  Recent observations have been discovering new ultra-faint dwarf galaxies as
small as $\sim20~{\rm pc}$ in half-light radius and $\sim3~{\rm km~s^{-1}}$ in
line-of-sight velocity dispersion. In these galaxies, dynamical friction on a
star against dark matter can be significant and alter their stellar density
distribution. The effect can strongly depend on a central density profile of
dark matter, i.e. cusp or core. In this study, I perform computations using a
classical and a modern analytic formulae and $N$-body simulations to study how
dynamical friction changes a stellar density profile and how different it is
between cuspy and cored dark matter haloes. This study shows that, if a dark
matter halo has a cusp, dynamical friction can cause shrivelling instability
which results in emergence of a stellar cusp in the central region
$\simeq2~{\rm pc}$. On the other hand, if it has a constant-density core,
dynamical friction is significantly weaker and does not generate a stellar cusp
even if the galaxy has the same line-of-sight velocity dispersion. In such a
compact and low-mass galaxy, since the shrivelling instability by dynamical
friction is inevitable if it has a dark matter cusp, absence of a stellar cusp
implies that the galaxy has a dark-matter core. I expect that this could be
used to diagnose a dark matter density profile in these compact ultra-faint
dwarf galaxies.
