Annu. Rev. Astron. Astrophys. 2001. 39: 137-174
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The difference between the matter distribution implied by the luminosity, and the distribution of mass implied by the rotation velocities, offers strong evidence that spiral galaxies are embedded in extended halos of dark matter. The physics of dark matter has been and will be one of the major issues to be studied by elementary particle physicists and astronomers.

6.1. Flat Rotation Curve in the Halo

When Rubin & Ford (1970) published the rotation curve of M31, formed from velocities of 67 HII regions, they noted that the mass continued to rise out to the last measured region, 24 kpc. They concluded "extrapolation beyond that distance is clearly a matter of taste". By 1978, Rubin et al. (1978) had learned that "rotation curves of high luminosity spiral galaxies are flat, at nuclear distances as great as 50 kpc" (H0 = 50 km s-1 Mpc-1). Flat HI rotation curves were first noticed (Roberts & Rots 1973) using a single dish telescope. However, it would be a few years before the observers and the theorists (Ostriker & Peebles 1973; Ostriker et al. 1974, Einasto et al. 1974) recognized each others' work, and collectively asserted that disk galaxies are immersed in extended dark matter halos.

Deeper and higher-resolution HI observations with synthesis telescopes reveal that for the majority of spiral galaxies, rotation curves remain flat beyond the optical disks (Bosma 1981a, b; Guhathakurta et al. 1988; van Albada et al. 1985; Begeman 1989). The Sc galaxy UGC 2885 has the largest known HI disk, with HI radius of 120 kpc for H0 = 50 km s-1 Mpc-1 (85 kpc for H0 = 70 km s-1 Mpc-1); the HI rotation curve is still flat (Roelfsema & Allen 1985).

The conclusion that a flat rotation curve is due to a massive dark halo surrounding a spiral disk requires that Newtonian gravitational theory holds over cosmological distances. Although proof of this assumption is lacking, most astronomers and physicists prefer this explanation to the alternative, that Newtonian dynamics need modification for use over great distances. For readers interested in such alternatives, see Milgrom (1983), Sanders (1996), McGaugh & de Blok (1998), de Blok & McGaugh (1998), Begeman et al. (1991), Sanders (1996), Sanders & Verheijen (1998). Non gravitational acceleration of halo gas rotation would be also an alternative, such as due to magneto-hydrodynamical force (Nelson 1998).

6.2. Massive Dark Halo

One of the best indicators of dark matter is the difference between the galaxy mass predicted by the luminosity and the mass predicted by the velocities. This difference, which also produces a radial variation of the mass-to-luminosity ratio (M / L), is a clue to the distribution of visible and dark (invisible) mass (e.g., Bosma 1981a, b; Lequeux 1983; Kent 1986, 1987; Persic & Salucci 1988, 1990; Salucci & Frenk 1989; Forbes 1992; Persic et al. 1996; Héraudeau & Simien 1997; Takamiya & Sofue 2000). Unfortunately, there is not yet a model independent procedure for determining the fraction of mass contained in the bulge, disk, and dark halo, and mass deconvolutions are rarely unique. Most current investigations assume that the visible galaxy consists of a bulge and a disk, each of constant M / L. Kent (1986, 1991, 1992) has used the "maximum-disk method" to derive averaged M / Ls in the individual components. Athanasoula et al. (1987) attempted to minimize the uncertainty between maximum and minimal disks by introducing constraints to allow for the existence of spiral structure. Even for our Galaxy, discussions of maximum or non-maximum disk persist (van der Kruit 2001).

Radial profiles of the surface-mass density (SMD) and surface luminosity can be used to calculate M / L directly. Forbes (1992) derived the radial variation in the ratio of the total mass to total luminosity involved within a radius, r, an 'integrated M / L'. Takamiya and Sofue (2000) determine the SMD directly from the rotation curves, which can be sandwiched by mass distributions calculated from rotation curves on both spherical and flat-disk assumptions by solving directly the formula presented by Binney and Tremaine (1987). A comparison of the SMD distributions with optical surface photometry shows that the radial distributions of the M / L ratio is highly variable within the optical disk and bulge, and increases rapidly beyond the disk, where the dark mass dominates.

Separation of halo mass from disk mass, whether dark or luminous, is an issue for more sophisticated observations and theoretical modeling. Weiner et al. (2000a, b) have used non-circular streaming motion to separate the two components using their theory that the streaming motion in a bar potential is sensitive to the halo mass.

6.3. The Extent of the Milky Way Halo

Interior to the Sun's orbit, the mass of the Galaxy is approx 1011 Modot. Although there is evidence that the halo rotation curve is declining beyond 17 kpc in a Keplerian fashion (Honma & Sofue 1997a), the mass distribution beyond the HI disk, e.g. at > 22 kpc, is still controversial. Interior to the distance of the Large Magellanic Cloud, the Galaxy mass may grow to 6 × 1011 Modot (Wilkinson & Evans 1999), which depends upon the assumed orbit of the Cloud. Interior to 200 kpc, the mass is at least 2 × 1012 Modot (Peebles 1995), matching masses for a set of Milky Way-like galaxies with masses inferred statistically from the velocities of their satellite galaxies (Zaritsky 1992).

A Milky Way halo which extends at least 200 kpc is getting close to the half-way distance between the Galaxy and M31, 350 kpc. And if halos are as large as those suggested by the gravitational distortion of background galaxies seen in the vicinity of foreground galaxies (Fischer et al. 2000; Hoekstra 2000), then the halo of our Galaxy may brush the equivalently large halo of M31.

6.4. Declining Rotation Curves

Few spirals exhibit a true Keplerian decline in their rotation velocities. Among peculiar rotation curves, declining rotation curves are occasionally observed, confirming the conventional belief that the mass distribution is truncated at about 1 to 3 optical radii (3-5 scale lengths) (Casertano 1983; Casertano & van Gorkom 1991; Barteldrees & Dettman 1994). Yet some galaxies exhibit Keplerian rotation curves well beyond the critical truncation radius (Honma & Sofue 1997a, b). While truncation is an important issue for those who wish to weaken the notion of a conspiracy of luminous and dark matter (Casertano & van Gorkom 1991), the issue is far from resolved.

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