Published in EOS Trans. AGU 73(43):63, October 27, 1992, Poster U21B-09, Abstract Only
Is the Earth Pear-shaped?
Of course not. The shape of the Earth (the geoid) is represented as a sum of a large number of spherical harmonic functions. The coefficient of each harmonic is determined through analysis of satellite orbits. The largest coefficient is degree zero, followed by the degree two, order zero, coefficient (a bi-axial ellipsoid, an effect of the centrifugal bulge). These dominate the other harmonics by two orders of magnitude. Thus, even with its rough topography, the Earth is often said to be as smooth as a billiard ball. Though not quite as round as a billiard ball, it is hardly "pear-shaped". Because early satellite orbit analysis found some power in the degree three order zero harmonic, the Earth was described as pear-shaped. Regardless of the small magnitude of the degree three order zero coefficient, subsequent analyses have found that there are many harmonics that are even larger than the degree three order zero. The Earth should no longer be said to be pear-shaped.
The r.m.s. value of the coefficients for each degree is invariant under change of axis. By changing the axis about which the coefficients are computed, we find that a considerable amount of the power in each degree resides in a small number of coefficients. This minimum entropy representation, which can be applied to any distribution, is ergonomic. Reducing the number of significant coefficients (2n + 1, for each degree n) aids in visualization. For degree two, we find that the shape of the Earth (exclusive of the centrifugal bulge) is best represented by a single order two harmonic (a tri-axial ellipsoid) whose axis is at 0N 75E. About this axis, the other four coefficients are nearly zero. From another axis, the coefficients are all about equal. Thus, the within-degree spectrum can be either completely ordered or random, for the same distribution, depending upon the axis chosen. Using this minimum entropy representation for degree three, we find that the shape of the Earth is well-represented by a degree three order two harmonic - which looks more like a tetrahedron than a pear.