A Statistical Analysis of the Earth's Internal Magnetic Field

The earth's internal magnetic field consists of the field of a large dipole and an irregular part which is of the order of a few percent of the main dipole field. The field may be represented by a series of spherical harmonics and 42 coefficients are numerically known. The coefficients which characterize the irregular part of the field are here statistically compared with a simple model for this field. It is assumed that an arbitrary number of dipoles are thrown at random inside a spherical shell and mean values are calculated for the probabilities with which the various spherical harmonics appear in the field produced. It is shown that a statistical agreement between the observed and calculated coefficients can be obtained only if the outer radius of the spherical shell is approximately $0.50R$ and is definitely not in excess of the radius of the earth's core, $0.55R$ ($R$, earth's radius). The result is independent of the number of dipoles producing the field. It seems that if the latter number is not very large, the result does not depend too critically upon the assumption of completely random distribution of the dipoles.