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 [190
]. 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; [113]).
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 [60] 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 [190] 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 [255] 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 [7] 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 [190], 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 [116], 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 [75
] estimate there to be 850 WDNS binaries within 3 kpc of the
Sun which will merge within
.
Population syntheses by Tauris & Sennels [232] 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.
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Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer http://www.livingreviews.org/lrr-2001-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |