  The tidal disruption of a star by a supermassive black hole, and the
subsequent accretion of the disrupted debris by that black hole, offers a
direct means to study the inner regions of otherwise-quiescent galaxies. These
tidal disruption events (TDEs) are being discovered at an ever-increasing rate.
We present a model for the evolution of the tidally-disrupted debris from a
partial TDE, in which a stellar core survives the initial tidal encounter and
continues to exert a gravitational influence on the expanding stream of
tidally-stripped debris. We use this model to show that the asymptotic fallback
rate of material to the black hole in a partial TDE scales as $\propto
t^{-2.26\pm0.01}$, and is effectively independent of the mass of the core that
survives the encounter; we also estimate the rate at which TDEs approach this
asymptotic scaling as a function of the core mass. These findings suggest that
the late-time accretion rate onto a black hole from a TDE either declines as
$t^{-5/3}$ if the star is completely disrupted or $t^{-9/4}$ if a core is left
behind. We emphasize that previous investigations have not recovered this
result due to the assumption of a Keplerian energy-period relationship for the
debris orbits, which is no longer valid when a surviving core generates a
time-dependent, gravitational potential. This dichotomy of fallback rates has
important implications for the characteristic signatures of TDEs in the current
era of wide-field surveys.
