Astron. Astrophys. 326, 113-129 (1997)

3. Applications

3.1. Some applications that take advantage of the geometric setting

There are at least three applications that can make use of the geometric method described above.

1. Studying the instability properties of individual orbits in fixed potentials with compact phase-spaces (in this case the formulas in (25) will, of course, have to be modified accordingly).
2. Since in Eq. (23) depends only on functions that are given by sums over the particle positions and velocities, it is constant (up to fluctuations) for systems in statistical equilibrium. In this case, it is therefore possible to replace time averages with phase-space averages (that is different realization of same density and velocity fields). This time however the averages are made over the full phase-space. The Ricci curvature therefore would give us a powerful tool of exploring that space at and around equilibrium solutions. Important questions such as the degree of chaos in a system, mixing time-scales, and the variation of these properties with particle numbers can then be tackled in the phase-space without making assumptions about the particle particle correlations. It is very important to compare such predictions with those of single particle integrations in fixed potentials to evaluate the role of discreteness in the evolution of gravitational systems. For as we shall see in the next section, systems with large softening parameters appear to have very different phase-space structures compared to those composed of point particles.
3. One can apply the method directly to the results of N -body simulations. The formula for calculating the Ricci curvature should be easily incorporated into large-N codes such as the TREECODE. This would help in interpreting the results of N -body simulations and lead to classification of galaxies according to their dynamical instability properties. Moreover, one can then study the direction of evolution of instability properties of realistic gravitational systems. For example it has been argued by Gerhard (1985) that elliptical galaxies start from chaotic states and evolve towards progressively more regular states which are then for long times indistinguishable from those of integrable systems. An alternative scenario is closer to the conventional dynamical interpretation of statistical mechanics. In this picture, a quasi-steady state is achieved when a system, although highly mixing, keeps its macroscopic parameters constant and is stable against perturbations to its statistical properties. The point being that, if the instability is present for most initial conditions of interest (that is if we have constant negative for long enough times), that would mean that the system is free to move in the region of interest and will tend towards a more probable state. This state would (by definition) contain more microscopic states compatible with it and the system would be free to move between them. This situation is more compatible with the interpretation of violent relaxation as leading to most probable end states compatible with a set of constraints (Saslaw 1985) since regular states cannot be very probable because they lie on subspaces of the dimensional subset of the phase-space where all possible states live ( stands for the number of classical integrals used in the reduction of the N -body problem, e.g., Whittaker 1937).

3.2. Specific application and model parameters

As a first test we apply the method described above to small N -body systems of 231 point particles. The small number enables us to integrate the equations of motion with high precision, on the other hand the number should be sufficient for the results to have some statistical significance. It is essential to check if the Ricci curvature method makes adequate predictions about the evolution of gravitational systems under these controlled conditions before applying it to realistic galaxy models where a host of auxiliary problems will arise. The small numbers however makes it harder to distinguish clearly between the predictions of the Ricci method and those of two body relaxation theory. A detailed comparison (including dependence of the results on N) is better left to another study.

We choose initial conditions in which the particles are arrayed into two sheets. An upper one with 11 lines consisting of 11 particles each and a lower one composed of 11 lines with 10 particles each. The lines of the upper and lower sheets are positioned in such a way that a line in the lower sheet lies at half the distance (in the plane of the sheet) between two lines in the upper sheet, so as to avoid direct contact of particles from two different sheets when the system evolves. The separation between the two sheets is taken to be equal to half the separation between lines in the same sheet (see the snapshots in Fig. 3 for example). Such configurations are artificial and are therefore likely to quickly and visibly evolve, hence saving us the trouble of long time integrations.

We use units in which the gravitational constant is unity, the masses of all particles are also taken as unity. Time in these units will be referred to as "physical time". Three scales are used to determine the separation between adjacent particles , where Scale takes either the value of 1, 10.8 or 100. In what we may call the "main models" we give the initial configurations a rigid body rotation corresponding to an angular momentum of 44275 units around the Z-axis (with zero initial velocities in the Z direction), or a random number generator is used to fix the X-Y velocities which give rise to a small angular momentum of 62.61 units. The three cases of correspond to energies of with corresponding initial virial ratios of respectively. Alternatively, some runs are started with a fixed initial virial ratio of 1 and with the same energies as above. The angular momentum is adjusted accordingly. We shall call these the "equilibrium models" although they do not start from a detailed dynamical equilibrium.

The integrations are performed using a variable order variable step size Adams method as implemented in the NAG routine D02CBF using a tolerance of . The energy is accordingly conserved to better than 10 digits (usually 12) for a few dynamical times. This accuracy is necessary for a first numerical test to eliminate the factor of serious numerical error from the interpretations of the results. For the same reason we integrate the equations for a relatively short time of about four and a half dynamical times ( (mean density) ), ) corresponding to about 2200 time units for the case (in a few cases we have performed longer time integrations). The un-softened force law is used, and softening is only introduced to study its effect.

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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