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In this form of the virial theorem, we must remember that < T> does not refer to the total kinetic energy, but just to the portion of the kinetic energy which involves the average motion. If we observe a stellar system, we have no way of knowing how to separate these components. Nor, as a practical matter, can we average a system's internal motions over times which are much longer than its dynamical relaxation time. The first problem can be solved easily by recasting the virial theorem, and we
Part i ; Idealized homogeneous systems
will do that straightaway. The second difficulty is often assumed away by supposing that the configuration we observe actually is the time average. The validity of this assumption will be discussed in Section 41. Whereas the previous derivation of the virial theorem treated the system as a continuum, this one starts with the basic equations of motion for discrete objects. This will avoid the need to separate the kinetic energy into mean flow and 'thermal' terms. Moreover, we shall generalize the situation by allowing the mass of each object to change with time. This represents mass loss from stars or galaxies, accretion and merging of galaxies, or, more speculatively, cosmologies in which mass varies. The equations of motion for each object are
using (9.4). The (a) and (/?) indices serve the purpose of the primed and unprimed variables in the continuous case. The masses now are functions of time. For simplicity the objects lose mass isotropically, although the anisotropic case can easily be described by denoting the mass loss in each direction by mx?>- Vmt;j«»oy>, (9.24) 2atZLZ 2dt^ « which is motivated by differentiating the inertia tensor, given the form of (9.14) as a clue.
The virial theorem
For the kinetic energy tensor we now have
for the inertia tensor /u = 5> < " ) x|" ) x$ I \
and for the mass variation tensor J 0 . = £m ( oo, then 2 + <W,j>=0.
The time averages will not, except fortuitously, be equal to the instantaneous average here, since TiS and Wtj are explicitly time dependent. However, this time dependence does not change the form of the virial theorem. Contracting and summing over the indices gives 2 + <W^>=0,
which is the most quoted version. Any number can play in the virial theorem. It applies to a satellite going around a planet as well as to a cluster of galaxies. In the simplest case, a two-body circular orbit, for example, we know from first principles that centrifugal and gravitational forces balance: m^v2jr = Gm(x)mifi)/r2. Multiplying through by r gives the virial theorem. In this case the time average is equal to the instantaneous value since the orbit is periodic and symmetric. When applied to clusters of many objects, the virial theorem gives the order-of-magnitude relation used in Section 2 between the size of the cluster and its velocity dispersion: v2 « GmN/R. About a half-century ago this relation was used to estimate the mass of several clusters of galaxies from their observed radii and velocity dispersions. An alternative estimate from the calibrated luminosity-mass relation of the galaxies gave a cluster mass about an order of magnitude less than the dynamical mass. The disagreement of these two estimates was called 'the mystery of the missing mass' and is discussed further in Section 42.1.
10 The grand description - Liouville's equation and entropy All for one, one for all, that is our device. Alexandre Dumas, Elder
Langevin's equation, the Fokker-Planck equation, the master equation, and Boltzmann's equation are all just partial descriptions of gravitating systems. Each is based on different assumptions, suited to different conditions. They all arise from physical, rather intuitive, approaches to the problem. But there is also a more general description from which our previous ones emerge as special cases. We know this must be true because Newton's equations of motion provide a complete description of all the orbits. The trouble with Newton's equations is that they are not very compact: N objects generate 6N equations. True, the total angular and linear momenta, and energy, are conserved, at least for isolated systems, but this is not usually a great simplification. By extending our imagination, we can cope with the problem. We previously imagined a six-dimensional phase space for the collisionless Boltzmann equation. Each point in this phase space represented the three position and three velocity (or momentum) coordinates of a single particle. It was a slight generalization of the twodimensional phase plane whose coordinates are values of a quantity and its first derivative resulting from a second order differential equation for that quantity. The terminology probably arose from the case of the harmonic oscillator where this plane gave the particular stage or phase in the recurring sequence of movement of the oscillator. Now consider a bigger 'phase space' having 6N dimensions. Each point represents not a single object, but the entire system of N objects. As the system evolves, the trajectory of its phase point traces out this evolution. So far, nothing new has been added except a pictorial representation of the dynamics. The next step is a great piece of intuitive insight due to Gibbs. Although only one system with a particular set of properties may really exist in nature, suppose there were many such systems, an ensemble. If all the objects in each of these systems had exactly the same positions and velocities at some time, the systems would all share the same phase point. More interestingly, consider such a Gibbs ensemble in which each system has a different internal distribution of positions and velocities (for the same number of objects). This ensemble is represented by a cloud of points in 6N-dimensional phase space. At
The grand description - Liouville's equation and entropy
any time, the probability density for finding a system in the ensemble within a particular range of 6N coordinates will be denoted by / , • • .,x m ,v ( 1 ) ,.. .,ym, f)dx(1>... d v w .
This shorthand notation avoids the boredom of writing millions of coordinates for, say, a globular cluster. The value o f / w is the fraction of systems in the ensemble with the desired range of velocities and positions. Thus the integral of/(fV) over all phase space is unity " /...dv(N> = l.
The next conceptual step is to assume, following Gibbs, that the probability distribution of all members of the ensemble is the same as the probability of finding a given set of coordinates in any one member of the ensemble. For ordinary statistical mechanics this is justified by supposing that all members of the ensemble are fairly similar and represent different microscopic realizations of systems with the same macroscopic (average) properties such as temperature and density. Then one appeals to the ergodicity of the ensemble. Some physicists find this intuitively obvious and note that it leads to many experimentally verified predictions. Others find it intuitively implausible, and thus all the more remarkable for seeming to be true. They have therefore sought rigorous proofs in statistical mechanics and generated a considerable industry. In gravitational (and other explicitly Hamiltonian) systems, the situation is perhaps more straightforward: the Gibbs concept implies the exact equations of motion of the system. To see this, we first determine how/(Af), considered as the probability of finding a given system in the ensemble, changes with time. Initially the probability that a given system has coordinates (xj,1',...,v(0N),t0) lying within a small 6N-dimensional volume, with boundary So, of phase space is f / < o N) (x ( o 1) ,...,vrfo)dx ( o 1) ...dvW. (10.3) J s,, At some later time, the coordinates x 0 , v0 in the system will evolve dynamically into x and v, the distribution function will become/ (Ar) (x (1) ,..., vm, t) and the boundary S o will change to S,. The probability that the evolved system now lies within S, is A(to)=
A(t)= I /•(x