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Fast and accurate treatment of collisions in the context of modern N-body planet formation simulations remains a challenging task due to inherently complex collision processes. We aim to tackle this problem with machine learning (ML), in particular via residual neural networks. Our model is motivated by the underlying physical processes of the data-generating process and allows for flexible prediction of post-collision states. We demonstrate that our model outperforms commonly used collision handling methods such as perfect inelastic merging and feed-forward neural networks in both prediction accuracy and out-of-distribution generalization. Our model outperforms the current state of the art in 20/24 experiments. We provide a dataset that consists of 10164 Smooth Particle Hydrodynamics (SPH) simulations of pairwise planetary collisions. The dataset is specifically suited for ML research to improve computational aspects for collision treatment and for studying planetary collisions in general. We formulate the ML task as a multi-task regression problem, allowing simple, yet efficient training of ML models for collision treatment in an end-to-end manner. Our models can be easily integrated into existing N-body frameworks and can be used within our chosen parameter space of initial conditions, i.e. where similar-sized collisions during late-stage terrestrial planet formation typically occur., Comment: 13 pages, 7 figures, 7 tables

In the past 15 years, the triaxial Schwarzschild orbit-superposition code by van den Bosch et al. (2008) has been widely applied to study the dynamics of galaxies. Recently, Quenneville et al. (2022) reported a bug in the orbit calculation of this code, specifically in the mirroring procedure that is used to speed up the computation. We have fixed the incorrect mirroring in DYNAMITE, which is the publicly-released successor of the triaxial Schwarzschild code by van den Bosch et al. (2008). In this study, we provide a thorough quantification of how this bug has affected the results of dynamical analyses performed with this code. We compare results obtained with the original and corrected versions of DYNAMITE, and discuss the differences in the phase-space distribution of a single orbit and in the global stellar orbit distribution, in the mass estimate of the central black hole in the highly triaxial galaxy PGC 46832, and in the measurement of intrinsic shape and enclosed mass for more than 50 galaxies. Focusing on the typical scientific applications of a Schwarzschild triaxial code, in all our tests we find that differences are negligible with respect to the statistical and systematic uncertainties. We conclude that previous results with the van den Bosch et al. (2008) triaxial Schwarzschild code are not significantly affected by the incorrect mirroring., Comment: 14 pages, 11 figures. Submitted to A&A

Although it is accepted that perfect-merging is not a realistic outcome of collisions, some researchers state that perfect-merging simulations can still be considered as quantitatively reliable representations of the final stage of terrestrial planet formation. Citing the work of Kokubo & Genda [ApJL, 714L, 21], they argue that the differences between the final planets in simulations with perfect-merging and those where collisions are resolved accurately are small, and it is, therefore, justified to use perfect-merging results as an acceptable approximation to realistic simulations. In this paper, we show that this argument does not stand. We demonstrate that when the mass lost during collisions is taken into account, the final masses of the planets will be so different from those obtained from perfect-merging that the latter cannot be used as a valid approximation. We carried out a large number of SPH simulations of embryo-embryo collisions and determined the amount of the mass and water lost in each impact. We applied the results to collisions in a typical perfect-merging simulation and showed that even when the mass-loss in each collision is as small as 10%, perfect-merging can, on average, overestimate the masses of the final planets by $\sim 35\%$ and their water-contents by more than 18%. Our analysis demonstrates that, while perfect-merging simulations are still a powerful tool in proving concepts, they cannot be used to make predictions, draw quantitative conclusions (especially about the past history of a planetary system) and serve as a valid approximation to, or in lieu of the simulations in which collisions are resolved accurately., Comment: 26 pages, 4 figures, Accepted for publication in The Astrophysical Journal

The last phase of the formation of rocky planets is dominated by collisions among Moon- to Mars-sized planetary embryos. Simulations of this phase need to handle the difficulty of including the post-impact material without saturating the numerical integrator. A common approach is to include the collision-generated material by clustering it into few bodies with the same mass and uniformly scattering them around the collision point. However, this approach oversimplifies the properties of the collision material by neglecting features that can play important roles in the final structure and composition of the system. In this study, we present a statistical analysis of the orbital architecture, mass, and size distributions of the material generated through embryo-embryo collisions and show how they can be used to develop a model that can be directly incorporated into the numerical integrations. For instance, results of our analysis indicate that the masses of the fragments follow an exponential distribution with an exponent of $-2.21\pm0.17$ over the range of $10^{-7}$ to $2\times 10^{-2}$ Earth-masses. The distribution of the post-impact velocities show that a large number of fragments are scattered toward the central star. The latter is a new finding that may be quite relevant to the delivery of material from the outer regions of the asteroid belt to the accretion zones of terrestrial planets. Finally, we present an analytical model for the 2D distribution of fragments that can be directly incorporated into numerical integrations., Comment: 10 pages, 13 figures, accepted for publication in MNRAS

Today there are numerous studies on asteroid mining. They elaborate on selecting the right objects, prospecting missions, potential asteroid redirection, and the mining process itself. For economic reasons, most studies focus on mining candidates in the 100-500m size-range. Also, suggestions regarding the design and implementation of space stations or even colonies inside the caverns of mined asteroids exist. Caverns provide the advantages of confined material in near-zero gravity during mining and later the hull will shield the inside from radiation. Existing studies focus on creating the necessary artificial gravity by rotating structures that are built inside the asteroid. Here, we assume the entire mined asteroid to rotate at a sufficient rate for artificial gravity and investigate its use for housing a habitat inside. In this study we present how to estimate the necessary spin rate assuming a cylindrical space station inside a mined asteroid and discuss the implications arising from substantial material stress given the required rotation rate. We estimate the required material strength using two relatively simple analytical models and apply them to fictitious, yet realistic rocky near-Earth asteroids., Comment: 16 pages, 7 figures

In this work we describe a genetic algorithm which is used in order to study orbits of minor bodies in the frames of close encounters. We find that the algorithm in combination with standard orbital numerical integrators can be used as a good proxy for finding typical orbits of minor bodies in close encounters with planets and even their moons, saving a lot of computational time compared to long-term orbital numerical integrations. Here, we study close encounters of Centaurs with Callisto and Ganymede in particular. We also perform n-body numerical simulations for comparison. We find typical impact velocities to be between $v_{rel}=20[v_{esc}]$ and $v_{rel}=30[v_{esc}]$ for Ganymede and between $v_{rel}=25[v_{esc}]$ and $v_{rel}=35[v_{esc}]$ for Callisto., Comment: 18 pages, 3 figures

It has been suggested that the comet-like activity of Main Belt Comets is due to the sublimation of sub-surface water-ice that is exposed when these objects are impacted by meter-sized bodies. We recently examined this scenario and showed that such impacts can in fact excavate ice and present a plausible mechanism for triggering the activation of MBCs (Haghighipour et al. 2016). However, because the purpose of that study was to prove the concept and identify the most viable ice-longevity model, the porosity of the object and the loss of ice due to the heat of impact were ignored. In this paper, we extend our impact simulations to porous materials and account for the loss of ice due to an impact. We show that for a porous MBC, impact craters are deeper, reaching to approximately 15 m implying that if the activation of MBCs is due to the sublimation of sub-surface ice, this ice has to be within the top 15 m of the object. Results also indicate that the loss of ice due to the heat of impact is negligible, and the re-accretion of ejected ice is small. The latter suggests that the activities of current MBCs are most probably from multiple impact sites. Our study also indicates that in order for sublimation from multiple sites to account for the observed activity of the currently known MBCs, the water content of MBCs (and their parent asteroids) needs to be larger than the values traditionally considered in models of terrestrial planet formation., Comment: 22 pages, 8 figures, 2 tables, accepted for publication in ApJ

We present recent improvements in the simulation of regolith sampling processes in microgravity using the numerical particle method smooth particle hydrodynamics (SPH). We use an elastic-plastic soil constitutive model for large deformation and failure flows for dynamical behaviour of regolith. In the context of projected small body (asteroid or small moons) sample return missions, we investigate the efficiency and feasibility of a particular material sampling method: Brushes sweep material from the asteroid's surface into a collecting tray. We analyze the influence of different material parameters of regolith such as cohesion and angle of internal friction on the sampling rate. Furthermore, we study the sampling process in two environments by varying the surface gravity (Earth's and Phobos') and we apply different rotation rates for the brushes. We find good agreement of our sampling simulations on Earth with experiments and provide estimations for the influence of the material properties on the collecting rate., Comment: 15 pages, 11 figures, accepted for publication in Planetary and Space Science

Modern graphics processing units (GPUs) lead to a major increase in the performance of the computation of astrophysical simulations. Owing to the different nature of GPU architecture compared to traditional central processing units (CPUs) such as x86 architecture, existing numerical codes cannot be easily migrated to run on GPU. Here, we present a new implementation of the numerical method smooth particle hydrodynamics (SPH) using CUDA and the first astrophysical application of the new code: the collision between Ceres-sized objects. The new code allows for a tremendous increase in speed of astrophysical simulations with SPH and self-gravity at low costs for new hardware. We have implemented the SPH equations to model gas, liquids and elastic, and plastic solid bodies and added a fragmentation model for brittle materials. Self-gravity may be optionally included in the simulations and is treated by the use of a Barnes-Hut tree. We find an impressive performance gain using NVIDIA consumer devices compared to our existing OpenMP code. The new code is freely available to the community upon request., Comment: 18 pages, 15 figures, accepted for publication in A&A

The problem of the formation of the Moon is still not explained satisfactorily. While it is a generally accepted scenario that the last giant impact on Earth between some 50 to 100 million years after the starting of the formation of the terrestrial planets formed our natural satellite, there are still many open questions like the isotopic composition which is identical for these two bodies. In our investigation we will not deal with these problems of chemical composition but rather undertake a purely dynamical study to find out the probability of a Mars-sized body to collide with the Earth shortly after the formation of the Earth-like planets. For that we assume an additional massive body between Venus and Earth, respectively Earth and Mars which formed there at the same time as the other terrestrial planets. We have undertaken massive n-body integrations of such a planetary system with 4 inner planets (we excluded Mercury but assumed one additional body as mentioned before) for up to tens of millions of years. Our results led to a statistical estimation of the collision velocities as well as the collision angles which will then serve as the basis of further investigation with detailed SPH computations. We find a most probable origin of the Earth impactor at a semi-major axis of approx. 1.16 AU., Comment: 6 pages, 12 figures, submitted to MNRAS

We present preliminary results of terrestrial planet formation using on the one hand classical numerical integration of hundreds of small bodies on CPUs and on the other hand -- for comparison reasons -- the results of our GPU code with thousands of small bodies which then merge to larger ones. To be able to determine the outcome of collision events we use our smooth particle hydrodynamics (SPH) code which tracks how water is lost during such events., Comment: 2 pages, 2 figures, IAU Symposium 314, submitted

As part of a national scientific network 'Pathways to Habitability' the formation of planets and the delivery of water onto these planets is a key question as water is essential for the development of life. In the first part of the paper we summarize the state of the art of planet formation - which is still under debate in the astronomical community - before we show our results on this topic. The outcome of our numerical simulations depends a lot on the choice of the initial distribution of planetesimals and planetary embryos after gas disappeared in the protoplanetary disk. We also take into account that some of these planetesimals of sizes in the order of the mass of the Moon already contained water; the quantity depends on the distance from the Sun - close-by bodies are dry, but starting from a distance of about 2 AU they can contain substantial amounts of water. We assume that the gas giants and terrestrial planets are already formed when we check the collisions of the small bodies containing water (in the order of a few percent) with the terrestrial planets. We thus are able to give an estimate of the respective contribution to the actual water content (of some Earth-oceans) in the mantle, in the crust and on the surface of Earth. In the second part we discuss in more detail how the formation of larger bodies after a collision may happen as the outcome depends on parameters like collision velocity, impact angle, and the materials involved. We present results obtained by SPH (Smooth Particle Hydrodynamics) simulations. We briefly describe this method and show different scenarios with respect to the formed bodies, possible fragmentation and the water content before and after the collision. In an appendix we discuss detection methods for extrasolar planets (close to 2000 such objects have been discovered so far)., Comment: 12 pages, 10 figures, submitted to Nonlinear Phenomena in Complex Systems, ISSN 1561-4085 (Print), ISSN 1817-2458 (Online)

We investigate the outcome of collisions of Ceres-sized planetesimals composed of a rocky core and a shell of water ice. These collisions are not only relevant for explaining the formation of planetary embryos in early planetary systems, but also provide insight into the formation of asteroid families and possible water transport via colliding small bodies. Earlier studies show characteristic collision velocities exceeding the bodies' mutual escape velocity which - along with the distribution of the impact angles - cover the collision outcome regimes 'partial accretion', 'erosion', and 'hit-and-run' leading to different expected fragmentation scenarios. Existing collision simulations use bodies composed of strengthless material; we study the distribution of fragments and their water contents considering the full elasto-plastic continuum mechanics equations also including brittle failure and fragmentation., Comment: Submitted to IAU Symposium 310 Proceedings 4 pages, 2 figures

Context. We investigate the effects of including material strength in multi-material planetesimal collisions. Aims. The differences between strengthless material models and including the full elasto-plastic model for solid bodies with brittle failure and fragmentation when treating collisions of asteroid-sized bodies as they occur frequently in early planetary systems are demonstrated. Methods. We study impacts of bodies of Ceres-mass with a solid rock impactor and a target with 30 weight-% water content as surface ice.} The initial impact velocities and impact parameters are varied between the escape velocity $v_\mathrm{esc}$ to about 6 $v_\mathrm{esc}$ and from head-on collisions to close fly-bys, respectively. We simulate the collisions using our own SPH code utilizing both strengthless material and the full elasto-plastic material model including brittle failure. Results. One of the most prominent differences is the higher degree of fragmentation and shattered debris clouds in the solid model. In most collision scenarios however, the final outcomes are very similar and differ primarily by the about one order of magnitude higher degree of fragmentation in the solid case. Also, the survivors tend to be of less mass in the solid case which also predicts a higher water loss than the strengthless hydro model. This may be an effect of the relatively low-energy impacts that cannot destroy the solid material instantly. As opposed to giant impacts we also observe an indication that some water ice gets transferred between the bodies., Comment: Submitted for publication in Astronomy & Astrophysics

We investigate the distribution of encounter velocities and impact angles describing collisions in the habitable zone of the early planetary system. Here we present a catalogue of collision characteristics for a particular mass ratio of the colliding bodies and seven different planetesimal masses ranging from a tenth of Ceres' mass to 10 times the mass of the Moon. We show that there are virtually no collisions with impact speeds lower than the surface escape velocity and a similar velocity-impact angle distribution for different planetesimal masses if velocities are normalized using the escape velocity. An additional perturbing Jupiter-like object distorts the collision velocity and impact picture in the sense that grazing impacts at higher velocities are promoted if the perturber's orbit is close to the habitable zone whereas a more distant perturber has more the effect of a mere widening of the velocity dispersion., Comment: Submitted to IAU Symposium 299 Proceedings 4 pages, 3 figures

We give a brief introduction to smoothed particle hydrodynamics methods for continuum mechanics. Specifically, we present our 3D SPH code to simulate and analyze collisions of asteroids consisting of two types of material: basaltic rock and ice. We consider effects like brittle failure, fragmentation, and merging in different impact scenarios. After validating our code against previously published results we present first collision results based on measured values for the Weibull flaw distribution parameters of basalt., Comment: Accepted and to be published in Astronomical Notes