The vector approach to the problem of physical libration of the Moon: the linearized problem |
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| Authors: | B P Kondratyev |
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| Institution: | 1.Udmurt State University,Izhevsk,Russia |
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| Abstract: | A flexible and informative vector approach to the problem of physical libration of the rigid Moon has been developed in which
three Euler differential equations are supplemented by 12 kinematic ones. A linearized system of equations can be split into
an even and odd systems with respect to the reflection in the plane of the lunar equator, and rotational oscillations of the
Moon are presented by superposition of librations in longitude and latitude. The former is described by three equations and
consists of unrestricted oscillations with a period of T
1 = 2.878 Julian years (amplitude of 1.855″) and forced oscillations with periods of T
2 = 27.201 days (15.304″), one stellar year (0.008″), half a year (0.115″), and the third of a year (0.0003″) (five harmonics
altogether). A zero frequency solution has also been obtained. The effect of the Sun on these oscillations is two orders of
magnitude less than that of the Earth. The libration in latitude is presented by five equations and, at pertrubations from
the Earth, is described by two harmonics of unrestricted oscillations (T
5 ≈ 74.180 Julian years, T
6 ≈ 27.347 days) and one harmonic of forced oscillations (T
3 = 27.212 days). The motion of the true pole is presented by the same harmonics, with the maximum deviation from the Cassini
pole being 45.3″. The fifth (zero) frequency yields a stationary solution with a conic precession of the rotation axis (previously
unknown). The third Cassini law has been proved. The amplitudes of unrestricted oscillations have been determined from comparison
with observations. For the ratio $
\frac{{\sin I}}
{{\sin \left( {I + i} \right)}} \approx 0.2311
$
\frac{{\sin I}}
{{\sin \left( {I + i} \right)}} \approx 0.2311
, the theory gives 0.2319, which confirms the adequacy of the approach. Some statements of the previous theory are revised.
Poinsot’s method is shown to be irrelevant in describing librations of the Moon. The Moon does not have free (Euler) oscillations;
it has oscillations with a period of T
5 ≈ 74.180 Julian years rather than T ≈ 148.167 Julian years. |
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