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Cosmological N‐body simulations have been essential for astronomers to study the formation of nonlinear structures and hypotheses of dark matter, dark energy, etc. The scale of the problem naturally leads to extreme scenarios with billions or even trillions of particles, thus demanding massive computational power and highly efficient algorithms. In this paper, we present swPHoToNs, a Particle‐Mesh (PM) and Fast Multipole Method‐ (FMM) based code that can perform cosmological simulations with trillions of particles efficiently on the Sunway TaihuLight supercomputer. Our design includes three novel optimizations: (1) a multilevel domain decomposition and dynamic load‐balancing scheme; (2) a pipeline strategy for tree traversal and gravity calculation; (3) optimizations for both computation and MPI kernels with consideration of the hardware features. We manage to conduct cosmological simulations which contain up to 1.6 trillion particles, obtaining a sustained performance of 56.3 PFlops with a weak‐scaling parallel efficiency of 80.9% and a computational efficiency of 44.9%. We present the details of implementing a highly efficient and scalable Cosmological N‐body simulation framework on the heterogeneous many‐core supercomputer Sunway TaihuLight. We manage to conduct cosmological simulations which contain up to 1.6 trillion particles, obtaining a sustained performance of 56.3 PFlops with a weak‐scaling parallel efficiency of 80.9% and a computational efficiency of 44.9%.

The Juno mission 1 has provided an accurate determination of Jupiter’s gravitational field 2 , which has been used to obtain information about the planet’s composition and internal structure. Several models of Jupiter’s structure that fit the probe’s data suggest that the planet has a diluted core, with a total heavy-element mass ranging from ten to a few tens of Earth masses (about 5 to 15 per cent of the Jovian mass), and that  heavy elements (elements other than hydrogen and helium) are distributed within a region extending to nearly half of Jupiter’s radius 3 , 4 . Planet-formation models indicate that most heavy elements are accreted during the early stages of a planet's formation to create a relatively compact core 5 – 7 and that almost no solids are accreted during subsequent runaway gas accretion 8 – 10 . Jupiter’s diluted core, combined with its possible high heavy-element enrichment, thus challenges standard planet-formation theory. A possible explanation is erosion of the initially compact heavy-element core, but the efficiency of such erosion is uncertain and depends on both the immiscibility of heavy materials in metallic hydrogen and on convective mixing as the planet evolves 11 , 12 . Another mechanism that can explain this structure is planetesimal enrichment and vaporization 13 – 15 during the formation process, although relevant models typically cannot produce an extended diluted core. Here we show that a sufficiently energetic head-on collision (giant impact) between a large planetary embryo and the proto-Jupiter could have shattered its primordial compact core and mixed the heavy elements with the inner envelope. Models of such a scenario lead to an internal structure that is consistent with a diluted core, persisting over billions of years. We suggest that collisions were common in the young Solar system and that a similar event may have also occurred for Saturn, contributing to the structural differences between Jupiter and Saturn 16 – 18 . An energetic head-on collision between a large impactor and the proto-Jupiter with a primordial compact core could have mixed the heavy elements within the deep interior, leading to a ‘diluted’ core for Jupiter.

Binary compact objects are the special binary systems, which were composed of compact objects (i.e., white dwarf, neutron star and black hole). They contribute a lot to the sources of gravitational waves. The study of binary compact objects in star clusters and galaxies can provide a theoretical guidance for gravitational wave detection and improve the probability of detection effectively. We simulate the formation and fraction of binary compact objects in a small globular cluster, via NBODY6++GPU, an efficient N-body simulation code. We obtain the fractions of white dwarf binaries and neutron star binaries at different ages. The results show that the dynamic interactions among stars can also result in some binary compact objects besides binary evolution.

A large number of N -body simulations of the giant-impact phase of planet formation, combined with the measured concentrations of highly siderophile elements in Earth’s mantle, reveal that the Moon must have formed at least 40 million years after the condensation of the first solids of the Solar System. Dating the new Moon The age of the Moon has been a focus for geochemists for at least the past three decades. A number of chronometers have been used to address the question but the results differ from method to method, in part because of the varying assumptions required in the calculation of the so-called model ages. Seth Jacobson et al . have used an alternative approach. They run a large number of numerical simulations, some based on early Moon-forming events, others later events. They then arrive at a model-independent correlation between the formation age of the Moon and the amount of mass accreted by the Earth since then, the so-called Late Veneer. The concentration of highly-siderophile (iron-loving) elements observed in the Earth's mantle provides a constraint on the timing and rules out an early Moon-forming event. Instead, the authors calculate that the Moon-forming impact must have occurred at least 40 million years after formation of the Solar System. According to the generally accepted scenario, the last giant impact on Earth formed the Moon and initiated the final phase of core formation by melting Earth’s mantle. A key goal of geochemistry is to date this event, but different ages have been proposed. Some 1 , 2 , 3 argue for an early Moon-forming event, approximately 30 million years (Myr) after the condensation of the first solids in the Solar System, whereas others 4 , 5 , 6 claim a date later than 50 Myr (and possibly as late as around 100 Myr) after condensation. Here we show that a Moon-forming event at 40 Myr after condensation, or earlier, is ruled out at a 99.9 per cent confidence level. We use a large number of N -body simulations to demonstrate a relationship between the time of the last giant impact on an Earth-like planet and the amount of mass subsequently added during the era known as Late Accretion. As the last giant impact is delayed, the late-accreted mass decreases in a predictable fashion. This relationship exists within both the classical scenario 7 , 8 and the Grand Tack scenario 9 , 10 of terrestrial planet formation, and holds across a wide range of disk conditions. The concentration of highly siderophile elements (HSEs) in Earth’s mantle constrains the mass of chondritic material added to Earth during Late Accretion 11 , 12 . Using HSE abundance measurements 13 , 14 , we determine a Moon-formation age of 95 ± 32 Myr after condensation. The possibility exists that some late projectiles were differentiated and left an incomplete HSE record in Earth’s mantle. Even in this case, various isotopic constraints strongly suggest that the late-accreted mass did not exceed 1 per cent of Earth’s mass, and so the HSE clock still robustly limits the timing of the Moon-forming event to significantly later than 40 Myr after condensation.

The development of methods and algorithms to solve the N-body problem for classical, collisionless, non-relativistic particles has made it possible to follow the growth and evolution of cosmic dark matter structures over most of the universe’s history. In the best-studied case—the cold dark matter or CDM model—the dark matter is assumed to consist of elementary particles that had negligible thermal velocities at early times. Progress over the past three decades has led to a nearly complete description of the assembly, structure, and spatial distribution of dark matter haloes, and their substructure in this model, over almost the entire mass range of astronomical objects. On scales of galaxies and above, predictions from this standard CDM model have been shown to provide a remarkably good match to a wide variety of astronomical data over a large range of epochs, from the temperature structure of the cosmic background radiation to the large-scale distribution of galaxies. The frontier in this field has shifted to the relatively unexplored subgalactic scales, the domain of the central regions of massive haloes, and that of low-mass haloes and subhaloes, where potentially fundamental questions remain. Answering them may require: (i) the effect of known but uncertain baryonic processes (involving gas and stars), and/or (ii) alternative models with new dark matter physics. Here we present a review of the field, focusing on our current understanding of dark matter structure from N-body simulations and on the challenges ahead.

An experimental and computational investigation of the space-charge effects occurring in ultrafast photoelectron spectroscopy from the gas phase is presented. The target sample CF3I is excited by ultrashort (100 fs) far-ultraviolet radiation pulses produced by a free-electron laser. The modification of the energy distribution of the photoelectrons, i.e. the shift and broadening of the spectral structures, is monitored as a function of the pulse intensity. The experimental results are compared with computational simulations which employ a Barnes-Hut algorithm to calculate the effect of individual Coulomb forces acting among the particles. In the presented model, a survey spectrum acquired at low radiation fluence is used to determine the initial energy distribution of the electrons after the photoemission event. The spectrum modified by the space-charge effects is then reproduced by N-body calculations that simulate the dynamics of the photoelectrons subject to the individual mutual Coulomb repulsion and to the attractive force of the positive ions. The employed numerical method accounts for the space-charge effects on the energy distribution and allows to reproduce the complete photoelectron spectrum and not just a specific photoemission structure. The simulations also provide information on the time evolution of the space-charge effects on the picosecond scale. Differences with the case of photoemission from solid samples are highlighted and discussed. The presented simulation procedure, although it omits the analysis of angular distribution, constitutes an effective simplified model that allows to predict and account for space-charge effects on the photoelectron energy spectrum in time-resolved photoemission experiments with high-intensity pulsed sources.

We explore the possibility that Dark Matter (DM) may be explained by a nonuniform background of approximately stellar mass clusters of Primordial Black Holes (PBHs) by simulating the evolution from recombination to the present with over 5000 realisations using a Newtonian N-body code. We compute the cluster rate of evaporation and extract the binary and merged sub-populations along with their parent and merger tree histories, lifetimes and formation rates, the dynamical and orbital parameter profiles, the degree of mass segregation and dynamical friction and power spectrum of close encounters. Overall, we find that PBHs can constitute a viable DM candidate, and that their clustering presents a rich phenomenology throughout the history of the Universe. We show that binary systems constitute about 9.5% of all PBHs at present, with mass ratios of q¯B=0.154 , and total masses of m¯T,B=303M⊙ . Merged PBHs are rare, about 0.0023% of all PBHs at present, with mass ratios of q¯B=0.965 with total and chirp masses of m¯T,B=1670M⊙ and m¯c,M=642M⊙ , respectively. We find that cluster puffing up and evaporation leads to bubbles of these PBHs of order 1 kpc containing at present times about 36% of objects and mass, with one-hundred pc-sized cores. We also find that these PBH sub-haloes are distributed in wider PBH haloes of order hundreds of kpc, containing about 63% of objects and mass, coinciding with the sizes of galactic halos. We find at last high rates of close encounters of massive Black Holes ( M∼1000M⊙ ), with ΓS=(1.2+5.9−0.9)×107yr−1Gpc−3 and mergers with ΓM=1337±41yr−1Gpc−3 .

The computational efficiency of FORTRAN, C and Python languages in N-body simulation is investigated. The potential of these languages to promote the research of N-body simulation is therefore shown by this paper. Our work utilizes Particle-Particle (PP) algorithm, which not only balances the accuracy and efficiency, but also simplifies the traditional numerical calculation. The experimental results show that the computational efficiency of the three is almost the same in the case of a small number of particles, but FORTRAN shows the highest computational efficiency in the case of a large number of particles. The efficiency of Python is the lowest among three languages. The result suggests that FORTRAN is the best choice for N-body simulation, and Python should be used after optimizing the algorithm or working on extreme high-performance computers.

Massive planets form within the lifetime of protoplanetary disks, and therefore, they are subject to orbital migration due to planet–disk interactions. When the first planet reaches the inner edge of the disk, its migration stops and consequently the second planet ends up locked in resonance with the first one. We detail how the resonant trapping works comparing semi-analytical formulae and numerical simulations. We restrict to the case of two equal-mass coplanar planets trapped in first-order resonances, but the method can be easily generalized. We first describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using series expansions up to different orders in eccentricity as well as a non-expanded Hamiltonian. Then we show that during convergent migration the planets evolve along these families of equilibrium points. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict precisely analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that in most cases the resonant configurations achieved by migration become unstable after the removal of the protoplanetary disk. Here we probe the stability of the resonances as a function of planetary mass. For this purpose, we fictitiously increase the masses of resonant planets, adiabatically maintaining the low-amplitude libration regime until instability occurs. We discuss two hypotheses for the instability, that of a low-order secondary resonance of the libration frequency with a fast synodic frequency of the system, and that of minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter nevertheless play the decisive role in the destabilization of resonant pairs. We show evidence that as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.

We investigate the survivability of solar system-like planetary systems during close encounters in stellar associations using a suite of 1980 N-body simulations. Each system is based on one of the possible five-planet resonant configurations proposed to represent the initial solar system architecture and is systematically scaled in both planetary mass and orbital compactness to explore the parameter space of observed exoplanetary architectures. Simulations explore a range of stellar encounter scenarios drawn from four distinct cluster environments. Our results show that system survival depends critically on the interplay between planetary mass and orbital scale: compact configurations are more resistant to external perturbations, while increased planetary mass improves resilience only up to a threshold, beyond which internal instabilities dominate. No system whose planets are twice as massive as the ones in the solar system survives stellar encounters. Systems that are at least an order of magnitude more compact than the solar system remain stable under typical encounter conditions. These findings place strong constraints on the initial architectures of planetary systems that can endure stellar-dense birth environments.