N-body simulation for self-gravitating collisional systems with a new SIMD instruction set extension to the x86 architecture, Advanced Vector eXtensions

We present a high-performance N-body code for self-gravitating collisional systems accelerated with the aid of a new SIMD instruction set extension of the x86 architecture: Advanced Vector eXtensions (AVX), an enhanced version of the Streaming SIMD Extensions (SSE). With one processor core of Intel Core i7-2600 processor (8 MB cache and 3.40 GHz) based on Sandy Bridge micro-architecture, we implemented a fourth-order Hermite scheme with individual timestep scheme (Makino and Aarseth, 1992), and achieved the performance of ∼20 giga floating point number operations per second (GFLOPS) for double-precision accuracy, which is two times and five times higher than that of the previously developed code implemented with the SSE instructions (Nitadori et al., 2006b), and that of a code implemented without any explicit use of SIMD instructions with the same processor core, respectively. We have parallelized the code by using so-called NINJA scheme (Nitadori et al., 2006a), and achieved ∼90 GFLOPS for a system containing more than N = 8192 particles with 8 MPI processes on four cores. We expect to achieve about 10 tera FLOPS (TFLOPS) for a self-gravitating collisional system with N ∼ 10⁵ on massively parallel systems with at most 800 cores with Sandy Bridge micro-architecture. This performance will be comparable to that of Graphic Processing Unit (GPU) cluster systems, such as the one with about 200 Tesla C1070 GPUs (Spurzem et al., 2010). This paper offers an alternative to collisional N-body simulations with GRAPEs and GPUs.     

Unexpected formation modes of the first hard binary in core collapse

The conventional wisdom for the formation of the first hard binary in core collapse is that three-body interactions of single stars form many soft binaries, most of which are quickly destroyed, but eventually one of them survives. We report on direct N-body simulations to test these ideas, for the first time. We find that the assumptions are incorrect in the majority of the cases: (1) quite a few three-body interactions produce a hard binary from scratch; (2) in many cases there are more than three bodies directly and simultaneously involved in the production of the first binary. The main reason for the discrepancies is that the core of a star cluster, at the first deep collapse, contains typically only five or so stars. Therefore, the homogeneous background assumption, which still would be reasonable for, say, 25 stars, utterly breaks down. There have been some speculations in this direction, but we demonstrate this result here explicitly, for the first time.     

On a property of zero-velocity curves in N-body ring-type systems

Zero-velocity curves are a useful tool in the investigation of various aspects of a dynamical system. These curves that distinguish the regions where the motion of a particle is permissible from the regions where this motion is not permitted, present some basic properties. In this paper, we prove that in symmetric ring-type systems where a small particle moves under the resultant gravitational field of N coplanar big bodies, of which ν=N−1 are arranged at equal distances among them on the periphery of a circle, a new property concerning these curves, exists. All the zero-velocity curves drawn in the space of the initial conditions (x₀,C) and concerning configurations with the same number of peripheral primaries but various mass parameters, pass through two different focal points, the position of which does not depend on the value of the mass parameter.     

Sixth- and eighth-order Hermite integrator for N-body simulations

We present sixth- and eighth-order Hermite integrators for astrophysical N-body simulations, which use the derivatives of accelerations up to second-order (snap) and third-order (crackle). These schemes do not require previous values for the corrector, and require only one previous value to construct the predictor. Thus, they are fairly easy to implement. The additional cost of the calculation of the higher-order derivatives is not very high. Even for the eighth-order scheme, the number of floating-point operations for force calculation is only about two times larger than that for traditional fourth-order Hermite scheme. The sixth-order scheme is better than the traditional fourth-order scheme for most cases. When the required accuracy is very high, the eighth-order one is the best. These high-order schemes have several practical advantages. For example, they allow a larger number of particles to be integrated in parallel than the fourth-order scheme does, resulting in higher execution efficiency in both general-purpose parallel computers and GRAPE systems.     

High-performance direct gravitational N-body simulations on graphics processing units

We present the results of gravitational direct N-body simulations using the commercial graphics processing units (GPU) NVIDIA Quadro FX1400 and GeForce 8800GTX, and compare the results with GRAPE-6Af special purpose hardware. The force evaluation of the N-body problem was implemented in Cg using the GPU directly to speed-up the calculations. The integration of the equations of motions were, running on the host computer, implemented in C using the 4th order predictor–corrector Hermite integrator with block time steps. We find that for a large number of particles (N  10⁴) modern graphics processing units offer an attractive low cost alternative to GRAPE special purpose hardware. A modern GPU continues to give a relatively flat scaling with the number of particles, comparable to that of the GRAPE. The GRAPE is designed to reach double precision, whereas the GPU is intrinsically single-precision. For relatively large time steps, the total energy of the N-body system was conserved better than to one in 10⁶ on the GPU, which is impressive given the single-precision nature of the GPU. For the same time steps, the GRAPE gave somewhat more accurate results, by about an order of magnitude. However, smaller time steps allowed more energy accuracy on the grape, around 10⁻¹¹, whereas for the GPU machine precision saturates around 10⁻⁶ For N  10⁶ the GeForce 8800GTX was about 20 times faster than the host computer. Though still about a factor of a few slower than GRAPE, modern GPUs outperform GRAPE in their low cost, long mean time between failure and the much larger onboard memory; the GRAPE-6Af holds at most 256k particles whereas the GeForce 8800GTX can hold 9 million particles in memory.     

Impact of the Mass Parameter on Particle Dynamics in a Ring Configuration of N Bodies

Parameters play a very important and determinative role in the dynamics of a dynamical system as well as in the formation of its particular characteristics. In this paper we investigate the way in which a large scale variation of the mass parameter, influences the behavior of a mass-less particle which moves in the vicinity of a ring arrangement of N-bodies. More precisely, we study the impact of this parameter on periodic motions and their characteristics.     

Size-frequency distributions of fragments from SPH/N-body simulations of asteroid impacts: Comparison with observed asteroid families

We investigate the morphology of size-frequency distributions (SFDs) resulting from impacts into 100-km-diameter parent asteroids, represented by a suite of 161 SPH/N-body simulations conducted to study asteroid satellite formation [Durda, D.D., Bottke, W.F., Enke, B.L., Merline, W.J., Asphaug, E., Richardson, D.C., Leinhardt, Z.M., 2004. Icarus 170, 243-257]. The spherical basalt projectiles range in diameter from 10 to 46 km (in equally spaced mass increments in logarithmic space, covering six discrete sizes), impact speeds range from 2.5 to 7 km/s (generally in 1 km/s increments), and impact angles range from 15° to 75° (nearly head-on to very oblique) in 15° increments. These modeled SFD morphologies match very well the observed SFDs of many known asteroid families. We use these modeled SFDs to scale to targets both larger and smaller than 100 km in order to gain insights into the circumstances of the impacts that formed these families. Some discrepancies occur for families with parent bodies smaller than a few tens of kilometers in diameter (e.g., 832 Karin), however, so due caution should be used in applying our results to such small families. We find that ∼20 observed main-belt asteroid families are produced by the catastrophic disruption of D>100 km parent bodies. Using these data as constraints, collisional modeling work [Bottke Jr., W.F., Durda, D.D., Nesvorný, D., Jedicke, R., Morbidelli, A., Vokrouhlický, D., Levison, H.F., 2005b. Icarus 179, 63-94] suggests that the threshold specific energy, , needed to eject 50% of the target body's mass is very close to that predicted by Benz and Asphaug [Benz, W., Asphaug, E., 1999. Icarus 142, 5-20].     

Jacobi Dynamics And The N-Body Problem With Variable Masses

An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

From planetesimals to terrestrial planets: N-body simulations including the effects of nebular gas and giant planets

We present results from a suite of N-body simulations that follow the formation and accretion history of the terrestrial planets using a new parallel treecode that we have developed. We initially place 2000 equal size planetesimals between 0.5 and 4.0 AU and the collisional growth is followed until the completion of planetary accretion (>100 Myr). A total of 64 simulations were carried out to explore sensitivity to the key parameters and initial conditions. All the important effect of gas in laminar disks are taken into account: the aerodynamic gas drag, the disk-planet interaction including Type I migration, and the global disk potential which causes inward migration of secular resonances as the gas dissipates. We vary the initial total mass and spatial distribution of the planetesimals, the time scale of dissipation of nebular gas (which dissipates uniformly in space and exponentially in time), and orbits of Jupiter and Saturn. We end up with 1-5 planets in the terrestrial region. In order to maintain sufficient mass in this region in the presence of Type I migration, the time scale of gas dissipation needs to be 1-2 Myr. The final configurations and collisional histories strongly depend on the orbital eccentricity of Jupiter. If today’s eccentricity of Jupiter is used, then most of bodies in the asteroidal region are swept up within the terrestrial region owing to the inward migration of the secular resonance, and giant impacts between protoplanets occur most commonly around 10 Myr. If the orbital eccentricity of Jupiter is close to zero, as suggested in the Nice model, the effect of the secular resonance is negligible and a large amount of mass stays for a long period of time in the asteroidal region. With a circular orbit for Jupiter, giant impacts usually occur around 100 Myr, consistent with the accretion time scale indicated from isotope records. However, we inevitably have an Earth size planet at around 2 AU in this case. It is very difficult to obtain spatially concentrated terrestrial planets together with very late giant impacts, as long as we include all the above effects of gas and assume initial disks similar to the minimum mass solar nebular.     

The N-gon is not a local minimum of U 2 I for N > 7

A theorem of Palmore's concerning coplanar central configurations of equal mass bodies was shown to be false for all even N 6 by Slaminka and Woerner. Using a variation of that argument I prove that Palmore's Theorem is false for all N 6.Northwestern University     

A method of symplectic integrations with adaptive time-steps for individual Hamiltonians in the planetary <Emphasis Type="Italic">N</Emphasis>-body problem

A new algorithm is developed for long-term integrations of the N-body problem. The method uses symplectic integrations of the Hamiltonian equations of motion for each body. This allows one to employ individual adaptive time-steps in computations. The efficiency of this technique is demonstrated by several tests performed for typical problems of Solar System dynamics.     

KAM tori for N-body problems: a brief history

We review analytical (rigorous) results about the existence of invariant tori for planetary many-body problems.     

A spectroscopic study of DD UMa: Ursa Major group member and candidate for BRITE

The Ursa Major group is a nearby stellar supercluster which, while not gravitationally bound, is defined by co-moving members. DD UMa is a δ Scuti star whose membership in the Ursa Major group is unclear.The objective of this study is to confirm the membership of DD UMa in the Ursa Major group, as well as perform a detailed spectral analysis of the star. Since DD UMa is a low-amplitude δ Scuti star, we performed a frequency analysis. We determined fundamental parameters, chemical abundances, and derive a mass and age for the star.For this study we observed DD UMa at the Okayama Astrophysical Observatory with the high-resolution spectrograph HIDES, between the 27th of February and the 4th March, 2009. Additional observations were extracted from the ELODIE archive in order to expand our abundance analysis. Group membership of DD UMa was assessed by examining the velocity of the star in Galactic coordinates. Pulsational frequencies were determined by examining line profile variability in the HIDES spectra. Stellar fundamental parameters and chemical abundances were derived by fitting synthetic spectra to both the HIDES and ELODIE observations.DD UMa is found to be a member of the extended stream of the Ursa Major group, based on the space motion of the star. This is supported by the chemical abundances of the star being consistent with those of Ursa Major group members. The star is found to be chemically solar, with T_eff = 7450 ± 150 K and logg = 3.98 ± 0.2. We found pulsational frequencies of 9.4 and 15.0 c/d. While these frequencies are insufficient to perform an asteroseismic study, DD UMa is a good bright star candidate for future study by the BRITE-constellation.     

nmagic: a fast parallel implementation of a χ2-made-to-measure algorithm for modelling observational data

We describe a made-to-measure (M2M) algorithm for constructing N -particle models of stellar systems from observational data (χ²M2M), extending earlier ideas by Syer & Tremaine. The algorithm properly accounts for observational errors, is flexible, and can be applied to various systems and geometries. We implement this algorithm in a parallel code nmagic and carry out a sequence of tests to illustrate its power and performance. (i) We reconstruct an isotropic Hernquist model from density moments and projected kinematics and recover the correct differential energy distribution and intrinsic kinematics. (ii) We build a self-consistent oblate three-integral maximum rotator model and compare how the distribution function is recovered from integral field and slit kinematic data. (iii) We create a non-rotating and a figure rotating triaxial stellar particle model, reproduce the projected kinematics of the figure rotating system by a non-rotating system of the same intrinsic shape, and illustrate the signature of pattern rotation in this model. From these tests, we comment on the dependence of the results from χ²M2M on the initial model, the geometry, and the amount of available data.     

Effects of Dispersion of Wave Packets in the Determination of Lifetimes of High-Degree Solar p Modes from Time – Distance Analysis: TON Data

We use the method of time – distance analysis to measure lifetimes of solar p modes in the range ℓ=100 − 600 and ν=3.0 − 4.5 mHz with data taken with the Taiwan Oscillation Network (TON). The lifetimes of p modes are determined by the changes in the amplitude and width of the cross-correlation function of a wave packet with the number of skips. The amplitude of the cross-correlation function decreases exponentially with the number of skips as in previous work. This decrease has been interpreted as the effect of the finite p-mode lifetime. In this study, we find that the width of the cross-correlation function increases with the number of skips. We interpret this phenomenon as the effect of the dispersion of the wave packet. We include this effect in the determination of the lifetime of the wave packet. The lifetime increases after the dispersion is taken into account. We also study the change in lifetime between solar minimum and maximum.