We explore the properties of test-particle orbits in "bumpy" spacetimes -
stationary, reflection-symmetric, asymptotically flat solutions of Einstein
equations that have a non-Kerr (anomalous) higher-order multipole-moment
structure but can be tuned arbitrarily close to the Kerr metric. Future
detectors should observe gravitational waves generated during inspirals of
compact objects into supermassive central bodies. If the central body deviates
from the Kerr metric, this will manifest itself in the emitted waves. Here, we
explore some of the features of orbits in non-Kerr spacetimes that might lead
to observable signatures. As a basis for this analysis, we use a family of
exact solutions proposed by Manko & Novikov which deviate from the Kerr metric
in the quadrupole and higher moments, but we also compare our results to other
work in the literature. We examine isolating integrals of the orbits and find
that the majority of geodesic orbits have an approximate fourth constant of the
motion (in addition to the energy, angular momentum and rest mass) and the
resulting orbits are tri-periodic to high precision. We also find that this
fourth integral can be lost for certain orbits in some oblately deformed
Manko-Novikov spacetimes. However, compact objects will probably not end up on
these chaotic orbits in nature. We compute the location of the innermost stable
circular orbit (ISCO) and find that the behavior of orbtis near the ISCO can be
qualitatively different depending on whether the ISCO is determined by the
onset of an instability in the radial or vertical direction. Finally, we
compute periapsis and orbital-plane precessions for nearly circular and nearly
equatorial orbits in both the strong and weak field, and discuss weak-field
precessions for eccentric equatorial orbits.Comment: 42 pages, 20 figures, accepted by Phys. Rev. D, v2 has minor changes
to make it consistent with published versio