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The puzzle of how galaxy disks could be stable presented a major obstacle to the development of our understanding of disk dynamics for many years. Superficially reasonable models of disk galaxies were found repeatedly, both in simulations (e.g. Miller et al. 1970, Hohl 1971, Zang & Hohl 1978, Combes & Sanders 1981, Sellwood 1981, Athanassoula & Sellwood 1986, Khoperskov et al. 2007, Dubinski et al. 2009) and in linear stability analyses (Kalnajs 1978, Aoki et al. 1979, Toomre 1981, Sawamura 1988, Vauterin & Dejonghe 1996, Pichon & Cannon 1997, Korchagin et al. 2005, Polyachenko 2005, Jalali 2007) to possess vigorous, global and disruptive bar-forming instabilities. Fig. 6 illustrates the global, disruptive nature of this instability. While it is premature to claim that this problem has been completely solved, it now seems that the stability of a massive disk galaxy requires only a dense bulge-like mass component near the center, and owes little to the inner density of a dark matter halo. Galaxies lacking a central mass concentration, however, are still believed to require significant inner halo mass for global stability.

Figure 6

Figure 6. The formation of a bar in an (unpublished) N-body simulation. The three orthogonal projections show only particles in the initially exponential disk, halo particles are omitted. The disk, which started with Q = 1.2, has unit mass and unit scale length and G = 1. One orbit at R = 2 takes 15 time units, where the central attractions of the disk and halo are nearly equal.

4.1. Mechanism for the Bar Mode

Toomre (1981) provided the most important step forward by elucidating the mechanism of the bar instability (see also BT08 Section 6.3). Linear bar-forming modes are standing waves in a cavity, akin to the familiar modes of organ pipes and guitar strings. Reflections in galaxies take place at the center and at the corotation radius, except that outgoing leading spiral waves incident on the corotation circle are super-reflected into amplified ingoing trailing waves (i.e. swing amplification), while also exciting an outgoing transmitted trailing wave. The feedback loop is closed by the ingoing trailing wave reflecting off the disk center into a leading wave, which propagates outwards because the group velocity of leading waves has the opposite sign to that of the corresponding trailing wave. The amplitude of the continuous wave-train at any point in the loop rises exponentially, because the circuit includes positive feedback.

Toomre supported this explanation with linear stability studies of two disks. The Gaussian disk, which has a low central density, manifests a set of modes in which the more slowly-growing, higher "overtones" display the kind of standing wave pattern to be expected from the superposition of ingoing trailing and outgoing leading waves. (The eigenfrequencies of these modes are shown below in Fig. 9.)

The other linear stability study he presented was for the inappropriately-named "Mestel" disk, 5 whose unusual stability properties were first described by Toomre's student Zang (1976) in an unpublished thesis, and later by Evans & Read (1998). This disk has the scale-free surface density profile Sigma = Vc2 / (2pi GR), with Vc being the circular orbital speed that is independent of R. Zang had carried through a global, linear stability analysis of this disk, with random motions given by a smooth distribution function. In order to break the scale-free nature of the disk, Zang introduced a central cutout, and later an outer taper, in the active mass density, replacing the removed mass by rigid components in order that the central attraction remained unchanged at all radii. The dominant linear instabilities he derived for the tapered disk were confirmed in N-body simulations by Sellwood & Evans (2001).

Zang (1976) showed that a full-mass Mestel disk is stable to bi-symmetric modes, even if Q < 1, provided the tapers are gentle enough. Evans & Read (1998) extended this important result to other power-law disks, finding they have similar properties. Such disks are not globally stable, however, because they suffer from lop-sided instabilities (see Section 5). By halving the active disk mass, with rigid mass preserving the overall central attraction, Toomre (1981) was well able to eliminate the lop-sided mode from the Mestel disk. In fact, he claimed that when Q = 1.5, the half-mass Mestel disk was globally stable, making it the only known model of a non-uniformly rotating disk that is stable to all small amplitude perturbations.

Cutting the feedback loop really does stabilize a disk (see next section), which seems to confirm Toomre's cavity mode mechanism. Despite this, Polyachenko (2004) pointed out that the strong emphasis on phenomena at corotation places the blame for the instability on a radius well outside where the resulting bar has its peak amplitude. He therefore proposed an alternative mechanism for the bar instability based upon orbit alignment (Lynden-Bell 1979, see also Section 9.1). Even though the mechanism was originally envisaged as a slow trapping process, Polyachenko (2004) argued it may also operate on a dynamical timescale, and he devised (Polyachenko 2005) an approximate technique to compute global modes that embodied this idea. While his method should yield the same mode spectrum as other techniques, his alternative characterization of the eigenvalue problem may shed further light on the bar-forming mechanism.

4.2. Predicted Stability

The fact that small-amplitude, bi-symmetric instabilities are so easily avoided in the Mestel disk, together with his understanding of bar-forming modes in other models, led Toomre (1981) to propose that stability to bar-formation merely required the feedback cycle through the center to be cut. He clearly hoped that a dense bulge-like mass component, which would cause an ILR to exist for most reasonable pattern speeds, might alone be enough to stabilize a cool, massive disk.

Unfortunately, this prediction appeared to be contradicted almost immediately by the findings of Efstathiou et al. (1982), whose N-body simulations formed similar bars on short timescales irrespective of the density of the central bulge component! They seemed to confirm previous conclusions (Ostriker & Peebles 1973) that only significantly sub-maximal disks can avoid disruptive bar-forming instabilities.

However, Sellwood (1989a) found that Toomre's prediction does not apply to noisy simulations because of non-linear orbit trapping. Simulations in which shot noise was suppressed by quiet start techniques did indeed manifest the tendency towards global stability as the bulge was made more dense, as Toomre's linear theory predicted. The collective response to shot noise from randomly distributed particles was of large enough amplitude for non-linearities to be important, and a noisy simulation of a linearly stable disk quickly formed a strong bar, consistent with the results reported by Efstathiou et al. (1982).

Since density variations in the distribution of randomly distributed particles are responsible for bar formation in this regime, the reduced shot noise level from larger numbers of particles must result in lower amplitude responses that ultimately should avoid non-linear trapping. The precise particle number required depends on the responsiveness of the disk, which is weakened by random motion, lower surface density, and by increased disk thickness or gravity softening. Efstathiou et al. (1982) employed merely 20000 particles, which was clearly inadequate to capture the linear behavior. However, robustly stable, massive disks have been simulated by Sellwood & Moore (1999) and Sellwood & Evans (2001) that employed only slightly larger particle numbers. Note that these latter models also benefitted from more careful set-up procedures to create the initial equilibrium.

Thus the stabilizing mechanism proposed by Toomre (1981) does indeed work in simulations of high enough quality, and presumably therefore also in real galaxy disks. Indeed, Barazza et al. (2008) found a decreased incidence of bars in galaxies having luminous bulges, and argued that their result supports Toomre's stabilizing mechanism.

Thus the absence of bars in a significant fraction of high-mass disk galaxies does not imply that the disk is sub-maximal. The old stability criteria proposed by Ostriker & Peebles (1973) and Efstathiou et al. (1982) apply only to disks that lack dense centers; indeed Evans & Read (1998) showed explicitly that the power-law disks are clear counter-examples to the simple bi-symmetric stability criterion proposed by Ostriker & Peebles (1973).

4.3. Residual Concerns

While all this represents real progress, a few puzzles remain. The most insistent is the absence of large, strong bars in galaxies like M33, which has a gently rising rotation curve. Although many spiral arms can be counted in blue images (Sandage & Humphreys 1980), the near IR view (Block et al. 2004) manifests an open two-arm spiral, suggesting that the disk cannot be far from maximal, and also reveals a mild bar near the center of the galaxy. Corbelli & Walterbos (2008) also found kinematic evidence for a weak bar. Perhaps the stability of this galaxy can be explained by some slightly larger halo fraction, or perhaps the weak bar has some unexpected effect, but there is no convincing study to demonstrate that this galaxy, and others like it, can support a 2-arm spiral without being disruptively unstable.

The second concern is that lop-sided instabilities appear in extended full-mass disks with flat or declining rotation curves, which is discussed next. A third concern, which is discussed in Section 9.2, is that the mechanism is unable to predict the presence or absence of a bar in a real galaxy.

5 Mestel (1963) solved the far greater challenge of a disk of finite radius that has an exactly flat rotation curve. Back.

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