Magnetospheric Models and Trajectory Computations
نویسندگان
چکیده
The calculation of particle trajectories in the Earth’s magnetic field has been a subject of interest since the time of Störmer. The fundamental problem is that the trajectory-tracing process involves using mathematical equations that have ‘no solution in closed form’. This difficulty has forced researchers to use the ‘brute force’ technique of numerical integration of many individual trajectories to ascertain the behavior of trajectory families or groups. As the power of computers has improved over the decades, the numerical integration procedure has grown more tractable and while the problem is still formidable, thousands of trajectories can be computed without the expenditure of excessive resources. As particle trajectories are computed and the characteristics analyzed we can determine the cutoff rigidity of a specific location and viewing direction and direction and deduce the direction in space of various cosmic ray anisotropies. Unfortunately, cutoff rigidities are not simple parameters due to the chaotic behavior of the cosmic-ray trajectories in the cosmic ray penumbral region. As the computational problem becomes more manageable, there is still the issue of the accuracy of the magnetic field models. Over the decades, magnetic field models of increasing complexity have been developed and utilized. The accuracy of trajectory calculations employing contemporary magnetic field models is sufficient that cosmic ray experiments can be designed on the basis of trajectory calculations. However, the Earth’s magnetosphere is dynamic and the most widely used magnetospheric models currently available are static. This means that the greatest uncertainly in the application of charged particle trajectories occurs at low energies. 1. Historical Background The integration of the equation of motion of a charged particle in a magnetic field is a problem that has no solution in a closed form. The first numerical efforts at integration of the equations of particle motion began with Störmer (1930) who utilized a dipole representation of the Earth’s magnetic field. (The legend is that there were rooms of students manually doing the computations.) The work of Störmer is summarized in his book ‘The Polar Aurora’ (Störmer, 1950). The first application of computers to obtain solutions for particle trajectories was done by Lemaitre and Vallarta (1936a, b) who used a ‘Bush differential analyzer’ (what would now be called an analog computer) to obtain solutions for entire families of trajectories. Their definitions and classic work on the ‘allowed cone of cosmic radiation’ are still in use (Vallarta, 1938, 1961, 1978). The problem of defining particle trajectories in a magnetic field was so difficult that ‘terella’ experiments (large vacuum chambers with scale size simulations of the Earth’s magnetic field and evaluation Space Science Reviews 93: 305–333, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. 306 D. F. SMART ET AL. of electron trajectories in the magnetic field) were a preferred method of approach for a number of years (Brunberg, 1953, 1956; Brunberg and Dattner, 1953). Researchers at the University of Chicago employing the AVIDAC computer at Argonne National Laboratories did the first application of the digital computer to the cosmic ray trajectory problem in the United States. Jory (1956) calculated a set of 663 particle orbits in a dipole magnetic field. Lust (1957) calculated some 1,500 orbits to better define the concept of impact zones. Kasper (1959) did a more extensive set of trajectory calculations on a digital computer, some 2000 trajectories in a dipole magnetic field. More advanced magnetic field models were utilized by McCracken and his co-workers who became very successful in the digital computer calculation of cosmic-ray trajectories in high order simulations of the geomagnetic field to describe observed cosmic ray phenomena. They calculated particle access to specific cosmic ray stations on the Earth to describe the cosmic ray anisotropy (McCracken et al., 1962, 1965, 1968). They also showed that the observed cosmic ray intensity could be well ordered by geomagnetic cutoff rigidities derived from cosmic ray trajectories calculated in high order simulation of the Earth’s magnetic field (Shea et al., 1965). They also demonstrated that the Earth’s internal magnetic field is evolving (quite rapidly on geologic time scales), and that updated cutoff rigidity calculations are necessary to explain the changes observed in some areas of the world (Shea and Smart, 1970, 1990; Mischke et al., 1979). However, the Earth’s geomagnetic field evolution is not uniform. Geomagnetic ‘jerks’ have been found in the Earth’s magnetic field (Langel et al., 1986; Macmillan, 1996). Gall and co-workers were the first to utilize magnetospheric models to improve the calculations of asymptotic directions and high-latitude cutoff rigidities. They obtained a better resolution of the asymptotic cones of acceptance and calculated the range of the daily variation in both asymptotic directions and cutoff rigidities at high latitudes (Gall et al., 1968; 1969; 1971a, b; Smart et al., 1969). It also became possible to delineate solar particle access to regions of the magnetosphere by tracing allowed particles (see Morfill and Scholer, 1973, for a review of this period.) However, it became evident that the early models of the magnetospheric fields were deficient in that they were unable to adequately explain the low-altitude earth-orbiting spacecraft observations of energetic particle access into the Earth’s high polar regions during very anisotropic solar particle events. (Gall and Bravo, 1973; Morfill and Quenby, 1971; Morfill and Scholer, 1972a, b; Thomas et al., 1974). The general result of these interchanges was a realization of the inadequacies of the early magnetospheric field models. Paulikas (1974) noted that the early magnetospheric model trajectory calculations could delineate the general regions of solar particle access to the magnetosphere, but were not capable (at that time) of resolving the fine spatial structure noted by polar orbiting spacecraft. The results obtained by the trajectory tracing calculations, particularly in the magnetospheric tail, during the 1970s and early 1980s were reflecting the topology of the MAGNETOSPHERIC MODELS AND TRAJECTORY COMPUTATIONS 307 cartoon-like magnetic fields in the early magnetospheric models and not the result of physical processes. While advances in computer technology over the past decades have allowed researchers to more fully utilize the trajectory-tracing technique for various cosmic ray analyses, this approach continues to present a formidable problem. As computers become more powerful, magnetic field models of increasing complexity, which better represent the Earth’s magnetic topology, are being developed. Consequently, these more complex geomagnetic field representations must be utilized for analyses of the higher precision measurements of cosmic radiation phenomena. As long as the measurement techniques increase in accuracy and as long as the geomagnetic field models continue to improve, the trajectory-tracing process will be used for cosmic radiation research. This paper presents the mathematical equations used in the trajectory tracing procedure, identifies the various geomagnetic field representations, explains the determination of the cutoff rigidity values, and summarizes how these calculations have been and continue to be used for cosmic radiation studies. 2. The Equations Involved 2.1. THE CHARGED PARTICLE EQUATION OF MOTION The equation of charged particle motion in a magnetic field may be written in vector form as
منابع مشابه
Lane Change Trajectory Model Considering the Driver Effects Based on MANFIS
The lane change maneuver is among the most popular driving behaviors. It is also the basic element of important maneuvers like overtaking maneuver. Therefore, it is chosen as the focus of this study and novel multi-input multi-output adaptive neuro-fuzzy inference system models (MANFIS) are proposed for this behavior. These models are able to simulate and predict the future behavior of a Dri...
متن کاملSVD analysis of the magnetospheric AE index time series and comparison with low-dimensional chaotic dynamics
The singular value decomposition (SVD) analysis is used at different stages in this paper in order to extract useful information concerning the underlying dynamics of the magnetospheric AE index. As a frame of reference we use the dynamics of the Lorenz system perturbed by external noise, white or colored. One of the critical results is that the colored noise can be differentiated from the whit...
متن کاملEnsemble downscaling in coupled solar wind-magnetosphere modeling for space weather forecasting
Advanced forecasting of space weather requires simulation of the whole Sun-to-Earth system, which necessitates driving magnetospheric models with the outputs from solar wind models. This presents a fundamental difficulty, as the magnetosphere is sensitive to both large-scale solar wind structures, which can be captured by solar wind models, and small-scale solar wind "noise," which is far below...
متن کاملAnalyzing the performance of different machine learning methods in determining the transportation mode using trajectory data
With the widespread advent of the smart phones equipping with Global Positioning System (GPS), a huge volume of users’ trajectory data was generated. To facilitate urban management and present appropriate services to users, studying these data was raised as a widespread research filed and has been developing since then. In this research, the transportation mode of users’ trajectories was identi...
متن کاملCaII Infrared triplet line models in Classical T Tauri stars
We study the formation of the Calcium II infrared triplet lines 8498Å, 8542Å and 8662Å, in the accreting magnetospheric flows of Classical T Tauri stars (CTTS), and present a grid of models for a large range of magnetospheric conditions. We apply our models to the interpretation of multi epoch observations of the CTTS DI Cep. We find that these lines form in the magnetospheric infall and that t...
متن کامل