ELF and VLF radio waves

This review covers developments in ELF and VLF radio-wave propagation research over the last 50 years of the Journal of Atmospheric and Solar-Terrestrial Physics. A review of such a large field, over such a long period, cannot be fully comprehensive and the authors have therefore covered important areas which have they themselves have found interesting. The survey begins with a review of work on natural and man made sources of ELF and VLF radiation. This is followed by sections on experimental and theoretical studies of unperturbed (ambient) ELF and VLF radio propagation. Schumann resonance research, which is currently undergoing a renaissance, is then reviewed. A review of research into transient perturbations of ELF and VLF propagation follows, extending from the early work on nuclear explosions up to the current work on sprites. The review concludes with a brief summary of the VLF navigation systems of the USSR and USA, (Alpha and Omega) whose development and life-span covered most of the last 50 years.     

A Non-Planar Circular Model for the 4/7 Resonance

An adiabatic approximation for the non-planar, circular, restricted 3BP is presented for the external resonance 4/7. It can be used as a model for resonant Kuiper belt objects. The Hamiltonian is truncated at the fourth order in eccentricities and inclinations. After averaging, we have a system of two degrees of freedom with two frequencies. Numerical calculations show that the ratio of these frequencies is ~10². Having introduced suitable canonical variables, we used the adiabatic approach introduced by Wisdom in a different context. We left slow variables frozen and after solving the pendulum problem for fast variables, we used the averaged effect of fast variables on slow variables. In this way we obtained the guiding trajectories for slow variables as contour lines of adiabatic invariant. We discuss the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds to the separatrix of fast pendulum motion, where the assumption of sharp division between fast and slow frequencies is not correct and the adiabatic theory fails. The model works well for e ~ 0.1 and can be used for finding the chaotic regions, but for e~ 0.17 it becomes unsatisfactory due to truncation and bad convergence of the Laplace expansion. Qualitatively it can, however, help us to understand how the protective mechanism works as the interplay of mean motion and Kozai–Lidov resonance.     

Extended Schubart Averaging

The purpose of this paper is the presentation of an integrator for the average motion of an asteroid in mean motion commensurability with Jupiter. The program is valid for any (p+q)/p mean motion commensurability (except whenq=0) and uses a double precision version of DE (Shampine and Gordon 1975) as propagator. The averaged equations of motion of the asteroid are evaluated in a non-singular way for any value of the eccentricities and the inclinations and the orbit of Jupiter is described by the most important terms in Longstop 1B (Nobiliet al. 1989). This integrator can be considered as an extension of the well known Schubart Averaging (Schubart 1978) in which Jupiter is moving on a fixed ellipse.     

The high-eccentricity libration theory revisted

Using the local asymmetric expansion of the disturbing function for the planar elliptic restricted three-body problem up to degree 1 ine ₁ we develop a small amplitude libration theory. We review the laws that characterize the mean-motion resonances of asteroids with Jupiter and we obtain other new laws. Special attention is paid to the second forced mode whose equations are reformulated and new consequences of this component are discussed. An analytical expression for the trajectories in the phase space is obtained. The predictions are compared with numerical experiments which confirm the results.     

Hamiltonian Stability of Spin–Orbit Resonances in Celestial Mechanics

The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε_* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε_* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics of Small Earth-Approachers on Low-Eccentricity Orbits and Implications for Their Origins

The population of Near-Earth Asteroids (NEAs) appears to be overabundant at sizes smaller than 50m, compared to a power-law extrapolation from kilometer-sized objects. Several of these small NEAs are also concentrated on low-eccentricity orbits, where a few larger Earth-crossers are observed, and are called Small Earth-Approachers (SEAs). Their source region as well as the dynamical mechanisms involved in their transport close to the Earth on low-eccentricity orbits have not yet been determined. In this paper, we present our numerical and statistical study of the production and dynamical evolution of these SEAs. We first show that three main sources of Earth-crossers which are, according to recent simulations, the 3/1 and ₆ resonances in the main belt, and the Mars-crosser population, are not able to produce as many bodies on SEAs-like orbits compared to other Earth-crossing orbits as has been inferred from observations. From these sources, SEAs-like orbits are reached through the interplay of two required mechanisms: secular resonances and planetary close approaches. However, the time spent on these orbits remains smaller than 1 Myr as confirmed by the study of the evolutions of 11 observed SEAs which also reveal the action of various mechanisms such as close approaches to planets and/or secular resonances. Therefore, our results present some mechanisms which can be responsible for their production but none that would preserve the lifetime of the SEAs sufficiently to enhance their abundance relative to other Earth-crossing orbits at the level observed. The overabundance of the SEA population, if real, remains a problem and could be related to the influence of collisional disruption and tidal splitting of Earth-crossers, as well as to observational biases that might account for a discrepancy between theory and observation.     

A Perturbative Treatment of The Co-Orbital Motion

We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the co-orbital motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.This revised version was published online in October 2005 with corrections to the Cover Date.     

Analytical investigation of non-linear stability of the Lagrangian pointL 4 around the commensurability 1:2

The problem of stability of the Lagrangian pointL ₄ in the circular restricted problem of three bodies is investigated close to the 1 : 2 commensurability of the long and short period libration. By stability we define boundedness of the solution for a given initial finite displacement from the equilibrium point as function of the mass parameter close to the commensurability. A rigorous treatment close to the resonance condition is possible using a transformation that diagonalizes the matrix related to the linear part of the equations of motion. The so obtained equations are further transformed to action angle type variables. Then using an isolated resonance approach, only the slowly varying terms are kept in the equations and two independent isolating first integrals can be found. These integrals finally enable us to solve the stability problem in an exact way. The so obtained results are compared to numeric integration of the equations of motion and are found to be in perfect agreement.     

舒曼共振：测量全球热带气温的最新方法

全球气候突变已成为全世界科学家、企业家和政治家共同关注的焦点问题之一。为了准确地预测全球温度的变化趋势和变化速度，我们必须对全球大气温度进行更为广泛更为精确的测量，研究和发展测量大气温度的新方法。由于全球年平均温度变化不过零点几度，相当于其绝对温度的千分之几，由直接测量全球数千个测点的气温来准确估算全球大气平均温度及其变化是困难的。因此，找出一些依赖于气温的涨落而非线性变化的物理参量是很有用的。这样，测量那些参量的明显变化可以准确地确定温度的细微变化。同地球一电离层谐振腔以及全球范围的闪电活动相联系的舒曼共振的振幅就是这样一种物理量。舒曼共振是一种全球性的电磁现象，它能敏感地反映整个热带大气的温度变化。这就是说，在全球任何一个背景噪声较低的地点设置一个舒曼共振台站，可由这一台站纪录的舒曼共振幅度的明显变化推断全球热带大气气温的细微变化。     

Thunderstorms,Lightning, Sprites and Magnetospheric Whistler-Mode Radio Waves

Thunderstorms and the lightning that they produce are inherently interesting phenomena that have intrigued scientists and mankind in general for many years. The study of thunderstorms has rapidly advanced during the past century and many efforts have been made towards understanding lightning, thunderstorms and their consequences. Recent observations of optical phenomena above an active lightning discharge along with the availability of modern technology both for data collection and data analysis have renewed interest in the field of thunderstorms and their consequences in the biosphere. In this paper, we review the electrification processes of a thunderstorm, lightning processes and their association with global electric circuit and climate. The upward lightning discharge can cause sprites, elves, jets, etc. which are together called transient luminous events. Their morphological features and effects in the mesosphere are reviewed. The wide spectrum of electromagnetic waves generated during lightning discharges couple the lower atmosphere with the ionosphere/magnetosphere. Hence various features of these waves from ULF to VHF are reviewed with reference to recent results and their consequences are also briefly discussed.