The spin, the nutation, and the precession of the Earth's axis revisited from a (numerical) mechanics perspective
Modeling the motion of the Earth's axis, i.e., its spin, nutation and its precession, is a prime example of our ongoing effort to simulate the behavior of complex mechanical systems. In fact, models of increasing complexity of this motion have been presented for more than 400 years leading to an increasingly accurate description. The objective of this paper is twofold namely, first, to provide a review of these efforts and, second, to provide an improved analysis, if possible, based on today's numerical possibilities. Newton himself treated the problem of the precession of the Earth, a.k.a. the precession of the equinoxes, in Liber III, Propositio XXXIX of his Principia [1]. He decomposed the duration of the full precession into a part due to the Sun and another part due to the Moon, which would lead to a total duration of 25,918 years. This agrees fairly well with the experimentally observed value. However, Newton does not really provide a concise rational derivation of his result. This task was left to Chandrasekhar in Chapter 26 of his annotations to Newton's book [2]. He follows an approach suggested by Scarborough [3] starting from Euler's equations for the gyroscope and by calculating the torques due to the Sun and to the Moon on a tilted spheroidal Earth. These differential equations can be solved approximately in an analytic fashion, yielding something close to Newton's more or less fortuitous result. However, the equations can also be treated more properly in a numerical fashion by using a Runge-Kutta approach allowing for a study of their general non-linear behavior. This paper will show how and discuss the outcome of the numerical solution. A comparison to actual measurements will also be attempted. When solving the Euler equations for the aforementioned case numerically it shows that besides the precessional movement of the Earth's axis there is also a nutational one present. However, as we shall show, if Scarborough's procedure is followed, the period of this nutation turns out to be roughly half a year with a very small amplitude whereas the observed (main) nutational period is much longer, namely roughly nineteen years, and much more intense amplitude-wise. The reason for this discrepancy is based on the assumption that the torques of both the Sun and the Moon are due to gravitational actions within the equinoctial plane. Whilst this is true for the Sun, the revolution of the Moon around the Earth occurs in a plane, which is inclined by roughly 5° w.r.t. the equinoctial. Moreover, this plane rotates such that the ascending and descending nodes of the moon precede with a period of roughly 18 years. If all of this is taken into account the analytically predicted nutation period will be of the order of the observed value [4, 5]. As in the case of the precession we will provide a more stringent analysis based on a numerical solution of the Euler equations, which leads beyond the results presented in Sect.12.10 of [5].