The motion of gas in the outskirts of galactic disks is responsive to the distribution of unseen matter in galactic halos. Warped disks and flat rotation curves together imply that halos are massive and nearly round.
There are at least two reasons why the kinematics of extended gas in disk galaxies provides important information on the gravitational potentials of these systems. First, atomic hydrogen may be detected far beyond the optical scales of galactic disks, so the gas provides the best constraints on invisible matter at large radii. Second, the gas may be assumed to be moving on closed orbits, which greatly simplifies the analysis of the observations.
The idea that unseen matter is needed to account for the rotation velocities of disk galaxies was first put forward by Freeman (1970):
If the H I rotation curve is correct, then 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 radical implications of this idea made it quite controversial; many questioned if dark matter on galactic scales is really necessary. Kalnajis (1983) showed that within the radii accessible to optical (emission line) 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 ~10% out to four times the disk's scale length r_0 (van Albada et al. 1985). Thus observations extending well beyond the optical radii of a disk galaxy are required to establish the presence of dark matter.
While optical spectroscopy can'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.
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 ~30 kpc for H_0 = 75 km/sec/Mpc (van Albada et al. 1985). Over such radii most galactic disks are slightly warped (see below), 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 ~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 dPhi (2) v_d (R) = R ---- dr 2 = 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
(4) r_core = (2^1.5 - 1)^(1/gamma) a .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 implies a disk mass-to-light ratio of M/L_B = 3.6, which is quite 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 ~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).
The same observations which allow us to establish the presence of dark halos also show that the outer regions of disk galaxies are often warped. In edge-on galaxies such as NGC 5907 such warps are seen as so-called `integral sign' contours bending symmetrically away from the plane of the disk (e.g. MB81, Fig. 8-30). In disks seen more nearly face-on warps are revealed by characteristic distortions of the iso-velocity contours (MB81, Fig. 8-27) such that the kinematic principal axes remain roughly perpendicular at all radii. Such distorted velocity maps can be modeled by treating the disk as a collection of concentric circular rings, each tilted by a small amount with respect to its neighbors (MB81, Fig. 8-28).
Warps are very common. For example, the Milky Way, M31, and M33 are all warped (e.g. Binney 1992); the warp of M33 is so strong that some lines of sight intersect the disk more than once (MB81, Fig. 8-29).
Unlike the spirals of disk galaxies, which occur in a profusion of forms, galactic warps obey some fairly simple rules (Briggs 1990):
To explain the warps observed in galactic disks, start by considering a ring which has been inclined by displacing particles in the z direction by an amount proportional to exp(i m phi), where phi is the angular coordinate in the disk plane and m = 1 is the azimuthal wavenumber. If the vertical displacements are small the particles will execute harmonic motions with vertical frequency kappa_z (see BT87, Ch. 3.2.3). The net result of the combined rotation and vertical oscillations is that the ring pattern rotates about the z axis with a pattern speed
(7) Omega_p = Omega - kappa_z/m ,where Omega is as usual the angular velocity of a circular orbit (Toomre 1983).
In a spherical potential kappa_z = Omega and so any warp pattern constructed out of non-coplanar rings will persist unchanged and unchanging. But in a flattened potential, kappa_z > Omega, so an m = 1 disturbance precesses backwards with pattern speed Omega_p = Omega - kappa_z. Unless this Omega_p is the same at all radii, the warp pattern will wind up with time, much like the winding-up of kinematic spiral density waves.
If warps are long-lived structures, affairs must be arranged so that Omega_p does not change with radius. Ways to do do this include
The longevity of warps can be plausibly explained if disk galaxies have misaligned halos (Toomre 1983, Dekel & Shlosman 1983). Such misalignments can arise if dark halos formed by collisionless gravitational clustering provide the potential wells in which visible galaxies accumulate dissipatively (White & Rees 1978), because dark and luminous components have very different dynamical histories. In a high-density universe, galaxies would be accreting matter right up to the present, and the late--arriving material is likely to prefer rather different principal planes (e.g. Binney 1990).
At large radii the halo presumably dominates the gravitational field and the tenuous outer disk should settle into the principal plane of the halo, while at small radii the gravitational attraction of adjacent rings of disk material tends to enforce a coplanar geometry. Thus a disk which is misaligned with respect to its halo should slowly precess as a unit out to a few times the disk scale length r_0, and then go smoothly but rather quickly over to the plane of the halo (e.g. Toomre 1983, Fig. 2). A numerical model which treats the disk as warped surface and the halo as a rigid potential finds a discrete warped mode which does not depend on the details of the disk edge (Sparke & Casertano 1988). This model does an impressive job of reproducing the warp of NGC 4013.
For this explanation to work several requirements must be met:
The picture of warps as products of misaligned halos might be refined in several ways:
The two major lines of evidence considered above both favor the impression that dark halos are many times more massive than the disks they contain, 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. 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 and warped disks 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 Wan 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; a recent preprint reports distortions of background galaxies attributed to weak gravitational lensing by halos extending to at least 100 kpc (Brainerd, Blandford, & Smail 1995).
Last modified: March 22, 1995