Excitation of planetary inclinations via dynamical bifurcations driven by misaligned disks

Primordial misalignments between protoplanetary disks and their host stars' spin axes have been proposed to be important origins of the widespread stellar obliquities observed in exoplanets. Recent works have further revealed nontrivial and rich dynamics in planetary systems driven by misaligned disks. These dynamics may have played critical roles in sculpting the architectures of exoplanetary systems and have left detectable imprints. We present a comprehensive analytical study of the dynamics driven by misaligned disks given its potential importance in explaining the dynamical evolution of exoplanetary systems at their early stages. We developed an analytical averaged model that includes the disk's full-space gravity, stellar quadrupole moment, and planetary interactions. We then investigated equilibria, their stability, and bifurcations based on different system configurations. We demonstrate that the dynamical bifurcation-induced effect—which generates large-amplitude librating mutual inclinations through separatrix-crossing behaviors at a saddle-center bifurcation—stems from the nonlinear inclination dependence of disk gravity. Crucially, the linear disk gravity model (which predicts constant nodal precession) adopted in prior studies fails to capture this effect. The introduction of an outer perturbing body in a hierarchical configuration suppresses the bifurcation-induced effect, quantified by a dimensionless parameter, ε_ ⋆ p (the stellar oblateness relative to external perturbation): as ε_ ⋆ p ̊ightarrow ∞, ψ_ ⋆0 crit ̊ightarrow 44.6^̧irc; and as ε_ ⋆ p ̊ightarrow 1, ψ_ ⋆0 crit ̊ightarrow 90^̧irc. Bifurcations are entirely inhibited when ε_ ⋆ p < 1. This mechanism also operates in compact multi-planet systems. We establish an approximate criterion to roughly distinguish their evolution patterns. Statistical analysis of Kepler multi-planet systems confirms that regimes producing coplanar multi-planet systems with high stellar obliquities are rare; this is consistent with the observed low obliquities.