asteroseismology

The development of gravitational wave detectors like LIGO [67], VIRGO [208], TAMA300 [301] and GEO600 [299] is opening up a new window of astronomical observations. With a central density on the order of ∼ 1015 g cm–3, neutron stars are among the most compact objects in the universe, and could be efficient sources of gravitational waves. The existence of such waves predicted by Einstein’s theory of general relativity was beautifully confirmed by the observations of the binary pulsar PSR 1913+16 by Russel Hulse and Joseph Taylor (who were awarded the Nobel Prize in 1993 [306]). To compare the importance of general relativistic effects in binary pulsars with those around ordinary stars like the Sun, let us remember that the pulsar’s periastron in PSR 1913+16 advances every day by the same amount as Mercury’s perihelion advances in a century!

12.5.1 Mountains on neutron stars

A spherical neutron star does not radiate gravitational waves, in accordance with Birkhoff’s theorem. This is still true for an axially-symmetric neutron star. However, a neutron star with nonaxial deformations, rigidly rotating with the angular frequency Ω, radiates gravitational waves and thus loses energy at a rate given by the formula

32 G E˙GR = − ----5I2ε2Ω6 , (291 ) 5 c
where ε is a dimensionless parameter characterizing deformations of the star. Pulsar timing data can be used to derive an upper bound on this parameter ε. Since the energy radiated away in gravitational waves can be at most equal to the loss of kinetic energy due to the spinning down of the pulsar, this implies
( )3 ∕2( )1∕2( )− 1∕2 ε < 3 × 10−9 -P--- --P˙-- -----I----- , (292 ) 1 ms 10− 19 1045 g cm2
where P, P˙ and I are, respectively, the pulsar’s period, period derivative and moment of inertia.

The parameter ε can be constrained, independent of the pulsar timing data, by direct observations with gravitational wave detectors [220]

h d ( f )− 2( I ) −1 ε < 0.237------------ ----- ----------- , (293 ) 10 −241 kpc 1 Hz 1045 g cm2
where h is the amplitude of a gravitational wave signal, f the neutron star’s spin frequency and d is the distance to the source. 78 radio pulsars have recently been observed using the LIGO detector [1]. The analysis of the data imposes upper limits on the maximum amplitude h of the gravitational waves emitted by these pulsars. The two inequalities (292View Equation) and (293View Equation) can be combined to put constraints on the moment of inertia I and on the parameter ε of a given pulsar. The example of the Crab pulsar [2Jump To The Next Citation Point] is shown in Figure 78View Image. Note that the constraint from the gravitational wave data has already reached the level of the spin-down limit.
View Image

Figure 78: The moment of inertia I vs. the deformation parameter ε for the Crab pulsar over the S5 run of the LIGO detector. The areas to the right of the diagonal lines are the experimentally excluded regions. The horizontal lines represent theoretical upper and lower limits on the moment of inertia. The lines labelled “uniform prior” and “restricted prior” are the results obtained respectively without and with prior information on the gravitational wave signal parameters. From [2].

The crust of a neutron star contains only a very small percentage of the mass of the star. Nevertheless, its elastic response to centrifugal, magnetic and tidal forces determines the overall shape of the star. The presence of mountains on the surface of the star leads to a nonvanishing value of the parameter ε. If the star is rotating around one of the principal axes of the inertia tensor, ε is given by [373Jump To The Next Citation Point]

I1 −-I2 ε = I , (294 )
where I1 and I2 are the moments of inertia of the star with respect to the principal axes orthogonal to the rotation axis. The topography of the surface of a neutron star depends on the elastic properties of the solid crust. In particular, the size of the highest mountain depends on the breaking strain σmax, beyond which the crust will crack. This parameter is believed to lie in the range 10–5 – 10–2, as argued by Smoluchowski [382]. Considering small perturbations in a Newtonian star composed of an incompressible liquid of density ρ, surrounded by a thin crust with a constant shear modulus μ, the deformation parameter ε is given by the formula [192]
9 μσ ΔR ε = ------max----, (295 ) 80 G ρ2 R3
where ΔR the thickness of the solid crust. Adopting the value μ = 1030 g cm–1 s–2 (see Section 7.1), ε can be written as
( ) ( )2 ( )3 ( ) ε ≃ 5 × 10− 5 σmax- 1.4-M-⊙- --R---- ΔR--- . (296 ) 10 −3 M 10 km 1 km
The size of the highest mountain of order ∼ εR can be roughly estimated from the inequality (292View Equation). For instance, the mountains on the surface of the Crab pulsar could be as high as a few meters, while in PSR 1957+20 they cannot exceed a few microns! A neutron star can also be deformed by its magnetic field (Section 6.4, see also [193Jump To The Next Citation Point] and references therein). The deformation depends on the configuration of the magnetic field and scales like
B2R4 ε ∝ ----2-. (297 ) GM
For a Newtonian star composed of an incompressible liquid, the deformation induced by an internal magnetic field is approximately given by [193]
( ) ( )4 ( ) −2 ^ 2 ε ≃ 10 −12 --R---- --M---- ---B--- , (298 ) 10 km 1.4M ⊙ 1012 G
where ^B is a suitably-averaged magnetic field. Comparing Equations (296View Equation) and (298View Equation) we conclude that for a canonical neutron star (M = 1.4 M ⊙, R = 10 km, ΔR = 1 km, ^ 12 B = 10 G) the deformations induced by the magnetic field are much smaller than those supportable by the elasticity of the crust. However, for B^ ∼ 1015 G, characteristic of magnetars, both deformations have comparable magnitudes. Moreover, it has been shown that neutron stars with large toroidal fields evolve into configurations, where the angular momentum of the star is orthogonal to the magnetic axis. Such configurations are associated with strong gravitational wave emission (see [105] and references therein).

12.5.2 Oscillations and precession

The presence of magnetic fields inside neutron stars or mountains on the surface are not the only mechanisms for the emission of gravitational waves. Time-dependent nonaxisymmetric deformations can also be caused by oscillations. For instance, a neutron star with a solid crust, rotating about some axis (i.e. not aligned with any principal axis of the stellar moment of inertia tensor) will precess. For a small wobble angle θ, the deformation parameter is given by [373]

ε = θ-I1 −-I3-, (299 ) 4 I
where I1, I2 and I3 are the principal moments of inertia (assuming I1 = I2), θ is the angle between the direction of the angular momentum and the stellar symmetry axis. Observational evidence of long-period precessions have been reported in PSR 1828–11 [387], PSR B1642–03 [371] and RXJ 0720.4–3125 [180].

A large number of different nonaxisymmetric neutron-star–oscillation modes exists, for instance, in the liquid surface layers (“ocean”), in the solid crust, in the liquid core and at the interfaces between the different regions of the star. These oscillations can be excited by thermonuclear explosions induced by the accretion of matter from a companion star, by starquakes, by dynamic instabilities growing on a timescale on the order of the oscillation period or by secular instabilities driven by dissipative processes and growing on a much longer timescale. Oscillation modes are also expected to be excited during the formation of the neutron star in a supernova explosion. The nature of these modes, their frequency, their growing and damping timescales depend on the structure and composition of the star (for a review, see, for instance, [28524412]).

12.5.3 Crust-core boundary and r-mode instability

Of particular astrophysical interest are the inertial modes or Rossby waves (simply referred to as r-modes) in neutron star cores. They can be made unstable by the radiation of gravitational waves on short timescales of a few seconds in the most rapidly-rotating neutron stars [17]. However, the growth of these modes can be damped. One of the main damping mechanisms is the formation of a viscous boundary Ekman layer at the crust-core interface [49] (see also [164] and references therein). It has been argued that the heat dissipated in this way could even melt the crust [265]. The damping rate depends crucially on the structure of bottom layers of the crust and scales like (δv∕v)2, where δv is the slippage velocity at the crust-core interface [263Jump To The Next Citation Point]. Let us suppose that the liquid in the core does not penetrate inside the crust, like a liquid inside a bucket. The slippage velocity in this case is very large δv ∼ v and as a consequence the r-modes are strongly damped. However, these assumptions are not realistic. First of all, the crust is not perfectly rigid, as discussed in Section 7. On the contrary, the crust is quite “soft” to shear deformations because μ ∕P ∼ 10−2, where μ is the shear modulus and P the pressure. The oscillation modes of the liquid core are coupled to the elastic modes in the crust, which results in much smaller damping rates [263163]. Besides, the transition between the crust and the core might be quite smooth. Indeed, neutron superfluid in the core permeates the inner crust and the denser layers of the crust could be formed of nuclear “pastas” with elastic properties similar to those of liquid crystals (see Section 7.2). The slippage velocity at the bottom of the crust could, therefore, be very small δv ≪ v. Consequently, the Ekman damping rate of the r-modes could be much weaker than the available estimates. If the crust were purely fluid, the damping rate would be vanishingly small. However the presence of the magnetic field would also affect the damping time scale and should be taken into account [289238]. Besides the character of the core oscillation modes is likely to be affected by coupling with the crust. The role of the crust in the dissipation of the r-mode instability is, thus, far from being fully understood. Finally, let us mention that by far the strongest damping mechanism of r-modes, due to a huge bulk viscosity, may be located in the inner neutron star core, provided it contains hyperons (see, e.g., [182264] and references therein).

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