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 , radiates gravitational waves and thus loses energy at a rate
given by the formula
The parameter can be constrained, independent of the pulsar timing data, by direct observations
with gravitational wave detectors [220]
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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 . If the
star is rotating around one of the principal axes of the inertia tensor,
is given by [373
]
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 , the deformation parameter is given by [373]
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, [285, 244, 12]).
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 , where
is
the slippage velocity at the crust-core interface [263
]. 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
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
, where
is the shear modulus and
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 [263, 163]. 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
. 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 [289, 238]. 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., [182, 264] and references
therein).
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