6.5. Explorations of Heating and Cooling Effects

The interpretations of multiwavelength observations and a full understanding of nonlinear feedback effects in the interstellar gas require simulations that include heating and cooling processes, and the exchange between thermal phases. However, it is extremely difficult to see how some of these effects should be treated. For example, the short timescales of atomic and molecular cooling processes relative to galaxy dynamical times presents one problem, though, in fact, we need not time-resolve rapid cooling. Another problem is the wide range of heating processes and scales associated with energy input from stellar activity, including: UV photoheating, winds from many different types of star, multiple supernova explosions in young clusters, etc. Finally, adiabatic cooling in expanding flows is probably one of the most important processes. In the turbulent interstellar gas, expansion flows are likely to be found on many length and time scales. Fortunately, we will usually only need to resolve the effects of the scale on the interaction driven waves.

Very little work has been done in this area; there is almost no literature to review. CS has begun some exploratory simulations with very simple models for the thermal processes. In this subsection we will very briefly describe some simulations of off-center collisions. We expect it will be some time before there is any standard or consensus method of modeling these processes, but the example below provides an illustration of their importance. The following description of the model is derived from Struck (1995).

The simulation is based on a quite standard smooth particle hydrodynamics (SPH) algorithm, which is essentially the same as that used in Struck-Marcell and Higdon (1993). 19,000 particles were used to represent the gas disk of the target galaxy and 3000 particles were used to simulate the companion disk. The disk particles were initialized on circular orbits in centrifugal balance with a fixed, rigid gravitational potential (i.e., the halo) with a rotation curve of the form v (r / )^1/n in the target disk, where v is the rotation velocity, and is the scale length of the potential. With the adopted value of n = 5, this rotation curve rises rapidly in the inner regions, but is nearly flat at large radii. A simple softened point-mass potential was used for the companion. These simulations did not explicitly include either a stellar disk or a separate stellar bulge component, and thus, represent late-type disk galaxies. The simulations do include a calculation of local self-gravity between gas particles on a scale comparable to the SPH smoothing length, as described in Struck-Marcell and Higdon (1993).

The simulation presented below used a simple step function cooling curve, with three steps. All gas elements are initialized to a single temperature T₀, assumed to lie in the range of about 5000-8000 K, corresponding to the warm interstellar gas. At temperatures between T₀ and 70 T₀ the cooling timescale _c is short relative to other timescales in this code, so it is assigned a value equal to a few times the typical numerical timestep. This high cooling rate represents the peak in the standard cooling curve due to hydrogen and helium line emission in an optically thin gas. At temperatures above 70 T₀ the cooling rate is decreased by a factor of 10. This decrease is meant to approximate the combined effects of heating and cooling, on the assumption that the hot gas is found near the sources of heating. On the other hand, the hot gas frequently cools rapidly by adiabatic expansion, so this part of the cooling function has a minor effect on the model results. At temperatures below T₀ the cooling rate is decreased by a factor of 20 from the value immediately above T₀. Cooling is weaker at these temperatures in the interstellar medium. Moreover, another reason for using this decreased rate is to inhibit the gas particles from cooling to negative temperatures, but to do so without having to time-resolve the low temperature cooling.

When a gas particle is located in a dense or high pressure region (cloud), so that its internal density exceeds a fixed critical value, it is assumed that stars are formed. The particle is then heated by increasing its temperature by a fixed multiple of T₀ at each timestep for a finite duration, or until it reaches a maximum T_max. While it is being heated it is not allowed to cool except by adiabatic expansion. It is clear from the results that even these very simple approximations are capable of representing phenomena that do not occur in an isothermal gas.

Initially the gas is below the critical density for SF at all locations. The gas density in the initial isothermal disk is also below, but near, the threshold density for axisymmetric gravitational instabilities, i.e. the value of the well-known Toomre Q parameter indicates stability. Particles out of the plane of the disk are not quite in centrifugal balance, and therefore settle into the disk at the beginning of the simulation in the target disk. The companion is so far below the threshold that it remains cold.

Figure 21a-c shows three orthogonal views (x-y, x-z, y-z) of three timesteps from this simulation (dimensionless length units are as in Struck-Marcell and Higdon 1993). In this figure the gas elements are placed in three temperature bins, and are color-coded to indicate both this, and which galaxy the particle originally belonged to. The coolest elements, with temperatures less than 10 times the initial value (i.e., < 50,000 K). Hot elements with temperatures greater than 300 times the initial and intermediate temperatures ("warm") are also indicated by color.

Figure 21. Particle snapshots of an SPH collisional simulation with heating and cooling as described in Section 6.6. Each row contains three orthogonal views - x-y, x-z, y-z from left to right - at one time. The three rows show the model at: a) thr time of impact, b) when the first ring has expanded significantly into the disk and c) when the second ring has almost reached the edge of the disk and a third ring has formed at the center. Note the formation of strong spokes inside the second ring in c). The color coding is as follows: For the target disk, red, green and cyan represent the "cool", "warm", and "hot" compnents. For the intruder galaxy, the same temperature sequence is represented by the color yellow, magenta and blue respectively. Only one in three SPH particles are show in these low resolution plot. See Color Plate X at the back of this issue.

Figure 21a shows the disk at the time of impact. The companion disk is setup perpendicular to the primary disk in the y - z plane. The orbit of the companion center is in the x - z plane and it impacts at a significant angle. The strong shocks in the two disks are shown in the first frame as a thin zone of hot, blue and cyan particles. The effects of the shock are visible in the other two views, where it is evident that the (yellow) disk does not simply punch through the primary disk. The rotation directions of the two disks can also be deduced from the small stochastic spirals in the unperturbed regions.

A significant part of the primary disk, and all of the companion disk are disrupted by the impact, and much gas is splashed out in, a bridge-plume. Nonetheless, most of the primary disk remains intact, and the ring forms as in collisionless encounters. The second set of frames (Figure 21b) shows a time when the first ring has nearly propagated through the disk. The ring is complete, but with azimuthal density variations and significant warping. Green (warm) and blue (hot) dots mark the sites of recent heating (i.e., star formation). The preponderance of green relative to blue indicates that the density is falling below the heating threshold. There is little heating behind the ring, where strong cooling results from postshock rarefaction.

The hot central regions and the blue-green are primarily the result of infall from the bridge. It is also a result of azimuthal compression in the off-center collision, as in Toomre's (1978) ring-to-spiral models. The fact that the infall is concentrated in one narrow sector is surprising. However, it appears that most of this infalling material originated in the half-disk where the rotation velocity was opposite the orbital direction in the companion, or from gas in the primary that it collided with (yellow and red appear quite well mixed in the bridge).

Even more spectacular is the reformation of the companion disk, which was almost totally disrupted in the collision. At this time the disk is accreting from a stream of material that includes gas from the companion that was orbiting in the same sense as companions orbit at the time of impact. This material spirals into a smaller disk, so there is much heating. The heating algorithm may not be all that realistic here, but the accretion processes may in fact be responsible for the enhanced activity in rings with attached companions.

The third set of frames (Figure 21c) is from a much later time when the companion has reached apoapse and begun to return. The ring at the edge of the disk is now the second ring, and, though the disk is warped, it is intrinsically noncircular. There are strong azimuthal variations in the heating (star formation) around the ring, a situation quite similar to that observed in the Cartwheel. A third ring is forming in the central regions. The star formation in this third ring is a result of the relatively high gas densities due to infall there, and gas driven inwards by collisions with infalling elements. This is not entirely realistic, because gas consumption has not been included, and the heating algorithm only looks at the total gas density, not what fraction of this gas is truly in a state conducive to star formation.

This timestep also illustrates how the most distant companion (G3 of Higdon 1993) in the Cartwheel system might have managed to be the intruder without having a very large relative velocity (despite its size, the observed outer ring is the second ring in this model). The scenario is especially compelling in the light of Higdon's recent discovery (private communication) that the intergalactic HI plume extends all the way to G3.

In this simulation, the most unstable wavelength in the pre-collision primary disk, and the scale on which self-gravity is computed, is quite small (0.1 in the simulation units). Thus it is not surprising that there are no strong spokes in the wake of the first ring (except for the "accretion arm"). Several strong spokes are apparent behind the second ring (which becomes the outer ring in the Cartwheel analogy). Because of their significant width in the third dimension, the density in the spokes does not exceed the star formation threshold. As noted above, this provides an alternative explanation for the rarity of the spoke phenomenon - it may take two cycles of wave compression and rarefaction to create them. The simulation shows that there are other complications, such as infall and the long-term effects of the "swing" in the off-center collision.

This simulation has not yet been analyzed in detail, nor have parameter dependencies been studied with additional simulations. More particles and spatial resolution in the multiphase gas disk would certainly be desirable, as would the inclusion of gas consumption and full selfgravity. Nonetheless, it illustrates some of the possible effects of the thermal terms and is the first model we have seen of gas exchange in a direct collision.