Generalized three-body problem and the instability of the core-halo objects in binary systems

The goal of this work is to construct a simplified model of the core-halo structures in binary systems, such as Thorne–Żytkov objects, hot Jupiters, protoplanets with large moons, red supergiants in binaries and globular clusters with central black hole. A generalized planar circular restricted three-body problem is investigated with one of the point masses, M, replaced with a spherical body of finite size. The mechanical system under consideration includes two large masses m and M, and a test body with small mass $\mu$. Mass $\mu$, initially, is placed at the geometric center of mass M, and shares its orbital motion. Only gravitational interactions are considered and the extended mass M is assumed to be rigid with rotational degrees of freedom neglected. Equations of motion are presented, and linear instability criteria are derived using quantifier elimination. The motion of the test mass $\mu$ is shown to be unstable due to the resonance between orbital and internal frequencies. In the framework of model, the central mass \mu can be ejected if resonance conditions are met during the evolution of the system. The above result is important for core-collapse supernova theory, with mass $\mu$ identified with the helium core of the exploding massive star. The instability cause off-center supernova "ignition" relative to the center of mass of the hydrogen envelope. The instability is also inevitable during the protoplanet growth, with hypothetical ejection of the rocky core from gas giants and formation of the "puffy planets" due to resonance with orbital frequency. Hypothetical central intermediate black holes of the globular clusters are also in unstable position with respect to perturbations caused by the Galaxy. As an amusing example, we note that the Earth–Moon or the Earth–Sun systems are stable in the above sense, with the test body $\mu$ being a hypothetical black hole created in the high-energy physics experiment.