Stellar Dynamics E NCYCLOPEDIA of a STRONOMY and a STROPHYSICS

Stellar Dynamics E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS Stellar Dynamics Stellar dynamics describes systems of many point mass particles whose mutual gravitational interactions determine their orbits. These particles are usually taken to represent stars in small GALAXY CLUSTERS with 2 3 about 10 –10 members, or in larger GLOBULAR CLUSTERS 4 6 with 10 –10 members or in GALACTIC NUCLEI with up to about 109 members or in galaxies containing as many as 1012 stars. Under certain conditions, stellar dynamics can also describe the motions of galaxies in clusters, and even the general clustering of galaxies throughout the universe Figure 1. The deflection of a star m2 by a more massive star m1, itself. This last case is known as the cosmological many- schematically illustrating two-body relaxation. body problem. The essential physical feature of all these examples is that each particle (whether it represents a star or random orbits, their mean free path to geometric collisions an entire galaxy) contributes importantly to the overall is R3 gravitational field. In this way, the subject differs from λG ≈ 1/nσ ≈ d (1) Nd3 CELESTIAL MECHANICS where the gravitational force of a 3 massive planet or star dominates its satellite orbits. Stellar where R is the radius of a spherical system containing N dynamical orbits are generally much more irregular and objects distributed approximately uniformly. chaotic than those of celestial mechanical systems. This has two easy physical interpretations. First, the Consequently, the description of stellar dynamical average number of times an object can move through its systems is usually concerned with the statistical properties own diameter before colliding is essentially the ratio of the of many orbits rather than with the detailed positions and cluster’s volume to that occupied by all the stars. Second, velocities of an individual orbit. Thus it is not surprising the number of cluster radii the object can traverse before that the KINETIC THEORY OF GASES developed by MAXWELL, colliding is essentially the ratio of the projected area of Boltzmann and others in the late 19th century was adapted the cluster to that of all the objects. In many astronomical by astrophysicists such as Jeans to stellar dynamics in the systems, these ratios are very large. As examples, 105 stars 11 early 20th century. Subsequent results in stellar dynamics in a globular cluster of 10 pc radius have λG/R ≈ 3 × 10 3 λ /R ≈ contributed to the first analyses of kinetic plasma physics and 10 galaxies in a cluster of 3 Mpc radius have G in the 1950s. Then rapid evolution of plasma theory 30. Therefore stellar dynamics is a good approximation in the second half of the 20th century, stimulated partly over a wide range of conditions. It may, however, break by prospects of controlled thermonuclear fusion, in turn down in the cores of realistic systems where only a few contributed to stellar dynamics. This was an especially objects dominate at the very center and orbits are more productive interdisciplinary interaction. regular. After describing basic physical processes such Although geometric collisions may be infrequent, as timescales, relaxation processes, dynamical friction gravitational encounters are common. These occur when and damping, this article derives the virial theorem one object passes by another, perturbing both orbits. and mentions some applications, discusses distribution Naturally, in a finite system all the objects are passing by functions and their evolution through the collisionless each other all the time, so this process is continuous. In a Boltzmann equation and the BBGKY hierarchy, and system which is already fairly stable, most perturbations outlines the thermodynamic descriptions of finite and are small. However, their cumulative effects over long times can be large and affect the evolution of the system infinite gravitating systems. The emphasis here is on significantly. To see how this works, we introduce the fundamental physics rather than on detailed models. fundamental notion of a ‘stellar dynamical relaxation time’. This is essentially the timescale for a dynamical Basic ideas quantity such as a particle’s velocity to change by an We start with simple ideas that are common to most stellar amount approximately equal to its original value. dynamical systems. For point masses to represent their As an illustration of the general principle, consider components, physical collisions must be rare. In a system two-body relaxation. Suppose, as in figure 1, that a d m m of objects, each with radius , this means that the total massive star 1, deflects a much less massive star 2. m v internal volume of all the objects must be much less than Initially, 2 moves with velocity perpendicular to the the volume over which they swarm. Two spherical objects distance b (the impact parameter) at which the undeflected d d m of radius will have an effective radius 2 for a grazing orbit would be closest to 1. σ = πd2 Gm b−2 collision whose cross section is therefore 4 . If there There is a gravitational acceleration 1 which is a number density n of these objects and they move on acts for an effective time 2bv−1 and produces a component Copyright © Nature Publishing Group 2001 Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998 and Institute of Physics Publishing 2001 Dirac House, Temple Back, Bristol, BS1 6BE, UK 1 Stellar Dynamics E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS of velocity within a system. For most systems, τR exceeds the lifetime Gm 2 1 of the system, so it cannot have been the primary process v ≈ (2) bv for a system’s formation and currently relaxed regular roughly spherical appearance. approximately perpendicular to the initial velocity. Since τ τ v v, the effects are linear and they give a scattering Systems in which R c are essentially ‘collisionless’ angle ψ ≈ v/v. A more exact version follows the orbits in the sense that their nearly smooth global gravitational in detail, but this is its physical essence. It shows that large force dominates the average orbits of their particles. Other individual velocity changes, v ≈ v, and large scattering systems in which strong local fluctuations in gravitational angles occur when two objects are so close that the forces dominate particle orbits are ‘collisional’, since these fluctuations produce large-angle scatterings of the gravitational potential energy, Gm m /b, approximately 1 2 orbits. (This standard terminology does not refer to equals the kinetic energy, m v2/2. Such close encounters 2 bodily collisions of objects.) Realistic astronomical are rare. Typical encounters in a spherical system of systems usually combine aspects of both of these idealized total mass M and radius R containing N objects have an categories. For example, local gas clouds, star clumps impact parameter about equal to the mean separation, b ≈ RN−1/3 m ≈ M/N or dark matter inhomogeneities may scatter stellar orbits . The average mass 1 , and the from their average smooth paths in GALAXIES. initial velocity is given by the approximate balance of the Even for maintaining a relaxed system near a state system’s total kinetic and potential energy, v2 ≈ GM/R. of equilibrium, gravitational encounters cannot be the (This last relation follows from the virial theorem below.) whole story. If they were, the constant average increase Therefore from equation (2) in the root-mean-square velocities of all the particles v would eventually cause them all to escape, defying the ≈ ψ ≈ N −2/3 v (3) conservation of energy. A second, balancing, process must operate. This is dynamical friction. for N 1, and most gravitational encounters involve Imagine yourself on a particle moving faster than little energy or momentum exchange. Exceptions may the average speed of the particles around it. In a occur in the centers of clusters, particularly among more stationary reference frame attached to your particle, the massive particles, where the core is relatively isolated and other particles will appear to stream by. More particles the effective value of N is low. will stream by in the direction you are moving toward Although most individual deflections are small, their than in other directions. The orbits of these excess cumulative effects are not. Over long times, uncorrelated particles will be deflected slightly by the gravity of your deflections from the nearly randomly moving orbits particle. This deflection causes a slight convergence of cause each particle’s velocity to randomly walk around the orbits behind your particle, opposite the direction of its original value. Thus the root-mean-square velocity its motion. On average, therefore, there is a statistical increases from its initial value in proportion to the square excess of particles behind yours. Their excess gravity root of the number of encounters, or the square root of decelerates the motion of your particle, and tends to the time. To estimate this change, we multiply the square reduce it to the average velocity. Conversely, if the speed of each velocity change from equation (2) by the number of your particle is slower than average, there will be a of encounters with a given velocity and impact parameter statistical excess of particles passing it in your direction during a unit time and then integrate over the whole range of motion. Their orbits will converge slightly in front of of velocities, impact parameters and times of interest. the motion of your particle, and their excess gravitational Finally setting ( v)2 = v2 gives the stellar dynamical force will speed up your particle. Thus dynamical relaxation time for the cumulative small changes to modify friction is a great leveller of velocities. The balance the velocity by an amount equal to its initial value. For between dynamical friction and two-body acceleration τ homogeneous systems near equilibrium with GM ≈ Rv2 keeps the system close to equilibrium.

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