The Lidov-Kozai Oscillation and Hugo von Zeipel

TL;DR

This paper reviews the Lidov-Kozai oscillation in celestial mechanics, highlighting that early 20th-century work by von Zeipel laid much of the foundational theory, suggesting a unified von Zeipel-Lidov-Kozai framework.

Contribution

It demonstrates that von Zeipel's early work already encompassed key aspects of the Lidov-Kozai oscillation, proposing a unified naming convention for the phenomenon.

Findings

Von Zeipel's work predates and covers fundamental aspects of Lidov-Kozai oscillation.

A comprehensive comparison of von Zeipel, Lidov, and Kozai's contributions.

Proposal to rename the phenomenon as von Zeipel-Lidov-Kozai oscillation.

Abstract

The so-called Lidov-Kozai oscillation is very well known and applied to various problems in solar system dynamics. This mechanism makes the orbital inclination and eccentricity of the perturbed body in the circular restricted three-body system oscillate with a large amplitude under certain conditions. It is widely accepted that the theoretical framework of this phenomenon was established independently in the early 1960s by a Soviet Union dynamicist (Michail L'vovich Lidov) and by a Japanese celestial mechanist (Yoshihide Kozai). A large variety of studies has stemmed from the original works by Lidov and Kozai, now having the prefix of "Lidov-Kozai" or "Kozai- Lidov." However, from a survey of past literature published in late nineteenth to early twentieth century, we have confirmed that there already existed a pioneering work using a similar analysis of this subject established in that period. This was accomplished by a Swedish astronomer, Edvard Hugo von Zeipel. In this monograph, we first outline the basic framework of the circular restricted three-body problem including typical examples where the Lidov-Kozai oscillation occurs. Then, we introduce what was discussed and learned along this line of studies from the early to mid-twentieth century by summarizing the major works of Lidov, Kozai, and relevant authors. Finally, we make a summary of von Zeipel's work, and show that his achievements in the early twentieth century already comprehended most of the fundamental and necessary formulations that the Lidov-Kozai oscillation requires. By comparing the works of Lidov, Kozai, and von Zeipel, we assert that the prefix "von Zeipel-Lidov-Kozai" should be used for designating this theoretical framework, and not just Lidov-Kozai or Kozai-Lidov.