The proper motion of stars in dwarf galaxies: distinguishing central density cusps from cores

We show that measuring the proper motion of similar to 2000 stars within a dwarf galaxy, with an uncertainty of 1 km s(-1) at most, can establish whether the dark matter (DM) density profile of the dwarf has a central core or cusp. We derive these limits by building mock star catalogues similar to those expected from future astrometric Theia-like missions and including celestial coordinates, radial velocity and proper motion of the stars. The density field of the DM halo of the dwarf is sampled from an extended Navarro-Frank-White (eNFW) spherical model, whereas the number density distribution of the stars is a Plummer sphere. The velocity field of the stars is set according to the Jeans equations. A Monte Carlo Markov chain algorithm applied to a sample of N greater than or similar to 2000 stars returns unbiased estimates of the eNFW DM parameters within 1 sigma per cent of the true values and with la relative uncertainties less than or similar to 20 per cent. The proper motions of the stars lift the degeneracy among the eNFW parameters which appears when the line-of-sight velocities alone are available. Our analysis demonstrates that, by estimating the log-slope of the mass density profile estimated at the half-light radius, a sample of N = 2000 stars can distinguish between a core and a cusp at more than 8 sigma. Proper motions also return unbiased estimates of the dwarf mass profile with 1 sigma uncertainties that decrease, on average, from 2.65 dex to 0.15 dex when the size of the star sample increases from N = 100 to N = 6000 stars. The measure of the proper motions can thus strongly constrain the distribution of DM in nearby dwarfs and provides fundamental contribution to understanding the nature and the properties of DM.