Many observed triple systems in our Universe are in a hierarchical configuration: two objects orbit each other in a relatively tight inner binary while the third object is on a much wider orbit. Furthermore, the secular approximation for the evolution of hierarchical triple body systems has been proven to be very useful in many astrophysical contexts, from planetary to triple-star systems and even black holes. In this approximation the orbits may change shape and orientation, on timescales longer than the orbital periods, but the semi-major axes are constant. In early studies of hierarchical secular three-body systems (Kozai 1962; Lidov 1962), the wider orbit was set to be circular and one of the tight binary members was assumed to be a test (massless) particle. In this situation, the component of the tight orbitâ€™s angular momentum along the total angular momentum is conserved, and the lowest order of the approximation (i.e., the quadrupole approximation) is valid. I will discussed recent developments that showed that considering systems beyond the test particle approximation, or circular orbits, requires the next level of approximation for a correct representation of the physics, called the octupole-level. This leads to qualitative different behavior of the system. In this case, the angular momenta component of the tight and wide orbits along the total angular momentum is not conserved. Most interestingly, at this level of approximation, for an eccentric wide orbit, the tight orbit can reach extremely high eccentricities and undergo chaotic flips of its orientation. This behavior has important implications to the evolution of many systems, and I will present some nominal examples, such as retrograde hot Jupiters, blue stragglers and low-mass X-ray binaries.

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