3.3 Gravitational wave pulsars

Some likely gravitational wave sources behave like the centrifuge example we used in the first paragraph of this section, only on a grander scale. Suppose a neutron star of radius R and mass M spins with a frequency f and has an irregularity, a deformation of its otherwise axially symmetric shape. We idealize this as a “bump” of mass m on its surface, although of course it will really be a distribution of mass leading to an asymmetrical quadrupole tensor. The moment of inertia of the bump will be mR2, and it is conventional to parameterize the bump in terms of the fractional asymmetry it creates in the moment of inertia tensor itself. If we idealize the star as having uniform density, then the spherical moment of inertia is 2M R2∕5, and so the bump has fractional asymmetry
𝜖 = 5-m--, m = 0.4𝜖M. (22 ) 2 M
The bump will emit gravitational radiation at frequency 2f because the star spins about its net center of mass, so it effectively has mass excesses on both sides of the star. The nonspherical velocity will be just vnonsph = 2πRf. The radiation amplitude will be, from Equation (9View Equation),
hbump ∼ (4∕5 )(2πRf )2𝜖M ∕r, (23 )
and the luminosity, from Equation (15View Equation) (assuming that roughly four comparable components of Qjk contribute to the sum),
L ∼ (16∕125 )(2πf )6𝜖2M 2R4. bump

The radiated energy would presumably come from the rotational energy of the star 2 M v ∕5. This would lead to a spindown of the star on a timescale

( ) −1 tspindown ∼ 1M v2 ∕Lbump ∼ 25-𝜖− 2f −1 M-- v− 3. (24 ) 5 32π R
It is believed that neutron star crusts are not strong enough to support fractional asymmetries larger than about −6 𝜖 ∼ 10 [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 × 1010 s, about 2500 yr) would require an exceptionally large asymmetry. If we take the neutron star’s radius to be 10 km, so that M ∕R ∼ 0.21 and the speed of any irregularity is v∕c ∼ 6.2 × 10 −3, then Equation (24View Equation) would require an asymmetry of 𝜖 ∼ 1.4 × 10−3. 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 𝜖 ∼ 7 × 10− 4 [12Jump To The Next Citation Point]. 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 (23View Equation) the Crab pulsar would, if its spindown is dominated by gravitational wave losses, produce an amplitude at the Earth of h ∼ 1.5 × 10−24, 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 [12Jump To The Next Citation Point]. 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 I = 1038 kg m2, one gets an upper limit on the ellipticity of −4 𝜖 ≤ 1.8 × 10, assuming the pulsar is at 2 kpc. This is a factor of 4.2 below the spindown limit [12Jump To The Next Citation Point]. 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 𝜖 ≤ 9 × 10−4, 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 [79Jump To The Next Citation Point] 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 [310270]. 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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