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Cosmic ray cutoff rigidities and associated solar-terrestrial phenomena
thesisposted on 2023-05-27, 17:08 authored by Shea, MA
GEOMAGNETIC CUTOFF RIGIDITIES HISTORICAL BACKGROUND. Experimental studies using mobile cosmic ray detectors such as ionization chambers and later neutron monitors (see, for example, Clay, 1934; Clay et al., 1936; Compton and Turner, 1937; Rose et al., 1956; Simpson et al., 1956; Katz, et al., 1958; Pomerantz and Agarwal, 1962) demonstrated that the cosmic ray intensity was a minimum in the earth's equatorial regions and increased to a maximum value as one approached the polar latitudes. These experimental results confirmed the predictions of theoretical studies by Störmer (1930) and Lemaitre and Vallarta (1936). The reason for this increase in cosmic radiation intensity between the equatorial and polar regions is that the earth's magnetic field provides a shield against the incoming charged particle nucleon radiation. Störmer (1930), Vallarta (1938) and Rossi (1940) recognized that the accurate determination of particle access to a specific location from a specified direction on the earth would require a series of calculations of charged particle orbits in a mathematical model of the earth's magnetic field. The general equation of particle motion in a magnetic field does not have a solution in closed form, even in a simple dipole field. To determine which particles are allowed at a specific/ geographic location, it is necessary to perform detailed and extensive numerical calculations of cosmic ray trajectories in a mathematical model of the earth's magnetic field. The initial calculations were made using a dipole approximation to the geomagnetic field. The method used was to launch a particle of negative charge and a specific rigidity in a specified direction from a specific location on the earth and numerically calculate the particle trajectory to a location far enough from the earth that the particle trajectory could be declared "allowed" or "forbidden". Since most of the cosmic ray particles are positively charged, the trajectory of a negative particle outward from the earth would be identical to the path of a similar positive particle moving from the interplanetary medium to the detection location. An "allowed" particle meant that the trajectory extended to interplanetary space; a "forbidden" particle meant that the particle trajectory intersected the solid earth or did not have enough momentum to escape the magnetic field (i.e. became trapped). Because of this labor-intensive effort, only a relatively few particle trajectories could be calculated in this manner. One of the most important aspects of the work to delineate charged particle access was to define the lowest rigidity a charged particle could possess and still arrive from a specified direction at a specific point within the magnetosphere. This rigidity value became known as the "cutoff rigidity". Strictly speaking the cutoff rigidity of any geographic location is a function of the zenith and azimuth angles of arrival, the altitude of the detection location, and the geomagnetic conditions at the time of the measurement. However, there is an additional important aspect that must be considered in the determination of cosmic ray cutoff rigidities — the cosmic ray penumbra. The cosmic ray penumbra is a region, identified by the work of Störmer, where there are alternating allowed and forbidden orbits. Particle trajectories are typically traced at successively decreasing rigidity values. The initial value is selected such that all particles above this value will be allowed at a specific location from the specified direction. As the rigidity value is decreased, a value is reached where the negative particle cannot escape the geomagnetic field. This is called the main cone cutoff or the upper cutoff value. As the particle rigidity decreases, alternating allowed and forbidden particle trajectories are identified until a value is reached below which all orbits are forbidden. The lowest allowed rigidity value is called the Störmer cutoff or lower rigidity cutoff value. The region between the upper cutoff rigidity and the lower cutoff rigidity is the cosmic ray penumbra. (See Paper 20 for a complete definition of cosmic ray cutoff terminology.) To accurately estimate the effect of the cosmic ray penumbra, which can extend over rigidity ranges for more than 3 GV, was extremely difficult using the initially available methods. Because of the complexities involved, calculated cutoff rigidities were usually determined for only the vertical direction at the desired location. Based upon the initial work of Störmer (1930) and Vallarta (1938), and with the onset of space research, new methods were developed to estimate vertical cutoff rigidity values. Quenby and Webber (1959) included nondipole terms for the geomagnetic field model. Quenby and Wenk (1962) employed field line calculations to estimate cutoff values for high latitudes, a modified Störmer method for equatorial latitudes, and a combination of the two methods for mid latitudes. Makino (1963) modified the Quenby and Webber approximations by introducing different penumbral corrections than those of Quenby and Wenk and by introducing an empirical eastward shift of the impact point of the particle. Several tables of world wide cutoff rigidity values were published (Quenby and Webber, 1959; Quenby and Wenk, 1962; Makin°, 1963). Although these values represented improvements over the values derived from the Störmer dipole equation or the eccentric dipole approximations of Vallarta (1935), inconsistencies were still noted when these values were utilized in cosmic ray data analyses (Kodama, 1960). With the availability of high speed digital computers in the early 1960s, it became tractable to use numerical methods to calculate a large number of trajectories of charged particles as they traversed the earth's magnetic field. In an effort to resolve an inconsistency related to the study of the relativistic solar particle events in November 1960, Freon and McCracken (1962) numerically calculated a series of cosmic ray trajectories in a high order mathematical model of the geomagnetic field. These calculations resulted in the determination of the vertical cutoff rigidity for a single location: Port aux Francais, Kerguelen Islands. This was the general status of cosmic radiation cutoff rigidities when I became interested in this type of research problem. CONTRIBUTIONS TO GEOMAGNETIC CUTOFF RIGIDITIES: INTERNAL GEOMAGNETIC FIELD MODEL CALCULATIONS Paper 1 represents the initial publication related to the numerical tracing of cosmic ray particles in a high order mathematical model of the geomagnetic field. Although published as a technical report by the Massachusetts Institute of Technology, this publication has been extensively cited in the scientific literature. Almost 40 years since it was printed, the authors still receive requests for this landmark publication. The mathematical field model utilized for these initial trajectory calculations was the 6th order Finch and Leaton (1957) representation of the internal geomagnetic field. The coefficients for this model were derived from the British Admiralty magnetic charts for 1955.0. Paper 1 presented the concept of particle tracing and defines the asymptotic direction of approach of a cosmic ray particle which arrives at some given point on the earth's surface as the direction from which the particle was moving prior to its entry into the geomagnetic field. A listing of the initial FORTRAN computer program to trace cosmic ray trajectories in a high degree simulation of the geomagnetic field was included together with extensive tables enabling users to verify their results when using the computer code. The report also included tables of variational coefficients, permitting calculation of time variations observed by a cosmic ray detector as a consequence of any arbitrary anisotropic flux of cosmic radiation outside the geomagnetic field. Although geomagnetic cutoff rigidities are not specifically mentioned, the particle tracing technique formed the basis of the extensive calculations required for geomagnetic cutoff calculations. The trajectory-tracing calculations were initiated at an altitude of 20 km above the surface of the earth. This altitude was selected as optimum since the geomagnetic field controls the particle path above the atmosphere. After the first nuclear interaction with atmospheric atoms, geomagnetic forces no longer control the particle path. High-energy particles generate a nuclear cascade that propagates through the atmosphere and can be detected on the surface of the earth by cosmic ray detectors such as neutron monitors. At the request of the organizers of the International Years of the Quiet Sun (IQSY), Paper 1 was expanded into IQSY Instruction Manual No. 10 (McCracken et al., 1965). It was later revised and included in the Annals of the IQSY (Paper 6) with supplementary tables published as a technical report (Report T2 in Table I). As the junior member of this team, I had the responsibility of the computer calculations. These calculations were very demanding for the computer capability at the time and were made on the high speed digital computers at the Massachusetts Institute of Technology and the Air Force Cambridge Research Center. I evaluated the computational results, and as each set of calculations was completed, I compiled the data for final analysis by my co-authors. Paper 2 was the initial paper summarizing the use of cosmic ray trajectory calculations to determine vertical cutoff rigidities for a selected set of locations along the route of a cosmic ray latitude survey. This paper was presented at the 8th International Cosmic Ray Conference and published in the non-refereed proceedings. The primary purpose of this presentation and publication was to acquaint the cosmic ray community with the powerful particle trajectory-tracing technique for the determination of geomagnetic cutoff rigidity values. Paper 3 provided the foundation for ...