  Fast radio bursts (FRBs) are mysterious transient phenomena. The study of
repeating FRBs may provide useful information about their nature due to their
re-detectability. The two most famous repeating sources are FRBs 121102 and
180916, with a period of 157 days and 16.35 days, respectively. Previous
studies suggest that the periodicity of FRBs is likely associated with neutron
star (NS) binary systems. Here we propose a new model that the periodic
repeating FRBs are due to the interaction of an NS with its planet in a highly
elliptical orbit. The periastron of the planet is very close to the NS so that
it would be partially disrupted by tidal force every time it passes through the
periastron. Fragments generated in the process will fall toward the NS and
finally collide with the compact star to give birth to observed FRBs. The model
can naturally explain the repeatability of FRBs with a period ranging from a
few days to several hundred days, but it generally requires that the
eccentricity of the planet orbit should be large enough. Taking FRBs 121102 and
180916 as examples, it is shown that the main features of the observed
repeating behaviors can be satisfactorily accounted for.
