Focusing of Paraxial Rays in a Magnetic Dipole Field

It is well known that a magnetic field of axial symmetry has a focusing effect for paraxial orbits. The field of the earth is of this type and it is shown that, if the cosmic rays consist of protons, a bundle of orbits parallel at infinite distance is focused on the surface of the earth if the energy is about to GeV. The co-latitude of the bundle at infinite distance is about 2.2 times the co-latitude of the focal point. The asymptote to the orbits is about 22° further to the east than the focal point. There is also an infinite series of lower energies exhibiting focusing properties.

The calculations refer to paraxial orbits, but a comparison with terrella measurements shows that the focal energy and the above relation between the co-latitudes is appropriate down to about 50° latitude.