Go to previous page Go up Go to next page

2.2 Globular cluster structure

The overall structure of a globular cluster can be described in terms of a roughly spherical N-body system with central densities in the range −1 10 to 6 3 10 M ⊙ ∕ pc, and an average of 4 3 10 M ⊙∕ pc. The important characteristic radii of a globular cluster are the core radius rc, the half-light radius rh, and the tidal radius rt. The core radius is defined to be the radius at which the surface brightness has dropped to half the central value. The half-light radius is the radius that contains half of the light of the cluster and the tidal radius is defined as the radius beyond which the external gravitational field of the galaxy dominates the dynamics. Theorists define rh to be the radius containing half the mass of the cluster. The half-mass radius is a three-dimensional theoretical construct, while the half-light radius is a two-dimensional observational construct. The tidal radius is always determined by some theoretical model. Typical values of these radii are 1.5 pc, 10 pc, and 50 pc, respectively [24Jump To The Next Citation Point172Jump To The Next Citation Point].

There are also important characteristic time scales that govern the dynamics of globular clusters. These are the crossing time tcross, the relaxation time trelax, and the evaporation time tevap. The crossing time is the typical time required for a star in the cluster to travel the characteristic size R of the cluster (typically taken to be the half-mass radius). Thus, tcross ∼ R ∕v, where v is a typical velocity (∼ 10 km/s). The relaxation time is the typical time for gravitational interactions with other stars in the cluster to remove the history of a star’s original velocity. This amounts to the time required for gravitational encounters to alter the star’s velocity by an amount comparable to its original velocity. Since the relaxation time is related to the number and strength of the gravitational encounters of a typical cluster star, it is related to the number of stars in the cluster and the average energy of the stars in the cluster. Thus, it can be shown that the mean relaxation time for a cluster is [24Jump To The Next Citation Point171]

0.1N- trelax ≃ lnN tcross. (1 )
For a globular cluster with 5 N = 10, a characteristic size of R ∼ rh ∼ 10 pc, and a typical velocity of v ∼ 10 km ∕s, the crossing time and relaxation time are tcross ∼ 106 yr and trelax ∼ 109 yr, although Binney and Tremaine use t ∼ 105 yr cross and consequently t 108 yr relax [24Jump To The Next Citation Point]. In real globular clusters, the relaxation time varies throughout the cluster and the median value is closer to 9 10 yr [24Jump To The Next Citation Point] as found in Figure 1.3 of Spitzer [222] and in Padmanabhan [172Jump To The Next Citation Point].

The evaporation time for a cluster is the time required for the cluster to dissolve through the gradual loss of stars that gain sufficient velocity through encounters to escape its gravitational potential. In the absence of stellar evolution and tidal interactions with the galaxy, the evaporation time can be estimated by assuming that a fraction γ of the stars in the cluster are evaporated every relaxation time. Thus, the rate of loss is dN ∕dt = − γN ∕trelax = − N ∕tevap. The value of γ can be determined by noting that the escape speed ve at a point x is related to the gravitational potential Φ (x ) at that point by v2 = − 2Φ (x) e. Consequently, the mean-square escape speed in a cluster with density ρ(x) is

∫ 2 3 ∫ 3 ⟨v2⟩ = -∫ρ(x)ve-d-x = − 2--ρ(x-)Φ-(x)-d-x = − 4W--, (2 ) e ρ(x) d3x M M
where W is the total potential energy of the cluster and M is its total mass. If the system is virialized (as we would expect after a relaxation time), then − W = 2K = M ⟨v2⟩, where K is the total kinetic energy of the cluster, and
2 2 ⟨ve⟩ = 4⟨v ⟩. (3 )
Thus, stars with speeds above twice the RMS speed will evaporate. Assuming a Maxwellian distribution of speeds, the fraction of stars with v > 2vrms is γ = 7.38 × 10 −3. Therefore, the evaporation time is
trelax- tevap = γ = 136 trelax. (4 )
Stellar evolution and tidal interactions tend to shorten the evaporation time (see Gnedin and Ostriker [83] and references therein for a thorough discussion of these effects). Using a typical trelax for a globular cluster, we see that tevap ∼ 1010 yr, which is comparable to the observed age of globular clusters.

The characteristic time scales of globular clusters differ significantly from each other: tcross ≪ trelax ≪ tevap.

When discussing stellar interactions during a given epoch of globular cluster evolution, it is possible to describe the background structure of the globular cluster in terms of a static model. These models describe the structure of the cluster in terms of a distribution function f that can be thought of as providing a probability of finding a star at a particular location in phase-space. The static models are valid over time scales which are shorter than the relaxation time so that gravitational interactions do not have time to significantly alter the distribution function. We can therefore assume ∂f∕∂t ∼ 0. The structure of the globular cluster is then determined by the collisionless Boltzmann equation,

v ⋅ ∇f − ∇ φ ⋅ ∂f = 0, (5 ) ∂v
where the gravitational potential φ is found from f with
2 ∫ 3 ∇ φ = 4π f (x,v,m ) d vdm. (6 )

The solutions to Equation (5View Equation) are often described in terms of the relative energy per unit mass ℰ ≡ Ψ − v2∕2 with the relative potential defined as Ψ ≡ − φ + φ0. The constant φ0 is chosen so that there are no stars with relative energy less than 0 (i.e. f > 0 for ℰ > 0 and f = 0 for ℰ < 0). A simple class of solutions to Equation (5View Equation),

f(ℰ ) = F ℰ 7∕2, (7 )
generates what are known as Plummer models. A convenient class of models which admits anisotropy and a distribution in angular momenta L is known as King–Michie models. The King–Michie distribution function is
( ) 2− 3∕2 -− L2- [ ℰ∕σ2 ] f(ℰ,L ) = ρ1(2πσ ) exp 2r2σ2 e − 1 , ℰ > 0, (8 ) a
with f = 0 for ℰ ≤ 0 and ρ1 being a constant. The velocity dispersion is determined by σ and the anisotropy radius ra is defined so that the velocity distribution changes from nearly isotropic at the center to nearly radial at ra. The King–Michie distribution can be generalized to multi-mass systems, and although not dynamically correct, they can be used for mass estimates. A good description of the construction of a multi-mass King–Michie model can be found in the appendix of Miocchi [161].
  Go to previous page Go up Go to next page