Stellar DynamicsENCYCLOPEDIA OF ASTRONOMY AND ASTROPHYSICSStellar DynamicsStellar dynamics describes systems of many pointmass particles whose mutual gravitational interactionsdetermine their orbits. These particles are usuallytaken to represent stars in smallGALAXY CLUSTERS withabout 102–103members, or in larger GLOBULAR CLUSTERSwith 104–106members or in GALACTIC NUCLEI with up toabout 109members or in galaxies containing as many as1012stars. Under certain conditions, stellar dynamics canalso describe the motions of galaxies in clusters, and eventhe general clustering of galaxies throughout the universeitself. This last case is known as the cosmological many-body problem.The essential physical feature of all these examplesis that each particle (whether it represents a star oran entire galaxy) contributes importantly to the overallgravitational field. In this way, the subject differs fromCELESTIAL MECHANICS where the gravitational force of amassive planet or star dominates its satellite orbits. Stellardynamical orbits are generally much more irregular andchaotic than those of celestial mechanical systems.Consequently, the description of stellar dynamicalsystems is usually concerned with the statistical propertiesof many orbits rather than with the detailed positions andvelocities of an individual orbit. Thus it is not surprisingthat theKINETIC THEORY OF GASES developed by MAXWELL,Boltzmann and others in the late 19th century was adaptedby astrophysicists such as Jeans to stellar dynamics in theearly 20th century. Subsequent results in stellar dynamicscontributed to the first analyses of kinetic plasma physicsin the 1950s. Then rapid evolution of plasma theoryin the second half of the 20th century, stimulated partlyby prospects of controlled thermonuclear fusion, in turncontributed to stellar dynamics. This was an especiallyproductive interdisciplinary interaction.After describing basic physical processes suchas timescales, relaxation processes, dynamical frictionand damping, this article derives the virial theoremand mentions some applications, discusses distributionfunctions and their evolution through the collisionlessBoltzmann equation and the BBGKY hierarchy, andoutlines the thermodynamic descriptions of finite andinfinite gravitating systems. The emphasis here is onfundamental physics rather than on detailed models.Basic ideasWe start with simple ideas that are common to most stellardynamical systems. For point masses to represent theircomponents, physical collisions must be rare. In a systemof objects, each with radius d, this means that the totalinternal volume of all the objects must be much less thanthe volume over which they swarm. Two spherical objectsof radius d will have an effective radius 2d for a grazingcollision whose cross section is therefore σ = 4πd2. If thereis a number density n of these objects and they move onFigure 1. The deflection of a star m2by a more massive star m1,schematically illustrating two-body relaxation.random orbits, their mean free path to geometric collisionsisλG≈ 1/nσ ≈R33Nd3d (1)where R is the radius of a spherical system containing Nobjects distributed approximately uniformly.This has two easy physical interpretations. First, theaverage number of times an object can move through itsown diameter before colliding is essentially the ratio of thecluster’s volume to that occupied by all the stars. Second,the number of cluster radii the object can traverse beforecolliding is essentially the ratio of the projected area ofthe cluster to that of all the objects. In many astronomicalsystems, these ratios are very large. As examples, 105starsin a globular cluster of 10 pc radius have λG/R ≈ 3 × 1011and 103galaxies in a cluster of 3 Mpc radius have λG/R ≈30. Therefore stellar dynamics is a good approximationover a wide range of conditions. It may, however, breakdown in the cores of realistic systems where only a fewobjects dominate at the very center and orbits are moreregular.Although geometric collisions may be infrequent,gravitational encounters are common. These occur whenone object passes by another, perturbing both orbits.Naturally, in a finite system all the objects are passing byeach other all the time, so this process is continuous. In asystem which is already fairly stable, most perturbationsare small. However, their cumulative effects over longtimes can be large and affect the evolution of the systemsignificantly. To see how this works, we introduce thefundamental notion of a ‘stellar dynamical relaxationtime’. This is essentially the timescale for a dynamicalquantity such as a particle’s velocity to change by anamount approximately equal to its original value.As an illustration of the general principle, considertwo-body relaxation. Suppose, as in figure 1, that amassive star m1, deflects a much less massive star m2.Initially, m2moves with velocity v perpendicular to thedistance b (the impact parameter) at which the undeflectedorbit would be closest to m1.There is a gravitational acceleration Gm1b−2whichacts for an effective time 2bv−1and produces a componentCopyright © Nature Publishing Group 2001Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998and Institute of Physics Publishing 2001Dirac House, Temple Back, Bristol, BS1 6BE, UK1Stellar DynamicsENCYCLOPEDIA OF ASTRONOMY AND ASTROPHYSICSof velocityv ≈2Gm1bv(2)approximately perpendicular to the initial velocity. Sincev  v, the effects are linear and they give a scatteringangle ψ ≈ v/v. A more exact version follows the orbitsin detail, but this is its physical essence. It shows that largeindividual velocity changes, v ≈ v, and large scatteringangles occur when two objects are so close that thegravitational potential energy, Gm1m2/b, approximatelyequals the kinetic energy, m2v2/2. Such close encountersare rare. Typical encounters in a spherical system oftotal mass M and radius R containing N objects have animpact parameter about equal to the mean separation,b ≈ RN−1/3. The average mass m1≈ M/N, and theinitial velocity is given by the approximate balance of thesystem’s total kinetic and potential energy, v2≈ GM/R.(This last relation follows from the virial theorem below.)Therefore from equation (2)vv≈ ψ ≈ N−2/3(3)for N  1, and most gravitational encounters involvelittle energy or momentum exchange. Exceptions mayoccur in the centers of clusters, particularly among moremassive