The Onset of Chaos in Permanently Deformed Binaries from Spin–Orbit and Spin–Spin Coupling

Permanently deformed objects in binary systems can experience complex rotation evolution, arising from the extensively studied effect of spin–orbit coupling as well as more nuanced dynamics arising from spin–spin interactions. The ability of an object to sustain an aspheroidal shape largely determines whether or not it will exhibit nontrivial rotational behavior. In this work, we adopt a simplified model of a gravitationally interacting primary and satellite pair, where each body's quadrupole moment is approximated by two diametrically opposed point masses. After calculating the net gravitational torque on the satellite from the primary, as well as the associated equations of motion, we employ a Hamiltonian formalism that allows for a perturbative treatment of the spin–orbit and retrograde and prograde spin–spin coupling states. By analyzing the resonances individually and collectively, we determine the criteria for resonance overlap and the onset of chaos, as a function of orbital and geometric properties of the binary. We extend the 2D planar geometry to calculate the obliquity evolution. This calculation indicates that satellites in spin–spin resonances undergo precession when inclined out of the plane, but they do not tumble. We apply our resonance overlap criteria to the contact binary system (216) Kleopatra, and find that its satellites, Cleoselene and Alexhelios, may plausibly be exhibiting chaotic rotational dynamics from the overlap of the spin–orbit and retrograde spin–spin resonances. While this model is, by construction, generalizable to any binary system, it will be particularly useful for studies of small bodies in the Solar System, whose irregular shapes make them ideal candidates for exotic rotational states.