Analytic calculation of dynamical friction for Plummer sphere in ultralight dark matter

The dynamical friction force acting on a spatially extended probe (globular clusters and dwarf galaxies) moving in the environment of ultralight bosonic dark matter in the state of the Bose-Einstein condensate is determined. Modeling the probe as a Plumer sphere of radius $l_p$, the radial and tangential components of the dynamic friction force are found in an analytic form, which reduce in the limit $l_p\to 0$ to the corresponding analytic expressions obtained in the literature in the case of a point probe. The dependence of the dynamical friction force on boson particle mass $m$ was analyzed and found to be non-monotonous in the interval ${10}^{-23}-{10}^{-21}$eV.