Precession, nutation, pole motion and variations of LOD of the Earth and the Moon Yuri Barkin, Hideo Hanada, Misha Barkin Sternberg Astronomical Institute, Moscow, Russia. National Astronomical Observatory of Japan, Mizusawa, Japan; Bauman Moscow Technical University, Moscow, Russia.

A schematic model of the Earth's mass with variable geometry Gravitational attractions of planetLunar-solar tides Volcanos retreat Winds post-glacial rebound melting ice oceanic load ocean currents Atmospheric pressure continental waters

Geometry and dynamical sense of Andoyer variables - canonical Andoyer variables - the projection of the angular momentum vector on the axis of inertia of the planet - the projection of the angular momentum vector on the fixed axis Z 2

First integrals of Euler - Liouville problem - the constancy of the vector the angular momentum of the rotational motion The equations of motion tasks in canonical Andoyer’s variables 3 The Earth Venus Mars Asteroids Base plane Andoyer’s plane

Earth’s model with slightly variable geometry masses The small temporal variations of the coefficients of the geopotential 4 Expressions of the second harmonic coefficients of the geopotential via centrifugal and axial moments of inertia

The Earth with variable geometry of masses Annual and semi-annual variation coefficients of the second harmonic of the geopotential (Moore et al, 2005) 5

Annual, semi-annual variations of coefficients of the second harmonic of the geopotential 6

The unperturbed circular Chandler polar motion of the Earth constant angle 0”25 The trajectory of the north pole of the Earth between unit = 3 m. The classic approach (astrometry) New theory of the Earth's rotation The conical motion of the axis of inertia in unperturbed Chandler motion (with a period of 432 days). 8 Base plane Andoyer’s plane

20 The Earth Venus Mars Asteroids Andoyer’s plane Base plane Secular variation of the geometry of the mass of the Earth and their impact on the variation coefficients of the geopotential and the rotation of the Earth

Found values are agree well with the values obtained from observations (Vondrak, 1999). Theory: Observations: Explanation of the secular drift of the poles of the Earth 22 Model:

M= M= M=8.8 Greenwich Trend and a observed secular drift of poles of the Earth 2323 (1) (2)

The internal structure of the Earth and the Moon

A theory of rotation of the non-sphericity of the Earth with an elastic mantle, variable outer shell and a liquid ellipsoidal core in the gravitational field of the moon and sun. As unperturbed rotational motion of the Earth taken not axial and conical motion axisymmetric Earth with respect to the angular momentum of the rotational motion. As the base we used the equation of motion in variables Andoyer. Taken into account the second harmonic of the power function for high-precision description of the orbital motion of the Earth and the Moon. An approximate solution of the problem of the rotation of the Earth is constructed using Construct the table of precession, nutation oscillations pole axis of rotation Earth and others. The good agreement between theory and previously built theories of the Earth's rotation (Kinoshita, 1977; Getino, Ferrandiz, 2001, and others) method of small parameter, Andoyer’s variables, as well as projections the angular velocity of the Earth and its core. It is assumed that the core is an ideal fluid undergoing a simple motion of the Poincare. Construct the table of precession, nutation oscillations pole axis of rotation Earth and others. The good agreement between theory and previously built theories of the Earth's rotation (Kinoshita, 1977; Getino, Ferrandiz, 2001, and others.) The rotation of the Earth with liquid core. Work content.

The main shells of the Earth and the Moon In this paper, we consider the two-layer model of the Earth and Moon: nonsphericity solid mantle and liquid ellipsoidal core. Objective: To construct an analytic theory of rotation of the Earth (and Moon). -Mantle - Liquid core - Rigid core The Earth system The Moon system

The two-layer model of the theory of the Earth's rotation mantle core The Earth The Moon Euler variables Andoyer’s variables Sasao,Ocubo,Saito (1980) Sevilla,Romero (1987) Getino, Ferrandiz ( ) Ferrandiz, Barkin (2000,2001) Applications to the theory of the Earth rotation Variables of the Moon physical librations

The Earth The Moon The dynamical ellipticities: The Moon three-layer system

Andoyer’s variables The projections of the angular velocity

Andoyer - Poincare variables,

The equations of rotational motion of the Earth in Andoyer’s variables - Poincare Kinetic energy Hamiltonian of problem

The equations of rotation of the solid Earth Andoyer variables. The unperturbed rotational motion of the Earth.

The overall structure of the expansion of the force function Киношита часть (1977) Новые слагаемые разложения Meeting at NAOJ in Mitaka (Tokyo, May 2013) N. Rambaux, H. Kinoshita, Yu. Barkin

Barkin, Kudryavtsev, Barkin, 2009 Kinoshita, 1977, Earth rotation theory Баркин, 1989, Теория вращения Луны Kudryavtsev S.M. (2007) Astronomy & Astrophysics Long - term harmonic development of lunar ephemeris

General structure of solution of the problem about the Moon physical librations frequencies of free oscillations of the Moon

Construction of the second harmonic expansions the gravitational potential of the Earth Developments of functions of spherical coordinates of the Moon

The perturbations of the first order in the rotational motion of the Earth in variables Andoyer.

, The perturbations of the first order. Variable The module of the angular momentum of the rotational movement of the Earth. The parameters of the Earth. Note that these formulas generalize similar formulas Kinoshita. Here the angle theta is small, but not zero.

..... Special functions of angles of nutations

Main results The new theory of libration of the Moon with elastic mantle and with ellipsoidal liquid core have been developed with two different approaches. Determination and explanation of the fourth mode of free physical libration of the Moon caused by the ellipsoidal liquid core.

Barkin Y., Hanada H., Ferrandiz J., Matsumoto K., Jin S., Barkin M. (2014) The theory of the physical libration of the Moon with a liquid core. Chapter 13. Taylor & Francis/CRC, USA. pp

Taylor & Francis/CRC, USA.

Trajectory of the pole of angular velocity of the Moon in its free libration in projection on the lunar surface 1 unit=1 arcs x To the Earth Mean radius of the Moon / km (Araki et al., 2010) Y Direct pole motion Eccentricity of trajectory

The trajectory of the end of the angular momentum vector projected onto the plane of the ecliptic with the free librations 1 unit =1 arcs Cassini’s node X Y

Space trajectory of the end of the angular momentum vector of the Moon in its free librations with respect to ecliptic reference system XYZ connected with moving mean node of orbit 1 unit =1 arcs 2000 X Y Z 2012

Forced variations of the LOD (of duration of day in seconds of time) of the Moon for model with ellipsoidal liquid core and without core and their difference: Effects of the ellipsoidal liquid core in duration of lunar day

№MizusawaMoons Moons - Mizusawa , , Table 1. Periods and amplitudes of variations of the variable in analytical theories Mizusawa, Moons and empirical theory of Rambaux-Williams. In ideal variant its well to confirm (or not) this big term with big period in years (and some others) directly, with using data of observations. New terms of forced librations in longitude with very big period years and amplitude 211”45 Theory of physical libration of the Moon is needed in development

1. Development of forced libration of the Moon for its models with a liquid core. 2. Development of the analytical theory of the rotation of the three-layer model of the Moon and the Earth A.On the base Prof. Getino, Prof/ Ferrandiz model (for the Earth) B.On the base of Prof. Vilke model (for planet)

Two approaches to construction a theory of rotation of the Moon and the Earth as three layered celestial bodies The main area of ​​ research involves the development of the analytical theory of the physical libration of the Moon with an elastic mantle and a liquid core and a solid core and a full account of the gravitational perturbation factors exerted by the Earth, the Sun and the major planets (Venus, Mars, Jupiter, Saturn, Neptune, Uranus ). In connection with the installation of the telescope on the lunar surface for precise determination of the parameters of orientation and rotation of the Moon (the predicted accuracy is about 0.001'') requirements increase to theories of the physical libration of To solve these problems, to determine the parameters of the Moon free libration (4 and 5 modes, etc.) due to liquid or solid core requires new dynamical studies of the perturbed rotational motion of the Moon based on its current two-and three layers models.

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