Annu. Rev. Astron. Astrophys. 2001. 39: 137-174
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There is a marked similarity of form, but not of amplitude, of disk and halo rotation curves for galaxies with different morphologies from Sa to Sc (Rubin et al. 1985). Thus the form of the gravitational potential in the disk and halo is not strongly dependent on the form of the optical luminosity distribution. Some moderate correlation is found between total luminosity and rotation velocity amplitude. Also, less luminous galaxies tend to show increasing outer rotation curve, while most massive galaxies have slightly declining rotation in the outmost part (Persic et al. 1996). On the other hand, form of central rotation curves depend on the total mass and galaxy types (Sofue et al. 1999): Massive galaxies of Sa and Sb types show a steeper rise and higher central velocities within a few hundred pc of the nucleus compared to less massive Sc galaxies and dwarfs. Dwarf galaxies generally show a gentle central rise.

7.1. Sa, Sb, Sc Galaxies

The maximum rotation velocities for Sa galaxies are higher than those of Sb and Sc galaxies with equivalent optical luminosities. Median values of Vmax decreases from 300 to 220 to 175 km s-1 for the Sa, Sb, and Sc types, respectively (Rubin et al. 1985).

Sb galaxies have rotation curves with slightly lower values of the maximum velocity than Sa (Rubin et al. 1982). The steep central rise at 100-200 pc is often associated with a velocity peak at radii r ~ 100 - 300 pc (Sofue et al. 1999a). The rotation velocity then declines to a minimum at r ~ 1 kpc, and is followed by a gradual rise to a broad maximum at r ~ 2 - 7 kpc, arising from the disk potential. The disk rotation curve has superposed amplitude fluctuations of tens of km s-1 due to spiral arms or velocity ripples. The outermost parts are usually flat, due to the massive dark halo. Some Sb galaxies show a slight outer decline, often no larger than the inner undulations (Honma & Sofue 1997a, b).

The rotation curve of the Milky Way Galaxy, a typical Sb galaxy, is shown in Fig. 2 (Clemens 1985; Blitz 1979; Brand and Blitz 1993; Fich et al. 1989; Merrifield 1992; Honma and Sofue 1995). The rotation curve of Sb galaxies, including the Milky Way, can be described as having: (a) a high-density core, including the massive black hole, which causes a non-zero velocity very close to the center; (b) a steep rise within the central 100 pc; (c) a maximum at radius of a few hundred pc, followed by a decline to a minimum at 1 to 2 kpc; then, (d) a gradual rise from to the disk maximum at 6 kpc; and (e) a nearly flat outer rotation curve.

Sc galaxies have lower maximum velocities than Sa and Sb (Rubin et al. 1980, 1985), ranging from leq 100 to ~ 200 km s-1. Massive Sc galaxies show a steep nuclear rise similar to Sb's. However, less-massive Sc galaxies have a more gentle rise. They also have a flat rotation to their outer edges. Low-surface brightness Sc galaxies have a gentle central rise with monotonically increasing rotation velocity toward the edge, similar to dwarf galaxies (Bosma et al. 1988).

7.2. Barred Galaxies

Large-scale rotation properties of SBb and SBc galaxies are generally similar to those of non-barred galaxies of Sb and Sc types. However, the study of their kinematics is more complicated than for non-barred spirals, because their gas tracers are less uniformly distributed (Bosma 1981a, 1996), and their iso-velocity contours are skewed in the direction toward the bar (Halpha, Peterson et al. 1978; HI, Sancisi et al. 1979; stellar absorption lines, Kormendy 1983). CO-line mapping and spectroscopy reveal high concentration of molecular gas in shocked lanes along a bar superposed by significant non-circular motions (Handa et al. 1990; Sakamoto et al. 1999).

Thus, barred galaxies show velocity jumps from ± ~ 30 - 40 km s-1 to geq 100 km s-1 on the leading edges of the bar, R ~ 2 - 5 kpc. Non-barred spirals can show velocity variation of about ± 10 - 20 km s-1, caused mainly by spiral arms. Compared with non-barred spirals, barred galaxies require a more complete velocity field to understand their kinematics. As discussed earlier, intensity-weighted velocities are underestimated compared to the circular velocities, which is particularly crucial for shock compressed molecular gas in the central regions (see Section 4.9).

This large velocity variation arises from the barred potential of several kpc length. Simulations of PV diagrams for edge-on barred galaxies show many tens of km s-1 fluctuations, superposed on the usual flat rotation curve (Athanassoula and Bureau 1999; Bureau and Athanassoula 1999; Weiner & Sellwood 1999). However, distinguishing the existence of a bar and quantifying it are not uniquely done from such limited edge-on information. For more quantitative results, two-dimensional velocity analyses are necessary (Wozniak & Pfenniger 1997). In these models, barred spirals contain up to 30% counterrotating stars; the orbits are almost circular and perpendicular to the bar. Pattern speeds for the bar have been determined from absorption line spectra (Buta et al. 1996; Gerssen 2000 and references therein).

Due to their kinematic complexity, barred galaxies have been observed considerably less than non-barred, even though they constitute a considerable fraction of all disk galaxies (Mulchaey & Regan 1997). However, high resolution optical observations, combined with HI and CO, have helped to stimulate the study of two-dimensional non-circular velocity fields (e.g. Wozniak & Pfenniger 1997, Hunter & Gottesman 1996; Buta et al. 1996). Gas streaming motions along the bar are an efficient way to transport gas to the nuclear regions (Sorensen et al. 1976; Schwartz 1981; Noguchi 1988; Wada & Habe 1992, 1995; Wada et al. 1998; Shlosman et al. 1990), and lead to enhanced star formation.

7.3. Low Surface Brightness Galaxies; Dwarf Galaxies

Until the last decade, observations of rotational kinematics were restricted to spirals with average or high surface brightness. Only within the past decade have low surface brightness (LSB) galaxies been found in great numbers (Schombert & Bothun 1988; Schombert et al. 1992); many are spirals. Their kinematics were first studied by de Blok et al. (1996) with HI, who found slowly rising curves which often continued rising to their last measured point. However, many of the galaxies are small in angular extent, so observations are subject to beam smearing. Recent optical rotation curves (Swaters 1999, 2001; Swaters et al. 2000; de Blok et al. 2001) reveal a steeper rise for some, but not all, of the galaxies studied previously at 21-cm. It is not now clear if LSB galaxies are as dominated by dark matter as they were previously thought to be; the mass models have considerable uncertainties.

Dwarf galaxies, galaxies of low mass, are often grouped with low surface brightness galaxies, either by design or by error. The two classes overlap in the low surface brightness/low mass region. However, some low surface brightness galaxies are large and massive; some dwarf galaxies have high surface brightness. Early observations showed dwarf galaxies to be slowly rotating, with rotation curves which rise monotonically to the last measured point (Tully et al. 1978; Carignan & Freeman 1985; Carignan & Puche 1990a, b; Carignan & Beaulieu 1989; Puche et al. 1990, 1991a, b; Lake et al. 1990; Broeils 1992). The dark matter domination of the mass of the dwarf galaxy NGC 3109 (Carignan 1985; Jobin & Carignan 1990) is confirmed by a reanalysis including Halpha Fabry-Perot data (Blais-Ouellette et al. 2000a, b). An exceptional case of a declining outer rotation curve has been found in the dwarf galaxy NGC 7793 (Carignan & Puche 1990a). Halpha velocity field observations of blue compact galaxies, with velocities less than 100 km s-1, show that rotation curves rise monotonically to the edges of the galaxies (Östlin et al. 1999).

Swaters (1999) derived rotation curves from velocity fields obtained with the Westerbork Synthesis Radio Telescope for 60 late-type dwarf galaxies of low luminosity. By an interactive analysis, he obtained rotation curves which are corrected for a large part of the beam smearing. Most of the rotation curve shapes are similar to those of more luminous spirals; at the lowest luminosities, there is more variation in shape. Dwarfs with higher central light concentrations have more steeply rising rotation curves, and a similar dependence is found for disk rotation curves of spirals (Fig. 5). For dwarf galaxies dominated by dark matter, as for LSB (and also HSB) spirals, the contributions of the stellar and dark matter components to the total mass cannot be unambiguously derived. More high quality observations and less ambiguous mass deconvolutions, perhaps more physics, will be required to settle questions concerning the dark matter fraction as a function of mass and/or luminosity.

Figure 5

Figure 5. The rotation curve slope between one and two scale lengths, plotted against the central concentration of light, Delta µR, which represents the difference between the observed central surface brightness and the extrapolated surface brightness of the exponential disk. For pure disk systems, Delta µR = 0. Filled and open circles are dwarfs (high and low accuracy), and triangles are spirals. [Courtesy of R. Swaters (1999)].

7.4. Large Magellanic Cloud

The LMC is a dwarf galaxy showing irregular optical morphology, with the enormous starforming region, 30 Dor, located significantly displaced from the optical bar and HI disk center. High-resolution HI kinematics of the Large Magellanic Cloud, Kim et al. (1998; see Westerlund 1999 review) indicate, however, a regular rotation around the kinematical center, which is displaced 1.2 kpc from the center of the optical bar as well as from the center of starforming activity (Fig. 6). The rotation curve has a steep central rise, followed by a flat rotation with a gradual rise toward the edge. This implies that the LMC has a compact bulge (but not visible on photographs), an exponential disk, and a massive halo. This dynamical bulge is 1.2 kpc away from the center of the stellar bar, and is not associated with an optical counterpart. The "dark bulge" has a large fraction of dark matter, with an anomalously high mass-to-luminosity (M / L) ratio (Sofue 1999). In contrast, the stellar bar has a smaller M / L ratio compared to that of the surrounding regions.

Figure 6

Figure 6. The HI velocity field of the LMC superposed on an Halpha image, and a position-velocity diagram across the kinematical major axis (Kim et al. 1998: Courtesy of S. Kim). The ellipse indicates the position of the optical bar. The thick line in the PV diagram traces the rotation curve, corrected for the inclination angle of 33°.

7.5. Irregular Galaxies: Interacting and Merging

Rotation curves for irregular galaxies are not straightforward. Some irregular galaxies exhibit quite normal rotation curves, such as observed for a ring galaxy NGC 660, amorphous galaxy NGC 4631 and NGC 4945 (Sofue et al. 1999a).

The interacting galaxy NGC 5194 (M51) shows a very peculiar rotation curve, which declines more rapidly than Keplerian at R ~ 8 - 12 kpc. This may be due to inclination varying with the radius, e.g. warping. Because the galaxy is viewed nearly face-on (i = 20°), a slight warp causes a large error in deriving the rotation velocity. If the galaxy's outer disk at 12 kpc has an inclination as small as i ~ 10°, such an apparently steep velocity decrease would be observed even for a flat rotation.

When galaxies gravitationally interact, they tidally distort each other, and produce the pathological specimens that had until recently defied classification. In an innovative paper, Toomre (1977; see also Toomre & Toomre 1972; Holmberg 1941) arranged eleven known distorted galaxies "in rough order of completeness of the imagined mergers" starting with the Antennae (NGC 4038/39) and ending with NGC 7252. Observers rapidly took up this challenge, and Schweizer (1982) showed that NGC 7252 is a late-stage merger, in which the central gas disks of the two original spirals still have separate identities.

There is now an extensive literature both observational and computational (Schweizer 1998 and references therein; Barnes & Hernquist 1992, 1996; Hibbard et al. 2000) that make it possible to put limits on the initial masses, the gas quantities, the time since the initial encounter, and the evolutionary history of the merger remnant. Equally remarkable, tidal tails can be used as probes of dark matter halos (Dubinski et al. 1999).

Many nearby galaxies are also products of mergers, and hence have been extensively studied. NGC 5128 (Cen A) has a long history of velocity observations, with few signs of being completely untangled yet (Halpha long slit, Graham 1979; Halpha Fabry-Perot, Bland-Hawthorn et al. 1997; CO, Phillips et al. 1987; HI, van Gorkom et al. 1990; PN, Hui et al. 1995).

The starburst dwarf galaxy NGC 3034 (M82) shows an exceptionally peculiar rotation property (Burbidge et al. 1964; Sofue et al. 1992). It has a normal steep nuclear rise and rotation velocities which have a Keplerian decline beyond the nuclear peak. This may arise from a tidal truncation of the disk and/or halo by an encounter with M81 (Sofue 1998).

The past and present history of the Milky Way and the Local Group is written in the warp of the Milky Way (Garcia-Ruiz et al. 2000), in the tidal disruption of the Sagittarius dwarf (Ibata et al. 1995), in the mean retrograde motion of the younger globular clusters (Zinn 1993), in the tidal streams in the halo (Lynden-Bell and Lynden-Bell, 1995), and in the orbits of the Magellanic Clouds and the Magellanic stream (Murai & Fujimoto 1980; Gardiner & Noguchi 1996). For an excellent discussion "Interactions and Mergers in the Local Group" with extensive references, see Schweizer (1998).

7.6. Polar Ring Galaxies

Polar ring galaxies provide an unique opportunity to probe the rotation and mass distribution perpendicular to galaxy disks, and hence the three dimensional distribution of the dark matter (Schweizer et al. 1983; Combes & Arnaboldi 1996; Sackett & Sparke 1990, et al 1994; van Driel et al. 1995). The conclusion of Schweizer et al. (1983) from emission line velocities, that the halo mass is more nearly circular than flattened, has been contested by Sackett & Sparke (1990), based on both emission and absorption line data. However, the data are limited and velocity uncertainties are large, so the conclusions are not robust.

A major surprise comes from the study of the polar ring galaxy NGC 4650A. Arnaboldi et al. (1997; see also van Gorkom et al. 1987) discovered an extended HI disk coplanar with the ring, which twists from almost edge-on to more face-on at large radii. The K-band optical features and the HI velocities can be fit simultaneously with a model in which spiral arms are present in this polar disk. Hence the polar ring is a very massive disk. This result strengthens previous suggestions that polar ring galaxies are related to spirals (Arnaboldi et al. 1995; Combes & Arnaboldi 1996).

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