The theory of [the future motion of Luna 3] will pose a pretty problem in celestial mechanics, which the mathematicians may well choose to shirk by resorting to electronic computers.
Abstract
Luna 3 (or Lunik 3 in Russian sources) was the first spacecraft to perform a flyby of the Moon. Launched in October 1959 on a translunar trajectory with large semimajor axis and eccentricity, it collided with the Earth in late March 1960. The short, 6-month dynamical lifetime has often been explained through an increase in eccentricity due to the Lidov–Kozai effect. However, the classical Lidov–Kozai solution is only valid in the limit of small semi-major axis ratio, a condition that is satisfied only for solar (but not for lunar) perturbations. We undertook a study of the dynamics of Luna 3 with the aim of assessing the principal mechanisms affecting its evolution. We analyze the Luna 3 trajectory by generating accurate osculating solutions, and by comparing them to integrations of singly and doubly averaged equations of motion in vectorial form. Lunar close encounters, which cannot be reproduced in an averaging approach, decisively affect the trajectory and break the doubly averaged dynamics. Solar perturbations induce oscillations of intermediate period that affect the geometry of the close encounters and cause the singly averaged and osculating inclinations to change quadrants (the orbital plane “flips”). We find that the peculiar evolution of Luna 3 can only be explained by taking into account lunar close encounters and intermediate-period terms; such terms are averaged out in the Lidov–Kozai solution, which is not adequate to describe translunar or cislunar trajectories. Understanding the limits of the Lidov–Kozai solution is of particular significance for the motion of objects in the Earth–Moon environment and of exoplanetary systems.
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Notes
King-Hele, op. cit., p. 685.
Thus, the mission achieved two firsts: the first view of the lunar far side, and the first gravity assist maneuver.
That is to say that the orbital elements of the Moon and of the spacecraft are referred to the Earth’s center, and those of the Sun are referred to the Earth–Moon barycenter.
GC61 seems to be the only original public source of ephemerides for Luna 3. No Two-Line Elements (TLEs) for the spacecraft are present in the US Space Object Catalog. The mission predates the establishment of the catalog, and no TLEs seem to have been derived or added to the catalog a posteriori. Moreover, no Luna 3 ephemerides are present in the JPL HORIZONS system (Park 2019). State vectors identical to those obtained by numerical integration in GC61 are given by King-Hele et al. (1987, p. 4).
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Acknowledgements
Parts of this work were presented at the 2018 John L. Junkins Dynamical Systems Symposium and at the 2019 Meeting of the AAS Division on Dynamical Astronomy (DDA). We are grateful to Ivan Shevchenko and one anonymous reviewer for their insightful review of this article. Davide Amato thanks Jay McMahon for his indispensable support during the writing of this article, Giulio Baù for comments that improved the quality of the article, and Giovanni Valsecchi for helpful discussions at the 2019 DDA Meeting about averaged solutions in the presence of orbit crossings. Renu Malhotra acknowledges funding from NSF (Grant AST-1824869), and the Marshall Foundation of Tucson, AZ, USA. We acknowledge the use of software routines from the IAU SOFA Collection (IAU SOFA Board 2019) in the reduction of the Luna 3 ephemerides.
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Amato, D., Malhotra, R., Sidorenko, V. et al. Lunar close encounters compete with the circumterrestrial Lidov–Kozai effect. Celest Mech Dyn Astr 132, 35 (2020). https://doi.org/10.1007/s10569-020-09972-6
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DOI: https://doi.org/10.1007/s10569-020-09972-6