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Early Evolution of the Earth-Moon System with a Fast-Spinning Icarus 256 (2015) 138–146 Contents lists available at ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus Early evolution of the Earth–Moon system with a fast-spinning Earth Jack Wisdom ⇑, ZhenLiang Tian Massachusett Institute of Technology, Cambridge, MA 02139, USA article info abstract Article history: The isotopic similarity of the Earth and Moon has motivated a recent investigation of the formation of the Received 14 October 2014 Moon with a fast-spinning Earth (Cuk, M., Stewart, S.T., [2012]. Science, doi:10.1126/science.1225542). Revised 21 February 2015 Angular momentum was found to be drained from the system through a resonance between the Moon Accepted 22 February 2015 and Sun. They found a narrow range of parameters that gave results consistent with the current angular Available online 7 March 2015 momentum of the Earth–Moon system. However, a tidal model was used that was described as approx- imating a constant Q tidal model, but it was not a constant Q model. Here we use a conventional constant Keywords: Q tidal model to explore the process. We find that there is still a narrow range of parameters in which Earth angular momentum is withdrawn from the system that corresponds roughly to the range found earlier, Moon Dynamics but the final angular momentum is too low to be consistent with the Earth–Moon system. Exploring a broader range of parameters we find a new phenomenon, not found in the earlier work, that extracts angular momentum from the Earth–Moon system over a broader range of parameters. The final angular momentum is more consistent with the actual angular momentum of the Earth–Moon system. We develop a simple model that exhibits the phenomenon. Ó 2015 Elsevier Inc. All rights reserved. 1. Introduction (Cuk and Stewart, 2012; Canup, 2012). The problem is that such an impact would leave the Earth spinning much more rapidly than An isotopic crisis (Melosh, 2009) has inspired a recent scenario previously assumed and some mechanism of removing the excess for the formation of the Moon (Cuk and Stewart, 2012). Prior angular momentum must be found. research (Canup, 2004, 2008) into the giant impact scenario for One such mechanism was recently proposed by Cuk and the formation of the Moon has assumed that when the Moon Stewart (2012). In this scenario, the Moon is formed by a hard was formed the Earth–Moon system was left with the present impact by a smaller and faster impactor than previously consid- angular momentum. Indeed, this was thought to be a virtue of ered, excavating material that forms the Moon that is largely the model. However, in these simulations the Moon is predomi- derived from Earth’s mantle. The Moon is initially formed with nantly formed from the impactor. Recent isotopic measurements low eccentricity just outside the Roche radius, and evolves out- have shown that the Moon and Earth have essentially identical iso- wards due to the action of tides on the Earth. As it evolves it is cap- topic composition for chromium (Lugmair and Shukolyukov, tured into a resonance between the Moon and the Sun where the 1998), oxygen (Wiechert et al., 2001), tungsten (Touboul et al., period of precession of the lunar pericenter is one year. This reso- 2007), and titanium (Zhang et al., 2012). Thus to explain the iso- nance is called the evection resonance (see Touma and Wisdom, topic similarity of the Earth and Moon one has to appeal to an acci- 1998). After the system is captured into the evection resonance dentally similar impactor, or to some sort of reequilibration of the the eccentricity of the lunar orbit rises dramatically. Due to the isotopic signature after the impact but before the Moon formed eccentricity damping effect of tides raised in the Moon (Pahlevan and Stevenson, 2007). But reequilibration of isotopes (Goldreich, 1963), an equilibrium eccentricity is reached. In cases in the protolunar disk seems less likely because refractory isotopes where the evection resonance is stable, angular momentum is have been found to be similar for the Earth and Moon. The similar- drained from the system. The rotation rate of the Earth slows ity of the Earth and Moon suggests that the Moon forming impact and the semimajor axis of the lunar orbit decreases until the angu- was more severe than previously investigated and that the Moon lar orbital motion near pericenter matches the rotation rate of the formed from material derived directly from the Earth’s mantle Earth, at which point the evection resonance becomes unstable (Touma and Wisdom, 1998) and the system escapes from reso- nance. The angular momentum of the Earth–Moon system is left Corresponding author. ⇑ with a value close to the present angular momentum. E-mail addresses: [email protected] (J. Wisdom), [email protected] (Z. Tian). http://dx.doi.org/10.1016/j.icarus.2015.02.025 0019-1035/Ó 2015 Elsevier Inc. All rights reserved. J. Wisdom, Z. Tian / Icarus 256 (2015) 138–146 139 However, the tidal model used by Cuk and Stewart (2012) was the rate of evolution corresponded to Q 100, where Q is the dis- e ¼ e described as approximating a ‘‘constant Q’’ tidal model, but in fact sipation parameter for the Earth. Before we go on let’s criticize this it had little to do with a conventional constant Q tidal model. In expression. First, it is simply not the case that in any real tidal model this paper we show that when a conventional constant Q tidal the tidal torque is always aligned with the angular momentum vec- model is used we can reproduce the basic phenomenon reported tor. The tidal torque should involve the orbital elements (or the posi- by Cuk and Stewart (2012), but too much angular momentum is tion and velocity) of the Moon, and the tidal torque need not be lost. However, we have found an alternate scenario that can drain aligned with the angular momentum of the Earth. Second, the fact an appropriate amount of angular momentum from the system that the factor T0 is given numerically, rather than being given in over a wide range of parameters. terms of parameters of the system, is indicative of the fact that this expression is not derived from more basic physical principles. Third, the origin of the factor S is not clear; its form seems to be completely 2. Methods arbitrary. Next, Cuk and Stewart state that ‘‘the acceleration of the Moon corresponds to a torque that is equal and opposite in sign.’’ The physical model and numerical methods that we use are Since a torque is derived from a force by a cross product, and infor- similar to those used in Touma and Wisdom (1994, 1998), and also mation is lost in performing a cross product, it is not possible to similar to those used by Cuk and Stewart (2012). One difference uniquely determine the acceleration from the torque. Cuk and between our model and that of Cuk and Stewart is that we studied Stewart took the tidal acceleration to be perpendicular to the radius the rotation of the Moon (for reasons explained below); for Cuk vector to resolve the ambiguity. Again, in conventional tidal models and Stewart the Moon was a point mass. it is not necessarily the case that the tidal acceleration is perpendic- Our numerical model integrates the orbital evolution of the ular to the radius vector. Finally, the acceleration of the Moon due to Earth, Moon, and Sun. We integrate the full rotational dynamics tides raised in the Moon is given by the expression of an axisymmetric Earth and a triaxial Moon, with spin–orbit cou- pling of Earth and Moon. We include tides in the Earth and Moon, A0vr €r ^r; 2 using the constant-Q Darwin–Kaula model. We add intrinsic wob- ¼Àr13=2 ð Þ ble damping following Peale (1977). 3 where A0 2:7 10À in the same units as before, and v r is the The symplectic mapping method of Wisdom and Holman ¼  (1991) forms the framework for our numerical model. In this radial component of the velocity. This has the virtue (and sole pur- method, the Hamiltonian is split into parts that are separately effi- pose) of damping the orbital eccentricity. It is not clear whether the ciently solvable; the complete evolution is obtained by interleaving eccentricity dependence of this tidal acceleration corresponds to the pieces. We think of the splitting in terms of the introduction of any conventional tidal model. The factor A0 is, again, not given in delta functions into the Hamiltonian. The n-body Hamiltonian is terms of physical parameters. split into Kepler Hamiltonians plus the interaction Hamiltonian To illustrate these criticisms, consider the expressions for the using hierarchical Jacobi coordinates (see Sussman and Wisdom, popular constant Dt tide (see Touma and Wisdom, 1994, for a ped- 2001). During the ‘‘drift’’ step we evolve both the Kepler motion agogical introduction). In this tide (sometimes called the Mignard– and the free rigid body motion of the Earth and Moon. During Hut tide, Mignard (1979) and Hut (1981)), the response of the pla- the ‘‘kick’’ step the spin–orbit interactions are evaluated, and the net to the tide raising potential is a coherent tidal bulge that is tides are introduced by a tidal kick. delayed in time by a constant time interval Dt. The force on the We actually developed two versions of our code. In one version Moon due to tides that it raises on the Earth is the rotational configuration of each rigid body was specified by the 3k GM2 R5 rotation matrix that takes the body from its reference orientation 2 m e r2 t r2 F 10 r D 2r r v r x v ; 3 to its actual orientation by an active rotation.
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