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Astron. Astrophys. 363, 869-886 (2000)

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6. Numerical simulations

When a spiral galaxy hosts several counterrotating components, the two-stream instability generates an [FORMULA] (one-arm) spiral wave. The strongest amplification is for the one-arm instability leading with respect to the most massive component (Lovelace et al. 1997). After a few rotations, the arm starts to wind up and transforms into a trailing pattern. If the counterrotating components are of comparable mass, a stationary highly-wound structure forms, quite similar to an ensemble of circles of arc (Comins et al. 1997). The details of this rapid transformation are so far poorly known. In NGC 3593, two counterrotating stellar components coexist in the center (r[FORMULA]1.5kpc), and presumably throughout the galaxy; the counterrotating gas co-exists with directly rotating stars. Therefore the two-stream instability is expected to play a role in this galaxy, and possibly generates [FORMULA] spirals. Observations support the existence of a [FORMULA] spiral in NGC 3593, characterized by a variable pitch angle. To investigate the nature of these instabilities, we have performed N-body numerical simulations including both stars and gas. Contrary to previous numerical experiments, we assume here the particular physical conditions of NGC 3593. The mass ratio and the radial extent of the stellar and gaseous counterrotating components are taken from the observed values, and a mass model is built to reproduce the observed rotation curve. Our objective is to study the orientation with respect to the rotation of these instabilities (either leading or trailing ), their time evolution, and their pattern speed. Finally, the availability of high-quality data, especially concerning the gaseous response in the NGC 3593 disk, allows a detailed comparison between the model and the observations.

As detailed below, there is a range of possible mass models which fit the observations, the big unknown being the adopted M/L ratio. We study the critical influence that the amount of dark matter admitted in the disk may have in the evolution of the system, by running two extreme models: the disk-dominated and the halo-dominated models. However, even in the second extreme case, we keep the halo mass fraction [FORMULA]56[FORMULA], quite far from the unrealistic large value assumed by Comins et al 1997 ([FORMULA]75-80[FORMULA]). In contrast, our assumption agrees closely with the value derived by Courteau & Rix (1999) for the inner parts of a typical nonbarred High Surface Brightness (HSB) spiral (55-60[FORMULA]), whose disks are thought to be submaximal.

6.1. Numerical code

The gravity is solved via Fast Fourier Transforms (FFTs), using a physical 2D grid of 256x256 points (the latter is enlarged to a computational 512x512 grid to suppress Fourier images). Particles interact through a Newtonian potential softened on scales smaller than 60 pc. The stellar disk is represented by 105 particles, while the bulge and dark halo are modeled by rigid potentials. The gas is treated by a sticky particle scheme, which best accounts for the clumpy nature of the interstellar medium. There are 2[FORMULA]104 gas particles. We have adopted a simple collisional scheme for gas clouds, without a cloud mass spectrum (Combes & Gerin 1985). Clouds interact with each other via inelastic collisions, losing 35% of their radial relative velocity in the collision, both in radial and tangential directions. The collisional grid is 240x240, and the cell grid is 60 pc. The average number of collisions is such that a typical cloud experiences 2 collisions per rotation (this means that the average collisional time-scale is 40 Myr).

6.2. The Galaxy model

Let us first describe the disk-dominated NGC 3593 model. It is made of three components: a small bulge, of Plummer shape potential


with [FORMULA] = 3 109 [FORMULA] and b=0.6 kpc, a Toomre stellar disk of surface density


truncated at rdisk=6 kpc, with a mass of [FORMULA] = 1010 [FORMULA], and characteristic radius of d = 2.2 kpc; and finally, a dark matter halo of Plummer shape potential with a characteristic scale of 8 kpc, ensures a flat rotation curve. The dark halo mass inside 8 kpc is 9 109 [FORMULA]. The halo mass percentage inside r=rdisk is [FORMULA]34[FORMULA]. The rotation curve obtained, together with the contribution of the various components is plotted in Fig. 7a. The gas component initially has an exponential radial distribution, with a scale length of 3 kpc. Its total mass is 1.6 109 [FORMULA].

[FIGURE] Fig. 7. a Rotation curve obtained from the disk-dominated NGC 3593 mass model (full line V and [FORMULA]), and the various contributions from the bulge (B), disk (D) and dark halo (H) in dashes. b Same as a, but for the mass model dominated by the dark matter halo.

In the second, halo-dominated, model, the disk mass is divided by 2, [FORMULA] = 5 109 [FORMULA], and the gas mass as well, to keep the same gas fraction in the disk. The bulge is unchanged, but the dark halo mass is now 1.4 1010 [FORMULA] inside 8 kpc, and its characteristic radius is 5 kpc (see Fig. 7b). The latter implies that [FORMULA]56[FORMULA] of the total mass is contained in the halo for r=rdisk.

To reproduce the kinematical observations of B96, the majority of the stellar rotational velocities were launched in the direct sense (counterclockwise), but within r=1.1 kpc two-thirds of the stars were reversed in direction; outside this separation radius of 1.1 kpc (hereafter denoted rcrit), 10% of the stars were also reversed to take into account a likely continuation of the central counterrotating disk at large radius. The ionized gas map showing that new stars are being formed along the one arm spiral supports this assumption. The entire gas component is launched counterrotating (clockwise).

6.3. Diagnostics

In addition to monitoring particle plots (Fig. 8a,b) we have computed the intensity of the various m components by Fourier analysis of the total potential at the different time steps of simulations. If the latter is decomposed as [FORMULA](r,[FORMULA]) = [FORMULA](r) + [FORMULA](r) cos (m [FORMULA]-[FORMULA]), we define the intensities of the various components by their maximal contribution to the tangential force, normalized by the radial force [FORMULA]r, through


[FIGURE] Fig. 8. a Particle plots of the stars (left) and the gas (right) for the disk-dominated run, at successive epochs 200 to 800 Myr by 200 Myr. Particles are plotted until a radius of 6.25 kpc. The majority of stars rotate in the direct sense (counterclockwise), while the gas is retrograde. b The halo-dominated run, at successive epochs 200, 400 Myr then 1400, 1600 Myr.

The maxima of these intensities over the radii are plotted for instance in Fig. 9ab.

[FIGURE] Fig. 9a and b. Intensity of the [FORMULA] (P1, solid line) and [FORMULA] (P2, dash) Fourier components of the potential (more exactly the corresponding components of the tangential force normalized by the radial force), a) for the mass model with dominating disk of NGC 3593. The dotted line displays the energy dissipated in gas cloud collisions, in arbitrary units (Tdis); b) for the mass model with dominating halo.

Over a run period of 1600 Myr, the Fourier analysis of the surface densities were done for the stars, the gas and the total density, to determine as a function of radius, the strength of the [FORMULA] components. These were stored every 10 Myr, in order to Fourier transform the result F(r, t) in the time dimension, and get the power as a function of pattern speed for different epochs, F(r, [FORMULA]). Expressed as a function of the frequency of the m perturbation ([FORMULA]), the pattern speed is [FORMULA]=[FORMULA].

6.4. Results

6.4.1. Disk-dominated model

The first run is the disk-dominated model, described in Fig. 7a. The Toomre parameter Q is chosen equal to 1 in the disk-dominated model for the stars and the gas, so as to make the disk barely stable with respect to axisymmetric instabilities. We will distinguish three epochs, based on the existence of distinct evolutionary patterns and different dominant modes (see Fig. 9a).

During a transitory regime (0-500 Myr) the particle plots in Fig. 8a reveal a rather violent lopsided instability in the total potential, with a clear trailing one-arm pattern in the gas. This [FORMULA] gas arm is nearly stationary, i.e. [FORMULA]0. The fast lopsided instability has a [FORMULA]=-220 km/s/kpc at the end of the transitory regime.

During the second phase, taking place for T=500-1000 Myr, an [FORMULA] instability is superposed, and it ends up surpassing (for T[FORMULA]500Myr) in strength the [FORMULA] component (see Fig. 9a). In the particle plots, these take the shape of oval/barred perturbations in the stars. In the radial distribution of [FORMULA] this manifests into two distinct peaks centered at 1 and 3 kpc. Fig. 9a shows that each development of an [FORMULA] or 2 instability generates a delayed burst of cloud-cloud collisions in the gas.

In Fig. 10a, the power spectrum of the total density reveals two distinct waves, with opposite pattern speeds, which correspond to the two radial peaks of [FORMULA] already observed in the total potential. The central wave is retrograde, and corresponds to the bar/oval that is rotating in the same sense as the gas and majority of the stars in the central disk (r[FORMULA]rcrit). Its pattern speed is [FORMULA] = -55 km/s/kpc. The outer one is direct with respect to the main stellar disk, and it counterrotates with respect to the gas, with a pattern speed of [FORMULA] = +35 km/s/kpc. The latter wave corresponds to the peak of the [FORMULA] curve, which is the most current case for [FORMULA] patterns in barred galaxies (e.g. Combes & Elmegreen 1993). There is an ILR at 1kpc, and corotation at 4 kpc. The central pattern, at [FORMULA] = -55 km/s/kpc, has no ILR, a corotation at 2.2 kpc and an OLR at 4kpc. In addition, during this second phase, the [FORMULA] power spectrum of the gas density shows a slow pattern with [FORMULA]= -55 km/s/kpc (Fig. 10a), suggesting that the [FORMULA] wave might be excited by an harmonic of it. The [FORMULA] mode is trailing for the gas at all radii.

[FIGURE] Fig. 10. a Pattern speed as a function of radius, in units of 100km/s/kpc, for the [FORMULA] mode, total density for T=500-1000 Myr (top ) [FORMULA] mode, gas for T=500-1000 Myr (middle ) and for [FORMULA] mode, total density, for T=0-1600 Myr (bottom )in the disk-dominated run. b Pattern speed as a function of radius, in units of 100km/s/kpc, for the [FORMULA] mode, gas density for T=400-1200 Myr (top ) and [FORMULA] mode, gas also for T=400-1200 Myr (middle ), and the [FORMULA] mode, total density for T=0-1600 Myr (bottom ), for the halo-dominated run.

During the last stage T=1000-1600 Myr (not shown), the stellar disk heats up progressively and the two bars dissolve in the end, due to angular momentum exchanges which end up by the mutual annihilation of the two modes (see also Friedli 1996 for a similar case). The corresponding gas response is a multi-arm stochastic pattern. The inclusion of a massive counterrotating component speeds up the evolution of the disk instabilities due to angular momentum exchanges and enhanced dynamical friction between the two disks. This shorter time-scale is to be compared with the case where no counterrotation is present. The only remaining instability of the disk at this final stage is a lopsided oscillation with [FORMULA] = -220 km/s/kpc (see Fig. 10a). During the entire run, the lopsided instability affects mainly the inner region (r[FORMULA]rcrit).

6.4.2. Halo-dominated model

In the halo-dominated model the higher halo mass fraction (56%) determines the stability of the disk and the growth-rate of the different modes. Based on the development and time-scale of the different disk instabilities, we will distinguish three epochs, as we did in the previous section.

During the first 400 Myr, a one-arm regular structure forms with [FORMULA]0, leading with respect to the gas rotation at all radii, and trailing with respect to the main stellar stream in the outer parts of the disk for r[FORMULA]rcrit. This might appear in contradiction with the winding sense predicted by the linear theory, but since the central stellar disk is counterrotating, the wave is indeed leading with respect to this main component.

After this regime, the more persistent [FORMULA] wave changes its orientation and it becomes leading with respect to the directly rotating main disk (r[FORMULA]rcrit), and trailing for the gas at all radii (this corresponds to T=500-1200Myr). An [FORMULA] mode is also growing in the gas response, superposed on the [FORMULA] mode. The two modes [FORMULA] and [FORMULA] are harmonics of each other and share a pattern speed of [FORMULA] = -41 km/s/kpc (Fig. 10b), very close to the maximum of the [FORMULA] curve. The corotation is at 3 kpc. A gas ring forms at the unique ILR of the [FORMULA] perturbation.

Finally, the [FORMULA] intensity grows suddenly at 1200 Myr, where a characteristic lopsidedness is observed in the potential (Fig. 9b). The fast [FORMULA] pattern ([FORMULA]=-200km/s/kpc) corresponds to a slight off-centering of the disk matter with respect to the fixed bulge and dark halo. The disk center then oscillates with a period of 30 Myr and maximum amplitude of 180 pc.

6.5. Comparison with previous numerical simulations

Thakar & Ryden (1996, 1998) studied the formation of counter-rotating disks through two main mechanisms: either gradual and secular gas infall or a gas-rich dwarf merger with a primary spiral disk. In their first simulations they used for the gas a sticky-particles code not dissipative enough to see detailed structures in the gas response. Later, they used a more dissipative SPH-code, that allowed thinner disks to form, and that was more responsive to instabilities. The structures are essentially [FORMULA] spirals. Although the spirals are asymmetrical, and certainly mixed with [FORMULA], the pure [FORMULA] structures in the transient phases are never seen.

Comins et al. (1997) and Friedli (1996) simulated purely stellar counter-rotating systems, but with initial conditions favoring more the [FORMULA] or [FORMULA] instabilities, respectively. Comins et al. (1997) embed their galaxies in dark unresponsive haloes containing an unrealistic high mass fraction (75-80%), which suppress completely the bar instability. They observe the leading one-arm expected from the linear theory, during a transient phase, but then the arms shift to trailing, and to fragmented arm structures, with very small pitch angle. The first phases of our halo-dominated run share many features with these models (see our Fig. 11 in particular). Counter-rotating bars, on the contrary, were observed by Friedli (1996), similarly to our disk-dominated model; the inclusion of gas in the model developed here explains naturally the formation of rings at the ILR of the bar.

[FIGURE] Fig. 11. A zoomed view (displayed radius r=3.1kpc) for the particle plots of the stars (left) and the gas (right) in the halo-dominated run, corresponding to the transitory regime at t=350Myr.

In all simulations of counter-rotating disks, the life-time of instabilities is shorter than in normal direct disks, because of the enhanced heating due to the annihilation of the two opposite angular momenta. The counter-rotating bars live only 1-2 Gyr, even in the absence of gas (Friedli 1996), and with gas being driven to the center, their life-time is even shorter (Fig. 9).

In the two mass models of our numerical simulations, there is a slight off-centering of the disk, that produces a strong [FORMULA] component, corresponding to the oscillation of the center of mass of the disk with respect to the halo and the bulge. This oscillation is very rapid, with an angular frequency of about 200 km/s/kpc, or a period of 30 Myr. The pure [FORMULA] modes are rare in spiral galaxies, although it is frequent that they are coupled with [FORMULA] patterns, since more than 50% of galaxies reveal lopsidedness (e.g. Richter & Sancisi 1994). The counter-rotation phenomenon is one of the essential mechanisms to produce them, but they can be also triggered by interactions with companions, (Weinberg 1994; Lovelace et al. 1999) and develop in nearly keplerian disks around central massive black-holes (Miller & Smith 1992; Taga & Iye 1998; Bacon et al. 2000).

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Online publication: December 5, 2000