Analytical Model for the Tidal Evolution of the Evection Resonance and the Timing of Resonance Escape

A high-angular momentum giant impact with the Earth can produce a Moon with a silicate isotopic composition nearly identical to that of Earth’s mantle, consistent with observations of terrestrial and lunar rocks. However, such an event requires subsequent angular momentum removal for consistency with the current Earth-Moon system. The early Moon may have been captured into the evection resonance, occurring when the lunar perigee precession period equals 1 year. It has been proposed that after a high- angular momentum giant impact, evection removed the angular momentum excess from the Earth-Moon pair and transferred it to Earth’s orbit about the Sun. However, prior N-body integrations suggest this result depends on the tidal model and chosen tidal parameters. Here, we examine the Moon’s encounter with evection using a complementary analytic description and the Mignard tidal model. While the Moon is in resonance, the lunar longitude of perigee librates, and if tidal evolution excites the libration amplitude sufficiently, escape from resonance occurs. The angular momentum drain produced by formal evection depends on how long the resonance is maintained. We estimate that resonant escape occurs early, leading to only a small reduction (~ few to 10%) in the Earth-Moon system angular momentum. Moon formation from a high-angular momentum impact would then require other angular momentum removal mechanisms beyond standard libration in evection, as have been suggested previously.


Appendix C -Mignard Tidal Model
Mignard first derives the force due to a second-order tidal distortion raised on the Earth by the Moon in the vector form, is the tidal Love number for the Earth, vectors , are the position and velocity of the Moon of mass , and is the Earth's spin vector, which, for simplicity, we will assume is perpendicular to the lunar orbit plane. The radial, , and tangential, , force components are then substituted into Gauss' form of the Lagrange equations (e.g., Brouwer and Clemence, 1961), where ≡ (1 − 2 ) and the rates are then averaged over an orbit to give the tidal changes in semi-major-axis and eccentricity.

Appendix D -Permanent Figure Torque
Consider a Moon with principal moments of inertia ≥ ≥ , where is the moment about its spin axis, assumed to be normal to its orbit plane, and is the moment about the Moon's long axis. The instantaneous value of the permanent figure (pf) torque is given by Danby (1992; see also Murray and Dermott, 1999) where r is the Earth-Moon distance, is the angle between the long axis of the Moon and the Earth-Moon line, i.e, = − , where is the angular position of the Moon's long axis with respect to the perigee, , and is the true anomaly (e.g., Goldreich and Peale, 1966a,b). We set = + , which for synchronous rotation is = + , where is the value of at perigee. If T is then averaged over an orbit, one obtains (e.g., Goldreich and Peale, 1966a,b), 〈T 〉 = − 3 2 2 ( − ) ( ) sin 2 (D2) where ( ) = 1 − 5 2 ⁄ + 13 2 16 ⁄ is a so-called Hansen polynomial. This torque leads to further contributions to semi-major axis and eccentricity variations, ̇′ and ̇.
Analogous to eqn. (4.2), conservation of angular momentum requires (D3) This must (nearly) balance the tidal torque to ensure synchronous stability so that ′ + ′ , =̇/Ω ⊕ , and the off-set angle adopts the value needed to accomplish this. A major difference between a torque on the permanent figure of the Moon and a torque on a tidal distortion is that the former is not accompanied by energy dissipation due to planetary flexing. Accordingly, the combination of orbital energy and spin energy of the Moon is also conserved under its action, i.e., ( ′ 2 2 ⁄ − 2 ′)/ | = 0 ⁄ . Taking the derivatives and rearranging yields an additional condition, where the final expression sets ′ = /Ω ⊕ = ′−3/2 . Approximating ̇′ +̇, ′ ≈ 0 due to the smallness of ̇ compared to either spin acceleration, we conclude that where ′ ′ 3 2 ⁄ = 1, which in combination with (4.9a,b) gives ̇′ ′ ⁄ = [ 1 ( ) − 2 ( )] ′ 8 ⁄ ; ̇= [ 1 ( ) − 2 ( )]/ ′8 (D10a,b) The above rates are valid so long as synchronous rotation can be maintained. However, | sin 2 | has a maximum value of unity, and so from eqns. (4.2) and (D2) there is a minimum value required for ( − )/ , where the final expression sets ′ ′ 3/2 = 1. This criterion reads where Δ ≈ 4 min for the current Moon (Williams and Boggs, 2015). If violated, the synchronous lock is broken.
The above estimate considers whether the permanent figure torque is sufficient to maintain synchronous rotation against the competing tidal torque. Goldreich (1966) considered an initial rotation faster than n, and found that this rate would decrease, librate about synchronous rotation, and ultimately damp to the synchronous state if ( − ) ≳ 7.5 2 2 (D12) For a shape similar to that of the current Moon, with ( − ) = 2.28 × 10 −4 ⁄ , eqn. (D11b) implies that synchronous lock could be maintained at the time the Moon encounters evection (i.e., ′ ∼ 7) for an initially low eccentricity ( < 0.095), but that non-synchronous rotation would ensue as became large. Eqn. (D12) implies that the ( − )⁄ of the current Moon would be sufficient to establish synchronous rotation for < 0.0018. Of course, the current ( − )/ value may not have pertained to the early Moon, and so it is prudent to consider both synchronous and non-synchronous cases.
For the case of a non-synchronously rotating Moon without permanent figure torques, eqns. (6.10) and (6.12) in the main text provide the partial derivatives needed to evaluate whether the libration amplitude grows or damps. Analogous expressions can be developed for synchronous lunar rotation maintained by a permanent figure torque, with (̇)⁄ replaced by ( (1 − ) 1/2 − 1.

Appendix E -Additional Zero Libration Evolutions
Here we show additional zero-libration evolutions as considered in Section 5. Note that in these and the other evolutions in the main text we ignore the potential for tidal disruption when the lunar perigee is interior to the Roche limit, which can occur for low cases. Figure A1 displays tracks for = 10 with different starting values of the system angular momentum, 0 ′ , corresponding to varied initial Earth spin rates, 0 ′ , following a lunar forming impact. Changing 0 ′ alters the encounter distance for the resonance as in Figure 4. For lower 0 ′ , the resonance occurs closer to the Earth and the stall in the Moon's orbital expansion occurs at smaller ′ and  . However, all cases eventually converge on the same end state in the limiting case that the Moon remains in resonance throughout its whole evolution (which as we show in Section 6 is unlikely to occur, as much earlier resonance escape is predicted). Accordingly, the higher the starting 0 ′ , the greater the angular momentum decay, Δ ′ = 0 ′ − ′ , and evolutionary tracks for high 0 ′ are reminiscent of those shown in CS12. for other values of A. As increases, the stationary state eccentricity is suppressed by progressively stronger lunar tides. This in turn weakens the tidal torque (due to the larger lunar periapsis), prolonging the evolutionary time scale. Figure A3 displays a synchronous evolution with = 10, 0 = 2 contrasted to the non-synchronous evolution shown in Figure 4 in the main text, shown in grey. Here we have set ′ ′ 3 2 ⁄ = 1, and modified the expressions for tidal changes in and  to include the permanent figure torques as in eqns. (D8) and (D9).
The non-synchronous track acquires higher maximum values for a and but these then decrease somewhat more rapidly than in the synchronous case.