  We constrain the parameters of a self-interacting massive dark matter scalar
particle in a condensate using the kinematics of the eight brightest dwarf
spheroidal satellites of the Milky Way. For the case of a repulsive
self-interaction the condensate develops a mass density profile with a
characteristic scale radius that is closely related to the fundamental
parameters of the theory. We find that the velocity dispersion of dwarf
spheroidal galaxies suggests a scale radius of the order of 1 kpc, in tension
with previous results found using the rotational curve of
low-surface-brightness and dwarf galaxies. The new value is however favored
marginally by the constraints coming from the number of relativistic species at
Big-Bang nucleosynthesis. We discuss the implications of our findings for the
particle dark matter model and argue that while a single classical coherent
state can correctly describe the dark matter in dwarf spheroidal galaxies, it
cannot play, in general, a relevant role for the description of dark matter in
bigger objects.
