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A quasiharmonic Hamiltonian system is a system whose unperturbed part H0(J) is a linear function of the actions. That is, 10.1$$ H = \sum\limits_{i = 1}^N {w_i } J_i + R(q,J,\varepsilon ), $$ where ωi = const and R is a 2π-periodic function of the angles θi, analytical in ε and vanishing for ε = 0. This system is a paradigm of many conservative nonlinear systems. [ABSTRACT FROM AUTHOR]

The integration of the homological equation of the general perturbation theories of Chaps. 3 and 6 is only possible when the short-period frequencies $$ \nu _\mu ^* $$ (μ = 1..., M) obey the non-resonance condition 8.1$$ (h\nu *) = \sum\limits_{\mu = 1}^M {h_\mu } \nu _\mu ^* \ne 0 $$ for all h ≡ (h1..., hM) ∈ ZM appearing in the right-hand-side trigonometric polynomials. The strong restriction introduced by this condition is the very reason for which, in general, those theories cannot be extended to an arbitrarily high order. As discussed in Sect. 3.3.1, the set Dk ⊂ ZM of values of h grows with the order of approximation k and values of (hν*) smaller than any given limit may be formed as the set Dk grows. However, in the applications, we are often interested in a phase space domain where (hν*) = 0 for some h present in the given perturbation εR(θ, J; ε). We have, then, to extend canonical perturbation theories to such cases and learn how to construct formal solutions valid in the neighborhood of resonances. [ABSTRACT FROM AUTHOR]

The perturbation of a regular integrable Hamiltonian may lead to bifurcations in the phase portrait. Let us consider the case of only one resonance and the simple case of one degree of freedom. Let H0 be the Hamiltonian of a differential rotator: 9.1$$ H_0 = \frac{1} {2}\mathcal{V}_1^o (x_1^2 + y_1^2 ) + \frac{1} {8}\mathcal{V}_{11}^o (x_1^2 + y_1^2 )^2 $$ and let us assume that $$ \mathcal{V}_1^o $$ and $$ \mathcal{V}_{11}^o $$ have opposite signs, e.g. $$ \mathcal{V}_1^o < 0 $$ < 0 and $$ \mathcal{V}_{11}^o < 0 $$ > 0. In that case, H0 has a minimum on a circle of radius $$ \sqrt { - 4 - \nu _1^o /\nu _{11}^o } $$ on which the direction of motion changes. This is the classical case of a twist mapping and the Poincaré-Birkhoff theorem predicts that, when the rotator is perturbed, new centers and saddle points may appear in the phase portrait near the place where the frequency of the undisturbed rotator is zero (see [63]). [ABSTRACT FROM AUTHOR]

The actions Ji defined by the phase integrals Ji = 1/2π ∮ pidqi may become singular. The simplest example is given by the actions of a Hamiltonian depending on the squares of the momenta. In this case, pi is proportional to $$ \dot q_i $$ and, as a consequence, the integral ⊂ pidqi is proportional to ⊂ $$ \int {\dot q_i^2 } dt $$ dt and, thus, sign definite. In other words, the integration path is always circulated in the same direction and the sign of Ji may not be reversed (Fig. 7.1). Consequently, the equations of motion in this variable are singular at Ji = 0. [ABSTRACT FROM AUTHOR]

There is a straightforward way of introducing Lie series mappings into the study of Hamiltonian systems. Proposition 5.3.1 shows that the mapping defined by the series (5.25)-(5.26) is canonical and may be used in the construction of canonical perturbation theories. Thus, the reader interested only in such applications can go straight to those equations and skip the first two sections whose aim is to introduce Lie derivatives, Lie mappings and the relationship between Lie mappings and Jacobian canonical transformations. [ABSTRACT FROM AUTHOR]

In 1966, the Poincaré and von Zeipel-Brouwer theories were rejuvenated by Hori [53] through the introduction of canonical transformations expressed by Lie series mappings instead of the classical Jacobian transformations. The use of mathematical operations in perturbation theories with the same properties as Lie derivatives and Lie series was already current among physicists [13] and, around 1960, in at least one instance (Sérsic [85]), it was suggested that Lie series could be used to represent the canonical transformations of Celestial Mechanics. [ABSTRACT FROM AUTHOR]

Delaunay was the first astronomer to use the mechanics of Hamilton and Jacobi to obtain the approximated solution of the equations of motion of a celestial body. His lunar theory [22] is a pioneer work in many respects. We credit Delaunay with the introduction of the set of angle-action variables ℓ, g, h, L, G, H in which the Lagrange equations for the variation of the orbital elements under a perturbation are canonical. His theory of the motion of the Moon is not a collection of clever tricks, as other theories in the old Celestial Mechanics. Having obtained the variation equations in canonical form, his problem was to find the solutions of the differential equations defined by the Hamiltonian 4.1$$ H = H_0 (J) + \varepsilon \sum\limits_{h \in D} {A_h } (J)cos(h\theta ), $$ where the canonical variables are J ≡ (J1..., JN) and θ ≡ (θ1..., θN), ε is a small parameter and D ⊂ ZN. The technique adopted by Delaunay is methodologically very clear. He defined an operation and performed it, successively, almost 500 times. This operation starts with the choice of one argument (h1θ) in (4.1) and the consideration of the dynamical system defined by the abridged Hamiltonian 4.2$$ \mathcal{F}_1 = H_0 (J) + \varepsilon A_{h_1 } (J)cos(h_1 \theta ). $$ This system is integrable, since the angles θi are present only through the linear combination (h1θ). [ABSTRACT FROM AUTHOR]

The trajectories of systems with one degree of freedom are the curves H(q1, p1) = E. As shown in Sect. 1.8, the equations of the motion are given by 2.1$$ t + \alpha _1 = q_1^* = \frac{{\partial S}} {{\partial E}}\mathop = \limits^{def} F_{11} (q_1 ), $$ where α1 is a constant and S = S(q1, E) is the solution of the Hamilton-Jacobi equation. [ABSTRACT FROM AUTHOR]

Many general perturbation theories devised since the nineteenth century were founded on the powerful tools of Hamiltonian mechanics. They aimed at solving the specific problem of finding the solutions of the canonical system of 2N differential equations 3.1$$ \frac{{dq_i }} {{dt}} = \frac{{\partial H}} {{\partial p_i }}, \frac{{dp_i }} {{dt}} = - \frac{{\partial H}} {{\partial q_i }}, $$ (i = 1, 2..., N), where the Hamiltonian H is the energy of the system and may be written as 3.2$$ H = \mathcal{H}_0 (q,p) + \mathcal{R}(q,p,\varepsilon ) $$ with q ≡ (q1..., qN), p ≡ (p1..., pN). $$ \mathcal{H}_0 $$ is the Hamiltonian of a separable system and $$ \mathcal{R} $$ is a disturbing potential, analytical in some small parameter ε, and vanishing for ε = 0. [ABSTRACT FROM AUTHOR]

Astronomers in the nineteenth century found that the form of the Lagrange-Laplace equations for the perturbed Keplerian motion becomes very simple when the set of variables known as Delaunay variables, 1.1$$ \begin{gathered} \ell = mean anomaly, L = \sqrt {\mu a} , \hfill \\ g = argument of the periapis, G = L\sqrt {1 - e^2 } , \hfill \\ h = longtitude of the node, H = G cos i, \hfill \\ \end{gathered} $$ is used (see [15]). Here, μ is the product of the gravitational constant and the mass of the central body, a the semi-major axis, e the orbital eccentricity and i the inclination of the orbit over the reference plane. [ABSTRACT FROM AUTHOR]

Geomagnetic indices play a significant role in describing the magnetic configuration of the Earth's magnetosphere. In the past 15 years, they have become a key parameter in Space Weather research, being commonly used to detect and describe Space Weather events. Research is currently being carried out into using them for forecasting. The objective of this paper is to contribute to a better understanding of the usefulness, potential and limitations of geomagnetic indices in Space Weather research and applications [ABSTRACT FROM AUTHOR]

The European Commission Directorate General Transport and Energy published in 2004 a summary report of research on aircrew dosimetry carried out by the EURADOS working group WG5 (European Radiation Dosimetry Group, http://www.eurados.org/). The aim of the EURADOS working group WG5 was to bring together, in particular from European research groups, the available, preferably published, experimental data and results of calculations, together with detailed descriptions of the methods of measurement and calculation. The purpose is to provide a dataset for all European Union Member States for the assessment of individual doses and/or to assess the validity of different approaches, and to provide an input to technical recommendations by the experts and the European Commission. Furthermore EURADOS (European Radiation Dosimetry Group, http://www.eurados.org/) started to coordinate research activities in model improvements for dose assessment of solar particle events. Preliminary results related to the European research project CONRAD (Coordinated Network for Radiation Dosimetry) on complex mixed radiation fields at workplaces are presented. The major aim of this work is the validation of models for dose assessment of solar particle events, using data from neutron ground level monitors, in-flight measurement results obtained during a solar particle event and proton satellite data. The radiation protection quantity of interest is effective dose, E (ISO), but the comparison of measurement results obtained by different methods or groups, and comparison of measurement results and the results of calculations, is done in terms of the operational quantity ambient dose equivalent, H* (10). This paper gives an overview of aircrew radiation exposure measurements during quiet and solar storm conditions and focuses on dose results using the EURADOS In-Flight Radiation Data Base and published data on solar particle events [ABSTRACT FROM AUTHOR]

The harsh radiation environment in the inner magnetosphere up to geosynchronous orbit is of major concern to an ever increasing amount of space hardware. While the average or quiescent conditions of the energetic particle population are fairly well characterized, the dynamics during magnetic storms are severely under-sampled. The description of the energetic electron and proton radiation environment at all points in space, which can provide reliable environmental data for locations of satellites that do not carry any energetic particle instrumentation is not trivial. The following approaches, theoretical, empirical, and model/data assimilation are examined in this paper [ABSTRACT FROM AUTHOR]

The solar wind effects on Earth environment are studied for their basic science value as well as for their crucial practical impact on human technological systems. Increased dissipation of solar wind energy in the near-Earth environment is a significant source of consequent perturbations in the upper atmosphere and ionosphere. This chapter addresses the ionospheric manifestation of geomagnetic storms induced by solar wind. Changes in the electron density distribution at the ionospheric F region heights above Europe during strong-to-severe geomagnetic storms, which occurred over present solar cycle, have been analysed. As for the seasonal preference, during storm main phase only negative phases dominate in summer, while during winter occurrence of both negative and positive phases is probable. Enhancements of electron density have been sometimes observed several hours before the onset of geomagnetic storm. Also the existence of few-hours-long periods during storm main phase, when the deviation of the electron density from median was insignificant, has been observed. Independent of the sign of the storm effect on F2 region ionisation, the effect on electron density at the F1 region heights at European higher middle latitudes has been found negative, if any at all. The F1 region response to magnetic disturbances also shows substantial summer/winter asymmetry. The stormy high latitude F region is most variable compared with middle and lower middle latitudes, being strongly influenced by magnetospheric processes, in particular, strong electric fields, which are usually present during geomagnetic storms. Several specific features of the storm-time high latitude ionosphere will briefly be mentioned including behaviour of ionospheric scintillations. The comparative analysis illustrates that the improved IRI-2001 model with the activated STORM option provides better description of the ionisation distribution above Europe under geomagnetic storm conditions. Nevertheless, our results show that model not always estimates correctly the storm phase and the magnitude of the effects on F region electron density [ABSTRACT FROM AUTHOR]

An analysis of the F2-layer short-term forecast problem has been done. Both objective and methodological problems prevent us from a deliberate F2-layer forecast issuing at present. An empirical approach based on statistical methods may be recommended for practical use. A forecast method based on a new aeronomic index (a proxy) AI has been proposed and tested over selected 64 severe storm events. The method provides an acceptable prediction accuracy both for strongly disturbed and quiet conditions. The problems with the prediction of the F2-layer quiet-time disturbances as well as some other unsolved problems are discussed [ABSTRACT FROM AUTHOR]

Global Navigation Satellite Systems (GNSS) are widely used to measure positions with accuracies ranging from a few mm to about 20 m. The effect of the Earth ionosphere on GNSS signal propagation is one of the main error sources which limits the accuracy and the reliability of GNSS applications. In particular, disturbed Space Weather conditions can be the origin of strong variability in the ionosphere Total Electron Content (TEC) which itself degrades the accuracy of GNSS applications. Space Weather effects on GNSS depend very much on the type of application. In this paper, we discuss the effects of Space Weather conditions on differential positioning with the Differential GPS (DGPS) technique and on relative positioning with the Real Time Kinematic (RTK) technique. We show that DGPS is affected by medium to large-scale gradients in TEC mainly observed at solar maximum when RTK will be degraded by smaller-scale ionospheric variability due to scintillations, TEC noise-like behaviour and Travelling Ionospheric Disturbances [ABSTRACT FROM AUTHOR]

Thermospheric density models are a main source of error in the orbit determination and prediction of low Earth satellites. The empirical models that are in wide use today show large systematic errors when compared with data derived from accelerometers and spacecraft tracking. This accuracy limit is inherent in their model formulation, which is based on an imperfect correlation of observed thermosphere density with a limited set of certain space weather proxy indices. It has been demonstrated that a substantially higher accuracy can be reached by model calibration using concurrent observations of satellite drag. Such drag observations can be obtained by processing freely available Two-Line Element (TLE) data, which are used for representing and distributing satellite orbit trajectories. Several aspects of this data processing require specific attention. These include the selection of suitable space objects, determining their ballistic coefficients, and taking into account thermospheric winds and radiation pressure accelerations [ABSTRACT FROM AUTHOR]

As a consequence of alterations of atmospheric chemical composition due to anthropogenic emissions, Earth's ionosphere and thermosphere are expected to change. A number of authors tried to detect signs of global change in their ionospheric data, but many findings remain controversial. We briefly review long-term trends observed in the critical frequencies of the ionospheric E and F2 layers as well as in the height of the F2-peak, i.e. the layer of maximum electron density. Using 48 years of F2-layer critical frequency data from Sodankylä, Finland, we demonstrate how the sign and amplitude of the detected trends depend upon the choice of model fitted to the data and suggest a method to choose the best possible model [ABSTRACT FROM AUTHOR]

In order to determine the influence of mankind on climate change it is important to understand the natural causes of climate variability. A natural effect that has been hard to understand physically is an apparent link between climate and solar activity. From historical and geological records there are strong indications that the sun has played an important role in the past climate of the Earth, but the physical mechanism is currently unknown. Whatever mechanism caused those earlier changes would most likely also be operating today and may have been active throughout the history of our planet. There have been several attempts to explain the link between solar activity and climate from variations in the sun's radiative output. These have tended to rely on simulations involving Global Climate Models (GCM), which are limited by our current understanding of the fundamental physics. In the following contribution, an outline of the current candidate mechanisms involving solar activity will be presented together with a description of the ESA funded project to study the Influence of Solar Activity cycles on Earth's Climate (ISAC) [ABSTRACT FROM AUTHOR]

Solar radiation is the fundamental energy source for the atmosphere and the global average equilibrium temperature of the Earth is determined by a balance between the energy acquired by the solar radiation absorbed and the energy lost to space by the emission of heat radiation. The interaction of this radiation with the climate system is complex but it is clear that any change in total solar irradiance (TSI) has the potential to influence climate. In the past, although many papers were written on relationships between sunspot numbers and the weather, the topic of solar influences on climate was often disregarded by meteorologists. This was due to a combination of factors of which the key was the lack of any robust measurements indicating that solar radiation did indeed vary. There was also mistrust of the statistical validity of the evidence and, importantly, no established scientific mechanisms whereby the apparent changes in the Sun might induce detectable signals near the Earth's surface. Another influence was a desire by the meteorological profession to distance itself from the Astrometeorology movement popular in the 19th century (anderson1999). Nowadays, with improved measurements of solar and climate parameters, evidence for an influence of solar variability on the climate of the lower atmosphere has emerged from the noise. This article provides a brief review of the observational evidence and an outline of the mechanisms whereby rather small changes in solar radiation may induce detectable signals near the Earth's surface is not possible to review here all potential mechanisms for solar-climate links. What is presented offers, necessarily, a personal perspective but, of the areas that are not covered, two may be pertinent: the effects of solar energetic particles on stratospheric composition (see e.g. jackman et al. 2005) and the possible influence of galactic cosmic rays on clouds through ionisation processes (see Marsh, this volume). [ABSTRACT FROM AUTHOR]

A factor for understanding the origin and evolution of life on Earth is the evolution of the Sun itself, especially the evolution of space climate and weather. Many aspects of the Sun's history remain to be understood. We reconsider constraints that knowledge of our own star implies for the emergence of life on Earth. This provides further insights into what may happen in other solar systems. Fortunately, particles emitted by the Sun in the past have left a record in geologic samples, but on this bases we cannot exclude earlier dates for the onset of life on Earth. A very early origin of life has to take into account the imprints of solar energetic particles during the first billion years (Gyr) after the formation of the Sun, approximately from 4.6 till 3.6 Gyr before the present (BP). Our review includes the isotopic fractionation of the noble gases, the depletion of volatile elements on the Moon and constraints for the origin of life on Europa, the icy moon of Jupiter [ABSTRACT FROM AUTHOR]

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Solar energetic particles (SEPs) in the interplanetary (IP) medium are transported under the influence of electromagnetic fields of the solar wind. These fields consists of the smooth background fields, which can be modeled by the MHD equations governing the expansion of the solar wind, and of the small-scale fluctuations (waves or turbulence) that scatter the particles in pitch angle and act as agents enabling their acceleration at IP shock waves. We review theoretical models of SEP transport and acceleration in the IP medium. We start from the simple analytical approaches (diffusion models), which assume quasi-isotropic particle distributions, and then continue to the more accurate numerical approaches based on the focused transport equation, not making this simplifying assumption. A careful analysis of two SEP events, an impulsive and a gradual one, is presented and the spatial scaling of their peak intensities, differential fluences and time-integrated net fluxes is discussed. We conclude that rather simple scaling laws for these quantities can be obtained for impulsive events but no simple scaling laws can be expected to govern the gradual SEP events [ABSTRACT FROM AUTHOR]

The tearing instability can play a significant role in reconnecting current layers, but it is well stabilized in many cases of interest. For this reason, quasi-stationary current layers can exist for a long time in astrophysical plasma, for example in the solar corona, in the Earth magnetospheric tail. [ABSTRACT FROM AUTHOR]

The inductive electric field is directed along the current inside a collisionless reconnecting current layer (RCL). This strong field does positive work on charged particles, thus increasing their energy impulsively, for example, in solar flares of flares in the accretion disk coronae of compact astrophysical objects. [ABSTRACT FROM AUTHOR]