More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits (1705.00527v4)

Abstract: The famous three-body problem can be traced back to Isaac Newton in 1680s. In the 300 years since this "three-body problem" was first recognized, only three families of periodic solutions had been found, until 2013 when \v{S}uvakov and Dmitra\v{s}inovi\'c [Phys. Rev. Lett. 110, 114301 (2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known Figure-eight family found by Moore in 1993, the 11 families found by \v{S}uvakov and Dmitra\v{s}inovi\'c in 2013, and more than 600 new families that have been never reported, to the best of our knowledge. With the definition of the average period $\bar{T} = T/L_f$, where $L_f$ is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law $ \bar{T}^* \approx 2.433 \pm 0.075$ for the considered case, where $\bar{T}^*= \bar{T} |E|^{3/2}$ is the scale-invariant average period and $E$ is its total kinetic and potential energy, respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere" can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm

• The paper reports the discovery of 695 families of periodic orbits in the Newtonian planar collisionless three-body system, significantly expanding the known solutions.
• Utilizing high-order Taylor series and Clean Numerical Simulation, the authors found a quasi Kepler's third law relating the scale-invariant average period to system energy.
• The findings expand the understanding of three-body dynamics, providing a robust dataset that can serve as a benchmark for numerical methods and inform astrodynamics research.

Exploration of New Families in the Three-Body Problem

The three-body problem, a classical unsolved problem in astrophysics and celestial mechanics, has intrigued researchers since the time of Newton. It involves predicting the motion of three celestial bodies interacting through gravitational forces. Due to its chaotic nature and sensitivity to initial conditions, finding periodic orbits within such a system has been challenging, leading to significant interest and research to uncover periodic solutions.

In a notable progression for this problem, Xiaoming Li and Shijun Liao have discovered 695 families of periodic orbits in the Newtonian planar, collisionless three-body system with equal mass and zero angular momentum. This remarkable work extends the understanding of the complexity and richness of the three-body problem by uncovering a multitude of previously unidentified periodic families, significantly expanding the catalog of known three-body periodic solutions.

Methodology and Results

The authors utilized a robust numerical approach, building upon and diverging from the strategies employed by \v{S}uvakov and Dmitra\v{s}inovi\'c in their 2013 paper, which identified 11 new families of periodic orbits. This paper's methodology centered on analyzing initial conditions based on isosceles collinear configurations.

Key to the success was the use of high-order Taylor series methods alongside grid search techniques. Multiple precision numerical simulations ensured the precision and reliability required for handling chaotic dynamics within the problem. The use of finer grid partitions and the Clean Numerical Simulation (CNS) for minimizing numerical noise led to the discovery of 243 more periodic orbits than when using standard double-precision solvers.

The authors classified these periodic orbits according to geometrical and algebraic symmetries, aiding in their methodical paper and comparison to existing solutions. Of the 695 families identified, more than 600 are reported as new, diversifying the landscape of known three-body problem solutions.

Implications and Future Research

The results have significant implications for both theoretical considerations and practical computations in celestial dynamics:

1. Generalized Kepler's Third Law: The researchers propose a quasi Kepler's third law $\bar{T}^* \approx 2.433 \pm 0.075$, relating the scale-invariant average period to system energy. This law suggests a deeper, potentially universal property linking periodic trajectories in three-body systems.
2. Complex Dynamics Exploration: The discovery lays a foundation for further exploration and mapping of the complex dynamics within the three-body problem. Future work could involve exploring other initial configurations and expanding numerical resolution, potentially leading to more families.
3. Database Creation: A comprehensive database that includes the newly identified periodic orbits could help enhance understanding and stimulate further research. Such a catalog would benefit fields such as astrodynamics, where periodic solutions can be relevant for mission design and understanding natural celestial phenomena.
4. Benchmark for Numerical Methods: The large number of identified conditions provides a robust test bed for validating and improving numerical approaches in solving chaotic differential equations, which are pivotal in studying dynamical systems.

In summary, this research expands the compendium of periodic solutions considerably and provides novel insight into the three-body problem's inherent complexity. It highlights not only the power of modern computational techniques but also sets a clear trajectory for ongoing explorations in the field of gravitational dynamics. As computational capabilities continue to advance, these discoveries may only be a glimpse of what lies beneath the chaotic surface of three-body systems.