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 .