We made N-body simulations with a self-gravitating stellar disk by using a two-dimensional logarithmic polar grid to calculate the potential. Typical grid geometry is 144 radial and 108 azimuthal cells. We also reproduced a few of our models with a 3-fold higher grid resolution to check the possible effects of grid softening. The standard particle number in our simulations was 500 000, distributed as an exponential disk with a cut-off radius at six disk scale lengths. Here we adopt a scaling where the scale length of the disk equals to 3 kpc. To test the effect of particle number we repeated some of the models with 200 000 and 1 000 000 particles. The value of the softening parameter was 1/8 of the disk scale length. The initial velocity distribution was created using the epicyclic approximation and corrected for asymmetric drift. The movement of particles was integrated by the time-centred leap-frog method for a time span corresponding to 15 gigayears. The length of a time step was 250 000 years, corresponding about 150 time steps per rotation period at the distance of one disk scale length, and to about 40 time steps very near the centre.
The gas component was modelled by inelastically colliding massless test particles. In each impact, the normal component of the relative velocity of the colliding particles was reduced to zero, i.e. the so-called coefficient of restitution is set to 0.0. On the other hand, the tangential component of the relative velocity was retained. In most simulations 20 000 particles were used, having a uniform surface density to the outer edge of the stellar disk. The usual value of the particle radius is 7.5 pc. This yields a collision frequency from 20 to 400 per particle during one gigayear. The lowest value is for systems before the formation of a bar or a global spiral structure and the highest for systems with strong bars and nuclear rings. Similar treatment of the gas component was used in Salo (1991), Byrd et al. (1994) and Salo et al. (1999).
We have followed the evolution of our models by classifying the ring structures and by measuring their major and minor axes. In addition to morphological analysis, we have studied the Fourier decomposition of disk surface density:
where r and are the polar coordinates, the disk surface density, the axisymmetric surface density and and are the Fourier amplitude and phase angles. This decomposition was done separately for the stellar and the gas components by 200 times per Gyr. From the temporal evolution of and we constructed amplitude spectra (see e.g. Masset & Tagger 1997) displaying the amplitude of signals at different pattern speeds and distances. This gave essential information about the evolution of the system, especially when several modes were present. The resonance radii were determined by plotting curves showing and with the amplitude spectra and measuring where the modes crossed these curves. Here and are the circular and epicycle frequencies, respectively. The most important resonances are the Lindblad resonances (), the ultraharmonic () or 4/1-resonances and the corotation resonance, where the pattern speed of the mode is equal to local circular frequency. These resonance radii were then compared with the sizes of the rings in selected timesteps. The amplitude spectra of the gas component were used to confirm whether the gas in the rings follows the modes of the stellar component or not. We do not include the amplitude spectra of the gas component in this article for two reasons: to limit the number of figures and because these spectra are very noisy due to small particle number.
We also used the Fourier decompositions to determine the timescales of bar and ring formation more accurately: due to computer disk space limitations, the particle positions were saved only by every 625 million years, which is not always accurate enough. On the other hand, the higher saving frequency of the Fourier decompositions makes more accurate timescale estimates possible. We compared the morphologies of the particle position images with those constructed from the Fourier components, and found that they give equivalent gas morphology. The spiral structure of the stellar component was often better resolved in the images constructed from the Fourier components. We noticed that certain relative orientation of bar and spiral modes with different pattern speeds can give a temporary illusion of a considerably longer bar component than the actual one. A similar phenomenon was also found by Debattista & Sellwood (2000).
In a few selected models, we inferred the actual shapes of the modes by the method we applied in Rautiainen & Salo (1999): first, we rotated the Fourier phase angles of the decomposition of Eq. 1 to a coordinate system rotating with the same angular speed as the mode (measured from the amplitude spectrum), we then constructed the component of the density distribution from the Fourier decomposition, and finally we summed and averaged the images into a single image showing the actual shape of the mode. Naturally, this method can be applied only when the pattern speed and the shape do not change much during the time interval used. Basically, the reconstructed images of the modes can include ghost images produced by other modes. To ensure that these do not interfere with our images, we checked the amplitude levels of the ghost images and found them to be below the lowest displayed contour level.
We have also studied the velocity dispersion of the gas component in our models. We did this by measuring the local radial velocity dispersion in a grid with 100 by 100 cells covering the disk. In calculating the average radial velocity dispersion we included the data only from the grid cells with at least ten particles.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000