## 2. The modelDue to the inherently local nature of swing amplification no global disk model is required. Instead, we use a sliding box scheme similar to the one used by Toomre & Kalnajs (1991). The central idea of the sliding box scheme is to examine the dynamics of a rectangular patch of a galactic disk that is periodically repeated such that when, for example, a particle leaves the patch at the rotationally leading border, it is re-fed at the trailing one. This resembles the periodic boundary conditions used in plasma and solid state physics, with the complication that in galactic dynamics the periodic continuations must share the general shear of the disk. In consequence, particles leaving the patch radially must be re-fed at the opposite edge at positions varying with time, and the difference in circular velocities between the inner and outer edges must be accounted for, so as to keep the accelerations continuous during the re-feeding process. In such a patch pseudo-Cartesian coordinates
and are introduced, where This leads to the epicyclic equations of motion where , are the acceleration components due to the spiral field. denotes Oort's first constant. The perturbation forces are calculated according to the work of Fuchs (1991) where the Boltzmann and Poisson equations were used to obtain the perturbation potential in the form where is the wave vector of the perturbation. Since swing amplification is most effective in the vicinity of , we use this choice throughout the following. In the case of free swing amplification one has for a sinusoidal excitation with an initial radial wave number and assuming positive . While one has to expect that excitations will not be sinusoidal in reality, the restriction to a single Fourier component should not be critical due to the linearity assumed in the derivation of Eq. (2) and because components with wave numbers significantly different from an optimal will either be not amplified strongly or damped out before reaching the zone of amplification. The integral equation determining the amplitude function derived by Fuchs (1991) is solved numerically. Its solution is shown in Fig. 1 for the standard parameters of our model, where is Toomre's critical wave number,
defined with the epicyclic frequency , the
gravitational constant and the unperturbed mass
surface density . Our choice of
gives , somewhat more
than what is found for the Milky Way in the solar neighbourhood
(Kuiken & Gilmore 1991).
Eq. (2) describes sinusoidal waves of constant circumferential wavelength. Due to the shear of the disk, the radial wavenumber grows with time. In this way an initially leading wave () is transformed into a trailing one and, as can be seen in Fig. 1, strongly amplified while "swinging by". With increasing phase mixing leads to a damping of the perturbation as the wavelength of drops beyond the typical epicycle size. A single swing amplification event from excitation to decay has a duration of a few years. In our model the ISM is approximated as a dissipative medium of
discrete clouds. Their motions are governed by Eqs. (1) as long as
they do not collide. Collision detection is performed on a grid of
width , where we chose .
If there are two or more clouds in one cell, we form pairs by a random
process and let a pair collide if the clouds that make it up are
approaching each other. The collisions treat clouds as "sticky"
spheres, i.e. their relative velocities are multiplied by a
factor after performing an elastic two-body
collision (Brahic 1977, Schwarz 1981). In such a system the cooling
rate depends on the collision rate and the inelasticy coefficient
The sticky particles are initially homogeneously distributed with a
density of and have a Gaussian velocity
distribution. We chose a radial velocity dispersion of
. The velocity dispersion in the Velocity dispersions of this order are typical of giant molecular
clouds (Clemens 1985). Our sticky particles, however, are not meant to
mimic GMCs themselves. Their number is much larger than that of GMCs,
they are stable, and of course their dynamical behavior can at best be
regarded as idealized from real clouds. Technically, they just
implement a dissipative medium. From the structure of the equations of
motion, we do not need to make any assumptions about the masses of the
clouds. Also, it is very hard to assign radii to the model clouds from
the sticky particles formalism. One might use
as a gross measure of the collision cross section. The details of the
modeling do not seem to be critical for our results, since the
dissipativity of the medium does not influence its dynamical evolution
very significantly except through cooling (see below). In this way,
our results would be largely unchanged if, for example, one increased
Introducing two-particle interactions like collisions requires some care in strictly two-dimensional calculations like the present one. Rybicki (1972) showed that in contrast to three-dimensional systems in two-dimensional self-gravitating sheets the relaxation time is always of the order of the crossing time. Interactions "softer" than gravity are, however, less affected by this limitation. In fact, one can choose of the ISM freely in our model by changing and the initial surface density since all interactions with impact parameters smaller than are equivalent. Checks with 3-D-models showed no significant deviations from the evolution described below. © European Southern Observatory (ESO) 1998 Online publication: February 16, 1998 |