Prospects in the orbital and rotational dynamics of the Moon with the advent of sub-centimeter lunar laser ranging

Kopeikin, Sergei M.; Pavlis, E. C.; Pavlis, D. E.; Brumberg, V. A.; Escapa, Alberto; Getino, Juan; Gusev, Alexander; Müller, Jürgen; Ni, Wei-Tou; Petrova, Natalia

doi:10.1016/j.asr.2008.02.014

Cited by 25 publications

(12 citation statements)

References 82 publications

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“…Kinoshita 1977;Touma and Wisdom 2001;Cottereau and Souchay 2009;D'Hoedt et al 2010), or natural and artificial satellites (e.g. Henrard 2005;Noyelles 2008;Celletti and Sidorenko 2008;Kopeikin et al 2008;Vilhena de Moraes et al 2009). Among these investigations we must underline those devoted to the Earth.…”

Section: Introductionmentioning

confidence: 99%

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

Escapa

2011

Celest Mech Dyn Astr

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We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l'Observatoire de Paris, pp 74-81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251-283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117-180, 1995), but considering the contribution of the convective terms. Specifically, we 123 100 A. Escapa derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261-274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide rai...

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Section: Introductionmentioning

confidence: 99%

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

Escapa

2011

Celest Mech Dyn Astr

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“…Currently, there are no experimental data restricting the asymptotic behavior of ω 0 ∼ ω 3 0 β which could help us to understand better the nature of the coupling function ω(Φ). This makes the parameter β a primary target for experimental study in the near-future gravitational experiments [198][199][200] including the advanced lunar laser ranging (LLR) [201][202][203] and gravitational wave detectors [91]. The background scalar field Φ 0 and the parameter of coupling ω 0 determine the observed numerical value of the universal gravitational constant…”

Section: E Post-newtonian Field Equationsmentioning

confidence: 99%

Covariant Equations of Motion of Extended Bodies with Arbitrary Mass and Spin Multipoles

Kopeikin

2018

Preprint

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Gravitational wave detectors allow to test general relativity and to study the internal structure and orbital dynamics of neutron stars and black holes in inspiraling binary systems with a potentially unlimited rigor. Currently, analytic calculations of gravitational wave signal emitted by inspiralling compact binaries are based on the numerical integration of the asymptotic post-Newtonian expansions of the equations of motion in a pole-dipole approximation that includes masses and spins of the bodies composing the binary. Further progress in the accurate construction of gravitational-wave templates of the compact binaries strictly depends on our ability to significantly improve theoretical description of gravitational dynamics of extended bodies by taking into account the higher-order (quadrupole, octupole, etc.) multipoles in equations of motion of the bodies both in the radiative and conservative approximations of general relativity and other viable alternative theories of gravity.This paper employs the post-Newtonian approximations of a scalar-tensor theory of gravity along with the mathematical apparatus of the Cartesian symmetric trace-free tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies having arbitrary distribution of mass and velocity of matter. We assume that spacetime manifold can be covered globally by a single coordinate chart which asymptotically goes over to the Minkowskian coordinate chart at spatial infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field in the neighborhood of each body is parametrized by an infinite set of mass and spin multipoles of the body as well as by the set of tidal gravitoelectric and gravitomagnetic multipoles of external N − 1 bodies. The origin of the local coordinates is set moving along the accelerated worldline of the center of mass of the corresponding body by an appropriate choice of the internal and external dipole moments of the gravitational field. Translational equations of motion of the body's center of mass and rotational equations of motion for its spin are derived by integrating microscopic equations of motion of body's matter and applying the method of asymptotic matching technique to splice together the post-Newtonian solutions of the field equations of the scalar-tensor theory of gravity for the metric tensor and scalar field obtained in the global and local coordinate charts.The asymptotic matching is also used for separating the post-Newtonian self-field effects from the external gravitational environment and constructing the effective background spacetime manifold. It allows us to present the equations of translational and rotational motion of each body in covariant form by making use of the Einstein principle of equivalence. This relaxes the slowmotion approximation and makes the covariant post-Newtonian equations of motion of extended bodies with weak self-gravity applicab...

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“…In addition, the scope of these roto-translatory studies is very wide, ranging from long-term dynamics surveys related to the evolution of solar systems bodies, usually connected with resonant conditions [1,2], to detailed numerical investigations of the spin-orbit dynamics, which may show chaos as soon as the orbital eccentricity departs, even slightly, from circular or when the angular momentum of a triaxial body is far from the principal directions with extrema moment of inertia (see reference [3] and references therein). 1 Recently, because of current technology improvements which make it possible to measure distances to the Moon with a precision approaching 1 mm, there is renewed interest in significantly improving the theoretical models of the orbital and rotational dynamics of the Earth-Moon system itself, and of satellites orbiting in this system [4]. At the same time, during the last few years, the roto-translatory dynamics of binary systems is another area of research in which classical models do require a full revision because the orbit no longer can be taken as a Keplerian ellipse (see reference [5], and references therein).…”

Section: Introductionmentioning

confidence: 99%

On Roto-Translatory Motion: Reductions and Radial Intermediaries

Ferrer

Lara

2012

J of Astronaut Sci

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The roto-translational dynamics of an axial-symmetric rigid body is discussed in a central gravitational field. The six-degree-of-freedom Hamiltonian problem is formulated as a perturbation of the Kepler motion and torque-free rotation. A chain of canonical transformations is used to reduce the problem. First, the elimination of the nodes reduces the problem to a system of four degrees of freedom. Then, the elimination of the parallax simplifies the resulting Hamiltonian, which is shaped as a radial intermediary plus a remainder. Some features of this integrable intermediary are pointed out. The normalized first order system in closed form is also given, thus completing the solution. Finally the full reduction of the radial intermediary is constructed using the Hamilton-Jacobi equation.

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Prospects in the orbital and rotational dynamics of the Moon with the advent of sub-centimeter lunar laser ranging

Cited by 25 publications

References 82 publications

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

Covariant Equations of Motion of Extended Bodies with Arbitrary Mass and Spin Multipoles

On Roto-Translatory Motion: Reductions and Radial Intermediaries

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