On mean elements in artificial satellite theory [CL]

http://arxiv.org/abs/2305.09303


The merits of a perturbation theory based on a mean to osculating transformation that is pure periodic in the fast angle are investigated. The exact separation of the purely short-period effects of the perturbed Keplerian dynamics from the long-period mean frequencies is achieved by a non-canonical transformation, which, therefore, cannot be computed by Hamiltonian methods. For this case, the evolution of the mean elements strictly adheres to the average behavior of the osculating orbit. However, due to the inescapable truncation of perturbation solutions, the fact that this theory confines the long-period variations of the semimajor axis into the mean variation equations, how tiny they may be, can have adverse effects in the accuracy of long-term semi-analytic propagations based on it

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M. Lara
Wed, 17 May 23
64/67

Comments: 26 pages, 6 figures, 2 Tables, submitted to Celestial Mechanics and Dynamical Astronomy

Reliable and Repeatable Transit Through Cislunar Space Using the 2:1 Resonant Spatial Orbit Family [EPA]

http://arxiv.org/abs/2304.13584


This work focuses on the identification of reliable and repeatable spatial (three-dimensional) trajectories that link the Earth and the Moon. For this purpose, this paper aims to extend the 2:1 resonant prograde family and 2:1 resonant retrograde family to three dimensions and to introduce spatial orbits that are not currently present in the literature. These orbits, named the 2:1 resonant spatial family, bifurcate from the two-dimensional families and smoothly transition between them in phase space. The stability properties of this new family of resonant orbits are discussed, and, interestingly, this family includes marginally stable members. Furthermore, this new family of orbits is applied to several engineering problems in the Earth-Moon system. First, this paper selects an appropriate member of 2:1 resonant spatial family on the basis of its stability properties and relationships with other multibody orbits in the regime. Next, this work combines this trajectory with momentum exchange tethers to transit payloads throughout the system in a reliable and repeatable fashion. Finally, this paper studies the process of aborting a catch and related recovery opportunities.

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A. Binder and D. Arnas
Thu, 27 Apr 23
42/78

Comments: 33 pages, 31 figures

Secular orbital dynamics of the innermost exoplanet of the $\upsilon$-Andromedæ system [EPA]

http://arxiv.org/abs/2304.09038


We introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. In particular, we refer to the motion of $\upsilon$-And $b$ which is the innermost planet among those discovered in the extrasolar system orbiting around the $\upsilon$-Andromedae A star. We preassign the orbits of the Super-Jupiter exoplanets $\upsilon$-And $c$ and $\upsilon$-And $d$ in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the Frequency Analysis and are injected in the equations describing the orbital dynamics of $\upsilon$-And $b$ under the gravitational effects exerted by those two external exoplanets (expected to be major ones in such an extrasolar system). We end up with a $2+3/2$ degrees of freedom Hamiltonian model; its validity is confirmed by the comparison with several numerical integrations of the complete $4$-body problem. Furthermore, the model is enriched by taking into account also the relativistic effects on the secular motion of the innermost exoplanet. We focus on the problem of the stability of $\upsilon$-And $b$ as a function of the parameters that mostly impact on its orbit, i.e. the initial values of its inclination and the longitude of its node. We study the evolution of its eccentricity, crucial to exclude orbital configurations with high probability of (quasi)collision with the central star in the long-time evolution of the system. Moreover, we also introduce a normal form approach, that further reduces our Hamiltonian model to a system with $2$ degrees of freedom, which is integrable because it admits a constant of motion related to the total angular momentum. This allows us to quickly preselect the domains of stability for $\upsilon$-And $b$, with respect to the set of the initial orbital configurations that are compatible with the observations.

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R. Mastroianni and U. Locatelli
Wed, 19 Apr 23
48/58

Comments: N/A

Accelerating universe at the end of time [CL]

http://arxiv.org/abs/2303.03418


We investigate whether an accelerating universe can be realized as an asymptotic late-time solution of FLRW-cosmology with multi-field multi-exponential potentials. Late-time cosmological solutions exhibit a universal behavior which enables us to bound the rate of time variation of the Hubble parameter. In string-theoretic realizations, if the dilaton remains a rolling field, our bound singles out a tension in achieving asymptotic late-time cosmic acceleration. Our findings go beyond previous no-go theorems in that they apply to arbitrary multi-exponential potentials and make no specific reference to vacuum or slow-roll solutions. We also show that if the late-time solution approaches a critical point of the dynamical system governing the cosmological evolution, the criterion for cosmic acceleration can be generally stated in terms of a directional derivative of the potential.

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G. Shiu, F. Tonioni and H. Tran
Mon, 10 Apr 23
30/36

Comments: 6 pages + appendix

The Hamiltonian for von Zeipel-Lidov-Kozai oscillations [EPA]

http://arxiv.org/abs/2304.01257


The Hamiltonian used in classical analyses of von Zeipel-Lidov-Kozai or ZLK oscillations in hierarchical triple systems is based on the quadrupole potential from a distant body on a fixed orbit, averaged over the orbits of both the inner and the outer bodies (“double-averaging”). This approximation can be misleading, because the corresponding Hamiltonian conserves the component of angular momentum of the inner binary normal to the orbit of the outer binary, thereby restricting the volume of phase space that the system can access. This defect is usually remedied by including the effects of the octopole potential, or by allowing the outer orbit to respond to variations in the inner orbit. However, in a wide variety of astrophysical systems nonlinear perturbations are comparable to or greater than these effects. The long-term effects of nonlinear perturbations are described by an additional Hamiltonian, which we call Brown’s Hamiltonian. At least three different forms of Brown’s Hamiltonian are found in the literature; we show that all three are related by a gauge freedom, although one is much simpler than the others. We argue that investigations of ZLK oscillations in triple systems should include Brown’s Hamiltonian.

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S. Tremaine
Wed, 5 Apr 23
32/62

Comments: 11 pages, 3 figures. To be published in Monthly Notices of the Royal Astronomical Society

Analytic frozen and other low eccentric orbits under J2 perturbation [EPA]

http://arxiv.org/abs/2212.09958


This work presents an analytical perturbation method to define and study the dynamics of frozen orbits under the perturbation effects produced by the oblatness of the main celestial body. This is done using a perturbation method purely based on osculating elements. This allows to characterize, define, and study the three existing families of frozen orbits in closed-form: the two families of frozen orbits close to the critical inclination, and the family of frozen orbits that appears at low values of eccentricity. To that end, this work includes the first and second order approximate solutions of the proposed perturbation method, including their applications to define frozen orbits, repeating ground-track orbits, and sun-synchronous orbits. Examples of application are also presented to show the expected error performance of the proposed approach.

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D. Arnas
Wed, 21 Dec 22
60/81

Comments: 41 pages, 19 figures

Analytic transformation from osculating to mean elements under J2 perturbation [EPA]

http://arxiv.org/abs/2212.08746


This work presents an analytical perturbation method to study the dynamics of an orbiting object subject to the term $J_2$ from the gravitational potential of the main celestial body. This is done using a power series expansion in the perturbation constant $J_2$ on all the variables of the system, and a time regularization based on the argument of latitude of the orbit. This enables the generation of analytic solutions without the need to control the perturbed frequency of the system. The resultant approach allows to approximate the dynamics of the system in osculating elements for orbits at any eccentricity, and to obtain the approximate analytical transformation from osculating to mean elements in these orbits. This includes near circular, elliptic, parabolic and hyperbolic orbits at any inclination. Several examples of application are presented to show the accuracy of the perturbation approach and their related transformations.

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D. Arnas
Tue, 20 Dec 22
17/97

Comments: 36 pages, 16 figures

An Effective Sign Switching Dark Energy: Lotka-Volterra Model of Two Interacting Fluids [CL]

http://arxiv.org/abs/2212.04429


One of the recent attempts to address the Hubble and $\sigma_8$ tensions is to consider the Universe started out not as a de Sitter-like spacetime, but rather anti-de Sitter-like. That is, the Universe underwent an “AdS-to-dS” transition at some point. We study the possibility that there are two dark energy fluids, one of which gave rise to the anti-de Sitter-like early Universe. The interaction is modeled by the Lotka-Volterra equations, commonly used in population biology. We consider “competition” models that are further classified as “unfair competition” and “fair competition”. The former involves a quintessence in competition with a phantom, and the second involves two phantom fluids. Surprisingly, even in the latter scenario it is possible for the overall dark energy to cross the phantom divide. The latter model also allows a constant $w$ “AdS-to-dS” transition, thus serving as a counter-example to the claim that such a dark energy must possess a singular equation of state. We also consider a “conversion” model in which a phantom fluid still manages to achieve “AdS-to-dS” transition even if it is being converted into a negative energy density quintessence. In these models, the energy density of the late time effective dark energy is related to the coefficient of the quadratic self-interaction term of the fluids, which is analogous to the resource capacity in population biology.

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Y. Ong
Fri, 9 Dec 22
69/75

Comments: N/A

A Lambert's Problem Solution via the Koopman Operator with Orthogonal Polynomials [CL]

http://arxiv.org/abs/2212.01390


Lambert’s problem has been long studied in the context of space operations; its solution enables accurate orbit determination and spacecraft guidance. This work offers an analytical solution to Lambert’s problem using the Koopman Operator (KO). In contrast to previous methods in the literature, the KO provides the analysis of a nonlinear system by seeking a transformation that embeds the nonlinear dynamics into a global linear representation. Our new methodology to solve for Lambert solutions considers the position of the system’s eigenvalues on the phase plane, evaluating accurate state transition polynomial maps for a computationally efficient propagation of the dynamics. The methodology used and multiple-revolution solutions found are compared in accuracy and performance with other techniques found in the literature, highlighting the benefits of the newly developed analytical approach over classical numerical methodologies.

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J. Pasiecznik, S. Servadio and R. Linares
Tue, 6 Dec 22
46/87

Comments: Conference Proceedings from the 73rd International Astronautical Congress

A Dynamical Systems Analysis of the Effects of the Launch Rate Distribution on the Stability of a Source-Sink Orbital Debris Model [EPA]

http://arxiv.org/abs/2212.01000


Future launches are projected to significantly increase both the number of active satellites and aggregate collision risk in Low Earth Orbit (LEO). In this paper, a dynamical systems theory approach is used to analyze the effect of launch rate distribution on the stability of the LEO environment. A multi-shell, three-species source-sink model of the LEO environment, referred to as MOCAT-3 for MIT Orbital Capacity Assessment Tool 3 Species, is used to study the evolution of the species populations. The three species included in the model are active satellites, derelict satellites, and debris. The model’s coefficients represent atmospheric drag, collision rate, mean satellite lifetime, post-mission disposal probability, and active debris removal rate. Solutions of the system of differential equations are computed, and an analysis of the stability of the equilibrium points is conducted for numerous launch rate distributions. The stability of the equilibrium points is used to test the sensitivity of the environment to run-away debris growth, known as Kessler syndrome, that occurs at the instability threshold. An analysis of the environment’s response to perturbations in launch rate and debris population is conducted. The maximum perturbation in the debris population from the equilibrium state, for which the system remains in a stable configuration, is calculated. Plots of the phase space about the equilibrium points are generated. The results will help to better understand the orbital capacity of LEO and the stability of the space environment, as well as provide improved guidelines on future launch plans to avoid detrimental congestion of LEO.

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C. Pasiecznik, A. D’Ambrosio, D. Jang, et. al.
Mon, 5 Dec 22
52/63

Comments: Conference Proceedings from the 73rd International Astronautical Congress

Regularization of the Hill four-body problem with oblate bodies [CL]

http://arxiv.org/abs/2209.13625


We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the fourth, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the small massive body passes through the zero value.

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E. Belbruno, M. Gidea and W. Lam
Thu, 29 Sep 22
40/70

Comments: N/A

Three-dimensional Lagrangian Coherent Structures in the Elliptic-Restricted Three-body Problem [EPA]

http://arxiv.org/abs/2209.11561


In the preliminary design of space missions it can be useful to identify regions of dynamics that drive the system’s behaviour or separate qualitatively different dynamics. The Lagrangian Coherent Structure (LCS) has been widely used in the analysis of dynamical systems, and generalises the concept of the stable and unstable manifolds to systems with arbitrary time-dependence. However, the use of three-dimensional LCS in astrodynamics has thus far been limited. This paper presents the application of a new numerical method introduced by the authors, DA-LCS, to astrodynamics systems using the Elliptic-Restricted Three-body Problem (ER3BP) as a test case. We are able to construct the full, three-dimensional LCS associated with the Sun-Mars ER3BP directly from the variational theory of LCS even for numerically challenging initial conditions. The LCS is analysed in detail, showing how it in this case separates regions of qualitatively different behaviour without any a priori knowledge. The paper then studies the effect of integration time and the parameterisation of the initial condition on the LCS found. We highlight how round-off errors arise from limits of floating-point arithmetic in the most challenging test cases and provide mitigating strategies for avoiding these errors practically.

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J. Tyler and A. Wittig
Mon, 26 Sep 22
5/62

Comments: 26 pages; submitted for publication at the journal of Celestial Mechanics and Dynamical Astronomy in September 2022

The Phase Space Structure in the vicinity of vertical Lyapunov orbits around L1,2 in a barred galaxy model [GA]

http://arxiv.org/abs/2209.10249


We study the phase space structure and the orbital diffusion from the vicinity of the vertical Lyapunov periodic orbits around the unstable Lagrangian points L1,2 in a 3D barred galaxy model. By perturbing the initial conditions of these periodic orbits, we detected the following five types of orbital structures in the 4D spaces of section: (i) Ring-like structures, sticky for large time intervals to the unstable invariant manifolds of the simple and double unstable vertical Lyapunov periodic orbits. (ii) 2D tori belonging to quasi-periodic orbits around stable periodic orbits existing in the region. They are associated either with vertical stable periodic orbits around L4,5 or with “stable anomalous” periodic orbits. (iii) Orbits sticky for large time intervals to these tori, forming “sticky tori”, before they slowly depart from them. (iv) Clouds of points that have a strong chaotic behavior. Such clouds of consequents have slow diffusion speeds, because they are hindered by the presence of the tori around the “stable anomalous” periodic orbits. (v) Toroidal zones consisting of points that stick for long time on the unstable invariant manifolds of the “unstable anomalous” periodic orbits. By continuing the integration, we find that eventually they become strongly chaotic, retaining however small diffusion speeds, due to the presence of the tori around the stable anomalous periodic orbits.

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M. M.Katsanikas and P. P.Patsis
Thu, 22 Sep 22
5/65

Comments: 13 pages, accepted for publication in MNRAS

Predicting the Stability of Hierarchical Triple Systems with Convolutional Neural Networks [SSA]

http://arxiv.org/abs/2206.12402


Understanding the long-term evolution of hierarchical triple systems is challenging due to its inherent chaotic nature, and it requires computationally expensive simulations. Here we propose a convolutional neural network model to predict the stability of hierarchical triples by looking at their evolution during the first $5 \times 10^5$ inner binary orbits. We employ the regularized few-body code TSUNAMI to simulate $5\times 10^6$ hierarchical triples, from which we generate a large training and test dataset. We develop twelve different network configurations that use different combinations of the triples’ orbital elements and compare their performances. Our best model uses 6 time-series, namely, the semimajor axes ratio, the inner and outer eccentricities, the mutual inclination and the arguments of pericenter. This model achieves an area under the curve of over $95\%$ and informs of the relevant parameters to study triple systems stability. All trained models are made publicly available, allowing to predict the stability of hierarchical triple systems $200$ times faster than pure $N$-body methods.

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F. Lalande and A. Trani
Mon, 5 Sep 22
27/53

Comments: 12 pages, 6 figures, accepted for publication in ApJ

Predicting the Stability of Hierarchical Triple Systems with Convolutional Neural Networks [SSA]

http://arxiv.org/abs/2206.12402


Understanding the long-term evolution of hierarchical triple systems is challenging due to its inherent chaotic nature, and it requires computationally expensive simulations. Here we propose a convolutional neural network model to predict the stability of hierarchical triples by looking at their evolution during the first $5 \times 10^5$ inner binary orbits. We employ the regularized few-body code TSUNAMI to simulate $5\times 10^6$ hierarchical triples, from which we generate a large training and test dataset. We develop twelve different network configurations that use different combinations of the triples’ orbital elements and compare their performances. Our best model uses 6 time-series, namely, the semimajor axes ratio, the inner and outer eccentricities, the mutual inclination and the arguments of pericenter. This model achieves an area under the curve of over $95\%$ and informs of the relevant parameters to study triple systems stability. All trained models are made publicly available, allowing to predict the stability of hierarchical triple systems $200$ times faster than pure $N$-body methods.

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F. Lalande and A. Trani
Mon, 5 Sep 22
15/53

Comments: 12 pages, 6 figures, accepted for publication in ApJ

A dynamical definition of the sphere of influence of the Earth [EPA]

http://arxiv.org/abs/2205.09340


The concept of sphere of influence of a planet is useful in both the context of impact monitoring of asteroids with the Earth and of the design of interplanetary trajectories for spacecrafts. After reviewing the classical results, we propose a new definition for this sphere that depends on the position and velocity of the small body for given values of the Jacobi constant $C$. Here we compare the orbit of the small body obtained in the framework of the circular restricted three-body problem, with orbits obtained by patching two-body solutions. Our definition is based on an optimisation process, minimizing a suitable target function with respect to the assumed radius of the sphere of influence. For different values of $C$ we represent the results in the planar case: we show the values of the selected radius as a function of two angles characterising the orbit. In this case, we also produce a database of radii of the sphere of influence for several initial conditions, allowing an interpolation.

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I. Cavallari, C. Grassi, G. Gronchi, et. al.
Fri, 20 May 22
62/65

Comments: 35 pages, 32 figures

Analytic Solution for Perturbed Keplerian Motion Under Small Acceleration Using Averaging Theory [EPA]

http://arxiv.org/abs/2204.06959


A novel approach is developed for analytic orbit propagation based on asymptotic expansion with respect to a small perturbative acceleration. The method improves upon existing first order asymptotic expansions by leveraging on linear systems and averaging theories. The solution starts with the linearization of Gauss planetary equations with respect to both the small perturbation and the six orbital elements. Then, an approximate solution is obtained in terms of secular and short period components. The method is tested on a low-thrust maneuver scenario consisting of a Keplerian orbit perturbed by a constant tangential acceleration, for which a solution can be obtained in terms of elliptic integrals. Results show that the positional propagation error is about one order of magnitude smaller with respect to state-of-the-art methods. The position accuracy for a LEO orbit, apart from pathological cases, is typically in the range of tens of meters for a dimensionless tangential acceleration of 1e-5 after 5 orbital periods propagation.

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G. Curzi and D. Modenini
Fri, 15 Apr 22
44/50

Comments: Article submitted for publication in Advances in Space Research

Design of optimal low-thrust manoeuvres for remote sensing multi-satellite formation flying in low Earth orbit [IMA]

http://arxiv.org/abs/2202.03998


This paper presents a strategy for optimal manoeuvre design of multi-satellite formation flying in low Earth orbit environment, with the aim of providing a tool for mission operation design. The proposed methodology for formation flying manoeuvres foresees a continuous low-thrust control profile, to enable the operational phases. The design is performed starting from the dynamic representation described in the relative orbital elements, including the main orbital perturbations effects. It also exploits an interface with the classical radial-transversal-normal description to include the maximum delta-v limitation and the safety condition requirements. The methodology is applied to a remote sensing mission study, Formation Flying L-band Aperture Synthesis, for land and ocean application, such as a potential high-resolution Soil Moisture and Ocean Salinity (SMOS) follow-on mission, as part of a European Space Agency mission concept study. Moreover, the results are applicable to a wide range of low Earth orbit missions, exploiting a distributed system, and in particular to Formation Flying L-band Aperture Synthesis (FFLAS) as a follow-on concept to SMOS.

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F. Scala, G. Gaias, C. Colombo, et. al.
Wed, 9 Feb 22
26/48

Comments: 19 pages

Spin-orbit coupling for close-in planets [EPA]

http://arxiv.org/abs/2112.14335


We study the spin evolution of close-in planets in multi-body systems and present a very general formulation of the spin-orbit problem. This includes a simple way to probe the spin dynamics from the orbital perturbations, a new method for computing forced librations and tidal deformation, and general expressions for the tidal torque and capture probabilities in resonance. We show that planet-planet perturbations can drive the spin of Earth-size planets into asynchronous or chaotic states, even for nearly circular orbits. We apply our results to Mercury and to the KOI-1599 system of two super-Earths in a 3/2 mean motion resonance.

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A. Correia and J. Delisle
Thu, 30 Dec 21
58/71

Comments: 12 pages, 9 figures

Spin-orbit coupling for close-in planets [EPA]

http://arxiv.org/abs/2112.14335


We study the spin evolution of close-in planets in multi-body systems and present a very general formulation of the spin-orbit problem. This includes a simple way to probe the spin dynamics from the orbital perturbations, a new method for computing forced librations and tidal deformation, and general expressions for the tidal torque and capture probabilities in resonance. We show that planet-planet perturbations can drive the spin of Earth-size planets into asynchronous or chaotic states, even for nearly circular orbits. We apply our results to Mercury and to the KOI-1599 system of two super-Earths in a 3/2 mean motion resonance.

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A. Correia and J. Delisle
Thu, 30 Dec 21
68/71

Comments: 12 pages, 9 figures

Nekhoroshev estimates for the orbital stability of Earth's satellites [EPA]

http://arxiv.org/abs/2112.06045


We provide stability estimates, obtained by implementing the Nekhoroshev theorem, in reference to the orbital motion of a small body (satellite or space debris) around the Earth. We consider a Hamiltonian model, averaged over fast angles, including the $J_2$ geopotential term as well as third-body perturbations due to Sun and Moon. We discuss how to bring the Hamiltonian into a form suitable for the implementation of the Nekhoroshev theorem in the version given by P\”oschel(1993) for the non-resonant' regime. The manipulation of the Hamiltonian includes i) averaging over fast angles, ii) a suitable expansion around reference values for the orbit's eccentricity and inclination, and iii) a preliminary normalization allowing to eliminate particular terms whose existence is due to the non-zero inclination of the invariant plane of secular motions known as theLaplace plane’. After bringing the Hamiltonian to a suitable form, we examine the domain of applicability of the theorem in the action space, translating the result in the space of physical elements. We find that the necessary conditions for the theorem to hold are fulfilled in some non-zero measure domains in the eccentricity and inclination plane (e, i) for a body’s orbital altitude (semi-major axis) up to about 20000 km. For altitudes around 11000 km we obtain stability times of the order of several thousands of years in domains covering nearly all eccentricities and inclinations of interest in applications of the satellite problem, except for narrow zones around some so-called `inclination-dependent’ resonances. On the other hand, the domains of Nekhoroshev stability recovered by the present method shrink in size as the semi-major axis a increases (and the corresponding Nekhoroshev times reduce to hundreds of years), while the stability domains practically all vanish for a > 20000 km.

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A. Celletti, I. Blasi and C. Efthymiopoulos
Tue, 14 Dec 21
39/98

Comments: N/A

On the stability of isothermal shocks in black hole accretion disks [HEAP]

http://arxiv.org/abs/2112.01665


Most black holes possess accretion disks. Models of such disks inform observations and constrain the properties of the black holes and their surrounding medium. Here, we study isothermal shocks in a thin black hole accretion flow. Modelling infinitesimal molecular viscosity allows the use of multiple-scales matched asymptotic methods. We thus derive the first explicit calculations of isothermal shock stability. We find that the inner shock is always unstable, and the outer shock is always stable. The growth/decay rates of perturbations depend only on an effective potential and the incoming–outgoing flow difference at the shock location. We give a prescription of accretion regimes in terms of angular momentum and black hole radius. Accounting for angular momentum dissipation implies unstable outer shocks in much of parameter space, even for realistic viscous Reynolds numbers of the order $\approx 10^{20}$.

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E. Hester, G. Vasil and M. Wechselberger
Mon, 6 Dec 21
28/61

Comments: 26 pages

Dynamics around Non-Spherical Symmetric Bodies: I. The case of a spherical body with mass anomaly [EPA]

http://arxiv.org/abs/2112.01817


The space missions designed to visit small bodies of the Solar System boosted the study of the dynamics around non-spherical bodies. In this vein, we study the dynamics around a class of objects classified by us as Non-Spherical Symmetric Bodies, including contact binaries, triaxial ellipsoids, spherical bodies with a mass anomaly, among others. In the current work, we address the results for a body with a mass anomaly. We apply the pendulum model to obtain the width of the spin-orbit resonances raised by non-asymmetric gravitational terms of the central object. The Poincare surface of section technique is adopted to confront our analytical results and to study the system’s dynamics by varying the parameters of the central object. We verify the existence of two distinct regions around an object with a mass anomaly: a chaotic inner region that extends beyond the corotation radius and a stable outer region. In the latter, we identify structures remarkably similar to those of the classical restrict and planar 3-body problem in the Poincare surface of sections, including asymmetric periodic orbits associated with 1:1+p resonances. We apply our results to a Chariklo with a mass anomaly, obtaining that Chariklo rings are probably related to first kind periodic orbits and not with 1:3 spin-orbit resonance, as proposed in the literature. We believe that our work presents the first tools for studying mass anomaly systems.

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G. Madeira, S. Winter, T. Ribeiro, et. al.
Mon, 6 Dec 21
34/61

Comments: Accepted 2021 December 02. Received 2021 November 10; in original form 2021 August 06 – MNRAS

The spatial Hill four-body problem I — An exploration of basic invariant sets [EPA]

http://arxiv.org/abs/2112.00135


In this work, we perform a first study of basic invariant sets of the spatial Hill’s four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter mu in such a way that the classical Hill’s problem is recovered when mu = 0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter mu by applying a classical predictor-corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill’s three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill’s three-body problem; these families have some desirable properties from a practical point of view.

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J. Burgos-Garcia, A. Bengochea and L. Franco-Perez
Thu, 2 Dec 21
41/61

Comments: N/A

Proper Elements for Space Debris [EPA]

http://arxiv.org/abs/2111.13835


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.

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A. Celletti, G. Pucacco and T. Vartolomei
Tue, 30 Nov 21
54/105

Comments: 42 pages, 15 figures, submitted to Celestial Mechanics and Dynamical Astronomy

New results on orbital resonances [EPA]

http://arxiv.org/abs/2111.09289


Perturbative analyses of planetary resonances commonly predict singularities and/or divergences of resonance widths at very low and very high eccentricities. We have recently re-examined the nature of these divergences using non-perturbative numerical analyses, making use of Poincar\’e sections but from a different perspective relative to previous implementations of this method. This perspective reveals fine structure of resonances which otherwise remains hidden in conventional approaches, including analytical, semi-analytical and numerical-averaging approaches based on the critical resonant angle. At low eccentricity, first order resonances do not have diverging widths but have two asymmetric branches leading away from the nominal resonance location. A sequence of structures called “low-eccentricity resonant bridges” connecting neighboring resonances is revealed. At planet-grazing eccentricity, the true resonance width is non-divergent. At higher eccentricities, the new results reveal hitherto unknown resonant structures and show that these parameter regions have a loss of some — though not necessarily entire — resonance libration zones to chaos. The chaos at high eccentricities was previously attributed to the overlap of neighboring resonances. The new results reveal the additional role of bifurcations and co-existence of phase-shifted resonance zones at higher eccentricities. By employing a geometric point of view, we relate the high eccentricity phase space structures and their transitions to the shapes of resonant orbits in the rotating frame. We outline some directions for future research to advance understanding of the dynamics of mean motion resonances.

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R. Malhotra
Thu, 18 Nov 21
27/92

Comments: To appear in Proceedings of IAU Symposium 364. Recording of related invited talk at the symposium is available on youtube at: this https URL

Analysis of the PPN Two-Body Problem Using Non-Osculating Orbital Elements [CL]

http://arxiv.org/abs/2111.07207


The parameterised post-Newtonian (PPN) formalism is a weak-field slow-motion approximation for both GR and some of its generalisations. It permits various parameterisations of the motion, among which are the Lagrange-type and Gauss-type orbital equations. Often, these equations are developed under the Lagrange constraint, which makes the evolving orbital elements parameterise instantaneous conics tangent to the orbit. Arbitrary mathematically, this choice of a constraint is convenient under perturbations dependent only on positions. Under perturbations dependent also on velocities (like relativistic corrections) the Lagrange constraint unnecessarily complicates solutions that can be simplified by introducing a freedom in the orbit parameterisation, which is analogous to the gauge freedom in electrodynamics and gauge theories. Geometrically, this freedom is the freedom of nonosculation, i.e. of the degree to which the instantaneous conics are permitted to be non-tangent to the actual orbit. Under the same perturbation, all solutions with different degree of nonosculation look mathematically different, though describe the same physical orbit. While non-intuitive, the modeling of an orbit with a sequence of nontangent instantaneous conics can at times simplify calculations. The appropriately generalised (“gauge-generalised”) Lagrange-type equations, and their applications, appeared in the literature hitherto. We in this paper derive the gauge-generalised Gauss-type equations and apply them to the PPN two-body problem. Fixing the gauge freedom in three different ways (i.e. modeling an orbit with non-osculating elements of three different types) we find three parameterisations of the PPN two-body dynamics. These parameterisations render orbits with either a fixed non-osculating semimajor axis, or a fixed non-osculating eccentricity, or a fixed non-osculating argument of periastron.

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P. Gurfil and M. Efroimsky
Tue, 16 Nov 21
73/97

Comments: N/A

Closed form perturbation theory in the restricted three-body problem without relegation [EPA]

http://arxiv.org/abs/2110.14489


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.

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I. Cavallari and C. Efthymiopoulos
Thu, 28 Oct 21
72/76

Comments: N/A

Closed-form perturbation theory in the Sun-Jupiter restricted three body problem without relegation [EPA]

http://arxiv.org/abs/2110.13131


We present a closed-form normalization method suitable for the study of the secular dynamics of small bodies inside the trajectory of Jupiter. The method is based on a convenient use of a book-keeping parameter introduced not only in the Lie series organization but also in the Poisson bracket structure employed in all perturbative steps. In particular, we show how the above scheme leads to a redefinition of the remainder of the normal form at every step of the formal solution of the homological equation. An application is given for the semi-analytical representation of the orbits of main-belt asteroids.

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I. Cavallari and C. Efthymiopoulos
Tue, 26 Oct 21
88/109

Comments: N/A

Using GPUs and the Parameterization Method for Rapid Search and Refinement of Connections between Tori in Periodically Perturbed Planar Circular Restricted 3-Body Problems [CL]

http://arxiv.org/abs/2109.14814


When the planar circular restricted 3-body problem is periodically perturbed, most unstable periodic orbits become invariant tori. However, 2D Poincar\’e sections no longer work to find their manifolds’ intersections; new methods are needed. In this study, we first review a method of restricting the intersection search to only certain manifold subsets. We then implement this search using Julia and OpenCL, representing the manifolds as triangular meshes and gaining a 30x speedup using GPUs. We finally show how to use manifold parameterizations to refine the approximate connections found in the mesh search. We demonstrate the tools on the planar elliptic RTBP.

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B. Kumar, R. Anderson and R. Llave
Fri, 1 Oct 21
19/65

Comments: 20 pages, 6 figures

Computation and Analysis of Jupiter-Europa and Jupiter-Ganymede Resonant Orbits in the Planar Concentric Circular Restricted 4-Body Problem [CL]

http://arxiv.org/abs/2109.14815


Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In this study, we compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-body problem (CCR4BP). Motivated by the 2:1 Laplace resonance between Europa and Ganymede’s orbits, we then attempt the continuation of a Jupiter-Europa 3:4 resonant orbit from the CCR4BP into the Jupiter-Ganymede PCRTBP. We strongly believe that the resulting dynamical object is a KAM torus lying near but not on the 3:2 Jupiter-Ganymede resonance.

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B. Kumar, R. Anderson, R. Llave, et. al.
Fri, 1 Oct 21
35/65

Comments: 20 pages, 18 figures

High-Order Resonant Orbit Manifold Expansions For Mission Design In the Planar Circular Restricted 3-Body Problem [CL]

http://arxiv.org/abs/2109.14800


In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections. In this paper, we study hyperbolic resonant periodic orbits in the planar circular restricted 3-body problem, and transfer trajectories between them, by: 1) determining where to search for resonant periodic orbits; 2) developing and implementing a parameterization method for accurate computation of their invariant manifolds as Taylor series; and 3) developing a procedure to compute intersections of the computed stable and unstable manifolds. We develop and implement algorithms that accomplish these three goals, and demonstrate their application to the problem of transferring between resonances in the Jupiter-Europa system.

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B. Kumar, R. Anderson and R. Llave
Fri, 1 Oct 21
46/65

Comments: 17 pages, 8 figures

On the stability of satellites at unstable libration points of sun-planet-moon systems [EPA]

http://arxiv.org/abs/2109.12612


The five libration points of a sun-planet system are stable or unstable fixed positions at which satellites or asteroids can remain fixed relative to the two orbiting bodies. A moon orbiting around the planet causes a time-dependent perturbation on the system. Here, we address the sense in which invariant structure remains. We employ a transition state theory developed previously for periodically driven systems with a rank-1 saddle in the context of chemical reactions. We find that a satellite can be parked on a so-called time-periodic transition state trajectory — which is an orbit restricted to the vicinity of the libration point L2 for infinitely long time — and investigate the stability properties of that orbit.

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J. Reiff, J. Zatsch, J. Main, et. al.
Tue, 28 Sep 21
19/89

Comments: Main article has 15 pages, 6 figures. arXiv admin note: text overlap with arXiv:2011.04029

Tadpole type motion of charged dust in the Lagrange problem with planet Jupiter [EPA]

http://arxiv.org/abs/2109.00800


We investigate the dynamics of charged dust interacting with the interplanetary magnetic field in a Parker spiral type model and subject to the solar wind and Poynting-Robertson effect in the vicinity of the 1:1 mean motion resonance with planet Jupiter. We estimate the shifts of the location of the minimum libration amplitude solutions close to the location of the L4 and L5 points of the classical – gravitational – problem and provide the extension of the ‘librational regimes of motion’ and the width of the resonance in dependency of the nongravitational parameters related to the dust grain size and surface potential of the particles. Our study is based on numerical simulations in the framework of the spatial, elliptic restricted three-body problem and semi-analytical estimates obtained by averaging of Gauss’ planetary equations of motion.

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C. Lhotka and L. Zhou
Fri, 3 Sep 21
3/52

Comments: Accepted manuscript in Communications in Nonlinear Science and Numerical Simulation Available online 1 September 2021, 106024, 32 pages, 14 figures

Rapid and Accurate Computation of Whiskered Tori and their Manifolds Near Resonances in Periodically Perturbed Planar Circular Restricted 3-Body Problems [CL]

http://arxiv.org/abs/2105.11100


When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable resonant 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 as well as 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 on the planar elliptic RTBP.

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B. Kumar, R. Anderson and R. Llave
Tue, 25 May 21
52/75

Comments: 34 pages, 6 figures

A study of the 1/2 retrograde resonance: Periodic orbits and resonant capture [EPA]

http://arxiv.org/abs/2104.05795


We describe the families of periodic orbits in the 2-dimensional 1/2 retrograde resonance at mass ratio 0.001, analyzing their stability and bifurcations into 3-dimensional periodic orbits. We explain the role played by periodic orbits in adiabatic resonance capture, in particular how the proximity between a stable family and an unstable family with a nearly critical segment, associated with Kozai separatrices, determines the transition between distinct resonant modes observed in numerical simulations. Combining the identification of stable, critical and unstable periodic orbits with analytical modeling, resonance capture simulations and computation of stability maps helps to unveil the complex 3-dimensional structure of resonances.

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M. Morais, F. Namouni, G. Voyatzis, et. al.
Wed, 14 Apr 2021
20/67

Comments: Accepted for publication in Celestial Mechanics and Dynamical Astronomy

Surface Gravity of Rotating Dumbbell Shapes [EPA]

http://arxiv.org/abs/2102.11990


We investigate the problem of determining the shape of a rotating celestial object – e.g., a comet or an asteroid – under its own gravitational field. More specifically, we consider an object symmetric with respect to one axis – such as a dumbbell – that rotates around a second axis perpendicular to the symmetry axis. We assume that the object can be modeled as an incompressible fluid of constant mass density, which is regarded as a first approximation of an aggregate of particles.
In the literature, the gravitational field of a body is often described as a multipolar expansion involving spherical coordinates (Kaula, 1966). In this work we describe the shape in terms of cylindrical coordinates, which are most naturally adapted to the symmetry of the body, and we express the gravitational potential generated by the rotating body as a simple formula in terms of elliptic integrals. An equilibrium shape occurs when the gravitational potential energy and the rotational kinetic energy at the surface of the body balance each other out. Such an equilibrium shape can be derived as a solution of an optimization problem, which can be found via the variational method. We give an example where we apply this method to a two-parameter family of dumbbell shapes, and find approximate numerical solutions to the corresponding optimization problem.

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W. Lam, M. Gidea and F. Zypman
Thu, 25 Feb 21
6/50

Comments: N/A

A Rapid Method For Orbital Coverage Statistics With $\mathbf{J_2}$ Using Ergodic Theory [EPA]

http://arxiv.org/abs/2102.03459


Quantifying long-term statistical properties of satellite trajectories typically entails time-consuming trajectory propagation. We present a fast, ergodic\cite{Arnold} method of analytically estimating these for $J_2-$perturbed elliptical orbits, broadly agreeing with trajectory propagation-derived results. We extend the approach in Graven and Lo (2019) to estimate: (1) Satellite-ground station coverage with limited satellite field of view and ground station elevation angle with numerically optimized formulae, and (2) long-term averages of general functions of satellite position. This method is fast enough to facilitate real-time, interactive tools for satellite constellation and network design, with an approximate $1000\times$ GPU speedup.

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A. Graven, A. Barr and M. Lo
Tue, 9 Feb 21
8/87

Comments: 18 pages, 8 figures, presented at the 2021 AAS Space Flight Mechanics Meeting

On Asteroid Retrieval Missions Enabled by Invariant Manifold Dynamics [EPA]

http://arxiv.org/abs/2101.07610


In recent years, the retrieval of entire asteroids has received significant attention, with many approaches leveraging the invariant manifolds of the Circular-Restricted Three-body Problem to capture an asteroid into a periodic orbit about the $L_1$ or $L_2$ points of the Sun-Earth system. Previous works defined an `Easily Retrievable Object’ (ERO) as any Near-Earth Object (NEO) which is retrievable using these invariant manifolds with an impulsive $\Delta v$ of less than $500$ m/s. We extend the previous literature by analysing the Pareto fronts for the EROs discovered for the first time, using high-performance computing to lift optimisation constraints used in previous literature, and modifying the method used to filter unsuitable NEOs from the NEO catalogue. In doing so, we can demonstrate that EROs have approximately the same transfer cost for almost any possible transfer time, including single impulse transfers, which could offer significant flexibility to mission designers. We also identify $44$ EROs, of which $27$ are new, and improve on previously-known transfer solutions by up to $443$ m/s, including $17$ new capture trajectories with $\Delta v$ costs of less than $100$ m/s.

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J. Tyler and A. Wittig
Wed, 20 Jan 21
23/61

Comments: Preprint submitted to Acta Astronautica on December 17, 2020

Semi-analytical estimates for the orbital stability of Earth's satellites [CL]

http://arxiv.org/abs/2101.05340


Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. i) We demonstrate the long term stability of the semimajor axis within the framework of the $J_2$ problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $H_{J_2}$ . ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $H_{gls}$), after a suitable reduction of the Hamiltonian to the Laplace plane. iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $H_{J_2}$ and $H_{gls}$ models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the $H_{J_2}$ model is non-convex, but satisfies a ‘three-jet’ condition, while the $H_{gls}$ model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.

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I. Blasi, A. Celletti and C. Efthymiopoulos
Fri, 15 Jan 21
49/60

Comments: 35 pages

Space Occupancy in Low-Earth Orbit [EPA]

http://arxiv.org/abs/2012.09240


With the upcoming launch of large constellations of satellites in the low-Earth orbit (LEO) region it will become important to organize the physical space occupied by the different operating satellites in order to minimize critical conjunctions and avoid collisions. Here, we introduce the definition of space occupancy as the domain occupied by an individual satellite as it moves along its nominal orbit under the effects of environmental perturbations throughout a given interval of time. After showing that space occupancy for the zonal problem is intimately linked to the concept of frozen orbits and proper eccentricity, we provide frozen-orbit initial conditions in osculating element space and obtain the frozen-orbit polar equation to describe the space occupancy region in closed analytical form. We then analyze the problem of minimizing space occupancy in a realistic model including tesseral harmonics, third-body perturbations, solar radiation pressure, and drag. The corresponding initial conditions, leading to what we call minimum space occupancy (MiSO) orbits, are obtained numerically for a set of representative configurations in LEO. The implications for the use of MiSO orbits to optimize the design of mega-constellations are discussed.

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C. Bombardelli, G. Falco, D. Amato, et. al.
Fri, 18 Dec 20
47/78

Comments: Submitted to The Journal of Guidance, Control, and Dynamics

Minimum-Time Earth-to-Mars Interplanetary Orbit Transfer Using Adaptive Gaussian Quadrature Collocation [CL]

http://arxiv.org/abs/2012.04116


The problem of minimum-time, low-thrust, Earth-to-Mars interplanetary orbital trajectory optimization is considered. The minimum-time orbital transfer problem is modeled as a four-phase optimal control problem where the four phases correspond to planetary alignment, Earth escape, heliocentric transfer, and Mars capture. The four-phase optimal control problem is then solved using an adaptive Gaussian quadrature collocation method. The following three models are used in the study: (1) circular planetary motion; (2) elliptic planetary motion; and (3) elliptic planetary motion with gravity perturbations. Results for all three cases are provided, and one particular case is studied in detail to show the key features of the optimal solutions. It was found that the minimum times for cases (1), (2), and (3) are, respectively, 215 d, 196 d, and 198 d with departure dates, respectively, of 1 July 2020, 30 June 2020, and 28 June 2020. Finally, the problem formulation developed in this study is compared against prior work on an Earth-to-Mars interplanetary orbit transfer where it is found that the results of this research show significant improvement in transfer time relative to the prior work.

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B. Holden, S. He and A. Rao
Wed, 9 Dec 20
24/80

Comments: 29 pages, 10 figures

Fast error-safe MOID computation involving hyperbolic orbits [IMA]

http://arxiv.org/abs/2011.12148


We extend our previous algorithm computing the minimum orbital intersection distance (MOID) to include hyperbolic orbits, and mixed combinations ellipse–hyperbola. The MOID is computed by finding all stationary points of the distance function, equivalent to finding all the roots of an algebraic polynomial equation of 16th degree. The updated algorithm carries about numerical errors as well, and benchmarks confirmed its numeric reliability together with high computing performance.

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R. Baluev
Wed, 25 Nov 2020
52/65

Comments: 15 pages, 5 figures, 2 tables; accepted by Astronomy & Computing

Ejection-collision orbits in two degrees of freedom problems in celestial mechanics [CL]

http://arxiv.org/abs/2008.06526


In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection-collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.

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M. Alvarez-Ramírez, E. Barrabés, M. Medina, et. al.
Tue, 18 Aug 20
-998/70

Comments: N/A

Multi-Fidelity Modelling of Low-Energy Trajectories for Space Mission Design [EPA]

http://arxiv.org/abs/2006.03872


The proposal of increasingly complex and innovative space endeavours poses growing demands for mission designers. In order to meet the established requirements and constraints while maintaining a low fuel cost, the use of low-energy trajectories is particularly interesting. These allow spacecraft to change orbits and move with little to no fuel, but they are computed using motion models of a higher fidelity than the commonly used two-body problem (2BP). For this purpose, perturbation methods that explore the third-body effect are especially attractive, since they can accurately convey the system dynamics of a three-body configuration with a lower computational cost, by employing mapping techniques or exploring analytical approximations.
The focus of this work is to broaden the knowledge of low-energy trajectories by developing new mathematical tools to assist in mission design applications. In particular, novel motion models based on the third-body effect are conceived. One application of this study focuses on the trajectory design for missions to near-Earth asteroids. Two different projects are explored: one is based on the preliminary design of separate rendezvous and capture missions to the invariant manifolds of libration point $L_2$. This is achieved by studying two recently discovered asteroids and determining dates, fuel cost and final control history for each trajectory. The other covers a larger study on asteroid capture missions, where several bodies are regarded as potential targets. The candidates are considered using a multi-fidelity design framework that filters through the trajectory options using models of motion of increasing accuracy, so that a final refined, low-thrust solution is obtained. The trajectory design hinges on harnessing Earth’s gravity by exploiting encounters outside its sphere of influence, the named Earth-resonant encounters.

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R. Neves
Tue, 9 Jun 20
88/94

Comments: PhD Thesis; including post-submission corrections to Section 3.2 (Eqs. 3.11-3.15)

Science orbits in the Saturn-Enceladus circular restricted three-body problem with oblate primaries [EPA]

http://arxiv.org/abs/2004.04708


This contribution investigates the properties of a category of orbits around Enceladus. The motivation is the interest in the in situ exploration of this moon following the detection on behalf of Cassini of plumes of water and organic compounds close to its south pole. In a previous investigation, a set of heteroclinic transfers were designed between Halo orbits around the equilibrium points L1 and L2 of the circular restricted three-body problem with Saturn and Enceladus as primaries. The kinematical and geometrical characteristics of those trajectories makes them good candidates as science orbits for the extended observation of the surface of Enceladus: they are highly inclined, they approach the moon and they are maneuver free. However, the low heights above the surface and the strong perturbing effect of Saturn impose a more careful look at their dynamics, in particular regarding the influence of the polar flattening of the primaries. Therefore, those solutions are here reconsidered by employing a dynamical model that includes the effect of the oblateness of Saturn and Enceladus, individually and in combination. Substitutes of the Halo orbits around the equilibrium points L1 and L2 and their stable and unstable hyperbolic invariant manifolds are obtained in the perturbed models, and maneuver-free heteroclinic transfers are identified in the new framework. A systematic comparison with the corresponding solutions of the unperturbed problem shows that qualitative and quantitative features are not significantly altered when the oblateness of the primaries is taken into account, and that J2 of Saturn plays a larger role than the oblateness of Enceladus. From a mission perspective, the results confirm the scientific value of the solutions obtained in the classical circular restricted three-body problem and suggests that this simpler model can be used in a preliminary feasibility analysis.

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F. Salazar, A. Alkhaja, E. Fantino, et. al.
Fri, 10 Apr 20
47/56

Comments: N/A

Assessing and Minimizing Collisions in Satellite Mega-Constellations [EPA]

http://arxiv.org/abs/2002.00430


We aim to provide satellite operators and researchers with an efficient means for evaluating and mitigating collision risk during the design process of mega-constellations. We first establish a baseline for evaluating various techniques for close-encounter prediction and collision-probability calculation (Hoots et al. 1984, Gronchi 2005, JeongAhn and Malhotra 2015) by carrying out brute-force numerical simulations and using a sequence of filters to greatly reduce the computational expense of the algorithm. Next, we estimate conjunction events in the orbital environment following the anticipated deployments of the OneWeb LEO and SpaceX Starlink mega-constellations. As a final step, we investigate Minimum Space Occupancy (MiSO) orbits (Bombardelli et al. 2018), a generalization of the well-known frozen orbits that account for the perturbed-Keplerian dynamics of the Earth-Moon-Sun-satellite system. We evaluate the ability of MiSO configurations of the proposed mega-constellations, as suggested by Bombardelli et al. 2018, to reduce the risk of endogenous (intra-constellation) collisions. The results indicate that the adoption of the MiSO orbital configuration can significantly reduce risk with nearly indistinguishable adjustments to the nominal orbital elements of the constellation satellites.

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N. Reiland, A. Rosengren, R. Malhotra, et. al.
Tue, 4 Feb 20
8/52

Comments: 32 pages, 19 figures; submitted for publication in Adv. Space Res.; comments welcome

Global time-regularization of the gravitational $N$-body problem [CL]

http://arxiv.org/abs/2001.01221


This work considers the gravitational $N$-body problem and introduces time-reparametrization functions that allow to define globally solutions of the $N$-body equations. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-reparametrization. In the new fictitious time $\tau$, it is then proved that any solution exists for all $\tau \in \mathbb{R}$, and that it is uniquely extended as a holomorphic function to a strip of fixed width. As a by-product, a global power series representation of the solutions of the $N$-body problem is obtained. Noteworthy, our global time-regularization remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new time-regularization functions for some $N$-problems with close encounters.

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M. Antoñana, P. Chartier, J. Makazaga, et. al.
Tue, 7 Jan 20
25/71

Comments: N/A

Disassociation energies for the finite density N-body problem [EPA]

http://arxiv.org/abs/1912.09032


This paper considers the energy required for collections of finite density bodies to undergo escape under internal gravitational interactions alone. As the level of the system energy is increased there are different combinations of components that can escape, until the total energy becomes positive, when the entire system can undergo mutual disruption. The results are also defined for bodies modeled as a continuum. These results provide rigorous constraints for the disruption of rubble pile asteroids when only considering gravitational interaction effects, with the energy provided by rotation of an initial collection of the system. These issues are considered for discrete particles in the N body problem and for size distributions of discrete particles in the continuum limit.

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D. Scheeres
Fri, 20 Dec 19
8/63

Comments: Accepted in Celestial Mechanics and Dynamical Astronomy, Topical Collection of the 50th Anniversary Issue

Resonances in the Earth's Space Environment [EPA]

http://arxiv.org/abs/1912.04593


We study the presence of resonances in the region of space around the Earth. We consider a massless body (e.g, a dust particle or a small space debris) subject to different forces: the gravitational attraction of the geopotential, the effects of Sun and Moon. We distinguish different types of resonances: tesseral resonances are due to a commensurability involving the revolution of the particle and the rotation of the Earth, semi-secular resonances include the rates of variation of the mean anomalies of Moon and Sun, while secular resonances just depend on the rates of variation of the arguments of perigee and the longitudes of the ascending nodes of the perturbing bodies. We characterize such resonances, giving precise statements on the regions where the resonances can be found and provide examples of some specific commensurability relations.

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A. Celletti, C. Gales and C. Lhotka
Wed, 11 Dec 19
64/69

Comments: 38 pages, 9 figures, submitted to Communications in Nonlinear Science and Numerical Simulation

Embedded operator splitting methods for perturbed systems [EPA]

http://arxiv.org/abs/1912.03255


It is common in classical mechanics to encounter systems whose Hamiltonian $H$ is the sum of an often exactly integrable Hamiltonian $H_0$ and a small perturbation $\epsilon H_1$ with $\epsilon\ll1$. Such near-integrability can be exploited to construct particularly accurate operator splitting methods to solve the equations of motion of $H$. However, in many cases, for example in problems related to planetary motion, it is computationally expensive to obtain the exact solution to $H_0$.
In this paper we present a new family of embedded operator splitting (EOS) methods which do not use the exact solution to $H_0$, but rather approximate it with yet another, embedded operator splitting method. Our new methods have all the desirable properties of classical methods which solve $H_0$ directly. But in addition they are very easy to implement and in some cases faster. When applied to the problem of planetary motion, our EOS methods have error scalings identical to that of the often used Wisdom-Holman method but do not require a Kepler solver, nor any coordinate transformations, or the allocation of memory. The only two problem specific functions that need to be implemented are the straight-forward kick and drift steps typically used in the standard second order leap-frog method.

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H. Rein
Mon, 9 Dec 19
5/53

Comments: 7 pages, 4 figures, submitted to MNRAS

Charged dust close to outer mean-motion resonances in the heliosphere [EPA]

http://arxiv.org/abs/1911.02778


We investigate the dynamics of charged dust close to outer mean-motion resonances with planet Jupiter. The importance of the interplanetary magnetic field on the orbital evolution of dust is clearly demonstrated. New dynamical phenomena are found that do not exist in the classical problem of uncharged dust. We find changes in the orientation of the orbital planes of dust particles, an increased amount of chaotic orbital motions, sudden ‘jumps’ in the resonant argument, and a decrease in time of temporary capture due to the Lorentz force. Variations in the orbital planes of dust grain orbits are found to be related to the angle between the orbital angular momentum and magnetic axes of the heliospheric field and the rotation rate of the Sun. These variations are bound using a simplified model derived from the full dynamical problem using first order averaging theory. It is found that the interplanetary magnetic field does not affect the capture process, that is still dominated by the other non-gravitational forces. Our study is based on a dynamical model in the framework of the inclined circular restricted three-body problem. Additional forces include solar radiation pressure, solar wind drag, the Poynting-Robertson effect, and the influence of a Parker spiral type interplanetary magnetic field model. The analytical estimates are derived on the basis of Gauss’ form of planetary equations of motion. Numerical results are obtained by simulations of dust grain orbits together with the system of variational equations. Chaotic regions in phase space are revealed by means of Fast Lyapunov Chaos Indicators.

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C. Lhotka and C. Gales
Mon, 11 Nov 19
81/105

Comments: 31 pages, 13 figures, 1 table

Periodic orbits of the retrograde coorbital problem [EPA]

http://arxiv.org/abs/1910.09042


Asteroid (514107) Kaepaokaawela is the first example of an object in the 1/1 mean motion resonance with Jupiter with retrograde motion around the Sun. Its orbit was shown to be stable over the age of the Solar System which implies that it must have been captured from another star when the Sun was still in its birth cluster. Kaepaokaawela orbit is also located at the peak of the capture probability in the coorbital resonance. Identifying the periodic orbits that Kaepaokaawela and similar asteroids followed during their evolution is an important step towards precisely understanding their capture mechanism. Here, we find the families of periodic orbits in the two-dimensional retrograde coorbital problem and analyze their stability and bifurcations into three-dimensional periodic orbits. Our results explain the radical differences observed in 2D and 3D coorbital capture simulations. In particular, we find that analytical and numerical results obtained for planar motion are not always valid at infinitesimal deviations from the plane.

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M. Morais and F. Namouni
Tue, 22 Oct 19
13/91

Comments: 7 pages, to appear in Monthly Notices of the Royal Astronomical Society

Real-Time Thermospheric Density Estimation Via Two-Line-Element Data Assimilation [CL]

http://arxiv.org/abs/1910.00695


Inaccurate estimates of the thermospheric density are a major source of error in low Earth orbit prediction. To improve orbit prediction, real-time density estimation is required. In this work, we develop a reduced-order dynamic model for the thermospheric density by computing the main spatial modes of the atmosphere and deriving a linear model for the dynamics. The model is then used to estimate the density using two-line element (TLE) data by simultaneously estimating the reduced-order modes and the orbits and ballistic coefficients of several objects using an unscented Kalman filter. Accurate density estimation using the TLEs of 17 objects is demonstrated and validated against CHAMP and GRACE accelerometer-derived densities. Finally, the use of the model for density forecasting is shown.

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D. Gondelach and R. Linares
Thu, 3 Oct 19
25/59

Comments: 24 pages, 13 figures, preprint

Chaotic transport of navigation satellites [EPA]

http://arxiv.org/abs/1909.11531


Navigation satellites are known from numerical studies to reside in a dynamically sensitive environment, which may be of profound importance for their long-term sustainability. We derive the fundamental Hamiltonian of GNSS dynamics and show analytically that near-circular trajectories lie in the neighborhood of a Normally Hyperbolic Invariant Manifold (NHIM), which is the primary source of hyperbolicity. Quasi-circular orbits escape through chaotic transport, regulated by the NHIM’s stable and unstable manifolds, following a power-law escape time distribution $P(t) \sim t^{-\alpha}$, with $\alpha \sim 0.8 – 1.5$. Our study is highly relevant for the design of satellite disposal trajectories, using manifold dynamics.

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I. Gkolias, J. Daquin, D. Skoulidou, et. al.
Thu, 26 Sep 19
23/61

Comments: Accepted for publication in Chaos: An Interdisciplinary Journal of Nonlinear Science

Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance [EPA]

http://arxiv.org/abs/1909.09462


Extrasolar systems with planets on eccentric orbits close to or in mean-motion resonances are common. The classical low-order resonant Hamiltonian expansion is unfit to describe the long-term evolution of these systems. We extend the Laplace-Lagrange secular approximation for coplanar systems with two planets by including (near-)resonant harmonics, and realize an expansion at high order in the eccentricities of the resonant Hamiltonian both at orders one and two in the masses. We show that the expansion at first order in the masses gives a qualitative good approximation of the dynamics of resonant extrasolar systems with moderate eccentricities, while the second order is needed to reproduce more accurately their orbital evolutions. The resonant approach is also required to correct the secular frequencies of the motion given by the Laplace-Lagrange secular theory in the vicinity of a mean-motion resonance. The dynamical evolutions of four (near-)resonant extrasolar systems are discussed, namely GJ 876 (2:1 resonance), HD 60532 (3:1), HD 108874 and GJ 3293 (close to 4:1).

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M. Sansottera and A. Libert
Mon, 23 Sep 19
38/46

Comments: 21 pages, 7 figures

Mathematical theory of physical vector fields [HEAP]

http://arxiv.org/abs/1909.04836


We study the dynamics and statistics of real vector fields in flat (n+1)-dimensional space-time with an emphasis on the field topology and stochasticity in physical applications such as stochastic magnetic and velocity fields in cosmological systems. We show that the natural field topology defined by the metric, induced by the Euclidean vector norm, is physically implausible. However, any vector field corresponds to a dynamical system with a topology in the corresponding phase space. This phase space topology, unlike the natural topology in Euclidean space, is defined using open balls that contain nearby vectors in both vector space and real space and hence is more physical. In addition, it is preserved under time translation if certain conditions including time reversal invariance are satisfied by the field. If these mathematical conditions are not satisfied, therefore, the field’s topology can spontaneously change as the field evolves in time. In this context, similar to topological entropy, which measures the complexity of a dynamical system in the phase space, a simple quantity is defined for a vector field which measures its spatial complexity in real space. For stochastic fields, this spatial complexity can be taken as a measure of the field’s stochasticity level. Generalizing a previous work based on renormalization group invariance, we show that corresponding to any arbitrary vector field, there exists a scalar field whose properties provide a means to quantify the vector field’s spatial complexity, stochasticity level and dissipation rate.

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A. Jafari and E. Vishniac
Thu, 12 Sep 19
62/84

Comments: N/A

On the accuracy of symplectic integrators for secularly evolving planetary systems [EPA]

http://arxiv.org/abs/1908.03468


Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20Myr integrations of the Solar System using various symplectic integrators.
We find that the specific choice of metric for determining a simulation’s accuracy is important. Only looking at metrics related to integrals of motions such as the energy error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the accuracy of secular frequencies compared to the standard Wisdom-Holman method without symplectic correctors, despite the fact that the energy error is three orders of magnitudes smaller. We present a framework to trace the origin of this apparent paradox to one term in the shadow Hamiltonian. Specifically, we find a term that leads to negligible contributions to the energy error but introduces non-oscillatory errors that result in artificial periastron precession. This term is the dominant error when determining secular frequencies of the system. We show that higher order symplectic methods such as the Wisdom-Holman method with a modified kernel or the SABAC family of integrators perform significantly better in secularly evolving systems because they remove this specific term.

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H. Rein, G. Brown and D. Tamayo
Mon, 12 Aug 19
30/42

Comments: 13 pages, 7 figures, submitted

Stable attitude dynamics of planar helio-stable and drag-stable sails [EPA]

http://arxiv.org/abs/1907.06908


In this paper the planar orbit and attitude dynamics of an uncontrolled spacecraft is studied, taking on-board a deorbiting device. Solar and drag sails with the same shape are considered and separately studied. In both cases, these devices are assumed to have a simplified pyramidal shape that endows the spacecraft with helio and drag stable properties. The translational dynamics is assumed to be planar and hence the rotational dynamics occurs only around one of the principal axes of the spacecraft. Stable or slowly-varying attitudes are studied, subject to disturbances due to the Earth oblateness effect and gravity gradient torques, and either solar radiation pressure or atmospheric drag torque and acceleration. The results are analysed with respect to the aperture of the sail and the center of mass – center of pressure offset.

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N. Miguel and C. Colombo
Wed, 17 Jul 19
71/75

Comments: 28 pages, 31 figures

Prediction of System Evolution by Learning Machine [CL]

http://arxiv.org/abs/1905.08313


The orthodox approach for understanding a dynamical system is to establish its equation of motion, by which one can unveil its dynamical behavior at a given system parameter set, and reveal how the dynamic behavior evolves as the system parameters change. Here we show that this task can be fulfilled with a learning machine in a model-free way. We find that, based only on a segmental time series of a state variable recorded at present stage, the dynamics exhibited by the learning machine at different training stages can be mapped to the dynamics of the target system along a particular path in its parameter space following an appropriate training strategy that monotonously decreases the cost. This path is important, because along which the primary dynamical properties of the target system will emerge subsequently, in the simple-to-complex order, matching closely to the evolution of a natural system. A theoretical framework is proposed to explain the underlying mechanism. This revealed function of the learning machine opens up a novel way to probe the global dynamical properties of a black-box system without the equation of motion established artificially, and as such it might have huge potential applications. As an application example, this method is applied to infer what dynamical stages a variable star has experienced and how it will evolve in future by using the light curve observed presently.

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H. Zhao
Wed, 22 May 19
48/59

Comments: 16 pages, 10 figures

Accurate modelling of the low-order secondary resonances in the spin-orbit problem [EPA]

http://arxiv.org/abs/1904.07047


We provide an analytical approximation to the dynamics in each of the three most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating from the synchronous primary resonance in the gravitational spin-orbit problem. To this end we extend the perturbative approach introduced in Gkolias et. al. (2016), based on normal form series computations. This allows to recover analytically all non-trivial features of the phase space topology and bifurcations associated with these resonances. Applications include the characterization of spin states of irregular planetary satellites or double systems of minor bodies with irregular shapes. The key ingredients of our method are: i) the use of a detuning parameter measuring the distance from the exact resonance, and ii) an efficient scheme to `book-keep’ the series terms, which allows to simultaneously treat all small parameters entering the problem. Explicit formulas are provided for each secondary resonance, yielding i) the time evolution of the spin state, ii) the form of phase portraits, iii) initial conditions and stability for periodic solutions, and iv) bifurcation diagrams associated with the periodic orbits. We give also error estimates of the method, based on analyzing the asymptotic behavior of the remainder of the normal form series.

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I. Gkolias, C. Efthymiopoulos, A. Celletti, et. al.
Tue, 16 Apr 19
67/88

Comments: Accepted for publication in Communications in Nonlinear Science and Numerical Simulation

Attitude and orbit coupling of planar helio-stable solar sails [CL]

http://arxiv.org/abs/1904.00436


The coupled attitude and orbit dynamics of solar sails is studied. The shape of the sail is a simplified quasi-rhombic-pyramid that provides the structure helio-stablility properties. After adimensionalisation the system is put in the form of a fast-slow dynamical system where the different time scales are explicitely related to the physical parameters of the system. The orientation of the body frame with respect to the inertial orbit frame is a fast phase that can be averaged out. This gives rise to a simplified formulation that only consists of the orbit dynamics perturbed by a flat sail with fixed attitude perpendicular to the direction of the sunlight. The results are exemplified using numerical simulations.

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N. Miguel and C. Colombo
Tue, 2 Apr 19
7/90

Comments: 31 pages, 23 figures

Hybrid Symplectic Integrators for Planetary Dynamics [EPA]

http://arxiv.org/abs/1903.04972


Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.

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H. Rein, D. Hernandez, D. Tamayo, et. al.
Wed, 13 Mar 19
107/125

Comments: Accepted for publication in MNRAS, 9 pages, 2 figures

Equilibrium points and basins of convergence in the triangular restricted four-body problem with a radiating body [CL]

http://arxiv.org/abs/1812.08641


The dynamics of the four-body problem have attracted increasing attention in recent years. In this paper, we extend the basic equilateral four-body problem by introducing the effect of radiation pressure, Poynting-Robertson drag, and solar wind drag. In our setup, three primaries lay at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we find that in all cases the libration points are unstable if the radiation factor is larger than 0.01 and hence able to destroy the stability of the libration points in the restricted four-body problem composed by Sun, Jupiter, Trojan asteroid and a test (dust) particle. Also, we conclude that the number of fixed points decreases with the increase of the radiation factor.

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J. Osorio-Vargas, G. González and F. Dubeibe
Fri, 21 Dec 18
6/72

Comments: 15 pages, 10 figures, Preprint submitted to Elsevier

Element sets for high-order Poincaré mapping of perturbed Keplerian motion [CL]

http://arxiv.org/abs/1810.11684


The propagation and Poincar\’e mapping of perturbed Keplerian motion is a key topic in celestial mechanics and astrodynamics, e.g. to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits over many revolutions. For this, the method uses the high-order Taylor expansion of a Poincar\’e or stroboscopic map, which is accurate close to the expansion point. In this paper, we investigate the performance of the HOTM method using different element sets for building the high-order map. The element sets investigated are the classical orbital elements, modified equinoctial elements, Hill variables, cylindrical coordinates and Deprit’s ideal elements. The performances of the different coordinate sets are tested by comparing the accuracy and efficiency of mapping low-Earth and highly-elliptical orbits perturbed by $J_2$ with numerical propagation. The accuracy of HOTM depends strongly on the choice of elements and type of orbit. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and centre manifolds of Poincar\’e maps.

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D. Gondelach and R. Armellin
Tue, 30 Oct 18
23/73

Comments: Pre-print of journal article

Nonlinear librations of distant retrograde orbits: a perturbative approach — The Hill problem case [CL]

http://arxiv.org/abs/1807.06095


The non-integrability of the Hill problem makes that its global dynamics must be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem close to the origin, and the libration point dynamics have been thoroughly investigated by perturbation methods. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes to progress in the perturbation approach beyond the lower orders of the solution really difficult. We propose an alternative perturbation approach that allows us to provide a very simple low order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.

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M. Lara
Wed, 18 Jul 18
39/90

Comments: 42 pages, 16 figures

On the co-orbital motion in Three-Body Problem: Existence of quasi-periodic Horseshoe-shaped orbits [CL]

http://arxiv.org/abs/1806.07262


Janus and Epimetheus are two co-orbital moons of Saturn (the co-orbital motion is associated with trajectories in 1:1 mean-motion resonance) which exhibit a peculiar dynamics associated with horseshoe-shaped trajectories. As they orbit Saturn on quasi-coplanar and quasi-circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four years the bodies are getting closer and their mutual gravitational influence leads to a swapping of the orbits. The outer moon becoming the inner one and vice-versa, this behavior generates horseshoe-shaped trajectories depicted in an adequate rotating frame.
In spite of analytical theories developed to describe their long-term dynamics as well as the indications provided by some numerical investigations, so far no rigorous long time stability results have been obtained even in the restricted three-body problem.
Following the idea of Arnol’d (1963) (but in a much more tricky context as it is 1:1-resonant, while Arnol’d situation relies on non-resonant Kepler orbits), we provide a rigorous proof of the existence of 2 dimensional-elliptic invariant tori associated with the Janus and Epimetheus horseshoe motion in the planar three-body problem using KAM theory.

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L. Niederman, A. Pousse and P. Robutel
Wed, 20 Jun 18
18/58

Comments: 40 pages, 5 figures

The Mid Pleistocene Transition from a Budyko-Sellers Type Energy Balance Model [EPA]

http://arxiv.org/abs/1806.03900


A conceptual model of the Plio-Pleistocene glacial cycles is developed based on the Budyko-Sellers type energy balance model. The model is shown to admit a phenomenon like the Mid-Pleistocene transition, capturing the essence of the albedo and the temperature precipitation feedbacks.

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E. Widiasih, M. Stuecker and S. Baek
Tue, 12 Jun 18
52/79

Comments: N/A

New general parametrization of quintessence fields and its observational constraints [CL]

http://arxiv.org/abs/1803.09204


We present a new parameterization of quintessence potentials for dark energy based directly upon the dynamical properties of the equations of motion. Such parameterization arises naturally once the equations of motion are written as a dynamical system in terms of properly defined polar variables. We have identified two different classes of parameters, and we dubbed them as dynamical and passive parameters. The dynamical parameters appear explicitly in the equations of motion, but the passive parameters play just a secondary role in their solutions. The new approach is applied to the so-called thawing potentials and it is argued that only three dynamical parameters are sufficient to capture the evolution of the quintessence fields at late times. This work reconfirms the arbitrariness of the quintessence potentials as the recent observational data fail to constrain the dynamical parameters.

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N. Roy, A. Gonzalez-Morales and L. Urena-Lopez
Fri, 6 Apr 18
47/49

Comments: 15 pages, 10 figures

Dynamical systems applied to cosmology: dark energy and modified gravity [CL]

http://arxiv.org/abs/1712.03107


The Nobel Prize winning confirmation in 1998 of the accelerated expansion of our Universe put into sharp focus the need of a consistent theoretical model to explain the origin of this acceleration. As a result over the past two decades there has been a huge theoretical and observational effort into improving our understanding of the Universe. The cosmological equations describing the dynamics of a homogeneous and isotropic Universe are systems of ordinary differential equations, and one of the most elegant ways these can be investigated is by casting them into the form of dynamical systems. This allows the use of powerful analytical and numerical methods to gain a quantitative understanding of the cosmological dynamics derived by the models under study. In this review we apply these techniques to cosmology. We begin with a brief introduction to dynamical systems, fixed points, linear stability theory, Lyapunov stability, centre manifold theory and more advanced topics relating to the global structure of the solutions. Using this machinery we then analyse a large number of cosmological models and show how the stability conditions allow them to be tightly constrained and even ruled out on purely theoretical grounds. We are also able to identify those models which deserve further in depth investigation through comparison with observational data. This review is a comprehensive and detailed study of dynamical systems applications to cosmological models focusing on the late-time behaviour of our Universe, and in particular on its accelerated expansion. In self contained sections we present a large number of models ranging from canonical and non-canonical scalar fields, interacting models and non-scalar field models through to modified gravity scenarios. Selected models are discussed in details and interpreted in the context of late-time cosmology.

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S. Bahamonde, C. Boehmer, S. Carloni, et. al.
Mon, 11 Dec 17
40/62

Comments: 176 pages, 35 figures, 29 tables. Comments and suggestions for references are welcome, please note Section 1.4 `Note to the arXiv reader’

Chaotic dynamics in the (47171) Lempo triple system [EPA]

http://arxiv.org/abs/1710.08401


We investigate the dynamics of the (47171) Lempo triple system, also known by 1999TC$_{36}$. We derive a full 3D $N$-body model that takes into account the orbital and spin evolution of all bodies, which are assumed triaxial ellipsoids. We show that, for reasonable values of the shapes and rotational periods, the present best fitted orbital solution is chaotic and unstable in short time-scales. The formation mechanism of this system is unknown, but the orbits can be stabilised when tidal dissipation is taken into account. The dynamics of this system is very rich, but depends on many parameters that are presently unknown. A better understanding of this systems thus requires more observations, which also need to be fitted with a complete model like the one presented here.

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A. Correia
Tue, 24 Oct 17
55/68

Comments: 43 pages, 8 figures

Dynamics of resonances and equilibria of Low Earth Objects [EPA]

http://arxiv.org/abs/1710.02519


The nearby space surrounding the Earth is densely populated by artificial satellites and instruments, whose orbits are distributed within the Low-Earth-Orbit region (LEO), ranging between 90 and 2 000 $km$ of altitude. As a consequence of collisions and fragmentations, many space debris of different sizes are left in the LEO region. Given the threat raised by the possible damages which a collision of debris can provoke with operational or manned satellites, the study of their dynamics is nowadays mandatory. This work is focused on the existence of equilibria and the dynamics of resonances in LEO. We base our results on a simplified model which includes the geopotential and the atmospheric drag. Using such model, we make a qualitative study of the resonances and the equilibrium positions, including their location and stability. The dissipative effect due to the atmosphere provokes a tidal decay, but we give examples of different behaviors, precisely a straightforward passage through the resonance or rather a temporary capture. We also investigate the effect of the solar cycle which is responsible of fluctuations of the atmospheric density and we analyze the influence of Sun and Moon on LEO objects.

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A. Celletti and C. Gales
Tue, 10 Oct 17
19/70

Comments: 39 pages, 10 figures

Dynamical Evolution Induced by Planet Nine [EPA]

http://arxiv.org/abs/1710.01804


The observational census of trans-Neptunian objects with semi-major axes greater than ~250 AU exhibits unexpected orbital structure that is most readily attributed to gravitational perturbations induced by a yet-undetected, massive planet. Although the capacity of this planet to (i) reproduce the observed clustering of distant orbits in physical space, (ii) facilitate dynamical detachment of their perihelia from Neptune, and (iii) excite a population of long-period centaurs to extreme inclinations is well established through numerical experiments, a coherent theoretical description of the dynamical mechanisms responsible for these effects remains elusive. In this work, we characterize the dynamical processes at play, from semi-analytic grounds. We begin by considering a purely secular model of orbital evolution induced by Planet Nine, and show that it is at odds with the ensuing stability of distant objects. Instead, the long-term survival of the clustered population of long-period KBOs is enabled by a web of mean-motion resonances driven by Planet Nine. Then, by taking a compact-form approach to perturbation theory, we show that it is the secular dynamics embedded within these resonances that regulates the orbital confinement and perihelion detachment of distant Kuiper belt objects. Finally, we demonstrate that the onset of large-amplitude oscillations of orbital inclinations is accomplished through capture of low-inclination objects into a high-order secular resonance and identify the specific harmonic that drives the evolution. In light of the developed qualitative understanding of the governing dynamics, we offer an updated interpretation of the current observational dataset within the broader theoretical framework of the Planet Nine hypothesis.

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K. Batygin and A. Morbidelli
Fri, 6 Oct 17
48/51

Comments: 22 pages, 13 figures, accepted for publication in the Astronomical Journal

On the planar central configurations of rhomboidal and triangular four- and five-body problems [EPA]

http://arxiv.org/abs/1708.07664


We consider a symmetric five-body problem with three unequal collinear masses on the axis of symmetry. The remaining two masses are symmetrically placed on both sides of the axis of symmetry. Regions of possible central configurations are derived for the four- and five-body problems. These regions are determined analytically and explored numerically. The equations of motion are regularized using Levi-Civita type transformations and then the phase space is investigated for chaotic and periodic orbits by means of Poincar\’e surface of sections.

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M. Shoaib, A. Kashif and I. Szucs-Csillik
Mon, 28 Aug 17
34/46

Comments: 22 figures

Superspace and global stability in General Relativity [CL]

http://arxiv.org/abs/1703.06615


A framework is developed enabling the global analysis of the stability of cosmological models using the local geometric characteristics of the infinite-dimensional superspace, i.e. using the generalised Jacobi equation reformulated for pseudo-Riemannian manifolds. We give a direct formalism for dynamical analysis in the superspace, the requisite equation pertinent for stability analysis of the universe by means of generalized covariant and Fermi derivative is derived. Then, the relevant definitions and formulae are retrieved for cosmological models with a scalar field.

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A. Gurzadyan and A. Kocharyan
Wed, 22 Mar 2017
16/65

Comments: 7 pages, 2 figures, to appear in IJMPD April issue

Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces [EPA]

http://arxiv.org/abs/1703.05825


In a realistic scenario, the evolution of the rotational dynamics of a celestial or artificial body is subject to dissipative effects. Time-varying non-conservative forces can be due to, for example, a variation of the moments of inertia or to tidal interactions. In this work, we consider a simplified model describing the rotational dynamics, known as the spin-orbit problem, where we assume that the orbital motion is provided by a fixed Keplerian ellipse. We consider different examples in which a non-conservative force acts on the model and we propose an analytical method, which reduces the system to a Hamiltonian framework. In particular, we compute a time parametrisation in a series form, which allows us to transform the original system into a Hamiltonian one. We also provide applications of our method to study the rotational motion of a body with time-varying moments of inertia, e.g. an artificial satellite with flexible components, as well as subject to a tidal torque depending linearly on the velocity.

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I. Gkolias, C. Efthymiopoulos, G. Pucacco, et. al.
Mon, 20 Mar 2017
41/47

Comments: Accepted for publication in Communications in Nonlinear Science and Numerical Simulation

Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque [CL]

http://arxiv.org/abs/1703.01189


Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury’s spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.

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M. Bartuccelli, J. Deane and G. Gentile
Mon, 6 Mar 17
38/47

Comments: 32 pages, 8 figures, 5 tables

Dynamics and evolution of planets in mean-motion resonances [EPA]

http://arxiv.org/abs/1702.02494


In some planetary systems the orbital periods of two of its members present a commensurability, usually known by mean-motion resonance. These resonances greatly enhance the mutual gravitational influence of the planets. As a consequence, these systems present uncommon behaviours and their motions need to be studied with specific methods. Some features are unique and allow us a better understanding and characterisation of these systems. Moreover, mean-motion resonances are a result of an early migration of the orbits in an accretion disk, so it is possible to derive constraints on their formation. Here we review the dynamics of a pair of resonant planets and explain how their orbits evolve in time. We apply our results to the HD45365 planetary system

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A. Correia, J. Delisle and J. Laskar
Thu, 9 Feb 17
24/67

Comments: invited review; comments and feedback welcome

An Analytic Criterion for Turbulent Disruption of Planetary Resonances [EPA]

http://arxiv.org/abs/1701.07849


Mean motion commensurabilities in multi-planet systems are an expected outcome of protoplanetary disk-driven migration, and their relative dearth in the observational data presents an important challenge to current models of planet formation and dynamical evolution. One natural mechanism that can lead to the dissolution of commensurabilities is stochastic orbital forcing, induced by turbulent density fluctuations within the nebula. While this process is qualitatively promising, the conditions under which mean motion resonances can be broken are not well understood. In this work, we derive a simple analytic criterion that elucidates the relationship among the physical parameters of the system, and find the conditions necessary to drive planets out of resonance. Subsequently, we confirm our findings with numerical integrations carried out in the perturbative regime, as well as direct N-body simulations. Our calculations suggest that turbulent resonance disruption depends most sensitively on the planet-star mass ratio. Specifically, for a disk with properties comparable to the early solar nebula with $\alpha=0.01$, only planet pairs with cumulative mass ratios smaller than $(m_1+m_2)/M\lesssim10^{-5}\sim3M_{\oplus}/M_{\odot}$ are susceptible to breaking resonance at semi-major axis of order $a\sim0.1\,$AU. Although turbulence can sometimes compromise resonant pairs, an additional mechanism (such as suppression of resonance capture probability through disk eccentricity) is required to adequately explain the largely non-resonant orbital architectures of extrasolar planetary systems.

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K. Batygin and F. Adams
Mon, 30 Jan 2017
15/41

Comments: 12 pages, 6 figures, accepted to AJ

The structure of invariant tori in a 3D galactic potential [CL]

http://arxiv.org/abs/1009.1993


We study in detail the structure of phase space in the neighborhood of stable periodic orbits in a rotating 3D potential of galactic type. We have used the color and rotation method to investigate the properties of the invariant tori in the 4D spaces of section. We compare our results with those of previous works and we describe the morphology of the rotational, as well as of the tube tori in the 4D space. We find sticky chaotic orbits in the immediate neighborhood of sets of invariant tori surrounding 3D stable periodic orbits. Particularly useful for galactic dynamics is the behavior of chaotic orbits trapped for long time between 4D invariant tori. We find that they support during this time the same structure as the quasi-periodic orbits around the stable periodic orbits, contributing however to a local increase of the dispersion of velocities. Finally we find that the tube tori do not appear in the 3D projections of the spaces of section in the axisymmetric Hamiltonian we examined.

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M. Katsanikas and P. Patsis
Mon, 9 Jan 17
21/52

Comments: 26 pages, 34 figures, accepted for publication in the International Journal of Bifurcation and Chaos

Chains of rotational tori and filamentary structures close to high multiplicity periodic orbits in a 3D galactic potential [CL]

http://arxiv.org/abs/1103.3981


This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis and Zachilas 1994] in order to visualize them. As examples we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

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M. Katsanikas, P. Patsis and A. Pinotsis
Mon, 9 Jan 17
26/52

Comments: 12 pages,22 figures, Accepted for publication in the International Journal of Bifurcation and Chaos

The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation [CL]

http://arxiv.org/abs/1012.2463


We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D “confined tori”. Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

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M. Katsanikas, P. Patsis and G. Contopoulos
Mon, 9 Jan 17
42/52

Comments: 10 pages, 14 figures, accepted for publication in the International Journal of Bifurcation and Chaos

Instabilities and stickiness in a 3D rotating galactic potential [CL]

http://arxiv.org/abs/1201.2108


We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four dimensional spaces of section we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in the 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed.

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M. Katsanikas, P. Patsis and G. Contopoulos
Mon, 9 Jan 17
44/52

Comments: 29pages, 25 figures, accepted for publication in the International Journal of Bifurcation and Chaos

Dynamical models and the onset of chaos in space debris [EPA]

http://arxiv.org/abs/1612.08849


The increasing threat raised by space debris led to the development of different mathematical models and approaches to investigate the dynamics of small particles orbiting around the Earth. Such models and methods strongly depend on the altitude of the objects above Earth’s surface, since the strength of the different forces acting on an Earth orbiting object (geopotential, atmospheric drag, lunar and solar attractions, solar radiation pressure, etc.) varies with the altitude of the debris.
In this review, our focus is on presenting different analytical and numerical approaches employed in modern studies of the space debris problem. We start by considering a model including the geopotential, solar and lunar gravitational forces and the solar radiation pressure. We summarize the equations of motion using different formalisms: Cartesian coordinates, Hamiltonian formulation using Delaunay and epicyclic variables, Milankovitch elements. Some of these methods lead in a straightforward way to the analysis of resonant motions. In particular, we review results found recently about the dynamics near tesseral, secular and semi-secular resonances.
As an application of the above methods, we proceed to analyze a timely subject namely the possible causes for the onset of chaos in space debris dynamics. Precisely, we discuss the phenomenon of overlapping of resonances, the effect of a large area-to-mass ratio, the influence of lunisolar secular resonances.
We conclude with a short discussion about the effect of the dissipation due to the atmospheric drag and we provide a list of minor effects, which could influence the dynamics of space debris.

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A. Celletti, C. Efthymiopoulos, F. Gachet, et. al.
Fri, 30 Dec 16
39/64

Comments: 39 pages, 4 figures

Stellar and planetary Cassini states [EPA]

http://arxiv.org/abs/1612.07564


Cassini states correspond to equilibria of the spin axis of a body when its orbit is perturbed. They were initially described for satellites, but the spin axis of stars and planets undergoing strong dissipation can also evolve into some equilibria. For small satellites, the rotational angular momentum is usually much smaller than the total angular momentum, so classical methods for finding Cassini states rely on this approximation. Here we present a more general approach, which is valid for the secular quadrupolar non-restricted problem with spin. Our method is still valid when the precession rate and the mutual inclination of the orbits are not constant. Therefore, it can be used to study stars with close-in companions, or planets with heavy satellites, like the Earth-Moon system.

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A. Correia
Fri, 23 Dec 16
8/60

Comments: 8 pages, 6 figures

Dynamical system modeling fermionic limit [CL]

http://arxiv.org/abs/1612.05442


The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

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D. Bors and R. Stanczy
Mon, 19 Dec 16
46/54

Comments: 12 pages, 2 figures

On the period of the periodic orbits of the restricted three body problem [CL]

http://arxiv.org/abs/1611.07550


We will show that the period $T$ of a closed orbit of the planar circular restricted three-body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, $2 T=k\pi+\int_\Omega g$ where $k$ is an integer, $\Omega$ is the region enclosed by the periodic orbit and $g:\mathbb{R}^2\to \mathbb{R}$ is a function that only depends on the constant $C$ known as the Jacobian integral; it does not depend on $\Omega$. This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application, we prove that there is a neighborhood around $L_4$ such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for $L_5$.

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O. Perdomo
Thu, 24 Nov 16
16/54

Comments: 5 figures

Charged dust grain dynamics subject to solar wind, Poynting-Robertson drag, and the interplanetary magnetic field [EPA]

http://arxiv.org/abs/1608.07040


We investigate the combined effect of solar wind, Poynting-Robertson drag, and the frozen-in interplanetary magnetic field on the motion of charged dust grains in our solar system. For this reason we derive a secular theory of motion by the means of averaging method and validate it with numerical simulations of the un-averaged equations of motions. The theory predicts that the secular motion of charged particles is mainly affected by the z-component of the solar magnetic axis, or the normal component of the interplanetary magnetic field. The normal component of the interplanetary magnetic field leads to an increase or decrease of semi-major axis depending on its functional form and sign of charge of the dust grain. It is generally accepted that the combined effects of solar wind and photon absorption and re-emmision (Poynting-Robertson drag) lead to a decrease in semi-major axis on secular time scales. On the contrary, we demonstrate that the interplanetary magnetic field may counteract these drag forces under certain circumstances. We derive a simple relation between the parameters of the magnetic field, the physical properties of the dust grain as well as the shape and orientation of the orbital ellipse of the particle, which is a necessary conditions for the stabilization in semi-major axis.

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C. Lhotka, P. Bourdin and Y. Narita
Fri, 26 Aug 16
25/47

Comments: 32 pages, 5 figures, 2 tables

Secular and tidal evolution of circumbinary systems [EPA]

http://arxiv.org/abs/1608.03484


We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin-orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star-planet-satellite systems and triple stellar systems.

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A. Correia, G. Boue and J. Laskar
Fri, 12 Aug 16
33/38

Comments: 40 pages, 18 figures

On dynamical systems approaches and methods in $f(R)$ cosmology [CL]

http://arxiv.org/abs/1607.05715


We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in $f(R)$-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, $f(R) = R + \alpha R^2$, $\alpha > 0$, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to $f(R)$ cosmology and discuss their advantages and disadvantages.

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A. Alho, S. Carloni and C. Uggla
Wed, 20 Jul 16
4/66

Comments: 36 pages, 7 figures

Near Periodic solution of the Elliptic RTBP for the Jupiter Sun system [CL]

http://arxiv.org/abs/1606.01819


Let us consider the elliptic restricted three body problem (Elliptic RTBP) for the Jupiter Sun system with eccentricity $e=0.048$ and $\mu=0.000953339$. Let us denote by $T$ the period of their orbits. In this paper we provide initial conditions for the position and velocity for a spacecraft such that after one period $T$ the spacecraft comes back to the same place, with the same velocity, within an error of 4 meters for the position and 0.2 meters per second for the velocity. Taking this solution as periodic, we present numerical evidence showing that this solution is stable. In order to compare this periodic solution with the motion of celestial bodies in our solar system, we end this paper by providing an ephemeris of the spacecraft motion from February 17, 2017 to December 28, 2028.

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O. Perdomo
Tue, 7 Jun 16
4/80

Comments: 1 figure. Youtube Link for the motion described in this paper at this https URL

The trunkenness of a volume-preserving vector field [CL]

http://arxiv.org/abs/1606.01639


We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.

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A. Rechtman and P. Dehornoy
Tue, 7 Jun 16
56/80

Comments: N/A

Poynting-Robertson drag and solar wind in the space debris problem [EPA]

http://arxiv.org/abs/1605.06965


We analyze the combined effect of Poynting-Robertson and solar wind drag on space debris. We derive a model within Cartesian, Gaussian and Hamiltonian frameworks. We focus on the geosynchronous resonance, although the results can be easily generalized to any resonance. By numerical and analytical techniques, we compute the drift in semi-major axis due to Poynting-Robertson and solar wind drag. After a linear stability analysis of the equilibria, we combine a careful investigation of the regular, resonant, chaotic behavior of the phase space with a long-term propagation of a sample of initial conditions. The results strongly depend on the value of the area-to-mass ratio of the debris, which might show different dynamical behaviors: temporary capture or escape from the geosynchronous resonance, as well as temporary capture or escape from secondary resonances involving the rate of variation of the longitude of the Sun. Such analysis shows that Poynting-Robertson and solar wind drag must be taken into account, when looking at the long-term behavior of space debris. Trapping or escape from the resonance can be used to place the debris in convenient regions of the phase space.

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C. Lhotka, A. Celletti and C. Gales
Tue, 24 May 16
13/73

Comments: Accepted article in Monthly Notices of the Royal Astronomical Society this http URL&ijkey=Zn1L2sHThSXlmq4. MNRAS doi:10.1093/mnras/stw927 first published online April 26, 2016

Existence and Stability the Lagrangian point $L_4$ for the Earth-Sun system under a relativistic framework [EPA]

http://arxiv.org/abs/1605.04527


It is well known that, from the Newtonian point of view, the Lagrangian point $L_4$ in the circular restricted three body is stable if $\mu< \frac{1}{18}(9-\sqrt{19})\approx 0.03852$. In this paper we will provide a formula that allows us to compute the eigenvalues of the matrix that determines the stability of the equilibrium points of a family of ordinary differential equations. As an application we will show that, under the relativistic framework, the Lagrangian point $L_4$ is also stable for the Sun-Earth system. Similar arguments show the stability for $L_4$ not only for the Sun-Earth system but for systems coming from a range of values for $\mu$ similar to those in the Newtonian restricted three body problem.

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O. Perdomo
Tue, 17 May 16
25/65

Comments: This paper is a modification of the previous paper arXiv:1601.00924. The main difference between the paper is that the old one focuses on the stability near the critical value for mu and this new version focuses on providing a mathematical proof for the stability

Relative Equilibria in the Spherical, Finite Density 3-Body Problem [CL]

http://arxiv.org/abs/1605.01809


The relative equilibria for the spherical, finite density 3 body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical 5 relative equilibria for the point-mass 3-body problem. None of the identified relative equilibria exist or are stable over all values of angular momentum. The stability and bifurcation pathways of these relative equilibria are mapped out as the angular momentum of the system is increased. This is done under the assumption that they have equal and constant densities and that the entire system rotates about its maximum moment of inertia. The transition to finite density greatly increases the number of relative equilibria in the 3-body problem and ensures that minimum energy configurations exist for all values of angular momentum.

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D. Scheeres
Mon, 9 May 16
15/48

Comments: Accepted for publication in the Journal of Nonlinear Science

Supersymmetric Theory of Stochastic ABC Model: A Numerical Study [CL]

http://arxiv.org/abs/1604.08609


In this paper, we investigate numerically the stochastic ABC model, a toy model in the theory of astrophysical kinematic dynamos, within the recently proposed supersymmetric theory of stochastics (STS). STS characterises stochastic differential equations (SDEs) by the spectrum of the stochastic evolution operator (SEO) on elements of the exterior algebra or differentials forms over the system’s phase space, X. STS can thereby classify SDEs as chaotic or non-chaotic by identifying the phenomenon of stochastic chaos with the spontaneously broken topological supersymmetry that all SDEs possess. We demonstrate the following three properties of the SEO, deduced previously analytically and from physical arguments: the SEO spectra for zeroth and top degree forms never break topological supersymmetry, all SDEs possesses pseudo-time-reversal symmetry, and each de Rahm cohomology class provides one supersymmetric eigenstate. Our results also suggests that the SEO spectra for forms of complementary degrees, i.e., k and dim X -k, may be isospectral.

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I. Ovchinnikov, Y. Sun, T. Ensslin, et. al.
Mon, 2 May 16
39/49

Comments: Revtex 4-1, 9 pages, 3 figures

Second-order variational equations for N-body simulations [EPA]

http://arxiv.org/abs/1603.03424


First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO).
In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton’s method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection.
We provide an implementation of first and second-order variational equations for the publicly available REBOUND integrator package. Our implementation allows the simultaneous integration of any number of first and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.

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H. Rein and D. Tamayo
Mon, 14 Mar 16
32/47

Comments: 11 pages, accepted for publication in MNRAS, code available at this https URL, figures can be reproduced interactively with binder at this http URL

On The Big Bang Singularity in $k=0$ FLRW Cosmologies [CL]

http://arxiv.org/abs/1602.02456


In this brief paper, we consider the dynamics of a spatially flat FLRW spacetime with a positive cosmological constant and matter obeying a barotropic equation of state. By performing a change of variables on the Raychaudhuri equation, we are able to compactify the big bang singularity to a finite point. We then use Chetaev’s instability theorem to prove that such a model is always past asymptotic to a big bang singularity assuming only the weak energy condition, which is more general than the strong energy condition used in the classical singularity theorems of cosmology.

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I. Kohli
Tue, 9 Feb 16
23/63

Comments: N/A

Scale dynamical origin of modification or addition of potential in mechanics. A possible framework for the MOND theory and the dark matter [CL]

http://arxiv.org/abs/1601.01130


Using our mathematical framework developed in \cite{cresson-pierret_scale} called \emph{scale dynamics}, we propose in this paper a new way of interpreting the problem of adding or modifying potentials in mechanics and specifically in galactic dynamics. An application is done for the two-body problem with a Keplerian potential showing that the velocity of the orbiting body is constant. This would explain the observed phenomenon in the flat rotation curves of galaxies without adding \emph{dark matter} or modifying Newton’s law of dynamics.

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F. Pierret
Thu, 7 Jan 16
35/36

Comments: N/A

Bifurcations of lunisolar secular resonances for space debris orbits [EPA]

http://arxiv.org/abs/1512.02178


Using bifurcation theory, we study the secular resonances induced by Sun and Moon on space debris orbits around the Earth. In particular, we concentrate on a special class of secular resonances, which depends just on the debris’ orbital inclination. This class is typically subdivided into three distinct types of secular resonances: those occurring at the critical inclination, those corresponding to polar orbits and a third type resulting from a linear combination of the rates of variation of the argument of perigee and the longitude of the ascending node.
The model describing the dynamics of space debris includes the effects of the geopotential, as well as Sun’s and Moon’s attractions, and it is defined in terms of suitable action-angle variables. We consider the system averaged over both the mean anomaly of the debris and those of Sun and Moon. Such multiply-averaged Hamiltonian is used to study the lunisolar resonances which depend just on the inclination.
Borrowing the technique from the theory of bifurcations of Hamiltonian normal forms, we study the birth of periodic orbits and we determine the energy thresholds at which the bifurcations of lunisolar secular resonances take place. This approach gives us physically relevant information on the existence and location of the equilibria, which help us to identify stable and unstable regions in the phase space. On the other hand, beside their physical interest, the study of inclination dependent resonances offers interesting insights from the dynamical point of view, since it sheds light on different phenomena related to bifurcation theory.

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A. Celletti, C. Gales and G. Pucacco
Tue, 8 Dec 15
66/71

Comments: 35 pages, 8 figures