A number of dwarf spheroidal (dSph) galaxies are known to contain a more extended, metal-poor population with a flattish velocity dispersion profile, and a more concentrated, metal-rich population with a velocity dispersion declining with radius. The two populations can be modelled with Michie–King distribution functions (DFs) in the isothermal and in the sharply truncated limits, respectively. We argue that the truncation of the metal-rich population can be traced back to the spatial distribution of the star-forming gas. Suppose δ is the exponent of the first non-constant term in the Taylor expansion of the total potential at the centre (δ= 1 for Navarro–Frenk–White or NFW haloes, δ= 2 for cored haloes). Then, we show that the ratio of the half-light radii of the populations Rδ/2h, 2/Rh, 1δ/2 must be smaller than the ratio of the line-of-sight velocity dispersions σlos, 2(Rh, 2)/σlos, 1(Rh, 1).
Specializing to the case of the Sculptor dSph, we develop a technique to fit simultaneously both populations with Michie–King DFs. This enables us to determine the mass profile of the Sculptor dSph with unprecedented accuracy in the radial range 0.2 < r < 1.2 kpc. We show that cored halo models are preferred over cusped halo models, with a likelihood ratio test rejecting NFW models at any significance level higher than 0.05 per cent. Even more worryingly, the best-fitting NFW models require concentrations with c ≲ 20, which is not in the cosmologically preferred range for dwarf galaxies. We conclude that the kinematics of multiple populations in dSphs provides a substantial new challenge for theories of galaxy formation, with the weight of available evidence strongly against dark matter cusps at the centre.