Instability of rotating Bose stars

Light bosonic (axion-like) dark matter may form Bose stars - clumps of nonrelativistic Bose-Einstein condensate supported by self-gravity. We study rotating Bose stars composed of condensed particles with nonzero angular momentum $l$. We analytically prove that these objects are unstable at arbitrary $l \ne 0$ if particle self-interactions are attractive or negligibly small. They decay by shedding off the particles and transporting the angular momentum to the periphery of the system until a Saturn-like configuration appears: one (or several) spin-zero Bose stars and clouds of diffuse particles orbit around the mutual center. In the case of no self-interactions we calculate the profiles and dominant instability modes of the rotating stars: numerically at $1 \leq l\leq 15$ and analytically at $l\gg 1$. Notably, their lifetimes are always comparable to the inverse binding energies; hence, these objects cannot be considered long-living. Finally, we numerically show that in models with sufficiently strong repulsive self-interactions the Bose star with $l=1$ is stable.