6.1 Mass Models of Spiral Galaxies

In the 1970s, the idea of dark halos around spirals was not too alarming, since the presence of DM in clusters had been strongly suspected for decades, and there were other reasons for believing spirals might have such halos (e.g. Ostriker and Peebles 1973). A consensus rapidly developed supporting the idea that spirals were surrounded by large amounts of DM. This unusual agreement amongst astronomers was offset by considerable differences of opinion concerning the details of the mass distribution around spirals. The issues are still not resolved and center on the question of how to construct mass models from observational data. It is helpful at this stage to describe some of the debatable aspects of mass decompositions since many of the conclusions summarized later in this Section are somewhat dependent on the mass models. Further details are given by Casertano and van Albada (1990) and Rubin (1991).

The traditional method of decomposing spiral galaxy rotation curves into disk and halo components has been to boost the mass-to-light ratio of the stellar disk to the highest value consistent with the inner rotation curve. The discrepancy at large radii between the velocity produced by such a ``maximum disk'' and the observed rotation curve is attributed to DM (e.g. van Albada et al. 1985; Kent 1986, 1987a, b, 1988; Sancisi and van Albada 1987).

Apparent support for the reliability of this technique came from observations of the inner regions of spirals where minor features in the rotation curve seemed to correlate with the light distribution. This was interpreted as indicating that the disk was making a significant contribution to the overall rotation in this regime (Kent 1986, 1987b; Casertano and van Albada 1990). Based on more recent observations, Freeman (DMW) also uses this phenomenon to argue in favor of the maximum disk hypothesis.

However, other considerations point in a different direction. For instance, maximum disk solutions formally require hollow halos in most galaxies. (The disk is providing all the observed rotation velocity near the galactic center, so the halo has zero mass in its inner regions.) This is clearly unphysical and more realistic models require that the disk in many galaxies has a mass somewhat below the value derived from the maximum disk hypothesis.

Another difficulty with maximum disks is that some galaxies have similar rotation curves, despite having very different light profiles (van der Kruit 1992). This behaviour is difficult to understand in the context of maximum disk models, since the rotation curve is expected to reflect predominantly the distribution of visible material.

The shape of optical rotation curves also lies at the center of recent debates. Burstein and Rubin (1985) suggested that such curves could be classified into three distinct mass types based on overall shape. These authors claimed that their mass types did not correlate with the galaxy light distribution, so that they were reflecting underlying differences in halo properties. Persic and Salucci (1991b) also argued that differences in optical rotation curves reflect changes in halo parameters from one galaxy to another.

Forbes and Whitmore (1989) found that the Burstein-Rubin mass types did show a correlation with Hubble type and luminosity, suggesting that the rotation curve differences could be produced by the gravitational effects of the disk alone. It is therefore not clear whether the Burstein and Rubin (1985) mass-types are reflecting the change in the concentration of visible material along the Hubble sequence of spiral galaxies, or whether the differences are produced by the halos. However, the evidence that halos dominate the inner dynamics of dwarf spirals (see Section 5 above) suggests that in at least some systems rotation curves predominantly reflect halo properties.

The presence of distinct spiral arms in many disk galaxies has been used to place constraints on the amount of DM in their halos. In particular, it has been argued that the number of spiral arms is related to the mass fraction in the halo. Swing-amplification theory predicts that the dominant mode in a disk surrounded by a dark halo is given by

(6.1)

(Toomre 1981), where m represents the number of spiral arms. Since most bright spirals have two dominant arms, this expression is consistent with the finding that the disk and halo have comparable masses within the optical radius of such galaxies (see Section 6.2).

Athanassoula, Bosma and Papaioannou (1987) exploited this relation to place constraints on the halos of a sample of spirals. They noted that standard disk-halo mass decompositions typically allowed a large range of disk masses from no disk at all up to the maximum disk value. Athanassoula et al. (1987) showed that by demanding the m = 2 mode was dominant, the range of permissable M/L values for the disk could be constrained. Their results further suggested that halos around early type spirals were more centrally concentrated than those around later types. One caveat to this argument is that hot disks may be able to support spiral structure even in the presence of appreciable DM halos (Bertin et al. 1989).

Flat rotation curves around spirals have usually been interpreted as indicating ``isothermal'' dark halos. Since V² M/R from the virial theorem, constant V implies M R or R^-2. Many mass models therefore assume that the dark halo can be modelled by a modified isothermal sphere with a density distribution given by

(6.2)

where V_max is the asymptotic rotation velocity due to the halo and r_c is the core radius.

Lake and Feinswog (1989) took a critical look at this procedure and reached some dramatic conclusions. Using 37 optical rotation curves and 16 HI rotation curves of spiral galaxies they examined constraints on the density profiles of the dark halos. If an isothermal halo was assumed, Lake and Feinswog (1989) found that the central density of the halo ( r_c^-2 V_max²) was often well-determined, but the individual values of r_c and V_max generally were not. More significantly, they found that density profiles that tended to R^-3 or R^-4 at large radii also gave adequate fits to the data. Similar results were obtained by Bahcall, Schmidt and Soneira (1982), who also found that density profiles were not well-constrained by rotation curves.

While this last result undermines the usual view that dark halos are isothermal, in some ways it is not surprising. In many mass decompositions the halo core radius is a significant fraction of the galactocentric distance to the edge of the rotation curve. The asymptotic behavior is therefore not well constrained by such data. Even in galaxies where the rotation curve seems to be reaching the edge of the halo, the value of r_c is still a sizeable portion of the halo radius. The full consequences of the work of Lake and Feinswog (1989) do not seem to have been absorbed, but it may be that parametric forms of the halo density distribution are not particularly meaningful.

While a good deal has been learnt about the DM around spirals, it is clearly desirable that new mass modelling techniques be devised so that the remaining uncertainties can be reduced. Casertano (DMW) is investigating methods to maximize the amount of information that can be obtained from HI observations of spiral galaxies. The idea is to model the full HI distribution to extract kinematic information, rather than the usual technique of fitting a tilted ring model to the two-dimensional velocity field (e.g. Bosma 1978). This work is still in the preliminary stage, but seems to have a good deal of potential.