Giant Impact Simulations Research Articles

Accelerating Giant-impact Simulations with Machine Learning

Constraining planet-formation models based on the observed exoplanet population requires generating large samples of synthetic planetary systems, which can be computationally prohibitive. A significant bottleneck is simulating the giant-impact phase, during which planetary embryos evolve gravitationally and combine to form planets, which may themselves experience later collisions. To accelerate giant-impact simulations, we present a machine learning (ML) approach to predicting collisional outcomes in multiplanet systems. Trained on more than 500,000 N-body simulations of three-planet systems, we develop an ML model that can accurately predict which two planets will experience a collision, along with the state of the postcollision planets, from a short integration of the system’s initial conditions. Our model greatly improves on non-ML baselines that rely on metrics from dynamics theory, which struggle to accurately predict which pair of planets will experience a collision. By combining with a model for predicting long-term stability, we create an ML-based giant-impact emulator, which can predict the outcomes of giant-impact simulations with reasonable accuracy and a speedup of up to 4 orders of magnitude. We expect our model to enable analyses that would not otherwise be computationally feasible. As such, we release our training code, along with an easy-to-use user interface for our collision-outcome model and giant-impact emulator (https://github.com/dtamayo/spock).

Formation of super-Mercuries via giant impacts

ABSTRACT During the final stage of planetary formation, different formation pathways of planetary embryos could significantly influence the observed variations in planetary densities. Of the approximately 5000 exoplanets identified to date, a notable subset exhibits core fractions reminiscent of Mercury, potentially a consequence of high-velocity giant impacts. In order to better understand the influence of such collisions on planetary formation and compositional evolution, we conducted an extensive set of smoothed particle hydrodynamics giant impact simulations between two-layered rocky bodies. These simulations spanned a broad range of impact velocities from 1 to 11 times the mutual escape velocity. We derived novel scaling laws that estimate the mass and core mass fraction of the largest post-collision remnants. Our findings indicate that the extent of core vaporization markedly influences mantle stripping efficiency at low impact angles. We delineate the distinct roles played by two mechanisms – kinetic momentum transfer and vaporization-induced ejection – in mantle stripping. Our research suggests that collisional outcomes for multilayered planets are more complex than those for undifferentiated planetesimal impacts. Thus, a single universal law may not encompass all collision processes. We found a significant decrease in the mantle stripping efficiency as the impact angle increases. To form a 5 M⊕ super-Mercury at 45°, an impact velocity over 200 km s−1 is required. This poses a challenge to the formation of super-Mercuries through a single giant impact, implying that their formation would favour either relatively low-angle single impacts or multiple collisions.

The Influence of Equation of State on the Giant Impact Simulations

AbstractWe explore the role of various equations of state (EoS) in controlling the composition of the Moon formed by a giant impact (GI) using a density‐independent SPH code. A limitation in our previous model Hosono et al. (2019), https://doi.org/10.1038/s41561-019-0354-2 is improved by replacing the EoS of the solid from Tillotson EoS to M‐ANEOS, and we also explored two recently proposed EoSs by Stewart et al. (2020), https://doi.org/10.1063/12.0000946 and Wissing and Hobbs (2020a), https://doi.org/10.1051/0004-6361/201935814; Wissing and Hobbs (2020b), https://doi.org/10.1051/0004-6361/201936227. The goal is to investigate to what extent we can explain the observed composition of the Moon including the similarity in the isotopic composition and the dissimilarity in the FeO/(FeO + MgO) ratio as compared to that of Earth by the different types of EoS assuming the conventional collision conditions. We found that changing the EoS for solids from Tillotson to M‐ANEOS EoS resolves the issues of latent heat, but its effect on the composition of the disk is small compared to the influence of the hard‐sphere EoS of magma ocean in controlling the composition of the disk. Similarly, two recently proposed EoSs have small effects on the composition of the disk in comparison to the model where the hard‐sphere EoS is used for preexisting magma ocean. We attribute this difference to a fundamental difference in thermodynamic behavior of silicate melts captured by the hard‐sphere EoS and by newly proposed EoSs; in the hard‐sphere model of silicate melts, configurational entropy dominates in free energy, whereas in the newly proposed model, entropy is dominated by vibrational entropy similar to entropy of solids.

Dealing with density discontinuities in planetary SPH simulations

ABSTRACT Density discontinuities cannot be precisely modelled in standard formulations of smoothed particles hydrodynamics (SPH) because the density field is defined smoothly as a kernel-weighted sum of neighbouring particle masses. This is a problem when performing simulations of giant impacts between proto-planets, for example, because planets typically do have density discontinuities both at their surfaces and at any internal boundaries between different materials. The inappropriate densities in these regions create artificial forces that effectively suppress mixing between particles of different material and, as a consequence, this problem introduces a key unknown systematic error into studies that rely on SPH simulations. In this work, we present a novel, computationally cheap method that deals simultaneously with both of these types of density discontinuity in SPH simulations. We perform standard hydrodynamical tests and several example giant impact simulations, and compare the results with standard SPH. In a simulated Moon-forming impact using 107 particles, the improved treatment at boundaries affects at least 30${{\ \rm per\ cent}}$ of the particles at some point during the simulation.

The energy budget and figure of Earth during recovery from the Moon-forming giant impact

Quantifying the energy budget of Earth in the first few million years following the Moon-forming giant impact is vital to understanding Earth's initial thermal state and the dynamics of lunar tidal evolution. After the impact, the body was substantially vaporized and rotating rapidly, very different from the planet we know today. The subsequent evolution of Earth's energy budget, as the body cooled and angular momentum was transferred during lunar tidal recession, has not been accurately calculated with all relevant energy components included. Here, we use giant impact simulations and planetary structure models to calculate the energy budget at stages in Earth's evolution. We show that the figure and internal structure of Earth changed substantially during its post-impact evolution and that changes in kinetic, potential, and internal energy were all significant. These changes have important implications for the dynamics of tidal recession and the thermal structure of early Earth.

Enhanced Mixing in Giant Impact Simulations with a New Lagrangian Method

Abstract Giant impacts (GIs) are common in the late stage of planet formation. The Smoothed Particle Hydrodynamics (SPH) method is widely used for simulating the outcome of such violent collisions, one prominent example being the formation of the Moon. However, a decade of numerical studies in various areas of computational astrophysics has shown that the standard formulation of SPH suffers from several shortcomings such as artificial surface tension and its tendency to promptly damp turbulent motions on scales much larger than the physical dissipation scale, both resulting in the suppression of mixing. In order to estimate how severe these limitations are when modeling GIs we carried out a comparison of simulations with identical initial conditions performed with the standard SPH as well as with the novel Lagrangian Meshless Finite Mass (MFM) method using the multimethod code, GIZMO. We confirm the lack of mixing between the impactor and target when SPH is employed, while MFM is capable of driving vigorous subsonic turbulence and leads to significant mixing between the two bodies. Modern SPH variants with artificial conductivity, a different formulation of the hydro force or reduced artificial viscosity, do not improve mixing as significantly. Angular momentum is conserved similarly well in both methods, but MFM does not suffer from spurious transport induced by artificial viscosity, resulting in a slightly higher angular momentum of the protolunar disk. Furthermore, SPH initial conditions unphysically smooth the core-mantle boundary, which is easily avoided in MFM.

Unconvergence of very-large-scale giant impact simulations

Abstract The giant impact (GI) hypothesis is one of the most important hypotheses both in planetary science and in geoscience, since it is related to the origin of the Moon and also the initial condition of the Earth. A number of numerical simulations have been done using the smoothed particle hydrodynamics (SPH) method. However, GI hypothesis is currently in a crisis. The “canonical” GI scenario failed to explain the identical isotope ratio between the Earth and the Moon. On the other hand, little has been known about the reliability of the result of GI simulations. In this paper, we discuss the effect of the resolution on the results of the GI simulations by varying the number of particles from 3 × 103 to 108. We found that the results does not converge, but show oscillatory behaviour. We discuss the origin of this oscillatory behaviour.

Numerical aspects of giant impact simulations

In this paper, we present solutions to three short comings of smoothed particles hydrodynamics (SPH) encountered in previous work when applying it to giant impacts. First we introduce a novel method to obtain accurate SPH representations of a planet's equilibrium initial conditions based on equal area tessellations of the sphere. This allows one to imprint an arbitrary density and internal energy profile with very low noise which substantially reduces computation because these models require no relaxation prior to use. As a consequence one can significantly increase the resolution and more flexibly change the initial bodies to explore larger parts of the impact parameter space in simulations. The second issue addressed is the proper treatment of the matter/vacuum boundary at a planet's surface with a modified SPH density estimator that properly calculates the density stabilizing the models and avoiding an artificially low-density atmosphere prior to impact. Further we present a novel SPH scheme that simultaneously conserves both energy and entropy for an arbitrary equation of state. This prevents loss of entropy during the simulation and further assures that the material does not evolve into unphysical states. Application of these modifications to impact simulations for different resolutions up to 6.4 × 106 particles show a general agreement with prior result. However, we observe resolution-dependent differences in the evolution and composition of post-collision ejecta. This strongly suggests that the use of more sophisticated equations of state also demands a large number of particles in such simulations.

The giant impact simulations with density independent smoothed particle hydrodynamics

At present, the giant impact (GI) is the most widely accepted model for the origin of the Moon. Most of the numerical simulations of GI have been carried out with the smoothed particle hydrodynamics (SPH) method. Recently, however, it has been pointed out that standard formulation of SPH (SSPH) has difficulties in the treatment of a contact discontinuity such as a core–mantle boundary and a free surface such as a planetary surface. This difficulty comes from the assumption of differentiability of density in SSPH. We have developed an alternative formulation of SPH, density independent SPH (DISPH), which is based on differentiability of pressure instead of density to solve the problem of a contact discontinuity. In this paper, we report the results of the GI simulations with DISPH and compare them with those obtained with SSPH. We found that the disk properties, such as mass and angular momentum produced by DISPH is different from that of SSPH. In general, the disks formed by DISPH are more compact: while formation of a smaller mass moon for low-oblique impacts is expected with DISPH, inhibition of ejection would promote formation of a larger mass moon for high-oblique impacts. Since only the improvement of core–mantle boundary significantly affects the properties of circumplanetary disks generated by GI and DISPH has not been significantly improved from SSPH for a free surface, we should be very careful when some conclusions are drawn from the numerical simulations for GI. And it is necessary to develop the numerical hydrodynamical scheme for GI that can properly treat the free surface as well as the contact discontinuity.

GIANT IMPACT: AN EFFICIENT MECHANISM FOR THE DEVOLATILIZATION OF SUPER-EARTHS

Mini-Neptunes and volatile-poor super-Earths coexist on adjacent orbits in proximity to host stars such as Kepler-36 and Kepler-11. Several post-formation processes have been proposed for explaining the origin of the compositional diversity: the mass loss via stellar XUV irradiation, degassing of accreted material, and in-situ accumulation of the disk gas. Close-in planets are also likely to experience giant impacts during the advanced stage of planet formation. This study examines the possibility of transforming volatile-rich super-Earths / mini-Neptunes into volatile-depleted super-Earths through giant impacts. We present the results of three-dimensional giant impact simulations in the accretionary and disruptive regimes. Target planets are modeled with a three-layered structure composed of an iron core, silicate mantle and hydrogen/helium envelope. In the disruptive case, the giant impact can remove most of the H/He atmosphere immediately and homogenize the refractory material in the planetary interior. In the accretionary case, the planet can retain more than half of the gaseous envelope, while a compositional gradient suppresses efficient heat transfer as its interior undergoes double-diffusive convection. After the giant impact, a hot and inflated planet cools and contracts slowly. The extended atmosphere enhances the mass loss via both a Parker wind induced by thermal pressure and hydrodynamic escape driven by the stellar XUV irradiation. As a result, the entire gaseous envelope is expected to be lost due to the combination of those processes in both cases. We propose that Kepler-36b may have been significantly devolatilized by giant impacts, while a substantial fraction of Kepler-36c's atmosphere may remain intact. Furthermore, the stochastic nature of giant impacts may account for the large dispersion in the mass--radius relationship of close-in super-Earths and mini-Neptunes.

Melting and mixing states of the Earth's mantle after the Moon-forming impact

The Earth's Moon is thought to have formed by an impact between the Earth and an impactor around 4.5 billion years ago. This impact could have been so energetic that it could have mixed and homogenized the Earth's mantle. However, this view appears to be inconsistent with geochemical studies that suggest that the Earth's mantle was not mixed by the impact. Another outcome of the impact is that this energetic impact melted the whole mantle, but the extent of mantle melting is not well understood even though it must have had a significant effect on the subsequent evolution of the Earth's interior and atmosphere. To understand the initial state of the Earth's mantle, we perform giant impact simulations using smoothed particle hydrodynamics (SPH) for three different models: (a) standard: a Mars-sized impactor hits the proto-Earth, (b) fast-spinning Earth: a small impactor hits a rapidly rotating proto-Earth, and (c) sub-Earths: two half Earth-sized planets collide. We use two types of equations of state (MgSiO3 liquid and forsterite) to describe the Earth's mantle. We find that the mantle remains unmixed in (a), but it may be mixed in (b) and (c). The extent of mixing is most extensive in (c). Therefore, (a) is most consistent and (c) may be least consistent with the preservation of the mantle heterogeneity, while (b) may fall between. We determine that the Earth's mantle becomes mostly molten by the impact in all of the models. The choice of the equation of state does not affect these outcomes. Additionally, our results indicate that entropy gains of the mantle materials by a giant impact cannot be predicted well by the Rankine–Hugoniot equations. Moreover, we show that the mantle can remain unmixed on a Moon-forming timescale if it does not become mixed by the impact.

Joint inversion of seismic and gravity data for lunar composition and thermal state

SUMMARY We invert the Apollo lunar seismic data set, together with lunar mass and moment of inertia, directly for the chemical composition and thermal state of the Moon. The lunar mantle and crust are modelled in the chemical system CaO‐FeO‐MgO‐Al2O3‐SiO2. The stable minerals, their seismic properties, and the bulk density are computed by Gibbs free energy minimization. Voigt‐Reuss‐Hill averaging is then used to compute seismic-wave velocity profiles, from which traveltimes are estimated, while mass and moment of inertia are obtained by integration of the density profile. Given this scheme, the data are jointly inverted using a Markov chain Monte Carlo algorithm, from which a range of compositions and temperatures fitting data within uncertainties are obtained. The analysis constrains the range of compositions, thermal states, mineralogy and physical structure of the lunar interior that are consistent with data. Additionally, the analysis provides estimates for the size and density of the lunar core. The inferred lunar compositions have lower Mg#s (∼83) than the Earth’s mantle (∼89), suggesting that the Moon was derived from material other than the Earth’s mantle. This supports giant impact simulations of lunar origin that show that more than 80 per cent of the material making up the Moon is derived from the impactor.

Dynamics of Lunar Formation

▪ Abstract The giant impact theory is the leading hypothesis for the origin of the Moon. This review focuses on dynamical aspects of an impact-induced lunar formation, in particular those areas that have advanced considerably in the past decade, including (a) late-stage terrestrial accretion, (b) giant impact simulations, (c) protolunar disk evolution and lunar accretion, and (d) the origin of the initial lunar inclination. In all, recent developments now provide a reasonably consistent dynamical account of the origin of the Moon through a late giant impact with Earth, and suggest that the impact-generation of satellites is likely to be a common process in late-stage solid planet formation.

Co-accretion of the Earth–Moon system after the giant impact: reduction of the total angular momentum by lunar impact ejecta

Though the Moon is considered to have been formed by the so-called giant impact, the mass of the Earth immediately after the impact is still controversial. If the Moon was formed during the Earth's accretion, a subsequent accretion of residual heliocentric planetesimals onto the protoearth and the protomoon must have occurred. In this co-accretion stage, a significant amount of lunar-impact-ejecta would be ejected to circumterrestrial orbits, since the mean impact velocity of the planetesimals with the protomoon is much larger than the escape velocity of the protomoon. Orbital calculations of test particles ejected from the protomoon, whose semimajor axis is smaller than that of the present Moon, reveal that most of the particles escaping from the protomoon also escape from the Hill sphere of the protoearth and reduce the planetocentric angular momentum of the primordial Earth–Moon system. Using the results of the ejecta simulations, we investigate the evolution of the mass ratio and the total angular momentum (Earth's spin angular momentum + Moon's orbital angular momentum) of the Earth–Moon system during the co-accretion. We find that the mass of the protomoon is almost constant or rather decreases and the total angular momentum decreases significantly, if the random velocity of planetesimals is as large as the escape velocity of the protoearth. On the other hand, if the random velocity is the half of the escape velocity of the protoearth, the mass ratio is kept to be almost as large as the present value and the decrease of the total angular momentum is not so significant. Comparing with the results of giant impact simulations, we find that the mass of the protoearth immediately after the Moon-forming impact was 0.7–0.8 times the present value if the impactor-to-target mass ratio was 3:7, whereas the giant impact occurred almost in the end of the Earth's accretion if the impactor-to-target mass ratio was 1:9.

Terrestrial accretion rates and the origin of the Moon

The former decay of 182Hf to 182W ( T 1/2=9 Myr) has resulted in variations in W isotope composition that reflect early solar system time-integrated Hf/W ratios. The bulk silicate Earth (BSE) has non-chondritic Hf/W because of core formation, yet has a chondritic W isotopic composition. This is inconsistent with models that involve the completion of terrestrial accretion and core formation within the first 10 Myr of solar system history, such as early heterogeneous accretion of silicate and metal from a fractionated partially condensed nebula. Protracted accretion of material that has, on average, chondritic compositions with respect to Hf–W is more in accord with the chondritic W isotopic composition of the BSE. Most early-formed low Hf/W metal and high Hf/W silicate that was added to the Earth during accretion must have largely equilibrated isotopically with the growing BSE, otherwise its W isotopic composition would not be chondritic. Within this framework, both the W and Pb isotope data for the Earth can be modeled with homogeneous accretion and continuous core formation at exponentially decreasing rates. The accretionary mean life would need to be between 25 and 40 Myr assuming the Hf/W ratio of the BSE is ∼15 and 238U/ 204Pb of the total Earth is ∼0.7. Such models do not emulate late stage major impacts such as probably formed the Moon. There is now considerable evidence that the Moon formed no earlier than ∼50 Myr after the start of the solar system. A collision at or before 50 Myr between a near Earth-sized proto-Earth and a Mars-sized impactor, here named Theia, would not yield chondritic W for the present day BSE, unless there was also significant subsequent accretion. The recent suggestion that the proto-Earth to Theia mass ratio was more like 7:3 and that the proto-Earth was <65% formed before the collision is easily reconciled with W isotope data. The Pb isotope data for the average BSE can be modeled with the same accretion parameters provided that the proto-Earth was >50% formed by the time of the impact, Theia adding at least a further 20% of the Earth’s mass. Less than 25% of additional material would be accreted after the collision. If the proposed 238U/ 204Pb of 0.7 for the total Earth is grossly incorrect or if smoothly decreasing rates of accretion before and after the impact are inadequate approximations, these figures would need to be changed. Isotopic equilibration with the W in the silicate portion of the Earth is harder to envisage for a very large impactor with a distinct metal core that immediately coalesces with the Earth’s core, as in Giant Impact simulations. However, the silicate portion of the Earth would not need to isotopically equilibrate with the total mass of Theia during the Moon-forming collision if Theia, like the proto-Earth, had been accreting relatively slowly. The chondritic initial W isotopic composition of the Moon provides supporting evidence that Theia grew slowly. A possible explanation for this slow growth of Theia and the proto-Earth is that they grew in close proximity, competing for, and perturbing the trajectories of accreting material. Close proximity would also increase the probability of a collision between the proto-Earth and Theia and explain why the Earth and Moon share the same O and Cr isotopic compositions.