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Astron. Astrophys. 327, 930-946 (1997)

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2. Initial conditions and computational method

We have performed a series of N-body simulations with the purpose of studying the formation of central dominant galaxies and in particular of cD galaxies. In all cases we used 45000 particles representing at the onset 50 identical galaxies of 900 particles each. This is somewhat higher or of the same order as the corresponding number used by Funato et al. (1993) and by Bode et al. (1994) and, as shown by García-Gómez et al. (1996), is sufficient for our purposes. The particles in a given galaxy were initially taken to follow a Plummer distribution of core radius 0.2 and unit mass. For all the simulations, except for Vh, the radial distances from the center of the group to the galaxy centers were picked at random between 0 and [FORMULA]. For simulation Vh the central part of the sphere contained no galaxy, i.e. the radial distances were picked between 0.5 [FORMULA] and [FORMULA]. In the case of non-spherical initial conditions the X coordinate of the distance of each galaxy from the group center is multiplied by some appropriate constant. The essential information on the initial conditions can be found in Table 1. Column 1 gives the run identifier and Column 2 the radius of the sphere initially containing all the galaxies. In Column 3 we can find the ratio of initial kinetic energy to the absolute value of the potential energy of the galaxies seen as point masses. A value of 0.0 for this ratio stands for collapsing systems, while a value of 0.5 stands for initially virialised systems. In the latter case the velocity dispersion of the bulk motion of the galaxies is taken to be isotropic. Column  4 gives the ratio of the initial velocity dispersion of the galaxies within the cluster to the velocity dispersion of the particles within a galaxy and, finally, Column  5 gives the axial ratios of the system.


Table 1. Initial conditions

Runs C1, C2, Cp and Co ("p" for prolate and "o" for oblate) are initially collapsing systems, while runs V, Vh, Vc1 and Vc2 ("h" for hollow and "c" for compact) are initially virialised systems. We shall, for brevity, often refer to the former simply as "collapsing", rather than "initially collapsing", and to the latter simply as "virialised". Runs V and Vh have similar initial global conditions but, for reasons which will be discussed later, run Vh was initially depleted of galaxies in the central part. Since the number of galaxies in all the runs is the same, this means that there are more galaxies in the outer parts of the group in run Vh than in run V. Runs Vc1 and Vc2 are also virialised systems, but in these cases the initial radius of the sphere containing the galaxies is half that of run V. In all these initial conditions all the mass is initially bound to galaxies. Simulations where a fraction of the mass is in a common halo will be discussed in a future paper.

For the collapsing simulations C1 and C2 and the virialised ones Vc1 and Vc2 we used the same global initial conditions, but different initial seeds, in order to check for a possible influence of the realisations on the final results. The collapsing systems of run Cp and run Co were initially anisotropic systems and were performed with the aim of studying the possible influence of the initial shape of the system on the final properties of the central galaxy. Run Cp is an initially prolate system where the initial size of the X axis is doubled, while run Co is an initially oblate system for which the initial size of the X axis is halved.

We followed the evolution of these groups using a version of the Barnes and Hut treecode (Barnes & Hut 1986), particularly adapted for a Cray computer (Hernquist 1988). The time step was taken to be equal to 0.0075 and the softening length equal to 0.05, which is of the order of the mean interparticle distance in the initial galaxy. This ensured an energy conservation better than [FORMULA]. Each simulation was continued for 4000 steps, i.e. a total time in simulation units of 30. One simulation lasted about 150 hours on a CRAY 2L. In this paper we will use the computer units [FORMULA], where [FORMULA] is the initial mass in each galaxy, [FORMULA] is their initial radius and G is the gravitational constant. To compare with the observations these can be converted to real units by assigning a mass and a radius to each galaxy. In the following we will assume [FORMULA] and [FORMULA] kpc, which gives [FORMULA] years and [FORMULA] for the units of time and velocity respectively. It is obvious that this choice, albeit reasonable, is not unique, and that other neighbouring values would have been equally well acceptable. This should be kept in mind when comparing with observational data, hence agreements to within a factor of two should be considered quite satisfactory.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998