Annu. Rev. Astron. Astrophys. 2001. 39: 137-174
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5. DISK ROTATION CURVES

A disk rotation curve manifests the distribution of surface mass density in the disk, attaining a broad maximum at a radius of about twice the scale radius of the exponential disk. For massive Sb galaxies, the rotation maximum appears at a radius of 5 or 6 kpc, which is about twice the scale length of the disk. Beyond the maximum, the rotation curve is usually flat, merging with the flat portion due to the massive dark halo. Superposed on the smooth rotation curve are fluctuations of a few tens of km s-1 due to spiral arms or velocity ripples. For barred spirals, the fluctuations are larger, of order 50 km s-1, arising from non circular motions in the oval potential.

5.1. Statistical Properties of Rotation Curves

The overall similarity of shapes of rotation curves for spiral galaxies has led to a variety of attempts to categorize their forms, and to establish their statistical properties. Kyazumov (1984) cataloged rotation curve parameters for 116 normal S and Ir galaxies, and categorized the shapes. Rubin et al. (1985) formed families of Sa, Sb, and Sc synthetic rotation curves as a function of luminosity, from the galaxies they had observed. Casertano & van Gorkom (1991), using HI velocities, studied rotation curves as a function of luminosity.

Mathewson et al. (1992, 1996) used their massive set of Halpha rotation curves together with optical luminosity profiles for 2447 southern galaxies, to examine the Tully-Fisher (1977) relation. For a subset of 1100 optical and radio rotation curves, Persic et al. (1995, 1996) fit the curves by a formula, which is a function of total luminosity and radius, comprising both disk and halo components. Both the forms and amplitudes are functions of the luminosity, and the outer gradient of the RC is a decreasing function of luminosity. Their formula does not contain any free parameters, and they call it universal rotation curve. Courteau (1997) obtained optical long-slit rotation curves for 304 Sb-Sc northern UGC galaxies for Tully-Fisher applications, and fitted them empirically by a simple function for the purpose to calculate line widths. Roscoe (1999) has also empirically parameterized outer rotation curves by an extremely simple power law of radius.

Universal rotation curves reveal the following characteristics. Most luminous galaxies show a slightly declining rotation curves in the outer part, following a broad maximum in the disk. Intermediate galaxies have nearly flat rotation from across the disk. Less luminous galaxies have monotonically increasing rotation velocities across the optical disk. While Persic et al. conclude that the dark-to-luminous mass ratio increases with decreasing luminosity, mass deconvolutions are far from unique.

A study of 30 spirals in the Ursa Major Cluster (Verheijen 1997) showed that 1/3 of the galaxies (chosen to have kinematically unperturbed gas disks) have velocity curves which do not conform to the universal curve shape. Like humans, rotation curves have their individualities, but they share many common characteristics. These common properties are meaningful in some situations: in other circumstances their use may be misleading. It is important to apply the common properties only in appropriate situations, e.g., for outer disk and halo beyond ~ 0.5 optical radii, corresponding to several kpc for Sb and Sc galaxies. Inner rotation curves have greater individuality (Sofue et al. 1999a).

5.2. Environmental Effects in Clusters

A variety of physical mechanisms can alter the internal kinematics of spirals in clusters, just as these mechanism have altered the morphology of galaxies in clusters (Dressler 1984; Cayatte et al. 1990). Gas stripping (REF), star stripping, galaxy-galaxy encounters, and interaction with the general tidal field are all likely to occur. Early studies of optical rotation curves for galaxies in clusters (Burstein et al. 1986; Rubin et al. 1988; Whitmore et al. 1988, 1989) detected a correlation between outer rotation-velocity gradients and distances of galaxies from the cluster center. Inner cluster galaxies show shallower rotation curves than outer cluster galaxies, for distances 0.25 to 5 Mpc from cluster centers. These authors suggest that the outer galaxy mass is truncated in the cluster environment. Later studies have failed to confirm this result (Amram et al. 1992, 1993, 1996; Sperandio et al. 1995).

A study of rotation curves for 81 galaxies in Virgo (Rubin et al. 1999, Rubin & Haltiwanger 2001) shows that about half (43) have rotation curves identified as disturbed. Abnormalities include asymmetrical rotation velocities on the two sides of the major axis, falling outer rotation curves, inner velocity peculiarities, including velocities hovering near zero at small radii, and dips in mid-disk rotation velocities. Kinematic disturbance is not correlated with morphology, luminosity, Hubble type, inclination, maximum velocity, magnitude, or local galaxy density.

Virgo spirals with disturbed kinematics have a Gaussian distribution of systemic velocities which matches that of the cluster ellipticals; spirals with regular rotation show a flat distribution. Both ellipticals and kinematically disturbed spirals are apparently in the process of establishing an equilibrium population. Halpha emission extends farther in the disturbed spirals; the gravitational interactions have also enhanced star formation. The distribution on the sky and in systemic velocity suggests that kinematically disturbed galaxies are on elongated orbits which carry them into the cluster core, where galaxy-cluster and galaxy-galaxy interactions are more common and stronger. Self-consistent N-body models that explore the first pass of two gravitationally interacting disk galaxies (Barton et al. 1999) produce rotation curves with mid-region velocity dips matching those observed. Models of disk galaxies falling for the first time into the cluster mean field (Valluri 1993) show m=1 (warp) and m=2 (bar and spiral arms) perturbations.

5.3. Lopsided Position-Velocity Diagrams

There is increased interest in galaxies with kinematically lopsided HI profiles (Baldwin 1980; Sancisi 2001), which can arise from a large-scale asymmetry of HI gas distribution in the spiral disk. Of 1700 HI profiles, at least 50% show asymmetries (Richter & Sancisi 1994); recent work (Haynes et al. 1998) confirms this fraction. Because HI profiles result from an integration of the velocity and the HI distribution, resolved HI velocity fields offer more direct information on kinematic lopsidedness. From resolved HI velocity fields, Swaters et al. (1999) also estimate the disturbed fraction to be at least 50%.

As noted above, about 50% of Virgo spirals show optical major axis velocity disturbances; how this figure translates into lopsided HI profiles is presently unclear. The field spirals studied earlier by Rubin et al. (1985), chosen to be isolated and without obvious morphological peculiarities, have rotation curves which are very normal (74%; Rubin et al. 1999). Yet the sample of optical rotation curves for galaxies in the Hickson groups (Rubin et al. 1991) shows noticeably lopsided rotation curves for geq 50%. Further studies are needed to establish the frequency of lopsidedness as a function of luminosity, morphology, HI content, resolution, sensitivity, extent of the observations, and environment.

5.4. Counterrotating Disks and Other Kinematic Curiosities

Only a handful of galaxies are presently known to have counterrotating components over a large fraction of their disks (Rubin 1994b). The disk of E7/S0 NGC 4550, (Rubin et al. 1992; Keneey & Faundez 2000) contains two hospital stellar populations, one orbiting programmed, one retrograde. This discovery prompted modification of computer programs which fit only a single Gaussian to integrated absorption lines in galaxy spectra (Rix et al. 1992). In NGC 7217 (Sab), 30% of the disk stars orbit retrograde (Merrifield & Kuijen 1994). The bulge in NGC 7331 (Sbc) may (Prada et al. 1996) or may not (Mediavilla et al. 1998) counterrotate with respect to the disk. Stars in NGC 4826 (Sab; the Black Eye or Sleeping Beauty) orbit with a single sense. Gas extending from the nucleus through the broad dusty lane rotates prograde, but reverses its sense of rotation immediately beyond; radial infall motions are present where the galaxy velocities reverse (Rubin et al. 1965; Braun et al. 1994; Rubin 1994a; Walterbos et al. 1994; Rix et al. 1995; Sil'chenko 1996).

However, galaxies with extended counterrotating disks are not common. A peculiar case is the early type spiral NGC 3593, which exhibits two cold counterrotating disks (Bertola et al. 1996). Of 28 S0 galaxies examined by Kuijken (1996), none have counterrotating components accounting for more than 5% of the disk light (see also Kannappan & Fabricant 2000). Formation mechanisms for counterrotating disks can be devised (Thaker & Ryden 1998), although cases of failure are reported only anecdotally (Spergel, private communication). While such galaxies are generally assumed to be remnants of mergers, models show that generally the disk will heat up and/or be destroyed in a merger.

In an effort to circumvent the problem of disk destruction in a merger, Evans & Collett (1994) devised a mechanism for producing equal numbers of prograde and retrograde stellar disk orbits, by scattering stars off a bar in a galaxy whose potential slowly changes from triaxial to more axisymmetric. In an even more dramatic solution by Tremaine & Yu (2000; see also Heisler et al. 1982; van Albada et al. 1982) polar rings and/or counterrotating stellar disks can arise in a disk galaxy with a triaxial halo. As the pattern speed of the initially retrograde halo changes to prograde due to infalling dark matter, orbits of disk stars caught at the Binney resonance can evolve from prograde to retrograde disk orbits. If, instead, the halo rotation decays only to zero, stars with small inclinations are levitated (Sridhar & Touma 1996) into polar orbits. This model predicts that stellar orbits in a polar ring will be divided equally into two! counterrotating streams, making this a perfect observing program for a very large telescope.

As spectral observations obtain higher resolution and sensitivity, emission from weaker components is measured. VLA observations of NGC 2403 (Fraternali et al. 2000) reveal a normal rotation, plus more slowly rotating HI extensions, the "beard". HI detected in the forbidden velocity quadrants can be due to an infall of gas from an extended HI halo with slower rotation, perhaps from a galactic fountain flow (Schaap et al. 2000). Improved instrumentation permits detection of weaker features, so we glimpse the kinematic complexities which exist in minor populations of a single galaxy.

Edge-on and face-on spirals are fine laboratories for studying vertical kinematics. In the edge on NGC 891, HI extends about 5 kpc above the plane, where it rotates at about 25 km s-1, more slowly than the disk (Swaters et al. 1997). Slower rotation is also observed in the CO halo in the edge on dwarf M82 (Sofue et al. 1992). Face on galaxies like M101 and NGC 628 show often extended HI disk showing different kinematical properties from the disk (Kamphuis et al. 1991; Kamphuis & Brigg 1992).

5.5. Rotation of High Redshift Galaxies

Only recently have rotation curves been obtained for distant galaxies, using HST and large-aperture ground-based telescopes with sub-arc second seeing. We directly observe galaxy evolution by studying galaxies closer to their era of formation. Rotation velocities for moderately distant spirals, z approx 0.2 to 0.4, (Bershady 1997, et al. 1999, Simard & Prichet 1998, Kelson et al. 2000a) have already been surpassed with Keck velocities reaching z approx 1 (Vogt et al. 1993, 1996, 1997; Koo 1999), for galaxies whose diameters subtend only a few seconds of arc. The rotation properties are similar to those of nearby galaxies, with peak velocities between 100 to 200 km s-1, and flat outer disk velocities.

Regularly rotating spiral disks existed at z approx 1, when the universe was less than half of its present age. The Keck rotation velocities define a TF relation (i.e., the correlation of rotation velocity with blue magnitude) which matches to within leq 0.5 magnitudes that for nearby spirals. Spiral galaxy evolution, over the last half of the age of the universe, has not dramatically altered the TF correlation.

5.6. Rotation Velocity as a Fundamental Parameter of Galaxy Dynamics and Evolution

The maximum rotation velocity, reached at a few galactic-disk scale radii for average and larger sized spiral galaxies, is equivalent to one-half the velocity width of an integrated 21 cm velocity profile. The Tully-Fisher relation (1977; Aaronson et al. 1980; Aaronson & Mould 1986), the correlation between total velocity width and spiral absolute magnitude, represents an oblique projection of the fundamental plane of spiral galaxies, which defines a three-dimensional relation among the radius, rotation velocity, and luminosity (absolute magnitude) (Steinmetz & Navarro 1999; Koda et al. 2000a, b). The shape of a disk rotation curve manifests the mass distribution in the exponential-disk, which is the result of dissipative process of viscous accreting gas through the proto-galactic disk evolution (Lin & Pringle 1987).

As such, it emphasizes the essential role that rotation curves play in determining the principal galactic structures, and in our understanding of the formation of disk galaxies (e.g. Mo et al. 1998). For elliptical galaxies, the three parameter (half-light radius, surface brightness, and central velocity dispersion) fundamental plane relation (Bender et al. 1992; Burstein et al. 1997) is a tool for studying elliptical galaxy evolution, analogous to the spiral TF relation. Keck spectra and HST images of 53 galaxies in cluster CL1358+62 (z = 0.33) define a fundamental plane similar to that of nearby ellipticals (Kelson et al. 2000b). Ellipticals at z = 0.33 are structurally mature; data for more distant ellipticals should be available within several years.

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