Secular dynamics of hierarchical (2+2) four-body systems
We study the secular dynamics of a four-body hierarchical system consisting of two binaries orbiting each other's center of mass. We first expand the equations of motion in powers of the ratio of the semi-major axes of the inner binary and the mutual binary, keeping terms up to fourth (or octupole) order in this ratio. These equations are expressed in terms of their orbital elements, turning them into the Lagrange planetary equations, and these are subsequently subjected to a triple-averaging procedure to determine the long-term evolution of the orbital elements. The resulting secular equations at quadrupole order with the fourth body's mass equal to zero, are responsible for the classic Kozai-Lidov oscillations of inclination and eccentricity. We numerically solve the secular equations, examining dynamics under different initial conditions. We find that four-body systems can evolve towards higher eccentricities and undergo orbital flips not possible in their three-body system counterparts.