Applications to stellar and galactic dynamics

Computation and a wealth of new observational techniques have reinvigorated dynamical studies of galaxies and star clusters. These objects are examples of the gravitationaln-body problem withn in the range from a few hundred to 10¹¹. Relaxation effects dominate at the low end and are completely negligible at the high end. The gravitationaln-body problem is chaotic, and the principal challenge in doing physics where that problem is involved (whether computationally or with analytic theory) is to ensure that chaos has not vitiated the results. Enforcing a Liouville theorem accomplishes this with collision-free large-n problems, but equivalent recipes are not in common use for smallern. We describe some important insights and discoveries that have come from computation in stellar dynamics, discuss chaos briefly, and indicate the way the physics that comes up in different astronomical contexts is addressed in numerical methods currently in use. Graphics is a vital part of any computational approach. The long range prospects are very promising for continued high scientific productivity in stellar dynamics.     

Magnetic Field Line Transport in Anisotropic Magnetic Turbulence: Anomalous, Quasilinear, and Percolative Regimes Versus the Kubo Number

The transport of plasma and of energetic particles because of magnetic turbulence is relevant to many space plasmas, ranging from the planetary magnetospheres to the solar corona and to the heliosphere. Various transport regimes for magnetic field lines can be obtained depending on the Kubo number. Here we show, by means of a numerical simulation, that the Kubo number also determines the level of chaos of the field lines. Weak chaos, closed magnetic surfaces, and anomalous transport regimes are obtained for R≪ 1; widespread chaos, destroyed magnetic surfaces, and quasilinear scaling of the diffusion coefficient for R ≳ 0.3; and global stochasticity as well as percolation scaling of the diffusion coefficient for R≫ 1. This revised version was published online in July 2006 with corrections to the Cover Date.     

Characterization of intermittent chaos and oscillating frequencies in some pulsating stars

If one zone models are available for some pulsating stars and the motion remains close to adiabaticity, we show the existence of odd frequencies in the power spectrum of these pulsating variable stars, and the frequency of free oscillation thus becomes dependent on the amplitude. We find 3 appearing in the power spectrum of the white dwarf PG 1351+489. Furthermore, the intrinsic stochasticity in the basic equations describing the structure of stars is discussed in this paper. We found that the pulsation of intermittent chaos can occur spontaneously in the basic equations of a star within certain intervals of values of its control parameters, and the intermittency can be used as an interpretation of pulsation in variable stars.     

Exploiting Unstable Periodic Orbits of a Chaotic Invariant Set for Spacecraft Control

The purpose of this work is to show that chaos control techniques (OGY, in special) can be used to efficiently keep a spacecraft around another body performing elaborate orbits. We consider a satellite and a spacecraft moving initially in coplanar and circular orbits, with slightly different radii, around a heavy central planet. The spacecraft, which is the inner body, has a slightly larger angular velocity than the satellite so that, after some time, they eventually go to a situation in which the distance between them becomes sufficiently small, so that they start to interact with one another. This situation is called as an encounter. In previous work we have shown that this scenario is a typical situation of a chaotic scattering for some well-defined range of parameters. Considering this scenario, we first show how it is possible to find the unstable periodic orbits that are located in the chaotic invariant set. From the set of unstable periodic orbits, we select the ones that can be combined to provide the desired elaborate orbit. Then, chaos control technique based on the OGY method is used to keep the spacecraft in the desired orbit. Finally, we analyze the results and make considerations regarding a realistic scenario of space exploration.     

Temporal behaviour of pressure in solar coronal loops

The temporal evolution of pressure in solar coronal loops is studied using the ideal theory of magnetohydrodynamic turbulence in cylindrical geometry. The velocity and the magnetic fields are expanded in terms of the Chandrasekhar-Kendall (C-K) functions. The three-mode representation of the velocity and the magnetic fields submits to the investigation of chaos. When the initial values of the velocity and the magnetic field coefficients are very nearly equal, the system shows periodicities. For randomly chosen initial values of these parameters, the evolution of the velocity and the magnetic fields is nonlinear and chaotic. The consequent plasma pressure is determined in the linear and nonlinear regimes. The evidence for the existence of chaos is established by evaluating the invariant correlation dimension of the attractorD ₂, a fractal value of which indicates the existence of deterministic chaos.     

Thermal and topographic tests of Europa chaos formation models from Galileo E15 observations

Stereo topography of an area near Tyre impact crater, Europa, reveals chaos regions characterised by marginal cliffs and domical topography, rising to 100-200 m above the background plains. The regions contain blocks which have both rotated and tilted. We tested two models of chaos formation: a hybrid diapir model, in which chaos topography is caused by thermal or compositional buoyancy, and block motion occurs due to the presence of near-surface (1-3 km) melt; and a melt-through model, in which chaos regions are caused by melting and refreezing of the ice shell. None of the hybrid diapir models tested generate any melt within 1-3 km of the surface, owing to the low surface temperature. A model of ocean refreezing following melt-through gives effective elastic thicknesses and ice shell thicknesses of 0.1-0.3 and 0.5-2 km, respectively. However, for such low shell thicknesses the refreezing model requires implausibly large lateral density contrasts (50-100 kg m⁻³) to explain the elevation of the centres of the chaos regions. Although a global equilibrium ice shell thickness of ≈2 km is possible if Europa's mantle resembles that of Io, it is unclear whether local melt-through events are energetically possible. Thus, neither of the models tested here gives a completely satisfactory explanation for the formation of chaos regions. We suggest that surface extrusion of warm ice may be an important component of chaos terrain formation, and demonstrate that such extrusion is possible for likely ice parameters.     

Symmetries in the Optimal Control of Solar Sail Spacecraft

The theory of optimal control is applied to obtain minimum-time trajectories for solar sail spacecraft for interplanetary missions. We consider the gravitational and solar radiation forces due to the Sun. The spacecraft is modelled as a flat sail of mass m and surface area A and is treated dynamically as a point mass. Coplanar circular orbits are assumed for the planets. We obtain optimal trajectories for several interrelated problem families and develop symmetry properties that can be used to simplify the solution-finding process. For the minimum-time planet rendezvous problem we identify different solution branches resulting in multiple solutions to the associated boundary value problem. We solve the optimal control problem via an indirect method using an efficient cascaded computational scheme. The global optimizer uses a technique called Adaptive Simulated Annealing. Newton and Quasi-Newton Methods perform the terminal fine tuning of the optimization parameters.     

Observational constraints on the identification and distribution of chaotic terrain on icy satellites

We outline the observational constraints required to identify chaos regions on Europa. Large incidence angle, rather than high resolution, appears to be the primary observational requirement for identifying chaos. At incidence angles >70°, chaos can be identified on Europa at image resolutions as low as 1.5 km/pixel. Similar images obtained at moderate or low incidence angles (<50°) require image resolutions upwards of ～250 m/pixel to identify chaos. If global images of Europa can be acquired at high incidence angles, the majority of its chaotic terrain can be identified, helping to constrain models of chaos formation and distribution. Furthermore, our results indicate that the areal coverage of chaos may be more uncertain than previously reported, representing as little as 10% or as much as 50% of the non-polar regions of Europa. These guidelines will aid in the development of optical instruments for future Europa missions, as well as other icy bodies, such as Triton.     

Chaos induced by De Haerdtl inequality in the Galilean system

This paper aims at studying the long-term orbital consequences of the perturbations related to De Haerdtl inequality, a current quasi-commensurability between the Galilean satellites of Jupiter Ganymede and Callisto. We used the method of Frequency Map Analysis to detect a chaotic behavior in a 5-bodies system where every inequality has been dropped, except of De Haerdtl one. We also used Frequency Analysis to draw the behavior of the arguments likely to become resonant, in several numerical integrations. We show that De Haerdtl inequality might have induced chaos in the past if Ganymede's and Callisto's eccentricities have been higher than 4×¹⁰⁻³. Moreover, we enlight the influence of Jupiter's obliquity on this chaos. We also enlight some aspects of this chaotic behavior, showing for instance stable chaos and single resonances. The main result of this study is that De Haerdtl inequality should be taken into account in every study of the long term orbital evolution of the Galilean satellites.     

On periodic orbits and resonance in extrasolar planetary systems

We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω ₂ ? ω ₁) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω ₂ ? ω ₁) librates or rotates are not different, but are closely connected to the exact periodic motion.     

Galactic chaos driven by an oscillating massive particle

The effect of an oscillating massive particle on the motion of stars in a spherical Plummer gravitational system is examined for chaotic behaviour for ratios of satellite to parent galaxy masses ranging from .001 to .15. Thee-folding times for chaos are calculated for non-zero angular momentum orbits and discussed in relation to the time-scales for dynamical friction.     

A minimal dynamical model for tidal synchronization and orbit circularization

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.     

Approximate General Solution Of The Two-Dimensional Duffing Dynamical System

This paper presents the approximate general solution of the triple well, double oscillator non-linear dynamical system. This system is non-integrable and the approximate general solution is calculated by application of the Last Geometric Theorem of Poincaré (Birkhoff, 1913, 1925). The original problem, known as the Duffing one, is a 1 degree of freedom system that, besides the conservative force component, includes dumping and external forcing terms (see details in the web site: http://www.uncwil.edu/people/hermanr/chaos/ted/chaos.html). The problem considered here is a 2 degree of freedom, autonomous and conservative one, without dumping, and of axisymmetric potential. The space of permissible motions is scanned for identification of all solutions re-entering after from one to nine oscillations and the precise families of periodic solutions are computed, including their stability parameter, covering all cases with periods T corresponding to 4osc/T. Seven sub-domains of the space of solutions were investigated in detail by zooming, an operation that proved the possibility to advance the accuracy of the approximate general solution to the level permitted by the integration routine. The approximation of the general solution, although impressive, provides clear evidence of the complexity of the problem and the need to proceed to larger period families. Nevertheless, it allows prediction of the areas where chaos and order regions in the Poincaré surfaces of section are to be expected. Examples of such surfaces of sections, as well as of types of closed solutions, are given. Two peculiar points of the space of solutions were identified as crossing, or source points from which infinite families of periodic solutions emanate. The morphology and stability of solutions of the problem are studied and discussed.     

Observation of Hamiltonian chaos in wave–particle interaction

The motion of charged particle in longitudinal waves is a paradigm for the transition to large scale chaos in Hamiltonian systems. Recently a test cold electron beam has been used to observe its non-self-consistent interaction with externally excited wave(s) in a specially designed Traveling Wave Tube (TWT). The velocity distribution function of the electron beam is recorded with a trochoidal energy analyzer at the output of the TWT. An arbitrary waveform generator is used to launch a prescribed spectrum of waves along the slow wave structure (a 4 m long helix) of the TWT. The resonant velocity domain associated to a single wave is observed, as well as the transition to large scale chaos when the resonant domains of two waves and their secondary resonances overlap. This transition exhibits a “devil’s staircase” behavior when increasing the excitation amplitude in agreement with numerical simulation. A new strategy for control of chaos by building barriers of transport which prevent electrons to escape from a given velocity region as well as its robustness are also successfully tested. Thus generic features of Hamiltonian chaos have been experimentally observed.     

Wavelet analysis of the standard map: Structure and scaling

Wavelet analysis is applied to distributions of points generated by iterating the standard map. The initial condition is chosen so that the points fill the largest chaotic region. When the standard map parameterk=1.3, the distribution of points contains many voids corresponding to islands in the chaotic region. The wavelet transform is dominated by contributions from these islands. Fork=10 the chaos fills phase space and no structure is apparent; the wavelet transform reveals statistical fluctuations in the distribution of points.     

Period and phase of the 88-year solar cycle and the Maunder minimum: Evidence for a chaotic sun

The problem of whether the solar dynamo is quasi-periodic or chaotic is addressed by examining 1500 years of sunspot, geomagnetic and auroral activity cycles. We find sub-harmonics of the fundamental solar cycle period during the years preceding the Maunder minimum and loss of phase of the subharmonic on emergence from it. These phenomena are indicative of chaos. They indicate that the solar dynamo is chaotic and is operating in a region close to the transition between period doubling and chaos. Since Maunder type minima reoccur irregularly for millennia, it appears that the Sun remains close to this transition to and from chaos. We postulate this as a universal characteristic of solar type stars caused by feedback in the dynamo number.     

Formation and evolution of the chaotic terrains by subsidence and magmatism: Hydraotes Chaos, Mars

The origin of the martian chaotic terrains is still uncertain; and a variety of geologic scenarios have been proposed. We provide topographic profiles of different chaos landscapes, notably Aureum and Hydraotes Chaos, showing that an initial shallow ground subsidence occurred at the first step of the chaos formation. We infer that the subsidence was caused by intrusion of a volcanic sill; which could have produced consequent melting as well as release of ground water from disrupted aquifer. Signs of a volcanic activity are observed on the floor of Hydraotes Chaos, a complex and deep depression located at the junction of three channels. The volcanic activity is represented by small, 0.5 to 1.5 km diameter, rounded cones with summit pits. The cone's size and morphology, as well as the presence of possible surrounding lava flows, suggest that they are primary volcanic cones similar to terrestrial cinder cones. The identification of volcanic activity on the deepest chaos, where the lower crustal thickness and the faults/fractures system contributed to the magma rising, reveals that magmatic activity, proved by the cones, and possibly help by structural activity, has been a major factor in the formation of chaotic terrains.     

Wisdom system: Dynamics in the adiabatic approximation

A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place. Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos.     

Impact penetration of Europa's ice crust as a mechanism for formation of chaos terrain

Abstract— Ice thickness estimates and impactor dynamics indicate that some impacts must breach Europa's ice crust; and outcomes of impact experiments using ice‐over‐water targets range from simple craters to chaos‐like destroyed zones, depending on impact energy and ice competence. First‐order impacts‐into thick ice or at low impact energy‐produce craters. Second‐order impacts punch through the ice, making holes that resemble raft‐free chaos areas. Third‐order impacts‐into thinnest ice or at highest energy‐produce large irregular raft‐filled zones similar to platy chaos. Other evidence for an impact origin for chaos areas comes from the size‐frequency distribution of chaos+craters on Europa, which matches the impact production functions of Ganymede and Callisto; and from small craters around the large chaos area Thera Macula, which decrease in average size and density per unit area as a function of distance from Thera's center. There are no tiny chaos areas and no craters >50 km diameter. This suggests that small impactors never penetrate, whereas large ones (ÜberPenetrators: >2.5 km diameter at average impact velocity) always do. Existence of both craters and chaos areas in the size range 2–40 km diameter points to spatial/temporal variation in crust thickness. But in this size range, craters are progressively outnumbered by chaos areas at larger diameters, suggesting that probability of penetration increases with increasing scale of impact. If chaos areas do represent impact sites, then Europa's surface is older than previously thought. The recalculated resurfacing age is 480 (‐302/+960) Ma: greater than prior estimates, but still very young by solar system standards.