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2.2 Globular cluster structure

The overall structure of a globular cluster can be described in terms of an N-body system. These systems, each containing between 104 and 107 stars, have central densities in the range 10- 1 to 106M /pc3 o., with an average of 104M /pc3 o .. 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 construct, while the half-light radius is two-dimensional. 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 [19Jump To The Next Citation Point117Jump 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 timerequired 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 [19Jump To The Next Citation Point116]

t - ~ 0.1N-t . (1) relax lnN cross
For a globular cluster with N = 105, a characteristic size of R ~ rh ~ 10 pc, and a typical velocity of v ~ 10 km/s, the crossing time and relaxation time are 5 tcross ~ 10 yr and 8 trelax ~ 10 yr [19Jump To The Next Citation Point]. It should be noted, however, that Figure 1.3 of Spitzer [152] gives a median trelax ~ 1 Gyr and Padmanabhan gives t ~ 109 yr relax [117Jump 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 g of the stars in the cluster are evaporated every relaxation time. Thus, the rate of loss is dN/dt = - gN/trelax = - N/tevap. The value of g can be determined by noting that the escape speed ve at a point x is related to the gravitational potential P(x) at that point by v2e = - 2P(x). Consequently, the mean-square escape speed in a cluster with density r(x) is

integral 2 3 integral 3 <v2e> = - integral r(x)ved-x-= - 2-r(x)P(x)d--x-= - 4W--, (2) r(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 2 - W = 2K = M <v >, where K is the total kinetic energy of the cluster, and
2 2 &lt;ve&gt; = 4&lt;v &gt;. (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 g = 7.38× 10-3. Therefore, the evaporation time is
trelax tevap = -----= 136 trelax. (4) g
Stellar evolution annd tidal interactions tend to shorten the evaporation time. See Gnedin and Ostriker [59] and references therein for a thorough discussion of these effects. For a globular cluster, tevap ~ 1010 yr, which is comparable to the observed age of globular clusters.

The characteristic time scales 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,

@f v . \~/ f - \~/ f .@v- = 0, (5)
where the gravitational potential f is found from f with
integral 2 3 \~/ f = 4p 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 =_ Y - v2/2 with the relative potential defined as Y =_ - f + f0. The constant f0 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),

7/2 f(E ) = F E , (7)
generate what are known as Plummer models. A convenient class of models which admit anisotropy and a distribution in angular momenta L are known as King-Michie models. The King-Michie distribution function is:
( - L2 ) [ 2 ] f (E,L) = r1(2ps2) -3/2exp --2-2- eE/s - 1 , E &gt; 0, (8) 2ras
with f = 0 for E < 0 and r1 a constant. The velocity dispersion is determined by s and the anisotropy radius ra is defined so that the velocity distribution changes from nearly isotropic at the center to nearly radial at ra.

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