Ramon Brasser

Context. After the formation of the Moon, the terrestrial planets were pummelled by impacts from planetesimals left over from terrestrial planet formation. Most lunar crater chronologies are fitted with an exponentially decreasing impact rate with an e-folding time of about 150 Myr. Dynamical simulations consisting of leftover planetesimals and the asteroid belt should reproduce this impact rate. Aims. This work attempts to reproduce the impact rates set by modern crater chronologies using leftover planetesimals from three different dynamical models of terrestrial planet formation. Methods. I ran dynamical simulations for 1 billion years using leftover planetesimals from the grand tack, depleted disc, and implantation models of terrestrial planet formation with the CPU version of the Gravitational ENcounters with GPU Acceleration (GENGA) N-body integrator. I fitted the cumulative impacts on the Earth and Mars as well as the fraction of remaining planetesimals using a function that is a sum of exponentials with different weighing factors and e-folding times. Results. Most fits require three or four terms. The fitted timescales cluster around τ₁ = 10 million years (Myr), τ₂ = 35 Myr, τ₃ = 100 Myr, and τ₄ > 200 Myr. I attribute them to dynamical losses of planetesimals through different mechanisms: high-eccentricity Earth-crossers and the ν₆ secular resonance, Earth-crossers, Mars-crossers, and objects leaking onto Mars-crossing orbits from beyond Mars. I placed a constraint on the initial population using the known Archean terrestrial spherule beds, and I conclude that the Archean impacts were mostly created by leftover planetesimals. The inferred mass in leftover planetesimals at the time of the Moon’s formation was about 0.015 Earth masses. Conclusions. The third time constant, τ₃, is comparable to that of modern crater chronologies. As such, the crater chronologies are indicative of impacts by an ancient population of Mars-crossers. The initial perihelion distribution of the leftovers is a major factor in setting the rate of decline: to reproduce the current crater chronologies, the number of Earth-crossers at the time of the Moon’s formation had to be at most half that of the Mars-crossers. These results together place constraints on dynamical models of terrestrial planet formation.

Context. The theory of planet formation through pebble accretion has gained in popularity over the past decade. Recent studies claim that pebble accretion could potentially explain the mass and orbits of the terrestrial planets in the Solar system, the size and water contents of the planets in the TRAPPIST-1 system, and the formation of super-Earth systems at small orbital radii. However, all these studies start with planetary embryos much larger than those expected from the streaming instability. Aims. We analyse the formation of terrestrial planets around stars with masses ranging from 0.09 to 1.00 M_⊙ through pebble accretion, starting from small planetesimals with radii between 175 and 450 km. Methods. We performed numerical simulations using a modified version of the N-body simulator SyMBA, which includes pebble accretion, type I and II migration, and eccentricity and inclination damping. We analysed two different prescriptions for the pebble accretion rate. Results. We find that Earth-like planets are consistently formed around 0.49, 0.70, and 1.00 M_⊙ stars, irrespective of the pebble accretion model that is used. However, Earth-like planets seldom remain in the habitable zone, for they rapidly migrate to the inner edge of the disc. Furthermore, we find that pebble accretion onto small planetesimals cannot produce Earth-mass planets around 0.09 and 0.20 M_⊙ stars, challenging the proposed narrative of the formation of the TRAPPIST-1 system. Conclusions. Although we have the ability to explain the formation of Earth-mass planets around Sun-like stars, we find a low likelihood of Earth-like planets remaining in the habitable zone. Further research is needed to determine if models with a lower pebble mass flux or with additional migration traps could produce more Solar System-like planetary systems.

Context. The impact and cratering rates onto the regular satellites of the giant planets are subject to great uncertainties. Aims. We aim to compute the impact rates for objects with a diameter D_i > 1 km onto the regular satellites of Jupiter, Saturn, and Uranus using dynamical simulations of the evolution of the outer Solar System coupled with the best estimates of the current population of objects beyond Neptune, and their size-frequency distribution. Methods. We analyse the last 3.5 billion years of evolution of the outer Solar System from our database of simulations and combine this with observational constraints of the population beyond Neptune to compute the flux of objects entering the Centaur region. The initial conditions of these simulations resemble the current population. We obtain an improved estimate of the impact probability of a Centaur with the satellites from enacting simulations of planetesimals flying past the satellites on hyperbolic orbits, which agree with literature precedents. Results. Our impact rate of objects D_i > 1 km with Jupiter is 0.001 yr⁻¹, which is 3–6 times lower than previous estimates. Both our impact probabilities with the satellites scaled to the giant planets and leakage rate of objects from beyond Neptune into the Centaur region are consistent with earlier literature estimates. However, our absolute impact probabilities with the giant planets are lower. We attribute this difference to whether the impact probabilities are computed over the whole age of the Solar System including planet migration, or over a shorter interval closer to the present. Conclusions. Our lower impact rate compared to earlier literature estimates is due to basing our results on the flux of objects coming in from beyond Neptune rather than relying on the current observed impact rate with Jupiter. We stress the importance of clearly stating all parameters and assumptions in future studies to enable meaningful comparisons.

Context. Terrestrial planet formation (TPF) is a difficult problem that has vexed researchers for decades. Numerical models are only partially successful at reproducing the orbital architecture of the inner planets, but have generally not considered the effect of the growth of the giant planets. Cosmochemical experiments suggest that the nucleosynthetic isotopic composition of bodies from beyond Jupiter is different from that of the inner Solar System. This difference could have implications for the composition of the terrestrial planets. Aims. I aim to compute how much material from the formation region of the gas giants ends up being implanted in the inner Solar System due to gas drag from the protoplanetary disc, how this implantation alters the feedstocks of the terrestrial planets, and whether this implantation scenario is consistent with predictions from cosmochemistry. Methods. I dynamically model TPF as the gas giants Jupiter and Saturn are growing using the Graphics Processing Unit (GPU) software Gravitational ENcounters with GPU Acceleration (GENGA). The evolution of the masses, radii, and orbital elements of the gas giants are precomputed and read and interpolated within GENGA. The terrestrial planets are formed by planetesimal accretion from tens of thousands of self-gravitating planetesimals spread between 0.5 au and 8.5 au. The total mass of the inner planetesimal disc and outer disc are typically 2 and 3 Earth masses (M_⊕) respectively, and the composition of the planetesimals changes from non-carbonaceous-like to carbonaceous-like at a prescribed distance, ranging from 2 au to 5 au. Results. Here, I report on the first 5 million years of evolution. At this time approximately 20% of the mass of planetesimals in the Jupiter-Saturn region is implanted in the inner Solar System, which could be more than the cosmochemical models predict; this amount can be reduced by reducing the total mass of the outer planetesimal disc, and the results suggest a mass of 1 M_⊕ could suffice. The mass-weighted fraction of outer Solar System material implanted in the inner Solar System shows a flat or bimodal distribution beyond 0.7 au, with an occasional peak near 0.9 au. The planetesimals that remain in the inner Solar System have a mixed composition, which could have implications for late accretion. Conclusions. The growing gas giants scattered the planetesimals in their vicinity into the inner Solar System, which changed the isotopic composition of the terrestrial planets. The inner planetesimal disc may not have extended much farther than 2 au, otherwise embryos do not grow fast enough to produce Mars analogues. This could mean that the region of the current asteroid belt never contained much mass to begin with. The implantation scenario could also explain the existence of active asteroids in the main belt.

Context. The mass-radius distribution of currently known exoplanets suggests a fascinating variety in terms of their chemical compositions. Still, the interior structures and compositions of rocky exoplanets are essentially inaccessible to observations. Here, we present a model that combines planet formation simulations with equilibrium condensation models to estimate the interior composition and structure of exoplanets around K-dwarf stars. Aims. In a previous study, we reproduced the observed population of close-in super-Earths around K-dwarf stars with suitable initial conditions. However, the masses and radii of these simulated planets are not self-consistent, since the radius information is postprocessed based on an assumed constant average density (3 g cm⁻³). In this work, we have incorporated compositions into the N-body simulations using the chemistry of the protoplanetary disk from equilibrium condensation sequences, together with thermodynamics of the various mineral phases. This has allowed us to develop models of likely interior structures, bulk densities, and, thus, radii for rocky exoplanets. Methods. We selected the outcomes of ten N-body simulations around a 0.8 M_⊙ star that best reproduced the observed population. We employed stellar abundances from six K-dwarf stars with a range of metallicities. Based on the condensation sequence and the temperature-pressure profile of a disk, we assigned bulk elemental compositions to planetesimals and then tracked materials accreted onto planets during their formation. Results. We have obtained a set of planets with more realistic radii determinations than those purely based on one preset density for all bodies. We formed various types of planets: i) Ca- and Al-rich(er), ii) Mg-depleted, iii) fully oxidized core-less planets, iv) planets similar to Earth or Mars in composition and core size, and v) planets with different core mass fractions (from significantly smaller than the Martian core to significantly larger than the Earth’s core); however, we do not have Mercury-like planet with a huge core.

We aim to compute the impact rates for objects with a diameter of 1 km onto the regular satellites of Jupiter, Saturn and Uranus using our latest dynamical simulations of the evolution of outer solar system coupled with the best estimates of the current population of objects beyond Neptune and their size-frequency distribution. We use the outcome of the last 3.5~Gyr of evolution of the outer solar system from our database of simulations and combine this with observational constraints of the population beyond Neptune to compute the flux of objects entering the Centaur region, with uncertainties. The initial conditions resemble the current population rather than a near-circular, near-planar disc usually assumed just before the onset of giant planet migration. We obtain a better estimate of the impact probability of a Centaur with the satellites from enacting simulations of planetesimals flying past the satellites on hyperbolic orbits, which agree with literature precedents. We find that our impact rate of objects greater than 1 km in diameter with Jupiter is 0.0012/yr, which is a factor of 3--6 lower than previous estimates of 0.0044/yr from Nesvorny et al. (2023) and 0.0075/yr from Zahnle et al. (2003). On the other hand our impact probabilities with the satellites scaled to the giant planets are consistent with these earlier literature estimates, as is the leakage rate of objects from beyond Neptune into the Centaur region. However, our absolute impact probabilities with the giant planets are lower. We attribute this to our choice of initial conditions.

Terrestrial planet formation (TPF) is a difficult problem that has vexed researchers for decades. Numerical models are only partially successful at reproducing the orbital architecture of the inner planets, but have generally not considered the effect of the growth of the giant planets. I dynamically model TPF as the gas giants Jupiter and Saturn are growing using GENGA. The evolution of the masses, radii and orbital elements of the gas giants are precomputed and read and interpolated within GENGA. The terrestrial planets are formed by planetesimal accretion from tens of thousands of self-gravitating planetesimals spread between 0.5 au and 8.5 au. The total mass of the inner planetesimal disc and outer disc are typically 2 and 3 Earth masses respectively, and the composition of the planetesimals changes from non-carbonaceous-like to carbonaceous-like at a prescribed distance, ranging from 2 au to 5 au. After 5 Myr of evolution approximately 10% to 25% of the mass of planetesimals in the Jupiter-Saturn region is implanted in the inner solar system, which is more than what cosmochemical models predict. The implantation initially sets up a composition gradient in the inner solar system, with the fraction of outer solar system material increasing with increasing distance to the Sun. The planetesimals that remain in the inner solar system have a mixed composition. The growth of the gas giants scatters planetesimals in their vicinity into the inner solar system, which changes the isotopic composition of the terrestrial planets. The planetesimal disc in the vicinity of the gas giants may not have been very massive, ~1 ME. The inner planetesimal disc may not have extended much farther than 2 au otherwise embryos do not grow fast enough to produce Mars analogues. This could mean that the region of the current asteroid belt never contained much mass to begin with.

Context. Graphics processing unit (GPU) computing has become popular due to the enormous calculation potential that can be harvested from a single card. The N-body integrator Gravitational ENcounters with GPU Acceleration (GENGA) is built to harvest the computing power from such cards, but it suffers a severe performance penalty on consumer-grade Nvidia GPUs due to their artificially truncated double precision performance. Aims. We aim to speed up GENGA on consumer-grade cards by harvesting their high single-precision performance. Methods. We modified GENGA to have the option to compute the mutual long-distance forces between bodies in single precision and tested this with five experiments. First, we ran a high number of simulations with similar initial conditions of on average 6600 fully self-gravitating planetesimals in both single and double precision to establish whether the outcomes were statistically different. These simulations were run on Tesla K20 cards. We supplemented this test with simulations that (i) began with a mixture of planetesimals and planetary embryos, (ii) planetesimal-driven giant planet migration, and (iii) terrestrial planet formation with a dissipating gas disc. All of these simulations served to determine the accuracy of energy and angular momentum conservation under various scenarios with single and double precision forces. Second, we ran the same simulation beginning with 40 000 self-gravitating planetesimals using both single and double precision forces on a variety of consumer-grade and Tesla GPUs to measure the performance boost of computing the long-range forces in single precision. Results. We find that there are no statistical differences when simulations are run with the gravitational forces in single or double precision that can be attributed to the force prescription rather than stochastic effects. The accumulations in uncertainty in energy are almost identical when running with single or double precision long-range forces. However, the uncertainty in the angular momentum using single rather than double precision long-range forces is about two orders of magnitude greater, but still very low. Running the simulations in single precision on consumer-grade cards decreases running time by a factor of three and becomes within a factor of three of a Tesla A100 GPU. Additional tuning speeds up the simulation by a factor of two across all types of cards. Conclusions. The option to compute the long-range forces in single precision in GENGA when using consumer-grade GPUs dramatically improves performance at a little penalty to accuracy. There is an additional environmental benefit because it reduces energy usage.

GPU computing is popular due to the calculation potential of a single card. The N-body integrator GENGA is built to for this, but it suffers a performance penalty on consumer-grade GPUs due to their truncated double precision (FP64) performance. We aim to speed up GENGA on consumer-grade cards by harvesting their high single-precision performance (FP32). We modified GENGA to be able to compute the long-distance forces between bodies in FP32 precision and tested this with 5 experiments. We ran simulations with similar initial conditions of 6600 planetesimals in both FP32 and FP64 precision. We also ran simulations that i) began with a mixture of planetesimals and planetary embryos, ii) planetesimal-driven giant planet migration, and iii) terrestrial planet formation with a gas disc. Second, we ran the same simulation beginning with 40 000 planetesimals using both FP32 and FP64 precision forces on a variety of consumer-grade and Tesla GPUs to measure the performance boost of FP32 computing. There are no statistical differences when running in FP32 or FP64 precision that can be attributed to the force prescription rather than stochastic effects. The uncertainties in energy are almost identical when using both precisions. However, the uncertainty in the angular momentum using FP32 rather than FP64 precision long-range forces is about two orders of magnitude greater, but still very low. Running the simulations in single precision on consumer-grade cards decreases running time by a factor of three and becomes within a factor of three of a Tesla A100 GPU. Additional tuning speeds up the simulation by a factor of two across all types of cards. The option to compute the long-range forces in single precision in GENGA when using consumer-grade GPUs dramatically improves performance at a little penalty to accuracy. There is an additional environmental benefit because it reduces energy usage.

We present recent updates and improvements of the graphical processing unit (GPU) N-body code GENGA. Modern state-of-the-art simulations of planet formation require the use of a very high number of particles to accurately resolve planetary growth and to quantify the effect of dynamical friction. At present the practical upper limit is in the range of 30,000–60,000 fully interactive particles; possibly a little more on the latest GPU devices. While the original hybrid symplectic integration method has difficulties to scale up to these numbers, we have improved the integration method by (i) introducing higher level changeover functions and (ii) code improvements to better use the most recent GPU hardware efficiently for such large simulations. We added treatments of non-Newtonian forces such as general relativity, tidal interaction, rotational deformation, the Yarkovsky effect, and Poynting–Robertson drag, as well as a new model to treat virtual collisions of small bodies in the solar system. We added new tools to GENGA, such as semi-active test particles that feel more massive bodies but not each other, a more accurate collision handling and a real-time openGL visualization. We present example simulations, including a 1.5 billion year terrestrial planet formation simulation that initially started with 65,536 particles, a 3.5 billion year simulation without gas giants starting with 32,768 particles, the evolution of asteroid fragments in the solar system, and the planetesimal accretion of a growing Jupiter simulation. GENGA runs on modern NVIDIA and AMD GPUs.