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2.3 Globular cluster evolution

An overview of the evolution of globular clusters can be found in Hut et al. [116Jump To The Next Citation Point], Meylan and Heggie [157Jump To The Next Citation Point], and Meylan [156]. We summarize here the aspects of globular cluster evolution that are relevant to the formation and concentration of relativistic binaries. The formation of globular clusters is not well understood [76] and the details of the initial mass function (IMF) are an ongoing field of star cluster studies. The Kroupa mass function [138] is the most common IMF currently used (see [137] for a discussion of the local IMF). It has the form
dN ∝ m −αidm, (9 )
α0 = +0.3 ± 0.7, 0.01 ≤ m ∕M ⊙ < 0.08, α1 = +1.3 ± 0.5, 0.08 ≤ m ∕M ⊙ < 0.50, (10 ) α2 = +2.3 ± 0.3, 0.50 ≤ m ∕M ⊙ < 1.00, α3 = +2.3 ± 0.7, 1.00 ≤ m ∕M ⊙.
Some older work uses the Salpeter IMF which assumes a single value of α in Equation (9View Equation) for all masses. Once the stars form out of the initial molecular cloud the system is not virialized (i.e. it does not satisfy Equation (5View Equation)), and it will undergo what is known as violent relaxation as the protocluster first begins to collapse. During violent relaxation, the total energy of individual stars can change as the local gravitational potential changes. The process of violent relaxation is a collisionless process and it occurs rapidly over a timescale given by a few crossing times. During violent relaxation, the energy per mass of a given star changes in a way that is independent of the mass of the star [24Jump To The Next Citation Point]. Thus, the more massive stars will have more kinetic energy. These stars will then lose their kinetic energy to the less massive stars through stellar encountersUpdateJump To The Next Update Information which leads towards equipartition of energy. Through virialization, this tends to concentrate the more massive stars in the center of the cluster – a process known as mass segregation. The process of mass segregation for stars of mass mi occurs on a timescale given by ti ∼ trelax⟨m ⟩∕mi.

The higher concentration of stars in the center of the cluster increases the probability of an encounter, which, in turn, decreases the relaxation time. Thus, the relaxation time given in Equation (1View Equation) is an average over the whole cluster. The local relaxation time of the cluster is given in Meylan and Heggie [157Jump To The Next Citation Point] and can be described by

0.065⟨v2⟩3∕2 tr = ------------, (11 ) ρ⟨m ⟩G2 ln Λ
where ρ is the local mass density, ⟨v2⟩ is the mass-weighted mean square velocity of the stars, and ⟨m ⟩ is the mean stellar mass. The Coulomb logarithm, ln Λ, is the logarithm of the ratio of the maximum to minimum expected impact parameters in the cluster. Typical values of Λ range between 0.4N [157172Jump To The Next Citation Point] and 0.1N [79]. Binney and Tremaine provide a range of values for ln Λ from 10.1 in the center of the cluster to 12 at rh. Note that in the central regions of the cluster, the value of t r is much lower than the average relaxation time. This means that in the core of the cluster, where the more massive stars have concentrated, there are more encounters between these stars.

The concentration of massive stars in the core of the cluster will occur within a few relaxation times, t ∼ trelax ∼ 109 yr. This time is longer than the lifetime of low metallicity stars with M ≥ 2 M ⊙ [207]. Consequently, these stars will have evolved into carbon-oxygen (CO) and oxygen-neon (ONe) white dwarfs, neutron stars, and black holes. After a few more relaxation times, the average mass of a star in the globular cluster will be around 0.5 M ⊙ and these degenerate objects will once again be the more massive objects in the cluster, despite having lost most of their mass during their evolution. Thus, the population in the core of the cluster will be enhanced in degenerate objects. Any binaries in the cluster that have a gravitational binding energy significantly greater than the average kinetic energy of a cluster star will act effectively as single objects with masses equal to their total mass. These objects, too, will segregate to the central regions of the globular cluster [236]. The core will then be overabundant in binaries and degenerate objects.

The core would undergo what is known as core collapse within a few tens of relaxation times unless these binaries release some of their binding energy to the cluster. In core collapse, the central density increases to infinity as the core radius shrinks to zero. An example of core collapse can be seen in the comparison of two cluster evolution simulations shown in Figure 4View Image [126Jump To The Next Citation Point]. Note the core collapse when the inner radius containing 1% of the total mass dramatically shrinks after t ∼ 15 trelax. Since these evolution syntheses are single-mass Plummer models without binary interactions, the actual time of core collapse is not representative of a real globular cluster.

The static description of the structure of globular clusters using King–Michie or Plummer models provides a framework for describing the environment of relativistic binaries and their progenitors in globular clusters. The short-term interactions between stars and degenerate objects can be analyzed in the presence of this background. Over longer time scales (comparable to t relax), the dynamical evolution of the distribution function as well as population changes due to stellar evolution can alter the overall structure of the globular cluster. We will discuss the dynamical evolution and its impact on relativistic binaries in Section 5.

Before moving on to the dynamical models and population syntheses of relativistic binaries, we will first look at the observational evidence for these objects in globular clusters.

View Image

Figure 4: Lagrange radii indicating the evolution of a Plummer model globular cluster for an N-body simulation and a Monte Carlo simulation. The radii correspond to radii containing 0.35, 1, 3.5, 5, 7, 10, 14, 20, 30, 40, 50, 60, 70, and 80% of the total mass. Figure taken from Joshi et al. [126Jump To The Next Citation Point].

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