1.4.3 Mass Modelling Methods
Early ways of dealing with rotation curves were to take the observed form, and to ``invert'' it to derive a density law for the disk. Such a method is only valid if the mass resides indeed in the disk. If that is the case, for extended HI rotation curves, two major conclusions can be drawn : the mass-to-light ratio increases to very high numbers in the outer parts, and the ratio of HI gas mass to total mass stays roughly constant. (cf. Bosma 1978, Bosma & Van der Kruit 1979). In the early 70s it was not yet accepted that low mass stars (red dwarfs) could not account for the bulk of the mass in the disk; in the 90s the idea that dark matter is baryonic and residing in the disk resurfaced in the form of very cold gas, most of it undetectable (Pfenniger & Combes 1994). This idea was directly based on the constancy of the ratio of HI gas mass to total mass in the outer parts.
The inversion method leads to models which cannot be further analyzed. A more fruitful alternative is thus to consider a ``realistic'' mass distribution of disk and bulge, and to attribute the rotation curve discrepancy in the outer parts to an extended dark halo. Apart from specifying the bulge/disk decomposition, such a method requires a postulate concerning the mass-to-light ratio of the disk (and the bulge). An early application of this way of modelling was done by Kalnajs (1983), who demonstrated that no dark halo is necessary when the rotation curve does not extend far enough. As a rule, HI data extend at least twice as far as the ``easy visible disk'', and far enough to establish the discrepancy between the expected and the observed rotation curve, but optical (H) data usually do not extend far enough to reach this conclusion. This was further established by Kent (1986, 1987, 1988).
The question of disk stability in the presence of dark halos has been pursued vigourously. In particular, the competing influence of velocity dispersions in the disk on suppressing the bar instability has been adressed by Athanassoula & Sellwood (1986), who find that both a massive dark halo and high disk velocity dispersions slow down the developement of a bar. Nevertheless, it is the initial velocity dispersion in the disk which determines the axial ratio of the bar (Athanassoula 1983). Toomre (1981) examined the question of spiral structure, and identified a mechanism called swing amplification, which could lead to disks with strong spiral structure. The presence of a dark matter halo is to lessen the dynamical influence of the disk, and for small disk/halo ratios the amplification may be suppressed altogether. Athanassoula et al. (1987) have used this theory to try to get limits on the possible mass-to-light ratios for the disk.