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Titel |
To development of analytical theory of rotational motion of the Moon |
VerfasserIn |
Yu. V. Barkin, J. M. Ferrandiz, J. F. Navarro |
Konferenz |
EGU General Assembly 2009
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Medientyp |
Artikel
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Sprache |
Englisch
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Digitales Dokument |
PDF |
Erschienen |
In: GRA - Volume 11 (2009) |
Datensatznummer |
250021998
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Zusammenfassung |
Resume. In the work the analytical theory of forced librations of the Moon considered as a
celestial body with a liquid core and rigid non-spherical mantle is developed. For
the basic variables: Andoyer, Poincare and Eulerian angles, and also for various
dynamic characteristics of the Moon the tables for amplitudes, periods and phases of
perturbations of the first order have been constructed. Resonant periods of free
librations have been estimated. The influence of a liquid core results in decreasing
of the period of free librations in longitude approximately on 0.316 day, and in
change of the period of free pole wobble of the Moon on 25.8 days. In the first
approximation the liquid core does not render influence on the value of Cassini’s
inclination and on the period of precession of the angular momentum vector. However it
causes an additional "quasi-diurnal” librations with period about 27.165 days. In
comparison with model of rigid non-spherical of the Moon the presence of a liquid core
should result in increase of amplitudes of the Moon librations in longitude on 0.06
%.
1 Development of analytical theory of rotational motion of the Moon with liquid core
and rigid mantle. The work has been realized in following stages. 1. Canonical equations of
rotation of the Moon with liquid core and elastic mantle in Andoyer and Poincare variables
have been constructed. Developments of second harmonic of force function of the Moon in
pointed variables have been obtained for accurate trigonometric presentation of perturbations
of the Moon orbital motion. 2. Two approaches (two methods) of construction of analytical
theory have been developed. These approaches use different principles for eliminating of
singularities for axial rotation of the Moon. One is based on direct application of Andoyer
variables by changing of notations of moments of inertia [1]. Second is based on
application of Poincare elements. For comparison both approaches are developed. 3. The
main equation for determination of Cassini’s inclination and its solution has been
obtained in the case of accurate orbit of the Moon. An dynamical explanation of
Cassini’s laws has been done for model of the Moon with liquid core [2]. 4. Compact
formulae for perturbations of the first (and second) order have been constructed for
general used variables and for different kinematical and dynamical characteristics of
the Moon (23 variables and characteristics: Andoyer-Poincare variables, classical
variables, components of angular velocity and angular momentums of the Moon and its
core). 5. Analytical formulae for 4 periods of free librations of the Moon have been
constructed: for librations in longitude, in pole wobble, for free precession, and
“quasi-diurnal” librations, caused by the liquid core. 6. The dynamical effects in the Moon
rotation, caused by secular orbital perturbations of the Earth and Sun, have been
studied.
2 Structure perturbations of the first order and their tabulation. For example,
perturbations (periodic and of mixed type) in inclination Ïand in node h of angular
momentum of the Moon are determined by formulae: Ï = Ï0 + -
νÏν(1) cosθv,
h = Ï + -
-¥Î½-¥hν(1) sinθν. Here Ï0 = 1033-²50” is the Cassini’s inclination of the
Moon; Ïν(1), hν(1)are constant coefficients; θv = v1lM + v2lS + v3F + v4D,
ν = (v1,v2,v3,v4)Tare combinations of known classical arguments of the Moon orbital
theory; v1,v2,v3 and v4 are integer.
3 Influence of the liquid core and its ellipticity É on amplitudes of the Moon
forced and free librations. An influence of the liquid core and its ellipticity is
determined by positive correction to amplitudes of librations for model of the rigid Moon.
If the amplitudes of librations of rigid Moon we note as 1, so the corresponding
amplitudes of librations of the Moon with the liquid core will be characterized
by parameter 1 + L, where correction for liquid core is determined by formula
L = Cc(1- É2)-C - Cc-C = 0.5996 -
10-3, where Cand Ccis the polar moments of
inertia of the Moon and its core;É = (a2 - b2)- (a2 + b2)- (a - b)-a is an ellipticity of
equatorial ellipse of core cavity with semi-axes a and b. So all amplitudes of librations in
longitude due to the liquid core are increased on 0.06%. A small effect of ellipticity has more
smaller order. Here as example we present formula for perturbations of the first order of the
Moon in longitude:
(1) 21-+-L
λ = 6n0 I C22Ã
- - D (1) (Ï )- D(-1) (Ï )
à (- 1)ν5-ν1.ν2.ν3+2.ν4.ν5--0----ν1.ν2.ν3-2.ν4.ν25-0-sin(v1lM + v2lS + v3F + v4D )
-¥Î½-¥>0 ν5 (v1nM + v2nS + v3nF + v4nD)
I = C-(mr2) is the dimensionless moment of inertia of the Moon (m and rare it’s the mass
and mean radius). Kinoshita’s inclination functions Dν1.ν2.ν3.ν4.ν5(±1)(Ï
0) are
determined by known formulae through the value of Cassini’s angleÏ = 1033-²50”.
v1nM + v2nS + v3nF + v4nD = Ëθv1,v2,v3,v4 are derivatives with respect to the time of
corresponding linear combinations of classical arguments of lunar orbit theory;
nM,nS,nF and nD are velocities of changes of these arguments; C22 is the selenopotential
coefficient; n02 = fm--a3, a is an unperturbed value of semi-axis major of lunar orbit, fis a
gravitational constant. The perturbations of the first order for others variables and considered
dynamical characteristics have the structure similar to the formula for Ëλ(1). In given table 1
we present amplitudes of forced librations in longitude of intermediate Andoyer plane
λν1,ν2,ν3,ν4 (in arc seconds) and perturbations of angular velocity of the Moon axial
rotation Ïν1,ν2,ν3,ν4 (in units10-4nF). Tν1,ν2,ν3,ν4are periods of corresponding
perturbations.
Table 1. Main perturbations in the Moon librations in longitude.
ν1 ν2 ν3 ν4 Tν1,ν2,ν3,ν4 λν1,ν2,ν3,ν4
0 1 0 0 365.26 81”02
1 0 0 0 27.555 -15”65
1 -1 0 -1 -3232.9 9”85
2 0 0 -2 205.89 9”69
1 0 0 -2 31.81 4”15
1 0 0 -1 411.78 -2”98
2 0 -2 0 -1095.2 -1”86
2 -1 0 -2 471.89 0”74
0 0 0 2 14.77 -0”61
The results of tabulations of amplitudes of perturbations in the Moon rotation give good
agreement with earlier constructed theories for its rigid model. Barkin’s work partially was
financially accepted by Spanish grants, Japanese-Russian grant N-07-02-91212 and by RFBR
grant N 08-02-00367.
References
[1] Barkin, Yu. (1987) An Analytical Theory of the Lunar Rotational Motion. In:
Figure and Dynamics of the Earth, Moon and Planets/ Proceedings of the Int. Symp.
(Prague, Czechoslovakia, Sept. 15-20, 1986)/ Monogr. Ser. of UGTK, Prague. pp.
657-677.
[2] Ferrandiz, J., Barkin, Yu. (2003) New approach to development of Moon
rotation theory. Procced. of Inter. Conf. “Astrometry, Geodynamics and Solar System
Dynamics”. Journees 2003 (Sept. 22-25, 2003, St. Peters., Russia). IPA RAS, 199-200. |
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