11.1. Bar Formation
The bar instability was discovered in early Nbody simulations of rotating stellar disks (Miller & Prendergast 1968; Hockney & Hohl 1969). Because of these results, Kalnajs (1971, 1972) studied the stability of disks with respect to bar modes through a linear analysis, and made predictions about the eigenvalues and growth rates of the normal modes for a given density and velocity distribution. These have been verified by simulations in the linear regime (e.g. Sellwood and Athanassoula 1986).
Bars can be considered as longlived modes, made by the superposition of leading and trailing waves, i.e. forming a standing wave. As such, the bar mode can grow through swing amplification, as outlined by Toomre (1981) for spiral density waves. The amplification of waves relies on the corotation region (CR), which separates the galaxy into two regions where the waves have opposite signs of energy and angular momentum (negative inside and positive outside CR). At CR a wave will be partially reflected and transmitted; the transmitted wave will carry energy of opposite sign as the incident wave, so that the reflected wave must have an increased amplitude to ensure conservation. The corotation amplifier, coupled with a feedback cycle that reflects the waves back to CR, can explain the growth of modes. Several feedback cycles were proposed, such as the WASER (Mark 1974b) based on longtrailing waves, while the swing involves the feedback of short leading waves. In the WKB theory, waves are, however, evanescent around CR, and tunnel through a forbidden zone (Lin & Shu 1964); the exponential decrease of wave amplitude in this region kills the amplifier, and the gain of the feedback cycles proposed is of the order of unity. Actual amplitude gain over a cycle relies on another kind of amplification, a positive feedback first identified by Goldreich & LyndenBell (1965) and Julian & Toomre (1966), and detailed by Toomre (1981), with the help of numerical simulations.
The amplification is due to a conspiracy between differential rotation, epicyclic oscillation, and selfgravity. Trailing density waves propagate radially towards the center, while leading waves propagate outwards. The leading wave packet becomes more and more open while traveling, due to differential shear, until it turns into a trailing wave. During this swing from leading to trailing, particles running on their epicyclic motion closely follow the wave, and strongly interact with it. Selfgravity contributes to gather particles, and amplify their density contrast. The wave energy is amplified at the expense of the rotational energy.
The trailing waves traveling inwards can be reflected in the center, while the leading waves give rise to trailing reflected waves, and transmitted waves at corotation. The reflection in the center occurs only if a wave can travel there without being damped at the inner Lindblad resonance. The problem of a possible Landau damping of waves at the inner resonance has long suggested that bars can only develop without this resonance. The pattern speed should then be high enough to prevent the resonance. This appears to be verified in Nbody simulations in the linear regime, at the beginning of bar formation. But it does not seem to be the rule in the nonlinear regime in Nbody simulations, nor in the observations, when some hint can be gained of the bar pattern speed.
Another point of view to better understand the Nbody problem is in terms of stellar orbits, and families of periodic orbits as will be described in the next section. Periodic orbits are closed orbits in the frame rotating with the bar. The stable ones trap regular orbits around them. They are thus the skeleton of the orbital structure of the disk. Periodic orbits are the fixed points that depend essentially on the symmetry of the potential, and not on the detailed mass distribution. In the potential of a rotating bar, the main family of orbits is elongated along the bar, supporting it, and we can understand under which conditions a selfgravitating system will become barred. When the mass concentration towards the center is strong enough, the elongated orbits will be replaced by periodic orbits perpendicular to the bar, and we can predict the dissolution of the bar. This approach can help to determine the pattern speed of realistic selfconsistent bars. Such an approach has been developed by Contopoulos and collaborators (e.g. Contopoulos & Papayannopoulos 1980).
The consideration of nearresonant orbits aligned with the bar led LyndenBell (1979) to propose that bar instability could come from a kind of Jeans instability, trapping all elongated orbits and aligning their major axes. He studied the conditions under which an elongated closed orbit in the bar rotating frame will be forced to align with the bar, and therefore reinforce it, because of gravitational torques. He concluded that for this to occur, the precession rate of elongated orbits (  / 2) must increase with specific angular momentum, a condition that is fulfilled only in the central parts of galaxies where the velocity curve is rising. The pattern speed of the bar in this scenario must be lower than the peak of the  / 2 axisymmetric curve, which is not the case at the beginning of the bar instability in Nbody simulations.
The development of the instability has now been followed through a wide series of Nbody simulations (e.g. Sellwood 1981; Combes & Sanders 1981; Sellwood & Wilkinson 1993). In an initially axisymmetric stellar disk, first a transient twoarmed spiral wave develops; since it is trailing, it transfers angular momentum outwards (LyndenBell & Kalnajs 1972). The bar then forms in two steps: first a short and weak bar forms, rotating with a high pattern speed which is always higher than the maximum of the precession rate  / 2. The bar, as a wave inside its corotation, has a negative angular momentum, and is amplified through the outwards transfer provided by the spiral arms. Then, the bar slows down, with a growing intensity, trapping more and more particles in its potential well. This can be understood in the frame of density wave theory as well as in stellar orbit theory. At the beginning, the perturbation is linear; for the swing amplifier to work, there should be no inner Lindblad resonance. This is fulfilled if the bar pattern speed is well over the  / 2 curve, justifying the fast rotation at the start.
In parallel, we can consider that the bar traps more particles in extending its length. Those particles, at larger radii, have lower precession rates, and it is likely that the global equalized rate, i.e. the pattern speed, will be lowered by the adjunction of these particles. As the pattern speed decreases, the bar loses angular momentum through the spiral wave.
t = 200 Myr
 t = 400 Myr
 t = 600 Myr

t = 800 Myr
 t = 1 Gyr
 t = 1.2 Gyr

Periodic orbits are parallel to the bar only inside corotation, as we shall see below. As _{b} decreases, corotation propagates outwards, and the bar extension could be higher. Bar formation by trapping of orbits is illustrated in the Nbody simulation of Figure 57.