There are also important characteristic time scales that govern the dynamics of globular clusters. These are the crossing time , the relaxation time , and the evaporation time . The crossing time is the typical time required for a star in the cluster to travel the characteristic size of the cluster (typically taken to be the half-mass radius). Thus, , where 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 [24, 171]

For a globular cluster with , a characteristic size of , and a typical velocity of , the crossing time and relaxation time are and , although Binney and Tremaine use and consequently [24]. In real globular clusters, the relaxation time varies throughout the cluster and the median value is closer to [24] as found in Figure 1.3 of Spitzer [222] and in Padmanabhan [172].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 of the stars in the cluster are evaporated every relaxation time. Thus, the rate of loss is . The value of can be determined by noting that the escape speed at a point is related to the gravitational potential at that point by . Consequently, the mean-square escape speed in a cluster with density is

where is the total potential energy of the cluster and is its total mass. If the system is virialized (as we would expect after a relaxation time), then , where is the total kinetic energy of the cluster, and Thus, stars with speeds above twice the RMS speed will evaporate. Assuming a Maxwellian distribution of speeds, the fraction of stars with is . Therefore, the evaporation time is 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 for a globular cluster, we see that , which is comparable to the observed age of globular clusters.The characteristic time scales of globular clusters differ significantly from each other: .

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 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 . The structure of the globular cluster is then determined by the collisionless Boltzmann equation,

where the gravitational potential is found from withThe solutions to Equation (5) are often described in terms of the relative energy per unit mass with the relative potential defined as . The constant is chosen so that there are no stars with relative energy less than 0 (i.e. for and for ). A simple class of solutions to Equation (5),

generates what are known as Plummer models. A convenient class of models which admits anisotropy and a distribution in angular momenta is known as King–Michie models. The King–Michie distribution function is with for and being a constant. The velocity dispersion is determined by and the anisotropy radius is defined so that the velocity distribution changes from nearly isotropic at the center to nearly radial at . 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].http://www.livingreviews.org/lrr-2006-2 |
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