Classical and Quantum Kepler's Third Law of N-Body System

Abstract

Inspired by amazing result obtained by Semay \cite{semay-1}, this study revisits generalised Kepler's third law of an n-body system from the perspective of dimension analysis. To be compatible with Semay's quantum n-body result, this letter reports a conjecture which had not be included in author's early publication \cite{sun2018} but formulated in the author's research memo. The new conjecture for quantum N-body system is proposed as follows: $T_q|E_q{|}^{3/2}=\frac{\pi }{\sqrt{2}}G{\left[\frac{{\left({\sum }_{i=1}^N{\sum }_{j=i+1}^N{m}_i{m}_j\right)}^3}{{\sum }_{k=1}^N{m}_k}\right]}^{1/2}$. This formulae is, of course, consistent with the Kepler's third law of 2-body system, and exact same as Semay's quantum result for identical bodies.

arXiv comment2 pages, 2 figures

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