Analytical solutions for weak black hole kicks
  The black hole is modeled by a combined gravitational potential of the bulge,
disk and halo and is subjected to an initial weak kick. The resulting
differential equations are set up, and shown to possess analytical solutions.
The effects of black hole accretion and dynamical friction are also
incorporated into this analytical framework. The resultant frequencies and
amplitudes are computed, and are compared with the ones obtained from numerical
simulations. Within the valid range of parameters of the analytical model, the
two sets of results are shown to be in reasonable agreement. It is shown that
this model reproduces the linear dependence of the amplitude on the initial
kick velocity, and the constant of proportionality is close to that obtained
from the simulations. The analytical treatment presented is quite general, and
its applications to other areas are also indicated.
