precession, nutation, pole motion and variations of lod of the earth and the moon yuri barkin, hideo...
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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 !