Rotation curves of disk galaxies rise steeply in their inner regions and then remain roughly flat out to the last point measured. To explain these observations within the framework of Newtonian gravity, a considerable amount of unseen mass is required.
The idea that unseen matter is needed to account for the rotation velocity curves of disk galaxies was put forward by Freeman (1970):
there must be undetected matter beyond the optical extent of NGC 300; its mass must be at least of the same order as the mass of the detected galaxy.The implications of this idea made it controversial; many wondered if dark matter on galactic scales is unavoidable. Kalnajis (1983) showed that within the radii accessible to optical spectroscopy, the rotation curves of disk galaxies could be well-fit without any dark matter. Indeed, a truncated exponential disk and a de Vaucouleurs bulge can be combined to yield a rotation curve which is flat to about 10% out to four times the disk's scale length r_0 (van Albada et al. 1985). Thus neutral-hydrogen (HI) observations like Freeman's, extending well beyond the optical radii of galaxies, were required to establish the presence of dark matter.
While optical spectroscopy couldn't prove that dark halos exist, rotation curves measured using optical emission lines were important in posing the problem of dark matter in disk galaxies. Results of optical studies are reviewed by Rubin (1983). These show that most disk galaxies have rotation curves which rise at small radii and then level off. For galaxies of a given morphological type (e.g. Sc) the shape of the rotation curve shows a systematic trend with luminosity: low-luminosity galaxies show a fairly gradual rise, while in high-luminosity galaxies the rotation curve rises sharply and then levels off or even drops slightly.
Radio-synthesis velocity maps of neutral hydrogen emission provide the most convincing evidence for dark matter (e.g. Carignan & Freeman 1985). In NGC 3198 the rotation curve has been measured to 11 r_0, or about 30 kpc for H_0 = 75 km/sec/Mpc (van Albada et al. 1985). Over such radii most galactic disks are slightly warped, and NGC 3198 is no exception. The warped disk is modeled as a set of concentric circular rings with inclinations varying fairly smoothly from 72 degrees at the center to 76 degrees at the edge. The fitted circular velocity of each ring, plotted against its radius, defines the rotation curve. After rising fairly slowly to a peak value of 157 km/sec at about 3 r_0, the rotation curve does not fall significantly below 150 km/sec out to the last point measured.
The rotation curve of NGC 3198 can be modeled by combining the gravitational forces of
The disk has the usual surface-density profile,
(1) Sigma(R) = Sigma_0 exp(-R/r_0) ,where Sigma_0 is the central surface density and r_0 is the disk scale length. For such a mass distribution the circular speed in the plane of the disk is
2 2 (2) v_d (R) = 4 pi G Sigma_0 r_0 y (I_0(y)K_0(y) - I_1(y)K_1(y)) ,where y = R/(2r_0) and I_n(y) and K_n(y) are modified Bessel functions of the first and second kinds (BT87, Ch. 2.6.3(b)).
The halo is assigned the space density profile
rho_h(0) (3) rho_h(r) = --------------- , 1 + (r/a)^gammawhere r is the spherical radius, a sets the halo length scale, and -gamma is the asymptotic slope of the profile at large r. The core radius of the halo, defined by rho_h(r_core) = rho_h(0)/2^1.5 (roughly, the 3-D equivalent of the `half central surface density' definition used observationally) is
1.5 (1/gamma) (4) r_core = a (2 - 1) .Since the halo is assumed to be spherical the circular speed is
2 M_h(r) 4pi G / 2 (5) v_h (r) = G ------ = ----- | dr r rho_h(r) . r r /
The rotation curve in the plane of the disk is found by adding the disk and halo contributions in quadrature:
2 2 1/2 (6) v_c(R) = (v_d (R) + v_h (R)) .Assuming that the observed disk has a constant M/L ratio, the disk scale length is r_0 = 2.68 kpc. Good fits to the observed rotation curve can be obtained for a wide range of model parameters; even more so than in fitting luminosity profiles, the problem of modeling galactic rotation curves is severely underconstrained. The model with the maximum disk mass has a disk mass-to-light ratio of M/L_B = 3.6, consistent with a disk-type stellar population. But even this choice does not `nail down' the halo model; the slope and scale parameters may be varied in a correlated fashion over the range 1.9 < gamma < 2.9, 7 < a < 12 kpc. The halo core radius is somewhat better determined, with `subjective' 1-sigma limits of r_core = 12.5 +/- 1.5 kpc. Within the last point measured at 30kpc the total disk mass is 3 10^10 solar masses and the total halo mass is about 4 times greater. The detected neutral hydrogen amounts to 5 10^9 solar masses, or about 15% of the maximum disk mass (van Albada et al. 1985).
Rotation curves imply the presence of significant amounts of dark matter but provide little constraint on the shape of the dark matter distribution. A highly flattened and dynamically cold dark matter distribution would run afoul of the same stability problem afflicting ``bare'' disk galaxies, but few other a priori constraints are available.
Gas orbits in the disk plane provide some information about halo shapes. The gas settles onto closed, non-intersecting orbits. In an axisymmetric potential such orbits will be circular, but a non-axisymmetric halo will force the gas onto non-circular orbits. Non-axisymmetric halos may cause oval distortions in some disk galaxies (Kormendy 1982) and kinematic asymmetries in HI maps (e.g. MB87, Fig. 8-31). Asymmetries in the Milky Way's HI kinematics have been attributed to a rotating triaxial halo (Blitz & Spergel 1991). But non-axisymmetric halos will also increase the scatter in the Tully & Fisher (1977) relationship between luminosity and circular velocity; by comparing the observed and predicted scatter, Franx & de Zeeuw (1992) conclude that the potentials of typical disk galaxies can be no flatter than about 0.9:1 in the disk plane. Stronger limits are available in special cases; for example, the potential of the E/S0 galaxy IC 2006 is axisymmetric to better than 0.98:1 (Franx, van Gorkom, & de Zeeuw 1994).
The three-dimensional shape of a disk galaxy's potential can only be inferred from tracers which travel far from the plane of the disk. Polar-ring galaxies are therefore important for discussions of halo shapes; these are S0 galaxies with rings or disks of gas and stars in polar orbits. Such rings probably form via mergers or mass transfers (Schweizer, Whitmore, & Rubin 1983).
Early studies established that the central objects are fairly normal S0 galaxies seen roughly edge-on; in particular, these objects exhibit the rapid rotation characteristic of S0 disks. Observed rotation velocities in the disk plane and in the ring plane are comparable. Simple models in which the potential is halo-dominated then imply that the dark halos of these systems are nearly spherical (Whitmore, McElroy, & Schweizer 1987).
However, more sophisticated models including the gravitational potentials of the central galaxy and polar ring appear to rule out a spherical halo for the polar-ring galaxy NGC 4650A. These models fit the ring with a mass significantly greater than the observed HI mass (Sackett & Sparke 1990); presumably the stellar component of the ring accounts for the difference, though the blue colors of this population (Whitmore et al. 1987) imply a relatively low mass-to-light ratio. Best-fit halo models have minor axis ratios between 0.3:1 and 0.4:1 (Sackett et al. 1994). This is comparable to the axial ratio of the central S0; in this case the dark matter appears to be as strongly flattened as the luminosity!
Besides the apparent coincidence of shapes just mentioned, yet another coincidence demands explanation: Bahcall & Casertano (1985) argue that ``at least one parameter of a two-component mass model must be finely tuned in order to reproduce the observed flatness of... rotation curve[s]''. At small radii, the rotation velocities of disk galaxies are set by the luminous components, while at large radii they are set by the dark matter. It appears that these two components must ``conspire'' to produce rotation curves which remain flat to many times the disk scale length, r_0 (van Albada & Scancisi 1986).
Recent studies have somewhat blunted the teeth of this conspiracy. For one thing, rotation curves are not invariably flat; low-luminosity disk galaxies exhibit gently rising rotation curves, while high-luminosity disk galaxies have rotation curves which peak at about 2 r_0 and gently decline further out (Ruben et al. 1985, Casertano & van Gorkom 1991). Moreover, at least part of the proposed conspiracy may be explained by the dynamical response of a pre-existing dark halo as a disk slowly grows inside it (Blumenthal et al. 1986, Barnes 1987, Athanassoula 1988, Ryden 1991).
But the shapes of halos may provide fresh evidence for conspiracy theorists. Numerical simulations of clustering in a universe full of Cold Dark Matter (Blumenthal et al. 1984) generally yield strongly triaxial halos with minor axis ratios no flatter than about 0.4:1 (Frenk et al. 1988, Dubinski & Carlberg 1991). Halos as flat as the one attributed to NGC 4650A seem to be excluded and may even be dynamically unstable. Moreover, while the gravity of the luminous material may pull the halo in radially, it's much less effective at flattening a dynamically hot mass distribution (e.g. Barnes 1987). Highly flattened yet axisymmetric halos would be easier to explain if dark matter is baryonic and dissipative (e.g. Pfenniger, Combes, & Martinet 1994).
The evidence indicates that dark halos are many times more massive than visible galaxies, that the dark matter is much more extended than the luminous component, that the core radii of dark halos are relatively large, and that halos are somewhat but not extremely flattened (e.g. Trimble 1987, Ashman 1992). These conclusions echo those previously reached -- on the basis of generally weaker evidence -- for halos of elliptical galaxies.
Nonetheless, the above explanations for flat rotation curves would be in trouble if the gas was the only tracer providing evidence for massive halos. But in fact, there is ample evidence for massive halos from other observations (BT87, Ch. 10). In the Milky Way, maximum stellar velocities in the solar neighborhood imply that our galaxy has a potential well deeper than the visible matter alone can generate; velocities of globular clusters and satellite galaxies do not show the fall-off with radius expected in a Keplerian potential, and the tidal forces required to explain the truncations of these systems and the tearing-off of the Magellanic Stream likewise demand more mass than is seen in stars and gas. The present motion of the Milky Way and M31 toward each other implies a total mass at least ten times the luminous mass of these galaxies, and the orbits of binary galaxies seem to be inconsistent with Keplerian potentials. Finally, not only gas and stars but even light seems to respond to the gravitational field of the dark matter in galaxy halos; distortions of background galaxies may be attributed to weak gravitational lensing by halos extending to at least 100 kpc (Brainerd, Blandford, & Smail 1996).
Last modified: April 14, 1997