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Astrophys Space Sci (2020) 365:144
https://doi.org/10.1007/s10509-020-03855-w
ORIGINAL ARTICLE
The orbital evolution of the Sun–Jupiter–Saturn–Uranus–Neptune
system on long time scales
Alexander Perminov1·Eduard Kuznetsov1
Received: 22 January 2020 / Accepted: 12 August 2020 / Published online: 27 August 2020
© Springer Nature B.V. 2020
Abstract The averaged semi-analytical motion theory of
the four-planetary problem is constructed up to the third or-
der in planetary masses and the sixth degree in the orbital ec-
centricities and inclinations. The second system of Poincaré
elements and the Jacobi coordinate system are used for the
construction of the Hamiltonian expansion. The averaged
Hamiltonian is obtained in the third approximation by the
Hori–Deprit method. All analytical transformations are per-
formed by using CAS Piranha.
The constructed equations of motion in averaged ele-
ments are numerically integrated by the different methods
for the giant planets of the Solar System over a time inter-
val of up to 10 Gyr. The planetary motion is quasi-periodic,
and the short-term perturbations of the orbital elements con-
serve small values in the modeling process. The compari-
son of obtained amplitudes and periods of the change of the
orbital elements with numerical motion theories shows an
excellent agreement with them. The properties of the plane-
tary motion are given. The short-periodic perturbations and
the precision of the integration are estimated.
Keywords Celestial mechanics ·Methods: analytical ·
Methods: numerical ·Planet–star interactions ·Planets and
satellites: dynamical evolution and stability ·Planets and
satellites: individual: Jupiter, Saturn, Uranus, Neptune
BA. Perminov
perminov12@yandex.ru
E. Kuznetsov
eduard.kuznetsov@urfu.ru
1Ural Federal University, 51 Lenin Avenue, 620000 Ekaterinburg,
Russia
1 Introduction
This article is related to the problem of the study of the or-
bital evolution of planetary systems. The averaged equations
of motion are constructed analytically up to the third degree
in the small parameter for the case of the four-planetary sys-
tem. The ratio of the sum of planetary masses to the mass of
the star plays the role of the small parameter in the problem
(from now on denoted μ).
The orbital evolution of the four-planet Sun–Jupiter–
Saturn–Uranus–Neptune system is considered in this article.
The mass of the Solar System’s giant planets is smaller than
the mass of the Sun to three orders (the value of the small pa-
rameter). The mass of the terrestrial planets is smaller than
the giant planets’ one by three orders also. Therefore the in-
fluence of the terrestrial planets on the orbital motion of the
giant planets is insignificant. On the contrary, catastrophic
changes in the motion of the giant planets can lead to the de-
struction of the whole Solar System. So, the giant-planetary
approximation of the Solar System is sufficient for the in-
vestigation of the dynamical evolution and stability on long
time scales.
It is convenient to use the equations of motion in aver-
aged orbital elements for the study of dynamical evolution
on long time scales. The use of averaged elements allows
one to sufficiently increase the integration step of the equa-
tions of motion and, consequently, reduce the integration
time. The Solar System’s orbital evolution on cosmogonic
time scales was first studied analytically in the 19th century.
The modern averaging theories are developed based on ideas
of Lagrange, Laplace, and Gauss.
The main point of averaging methods is the elimination
from the equations of motion all short-periodic terms whose
periods are comparable with or less than the circulation pe-
riods of the planets. The fast variables of the problem give
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