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. Therefore R and a Maxwellian distribution of velocities, this is must also be the approximate timescale for relaxation by dynamical friction. Mathematically, a second-order non- 0.2N linear partial differential–integral equation, known as the τR ≈ τc (4) ln(N/2) Fokker–Planck equation, describes this balance, and it is usually solved numerically. − / where τc = R/v ≈ (Gρ)¯ 1 2 is the dynamical crossing time Dynamical friction also plays another important role for a particle with average velocity and ρ¯ is the average in stellar dynamics. If a massive object M (such as a high- density. mass star in a galaxy, or a large black hole in a galactic In globular clusters, galaxies and rich clusters of nucleus or even a small galaxy in the halo of another galaxies N 103 so τR τc. Cumulative small deﬂections larger galaxy) moves rapidly through a ﬁeld of much do not amount to much over any one orbital period, but less massive objects, m, dynamical friction will cause it their secular effect can dominate after many orbits. The to lose kinetic energy, fall deeper into the surrounding relaxation time τR will vary depending on the geometry, gravitational well, speed up while falling, lose more range of masses and velocity and density distributions energy and consequently spiral into the center of the larger
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 2 Stellar Dynamics E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS system. This helps promote the merging of galaxies, Violent relaxation occurs in the extreme limit when for example (see GALAXIES: INTERACTIONS AND MERGERS). The collective modes are so chaotic on all scales that each of relaxation time τR of equation (4) is decreased signiﬁcantly them lasts for only a short time but is quickly replaced by a factor of order m/M. This is because the large by another such mode elsewhere in the system. None of mass difference induces gravitational excitations, known these modes correlates with one another. Each star moves as grexons (also from the Latin ‘grex’ meaning ‘herd’ or in a mean ﬁeld which changes quickly in time as well as ‘ﬂock’): a collective gravitational excitation in the wake in space. Consequently, the energy of each individual of a massive particle moving through a system of less star along its orbit is not conserved; only the energy massive particles. It generally takes the form of a region of the entire system remains constant. If this process of statistical overdensity among the less massive particles could continue indeﬁnitely, the velocity distribution of passing through the wake. These gravitating collective the stars would generally become similar to the Maxwell– modes of many small masses interact with each other, Boltzmann distribution for a perfect gas after a timescale enhancing their lifetimes and producing a larger effective τc. statistical condensation behind M through which particles However, it does not. On this same gravitational free- of mass m move but linger longer than they would for fall timescale, τc, damping mechanisms destroy the ideal M ≈ m. These grexons arise when M m and amplify conditions for violent relaxation. In stellar dynamics, as the gravitational drag. in other systems, damping mechanisms reduce departures Other forms of collective excitations can also amplify from an equilibrium state. Most damping is characterized relaxation in systems which are initially far from by dynamical dissipation—the roughly random transfer equilibrium. This collective relaxation—of which ‘violent of relatively ordered (low-entropy) energy into relatively relaxation’ is an extreme case—is much faster than τR and disordered (high-entropy) energy. The transfer may closer to the crossing timescale τc. Let us start by thinking occur among ‘particles’, as when a stellar cluster or a about a relatively simple case. Suppose one star of mass m forming cluster of galaxies ejects a high-energy member initially has zero velocity at the edge of an already relaxed so that the remaining cluster becomes more tightly bound. cluster of stars with total mass M and radius R. Details of Another example of particle dissipation is the phase its orbit would depend on the density distribution of the mixing of orbits. Generally this involves interactions of cluster and the building up of small perturbations, but the stars at slightly different phases of the same, otherwise star would roughly follow the equation of motion mr¨ ≈ unperturbed, orbit. As a simple illustration, suppose an GmMR−2. Sincer ¨ ≈ R/t2, this would give a characteristic isolated spherical cluster of stars, initially completely at infall timescale ∼(Gρ)¯ −1/2, which is just the crossing time rest, starts falling together. The stars will not simply all τc. We may think of this as the star being scattered by the plunge together into the center. Instead, stars at different average force of the entire cluster, i.e. by the ‘mean ﬁeld’. distances from the center, i.e. at different phases of their As the star falls inward, the small discrete perturbations of radial orbits, will perturb each other. These perturbations nearby stars will also scatter it, and this ‘ﬂuctuating ﬁeld’ can be large among stars with low relative velocities, will impart some net angular momentum to the star so that and can occur on the dynamical crossing timescale τc. it is unlikely to reach the exact center of the cluster. Instead, Most stars will acquire some angular momentum from it will join the other stars in their complicated orbits. The these perturbations, although the cluster’s net angular net relaxed orbit of the star comes from the joint effects of momentum remains zero. This further mixes the phase of the smooth global mean ﬁeld and the two-body relaxation the orbits, dissipating some of the initially highly ordered of the local ﬂuctuating ﬁeld. radial infall velocities into more random transverse Now let us make the process a little more exciting. velocities. Eventually, a component of random kinetic Suppose the cluster is not itself relaxed. Then it energy builds up as a form of heat whose effective pressure will not be quasi-static. Parts of it will be falling resists further collapse of the cluster on the free-fall together, interpenetrating, separating, streaming in timescale τc. Similar phase mixing of orbits, combined different directions and generally ﬂopping around. These with heat generated by collective relaxation itself, can represent larger-scale ﬂuctuations—collective modes— occur to damp more general collective motions such as which will scatter the incoming star in addition to its those of violent relaxation. More gentle phase mixing can scattering by the global mean ﬁeld and the shorter- also occur in relaxed galaxies over longer timescales to mix scale ﬂuctuations of nearby individual stars. Moreover, up streaming motions of stars. the large-scale ﬂuctuations will also be scattering and Dissipation and damping may also result when interacting with each other. With high-amplitude large- particles and waves interact. The waves are moving mass collective modes ﬂuctuating over a wide range of periodic density perturbations. The best-known case lengthscales, and on corresponding timescales shorter was ﬁrst found for plasmas by Landau and then than τc for the entire cluster, individual orbits will relax on applied to stellar dynamics. It occurs even in idealized timescales closer to τc than τR. This is collective relaxation, collisionless systems where particles do not scatter one and it may apply to the formation of clusters of stars, another signiﬁcantly. The physical reason for collisionless galaxies and clusters of galaxies, particularly if they build damping arises from the detailed interaction of a wave up by the merging of smaller systems. with the orbits of background stars which are not part of
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 3 Stellar Dynamics E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS the wave. Thus this process would not show up in a pure for the ith components of position and velocity, where the continuum approximation. What happens is that some of gravitational potential the background stars will move a bit faster than the wave, m(β) others a bit slower, even though their average background ϕ = G |x(α) − x(β)| (6) number density is constant. The wave exerts a force on β =α these stars and thus exchanges energy with them. On average, the wave gains energy from the fast-moving stars, results from all the other particles. This derivation is which therefore amplify the wave, and loses energy to the general enough to let the particle masses vary with time, slow-moving stars, which damp the wave. The net result by mass loss or accretion, isotropically as we will assume, depends on whether there are more fast or slow stars with or even anisotropically with a net ‘rocket effect’. v = ω/k (α) velocities near the phase velocity of the wave. Now multiply equation (5) by xj and sum over α Moreover, the strength of the growth or damping depends to take its lowest-order position moment. Since α and β on the total net number of fast or slow stars involved. are just dummy indices, we could just as validly multiply A detailed analysis of this result is complex but, for a (β) through by xj . Equivalently, and more elegantly, we roughly Maxwellian velocity distribution of background could represent the right-hand side as half the sum of both stars, damping is greatest for waves whose phase velocity these multiplications, i.e. as ω/k is near the velocity dispersion v of the background stars. These waves decay on a timescale λ/ v during (x(α) − x(β))(x(α) − x(β)) G (α) (β) i i j j which the stars are able to move through the wave. Wij =− m m (7) |x(α) − x(β)|3 Phase mixing, Landau damping and other processes 2 α β =α such as trapping by clusters, tidal disruption and small- α β angle scattering all combine with violent relaxation into a which is symmetric to an interchange of the and form of collective relaxation which randomizes velocities particles. This is known as the potential energy tensor, W on a timescale ∼(Gρ)¯ −1/2, provided that the system starts ij . out very far from its eventual quasi-equilibrium state. Rewriting the left-hand side of equation (5) multiplied x(α) What is left is a system in quasi-equilibrium with a roughly by j gives Maxwellian velocity distribution. No direct collisions are responsible for this end result—unlike the case of a perfect x(α) d (m(α)v(α)) = d m(α)x(α)v(α) − m(α)v(α)v(α) j t i t j i i j gas—only the non-linear encounters among particles and α d d α α collective modes. d 1 (α) (α) (α) (α) (α) = m [(xj vi + xi vj ) dt 2 The virial theorem α (α) (α) (α) (α) (α) (α) (α) This is perhaps the most astronomically important + (xj vi − xi vj )] − m vi vj . (8) dynamical property of a quasi-equilibrium system. α Therefore we describe the virial theorem in some detail. The terms in parentheses separated by a minus sign With it, we can estimate the system’s total mass from are antisymmetric, and since all the other contributions observations of the positions and velocities of its particles. to the moment of equation (5) are symmetric, these The virial theorem is essentially just a position moment antisymmetric terms must be zero. This proves of the self-consistent gravitational equations of motion that angular momentum is conserved, since these within the system. To illustrate it in the simplest (α) antisymmetric terms are just the total angular momentum case, consider a satellite of mass m in a circular orbit (β) of the isolated system. The symmetric terms in equation (8) around a much more massive object of mass m . The may be written as balance between centrifugal and gravitational forces gives m(α)v2/r = Gm(α)m(β)/r2. Multiplying through by r gives (α) 1 d (α) (α) (α) (α) (α) (α) (α) (α) 2K + W = 0 where K = m v2/2 is the kinetic energy and m (xj vi + xi vj ) − m vi vj (α) (β) 2 dt W = Gm m /r2 is the gravitational potential energy. α α 2 In this circular orbit the instantaneous value and the time 1 d (α) (α) (α) = m xi xj t 2 average of the energy are identical. For more complicated 2 d α conﬁgurations, we might suspect that this result will still 1 d (α) (α) (α) (α) (α) (α) hold for the more general time averages of K and W.It − m˙ xi xj − m vi vj . (9) 2 dt does. α α The most general Newtonian equations of motion for Now each of the three summations over α on the right- a system of gravitating particles are hand side of equation (9) has a physical interpretation. The ∂ϕ ﬁrst summation is the usual inertia tensor Iij , the second d (m(α)v(α)) = F (α) = m(α) t i i (α) we shall call the mass variation tensor Jij and the third is d ∂xi Tij (α) (β) twice the kinetic energy tensor which incorporates all (α) (β) xi − xi =−Gm m (5) the motions of all the particles. Combining equations (7) |x(α) − x(β)|3 β =α and (9) for the moment of equation (5) then gives the tensor
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 4 Stellar Dynamics E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS virial theorem for a system of gravitationally interacting masses are not identical. Other factors which inﬂuence particles: γ when trying to deduce M are projection effects 1 d2Iij 1 d in transverse positions, loss of one position and two − Jij = 2Tij + Wij . (10) 2 dt 2 2 dt velocity components, departures of time averages from This is an exact result, without any approximations the instantaneous snapshots of observations, questions of so far. It provides an overall constraint on the system’s cluster membership and selection of a limited number of evolution. Further constraints follow from taking higher- particles for observation. Each of these effects depends on order moments of the equations of motion. Multiplying the particular circumstances of an individual cluster. Their γ equation (5) by xmvn and summing gives the (m + n)th combination introduces considerable uncertainty into , combined spatial and velocity moment. This forms an but this can often be estimated with computer simulations. inﬁnite sequence of virial moments. The more moments Virial estimates of the masses of CLUSTERS OF GALAXIES are used, the tighter are the constraints on the system’s have been made since the 1930s. A succession of more evolution. All the moments taken together usually become accurate results with more complete samples and better equivalent to a complete solution of the original equations simulations, which continues to the present, revealed clear of motion (5). discrepancies between the virial mass of many clusters In practice, most astronomical calculations are and their mass estimated from the luminosities of their content with a simpliﬁed version of the lowest-order virial individual galaxies. Unless cluster formation proceeded (α) equation (5). If the mass loss ratem ˙ = 0, then Jij = 0 by a radically different route from the condensation and the form reduces to that quoted in many texts. For processes discussed currently, this discrepancy suggests the special rate of mass loss proportional to the mass, that the amount of ‘DARK MATTER’ in clusters is about ﬁve m˙ (α) = f(t)m(α)(t), the mass loss tensor is proportional times the amount of luminous matter in galaxies. This to the moment of inertia tensor. A great simpliﬁcation gives roughly one-fourth of the total mass needed for a follows by taking the time average of the virial theorem. closed Einstein–Friedmann universe. The form of this For example, the time average of the ﬁrst term gives dark matter and its total amount are two of the main problems of modern astronomy. τ d2I 1 dI˙ 1 = lim dt = lim [I(τ)˙ − I(˙ 0)]. (11) Distribution functions dt 2 τ→∞ τ dt τ→∞ τ 0 Rather than dealing with the positions and motions of each identiﬁable particle individually, stellar dynamics This time average can be zero either if the system is often ﬁnds a less detailed description based on distribution localized in position and velocity space so that I(τ)˙ has functions to be more useful and solvable. These an upper bound for all τ or if the orbits are periodic so that distributions represent the probability that any arbitrary I(τ)˙ = I(˙ 0). Similarly the time average of Jij can be zero if particle in the system has a particular property of interest. m(t)˙ does not increase as fast as t for t →∞. The terms on Usually stellar dynamics is concerned with the single- the right-hand side of equation (10) remain non-zero over particle distribution f(r, v) for the number of particles any time average, so having position r and velocity v. Integrated over all velocities, this gives the density as a function of position. 2 Tij + Wij =0. (12) Integrated over all positions, this gives the velocity The usual (contracted) form of the virial theorem is distribution. Integrated over both position and velocity, it obtained by setting i = j and summing over i = 1, 2, gives the total number of particles in the system. When 3 to give f(r, v) is normalized to this total number, it becomes 2 T + W =0 (13) a probability distribution. Then f(r, v) dr dv is the probability that a particle is in the differential volume dr T W where and are the entire kinetic and potential energies around r and has a velocity in the range dv around v. of the system. Other types of distribution functions are also useful. The virial theorem describes reasonably bound, f(r , r , v , v ) For example, the two- particle distribution 1 2 1 2 quasi-stable clusters of stars and galaxies. Its usual describes the probability that of any two particles one is application is to estimate the mass of these clusters by r v in the volume d 1 with velocity in the range d 1 and the writing equation (13) in the approximate form r v other is simultaneously in d 2 with velocity in d 2. This GM two-particle distribution may be written as the product of V 2 = γ . R (14) two single-particle distributions plus terms representing any correlations that may be present between particles in This is similar to the form for a circular satellite orbit the two volumes or the two velocity ranges. Similarly, discussed earlier, but now V 2 is the velocity dispersion a hierarchy of three-, four-, etc particle distributions and of objects in the cluster, R is the cluster’s radius and correlations may be built up. Eventually it will reach the M is its total mass. The constant γ is usually of N-particle distribution which gives a complete description order unity and depends on the precise operational of the positions and velocities of all N particles in the deﬁnitions of V , R and M, especially if the particle system. This would then be equivalent to specifying the
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 5 Stellar Dynamics E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS position and velocity of every particle—the most detailed of hydrodynamics. However, because short-range description possible. A related form of distribution atomic interactions dominate ﬂuids, it is a much better function, f(N), concentrates on cells rather than particles approximation to truncate the moment equations at low and speciﬁes the probability that a cell of given size and order for ﬂuids than it is in the stellar dynamical case. shape contains N particles. Moreover, the general stellar hydrodynamical equations All these distribution functions may generally have are anisotropic in their spatial and velocity coordinates. anisotropic dependences on position and velocity and Generalizations of the collisionless Boltzmann equa- will evolve with time unless they are in a stationary tion have been developed to incorporate local ﬂuctuations, equilibrium state. Anisotropic distribution functions and such as those described earlier, which lead to dynamical their properties have been widely explored in recent years. relaxation, evaporation and other instabilities. The most Their complexity provides useful descriptions of stellar useful of these is the Fokker–Planck description which rep- motions in both spiral and elliptical galaxies. resents the total gravitational ﬁeld as the sum of two parts: The simplest evolution for the one-particle distribu- the smooth average long-range ﬁeld and the local ﬂuctuat- tion function f(r, v,t)occurs in collisionless systems and ing ﬁeld due to near-neighbor particles. These ﬂuctuations follows the collisionless Boltzmann equation. Here colli- cause the particles to diffuse in velocity space as well as in sionless means that particle orbits proceed smoothly un- conﬁguration space. They also incorporate the dynamical der the inﬂuence of just the mean ﬁeld. There are no scat- friction effects, mentioned earlier, which prevent excessive velocities from being reached. The Fokker–Planck equa- terings by other particles or by local ﬂuctuations. Under tion provides a more accurate account of processes such these conditions f(r, v,t)satisﬁes an equation of continu- as evaporation, core collapse and oscillations which can ity. This is equivalent to its total time derivative, follow- occur in globular clusters. ing the motion through a 6-dimensional position–velocity The most general and rigorous description of stellar phase space, being zero: dynamics, and therefore the least solvable, is known as the Df ∂f BBGKY hierarchy (after the initials of its early developers = + v · ∇f +˙v · ∇vf = 0. (15) in other subjects, Born, Bogoliubov, Green, Kirkwood and Dt ∂t Yvon). It starts from Liouville’s equation which is an Equation (15) is the collisionless Boltzmann equation—one equation of continuity similar to equation (15) but in a N of the most useful descriptions of stellar dynamics. Notice 6 -dimensional (Gibbs) phase space rather than in the N that the gradients in position and in velocity space are 6-dimensional (Boltzmann) phase space. Here is the treated equivalently. Although, written this way, it looks number of physical particles in the system and each point N N like a fairly simple linear partial differential equation for in the Gibbs phase space of 3 position plus 3 velocity f(r, v,t) dimensions represents the state of the entire system of , it is really a complicated non-linear differentio- N integral equation. This is because the gravitation force, 6 particles at a given time. As the system evolves, its proportional to the accelerationv ˙ = ∇φ, is a function of the representative point moves continuously and smoothly through this phase space because there are no external density which is itself an integral of f(r, v,t)over velocity forces outside the system to perturb it. Thus the Liouville space. This relation is obtained from Poisson’s equation, continuity equation is exact. It is a ﬁrst-order linear ∇2φ(r,t) =−4πGρ(r,t), and together with the relevant differential equation. The problem is that it has 6N initial and boundary conditions leads to self-consistent variables, plus time. In fact, it is just a condensed way solutions. of representing the orbits of all the particles in the system. Very few solutions are known in closed analytic To make this description useful, it is necessary to form; most are results of numerical integrations or successively integrate out all but 6(N − 1),6(N − 2),...,6 perturbation theory. The best-known solution occurs of the 6N variables in the full distribution function. This for an idealized homogeneous, isotropic, uncorrelated leaves a set of N coupled non-linear integro-differential equilibrium distribution. It is the Maxwell–Boltzmann equations for all the N, N − 1, N − 2,..., one-particle distribution with a Gaussian distribution of velocities. distribution functions—the BBGKY hierarchy. The entire Solutions of equation (15) also provide a zero-order set is equivalent to the equations of motion for all approximation for systems of particles such as globular N particles. Truncating the lowest-order equation for clusters and rich clusters of galaxies. Their corresponding f(r, v,t) f(r , r , v , v ,t) by neglecting its coupling to 1 2 1 2 spatial density is spherically symmetric, decreasing with is equivalent to the collisionless Boltzmann equation. radius. Such isothermal spheres have to be truncated to Retaining this coupling, but neglecting any higher-order describe clusters of ﬁnite mass. Truncation by tidal cutoffs, coupling, gives essentially the Fokker–Planck equation. evaporation, inﬂow, etc for realistic systems will modify This formalism has been much studied and gives great their distribution functions. insight into the physical nature and relations among Spatial and velocity moments of the collisionless different stellar dynamical descriptions. It has found Boltzmann equation, similar to those leading to the important applications in understanding the linear growth virial theorem and its generalizations described earlier, of two- and three-particle correlation functions in galaxy yield the Jeans equations of ‘stellar hydrodynamics’. clustering, as well as for plasma physics and the theory of These are analogous to the usual ﬂuid equations imperfect ﬂuids.
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Thermodynamic descriptions system grows cooler. It is essentially the same effect as Particle and distribution function descriptions are math- adding energy to a satellite orbit around a massive body, ematically complex because they contain much detailed and so it should be, since the virial theorem applies to both microscopic information about a stellar dynamical system. cases. Thermodynamic descriptions are mathematically much Applied to ﬁnite, isolated spherical clusters of stars, simpler because they deal mainly with macroscopic infor- gravitational thermodynamics has been especially useful mation which averages over the microscopic detail. This in elucidating their global instabilities. The isothermal relative simplicity has provided considerable physical in- sphere, where all particles have the same mass and sight into the behavior of stellar dynamical systems. temperature, provides a relatively simple illustration. Its Since the macroscopic properties generally change density is obtained by solving the collisionless Boltzmann with time (e.g. even in virialized clusters, particles equation and Poisson’s equation for these conditions, or, slowly evaporate), stellar dynamical systems cannot be more easily but less accurately, by solving the gravitational in equilibrium. Therefore it might be thought that hydrostatic equations with an isothermal gas equation of thermodynamics, which is primarily an equilibrium state. It has a central core, and density ρ ∝ r−2 in the theory, could not apply. Often, however, the timescales for outer parts, and is usually truncated to keep its mass ﬁnite. departures from equilibrium are very long, and singular (Other stellar clusters with more general equations of state states are essentially unattainable over shorter times. As similar to polytropic stars can also be manufactured to a result, thermodynamics provides a good approximation provide simple models.) for the periods of interest. The system may be in quasi- The stability of such an isothermal sphere depends equilibrium, during which it evolves through a sequence on the relation between its total energy and its total of equilibrium states. In simple systems, each state entropy. Systems of the same total energy may have can be described well in terms of average macroscopic different entropies depending on their size and internal thermodynamic variables such as temperature, pressure, distribution. If the entropy has a local maximum for a chemical potential, internal energy, volume, total number given energy, then that conﬁguration is stable to small of particles and entropy, but these quantities change on a ﬂuctuations (but possibly metastable to large changes). timescale which is slow compared with the timescale for If the entropy does not have a local maximum, then the a local microscopic conﬁguration of particle positions and system of a given energy can redistribute itself internally velocities to change. to increase its entropy, and the situation is unstable. Such Symmetry properties of different physical systems analysis, originally done by Antonov, for an isothermal strongly inﬂuence their thermodynamic descriptions. For sphere conﬁned to a spherical box shows that, if the example, inﬁnite statistically homogeneous systems— ratio of the central to the boundary density exceeds 709, which may represent galaxy clustering—have rotational the system becomes unstable. It tends to evolve away and translational symmetry everywhere. However, from the isothermal sphere density distribution into a ﬁnite spherical systems have rotational symmetry only denser central core surrounded by a much less dense halo. at their center and translational symmetry nowhere. Similar core–halo evolution is also found in numerical Consequently these inﬁnite systems are described by a simulations. Its underlying dynamical mechanism is the grand canonical ensemble in which energy and particles slow evaporation of stars which have accumulated just can move across boundaries. Finite, isolated clusters, on enough energy to escape by their interactions with the the other hand, are described either by a microcanonical ﬂuctuating gravitational ﬁeld of neighboring stars. Since ensemble with no transport across boundaries or by a the total gravitational energy of the isolated sysem is canonical ensemble with only energy transport. (An conserved, its core becomes denser with its energy more ensemble is a collection of systems with the same negative in order to compensate the positive total energy average macroscopic properties but different microscopic of the escaping stars. conﬁgurations consistent with the macroscopic averages.) An inﬁnite statistically homogeneous system pro- This leads to differences in their distribution functions and vides a contrasting application of gravitational thermo- ﬂuctuation spectra. dynamics. The cosmological many-body problem is an Thermodynamic behavior in systems dominated by example. In its simplest case, we start with a uniform gravity is often ‘counterintuitive’, although really just in random (Poisson) distribution of identical point masses the sense that they behave differently from more familiar throughout the universe and ask how this distribution systems of particles without long-range, unshielded, changes as the universe expands. In the usual Einstein– attractive forces. For example, if one removes energy Friedmann cosmological models, the cosmological expan- from a self-gravitating cluster of stars, it becomes hotter. sion exactly compensates the smooth long-range compo- Adding energy makes it cooler. Thus it has a negative nent of the gravitational ﬁeld. This leaves only the rela- speciﬁc heat. In fact, this follows simply from the virial tively local ﬂuctuations caused by the discreteness of the theorem for a bound system. Slowly adding energy particle gravitational ﬁelds and by any clustering. These makes the system’s gravitational well less negative. To ﬂuctuations can be described by an equation of state which maintain their changing quasi- equilibrium state, the incorporates the gravitational interaction into thermody- particle velocities must decrease on average, and the namic quantities such as the internal energy and pressure.
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Because the universe generally expands more slowly than dimensions, using COUNTS IN CELLS as well as VOID and near- the timescale for particle conﬁgurations within local ﬂuc- neighbor statistics, are also in very good agreement with tuations to change, it undergoes a quasi-equilibrium evo- equation (16). The currently observed value of b is about lution. At any time an equilibrium thermodynamic state 0.75. provides a good description of the clustering. Its thermo- The BBGKY kinetic hierarchy, mentioned earlier, has dynamic quantities change slowly as the universe expands also been used to examine hierarchial galaxy clustering. adiabatically. It is partially solvable for the two- and three-particle Inﬁnite statistically homogeneous systems have a correlation functions in the linear regime. However, it uniform density when averaged over sufﬁciently large generally contains less usable observational information scales or over an ensemble of smaller scales. These inﬁnite than the distribution functions. There are also many more systems are characterized by the ﬂuctuations over various complicated models of galaxy clustering, involving hot scales of macroscopic quantities around their averages. and cold dark matter, various forms of initial density and The ﬂuctuations most closely related to observations are velocity perturbations, biases between the galaxies and the particle distribution functions f(N,V) which give dark matter, etc, but their detailed applicability to our the probability for ﬁnding N particles (e.g. galaxies) universe remains unclear. in a randomly placed volume of size V . Applying Many applications of these main physical descrip- thermodynamic ﬂuctuation theory to the cosmological tions of stellar dynamics—particle orbits, kinetic theory, many-body equation of state gives a relatively simple distribution functions and thermodynamics—have devel- formula for this distribution: oped over the last century. They have helped provide an understanding of the formation, relaxation and dynami- ¯ N(1 − b) N− −N(¯ −b)−Nb f(N,V)= [N(¯ 1 − b) + Nb] 1 e 1 . (16) cal evolution of star clusters within galaxies. They help N! explain the streaming motions and arms of spiral galaxies, as well as the triaxial shapes of elliptical galaxies. Com- N¯ =¯nV Here is the average number in the volume bined with the possible existence and effects of massive V n¯ for the average number density . The quantity black holes in galactic nuclei, stellar dynamics helps de- b =−W/ K 2 is the ratio of the gravitational correlation scribe the evolution of the nuclei and the feeding of their energy to twice the kinetic energy of random motions, black holes. On a larger scale, stellar dynamics gives a V averaged over all volumes of size having a particular background for understanding the formation, relaxation W r−1 shape. The correlation energy, , is the integral of the and evolution of galaxy clusters, incorporating effects of interparticle gravitational potential multiplied by the two- their internal dark matter. Even more generally, the princi- ξ(r) ξ(r) particle correlation function over the volume. is ples of stellar dynamics play an important role in account- the average excess over the random Poisson probability ing for the structure of matter over the largest scales in the for ﬁnding a particle in a differential volume element at a universe. r distance from another particle. Although stellar dynamics is an old and well- In a completely uncorrelated initial Poisson distribu- established subject, it is often rejuvenated by new insights ξ(r) = W = b = tion, 0, 0 and consequently 0. In this limit, and new applications. Some interesting areas for future the distribution function of equation (16) does indeed re- further developments include the detailed links between duce to the standard Poisson form. As the system evolves, dynamics and thermodynamics, implications for galaxy regions where near-neighbor points happen to be closer clustering, galaxy and cluster merging, distributions than average cluster as a result of their enhanced gravity. around black holes in galactic nuclei, counterstreaming in These clusters subsequently cluster themselves and a hier- galaxies and the development of stellar motions in large archy of non-linear clustering, with a wide range of ampli- star-forming regions. tudes and scales, represented by the increasing value of b, gradually builds up. In the Einstein–Friedmann universe, Bibliography this non-linear evolution of b can be calculated analytically Binney J J and Tremaine S 1987 Galactic Dynamics and it asymptotically approaches unity as the universe ex- (Princeton, NJ: Princeton University Press) pands. This asymptotic limit represents bound clustering Chandrasekhar S 1960 Principles of Stellar Dynamics = on all scales, and is strictly reached only for 0 1. (New York: Dover) A velocity distribution function can also be derived Ogorodnikov K F 1965 Dynamics of Stellar Systems from equation (16). Compared with the Maxwell– (New York: Pergamon) Boltzmann distribution for ﬁnite isothermal spheres, the Saslaw W C 1985 Gravitational Physics of Stellar and Galactic velocity distribution function for the cosmological case Systems (Cambridge: Cambridge University Press) is much broader. This is because it includes all levels Saslaw W C 2000 The Distribution of the Galaxies: of clustering, from isolated ﬁeld galaxies to the richest Gravitational Clustering in Cosmology (Cambridge: bound clusters. Numerical computer simulations of Cambridge University Press). the cosmological many-body system verify equation (16) as well as its associated velocity distribution function. William C Saslaw Observations of galaxies on the sky and in three
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