3.4 The population of relativistic binaries

Although no radio pulsar has so far been observed in orbit around a black hole companion, we now know of several double neutron star and neutron star–white dwarf binaries which will merge due to gravitational wave emission within a reasonable timescale. The current sample of objects is shown as a function of orbital period and eccentricity in Figure 18View Image. Isochrones showing various coalescence times τg are calculated using the expression
( )8∕3 ( ) −2∕3( )− 1 τ ≃ 9.83 × 106 yr Pb- m1--+-m2- -μ-- (1 − e2)7∕2 , (7 ) g hr M ⊙ M ⊙
where m1 and m2 are the masses of the two stars, μ = m1m2 ∕(m1 + m2) is the “reduced mass”, Pb is the binary period and e is the eccentricity. This formula is a good analytic approximation (within a few percent) to the numerical solution of the exact equations for τg [283284].
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Figure 18: The relativistic binary merging plane. Top: Orbital eccentricity versus period for eccentric binary systems involving neutron stars. Bottom: Orbital period distribution for the massive white dwarf–pulsar binaries. Isocrones show coalescence times assuming neutron stars of 1.4M ⊙ and white dwarfs of 0.3M ⊙.

In addition to tests of strong-field gravity through observations of relativistic binary systems (see Section 4.4), estimates of their Galactic population and merger rate are of great interest as one of the prime sources for current gravitational wave detectors such as GEO600 [124], LIGO [208], VIRGO [375] and TAMA [352]. In the following, we review empirical determinations of the population sizes and merging rates of binaries where at least one component is visible as a radio pulsar.

3.4.1 Double neutron star binaries

As discussed in Section 2.2, double neutron star (DNS) binaries are expected to be rare. This is certainly the case; as summarized in Table 1, only around ten DNS binaries are currently known. Although we only see both neutron stars as pulsars in J0737–3039 [242Jump To The Next Citation Point], we are “certain” of the identification in five other systems from precise component mass measurements from pulsar timing observations (see Section 4.4). The other systems listed in Table 1 have eccentric orbits, mass functions and periastron advance measurements that are consistent with a DNS identification, but for which there is presently not sufficient component mass information to confirm their nature. One further DNS candidate, the 95-ms pulsar J1753–2243 (see Table 3), has recently been discovered [189Jump To The Next Citation Point]. Although the mass function for this pulsar is lower than the DNS systems listed in Table 1, a neutron star companion cannot be ruled out in this case. Further observations should soon clarify the nature of this system. We note, however, that the 13.6-day orbital period of this system means that it will not contribute to gravitational wave inspiral rate calculations discussed below.

Table 1: Known and likely DNS binaries. Listed are the pulse period P, orbital period Pb, orbital eccentricity e, characteristic age τ c, and expected binary coalescence timescale τ g due to gravitational wave emission calculated from Equation (7View Equation). To distinguish between definite and candidate DNS systems, we also list whether the masses of both components have been determined via the measurement of two or more post-Keplerian parameters as described in Section 4.4.
J0737–3039 J1518+4904 B1534+12 J1756–2251 J1811–1736
P [ms] 22.7/2770 40.9 37.9 28.5 104.2
Pb [d] 0.102 8.6 0.4 0.32 18.8
e 0.088 0.25 0.27 0.18 0.83
log10(τc/[yr]) 8.3/7.7 10.3 8.4 8.6 9.0
log10(τg/[yr]) 7.9 12.4 9.4 10.2 13.0
Masses measured? Yes No Yes Yes Yes
B1820–11 J1829+2456 J1906+0746 B1913+16 B2127+11C
P [ms] 279.8 41.0 144.1 59.0 30.5
P b [d] 357.8 1.18 0.17 0.3 0.3
e 0.79 0.14 0.085 0.62 0.68
log10(τc/[yr]) 6.5 10.1 5.1 8.0 8.0
log10(τg/[yr]) 15.8 10.8 8.5 8.5 8.3
Masses measured? No No Yes Yes Yes

Despite the uncertainties in identifying DNS binaries, for the purposes of determining the Galactic merger rate, the systems for which τg is less than τu (i.e. PSRs J0737–3039, B1534+12, J1756–2251, J1906+0746, B1913+16 and B2127+11C) are primarily of interest. Of these PSR B2127+11C is in the process of being ejected from the globular cluster M15 [295290] and is thought to make only a negligible contribution to the merger rate [286Jump To The Next Citation Point]. The general approach with the remaining systems is to derive scale factors for each object, construct the probability density function of their total population (as outlined in Section 3.2.1) and then divide these by a reasonable estimate for the lifetime. Getting such estimates is, however, difficult. It has been proposed [181] that the observable lifetimes for these systems are determined by the timescale on which the current orbital period is reduced by a factor of two [10]. Below this point, the orbital smearing selection effect discussed in Section 3.1.3 will render the binary undetectable by current surveys. More recent work [76Jump To The Next Citation Point] has suggested that a significant population of highly eccentric binary systems could easily evade detection due to their short lifetimes before gravitational wave inspiral. If this selection effect is significant, then the merger rate estimates quoted below could easily be underestimated by a factor of a few.

The results of the most recent DNS merger rate estimates of this kind [191Jump To The Next Citation Point] are summarised in the left panel of Figure 19View Image. The combined Galactic merger rate, dominated by the double pulsar and J1906+0746 is found to be 118+174Myr −1 −79, where the uncertainties reflect the 95% confidence level using the techniques summarised in Section 3.2.2. Extrapolating this number to include DNS binaries detectable by LIGO in other galaxies [286], the expected event rate is +73 −3 −1 49−33 × 10 yr for initial LIGO and 265+3−91078 yr− 1 for advanced LIGO. Future prospects for detecting gravitational wave emission from binary neutron star inspirals are therefore very encouraging, especially if the population of highly eccentric systems is significant [76]. Since much of the uncertainty in the rate estimates is due to our ignorance of the underlying distribution of double neutron star systems, future gravitational wave detection could ultimately constrain the properties of this exciting binary species.

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Figure 19: The current best empirical estimates of the coalescence rates of relativistic binaries involving neutron stars. The individual contributions from each known binary system are shown as dashed lines, while the solid line shows the total probability density function on a logarithmic and (inset) linear scale. The left panel shows the most recent analysis for DNS binaries [191Jump To The Next Citation Point], while the right panel shows the equivalent results for NS–WD binaries [193Jump To The Next Citation Point]. Figures provided by Chunglee Kim.

Although the double pulsar system J0737–3039 will not be important for ground-based detectors until its final coalescence in another 85 Myr, it may be a useful calibration source for the future space-based detector LISA [215Jump To The Next Citation Point]. It is calculated [178Jump To The Next Citation Point] that a 1-yr observation with LISA would detect (albeit with S∕N ∼ 2) the continuous emission at a frequency of 0.2 mHz based on the current orbital parameters. Although there is the prospect of using LISA to detect similar systems through their continuous emission, current calculations [178] suggest that significant (S∕N > 5) detections are not likely. Despite these limitations, it is likely that LISA observations will be able to place independent constraints on the Galactic DNS binary population after several years of operation.

3.4.2 White dwarf–neutron star binaries

Although the population of white dwarf–neutron star (WDNS) binaries in general is substantial, the fraction which will merge due to gravitational wave emission is small. Like the DNS binaries, the observed WDNS sample suffers from small-number statistics. From Figure 18View Image, we note that only three WDNS systems are currently known that will merge within τu, PSRs J0751+1807 [238Jump To The Next Citation Point], J1757–5322 [103Jump To The Next Citation Point] and J1141–6545 [186Jump To The Next Citation Point]. Applying the same techniques as used for the DNS population, the merging rate contributions of the three systems can be calculated [193Jump To The Next Citation Point] and are shown in Figure 19View Image. The combined Galactic coalescence rate is +5 −1 4−3 Myr (at 68% confidence interval). This result is not corrected for beaming and therefore should be regarded as a lower limit on the total event rate. Although the orbital frequencies of these objects at coalescence are too low to be detected by LIGO, they do fall within the band planned for LISA [215]. Unfortunately, an extrapolation of the Galactic event rate out to distances at which such events would be detectable by LISA does not suggest that these systems will be a major source of detection [193Jump To The Next Citation Point]. Similar conclusions were reached by considering the statistics of low-mass X-ray binaries [81].

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