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Vertical Angular Momentum Constraint on Lunar Formation and Orbital History Vertical angular momentum constraint on lunar formation and orbital history ZhenLiang Tiana,b,1,2 and Jack Wisdomc,1,2 aDepartment of Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064; bDepartment of Earth and Space Sciences, Southern University of Science and Technology, Shenzhen 518055, China; and cDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139 Contributed by Jack Wisdom, May 4, 2020 (sent for review February 25, 2020; reviewed by Robin M. Canup and Scott Tremaine) The Moon likely formed in a giant impact that left behind a fast- A New Constraint: Vertical AM rotating Earth, but the details are still uncertain. Here, we exam- For a test particle around a central mass with large eccentricity ine the implications of a constraint that has not been fully (e) and inclination (i), nonresonant perturbation from a mas- exploited: The component of the Earth–Moon system’s angu- sive exterior body moving in a circular orbit induces oscillations lar momentum that is perpendicular to the Earth’s orbital plane in e and i, the Lidov–Kozai oscillations (18, 19). In the nonti- is nearly conserved in Earth–Moon history, except for possible dal Lidov–Kozai problem, the Hamiltonian governing the system intervals when the lunar orbit is in resonance with the Earth’s evolution is averaged over the orbital periods of both the test motion about the Sun. This condition sharply constrains the particle and the perturber. This leads to a conservation of the postimpact Earth orientation and the subsequent lunar orbital semimajor axis (a) of the test particle, and the component of the history. In particular, the scenario involving an initial high- orbital AM of the test particle that is perpendicular to the orbital obliquity Earth cannot produce the present Earth–Moon sys- orb plane of the massive perturber, which we refer to as Lz . Tak- tem. A low-obliquity postimpact Earth followed by the evection ing the orbital plane of the massive perturber as the reference limit cycle in orbital evolution remains a possible pathway for orb p 2 plane for inclination, then Lz / a(1 − e ) cos i. The conser- producing the present angular momentum and observed lunar orb composition. pvation of a and Lz implies that as e and i oscillate, the quantity 1 − e2 cos i is conserved (the Lidov–Kozai constant). moon formation j orbital evolution j angular momentum The motion of the Earth–Moon system has much richer dynamics than the three-body problem. Perturbations to the sys- tem come from not only the Moon–Sun interaction, but also ecent giant-impact simulations (1–3) aimed at producing a the Earth’s oblateness, which leads to precession of the Earth’s RMoon with an Earth-like isotopic composition (4–8) would spin axis and contributes to lunar-orbit precession; the Moon’s leave the postimpact Earth rotating too fast. Some subsequent permanent triaxial figure, which allows the Moon to maintain process must be responsible for draining the excess angular synchronous rotation; and the Moon’s oblateness and triaxiality, momentum (AM). which dictates the equilibrium obliquity of the Moon (the Cassini Cuk´ and Stewart (1) proposed that the excess AM could be states)—not to mention the tides on the Earth and Moon. These drained by the lunar orbit’s temporary capture into the evection processes modify the orbital AM of the Moon and the rotational resonance (9). Wisdom and Tian (10) found that the evection AM of the Earth. Nevertheless, there is a quantity analogous limit cycle can also do the job. While ref. 10 used the Darwin– to the Lidov–Kozai constant, the vertical component of the AM Kaula constant Q tidal model (SI Appendix, Model Comparison), of the Earth–Moon system (Lz ), that is conserved if resonances Rufu and Canup (11) found similar phenomena using the con- between the Sun and the Earth or Moon are not encountered. stant ∆t tidal model. Tian et al. (12) studied the consequences of For instance, the evection resonance and the evection limit cycle tidal heating in these resonance and near-resonance mechanisms and found that the evection resonance is unstable and therefore Significance unable to drain enough AM, whereas the evection limit cycle is stable and continues to drain AM. There are various scenarios for the formation of the Moon and Instead of starting the postimpact fast-spinning Earth with subsequent dynamical evolution of the Earth–Moon system, a small obliquity (Ie ), Cuk` et al. (13) proposed that the giant ◦ all of which are subject to a constraint that has not previously impact left the Earth in a high-obliquity (Ie = 65 to 80 ), fast- been fully exploited. Using this constraint, we demonstrate spinning state. They suggested that the subsequent evolution that the recently proposed high-obliquity scenario is not con- could not only drain the excess AM through an instability associ- sistent with the present Earth–Moon system. This constraint ated with the Laplace plane transition (LPT), but also solve the p will have to be taken into account in all future investigations long-standing puzzle of the present-day lunar inclination (i = of the formation and evolution of the Moon. 5◦, where i is the lunar inclination, and the p superscript denotes the present-day value) (9, 14, 15). When the Moon is close to Author contributions: Z.T. and J.W. designed research, performed research, analyzed the Earth, the lunar precessional motion is strongly affected by data, and wrote the paper.y the oblateness of the Earth, but when the Moon is far from the Reviewers: R.M.C., Southwest Research Institute; and S.T., Institute for Advanced Study.y Earth, the precessional motion is more strongly affected by solar The authors declare no competing interest.y perturbations. The LPT occurs when these two effects are com- This open access article is distributed under Creative Commons Attribution-NonCommercial- parable. The LPT instability occurs only when Ie reaches large NoDerivatives License 4.0 (CC BY-NC-ND).y values during the LPT (16, 17). The high Ie leads to nonzero Data deposition: The computer codes we used for the simulations in this paper are orbital eccentricities via Lidov–Kozai-like oscillations and tem- available at GitHub, https://github.com/zhenliangtian/em3d.y porary contraction of the lunar orbit, during which substantial 1 Z.T. and J.W. contributed equally to this work.y AM is removed from the Earth–Moon. 2 To whom correspondence may be addressed. Email: [email protected] or [email protected] However, there exists a nearly conserved quantity, which has This article contains supporting information online at https://www.pnas.org/lookup/suppl/ so far not been taken into account, that significantly constrains doi:10.1073/pnas.2003496117/-/DCSupplemental.y the possible evolutionary histories of the Earth–Moon system. First published June 22, 2020. 15460–15464 j PNAS j July 7, 2020 j vol. 117 j no. 27 www.pnas.org/cgi/doi/10.1073/pnas.2003496117 Downloaded by guest on September 29, 2021 (1, 10–12) can modify Lz . Such resonances or near-resonances are absent in scenarios like the LPT instability, which we term as nonresonant scenarios. The demonstration of the conservation of Lz for nonresonant scenarios exactly follows the derivation for the Lidov–Kozai problem (SI Appendix, SI Text). The Hamil- tonian is averaged over the orbital period of the Moon and the orbital period of the motion of the Earth about the Sun. The average over the lunar period leads to the conservation of the semimajor axis of the Moon. The average over the motion of the Earth about the Sun leads to the conservation of Lz . Tides between the Earth and Moon conserve Lz , but induce long-term changes in a and the other system parameters. The AM of the Earth–Moon system ~L is approximately ~L⊕ + ~L , where ~L⊕ is the rotational AM of the Earth and ~L is the orbital$ AM of the Moon. Here, we are neglecting the small$ rota- tional AM of the Moon. Taking the ecliptic as the reference plane, the component of AM perpendicular to this plane is Fig. 1. Lz versus a for a simulation in ref. 20, which evolves the Earth–Moon ⊕ Lz = L cos Ie + L cos i, [1] system with solar tides, Earth–Moon mutual tides, and Sun–Earth–Moon $ cross tides, in the full chaotically evolving planetary system, using the constant ∆t tidal model. The system begins with a conventional initial L⊕ ~L⊕ L where is the magnitude of , and is the magnitude of state, instead of a high-AM state. Lz declines by only 3.5% throughout the ⊕ $ ~L . We denote the scalar sum L + L by Ls . evolution. $In our model, which includes the full$ rotational and orbital dynamics of the Moon, interacting with an oblate, precessing Therefore, we can take Lz conservation as a strong constraint Earth on a near circular orbit about the Sun, with tidal interac- on the evolution of the Earth–Moon system. For nonresonant ATMOSPHERIC, tions between the Moon and Earth, we find that Lz is conserved scenarios of Earth–Moon formation and evolution (e.g., ref. 13), to a part in a thousand, as the Moon evolves from 5Re to 50Re p EARTH, the postimpact Lz value should be near Lz (at most a few per- AND PLANETARY SCIENCES (where we terminate the integrations; Re = 6,371 km). This is cent different). For models of Earth–Moon history that involve the case, even though the system passes through the instability resonances related to Earth’s motion about the Sun (e.g., refs.
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