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Issue 4, 2026
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Methodological notes


From Kepler's ellipses — to R-toroids

†  a, b
a Lomonosov Moscow State University, Shternberg State Astronomical Institute, Universitetskii prosp. 13, Moscow, 119234, Russian Federation
b The Central Astronomical Observatory of the Russian Academy of Sciences at Pulkovo, Pulkovskoe shosse 65/1, St. Petersburg, 196140, Russian Federation

An overview of new directions in modern celestial mechanics is given. The starting point is the Gaussian ring, the gravitational potential of which is obtained in analytical form. Instead of osculating Keplerian elements, a set of new variables is introduced: these are the Laplace and Hamilton vectors, as well as the vector of the orbital angular momentum of the body. In the linear approximation, the evolution equations in new variables are obtained, replacing the classical Lagrange equations. The method is tested on the exosystem HD 206893. The problem of confinement of rings around small celestial bodies is considered. Two new approaches based on two-dimensional (R-disk) and three-dimensional (R-toroid) generalizations of precessing Gauss rings are presented. The methods are tested by solving problems on the dynamics of stellar rings in the Galaxy and the secular evolution of orbits in various exoplanet systems.

Typically, an English full text is available in about 1 month from the date of publication of the original article.

Keywords: Kepler ellipses, Gaussian ring and its potential, orbital orientation vectors, confinement of rings around small celestial bodies, precession of Gauss rings: R-ring and R-toroid models, dwarf planet Haumea, two-planet problems, circumbinary exosystems
PACS: 02.30.Em, 95.10.Ce, 96.15.De, 96.30.Wr, 97.82.−j (all)
DOI: 10.3367/UFNe.2025.09.040024
URL: https://ufn.ru/en/articles/2026/4/f/
Citation: Kondratyev B P "From Kepler's ellipses — to R-toroids" Phys. Usp. 69 (4) (2026)

Received: 17th, August 2025, 10th, September 2025

Оригинал: Кондратьев Б П «От эллипсов Кеплера — к R-тороидам» УФН 196 422–435 (2026); DOI: 10.3367/UFNr.2025.09.040024

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