As the dynamics of galaxy interactions is too broad a subject to be discussed in any depth in a single review, I will concentrate only on a few topics proposed to me by the organisers. Since a pair should be “two things of a kind”, I will not discuss merging and merger remnants. I will also leave out discussions on M/L and hence halo mass determinations. Good reviews covering these subjects, and some of the subjects treated here, have been given by Toomre (1977), Tremaine (1981), White (1982, 1983a), van Albada (1988), Barnes (1990) and others. Here I will briefly review the dynamics of sinking satellites and the effect of companions on elliptical galaxies, then discuss recent work on interacting disk systems, and finally focus on my favourite interacting pair NGC 5194/5195.

A study of the expected frequency of merging galaxies is conducted, using the impulsive approximation. Results indicate that if we consider mergers involving galaxy pairs without halos in a single crossing time or orbital period, the expected frequency of mergers is two orders of magnitude below the observed value for the present epoch. If we consider mergers invovling several orbital periods or crossing times, the expected frequency goes up by an order of magnitude. Preliminary calculation indicate that if we consider galaxy mergers between pairs with massive halos, the merger is very much hastened.

Detailed spectroscopic and imaging observations of colliding elliptical galaxies have revealed unmistakable diagnostic signatures of the tidal interactions. It is possible to compare both the distorted luminosity distributions and the disturbed internal rotation profiles with numerical simulations in order to model the strength of the tidal gravitational field acting within a given pair of galaxies. Using the best-fit numerical model, one can then measure directly the mass of a specific interacting binary system. This technique applies to individual pairs and therefore complements the classical methods of measuring the masses of galaxy pairs in well-defined statistical samples. The “personalized” modeling of galaxy pairs also permits the derivation of each binary’s orbit, spatial orientation, and interaction timescale. Similarly, one can probe the tides in less-detailed observations of disturbed galaxies in order to estimate some of the physical parameters for larger samples of interacting galaxy pairs. These parameters are useful inputs to the more universal problems of (1) the galaxy merger rate, (2) the strength and duration of the driving forces behind tidally-stimulated phenomena (e.g., starbursts and maybe QSOs), and (3) the identification of long-lived signatures of interaction/merger events.

We present numerical experiments on dynamical friction in pairs of elliptical galaxies of unequal mass. We confirm that the self-gravity of the response is not important and show the drastic effect of the deformability of the companion which reduces the decay time by more than a factor 2. Almost the same amount of orbital energy is dissipated within the satellite than within the large galaxy. Finally, we discuss the importance of distant encounters for the dynamical evolution of systems of galaxies.

Recent observational studies (Franx and Illingworth 1987; Jedrzejewski and Schechter 1988; Bender 1988; Illingworth and Franx 1989) have shown that some elliptical galaxies have a small region near the center that rotates in the opposite direction from the outer parts of the galaxy. Often the rotation in the central part is much faster than that in the outer part. A few other galaxies show a small region near the center that rotates in the same di rection as the rest of the galaxy, but much faster. Either way, the part near the center that shows a strange pattern of rotation (the “core”) has been interpreted as a distinct dynamical subsystem. Very briefly, the observational data seem to be that (1) anomalies show up in rotation curves near the centers of some elliptical galaxies and that (2) galaxies with these strange rotational properties do not show a photometric signature: there are no noticeable bumps in the brightness profile and no unusual shapes of isophotal contours that would suggest an excess of matter concentrated near the center. No strong color variations have been reported. The puzzle is to learn what we can about elliptical galaxies in general, and about galaxies with strange central regions in particular, from these observational facts.

It is explained why one can observe a leading spiral pattern in close pairs of galaxies where the direction of the orbital momentum of the satellite is opposite to the direction of the spin of the spiral galaxy.

Many carefully selected samples of interacting galaxies have been observed expensively in attempts to clarify whether interaction produces activity in galaxies. Because the sample members represent a wide range of encounter parameters and times, one can then study whether there are correlations between observable encounter features and, for example, Seyfert activity. On the other hand, in theoretical studies, simulations typically deal with either time-consuming detailed modelling of single galaxy pairs or tracing a few model encounters over time. We are extending the observational survey approach by combining it with a simulation survey.

Using the impulsive approximation to study the velocity changes of stars during disk-sphere collisions and a method due to Bottlinger to study the post collision orbits of stars, the formation of various types of interacting galaxies is studied as a function of the distance of closest approach between the two galaxies.

The M51 system (NGC 5194/5195) provides an excellent problem both in spiral structure and in galaxy interactions. We present an analytic study of a computer experiment on the excitation mechanisms for M51’s spiral arms and whether or not a halo is important for these mechanisms. This work extends previous numerical studies of the M51 system by including self-gravitation in a two component disk: ’gas’ and ’stars’, and a dark halo. The analytic study provides two new observational constraints: the time (≈ 70 — 84 million years ago) and position angle of perigalacticon (300°). By using these constraints and a simple conic approximation, the search for the companion’s possible orbit is greatly simplified. This requires fewer N-body simulations than a fully self-gravitating orbit search. Fig. 1 shows the dust lane spiral pattern of M51 overlaid on an optical photograph. The analytically determined direction of perigalacticon is indicated with a line.

The system AM2208-251 is a roughly edge-on spiral extending east-west with a smaller round compact E system about 60 arcsec east of the spiral nucleus along the major axis of the spiral (Fig. la). Further details are in A Catalog of Southern Peculiar Galaxies and Associations (Arp and Madore 1987).

We have numerically studied tidal effects of a massive perturber on a satellite galaxy. The model consists of a spherical satellite galaxy and a point mass perturber and the encounter is non-penetrating. A wide range of density ratios and eccentricities of the relative orbits have been considered. The disruption of the satellite galaxy has been observed when the numerical value of the fractional change in the energy is greater than two. The changes in the energy and angular momentum show smooth variation in the case of unbound orbits and irregular variation in the bound orbit cases. It is shown that for a constant pericentric distance, increasing the density ratio decreases the tidal effects; and for a given density ratio an increase in the eccentricity decreases the tidal effects.

Two-grid simulation method combining advantages of both polar and cartesian mesh-codes is described. In addition to stellar component reacting solely to gravitational forces, gas component is included with dissipatively colliding particles. This allows fairly realistic simulation of planar encounters where both systems contain star+gas disks.

Galaxies having the same luminosity may have very different bulge to disk ratios, while the mean bulge to disk ratio slowly increases with total luminosity (Schechter and Dressier, 1987, Sandage et al., 1985). Such a behaviour is expected if ellipticals and the spheroidal components of disk galaxies are produced by secondary accretion of galaxies by larger galaxies. This is illustrated using a simple toy model of the evolution of the mass function of galaxies due to galaxy mergers.

It is well known that galaxies associate in groups and clusters. Perhaps 40% of all galaxies are found in groups of 4 to 20 galaxies (e.g., Tully 1987). Although most groups appear to be so loose that the galaxy interactions within them ought to be insignificant, the apparently densest groups, known as compact groups appear so dense when seen in projection onto the plane of the sky that their members often overlap. These groups thus appear as dense as the cores of rich clusters. The most popular catalog of compact groups, compiled by Hickson (1982), includes isolation among its selection critera. Therefore, in comparison with the cores of rich clusters, Hickson’s compact groups (hereafter, HCGs) appear to be the densest isolated regions in the Universe (in galaxies per unit volume), and thus provide in principle a clean laboratory for studying the competition of very strong gravitational interactions. The $64,000 question here is then: Are compact groups really bound systems as dense as they appear? If dense groups indeed exist, then one expects that each of the dynamical processes leading to the interaction of their member galaxies should be greatly enhanced. This leads us to the questions: How stable are dense groups? How do they form? And the related question, fascinating to any theorist: What dynamical processes predominate in dense groups of galaxies?. If HCGs are not bound dense systems, but instead ID chance alignments (Mamon 1986, 1987; Walke & Mamon 1989) or 3D transient cores (Rose 1979) within larger looser systems of galaxies, then the relevant question is: How frequent are chance configurations within loose groups? In this review, I will answer these last four questions after comparing in some detail the methods used and the results obtained in the different studies of dense groups, while in the accompanying contribution, I will attempt to reconcile the recent observations of galaxy interactions in HCGs with a negative answer to the first question.

The physical nature of the apparently densest groups of galaxies, known as compact groups is a topic of some recent controversy, despite the detailed observations (Hickson, in these proceedings, and references therein) of a well-defined catalog of 100 isolated compact groups compiled by Hickson (1982). Whereas many authors have espoused the view that compact groups are bound systems, typically as dense as they appear in projection on the sky (e.g., Williams & Rood 1987; Sulentic 1987; Hickson & Rood 1988), others see them as the result of chance configurations within larger systems, either in ID (chance alignments: Mamon 1986; Walke & Mamon 1989), or in 3D (transient cores: Rose 1979). As outlined in the companion review to this contribution (Mamon, in these proceedings), the implication of Hickson’s compact groups (hereafter, HCGs) being dense bound systems is that they would then constitute the densest isolated systems of galaxies in the Universe and the privileged site for galaxy interactions.

In very old times, people counted – one, two, many. I want to show you now that they were right.

Let’s consider the motions of isolated bodies:

1. 1. N = 1 - simple motion;

2. 2. N = 2 - Keplerian orbits;

3. 3. N = 3 - this is the difficult problem. In general, this problem can be studied only by computer simulations.

Two methods are proposed for describing the distributions of the triplet configuration parameters characterizing a tendency to alignment and hierarchy: (1) obtaining a representative sample of configurations and determining its statistical parameters (moments and percentages); and (2) dividing the region of possible configurations of triple systems (Agekian and Anosova, 1967) into a set of segments and finding the probabilities for the configurations to find themselves in each of them.

It has been reported that galaxies in large regions (~102Mpc), including some clusters of galaxies, may be streaming coherently with velocities up to 600km/sec or more with respect to the rest frame determined by the microwave background radiation.) On the other hand, it is suggested that the dominant mass component of the universe is dark matter. Because we can only speculate the motion of dark matter from the galaxy motions, much attention should be paid to the correlation of velocities between the observed galaxies and cold dark matter. So we investigate whether such coherent large-scale streaming velocities are due to dark matter or only to baryonic objects which may be formed by piling up of gases due to some explosive events.

1. Recently Karachentsev’s group at SAO (6-meter Telescope Observatory) published a list of 84 triple systems of galaxies with their distances, radial (line of sight) velocities, and angular sizes (Karachentseva et al., 1988). This gives a new ground for studies of the dark matter problem which fills the gap between the large cosmic scales (White, 1987; Dekel and Rees, 1987, and Einasto et al., 1977) and the scale of individual galaxies (Erickson et al., 1987). The data on the typical velocity dispersions and linear dimension of the triplets indicate that they contain considerable amounts of dark matter (see also earlier work of Karachentseva et al. (1979). Numerical simulations show that the statistical characteristics of the Karachentsev triplets can be imitated by model ensembles of triple systems with dark matter masses Md = (1 – 3) x 1012 Mo, which is almost ten times greater than the typical mass of stellar galaxies estimated by the standard mass-to-luminosity ratio (Kiseleva and Chernin, 1988).