Analytic and Computer Studies of Electron Collection by a Spherical Satellite in the Earth’s Magnetic Field

If an attempt is made to produce an artificial aurora by ejecting a beam of electrons from a satellite, the experiment will be jeopardized if the satellite potential becomes so high as to prevent the escape of the beam electrons. The buildup of potential depends on the current collection properties of the satellite. The theoretical problem considered is that of the collection of electrons by a spherical satellite in the earth's magnetic field, as a function of the satellite potential, the satellite dimensions, and the magnetic field strength. At low or intermediate positive satellite potentials, the collected current is of the order of the ambient thermal electron current collected by an area equal to twice the cross-sectional area of the satellite. At large potentials, the current rises slowly, and is proportional to v", where V is the potential, and m is less than unity. The value of m has been estimated theoretically in this work, by two methods. One of these methods, namely, drift approximation theory to second order, leads to an equation, involving the form of the potential distribution, from which the value of m can be found. If the potential distribution is assumed proportional to rBn, where r is the radius in spherical coordinates, the value of m is found to be 2/(n+2). In the other method, the integrals of the exact equations of motion are analyzed to obtain rigorous bounds on the current collected. The current is found to be bounded by a value proportional to V l/2 , independent of the form of the potential distribution under reasonably general conditions. At low potentials, the current may be calculated by solving a self-consistent Poisson problem in which the charge densities depend on the form of the potential distribution. Methods for performing this calculation are discussed. The accuracy and stability of the numerical solution of the Poisson problem are shown to depend on the position of the boundary at which the potential is assumed to be zero.