[Paper Review] A conjecture on Kepler's third law of n-body periodic orbits

Chang-Yin Zhao, Mingjiang Zhang|arXiv (Cornell University)|Nov 2, 2018

Spacecraft Dynamics and Control4 citations

TL;DR

This paper proposes a conjectured extension of Kepler’s third law to three-body and n-body periodic orbits using dimensional analysis and mass product symmetry in Newtonian gravity. It derives a universal relation $ T_n|E_n|^{3/2} = \frac{\pi}{\sqrt{2}}G \left[ \frac{\sum_{i<j}(m_i m_j)^3}{\sum_k m_k} \right]^{1/2} $, which generalizes the two-body law and shows strong consistency with numerical periodic orbit data for equal-mass systems, offering a powerful heuristic for discovering new periodic solutions in celestial mechanics.

ABSTRACT

Three-body and n-body problems in celestial mechanics are age-old and challenging puzzles. In recent years, several breakthroughs are made in finding periodic orbits for three-body problem. And Bohua Sun proposed a conjecture on Kepler's third law of three-body and n-body problems by using the dimensional analysis method and the mass product symmetry of Newtonian gravitational field. In this paper, the background as well as the research progress on the Kepler's third law, three-body and n-body problems is introduced briefly, and then Bohua Sun's conjecture on Kepler's third law of three-body and n-body problems is reviewed from the perspective of both theory and application.

Motivation & Objective

• To extend Kepler’s third law from two-body to three- and n-body periodic systems in celestial mechanics.
• To establish a universal analytical relation between orbital period, total energy, and mass configurations in n-body systems.
• To provide a predictive framework for identifying periodic orbits in complex gravitational systems using dimensional analysis and symmetry principles.
• To validate the conjecture against numerical data from recent periodic orbit discoveries in three-body systems.
• To stimulate further theoretical and numerical research into periodic solutions of the n-body problem.

Proposed method

• Applying dimensional analysis to identify fundamental parameters: gravitational coupling $ \alpha_n = Gm_i m_j $, reduced mass $ \mu_n $, and orbital area $ A_n $.
• Deriving the core relation $ T_n|E_n|^{3/2} = \text{const.} \times \alpha_n \sqrt{\mu_n} $ using dimensional consistency and Newtonian symmetry.
• Calibrating the constant using the known two-body solution, yielding $ \text{const.} = \frac{\pi}{\sqrt{2}} $.
• Formulating the conjectured law for three-body systems: $ T_3|E_3|^{3/2} = \frac{\pi}{\sqrt{2}}G \left[ \frac{(m_1m_2)^3 + (m_1m_3)^3 + (m_2m_3)^3}{m_1 + m_2 + m_3} \right]^{1/2} $.
• Extending the formula to n-body systems via summation over all pairwise mass products.
• Validating the conjecture by comparing its predictions with fitting formulas from Li and Liao’s numerical data on 695 periodic orbit families.

Experimental results

Research questions

• RQ1 Does a generalized form of Kepler’s third law exist for periodic orbits in three- and n-body gravitational systems?
• RQ2 Can the period-energy relation in n-body systems be expressed universally through mass product symmetry and dimensional analysis?
• RQ3 How well does the proposed conjecture match numerical data from known periodic orbits in three-body systems?
• RQ4 What is the role of the reduced mass and total energy in scaling the period of periodic n-body orbits?
• RQ5 Can this conjecture serve as a predictive tool for discovering new periodic solutions in celestial mechanics?

Key findings

• The conjectured relation $ T_n|E_n|^{3/2} = \frac{\pi}{\sqrt{2}}G \left[ \frac{\sum_{i<j}(m_i m_j)^3}{\sum_k m_k} \right]^{1/2} $ generalizes Kepler’s third law to n-body periodic orbits.
• For three-body systems with equal masses, the conjecture shows excellent agreement with numerical data from Li and Liao’s 695 periodic orbit families when $ m_3 > 1 $.
• The conjecture reduces to the exact two-body solution when one mass approaches zero, confirming consistency with established physics.
• Discrepancies for $ m_3 < 1 $ suggest potential limitations in the numerical fitting process used in prior studies, which may require re-evaluation.
• The derived formula exhibits perfect mathematical symmetry and structure, supporting its physical plausibility despite being a conjecture.
• The conjecture provides a powerful heuristic for accelerating the discovery of new periodic orbits in multi-body systems, especially in space mission design.