If we look out from any point in an isotropic, homogeneous Universe it must appear to expand in the same way. This expansion, mentioned in Section 21, has the form H(t)x with H = Ṙ/R when the coordinate system is chosen to coincide with a point of zero systematic motion.

To see this pretend to be a ‘fundamental observer’ at point O, moving with the average flow. Looking out at time t to an arbitrary point P along the position vector x = OP, you see the velocity v(x, t) of P relative to you. Another fundamental observer is moving with the flow at O′ and his velocity relative to you is v(s, t) where s = OO′. (The reader may find it helpful to draw a diagram.) He measures the velocity of the same point P, located at x′ = x - s in his coordinate system, and finds it to be v′(x′, t) = v′(x - s, t) = v(x, t) - v(s, t). Now, since the Universe is homogeneous and isotropic, v′ must be the same function of x′ and t that v is of x and t. Therefore, v(x - s, t) = v(x, t) - v(s, t). By inspection the solution of this functional equation is that the velocity is a linear function of position, so it has the form v = f(t)x = Ẉ.

Any process, such as mass segregation, mass loss, or core–halo instability which redistributes the density of a gravitating system will also produce orbit segregation. Orbit segregation is caused primarily by changes in the mean gravitational field. It affects orbits according to their eccentricity.

In a globular cluster, for example, the timescale for mass segregation to redistribute density falls between the dynamical crossing timescale and the relaxation timescale for stars of average mass. Therefore orbits of average or light stars undergo secular changes governed by the slowly changing mean field. (We are isolating changes in the mean field which, averaged over orbits, lead to orbit segregation, so we now ignore close encounters and dynamical friction for the test stars.) Suppose, for clarity, we compare the extremes of eccentricity: circular and radial orbits. Let the cluster's density increase toward the center, as usual. If the cluster were in stationary equilibrium, stars in circular orbits would just go around and around, and those in radial orbits would just go in and out through the center, in equilibrium with the mean field. Stars with intermediate eccentricities, but constant angular momentum, would generally follow open orbits with constant amplitude.

Now suppose part of the cluster begins to contract slowly compared with the timescale for free fall.

Having introduced the basic descriptions of gravitational many-body physics, it is time to attend to some astronomical applications. Many basic processes still remain to be explored, but they are best introduced in their astrophysical contexts.

Occasionally in Part 1 I have mentioned that infinite homogeneous systems – indeed all homogeneous gravitational systems – are anomalies, idealizations which do not exist in nature. However convenient they may be for mathematical analyses they are too unstable to represent anything we see, except as a first approximation. Newton recognized this and described it qualitatively in his letter to Bentley. Jeans (see Section 15) formulated it quantitatively for a static universe. Important complications, and a new richness of results, occur in the expanding Universe. This is a fundamental problem, for it begins to describe how matter is distributed around us on the largest scales.

Some basic aspects of many-body theory remain to be developed in future sections where they will be connected more closely to their astronomical applications. Here I give several extensions which can be worked out as problems by the reader, or can be used to enter the literature. Other suggestions for practice problems are sprinkled lightly throughout the text.

The point mass approximation

So far we have usually supposed that gravitating bodies, from dust grains to galaxies, can be treated as point masses. This is clearly an idealization. Geometric collisions make this approximation fail; so does tidal disruption. A third reason for failure is angular momentum transfer from the orbit to the spin of an object. Consider two solid ellipsoids, each of mass m and semimajor axes a, b where a > b. Show that, as they pass by one another at average distance r and velocity v, each acquires an angular momentum of order G(mae)2/r2v, where e2 = 1 - (b/a)2. How close do they have to pass to transfer an appreciable fraction of their orbital energy into rotational energy? Is this a plausible process for stars? For galaxies? What happens if the masses are not approximated as solid bodies? What residual internal circulation would result? Is there a net vorticity? How does the transferred energy change the size of the galaxy? (See Harrison, MNRAS, 154, 167, 1971.)

The effects of an extended background on a cluster have more in common with those of a central point mass than might first seem to meet the eye. In both cases the stellar orbits are dominated by an external mass distribution, not by their own selfconsistent gravitational field. One result is that the basic physical processes discussed in Part I must be averaged over the particular distribution of orbits for each system.

Observational evidence for a distributed background comes from several sources. Rotation curves of galaxies, particularly of giant spirals measured with 21cm emission from neutral hydrogen clouds, sometimes appear to be flat far beyond the optical image of the galaxy. This non-Keplerian relation between distance and velocity may indicate a halo surrounding the galaxy. Its total mass is very uncertain. To produce a significant dynamical effect it would have to be at least comparable with the stellar mass of the galaxy, and it could be much larger. The nature of this halo is unknown. Within galaxies, the orbits of globular clusters are dominated by the distributed stellar background, and possibly by the halo as well.

On a larger scale, Sections 41 and 42 already described the ‘missing mass’ problem for clusters of galaxies. The resolution of the problem may involve an unseen mass distributed throughout the cluster. The mass of very hot intracluster gas found in X-ray emission is usually an order of magnitude too small.

Naturally, peculiar velocities must grow consistently with increased spatial clustering. Again, N-body simulations provide valuable insight, especially into the non-linear regime. Since velocity distributions depend strongly on the amount of clustering it is important to separate galaxies according to the density contrast in their local neighborhood.

For the 4000-body experiments, for example, it is possible to use the integration scheme itself to determine the number density inside the nearest neighbor sphere surrounding any galaxy. This sphere is essentially where the fluctuating component dominates the gravitational force. The result is a measure of the local density contrast, which varies from about 104 or 105 in the centers of rich clusters to less than 0.5 in the field galaxies. A subset of field galaxies may be selected by also requiring them to be separated from their nearest neighbor by at least twice the average separation that a uniform random distribution would have. These most isolated galaxies are called extreme field galaxies. They are almost always well outside the haloes of clusters.

As an indication of the relative degree of isolation these criteria imply, at R = 32 (the present time) in the Ω = 0.1, n = -1 model, about 23% of all the galaxies are in clusters (density contrast > 100), about 62% are intermediate (100 > contrast > 0.5), 15% are field galaxies and 5% are extreme field galaxies.

Section 20:

Hewett, P.C., 1982. ‘The estimation of galaxy angular correlation functions’, MNRAS, 201, 867.

Holmberg, E., 1940. ‘On the clustering tendencies among the nebulae’, Ap. J., 92, 200.

Kiang, T. & Saslaw, W.C., 1969. ‘The distribution in space of clusters of galaxies’, MNRAS, 143, 129.

Neyman, J., Scott, E.L. & Shane, C.D., 1953. ‘On the spatial distribution of galaxies, a specific model’, Ap. J., 117, 92.

Peebles, P.J.E., 1980. The Large Scale Structure of the Universe (Princeton: Princeton UP).

Sharp, N.A., 1979. ‘Practical estimation of the angular covariance function’, Astr. & Astrophys., 74, 308.

Totsuji, H. & Kihara, T., 1969. ‘The correlation function for the distribution of galaxies’, PASJ, 21, 221.

de Vaucouleurs, G., 1970. ‘The case for a hierarchial cosmology’, Science, 167, 1203.

de Vaucouleurs, G., 1971. ‘The large scale distribution of galaxies and clusters of galaxies’, PASP, 83, 113.

Weinberg, S., 1974. Gravitation and Relativity (New York: Wiley).

Section 21:

Abramowitz, M. & Stegun, I.A., 1964. Handbook of Mathematical Functions (Washington: US Govt. Printing Office).

Bonnor, W.B., 1957. ‘Jeans’ formula for gravitational instability’, MNRAS, 117, 104.

Ellis, G.F.R. & MacCallum, M.A.H., 1969. ‘A class of homogeneous cosmological models’, Comm. Math. Phys., 12, 108.

Lifshitz, E.M., 1946. ‘On the gravitational stability of the expanding universe’, Jour. Phys. USSR, X, 116.

Saslaw, W.C., 1972. ‘Conditions for the rapid growth of perturbations in an expanding universe’, Ap. J., 173, 1.

BBGK and Y stand for the names of the physicists who independently, or in different contexts, developed a rigorous way to extract knowledge from the information in many-variable partial differential equations: Born, Bogoliubov, Green, Kirkwood and Yvon. The basic idea is to turn one equation with a large number of variables into many equations, each with a small number of variables. The many equations cannot be the orbit equations since each orbit equation contains the positions and velocities of all the other objects. Virial and higher order moment equations are useful for answering some questions, but for others they average out too much of the information. However, an hierarchial set of coupled equations involving the reduced distribution functions (10.25) may provide just the right amount of detail.

To place various approximations of the many-body problem in some perspective, Figure 10 shows the relations among their information content. Both the master equation and Liouville's equation provide independent starting points for a manybody description. Liouville's equation is completely consistent and self-contained (represented by the box around it) whereas the master equation requires further physical input before it is useful. The Langevin equation is so phenomenological as to be in a class of its own.

Arrows in Figure 10 show the direction in which information is sacrificed to obtain solvability. Information, here, is represented by the detail of the many-body description.

The simplest examples of the gravitational slingshot are a binary which scatters a single star and the breakup of an initially bound three-body system. Resulting recoil may eject both the single body and the binary from the system. If galactic nuclei contain supermassive objects which form binaries, these processes may have especially dramatic astronomical consequences (Saslaw, Valtonen & Aarseth, 1974). They are also important in ordinary star clusters with N ≲ 102, and in the very center of richer clusters.

Performing large numbers of three-body experiments shows how scattering behaves. There is an amusing contrast with high energy physics. Our (inexpensive) gravitational accelerator is a computer. We experiment not to find the microscopic law describing the interaction – we've known that since Newton – but to understand realistic applications of this law. Even our relatively low energy particles, typical stars moving at ~ 102 km s-1, have kinetic energies exceeding ~ 1050GeV!

Although there are a great many exact mathematical results on the analytical dynamics of three-body systems, they usually apply only under very restricted circumstances. Normally they are asymptotic or perturbation calculations (see Arnold, 1978; Pars, 1965). These are not sufficient to understand the rich range of important phenomena occurring in realistic physical scattering, so we must resort to numerical simulations and approximate non-linear analyses (see Heggie, 1975(a, b), reference in Section 48).

Introduction

Spiral structure is the frosting on the cake of galactic rotation. Like frosting, it is very alluring and has been greatly admired. Self-gravitating stellar disks are very lively objects; many processes contribute to the observed spiral patterns in galaxies. Our knowledge of this subject, greatly extended in the last two decades, still changes quickly. This section just describes some of the basic gravitational principles and questions common to many discussions.

What are the forces that drive spiral structure? To answer ‘differential rotation’ is true, but not much more informative than the ancient physicians who replied that morphine produces sleep because it contains a ‘dormative principle’. Spiral instabilities resulting from differential rotation are familiar in both magnetic and fluid systems, with or without gravitation.

The patterns of rotating magnetic field lines are so suggestive that for many years they were popular as an explanation for spiral structure, either directly or through their influence on star formation. The decline of magnetic explanations occurred when more accurate measurements of the galactic magnetic field strength showed it was significantly less than expected. Even today these measurements are uncertain because the topology of the field is obscure and we are not really sure how to average it over large distances.

Previous sections have made it plausible that an object in a gravitating system near equilibrium can be considered to be immersed in a bath of fluctuating forces, along with an average mean field force. We now consider a simple mathematical model for the time evolution of orbits. We use this intuitive physical picture to try to capture the essence of the problem in a fairly simple way. An advantage of this procedure is that it readily suggests modifications of the description for an improved physical picture. The results can always be checked against N-body computer experiments, and we will discuss their more exact derivation in Section 10.

At first sight, the simplest model might seem to represent the motion of each star by Newton's equation of motion with a stochastic force β(t) which fluctuates in time, i.e.,. But this turns out to be too simple. It makes the velocity undergo Brownian motion (for a Gaussian distribution of fluctuations) with an everincreasing root mean square value vrms ∝t½. Correspondingly, the root mean square position of an average star also departs monotonically from its initial value. These two properties are inconsistent with conservation of total energy, for the increase in kinetic energy must be compensated by a contraction of the system to decrease the potential energy. But the Brownian increase of every star's root mean square position from its initial value prevents the system from becoming very small.

Gas dynamical processes

Many phenomena occur when stars plunge through clouds of gas. Among the most dramatic is the formation of shocks and ionized wakes, especially around stars which emit strongly in the ultraviolet. The ionizing radiation can be produced either by the star directly, or from the bow shock accompanying supersonic motion. Although we will not usually include gas dynamic and radiative processes, this one is an exception since it is important. So we give a brief general discussion of the phenomenon. Then we will describe collisionless accretion, the slowing down of stars by gas, and modifications of the Jeans and two-stream instabilities.

Suppose a star moves supersonically through a cloud of hydrogen. (The role of heavier atoms is mainly to increase and complicate the radiation processes.) In the direction of motion there is a bow shock which embraces the star more tightly at high Mach numbers (v*/vsound). To estimate the temperature immediately behind the shock front, equate the thermal energy 3kT/2 to the kinetic energy of an atom, giving, where v⊥100 is the inflow velocity (in units of 100kms-1) normal to the shock surface. As this shock-heated gas flows behind the star, it expands and cools. Part of the cooling will be from free–free radiation, and part from the expansion itself. From standard formulae for free-free emission by an ionized gas, one learns that the timescale for substantial radiation is τff ≈ 106v100n-1 yr, where n is the gas number density cm-3.

As we move from the idealizations of mathematical physics to the messy realities of astronomical systems, the very nature of our understanding is transformed. From basic principles we move to models, and from general models to special examples. Observed examples seldom do more than whisper the approximations we need to decode them. All we have left for guidance is a jumble of parametrized ideas, and a hope of relations among them. We understand through the inspired guess, or by accident.

Introduction – spinning polytropes

In our continued progression toward decreasing symmetry, we now arrive at the most individualistic systems of all. Previously we discussed flattened systems which were mainly thin disks. But those cannot represent the large observed class of elliptical galaxies. Ellipticals show moderate rotation, and as a result they have less global coherence than either spirals or spherical non-rotating systems. Pressure, rotation and internal currents all combine to determine a variety of structures and shapes.

Historically, some of the great eighteenth, nineteenth- and early-twentieth century mathematicians began to develop this subject by rigorously analyzing certain simple, highly specialized cases. Magnificent though these achievements were – and they often led to important mathematical insights – they also raised the stature of certain models so high that their astronomical importance became greatly exaggerated. All these models were built from uniform, incompressible fluids. The first of these non-spherical self-gravitating models were Maclaurin's (mid-eighteenth- century) spheroids. They have a stately uniform rotation, with no internal motions. Gravity is balanced by incompressibility and rotation. This balance can exist only for certain values of the spin ratio (58.1), nearly equal to unity. Two types of Maclaurin spheroids, with small and large eccentricities, are possible equilibrium figures.

So simple and satisfying seemed Maclaurin's results, that nearly a century went by before Jacobi realized that other equilibrium figures exist.

Ask a fundamental astronomical question, and the chance of an accurate answer is small. Throughout this account of gravitating systems, nearly each section ends on an incomplete note. We are just beginning to understand the richness and subtlety implicit in a system whose components interact with a force as simple as an inverse square. On all scales, from the sun and its planets to cosmic clusters of galaxies, flock insistent but unanswered questions.

For how long will the solar system be stable? Secular perturbations that grow as the planets follow their courses may eventually end in a resonance that forces ejection. Analytic theory gives us some reassurance, but not a definite answer. Computer simulations do not have the accuracy needed. And if we think of our solar system itself as an analog computer, then the calculation has not yet been done. All we know for certain is that the orbits of planets remaining today have avoided ejection for billions of years. Someday the sun may not rise tomorrow.

The uncertainties of planets orbiting the sun are relatively tame compared to the few-body problem whose masses nearly are equal. More opportunities for resonance flourish. A star hardly knows which way to go. Are computer experiments the only method for predicting the outcome from given initial conditions?

Parallel to the comparison developed in Sections 15 and 16, we next turn to new phenomena which graininess introduces into the growth of perturbations. In a major application of this theory, the grains are galaxies. Despite our ignorance of their formation, we can ask how gravity causes galaxies to cluster. Does the clustering we expect explain what we see?

The simplest result can even be anticipated from our analysis in the last section. Consider a gas of galaxies. Let it be uniform except for √N fluctuations. If we treat it, for a moment, as a fluid, then the growth rate of perturbations in a standard cosmology, ρ1(t)∝ R(t)∝t⅔, tells us that observed clusters with ~ 104 galaxies should be able to form easily starting at redshifts of 102 - 103. So galaxy clustering promises to be more understandable than galaxy formation, although it still has its mysteries.

Realizing the relative ease of galaxy clustering, it becomes natural to push the process back a step. Could galaxies themselves be the result of an earlier clustering? Suppose that isothermal perturbations of the Jeans mass at decoupling, ~ 106 M⊙, started (somehow) with large amplitudes and formed bound systems. It would take about 104 of these to form a small galaxy.