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It has been known for a long time that the light and hence the mass distribution in disks of spiral galaxies is not strictly axisymmetric, as for example in M101 or in NGC 1637 (Sandage 1961), where the isophotes are elongated in one half of the galaxy. Despite this, however, astronomers have largely tended to ignore this fact and to assume the disks to be axisymmetric because it is much simpler to study the dynamics of axisymmetric disks. This phenomenon was first highlighted in the pioneering paper by Baldwin, Lynden-Bell, & Sancisi (1980), where they detected an asymmetry in the spatial extent of the atomic hydrogen gas in the outer regions in the two halves of some galaxies, and gave these the apt name of `lopsided' galaxies. A galaxy is said to be lopsided if it displays a global non-axisymmetric spatial distribution of type m = 1 where m is the azimuthal wavenumber, or a cos phi distribution where phi is the azimuthal angle in the plane of the disk.

Surprisingly no further systematic work was done on this topic till mid-1990's. Since then there has been a resurgence in this field. The lopsided distribution has now been detected and studied also in the old stellar component as traced in the near-IR starting with the observations of Block et al. (1994) and Rix & Zaritsky (1995). This exciting new development of the imaging studies of spiral galaxies in the near-IR K-band (2.2 µ) was made possible by the development of the NICMOS 3 array. The dust extinction effects are negligible in the near-IR, hence these studies reveal the spatial distribution of the underlying old stellar population, which constitute the main mass component of the disk. These observations detected a non-axisymmetric m = 1 distribution of surface density of old stars in the inner/optical region of the disk. Rix & Zaritsky (1995) define A1, the fractional amplitude of the first azimuthal fourier component (m = 1) of surface brightness, to be the quantitative measure of disk lopsidedness. They find that A1 increases with radius. The average value measured between 1.5-2.5 disk scalelengths is large geq 0.1, and 30% of the galaxies studied show a higher lopsidedness (Zaritsky & Rix 1997). A similar high average value of disk lopsidedness was confirmed in a recent Fourier-analysis study of a much larger sample of 149 galaxies (Bournaud et al. 2005b).

The above analysis shows that nearly one third of the 149 galaxies exhibit 10% or more asymmetry in the amplitude of the m=1 Fourier component. Thus, lopsided distribution in the disk is a general phenomenon, and is stronger at larger radii. Hence it is important to understand the origin and dynamics of the lopsided distribution in spiral galaxies.

The lopsided distribution in the HI gas has been mapped spatially (Haynes et al. 1998), and also mapped kinematically for a few galaxies (Schoenmakers, Franx & de Zeeuw 1997, Swaters et al. 1999), and by global velocity profiles for a much larger sample (Richter & Sancisi 1994). Such an asymmetry has also been detected in dwarf galaxies (Swaters et al. 2002), and also in the star-forming regions in irregular galaxies (Heller et al. 2000). The asymmetry may affect all scales in a galaxy. While the large-scale lopsidedness is more conspicuous, the off-centering of nuclei is now often discovered at high spatial resolution. A prototype of this m = 1 nuclear distribution is the inner region of M31, where the central black hole is clearly off-centered with respect to its nuclear stellar disk (e.g., Tremaine 2001). This frequent nuclear m = 1 perturbation must play a central role in the fueling of the active galactic nucleus (AGN) in a galaxy.

The origin and the evolution of lopsidedness are not yet well-understood, though a beginning has been made to address these problems theoretically. Like any other non-axisymmetric perturbation, the lopsided distribution would also tend to get wound up by the differential rotation in the galactic disk within a few dynamical timescales. Since a large fraction of galaxies exhibit lopsidedness, it must be either a long-lived phenomenon or generated frequently. Tidal interaction (Beale & Davies 1969), and satellite galaxy accretion (Zaritsky & Rix 1997) have been suggested as the origin of the disk lopsidedness, these can occur frequently. Weinberg (1995) has shown that the tidal interaction between the Galaxy and the Large Magellanic Cloud (LMC) leads to a lopsided distortion of the Galaxy halo at resonance points between the LMC and the halo orbit frequencies, which in turn causes a lopsided distribution in the disk of the Galaxy. Since galaxy interactions are now known to be common, the origin of disk lopsidedness as attributed to the disk response to the tidal distortion in a halo has been proposed and studied by Jog (1997, 2002), and Schoenmakers et al. (1997). Some other possible mechanisms that have been suggested involve an off-center disk in a halo as in a dwarf galaxy (Levine & Sparke 1998), or gas accretion (Bournaud et al. 2005b), or treating it as a global, long-lived mode (Saha, Combes & Jog 2007).

The m = 1 distribution in the inner regions of some galaxies such as M31 has been modeled through analytical work and numerical simulations (Tremaine 1995, Statler 2001, Bacon et al. 2001, de Oliveira & Combes 2008). According to the various physical conditions in galaxy nuclei (such as the mass of the bulge, the mass of the nuclear disk and that of the central black hole, the presence of gas, etc.), an m = 1 mode is unstable, or an m = 1 excitation is very slowly damped and can persist for several hundreds of dynamical times. Such long-lived lopsided distribution is also seen in the centers of advanced mergers of galaxies (Jog & Maybhate 2006). Due to their persistence, the lopsided modes could play a significant role in the evolution of the central regions of galaxies, especially in the fueling of a central AGN.

An m = 1 perturbation in a disk leads to a shift in the center of mass in the disk, and this then acts as an indirect force on the original center of the disk. The disk is inherently supportive of an m = 1 mode, which is a particular feature only of a lopsided mode. This results in long-lasting global lopsided modes. While the m = 2 case corresponding to the two-armed spiral pattern or bars has been studied extensively, the m = 1 mode has not received comparable attention in the literature so far. This has to be redressed: first, m = 1 is common and the amplitude is even larger than for m = 2 (Jarrett et al. 2003), second, the m = 1 modes do not have an Inner Lindblad Resonance, or ILR (e.g., Block et al. 1994), and hence can allow transport of matter in the inner regions, and third, these appear to be long-lived.

The existence of long-lived lopsided modes is expected to have a significant impact on the dynamics of the galaxy, the star formation in it, and on the nuclear fueling etc. In the tidal picture, the disk lopsidedness can be used as a diagnostic to study the lopsidedness of the dark matter halo (Jog 1999). Similarly, higher-order (m = 2) disk asymmetry can allow one to study the ellipticity of the dark matter halo (Jog 2000, Andersen & Bershady 2002). Thus in addition to being interesting and challenging in itself, a study of disk lopsidedness can yield information about a number of other interesting properties of galaxies. Extra-planar gas and lopsidedness are frequently correlated (Sancisi et al 2008).

Recently, the lopsided distribution in galaxies in different environments such as groups and clusters and centers of mergers has been studied. These show different properties, such as the higher observed lopsided amplitudes in the group galaxies (Angiras et al. 2006). In future studies, these can act as important tracers of the dynamics of disks and dark matter halos in these settings.

In Section 2, we discuss the observational properties of lopsidedness as seen in HI and old stars, in the main disk as well as in the central region of galaxies. The various theoretical models proposed in the literature and their comparison is given in Section 3. The dynamics of lopsidedness in the central region of the near-Keplerian case region is discussed in Section 4. Section 5 discusses the observations and dynamical implications of lopsidedness in galaxies in a different setting, namely in groups, clusters and in centers of mergers. Section 6 discusses several related points including the comparison between m = 1 and 2 cases, the deduction of the halo asymmetry etc. Section 7 gives the effect of lopsidedness on the galaxy. Finally, Section 8 gives a brief summary and future directions for this field.

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