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The Spin, the Nutation and the Precession of the Earth's Axis Revisited

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The Spin, the Nutation and the Precession of the Earth's Axis Revisited The spin, the nutation and the precession of the Earth’s axis revisited The spin, the nutation and the precession of the Earth’s axis revisited: A (numerical) mechanics perspective W. H. M¨uller [email protected] Abstract Mechanical models describing the motion of the Earth’s axis, i.e., its spin, nutation and its precession, have been presented for more than 400 years. 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 decomposes the duration of the full precession into a part due to the Sun and another part due to the Moon, to predict a total duration of 26,918 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] starting from Euler’s equations for the gyroscope and 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 Newton’s result. However, they can also be treated numerically by using a Runge-Kutta approach allowing for a study of their general non-linear behavior. This paper will show how and explore the intricacies of the numerical solution. 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 nu- tation present. However, 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. This discrepancy can only be solved if the torques of both the Sun and the Moon are considered and if it is observed that the revolution of the Moon around the Earth occurs in a plane inclined by roughly 5◦ w.r.t. the equinoctial, whilst the ascending and descending nodes of the moon precede with a period of roughly 18 years. For the sake of brevity the corresponding numerical assessment will not be presented here. 1 Some historical background and the experimental evidence On first glance it may appear trivial, but the motion of the Earth or, more precise, the motion of its axis is quite complex and convoluted. In geoscience [4] it is customary to decompose it into a product of various movements, characterized by matrices, as follows: xIRF = P · N · S · W · xTRF : (1) The symbol xIRF and xTRF refer to the position of a point, i.e., coordinates including (Cartesian) base vectors when viewed from an Inertial and from a Terrestrial Reference Frame, IRF and TRF, respectively. The latter is co-moving with the Earth, so-to-speak “rigidly attached,” and it is located in its center of mass. The numerics in this paper 379 Proceedings of XLII International Summer School–Conference APM 2014 focuses exclusively on rigid Earth models. In this context the Earth will be idealized as a spheroid, i.e., an ellipsoid with two principal axes of equal length, and it is assumed that the mass distribution is such that the center of gravity is at the very center of the ellipsoid. W refers to the so-called wobble or polar motion matrix. Wobble is the movement that remains after the Earth is freed of all the actions resulting from external forces and torques (i.e., moments of the gravitational forces) exerted by the Sun, the moon, and the other planets. Its origin is internal mass motion of planet Earth, e.g., caused by winds or by the oceans. For the sake of brevity we will not engage in an in-depth discussion of the wobble phenomenon and its modeling in this paper. The matrices P , N, and S refer to the precession (of the equinoxes), the nutation, and to the spin of the Earth’s axis. The situation is illustrated in Fig. 1. In particular, P and N are the result of the aforementioned external gravitational torques. The observed precessional period of ca. 25,730 years [6] is essentially due to the combined gravitational pull of the Sun and of the Moon on the tilted (by ca. 23:4◦ w.r.t. the ecliptic) spheroidal Earth. The observation of the precession phenomenon is very old and goes back to the ancient Greeks, namely Hipparchos [7], who relied upon archaic but effective astronomical measurements. Newton was the first who provided a theoretically based value for the related precessional speed, namely 50000000012IV = 2:42·10−4 rad/year in Propositio XXXIX, Lemma IV of his Principia [1]. To be precise, the term “year” refers to a sidereal year. Newton’s explanations, however, are very much in need of getting used to. Luckily, Chandrasekhar explains them to us in modern terms in [2]. Figure 1: Illustration of Preces- Figure 2: Nutation angle data sets from [16]; note sion (P ), Nutation (N) and Ro- that ∆ sin ! '_∆t sin # and ∆ ! #_∆t in stan- tation (spin, R) of the Earth dard mechanics nomenclature for the Euler an- from [5]. gles, mas = milli arc seconds. Newton is also credited [7] for having touched upon the existence of a small nutation, N. However, its proper amount was quantified much later: Based on careful astronomical observations a nutation period of 18 years with an amplitude of 900 was announced in a paper by Bradley in 1748 [7]. Bradley associated the duration of the nutation period correctly with an inclination and precession of the Moon’s orbit w.r.t. the ecliptic. The accuracy of measurements improved over time and in 1898 Newcomb [8] presented a value of 900;210 = 4:47·10−5 rad for the amplitude. The duration of the orbital precession period, namely 18:6 years = 6;794 days, has, at least approximately, been known since the days of Stonehenge. However, note that this value refers only to the main, dominant period of the observed nutation. Nowadays the higher harmonics (periods as well as amplitudes) can also be measured most accurately as explained in [9]. Some of the shorter time-periods are immediately visible when studying Fig. 2. Their physical origin is manifold, luni-solar, planetary, and geodesic. Finally, the spin S in Eqn. (1), i.e., the daily rotation rate of the Earth, is essentially 380 The spin, the nutation and the precession of the Earth’s axis revisited characterized by completing a full rigid body turn of 2π within one sidereal day, i.e., 1 d = 86;164 s. However, slight variations of this rate have been observed and they are presented, e.g., in Fig. 3 of [4]. 2 KinematicsEngineering vs. geoscience terminology In technical mechanics, in particular, in context with 3D-rigid body-motion, it is customary to use the three Euler matrices in combination with the three Euler angles, namely the precession angle, ', the nutation angle, #, and the spin angle, . Here it is also customary IRF TRF to transform from the IRF base, el , into the TRF base, ei (t),(cf., [10]), by applying first the precessional, then the nutational and, finally, the spin Euler matrices: 0 1 0 1 0 1 cos ' sin ' 0 1 0 0 cos sin 0 O' =B− sin ' cos ' 0C ;O# =B0 cos # sin #C ;O =B− sin cos 0C ; ij @ A ij @ A ij @ A 0 0 1 0 − sin # cos # 0 0 1 (where the symbol O is a reminder of the generic Euclidian transformation of rational continuum mechanics) as follows: TRF # ' IRF −1 −1 # −1 ' ei (t) = Oij (t)Ojk(t)Okl(t)el ) S = O ; N = O ; P = O : (2) Clearly, the TRF-base is time-dependent, just like the two intermediate bases. This must be taken into consideration when performing time-derivatives as we shall see in the kinetics section shortly. It must also be mentioned that the three Euler transformations are applied such that the TRF-base coincides with the principal axes of the (rigid) spheroidal TRF Earth: e1=2=3 ≡ eI/II/III. It is customary to associate the principal axes I and II with the minimum and the intermediate value of the three principal moments of inertia, II = Imin and III, respectively. Axis III is related to the maximum moment of inertia IIII = Imax. Clearly for a spheroidal Earth we have II = III. Intuitively we may argue that the base TRF TRF vectors e1=2 point from the very center of the Earth to its equator and e3 to its North Pole. The precise intersection points along the equator are set apart by 90◦ but, for a spheroidal Earth model, arbitrary otherwise. We are now in a position to define the angular velocity vector in a rational manner: ' IRF _ # # IRF _ # ' IRF ! ='O _ 3l(t)el + #O1k(t)Okl(t)el + O3j(t)Ojk(t)Okl(t)el : (3) Note that angles and their temporal changes are considered to be positive or negative by applying the right-hand-rule correctly after each Euler rotation (see [10]). Eqn. (3) is obviously an IRF-based representation of the angular velocity vector. In context with the evaluation of the dynamical equations other representations are required. One is of partic- ular importance, namely the one after the second, intermediary Euler transformation: (2) (2) (2) # # IRF (2) _ _ ! = !i ei ; ei = Oik(t)Okl(t)el ;!i = (#; '_ sin #; '_ cos # + ) : (4) 3 KineticsEngineering vs.
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