  Librational motion in celestial mechanics is generally associated with the
existence of stable resonant configurations and signified by the existence of
stable periodic solutions and oscillation of critical (resonant) angles. When
such an oscillation takes place around a value different than 0 or $\pi$, the
libration is called asymmetric. In the context of the planar circular
restricted three-body problem (CRTBP), asymmetric librations have been
identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well
as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer
one. In this paper, we study asymmetric librations in the 3-dimensional space.
We employ the computational approach of Markellos (1978) and compute families
of asymmetric periodic orbits and their stability. Stable, asymmetric periodic
orbits are surrounded in phase space by domains of initial conditions which
correspond to stable evolution and librating resonant angles. Our computations
were focused on the spatial circular restricted three-body model of the
Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results
with numerical integrations of observed TNOs, which reveal that some of them
perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs
librators, we find that their libration seems to be related with the vertically
stable planar asymmetric orbits of our model, rather than the 3-dimensional
ones found in the present study.
