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Abstract: 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. PubDate: 2022-05-10

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Abstract: 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. PubDate: 2022-05-07

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Abstract: Abstract 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. PubDate: 2022-04-29

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Abstract: Abstract 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. PubDate: 2022-04-20

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Abstract: Abstract 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 \(30\, \) 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 \(3.3\, \) 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 \(30\, \) Myr. Our low-order simulations seem to match high-order methods over long timescales. PubDate: 2022-04-09

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Abstract: Abstract 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 \(L_2\) southern halo orbit family. PubDate: 2022-04-08

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Abstract: Abstract In this work, we study the existence of global families of symmetric periodic solutions of a generalized Sitnikov problem that bifurcate from equilibrium \(z=0\) . For global families emerging from a circular generalized Sitnikov problem, we study whether they continue for all values of eccentricity \(e\in [0,1)\) or ends in equilibrium. PubDate: 2022-04-05

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Abstract: Abstract 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. PubDate: 2022-04-04

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Abstract: Abstract 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. PubDate: 2022-03-29

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Abstract: Abstract 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. PubDate: 2022-03-28

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Abstract: Abstract 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. PubDate: 2022-03-28

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Abstract: Abstract 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. PubDate: 2022-03-24

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Abstract: Abstract 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. PubDate: 2022-03-18

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Abstract: Abstract 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. PubDate: 2022-03-17

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Abstract: Abstract 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. PubDate: 2022-03-08 DOI: 10.3847/1538-3881/abd414)

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Abstract: Abstract 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 \((N+1)\) -body problem. In this paper, we consider the restricted few-body-few-center problem (an intermediate problem between restricted \((N+1)\) -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 \((N+1)\) -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. PubDate: 2022-02-23 DOI: 10.1007/s10569-022-10065-9

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Abstract: Abstract 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 eccentricity 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 Lagrange family, the family of non-periodic solution and the 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 non-periodic solutions. After pointing this out, the authors find an exact formula for the bifurcation point in the non-periodic family and use it to show the mathematical existence of nonplanar 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 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. PubDate: 2022-02-05 DOI: 10.1007/s10569-022-10062-y

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Abstract: Abstract 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. PubDate: 2022-02-01 DOI: 10.1007/s10569-022-10063-x

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Abstract: Abstract 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 \(2g+h\) 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 \(2g+h\) resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, \(i_{L}\) . Since \(i_L\) is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination \(i_{p}\) , instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold \({\mathcal {C}}\) , where actual navigation satellites lie. Using \(i_p\) as a label, we compute its range of values for which \({\mathcal {C}}\) becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in \({\mathcal {C}}\) 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 \(a =\) 24,000 km and \(a=\) 30,000 km as a result of the sweeping of the \(2g+h\) 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. PubDate: 2022-01-20 DOI: 10.1007/s10569-021-10060-6

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Abstract: Abstract 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 \(n_{0}\) around a body with rotation rate c. The approximations are shown to be valid for values of angular rate ratio \(\varGamma = c/n_{0} > 1\) , with accuracy decreasing as \(\varGamma \rightarrow 1\) , and singularities at and near \(\varGamma = 1\) . 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. PubDate: 2022-01-20 DOI: 10.1007/s10569-022-10061-z