Although -body simulations have the
potential to provide the most detailed population syntheses of
relativistic binaries in globular clusters, there are very few
actual populations described in the literature. Most of the current
work that treats binaries in a consistent and detailed way is
limited to open clusters [128
, 80
, 95
] and is focused on a
particular outcome of the binary population, such as blue
stragglers in the case of Hurley et
al. [80], or brown dwarfs
in the case of Kroupa et
al. [95]. Portegies
Zwart et al. focus on photometric
observations of open clusters, but promise a more detailed look at
the binary population in a future spectroscopic paper [128].
In their comparison of
-body and Fokker-Planck
simulations of the evolution of globular clusters, Takahashi and
Portegies Zwart [155] followed
the evolution of
, and
systems with initial mass functions given by
Equation (9
) and initial density
profiles set up from King models. Although they allowed for
realistic stellar binary evolution in their comparisons, their
focus was on the structural evolution of globular clusters.
Consequently there is no binary population provided. Other
-body simulations suffer from this same
problem [125].
It is possible to generate a population
distribution for black hole binaries in globular clusters using the
-body simulations of Portegies Zwart and
McMillan [127] that were intended to
describe the population of black hole binaries that were ejected from globular clusters. Their
scenario for black hole binary ejection describes a population of
massive stars that evolves into black holes. The black holes then
rapidly segregate to the core and begin to form binaries. As the
black holes are significantly more massive than the other stars,
they effectively form a separate sub-system, which interacts solely
with itself. The black holes form binaries and then harden through
binary-single black hole interactions that occasionally eject
either the binary, the single black hole, or both.
They simulated this scenario using and
systems with 1% massive
stars. The results of their simulations roughly confirm a
theoretical argument based on the recoil velocity that a binary
receives during an interaction. Noting that each encounter
increases the binding energy by about 20% and that roughly
of this energy goes into binary recoil, the minimum
binding energy
of an ejected black hole binary is
At the end of this phase of black hole binary
ejection, there is a 50% chance that a binary remains in the
cluster with no other black hole to eject it. Thus, there should be
a stellar mass black hole binary remaining in about half of the
galactic globular clusters. The maximum binding energy of the
remaining black hole binary is and is also
given by Equation (29
). We can then
approximate the distribution in energies of the remaining black
hole binaries as being flat in
. The
eccentricities of this population will follow a thermal
distribution with
.
Dynamical Monte Carlo simulations can be used
to study the evolution of binary populations within evolving
globular cluster models. Rasio et
al. [134] have used a Monte
Carlo approach (described in Joshi et
al. [88, 89]) to study the
formation and evolution of NS-WD binaries, which may be progenitors
of the large population of millisecond pulsars being discovered in
globular clusters (see Section 3.3). In
addition to producing the appropriate population of binary
millisecond pulsars to match observations, the simulations also
indicate the existence of a population of NS-WD binaries (see
Figure 8
).
|
There is also great promise for the hybrid gas/Monte Carlo method being developed by Spurzem and Giersz [153]. Their recent simulation of the evolution of a cluster of 300,000 equal point-mass stars and 30,000 binaries yields a wealth of detail about the position and energy distribution of binaries in the cluster [58]. One expects that the inclusion of stellar evolution and a mass spectrum would produce similar detail concerning relativistic binaries.
Perhaps because of the paucity of observations
of double white dwarf binaries (there is only one candidate He-CO
binary [41]), there have been
few population syntheses of WD-WD binaries in globular clusters.
Sigurdsson and Phinney [151
] use Monte Carlo
simulations of binary encounters to infer populations using a
static background cluster described by an isotropic King-Michie
model. Their results are focused toward predicting the observable
end products of binary evolution such as millisecond pulsars,
cataclysmic variables, and blue stragglers. Therefore, there are no
clear descriptions of relativistic binary populations provided.
Davies and collaborators also use the technique of calculating
encounter rates (based on calculations of cross-sections for
various binary interactions and number densities of stars using
King-Michie static models) to determine the production of end
products of binary evolution [33
, 31
]. Although they also
do not provide a clear description of a population of relativistic
binaries, their results allow the estimation of such a
population.
Using the encounter rates of Davies and
collaborators [33, 31], one can follow the
evolution of binaries injected into the core of a cluster. A
fraction of these binaries will evolve into compact binaries which
will then be brought into contact through the emission of
gravitational radiation. By following the evolution of these
binaries from their emergence from common envelope to contact, we
can construct a population and period distribution for present day
globular clusters [16
]. For a globular
cluster with dimensionless central potential
, Davies [31
] followed the
evolution of 1000 binaries over two runs. The binaries were chosen
from a Salpeter IMF with exponent
, and the
common envelope evolution used an efficiency parameter
. One run was terminated after 15 Gyr and the
population of relativistic binaries which had been brought into
contact through gravitational radiation emission was noted. The
second run was allowed to continue until all binaries were either
in merged or contact systems. There are four classes of
relativistic binaries that are brought into contact by
gravitational radiation: high mass white dwarf-white dwarf binaries
with total mass above the Chandrasekhar mass; low
mass white dwarf-white dwarf binaries
with total mass
below the Chandrasekhar mass; neutron star-white dwarf binaries
; and neutron star-neutron star binaries
.A 1
The number of systems brought into contact at the end of each run is given in Table 3.
In the second run, the relativistic binaries had
all been brought into contact. In similar runs, this occurs after
another 15 Gyr. An estimate of the present-day period
distribution can be made by assuming a constant merger rate over
the second 15 Gyr. Consider the total number of binaries that
will merge to be described by . Thus, the merger rate is
. Assuming that the mergers are driven solely by
gravitational radiation, we can relate
to the
present-day period distribution. We define
to be the number of binaries with period less than
P, and thus
The merger rate is given by the number of mergers of each binary type per 1000 primordial binaries per 15 Gyr. If the orbits have been circularized (which is quite likely if the binaries have been formed through a common envelope), the evolution of the period due to gravitational radiation losses is given by [76]
whereFollowing this reasoning and using the numbers in
Table 3, we can determine the
present day population of relativistic binaries per 1000 primordial
binaries. To find the population for a typical cluster, we need to
determine the primordial binary fraction for globular clusters.
Estimates of the binary fraction in globular clusters range from
13% up to about 40% based on observations of either eclipsing
binaries [5, 165, 166] or luminosity
functions [138, 139].
Assuming a binary fraction of 30%, we can determine the number of
relativistic binaries with short orbital period for a typical cluster with
and the galactic globular cluster system with
[151] by
simply integrating the period distribution from contact
up to
The expected populations for an individual
cluster and the galactic cluster system are shown in Table 4
using neutron star masses of , white dwarf
masses of
and
, and
.
Although we have assumed the orbits of these
binaries will be circularized, there is the possible exception of
binaries, which may have a thermal distribution of
eccentricities if they have been formed through exchange
interactions rather than through a common envelope. In this case,
Equations (32
) and (33
) are no longer valid.
An integration over both period and eccentricity, using the
formulae of Pierro and Pinto [122], would be
required.
The small number of observed relativistic
binaries can be used to infer the population of dark progenitor
systems [18]. For
example, the low-mass X-ray binary systems are bright enough that
we see essentially all of those that are in the galactic globular
cluster system. If we assume that the ultracompact ones originate
from detached WD-NS systems, then we can estimate the number of
progenitor systems by looking at the time spent by the system in
both phases. Let be the number of ultracompact LMXBs and
be their typical lifetime. Also, let
be the number of detached WD-NS systems that will
evolve to become LMXBs, and
be the time spent during the
inspiral due to the emission of gravitational radiation until the
companion white dwarf fills its Roche lobe. If the process is
stationary, we must have
There are four known ultracompact
LMXBs [37] with orbital
periods small enough to require a degenerate white dwarf companion
to the neutron star. There are six other LMXBs with unknown orbital
periods. Thus, . The lifetime
is rather uncertain, depending upon the nature of
the mass transfer and the timing when the mass transfer would
cease. A standard treatment of mass transfer driven by
gravitational radiation alone gives an upper bound of
[131], but other
effects such as tidal heating or irradiation may shorten this to
[8, 134]. The value of
depends critically upon the evolution of the neutron
star-main-sequence binary, and is very uncertain. Both
and
depend upon the masses of the WD
secondary and the NS primary. For a rough estimate, we take the
mass of the secondary to be a typical He WD of mass
and the mass of the primary to be
. Rather than estimate the typical period of
emergence from the common envelope, we arbitrarily choose
. We can be certain that all progenitors have emerged
from the common envelope by the time the orbital period is this
low. The value of
can be determined by using
Equation (35
) and the radius of the
white dwarf as determined by Lynden-Bell and O’Dwyer [99].
Adopting the optimistic values of
and
, and evaluating Equation (37
) gives
. Thus, we find
, which
is within an order of magnitude of the numbers found through
dynamical simulations (Section 5.3.2)
and encounter rate estimations (Section 5.3.3).
Continuing in the spirit of small number
statistics, we note that there is one known radio pulsar in a
globular cluster NS-NS binary (B2127+11C) and about 50 known radio
pulsars in the globular cluster system as a whole (although this
number may continue to grow) [98]. We may estimate
that NS-NS binaries make up roughly
of the total
number of neutron stars in the globular cluster system. A lower
limit on the number of neutron stars comes from estimates of the
total number of active radio pulsars in clusters, giving
[97]. Thus, we
can estimate the total number of NS-NS binaries to be
. Not all of these will be in compact orbits, but we
can again estimate the number of systems in compact orbits by
assuming that the systems gradually decay through gravitational
radiation and thus