Periodic orbits in a three-dimensional galactic potential model via averaging theory
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Periodic orbits in a three-dimensional galactic potential model via averaging theory M. Alvarez-Ramírez1,a
, J. D. García-Saldaña2,b
, M. Medina1,c
1 Departamento de Matemáticas, UAM-Iztapalapa, 09340 Iztapalapa, Mexico City, Mexico 2 Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso
de Ribera 2850, Concepción, Chile Received: 5 August 2020 / Accepted: 24 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Using averaging theory, the existence of periodic solutions is established for an axially symmetric galaxy model described by a Hamiltonian endowed with a three-dimensional potential of two Miyamoto disks galaxies rotating around its z-axis.
1 Introduction The Milky Way is a gigantic spiral-shaped disk with a bright, central bulge containing over 100 billion stars that revolves around the central core, the galaxy moves continuously, likewise, our solar system is also in motion. The solar system is located about 3/4 of the way out from the center in one of the galaxy’s spiral arms. Since the first images of the Milky Way as individual stars obtained by Galileo Galilei, the knowledge of this galaxy has increasingly grown, but even now, is very far from being complete. Many scholars have directed their attention to its study, making it an extensive area of research by using all the available tools existing within their area of expertise. Galaxies are defined as large groups of stars together with dust and gas that are held together by the action of the gravity force. Since galaxies are very complex dynamical objects, it becomes important to study their dynamics. One way to do so is to consider it under the realm of the n-body problem for large n, integrating the equations of motion for systems up to thousands of stars by making use of extensive numerical computations [1], or carry out studies from a theoretical point of view. Another way to conduct research about the nature of dynamics of the Milky Way is to construct galaxy mass models obtained on simple and analytical functions for density and potential pairs. Disk models that are constrained purely by kinematic data only provide information about the global disk structure very near the galactic plane. These models reflect the three-dimensional structures of the galaxies and are constrained based on recent galactic observations and have been widely proposed on the literature, see [2]. Such dynamical models are simple and have about the same dynamical properties as those of the galaxy. It is well known that most galaxies are not exactly axially symmetric; however, an axially symmetric
a e-mail: [email protected] b e-mail: [email protected] (corresponding author) c e-mail: [email protected]
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model is a good approximation which allows us to simplify the computations for studying the motion of the galaxy, see, for instance, [10,14]. The majority of axially symmetric
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