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

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

Comments: 32 pages, 8 figures, 5 tables


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

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

Comments: invited review; comments and feedback welcome

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

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

Comments: 12 pages, 6 figures, accepted to AJ

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

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

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]

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

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]

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

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]

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

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]

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

Comments: 39 pages, 4 figures

Stellar and planetary Cassini states [EPA]

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

Comments: 8 pages, 6 figures

Dynamical system modeling fermionic limit [CL]

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

Comments: 12 pages, 2 figures

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

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

Comments: 5 figures

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

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

Comments: 32 pages, 5 figures, 2 tables

Secular and tidal evolution of circumbinary systems [EPA]

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

Comments: 40 pages, 18 figures

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

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

Comments: 36 pages, 7 figures

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

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

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]

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

Comments: N/A

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

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

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]

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

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]

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

Comments: Accepted for publication in the Journal of Nonlinear Science

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

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

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

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

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

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]

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

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]

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

Comments: N/A

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

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

Comments: 35 pages, 8 figures

Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance [CL]

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.

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A. Marchesiello and G. Pucacco
Thu, 3 Dec 15

Comments: 36 pages, 10 figures, accepted on International Journal of Bifurcation and Chaos. arXiv admin note: substantial text overlap with arXiv:1401.2855

On the relativistic Lagrange-Laplace secular dynamics for extrasolar systems [CL]

We study the secular dynamics of extrasolar planetary systems by extending the Lagrange-Laplace theory to high order and by including the relativistic effects. We investigate the long-term evolution of the planetary eccentricities via normal form and we find an excellent agreement with direct numerical integrations. Finally we set up a simple analytic criterion that allows to evaluate the impact of the relativistic effects in the long-time evolution.

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M. Sansottera, L. Grassi and A. Giorgilli
Fri, 23 Oct 15

Comments: 4 pages, 4 figures, Proceedings IAU Symposium No. S310 (Complex Planetary Systems)

Effective resonant stability of Mercury [CL]

Mercury is the unique known planet that is situated in a 3:2 spin-orbit resonance nowadays. Observations and models converge to the same conclusion: the planet is presently deeply trapped in the resonance and situated at the Cassini state $1$, or very close to it. We investigate the complete non-linear stability of this equilibrium, with respect to several physical parameters, in the framework of Birkhoff normal form and Nekhoroshev stability theory. We use the same approach adopted for the 1:1 spin-orbit case with a peculiar attention to the role of Mercury’s non negligible eccentricity. The selected parameters are the polar moment of inertia, the Mercury’s inclination and eccentricity and the precession rates of the perihelion and node. Our study produces a bound to both the latitudinal and longitudinal librations (of 0.1 radians) for a long but finite time (greatly exceeding the age of the solar system). This is the so-called effective stability time. Our conclusion is that Mercury, placed inside the 3:2 spin-orbit resonance, occupies a very stable position in the space of these physical parameters, but not the most stable possible one.

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M. Sansottera, C. Lhotka and A. Lemaitre
Fri, 23 Oct 15

Comments: 9 pages

Conic-Helical Orbits of Planets around Binary Stars do not Exist [CL]

Oks proposes the existence of stable planetary orbits around binary stars, in the shape of a helix on a conical surface whose axis of symmetry coincides with the interstellar axis. We show that planetary orbits initially meeting this description will not continue to do so as the binary pair rotates.

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G. Egan
Tue, 20 Oct 15

Comments: 3 pages

Multiscale functions, Scale dynamics and Applications to partial differential equations [CL]

Modeling phenomena from experimental data, always begin with a \emph{choice of hypothesis} on the observed dynamics such as \emph{determinism}, \emph{randomness}, \emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \emph{“With a finite set of data concerning a phenomenon, can we recover its underlying nature ?} From this problem, we introduce in this paper the definition of \emph{multi-scale functions}, \emph{scale calculus} and \emph{scale dynamics} based on the \emph{time-scale calculus} (see \cite{bohn}). These definitions will be illustrated on the \emph{multi-scale Okamoto’s functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \emph{scale regimes} whereas the equation is \emph{scale invariant}. A typical example of such an equation, is the \emph{Euler-Lagrange equation} and particularly the \emph{Newton’s equation} which will be discussed. Notably, we obtain a \emph{non-linear diffusion equation} via the \emph{scale Newton’s equation} and also the \emph{non-linear Schr\”odinger equation} via the \emph{scale Newton’s equation}. Under special assumptions, we recover the classical \emph{diffusion} equation and the \emph{Schr\”odinger equation}.

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J. Cresson and F. Pierret
Fri, 4 Sep 15

Comments: N/A

Finite-Time Singularities in $k=0$ FLRW Cosmologies [CL]

In this paper, we consider a spatially flat FLRW cosmological model with matter obeying a barotropic equation of state $p = w \mu$, $-1<w\leq1$, and a cosmological constant, $\Lambda$. We use Osgood’s criterion to establish three cases when such models admit finite-time singularities. The first case is for an arbitrary initial condition, with a negative cosmological constant, and phantom energy $w < -1$. We show that except for a very fine-tuned choice of the initial condition $\theta_{0}$, the universe will develop a finite-time singularity. The second case we consider is for a nonnegative cosmological constant, phantom energy, and the expansion scalar being larger than that of the flat-space de Sitter solution, and show that such models only expand forever for $\Lambda = 0$. In all other cases, the universe model develops a finite-time singularity. The final case we consider is for a nonnegative cosmological constant, a matter source with $-1 < w \leq 1$, and an expansion scalar that is asymptotically that of the de Sitter universe. We show that such models will only expand forever when $\Lambda = 0$, otherwise, they will develop a finite-time singularity. This is significant, since the inflationary epoch is a subset of this domain. However, as we show, the inclusion of a bulk viscosity term in the Einstein field equations eliminates this singularity, and the universe expands forever. This could have interesting implications for the role of bulk viscosity in dynamical models of the universe.

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I. Kohli
Thu, 9 Jul 15

Comments: arXiv admin note: text overlap with arXiv:1505.07770

Shadowing Lemma and Chaotic Orbit Determination [EPA]

Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. In a simple discrete model, the standard map, we tackle the problem of chaotic orbit determination when observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing orbit is computed by the least squares orbit determination. We test both the convergence of the orbit determination iterative procedure and the behaviour of the uncertainties as a function of the maximum number $n$ of map iterations observed. When the initial conditions belong to a chaotic orbit, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula. Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is added to the initial conditions as parameter to be estimated. The uncertainty of the dynamical parameter decreases like $n^a$ with $a<0$ but not large (of the order of unity). If only the initial conditions are estimated, their uncertainty decreases exponentially with $n$. If they belong to a non-chaotic orbit the computational horizon is much larger, if it exists at all, and the decrease of the uncertainty is polynomial in all parameters, like $n^a$ with $a\simeq 1/2$. The Shadowing Lemma does not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena have significant implications, which remain to be studied, in practical problems of orbit determination involving chaos, such as the chaotic rotation state of a celestial body and a chaotic orbit of a planet-crossing asteroid undergoing many close approaches.

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F. Spoto and A. Milani
Thu, 11 Jun 15

Comments: N/A

Rigorous treatment of the averaging process for co-orbital motions in the planetary problem [EPA]

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamiltonian, we estimate the size of the transformation that maps this Hamiltonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale.

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P. Robutel and L. Niederman
Wed, 10 Jun 15

Comments: N/A

On Singularities in Cosmic Inflation [CL]

In this paper, we examine a flat FLRW spacetime with a scalar field potential and show by applying Osgood’s criterion to the Einstein field equations that all such models, irrespective of the particular choice of potential develop finite-time singularities. That is, we show that solutions to the field equations rapidly diverge in finite time. This can have important implications for the role of inflation in cosmological models, since one of the implications of this is that within the inflationary epoch, a singularity develops in finite time, which would call into question the role of inflation in the dynamic evolution of our universe. We further point out that a possible reason for this behaviour is that the solutions to the field equations in such inflationary scenarios do not obey global existence and uniqueness properties, which is a typical characteristic of solutions that diverge in finite time.

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I. Kohli
Fri, 29 May 15

Comments: For submission to: Classical and Quantum Gravity

Capture of Planets Into Mean Motion Resonances and the Origins of Extrasolar Orbital Architectures [EPA]

The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this work, we employ an integrable Hamiltonian formalism for first order planetary resonances that allows both secondary bodies to have finite masses and eccentricities, and construct a comprehensive theory for resonant capture. Particularly, we derive conditions under which orbital evolution lies within the adiabatic regime, and provide a generalized criterion for guaranteed resonant locking as well as a procedure for calculating capture probabilities when capture is not certain. Subsequently, we utilize the developed analytical model to examine the evolution of Jupiter and Saturn within the protosolar nebula, and investigate the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets. Our calculations show that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e~0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.

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K. Batygin
Fri, 8 May 15

Comments: 22 pages, 15 figures, accepted for publication in MNRAS

Dynamical Evolution of Multi-Resonant Systems: the Case of GJ876 [EPA]

The GJ876 system was among the earliest multi-planetary detections outside of the Solar System, and has long been known to harbor a resonant pair of giant planets. Subsequent characterization of the system revealed the presence of an additional Neptune mass object on an external orbit, locked in a three body Laplace mean motion resonance with the previously known planets. While this system is currently the only known extrasolar example of a Laplace resonance, it differs from the Galilean satellites in that the orbital motion of the planets is known to be chaotic. In this work, we present a simple perturbative model that illuminates the origins of stochasticity inherent to this system and derive analytic estimates of the Lyapunov time as well as the chaotic diffusion coefficient. We then address the formation of the multi-resonant structure within a protoplanetary disk and show that modest turbulent forcing in addition to dissipative effects is required to reproduce the observed chaotic configuration. Accordingly, this work places important constraints on the typical formation environments of planetary systems and informs the attributes of representative orbital architectures that arise from extended disk-driven evolution.

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K. Batygin, K. Deck and M. Holman
Thu, 2 Apr 15

Comments: 15 pages, 7 figures, accepted to AJ

Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids [CL]

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat FLRW cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the `attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Pad\’e approximants and compare with the slow-roll approximation.

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A. Alho, J. Hell and C. Uggla
Wed, 25 Mar 15

Comments: 33 pages, 12 figures

The Astrophysics of Resonant Orbits in the Kerr Metric [CL]

This paper gives a complete characterization of resonant orbits in a Kerr spacetime. A resonant orbit is defined as a geodesic for which the longitudinal and radial orbital frequencies are commensurate. Our analysis is based on expressing the resonance condition in its most symmetric form using Carlson’s integrals. We provide a number of concise formulae for the dependence of resonances on the system parameters. Resonant effects may be observable during the in-spiral of a compact object into a super-massive black hole. When the slowly evolving orbital frequencies pass through a series of low-order resonances, rapid changes in the orbital parameters could produce measurable phase shifts in the emitted gravitational radiation (GW). Resonant orbits may also capture dust leading to electromagnetic emission. The KAM theorem indicates that, low order resonant orbits demarcate the regions where the onset of chaos could occur around a perturbed black-hole. We find that the 1/2 and 2/3 resonances occur at ~4 and 5.4 Schwarzschild radii (Rs) from the event horizon. For compact object in-spirals around super-massive black holes, this region lies within the sensitivity band of space-based GW detectors. For Sgr A*, length scales of ~41 and 55 microarcseconds and timescales of 50 and 79 min respectively should be associated with resonant effects, if Sgr A* is non-spinning. Spin decreases these values by up to ~32% and ~28%. These length-scales are potentially resolvable with VLBI measurements. We find that all low-order resonances are localized to the strong field region r < 50 Rs. This fact guarantees the validity of using approximations based on averaging to model the frequency evolution of a test object in region 50 Rs <r <1000 Rs. The systematic determination of the multipole moments of the central object by observing the orbit of a pulsar, free of chaotic effects, is thus possible.

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J. Brink, M. Geyer and T. Hinderer
Mon, 2 Feb 15

Comments: 25 Pages, 17 figures

A study of the main resonances outside the geostationary ring [CL]

We investigate the dynamics of satellites and space debris in external resonances, namely in the region outside the geostationary ring. Precisely, we focus on the 1:2, 1:3, 2:3 resonances, which are located at about 66 931.4 km, 87 705.0 km, 55 250.7 km, respectively. Some of these resonances have been already exploited in space missions, like XMM-Newton and Integral.
Our study is mainly based on a Hamiltonian approach, which allows us to get fast and reliable information on the dynamics in the resonant regions. Significative results are obtained even by considering just the effect of the geopotential in the Hamiltonian formulation. For objects (typically space debris) with high area-to-mass ratio the Hamiltonian includes also the effect of the solar radiation pressure. In addition, we perform a comparison with the numerical integration in Cartesian variables, including the geopotential, the gravitational attraction of Sun and Moon, and the solar radiation pressure.
We implement some simple mathematical tools that allows us to get information on the terms which are dominant in the Fourier series expansion of the Hamiltonian around a given resonance, on the amplitude of the resonant islands and on the location of the equilibrium points. We also compute the Fast Lyapunov Indicators, which provide a cartography of the resonant regions, yielding the main dynamical features associated to the external resonances. We apply these techniques to analyze the 1:2, 1:3, 2:3 resonances; we consider also the case of objects with large area-to-mass ratio and we provide an application to the case studies given by XMM-Newton and Integral.

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A. Celletti and C. Gales
Tue, 27 Jan 15

Comments: 30 pages, 10 figures

Introduction to the application of the dynamical systems theory in the study of the dynamics of cosmological models of dark energy [CL]

The theory of the dynamical systems is a very complex subject which has brought several surprises in the recent past in connection with the theory of chaos and fractals. The application of the tools of the dynamical systems in cosmological settings is less known in spite of the amount of published scientific papers on this subject. In this paper a — mostly pedagogical — introduction to the application in cosmology of the basic tools of the dynamical systems theory is presented. It is shown that, in spite of their amazing simplicity, these allow to extract essential information on the asymptotic dynamics of a wide variety of cosmological models. The power of these tools is illustrated within the context of the so called $\Lambda$CDM and scalar field models of dark energy. This paper is suitable for teachers, undergraduate and postgraduate students from physics and mathematics disciplines.

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R. Garcia-Salcedo, T. Gonzalez, F. Horta-Rangel, et. al.
Wed, 21 Jan 15

Comments: 15 pages, 2 figures

Low-energy capture of asteroids onto KAM tori [EPA]

We present a new method for engineering the artificial capture of asteroids. Based on theories of the chaos-assisted capture of natural satellites of the giant planets, we show how an unbound asteroid that passes close to a regular region of phase space can be easily moved onto the nearby KAM tori and essentially permanently captured with the Earth’s Hill sphere without closing the zero velocity curves. The method has the advantages of a relatively low delta-v requirement and no need for control strategies. An illustration of the method is given for an example asteroid trajectory, demonstrating that it is a viable strategy for the final capture stage of asteroids in the Earth’s neighbourhood.

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P. Verrier and C. McInnes
Thu, 15 Jan 15

Comments: 13 pages, 3 figures, accepted by the Journal of Guidance, Control, and Dynamics

The effect of Poynting-Robertson drag on the triangular Lagrangian points [EPA]

We investigate the stability of motion close to the Lagrangian equilibrium points L4 and L5 in the framework of the spatial, elliptic, restricted three- body problem, subject to the radial component of Poynting-Robertson drag. For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to beta a1 n1 , where beta is the ratio of the solar radiation over the gravitational force, and a1, n1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray (1994)) on the asymmetry of the stability indices of L4 and L5 to a more realistic force model. Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.

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C. Lhotka and A. Celletti
Fri, 5 Dec 14

Comments: 47 pages, 8 figures,

Deformation and tidal evolution of close-in planets and satellites using a Maxwell viscoelastic rheology [EPA]

In this paper we present a new approach to tidal theory. Assuming a Maxwell viscoelastic rheology, we compute the instantaneous deformation of celestial bodies using a differential equation for the gravity field coefficients. This method allows large eccentricities and it is not limited to quasi-periodic perturbations. It can take into account an extended class of perturbations, including chaotic motions and transient events. We apply our model to some already detected eccentric hot Jupiters and super-Earths in planar configurations. We show that when the relaxation time of the deformation is larger than the orbital period, spin-orbit equilibria arise naturally at half-integers of the mean motion, even for gaseous planets. In the case of super-Earths, these equilibria can be maintained for very low values of eccentricity. Our method can also be used to study planets with complex internal structures and other rheologies.

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A. Correia, G. Boue, J. Laskar, et. al.
Mon, 10 Nov 14

Comments: 16 pages, 13 figures, 2 tables

Earth–Mars Transfers with Ballistic Capture [EPA]

We construct a new type of transfer from the Earth to Mars, which ends in ballistic capture. This results in a substantial savings in capture $\Delta v$ from that of a classical Hohmann transfer under certain conditions. This is accomplished by first becoming captured at Mars, very distant from the planet, and then from there, following a ballistic capture transfer to a desired altitude within a ballistic capture set. This is achieved by manipulating the stable sets, or sets of initial conditions whose orbits satisfy a simple definition of stability. This transfer type may be of interest for Mars missions because of lower capture $\Delta v$, moderate flight time, and flexibility of launch period from the Earth.

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F. Topputo and E. Belbruno
Mon, 3 Nov 14

Comments: N/A

The phase-space of boxy-peanut and X-shaped bulges in galaxies I. Properties of non-periodic orbits [CL]

The investigation of the phase-space properties of structures encountered in a dynamical system is essential for understanding their formation and enhancement. In the present paper we explore the phase space in energy intervals where we have orbits that act as building blocks for boxy-peanut (b/p) and “{\sf X}-shaped” structures in rotating potentials of galactic type. We underline the significance of the rotational tori around the 3D families x1v1 and x1v1$^{\prime}$ that have been bifurcated from the planar x1 family. These tori play a multiple role: (i) They belong to quasi-periodic orbits that reinforce the local density. (ii) They act as obstacles for the diffusion of chaotic orbits and (iii) they attract a large number of chaotic orbits that become sticky to them. There are also bifurcations of unstable families (x1v2, x1v2$^{\prime}$). Their unstable asymptotic curves wind around the x1v1 and x1v1$^{\prime}$ tori generating orbits with hybrid morphologies between that of x1v1 and x1v2. In addition, a new family of multiplicity 2, called x1mul2, is found to be important for the peanut construction. Our work shows also that there are peanut-supporting orbits before the vertical ILR. Non-periodic orbits associated with the x1 family secure this contribution as well as the support of b/p structures at several other energy intervals. Non-linear phenomena associated with complex instability of single and double multiplicity families of periodic orbits show that these structures are not interrupted in regions where such orbits prevail. Depending on the main mechanism behind their formation, boxy bulges exhibit different morphological features. Finally our analysis indicates that “X” features shaped by orbits in the neighbourhood of x1v1 and x1v1$^{\prime}$ periodic orbits are pronounced only in side-on or nearly end-on views of the bar.

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P. Patsis and M. Katsanikas
Tue, 21 Oct 14

Comments: 22 pages, 24 figures, accepted for publication in the MNRAS

The phase-space of boxy-peanut and X-shaped bulges in galaxies II. The relation between face-on and edge-on boxiness [GA]

We study the dynamical mechanisms that reinforce the formation of boxy structures in the \textit{inner} regions, roughly in the middle, of bars observed nearly \textit{face-on}. Outer boxiness, at the ends of the bars, is usually associated with orbits at the inner, radial 4:1 resonance region and can be studied with 2D dynamics. However, in the middle of the bar dominate 3D orbits that give boxy/peanut bulges in the edge-on views of the models. In the present paper we show that 3D quasi-periodic, as well as 3D chaotic orbits sticky to the x1v1 and x1v1$^{\prime}$ tori, especially from the Inner Lindblad Resonance (ILR) region, have boxy projections on the equatorial plane of the bar. The majority of vertically perturbed 2D orbits, initially on the equatorial plane in the ILR resonance region, enhance boxy features in face-on bars. Orbits that build a bar by supporting sharp “{\sf X}” features in their side-on views at energies \textit{beyond} the ILR, may also have a double boxy character. If populated, the extent of the inner boxiness along the major axis is about the same with that of the peanut supporting orbits in the side-on views. At any rate these orbits do not obscure the observation of the boxy orbits of the ILR region in the face-on views, as they contribute more to the surface density at the sides of the bar than to their central parts.

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P. Patsis and M. Katsanikas
Tue, 21 Oct 14

Comments: 12 pages, 13 figures, accepted for publication in the MNRAS

Andoyer construction for Hill and Delaunay variables [IMA]

Andoyer variables are well known for the study of the rigid body dynamics. But these variables were derived by Andoyer through a procedure that can be also used to obtain the Delaunay variables of the Kepler problem in a direct way, without the use of Hamilton-Jacobi theory or non intuitive generating functions.

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J. Laskar
Tue, 23 Sep 14

Comments: 5 pages, 1 figure

Periodic orbits for 3 and 4 co-orbital bodies [EPA]

We investigate the natural families of periodic orbits associated with the equilibrium configurations of the the planar restricted $1+n$ body problem for the case $2\leq n \leq 4$ equal mass satellites. Such periodic orbits can be used to model both trojan exoplanetary systems and parking orbits for captured asteroids within the solar system. For $n=2$ there are two families of periodic orbits associated with the equilibria of the system: the well known horseshoe and tadpole orbits. For $n=3$ there are three families that emanate from the equilibrium configurations of the satellites, while for $n=4$ there are six such families as well as numerous additional connecting families. The families of periodic orbits are all of the horseshoe or tadpole type, and several have regions of neutral linear stability.

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P. Verrier and C. McInnes
Fri, 30 May 14

Comments: 14 pages, 18 figures, accepted by MNRAS

Analytical invariant manifolds near unstable points and the structure of chaos [CL]

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain,the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as i) the order of truncation of the series increases, and ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.

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C. Efthymiopoulos, G. Contopoulos and M. Katsanikas
Thu, 1 May 14

Comments: (in press)

Regularization of the big bang singularity with a time varying equation of state $w > 1$ [CL]

We study the classical dynamics of the universe undergoing a transition from contraction to expansion through a big bang singularity. The dynamics is described by a system of differential equations for a set of physical quantities, such as the scale factor $a$, the Hubble parameter $H$, the equation of state parameter $w$, and the density parameter $\Omega$. The solutions of the dynamical system have a singularity at the big bang. We study if these solutions can be uniquely extended through the singularity. In particular, we consider the model in which the contracting universe is dominated by a scalar field with a time varying equation of state $w$, which approaches a constant value $w_c$ near the singularity. We prove that, for $w_c > 1$, the singularity is regularizable only for a discrete set of $w_c$ values that satisfy a coprime number condition. Our result implies that the evolution of a bouncing universe through the big bang singularity does not have a continuous classical limit unless the equation of state is extremely fine-tuned.

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B. Xue and E. Belbruno
Tue, 11 Mar 14

Exploring Vacuum Energy in a Two-Fluid Bianchi Type I Universe [CL]

We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a two-fluid, non-tilted Bianchi Type I cosmological model. In our model, one of the fluids is a fluid with bulk viscosity, while the other fluid assumes the role of a cosmological constant and represents nonnegative vacuum energy. We begin by completing a detailed fixed-point analysis of the system which gives information about the local sinks, sources and saddles. We then proceed to analyze the global features of the dynamical system by using topological methods such as finding Lyapunov and Chetaev functions, and finding the $\alpha$- and $\omega$-limit sets using the LaSalle invariance principle. The fixed points found were a flat Friedmann-LeMa\^{\i}tre-Robertson-Walker (FLRW) universe with no vacuum energy, a de Sitter universe, a flat FLRW universe with both vacuum and non-vacuum energy, and a Kasner quarter-circle universe. We also show in this paper that the vacuum energy we observe in our present-day universe could actually be a result of the bulk viscosity of the ordinary matter in the universe, and proceed to calculate feasible values of the bulk viscous coefficient based on observations reported in the Planck data. We conclude the paper with some numerical experiments that shed further light on the global dynamics of the system.

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I. Kohli and M. Haslam
Tue, 11 Feb 14