  This work is the first in a series of studies aimed at understanding the
dynamics of highly eccentric binary neutron stars, and constructing an
appropriate gravitational-waveform model for detection. Such binaries are
possible sources for ground-based gravitational wave detectors, and are
expected to form through dynamical scattering and multi-body interactions in
globular clusters and galactic nuclei. In contrast to black holes, oscillations
of neutron stars are generically excited by tidal effects after close
pericenter passage. Depending on the equation of state, this can enhance the
loss of orbital energy by up to tens of percent over that radiated away by
gravitational waves during an orbit. Under the same interaction mechanism, part
of the orbital angular momentum is also transferred to the star. We calculate
the impact of the neutron star oscillations on the orbital evolution of such
systems, and compare these results to full numerical simulations. Utilizing a
Post-Newtonian flux description we propose a preliminary model to predict the
timing of different pericenter passages. A refined version of this model
(taking into account Post-Newtonian corrections to the tidal coupling and the
oscillations of the stars) may serve as a waveform model for such highly
eccentric systems.
