New insight of time-transformed symplectic integrator I: hybrid methods for hierarchical triples

Abstract: Accurate $N$-body simulations of multiple systems such as binaries and triples are essential for understanding the formation and evolution of interacting binaries and binary mergers, including gravitational wave sources, blue stragglers and X-ray binaries. The logarithmic time-transformed explicit symplectic integrator (LogH), also known as algorithmic regularization, is a state-of-the-art method for this purpose.However, we show that this method is accurate for isolated Kepler orbits because of its ability to trace Keplerian trajectories, but much less accurate for hierarichal triple systems. The method can lead to an unphysical secular evolution of inner eccentricity in Kozal-Lidov triples, despite a small energy error. We demonstrate that hybrid methods, which apply LogH to the inner binary and alternative methods to the outer bodies, are significantly more effective, though not symplectic. Additionally, we introduce a more efficient hybrid method, BlogH, which eliminates the need for time synchronization and is time symmetric. The method is implemented in the few-body code SDAR. We explore suitable criteria for switching between the LogH and BlogH methods for general triple systems. These hybrid methods have the potential to enhance the integration performance of hierarchial triples.