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

Abstract

ABSTRACT We show that measuring the proper motion of ∼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 ≳ 2000 stars returns unbiased estimates of the eNFW DM parameters within $10{{\ \rm per\, cent}}$ of the true values and with 1σ relative uncertainties ≲ 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 8σ. Proper motions also return unbiased estimates of the dwarf mass profile with 1σ 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.