  We analyze the behavior of axion dark matter before it falls into a galactic
gravitational potential well. The axions thermalize sufficiently fast by
gravitational self-interactions that almost all go to their lowest energy state
consistent with the total angular momentum acquired from tidal torquing. That
state is a state of rigid rotation on the turnaround sphere. It predicts the
occurrence and detailed properties of the caustic rings of dark matter for
which observational evidence had been found earlier. We show that the vortices
in the axion Bose-Einstein condensate (BEC) are attractive, unlike those in
superfluid $^4$He and dilute gases. We expect that a large fraction of the
vortices in the axion BEC join into a single big vortex along the rotation axis
of the galaxy. The resulting enhancement of caustic rings explains the typical
size of the rises in the Milky Way rotation curve attributed to caustic rings.
We show that baryons and ordinary cold dark matter particles are entrained by
the axion BEC and acquire the same velocity distribution. The resulting
baryonic angular momentum distribution gives a good qualitative fit to the
distributions observed in dwarf galaxies. We give estimates of the minimum
fraction of dark matter that is axions.
