The special concave toric domain for the rotating Kepler problem

Abstract

The Rotating Kepler Problem (RKP) arises as a fundamental model in celestial mechanics, appearing as a limiting case of the circular restricted three-body problem. It offers a tractable yet rich framework for studying periodic orbits, energy levels, and symplectic structures. In this work, we investigate the RKP for energy values less than or equal to the critical threshold -3/2. Using the Ligon-Schaaf and Levi-Civita symplectic regularizations, we identify a bounded component of the RKP phase space. Within this setting, we construct a special concave toric domain (SCTD) tailored to the RKP, which provides a concrete geometric framework for computing embedded contact homology (ECH) capacities below the critical energy. The SCTD enables a rigorous analysis of symplectic embedding problems and energy constraints in dynamical systems. Furthermore, we introduce a combinatorial tree structure, inspired by the Stern-Brocot tree, that encodes energy data and facilitates the computation of ECH capacities on the bounded component. These results advance the understanding of symplectic embeddings in celestial mechanics and provide new tools for the study of Hamiltonian dynamics in rotating systems.