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

## TL;DR

This paper revisits and extends Kepler's third law to N-body systems, proposing a new quantum conjecture consistent with previous results and emphasizing the role of dimension analysis.

## Contribution

It introduces a novel quantum N-body Kepler's third law conjecture based on dimension analysis, aligning with Semay's quantum results for identical bodies.

## Key findings

- Proposes a new quantum N-body Kepler's third law conjecture.
- The formula aligns with classical 2-body Kepler's law.
- Consistent with Semay's quantum results for identical bodies.

## 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: $T_q|E_q|^{3/2}=\frac{\pi}{\sqrt{2}} G\left[\frac{\left(\sum_{i=1}^N\sum_{j=i+1}^Nm_im_j\right)^3}{\sum_{k=1}^N m_k}\right]^{1/2}$. 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.