Semi-analytical calculations of the secular evolution of 4-body systems

Kıroğlu, Fulya
The observations of a large fraction of Sun-like stars in multiple-star systems and planet-forming circumstellar disks’ existence around them have triggered a renewed interest in the dynamical evolution and stability of planetary systems in binaries. In this thesis, we study the secular evolution of quadruple (N = 4) systems consisting of two planets around a member of a binary star system where the Kozai-Lidov mechanism plays a role. The standard Kozai-Lidov mechanism has been studied extensively for hierarchical triple systems in the literature and has a number of applications to the systems with cylindrical symmetry, i.e., circular binary orbits. In this mechanism, the conservation of the component of the angular momentum vector of a test particle along the symmetry axis restricts its orientation in space, i.e., prograde orbits cannot become retrograde. One way to break the cylindrical symmetry and thus to avoid this restriction is to make the perturber’s orbit eccentric and to go beyond the test particle approximation, which magnify the effects of high-order (octupole) terms in the disturbing function. These generalizations have been shown to cause large eccentricity excitations as well as orbit flips (i > 90◦ ) in 3-body systems. We investigate another way of removing the axial symmetry by adding one more body to triple systems. The presence of a fourth body allows visits to the parts of phase space unavailable to triples. Depending on the initial setup of the system, the fourth body may create effects similar to that of the high-order terms in the disturbing function in the 3-body problem. We observe that the addition of a second planet on a highly inclined orbit removes the cylindrical symmetry of the companion star on a circular orbit. This in turn induces dramatic changes in the orbital eccentricity of the inner planet and even flips its orientation. On the other hand, the fourth body may suppress the high-order effects present in triples by causing periapsis precession of the inner planet’s orbit at a faster rate. The strength of the coupling of the planets’ orbits determines the evolution and the stability of 4-body systems. In our work, we observe that especially weakly-coupled two-planet systems in binaries exhibit rich features. We calculate the secular interactions in these nearly Keplerian systems semi-analytically by combining two approximation methods: the Hamiltonian perturbation theory and the Gauss method.