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Astron. Astrophys. 344, 483-493 (1999)

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2. Methods

The 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 characteristics

We have used a modified version of GALAX (Thomasson 1991) which is described further in Lerner (1998). It is a two-dimensional N-body code of particle-mesh type using a logarithmic polar coordinate system for potential calculations combined with a Cartesian leapfrog algorithm for higher efficiency. The coordinate system is given by


where [FORMULA], S is a scale factor, and [FORMULA], [FORMULA] and [FORMULA] defines the size of the coordinate mesh.

We have set S to 0.5 and use a [FORMULA] grid with [FORMULA], [FORMULA] and [FORMULA]. The outer rim of the coordinate grid is, with this choice of parameters, located almost at 500 l.u. (length units ), while there is a small hole with [FORMULA] l.u. cut out in the centre. Forces acting on particles entering this central hole are calculated by direct summation of the contributions from the surface density matrix. Particles trying to leave the coordinate system are trapped at the outer perimeter. This simplistic treatment is of low importance since very few particles ([FORMULA]%) are expelled to such large distances from the galaxy.

2.2. Galaxy models

Our 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, [FORMULA], for each component is then given by


where R is a scale length. The values of M and R for the simulation presented in this paper are listed in Table 1.


Table 1. The basic parameters used for the simulation presented in this paper are given in this table including the mass, M, and the characteristic length scale, R, of each component.

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 [FORMULA] 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, [FORMULA], of about 300 timesteps and slows down to [FORMULA] timesteps at the end of the simulation as indicated in Table 2.


Table 2. Bar characteristics for a few selected timesteps. The characteristics presented are the length of the bar (major axis length), [FORMULA], and the rotation period, [FORMULA]. The column [FORMULA] represents time since formation of the bar. The corresponding co-rotation radius of the bar, [FORMULA], is also given in the table.

2.3. Analysis methods

We use the phase, [FORMULA], of the [FORMULA] Fourier component of the surface density mesh to define the size and formation time of the bar. A bar window consisting of 14 consecutive radial grid points ([FORMULA]) is used to define the average phase angle of the bar, [FORMULA]:


We then obtain the inner and outer radii of the bar, [FORMULA] and [FORMULA], 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 [FORMULA] and [FORMULA] by averaging over the 21 nearest values:


The value [FORMULA] is then used as an estimate of the bar length.

The bar formation time, [FORMULA], can now be defined as the first timestep for which [FORMULA] and [FORMULA] fulfil the criteria [FORMULA] and [FORMULA] 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, [FORMULA], is obtained from the bar pattern speed, [FORMULA], which also can be calculated from [FORMULA]. For each of the 14 radii in the bar window, pattern speed estimates are obtained from the slopes of linear fits to 21 consecutive [FORMULA]-values in the time domain. These values are then averaged together to produce [FORMULA], which gives us [FORMULA].

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).

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999