  Tidal disruption events (TDEs) provide a clue to the properties of a central
supermassive black hole (SMBH) and an accretion disk around it, and to the
stellar density and velocity distributions in the nuclear star cluster
surrounding the SMBH. Deviations of TDE light curves from the standard
occurring at a parabolic encounter with the SMBH depends on whether the stellar
orbit is hyperbolic or eccentric (Hayasaki et al. 2018) and the penetration
factor ($\beta$, tidal disruption radius to orbital pericenter ratio). We study
the orbital parameters of bound and unbound stars being tidally disrupted by
comparison of direct $N$-body simulation data with an analytical model.
Starting from the classical steady-state Fokker-Planck model of Cohn & Kulsrud
(1978), we develop an analytical model of the number density distribution of
those stars as a function of orbital eccentricity ($e$) and $\beta$. To do so
fittings of the density and velocity distribution of the nuclear star cluster
and of the energy distribution of tidally disrupted stars are required and
obtained from $N$-body data. We confirm that most of the stars causing TDEs in
a spherical nuclear star cluster originate from the full loss-cone region of
phase space, derive analytical boundaries in eccentricity-$\beta$ space, and
find them confirmed by $N$-body data. Since our limiting eccentricities are
much smaller than critical eccentricities for full accretion or full escape of
stellar debris, we conclude that those stars are only very marginally eccentric
or hyperbolic, close to parabolic.
