Excitation of Lunar Eccentricity by Planetary Resonances

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Science  12 Oct 2007:
Vol. 318, Issue 5848, pp. 244
DOI: 10.1126/science.1146984


The origin of the Moon's nonnegligible orbital eccentricity of 0.053 has no theoretical explanation. Lunar laser ranging indicates that tides on Earth are currently increasing the Moon's eccentricity. However, ocean tides were likely much weaker during the first billion years, allowing lunar tides to damp any primordial lunar eccentricity very early on. During the tidally driven expansion of its orbit, the Moon must have been affected by two substantial resonances related to Jupiter and Venus, passage through which may have generated today's lunar eccentricity.

The Moon is thought to have formed in a collision between Earth and another protoplanet (1), after which a circumterrestrial debris disk quickly accumulated into a satellite (2). Subsequent tidal interactions (3) expanded the lunar orbit from 3 to 60 Earth radii (RE). The effects of tides on lunar eccentricity (e) are complex: Whereas tides raised on Earth increase e, solid body tides within the Moon damp it (4).

Goldreich (4) derived analytical estimates of the relative importance of the tides within Earth and the Moon and found that the lunar e should be increasing. Because currently lunar free eccentricity, efree, is 0.053 (Materials and Methods), the two effects had to be close during much of the system's history if they were to prevent extreme damping or growth (2, 4).

Lunar laser ranging (5) confirmed the recession but found the increase in e to be significantly larger than the theory predicts. The most plausible explanation for this is that the ocean tides, which are known to provide over 95% of the Earth's tidal dissipation, cannot be modeled by simple tidal bulges, which lead the Moon in longitude.

Both the system's age and ocean-tide models (6) suggest that Earth's tides have become more important over time, probably because the ocean dissipation depends on the forcing frequency (i.e., length of day). If they were weaker than lunar tides during the initial rapid orbital expansion, any primordial e would have been greatly reduced during this crucial epoch, contradicting Goldreich's hypothesis (5).

If not primordial, where does the present e come from? Kaula and Yoder (7) noted that an otherwise minor periodic perturbation known as jovian evection (governed by the geocentric angle between the lunar perigee and the position of Jupiter) becomes resonant at the Earth-Moon distance of about 53RE and calculated that the resonance capture would have occurred if efree < 0.0076 when entering the resonance.

To test this hypothesis, we constructed a numerical integrator by using a symplectic algorithm [explicitly separating Keplerian motion from small perturbations (8)]. The lunar orbit was integrated directly, whereas the orbits of Earth and Jupiter were considered unperturbed. Earth tides were included directly by using a quadrupole “bulge” leading the Moon by a constant angle, whereas the satellite tides were ignored.

The integration with a constant-e Earth's orbit (fig. S1) matched the analytical results (7). However, it is not a good model for the resonance passage because we have ignored the variation of lunar orbital precession, which mirrors the planet-induced oscillations in Earth's eccentricity. To avoid directly integrating all eight planets, we treat the Earth's orbit as Keplerian but with slowly varying eccentricity and longitude of pericenter following a secular theory (Materials and Methods). Thus, modified simulations (fig. S2 and Fig. 1) show that the present orbital efree of the Moon can be generated by the resonance passage if efree ≥ 0.005 before the resonance.

Fig. 1.

Median postresonance efree for nine sets of numerical simulations plotted against their initial efree. Six sets of 30 runs with initial efree = 0.005 to 0.015 lasted 50 My, and the three sets of 15 runs with lower initial efree lasted 33 My. Each error bar plots the scatter of the 50% of the runs closest to the median. The two dashed lines show the free eccentricity of the Moon at 53RE extrapolated back from the present, assuming Earth's tidal quality factor QE = 12 (bottom) or QE = 35 (top).

We also considered other planetary resonances that the Moon must have encountered. The only other important resonance happens at 46.6RE and involves Venus (table S1). With use of numerical simulations, we found that the preresonance efree at 46.6RE had to be larger than about 0.001 if the Moon was to enter jovian resonance at 53RE with efree = 0.005 (fig. S3).

A lunar efree ∼10–3 could be a tidally damped remnant of a primordial e or a product of the lunar cataclysm (9) through basin-forming impacts or close encounters with massive (≅1000 km) objects. In either case, our results support the view that lunar tides were dominant over terrestrial ones during the first few 100 million years (My) of the system's history (4, 5), which would make high-e lunar orbit (10) beyond 20RE rather unlikely. Although our analysis cannot exclude solar resonances (11) inducing a high e closer to Earth, interactions with a circumterrestrial ring (12) likely made early lunar orbital evolution too fast for large eccentricity excitation (11).

Supporting Online Material

Materials and Methods

Figs. S1 to S3

Table S1



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