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Recent observations have been discovering new ultra-faint dwarf galaxies assmall as $\sim20~{\rm pc}$ in half-light radius and $\sim3~{\rm km~s^{-1}}$ inline-of-sight velocity dispersion. In these galaxies, dynamical friction on astar against dark matter can be significant and alter their stellar densitydistribution. The effect can strongly depend on a central density profile ofdark matter, i.e. cusp or core. In this study, I perform computations using aclassical and a modern analytic formulae and $N$-body simulations to study howdynamical friction changes a stellar density profile and how different it isbetween cuspy and cored dark matter haloes. This study shows that, if a darkmatter halo has a cusp, dynamical friction can cause shrivelling instabilitywhich 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 cuspeven if the galaxy has the same line-of-sight velocity dispersion. In such acompact and low-mass galaxy, since the shrivelling instability by dynamicalfriction is inevitable if it has a dark matter cusp, absence of a stellar cuspimplies that the galaxy has a dark-matter core. I expect that this could beused to diagnose a dark matter density profile in these compact ultra-faintdwarf galaxies.

Emergence of a stellar cusp by a dark matter cusp in a low-mass compact ultra-faint dwarf galaxy