FORMATION OF BLACK HOLE X-RAY BINARIES IN GLOBULAR CLUSTERS

In a dense stellar system, once a BH–WD binary is formed through a dynamical encounter, its further binary evolution could be affected by subsequent encounters with other stars. The cross section for an encounter between two objects of total mass m_tot with a distance of closest approach less than r_max is computed as

$$\begin{equation} \sigma = \pi r_{\rm max}^2 \left(1 + \frac{v_{\rm p}^2}{v_{\infty }^2}\right), \end{equation} \tag{ 1 }$$

where the second term accounts for gravitational focusing, with v²_p = 2Gm_tot/r_max and v_∞ the relative velocity at infinity. For strong interactions, r_max is usually on the order of the binary semimajor axis: r_max = ka, where k is of order unity (Hut & Bahcall 1983). In the limit of strong gravitational focusing, v²_p ≫ v²_∞ and

$$\begin{equation} \sigma = 2 \pi G k a m_{\rm tot} v_\infty ^{-2}. \end{equation} \tag{ 2 }$$

In our case, the first object is the BH–WD binary of mass m_BHWD = m_BH + m_WD while the second object is a core star of mass m_⋆. In GCs, the close approaches that we are interested in have v²_p ≫ v²_∞, and the BH mass is significantly more massive than both its WD companion and a typical core star. Thus, the rate at which a BH–WD binary undergoes a strong (binary-single) encounter is

$$\begin{eqnarray} \Gamma _{\rm BS} & = & \sigma n_{\rm c} v_\infty \nonumber\\ &\simeq & 2 \pi G k m_{\rm BH} n_{\rm c} a v_{\infty }^{-1} \nonumber \\ &\simeq & 0.1k \frac{m_{\rm BH}}{15\, M_\odot } \frac{n_{\rm c}}{10^5 \,{\rm pc}^{-3}} \frac{10\,{\rm km\, s}^{-1}}{v_{\infty }} \frac{a}{R_\odot }\, {\rm \ Gyr}^{-1} \nonumber \\ &= & 0.1 K \frac{a}{R_\odot }\, {\rm \ Gyr}^{-1}. \end{eqnarray} \tag{ 3 }$$

The final equal sign in Equation (3) defines the dimensionless parameter K. The timescale for a BH–WD binary to experience a strong encounter can be calculated as τ_BS = 1/Γ_BS ≈ 10 K⁻¹ R_☉/a Gyr: see Figure 1. Throughout this paper, we consider clusters with core number densities n_c near 10⁵ pc⁻³, velocity dispersions of ∼10 km s⁻¹, and BH masses of ∼15 M_☉: consequently, K is of order unity.

Zoom In Zoom Out Reset image size

Once formed, the orbit of a BH–WD binary starts to shrink due to gravitational radiation. The time τ_gw for the orbit to decay enough for MT to commence depends on the initial post-encounter binary semimajor axis a and eccentricity e. Applying Peters (1964) equations, we can find the maximum (post-encounter) semimajor axis a, as a function of post-encounter eccentricity e, for which a system will start MT within any specified time, e.g., 1 Gyr and 10 Gyr (see the dotted curves in Figure 1). Although almost all binaries start their orbital decay being eccentric, gravitational radiation quickly reduces the orbital eccentricity. For most initial semimajor axes less than a_max for the 10 Gyr merger time, the final eccentricity becomes negligibly small (less than 1%), and so we assume that at the start of MT all binaries are non-eccentric.

The condition for MT to start is that the WD fills its Roche lobe. For a BH–WD binary in which the BH is 25 times more massive than the WD, the Eggleton approximation (Eggleton 1983) gives a Roche lobe radius for the secondary that is 16% of the semimajor axis a. Thus, assuming a WD radius ∼0.01 R_☉, MT commences at a ∼ 0.06 R_☉.

In Figure 1, we also show the semimajor axis a_sep(e) such that τ_BS = τ_gw(e), assuming K = 1. BH–WD binaries with post-encounter semimajor axes a < a_sep(e) will become BH–WD X-ray binaries. BH–WD binaries with post-encounter semimajor axes a > a_sep(e) will have one or more strong encounters before they shrink to the point of merger due to gravitational radiation. In the next subsection, we consider the possible outcomes of such encounters.

Here, we consider the outcomes and their frequencies for encounters between a BH–WD and a single star which, as we found in Section 2.1, typically occurs only if a ≳ a_sep(e). An encounter between a soft binary and a third star can lead to ionization if the third star approaches with sufficient speed; specifically, for ionization to be energetically possible, the relative velocity at infinity v_∞ between the binary and the third star must exceed the binary's critical velocity v_c defined such that the total energy of the binary-single system is zero:

$$\begin{equation} v_{\rm c}^2 = \frac{m_{\rm BHWD}+m_\star }{m_{\rm BHWD}} \frac{m_{\rm WD}}{ m_{\star }}\frac{G m_{\rm BH}}{ a}. \end{equation} \tag{ 4 }$$

For m_BH = 15 M_☉, m_WD = 0.6 M_☉, and m_⋆ = 0.6 M_☉, the critical velocity v_c is less than 10 km s⁻¹ only if the semimajor axis a of the binary exceeds ∼3 × 10⁴ R_☉. Even for encounters between a BH–WD binary and a 15 M_☉ BH, the critical velocity v_c is less than 10 km s⁻¹ only for a ≳ 2000 R_☉. Consequently, in a typical cluster with velocity dispersion ∼10 km s⁻¹, ionization by itself cannot destroy BH–WD binaries that are able to start MT within 10 Gyr, as all such binaries are easily of sufficiently small semimajor axis (see Figure 1). Essentially, a BH–WD binary with a ≲ 2000 R_☉ can be conservatively classified as a hard binary.

A strong encounter between a hard binary and a third star would ultimately lead to preservation of the BH–WD binary, a companion exchange, or a physical collision. Such outcomes, which we now consider, could potentially prevent a BH–WD binary from ever reaching the MT stage. We note that if an encounter occurs with another binary, a possible outcome is also a hierarchically stable triple system; we consider this option in the next subsection.

The collision cross section for binary-single interactions is approximately (for a non-eccentric binary):

$$\begin{equation} \sigma _{\rm coll} \sim \pi a^2 \left(\frac{V_{\rm g}}{v_\infty }\right)^{2} \left(\frac{215 R}{a}\right)^{0.65}, \end{equation} \tag{ 5 }$$

where V²_g = Gm_tot/(2a) and the maximum radius R among the encounter participants is presumed to be less than a. This approximation was derived using results from Fregeau et al. (2004). Our definition of V_g follows that of Heggie et al. (1996). We note that V_g = v_c in the case of three equal mass stars. We have verified with numerical experiments for many mass combinations that the use of V²_g yields the appropriate scaling of the cross section with mass (provided v_∞ ≪ V_g).

Using Equations (2) and (5), we can now estimate the fraction of binary-single encounters that result in a collision:

$$\begin{equation} \frac{\sigma _{\rm coll}}{\sigma } = \frac{1}{4k} \left(\frac{215 R}{a}\right)^{0.65}, \end{equation} \tag{ 6 }$$

where k is the ratio of the maximum pericenter considered as an encounter to the semimajor axis a of the binary. Equation (6) implies that if a non-eccentric BH–WD has a ≲ 5 R_☉, then all encounters with R = 0.6 R_☉ main-sequence (MS) star at k < 2 will result in a merger. Similarly, we can estimate that for a non-eccentric BH–WD binary with a ≲ 15 R_☉, a merger will result for all encounters with k ⩽ 1. Consequently, we expect that most BH–WD binaries that have a > a_sep and are able to start MT within 10 Gyr (see Figure 1) will instead experience a merger if a binary-single encounter with an MS star occurs.

To examine the collision cross section more carefully, we perform scattering experiments as in Fregeau et al. (2004), but for our BH–WD binary with the separations from 1 to 100  R_☉ and fixed v_∞ = 10 km s⁻¹. We find the k_coll, defined by Equation (2) with σ = σ_coll and k = k_coll, varies from ∼1 for a = 1 R_☉ to ∼0.1 for a = 100 R_☉. For example, k_coll ≈ 0.24 for a = 10 R_☉, and, accordingly from Equation (3), the merger rate for such BH–WD binaries that are able to start MT within 10 Gyr exceeds 0.2 per Gyr. We note that effectively this rate could be a factor of few smaller as a fraction of collisions will result in a merger between a BH and an MS star. We have verified with hydrodynamical simulations that at least some of the collisions where an MS star collided with the BH will not strongly affect the initial BH–WD binary. We assume the BH–WD binary does not survive if an MS collides with the WD.

The relative fraction of WDs in the core, at the age of several to 14 Gyr, is f_WD ∼ 0.2 of the total core population, and the encounter rate between a BH–WD binary and a WD is Γ_BS,WD = f_WDΓ_BS. We also note that some simulations have shown that WDs can contribute up to 70% of the total core population, as was found in simulations performed for Fregeau et al. (2009b) without WD birth kicks. From Equation (6), we roughly estimate that a physical collision with an R = 0.01 R_☉ WD would occur only in binaries with a ≲ 0.1 R_☉, which is well below a_sep(e). A more likely outcome in such cases would be preservation or a companion exchange. Exchanges preferentially occur if the incoming star is more massive than pre-encounter companion and accordingly could happen if a BH–WD binary had a light WD companion. As a result of exchange, the post-encounter semimajor axis will be increased by the ratio of new companion mass to the old companion mass. We conclude that exchanges with WDs will not tend to make harder binaries.

To reiterate, we consider here only the cases when a BH–WD binary has a > a_sep(e) (because if a < a_sep, a merger occurs before a strong encounter). Therefore, the resulting binary will most likely never be able to reach MT, as its binary separation will exceed the value necessary for MT to start within 10 Gyr. If a binary is preserved, though, it will most likely then experience a subsequent encounter with a single star. Even if WDs make up as much as 70% of the core population, this subsequent encounter has a high probability of having an MS star as a participant and therefore to result in a merger.

We have shown that BH–WD binaries that can evolve to an LMXB within 10 Gyr and yet are wide enough to have an encounter with a single star (so their a > a_sep) will either experience a physical collision with destruction of the BH–WD binary, or, after an exchange, will become too wide to reach MT. As such, we conclude that strong encounters with single stars will not create BH–WD binaries with a < a_sep(e) and therefore will not lead to the direct formation of BH–WD X-ray binaries.

If a BH–WD has an encounter with another binary, a possible outcome is a hierarchically stable triple. For two identical binaries, the maximum impact parameter for which strong encounters still could occur is a bit larger than for a binary-single encounter (Mikkola 1983):

$$\begin{equation} \sigma _{\rm BB} \approx 4.6 \pi (a_1+a_2)^2 \frac{v_{\rm c}^2}{v_\infty ^2}. \end{equation} \tag{ 7 }$$

Here, v_c is the critical velocity of the two binaries with the masses m₁ and m₂:

$$\begin{equation} v_{\rm c}^2 = \frac{G}{\mu } \left(\frac{m_{11}m_{12}}{a_1} + \frac{m_{21}m_{22}}{a_2} \right), \end{equation} \tag{ 8 }$$

where the reduced mass μ = m₁m₂/(m₁ + m₂), where m₁₁, m₁₂, m₂₁, and m₂₂ are the masses of the binary components, and where a₁ and a₂ are the semimajor axes.

For hard binaries with equal masses and binary separations, ∼25% of all binary–binary encounters within maximum approach described by Equation (7) and v²_c/v²_∞ = 10 form a triple system (Mikkola 1983; Fregeau et al. 2004), with triple formation increasing with v²_c/v²_∞. For a larger mass ratio between binaries, or for a softer incoming binary of less mass, this fraction can be even higher.

We performed numerical experiments in which BH–WD binaries consisting of a 15 M_☉ BH and a 0.6 M_☉ WD encountered at v_∞ = 10 km s⁻¹ binaries consisting of 0.6 M_☉ companions. We record the fraction of encounters that result in a stable triple with an inner binary consisting of the initial BH and a companion 0.6 M_☉ star. We run the simulations for 3 values of BH–WD binary separation and 11–17 values of the incoming binary separations, performing 10,000 encounters for each. For the numerical integration, we use Fewbody (Fregeau et al. 2004), a small N-body integrator. The results demonstrate that periastra somewhat larger than those suggested by Equation(7) can result in a triple formation, and we choose the maximum possible periastron for the encounters to be fairly large, r_max,p = 20(a₁ + a₂) (that is, k = 20). The impact parameters for every encounter are distributed uniformly in area. The eccentricities are distributed thermally. The results are shown in Figure 2.

Zoom In Zoom Out Reset image size

We find that for almost all the encounters in which the incoming binary is wider than the BH-WD binary, the efficiency of triple formation is ∼20%–30% with k = 20. This gives

$$\begin{equation} \sigma _{\rm triples} \approx 120 \pi (a_1 + a_2)^2 \left(1 + \frac{G m_{\rm BH}}{10 (a_1+a_2) v_\infty ^2} \right). \end{equation} \tag{ 9 }$$

The formation rate of triple systems, for hard binaries, compared to binary-single encounters rate (3) with k = 2, is then

$$\begin{equation} \frac{\Gamma _{\rm triples}(a_1, a_2)}{\Gamma _{\rm BS}(a_1)} \sim 3\sqrt{2} f_{\rm wb} \left(1+\frac{a_2}{a_1} \right). \end{equation} \tag{ 10 }$$

Here, f_wb is the fraction of stars that are hard binaries, but still wider than the BH–WD binary. In deriving the above equation, we have applied equipartition of energy, so that the v_∞ of a typical binary in the core is $\sqrt{2}$ times less than that of a typical single star.

The overall binary fraction in observed dense, but not core-collapsed, clusters at the current epoch, is about a few percent (Albrow et al. 2001; Ivanova et al. 2005a; Milone et al. 2008). With f_wb = 5% and a flat distribution of binary separations between, e.g., 20 and 2000 R_☉, a BH–WD binary that has a = 20 R_☉ can form a triple about 30 times per Gyr. This exceeds the interaction rate with single stars.

Although these triples are stable in isolation, they can be destroyed during their next dynamical encounter. Before this next encounter occurs, the presence of the third component can significantly affect the eccentricity of the inner BH–WD binary via the Kozai mechanism (Kozai 1962), provided the inclination of the outer orbit is large enough (sin i₀ > (2/5)^1/2, or equivalently i₀ ≳ 39°, the Kozai angle). The maximum eccentricity that can be achieved via the Kozai mechanism is (e.g., Innanen et al. 1997; Eggleton & Kiseleva-Eggleton 2001)

$$\begin{equation} e_{\rm max} \simeq \sqrt{5/3 \sin ^2(i_0)-2/3}. \end{equation} \tag{ 11 }$$

In dynamically formed triples, the inclinations are distributed almost uniformly, and so the number of Kozai triples is expected to be proportional to the solid angle (e.g., Ivanova 2008, where, depending on the sample of runs, the triples affected by the Kozai mechanism constituted from 30% to 40% of all formed triples, while a uniform distribution predicts 37%). Thus, the fraction of the dynamically formed triples with a post-encounter inclination greater than i is f_i = 1 − sin i.

The BH–WD binaries that could become MT systems are only those that have a less than the maximum a_sep (≲80 R_☉ for e < 0.99). To find the formation rate of such triples, we first determine the minimum value of e_max that a triple with the specific binary semimajor axis a must achieve: this is found by inverting the numerically obtained a_sep(e) at k = 2. The obtained e_max then provides the fraction of formed triples that can bring the inner binary into MT via the Kozai mechanism:

$$\begin{equation} f_{i} = 1-\sin i = 1-\sqrt{3/5\ e_{\rm max}^2+2/5}. \end{equation} \tag{ 12 }$$

We define triple induced mass transfer (TIMT) systems to be those binaries that are brought into MT via the Kozai mechanism. We find that the fraction f_i of TIMT systems ranges between approximately 0.07 (for a = 15 R_☉) and 0.01 (for most semimajor axes, a ≳ 40 R_☉). Accounting for the triple formation rate, we find that all BH–WD binaries with 15 R_☉ ≲ a₁ ≲ 80 R_☉ could make it to the MT through the triple formation mechanism. At a conservative level, assuming even that all the binaries that have a binding energy 10 times more than the kinetic energy of an average object in the core are destroyed, we find that a BH–WD binary with a₁ ≳ 50 R_☉ has a 10%–40% chance to become a TIMT system within 1 Gyr (10% for a₁ ∼ 80 R_☉ and 40% for a₁ ∼ 50 R_☉); this chance is 100% for a₁ ≲ 35 R_☉. During several Gyr, all BH–WD binaries with a₁ ≲ 80 R_☉ can become a TIMT system at least once.

This TIMT system formation will be successful only if the time between subsequent encounters is longer than the time necessary to achieve e_max. The period of the cycle to achieve e_max is (Innanen et al. 1997; Miller & Hamilton 2002a)

$$\begin{equation} \tau _{\rm Koz} \simeq \frac{0.42 \ln (1/e_{\rm i}) }{\sqrt{\sin ^2(i_0) -0.4}} \left(\frac{m_{i1}+m_{i2}}{m_{\rm o}} \frac{b^3_{\rm o}}{a_1^3} \right) ^{1/2} \left(\frac{b^3_{\rm o}}{ G m_{\rm o}} \right)^{1/2}, \end{equation} \tag{ 13 }$$

where m_i1 and m_i2 are the companion masses of the inner binary, e_i is the initial eccentricity of the inner binary, m_o is the mass of the outer star, a_o is the initial semimajor axis for the outer orbit, and b_o = a_o(1 − e²_o)^1/2 is the semiminor axis of the outer orbit. Roughly, a_o will be of the order of magnitude of a₂. Considering the extreme case a₂ ≫ a₁, for which τ_Koz would be maximum, we find:

$$\begin{eqnarray} \frac{\tau _{\rm Koz}}{\tau _{\rm BB}} &\approx & 4\times 10^{-14} f_{\rm b} \frac{0.42 \ln (1/e_{\rm i}) }{\sqrt{\sin ^2(i_0) -0.4}} \left(\frac{a_{\rm 2}^2}{a_1 R_\odot } \right) ^{5/2} \nonumber\\ & &\times \left(\frac{m_{\rm BH}}{15\, M_{\odot }}\right)^{3/2} \frac{M_{\odot }}{m_{\rm o}} \left(1 - e_{\rm o}^2 \right)^{3/2} \frac{n_{\rm c}}{10^5 {\rm \,pc}^{-3}} \frac{10\,{\rm km\, s^{-1}}}{v_{\infty }} \nonumber \\ \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \frac{\tau _{\rm Koz}}{\tau _{\rm BS}} &\approx & 4\times 10^{-14} \frac{0.42 \ln (1/e_{\rm i}) }{\sqrt{\sin ^2(i_0) -0.4}} \left(\frac{a_{\rm 2}}{R_\odot } \right) ^{3} \left(\frac{R_\odot }{a_1} \right) ^{1/2}\nonumber\\ & &\times \left(\frac{m_{\rm BH}}{15\, M_{\odot }}\right)^{3/2} \frac{M_{\odot }}{m_{\rm o}} \left(1 - e_{\rm o}^2 \right)^{3/2} \frac{n_{\rm c}}{10^5 {\rm \,pc}^{-3}} \frac{10\,{\rm km\, s^{-1}}}{v_{\infty }} \nonumber\\ \end{eqnarray} \tag{ 15 }$$

Here, f_b is the binary fraction, τ_BB = Γ⁻¹_BB = (σ_BBf_bn_cv_∞)⁻¹, and τ_BS = 1/Γ_BS has been evaluated at k = 2.

An average dynamically formed triple would have e_o ≈ 0.9 (Ivanova 2008). For the purpose of the estimate, we adopt that a BH–WD binary would have its initial eccentricity distributed thermally, with an average e_i ∼ 2/3, and adopt i₀ = 54° (this is the mean inclination of dynamically formed Kozai affected triples). From Equation (15), we have then that triples with 15 R_☉ ≲ a₁ ≲ 80 R_☉ and a₂ ≲ 10⁵ R_☉ (where 4500 R_☉ is the boundary between hard and soft binaries) would have their Kozai time significantly shorter than their collision time with either binary or single stars. We conclude that once a potential TIMT system is formed, it will succeed in bringing its inner BH–WD to MT before its next encounter. Accordingly, all BH–WD binaries with a ≲ 80 R_☉ will become X-ray binaries via the TIMT mechanism.

It is well known that through multiple fly-by encounters hard binaries get harder (e.g., Hut 1983). The number of encounters necessary to change the binary energy, e.g., by a factor of 4, can be estimated as (Heggie & Hut 2003)

$$\begin{equation} N_{\rm hard} \approx \log (4)/\log (1+\delta), \end{equation} \tag{ 16 }$$

where δ is the relative change in the binary's binding energy caused by an encounter. This change, for an eccentric binary with a large mass ratio of one star to a companion and an encountering m₃ star, if occurred at the distance q (q ≫ a), can be roughly estimated as (e.g., Heggie 1975; Roy & Haddow 2003)

$$\begin{equation} \delta \approx \frac{m_3}{m_{\rm BH}} \left({\frac{a}{q}} \right)^{3/4} e^{-(q/a)^{3/4}}. \end{equation} \tag{ 17 }$$

Assuming that all hardening happened with the maximum δ ∼ m₃/m_bh ∼ 0.04, it can therefore be estimated that, the hardening of a 1000 R_☉ BH–WD binary to, e.g., 35 R_☉ should take about 100 encounters. As the collision time for a 1000 R_☉ BH–WD binary is only about 10⁷ yr, and is still only about 10⁸ yr for a 35 R_☉ BH–WD binary, it is plausible therefore that all 1000 R_☉ BH–WD binary can be hardened to 35 R_☉ within a few Gyr. We choose here 35 R_☉ as the critical separation at which triple formation starts to have 100% efficiency in making this binary an MT binary (see Section 2.3).

However, this is an idealistic picture. There are plenty of encounters that this binary would have undergone in order to reach 35 R_☉, and certainly not all of them will be simple fly-by hardening encounters. Some of the encounters would result in exchanges (where the lighter companion is often exchanged with a more massive incoming star, and the binary gets wider), mergers, or even binary ionizations, therefore reducing the chance for a binary to harden to the separation we are interested in.

We set up a numerical experiment, by taking a BH–WD binary with the specific initial binary semimajor axis a_i. At the collision rate predicted by the current binary semimajor axis and the adopted n_c = 10⁵ pc⁻³, we bombard the binary with single stars drawn from the simplified core's population, randomizing impact parameters and using Fewbody. Initial eccentricities of BH–WD binaries for this experiment are adopted to have a thermal distribution. A simplified core population is taken from Monte Carlo runs (Ivanova et al. 2008) and has only four different groups for the non-BH population: stars with masses 0.22, 0.506, 0.75, and 1.13 M_☉, with, respectively, 33%, 38%, 25%, and 4% contributions to the non-BH population. Then we vary the BH component, starting from the highest possible, where the number of BHs is 0.4% of all other stars in the core (this is about 100% of the initially formed BH population, see the discussion in Section 1, or f_BH,0.1 = 10), 0.04% (this corresponds to the case of f_BH,0.1 = 1), 0.004%, and no contribution at all (this correspond to the case when only one BH, which is in the considered BH–WD binary, is left). Here, one BH corresponds to 0.001% of the core population.

We consider the fraction of BH–WD binaries that would harden to 35 R_☉ within 10 Gyr as well as the average time it will take them. We also separately consider cases where we insist that the original BH–WD binary makes it to 35 R_☉, or simply any binary containing an initial BH would make it (allowing multiple exchanges). The latter case could correspond to the case when almost all the stellar non-BH population in the core are WDs (e.g., in some runs performed in Fregeau et al. 2009b; WDs fraction in the core reached 70%). The results are shown in the Table 1.

There are several trends in the results, all of them well expected theoretically. Indeed, as expected, the fraction of survived and hardened BH–WD binaries f_hard is well below 1. The fraction f_hard decreases with increasing a_i and increases when the BH population in the core drops, as other BHs contribute significantly into exchange reactions or ionization. The average time that it takes to harden a binary agrees with the analytic estimate above and is about 1–2 Gyr. Even though f_hard increases when the BH population drops, it does not increase by a factor comparable to the relative drop in the BH population. Accordingly, since the resulting formation rate of hardened BH–WD binaries (their production from initially wider binaries) Γ_hard ∝ f_BH,0.1f_hard, the highest Γ_hard occurs in an unlikely case when most of initially formed BHs are retained (f_BH,0.1 = 10). The fraction f_hard for f_BH,0.1 ≲ 1 does not vary much with f_BH,0.1.

We also study what fractions can be hardened to a = 80 R_☉, for our optimistic scenario when all BH–WD binaries with a = 80 R_☉ make it to the MT via TIMT. In this case, significant fractions of wide binaries can be successfully hardened.

In a binary that has been newly formed via exchange, the post-encounter semimajor axis will be a_post ∼ a_prem_BH/m_c where m_c is the lower mass companion replaced by a BH. The post-encounter eccentricities are distributed thermally with the mean e ∼ 2/3. The mass m_c would normally not exceed that of an average star in the core, and accordingly the post-encounter semimajor axis will be at least 25 times larger than pre-encounter semimajor axis.

Thus, in Figure 1, we see that only those binaries with pre-encounter semimajor axes a_pre ≲ 7 R_☉/25 ≈ 0.3 R_☉ could create binaries that will evolve to MT directly (though up to a_pre ≈ 80 R_☉/25 ≈ 3 R_☉ if the post-encounter eccentricity is as large as 0.99). For an MT binary formed via a triple, a_pre ≲ 35 R_☉/25 ≈ 1.4 R_☉. In both cases, a_pre is small enough to expect a merger if at least one companion in the pre-encounter binary was an MS star.

Therefore, to create a BH–WD binary with a_post ≲ 35 R_☉, only encounters with WD–WD binaries are relevant. The fraction of WD–WD binaries is just a few percent of the total binary population. From the simulations set in Ivanova et al. (2008, 2006), we find the fraction of the short-period WD–WD binaries at 10 Gyr in their "standard" GC (corresponds to our typical dense GC) to be f_b ≈ 0.8% (as the fraction of the all objects in the core). Adopting a_pre ≈ 1 R_☉, we find that the formation rate of potential BH–WD X-ray binaries through exchanges coupled with TIMT is Γ_exch,1 ≈ 2 × 10⁻³ per Gyr per BH.

Relatively wide hard BH–WD binaries (a ∼ 80–1000 R_☉) can be formed through encounters with both MS–WD and WD–WD binaries, as pre-encounter binaries can be up to 20 R_☉ (here, MS–WD binaries could be only those that have a ≳ 15 R_☉). Again, using the same set of simulations, we find that such binaries have a relative fraction f_b = 1.4% and an average a_pre ≈ 22 R_☉. The resulting formation rate is fairly similar to the formation via encounters with WD–WD binaries, Γ_exch,2 ≈ 0.06f_hard ≈ 2 × 10⁻³ per Gyr per BH.

In our "optimistic" case, TIMT is successful in all binaries with a_post ≲ 80 R_☉. Then, the fraction of binaries playing a role in direct exchanges (with semimajor axes a_pre ≲ 80R_☉/25 = 3.2R_☉) is f_b ∼ 2%. When averaged over all binaries in several numerical simulations, f_hard ∼ 25% (for f_BH,0.1 = 1). We find then Γ_exch,1 = 6 × 10⁻³ and Γ_exch,2 = 1.5 × 10⁻² per Gyr per BH.

Physical collisions between compact stars (NSs) and low-mass RGs can lead to a formation of NS–WD X-ray binaries (Ivanova et al. 2005b), which, during their persistent phase, are also known as ultracompact X-ray binaries (UCXBs). From observations of UCXBs in the Milky Way's galactic GCs, we know that UCXBs are formed at a high rate. The fraction of UCXBs among all LMXBs is much larger in GCs than in the field (Deutsch et al. 2000).

When natal kicks from Hobbs et al. (2005) are adopted for NSs formed via core collapse and reduced kicks are adopted for NS formed via electron capture supernova, the theoretically predicted UCXB formation rate through physical collisions is consistent with the observations of GCs both in the Milky Way and in external galaxies (Ivanova et al. 2005b).

Like in the case of UCXBs formation, it is plausible that BHs, if retained and not detached in a BHs subcluster from the rest of the stellar population, will experience physical collisions with RGs. The rate of such collisions, per BH, can be estimated as in Ivanova et al. (2005b):

$$\begin{eqnarray} \Gamma _{\rm BHRG}&\approx & 2 \pi G f_{\rm RG} f_{\rm p} m_{\rm BH} n_{\rm c} {\bar{R}}_{\rm rg} v_{\infty }^{-1} \nonumber\\ &\approx & 0.1 f_{\rm RG} f_{\rm p} \frac{m_{\rm BH}}{15\, M_\odot } \frac{n_{\rm c}}{10^5\,{\rm pc}^{-3}} \frac{\bar{R}_{\rm RG}}{R_\odot } \frac{10\,{\rm km/s}}{v_{\infty }} {\rm per \ Gyr}, \nonumber\\ \end{eqnarray} \tag{ 18 }$$

where ${\bar{R}_{\rm RG}}$ is the average radius of the RGs, $f_{\rm p}= r_{\rm p}/{\bar{R}_{\rm RG}}$ describes how close has to be an encounter in order to result in the formation of a binary compact enough to start the MT. For NSs, f_p ≈ 1.3 (Lombardi et al. 2006). However, for BHs the maximum periastron that leads to Roche lobe overflow of the RG at the closest approach is larger, f_p ≈ 5.⁶ The fraction f_RG of stars at the RG stage is typically ∼0.4% of non-degenerate single stars at the age 10–12 Gyr, with the minimum stellar mass in the population being 0.08 M_☉ and with the initial mass function as in Kroupa (2002). Mass segregation of stars in the core results in a higher f_RG. Analyzing the set of simulations from Ivanova et al. (2008), we find that this fraction is about 0.8% of the overall core population and ${\bar{R}_{\rm RG}}=3.7$ at 10 Gyr; in the simulations presented in Fregeau et al. (2009a), the post-processed f_RG ≈ 0.8% as well. In our relatively dense stellar cluster, with n_c = 10⁵ pc⁻³ and velocity dispersion of about 10 km s⁻¹, a BH of 15  M_☉ therefore could have ∼3 × 10⁻³f_p chance of a physical collisions during 1 Gyr. Therefore, physical collisions can provide a significant contribution to overall BH–WD X-ray binary formation only if all the collisions leading to the formation of bound binaries, with all of the periastra up to 5 R_RG, lead to X-ray binary formation.

Let us estimate what fraction of the formed bound binaries can successfully become BH–WD X-ray binaries. A simple estimate for the outcome of a BH–RG collision can be done using an energy argument, as in the standard common envelope consideration, using the α_CEλ formalism (Webbink 1984). Specifically, the final semimajor axis a_f is given by

$$\begin{equation} a_{\rm f} = {R_{\rm RG}} \frac{\alpha _{\rm CE} {\lambda }}{2} \frac{m_{\rm BH}}{m_{\rm RG}} \frac{ m_{\rm RG,core} }{ m_{\rm RG, env}} \, \end{equation} \tag{ 19 }$$

where m_RG,core and m_RG,env are, respectively, the core and envelope masses of the RG, and R_RG is the RG radius. The parameter λ introduced to characterize the donor envelope central concentration can be found directly from stellar models and is about 1 for low-mass RGs. The parameter α_CE is introduced as a measure of the energy transfer efficiency from the orbital energy into envelope expansion, and is bound to be below 1. The energy balance in Equation (19) assumes that the amount of the energy that is transferred from the orbital motion to the envelope expansion does not exceed the energy that is required to eject the envelope to infinity. The kinetic energy that the ejected gas has at infinity is neglected, as well as that two stars had initially at the infinity, as it is very small compared to the binding energy of the envelope.

It can be seen, if α_CE = 1, that the final binary separation for a case of a BH collision with a low-mass RG will result in the formation of a binary with a_f > 1.3 R_RG. (This minimum value is for a case of an early subgiant, when the core mass is minimum: e.g., the core mass is only ∼0.12 M_☉ for a 0.9 M_☉ RG of radius ∼2 R_☉ and metallicity Z = 0.005.) Comparison with the parameter space on Figure 1 shows that, if the post-collision eccentricity is small, only encounters with a subgiant satisfying R_RGm_RG,core/m_RG,env ≲ 0.66 R_☉ could lead directly to a BH–WD X-ray binary. This corresponds to a subgiant with m_RG,core ≈ 0.17 M_☉ (so that for a 0.9 M_☉ RG, the radius is then still only 2.75 R_☉). The time this RG spends as a subgiant before it reaches m_RG,core ≈ 0.17 M_☉ is only 135 Myr, or ∼25% of its RG lifetime. Assuming that the final separation after a physical collision does not exceed that predicted by simple energy conservation, the formation rates are then ∼2 × 10⁻³f_p per BH per Gyr. This rate is an upper limit, as it also is being assumed that all of the physical collisions that lead to the formation of bound systems will lead to stripping of the RG envelope as well. From smoothed particle hydrodynamic (SPH) simulations, we find that collisions with f_p ≳ 1 will lead to a formation of binaries with e ≲ 0.8, making f_p ≈ 1 the border line case separating those collisions that lead to the formation of BH–WD binaries able to start MT in isolation from those that could not (see also Figure 1).⁷ This non-TIMT channel therefore implies a formation rate of ∼4 × 10⁻⁴ per BH per Gyr.

For collisions with f_p ≳ 1, TIMT is the main mechanism that brings a formed binary to MT. We can estimate the maximum formation rate with optimistic assumptions. For that, we find what fraction of collision would lead to the formation of BH–WD binaries with a ≲ 80 R_☉, assuming that, even if a physical collision itself could not form a binary that will start MT in isolation, TIMT will succeed (accordingly, f_p = 5). Using Equation (19), we find that the RG should have R_RGm_RG,core/m_RG,env ≲ 9.6 R_☉. This occurs when a 0.9 M_☉ RG has m_RG,core ≈ 0.31 M_☉ and R_RG ≈ 17.8 R_☉, corresponding to almost 95% of the RG lifetime (accordingly, and roughly, all collisions with RGs can be counted toward TIMT influenced collisions). We conclude therefore that most optimistic formation rate via physical collisions, when TIMT is taken into account, is 1.5 × 10⁻² per BH per Gyr.

Three-body binary formation occurs if

• 1.  
  three objects will meet within some vicinity of each other a_v;
• 2.  
  in the case of such a meeting at least two of the objects will form a bound system;
• 3.  
  the formation of this bound system is not altered by physical collisions or strong tidal interactions between the objects due to finite size effects.

In the literature, usually only the first condition is meant when the generalized three-body binary formation rate is described (e.g., Binney & Tremaine 1987; Ivanova et al. 2005a). There are however additional limitations that make such treatments inapplicable to our case. The formation rate in Binney & Tremaine (1987), for example, is derived for equal masses only and considers only the formation of binaries which have a hardness ratio η (the ratio of their binding energy to the kinetic energy of an average object in the core) of 1. In Ivanova et al. (2005a), the formation rate is derived to take into account unequal masses and different energies; however, the treatment is applicable only when the resulting binaries are near the hard–soft boundary or a bit harder.

In their derivation of the formation rates of binaries with different energies, Ivanova et al. (2005a) approximate that the semimajor axis in the formed binary is the same as the size of the vicinity a_v where the three objects meet. As shown by numerical experiments in Aarseth & Heggie (1976), this assumption is approximately satisfied at least for equal masses, where a(1 + e)/a_v ≈ 1.

In our study, we are most interested in the formation rates where objects have a fairly large mass ratio and have high energies (η ≳ 100). Compared to η = 1 binaries, the decrease in the hard binary formation rate due to physical collisions and tidal effects could be significant. We therefore limit ourselves to the consideration of only degenerate objects meeting each other. On the other hand, the limitation of the considered cases to only very hard binaries eliminates the necessity to consider the second condition: Aarseth & Heggie (1976) showed that the probability of forming a binary strongly increases as vicinity decreases, P ∝ η⁻², and, even in the case of η = 2, 77% of three-body encounters resulted in binary formation. They have also shown that the average eccentricity of formed binaries is a bit higher than in the thermal distribution: 〈 e〉 = 0.77 for η = 2 cases, though this was the largest energy considered and the average values were slowly decreasing with hardness. We may expect that the average eccentricities could go as low as that of the thermal distribution.

The rate at which three objects meet in the same vicinity a_v is the product of the rate Γ₂(a_v, m₁, m₂) for two objects to meet in this neighborhood and the probability P₃ that during this event a third object will be in the vicinity. As previously done, the two-body encounter rate Γ₂(a_v, m₁, m₂) is standardly derived as a combination of geometric cross section and gravitational focusing:

$$\begin{equation} \Gamma _2 (a_{\rm v}, m_1, m_2) = \pi n_2 v_{\infty } a_{\rm v}^2 \left(1 + \frac{v_{\rm p}^2}{v_{\infty }^2}\right) \end{equation} \tag{ 20 }$$

where v²_p = 2G(m₁ + m₂)/a_v. Two objects spend in this vicinity about a time τ_v = 2a_v/v_p.

The probability P₃ that a third object will be within the same vicinity is then usually found assuming that a third object will geometrically sweep only a certain volume during τ_v (Binney & Tremaine 1987; Ivanova et al. 2005a). A "geometric" cross section however is good only for cases of a_v large enough so v_p ≲ v_∞. In the case, when we are interested in the encounters occurring within a small vicinity only, gravitational focusing must be taken into account. In some sense, this implies that we are looking for a probability that, at the same time, two objects will be gravitationally focused by a BH. The probability that a third object will pass within a_v during τ_v is then

$$\begin{equation} P_3 \simeq 2 \pi n_3 a_{\rm v}^3 \frac{v_{\rm p}}{v_{\infty }}. \end{equation} \tag{ 21 }$$

The three-body formation rate then is

$$\begin{equation} \Gamma _3 (a_{\rm v}) \simeq 2 \pi ^2 n_2 n_3 a_{\rm v}^5 \frac{v_{\rm p}^3}{v_{\infty }^2} \simeq 2^{5/2} \pi ^2 G^{3/2} n_2 n_3 a_v^{3.5} v_{\infty }^{-2} m_{\rm BH} ^{3/2}. \end{equation} \tag{ 22 }$$

For very hard binaries, this rate is significantly larger than the corresponding rate derived in Ivanova et al. (2005a) and can be written as

$$\begin{eqnarray} &\Gamma _3&(a_{\rm v}) \simeq 1.6\times 10^{-16} \,{\rm per\ Gyr} \nonumber \\ & &\times\left(\frac{n_{\rm c}}{10^5 \,{\rm pc}^{-3}}\right)^2 \left(\frac{a_v}{R_\odot }\right)^{3.5} \left(\frac{10\, {\rm km/s}}{v_{\infty }}\right)^{2} \left(\frac{m_{\rm BH}}{15\, M_\odot }\right)^{3/2}. \nonumber\\ \end{eqnarray} \tag{ 23 }$$

Therefore, even in a relatively dense stellar cluster, a BH of 15  M_☉ will have only about 2 × 10⁻⁹ three-body encounters within its 100 R_☉ vicinity during 1 Gyr. The formation rate necessary to explain BH–WD X-ray binaries formation is consistent only with encounters within ∼5000 R_☉, or η = 10. We also note that in this case v_p ≫ v_∞, the assumption used for the derivation, is no longer valid. Even in a denser cluster (n_c ∼ 10⁶ pc) with a somewhat smaller velocity dispersion, the rate given by the above estimate is still about an order of magnitude less than the observed formation rate.

We conclude that unless a significant fraction of small-vicinity three-body encounters lead to a formation of binaries much smaller than the vicinity where they met (10–50 times smaller), three-body binary formation cannot explain the observed rates of BH–WD X-ray binary formation.