4 Simulations

4.1 Performance of the code

It is a common practice to test a new N-body code by integrating the equations of motion of a simple system of particles known to be in a steady state, and studying such things as relaxation effects and conservation laws. Thus, we set up several N-body systems following King's phase-space distribution function. We set the central density parameter W₀=5 in all cases. The units were chosen so that the total mass M=1, the dispersion parameter $\sigma=0.762$, and the gravitational constant G=1; with this set of units, the total energy of the system is E=-1/2, and the global dynamical time t_D=1, so we are able to compare our results with those of Hernquist & Barnes ([1990]) and Huang et al. ([1993]). Only three runs are commented here, namely those with N=4096, N=15000, and N=100 000.

All the runs lasted 10 dynamical times. In particular, we ran the N=10⁵ model over seven PCs, with the following processors: four Pentium/166 MHz, one Pentium Pro/200 MHz, one Pentium II/233 MHz and one Pentium II/266 MHz. To compute the CPU time demanded by the whole integration, we took the maximum among the CPU times of the processors in each cycle of integration, and summed up all these maxima to get the total. (If the program were perfectly balanced, all processors would finish simultaneously their respective tasks in each cycle; unfortunately, this could not be accomplished at all times, mainly due to the fact that our machines were not entirely dedicated to the integration. However, the performance was always near the ideal, since the program continually compensates the different third-party loads of the machines by redistributing adequately the number of particles assigned to each processor.) The integration of the ten dynamical times of the N=10⁵ run consumed 9.5 hours of CPU; it involved 1000 time steps (100 time steps per dynamical time).

Table 1 compares the speed of our code with that used by Hultman & Källander ([1997]). In all these experiments, only one processor was used. The fourth and fifth columns give the CPU times when using a Pentium II-266 MHz processor running a Linux operating system, and a HP 735 Workstation, respectively. The last column shows the times reported by Hultman & Källander ([1997]), whose code was also run on a HP 735 Workstation. The (fixed) time steps in our simulations (third column) were equal to the shortest individual time steps of the experiments of Hultman & Källander ([1997]). We can see that, despite this last disadvantage, the performance of our code increases with N, and with decreasing $\theta$. This is probably due to a better run over the tree when computing accelerations.

 

 

Table 1: CPU times used to integrate King's spheres, in hours

N	$\theta$	$\Delta t$	PII 266	HP 735	HP 735 (H)
4626	1.0	0.0039	1.0	1.9	1.9
4626	0.1	0.0034	11.8	22.0	32.1
15238	1.0	0.0063	2.7	5.0	9.4

Figure 1: Standard deviation of relative particle energies $\sigma$ vs. time, for different number of particles. The straight lines have a slope of 0.5

In these preliminary simulations, all the standard tests were satisfactory (e.g., energy was conserved better than 3 10^-4 in all cases). However, a test we found not to agree with previous results (Hernquist & Barnes [1990]; Hultman & Källander [1997]). This can be seen in Fig. 1, which shows the temporal behaviour of the standard deviation $\sigma$ of relative particle energies

$$E_i=\frac{E_{\mathrm{f}i}-E_{0i}}{E_{0i}},$$ (1)

where E_fi and E_0i are the final and initial energies of particle i, respectively. If the changes in energy were driven merely by a random walk diffusion process, the slope of $\log\sigma$ vs. $\log t$ would be 0.5 (Hernquist & Barnes 1990; Hultman & Källander 1997). However, as can be seen from the figure, the slope depends on the number of particles N, so relaxation is playing a role aside the diffusion due to the random accelerations. No dependence on the aperture angle $\theta$ or the time step $\Delta t$ was found.

4.2 Colliding galaxies

Taking advantage of the speed of computation, we set up a pair of experiments in which two galaxies collide one another.

The first experiment was built with the aim of reproducing the Antennæ (NGC 4038/4039), a classical model to simulate since the pioneer work of Toomre & Toomre ([1972]). We therefore needed a model for spiral galaxies which remains stable at least during one dynamical time $t_{\rm D}$, i.e., a period which suffices to obtain only those features caused by the collision, and not those caused by intrinsic evolution. To this end, we first followed Hernquist's ([1993]) recipe for building compound galaxies. However, this model proved not to be sufficiently stable to our experiment: when isolated, it evolves significantly well before the time at which the collision would begin. In most of our runs, a $\chi^2$ test of this model yields $P(\chi^2)\simeq1$ at only $t\simeq0.2 t_{\rm D}$.

Therefore, we shifted to Barnes' ([1992]) model of compound galaxy. We used an exponential disc with $N_{\rm d}=3072$ particles, mass $M_{\rm d}=0.1875$, radial scale $R_{\rm d}=0.083$, vertical scale z₀=0.005, and radial and vertical velocity dispersions in the ratio $\sigma_{\rm R}/\sigma_z=2$. For the bulge, we set up a King's sphere with $N_{\rm b}=1024$ particles, central potential W₀=3, total mass $M_{\rm b}=0.0625$, and the scale of velocities $\sigma=1$. Finally, for the halo, we used a similar King's sphere but with $N_{\rm h}=16384$ particles, and total mass $M_{\rm b}=4$. Thus, the halo, disc and bulge masses are in the relation 16:3:1, respectively, and their total mass is $M_{\rm t}=4.5$. Fortunately, this compound galaxy proved to be stable at least during $1.75 t_{\rm D}$.

Figure 2: Model of NGC 4038/39, the Antennæ. Top: initial configuration of the experiment. Bottom: intermediate (t=2.9) state. The time and orientation were chosen in order to show the Antennæ as seen in the sky

Once obtained a satisfactory model for the galaxy, we set up the initial conditions for the encounter leading to the Antennæ. We built a galaxy with 20480 particles as before, replicated it, and put both copies on the apocenter of a binary elliptical orbit with eccentricity e=0.5 and pericentric distance $r_{\rm p}=0.5$, with the standard Antennæ inclinations for their angular momenta e.g., Barnes ([1988]). We set the softening parameter $\varepsilon=0.015$, the tree aperture angle $\theta=0.7$, and a time step of $\Delta=10^{-3}$. Figure 2 shows the initial conditions (t=0) and the snapshot for which the experiment best resembles the sky-view of NGC 4038/39 ( $t=2.9 t_{\rm D}$). This simulation was run over six PCs (all the abovementioned machines, except one of those with the Pentium 166 MHz processor). The final time $t=3.38 t_{\rm D}$ was achieved after 5.47 hours of CPU time. In terms of speed of the code (Dubinsky [1996]), i.e., the number of particles which could be evaluated per second, this simulation attained 2078 particles/s. (We disagree with the nomenclature here, because this is not really a measure of the speed of the code itself, but it depends on the number of processors involved. If we divide the speed of the code by the number of processors, we get for our code 346 particles per second per processor; Dubinski's ([1996]) example, as a comparison, attains 375 particles per second per processor when using 16 processors.)

As a second simulation, we built up a King sphere with W₀=12, total mass M=2, King radius r₀=10^-3, and N=512, and threw it against one of the above (initial) compound galaxies, with N=6144, in order to simulate the Cartwheel galaxy. The King's sphere was initially at 10 units away from the galaxy, i.e. at the outskirts of the halo. The relative velocity was V = 2 units, along the symmetry axis of the compound galaxy. Before choosing these parameters, a series of toy experiments with different masses, distances and velocities were run in order to achieve a good resemblance to the Cartwheel.

Figure 3: Model of the Cartwheel galaxy

Figure 3 shows the final outcome of the simulation, from a point of view similar to that of the Earth. The experiment was run on one Pentium Pro/200 MHz and one Pentium II/266 MHz, and required 7.56 hours of CPU.