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The standard mechanism to form rings, as we have described above, appears to require the presence of a bar. We can note however that in the presence of a steady spiral density wave, gravity torques will also drive the gas towards the center inside corotation, and outwards outside corotation, so that it could accumulate in rings at resonances. But the mechanism is not efficient, since spirals are not steady enough in the absence of bars. Recurrent spiral structure has been obtained in N-body simulations, but always with different pattern speeds (e.g., Sellwood & Carlberg 1984). First, not only do spiral instabilities heat the stellar medium, which becomes unresponsive, but also angular momentum is exchanged due to phase-shifts between density and potential, and rapid evolution occurs. On the contrary, a bar is very steady and robust in the stellar component.

The existence of rings in nonbarred galaxies is a common phenomenon (see section 9.1 and Buta 1995). Some of them could be explained by the steady tidal action of a bound companion, since the non-axisymmetric perturbation is then very similar to that of a bar (Combes 1988); these are described in section 17. Many cases could also be explained by the existence of a weak oval distortion, weak enough that the galaxy is classified SA. A very good example is the case of NGC 7217, an isolated spiral which appears even on near-infrared photographs very axisymmetric, and which may be one of the most axisymmetric galaxies ever observed (Buta et al. 1995b). Yet the very faint m = 2 component discovered in the deprojected image is sufficient to explain the formation of the three rings observed in the galaxy with resonances. The nuclear, inner and outer rings of NGC 7217 are well reproduced by a simulation of gas flow in the potential derived from the I-band image (Buta et al. 1995b). A noteworthy characteristic discovered by the latter authors is a large extended circular halo of light in the outer parts of the galaxy.

Another possibility for the ringed nonbarred galaxies is that the rings were formed while the galaxy was barred, but the bar has now been dissolved through either tidal interaction with a companion, or through gas accretion and mass concentration. This has been proposed in particular for NGC 7217 (Verdes-Montenegro et al. 1995; Athanassoula 1996), since about 20-30% of the stars have been observed in counter-rotation in this galaxy by Merrifield & Kuijken (1994). The progressive destruction of the bar could also explain the counter-rotation, through the librating/rotating separatrix crossing mechanism developed by Evans & Collett (1994): in the rotating frame of the bar, some particles were trapped librating about the bar. When the bar disappears, the trapped librating orbits are scattered almost equally into clockwise and anti-clockwise rotating loop orbits. Since not all particles were trapped with the bar, and due to the preferential sense of rotation, less than 50% of the stellar population is expected to be counter-rotating in the final nonrotating frame (which could correspond to the observed 25%).

If this mechanism is attractive to explain the counter-rotating phenomena, it raises however some problems for the survival of rings. The destruction of the bar must not be due to a violent event, because that would have destroyed also the rings; for this reason, the tidal action of a companion is not likely (see section 17). If it is the gas accretion that makes the galaxy axisymmetric, it must have occurred quite rapidly, otherwise the changing pattern speed of the decaying bar would have diluted the rings. But if the axisymmetrization is rapid, the velocity dispersion produced at the separatrix crossing is too high (Evans & Collett 1994).

15.1. Bar Destruction, Lens, and Oval Formation

Recent numerical simulations, and in particular those including the gas component, have promoted the idea that bars might be only a transient phase in the life of a galaxy. Bars can be easily destroyed by massive gas flow towards the center or by a satellite merger, and also can re-form since spiral disks are cooled down and made unstable to new perturbations by fresh gas accretion. The duty cycle of the strong bar phase could be of the order of 30% of the Hubble time, explaining the observed frequency of SB galaxies. This is in contrast with the view brought along by previous N-body simulations dealing with only the stellar component, where the bar is robust and lasts for more than a Hubble time.

The influence of a central mass concentration on bar strength was first studied for its relevance to AGN's: a sufficiently massive black hole in the nucleus of a barred galaxy can weaken and destroy the bar (Norman et al. 1985; Hasan and Norman 1990; Pfenniger & Norman 1990). The black hole acts as a scattering mass that can axisymmetrize the central disk. The bar favors elongated orbits in the center, with less angular momentum, which are more sensitive to the central mass than circular orbits. For high enough central masses, stochasticity develops in the stellar orbital structure, and the bar is no longer supported by the x1 family of orbits. Through orbit computations, Hasan & Norman (1990) have determined the percentage of the remaining regular x1 orbits, as a function of the black hole mass. This percentage falls significantly when the central mass reaches 5% of the total mass: the percentage of stochastic orbits is so large that the bar is significantly weakened, then destroyed for a mass fraction of 10%. Through self-consistent stellar simulations, Nishida & Wakamatsu (1996) have shown that a bar still forms with the same amplitude in a galaxy with an initial mass concentration as high as 5%, but its strength weakens in the long run, while a bar with no central mass concentration keeps its amplitude in a steady state through a Hubble time.

In normal spiral galaxies, there do not exist massive black holes as large as 5% of the total mass, i.e. 5 x 109 Msmsun. A very concentrated nucleus or bulge (of mass Mb, and size ab) could play the role of the scattering mass, when the scattered velocity is of the order of the rotational velocity provided by the disk, i.e., v approx (GMb / ab)1/2 approx Vrot, and the ratio of scale-lengths between the disk and the central mass concentration is high enough (more than 5 typically). According to a recent compilation of observed profiles of spiral galaxies (Courteau et al. 1996), this scale-length ratio is on average larger than 10; the condition of a required (M/r) ratio comparable for the bulge and the disk, i.e., (M/r)b approx (M/r)d, translates then in terms of the surface densities µb approx 10 µd, which is often observed in early-type galaxies. Another possibility is that the strong gas infall driven by the bar produces itself a sufficient mass concentration, as suggested by simulations (e.g., Friedli & Benz 1993). The fate of these gas concentrations, once star formation has occurred, is to contribute to the nucleus or bulge component anyway.

How fast does the bar destruction occur, and can we recognize such a phenomenon in the observations? In the case of a satellite merger, the event can be very sudden, and the effective dissolution of the bar occurs in about 20 Myr (Pfenniger 1991); it might be too rare to be observed, although it could be recognized through the presence of a close companion. In the case of gas infall towards the center, it depends essentially on the amount of gas present and the rate of accretion. If the latter is high enough, the decoupling of a secondary bar might be the first step in the dissolution of the primary one (Friedli & Martinet 1993). For a milder gas accretion, the bar dissolution can be a very long phenomenon, because of self-regulation. The processes could be at the origin of lens formation, and therefore easy to recognize (cf. Combes 1996). The regulation comes from the fact that the gas accretion destroying the bar is driven by the bar itself: in a first phase, the torques due to the strong initial bar drive the gas inwards, increasing significantly the central mass concentration. The central potential is modified, and so are the frequency curves (Omega, Omega - kappa / 2), until the development of two inner Lindblad resonances, and the appearance of the family of x2 perpendicular orbits in the center. This has the effect of weakening the bar in the very center. Progressively, if the central concentration builds up, stochasticity extends in the orbital structure, and the fading of the bar reduces the torques and the gas infall. Part of the gas coming from the outer parts of galaxies has now time to form stars in the disk, which re-establishes the mass balance between the nucleus and the disk. The central mass concentration loses its dynamical efficiency, and a new strong bar phase can occur through gravitational instability, which closes the cycle. To have several such cycles in a Hubble time requires however a substantial mass accretion, i.e., important gas reservoirs in the outer parts of galaxies (cf. Pfenniger et al. 1994).

The bar destruction process affects the orbital structure in such a way as to explain lens formation (Combes 1996). The main effect is to introduce stochasticity in the middle of the bar: close to the central mass concentration, the potential becomes axisymmetric and orbits remain regular; near the end of the bar, the bar potential is dominant, and the x1 periodic orbits remain almost unperturbed. The chaotic region develops in the middle, corresponding to intermediate energies in the rotating frame. These orbits do not support the bar, and may contribute to a lens component. The regular orbits still trapped in the bar near its end might be at the origin of ``ansae'' condensations, that are sometimes observed in weak bars.

Chaotic orbits are bounded only by their limiting energy curve; in the bar rotating frame, the effective potential is Phi (r) - 1/2 Omega 2 r2, and is maximum at corotation. This plays the role of the last boundary for the chaotic region inside CR. Orbits with higher energies will eventually escape. This could be the origin of the sharp cut-off of the lens (e.g., Kormendy 1984). The observation that the lens and the bar have typically the same maximum size (the bar fills the lens along its intrinsic major axis) suggests that the lens ends also near corotation. An exception could be ESO 565-11, where the bar and lens are misaligned by 60° (Buta et al. 1995a).

Self-consistent simulations with stars and gas of the slow bar destruction by a central mass concentration illustrate the lens formation (Figure 83). In this particular case, the central mass concentration is 1% and the total gas mass is 0.5% of the total mass. A very thick oval forms through dissolution of the stellar bar, with the ``ansae''-shape. The gas bar is thinner and longer. The half-destroyed bar remains for a 2 x 109 yrs time-scale.

Figure 83. Isophotes of the results of N-body simulations with stars (left) and gas (right), starting with axisymmetric conditions, and a mass concentration of 1%. The bar stays half-dissolved for a time-scale of Gyrs. Note the ``ansae'' in the stellar distribution.
Figure 83

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