Precession, nutation, pole motion and variations of LOD of the Earth and the Moon Yuri Barkin, Hideo Hanada, Misha Barkin Sternberg Astronomical Institute,

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. E-mail: [email protected]; 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 planet Lunar-solar tides

Volcanos

retreat

Winds

post-glacial rebound

melting ice oceanic load

oce

an c

urr

ents

Atmospheric pressure

con

tinen

tal waters

Geometry and dynamical sense of Andoyer variables

, , , , ,L G H l g h

, , , , ,G l g h

- canonical Andoyer variables

cosL G - the projection of the angular momentum vector on the axis of inertia of the planet

cosH G - the projection of the angular momentum vector on the fixed axis Z

2

dl K

dt L

dL K

dt l

dg K

dt G

dG K

dt g

dh K

dt H

dH K

dt h

2 2 2 2 21sin cos sin 2 sin cos sin 2 sin cos

2K G a l b l f l c e l d l

sin cos sin cos , , , , , ,G l l U L G H l g h t

cosL

G cos

H

G

First integrals of Euler - Liouville problem - the constancy of the vectorthe angular momentum of the rotational motion

0G G 0 0h h

The equations of motion tasks in canonical Andoyer’s variables

3

The EarthVenusMarsAsteroids

Base plane

Andoyer’s plane

Earth’s model with slightly variable geometry masses

22 20

,4

B AC

mr

22 20

,2

FS

mr

21 20

,E

Cmr

21 20

DS

mr

(0)2 2 2 ,J J J

(0)22 22 22 ,C C C

(0)22 22 22 ,S S S (0)

21 21 21,C C C (0)21 21 21,S S S

( ), ( ), ( ), ( ), ( ), ( );A t B t C t F t E t D t ( ), ( ), ( )P t Q t R t2 22 21 21 22( ), ( ), ( ), ( ), ( )J t C t C t S t S t

The small temporal variations of the coefficients of the geopotential

4

Expressions of the second harmonic coefficients of the geopotential viacentrifugal and axial moments of inertia

2 20 20

2,

2

C A BJ C

mr

The Earth with variable geometry of massesAnnual and semi-annual variation coefficients of the second harmonic of the geopotential

0 0128 22cos 32 12cos 2 227atmR M M

M n t

(Moore et al, 2005)

0 0182.5 278cos 7 12cos 2 102atmP M M

0 013.7 54cos 106 55cos 2 16atmQ M M

21 2 -11 ед. 10 кг м с (для , )atm atmP Q 5

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

101 ед. 10

6

The unperturbed circular Chandler polar motion of the Earth constant angle 0”25

The trajectory of the north pole of the Earth between 1990 - 1996. 1 unit = 3 m.

The classic approach(astrometry) 0

New theory of the Earth's rotation

0 0

The conical motion of the axis of inertiain unperturbed Chandler motion (with a period of 432 days).

g

l

l

8

Base plane

Andoyer’splane

2 2J J t

21 21C C t

21 21S S t

22 22C C t

22 22S S t

20

The EarthVenusMarsAsteroids

Andoyer’splane

Base plane

Secular variation of the geometry of the mass of theEarth and their impact on the variation coefficients

of the geopotential and the rotation of the Earth

211 CHT Cp

T I

211 CHT Sq

T I

921 0.2739 10 1/cy,C 11

21 1.0745 10 1/cyS

9356.97 10 (1/cy),p

91400.39 10 (1/cy)q

91445.17 10 (1/cy),Pv

9394.5 10 1/cy,p

9/ 1547.5 10 1/cyq

91596.99 10 (1/cy)Pv

Found values are agree well with the values obtainedfrom observations (Vondrak, 1999).

Theory:

Observations:

Explanation of the secular drift of the poles of the Earth

22

75.70º W 

Model:

0.10

0.05

0.00

-0.05

0.40

-0.10

-0.15

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05

1906M=8.8

1920

1930

1950

1960

1910

1940

1970

19801990

2000

2010

1960M=9.5

2010M=8.8

( )x

( )y

Gre

en

wic

h

Trend and a observed secular drift of poles of the Earth

23

(1)

(2)

The internal structure of the Earth and the Moon

3/cmm g 4.44 3/cmm g 3.269

2/N m 112.57 10

2/N m 111.80 10

cr km3480

mr km6371cr km330

mr km1738

cr

mr

2/N m 100.40 10

2/N m 111.80 10

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-precisiondescription 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 rotationEarth and others. The good agreement between theory and previously builttheories 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

, , , , , ; , , , , ,c c c c c cp q r p q r Euler variables

Andoyer’s variables

, , , , , ; , , , , ,c c c c c cL G H l g h L G H l g h

Sasao,Ocubo,Saito (1980)Sevilla,Romero (1987)Getino, Ferrandiz (1991-2001)Ferrandiz, Barkin (2000,2001)

Applications to the theoryof the Earth rotation

1 2, , , , P P I Variables of the Moon physical librations

, ,A B C

, ,c c cA B C

The Earth

The Moon

, , m m mA B C mantle

, , lc lc lcA B C liquid core

, , rc rc rcA B C rigid core

The dynamical ellipticities:

The Moon three-layer system

,C A

eA

,f f

ff

C Ae

A

,s s

ss

C Ae

A

f f s ss

s s s

A A C Ae

A A A

Andoyer’s variables , , , , ,G h l g

, ,G h , ,l g hThe projections of the angular velocity

sin sinG

p lA

sin cosG

q lB

cosG

rC

g

,G ,H ,h ,l g

2( ) cos ,G L l 2( ) sin ,G L l

,cc G ,cH ,ch ,c c cl g

2( ) cos ,c c c cG L l cccc lLG sin)(2

Andoyer - Poincare variables

, , , , , ,c c ch h z , , , , ,c c cH H Z

,

, , , , ,

, , , , ,c c c

c c c

d h h

dt H H

, , , , ,

, , , , ,c c c

c c c

d H H

dt h h

T U

22 22 2 2 2 2 2 2 22

21 2 3

11 1 1 2

2 4 4 2 4c c cA B C

T

22 22 2 2 2 2 2 2 22

21 2 3

11 1 1 2

2 4 4 2 4c cc c c c c c c c

cc c c c c c

A B C 2 2 2 22 2 2 2

1 2 3

1 1 1 14 4 2 2

c c c c c c cc c c c

c c

F E D

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

1 2 3 4 5  l l F D ν

Meeting at NAOJ in Mitaka (Tokyo, May 2013)

N. Rambaux, H. Kinoshita, Yu. Barkin

3

1 1 1 1 1 12 2;0 ;1 ;2 ;0 ;1 ;2sin cos sin cos sin

ah A A t A t a a t a t

r

ν ν ν ν ν ν ν ν

ν

3

1 1 1 1 1 12 2;0 ;1 ;2 ;0 ;1 ;2sin cos cos cos sin

ah b b t b t B B t B t

r

ν ν ν ν ν ν ν ν

ν

3

2 2 2 2 2 22 2 2;0 ;1 ;2 ;0 ;1 ;2cos sin 2 cos sin

ah b b t b t B B t B t

r

ν ν ν ν ν ν ν ν

ν

3

2 2 2 2 2 22 2 2;0 ;1 ;2 ;0 ;1 ;2cos cos 2 cos sin

ah A A t A t a a t a t

r

ν ν ν ν ν ν ν ν

ν

3

0 0 0 0 0 02 2 2;0 ;1 ;2 ;0 ;1 ;2

11 3sin cos sin

2

aA A t A t a a t a t

r

ν ν ν ν ν ν ν ν

ν

Barkin, Kudryavtsev, Barkin, 2009

1 1;0 ;0A Bν ν

2 2;0 ;0A Bν ν

Kinoshita, 1977, Earth rotation theory

Баркин, 1989, Теория вращения Луны

Kudryavtsev S.M. (2007) Astronomy & Astrophysics Long - term harmonic development of lunar ephemeris

1 2 3 4 5M Sl l F D ν

( ) ( )( ), ( ),iqt iq iqt iqe t e t z z ( ) ( )( ), ( )ist is ist ise t e t z z

( ) ( ) ( ) 2 ( )0 1 2 ... z z z z 2

1 2( , ) ...q r

21 2 ...,q q q 2

1 2 ...,r r r

( ) ( ) ( ) ( ) ( ) ( )0 1 2 3 ...p p p p p z z z z z z

General structure of solution of the problem about the Moon physical librations

0( ) ( , , , ,...)M St l l F D Z Z Z Z

( , , , , , , , , , , , ,..., )res p q r s M SP Q R S U U U U l l F D tZ

2;0 ;1 ;2 ...c c c cq q q q

, , ,p q r s frequencies of free oscillations of the Moon

Construction of the second harmonic expansionsthe gravitational potential of the Earth

3

2 (0) (0)11 3sin cos sin

2

aA a

r

ν ν ν ν

ν

0 0 0(0) 2;0 ;1 ;2A A A t A t ν ν ν ν

0 0 0(0) 2;0 ;1 ;2a a a t a t ν ν ν ν

Developments of functions of spherical coordinates of the Moon

1 2 3 4 5M Sl l F D ν

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

,

1 2 3 4 5 ,M S F Dn n n n n ν 0 0"125, 00 23 45

The perturbations of the first order. Variable .GThe 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.

20 ( ) (2 ) 2 3sin ,..., 22,

1( ) cos 1 cos

8

0 1 22 20,

1 1 1( , ) 3cos 1 sin 2 sin

6 2 4R t A A A ν ν ν ν

0 1 22 20,

1 1 1( , ) 3cos 1 sin 2 sin

6 2 4...........................................................................................

r t a a a ν ν ν ν

1 2 2 0 2 1( ) 22,

12sin cos sin sin 2

2r b b a a a a

ν ν ν ν ν ν ν

1 . . . . .

Special functions of angles of nutations

222

2 22

40.668479 10

2

C

J C

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. 315-376.

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 1737.15 +/- 0.01 km

(Araki et al., 2010)

Y

ωa = 27.85 m

ωb = 68.83 mωa

ωb

2

21 0.9145

ae

b

Direct pole motion

Eccentricity of trajectory

3"3072sinK

F

Wp

n

8"1732cosK

F

q

nW

The trajectory of the end of the angular momentum vector projected onto the plane of the ecliptic with the free librations

1 unit =1 arcs2000

2028

Cassini’s node

X

Y

Space trajectory of the end of the angular momentum vector of theMoon in its free librations with respect to ecliptic reference system XYZ connected with moving mean node of orbit

1 unit =1 arcs

2000

XY

Z2012

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: , ,LOD P lν ( , ),LOD P rν

.LOD ν

Effects of the ellipsoidal liquid core in duration of lunar day

Tν№ Mizusawa Moons Moons -Mizusawa

22 1 -3 2 -3 -67048,710 -211,452 --- ---

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 termwith big period in 183.564338 years (and some others) directly, with using data of observations.

New terms of forced librations in longitude with very big period 183.564338 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.

1 2( , , , , )P P I Z

2. Development of the analytical theory of the rotation of the three-layer model of the Moon and the EarthA. On the base Prof. Getino, Prof/ Ferrandiz model (for the Earth)B. On the base of Prof. Vilke model (for planet)

0 0 0 0 0 02 2;0 ;1 ;2 ;0 ;1 ;2cos sint t t t ν ν ν ν ν ν ν ν

ν

Z Z Z zZ z z

1 2 3 4M Sl l F D ν

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.

Спасибо за внимание !