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6.1. Relative strengths of lopsidedness (m = 1) and bars/spiral arms (m = 2)

Nearly all physical mechanisms, such as a tidal encounter or gas accretion, that can give rise to an m = 1 mode will also give rise to an m = 2 mode. Which of these two dominates depends on the detailed parameters of the system. For example, for a distant encounter, the m = 2 mode is stronger. A prograde encounter is likely to preferentially generate the m = 2 mode while a retrograde encounter favours the generation of a lopsided (m = 1) mode (Bournaud et al. 2005b).

It is well-known that a two-armed spiral pattern is supported as a kinematical feature over most of the galactic disk, or in the region of nearly flat rotation curve in general, since the pattern speed given by Omega - kappa / 2 is nearly constant in this case (Lindblad 1959). The density wave theory of spiral features is built around this idea (Rohlfs 1977).

The human eye/brain likes to notice bisymmetry. This is one reason why the two-armed spirals have received enormous amount of attention in the theory of galactic structure and dynamics. This is despite the fact that it has been known for a long time that a higher m value or a flocculent behaviour is more commonly observed in galaxies (see e.g., Elmegreen & Elmegreen 1982).

It should be noted that the early observational studies of spiral structure such as the Hubble atlas (Sandage 1961) used blue filter where the emission from the young stars stands out, and also the dust extinction plays a major role in the galaxy image produced. In contrast, the more recent studies using the near-IR band data, in particular in the Ks band avoid this problem and trace more accurately the underlying old stellar mass population (e.g., Block et al. 1994, Rix & Zaritsky 1995). Interestingly, in their recent study leading to the Large Galaxy Catalog, Jarrett et al. (2003) show that m = 1 is the most common mode seen in the near-IR. They argue that, as noted by Block et al. (1994), there is theoretical reason to believe that m = 1 modes should dominate the internal structure of spirals.

The question then is, why is there a prevalent notion even amongst professional astronomers that lopsidedness is unusual in a galaxy while a two-armed spiral is the norm, which is totally in contrast to the observed case. We think one reason for this wrong notion could be that the phase of lopsidedness is observed to be nearly constant (see Section 2.3). Thus the resulting isophotal contours are oval-shaped or egg-shaped, and their elongation is not striking in the inner or optical parts of a galaxy, so an untrained eye may miss it. 2 If there were a strong radial dependence of the phase, then the resulting one-armed structure as it occurs in M51 or NGC 4564 would be easy to see.

The amplitude of lopsidedness is found to increase with radius for the stars (Rix & Zaritsky 1995) and for HI gas (Angiras et al. 2006, 2007). This is the reason why even a low-sensitivity HI map reveals the lopsidedness easily - see e.g. the images of a face-on galaxy like M101 or an edge-on galaxy like NGC 891 (Baldwin et al. 1980). Thus it is not surprising that lopsidedness was first detected in the HI maps. The fact that it is now found to be equally ubiquitous in the older stars as well, is what makes its study even more important and challenging. For example, it points to a basic dynamical mechanism for the origin of lopsidedness, and not something based purely on the gas dynamical processes. An interesting feature regarding observational detection of lopsidedness, first noted by Rix & Zaritsky (1995), is that it does not get confused with the inclination angle, unlike the A2 values.

6.1.1. Observed amplitudes of m = 1 and m = 2 components

The observed amplitudes A1 and A2 of m = 1 and 2 respectively for stars are generally comparable, and both are much larger than the values for the higher m modes for the field galaxies (Rix & Zaritsky 1995, Bournaud et al. 2005b). Normally m = 2 is taken to denote a bar or a spiral arm in the inner or outer regions respectively, or it can also denote disk ellipticity. The amplitudes for m = 2 have been measured for larger samples (Laurikainen, Salo, & Rautiainen 2002, Buta et al. 2005, Bournaud et al. 2005b). The average values of A1 and A2 between 1.5 - 2.5 disk scalelengths have been measured for the 149 galaxies in the OSU catalog (see the Appendix in Bournaud et al. 2005b), and the two generally show a positive correlation (see Fig. 8 in that paper). This cannot be explained if the tidal encounters were the main generating mechanism since m = 2 spiral arms or bars are more easily triggered on direct orbits where Omega - kappa/2 is positive (e.g., Gerin et al. 1990). In contrast, a lopsided mode is more likely to be triggered or last longer in a retrograde orbit since the pattern speed of m = 1 asymmetries, Omega - kappa, is negative in a galactic disk. The simulations by Thomasson et al. (1989) shows that retrograde tidal encounters between galaxies lead to the formation of leading one-armed spiral arms, as in NGC 4622. Observationally very few galaxies show a leading arm. Pasha (1985) found that only 2 out of the 189 galaxies studied show a leading arm (NGC 3786, NGC 5426), and each of these happen to be in a pair, which agrees with the work by Thomasson et al. (1989). It is possible that the pattern speed of m = 1 arms in other galaxies is small - this has to be checked observationally. As discussed in Section 3.2.3 this can have a bearing on the mechanism for the origin of lopsidedness. For example, if the pattern speed is small, the global modes are long-lived (Saha et al. 2007) and do not need to be triggered frequently.

On the other hand, the galaxies in groups show a higher lopsidedness for the early-type galaxies, and show a comparable magnitude for all the lower modes, m = 1,2, and 3. The frequent and even concurrent tidal encounters in this setting are probably responsible for this (see Section 5.2).

The different modes could interact directly but this obvious line of research has not been followed up much. A non-linear coupling between m = 1,3 and m = 2 was proposed by Masset & Tagger (1997) for the central regions. Even though the m = 1 mode is largely seen in the outer galactic disk while the bars and the spiral structure (m = 2) are seen more in the inner parts of a galactic disk, they can still have a dynamical effect on each other. The heating due to a bar (m = 2) can suppress the further growth of a lopsided mode, as was seen in the numerical simulations for a purely exponential disk by Saha et al. (2007). Conversely, the presence of a lopsided mode can lead to a bar dissolution as has been studied by Debattista & Sambhus (2008). This topic needs more study and could shed an important light on the dynamical evolution of a disk due to various non-axisymmetric features.

6.2. Asymmetry in the dark matter halo

The study of disk asymmetry including lopsidedness has triggered many applications where the asymmetry is used as a tool to gain information about the shape and the density distribution in the dark matter halo.

The dark matter halo is generally assumed to have a spherical shape, for the sake of simplicity. This view has been challenged, and there have been studies which have used various tracers such as the polar rings (Sackett & Sparke 1990), warps (e.g. Ideta et al. 2000), gas flaring (Olling 1996, Becquaert & Combes 1997, Narayan, Saha & Jog 2005, Banerjee & Jog 2008), to deduce the shape of the dark matter halo in galaxies. A summary of topic is given in Natarajan (2002).

In the tidal picture of the origin of disk lopsidedness, the disk responds to a distorted dark matter halo. Thus we can use the observed disk asymmetry to deduce the asymmetry in the galactic plane of the dark matter halo, which is not visible directly. As discussed in detail in Section 3.2.1, the observed disk lopsidedness can be used to deduce the halo lopsidedness and indicates a few % lopsidedness for the halo in a typical galaxy. Similarly, on treating a self-consistent disk response, and using the disk ellipticity, one can deduce the ellipticity of the dark matter halo (Jog 2000). The idea of negative disk response (Jog 1999, 2000) has been applied by Bailin et al. (2007) for a more realistic radial variation of the ellipticity of the potential to show that the disk response circularizes the net potential in the central region of a triaxial halo.

The amplitude of lopsidedness is higher in the group galaxies and can therefore imply a higher distortion of the halo, ~ 10% as shown for the case of the Eridanus group galaxies (Angiras et al. 2006). The power in the various m modes is comparable (see Schoenmakers 2000, Angiras et al. 2007). The values of all three perturbation potentials derived epsilon1, epsilon2, epsilon3 are comparable (Section 5.1). This can be an important clue to the mechanism for generating lopsidedness in groups, and perhaps indicates the importance of multiple, simultaneous tidal interactions that can occur under the special conditions of a group environment.

The asymmetry in the dark matter halo of the Galaxy has been studied quantitatively as follows. The recent survey of atomic hydrogen gas in the outer Galaxy (Levine et al. 2006) has revealed a striking asymmetry in the thickness map of HI gas. The gas in the Northern part flares more with the thickness higher by a factor of ~ 2 compared to that in the South, at a galactocentric radial distance of 30 kpc. This has been modeled by Saha et al (2008), who obtain the vertical scaleheight for the galactic disk in the gravitational field of the dark matter halo by solving the vertical force equation and the Poisson equation together. This model shows that the above asymmetry is best explained by a lopsided dark matter halo, with a small elliptical distortion that is out of phase with the lopsidedness.

The centres of dark matter halos are predicted to show lopsidedness as based on the N-body simulations in the LambdaCDM cosmology (Gao & White 2006) and the size of asymmetry is larger for larger size halos as in clusters of galaxies, though these models need to be followed by direct predictions which can be checked against observations as stressed by these authors.

The Fourier harmonic technique developed mainly to study the disk asymmetry (Jog 1997, Schoenmakers et al. 1997) has now been applied to the kinematical data along the minor axis for the dwarf galaxy DDO 47 (Gentile et al. 2005). This study has shown that the velocity dispersion components are too small to arise due to a cusp. Hence the galaxy was deduced to have a genuine core-like density distribution of the dark matter halo in the central regions.

The asymmetry in the halo is expected to be long-lived because of its collisionless nature. However, a finite pattern speed can reduce this life-time to be ~ a few Gyr, or much less than the Hubble time (see the discussion in Section 3.2.3).

6.3. Comparison with warps

Spiral galaxies also show a bending of the midplane at large radii, this is known as the warp. A warp is also a global feature of type m = 1 except in the vertical direction (Binney & Tremaine 1987). Warps are extremely common and in fact all the main galaxies in the Local group, namely the Galaxy, M31, M33, LMC are warped. Warps are seen mainly in the HI gas, typically beyond a 4-5 disk scalelength radius (Briggs 1990) but in many cases are also seen in stars in a radial region somewhat inside of this (Reshetnikov & Combes 1998).

A tidal encounter in an arbitrary orientation can generate both lopsidedness as well as warps, as is known, see e.g. Weinberg (1995), and Bournaud et al. (2004). Individual galaxies often exhibit both these phenomena, and galaxies with an intermediate angle of inclination allows both to be seen easily as in NGC NGC 2841 (see Fig. 1b in the present paper). Both the features share some common properties - namely they are seen preferentially in the outer parts of a galactic disk, and their long-term maintenance against differential rotation is a problem. The disk self-gravity resists imposed perturbation in the inner parts, as denoted by the negative disk response (Section 3.2.1.c). The resulting net self-consistent disk response shows that the disk will exhibit lopsidedness only outside of ~ 2 disk scalelengths (Jog 1999, Jog 2000). A similar calculation for the perturbation along the vertical direction shows that the onset of warps in a galactic disk occurs only beyond 4-5 disk scalelengths (Saha & Jog 2006).

We note, however, that warps and lopsided distribution are physically different features. First, in a lopsided distribution the centre of mass is shifted with respect to the original centre of mass of the galaxy. On the other hand, in a standard m = 1 S-shaped warp, the mass distribution is symmetric with respect to the centre of mass and to the symmetry plane of z = 0. Second, the onset of warps is determined by a somewhat arbitrary threshold of a few degree lifting of the mid-plane away from its central value, which is set by the observational detection limits. On the other hand, a lopsided amplitude is well-determined quantitatively following the Fourier analysis - although the threshold value of what constitutes a lopsided galaxy is still fairly arbitrary (see Section 2.2).

6.4. Implications for high redshift galaxies

Lopsidedness is more likely at high redshift, since galaxy interactions are more frequent. There is multiple observational evidence of more asymmetric galaxies at high redshift (e.g. Simard et al 2002). Measuring the lopsidedness is however complex due to the clumpy/non-uniform background distribution in a galaxy (e.g. Elmegreen et al 2004), and the overall low spatial resolution.

2 On the other hand, a little training or awareness allows one to detect lopsidedness in galaxies easily, as the authors of this present article have found ! Back.

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