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

Goal of the presented research is to construct simplified model of the core-halo structures in binary systems. Examples are provided by Thorne-Zytkov objects, hot Jupiters, protoplanets with large moons, red supergiants in binaries and globular clusters with central black hole. Instability criteria due to resonance between internal and orbital frequencies in such a systems has been derived. To achieve assumed goals, generalized planar circular restricted three body problem is investigated with one of the point masses, $M$, replaced with spherical body of finite size. Mechanical system under consideration includes two large masses $m$ and $M$ and the test body with small mass $\mu$. Only gravitational interactions are considered. Equations of motion are presented, and linear instability criteria are derived using quantifier elimination. Motion of the test mass $\mu$ is shown to be unstable due to resonance between orbital and internal frequencies if $\frac{M}{d^3} < \frac{4}{3} \pi \rho < \frac{ M + 3 m \left( 1+\mu/M \right)^{-1}}{d^3}$, where $\rho$ is the central density of mass $M$, and $d$ distance between masses $m$ and $M$ (circular orbit diameter). The above result is important for core-collapse supernova theory, with mass $\mu$ identified with 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 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.