Introduction

The formation and the evolution of bars, the mutual influence of the bulge and the bar and the effects of vertical resonances, have been studied in a series of N-body simulations (105≤N≤5×105) over long time scales (T≥2000 Myr). A PM method is used with a 3-D polar grid having an exponential spacing in R (the central resolution is 0.2 kpc or less), and a linear spacing in φ and z. More detail will be given in a future paper. The particles are distributed into a bulge and a disc components in hydrostatic equilibrium (Satoh & Miyamoto 1976). The dimensionless parameters are the bulge-to-disc mass ratio µ = Mb/Md, and the bulge-to-disc scale length ratio β = b/a.

Some results

VELOCITY DISPERSION

At T = 0 the velocity ellipsoid is isotropic by construction, but it becomes rapidly anisotropic, such that σR ≥ σφ ≥ σz everywhere, and its size decreases with R. Due to heating by time-dependent perturbations, σR and σφ grow considerably, as well as σzin the bar region since vertical resonances exist. This large scale heating is more efficient than that induced by local perturbations.

BULGE EVOLUTION

The disc, and subsequently the bar, flattens the initially spherical bulge, which aligns itself with the bar, i.e. the bulge co-rotates with the bar. For example with µ = 0.18 and β = 1/12, the bulge axis ratios are 1 : 0.84 : 0.82 (T = 2000 Myr).

BAR SHAPE

The horizontal and the vertical bar ellipticity, εh and εv, are measured at the most eccentric contour line of the projected density.