Although Lynds and Toomre (1976) were mainly concerned with a purely kinematic description of the ring formation, they did include one model with massive selfgravitating coaxial rings and the results were qualitatively similar to the models involving massless particles. Theys and Spiegel (1977) performed the first N-body simulations of the ring formation using 7000 mass points. In their 2-dimensional models most of the stars were placed in a disk which quickly developed a bar instability since no massive halo was included in the calculation. Nevertheless, despite the unrealistic heating of the disk stars, rings were generated as a result of the collision. Theys and Spiegel also present the results of more realistic calculations involving massive selfgravitating coaxial rings. Some attempt was also made to include dissipation in the interaction of the rings. The edge-on views of these latter simulations are interesting in that they show the displacement of the second ring from the first in the direction perpendicular to the target disk plane. Also of interest is the structure of the intruder, which can be seen to undergo considerable distortion, especially along the direction of its trajectory.
More realistic N-body simulations were performed in the early 1980's with the advent of faster computers and the development of particle-mesh algorithms for solving the N-body problem. Two groups at the University of Manchester, UK, performed simulations of the ring problem. Huang and Stewart (1988) performed a small number of simulations of a rigid companion with a disk enclosed in a fully self-consistent (live) halo. R.A. James and P.N. Appleton, in a largely unpublished work (see for example, Appleton and James 1990) performed over 100 completely self-consistent simulations of the collision of a (live) companion with a disk and (live) spherical massive halo. The results of both groups was quite similar and can be summarized as follows: a) The formation of the first ring is relatively insensitive to the inclination of the companion's orbit. In fact, both Huang and Stewart (1988) and Appleton and James (1990) were able to produce respectable ring waves in disks with modest companions even when they approached the disk at a high angle of inclination (60-80 degrees). Huang and Stewart noted that the existence of the "live" halo accentuated the strength of the first ring and slowed its outward rate of propagation, as compared with a non-responsive "rigid" halo of the same mass and scale-size. b) Very rich and complicated structure was found to develop in the inner regions of the disk as second and multiple oscillations of the inner disk occurred. Here the structure was found to be very sensitive to the impact parameters. Amongst the complex structures found, Appleton and James report the formation of a single spiral structure inside the ring during slightly prograde collisions which might be termed "single-spoked". Such structures, which are likely resonances between the orbital motion of the intruder and the disk material are probably not a good explanation for the spokes in the Cartwheel, although they may explain the faint structures seen in IIHz4. c) Even in the case of relatively low mass intruders, it was rare to see evidence for more than two stellar rings. The generic result from the simulations was either the formation of a bar-like structure or the formation of an inner lens which merged with the second ring after a few rotation periods. The lack of development of higher order rings is probably a result of the poor resolution of these particle-mesh algorithms, as well as strong phase-mixing of the higher-order rings, as expected in the simple kinematic model. d) Very few slightly retrograde models of ring collisions have been explored, although there were clear indications from the work of Appleton and James (1990) that rich structures are possible in these cases. e) Toomre (1978) had shown through restricted 3-body models that as the impact site of the intruder was shifted from the center to the outer disk, the structure underwent a transition from ring-like behavior to spiral-like forms. N-body simulations have confirmed this early work, showing that well defined rings can be produced when the impact is within 40-60% of the gravitational scale-length of the target galaxy.
More recently, supercomputers have allowed simulations with the much needed resolution required to investigate the possible effects of self-gravity in the rings. A careful comparison was made between analytical models of the ring kinematics, based on the impulse approximation, and the behavior of a fully self-consistent N-body and SPH gas disk (Gerber 1993). The results indicate that only during the late stages of the evolution of the expanding ring does the ring behavior differ significantly from that of the simple kinematic models. It is this fact, now seen in retrospect, that has led to the remarkable success of the original Lynds and Toomre (1976) models. In fact it is becoming clear that the most important component in the late evolution of the ring galaxy is the development of instabilities in the gas, not the stars. Indeed, as we have indicated in earlier sections, pure N-body simulations may be less relevant than we first believed because most definitive ring galaxies are defined primarily as a result of the morphology of the young stars which have formed from gas in the systems. Except for the possibility of interesting interactions between the perturbed massive halo and the disk (hinted at by Huang and Stewart), the emphasis has now shifted from studying purely N-body collisions to those models which also include the potentially important effects of the gas. In some cases, (see Hernquist and Weil 1993) the behavior of the gravitational effects of gas clouds on the underlying stellar dynamics can be significant. We therefore defer further discussion of the more recent N-body models to the next section.