Astron. Astrophys. 347, 821-840 (1999)

2. Numerical experiments: descriptions

Simulations of galaxies of stars can be divided into two basic categories: global and local. The former have been done to simulate the global dynamics and the development of large-scale spiral and bending structures (Hohl 1971, 1972, 1978; Sellwood & Carlberg 1984; Grivnev 1985; Peter et al. 1993; Griv & Chiueh 1998). Certainly, some aspects of dynamical behavior of stellar systems can be studied by global simulations only (nonlinear effects, etc.). An obvious shortcoming of the global simulation approach is that the numbers of stars in a simulation is orders of magnitude smaller than in a typical galaxy. This might not permit revelation of the small-scale spiral structure (see Appendix A.1 of the present paper). Here is the mean epicyclic (Coriolis) radius (Larmor radius in magnetized plasmas, respectively). As a rule, in spiral galaxies kpc, and and .

On the other hand, to study some aspects of particulate disk dynamics when inhomogeneity is relatively weak, a different numerical approach may be taken: local N-body simulations. The latter galactic N-body experiments in a local or Hill's approximation has been pioneered by Toomre (1990) and Toomre & Kalnajs (1991). In these simulations dynamics of particles in small regions of the disk are assumed to be statistically independent of dynamics of particles in other regions. The local numerical model thus simulates only a small part of the system and more distant parts are represented as copies of the simulated region. Wisdom & Tremaine (1988) applied the same numerical technique in studying the equilibrium properties of planetary rings. In addition, Salo (1992, 1995) and Griv (1997) studied the dynamical behavior of collisional self-gravitating rings systems by using the same method. In contrast to global simulations, in local ones complicated effects of disk inhomogeneity and finite thickness may be studied separately. This is the main reason why in the present work the local N-body simulations are used. In our opinion, this simple model is useful for clarifying the physics of the phenomenon, and provides us with results which can serve as a convenient starting point for more complicated theory and numerical simulations.

In fact, Morozov (1981a) has already attempted to confirm the criterion (3) numerically. However, because of the very small number of model stars, , Morozov's results are subject to considerable uncertainties, and additional simulations are clearly required to settle the issue. Furthermore, for that number of particles, the two-body relaxation timescale is comparable to the crossing time, even with Morozov's modest softening parameter, raising some question about the applicability of his simulations to actual almost collisionless galaxies. Increasing the number density of model stars is definitely a more reliable procedure. This paper presents the results of such simulations.

Since Chandrasekhar's (1960) fundamental "molecular-kinetic" studies, in stellar systems such as the solar vicinity of our own Galaxy, binary star-star encounters are well recognized to have no influence on the evolution. That is, if other perturbations were absent the motion of a star in its orbit in the regular gravitational field of a galaxy would be determined at every moment of time by the initial conditions that prevailed when the star was "born." ⁴ Thus, in sufficiently dynamically hot and rarefied stellar systems of flat galaxies interparticle collisions can be neglected on the timescale of interest , where in galaxies yr. Then, the equations of motion for an individual star of unit mass in the inertial frame with the origin at the disk center have the form (Chandrasekhar 1960, chap. 3):

where the indicates time derivatives of with respect to time. Here and below r, , and z are the galactocentric cylindrical coordinates and the axis of the galactic rotation is along the z-axis. In the equations above, the gravitational potential has been divided into the smoothed part satisfying the equilibrium condition

and the fluctuating small perturbation with for all and t.

As has been mentioned, the local simulation has been developed by Wisdom & Tremaine (1988), Toomre (1990), Toomre & Kalnajs (1991), and Salo (1992, 1995). Following them, let us assume that the radial extent of any region of interest is much smaller than its distance from the center of rotation and any relative motion is only a small fraction of the full rotation velocity. In such a model, the linearized Newtonean Eqs. (6) and (7) in Hill's approximation can be rewritten in the suitable form:

In the equations above,

is the reference radius, , and is the first Oort constant of the differential rotation which is a measure of the shear strength. In actual galaxies and typically . In general, and are the forces due to interactions with other stars. The gravitational forces are

where is the position of the i-th particle, is the position of the j-th particle, and is the mass of a particle. The cutoff radius of the potential was introduced in order to avoid numerical difficulties caused by rare very close encounters between the model particles. This "softening" parameter reduces the interaction at short ranges and puts a lower limit on the size of the model stars, i.e., the stars in the system can no longer be considered as point-masses - they are in fact Plummer spheres with a scale size . In addition, a sufficiently high value of makes the two-dimensional system a "collisionless" one (see below). Of course, the linearized equations of motion (9) and (10) are valid only if . Such equations do not allow for investigation of nonlinear effects, the such as the well-known (in plasma physics) quasilinear collective-type relaxation.

The system of equations of motion (9)-(10) for N identical particles was integrated by the standard Runge-Kutta method of the fourth order. A rotating Cartesian coordinate system with origin at the reference position was chosen, the x axis pointing radially outward, and the y axis pointing in the direction of the rotation (for details see Toomre 1990 and Salo 1995). The particles were initially placed on nearly circular orbits with an anisotropic Schwarzschild distribution of small radial and azimuthal random velocities components. The last statement means that according to the set of equations (16) and (17) of Appendix A.1, the ratio of the velocity dispersions in the azimuthal and the radial directions (in the rotating frame we are using) is given by (Spitzer & Schwarzschild 1953)

This is close to that observed in the solar vicinity of the Galaxy. In conformity with observations we set the Gaussian distribution of small random velocities along each coordinate in momentum space both in the theory and in the numerical experiments. Thus, equilibrium is established in a simple manner in such disks, i.e., it is governed mainly by the balance between the centrifugal and gravitational forces. It is this metaequilibrium that is to be examined for stability by local simulations.

The initial distribution of stars () was generated by means of pseudo-random number generator placing particles uniformly in the box in real space. The box should be thought of as being embedded in a galactic disk which has a constant angular velocity gradient in the x direction, that is, the velocities obey initially a linear shear profile, the stationary solution , , where and are the random velocities in the x-direction and y-direction, respectively. To maintain the system under the shearing stress in a steady state, the cyclic boundary conditions are used in the form suggested by Wisdom & Tremaine (1988) and Salo (1995). ⁵ A star leaving the computational domain at one side will enter again at the opposite side with the suitable velocity components.

In all the experiments reported in this paper, we have taken the box to be rectangular, , , or , where is the ordinary Jeans-Toomre radial wavelength (Toomre 1964, 1990). The direction of the disk rotation was taken to be clockwise and units are such that . Time corresponds to a single revolution of the disk, and the orbital period is . All the particles move with the same constant Runge-Kutta time step . We did not included any artificial extra damping force on the right side in Eqs. (9) and (10) suggested by Toomre (1990) to reduce the computational time.

In our simulations (within the local simulation technique), a particle at () has images at (, ), where t is the time and the values of l, s, and p were chosen to be equal 1 (Wisdom & Tremaine 1988; Toomre 1990). Gravitational forces on a given target particle are calculated from all the other particles whose nearest image lies within the distance min (Salo 1995, Fig. 1 in his paper). Then, more distant images , , and do not contribute to gravitational forces.

Within the simple molecular-kinetic theory by Chandrasekhar (1960, chap. 2), the classical collisional relaxation time for a three-dimensional system,

should be replaced by the collisional relaxation time in a two-dimensional system (Rybicki 1971; Hohl 1973; Grivnev 1985):

Here c is the averaged velocity dispersion, is the minimum impact parameter, is the mass of a field particle, and is the two-dimensional (Eq. [12]) or the three-dimensional (Eq. [11]) number density of field particles. Also is the so-called Newton's (or Coulomb's in plasmas) logarithm, by means of which the long-range nature of the gravitational force is taken into account. In galaxies , and N is the total number of field particles (Binney & Tremaine 1987, pp. 187 and 420). Theis (1998) has presented semi-analytical calculations for the two-body relaxation in softened potentials based on a Plummer mass distribution and compared these calculations with N-body simulations. It has been shown that with respect to a Keplerian potential the increase of the relaxation time given by Eq. (11) in the modified potentials is generally less than one order of magnitude, typically only between 2 and 5, if the softening length is of the order of the mean interparticle distance. Consequently, we expect that the expressions (11) and (12) for the time of two-body relaxation in the case of the softenend potential we are using, are correct at least to the order of magnitude.

In contrast to three-dimensional models, the collisional relaxation time for exactly two-dimensional computer models being calculated from Eq. (12) is very short, of the same order as the rotation period (Rybicki 1971). However, Rybicki (1971) has already been pointed out that fortunately numerical calculations are themselves subject to further approximations, and a discretization effect minimizes the difficulty with relaxation time. Indeed, in contrast to the three-dimensional case (Eq. [11]), there is no Newton's logarithm in the expression (12) for the relaxation time via the binary encounters in a two-dimensional system. On the other hand, in a two-dimensional system, encounters with small impact parameters play the main role for collisional relaxation, ; consequently, in two-dimensional systems there is no problem with the maximum impact parameter (in plasma physics the upper limit is the Debye radius). It is natural therefore to set in Eq. (12) [Grivnev 1985]. Clearly, by choosing a sufficiently large value of , one can construct a two-dimensional model which is practically collisionless on the timescale of interest.

It is very important to realize that the numerical model of a galaxy should properly simulate almost collisionless systems. According to Eq. (12), a way to achieve this in a two-dimensional system is to reduce the gravitational attraction at short distances so that the relaxation time . Otherwise, as it was shown by Griv & Peter (1996), Griv & Chiueh (1996), Griv & Yuan (1996), and Griv et al. (1997a) in a disk with frequent collisions, in which , another secular dissipative-type instability may develop effectively. This dissipative instability may produce structures completely unrelated to the effects we would like to model (e.g., Sterzik et al. 1995).

Eq. (12) indicates that our two-dimensional N-body system will remain practically collisionless for more than several revolutions, if , , and . In this case, in the lowest approximation one does not need to include the effect of interparticle collisions in the calculation of the accelerations on the right-hand sides of Eqs. (9) and (10). Below we describe the results of simulations of different computer models containing a sufficiently large number of particles (and with ).

The value of was chosen to be , but the results are not sensitive to the choice of in the range . (Note that such a value of does not suppress the axisymmetric Jeans instability if random motions are completely absent; see Toomre 1990 for an explanation.) We did not find any dependence of critical stability criterion on the amplitude of the smoothing parameter (in the range ) as advocated recently by Romeo (1997).

Summarizing, similar to actual galaxies of stars our model is a collisionless one to a good approximation at least on a timescale of several .

In all the experiments the simulation had been performed up to a time , but we shall present snapshots for times only, since after the first rotation the system is always stabilized and no further rapid evolution is visually detectable. We performed a few runs for systems containing model stars and for smaller systems containing only ones. It was found that the results obtained for those systems are qualitatively indistinguishable: we did not detect in our experiments any dependence of the type , where is the amplitude of the density variations. The last is clearly inconsistent with Toomre's (1990) hypothesis that the spirals observed in local simulations can be explained by the swing-amplified particle noise ("spiral chaos in an orbiting patch" or "kaleidoscope of chaotic arm features" which are responses to the random density irregularities orbiting within the particulate disk). In the following discussion we advocate a way to describe the rapidly evolving structures, such as those reported in the simulations (Sect. 3), in terms of local true instabilities of Jeans-type perturbations.

We claim that our numerical results are insensitive to the value of N at least in the range and therefore two-body relaxation effects are not important. Also we did not find any difference between the results of simulations with or without applying the so-called quiet starts procedure to select the initial coordinates of particles. The methods of quiet starts were developed first in plasma simulations by Byers & Grewal (1970). Basically, by applying the method of quiet starts, one uses no random numbers in the initial conditions to suppress the noise level in a system. Such techniques have proven useful in obtaining realistic noise levels without the use of a large number of particles. Moreover, tests indicated that the results were insensitive to changes in other gross parameters (the area of unit cell, etc.). A test gives a good check on the numerical stability of the code as well as the accuracy of the program; the code conserves energy to within during the first 3 rotations of the system.

© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
helpdesk.link@springer.de