Dark matter in dwarf spheroidals - I. Models

This paper introduces a new two-parameter family of dwarf spheroidal (dSph) galaxy models. The mass distribution has a Plummer profile and falls like R−4 in projection, in agreement with the star-count data. The first free parameter controls the velocity anisotropy and the second controls the dark matter content. The dark matter distribution can be varied from one extreme of mass-follows-light through a near-isothermal halo with flat rotation curve to the other extreme of an extended dark halo with harmonic core. This family of models is explored analytically in some detail — the distribution functions, the intrinsic moments and the projected moments are all calculated. For the nearby Galactic dSphs, samples of hundreds of discrete radial velocities are becoming available. A technique is developed to extract the anisotropy and dark matter content from such data sets by maximizing the likelihood function of the sample of radial velocities. This is constructed from the distribution function and corrected for observational errors and the effects of binaries. Tests on simulated data sets show that samples of ∼1000 discrete radial velocities are ample to break the degeneracy between mass and anisotropy in the nearby dSphs. Interesting constraints can already be placed on the distribution of the dark matter with samples of ∼160 radial velocities (the size of the present-day data set for Draco). The Space Interferometry Mission or SIM allows very accurate differential astrometry at faint magnitudes. This can be used to measure the internal proper motions of stars in the nearby Galactic dSphs. Our simulations show that ∼100 proper motions are sufficient to demolish the mass-anisotropy degeneracy completely. The target stars in Draco are at magnitudes of V ∼ 19–20 and the required proper motion accuracy is 3–6 μas yr−1. The measurement of the proper motions of a sample of ∼100 stars uncontaminated with binaries will take about 400 h of SIM time, or under 2 per cent of the mission lifetime.