6.2.3. The Current Generation of Models: 1990s

Recently interest in ring galaxies has increased and there have been a number of new modeling studies published. These include: Gerber, Lamb and Balsara (1992), Hernquist and Weil (1993), Gerber (1993), Struck-Marcell and Higdon (1993), and Horellou and Combes (1994). Gerber, Lamb and Balsara (1994) also have made available to us a preprint of further simulations like those in Gerber (1993). All of these works use particle hydrodynamics algorithms, which are well-suited to modeling not only the target and companion galaxies, but also the empty space between them. This space and the galaxy edges are difficult to simulate with finite difference algorithms. All of these simulations are fully three-dimensional, which is one reflection of the increased computer power in recent years. Most of the models include a self-consistent computation of the gravity in either the stellar or gas components, or both. Up to this time, there had been no hydrodynamic modeling of individual ring systems, but such modeling is the goal of several of these papers. We will postpone consideration of Hernquist and Weil (1993) and Struck-Marcell and Higdon (1993) to the next section, since these papers focus on the Cartwheel and the question of the origin of "spokes" like those observed in the Cartwheel.

Gerber, Lamb, and Balsara (1992) and Gerber (1993) modeled the Arp 147 system, one of the original "empty" ring systems of Theys and Spiegel. These simulations involved the collision of an elliptical (gas-free) companion, with an equal-mass primary consisting of a collisionless dark-matter halo, and a disk with both stellar and gas components. An SPH algorithm was used to compute the gas dynamics, and the gravitational interactions were computed with a particle-mesh algorithm. These simulations confirmed the finding of earlier stellar dynamical calculations (e.g., Appleton and James, 1990), that an off-center target impact can displace the nucleus away from the disk center, and in fact out of the disk plane. (Note, however, that after the collision this plane becomes very warped.) They found that the "nucleus can appear buried in one edge of the ring depending on the orientation relative to the observer", as observed in Arp 147 (see also photometry of Schultz et al. 1990).

The authors also described the formation of a "partial ring" in both gas and star components in the off-center impact models. This partial ring seems to be the same phenomenon as the banana or crescent wave discussed above and can be viewed as a confirmation of the basic theory, even though the three-dimensional structure of the numerical results was not considered in the earlier work. It appears that either the calculations were not run long enough, and the target disk was not large enough relative to the impact radius, to show the development of a complete ring as predicted by the theory.

A final interesting conclusion of this study was "that regions of high volume gas densities do not necessarily coincide with regions of high surface gas (and star) densities". This is an intrinsically three-dimensional result, not predicted by the planar theory. The theory does predict that the highest surface density regions will include material from the largest range of disk radii. If the disk is severely warped as in these simulations we can understand how this portion of the ring is the most three-dimensional, with gas clouds and stars executing their radial epicyclic oscillations in a range of vertically displaced planes. Gerber, Lamb and Balsara (1992) found that the strongest shocks and the highest gas volume densities were symmetrically located on both sides of the ring relative to the nucleus or surface density peak. This is in accord with the observations of Arp 147 which show bright knots symmetrically placed in the ring (see Gerber, Lamb and Balsara 1992). This distribution of knots appears peculiar to Arp 147.

Disk warping was also studied by Horellou and Combes (1994). Their work also consisted of three-dimensional N-body simulations, with a gas component in the disk. In this study, the gas dynamics was simulated by particles that were identified with individual molecular clouds, and which experienced inelastic collisions with other `clouds'. Only head-on collisions between equal mass targets and companions were considered. The target galaxy had a bulge component of mass equal to the disk. However, in contrast to other recent studies, no dark halo component was included. The companion consisted of a rigid Plummer sphere.

Collisions in two velocity regimes were considered: fast collisions with relative velocities of 2000 km s^-1, and slow collisions with relative velocities of 320 km s^-1. In the former case disk particles received velocity increments that were in good agreement with the predictions of the impulse approximation. Ring density amplitudes were modest but significant in both stars and gas, and there was little disk warping. In the latter case, ring (surface density) amplitudes were much larger, warping was severe. In this case, a significant fraction of the gas was "removed" from the central regions of the target disk. In the last time slice shown in this paper the gas is distributed in a continuous bridge. A view at a later time, showing how much gas actually escaped, would have been useful. Nonetheless, these simulations provide extremely interesting results on previously unexplored regions of parameter space.

Gerber (1993) and Gerber, Lamb and Balsara (1994) present additional simulations like the one used to model Arp 147. The Gerber, Lamb and Balsara (1992) results concerning disk warping and gas volume versus surface density in the ring as a function of azimuth are confirmed and generalized in these later works. Many more modeling details and a much more extensive analysis of the results are also given. Of particular interest for observational comparisons are star and gas density profiles as functions of time (see previous section). Many of the results presented in Gerber (1993) match the appearance of several well-known ring systems (in the tradition of Toomre 1978 and Appleton and James 1990). The simulations should provide a good basis for detailed modeling efforts when kinematic data become available.

In sum, this generation of hydrodynamical models has provided considerable improvement over the earlier work, and promises a great deal more. This is especially true in the case of asymmetric collisions, and the three-dimensional dynamical evolution of the target disk after the collision. However, the exploration of the orbital parameter space is still at an early stage. For example, it would be very interesting to have an extensive sequence of N-body simulations, densely covering the range of impact parameters from zero to some value greater than the disk radius. Ideally, the target would have a simple disk-halo structure to facilitate a detailed comparison with kinematic models (and thus KIA theory), like that carried out on a coarse grid by Gerber and Lamb (1994) and Gerber (1993). This would provide a firm basis for interpreting a further simulation sequence with the gas component included. Such a project should yield fascinating new results on wave structures, including systematics of the development of different spiral modes, their importance relative to the ring mode, the appearance of bars, and the structure and evolution of large-scale shocks. With a dense model grid some systematic understanding of these waves and their evolution should be obtainable.

All of the simulation studies described above assumed an isothermal equation of state with the exception of Struck-Marcell and Higdon (1993), which contains a minimal description of simulations of an adiabatic gas, together with a simple approximation to radiative cooling. To date, thermal effects are almost completely unexplored. The work of Appleton and Struck-Marcell (1987b) and Struck-Marcell and Appleton (1987) can be viewed as an initial attempt to study some of these effects at the level of interactions between typical clouds. However, there was no attempt in these studies to model a multiphase interstellar medium. Yet, the few available observational studies of the spatial distribution of HI (warm) gas and HII regions in rings, indicate interesting multiphase dynamics. This is also to be expected from detailed studies of nearby density wave galaxies cited in the previous subsection. This is clearly the frontier to be explored with the next generation of simulation codes.