Celestial Mechanics and Dynamical Astronomy

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Article
Machine learning (ML) is the branch of computer science that studies computer algorithms that can learn from data. It is mainly divided into supervised learning, where the computer is presented with examples of entries, and the goal is to learn a general rule that maps inputs to outputs, and unsupervised learning, where no label is provided to the learning algorithm, leaving it alone to find structures. Deep learning is a branch of machine learning based on numerous layers of artificial neural networks, which are computing systems inspired by the biological neural networks that constitute animal brains. In asteroid dynamics, machine learning methods have been recently used to identify members of asteroid families, small bodies images in astronomical fields, and to identify resonant arguments images of asteroids in three-body resonances, among other applications. Here, we will conduct a full review of available literature in the field and classify it in terms of metrics recently used by other authors to assess the state of the art of applications of machine learning in other astronomical subfields. For comparison, applications of machine learning to Solar System bodies, a larger area that includes imaging and spectrophotometry of small bodies, have already reached a state classified as progressing. Research communities and methodologies are more established, and the use of ML led to the discovery of new celestial objects or features, or new insights in the area. ML applied to asteroid dynamics, however, is still in the emerging phase, with smaller groups, methodologies still not well-established, and fewer papers producing discoveries or insights. Large observational surveys, like those conducted at the Zwicky Transient Facility or at the Vera C. Rubin Observatory, will produce in the next years very substantial datasets of orbital and physical properties for asteroids. Applications of ML for clustering, image identification, and anomaly detection, among others, are currently being developed and are expected of being of great help in the next few years.
 
Article
Let us consider the restricted three-body problem. Analysis of the orbital motion of a spacecraft around planets or moons is presented taking into account the nonsphericity of the primaries and the perturbations coming from a third body in an elliptical and inclined orbit. In the specific case of a spacecraft designed to explore a planet, moon or asteroid, it is noteworthy the increasing use of the averaging methods. This is a very powerful technique to simulate, very fast, the main effects caused by the disturbers on the dynamics of the spacecraft. In this work, we focus on the averaged methods applied in different conditions. Some comparisons are presented between the single-averaged, double-averaged models and the complete model, that is, the unaveraged model based on direct integration of the Cartesian (x, y, z) coordinates. This unaveraged model is quite necessary as it provides all the requirements to validate the performance and evaluate the usefulness of the averaged models for each specific problem. In the first part of this paper, we describe briefly some well-known techniques to obtain the averaged model considering the nonsphericity of the primary as well as the perturbation due to the third body. On the other hand, this is a opportunity to mention some misprints and typos problems, in the literature related to this subject. We compared the performance of single- and double-averaging models, keeping the x–y–z unaveraged model as the baseline of reference. The case of a high lunar orbit (Nie et al. in Celest Mech Dyn Astron 131(29):1–31, 2019) considering the perturbation of the Earth seems to be instructive. Single-average model is more accurate than the double-average model in the analysis of the eccentricity evolution, but in some cases of the inclination evolution, the three models agree and the average models are both very accurate. When comparing the results, eventual typos were detected in some works related to the literature of this subject. In the second part of this paper, we detached some aspects of the dynamics of a probe around Mercury (Sect. 5) involved in frozen orbit (FO) and in "quasi-frozen orbit", (quasi-FO). Due to the interesting gravitational field of the planet and its proximity to the Sun, this is an important problem. Recently, many papers, not only on pure dynamics but on gravitational field of Mercury, have been published, according to references listed in this work. An exhaustive investigation on FO using double-averaging model was reported in Tresaco et al. (Celest Mech Dyn Astron 130(9):1–26, 2018). In this paper we revisit this problem, using x–y–z-model as a primary source of results. After a number of experiments, it was possible to use confidently the single averaging in many cases, for instance, in searching "quasi-FO" for Mercury planet. Although we do not include the effect of the radiation pressure, a number of our simulations can be compared with those given in Tresaco et al. (2018).
 
Article
Collisions are one of the key processes shaping planetary systems. Asteroid families are outcomes of such collision still identifiable across our solar system. The families provide a unique view of catastrophic disruption phenomena and have been in the focus of planetary scientists for more than a century. Most of them are located in the main belt, a ring of asteroids between Mars and Jupiter. Here we review the basic properties of the families, discuss some recent advances, and anticipate future challenges. This review pays more attention to dynamic aspects such as family identification, age determination, and long-term evolution. The text, however, goes beyond that. Especially, we cover the details of young families that see the major advances in the last years, and we anticipate it will develop even faster in the future. We also discuss the relevance of asteroid families for water-ice content in the asteroid belt and our current knowledge on links between families and main-belt comets. query Please check the edit made in the article title.
 
Article
Mercury’s motion has been studied using numerical methods in the framework of a model including only the non-relativistic Newtonian gravitational interactions of the solar system: eight major planets and Pluto in translation around the Sun. Since the true trajectory of Mercury is an open, non-planar curve, special attention to the exact definition of the advance of Mercury’s perihelion has been given. For this purpose, the concepts of an extended and a geometrical perihelion have been introduced. In addition, for each orbital period, a mean ellipse was fitted to the trajectory of Mercury. I have shown that the perihelion advance of Mercury deduced from the behavior of the Laplace–Runge–Lenz vector, as well as from the extended and geometrical perihelion advance depend on the fitting time interval and for intervals of the order of 1 000 years converge to a value of 532.1″\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\prime \prime }$$\end{document} per century. The behavior of the perihelia, either extended or geometrical, is strongly impacted by the influence of Jupiter. The advance of the extended perihelion depends on the time step used in the calculations, while the advance of the geometrical perihelion and that deduced by the rotation of the Laplace–Runge–Lenz vector depends only slightly on it.
 
Article
The region where the main asteroid belt is now located may have started empty, to become populated early in the history of the Solar system with material scattered outward by the terrestrial planets and inward by the giant planets. These dynamical pathways toward the main belt may still be active today. Here, we present results from a data mining experiment aimed at singling out present-day members of the main asteroid belt that may have reached the belt during the last few hundred years. Probable newcomers include 2003 BM1, 2007 RS62, 457175 (2008 GO98), 2010 BG18, 2010 JC58, 2010 JV52, 2010 KS6, 2010 LD74, 2010 OX38, 2011 QQ99, 2013 HT149, 2015 BH103, 2015 BU525, 2015 RO127, 2015 RS139, 2016 PC41, 2016 UU231, 2020 SA75, 2020 UO43, and 2021 UJ5, all of them in the outer belt. Some of these candidates may have been inserted in their current orbits after experiencing relatively recent close encounters with Jupiter. We also investigated the likely source regions of such new arrivals. Asteroid 2020 UO43, if real, has a non-negligible probability of having an origin in the Oort cloud or even interstellar space. Asteroid 2003 BM1 may have come from the neighborhood of Uranus. However, most newcomers -including 457175, 2011 QQ99, and 2021 UJ5- might have had an origin in Centaur orbital space. The reliability of these findings is assessed within the context of the uncertainties of the available orbit determinations.
 
SSB-Earth vector difference between INPOP19a and INPOP10e in Cartesian coordinates. Left: before correction by a constant vector. Right: After correction by a constant vector based on the mean of the difference. We find dx =\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 84.3 km, dy =\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 44.8 km, dz =\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 3.6 km
Density plot of the residuals in the (AL,AC) plane expressed in milliarcsecond. Left panel: before the adjustment of the initial conditions. Right pane: after the adjustment of the initial conditions. The colourbar and the axis range were chosen to be directly comparable with Fig. 19 of Gaia Collaboration et al. (2018b). The white arrows point the over-density created by residuals of asteroids with G<13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G<13$$\end{document}
Density plot of the residuals in AL and AC with respect to the magnitude G after the adjustment of the initial conditions
Density plots of the mean residuals (first row) per transit in AL and AC with respect to the magnitude G after the adjustment of the initial conditions and density plots of the standard deviation of the residuals (second row) per transit in AL and AC with respect to the magnitude G after the adjustment of the initial conditions
Article
We used the INPOP19a planetary ephemerides to perform the orbital adjustment of 14099 asteroids based on Gaia-DR2 observations and compare for 23 of them the resulting orbits to radar data. As Gaia-DR2 has been processed using the planetary ephemeris INPOP10e, the primary goal of this paper is to confirm the portability of the data when using an updated version of the solar system model. In particular, we point out the fact that the Gaia satellite positions—provided with respect to the INPOP10e solar system barycenter—must be corrected when using another planetary ephemeris. We also present a convenient least square formalism that only handles small matrices and allows the adjustment of global parameters, such as masses. In order to check the consistency of the Gaia observations with other types of observations, we perform an orbital adjustment in combining Gaia and radar range observations for 23 objects, together with a careful post-fit analysis including an estimation of the Gaia systematic errors. Finally, we show that to ensure the combined use of Gaia angular DR2 observations and radar ranging, a more developed than firstly proposed dynamical modeling is required together with the addition of the systematic Gaia bias in the fit procedure. These results give promising directions for the next Gaia delivery, Gaia-DR3.
 
Article
We present FCIRK16, a 16th-order implicit symplectic integrator for long-term high-precision Solar System simulations. Our integrator takes advantage of the near-Keplerian motion of the planets around the Sun by alternating Keplerian motions with corrections accounting for the planetary interactions. Compared to other symplectic integrators (the Wisdom and Holman map and its higher-order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high-precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of roundoff errors. In addition, unlike typical explicit symplectic integrators for near-Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non-necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point-mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high-order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required.
 
Article
The satellite laser ranging (SLR) echo signals of the multi-reflector defunct spacecraft contain multiple distance information corresponding to the corner cube reflectors (CCRs) in different installation positions. The relative distances between different CCRs and their variations depend on the rotation state of the object. However, the rotation of many defunct spacecraft is relatively slow over a short observation time span. In this case, estimating the effective rotation state is still difficult due to the limited amount of information. In this paper, a slow-rotating multi-reflector defunct spacecraft CZ-2C R/B (NORAD ID 31114) is taken as an example to develop a new method to estimate the rotation state through laser ranging measurements from a single short arc. The reference vector method is used in new attitude representation. The reduced expressions between the relative distances, their first derivative and rotation state are derived. Through the state estimation method combining grid search and differential corrections, the reduced parameters are estimated efficiently and accurately. In addition, the consistent orientation of maximum principal axis of inertia of CZ-2C R/B are obtained, which provides effective information for the subsequent research.
 
Article
A stochastic optimization algorithm for analyzing planar central and balanced configurations in the n-body problem is presented. We find a comprehensive list of equal mass central configurations satisfying the Morse equality up to n=12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=12$$\end{document}. We show some exemplary balanced configurations in the case n=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=5$$\end{document}, as well as some balanced configurations without any axis of symmetry in the cases n=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=4$$\end{document} and n=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=10$$\end{document}.
 
Article
For the 1+5-body problem, we study the relationship between the masses of 5 satellites and given symmetric configurations, where the symmetry axis contains one satellite. Under certain assumptions, we find analytically some central configurations for suitable positive masses. Also, we prove that for certain symmetric configuration of these satellites, there exists a one-parameter family of mass vectors for which such configuration is a central configuration. Furthermore, we present some numerical results for configurations and derive the positive masses for these satellites, such that these configurations are central configurations.
 
Article
Periodic orbits and their invariant manifolds are known to be useful for transportation in space, but a large portion of the related research goes toward a small number of periodic orbit families that are relatively simple to compute. In this study, motivated by a search for new and lesser-known families of useful periodic orbits, the bifurcation diagram near Europa is explored and 400 bifurcation points are found. Families are generated for 74 of these and provided in a publicly accessible database. Of these 74 generated families, those that also appear to exist in a model perturbed by certain zonal harmonics of Jupiter and Europa are identified. Differential corrections techniques are discussed, and a new method for natural parameter continuation in the three-body problem is presented. Periodic orbits with particularly useful geometric and stability properties for science purposes are highlighted.
 
The meridional section of the spheroid of unit mass with the eccentricity ε=0.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0.4$$\end{document} and density (25). On the left, all the points of the section are colored; on the right, only a few equidensites. The color of a point corresponds to the logarithm of the density
The meridional section of the spheroid of unit mass with the eccentricity ε=0.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0.4$$\end{document} and constant density parameters f=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=0$$\end{document}, g=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g = 1$$\end{document}, h=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=2$$\end{document}. On the left, all the points of the section are colored; on the right, only a few equidensites. The color of a point corresponds to the logarithm of the density
The meridional section of the spheroid of unit mass with the eccentricity ε=0.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0.4$$\end{document} and density parameters f=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f = 0$$\end{document}, g=χ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g = \chi ^{2}$$\end{document}, h=χ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h = \chi ^{-2}$$\end{document}. On the left, all the points of the section are colored; on the right, only a few equidensites. The color of a point corresponds to the logarithm of the density
The meridional section of the spheroid of unit mass with the eccentricity ε=0.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0.4$$\end{document} and density (27). On the left, all the points of the section are colored; on the right, only a few equidensites. The color of a point corresponds to the logarithm of the density. The density inside the ellipsoid u⩽1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \leqslant 1$$\end{document} is constant
Article
A closed-form expression is obtained for the density of a simple layer, equipotential to an oblate level ellipsoid of revolution in an outer space. The potential of any level spheroid of positive mass with the inward direction of the attracting force on its surface can be represented in this way. A family of density functions defined within the whole volume of a level ellipsoid of revolution is found. Several density examples are considered.
 
Article
The creep tide theory is used to establish the basic equations of the tidal evolution of differentiated bodies formed by aligned homogeneous layers in co-rotation. The mass concentration of the body is given by the fluid Love number kf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_f$$\end{document}. The formulas are given by series expansions valid for high eccentricity systems. They are equivalent to Darwin’s equations, but formally more compact. An application to the case of Enceladus, with kf=0.942\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_f=0.942$$\end{document}, is discussed.
 
Reference frames and angles. (e^,k×e^,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\hat{\mathbf {e}}}}, \mathbf {k}\times {{\hat{\mathbf {e}}}}, \mathbf {k})$$\end{document} is a Cartesian frame such that k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {k}$$\end{document} is a unit vector normal to the orbital plane, e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {e}$$\end{document} is the Laplace–Runge–Lenz vector (where e^=e/e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\hat{\mathbf {e}}}}= \mathbf {e}/ e$$\end{document} gives the direction of the pericentre), ϖ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varpi $$\end{document} is the argument of the pericentre and υ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon $$\end{document} is the true anomaly. (p,q,s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {p},\mathbf {q},\mathbf {s}$$\end{document}) is a Cartesian frame, such that s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}$$\end{document} is a unit vector normal to the equatorial plane of the central body, p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {p}$$\end{document} is a unit vector along the line of nodes of the two reference planes, and θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is the angle between them. Note that, although s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}$$\end{document} gives the direction of the spin axis, the vectors p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {p}$$\end{document} and q=s×p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {q}= \mathbf {s}\times \mathbf {p}$$\end{document} follow the orbital plane and not the rotation
Article
We revisit the two-body problem, where one body can be deformed under the action of tides raised by the companion. Tidal deformation and consequent dissipation result in spin and orbital evolution of the system. In general, the equations of motion are derived from the tidal potential developed in Fourier series expressed in terms of Keplerian elliptical elements, so that the variation of dissipation with amplitude and frequency can be examined. However, this method introduces multiple index summations and some orbital elements depend on the chosen frame, which is prone to confusion and errors. Here, we develop the quadrupole tidal potential solely in a series of Hansen coefficients, which are widely used in celestial mechanics and depend just on the eccentricity. We derive the secular equations of motion in a vectorial formalism, which is frame independent and valid for any rheological model. We provide expressions for a single average over the mean anomaly and for an additional average over the argument of the pericentre. These equations are suitable to model the long-term evolution of a large variety of systems and configurations, from planet–satellite to stellar binaries. We also compute the tidal energy released inside the body for an arbitrary configuration of the system.
 
Article
Proper elements are quasi-integrals of motion of a dynamical system, meaning that they can be considered constant over a certain timespan, and they permit to describe the long-term evolution of the system with a few parameters. Near-Earth objects (NEOs) generally have a large eccentricity, and therefore they can cross the orbits of the planets. Moreover, some of them are known to be currently in a mean-motion resonance with a planet. Thus, the methods previously used for the computation of main-belt asteroid proper elements are not appropriate for such objects. In this paper, we introduce a technique for the computation of proper elements of planet-crossing asteroids that are in a mean-motion resonance with a planet. First, we numerically average the Hamiltonian over the fast angles while keeping all the resonant terms, and we describe how to continue a solution beyond orbit-crossing singularities. Proper elements are then extracted by a frequency analysis of the averaged orbit-crossing solutions. We give proper elements of some known resonant NEOs and provide comparisons with non-resonant models. These examples show that it is necessary to take into account the effect of the resonance for the computation of accurate proper elements.
 
Article
This paper compares the continuum evolution for density equation modelling and the Gaussian mixture model on the 2D phase space long-term density propagation problem in the context of high-altitude and high area-to-mass ratio satellite long-term propagation. The density evolution equation, a pure numerical and pointwise method for the density propagation, is formulated under the influence of solar radiation pressure and Earth’s oblateness using semi-analytical methods. Different from the density evolution equation and Monte Carlo techniques, for the Gaussian mixture model, the analytical calculation of the density is accessible from the first two statistical moments (i.e. the mean and the covariance matrix) corresponding to each sub-Gaussian distribution for an initial Gaussian density distribution. An insight is given into the phase space long-term density propagation problem subject to nonlinear dynamics. The efficiency and validity of the density propagation are demonstrated and compared between the density evolution equation and the Gaussian mixture model with respect to standard Monte Carlo techniques.
 
Article
We investigate how the temporal evolution of the rotation axis of a hypothetical exo-Earth is affected by the presence of a satellite, an exo-Moon. Namely, we study analytically and numerically how the range of the nutation angle of an exo-Earth changes if an exo-Moon is added to a system comprised of an exo-Sun, the exo-Earth and exoplanets. We say that the impact of an exo-Moon is stabilising if upon including the exo-Moon the range of the nutation angle decreases, and destabilising otherwise. The problem is considered in a general set-up. The exo-Earth is supposed to be rigid, axially symmetric and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. Assuming the orbits of the celestial bodies to be quasi-periodic, we apply time averaging over fast variables associated with order one frequencies to study rotation of the exo-Earth at times large relative the respective periods. Non-resonant frequencies are assumed. For a system comprised of the exo-Sun and exoplanets in the absence of small orbital frequencies, the system is integrable, which allows to calculate the range of the nutation angle as a function of initial conditions. Using these expressions, we identify a class of systems for which we prove analytically that the impact of the exo-Moon is stabilising and a class where it is destabilising. Namely, if the orbits of the planets are circular and their orbital planes coincide then the impact is destabilising. The impact is stabilising if the angle between orbital planes of the exo-Moon and the exo-Earth vanishes. We also investigate numerically how the impact of the exo-Moon in a particular system comprised of a star and two planets varies on modifying parameters of the orbits of the exo-Moon and the second planet, and the initial nutation angle.
 
Article
We compare the performance of four symplectic integration methods with leading order symplectic corrector in simulations of the Solar System. These simulations cover 10 Gyr. They are longer than the astrophysical predicted future of the present-day Solar System, thus this work is mainly a study of the integration methods. For the outer Solar System simulation, where the used stepsize was 100 days, the energy errors do not show any secular evolution. The maximum errors show a dependence on the method. The simulations of the full Solar System from Mercury, and including Pluto as a test particle, were calculated with a stepsize of 7 days. The energy errors behave somewhat differently having a small secular behavior. This may due to the short timestep and the short period of the planet Mercury or some small round off error produced by the code. Comparison of the eccentricity evolution’s within simulations show that some planets are dynamically strongly coupled. Venus and Earth form a dynamical pair, also Jupiter and Saturn form a dynamical pair. The FFT of the analysis of the simulations suggests that all the giant planets form a single dynamical quadruple system. The orbit of Mercury is possibly unstable. Each simulation is stopped when Mercury is expelled. All the methods show similar results for times less than $$30\, $$ 30 Myr in the way that the results for orbital elements are same within plotting precision. Inclusion of Mercury in simulations shortens the Solar System e-folding time to $$3.3\, $$ 3.3 Myr. It is clear that chaos has a strong effect in the evolution of orbital elements, especially eccentricities. This is easily seen in Mercury’s orbit when the simulation time exceeds at least $$30\, $$ 30 Myr. Our low-order simulations seem to match high-order methods over long timescales.
 
Article
Local toroidal coordinate systems are introduced to characterize relative motion near a periodic orbit with an oscillatory mode in the circular restricted three-body problem. These coordinate systems are derived from a first-order approximation of invariant tori relative to a periodic orbit and supply a geometric interpretation that is consistent across distinct periodic orbits. First, the local toroidal coordinate sets are used to rapidly generate first-order approximations of quasi-periodic relative motion. Then, geometric properties of these first-order approximations are used to predict the minimum and maximum separation distances between a spacecraft following quasi-periodic motion relative to another spacecraft located on a periodic orbit. Implementation of the local toroidal coordinate systems and associated geometric analyses is demonstrated in the context of spacecraft formations operating near members of the Earth–Moon L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} southern halo orbit family.
 
Relationship between the graph of the function φ(Mπ,e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (M\pi ,e)$$\end{document} and the global families of solutions bifurcating from z=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=0$$\end{document}
Numerical calculation of the function φ(8π,e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (8\pi ,e)$$\end{document} and its relationship with global families
Numerical calculation of the function φ(Mπ,e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (M\pi ,e)$$\end{document}(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Big ($$\end{document}diferent values of M and λ∈0,964)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \left( 0,\frac{9}{64}\right) \Big )$$\end{document}
Graphical representation of the regions where inequality (29) holds and does not hold
Article
In this work, we study the existence of global families of symmetric periodic solutions of a generalized Sitnikov problem that bifurcate from equilibrium z=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=0$$\end{document}. For global families emerging from a circular generalized Sitnikov problem, we study whether they continue for all values of eccentricity e∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e\in [0,1)$$\end{document} or ends in equilibrium.
 
Article
Orbital resonances can be leveraged in the mission design phase to target planets at different energy levels. On the other side, precise models are needed to predict possible threatening returns of natural and artificial objects closely approaching a target planet. To this aim, we propose a semi-analytic extension of the b-plane resonance model to account for perturbing effects inside the planet's sphere of influence. We compute the actual values of the perturbing coefficients by means of precise numerical simulations, whereas their expression stems from the properties of hyperbolic trajectories and asymptotic planetocentric velocity vectors. We apply the proposed b-plane model to design ballistic resonant flybys by solving a multilevel mixed-integer nonlinear optimization problem.
 
Article
We propose a closed-form normalization method suitable for the study of the secular dynamics of small bodies in heliocentric orbits perturbed by the tidal potential of a planet with orbit external to the orbit of the small body. The method makes no use of relegation, thus circumventing all convergence issues related to that technique. The method is based on a convenient use of a book-keeping parameter keeping simultaneously track of all the small quantities in the problem. The book-keeping affects both the Lie series and the Poisson structure employed in successive perturbative steps. In particular, it affects the definition of the normal form remainder at every normalization step. We show the results obtained by assuming Jupiter as perturbing planet, and we discuss the validity and limits of the method.
 
Equilateral chain
Triangular bipyramid configuration
Article
We consider central configurations of the strictly spatial five-body problem with a homogeneous potential which are equilateral chains, i.e., configurations with four sequential equilateral edges containing all five vertices. First, we prove that any such configuration must be a triangular bipyramid with an equilateral triangle base. Furthermore, we show that the masses located at the vertices of the triangle must be equal and the masses of the other two particles which are off the base also must be equal. We also found that a particular triangular bipyramid configuration with fixed masses is a central configuration for a range of homogenous potentials generalizing the Newtonian potential. Finally, we conclude that there is a unique triangular bipyramid central configuration with equal masses for these same homogenous potentials.
 
Article
Four small moons (Styx, Nix, Kerberos and Hydra) are at present known to orbit around the barycenter of Pluto and Charon, which are themselves considered a binary dwarf planet due to their relatively high mass ratio. The central, non-axisymmetric potential induces moon orbits inconvenient to be described by Keplerian osculating elements. Here, we report that observed orbital variations may not be the result of orbital eccentricities or observational uncertainties, but may be due to forced oscillations caused by the central binary. We show, using numerical integration and analytical considerations, that the differences reported on their orbital elements may well arise from this intrinsic behavior rather than limitations on our instruments.
 
Article
High-fidelity representations of the gravity field underlie all applications in astrodynamics. Traditionally these gravity models are constructed analytically through a potential function represented in spherical harmonics, mascons, or polyhedrons. Such representations are often convenient for theory, but they come with unique disadvantages in application. Broadly speaking, analytic gravity models are often not compact, requiring thousands or millions of parameters to adequately model high-order features in the environment. In some cases these analytic models can also be operationally limiting—diverging near the surface of a body or requiring assumptions about its mass distribution or density profile. Moreover, these representations can be expensive to regress, requiring large amounts of carefully distributed data which may not be readily available in new environments. To combat these challenges, this paper aims to shift the discussion of gravity field modeling away from purely analytic formulations and toward machine learning representations. Within the past decade there have been substantial advances in the field of deep learning which help bypass some of the limitations inherent to the existing analytic gravity models. Specifically, this paper investigates the use of physics-informed neural networks (PINNs) to represent the gravitational potential of two planetary bodies—the Earth and Moon. PINNs combine the flexibility of deep learning models with centuries of analytic insight to learn new basis functions that are uniquely suited to represent these complex environments. The results show that the learned basis set generated by the PINN gravity model can offer advantages over its analytic counterparts in model compactness and computational efficiency.
 
Article
We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.
 
Article
Proper elements are quasi-invariants of a Hamiltonian system, obtained through a normalization procedure. Proper elements have been successfully used to identify families of asteroids, sharing the same dynamical properties. We show that proper elements can also be used within space debris dynamics to identify groups of fragments associated to the same break-up event. The proposed method allows to reconstruct the evolutionary history and possibly to associate the fragments to a parent body. The procedure relies on different steps: (i) the development of a model for an approximate, though accurate, description of the dynamics of the space debris; (ii) the construction of a normalization procedure to determine the proper elements; (iii) the production of fragments through a simulated break-up event. We consider a model that includes the Keplerian part, an approximation of the geopotential, and the gravitational influence of Sun and Moon. We also evaluate the contribution of Solar radiation pressure and the effect of noise on the orbital elements. We implement a Lie series normalization procedure to compute the proper elements associated to semi-major axis, eccentricity and inclination. Based upon a wide range of samples, we conclude that the distribution of the proper elements in simulated break-up events (either collisions and explosions) shows an impressive connection with the dynamics observed immediately after the catastrophic event. The results are corroborated by a statistical data analysis based on the check of the Kolmogorov-Smirnov test and the computation of the Pearson correlation coefficient.
 
Article
We present fully three-dimensional equations to describe the rotations of a body made of a deformable mantle and a fluid core. The model in its essence is similar to that used by INPOP19a (Integration Planétaire de l’Observatoire de Paris) Fienga et al. (INPOP19a planetary ephemerides. Notes Scientifiques et Techniques de l’Institut de Mécanique Céleste, vol 109, 2019), and by JPL (Jet Propulsion Laboratory) (Park et al. The JPL Planetary and Lunar Ephemerides DE440 and DE441. Astron J 161(3):105, 2021. doi:10.3847/1538-3881/abd414), to represent the Moon. The intended advantages of our model are: straightforward use of any linear-viscoelastic model for the rheology of the mantle; easy numerical implementation in time-domain (no time lags are necessary); all parameters, including those related to the “permanent deformation”, have a physical interpretation. The paper also contains: (1) A physical model to explain the usual lack of hydrostaticity of the mantle (permanent deformation). (2) Formulas for free librations of bodies in and out-of spin-orbit resonance that are valid for any linear viscoelastic rheology of the mantle. (3) Formulas for the offset between the mantle and the idealised rigid-body motion (Peale’s Cassini states). (4) Applications to the librations of Moon, Earth, and Mercury that are used for model validation.
 
Article
Variational methods have been successfully applied to construct many different classes of periodic solution of N-body problem and N-center problem. However, until now, it is still challenging to apply variational methods to restricted (N+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N+1)$$\end{document}-body problem. In this paper, we consider the restricted few-body-few-center problem (an intermediate problem between restricted (N+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N+1)$$\end{document}-body problem and N-center problem) with symmetric torque-free primaries, identify its binary-syzygy sequence that can be realized by minimizers of the Lagrangian action functional, and construct its periodic solutions within certain topological classes. At the end, we further reveal similar results for restricted (N+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N+1)$$\end{document}-body problem with the general rhomboidal primary system. In order to achieve the above aim, we also demonstrate the asymptotic behavior of the massless particle near two-body collision with a moving primary and establish partial Sundman–Sperling estimates of the massless particle near multi-body collision.
 
Article
Given two positive real numbers $M$ and $m$ and an integer $n>1$, it is well known that we can find a family of solutions of the $(n+1)$-body problem where one body with mass $M$ stays at the origin and the other $n$ bodies, all with the same mass $m$, move on the $x$-$y$ plane following ellipses with eccentri\-city $e$. These periodic solutions were discovered by Lagrange and can be described analytically. In this paper, we prove the existence of periodic solutions of the $(n+1)$-body problem, they are not-trivial in the sense that none of the bodies follows conics. Besides showing the existence of these periodic solutions, we point out a trivial family of non-periodic solutions for the $(n+1)$-body problem that are easy to describe. In this way, we are considering three families of solutions of the $(n+1)$-body problem: The \textit{Lagrange} family, the family of \textit{non-periodic} solution and the \textit{non-trivial solutions}. The authors surprisingly discovered that a numerical solution of the $4$-body problem $-$ the one displayed on the video http://youtu.be/2Wpv6vpOxXk $-$ is part of a family of periodic solutions (those that we are calling the non-trivial) that does not approach a solution in the Lagrange family but it approaches a solution in the family that we are calling \textit{non-periodic} solutions. After pointing this out, the authors find an exact formula for the bifurcation point in the \textit{non-periodic} family and use it to show the mathematical existence of non planar periodic solutions of the $(n+1)$-body problem for any pair of masses $M,m$ and any integer $n>1$, (the family that we are calling \textit{non-trivial}). As a particular example, we find a non-trivial solution of the $4$-body problem where three bodies with mass $3$ moving around a body with mass $7$ that moves up and down.
 
Article
Rapid trajectory design in multi-body systems often leverages individual arcs along natural dynamical structures that exist in an approximate dynamical model. To reduce the complexity of this analysis in a chaotic gravitational environment, a motion primitive set is constructed to represent the finite geometric, stability, and/or energetic characteristics exhibited by a set of trajectories and, therefore, support the construction of initial guesses for complex trajectories. In the absence of generalizable analytical criteria for extracting these representative solutions, a data-driven procedure is presented. Specifically, k-means and agglomerative clustering are used in conjunction with weighted evidence accumulation clustering, a form of consensus clustering, to construct sets of motion primitives in an unsupervised manner. This data-driven procedure is used to construct motion primitive sets that summarize a variety of periodic orbit families and natural trajectories along hyperbolic invariant manifolds in the Earth–Moon circular restricted three-body problem.
 
Article
This paper explores the problem of analytically approximating the orbital state for a subset of orbits in a rotating potential with oblateness and ellipticity perturbations. This is done by isolating approximate differential equations for the orbit radius and other elements. The conservation of the Jacobi integral is used to make the problem solvable to first order in the perturbations. The solutions are characterized as small deviations from an unperturbed circular orbit. The approximations are developed for near-circular orbits with initial mean motion n0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{0}$$\end{document} around a body with rotation rate c. The approximations are shown to be valid for values of angular rate ratio Γ=c/n0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma = c/n_{0} > 1$$\end{document}, with accuracy decreasing as Γ→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma \rightarrow 1$$\end{document}, and singularities at and near Γ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma = 1$$\end{document}. Extensions of the methodology to eccentric orbits are discussed, with commentary on the challenges of obtaining generally valid solutions for both near-circular and eccentric orbits.
 
Article
Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the 2g+h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2g+h$$\end{document} lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the 2g+h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2g+h$$\end{document} resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, iL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_{L}$$\end{document}. Since iL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_L$$\end{document} is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination ip\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_{p}$$\end{document}, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}$$\end{document}, where actual navigation satellites lie. Using ip\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_p$$\end{document} as a label, we compute its range of values for which C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}$$\end{document} becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}$$\end{document} allows to explain the role of the initial phase h noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between a=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a =$$\end{document} 24,000 km and a=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=$$\end{document} 30,000 km as a result of the sweeping of the 2g+h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2g+h$$\end{document} resonance by the and resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon’s orbit with respect to the ecliptic. The role of the phases in the structures observed in the FLI maps is also clarified. Finally, (v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.
 
Article
When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we (1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; (2) implement continuation by both perturbation and rotation numbers; (3) compute Fourier–Taylor parameterizations of the stable and unstable manifolds; (4) regularize the equations of motion; and (5) globalize these manifolds. Our methodology improves on efficiency and accuracy compared to prior studies and applies to a variety of periodic perturbations. We demonstrate the tools near resonances in the planar elliptic RTBP.
 
Article
This study aims to explore more solutions for low-energy transfers to lunar distant retrograde orbits (DROs) from the vicinity of the Earth. Millions of transfer trajectories from a lunar free-return orbit (LFO) to a prescribed DRO were computed using multiple powered lunar flybys (PLFs) and weak-stability-boundary (WSB)-like ballistic transfer. We proposed a two-step design method, consisting of a database creation and trajectory patching, to construct low-energy LFO–DRO transfers in planar bicircular restricted four-body dynamics with the Sun, Earth, and Moon as primary bodies. The parallel computation technique allows the computation of millions of solutions with times of flight (TOFs) up to 135 days and a total velocity impulse (Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}V) of no more than 350 m/s, although this design method requires substantial computational load. These solutions help us identify key flight information, such as the Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}V–TOF Pareto fronts and launch windows for rendezvous with a station in a DRO. Low-energy transfers to a DRO can be achieved by exploiting single or multiple PLFs and WSB-like ballistic arcs at the expense of elongated TOFs. Moreover, triple PLFs render many more options for the spacecraft to accomplish the rendezvous of DROs. The WSB-like ballistic arcs to a DRO in this study exhibit new features compared with conventional and traditional WSB concepts.
 
Article
We consider the planar circular restricted three-body problem, as a model for the motion of a spacecraft relative to the Earth–Moon system. We focus on the collinear equilibrium points \(L_1\) and \(L_2\). There are families of Lyapunov periodic orbits around either \(L_1\) or \(L_2\), forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either \(L_1\) or \(L_2\), as well as heteroclinic orbits between the Lyapunov manifold around \(L_1\) and the one around \(L_2\). The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, which gives the future asymptotic of a homoclinic orbit as a function of the past asymptotic. In contrast to the more customary Melnikov theory, we do not need to assume that the asymptotic orbits have a special nature (periodic, quasi-periodic, etc.). We add a non-conservative, time-dependent perturbation, as a model for a thrust applied to the spacecraft for some duration of time, or for some other effect, such as solar radiation pressure. We compute the first-order approximation of the perturbed scattering map, in terms of fast convergent integrals of the perturbation along homoclinic/heteroclinic orbits of the unperturbed system. As a possible application, this result can be used to determine the trajectory of the spacecraft upon using the thrust.
 
Article
The equilibrium figure of an inviscid tidally deformed body is the starting point for the construction of many tidal theories such as Darwinian tidal theories or the hydrodynamical Creep tide theory. This paper presents the ellipsoidal equilibrium figure when the spin rate vector of the deformed body is not perpendicular to the plane of motion of the companion. We obtain the equatorial and the polar flattenings as functions of the Jeans and the Maclaurin flattenings, and of the angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} between the spin rate vector and the radius vector. The equatorial vertex of the equilibrium ellipsoid does not point toward the companion, which produces a torque perpendicular to the rotation vector, which introduces terms of precession and nutation. We find that the direction of spin may differ significantly from the direction of the principal axis of inertia C, so the classical approximation Iω≈Cω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {I}}\varvec{\omega } \approx C\varvec{\omega }$$\end{document} only makes sense in the neighborhood of the planar problem. We also study the so-called Cassini states. Neglecting the short-period terms in the differential equation for the spin direction and assuming a uniform precession of the line of the orbital ascending node, we obtain the same differential equation as that found by Colombo (Astron J 71:891, 1966). That is, a tidally deformed inviscid body has exactly the same Cassini states as a rotating axisymmetric rigid body, the tidal bulge having no secular effect at first order.
 
Evolution of spacecraft proper time with global coordinate time where δt≡t-τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta t \equiv t - \tau $$\end{document}
Relative frequency of quadratic invariant Eq. (20) for Mercury Planetary Orbiter (top), Molniya (middle) and Parker Solar Probe (bottom)-like objects. The individual orbits are propagated with integration time step h which is calculated with respect to the orbital period T. We define δI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta {\mathcal {I}}$$\end{document} according to δI≡I-c2/c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta {\mathcal {I}} \equiv \left( {\mathcal {I}} - c^2\right) /c^2$$\end{document} where I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}$$\end{document} is calculated at each numerical integration step. Hence, for the perturbed motion of test particles subject to a locally measured radiation-like four-force, GRAPE preserves I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}$$\end{document} at the order of 10-32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-32}$$\end{document}
Article
Spacecraft propagation tools describe the motion of near-Earth objects and interplanetary probes using Newton’s theory of gravity supplemented with the approximate general relativistic n -body Einstein–Infeld–Hoffmann equations of motion. With respect to the general theory of relativity and the long-standing recommendations of the International Astronomical Union for astrometry, celestial mechanics and metrology, we believe modern orbitography software is now reaching its limits in terms of complexity. In this paper, we present the first results of a prototype software titled General Relativistic Accelerometer-based Propagation Environment (GRAPE). We describe the motion of interplanetary probes and spacecraft using extended general relativistic equations of motion which account for non-gravitational forces using end-user supplied accelerometer data or approximate dynamical models. We exploit the unique general relativistic quadratic invariant associated with the orthogonality between four-velocity and acceleration and simulate the perturbed orbits for Molniya, Parker Solar Probe and Mercury Planetary Orbiter-like test particles subject to a radiation-like four-force. The accuracy of the numerical procedure is maintained using a 5-stage, $$10^\mathrm{th}$$ 10 th -order structure-preserving Gauss collocation symplectic integration scheme. GRAPE preserves the norm of the tangent vector to the test particle worldline at the order of $$10^{-32}$$ 10 - 32 .
 
Article
This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.
 
Article
Ballistic capture is a phenomenon by which a spacecraft approaches its target body, and performs a number of revolutions around it, without requiring manoeuvres in between. Capture orbits are characterized by specific dynamics, defining regions that guide transport phenomena. Because of the limitations associated with existing approaches, the development of heuristics informed by Lagrangian Coherent Structures appears desirable. In fact, such structures identify transport barriers in dynamical systems, separating regions with qualitatively different dynamics. In this work, different flow-informed approaches are presented, and their relations with ballistic capture are discussed. A new heuristic, the time-varying strainline, is introduced. This new tool is applied to compute ballistic capture orbits around Mars. Different degrees of model fidelity have been investigated, mainly in order to test the robustness of the proposed technique with respect to different features of the underlying dynamical model. We show that time-varying strainlines are useful in identifying ballistic capture orbits.
 
Plot of the surface z=Ω(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=\Omega (x,y)$$\end{document}. In the base of the parallelepiped, we have the (x, y)-plane, and in the height, we have the z-axis
The curves of zero velocity for different values of C in the (x, y)-plane
The curves of zero velocity and their Hill’s regions. The motion is allowed in the white regions
Article
We study the dynamics of the circular restricted 4-body problem with three primaries with equal masses at the collinear configuration of the 3-body problem with an infinitesimal mass. We calculate the equilibrium points and study their linear stability. By applying the Lyapunov theorem, we prove the existence of periodic orbits bifurcating from the equilibrium points and, further, prove that they continue in the full 4-body problem. Moreover, we prove analytically the existence of Hill and of comet-like periodic orbits.
 
Article
Poincaré maps are regularly used to facilitate rapid and informed trajectory design within multi-body systems. However, maps that capture a general set of spatial trajectories are often higher-dimensional and, as a result, challenging for a human to analyze. This paper addresses this challenge by employing techniques from data mining. Specifically, distributed clustering, dimension reduction and classification are used in combination to construct a data-driven approach to autonomously group higher-dimensional crossings on a Poincaré map according to the geometry of the associated trajectories generated over a short time interval. This procedure is demonstrated using a periapsis map that captures spatial trajectories at a single energy level in the Sun-Earth circular restricted three-body problem. Arcs along hyperbolic invariant manifolds associated with families of tori in the \(L_{1}\) and \(L_{2}\) gateways are also projected onto this clustering result to rapidly extract their fundamental geometries. Together, these examples demonstrate the potential for the presented data-driven approach to facilitate analysis of a complex solution space reflected on a higher-dimensional Poincaré map.
 
Article
We performed numerical orbit integrations of the equal-mass free-fall three-body problem. We provided numerical evidence of the existence of triple collision orbits concentrating on the shape of triangles formed by the three bodies. In the process, in the negative side, we confirmed that the shape of the triangle of non-triple collision orbits deviates from equilateral as the orbit approaches the instant of supposed triple collision. In the positive side, the orbit integrations have been done along the three binary collision curves on which the orbit perform binary collision. These curves meet at the triple collision point (TCP). Three binary collisions have the three base edges, whose direction defines the edges of the triple collision orbit. The triangle defined by these three edges precisely coincides with the equilateral triangles of the triple collision orbit.
 
Article
We introduce six quantities that generalize the equinoctial orbital elements when some or all the perturbing forces that act on the propagated body are derived from a potential. Three of the elements define a non-osculating ellipse on the orbital plane, other two fix the orientation of the equinoctial reference frame, and the last allows us to determine the true longitude of the body. The Jacobian matrices of the transformations between the new elements and the position and velocity are explicitly given. As a possible application, we investigate their use in the propagation of Earth’s artificial satellites, showing a remarkable improvement compared to the equinoctial orbital elements.
 
Article
The Cassini spacecraft discovered many close-in small satellites in Saturnian system, and several of them exhibit exotic orbital states due to interactions with Mimas and the oblateness of the planet. This work is devoted to Methone, which is currently involved in a 15:14 mean-motion resonance with Mimas. We give an in-deep study on the current orbit of Methone by analyzing and identifying the short, resonant and long-term gravitational perturbations on its orbit. In addition, we perform numerical integrations of full equations of motion of ensembles of close-in small bodies orbiting the non-central field of Saturn. Spectral analyses of the orbits and interpretation of them in dynamical maps allow us to describe the orbit and the dynamics of Methone in view of resonant and long-term dynamics. We show that the current geometric orbit of Methone is aligned with Mimas’ due to a forced resonant component in eccentricity, leading to simultaneous oscillations of several critical angles of the expanded disturbing function. Thus, we explain the simultaneous oscillations of four critical arguments associated to the resonance. The mapping of the Mimas–Methone resonance shows that the domains of the 15:14 Mimas–Methone resonance are dominated by regular motions associated to the Corotation resonance located at eccentricities lower than \(\sim \, 0.015\) and osculating semi-major axis in the interval 194,660–194,730 km. Methone is currently located deeply within this site.
 
Algorithm scheme
Res C continuation performance. Solid lines represent the adaptive step methods, while dashed lines indicate the corresponding fixed step methods
Article
Orbit generation in non-Keplerian environments poses some challenges related to the complex dynamical nature in which such trajectory exist. The absence of a parametric representation of the orbits requires an iterative approach to define families. Simple methods exist to fulfill such task, however, they are based on local information and prone to convergence/speed problems. A polynomial-based scheme is proposed to improve the search of the solutions along the orbital families, enhancing the overall speed of the process, while avoiding convergence issues. The scheme is tested in the framework of Earth–Moon system, and performances are discussed and compared to classical approaches.
 
Evolution of the mean motion coefficient multiplying n J 3 2 for the TOPEX orbital configuration. Left: e = 0.001. Right: I = 66.04 •
Osculating elements and corresponding mean elements after second-order corrections
Root sum square error of the Cartesian coordinates (in m) provided by different truncations {I,S,D}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{I,S,D\}$$\end{document} (for Inverse, Secular, Direct) of the two tested versions of Brouwer’s J2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_2$$\end{document} solution; numbers on the right side of the labels denote the maximum error. From top to bottom: TOPEX, PRISMA, and GTO orbits. Abscissas are days
Evolution of the mean motion coefficient multiplying nJ23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$nJ_2^3$$\end{document} for the TOPEX orbital configuration. Left: e=0.001\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e=0.001$$\end{document}. Right: I=66.04∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I=66.04^{\circ }$$\end{document}
Article
Brouwer’s solution to the artificial satellite problem is revisited. We show that the complete Hamiltonian reduction is rather achieved in the plain Poincaré’s style, through a single canonical transformation, than using a sequence of partial reductions based on von Zeipel’s alternative for dealing with perturbed degenerate Hamiltonian systems. Beyond the theoretical interest of the new approach as regards the complete reduction of perturbed Keplerian motion, we also show that a solution based on a single set of corrections may yield computational benefits in the implementation of an analytic orbit propagator.
 
Article
We numerically investigate triple collision orbits of the free-fall three-body system which has no double collisions before three bodies collide. Triple collision is an important property of the three-body system. Tanikawa, Saito, Mikkola (Celest Mech Dyn Astron 131(6):24, 2019) obtained 11 triple collision orbits without double collision for the free-fall three-body problem. In this paper, we present 1658 triple collision orbits including the Lagrange’s homothetic solution, 11 ones found by Tanikawa et al. (2019) and 1646 new triple collision orbits. The symbol sequences of these 1646 new triple collision orbits have digits that range between 1 and 120. With our high-precision results, numerical evidences of the asymptotic property of triple collision orbits are given.
 
Article
We are dealing with the averaged model used to study the secular effects in the motion of a body of the negligible mass in the context of a spatial restricted elliptic three-body problem. It admits a two-parameter family of equilibria (stationary solutions) corresponding to the motion of the third body in the plane of primaries motion, so that the apse line of the orbit of this body is aligned with the apse lines of the primaries’ orbits. The aim of our investigation is to analyse the stability of these equilibria. We show that they are stable in the linear approximation. The Arnold–Moser stability theorem provides sufficient conditions under which this means stability in a nonlinear sense too. These conditions are violated for parameters of the problem that belong to a set formed by a finite number of analytic curves in the parameters’ plane. As it turned out, in the system under consideration, violation of these conditions in some cases actually leads to an instability.
 
Article
We propose the general method of proving the bifurcation of new solutions from relative equilibria in N -body problems. The method is based on a symmetric version of Lyapunov center theorem. It is applied to study the Lennard–Jones 2-body problem, where we have proved the existence of new periodic or quasi-periodic solutions.
 
Top-cited authors
Iwan P Williams
  • Queen Mary, University of London
Al Conrad
  • The University of Arizona
D. Hestroffer
  • Institut de Mécanique Céleste et de Calcul des Éphémérides
Toshio Fukushima
  • National Astronomical Observatory of Japan
Regis Courtin
  • Observatoire de Paris