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Astron. Astrophys. 331, 485-492 (1998)

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2. The model

Due 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 [FORMULA] and [FORMULA] are introduced, where r, [FORMULA] are galactocentric polar coordinates, [FORMULA] is the distance of the center of the patch from the center of the galaxy, [FORMULA] is the angular velocity at [FORMULA], and t denotes the time. After inserting these new coordinates into the equations of motion derived from the Lagrangian [FORMULA] with the potential [FORMULA], we linearize the equations with respect to x and y. The linearization is valid under the assumption that (a) the radial extent of the patch is small compared to the scales over which the basic state of the galaxy varies significantly and (b) the peculiar motions are much smaller than the circular velocities. Note that no assumption about the circumferential extent of the patch is necessary.

This leads to the epicyclic equations of motion


where [FORMULA], [FORMULA] are the acceleration components due to the spiral field. [FORMULA] 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 [FORMULA] is the wave vector of the perturbation. Since swing amplification is most effective in the vicinity of [FORMULA], we use this choice throughout the following. In the case of free swing amplification one has [FORMULA] for a sinusoidal excitation with an initial radial wave number [FORMULA] and assuming positive [FORMULA]. 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 [FORMULA] will either be not amplified strongly or damped out before reaching the zone of amplification.

The integral equation determining the amplitude function [FORMULA] derived by Fuchs (1991) is solved numerically. Its solution is shown in Fig. 1 for the standard parameters of our model,


where [FORMULA] is Toomre's critical wave number, defined with the epicyclic frequency [FORMULA], the gravitational constant [FORMULA] and the unperturbed mass surface density [FORMULA]. Our choice of [FORMULA] gives [FORMULA], somewhat more than what is found for the Milky Way in the solar neighbourhood (Kuiken & Gilmore 1991). Q is the Toomre stability parameter, [FORMULA] is the amplitude of the initial density perturbation, and [FORMULA] denotes Oort's second constant. With these parameters one has [FORMULA] with a locally slightly falling rotation curve. The only assumption about the distance to the center of the disk is [FORMULA]. While the choice of [FORMULA], [FORMULA], [FORMULA], and Q is not meant to accurately model the solar neighborhood, we regard it as representative for the outer regions of giant spirals. Our results are generic in that they do not qualitatively change under reasonable modifications of these parameters. For example, as long as [FORMULA] is comparable to [FORMULA], it is unimportant whether the rotation curve is flat, rising or falling.

[FIGURE] Fig. 1. Amplitude of the potential perturbation, given in arbitrary units. The initial amplitude of the potential perturbation at [FORMULA] and the minimum at about [FORMULA] correspond to density perturbations of roughly [FORMULA] and [FORMULA] amplitude, respectively. Since the radial wave number increases linearly with time, this also shows the temporal development of the wave. The interval of wave numbers shown here, [FORMULA], corresponds to a time interval of [FORMULA] years.

Eq. (2) describes sinusoidal waves of constant circumferential wavelength. Due to the shear of the disk, the radial wavenumber [FORMULA] grows with time. In this way an initially leading wave ([FORMULA]) is transformed into a trailing one and, as can be seen in Fig. 1, strongly amplified while "swinging by". With increasing [FORMULA] phase mixing leads to a damping of the perturbation as the wavelength of [FORMULA] drops beyond the typical epicycle size. A single swing amplification event from excitation to decay has a duration of a few [FORMULA] 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 [FORMULA], where we chose [FORMULA]. 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 [FORMULA] 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 f. To have a clear signature of the collisonal nature of the cloudy medium, we have selected a high collision rate (about [FORMULA]) which in turn requires a comparatively low f of 0.2 to prevent the system from cooling down too quickly. For the implication of this choice, see the comprehensive investigation by Jungwiert & Palous (1996). However, as long as the total cooling rate is kept constant, our model is much less sensitive to the choice of f than theirs. With these parameters, an unperturbed system of clouds with Gaussian velocity distribution cools down exponentially with a characteristic time of about [FORMULA]. Collisions are checked for once every step, where one time step equals [FORMULA].

The sticky particles are initially homogeneously distributed with a density of [FORMULA] and have a Gaussian velocity distribution. We chose a radial velocity dispersion of [FORMULA]. The velocity dispersion in the y -direction was set according to the epicyclic ratio [FORMULA], resulting in a total initial velocity dispersion of [FORMULA]. As long as [FORMULA] is low enough (say, [FORMULA]), the behaviour of the model is not strongly influenced by this choice.

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 [FORMULA] 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 f and lowered the initial density accordingly to keep the collision rate constant.

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 [FORMULA] of the ISM freely in our model by changing [FORMULA] and the initial surface density since all interactions with impact parameters smaller than [FORMULA] are equivalent. Checks with 3-D-models showed no significant deviations from the evolution described below.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998