Globular Cluster Evolution

After a globular cluster has formed, and survived the tidal forces which threaten it in its early development, there are two major processes at play: mass segregation (or core-collapse) and evaporation.

1.3.3.1 Mass Segregation and Core-Collapse

In a globular cluster of sufficient age, encounters between stars will have redis-tributed the initial star kinetic energies, trying to drive the stars in a GC towards equipartition. As this happens, the most massive stars slow down and move towards the centre of the cluster, while less massive stars, which have higher velocities for the same kinetic energy, move towards the outer region. This process is called mass segregation (e.g. Elson et al. 1987; Heggie & Hut 2003). The central stars then occupy a smaller volume, making the core region more densely crowded. In turn, the shrinking of the core speeds up the energy transfer process, so the core contracts further and the GC halo expands more quickly, in a process known as core-collapse (e.g. Elson et al. 1987; Hut et al. 1992; Heggie & Hut 2003).⁴

Core-collapse can be delayed by the presence of a even a small population of primordial binary systems. When a star passes close to a binary system, the binary pair’s orbit tends to contract, releasing energy. Once the energy released from pri-mordial binaries contained within a cluster is exhausted (i.e. the binaries are all

4The phenomenon in which removal of energy from the core leads to stars dropping into lower orbits, moving faster, and therefore having more energy is sometimes referred to as gravothermal instability (Ashman & Zepf, 1998). Note that this can also happen in single mass models and is not dependent on mass segregation.

18 Chapter 1. An Introduction to Globular Clusters

disrupted or ejected), core-collapse proceeds (Heggie & Hut, 2003).

By contrast, tidal shocks caused by the GC repeatedly passing through the plane of a spiral galaxy tends to speed up core-collapse. As the GC passes through the disk (or the galactic bulge), the gravitational shock gives energy preferentially to stars in the outer parts of the cluster (Ostriker et al., 1972). This allows more stars to escape (see Section 1.3.3.2), but also adds to the velocity dispersion in the core and accelerates core-collapse (Gnedin et al., 1999).

The fact that GCs do not contain infinitely dense cores indicates that something must happen to reverse the process. It turns out that the system is able to sustain itself (see Heggie & Hut 2003). The high stellar densities in the central region of a core-collapsed cluster lead to frequent interactions, including three body in-teractions in which two single stars form a binary system, with a third star acting as a catalyst. In such an encounter, the single (catalyst) star gains kinetic energy, and the newly formed binary system is then more likely to undergo further interac-tions. When another single star passes close to the binary, the binary system’s orbit shrinks, and the binary is said to ‘harden’. The binary system loses gravitational energy, while the kinetic energy of the passing star increases. Similarly, as core density rises, existing binary systems are also more likely to interact with passing single (or binary) stars. In fact, mass segregation causes primordial binaries to ac-cumulate in the cluster core, increasing the efficiency of binary-binary interactions.

Close encounters between binary systems may dissolve one of the binaries, leaving a binary and two single stars, or may harden both binary systems, leading to the release of energy. The increase in available energy means that core-collapse hap-pens in reverse; stars move outwards from the core, lowering the core density, and the core expands further. This is sometimes known as a post-core-collapse bounce phase (Heggie & Hut, 2003).

Observationally, the evolutionary stage of a GC can be determined from its ra-dial profile. As explained in Section 1.2.2, the rara-dial profile (whether constructed in terms of surface brightness or stellar density) demonstrates that, in general, the stel-lar density increases with decreasing distance from the core. In a pre-core-collapse cluster, this increase continues down to a certain distance from the core, and then flattens. Recent models have shown that a cluster which has undergone post-core-collapse bounce may be indistinguishable, based on its radial profile, from a pre-collapse cluster (Heggie & Giersz, 2009). A GC that is undergoing core-pre-collapse, or in a post-core-collapse bounce phase can be identified by a continual increase in luminosity all the way to the GC centre. It should be noted, however, that such a ‘cusp’ can also indicate the presence of a central black hole. This is discussed

1.3 The Life of a Globular Cluster 19

further in Section 1.4.

Numerically, as explained in Section 1.2.2, the dynamical status of a GC can be described in terms of its concentration parameter, c= log₁₀(rt/rc). A concentration parameter of c& 2 − 2.5 is considered to be on the verge of collapsing, undergoing core-collapse, or post-core-collapse (Meylan & Heggie, 1997).

There is, as yet, no clear way to determine whether a cluster is about to undergo core-collapse, in the process of doing so, or in a post-collapse bounce phase, either from the slope of the cusp in a radial profile, or from its concentration parameter.

It is possible, however, to distinguish whether GCs are in this group of phases or not. Fitting radial profiles to single-mass King models, which are characterised by values of r_t and r_c, can give an indication of whether the entire cluster can be defined using one value of c, or if different values are needed to describe the core and outer regions. If multiple King models are needed to reproduce the radial profile of a cluster, it can be thought of as in one of the core-collapse phases. This is investigated in the case of NGC 6752, in Chapter 4.

1.3.3.2 Evaporation

While mass segregation and core-collapse dominate the dynamics of the central region of the cluster, the overall size of the cluster is affected by evaporation.

Evaporation occurs when stars in the outer part of the cluster have enough ki-netic energy to be removed entirely from the cluster’s gravitational influence (i.e.

its velocity exceeds the escape velocity of the cluster, vstar > vesc). It is one of the ways in which a GC can be destroyed. In isolation, this will happen through two-body relaxation as a cluster tries to move towards equipartition (Spitzer, 1940), as previously discussed (see Section 1.3.3.1), but the presence of an external gravita-tional field makes the process happen quicker. In Section 1.2.2, I defined the tidal radius as the radial distance at which the gravitational force of the GC is equal to its host galaxy. As a GC orbits a galaxy, stars outside this radius will be captured by the galaxy and stripped away. The external gravitational field can be introduced in a more dramatic way in the form of tidal shocking.

Tidal shocking, also known as disk shocking, has already been mentioned in the context of its affect on core-collapse rates. It occurs when the GC passes through the disk (Ostriker et al., 1972) or bulge (Gnedin & Ostriker, 1997) of a galaxy and is well known to increase the rate at which stars are lost from the outskirts of the cluster (Kundic & Ostriker, 1995). This tidal stripping effect is self-limiting;

as a GC loses mass and becomes more compact, the importance of tidal shocks

20 Chapter 1. An Introduction to Globular Clusters

diminishes.

The time-scales on which evaporation and tidal shocking occur can be inter-preted as indicating the characteristic time-scale over which a GC will be destroyed.

There are three key time-scales to consider in the context of GCs: the crossing time, relaxation time and evaporation time.

The crossing time of a cluster is the typical time taken for a star to travel a distance equal to the characteristic size of the cluster. This is usually taken to be the half-mass radius, r_m(see Section 1.2.2). The crossing time is given by

t_cross≈ r_m v ,

where v is the typical velocity of a cluster member (≈ 10 km/s; Benacquista 2006).

Typically, t_cross≈ 10⁵years (Padmanabhan, 2001).

The relaxation time of a gravitationally bound system is defined as the charac-teristic time taken for the component stars to interact with one another enough that they lose all history of their original velocity. Once this occurs, in theory, the stars all have equal kinetic energy. It is related to the number and strength of the interac-tions taking place within the cluster, so depends on the average length of time that a star takes to cross the system, as well as the number of stars contained within the system. The mean relaxation time can be shown to be

t_relax≈ 0.1N lnN t_cross.

(Binney & Tremaine, 1987). In a GC, the relaxation time is typically less than 10¹⁰years, at least out to the half-mass radius. By contrast, the relaxation time of an elliptical galaxy is generally longer than this, except in the innermost few parsecs (Elson et al., 1987), so the relaxation process is far more important in the evolution of a GC than a galaxy. Of course, stars moving towards the gravitational centre of the cluster, as well as interactions between individual stars mean that GCs cannot ever be truly ‘relaxed’.

The evaporation time of a globular cluster is the time taken for a cluster to dis-solve by losing stars whose kinetic energy is high enough for them to escape. This high-end tail of the distribution of speeds is replenished by relaxation. The evapo-ration time is given by

t_evap= t_relax γ ^,

whereγ is the fraction of stars which escape in one relaxation time. For an isolated system, assuming that the system is virialised, this turns out to give an evaporation