### 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 18. Isochrones showing various coalescence times are calculated using
the expression
where and are the masses of the two stars, is the so-called
“reduced mass”, is the binary period and is the eccentricity. This formula is a good
analytic approximation (within a few percent) to the numerical solution of the exact equations for
[245, 246].
In addition to testing general relativity through observations of these 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 [112], LIGO [46], VIRGO [131] and TAMA [225]. In
the following, we review recent empirical determination 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; despite extensive searches, only five certain DNS binaries are currently known: PSRs
J07373039 [44], B1534+12 [343], J17562251 [93], B1913+16 [129] and B2127+11C [257].
Although we only see both neutron stars as pulsars in J07373039 [198], we are “certain” of the
identification in the other four systems from precise component mass measurements from pulsar timing
observations (see Section 4.4). The spin and orbital parameters, timescales and mass constraints for
these systems are listed in Table 1, along with three other binaries with 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.

Update
Despite the uncertainties in identifying DNS binaries, for the purposes of determining the Galactic merger rate, the
systems
for which is less than (i.e. PSRs J07373039, B1534+12, J17562251, 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 [257, 252] and is thought to make only a negligible contribution to the
merger rate [248]. 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. It is generally
accepted [151] 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 [9]. Below this point, the orbital
smearing selection effect discussed in Section 3.1.3 will render the binary undetectable by current
surveys.
The results of the most recent study of this kind [147, 148], which take into account the discovery of
PSR J07373039, are summarised in the left panel of Figure 19. The combined Galactic merger rate,
dominated by the double pulsar, is found to be , 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 [248], the expected event rate is for
initial LIGO and for advanced LIGO. Future prospects for detecting gravitational wave
emission from binary neutron star inspirals are therefore very encouraging. Since much of the uncertainty in
the rate estimates is due to our ignorance of the underlying distribution of double neutron star
systems, future detection rates could ultimately constrain the properties of this exciting binary
species.
Although the double pulsar system J07373039 will not be important for ground-based detectors
until its final coalescence in another , it may be a useful calibration source for the future
space-based detector LISA [224]. Recent calculations [146] show that a observation with LISA would
detect (albeit with ) the continuous emission at a frequency of based on the current
orbital parameters. Although there is the prospect of using LISA to detect similar systems systems
through their continuous emission, current calculations [146] suggest that significant ()
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 18, we note that only three WDNS
systems are currently known that will merge within , PSRs J0751+1807 [192], J17575322 [90] and
J11416545 [154]. Applying the same techniques as used for the DNS population, the merging rate
contributions of the three systems can be calculated [157] and are shown in Figure 19. The combined
Galactic coalescence rate is (at 68% confidence interval). 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 [224]. 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 [157]. Similar conclusions were reached by considering the statistics of low-mass X-ray
binaries [69].