arXiv subjects

Ataru Tanikawa

Publications and source records attributed to Ataru Tanikawa.

At least 73 records · Page 4Linked to original sources

arXiv2018

Property of Young Massive Clusters in a Galaxy-Galaxy Merger Remnant

We investigate the properties of young massive clusters (YMCs) in a galaxy-galaxy merger remnant by analyzing the data obtained by a gas rich major merger simulations in Matsui et al. 2012. We found that the YMCs are distributed at a few $\rm kpc$ and at $\sim 10~{\rm kpc}$ from the galactic center, in other words, there are two components of their distribution. The former are formed in filamentary and turbulent gas generated at a few $\rm kpc$ from the center because of galaxy encounters, and the latter are formed in tidal tails which are far from the center. The YMCs are much less concentrated than galaxy stars. The mass function of the YMCs is $dN/dM \propto M^{-2}$. Most of YMCs are formed from the second encounter to the final coalescence phase of the galactic cores, and their formation rate is especially high at final coalescence phase. Most of them consists of single stellar population in age, but YMCs with multi stellar populations in age are also formed. The multiple populations are produced by the following process: a YMC captures dense gas, and another generation stars form within the cluster. There are several YMCs formed in an isolated disk before the encounter of galaxies. These candidates contain stars with various age by capturing dense gas and forming stars. YMCs in a merger remnant, have various orbits, but large fraction of candidates have circular orbits.

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arXiv2018

Three-Dimensional Simulation of Double-Detonations in the Double-Degenerate Model for Type Ia Supernovae and Interaction of Ejecta with a Surviving White Dwarf Companion

We study the hydrodynamics and nucleosynthesis in the double-detonation model of Type Ia supernovae (SNe~Ia) and the interaction between the ejecta and a surviving white dwarf (WD) companion in the double-degenerate scenario. We set up a binary star system with $1.0M_\odot$ and $0.6M_\odot$ carbon-oxygen (CO) WDs, where the primary WD consists of a CO core and helium (He) shell with $0.95$ and $0.05M_\odot$, respectively. We follow the evolution of the binary star system from the initiation of a He detonation, ignition and propagation of a CO detonation, and the interaction of SN ejecta with the companion WD. The companion (or surviving) WD gets a flung-away velocity of $\sim 1700$~km~s$^{-1}$, and captures $^{56}$Ni of $\sim 0.03M_\odot$, and He of $3 \times 10^{-4}M_\odot$. Such He can be detected on the surface of surviving WDs. The SN ejecta contains a `companion-origin stream', and unburned materials stripped from the companion WD ($\sim 3 \cdot 10^{-3}M_\odot$), although the stream compositions would depend on the He shell mass of the companion WD. The ejecta has also a velocity shift of $\sim 1000$~km~s$^{-1}$ due to the binary motion of the exploding primary WD. These features would be prominent in nebular-phase spectra of oxygen emission lines from the unburned materials like SN~2010lp and iPTF14atg, and of blue- or red-shifted Fe-group emission lines from the velocity shift like a part of sub-luminous SNe~Ia. We expect SN~Ia counterparts to the D$^6$ model would leave these fingerprints for SN~Ia observations.

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arXiv2018

Metal Pollution of Low-Mass Population III Stars through Accretion of Interstellar Objects like `Oumuamua

We calculate accretion mass of interstellar objects (ISOs) like `Oumuamua onto low-mass population III stars (Pop.~III survivors), and estimate surface pollution of Pop.~III survivors. An ISO number density estimated from the discovery of `Oumuamua is so high ($\sim 0.2$~au$^{-3}$) that Pop.~III survivors have chances at colliding with ISOs $\gtrsim 10^5$ times per $1$~Gyr. `Oumuamua itself would be sublimated near Pop.~III survivors, since it has small size, $\sim 100$~m. However, ISOs with size $\gtrsim 3$~km would reach the Pop.~III survivor surfaces. Supposing an ISO cumulative number density with size larger than $D$ is $n \propto D^{-α}$, Pop.~III survivors can accrete ISO mass $\gtrsim 10^{-16}M_\odot$, or ISO iron mass $\gtrsim 10^{-17}M_\odot$, if $α< 4$. This iron mass is larger than the accretion mass of interstellar medium (ISM) by several orders of magnitude. Taking into account material mixing in a convection zone of Pop.~III survivors, we obtain their surface pollution is typically [Fe/H] $\lesssim -8$ in most cases, however the surface pollution of Pop.~III survivors with $0.8M_\odot$ can be [Fe/H] $\gtrsim -6$ because of the very shallow convective layer. If we apply to Pop.III survivors located at the Galactocentric distance of 8 kpc, the dependence of the metal pollustion is as follows. If $α> 4$, Pop.~III survivors have no chance at colliding with ISOs with $D \gtrsim 3$~km, and keep metal-free. If $3 < α< 4$, Pop.~III survivors would be most polluted by ISOs up to [Fe/H] $\sim -7$. If $α< 3$ up to $D \sim 10$~km, Pop.~III survivors could hide in metal-poor stars so far discovered. Pop.~III survivors would be more polluted with decreasing the Galactocentric distance. Although the metal pollution depends on $α$ and the Galactocentric distance, we first show the importance of ISOs for the metal pollution of Pop.~III survivors.

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arXiv2018

Fortran interface layer of the framework for developing particle simulator FDPS

Numerical simulations based on particle methods have been widely used in various fields including astrophysics. To date, simulation softwares have been developed by individual researchers or research groups in each field, with a huge amount of time and effort, even though numerical algorithms used are very similar. To improve the situation, we have developed a framework, called FDPS, which enables researchers to easily develop massively parallel particle simulation codes for arbitrary particle methods. Until version 3.0, FDPS have provided API only for C++ programing language. This limitation comes from the fact that FDPS is developed using the template feature in C++, which is essential to support arbitrary data types of particle. However, there are many researchers who use Fortran to develop their codes. Thus, the previous versions of FDPS require such people to invest much time to learn C++. This is inefficient. To cope with this problem, we newly developed a Fortran interface layer in FDPS, which provides API for Fortran. In order to support arbitrary data types of particle in Fortran, we design the Fortran interface layer as follows. Based on a given derived data type in Fortran representing particle, a Python script provided by us automatically generates a library that manipulates the C++ core part of FDPS. This library is seen as a Fortran module providing API of FDPS from the Fortran side and uses C programs internally to interoperate Fortran with C++. In this way, we have overcome several technical issues when emulating `template' in Fortran. By using the Fortran interface, users can develop all parts of their codes in Fortran. We show that the overhead of the Fortran interface part is sufficiently small and a code written in Fortran shows a performance practically identical to the one written in C++.

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arXiv2018

Tidal Disruption of a White Dwarf by a Black Hole: The Diversity of Nucleosynthesis, Explosion Energy, and the Fate of Debris Streams

We run a suite of hydrodynamics simulations of tidal disruption events (TDEs) of a white dwarf (WD) by a black hole (BH) with a wide range of WD/BH masses and orbital parameters. We implement nuclear reactions to study nucleosynthesis and its dynamical effect through release of nuclear energy. The released nuclear energy effectively increases the fraction of unbound ejecta. This effect is weaker for a heavy WD with 1.2 $\mathrm{M}_{\odot}$, because the specific orbital energy distribution of the debris is predominantly determined by the tidal force, rather than by the explosive reactions. The elemental yield of a TDE depends critically on the initial composition of a WD, while the BH mass and the orbital parameters also affect the total amount of synthesized elements. Tanikawa et al. (2017) find that simulations of WD-BH TDEs with low resolution suffer from spurious heating and inaccurate nuclear reaction results. In order to examine the validity of our calculations, we compare the amounts of the synthesized elements with the upper limits of them derived in a way where we can avoid uncertainties due to low resolution. The results are largely consistent, and thus support our findings. We find particular TDEs where early self-intersection of a WD occurs during the first pericentre passage, promoting formation of an accretion disc. We expect that relativistic jets and/or winds would form in these cases because accretion rates would be super-Eddington. The WD-BH TDEs result in a variety of events depending on the WD/BH mass and pericentre radius of the orbit.

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arXiv2018

High-resolution hydrodynamic simulation of tidal detonation of a helium white dwarf by an intermediate mass black hole

We demonstrate tidal detonation during a tidal disruption event (TDE) of a helium (He) white dwarf (WD) with $0.45M_\odot$ by an intermediate mass black hole (IMBH) by extremely high-resolution simulations. Tanikawa et al. (2017) have showed tidal detonation in previous studies results from unphysical heating due to low-resolution simulations, and such unphysical heating occurs in 3-dimensional (3D) smoothed particle hydrodynamics (SPH) simulations even with $10$ million SPH particles. In order to avoid such unphysical heating, we perform 3D SPH simulations up to $300$ million SPH particles, and 1D mesh simulations using flow structure in the 3D SPH simulations for 1D initial conditions. The 1D mesh simulations have higher resolution than the 3D SPH simulations. We show tidal detonation occurs, and confirm this result is perfectly converged with different space resolution in both 3D SPH and 1D mesh simulations. We find detonation waves independently arises in leading parts of the WD, and yield large amounts of $^{56}$Ni. Although detonation waves are not generated in trailing parts of the WD, the trailing parts receive detonation waves generated in the leading parts, and leave large amounts of Si group elements. Eventually, this He WD TDE would synthesize $^{56}$Ni of $0.30M_\odot$ and Si group elements of $0.08M_\odot$, and could be observed as a luminous thermonuclear transient comparable to type Ia supernovae.

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arXiv2018

Tidal double detonation: a new mechanism for a thermonuclear explosion of a white dwarf induced by a tidal disruption event

We suggest "tidal double detonation": a new mechanism for a thermonuclear explosion of a white dwarf (WD) induced by a tidal disruption event (TDE). Tidal detonation is also a WD explosion induced by a TDE. In this case, helium (He) and carbon-oxygen (CO) detonation waves incinerate He~WD and CO~WD, respectively. On the other hand, for tidal double detonation, He detonation is first excited in the He shell of a CO~WD, and drives CO detonation in the CO core. We name this mechanism after the double detonation scenario in the context of type Ia supernovae. In this paper, we show tidal double detonation occurs in shallower encounter of a CO~WD with an intermediate mass black hole (IMBH) than simple tidal detonation, performing numerical simulations for CO~WDs with $0.60M_\odot$ with and without a He shell. We expect tidal double detonation spreads opportunity to WD~TDEs illuminating IMBHs.

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arXiv2017

The Detection Rates of Merging Binary Black Holes Originating from Star Clusters and Their Mass Function

Advanced LIGO achieved the first detection of the gravitational wave, which was from a merging binary black hole (BBH). In the near future, more merger events will be observed, and the mass distribution of them will become available. The mass distribution of merger events reflects the evolutionary path of BBHs: dynamical formation in dense star clusters or common envelope evolution from primordial binaries. In this paper, we estimate the detection rate of merging BBHs which dynamically formed in dense star clusters by combining the results of $N$-body simulations, modeling of globular clusters, and cosmic star-cluster formation history. We estimate that the merger rate density in the local universe within the redshift of 0.1 is 13--57 Gpc$^{-3}$ yr$^{-1}$. We find that the detection rate is 0.23--4.6 per year for the current sensitivity limit and that it would increase to 5.1--99 per year for the designed sensitivity which will be achieved in 2019. The distribution of merger rate density in the local universe as a function of redshifted chirp mass has a peak close to the low-mass end.The chirp mass function of the detected mergers, on the other hand, has a peak at the high-mass end, but is almost flat. This difference is simply because the detection range is larger for more massive BBHs.

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arXiv2017

Does Explosive Nuclear Burning occur in Tidal Disruption Events of White Dwarfs by Intermediate Mass Black Holes ?

We investigate nucleosynthesis in tidal disruption events (TDEs) of white dwarfs (WDs) by intermediate mass black holes (IMBHs). We consider various types of WDs with different masses and compositions by means of 3 dimensional (3D) smoothed particle hydrodynamics (SPH) simulations. We model these WDs with different numbers of SPH particles, $N$, from a few $10^4$ to a few $10^7$, in order to check mass resolution convergence, where SPH simulations with $N>10^7$ (or a space resolution of several $10^6$ cm) have unprecedentedly high resolution in this kind of simulations. We find that nuclear reactions become less active with increasing $N$, and that these nuclear reactions are excited by spurious heating due to low resolution. Moreover, we find no shock wave generation. In order to investigate the reason for the absence of a shock wave, we additionally perform 1 dimensional (1D) SPH and mesh-based simulations with a space resolution ranging from $10^4$ to $10^7$ cm, using characteristic flow structure extracted from the 3D SPH simulations. We find shock waves in these 1D high-resolution simulations. One of these shock waves triggers a detonation wave. However, we have to be careful of the fact that, if the shock wave emerged at a bit outer region, it could not trigger the detonation wave due to low density. Note that the 1D initial conditions lack accuracy to precisely determine where a shock wave emerges. We need to perform 3D simulations with $\lesssim 10^6$ cm space resolution in order to conclude that WD TDEs become optical transients powered by radioactive nuclei.

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arXiv2016

Unconvergence of Very Large Scale GI Simulations

The giant impact (GI) is one of the most important hypotheses both in planetary science and 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 $3 \times10^3$ to $10^8$. We found that the results does not converge, but shows oscillatory behaviour. We discuss the origin of this oscillatory behaviour.

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arXiv2016

Implementation and performance of FDPS: A Framework Developing Parallel Particle Simulation Codes

We present the basic idea, implementation, measured performance and performance model of FDPS (Framework for developing particle simulators). FDPS is an application-development framework which helps the researchers to develop particle-based simulation programs for large-scale distributed-memory parallel supercomputers. A particle-based simulation program for distributed-memory parallel computers needs to perform domain decomposition, redistribution of particles, and gathering of particle information for interaction calculation. Also, even if distributed-memory parallel computers are not used, in order to reduce the amount of computation, algorithms such as Barnes-Hut tree method should be used for long-range interactions. For short-range interactions, some methods to limit the calculation to neighbor particles are necessary. FDPS provides all of these necessary functions for efficient parallel execution of particle-based simulations as "templates", which are independent of the actual data structure of particles and the functional form of the interaction. By using FDPS, researchers can write their programs with the amount of work necessary to write a simple, sequential and unoptimized program of O(N^2) calculation cost, and yet the program, once compiled with FDPS, will run efficiently on large-scale parallel supercomputers. A simple gravitational N-body program can be written in around 120 lines. We report the actual performance of these programs and the performance model. The weak scaling performance is very good, and almost linear speedup was obtained for up to the full system of K computer. The minimum calculation time per timestep is in the range of 30 ms (N=10^7) to 300 ms (N=10^9). These are currently limited by the time for the calculation of the domain decomposition and communication necessary for the interaction calculation. We discuss how we can overcome these bottlenecks.

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arXiv2016

The critical mass ratio of double white dwarf binaries for violent merger-induced Type Ia supernova explosions

Mergers of carbon-oxygen (CO) white dwarfs (WDs) are considered as one of the potential progenitors of type Ia supernovae (SNe Ia). Recent hydrodynamical simulations showed that the less massive (secondary) WD violently accretes onto the more massive (primary) one, carbon detonation occurs, the detonation wave propagates through the primary, and the primary finally explodes as a sub-Chandrasekhar mass SN Ia. Such an explosion mechanism is called the violent merger scenario. Based on the smoothed particle hydrodynamics (SPH) simulations of merging CO WDs, we derived more stringent critical mass ratio (qcr) leading to the violent merger scenario than the previous results. We conclude that this difference mainly comes from the differences in the initial condition, synchronously spinning of WDs or not. Using our new results, we estimated the brightness distribution of SNe Ia in the violent merger scenario and compared it with previous studies. We found that our new qcr does not significantly affect the brightness distribution. We present the direct outcome immediately following CO WD mergers for various primary masses and mass ratios. We also discussed the final fate of the central system of the bipolar planetary nebula Henize 2-428, which was recently suggested to be a double CO WD system whose total mass exceeds the Chandrasekhar-limiting mass, merging within the Hubble time. Even considering the uncertainties in the proposed binary parameters, we concluded that the final fate of this system is almost certainly a sub-Chandrasekhar mass SN Ia in the violent merger scenario.

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arXiv2015

A systematic study of carbon-oxygen white dwarf mergers: mass combinations for Type Ia supernovae

Mergers of two carbon-oxygen (CO) white dwarfs (WDs) have been considered as progenitors of Type Ia supernovae (SNe Ia). Based on smoothed particle hydrodynamics (SPH) simulations, previous studies claimed that mergers of CO WDs lead to an SN Ia explosion either in the dynamical merger phase or stationary rotating merger remnant phase. However, the mass range of CO WDs that lead to an SN Ia has not been clearly identified yet. In the present work, we perform systematic SPH merger simulations for the WD masses ranging from $0.5~M_{\odot}$ to $1.1~M_{\odot}$ with higher resolutions than the previous systematic surveys and examine whether or not carbon burning occurs dynamically or quiescently in each phase. We further study the possibility of SN Ia explosion and estimate the mass range of CO WDs that lead to an SN Ia. We found that when the both WDs are massive, i.e., in the mass range of $0.9~M_{\odot} {\le} M_{1,2} {\le} 1.1~M_{\odot}$, they can explode as an SN Ia in the merger phase. On the other hand, when the more massive WD is in the range of $0.7~M_{\odot} {\le} M_{1} {\le} 0.9~M_{\odot}$ and the total mass exceeds $1.38~M_{\odot}$, they can finally explode in the stationary rotating merger remnant phase. We estimate the contribution of CO WD mergers to the entire SN Ia rate in our galaxy to be of ${\lt} 9\%$. So, it might be difficult to explain all galactic SNe Ia by CO WD mergers.

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arXiv2015

Hydrodynamical evolution of merging carbon-oxygen white dwarfs: their pre-supernova structure and observational counterparts

We perform smoothed particle hydrodynamics (SPH) simulations for merging binary carbon-oxygen (CO) white dwarfs (WDs) with masses of $1.1$ and $1.0$ $M_\odot$, until the merger remnant reaches a dynamically steady state. Using these results, we assess whether the binary could induce a thermonuclear explosion, and whether the explosion could be observed as a type Ia supernova (SN Ia). We investigate three explosion mechanisms: a helium-ignition following the dynamical merger (`helium-ignited violent merger model'), a carbon-ignition (`carbon-ignited violent merger model'), and an explosion following the formation of the Chandrasekhar mass WD (`Chandrasekhar mass model'). An explosion of the helium-ignited violent merger model is possible, while we predict that the resulting SN ejecta are highly asymmetric since its companion star is fully intact at the time of the explosion. The carbon-ignited violent merger model can also lead to an explosion. However, the envelope of the exploding WD spreads out to $\sim 0.1R_\odot$; it is much larger than that inferred for SN 2011fe ($< 0.1R_\odot $) while much smaller than that for SN 2014J ($\sim 1R_\odot$). For the particular combination of the WD masses studied in this work, the Chandrasekhar mass model is not successful to lead to an SN Ia explosion. Besides these assessments, we investigate the evolution of unbound materials ejected through the merging process (`merger ejecta'), assuming a case where the SN Ia explosion is not triggered by the helium- or carbon-ignition during the merger. The merger ejecta interact with the surrounding interstellar medium, and form a shell. The shell has a bolometric luminosity of more than $2 \times 10^{35}$ ergs$^{-1}$ lasting for $\sim 2 \times 10^4$ yr. If this is the case, Milky Way should harbor about $10$ such shells at any given time.

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arXiv2014

Merger Criteria of Multiple Massive Black Holes and the Impact on the Host Galaxy

We perform N-body simulations on a multiple massive black hole (MBH) system in a host galaxy to derive the criteria for successive MBH merger. The calculations incorporate the dynamical friction by stars and general relativistic effects as pericentre shift and gravitational wave recoil. The orbits of MBHs are pursed down to ten Schwarzschild radii (~ 1 AU). As a result, it is shown that about a half of MBHs merge during 1 Gyr in a galaxy with mass $10^{11}M_{\odot}$ and stellar velocity dispersion 240 km/s, even if the recoil velocity is two times as high as the stellar velocity dispersion. The dynamical friction allows a binary MBH to interact frequently with other MBHs, and then the decay of the binary orbits leads to the merger through gravitational wave radiation, as shown by Tanikawa & Umemura (2011). We derive the MBH merger criteria for the masses, sizes, and luminosities of host galaxies. It is found that the successive MBH mergers are expected in bright galaxies, depending on redshifts. Furthermore, we find that the central stellar density is reduced by the sling-shot mechanism and that high-velocity stars with ~ 1000 km/s are generated intermittently in extremely radial orbits.

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arXiv2013

Few-Body Modes of Binary Formation in Core Collapse

At the moment of deepest core collapse, a star cluster core contains less than ten stars. This small number makes the traditional treatment of hard binary formation, assuming a homogeneous background density, suspect. In a previous paper, we have found that indeed the conventional wisdom of binary formation, based on three-body encounters, is incorrect. Here we refine that insight, by further dissecting the subsequent steps leading to hard binary formation. For this purpose, we add some analysis tools in order to make the study less subjective. We find that the conventional treatment does remain valid for direct three-body scattering, but fails for resonant three-body scattering. Especially democratic resonance scattering, which forms an important part of the analytical theory of three-body binary formation, takes too much space and time to be approximated as being isolated, in the context of a cluster core around core collapse. We conclude that, while three-body encounters can be analytically approximated as isolated, subsequent strong perturbations typically occur whenever those encounters give rise to democratic resonances. We present analytical estimates postdicting our numerical results. If we only had been a bit more clever, we could have predicted this qualitative behaviour.

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arXiv2013

Dynamical evolution of stellar-mass black holes in dense stellar clusters: estimate for merger rate of binary black holes originating from globular clusters

We have performed N-body simulations of globular clusters (GCs) in order to estimate a detection rate of mergers of Binary stellar-mass Black Holes (BBHs) by means of gravitational wave (GW) observatories. For our estimate, we have only considered mergers of BBHs which escape from GCs (BBH escapers). BBH escapers merge more quickly than BBHs inside GCs because of their small semi-major axes. N-body simulation can not deal with a GC with the number of stars N ~ 10^6 due to its high calculation cost. We have simulated dynamical evolution of small-N clusters (10^4 <~ N <~ 10^5), and have extrapolated our simulation results to large-N clusters. From our simulation results, we have found the following dependence of BBH properties on N. BBHs escape from a cluster at each two-body relaxation time at a rate proportional to N. Semi-major axes of BBH escapers are inversely proportional to N, if initial mass densities of clusters are fixed. Eccentricities, primary masses, and mass ratios of BBH escapers are independent of N. Using this dependence of BBH properties, we have artificially generated a population of BBH escapers from a GC with N ~ 10^6, and have estimated a detection rate of mergers of BBH escapers by next-generation GW observatories. We have assumed that all the GCs are formed 10 or 12Gyrs ago with their initial numbers of stars N_i=5 x 10^5 -- 2 x 10^6 and their initial stellar mass densities inside their half-mass radii ρ_h,i=6 x 10^3 -- 10^6M_sun pc^-3. Then, the detection rate of BBH escapers is 0.5 -- 20 yr^-1 for a BH retention fraction R_BH=0.5. A few BBH escapers are components of hierarchical triple systems, although we do not consider secular perturbation on such BBH escapers for our estimate. Our simulations have shown that BHs are still inside some of GCs at the present day. These BHs may marginally contribute to BBH detection.

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arXiv2012

Phantom-GRAPE: numerical software library to accelerate collisionless $N$-body simulation with SIMD instruction set on x86 architecture

(Abridged) We have developed a numerical software library for collisionless N-body simulations named "Phantom-GRAPE" which highly accelerates force calculations among particles by use of a new SIMD instruction set extension to the x86 architecture, AVX, an enhanced version of SSE. In our library, not only the Newton's forces, but also central forces with an arbitrary shape f(r), which has a finite cutoff radius r_cut (i.e. f(r)=0 at r>r_cut), can be quickly computed. Using an Intel Core i7--2600 processor, we measure the performance of our library for both the forces. In the case of Newton's forces, we achieve 2 x 10^9 interactions per second with 1 processor core, which is 20 times higher than the performance of an implementation without any explicit use of SIMD instructions, and 2 times than that with the SSE instructions. With 4 processor cores, we obtain the performance of 8 x 10^9 interactions per second. In the case of the arbitrarily shaped forces, we can calculate 1 x 10^9 and 4 x 10^9 interactions per second with 1 and 4 processor cores, respectively. The performance with 1 processor core is 6 times and 2 times higher than those of the implementations without any use of SIMD instructions and with the SSE instructions. These performances depend weakly on the number of particles. It is good contrast with the fact that the performance of force calculations accelerated by GPUs depends strongly on the number of particles. Substantially weak dependence of the performance on the number of particles is suitable to collisionless N-body simulations, since these simulations are usually performed with sophisticated N-body solvers such as Tree- and TreePM-methods combined with an individual timestep scheme. Collisionless N-body simulations accelerated with our library have significant advantage over those accelerated by GPUs, especially on massively parallel environments.

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