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 FokkerPlanck 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 subsystem, which interacts solely with itself. The black holes form binaries and then harden through binarysingle 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
where is the average mass of a globular cluster star and is the dimensionless central potential. After most binaries are ejected, . After a few gigayears, nearly all of the black holes were ejected.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 NSWD 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 NSWD 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 pointmass 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 HeCO binary [41]), there have been few population syntheses of WDWD 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 KingMichie 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 crosssections for various binary interactions and number densities of stars using KingMichie 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 dwarfwhite dwarf binaries with total mass above the Chandrasekhar mass; low mass white dwarfwhite dwarf binaries with total mass below the Chandrasekhar mass; neutron starwhite dwarf binaries ; and neutron starneutron 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 presentday 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 presentday period distribution. We define to be the number of binaries with period less than P, and thus
soThe 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]
where is given by with the “chirp mass” .Following 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 value of can be determined by using the Roche lobe radius of Eggleton [42], and stellar radii as determined by LyndenBell and O’Dwyer [99].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 lowmass Xray 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 WDNS 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 WDNS 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
The time spent in the inspiral phase can be found from integrating Equation (32) to get where is the period at which the progenitor emerges from the common envelope and is the period at which RLOF from the white dwarf to the neutron star begins. Thus, the number of detached progenitors can be estimated fromThere 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 starmainsequence 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 LyndenBell 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 NSNS 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 NSNS 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 NSNS 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
where is the number of systems in compact orbits (), is the time spent as a compact system, and is the typical time for a globular cluster NSNS binary to coalesce due to gravitational radiation inspiral. Adopting the coalescence time of B2127+11C as typical, [129], and integrating Equation (37) for two neutron stars, we find . Again this value compares favorably with the values found from encounter rate estimations.