The faintest known galaxies in the Universe are the dwarf spheroidals. These objects are of particular interest since they appear to be the least luminous stellar systems that contain appreciable amounts of DM. Moreover, as discussed in Section 10, they are valuable laboratories for constraining the nature of the DM.
The first hints that dwarf spheroidals were surrounded by dark halos came in the early 1980s. Faber and Lin (1983) used the large tidal radii of local dwarf spheroidals to infer the presence of DM. (Hodge and Michie (1969) had previously noted that the tidal radius of Ursa Minor was greater than expected, but dark halos were not in vogue at that time and these authors suggested that the dwarf spheroidal might be disintegrating.) From equation (3.1) it is apparent that, by assuming a mass model for the Milky Way, the mass of a given dwarf spheroidal can be obtained from its tidal radius. A comparison with the visible mass in stars then indicates whether or not DM is present. However, as noted in Section 3, obtaining reliable values for the tidal radii of these galaxies is somewhat problematic.
At about the same time, Aaronson (1983) inferred the presence of significant amounts of DM from the velocity dispersions of stars in Draco. These results were subsequently confirmed by Aaronson and Olszewski (1987) who obtained similar results for Ursa Minor. There are various methods of estimating the dynamical mass from the velocity dipsersion, all of which are essentially forms of the virial theorem. The use of velocity dispersions has subsequently proved extremely successful and has become much more practical thanks to recent advances in instrumentation.