Although no radio pulsar in orbit around a black hole companion has so far been observed, 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 time-scale. The merging time of a binary system containing two compact objects due to the emission of gravitational radiation can be calculated from the following formula which requires only the component masses and current orbital period and eccentricity e :
Here are the masses of the two stars and . This formula is a good analytic approximation (within a few percent) to the numerical solution of the exact equations for in the original papers by Peters & Mathews [187, 188]. In the following subsections we review current knowledge on the population sizes and merging rates of such binaries where one component is visible as a radio pulsar.
Table 1: Known DNS binaries and candidates. Listed are the pulse period P, the orbital period , the orbital eccentricity e, the pulsar characteristic age , and the expected binary coalescence time-scale due to gravitational wave emission calculated from Equation (6). Cases for which is a factor of 100 or more greater than the age of the Universe are listed as . To distinguish between definite and candidate DNS systems, we also list whether the masses of both components have been determined.
Also listed in Table 1 are three further DNS candidates with eccentric orbits and large mass functions but for which there is presently not sufficient component mass information to confirm their nature.
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 B1534+12, 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 [195, 192] and is thought to make only a negligible contribution to the merger rate . The general approach with the remaining two systems is to derive scale factors for each object (as outlined in § 3.2.1) and then divide these by a reasonable estimate for the lifetime. In what follows we summarize the main studies of this kind. The most comprehensive investigation of the DNS binary population to date is the recent study by Kalogera et al. (hereafter KNST; ).
As discussed in § 3.2.1, scale factors are dependent on the assumed pulsar distribution. The key parameter here is the scale height of the population with respect to the Galactic plane which itself is a function of the velocity distribution of the population. KNST examined this dependence in detail and found scale heights in the range 0.8-1.7 kpc. Based on this range, KNST revised earlier scale factor estimates  to 145-200 for B1534+12 and 45-60 for PSR B1913+16. As mentioned in § 3.2.2 scale factors calculated from a small sample of objects are subject to a significant bias. KNST find the bias in their sample to be anywhere between 2 and 200. This boosts the scale factors to the range 190-40000 for B1534+12 and 90-12000 for B1913+16.
The above scale factors also require a beaming correction. As noted in § 3.2.3, current radio pulsar beaming models vary considerably. Fortunately, for the two pulsars under consideration, detailed studies of the beam sizes [9, 119, 262] lead KNST to conclude that both pulsars beam to only about a sixth of the entire sky. The beaming-corrected numbers suggest a total of between 1680 and 312,000 active DNS binaries in our Galaxy. Many of these systems will be extremely faint objects. These estimates are dominated by the small-number bias factor. KNST's study highlights the importance of this effect.
Some debate exists about what is the most reasonable estimate of the lifetime. Phinney  defines this as the sum of the pulsar's spin-down age plus defined above. A few years later, van den Heuvel and myself argued  that a more likely estimate can be obtained by appealing to steady-state arguments where we expect sources to be created at the same rate at which they are merging. The mean lifetime was then found to be about three times the current spin-down age. This argument does, however, depend on the luminosity evolution of radio pulsars which is currently only poorly understood. Arzoumanian, Cordes & Wasserman  used kinematic data to constrain the most likely ages of the DNS binaries. They note that the remaining detectable lifetime should also take account of the reduced detectability at later epochs due to acceleration smearing as the DNS binary becomes more compact due to gravitational wave emission. KNST concluded that the lifetimes are dominated by the latter time-scale which, following Arzoumanian et al., they took to be the time for the orbital period to halve. The resulting lifetimes are yr for B1534+12 and yr for B1913+16.
Taking these number and lifetime estimates, KNST find the Galactic merger rate of DNS binaries to range between and . Extrapolating this number out to include DNS binaries detectable by LIGO in other galaxies á la Phinney , KNST find the expected event rate to be for LIGO-I and 2-1300 for LIGO-II. Thus, despite the uncertainties, it seems that the prospects for detecting gravitational-wave emission from DNS inspirals in the near future are most promising.
PSR B2303+46 has a long orbital period and does not contribute significantly to the overall merger rate of WDNS binaries. The new discovery of PSR J1141-6545 , which will merge due to gravitational-wave emission within 1.3 Gyr, is suggestive of a large population of similar binaries. This is particularly compelling when one considers that the radio lifetime of the visible pulsar is only a fraction of total lifetime of the binary before coalescence due to gravitational-wave emission. Edwards & Bailes  estimate there to be 850 WDNS binaries within 3 kpc of the Sun which will merge within .
Population syntheses by Tauris & Sennels  suggest that the formation rate of WDNS binaries is between 10-20 times that of DNS binaries. Based on the merging rate estimates for DNS binaries discussed in the previous section, this translates to a merging rate of WDNS binaries of between and . In summary, although statistics are necessarily poor at this stage, coalescing WDNS binaries look to be very promising sources for gravitational wave detectors.
|Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer
© Max-Planck-Gesellschaft. ISSN 1433-8351
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