Discovery of 10,059 new three-dimensional periodic orbits of general three-body problem

Technical Summary: Discovery of 10,059 New Three-Dimensional Periodic Orbits of the General Three-Body Problem

Problem Statement
The general three-body problem (3BP), involving three finite masses interacting via gravitational forces, has remained a central challenge in celestial mechanics since Newton's formulation in the 1680s. While the problem is inherently three-dimensional (3D), the vast majority of discovered periodic orbits over the last three centuries have been restricted to two-dimensional (planar) cases. The difficulty in finding 3D periodic orbits stems from the system's chaotic nature, as revealed by Poincaré, which implies a lack of closed-form solutions and extreme sensitivity to initial conditions. Furthermore, traditional numerical algorithms (e.g., Runge-Kutta in double precision) suffer from numerical noise that rapidly pollutes chaotic trajectories, making the long-term integration required to verify periodicity unreliable. Consequently, very few 3D periodic orbits of the general 3BP (with three finite masses) have been identified, with most existing studies focusing on restricted cases (e.g., one massless body) or spatial isosceles configurations.

Methodology
To overcome the limitations of traditional numerical methods, the authors employed a high-accuracy numerical strategy centered on the Clean Numerical Simulation (CNS) method. The approach involved the following steps:

1. Initial Configuration: The study utilized a specific symmetric initial configuration where the three bodies start at positions $r_1 = (-1, 0, 0)$, $r_2 = (1, 0, 0)$, and $r_3 = (0, 0, z_0)$. The initial velocities were defined to satisfy zero total momentum, with $v_1 = (v_x, v_y, v_z)$, $v_2 = (v_x, v_y, -v_z)$, and $v_3$ calculated accordingly. The system was defined by four unknown variables: $z_0, v_x, v_y, v_z$.
2. Grid Search: A grid search method was applied to the parameter space ($z_0 \in (0, 1]$, $v_x \in (0, 1]$, $v_y \in (0, 1]$) with $v_z = 0$. For each mass ratio case, $10^9$ initial conditions were tested using the DOP853 solver to identify candidates where the "return proximity function" $\delta(T) = |X(T) - X(0)|$ was less than $10^{-1}$.
3. Refinement via CNS and Newton-Raphson: Candidates were refined using a numerical approach based on the Newton-Raphson method combined with CNS. CNS reduces numerical noise to extremely low levels, allowing for accurate integration over long time intervals. A 3D periodic orbit was confirmed when the return proximity function $\delta(T)$ was reduced to less than $10^{-10}$.
4. Stability Analysis: The linear stability of the discovered orbits was evaluated using Floquet theory by computing the monodromy matrix. Orbits were deemed linearly stable if all eigenvalues of the monodromy matrix lay on the unit circle.
5. Mass Parameters: The study focused on cases where $m_1 = m_2 = 1$ and $m_3 = 0.1n$ for integers $1 \le n \le 20$. The authors note that the method is theoretically valid for arbitrary mass values.

Key Contributions and Results
Using a national supercomputer (Tianhe-2), the authors successfully discovered 10,059 new three-dimensional periodic orbits for the general three-body problem in the specific cases where $m_1 = m_2 = 1$ and $m_3 = 0.1n$ (with $1 \le n \le 20$). The specific findings include:

• Total Discovery: 10,059 distinct 3D periodic orbits were identified across the tested mass configurations ($m_3 = 0.1n$ for $1 \le n \le 20$).
• Linear Stability: Of the total discoveries, 1,996 orbits (approximately 20%) were found to be linearly stable.
• Choreographic Orbits (Equal Masses): In the specific case of three equal masses ($m_1 = m_2 = m_3 = 1$), the authors discovered 21 "choreographic" periodic orbits. In these orbits, all three bodies move periodically along a single closed orbit. Unlike the famous planar Figure-8 orbit, these are fully three-dimensional.
• "Piano-Trio" Orbits (Two Equal Masses): In cases where $m_1 = m_2 = 1$ and $m_3 \neq 1$, the authors discovered 273 unique orbits specifically within the 19 cases where $m_3 = 0.1n$ and $n \neq 10$ (i.e., $1 \le n \le 20$ but $n \neq 10$). In these configurations, the two equal-mass bodies move along a single closed orbit, while the third body traces a distinct, separate closed orbit. The authors term these "piano-trio" orbits, analogous to a musical trio of two violins and one piano.
• Data Availability: The initial conditions, periods, and stability data for all discovered orbits are provided in supplementary materials and on a public GitHub repository. High-precision initial conditions (up to 70 significant digits) are available for specific stable orbits.

Significance and Claims
The paper asserts that, to the best of the authors' knowledge, none of these 10,059 3D periodic orbits have been reported previously, marking a significant expansion of the known solution space for the general three-body problem.

The authors highlight the scientific significance of these findings in the context of chaos theory. Citing Poincaré, they argue that periodic orbits serve as "the only opening through which we can try to penetrate in a place which, up to now, was supposed to be inaccessible." By providing a large dataset of 3D periodic orbits, the work offers new opportunities to better understand the chaotic properties of the three-body problem.

Furthermore, the authors emphasize the robustness of their numerical approach. They note that because the CNS method allows for arbitrary accuracy (limited only by computational resources), the discovered orbits can be treated with a level of precision comparable to closed-form solutions from a physical standpoint. The methodology is presented as scalable, suggesting that an arbitrarily large number of 3D periodic orbits can be generated for arbitrary mass configurations, potentially extending to systems with three unequal masses via continuation methods.