The radiated energy would presumably come from the rotational energy of the star . This would lead to a spindown of the star on a timescale

It is believed that neutron star crusts are not strong enough to support fractional asymmetries larger than about [370], and realistic asymmetries may be much smaller. From these considerations one can estimate the likelihood that the observed spindown timescales of
pulsars are due to gravitational radiation. In most cases, it seems that gravitational wave losses
could account for a substantial amount of the spindown: the required asymmetries are much
smaller than 10^{–4}, often smaller than 10^{–7}. But an interesting exception is the Crab pulsar,
PSR J0534+2200, whose young age and consequently short spindown time (measured to be
8.0 × 10^{10} s, about 2500 yr) would require an exceptionally large asymmetry. If we take the
neutron star’s radius to be 10 km, so that and the speed of any irregularity
is , then Equation (24) would require an asymmetry of .
Of course, we have made a lot of approximations to get here, only keeping our estimates of
amplitudes and energies correct to within factors of two, but a more careful calculation reduces this
only by a factor of two to [12]. What makes this interesting is the fact that an
asymmetry this large would produce radiation detectable by first-generation interferometers.
Conversely, an upper limit from first-generation interferometers would provide direct observational
limits on the asymmetry and on the fraction of energy lost by the Crab pulsar to gravitational
waves.

From Equation (23) the Crab pulsar would, if its spindown is dominated by gravitational wave losses,
produce an amplitude at the Earth of , if its distance is 2 kpc. Is this detectable when
present instruments are only capable of seeing millisecond bursts of radiation at levels of 10^{–21}? The answer
is yes, if the observation time is long enough. Indeed, the latest LIGO observations have not detected any
gravitational waves from the Crab pulsar, which has been used to set an upper limit on the asymmetry
in its mass distribution [12]. The limit depends on the model assumed for the pulsar. If one
assumes that gravitational waves are produced at exactly twice the pulsar spin frequency and
uses the inferred values of the pulsar orientation and polarization angle, then for a canonical
value of the moment-of-inertia , one gets an upper limit on the ellipticity of
, assuming the pulsar is at 2 kpc. This is a factor of 4.2 below the spindown limit [12].
If, however, one assumes that gravitational waves are emitted at a frequency close, but not
exactly equal, to twice the spin frequency and one uses a uniform prior for the orientation and
polarization angle, then one gets , which is 0.8 of the limit derived from the spin-down
rate.

Indeed, even signals weaker than the amplitude determined by the Crab spindown rate will be observable by present detectors, and these may be coming from a larger variety of neutron stars, in particular low-mass X-ray binary systems (LMXBs). The neutron stars in them are accreting mass and angular momentum, so they should be spinning up. Observations suggest that most neutron stars are spinning at speeds between about 300 and 600 Hz, far below their maximum, which is greater than 1000 Hz. The absence of faster stars suggests that something stops them from spinning up beyond this range. Bildsten suggested [79] that the limiting mechanism may be the re-radiation of the accreted angular momentum in gravitational waves, possibly due to a quadrupole moment created by asymmetrical heating induced by the accreted matter. Another possible mechanism [286] is that a “bump” of the kind we have treated is formed by accreting matter channeled onto the surface by the star’s magnetic field. It is also possible that accretion drives an instability in the star that leads to steady emission [310, 270]. In either case, the stars could turn out to be long-lived sources of gravitational waves. This idea, which is a variant of one proposed long ago by Wagoner [383], is still speculative, but the numbers make a plausible case. We discuss it in more detail in Section 7.3.5.

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