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Tidal evolution of the Moon from a high-obliquity, high-angular-momentum Earth

Nature volume 539, pages 402–406 (2016)Cite this article

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Abstract

In the giant-impact hypothesis for lunar origin, the Moon accreted from an equatorial circum-terrestrial disk; however, the current lunar orbital inclination of five degrees requires a subsequent dynamical process that is still unclear1,2,3. In addition, the giant-impact theory has been challenged by the Moon’s unexpectedly Earth-like isotopic composition4,5. Here we show that tidal dissipation due to lunar obliquity was an important effect during the Moon’s tidal evolution, and the lunar inclination in the past must have been very large, defying theoretical explanations. We present a tidal evolution model starting with the Moon in an equatorial orbit around an initially fast-spinning, high-obliquity Earth, which is a probable outcome of giant impacts. Using numerical modelling, we show that the solar perturbations on the Moon’s orbit naturally induce a large lunar inclination and remove angular momentum from the Earth–Moon system. Our tidal evolution model supports recent high-angular-momentum, giant-impact scenarios to explain the Moon’s isotopic composition6,7,8 and provides a new pathway to reach Earth’s climatically favourable low obliquity.

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Figure 1: Numerical simulation of the Moon’s early tidal evolution from Earth with initial obliquity of 70° and spin period of 2.5 h.
Figure 2: Lunar inclination and Earth’s obliquity for various simulations.
Figure 3: Similar to Fig. 1, but with different initial obliquities for Earth.
Figure 4: Numerical integration of the later phase of lunar tidal evolution, assuming a lunar inclination of 30° at 25RE and the current shape of the Moon.

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Acknowledgements

This work was supported by NASA’s Emerging Worlds programme, award NNX15AH65G.

Author information

Authors and Affiliations

  1. Carl Sagan Center, SETI Institute, 189 North Bernardo Avenue, Mountain View, 94043, California, USA

    Matija Ćuk

  2. Department of Astronomy, University of Maryland, Physical Sciences Complex, College Park, 20742, Maryland, USA

    Douglas P. Hamilton

  3. Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge, 02138, Massachusetts, USA

    Simon J. Lock

  4. Department of Earth and Planetary Sciences, University of California Davis, One Shields Avenue, Davis, 95616, California, USA

    Sarah T. Stewart

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  1. Matija Ćuk
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  4. Sarah T. Stewart
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Contributions

M.Ć. designed the study, wrote the software, analysed the data and wrote the paper. Co-authors contributed ancillary calculations, discussed the results and wider implications of the work, and helped edit and improve the paper.

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Correspondence to Matija Ćuk.

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The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks D. Stevenson and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Semi-analytical model of the lunar tidal evolution.

Evolution of lunar inclination (a) and obliquity (b) as the Moon evolves from 25RE to 60RE using our semi-analytical model (Methods subsection ‘Damping of lunar inclination by obliquity tides’). Initial inclinations were chosen so that the final lunar inclination was the current value of about 5°, while the obliquity was calculated assuming the Moon was in a Cassini state (jumps between 30RE and 35RE are due to the transitions between Cassini states 1 and 2)18. Love numbers were set at their current values (k2,E = 0.3, k2,M = 0.024)35, and the current lunar shape was assumed. The black and red lines plot the solutions for QM = 10,000 and QM = 38 (current value), respectively, while QE was in the range 33–35 range (it was adjusted so that a semimajor axis of 60RE was reached after 4,500 Myr). The blue line plots a history assuming QM = 100 interior to 40RE, and QM = 38 after the Moon passes that distance. The black line closely resembles the results of studies11,12 that neglected lunar obliquity tides, while the other two curves indicate that the past lunar inclination must have been much larger owing to lunar obliquity tides.

Extended Data Figure 2 Lunar tidal evolution following planetesimal encounters.

Evolution of lunar inclination (a) and eccentricity (b) following the excitation of the lunar inclination by encounters with planetesimals as proposed by ref. 3, using our semi-analytical model. The two sets of initial conditions are for the state a = 47RE, i = 5.8° featured in ref. 3 (Fig. 1), and a possible outcome with a more excited inclination i = 10° (also at a = 47RE). The initial eccentricities were estimated as e = 2sini. The black lines show the evolution assuming QM = 38 and QE = 34, with the current Love numbers. The red line plots the evolution for QM = 100 and the blue line the evolution for QE = 20. The circle symbol plots the current inclination and eccentricity of the lunar orbit. No combination of tidal parameters can simultaneously match both the current lunar inclination and eccentricity at the same time. A small QM combined with high e also keeps the Moon from reaching 60RE (top two lines). Decreasing QE also does not help, as stronger Earth tides further increase the lunar eccentricity (blue line).

Extended Data Figure 3 A snapshot of the simulation shown in Fig. 1 taken at 12.8 Myr.

The left-hand panels show eccentricity (a), lunar inclination and Earth’s obliquity with respect to the ecliptic (b), and the argument of perigee of lunar orbit, with the ecliptic as fundamental plane (c) versus time over a 500-year period, while the right-hand panels plot lunar eccentricity (d) and inclination (e) against the Moon’s argument of perigee (over the whole period of 30,000 yr). The eccentricity is clearly correlated with the argument of perigee, as expected for Kozai-type perturbations24,49. Rapid mutual precession of Earth’s spin axis and the plane of the Moon’s orbit, which are substantially inclined to one another (and to the ecliptic plane), clearly affects both the eccentricity and the perigee precession. This secular behaviour is periodic, with exactly three inclination cycles per one eccentricity cycle, which corresponds to half of the period of precession of the argument of perigee.

Extended Data Figure 4 A snapshot of QE/k2,E = 200 simulation shown in Fig. 1 (black line) at 34.6 Myr.

At this time, eccentricity excitation (a) and the associated variation of Earth’s obliquity (b) are not due to Kozai perturbations, but owing to the slow-varying near-resonant argument Ψ = 3Ω + 2ω − 3γ (c), where Ω and γ are longitudes of the lunar ascending node and Earth’s vernal equinox, respectively, and ω is the Moon’s argument of perigee. This near-resonant interaction is responsible for the substantial reduction of Earth’s obliquity seen in Fig. 1.

Extended Data Figure 5 Early tidal evolution of the Moon with QE/k2,E = 200 throughout the simulations.

Black lines are for the case with QM/k2,M = 200, while grey lines plot the simulation with QE/k2,E = 50. The most notable aspects of these simulations are low final obliquities of Earth (e) and a final AM of the Earth–Moon system (b) in excess of the current value of 0.35. a, c and d plot the semimajor axis, eccentricity and inclination of the Moon.

Extended Data Figure 6 Early tidal evolution of the Moon with Earth initially having a 2-h spin period.

This is equivalent to the system having twice the current AM. The grey lines plot a simulation in which the tidal properties of Earth and the Moon were QE/k2,E = QM/k2,M = 100 throughout. The black line shows a simulation branching at 30 Myr by changing QE/k2,E to 200. Although the final obliquity of Earth (e) is correct, the final AM of the Earth–Moon system (b) somewhat exceeds the current value of 0.35 × . a, c and d plot the semimajor axis, eccentricity and inclination of the Moon.

Extended Data Figure 7 Map of lunar rotational dynamics close to the Cassini state transition.

Outcomes of 512 simulations probing the end states of initially very fast and very slow lunar rotations for 16 different lunar semimajor axes a and 16 different lunar inclinations i. Simulations were run for 1 Myr, except for the rightmost three columns, which were followed for 3 Myr. Each a−i field is described by two symbols, one each for initial rotations of 127 rad yr−1 and 381 rad yr−1. Green and blue boxes indicate synchronous rotation in Cassini states 1 and 2, respectively. Crosses indicate non-synchronous rotation with stable obliquity, with large orange crosses indicating sub-synchronous rotation, and small magenta crosses plotting super-synchronous states. Red crosses signify variations in obliquity above 1° during the last 50 kyr of the simulation (indicating excited or chaotic spin axis precession).

Extended Data Figure 8 Lunar obliquity close to the Cassini state transition.

Obliquities for four ‘slices’ in inclination (at 5°, 10°, 15° and 20°) from the grid of short simulations shown in Extended Data Fig. 7 (solid red and magenta lines with points; the obliquities and inclinations are in the same order at far left and far right). When two different simulations for the same a and i differed in outcome, we chose the solution within the Cassini state, if available. The blue dashed lines plot the relevant Cassini states calculated using analytical formulae, while the black dashed line at 58.15° plots the upper limit for stable obliquities in the relevant Cassini state. Although the numerical and analytical results agree at the smallest and largest semimajor axes, the large discrepancies in between are due to non-synchronous rotations being dominant at the Cassini state transition.

Extended Data Figure 9 The Moon’s wobble as it approaches the annual resonance in Fig. 4.

The rotation rate around the longest axis of the Moon (a) and the angle between the longest axis and Earth (b) during the first phase of lunar tidal evolution (red points in Fig. 4) within 29.7RE, where we accelerated the tidal evolution by a factor of a hundred. The wobble is clearly building up as the Moon is approaching the resonance between its free wobble and Earth’s orbital period at about 29.7RE. The growth in lunar libration angle is more influenced by increasing lunar obliquity (Fig. 4b) than the increase in the amplitude of the wobble.

Extended Data Figure 10 Passage through the annual resonance of the lunar free wobble in Fig. 4.

Lunar obliquity (a), spin rate (b), rotation rates around the longest and shortest principal axes (c), and the angle between the Moon’s longest axis and Earth (d) during the first 1 Myr of the ‘blue’ segment of lunar tidal evolution in Fig. 4 (which was simulated at the nominal rate for tidal evolution). The free wobble (tracked by grey points in c) experiences a resonance at about 330 kyr, breaking the Moon’s synchronous rotation. Since the Moon is close to the Cassini state transition, it cannot evolve back into Cassini state 1 and it settles into a non-synchronous high-obliquity state58.

Supplementary information

Animation of relative orientations of Earth's spin and the Moon's orbit during the Laplace plane transition, following the simulation plotted in black in Fig. 1.

The system is seen from the direction of Earth's vernal equinox, the blue arrow is plotted along Earth's spin axis and points to the north, while the lunar orbit is plotted in red. Initially the Earth has a high obliquity, the Moon has low inclination and the Laplace plane is close to Earth's equator. As the animation progresses and the lunar orbit grows due to tidal dissipation, the Laplace plane shifts to the ecliptic plane (horizontal in this view). The moon acquires a large inclination during the Laplace plane transition, while Earth's obliquity decreases. The labels show time, Earth's spin period, total angular momentum (scaled to the present value) and angular momentum of the Earth-Moon system where only the ecliptic component (i.e. that along the vertical axis) of the lunar orbital momentum is taken into account. Unlike total angular momentum, this ecliptic component will be conserved during the Cassini state transition. (AVI 25361 kb)

Animation of relative orientations of lunar figure and orbit during the Cassini state transition, following the simulation plotted in Fig. 4.

The Moon is seen from the direction of the ascending node of lunar orbit, with the ecliptic plane (i.e. the Moon's Laplace plane at this time) parallel to the horizontal axis. The red arrow shows the orientation of the Moon's orbit normal. At first the Moon's orbit normal and spin axis are on the same side of the normal to the ecliptic, indicating that the Moon is in Cassini state 1. Once the Cassini state 1 is destabilized, after some wobbling, the Moon settles in a non-synchronous state somewhat similar to the Cassini state 2 (with the orbit normal and the spin axis being on opposite sides of the normal to the ecliptic). During this time both the inclination and obliquity (which is forced by inclination) are being damped by strong obliquity tides. At the semimajor axis of 35.1 Earth radii, the Moon becomes synchronous again and enters the Cassini state 2, where it stays for the rest of the simulation (this event is visible as a 5-degree jump in obliquity). (AVI 18922 kb)

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Ćuk, M., Hamilton, D., Lock, S. et al. Tidal evolution of the Moon from a high-obliquity, high-angular-momentum Earth. Nature 539, 402–406 (2016). https://doi.org/10.1038/nature19846

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