Arnold diffusion: an overview through dynamical astronomy

The aim of this work is to review the fundamental ideas behind the stability problem, emphasizing the differences between two well-known mechanisms that could lead to chaos, namely overlap of resonances and Arnold diffusion. Here we restrict the discussion to multidimensional autonomous Hamiltonian systems which are of major relevance in Dynamical Astronomy. Arnold diffusion is reviewed in a standard mathematical language, by means of different tools such as heuristic reasoning, graphic and geometrical considerations and numerical experiments. In this direction the pioneer work due to Chirikov [PhR 52 (1979) 263] is followed, but including additional notes, further examples and useful discussions that may well illuminate the understanding of Arnold diffusion. We also summarize the main difficulties when coping with this instability, from both the analytical and numerical sides of the problem. The discussion whether Arnold diffusion could play any role in the dynamical evolution of, for instance elliptical galaxies, is also included.     

On the connection between the Nekhoroshev theorem and Arnold diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschlé et al. (Science 289:2108–2110, 2000). A resonant normal form is constructed by a computer program and the size of its remainder ||R _opt || at the optimal order of normalization is calculated as a function of the small parameter ${\epsilon}$ . We find that the diffusion coefficient scales as ${D \propto ||R_{opt}||^3}$ , while the size of the optimal remainder scales as ${||R_{opt}|| \propto {\rm exp}(1/\epsilon^{0.21})}$ in the range ${10^{-4} \leq \epsilon \leq 10^{-2}}$ . A comparison is made with the numerical results of Lega et al. (Physica D 182:179–187, 2003) in the same model.     

Local and global diffusion in the Arnold web of a priori unstable systems

Using the numerical techniques developed by Froeschlé et al. (Science 289 (5487): 2108–2110, 2000) and by Lega et al. (Physica D 182: 179–187, 2003) we have studied diffusion and stochastic properties of an a priori unstable 4D symplectic map. We have found two different kinds of diffusion that coexist for values of the perturbation below the critical value for the Chirikov overlapping of resonances. A fast diffusion along some resonant lines that exist already in the unperturbed case and a slow diffusion occurring in regions of the phase space far from such resonances. The latter one, although the system does not satisfy the Nekhoroshev hypothesis, decreases faster than a power law and possibly exponentially. We compare the diffusion coefficient to the indicators of stochasticity like the Lyapunov exponent and filling factor showing their behavior for chaotic orbits in regions of the Arnold web where the secondary resonances appear, or where they overlap.     

A numerical study of the size of the homoclinic tangle of hyperbolic tori and its correlation with Arnold diffusion in Hamiltonian systems

Using a three degrees of freedom quasi-integrable Hamiltonian as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to single resonances. We measure an exponential dependence of the splitting of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. We also detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. The variation of the size of the homoclinic tangle as well as the topological transitions turn out to be correlated to the speed of Arnold diffusion.     

A mathematical interpretation of diffusion processes with concentration-dependent diffusion coefficient

The integral balance method has been used to obtain an approximate analytical solution of a nonlinear boundary value problem which arises in the theory of diffusion with a concentration-dependent coefficient. It is the purpose of this paper to give an interpretation of the supposition of interface reactions which obey the law of kinetic mass action.Nomenclature C(Z,t) concentration - C ₀ concentration at initial time - D diffusivity - D ₀ diffusivity at initial time - F(t) a function of time - K ₀ half-order reaction rate constant - k ₁ first-order reaction rate constant - k ₂ second-order reaction rate constante - L characteristic length - n parameter - t time - Z space variable Dimensionless variables and similarity criteria nondimensional half-order reaction rate constant - nondimensional first-order reaction rate constant - nondimensional second-order reaction rate constant - x=Z/L dimensionless space variable - F ₀=D ₀ t/L ² Fourier number - g(F ₀)=[F(t)–C ₀]/C ₀ a function of generalized time - (x, F ₀)=[C(x,t)–C ₀]/C ₀ dimensionless concentration - <(F ₀)> dimensionless average concentration     

A Global network of small automated telescopes

Globalizing observations means increasing awareness and participation in astronomical data collection. This involves the ‘astroeconomics’ of observing facilities and their access. How and to what extent can such facilities be ‘globally networked’? This article concentrates on information, costs and values, with a special regard to the low-cost end of the distribution. The inferences are applicable to a wide range of real observing situations.     

A possible example of giant convective cells delineated by magnetic fields

A remarkable north-south pattern of symmetry in the location of filaments on 11 June, 1972 together with an analysis of surface magnetic field harmonics by Altschuler et al. (1974) is offered as evidence for the existence of giant convective cells. Both data suggest a longitudinal wave number of 5, a value which seems to exclude this symmetry being due to effects of a solar -dynamo. The lifetime of these cells was about 2–4 months. This was apparently the only instance from 1959 to 1973 of such cells being manifested in magnetic field patterns which were evident by harmonic analyses or by filament inspections.     

Analysis of the Chaotic Behaviour of Orbits Diffusing along the Arnold Web

In a previous work [Guzzo et al. DCDS B 5, 687–698 (2005)] we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems. In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally, using the LLCNs we study the ergodicity of the chaotic part of the Arnold web.     

A model of diffusion produced by a cellular surface flow

A model of diffusion induced by the joint action of random cells of two different sizes (granulation and supergranulation) is developed. The basic properties of the model are defined by the dimensionless ratio = (the life time)/(characteristic size/velocity) constructed for these two types of cells. An analytical expression for the diffusivity as a function of is derived. It is estimated that for the standard parameters used the contribution of granulation to the joint diffusion is small. A comparison between the model and numerical simulations of the kinematic diffusion of magnetic fields on the solar surface is made.     

Forced diffusion: A mechanism to create dust clouds

Photodesorption from dust grains by an anisotropic radiation field will lead to much greater momentum transfer than that associated with radiation pressure. This can cause the segregation of dust to the centres of moderately dense interstellar clouds.     

Mean free paths and diffusion coefficients for energetic protons at small heliodistances calculated using Helios 1 and 2 data

Pitch angle scattering of energetic particles (100 MeV) in the interplanetary medium are studied using Helios 1 and 2 magnetometer and plasma data during 1976 near the minimum of solar activity. An IMF configuration was used in the computer experiments which allowed the pitch angle diffusion coefficient, D and hence the parallel mean free path, to be determined. The radial mean free path was found to vary as _r r ^-0.9 between 0.4 and 1 AU, but between 0.3 and 0.4 AU it decreases significantly. To reconcile our value of _r at 1 AU, lying between 0.01 and 0.02 AU, with the average prompt solar proton event profile, an increasing value of _r at lower radial distances would be required.     

A universal control system for small telescopes

An open loop stepper motor controller has been designed specifically for the control of small telescopes. The stepper motor driver is controlled by an IBM-type PC with TURBO PASCAL being used to develop the control software. The software has been set up to be driven either by a mouse or the keyboard. A configuration file contains information specific to the geographical location of the telescope and the mechanical details of the telescope drive such as the type of stepper motors used and the gearing ratios on the two axes. This feature allows the system to be easily attached to a variety of telescopes.     

A small contribution to the giant problem

We present a simple analytic model of a composite polytropic star, which exhibits a limiting Scho¨nberg–Chandrasekhar core mass fraction strongly analogous to the classic numerical result for an isothermal core, a radiative envelope and a μ-jump (i.e. a molecular weight jump) at the interface. Our model consists of an n _c = 5 core, an n _e = 1 envelope and a μ-jump by a factor ≥ 3; the core mass fraction cannot exceed 2/π. We use the classic U V plane to show that composite models will exhibit a Scho¨nberg–Chandrasekhar limit only if the core is 'soft', i.e. has n _c ≥ 5, and the envelope is 'hard', i.e. has n _e < 5; in the critical case ( n _c = 5), the limit only exists if the μ-jump is sufficiently large, ≥ 6/( n _e + 1).     

A diffusion model for cosmic ray propagation in the Galaxy

A diffusion model for the propagation of relativistic nuclear cosmic rays in the Galaxy is developed. The model has two nonstandard features: The escape of cosmic-ray particles from the Galaxy is simulated by a term in the diffusion equations, rather than the imposition of boundary conditions on the diffusion solution at the surface of the confinement region. And an age-dependent, locally-averaged effective gas distribution is employed in the diffusion equations. The model simulates free-particle outflow at the Galactic boundary. The model is fit to chemical composition data in the 0.3–5 GeV per nucleon range. It is then consistent with the large-scale Galactic -ray data, radio halo data, energy constraints on the assumed supernova sources, and, when extended to very high energies, cosmic-ray anisotrophy data. From the fit we conclude that the cosmic rays are confined in a large flattened or quasis-pherical halo with a scale height in the range 3–6 kpc and an average Galactic escape time of 10⁸ yr.     

A photoelectric lightcurve survey of small main belt asteroids

A survey to obtain photoelectric lightcurves of small main-belt asteroids was conducted from November 1981 to April 1982 using the 0.91- and 2.1-m telescopes at the University of Texas McDonald Observatory. A total of 18 main-belt asteroids having estimated diameters under 30 km were observed with over half of these being smaller than 15 km. Rotational periods were determined or estimated from multiple nights of observation for nearly all of these yielding a sample of 17 small main-belt asteroids which is believed to be free of observational selection effects. All but two of these objects were investigated for very short periods in the range of 1 min to 2 hr using power spectrum analysis of a continuous set of integrations. No evidence for such short periods was seen in this sample. Rotationally averaged B(1,0) magnitudes were determined for most of the surveyed asteroids, allowing diameter estimates to be made. Imposing the suspected selection effects of photographic photometry on the results of this survey gives excellent agreement with the results from that technique. This shows that the inability of photographic photometry to obtain results for many asteroids is indeed due to the rotational parameters of those asteroids.     

A space Fresnel Imager for ultra-violet astrophysics: example on accretion disks

The Fresnel Diffractive Imager concept is proposed for space borne astronomical imaging at Ultra-Violet wavelengths, using diffractive focalization. The high angular resolution and high dynamic range provided by this new concept makes it an ideal tool to resolve circumstellar structures such as disks or jets around bright sources, among them, pre-main sequence stars and young planetary disks. The study presented in this paper addresses the following configuration of Fresnel diffractive imager: a diffractive array 4 m large, with 696 Fresnel zones operating in the ultra-violet domain. The diffractive arrays are opaque foils punched with a large number of void subapertures with carefully designed shapes and positions. In the proposed space missions, these punched foils would be deployed in space. Depending on the size of the array and on the working spectral band, the focal length of such imagers will range from a few kilometers to a few tens of kilometers. Thus, such space mission requires a formation flying configuration for two satellites around the L2 Sun-Earth Lagragian point. In this article, we investigate numerically the potential of Fresnel arrays for imaging circumstellar dust environments. These simulations are based upon simple protostellar disk models, and on the computed optical characteristics of the instrument. The results show that protoplanetary disks at distances up to a few thousand parsecs can be successfully studied with a 4 m aperture Fresnel imager in the UV.