5. Time evolution
Table 1 gives an overview of the simulations described in this paper. Two series of simulations have been realised: the first one with moderate number of particles and spatial resolution (series "s", small simulations), and the other one pushing these quantities such as to exploit about half of the memory resources of the most powerful computers available at Geneva University (series "l", large simulations). In each series, we first integrated the initial axisymmetric model for 5 Gyr keeping the gaseous component fixed (simulations "xx"), and then we relaxed the gas particles at different intermediate times . The gas is not evolved from the beginning because the non-inclusion of star formation produces an excessive accumulation of gas in the central region due to the torques of the non-axisymmetric gravitational potential, raising the rotation curve and leading to a premature destruction of the bar (Friedli & Benz 1993) or even preventing its formation. Instead of introducing an artificial gas recycling procedure, the time consuming gas-live simulations of the l-series were integrated only over a few 100 Myr. To damp the initial disequilibrium of the gas owing to its circular kinematics in the already barred potential, the non-axisymmetric part of the potential is progressively and linearly brought to its nominal value in 75 Myr, i.e. roughly half a rotation period of the bar.
In the l-series, the gas has been released at two different times after the formation of the bar, at and 2.4 Gyr. In the first case, two values of the sound speed have been explored, and 20 km s-1 (simulations l10 and l20 respectively), and in the second case only km s-1 has been retained (simulation l10´). In the s-series, many runs with live gas have been performed, releasing the gas every 400 Myr and each time with two different sound speeds, but only the one mentioned in Table 1 will be discussed here. The adiabatic index of the gas is set to that of neutral hydrogen, i.e. . Taking the vertical gravitational resolution as a lower physical limit for the SPH smoothing length, the maximum gaseous density that can be reasonably modelised in the l-simulations is pc-3 (see the previous sections for the meaning of the symbols), which is at least one order of magnitude below the density of the Galactic molecular gas, but sufficient to describe the warmer neutral phase with a typical sound speed of 10 km s-1. However, Cowie (1980) has argued that a system of molecular clouds may be treated as a classical fluid with a sound speed equal to the mass averaged cloud velocity dispersion, which also amounts to 10-20 km s-1 in the Milky Way. Hence our simulations are expected to display properties of both the HI and H2 medium.
The SPH particles all have the same time-independent mass. The number of particles in each luminous component is always such that the mass per particle is the same as for the gas component to minimise relaxation effects, and the number of DH particles corresponds to a mass per particle three times larger than for the other components. The typical number of SPH particles within kpc (in initial units) and the corotation circle in the high resolution simulations is and respectively.
The simulations are run in a completely self-consistent way, without imposing any symmetry and taking into account all gravitational interactions between the mass components. In particular, the gas feels his own gravity and interacts with the stellar spiral arms. Moreover the bar parameters are not arbitrarily chosen but automatically and naturally adjust according to realistic dynamical constraints. The calculations have been done on Sparc Ultra and Silicon Graphics computers, each with over 1 Gbyte central memory.
Fig. 8 shows the whole face-on evolution of the simulations lxx and sxx with rigid gas component. Although the initial conditions of these simulations are drawn from exactly the same phase-space density function and the average number of particles per grid cell is about the same, the evolution clearly depends on the adopted resolution and number of particles: in simulation lxx the bar forms much more rapidly than in simulation sxx, around Gyr instead of 3.2 Gyr in the latter, and is of larger extent. Moreover, the contours of the bar become rounder close to the centre, as observed in many external barred galaxies (e.g. M100 in the near-IR), and the surrounding disc tends to a more flattened radial profile. An explanation for the delay of the bar formation could be that higher resolution simulations can catch smaller density fluctuations and hence favour the growth rate of asymmetries. It is not clear whether a convergence of properties with increasing resolution has been achieved in simulation lxx.
Another relevant dynamical aspect distinguishing the large simulations from those of the s-series is the offcentring of the stellar bar (Fig. 9). At Myr, the density centre starts to deviate from the global centre of mass and wanders around it. The maximum amplitude of the displacement reaches pc at Gyr, and the revolution frequency of the density centre amounts to km s- 1 kpc-1. Offcentred bars are commonly observed in external galaxies (see Colin & Athanassoula 1989; Block et al. 1994; other references in Levine & Sparke 1998) and reported in numerical simulations of galactic discs (e.g. Miller & Smith 1992). At least half of all spiral galaxies have lopsided light distribution (Schoenmakers 1999; see also Rudnick & Rix 1998). Fig. 9 confirms Weinberg's (1994) conclusion on the persistent nature of the phenomenon. However, it might be that the polar grids used to compute the gravitational forces amplify the density centre offcentring, as they artificially produce an exponential instability of the position of the centre of mass (Pfenniger & Friedli 1993). The same simulation repeated with a three-dimensional Cartesian grid could result in a lower amplitude oscillation.
The pattern speed of the bar in simulation lxx decreases roughly exponentially from 50 km s-1 kpc-1 at Gyr to 30 km s-1 kpc-1 at Gyr (see Fig. 10). The face-on axis ratio of the bar is about 0.6, i.e. close to the upper limit derived from the lower resolution simulations of Paper I.
The phase space coordinates of the stellar particles have been extracted from the simulations every 25 Myr. Adjusting the location of the observer (Sun) using the COBE/DIRBE dust subtracted K-band map as in Paper I is complicated by the density centre offcentring: in addition to the bar inclination angle, the relative galactocentric distance of the observer and the mass-to-light ratio, further parameters are needed for the direction of the Galactic centre in the models. Here we have simply applied the standard method of Paper I to models with weakly offcentred bars, assuming that the Galactic centre lies at the centre of mass and taking a finer Cartesian grid with to compare the data and model fluxes. In the notations of Paper I (see also Sect. 6) and for , the best fit location parameters for model lxxt1950 are and , with a mean quadratic relative residual %. A comparison of this model to the COBE data is shown in Fig. 11.
The time evolution of the gas flow in simulations l10, l20 and l10´ is illustrated in Figs. 12 and 13. In each simulation the gas flow rapidly becomes non-axisymmetric, forming transient spiral arms, shock fronts and a nuclear ring of orbits accumulated near the inner Lindblad resonance. Two kinds of spiral structure can a priori be distinguished in the bar region (see for instance frame Myr of simulation l10 in Fig. 12 or frame Myr in Fig. 13):
This classification should only be considered as a first order guide. The main distinction between the two structures is that the axis shocks intersect or brush the nuclear ring, while the lateral arms pass away from it.
Increasing the sound speed, i.e. the pressure forces relative to the gravitational forces, yields as expected a smoother gas distribution and softens the sharp corners in the spiral arms (see Fig. 12). The nuclear ring slightly shrinks and the axis shocks occur closer to the major axis of the bar, in agreement with the SPH results of Englmaier & Gerhard (1997). Note that some sub-structures also seem to develop within the nuclear ring for km s-1.
The gas flow never reaches a stationary state and is most of the time asymmetric, as is often observed in external disc galaxies. In particular, the lateral arms are rarely symmetric both in position and surface density, and the axis shocks considerably vary in shape. Sometimes, only one lateral arm or axis shock can be distinguished, and sometimes these structures are doubled, appearing twice on the same side of the bar (see e.g. l10t2175 in Fig. 12 for double lateral arms). It may even happen, as is the case in frame Myr of simulation l10 (Fig. 12), that the axis shocks receive a sufficient impulse from the lateral arms to transform themselves into such arms. The nuclear ring, owing to its gravitational coupling to the stellar components, intimately follows the bar density centre in its oscillations around the centre of mass. The inclusion of gas prevents the pattern speed of the bar to slow down (Fig. 10; see also Friedli & Benz 1993; Berentzen et al. 1998): after a substantial initial readjustment, remains very nearly constant over the short duration of the live gas simulations.
In the s-series, the moderate bar considerably weakens once the gas is switched-on. However, in simulation s10 it survives for a sufficiently long time to reveal an interesting asymmetric process in the gas flow (Fig. 14): the gas seems to rarefy alternatively on each side of the bar intermediate axis according to a cycle of roughly 125 Myr. As a large amount of gas flows on one side, the other appears almost devoided of gas, and when the gas of the former side reaches the axis shock, the situation reverses and the previously depleted side is filled. During this cycle, the gas density on the bar flanks can vary from single to triple.
The non-stationarity of the gas flow in the l-simulations is not a simple unachieved readjustment due to non-equilibrium in the initial conditions and the rather short integration time, since the smaller simulations with live gas, integrated much longer in time, themselves always remain time-dependent. Outside the bar region, the stellar spiral arms have a differential pattern speed, in general lower than the bar (see e.g. Sellwood & Sparke 1988), and drive the gaseous arms which repeatedly wound and dissolve. Many SPH simulations carried in rigid barred potentials seem to finally settle in a stationary spiral structure rotating at the same angular speed as the bar. Such results are an artificial consequence of the imposed fixed background potential.
© European Southern Observatory (ESO) 1999
Online publication: April 28, 1999