Dynamical structures of misaligned circumbinary planets under hierarchical three-body systems

All circumbinary planets (CBPs) currently detected are located in almost co-planar configurations with respect to the binary orbit, due to the fact that CBPs with higher misalignment are more difficult to detect. However, observations of polar circumbinary gas and debris disks in recent years and long-term orbital stability of inclined planets indicate that it is possible to form misaligned CBPs around eccentricity binaries (even polar CBPs). In this work we focus on the dynamical structures of CBPs in a wide range of parameters in order to provide a guidance for the space where the binary can host planets for a long enough time. To this end, the dynamical model is approximated as a hierarchical three-body problem, and the secular approximation is formulated up to the hexadecapolar order in semimajor axis ratio. Dynamical maps show that there are complex structures in the parameter space. A web of secular resonances is produced in the entire parameter space and it can well explain those numerical structures arising in dynamical maps. Based on perturbative treatments, an adiabatic invariant is introduced and thus dynamical structures can be explored by analysing phase portraits. It is found that (a) the quadrupole-order resonance (nodal resonance) is responsible for the distribution of V-shape region, and high-order and secondary resonances dominate those structures inside or outside V-shape region, and (b) the secondary 1:1 resonance is the culprit causing symmetry breaking of dynamical structures inside polar region.