On the elliptic case of the restricted problem of three bodies and the remote history of the Earth-Moon system

Ziegelhoeffer A.; De Jong J.W.; Ferrari R.; Turi Nagy L.

Icarus 1(1-6): 455-458


ISSN/ISBN: 0019-1035
DOI: 10.1016/0019-1035(62)90047-7
Accession: 061062492

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The aim of this paper has been to demonstrate that if the relative orbit of two finite masses in the restricted problem of three bodies is eccentric, no vis viva integral or the associated surfaces of zero velocity exist—no matter how small the orbital eccentricity of the finite masses may be. The existence of the well-known Jacobi integral is shown to be a singular property of the "circular" three-body problem, having no analogy in the eccentric case. As a consequence, it is pointed out that (inasmuch as the eccentricity of the terrestrial orbit around the Sun is finite, and has without doubt been so since the formation of our planet) the closed nature of the surface of zero velocity surrounding the present lunar orbit around the Earth, based on the present value of its Jacobi constant, does not guarantee that the Moon may not recede from the Earth to an arbitrary distance in the future, nor that it could not have approached the Earth from an indefinite distance in the past. Thus the celebrated argument which Darwin and Hill advanced in the last century to assert that the Moon must have been a permanent satellite of our Earth since the days of its formation appears to be invalid.