Orbits that respect at least three isolating integrals of motion have very special structures in phase space. The main characteristics of this structure are reviewed, and the concrete examples that are provided by orbits in Stäckel potentials, are discussed. Many orbits in general potentials admit three approximate isolating integrals and closely resemble orbits in Stäckel potentials. If the potential is that of an elliptical galaxy with negligible figure rotation, the overall orbital stucture of the potential differs from that of a Stäckel potential only by the presence of a few unimportant families of resonant orbits. However, this elegant picture is shattered by the introduction of non-negligible figure rotation: though substantial regions of phase space may still be occupied by orbits that individually resemble orbits in Stäckel potentials, the overall orbital structure is radically changed by figure rotation, and in a rotating potential significant portions of phase space are given over to chaotic orbits, quite unlike orbits in Stäckel potentials.

Orbits in the inner kpc of a triaxial galaxy are discussed, taking into account the effect of a central density concentration like a massive black hole, a dense stellar nucleus, or a de Vaucouleurs-type cusp. Since the box orbits that form the backbone of a triaxial galaxy pass arbitrarily close to the centre after long enough time, they will eventually be subjected to large-angle deflections by a central point mass, and the triaxiality of the inner part of the system will thereby be destroyed. A 108 M⊙ black hole is estimated to affect box orbits out to 1kpc in a Hubble time, while a similar influence of the observed (extended) nucleus in M31 reaches out to 500pc in the bulge. Regular box orbits persist, however, in systems with singular central density profiles such as implied by carrying the r1/4 law all the way to the centre. This result can be approximately understood in terms of the frequency ratio Ωr/Ωθ remaining close to the harmonic value of 2 for many orbits in the corresponding spherical potential. Finally, I discuss observable consequences of the box orbit scattering process and future work, and use the presence of isophote twists in the central parts of a number of elliptical galaxies to obtain approximate upper limits on the masses of the black holes these systems may contain.

A survey of the possible variety of sharp-edged, caustic features that may arise in the collision of galaxies with very different masses and sizes (Hernquist and Quinn 1986a) has shown that in general shells are morphologically very complex. It is therefore not easy to determine the history of the collision that produced the shells nor the properties of the galaxies involved. However, a small number of shell galaxies (notably NGC 3923) have a sufficiently simple and orderly shell distribution that we believe the shells were formed by a chance very symmetric and simple encounter. In such cases we are presented with a unique opportunity to investigate the potential well of an elliptical galaxy over a large range in radius (≃ 0.5re − 20re). An analysis of the NGC 3923 shell system (Hernquist and Quinn 1986b) has shown that a large amount of dark matter is present (Mdark≃ 40Mluminous, r < 17re).

Over the last 5 years, considerable progress has been made in our ability to construct self-gravitating stellar equilibria. One of these new methods is essentially a variant of Eddington's (1916) method. Two other key approaches are logical extensions of Schwarzschild's Linear Programming method, and can be applied to nonspherical models as well. These methods are reviewed below.

The application of these methods to galaxies has yielded a few very interesting results within the last year or two. The methods described below unambiguously establish M/L's for M87 and M32 within about 30 arc seconds. They strongly support Tonry's contention that the nucleus of M32 contains a large invisible mass, possibly a 107M⊙ black hole. They also suggest that observational recovery of the projected velociy distribution might permit the observer to distinguish between a massive halo and an increasingly tangential velocity distribution function.

Non–spherical dynamical models for galaxies, and the methods for their construction, are reviewed. The theory for two–integral axisymmetric models is reasonably well developed. Stäckel models give considerable insight in the structure of both three–integral axisymmetric models and non–rotating triaxial systems. Triaxial galaxies with appreciable figure rotation require much further study. Applications to elliptical galaxies and the bulges of disk galaxies are discussed.

N-body simulations are a useful tool for constructing equilibrium models of elliptical galaxies and for the exploration of their kinematical properties, in particular the tumbling rate of the figure about some axis and the internal streaming. As yet little is known about these, except that there is a large variety of possible equilibrium models. It is easy to make triaxial systems that tumble about the short axis, with internal streaming aligned with the rotation axis of the figure. Attempts to construct systems with figure rotation and internal streaming in opposite directions have not been successful. Use of current simulation codes for detailed studies of particle orbits is limited to several (about 10) dynamical times due to non-physical fluctuations in the force field.

Stars, which contain more than 90% of the visible matter of the Universe, are organized into systems comprising a hierarchy of different masses and scales: from galactic stellar clusters to clusters of galaxies themselves. Most of the systems in this hierarchic sequence are collisionless: the characteristic time between collisions is much more than typical dynamical times (e.g., the period of oscillations). Various conditions of formation of collisionless gravitating systems leave traces on their main characteristic parameters—such as the angular momentum, the degree of anisotropy, the kind of dependence of the stellar density on coordinates, and so on. The lifetime of a system with a certain set of parameters depends essentially on whether these parameters belong to a region in which the system is stable, or not. If they do, the system may then exist practically without changes for an arbitrarily long time (in terms of the dynamical time scale); if they do not, it must quickly adjust to another, stable state.

The idea that dynamical instabilities might play an important role in determining the equilibrium structure of elliptical galaxies is a startling one, especially to those of us who are accustomed to associating instabilities with rapidly-rotating systems like disk galaxies. The shock is even greater when we learn that these instabilities have been taken seriously by workers in the Soviet Union for a long time. As Dr. Polyachenko points out in his talk, instabilities affecting spherical, non-rotating galaxies were being discussed by Soviet astronomers as long ago as 1972. Much of this work has recently become more accessible through the publication of an English-language version of Fridman and Polyachenko's monograph, Physics of Gravitating Systems (Fridman and Polyachenko 1984). In the West, it appears that only two people were prescient enough to systematically test the stability of spherical models before learning of the Soviet work (Hénon 1973; Barnes 1985). In particular, Barnes discovered independently that a spherical system composed of predominantly radial orbits evolves rapidly into a bar. Subsequent work has demonstrated that even some mildly anisotropic models can be unstable in this way.

Three ways have been proposed to make elliptical galaxies: cooling by gas dynamical processes at late epochs, Compton cooling at early times and merging. These theories must address a variety of observational constraints, the most severe being the problems of slow rotation and high central phase densities. I look at some aspects of all three theories with particular attention to key numerical simulations and observations that can distinguish between the scenarios.

Dissipationless formation mechanisms envisage elliptical galaxies as arising from the collective relaxation of an aggregate of stars. Their key ingredients are thus a set of initial conditions derived from consideration of prior evolution, and a treatment of the relaxation process. I review numerical studies of violent relaxation carried out over the last decade and purely theoretical treatments going back twice as far. Relaxation is always incomplete, and as a result the final structure of a “galaxy” depends sensitively on the initial conditions assumed. The viability of dissipationless formation thus rests on the identification of plausible stellar initial conditions which relax to the present structure. I discuss the extent to which such initial conditions are compatible with current ideas on the origin of structure in the universe.

The evidence for dissipation in elliptical galaxies indicates neither the epoch of formation nor the rate of radiation. The hypotheses for the formation of ellipticals include mergers of pre-existing, mostly stellar, disk galaxies; accumulation of gassy fragments that subsequently turn into stars; and the dynamical collapse of a distinct protogalactic gas cloud with simultaneous star formation. Mergers of purely stellar disks seem unlikely, because the phase space density of disks is everywhere far below that of the cores of normal ellipticals. Allowing a few percent of the mass of the galaxy to dissipate into the core and turn into stars could remove this difficulty. In the Hubble sequence of galaxies, ellipticals are characterized by their low angular momentum content. As a start to understanding the general problem for galaxy formation and angular momentum acquisition in the presence of dissipation, a cosmological N-body experiment containing both a dominant collisionless component and an isothermal gas is described. The collisionless component clusters in the usual hierarchical manner appropriate to the spectrum of fluctuations. In contrast, the gas fragments only when the Jeans mass drops below the turnaround mass. The fragments subsequently shrink, becoming distinct entities with relatively low chances of being quickly incorporated in a larger unit. Gravitational torques transfer angular momentum outward in the dissipating gas, placing most of the gas angular momentum at large radii in the protogalaxy. The distant, high angular momentum gas has a relatively long infall time onto the galaxy. The gas may continue to rain down for some time if the galaxy remains undisturbed, or, the growth of clustering may strip the gas off, leaving a low angular momentum system.

This is the first IAU symposium devoted specifically to elliptical galaxies. I would like to congratulate the organizers for initiating a symposium on such a well-defined topic with so many interesting puzzles. I have been impressed at this meeting by the great strides made in this subject over the last ten years, both in the quality and quantity of data and in the sophistication of the theoretical models. In addition, I'm sure that I express the sentiments of the great majority of the participants when I say what a pleasure it was to attend a meeting on galaxies in which there was not a single paper on spiral structure!

Rather than attempting to give a general summary, I have chosen to concentrate on four specific topics which reflect some of my personal impressions of the meeting.

In the poster as presented at the meeting, I described global morphological properties of elliptical galaxies, based on the data from a CCD surface photometry survey of ~ 200 ellipticals and ~ 50 SO's (Djorgovski 1985). In this brief summary, I emphasize two points: (1) there is a very weak and very noisy trend of radial shape with luminosity, in the sense that more luminous galaxies are less concentrated, and (2) there is no preference for low-luminosity ellipticals to show boxy isophotes, and they differ in that respect from the bulges.

The correlation between the effective radius re and surface brightness μe for elliptical galaxies is a fundamental scaling law that theories of galaxy formation must explain. When re and μe are derived by fitting two-parameter fitting functions such as the de Vaucouleurs r1/4 law to brightness profiles, the errors in the parameters are strongly coupled. The purpose of this paper is to rederive the μe(log re) relations for ellipticals and bulges, taking account of the coupling in the errors and using only high-accuracy CCD data. Our preliminary conclusions are: (1) The coupled errors are too small to affect significantly the correlation derived for elliptical galaxies. (2) The correlation for bulges is not very different from that for ellipticals, but the galaxy sample is small and the errors in the parameters are large due to the inherent uncertainty in bulge-disk decomposition.

The core structure of elliptical galaxies is determined by two parameters: total luminosity L and central luminosity density ρC. Scatter in ρC at any L may imply dissipational formation of core structure; once this scatter is accounted for, however, the pure L dependence of the core parameters can be isolated and perhaps used as a metric distance indicator.

Surface brightness profiles of bright ellipticals (MV < −17, Ho = 100) in rich clusters and the field are reduced to structural parameters. Most fitting functions are found to be inadequate in describing the overall shape of profiles with only the r1/4 law providing a good match over the range of surface brightness from 19 to 24 mag arcsec−2. The systematic deviations in structure of first-ranked ellipticals from normal ellipticals (i.e. enlarged characteristic radii and shallow profile slopes) are well explained by comparison to the N-body simulations of merging systems from Duncan, Farouki, and Shapiro (1982).

What are the two-dimensional distributions of projected luminosity in the brightest ellipticals in Abell clusters (hereafter called “El” s)? If we treat isophotes as ellipses, how do their properties vary as functions of surface brightness? Are the isophote parameters correlated with a galaxy's global properties? Are they correlated with any properties of the surrounding cluster? Are Els morphologically similar to other cluster or field ellipticals? This paper describes work in progress to the answers to these questions, and illustrates an interesting result.

The nature and existence of a second parameter needed to characterize the family of normal elliptical galaxies has been much discussed. The need for a second parameter has been demonstrated by the correlation of the residuals from the well–known magnitude–velocity dispersion relation for ellipticals with other observables such as ellipticity, mass–to–light ratio and Mg line strength or metallicity. Here, evidence for a correlation between residuals in the V26 – log σ relation and the strength of Hβ is presented, suggesting that variations in the stellar populations in Virgo elliptical cores may be an important secondary parameter.

We have analyzed luminosity profiles of E galaxies in six clusters of galaxies. We have found a relationship between radius, luminosity and surface brightness for galaxies in each of the clusters. Moreover, it seems that there is a dependence of the zero point of the relation with environment. This relationship implies that there is not a universal luminosity profile for elliptical galaxies.

We have obtained photographic surface photometry for a large sample of elliptical galaxies. The plates were obtained using the KPNO 0.9 m telescopes with Carnegie image intensifiers and IIIaJ plates. The f/13.5 secondaries were used to obtain a large plate scale: 0.33 arcsec per 20 micron pixel. In addition to the galaxy images, flat fields on cloudy skies and geometrical distortion calibration images of astrometric clusters were obtained. All plates were calibrated with the standard KPNO spot sensitometers. The sample of galaxies contains 75 ellipticals of total mB greater than 13 and north of −300 declination. The plates were digitized on the KPNO PDS microdensitometer by Dr. E. Malumuth. At the present time about 50% of the sample is reduced.