5. 2-D simulations with an energy source
As already stated, the ISM is a dissipative medium. Structures radiate their energy in a time-scale much shorter than the GMCs typical life time inferred from observations. Therefore we are led to provide an energy source in the system to obtain a life time longer than the free fall time.
5.1. The choice of the energy source
Different types of sources are invoked to provide the necessary energy input: stellar winds, shock waves, heating from a thermostat, galatic shear... We have tested several possibilities.
Our first idea was to model an interaction with a thermostat by adding periodically a small random component to the velocity field. We have tried to add it either to all the particles or only to the particle in the cool regions. If enough energy is put into this random injection, the mono-clump collapse is avoided. However the resulting final state is not fractal, nor does it show density structures on several different scales. It consists in a few condensed points, like "blackholes", in a hot homogeneous phase.
The second idea is to model a generic large scale injection by introducing a large scale random force field. We have tried stationary and fluctuating fields. It is possible in this scheme, by tuning the input and the dissipation, to lengthen the medium life time in an "interesting" inhomogenous state by a factor 2 or 3. However the final state, after a few dynamical times, is the same as for thermal input.
We would like to emphasize that, in all the cases just mentioned, the existence of super-elastic collisions at very short distances, are not at all able to avoid the collapse of "blackholes". They are indeed beaten by inelastic collisions which happen more frequently. Thus the density cutoff introduced by superelastic collision, is not able all by itself to sustain or destroy clumps.
Only one of the schemes we have tested avoids the blackhole/homogeneity duality. In this scheme, energy is provided by the galactic shear. It produces an inhomogeneous state whose life time does not appear to be limited by a final collapse in the simulations. We now describe this scheme in detail.
5.2. Modelisation of the galactic shear
The idea is to consider the simulation box as a small part of a bigger system in differential rotation. The dynamics of such a sub-system has been studied in the field of planetary systems formation (Wisdom and Tremaine 1988). Toomre (1981) and recently Hubert & Pfenniger (1999) have applied it to galactic dynamics.
The simulation box is a rotating frame with angular speed which is the galactic angular speed at the center of the box. Coriolis force acts on the particles moving in the box. An additional external force arises from the discrepancy between the inertial force and the local mean gravitational attraction of the galaxy. The balance between these two forces is achieved only at points that are equally distant from the galactic center as the center of the box. Let us consider a 2-dimensional case. If y is the orthoradial direction, and x the radial direction the equations of motion are written:
The are the projections of the internal gravitational forces. The shear force acting in the radial direction is able to inject into the system energy taken from the rotation of the galaxy as a whole.
Some modifications must also be brought to the boundary condition. It is not consistent with the differential rotation to keep a strict spatial periodicity. To take differential rotation into account, layers of cells at different radii must slide according to the variation of the galactic angular speed between them. One particle still interacts with the closest of the replicas of another particle (including the particle itself). This is sketched on Fig. 4.
We have computed simulations in two dimensions with 10609 particles. The direction orthogonal to the galactic disk is not taken into account. A tuning between dissipation rate and shear strength is necessary to obtain a structured but non collapsing configuration. However it is not a fine tuning at all. We found out that, in a range of values of the shear, the dissipation adapts itself to compensate the energy input so that we did not encounter any limitation in the life time of the medium. The simulations have been carried out over more than 20 free-fall times. Snapshots from a simulation are plotted in Fig. 5.
5.3.1. Formation of persistent structures
We can see that tilted stripes appear in the density field. This structure appears also in simulations by Toomre (1981) and Hubert and Pfenniger (1999). The new feature is the intermittent appearance of dense clumps within the stripes. Theses clumps remain for a few free fall times of the total system, then disappear, torn apart by the shear. This phenomenon is unknown in the simple free fall simulations where a clump, once formed, gets denser and more massive with time. It also appeared that the tearing of the clumps by the shear happens only if super-elastic collisions enact an efficient density cut-off. If we use elastic collisions only, below the dissipation scale, thus enacting a less efficient density cut-off, the clumps persist. However they do not collapse to the dense blackhole states encountered with different large scale energy input schemes. So the combination of the galactic shear and super-elastic collisions below the dissipation scale is necessary to obtain destroyable clumps, while the shear alone produces rather stable clumps, and the super-elastic collisions alone cannot avoid complete collapse.
While we cannot consider the density field as a fractal struture, it is indeed the beginning of a hierarchical fragmentation since we have two levels of structures. However the stripes are mainly the result of the shear. On the other hand the clumps are the simple gravitational fragmentation of the stripes. These two structures are not created by the exact same dynamical processes.
It must also be noted that the shear plays a direct role at large scale (of the order of 10 pc), where the galactic tidal forces are comparable to the self-gravity forces of the corresponding structures. But since the structures are fragmenting hierarchically, it is expected that its effect propagates and cascades down to the smallest scales.
5.3.2. Fractal analysis
The fractal dimension of the medium is plotted at different times in Fig. 6. Two characteristic scales are clearly visible. At large scale, the scale of the stripes, the dimension is about 1.9 with some fluctuations in time. At small scale, the dimension fluctuates strongly due to the appearance and disappearance of clumps. This shows through the appearance of a bump at the clump size. This size is about the dissipation scale. It is important to mention that the small scale range, while under the cutoff of the initial homogeneous condition, is not under our dynamical resolution. The mean inter-particle distance decreases from its initial value and is consistently followed by the N-body dynamics.
5.3.3. Effect of the initial conditions
We have already emphasized that in GMC simulations, the resulting structure should be independent of the initial conditions. We have performed the simulation for various values of the exponant of the power spectrum. In the range the long time behaviour is the one described above, thus satisfying the required independence. However, for more steep spectrum value like -2, the medium tend to collapse early into large clumps, thereby preventing the shear to act efficiently.
We see that the behaviour is independent of the initial conditions as long as the continuity of the medium at scale larger than the GMC size (the clump size in the simulation) is not broken so that the shear can act to sustain the GMC against collapse.
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000