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. vation√ 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 | orbital evolution | angular momentum The motion of the Earth–Moon system has much richer dynamics than the three-body problem. Perturbations to the system 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 | PNAS | July 7, 2020 | vol. 117 | no. 27 www.pnas.org/cgi/doi/10.1073/pnas.2003496117 Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 ~ ~ inadWisdom and Tian C where AM, reference the modify can 10–12) (1, L of of conservation motion in conserve the changes the Moon to and over leads Earth the the Sun average between of The the The conservation about Sun. Moon. the Earth the the the to about of leads Earth axis period the the semimajor lunar and of the Moon motion over the the average of of period orbital period the orbital over averaged is Appendix, tonian (SI problem Lidov–Kozai the for conservation the as of term of demonstration we which The instability, scenarios. 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We general.) are more Moon is and tion Earth the that stated between addition, tides in 20, and 16 refs. and that ognized effects, additional which that, about these Moon, shows raised results all the tides those with of of include even reexamination dynamics not A rotational the did here.) the included by it are and Earth (But Moon Moon. the the and the on on Earth, affect the raised on that tides Sun Earth Sun the from the from interactions between tides evolv- direct interactions cross-tidal chaotically also tidal but the only Moon, and of not and eccentric all with evolving by planets, that fully perturbed ing than a orbit, complete included more Earth ways, model inclined of That number here. a a in used in evolution tran- was, the spin) that investigated (lunar (20) model state instability Wisdom Cassini and the the Touma through and sitions. passes LPT system the with the associated 5R though from even evolves integrations; case, Moon the the the terminate as thousand, we a interac- (where that in tidal find part with we a Sun, Earth, precessing to and the oblate, Moon about the orbit an between circular tions with near interacting a on Moon, Earth the of dynamics $ s p $ e e p h Mo h at–onsystem Earth–Moon the of AM The pr rmtednmclevolution, dynamical the from Apart normdl hc nldstefl oainladorbital and rotational full the includes which model, our In −3 ednt h clrsum scalar the denote We . where , L 0 = ≈ 0 = z L C 3.5% o orsnn cnro xcl olw h derivation the follows exactly scenarios nonresonant for r .345L L .3308 e p .. .%of 2.3% i.e., , ⊕ h hnein change the , stemgiueof magnitude the is a ~ L Fg ) Nt htrf.1,2,ad2 led rec- already 21 and 20, 16, refs. that (Note 1). (Fig. r M ⊕ L n h te ytmparameters. system other the and L and z e z sterttoa Mo h at and Earth the of AM rotational the is R scnevdi h averaged the in conserved is sa oto order of most at is e 2 L L stepeetdyvalue, present-day the is z p z L 0 = = s p S11 L L h aeipcso h onaemuch are Moon the on impacts late The . L z z .339L L C ⊕ uhrsnne rnear-resonances or resonances Such . . ⊕ e ( L S2) Fig. and cos sErhslretpicplmoment, principal largest Earth’s is is ~ L L r I ⊕ ~ ∆ L ). ⊕ e ⊕ spretyaindt h verti- the to aligned perfectly is L + + IApni,S Text SI Appendix, SI and , ⊕ L 10 L ≈ $ $ −3 L ~ L L C R cos by z L z L e e p sapproximately is L u nuelong-term induce but , $ L a tl osre to conserved still was r ,7 m.Ti is This km). 6,371 = r · L z L i .TeHamil- The Text). 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(e.g., postresonance Sun way. the same the 10–12), about and motion involve 1 that Earth’s history to Earth–Moon related of resonances models For different). cent nonresonant 13), ref. For (e.g., postimpact evolution system. the and Earth–Moon formation Earth–Moon the of of scenarios evolution the on the using system, state. planetary evolution. high-AM a evolving of instead chaotically state, Sun–Earth–Moon full and the tides, constant in mutual Earth–Moon tides, tides, cross solar with system 1. 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L e au hudb near be should value fe h rtse,A and AM step, first the After . z PNAS cnitn (i.e., -consistent i p = I e e L | 5 I Actually, . , ◦ e z hn ntescn tp hycnetae on concentrated they step, second the in Then, . uy7 2020 7, July L , L osraina togconstraint strong a as conservation i s z r and , dqaeyrpeet ninitial an represents adequately ) elnsb ny35 hogotthe throughout 3.5% only by declines ihtecntan of constraint the With . I e I L e and z ilices ntescn tp othe so step, second the in increase will L L L a hudb near be should s I s | z e 3 = and , i I r fkyitrs.We interest. key of are agsi .4t 2.84L to 1.94 in ranges = e L o.117 vol. is S1, Figs. Appendix, (SI e ls otepeetvalues, present the to close get I z p L e .5R IAppendix, SI i z p a otafwper- few a most (at . and nta conditions initial ) e utotiethe outside just , I e | and , i o 27 no. ∗ a epro- be can i the , einves- We a L L ≤ z p L z | 25R s L Model nthe in z = from 15461 values e L and L z p s p z ,

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES ref. 10 need to be generalized, they give a rough estimate of the value of e for da/dt = 0 that is consistent with our simulations and tidal parameters. During the phase in which da/dt ≈ 0, Ls declines. We can cal- culate the rate of decline if we assume da/dt = 0. In this case, the changing part of Ls is predominantly the rotational AM of the Earth. The rate of change of AM is the component of the tidal torque on the spin axis of the Earth. Though the tidal torque depends on many factors, the leading term, T0, sets the timescale and depends only on parameters and the semimajor axis. This term is the same for the constant Q tidal model and the constant ∆t tidal model (20):

2 5 3 k2e GMm Re T0 = − 6 , [2] 2 Qe a

where Mm is the mass of the Moon. Using the expression for the reference AM Lr , we find Fig. 2. Details of LPT [(A) angular momentum, (B) lunar eccentricity, (C) the d L 3 1 k M 2 R 9/2 angle between the ascending node of the lunar orbit and the ascending s = − 2e m e n, [3] node of the Earth’s equator, (D) lunar semimajor axis, (E) lunar inclination, dt Lr 2 λ Qe Me a and (F) Earth’s obliquity], with Lz-consistent initial conditions (in [Ie, Ls]): ◦ ◦ ◦ (61 , 0.7Lr [blue]), (65 , 0.8Lr [red]), (70 , 0.98Lr [black]). Parameters are where λ = 0.3308 is the present moment of inertia of the Earth as follows: Q /k = 100, Q /k = 100. In A, the top lines are L (=L⊕ + 2 e 2e m 2m s divided by Me Re , and n is the mean motion of the lunar orbit. ⊕ L ), and the horizontal line is Lz (=Lz + Lz ). The fact that the Lz line is Evaluating this expression for a = 18Re , we find, independent $ $ horizontal indicates that it is conserved by the evolution. Most of the decline of tidal models, a decline of about 7.9 × 10−3 per million years in Ls occurs when both e is nonzero and h oscillates. For the blue curves, a (My). The decline found in the simulations is comparable to this, and d mark the beginning and end of the LPT. about 7.8 × 10−3 per My. The agreement is excellent. This suc- cess allows us to generalize our simulation results to other tidal parameters. The rate of decline of Ls /Lr is simply proportional 57.6◦ L 61◦ L 65◦ we take four sampling points: ( , 0.63 r ), ( , 0.7 r ), ( , to k2e /Qe . With a larger Qe , we can expect that it would take L 70◦ L I 0.8 r ), and ( , 0.98 r ). A larger initial e corresponds to a longer to leave the LPT, but that the ending value of Ls /Lr larger initial Ls . We take Qe /k2e = 100 and Qm /k2m = 100, the would be roughly the same (see below). values used in ref. 13, where k2 is the potential Love number, and Termination of the LPT is marked by a change in the behav- Q is the tidal quality factor. ◦ ior of the angle between the ascending node of the lunar orbit In the case (57.6 , 0.63Lr ), Ie does not get large enough for on the ecliptic (Ω) and the ascending node of the Earth’s equa- the instability during the LPT (17), so the AM is not decreased. ◦ ◦ ◦ tor on the ecliptic (h0); we denote this angle by h (=Ω − h0). Results for the (61 , 0.7Lr ), (65 , 0.8Lr ), and (70 , 0.98Lr ) ◦ During the LPT, h oscillates about 0; after exit from the LPT, cases are shown in Fig. 2. In the (70 , 0.98Lr ) case, there are a h circulates through all angles (Fig. 2). The point of transition lot of sudden, large excursions in eccentricity, and it ends with an from oscillation to circulation of h is well defined. At this point, unbound orbit. All of the cases end with a high Ls ∼ 0.5Lr , 45% larger than the present value of 0.345Lr . The Earth’s obliquity after the LPT, around 50◦, is too large to produce the present ◦ Ie of 23.4 in the later evolution (SI Appendix, Low-e Phases of Evolution). The post-LPT inclination being high (∼ 34◦) may not be a problem, since it can be damped during the subsequent Cassini-state transition, provided that the lunar magma ocean has not solidified by that point (29). However, conservation of Lz implies that if the inclination is damped, then the obliquity must increase, making it even more difficult to produce the present Ie . Since these initial conditions are representative of all possible combinations of post-giant-impact Ie and Ls , these results show p that, with Lz = Lz , the high-obliquity scenario does not work to produce the present-day AM and Earth’s obliquity, at least for the tidal parameters used (Qe /k2e = 100 and Qm /k2m = 100). Next, we show that this is the case regardless of the tidal model and tidal parameters.

Characteristics of the High-Ie Scenario The decrease in Ls (AM scalar sum) occurs predominantly during the LPT instability, during which the lunar eccentricity is nonzero and the semimajor axis, a, stalls. The rate of change of a (da/dt) is a competition between tides on Earth (which tend to a a Fig. 3. The stability diagram for a = 18Re plots the color corresponding to increase ) and tides on the Moon (which tend to decrease ). At the value of L /L if the circular orbit is linearly unstable. The black line zero e, da/dt is positive. But da/dt decreases as e is increased s r marks the boundary between oscillating h and circulating h. Ls/Lr can only (e.g., ref. 10). There is an e at which da/dt is zero. The value of decrease if the system is in the unstable region with h oscillating. There is e at this point depends on a ratio of the tidal parameters of the a small “disconnected” arc-like unstable region with smaller Ie and i that is Earth and Moon and the tidal model. Though the expressions in not shown, because it cannot be reached by tidal evolution.

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L o all for , deter- h h obe to cos y ˙ s /L ) < 0 = 33 29 43 38 47 I e min to ω 0 e ◦ ◦ ◦ ◦ ◦ r vlto,adtemrsadcrepn otepit nFg ihthe with 2 Fig. in points the to correspond a–d labels. same marks the and bound- evolution, initial the with marks simulation line the black The track. between common ary a onto merge conditions, of hw nFg ,wt nta odtos(n[I (in conditions initial 0.8L with 2, Fig. in shown 4. Fig. L uinbgn nteuprrgt netesse nesthe enters system the Once axis right. semimajor upper the instability, the LPT of in value begins the lution indicate we so simulations, with that see We 2). (Fig. L simulations our in obtained predicted the that Notice of ble values other for results The to evolution tion). subsequent present for the required reach is that obliquity about post-LPT is reached obliquity Earth eue t uwr vlto,a niae ytecag of change the by indicated then as axis evolution, semimajor The outward system direction. its changes The resumes plot yellow. between the boundary on to the jectory orange until left are the h to colors diagonally the down evolves phase, this ing the Whenever in reduction significant There instability. the further LPT no the once is leaves soon But and course left). changes system lower the between toward the boundary (diagonally in instability (16) Ida of and Atobe reduce by torques found in those oscillations of large These diagram). system in constant the variations roughly point, ing large this undergo At to eccentricity. begins nonzero instabil- develops of tongues and colored ity the enters system the instability, the LPT the on large through like with evolution look begins tidal would the instability what LPT suggest they system, aged models. tidal and inclination parameters lunar tidal present-day The of regardless system, Moon s min z p silto n iclto srahd tti on,tetra- the point, this At reached. is circulation and oscillation xmnto ftesmltoscnrsti itr Fg 4). (Fig. picture this confirms simulations the of Examination aver- nontidal the for are diagrams stability the though Even oeta h he iuain,tog tre ihdfeetinitial different with started though simulations, three the that Note a. h high-I the , r e i sur],ad(70 and [square]), oscillating , and − i versus I e I e min ln.Tesmmjrai sntcntn nthe in constant not is axis semimajor The plane. h |h siltn t h ih)and right) the (to oscillating I e | e r h minimum the are < cnroi o bet rdc h rsn Earth– present the produce to able not is scenario whenever L 0. s h /L I 23.4 03 e ein,sc steclrdrgo nFg 3. 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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES color to red. In this ﬁnal phase, the stability diagram at ﬁxed a no and 0.404Lz in ref. 12, both with Qe = 400. It was then thought longer applies. The eccentricity damps to zero, and Ls no longer that the subsequent evolution would decrease Lz , but the Lz declines substantially. constraint rules out this possibility. It was found in ref. 12 that a smaller Lz can be produced with a Lz Constraint on Resonant Scenarios larger Qe (0.436, 0.404, and 0.389 Lr for Qe = 300, 400, and 500). 3 4 The evection resonance and the evection-limit cycle involve a With a large Qe (10 to 10 ) in the early history of the Earth (31), resonance between the precession of the lunar pericenter and the evection limit cycle remains a possible mechanism to drain the motion of the Earth around the Sun. Substantial AM can be the excess AM from a fast-spinning Earth. lost in the form of a decrease in Lz (1, 10–12). However, Lz is conserved in the postresonance evolution. The late-accretional Code Availability −3 impacts can only modify Lz by up to 8 × 10 Lr (either increase The computer codes we used for the simulations in this paper are or decrease). Even the presence of solar tides and planetary available at GitHub, https://github.com/zhenliangtian/em3d. perturbations can only decrease Lz as much as 0.01Lr . There- fore, for a scenario of evection resonance or evection limit cycle Conclusion to be a possible representation of the Earth–Moon history, the The Lz constraint places limits on the possible orbital histories of p postresonance state should have its Lz close to Lz (0.339Lr ): the Earth–Moon system and thus limits the details of the Moon- 0.331Lr < Lz < 0.357Lr . forming giant impact. For a high-AM impact (1–3), which is able The evection resonance (1) and the near-resonance described to produce a Moon with an Earth-like composition, the impact in ref. 11 both involve high lunar eccentricities (> 0.5). Such high geometry is constrained to cases where the postimpact Earth has eccentricities will lead to severe heating in the Moon and cause a small to medium obliquity. these mechanisms to quickly exit with little AM decreased (12). The evection limit cycle (near-resonance) described in refs. 10 ACKNOWLEDGMENTS. This research was initiated in anticipation of the Moon Workshop, held October 14–16, 2019, at the American University of and 12 leads to lower eccentricities and is thermally stable (12). Beirut, Lebanon, organized by Jihad Touma. We thank Francis Nimmo for However, the post-limit cycle minimum Lz is 0.393Lr in ref. 10 helpful discussions.

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