The high mutual inclination and the initial total angular momentum

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The dynamical constraint from the lunar orbital evo-lution

To find a set of impact parameters that can produce a massive and iron-depleted proto-lunar disk, we also need the dynamic constraint of the total angular momen-tum, which includes the rotational angular momentum of the Earth and the Moon, and the orbital angular momentum of the lunar motion. We know that the Moon is almost in a circular cycle revolving around the Earth with an eccentricity of 0.0549 and a slight inclination of 5.14 to the ecliptic plane (Fig. 1.1). From these real mea-surements, we can calculate the current total angular momentum of the Earth-Moon system. However, this value may not be the total angular momentum at the time when the Moon formed. Therefore, we need to find a way to recover its evolving history and obtain the total angular momentum at the time when it completed the accretion and detached from the proto-lunar disk. This is not an easy task since the Moon is continuously affected by the tidal interaction exerted by the Earth and Sun to recess its orbit.

The lunar orbital evolution

We can partially solve this problem by celestial mechanics and an essential contribu-tion has been made by Touma and Wisdom (1994) and Touma and Wisdom (1998). As these calculations are extremely complex due to the employment of generalized coordinates like action-angle variables, only a general frame is outlined. Starting with the derivation of an analytic Hamiltonian for the Earth-Moon-Sun system, Touma and Wisdom (1994) made several assumptions like the lunar orbit is a circle to reduce the problem into a manageable size. Together with averaging over the short orbital time scale, the remaining degrees of freedom in the Hamiltonian is the obliquity of the Earth to the ecliptic plane, the mutual inclination of the lunar orbit to the Earth’s equator, the inclination of lunar orbit to the ecliptic plane, and the precession rate of the lunar orbit. These variables are constantly changing due to the Earth’s oblateness and tidal interactions. Although the physical origin of the tidal torque can easily be understood as the presence of a phase lag due to the mis-match between the orbital and rotational motion, for practical implementations, we need to employ different tidal models to approximate it as a function of the orbital elements of the Earth and the Moon. Fortunately, the detailed evolving history of the lunar orbit shows indifference to a particular model. The most striking finding is that the mutual inclination has a value of 12 when approaching the Earth (Goldre-ich, 1966; Touma and Wisdom, 1994). However, it disagrees with the giant impact theory which predicts the formation of the Moon is near to the Earth’s equator and thus the initial value is close to zero.

The high mutual inclination and the initial total angular mo-mentum

The high mutual inclination problem has been re-investigated by Touma and Wis-dom (1998). With direct numerical simulations, they discover two new phenomena called evection and eviction resonance that have a dramatic impact on the dynamic evolution of the Earth-Moon system. Resonance is a phenomenon of describing increased oscillation amplitude when the frequency of a periodically applied force matches the natural frequency of the system. We can exemplify this interesting phe-nomenon by the vibration of a harmonic oscillator. If the driving frequency is equal to the natural frequency of the oscillator, the resonance is achieved and the ampli-tude of the oscillation increases dramatically. This concept can be extended into the planetary science where resonance means a commensurability amongst the frequen-cies of precession or orbital motions between two planets or stars. If it happens, a periodic gravitational influence will excite the orbital eccentricity or inclination to high values just like the vibration amplitude of an oscillator. The continuous action from the Earth’s tidal torque will make the lunar orbital expand and the precession period of the perigee increase (see Figure 13 in Touma and Wisdom, 1998). When this period is close to the orbital period (about one year) of the Earth, the evection resonance happens, and pumps up the eccentricity of the lunar orbit until the tides induced by the Moon and by the Earth cancel out so that the Moon stops moving outward. Then the lunar tide becomes slightly stronger than the tide in the Earth causing the contraction of the lunar orbit. During this stage, the spin rate of the Earth declines in a steady manner until escaping from the evection resonance. Af-ter that, the Moon encounters a mixed evection-inclination resonance twice which excites the mutual inclination to 12 .
Touma and Wisdom (1998) found the Moon escapes the evection resonance very fast and only a tiny amount of angular momentum is removed from the Earth-Moon system. However, Ćuk and Stewart (2012) suggest the contraction period is prolonged causing a significant de-spinning of the Earth and efficient removal of the angular momentum to the heliocentric orbit of the Earth. They suggest the initial total angular momentum at the time of the giant impact can be twice as large as the current value. By employing a different tidal model, Wisdom and Tian (2015) and Ward et al. (2020) found too much or too little angular momentum was removed from the Earth-Moon system if the initial angular momentum is high, which is inconsistent with the present Earth-Moon system. Different results obtained from applying different tidal models suggest more work is needed to investigate whether the evection resonance is a viable mechanism to remove the excess angular momentum. Ćuk et al. (2016) proposed a new origin scenario for the Moon. In this model, the Moon is accreted from the disk generated by a high-angular momentum impact with an initial high-obliquity (70 ) proto-Earth.
Figure 1.1: The dynamic evolution of the Moon. (a) The lunar orbital elements at present. The inclination of the lunar orbit to the ecliptic plane is 5 . The mutual inclination of the Moon’s orbit to the Earth’s equator varies from 18 to 28 . (b) The evolution of the mutual inclination as a function of radius. The mutual inclination must be at least 12 when the Moon forms. However, the giant impact theory predicts the moon formed at the Earth’s equator predicting an initial value of 0 . This discrepancy can be reconciled with the Moon-Sun resonance (Touma and Wisdom, 1998), the disk-satellite resonance(Ward and Canup, 2000) or the Laplace plane transition (Ćuk et al., 2016), which can excite the mutual inclination to a high value (see the black dashed line in b). This figure is reproduced with permission from Touma and Wisdom (1994). (c) Ćuk and Stewart (2012) have shown the Earth’s rotational rate slows down due to the Moon-Sun resonance which can be used to drain away the excess angular momentum. They suggest an initial high angular momentum at the time of the giant impact is dynamically feasible. However, subsequent studies using different tidal models found too much (Wisdom and Tian, 2015) or too little (Ward et al., 2020) angular momentum be removed from the Earth-Moon system, raising the question whether the Moon-Sun resonance is still a viable mechanism. This figure is reproduced with permission from Ćuk and Stewart (2012).

The giant impact theory 

transition will excite the lunar inclination to 30 and reduce the obliquity of the Earth to 20 . This process will also remove the excess angular momentum from the Earth-Moon system and transfer to the heliocentric orbit of the Earth. Once the Moon has passed through the Laplace phase transition, it will undergo the Cassini state transition to achieve the spin-orbit resonance causing a large lunar obliquity (over 30 ). The high-obliquity tides in the Moon, in return, will strongly damp the lunar inclination to reach the present value of 5 . However, Tian and Wisdom (2020) found the vertical component of the total angular momentum in the Earth-Moon system, which is perpendicular to the Earth’s orbital plane, is almost conserved. They suggest an initial high-obliquity (70 ) Earth with a high angular momentum proposed in Ćuk et al. (2016) would result in too large vertical angular momentum that is inconsistent with the present observations.

Summary

The total angular momentum at the time the Moon formed is an essential part of constraining the impact conditions. If there is no mechanism to remove or add an-gular momentum to the Earth-Moon system, its initial angular momentum is the same as the present Earth-Moon system. Ćuk and Stewart (2012) proposed the Moon-Sun resonance would remove a large amount of angular momentum from the Earth-Moon system. Therefore, the initial angular momentum could be up to two times higher than the present observation. However, Wisdom and Tian (2015) and Ward et al. (2020) found the resonance appears to remove too much to too little angular momentum if the initial angular momentum is high, causing inconsistency with the present Earth-Moon system. Ćuk et al. (2016) suggest that the instability associated with the Laplace plane transition would remove considerable angular mo-mentum from the Earth-Moon system. However, Tian and Wisdom (2020) suggest an initial high obliquity Earth with a high angular momentum proposed in Ćuk et al. (2016) cannot produce the present Earth-Moon system.

The giant impact theory

Material injection mechanism

Before we are going to discuss in detail the impact process and its geochemical im-plications, a concise introduction of the material injection mechanism is given. The energetic collision between the impactor and the proto-Earth creates powerful shock waves, which will compress constitutive materials very rapidly to a high pressure and temperature condition. When the shock wave reaches the free space assumed to be a vacuum, to sustain the zero pressure interface a rarefaction wave or a relief wave must be produced. Then it reflects from the free space and travels into the compressed materials to make them expand (Forbes, 2013).
Figure 1.2: In the canonical model (a), the graze collision of a Mars-sized projectile with proto-Earth at a velocity of 10 km/s can generate a disk (b) and leave the total angular momentum of post-impact structure close to the present Earth-Moon system. The disk material is made of silicates liquids with 20 wt% vapor and primarily derived from the impactor, from which the Moon is accreted. Thus the Moon should have a distinct isotope composition compared to the Earth (Canup and Asphaug, 2001). Canup and Esposito (1996) suggest a large amount of mass must be injected directly beyond the Roche limit to form a single lunar-mass satellite. In (b), all materials outside of the Roche limit are represented by a single Moon as the accretion process is very fast on a timescale of years. Due to the Jeans instability the high density magma disk inside the Roche limit will clump and then be sheared apart by the tidal force, which results in an effective viscosity and is much larger than the molecular viscosity (Stevenson, 1987). The entropy file in (e) represented by the blue dashed line is typical for the canonical impact, which indicates the entire structure is thermally stratified. In the high-angular momentum impact model (d), a smaller impactor with a high velocity of 30 km/s hitting a rapid spinning proto-Earth can produce a disk with enough material mixing between colliding bodies to account for the measured isotopic ratios (Ćuk and Stewart, 2012). The post-impact structure is highly thermally stratified (red line in (e)) and the silicate transitions smoothly from liquid to the supercritical fluid to vapor (Lock et al., 2018), which was named as synestia. Besides, there is a smooth change in the angular velocity from the corotating inner region to the sub-Keplerian disk-like region (f).

The giant impact theory 

energy and gravitational potential energy. For the high-angular momentum impact model, we need to include the rotational energy from the pre-impact spin to the total energy as well, which will make the injection more easily.
If the total energy of the injected materials is negative, they will follow a Keplerian orbit with the periapse on the Earth if there are no other materials block their pathway to inject. After one orbital period they will fall back and re-impact with the Earth. In contrast, if the injected materials have a positive total energy, they will escape the Earth’s gravitational field. In order to form a proto-lunar disk, we need other mechanisms to lift the periapse of some parts of shocked materials with a negative total energy above the Earth.
One possible mechanism is the gravitational torque (Stevenson, 1987), which pumps the angular momentum into the injected materials to make their orbits lift and avoid to re-impact with the Earth. This mechanism is very similar to the current recession of the lunar orbit due to the tidal torque exerted by the Earth. Another mechanism is the pressure gradient, which becomes important if there is significant vaporization happening during the giant impact. As compression is an irreversible process, the shocked materials will gain entropy in the course to reach the peak pressure and temperature conditions. Then an expansion is followed due to relief waves. As the decompression process happens very fast, convection, viscous dissipation and radiation play a very limited role. We can safely assume this process is isentropic. If the decompressed materials intersect with the liquid-vapor dome, it will turn into the liquid-vapor mixture. The significant volume change from the condensed phase (liquid or solid) to the vapor phase upon vaporization causes an abrupt increase in the particle velocity. Then the outflow materials is subjected to the pressure gradients which is able to lift the periapse of injected materials (see Figure 6 in Stevenson, 1987). Due to the complex impact geometry, how much materials are injected into the proto-lunar disk and which mechanism is dominant can only be determined from hydrodynamic simulations.

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The canonical impact model

As the laboratory-scale experiments are not able to simulate such planetary-scale impacts, our understanding of the giant impact mostly comes from hydrodynamic simulations (Fig. 1.2). In this section, I summarize the outcome for the canonical impact modeling. The high angular momentum impact scenario will be analysed in the next section. For simulation results, we are more interested in what kind of impact parameters will produce the present Earth-Moon system, which includes the total mass (MT) as a sum of the impactor and the proto-Earth, the impact-to-total-mass ratio ( ), impact angle (b) and the total angular momentum (Limp). The simulation details are well beyond my interests.
Benz and Cameron pioneered the application of the SPH method to the giant impact simulations. Interestingly, a series of papers were published in Icarus with the same title ’The origin of the Moon and the Single Impact Hypothesis’ (Benz et al., 1986, 1987, 1989; Cameron, 1997; Cameron and Benz, 1991). Due to the limit on the computational speed, all simulations except for in Cameron (1997) have a low resolution. For instance, the total number of particles in the generated proto-lunar disk is only 30 (Benz et al., 1987). From these simulations, they conclude that 1. the low-mass impactors with the impact-to-total-mass ratio less than 0.12 produces iron-rich disk; 2. the gravitational torque is more important than the pressure gradient for ma-terials emplacement, which still holds even in the present high-resolution hy-drodynamic simulations.
Cameron (1997) first performed the high-resolution simulations with a total number of particles around N 104. The most successful impact to produce the current Earth-Moon system is with =0.3 and MT=0.65ME, where ME is the mass of the Earth. This case is also called the early-Earth impact scenario and the Earth needs to acquire extra 0.35ME by the late accretion. Since the Moon will receive a proportional amount of material as well and there is no mechanism available to filter out iron in these materials, the initial iron-depleted Moon will become iron-rich again causing an inconsistency with the geochemical observations. Therefore, the early-Earth impact scenario is not favorable. Canup et al. (2001) re-examined the simulation results in Cameron (1997) and proposed a scaling law to describe the results of satellite-forming impact simulations. They found the disk mass tends to increase and iron content decreases with increasing b for 0:4 < b < 0:8, which is independent of MT. The maximum yielding of the massive and iron-depleted disk is at b 0:8. Canup and Asphaug (2001) were guided by this trend and revisited the small impactor case with < 0:12. A total of 36 impact simulations were run with = 0:108 0:115; b = 0:70 1:0; Limp = LEM and MT = ME, where LEM is the total angular momentum of the present Earth-Moon system. They found a massive proto-lunar disk can be generated to allow the accretion of a single moon. They also confirm most of the materials in the lunar disk is from the impactor with a mass fraction of 0.6-0.74, indicating that the Moon will inevitably have a different oxygen isotope ratio compared to the Earth’s mantle.

The high-angular momentum impact model

Ćuk and Stewart (2012) have proposed a new giant impact scenario which can produce a Moon being isotopically similar to the Earth. In this model, a small impactor with a mass around 0.026-0.1 ME but with a high velocity (30 km/s) hits a rapidly spinning Earth. It results in a vapor-dominated disk with enough material mixing between colliding bodies to account for the measured isotopic ratios. The resulted disk has a much high angular momentum and more massive than the one generated by the canonical impact model. However, the successful Moon-forming impact leaves the Earth-Moon system with an excess angular momentum. Ćuk and Stewart (2012) propose the Earth-Moon system can lose angular momentum by the orbital resonance between the Sun and Moon, which is still under debate.

The behaviour of iron during giant impacts 

Lock et al. (2018) found the post-impact structure is highly thermally stratified and the silicate transitions smoothly from the liquid phase to the supercritical phase, then to the vapor phase. This special structure was named as synestia. There is also a smooth change in the angular velocity from the corotating inner region to the sub-Keplerian disk-like outer region. The outer part of the disk-like region is likely to be well mixed due to the falling condensates and vertical fluid motion (Lock et al., 2018). However, a whole mixing in the synestia may be difficult. On the one hand, synestia is thermally stratified, meaning the outer part is hot and has a lower density. Lifting denser materials from the inner part to the outer part needs to overcome the gravitational force. On the other hand, the monotonic increase of the specific angular momentum from the inside-out creates a barrier. The exchange of a large amount of materials between the inner part and outer part in the synestia will decrease the angular momentum of the outer part, result in the collapse of the disk-like region and leave too little materials in the disk (Melosh, 2014).

Summary

The giant impact model is the leading theory to explain the formation of the Moon. The classic canonical impact model fails to explain the Moon’s isotopic similarity to the Earth, while the high-angular momentum impact model has problems with the removal of the excess angular momentum to match the present Earth-Moon system.

The behaviour of iron during giant impacts

Both the canonical and high-angular momentum impact models predict the im-pactor’s core merge rapidly into the proto-Earth’s core at a timescale of hours (Kraus et al., 2015). Since the chemical exchange between the sinking iron from the im-pactor and the silicates in the magma ocean requires a much longer time, there is a very limited chemical equilibration between the impactor’s core and the Earth’s silicate mantle. However, the hafnium-tungsten isotope studies (Kruijer et al., 2015; Touboul et al., 2015) have suggested a substantial level of metal-silicate equilibra-tion is needed to remove siderophile elements almost entirely from the rocky mantle to explain the small excess of 182W of the Earth relative to the Moon. There are two possible mechanisms to enhance equilibration. The first one is to mechanically break the impactor’s core into small pieces since the smaller metal pieces would have longer falling time and shorter chemical equilibration time. Dahl and Steven-son (2010) show only iron fragments less than 10 km in diameter would equilibrate with silicates. Kendall and Melosh (2016) have futher revised this value by perform-ing hydrodynamic simulations and found iron blobs with a radius of 100 km is in full equilibrium with the magma ocean. It should be noted that all Moon-forming impact simulations do not include the strength of materials, causing the overesti-mation of the size of fragmented iron core (Barr, 2016). The second mechanism is vaporisation. After cooling down, the vaporised materials would condensate into a distribution of small droplets on a centimetre level, and thus enhance the metal-silicate equilibration. Kraus et al. (2015) developed an experimental technique to determine the shock pressure required for vaporization of iron along the principal Hugoniot line. They found the starting vaporization pressure is around 415 GPa compared to 817 GPa from the previous theoretical estimate using ANEOS (Pier-azzo et al., 1997). Therefore, previous hydrodynamic simulations with ANEOS may underestimate the production of iron vapor.
Before the giant impact, the proto-Earth’s core may grow with stable compositional stratification because higher abundances of light elements would be incorporated into the liquid metal as a result of the increasing metal-silicate equilibration pressure and temperature during accretion (Jacobson et al., 2017). The stable stratification would inhibit the outer-core convection and prevent from generating a geodynamo. Jacobson et al. (2017) proposed that an energetic giant impact may pump enough energy to homogenise the core as long as at least 4% of the total energy is deposited in the core. However, it remains as an open question on the exact amount of energy distributed into the Earth’s core during the giant impact, which requires significant numerical simulations to clarify.

The role of equations of state

During the giant impact, the colliding interface and silicates in the disk can be heated up to 104 K and 7000-8000 K, respectively, even in the canonical impact model (Canup, 2004a). More surprisingly, the core of the impactor can reach as high as 45000 K (Canup, 2004a). It remains unclear whether these reported temperatures are physically reasonable and not caused by the equations of state of iron and silicates used in these simulations. For any giant impact simulation, the equations of state (EOS) are needed to describe the thermodynamic properties over a wide range of temperature, pressures and density. It will inevitably affect the energy distribution (Nakajima and Stevenson, 2015) and alter the after-impact dynamic evolution of the Earth such as the core-mantle equilibration and homogenising the compositional strafication of the Earth’s core.

Table of contents :

1.1 Introduction
1.2 The possible origin scenarios for the Moon
1.3 The geochemical constraints
1.4 The dynamical constraint from the lunar orbital evolution
1.4.1 The lunar orbital evolution
1.4.2 The high mutual inclination and the initial total angular momentum
1.4.3 Summary
1.5 The giant impact theory
1.5.1 Material injection mechanism
1.5.2 The canonical impact model
1.5.3 The high-angular momentum impact model
1.5.4 Summary
1.6 The behaviour of iron during giant impacts
1.7 The role of equations of state
1.8 The phase diagram of iron
1.9 Proposed research
2.1 Introduction
2.2 Deriving classical molecular dynamics
2.2.1 The large band gap system
2.2.2 The small band gap and metallic system
2.2.3 Summary
2.3 Finite-temperature density functional theory
2.3.1 Finite-temperature canonical-ensemble theory
2.3.2 Kohn-Sham formulation
2.4 Ab initio spin dynamics
2.4.1 Spin-polarized DFT at zero temperature
2.4.2 Spin dynamics at high temperature
2.4.3 Approximate paramagnetism by either non-magnetism or ferromagnetism?
3.1 Introduction
3.2 Simulation details
3.2.1 Ab initio molecular dynamics
3.2.2 Construction of the spinodal line
3.2.3 Structural analysis
3.2.4 The mean-square displacements
3.2.5 Velocity autocorrelation function
3.2.6 Entropy calculations
3.2.7 Viscosity
3.2.8 Electrical and thermal conductivity
3.3 Results and discussion
3.3.1 The critical point
3.3.2 Static structure
3.3.3 Speciation
3.3.4 Velocity autocorrelation function
3.3.5 Diffusion
3.3.6 Viscosity
3.3.7 Electrical and thermal conductivity
3.3.8 Hugoniot lines
3.3.9 Vaporization of small planetesimals
3.3.10 Vaporization during giant impacts
3.4 Conclusions

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