## 2. MethodsThe work presented in this paper is based on computer simulations of two-dimensional stellar discs. We will briefly summarize the general characteristics of the computer program as well as the galaxy model we have used in Sects. 2.1 and 2.2, respectively. Sect. 2.3 then describes how some bar parameters are determined. ## 2.1. Simulation characteristicsWe have used a modified version of GALAX (Thomasson 1991) which is
described further in Lerner (1998). It is a two-dimensional
where , We have set ## 2.2. Galaxy modelsOur model galaxy consists of three components: a disc, a bulge and a halo. Since the simulation is two-dimensional, the halo and the bulge components are represented by fixed potentials. The disc, which is purely stellar, is simulated by 1 004 640 particles of equal mass. We have used Kuzmin discs/Plummer spheres for all three components of the galaxies. The potential, , for each component is then given by where
When a simulation starts, the stellar particles are smoothly distributed on circles in the disc in accordance with the density model chosen for the disc. The initial disc is axisymmetric and the stars are placed on circular orbits with zero velocity dispersion. The stars are distributed throughout the radial interval l.u. A correction potential is then calculated and included in the simulation to give the initial disc the correct Kuzmin potential. In order to have reasonable relaxation times for our rather long simulation run, a softening length of 2 l.u. has been used. As a measure of the quality of our simulation, we have calculated the total changes in angular momentum and in energy and they remain below 0.05% and 2.3%, respectively. One timestep in our simulation corresponds to half a million years and a 20 000 timestep long simulation thus represents 10 billion years of galactic evolution. The bar starts with a rotation period, , of about 300 timesteps and slows down to timesteps at the end of the simulation as indicated in Table 2.
## 2.3. Analysis methodsWe use the phase, , of the
Fourier component of the surface
density mesh to define the size and formation time of the bar. A
We then obtain the inner and outer radii of the bar, and , from the condition We sample the surface density every 10th timestep and in order to make the bar length estimates smoother, we form the running averages and by averaging over the 21 nearest values: The value is then used as an estimate of the bar length. The bar formation time, , can now be defined as the first timestep for which and fulfil the criteria and for all subsequent timesteps. The choice of the bar window is done by testing all possible windows and selecting the one giving the earliest bar formation time. The bar rotation period, , is obtained from the bar pattern speed, , which also can be calculated from . For each of the 14 radii in the bar window, pattern speed estimates are obtained from the slopes of linear fits to 21 consecutive -values in the time domain. These values are then averaged together to produce , which gives us . The values for the parameters where chosen after a number of simulations had been studied. For a further discussion on the choice of parameters, see Lerner (1998). © European Southern Observatory (ESO) 1999 Online publication: March 18, 1999 |