  The high-inclination population in the 4:7 mean motion resonance (MMR) with
Neptune has also substantial eccentricities ($e\gtrsim0.1$), with more inclined
objects tending to occupy more eccentric orbits. For this high-order resonance,
there are two different resonant modes. The principal one is the
eccentricity-type mode, and we find that libration is permissible for orbits
with $e\ge e_c^0$, where the critical eccentricity $e_c^0$ increases as a
function of increasing inclination $i$. Correspondingly, we introduce a
limiting curve $e_c^0(i)$, which puts constraints on the $(e, i)$ distribution
of possible 4:7 resonators. We then perform numerical simulations on the
sweep-up capture and long-term stability of the 4:7 MMR, and the results show
that the simulated resonators are well-constrained by this theoretical limiting
curve. The other 4:7 resonant mode is the mixed-$(e, i)$-type, and we show that
stable resonators should exist at $i\gtrsim20^{\circ}$. We predict that the
intrinsic number of these mixed-$(e, i)$-type resonators may provide a new clue
into the Solar system's evolution, but, so far, only one real object has been
observed resonating in this mode.
