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5.3 Population syntheses

Over the last ten years, there have been several works addressing binary populations in globular clusters [19Jump To The Next Citation Point22Jump To The Next Citation Point45Jump To The Next Citation Point43Jump To The Next Citation Point444682Jump To The Next Citation Point112Jump To The Next Citation Point120Jump To The Next Citation Point160177184189Jump To The Next Citation Point196198Jump To The Next Citation Point208215Jump To The Next Citation Point219Jump To The Next Citation Point227Jump To The Next Citation Point]. These have been derived from both dynamical simulations and static models. Although the motivations have been varied, it is often possible to extract information about the resulting populations of relativistic binaries. Despite the differing models and population synthesis techniques, the predicted populations are in rough agreement. Here, we summarize the different techniques and their predictions for relativistic binaries in globular clusters.

5.3.1 N-body simulations

Although N-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 [190Jump To The Next Citation Point113Jump To The Next Citation Point112Jump To The Next Citation Point139Jump To The Next Citation Point149] and is focused on a particular outcome of the binary population, such as blue stragglers [113], brown dwarfs [139], initial binary distributions [140], or white dwarf CMD sequences [112]. Portegies Zwart et al. focus on photometric observations of open clusters [190] and on spectroscopy [186]. In their comparison of N-body and Fokker–Planck simulations of the evolution of globular clusters, Takahashi and Portegies Zwart [227] followed the evolution of N = 1 K, 16 K, and 32 K systems with initial mass functions given by Equation (9View Equation) 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 N-body simulations suffer from this same problem [188]. On the other hand, recent work by Shara and Hurley has focused specifically on white dwarf binary populations in globular clusters and has produced a detailed table of close white dwarf binaries that were generated in their simulation [215].

It is possible to generate a population distribution for black hole binaries in globular clusters using the N-body simulations of Portegies Zwart and McMillan [189] 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 N = 2048 and N = 4096 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 1/3 of this energy goes into binary recoil, the minimum binding energy Eb min of an ejected black hole binary is

Eb min ∼ 36W0 Mbh-kT, (30 ) ⟨M ⟩
where ⟨M ⟩ is the average mass of a globular cluster star and W = ⟨M ⟩|φ |∕kT 0 0 is the dimensionless central potential. After most binaries are ejected, ⟨M ⟩ ∼ 0.4 M ⊙. 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 Eb min and is also given by Equation (30View Equation). We can then approximate the distribution in energies of the remaining black hole binaries as being flat in log(Eb). The eccentricities of this population will follow a thermal distribution with P (e) = 2e.

5.3.2 Monte Carlo simulations

Dynamical Monte Carlo simulations can be used to study the evolution of binary populations within evolving globular cluster models. Rasio et al. [198Jump To The Next Citation Point] have used a Monte Carlo approach (described in Joshi et al. [125126]) 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 9View Image).

View Image

Figure 9: Results of the Monte Carlo simulation of NS–WD binary generation and evolution in 47 Tuc. Each small dot represents a binary system. The circles and error bars are the 10 binary pulsars in 47 Tuc with well measured orbits. Systems in A have evolved through mass transfer from the white dwarf to the neutron star. Systems in B have not yet evolved through gravitational radiation to begin RLOF from the white dwarf to the neutron star. Systems in C will not undergo a common envelope phase. Figure taken from Rasio et al. [198Jump To The Next Citation Point].

The tail end of the systems in group B of Figure 9View Image represents the NS–WD binaries that are in very short period orbits and are undergoing a slow inspiral due to gravitational radiation. These few binaries can be used to infer an order of magnitude estimate on the population of such objects in the galactic globular cluster system. If we consider that there are two binaries with orbital period less than 2000 s out of 6 ∼ 10 M ⊙ in 47 Tuc, and assume that this rate is consistent throughout the globular cluster system as a whole, we find a total of ∼ 60 such binaries. Although this estimate is quite crude, it compares favorably with estimates arrived at through the encounter rate population syntheses, which are discussed in Section 5.3.3.

More recent applications of the Monte Carlo simulations that have focused on the properties of binaries include Fregeau et al. [62] who look at the production of blue stragglers and other collision products as a result of binary interactions in globular clusters and Ivanova et al. [120Jump To The Next Citation Point] who have studied the evolution of binary fractions in globular clusters. The latter work demonstrates the gradual burning of binaries in the core that delays the collapse of the core. In addition, they have also shown the build-up of short period white dwarf binaries in the core through dynamical interactions (see Figure 10View Image).

There is also great promise for the hybrid gas/Monte Carlo method being developed by Spurzem and Giersz [224]. 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 [81]. Further improvements on their code have resulted in direct integration of the binary-binary and binary-single interactions [82]. As a result, they have been able to produce empirical cross-sections for eccentricity variations during interactions.

View Image

Figure 10: Binary period distributions from the Monte Carlo simulation of binary fraction evolution in 47 Tuc. The bottom panel indicates the period distribution for binaries containing at least one white dwarf. Nb is the total number of binaries and Nb,p is the number of binaries per bin. Figure taken from Ivanova et al. [120Jump To The Next Citation Point].

5.3.3 Encounter rate techniques

One method for exploring the production of relativistic binary populations in globular clusters involves determining the encounter rate expected between different classes of objects in a globular cluster. Sigurdsson and Phinney [219Jump To The Next Citation Point] 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. The work of Ivanova et al. [120] also uses this technique to determine the evolution of binary fractions, but they also do not provide sufficient detail of the population to distinguish the relativistic binaries from other binaries in the simulation. There is promise to produce a more detailed description of ultracompact X-ray binaries consisting of a white dwarf and a neutron star using encounter rates [122Jump To The Next Citation Point].

Davies and collaborators 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 [45Jump To The Next Citation Point43Jump To The Next Citation Point]. 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 [4543Jump To The Next Citation Point], 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 [19Jump To The Next Citation Point]. For a globular cluster with dimensionless central potential W0 = 12, Davies [43] followed the evolution of 1000 binaries over two runs. The binaries were chosen from a Salpeter IMF with exponent α = 2.35, and the common envelope evolution used an efficiency parameter αCE = 0.4. 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: low mass WD–WD binaries (WD2 a) with total mass below the Chandrasekhar mass; high mass WD–WD binaries (2 WD b) with total mass above the Chandrasekhar mass; NS–WD binaries (NW); and NS–NS binaries 2 NS. The number of systems brought into contact at the end of each run is given in Table 3.

WD2 a
WD2 b

15 Gyr 11 0 10 1
∞ 57 0 74 18

Table 3: Number of relativistic binaries brought into contact through binary interactions.

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 n (t). Thus, the merger rate is η = − dn∕dt. Assuming that the mergers are driven solely by gravitational radiation, we can relate n(t) to the present-day period distribution. We define n (P) to be the number of binaries with period less than P, and thus

η = − dn- = − dn- dP-, (31 ) dt dP dt
dn-= -−-η--. (32 ) dP dP ∕dt

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 [107]

dP- = − k P −5∕3, (33 ) dt 0
where k 0 is given by
96 G5 ∕3 k0 = --(2π)8∕3--5--ℳ5 ∕3, (34 ) 5 c
with the “chirp mass” ℳ5 ∕3 ≡ M1M2 (M1 + M2 )−1∕3.

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 [4242243] or luminosity functions [204205]. Assuming a binary fraction of 30%, we can determine the number of relativistic binaries with short orbital period (P < P ) orb max for a typical cluster with 106 M ⊙ and the galactic globular cluster system with 7.5 10 M ⊙ [219] by simply integrating the period distribution from contact Pc up to Pmax,

∫ Pmax-η- 5∕3 N = P k P dP . (35 ) c 0
The value of Pc can be determined by using the Roche lobe radius of Eggleton [57],
2∕3 RL = ------0.49q---------a, (36 ) 0.6q2∕3 + ln (1 + q1∕3)
and stellar radii as determined by Lynden-Bell and O’Dwyer [147Jump To The Next Citation Point].

The expected populations for an individual cluster and the galactic cluster system are shown in Table 4 using neutron star masses of 1.4M ⊙, white dwarf masses of 0.6M ⊙ and 0.3M ⊙, and Pmax = 2000 s.

WD2 a

Cluster 5.6 4.0 0.5
System 176.5 125.2 16

Table 4: Encounter rate estimates of the population of relativistic binaries in a typical globular cluster and the galactic globular cluster system.

Although we have assumed the orbits of these binaries will be circularized, there is the possible exception of NS2 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 (33View Equation) and (34View Equation) are no longer valid. An integration over both period and eccentricity, using the formulae of Pierro and Pinto [179], would be required.

5.3.4 Semi-empirical methods

The small number of observed relativistic binaries can be used to infer the population of dark progenitor systems [22]. 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 NX be the number of ultracompact LMXBs and TX be their typical lifetime. Also, let Ndet be the number of detached WD–NS systems that will evolve to become LMXBs, and T det 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

NX--= Ndet. (37 ) TX Tdet
The time spent in the inspiral phase can be found from integrating Equation (33View Equation) to get
-3--( 8∕3 8∕3) Tdet = 8k P0 − P c , (38 ) 0
where P0 is the period at which the progenitor emerges from the common envelope and Pc is the period at which RLOF from the white dwarf to the neutron star begins. Thus, the number of detached progenitors can be estimated from
NX 3 ( 8∕3 ) Ndet = ------- P 0 − P 8c∕3 . (39 ) TX 8k0

There are four known ultracompact LMXBs [48] 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, 4 ≤ NX ≤ 10. The lifetime TX 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 TX ∼ 109 yr [195], but other effects such as tidal heating or irradiation may shorten this to T ∼ 107 yr X [7198]. The value of P 0 depends critically upon the evolution of the neutron star–main-sequence binary, and is very uncertain. Both k0 and Pc depend upon the masses of the white dwarf secondary and the neutron star primary. For a rough estimate, we take the mass of the secondary to be a typical He white dwarf of mass 0.4 M ⊙ and the mass of the primary to be 1.4M ⊙. Rather than estimate the typical period of emergence from the common envelope, we arbitrarily choose P = 2000 s 0. 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 Pc can be determined by using Equation (36View Equation) and the radius of the white dwarf as determined by Lynden-Bell and O’Dwyer [147]. Adopting the optimistic values of NX = 10 and TX = 107 yr, and evaluating Equation (38View Equation) gives T ∼ 107 yr det. Thus, we find N ∼ 1– 10 det, which is within an order of magnitude of the numbers found through dynamical simulations (see Section 5.3.2) and encounter rate estimations (see Section 5.3.3).

Current production of ultracompact WD–NS binaries is more likely to arise through collisions of neutron stars with lower mass red giant stars near the current turn-off mass. The result of such a collision is a common envelope that will quickly eject the envelope of the red giant and leave behind the core in an eccentric orbit. The result of the eccentric orbit is to hasten the inspiral of the degenerate core into the neutron star due to gravitational radiation [176Jump To The Next Citation Point]. Consequently, Tdet can be significantly shorter [122]. Adopting a value of 6 Tdet ∼ 10 gives Ndet < 100.

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) [144Jump To The Next Citation Point]. We may estimate that NS–NS binaries make up roughly 1∕50 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 NNS2 ∼ 105 [142]. Thus, we can estimate the total number of NS–NS binaries to be ∼ 2000. 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

N T --compact = --compact, (40 ) NNS2 Tcoalesce
where Ncompact is the number of systems in compact orbits (Porb < 2000 s), Tcompact is the time spent as a compact system, and T coalesce is the typical time for a globular cluster NS–NS binary to coalesce due to gravitational radiation inspiral. Adopting the coalescence time of B2127+11C as typical, 8 Tcoalesce = 2 × 10 yr [192], and integrating Equation (38View Equation) for two 1.4 M ⊙ neutron stars, we find Ncompact ∼ 25. Again this value compares favorably with the values found from encounter rate estimations.

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