Effect of Orbital Characteristic of Inclined Third-body on Motion of Secondary-body for a Hierarchical Triple Systems

The influence of a third-body's orbital elements on the second-body's motion in a hierarchical triple system is a crucial problem in astrophysics. Most prolonged evaluation studies have focused on a distant zero-inclined third-body. This study presents a new perspective on second-body motion equations that addresses a perturbing-body in an elliptic orbit derived with consideration of the axial-tilt (obliquity) of the primary. The proposed model is compared by the dual-averaged method and the N-body problem algorithm. After validation, a generalized three-body model is derived to investigate the effects of the third-body's orbital elements on secondary-body motion behavior. The proposed model considers short-time oscillations that affect secular evaluation and applies to exoplanets with all the primary and third body eccentricities, inclinations, and mass ratios. It is shown that the obliquity of the primary (or third-body's inclination) must be considered for precise long-term assessment, even in highly-hierarchical systems.