Ring Formation from an Oscillating Black Hole
  Massive black hole (BH) mergers can result in the merger remnant receiving a
"kick", of order 200 km s$^{-1}$ or more, which will cause the remnant to
oscillate about the galaxy centre. Here we analyze the case where the BH
oscillates through the galaxy centre perpendicular or parallel to the plane of
the galaxy for a model galaxy consisting of an exponential disk, a Plummer
model bulge, and an isothermal dark matter halo. For the perpendicular motion
we find that there is a strong resonant forcing of the disk radial motion near
but somewhat less than the "resonant radii" $r_R$ where the BH oscillation
frequency is equal one-half, one-fourth, (1/6, etc.) of the radial epicyclic
frequency in the plane of the disk. Near the resonant radii there can be a
strong enhancement of the radial flow and disk density which can lead to shock
formation. In turn the shock may trigger the formation of a ring of stars near
$r_R$. As an example, for a BH mass of $10^8 M_\odot$ and a kick velocity of
150 km s$^{-1}$, we find that the resonant radii lie between 0.2 and 1 kpc. For
BH motion parallel to the plane of the galaxy we find that the BH leaves behind
it a supersonic wake where star formation may be triggered. The shape of the
wake is calculated as well as the slow-down time of the BH.
  The differential rotation of the disk stretches the wake into ring-like
segments.
