The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation

In the method of variation of parameters, we express the Cartesian coordinates or the Euler angles via the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants.” Out of convenience, the Lagrange constraint is often imposed. It nullifies the convective term and thereby guarantees that the functional dependence of the velocity upon the time and “constants” stays unaltered under perturbation. “Constants” obeying this condition are called osculating elements. Otherwise, they are simply called orbital or rotational elements. When the dynamical equations for the elements are demanded to be symplectic, the orbital elements are called Delaunay elements, and the rotational elements are called Andoyer elements. These sets of elements share a feature not readily apparent: in some situations the standard equations render these elements non-osculating. In orbital mechanics, the planetary equations yield non-osculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function. (Efroimsky & Goldreich 2003, 2004; Efroimsky 2005) It complicates both the Largangeand Delaunaytype planetary equations and destroys the canonicity of the Delaunay ones. Similarly, in attitude dynamics, the Andoyer elements come out non-osculating when the perturbation depends upon the angular velocity. For example, since a switch to a non-inertial frame is an angular-velocity-dependent perturbation, then amendment of the dynamical equations by only adding extra terms to the Hamiltonian makes the equations render non-osculating Andoyer elements. To make them osculating, extra terms must be added to the equations (and then these equations will no longer be symplectic). Calculations in non-osculating variables are mathematically valid, but their physical interpretation is problematic. Non-osculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A non-osculating i may differ much from the real inclination of the orbit, given by the osculating i .) In the attitude case, non-osculating Andoyer variables correctly describe perturbed spin but lack simple physical meaning, i.e., no longer enjoy simple interconnection with the angular momentum. The customary expressions for the spin-axis’ orientation angles in terms of the Andoyer elements will no longer be valid if the elements are non-osculating – circumstance ignored in the Kinoshita-Souchay (KS) theory which tacitly employs non-osculating Andoyer variables. Thought the KS theory is mathematically perfect and yields the correct figure axis, its predictions for the instantaneous spin axis need correction of order milliarcseconds. 1 1 The Hamiltonian approach to rotational dynamics 1.1 Historical preliminaries Perturbed rotation of a rigid body has long been among the key topics of both spacecraft engineering (Giacaglia & Jefferys 1971, ; Zanardi & Vilhena de Moraes 1999) and planetary astronomy (Kinoshita 1977; Laskar & Robutel 1993; Touma & Wisdom 1994; Mysen 2004). While free spin (the Euler-Poinsot problem) permits an analytic solution in terms of the elliptic Jacobi functions, the perturbed case typically involves numerics. Perturbation may come from a physical torque, or from an inertial torque caused by the frame noninertiality, or from nonrigidity (Getino & Ferrandiz 1990; Escapa, Getino & Ferrandiz 2001, 2002). The free-spin Hamiltonian, expressed through the Euler angles and their conjugate momenta, is independent from one of the angles, which reveals an internal symmetry of the problem. In fact, this problem possesses an even richer symmetry (Deprit & Elipe 1993), whose existence indicates that the unperturbed Euler-Poinsot problem can be reduced to one degree of freedom. The possibility of such reduction is not readily apparent and can be seen only under certain choices of variables. These, in analogy with the orbital mechanics, are called rotational elements. Then the forcedrotation case should be treated as a perturbation expressed through those elements. Rotational elements are often chosen as the Andoyer variables (Andoyer 1923, Giacaglia & Jefferys 1971, Kinoshita 1972), though other sets of canonical elements have appeared in the literature (Richelot 1850; Serret 1866; Peale 1973, 1976; Deprit & Elipe 1993; Fukushima & Ishizaki 1994) After a transition to rotational elements is performed within the undisturbed Euler-Poinsot setting, the next step is to extend this method to a forced-rotation case. To this end, one will have to express the torques via the elements. On completion of the integration, one will have to return back from the elements to the original, easily measurable, quantities – i.e., to the Euler angles and their time derivatives. 1.2 The Kinoshita-Souchay theory of rigid-Earth rotation The Hamiltonian approach to spin dynamics has found its most important application in the theory of Earth rotation. The cornerstone work on this topic was carried out by Kinoshita (1977) who switched from the Euler angles of the Earth figure to the Andoyer variables, and treated their dynamics by means of the Hori method. Then he translated the results of this development back into the language of Euler’s angles and provided the nutation spectrum. Later his approach was furthered to a much higher precision by Kinoshita & Souchay (1990). 1.3 Subtle points to be re-examined When one is interested only in the orientation of the rotator, it is sufficient to have expressions for the Euler angles through the elements. However, when one needs to know also the instantaneous angular velocity, he will need expressions for the Euler angles’ time derivatives via the elements. This poses the following question: if we write down the expressions for the Euler angles’ derivatives via the canonical elements in the free-spin case, will these expression stay valid under perturbation? In the parlance of orbital mechanics, this question may be formulated like this: are the canonical elements always osculating? As 1 Some authors use the term “Serret-Andoyer elements.” This term is not exact, because the set of elements introduced by Richelot (1850) and Serret (1862) differs from the set employed by Andoyer (1923). 2 we shall demonstrate below, under angular-velocity-dependent perturbations the osculation condition is incompatible with the canonicity demand; and therefore expression of the angular velocity via the canonical elements will, under such types of perturbations, become nontrivial. In 2004 the above question acquired a special relevance to the Earth-rotation theory. While the thitherto available observation means rendered the orientation of the Earth figure (Kinoshita 1978), a technique based on ring laser gyroscope provided a direct measurement of the instantaneous angular velocity of the Earth. (Schreiber et al 2004) Normally, rotational elements are chosen to have evident physical interpretation. For example, the Andoyer variable G coincides with the absolute value of the body’s spin angular momentum, while two other variables, H and L , are chosen to coincide, correspondingly, with the Z-component of the angular momentum in the inertial frame, and with its z-component in the body frame. The other Andoyer elements, g , l , h , too, bear some evident meaning. Hence another important question: will the canonical rotational elements preserve their simple physical meaning also under disturbance? The latter question is merely a re-formulation of the former one. Indeed, on the one hand, the interrelation between the angular-momentum components and the angular-velocity components is invariant under perturbations. On the other hand, below we shall prove that under angular-velocity-dependent perturbations the dependence of the angular velocity upon the elements changes it functional form (i.e., that under such perturbations the elements lose osculation). Therefore, under angular-velocity-dependent perturbations the dependence of the angular-momentum components upon the elements will, too, alter its form. Hence, under this type of perturbations, the elements will not only lose the osculation property but will also lose the simple physical meaning they used to enjoy in the undisturbed setting. 2 The canonical perturbation theory in orbital and attitude dynamics 2.1 Kepler and Euler In orbital dynamics, a Keplerian conic, emerging as an undisturbed two-body orbit, is regarded to be a “simple motion,” so that all the other available motions are conveniently considered as distortions of such conics, distortions implemented through endowing the orbital constants Cj with their own time dependence. Points of the orbit can be contributed by the “simple curves” either in a non-osculating fashion, as in Fig.1, or in the osculating manner, as in Fig.2. The disturbances, causing the evolution of the motion from one instantaneous conic to another, are the primary’s oblateness, the gravitational pull of other bodies, the atmospheric and radiation-caused drag, and the non-inertiality of the reference system. Similarly, in rotational dynamics, a complex spin can be presented as a sequence of instantaneous configurations borrowed from a family of some “simple rotations.” It will be most natural to employ in this role the motions exhibited by an undeformable free top with no torques acting thereupon. Each such undisturbed “simple motion” will be a trajectory on the three-dimensional manifold of the Euler angles. For the lack of a better term, we shall call 2 Here one opportunity will be to employ in the role of “simple” motions the non-circular Eulerian cones described by the actual triaxial top, when it is unforced. Another opportunity will be to use, as “simple” motions, the circular Eulerian cones described by a dynamically symmetrical top (and to treat its actual triaxiality as another perturbation). The main result of our paper will be invariant under this choice. 3 Fig.1. The perturbed orbit is a set of points belonging to a sequence of confocal instantaneous ellipses that are not supposed to be tangent or even coplanar to the orbit. As a result, the physical velocity ~̇r (tangent to the orbit) differs from the Keplerian velocity ~g (tangent to the ellipse). To parameterise the depicted sequence of non-osculating ellipses, and to single it out of the other sequences, it is suitable to employ the difference between ~̇r and ~g, expressed as a function of time and six (non-osculating) orbital elements: ~ Φ(t , C1 , . . . , C6) = ~̇r(t , C1 , . . . , C6) − ~g(t , C1 , . . . , C6) . Since ~̇r = ∂~r ∂t + 6 ∑ j=1 ∂Cj ∂t Ċj = ~g + 6 ∑ j=1 ∂Cj ∂t Ċj , then the difference ~ Φ is the convective term ∑ (∂~r/∂Cj) Ċj that emerges whenever the instantaneous ellipses are being gradually altered by the perturbation (and the orbital elements become time-dependent). In the literature, ~ Φ(t , C1 , . . . , C6) is called gauge function or gauge velocity or, simply, gauge. Fig.2. The orbit is represented by a sequence of confocal instantaneous ellipses that are tangent to the orbit, i.e., osculating. Now, the physical velocity ~̇r (tangent to the orbit) will coincide with the Keplerian velocity ~g (tangent to the ellipse), so that their difference ~ Φ(t C1 , . . . , C6) vanishes everywhere: ~ Φ(t, C1 , . . . , C6) ≡ ~̇r(t , C1 , . . . , C6)− ~g(t , C1 , . . . , C6) = 6 ∑ j=1 ∂Cj ∂t Ċj = 0 . This equality, called Lagrange constraint or Lagrange gauge, is the necessary and sufficient condition of osculation. 4 these unperturbed motions “Eulerian cones,” implying that the loci of the rotational axis, which correspond to each such non-perturbed spin state, make closed cones (circular, for an axially symmetrical rotator; and elliptic for a triaxial one). Then, to implement a perturbed motion, we shall have to go from one Eulerian cone to another, just as in Fig. 1 and 2 we go from one Keplerian ellipse to another. Hence, similar to those pictures, a smooth “walk” over the instantaneous Eulerian cones may be osculating or non-osculating. The torques, as well as the actual triaxiality of the top and the non-inertial nature of the reference frame will then act as perturbations causing this “walk.” Perturbations of the latter two types depend not only upon the rotator’s orientation but also upon its angular velocity. 2.2 Delaunay and Andoyer In orbital dynamics, we can express the Lagrangian of the reduced two-body problem via the spherical coordinates qj = { r , φ , θ } , then derive their conjugated momenta pj and the Hamiltonian H(q, p) , and then carry out the Hamilton-Jacobi procedure (Plummer 1918), to arrive to the Delaunay variables {Q1 , Q2 , Q3 ; P1 , P2 , P3 } ≡ {L , G , H ; lo , g , h } = (1) {√μa , √