A critical reassessment of particle Dark Matter limits from dwarf
  satellites
  Dwarf satellite galaxies are ideal laboratories for identifying particle Dark
Matter signals. When setting limits on particle Dark Matter properties from
null searches, it becomes however crucial the level at which the Dark Matter
density profile within these systems is constrained by observations. In the
limit in which the spherical Jeans equation is assumed to be valid for a given
tracer stellar population, we study the solution of this equation having the
Dark Matter mass profile as an output rather than as a trial parametric input.
Within our new formulation, we address to what level dwarf spheroidal galaxies
feature a reliable mass estimator. We assess then possible extrapolation of the
density profiles in the inner regions and -- keeping explicit the dependence on
the orbital anisotropy profile of the tracer population -- we derive general
trends on the line-of-sight integral of the density profile squared, a quantity
commonly dubbed $J$-factor and crucial to estimate fluxes from prompt Dark
Matter pair annihilations. Taking Ursa Minor as a study case among Milky Way
satellites, we perform Bayesian inference using the available kinematical data
for this galaxy. Contrary to all previous studies, we avoid marginalization
over quantities poorly constrained by observations or by theoretical arguments.
We find minimal $J$-factors to be about 2 to 4 times smaller than commonly
quoted estimates, approximately relaxing by the same amount the limit on Dark
Matter pair annihilation cross section from gamma-ray surveys of Ursa Minor. At
the same time, if one goes back to a fixed trial parametric form for the
density, e.g. using a NFW or Burkert profile, we show that the minimal $J$ can
hardly be reduced by more than a factor of 1.5.
