Comment On ‘The Earth's Equatorial Principal Axes and Moments of Inertia’ By H. S. Liu and B. F. Chao

Abstract

Recently Liu & Chao (1991) (hereafter cited as ‘L&C’) published a research note stating that prior studies about the orientation of the principal axes of the Earth ‘fell short in providing explicit derivation and expressions’ and that discussions on the subject matter were made, ‘usually in passing’. One can only interpret these remarks as inferring that no previous investigator has derived and published values for the angle between the first principal axis of the Earth and a ‘geographical’ (terrestrial would be a more appropriate word) reference frame. This inquiry appears to be the final objective of their analysis from which everything else evolves, as their summary emphatically demonstrates. The basic theory (equations 2a-2e in L&C) to which the authors parenthetically add ‘complete detailed derivation of the above with explicit normalization factors’ while directing the reader to another reference written by one of the co-authors, was fully elaborated in the seminal book by Hotine (1969, p. 161), well before the availability of reliable artificial satellite data. The coverage of the subject in this treatise does not fall under the author’s imprecise criticism of ‘skimpy, if treated at all’. Hotine arrives at very general equations which are presented, coincidentally, in complex form, the notation ultimately favoured by L&C. Later Bursa (1977) formulated, rather exhaustively, the mathematical problem addressed by L&C with complete explicit derivations and final angular expressions to determine the Earth’s principal directions. This was followed by another short and revealing contribution (Bursa 1983a) (hereafter referred to as ‘B83A’) which is crucial in this context. In fact, equation (6) in B83A [and also in Bursa (1983b)l is practically identical to the central equation in L&C (number Sb), except for an elementary switch to complex notation. It is difficult to understand why L&C also fail to mention the most obvious and rigorous alternative for determining this and other related dynamical parameters: diagonalization of the Earth’s 3 x 3 symmetric inertia tensor by using well known eigentheory procedures. At the core of the problem is the straightforward relationship between the inertia tensor of any arbitrary body and the normalized coefficients of degree 2 and order 2 (i.e. C,,, S,,, n=2, m = 0, . . . , n) of its spherical harmonic expansion of the gravitational potential. By diagonalizing the inertia matrix, not only the orientation of the principal axes (normalized eigenvectors) with respect to a coefficient-implied terrestrial coordinate system can be obtained, but also, as a by-product, the three principal moments of inertia