Theoretical lower limits on sizes of ultra faint dwarf galaxies from
  dynamical friction
  Dwarf spheroidal galaxies are the smallest known stellar systems where under
Newtonian interpretations, a significant amount of dark matter is required to
explain observed kinematics. In fact, they are in this sense the most heavily
dark matter dominated objects known. That, plus the increasingly small sizes of
the newly discovered ultra faint dwarfs, puts these systems in the regime where
dynamical friction on individual stars starts to become relevant. We calculate
the dynamical friction timescales for pressure supported isotropic spherical
dark matter dominated stellar systems, yielding $\tau_{DF} =0.93 (r_{h}/10
pc)^{2} (\sigma/ kms^{-1}) Gyr$, { where $r_{h}$ is the half-light radius}. For
a stellar velocity dispersion value of $3 km/s$, as typical for the smallest of
the recently detected ultra faint dwarf spheroidals, dynamical friction
timescales becomes smaller than the $10 Gyr$ typical of the stellar ages for
these systems, for $r_{h}<19 pc$. Thus, this becomes a theoretical lower limit
below which dark matter dominated stellar systems become unstable to dynamical
friction. We present a comparison with structural parameters of the smallest
ultra faint dwarf spheroidals known, showing that these are already close to
the stability limit derived, any future detection of yet smaller such systems
would be inconsistent with a particle dark matter hypothesis.
