A nonlinear analysis of the moon's physical libration in longitude

The Euler equations for the forced physical librations of the Moon have already been solved by using a digital computer to perform the semi-literal mathematical manipulations. Very near resonance, the computer solution for the physical libration in longitude is complemented by the solution of the appropriate Duffing equation with a dissipation term. Because of its apparent proximity to a resonant frequency, the term whose argument is 2 - twice the mean angular distance of the Moon's perigee from the ascending node of its orbit - is especially important. Its phase, which soon should be measurable, is related to the Moon's anelasticity. The term's frequency, in units of the sidereal month, increases as the semi-major axis of the Moon's orbit about the Earth increases. Using the Moon's mechanical ellipticity of Koziel and the rate of increase of the semi-major axis of MacDonald, it is estimated that the 2 term will cross the resonant frequency in 130 million years and, if the rate of energy dissipation is sufficiently low, a transient libration will be induced.