Kepler's third law of n-body periodic orbits in a Newtonian gravitation field

Abstract

This study considers the periodic orbital period of an n-body system from the perspective of dimension analysis. According to characteristics of the n-body system with point masses $(m_1,m_2,...,m_n)$, the gravitational field parameter, $\alpha \sim G{m}_i{m}_j$, the n-body system reduction mass $M_n$, and the area, $A_n$, of the periodic orbit are selected as the basic parameters, while the period, $T_n$, and the system energy, $|E_n|$, are expressed as the three basic parameters. Using the Buckingham $\pi$ theorem, We obtained an epic result, by working with a reduced gravitation parameter ${\alpha }_n$, then predicting a dimensionless relation $T_n|E_n{|}^{3/2}=\text{const}\times {\alpha }_n\sqrt{{\mu }_n}$ (${\mu }_n$ is reduced mass). The const$=\frac{\pi }{\sqrt{2}}$ is derived by matching with the 2-body Kepler's third law, and then a surprisingly simple relation for Kepler's third law of an n-body system is derived by invoking a symmetry constraint inspired from Newton's gravitational law: $T_n|E_n{|}^{3/2}=\frac{\pi }{\sqrt{2}}G{\left(\frac{{\sum }_{i=1}^n{\sum }_{j=i+1}^n(m_i{m}_j{)}^3}{{\sum }_{k=1}^n{m}_k}\right)}^{1/2}$. This formulae is, of course, consistent with the Kepler's third law of 2-body system, but yields a non-trivial prediction of the Kepler's third law of 3-body: $T_3|E_3{|}^{3/2}=\frac{\pi }{\sqrt{2}}G{\left[\frac{(m_1{m}_2{)}^3+(m_1{m}_3{)}^3+(m_2{m}_3{)}^3}{m_1+m_2+m_3}\right]}^{1/2}$. A numerical validation and comparison study was conducted. This study provides a shortcut in search of the periodic solutions of three-body and n-body problems and has valuable application prospects in space exploration.

6 pages, 11 figues

References in corpus (4)

• Three Classes of Newtonian Three-Body Planar Periodic Orbits
• More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits
• Topological Dependence of Kepler's Third Law for Collisionless Periodic Three-Body Orbits with Vanishing Angular Momentum and Equal Masses
• Gravitational Waves from Periodic Three-Body Systems

Cited by in corpus (5)

• Three-body problem -- from Newton to supercomputer plus machine learning
• Universal scaling law of an origami paper spring
• Quantum support to BoHua Sun's conjecture
• Quasi Kepler's third law for quantum many-body systems
• Classical and Quantum Kepler's Third Law of N-Body System