The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem |
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| Authors: | B P Kondratyev |
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| Institution: | 1.Pulkovo Astronomical Observatory,Russian Academy of Sciences,St. Petersburg,Russia;2.Udmurt State University,Izhevsk,Russia |
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| Abstract: | By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method
on small parameters we derive the closed system of nine differential equations with terms of the first and second order of
smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude,
though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent
from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and
two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are
periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory
a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical
libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s
spin axis. The character of nonlinear solutions near the basic forcing frequency Ω1, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by
the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients
in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior
period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies.
Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially
improves and supplements a linear picture of the physical libration. |
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