Giant Impact Scenarios: Earth-Moon System Formation
3. GIANT IMPACT SCENARIOS
Impact origin studies aim to identify the collisional scenarios that account for the properties of the Earth-Moon system.
The goal includes finding plausible scenarios based on our understanding of Earth’s formation.
It has proven challenging, leading to various impact models.
3.1 General Constraints and Methods
3.1.1 Constraints
The current angular momentum of the Earth-Moon system is defined as:
L_{EM} = 3.5 imes 10^{41} ext{ g cm}^2 ext{ s}^{-1}
Tidal interactions between the Earth and Moon lead to:
Slowing of Earth's spin.
Expansion of the Moon's orbit, while conserving angular momentum.
The initial terrestrial day length was approximately 5 hours after a newly-formed Moon was orbiting just outside the Roche limit, assuming the primordial system's angular momentum was comparable to LEM.
Oblique collisions are seen as a natural source of rapid early spin for Earth.
Angular momentum delivered by an impactor of mass Mi = MT (where M_T represents total colliding mass) is expressed as follows (Canup 2004b):
\begin{align} Li = b MT \left(\frac{5}{3}\right)^{\frac{f(\gamma)}{\sqrt{2G}} \left(\frac{4}{3}\right)^{\frac{1}{3}} \left(\frac{vi}{v{esc}}\right)^{1.3} L_{EM} b\right)
\end{align}
Definitions of symbols used in the equation:
b = \sin \theta (scaled impact parameter), where \theta is the impact angle (0 degrees for head-on impacts).
vi and v{esc} denote the impact and mutual escape velocities, respectively.
Average density of the impactor and target is assumed equal.
Function f(\gamma) is defined as:
f(\gamma) = \gamma (1 - \gamma) \sqrt{\gamma \frac{1}{3}} \left(1 - \gamma \frac{1}{3}\right).
It's conjectured that the initial angular momentum of the Earth-Moon system after Moon formation needed to be close to LEM, given subsequent minor alterations in angular momentum due to:
Mass escape during lunar accretion,
Direct solar tides,
Effects from smaller later impacts,
Short-term capture into the evection resonance.
Identified dynamical mechanisms may remove substantial angular momentum from the Earth-Moon system after the formation of the Moon, indicating initial system angular momentum could have been as high as 2 to 3 LEM (Ćuk & Stewart 2012; Wisdom and Tian 2015; Ćuk et al. 2016; Tian et al. 2017; Rufu and Canup 2020).
A successful single impact scenario must result in a disk with adequate mass and angular momentum to later form a lunar-mass Moon (mass M_L = 0.0123 M), orbiting outside of the Roche limit, which is given as:
a_R = 2.9 R (for lunar-density materials).
From conservation of mass and angular momentum, the mass of the moon (MM) that can be formed at a distance (aM = Xa aR) is approximated:
\frac{MM}{MD} \approx C1 \left(\frac{LD}{MD\sqrt{G M} aR}\right)^{-C2-C3\left(\frac{M{esc}}{MD}\right)}
where:
M{esc} is the mass that escapes during moon accretion, and constants C1, C2, C3 depend on X_a and specific angular momenta of escaping material.
This equation approximates whether a disk created by an impact can result in a lunar-mass Moon, invalid if MM/MD > 1, corresponding to disks with specific angular momenta too high for final moon formation.
Results from early N-body lunar accretion simulations suggest:
= 1.3 a_R (Ida et al. 1997; Kokubo et al. 2000).
More recent hybrid simulations found:
= 2.15 a_R (Salmon and Canup 2012), indicating less efficient accretion.
An initial disk mass between about 1.3 and 3 lunar masses is required to yield a lunar-mass Moon under typical post-impact conditions.
3.1.2 Methods
Ejecta on circum-planetary orbits arise from:
Vaporization-related pressure gradients.
Gravitational torques from interactions among ejected material or between the ejecta and the post-impact planet’s distorted shape.
Producing disks involves:
3D hydrodynamical modeling, including phase changes and self-gravity.
The most used method is Smoothing Particle Hydrodynamics (SPH):
Objects are represented by numerous particles evolved by
Self-gravity,
Pressure forces,
Shock dissipation (Benz et al. 1989).
Each particle is ascribed a composition (silicate or iron) and a corresponding equation of state.
Limited simulations have also applied grid-based Eulerian codes (Wada et al. 2006; Canup et al. 2013).
3.2 Canonical Impact
An impactor approximately Mars-mass colliding with Earth at low speed (similar to v_{esc}) can produce:
An iron-depleted Moon,
A planet-disk system with angular momentum L_{EM} (Canup and Asphaug 2001; Canup 2004a,b, 2008, 2014).
Impactor mass and velocity are common in terrestrial planet accretion models:
Collision angle needed is between 40 to 50 degrees (Canup 2004, 2008) centered on the average angle (45 degrees) for random impacts, thus around 20% fall in this range.
A central difficulty for canonical impacts is explaining isotopic similarities between Earth and Moon.
Typically, 70-80% of the disk comes from the impactor's mantle, resulting in:
f_T = -70 ext{ to } -90 ext{ percent} (e.g., Fig. 1 in Canup 2014).
This discrepancy yields greater compositional differences than the required |f_T| < 5 ext{%} for Mars-like hypothetical impactors.
Identifying canonical impacts, pre-impact rotation affects disk portion from the impactor modestly:
This percentage varies from 60 to 90% with pre-impact rotation, with fT ≤ -50 ext{%} under the condition that L{EM} remains intact (Canup 2008).
Research by Hosono et al. (2019) indicates:
A canonical impact can yield a disk mainly from Earth's mantle if:
Earth's surface magma ocean existed at the time of impact.
Utilized a modified SPH code.
Their adjusted simulations yielded massive disks (MD ≥ MM), primarily Earth-dominated, averaging only 30% impactor material, or f_T = -17 ext{%}.
In contrast, standard SPH and hard-sphere EOS simulations led to impactor-dominated disks.
A successful canonical Moon-forming impact necessitates either:
An impactor with Earth-like isotopic composition,
Vapor mixing between the impact disk and planet effectively erasing the impactor's compositional signature.
The latter process is termed equilibration and is deeply uncertain (Pahlevan and Stevenson 2007; §4.2).
3.3 Hit-and-Run Impact
Less oblique, higher-velocity impacts produce disks with more target material:
Increased vi results in substantial escaper angular momentum, enabling larger values of Li, thus potentially larger impactors.
Reufer et al. (2012) define a hit-and-run category with impact angles of 30 to 40 degrees, where:
1.2 < \frac{vi}{v{esc}} < 1.4
Lead to massive disks, achieving 40-60% from the impactor, and giving -50 < f_T < -35 (e.g., Fig. 1 in Canup 2014).
While this reduces the planet-disk compositional difference, it remains larger than needed to validate the isotopic similarities, particularly for Mars-like impactors.
The most effective hit-and-run impacts producing sufficient mass to generate a lunar-mass Moon and maintaining angular momentum around 1.3 ext{ to } 1.4 L_{EM} require mechanisms to extract angular momentum afterward (§4.7).
3.4 Fast-Spinning Earth Impact
Recent studies focus on impacts yielding significantly higher angular momentum systems to rationalize isotopic similarities between Earth and Moon (Ćuk and Stewart 2012; Canup 2012; Lock et al. 2018).
Two principal configurations generate post-impact planet-disk systems with roughly similar amounts of material from both bodies, i.e., |f_T| in the range 1 to 10%:
An impact with a proto-Earth spinning with angular momentum far exceeding LEM.
A collision between two half-Earth mass bodies (§3.5).
Both configurations necessitate a subsequent adjustment mechanism to achieve current angular momentum levels (§4.7).
N-body simulations indicate that high-energy impacts likely result in rapid rotation, suggesting many Earth-like planets formed quickly due to giant impacts, excluding fragmentation.
Ćuk and Stewart (2012) scrutinized impacts into rapidly rotating targets with angular momentum between around 2.0 to 3.0 LEM before Moon formation, with the upper limit being near spin stability.
Fast impactors (Mi = 0.026 ext{ to } 0.1M, vi = 1.5 ext{ to } 3 v{esc}) with retrograde angles (0 to -20 degrees) yield disks with |fT| between 4 and 15%, mostly from the target’s mantle.
The fast-spinning Earth impact scenario maintains specific energy one order of magnitude above a canonical impact, with the resultant disk regions largely vapor, contrasting with canonical impacts led by molten materials (leading to different post-impact characteristics and evolutions) (§4.4).
3.5 Half-Earth Impact
If the mass of the Moon-forming impactor is much smaller than M, the final planet derives mostly from the target (near unity)
A disk needing minimal compositional variance from the planet means that F_{P,tar}
ightarrow 1, established by equation (3.3).The challenge extends to ensuring the entirety of silicate compositions in both disk and planet matches, requiring:
\frac{F{P,tar}}{F{D,tar}} \rightarrow 1
For completely symmetrical collisions involving half-Earth mass bodies, the concluding disk and planet both contain equal proportions of their respective contributions (50% target and 50% impactor).
Notably, then:
F{P,tar} = F{D,tar} = 0.5,
\frac{F{P,tar}}{F{D,tar}} = 1, leading to f_T = 0.
Some collisions with impactors exceeding 40% of Earth's mass can yield minimal compositional differences between disk and planet (up to 1 to 10% difference, as suggested by Canup 2012).
Successful scenarios accommodate various impact speeds (1 < \frac{vi}{v{esc}} < 1.6) and impact parameters (0.35 < b < 0.7), regardless of the proto-Earth's pre-impact spin state.
Such impacts are akin to processes responsible for the Pluto-Charon binary (Canup 2005).
A half-Earth impact can also create a larger disk mass than other scenarios (approaching MD ~ 2 to 5M_L), which assists in forming a lunar-sized Moon in case of inefficient accretion (Lock et al. 2018, Fig. 7).
The produced Earth-disk system aligns angular momentum between 1.8 and 2.7 LEM, necessitating subsequent momentum reduction.
3.6 General High-Angular Momentum/High-Energy Impact
Scenarios of fast-spinning Earth and half-Earth take their positions as extreme cases of high-angular momentum impacts yielding near-equivalent silicate compositions in resulting disk-planet systems.
Conversely, a broader array of high angular momentum impacts remains viable if disk-planet compositional differences moderate upon equilibration.
Lock and Stewart (2017) found numerous giant impacts yielding a synestia, a partially vaporized, highly rotating planetary body with internal energy exceeding stationary objects (refer to §4.4).
Such equilibration within a synestia is potentially achievable via multiphase convective and turbulent mixing, especially in high-entropy regions (Lock et al. 2018).
A notable percentage of impacts producing post-impact bodies with angular momentum greater than 1.7 LEM (including fast-spinning Earth and half-Earth impacts) likely result in synestias, while impacts generating lower angular momentum systems might also yield synestias under very high-energy criteria (§4.4).
Many Earth-like planets likely experienced a sequence of synestia-producing impacts during their late accretion phase (Quintana et al. 2016; Lock and Stewart 2017).
3.7 Multiple Impacts
Simulations indicate that Earth endured multiple planetary-scale impacts throughout its final accretion phase (Agnor et al. 1999).
The hypothesis that the Moon formed from cumulative effects of numerous collisions was initially presented by Ringwood (1989) and later refined with contemporary methodologies.
Rufu et al. (2017) propose a multiple impact model where:
The Moon emerges from a series of medium- to large-size collisions involving impactors with mass ranging from 10^{-2} to 10^{-1} M.
Each impact results in a sub-lunar mass satellite migrating outward due to tidal interactions:
It initially migrates faster and slows down as it moves further from the proto-Earth.
A subsequent impact can yield a new inner moonlet, which through tidal expansion may merge with the preceding outer moon.
This chain process facilitates the Moon's assembly from smaller moonlets produced by several impacts.
Analyzing over 10% of the simulation trajectories leads to a final Earth rotating at a suitable rate (Rufu et al. 2017; §5.4).
Comparatively, the Rufu et al. model includes smaller impactors and a wider spectrum of impact angles (head-on to grazing) and velocities (1 - 4 v_{esc}).
Higher proportions of target materials arise from nearly head-on or high-velocity impacts, increasing the chance for compositional matches between Earth and Moon.
Increased impactor and moonlet counts reduce compositional variances between Earth and Moon, expressed as \sqrt{N}, aligning both closer to the mean planetesimal neighborhood.
Monte Carlo simulations estimate sequences of around N ~ 20 to 30 impacts can generate a lunar-sized satellite; substantial fractions of these histories yield isotopically comparable Earth and Moon pairs.
4. POST-IMPACT EVOLUTION
This section examines the post-impact evolution of Earth and its satellite material, emphasizing a multidisciplinary perspective.
The evolution must deliver an Earth-Moon system that meets various dynamical and compositional constraints.
4.1 Mixing During a Moon-Forming Impact and Earth's Mantle Heterogeneity
The relevant geochemical system for early Earth studies involves Hf-W isotopes.
Conventionally, Earth’s present mantle indices are noted as:
^{182}W = 0,
while chondrites reveal ^{182}W = -1.9 (Schoenberg et al. 2002; Kleine et al. 2002; Yin et al. 2002).
Recent findings unveil some Archean Earth samples display positive W anomalies relative to BSE (e.g., ^{182}W = 20 ext{ ppm} by Willbold et al. 2011; Touboul et al. 2012) and modern flood basalts exhibit even greater anomalies (e.g., ^{182}W > 50 ext{ ppm}, Rizo et al. 2016), while others present negative deviations (Mundl et al. 2017).
Such early W discrepancies might correlate with seismic anomalies observed at the core-mantle boundary, like the LLSVPs (large low-shear velocity provinces, Garnero and McNamara 2008).
Besides tungsten, other isotopic systems may also reflect early ancient mantle heterogeneity preservation.
Mukhopadhyay (2012) notes Ne (neon) ratios, like ^{20}Ne/^{22}Ne and ^{129}Xe/^{130}Xe (with 129Xe being the decay product of ^{129}I with half-life t_{1/2} = 16 ext{ Myr}), can suggest preservation linked to their separation since approximately 4.45 Ga.
The similarity of ^{20}Ne/^{22}Ne ratios in fluid inclusions of plume-related formations to solar metrics (Yokochi and Marty 2004) points towards the growing Earth’s mantle capturing nebular gas while the solar nebula persisted, without complete homogenization.
The Moon-forming impact is generally deemed to have transpired roughly 60-70 Myr after the initial solid formation of the solar system (§2.5).
Nonetheless, uncertainties concerning metal-silicate equilibration during Earth's accretion lead studies to suggest a formation window extending broadly from 10 to 175 Myr (Fischer and Nimmo 2018).
If the Moon's formation followed the development of the recognized mantle heterogeneities, it implies that the Moon-forming instance(s) did not entirely homogenize Earth’s mantle.
Canonical and multiple impact models align with theories concerning a non-fully mixed mantle.
Conversely, the significantly high-energy fast-spinning Earth impact might conserve some mantle heterogeneities contrary to the impact point (refer to Figure 3b).
A half-Earth impact is reputedly likely to dynamically mix the mantle, potentially erasing such heterogeneities (Nakajima and Stevenson 2015; Fig. 3c).
Further investigation is essential for a comprehensive understanding of this complex domain.