4.2 Constraining the Dark Matter Distribution

While the above considerations indicate significant amounts of DM and high DM densities in the dwarf spheroidals, there is considerable uncertainty in the quoted values. In most galaxies, these errors are dominated by uncertainties in the structural parameters of the galaxies, rather than measurement errors. The traditional method for obtaining the mass-to-light ratio of dwarf spheroidals is the ``core-fitting'' technique. This makes the assumption that the mass and light profiles are the same. For dwarf spheroidals this assumption is almost certainly wrong, but the exercise is still instructive. The central density and mass-to-light ratio are given by

(4.1)

and

(4.2)

respectively (Richstone and Tremaine 1986). Here ₀ is the central projected velocity dispersion and ₀ is the central surface brightness. The core radius r_c is defined as the radius at which the surface brightness has fallen to one half the central value. Values for (M/L)₀ and ₀ (derived from equations [4.1] and [4.2]) for the local dwarf spheroidals studied to date are given in Table 1. With the exception of the results for Sextans which come from Suntzeff et al. (1992), this table is adapted from Pryor (1992) who based the derived quantities on the observations summarized above.

Table 1.
Properties of Dwarf Spheroidal Galaxies

		R	0		0
Galaxy	MV	(kpc)	(km s-1)	(M/LV)0	(M pc-3)
Fornax	-12.4	145	10.0	5.7	0.073
Sculptor	-11.4	78	7.0	11	0.41
Carina	-9.2	92	8.8	53	0.50
Sextans	-9.1	88	6.8	30	0.07
Draco	-8.9	75	10.5	94	1.3
Ursa Minor	-8.9	69	10.5	83	1.0

An alternative to the core-fitting method is to use ₀ and r_c to obtain a total dynamical mass (Illingworth 1976), under the same assumption that mass traces light. Detailed fitting of the surface brightness profile is also required (cf. Mateo et al. 1991) and in practice use of the expression in (4.2) is preferred.

It is apparent from equations (4.1) and (4.2) that in order to derive the quantities of interest for a given dwarf spheroidal, an accurate velocity dispersion, core radius and distance must be obtained. Pryor (1992) summarizes the various measurement errors and shows how they propagate into the final determination of ₀ and (M/L)₀. In a worst-case scenario, it appears that measurement errors could inflate the derived mass-to-light ratio above its true value, but not sufficiently to remove the need for DM in the dwarf spheroidals identified as having dark halos. However, while the case for DM in the local dwarf spheroidals is compelling, the values of (M/L)₀ and ₀ are not well-determined. This is not a result of measurement uncertainties, but arises because the mass distribution of the dark halos is unknown. In other types of galaxy, the dark halos are observed to be more extended than the visible component, so it seems reasonable that this is also the case in dwarf spheroidals.

Increasing the core radius of the dark halo and making extreme assumptions about the stellar orbits can reduce the central density by as much as a factor of twenty for Draco and Ursa Minor (Pryor and Kormendy 1990; Lake 1990a). This uncertainty is large, although the Universe would have to be somewhat capricious for the central densities to be so much lower than the estimate given by equation (4.1). Even if ₀ is lower than the core-fitting value, the total amount of DM in dwarf spheroidals is not reduced so dramatically. This is because in order to reduce ₀ one has to assume that the dark halo is much more extended than the visible galaxy. The DM therefore gets pushed to larger radii rather than removed altogether.

There are also limits on the minimum central density _min that can be obtained from the data in a model-independent fashion (Merritt 1987; Pryor and Kormendy 1990). The ratio ₀ / _min, where ₀ is still given by equation (4.1), depends on the concentration (the ratio of tidal radius to core radius) of the galaxy and the quality of the data. In Ursa Minor, for instance, Pryor (1992) finds that ₀ / _min 8. While this illustrates that there is still a good deal of uncertainty in the central density, such limits do reduce the possible parameter space. They are also sufficient to show that the dark halos of dwarf spheroidals have significantly higher central densities than those around bright galaxies (see Sections 5 and 6).

A new technique for obtaining the gravitational potential of spherical systems has recently been explored by Dejonghe and Merritt (1992) and Merritt and Saha (1992). The above discussion illustrates the need for such a technique if we are to get a better idea of the form of DM halos around dwarf galaxies. Merritt and Saha (1992) show that line-of-sight velocities and projected radii provide more powerful probes of the gravitational potential than was previously realized. They further show how to compute likelihood intervals for the gravitational potential based on such a data set. This work is still somewhat preliminary, but may prove to be extremely important.