Analytic contributions to the Störmer problem

The purpose of the paper is to furnish two analytic contributions to the Störmer problem (motion of a charged particle in a magnetic dipole field). The fundamental role in these considerations is played by a quantity which depends on the constant velocity of the particle: by treating this quantity or its reciprocal as a small parameter, power series solutions can be obtained for the orbit projection upon a meridian plane valid for both low-energy and high-energy particles.For high-energy particles, the zero-order approximation is in general an ellipse, but will reduce to a straightline for particular values of the integration constants.For low-energy particles, the zero-order approximation is given as an infinite series in the variablew=sin², being the latitude, and it is shown that this basic orbit is not a magnetic line of force.Leading terms of the expansions have been given for the first-order approximations in both cases. Higher-order approximations can be obtained recurrently by solving linear, first-order differential equations having the same integrating factor, which depends on the zero-order approximation alone.The method is suitable for extensive numerical work in conjunction with a computer program of the formac class.