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Many galaxies have apparently lop-sided disks. The treatment here will not go into detail, since Jog & Combes (2009) have recently reviewed both the observational data and theory.

Both theoretical and simulation work on m = 1 distortions to an axisymmetric disk require special care, since the absence of rotational symmetry can lead to artifacts unless special attention is paid to linear momentum conservation. Rigid mass components present particular difficulties, since they should not be held fixed, and extensive mass components are unlikely to respond as rigid objects.

As noted above, Zang (1976) found that the dominant instability of the centrally cut-out Mestel disk was not the usual bar instability, but a lop-sided mode, which persists in a full mass disk no matter how large a degree of random motion or gentle the cutouts. This surprising finding was confirmed and extended to general power-law disks by Evans & Read (1998). A lop-sided instability dominated simulations (Sellwood 1985) of a model having some resemblance to Zang's, in that it had a dense massive bulge and no extended halo, while Saha et al. (2007) reported similar behavior in simulations of a bare exponential disk. Lovelace et al. (1999) found pervasive lop-sided instabilities near the disk center in a study of the collective modes of a set of mass rings.

Various mechanisms have been proposed to account for this instability. Baldwin et al. (1980) and Earn & Lynden-Bell (1996) explored the idea that long-lived lop-sidedness could be constructed from cooperative orbital responses of the disk stars, along the lines discussed for bars by Lynden-Bell (1979). Tremaine (2005) discussed a self-gravitating secular instability in near-Keplerian potentials. 6 A more promising mechanism is a cavity mode, similar to that for the bar-forming instability (Dury et al. 2008): the mechanism again supposes feedback to the swing-amplifier, which is still vigorous for m = 1 in a full-mass disk.

Feedback through the center cannot be prevented by an ILR for m = 1 waves, since the resonance condition Omegap = Omegaphi- OmegaR (eq. 5) is satisfied only for retrograde waves. But the lopsided mode can be stabilized by reducing the disk mass, which reduces the X parameter (eq. 12) until amplification dies for m = 1 (Toomre 1981). Sellwood & Evans (2001) showed that, together with a moderate bulge, the dark matter required for a globally stable disk need not be much more than a constant density core to the minimum halo needed for a flat outer rotation curve.

A qualitatively different lop-sided instability is driven by counter-rotation. This second kind of m = 1 instability was first reported by Zang & Hohl (1978) in a series of N-body simulations designed to explore the suppression of the bar instability by reversing the angular momenta of a fraction of the stars; they found that a lop-sided instability was aggravated as more retrograde stars were included in their attempts to subdue the bar mode. Analyses of various disk models with retrograde stars (Araki 1987, Sawamura 1988, Dury et al. 2008) have revealed that the growth rates of lop-sided instabilities increase as the fraction of retrograde stars increases. Merritt & Stiavelli (1990) and Sellwood & Valluri (1997) found lop-sided instabilities in simulations of oblate spheroids with no net rotation. Their flatter models had velocity ellipsoids with a strong tangential bias, whereas Sellwood & Merritt (1994) found that disks with half the stars retrograde, together with moderate radial motion were surprisingly stable.

Weinberg (1994) pointed out that lop-sided distortions to near spherical systems can decay very slowly, leading to a protracted period of "sloshing". This seiche mode in a halo, which decays particularly slowly in mildly concentrated spherical systems, could provoke lop-sidedness in an embedded disk (Kornreich et al. 2002, Ideta 2002).

If lop-sidedness is due to instability, then the limited work so far suggests that it would imply a near-maximum disk. But lop-sidedness in the outer parts could also be caused by tidal interactions, or simply by asymmetric disk growth, with the effects of differential rotation being mitigated perhaps by the cooperative orbital responses discussed by Earn & Lynden-Bell (1996).

6 The "sling amplification" mechanism proposed by Shu et al. (1990) applies only to gaseous accretion disks, since it relies on sound waves propagating outside the OLR. Back.

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