  We study eccentric equatorial orbits of a test-body around a Kerr black hole
under the influence of gravitational radiation reaction. We have adopted a well
established two-step approach: assuming that the particle is moving along a
geodesic (justifiable as long as the orbital evolution is adiabatic) we
calculate numerically the fluxes of energy and angular momentum radiated to
infinity and to the black hole horizon, via the Teukolsky-Sasaki-Nakamura
formalism. We can then infer the rate of change of orbital energy and angular
momentum and thus the evolution of the orbit. The orbits are fully described by
a semi-latus rectum $p$ and an eccentricity $e$. We find that while, during the
inspiral, $e$ decreases until shortly before the orbit reaches the separatrix
of stable bound orbits (which is defined by $p_{s}(e)$), in many
astrophysically relevant cases the eccentricity will still be significant in
the last stages of the inspiral. In addition, when a critical value
$p_{crit}(e)$ is reached, the eccentricity begins to increase as a result of
continued radiation induced inspiral. The two values $p_{s}$, $p_{crit}$ (for
given $e$) move closer to each other, in coordinate terms, as the black hole
spin is increased, as they do also for fixed spin and increasing eccentricity.
Of particular interest are moderate and high eccentricity orbits around rapidly
spinning black holes, with $p(e) \approx p_{s}(e)$. We call these
``zoom-whirl'' orbits, because of their characteristic behaviour involving
several revolutions around the central body near periastron. Gravitational
waveforms produced by such orbits are calculated and shown to have a very
particular signature. Such signals may well prove of considerable astrophysical
importance for the future LISA detector.
