Application of a massively parallel computer to the N-body problem

Parallel processor computers represent a new technology that has recently become available for astronomical applications. We have implemented an N-body code on a TMC Connection Machine CM-2 in order to investigate the advantages of a massively parallel computer over serial machines, including conventional supercomputers. For collisionless problems following N stars, a direct integration code scales as O(N²) on serial machines and on the CM-2 as O(log(N)) for small N and O(N log(N)) for large N. The CM-2 outperforms workstations for N>50 and supercomputers for N>4000.     

A momentum-conserving N-body scheme with individual time steps

We present an N-body code called Taichi for galactic dynamics and controlled numerical experiments. The code includes two high-order hierarchical multipole expansion methods: the Barnes-Hut (BH) tree and the fast multipole method (FMM). For the time integration, the code can use either a conventional adaptive KDK or a Hamiltonian splitting integrator. The combination of FMM and the Hamiltonian splitting integrator leads to a momentum-conserving N-body scheme with individual time steps. We find Taichi performs well in the typical applications in galactic dynamics. In the isolated and interacting galaxies tests, the momentum conserving scheme produces the same result as a conventional BH tree code. But for similar force accuracies, FMM significantly speeds up the simulations compared to the monopole BH tree. In the cold collapse test, we find the inner structure after relaxation can be sensitive to the force accuracies. Taichi is ready to incorporate special treatment of close encounters thanks to the Hamiltonian splitting integrator, suitable for studying dynamics around central massive bodies.     

Parallel Tree–SPH: A Tool for Galaxy Formation

We describe a new implementation of a parallel Tree-SPH code with the aim of simulating galaxy formation and evolution. The code has been parallelized using SHMEM, a Cray proprietary library to handle communications between the 256 processors of the Silicon Graphics T3E massively parallel supercomputer hosted by the Cineca Super-computing Center (Bologna, Italy). The code combines the smoothed particle hydrodynamics (SPH) method to solve hydrodynamical equations with the popular Barnes and Hut (1986) tree-code to perform gravity calculation with a N × log N scaling, and it is based on the scalar Tree-SPH code developed by Carraro et al. (1998). Parallelization is achieved by distributing particles along processors according to a workload criterion. Benchmarks of the code, in terms of load balance and scalability, are analysed and critically discussed against the adiabatic collapse of an isothermal gas sphere test using 2 × 10⁴ particles on eight processors. The code turns out to be balanced at more than 95% level. If the number of processors is increased, the load balance worsens slightly. The deviation from perfect scalability at increasing number of processors is negligible up to 64 processors. Additionally we have incorporated radiative cooling, star formation, feedback and an algorithm to follow the chemical enrichment of the interstellar medium. This revised version was published online in July 2006 with corrections to the Cover Date.     

Magnetofrictional Extrapolations of Low and Lou’s Force-Free Equilibria

We present a careful investigation of the magnetofrictional relaxation and extrapolation technique applied to the reconstruction of two test fields. These fields are taken from the family of nonlinear force-free magnetic equilibria constructed by Low and Lou (Astrophys. J. 352, 343, 1990), which have emerged as standard tests for extrapolation techniques in recent years. For the practically relevant case that only the field values in the bottom plane of the considered volume (vector magnetogram) are used as input information (i.e., not including the knowledge about the test field at the side and top boundaries), the test field is reconstructed to a higher accuracy than obtained previously. Detailed diagnostics of the reconstruction accuracy show that the implementation of fourth-order spatial discretization was essential to reach this accuracy for the given test fields and to achieve near machine precision in satisfying the solenoidal condition. Different variants of boundary conditions are tested, which all yield comparable accuracy. In its present implementation, the technique yields a scaling of computing time with total number of grid points only slightly below N ^5/3, which is too steep for applications to large (≥1024²) magnetograms, except on supercomputers. Directions for improvement are outlined.     

Performance tuning of N-body codes on modern microprocessors: I. Direct integration with a hermite scheme on x86_64 architecture

The main performance bottleneck of gravitational N-body codes is the force calculation between two particles. We have succeeded in speeding up this pair-wise force calculation by factors between 2 and 10, depending on the code and the processor on which the code is run. These speed-ups were obtained by writing highly fine-tuned code for x86_64 microprocessors. Any existing N-body code, running on these chips, can easily incorporate our assembly code programs.In the current paper, we present an outline of our overall approach, which we illustrate with one specific example: the use of a Hermite scheme for a direct N² type integration on a single 2.0 GHz Athlon 64 processor, for which we obtain an effective performance of 4.05 Gflops, for double-precision accuracy. In subsequent papers, we will discuss other variations, including the combinations of N log N codes, single-precision implementations, and performance on other microprocessors.     

A fast method for finding bound systems in numerical simulations: Results from the formation of asteroid binaries

We present a new code (companion) that identifies bound systems of particles in O(NlogN) time. Simple binaries consisting of pairs of mutually bound particles and complex hierarchies consisting of collections of mutually bound particles are identifiable with this code. In comparison, brute force binary search methods scale as O(N²) while full hierarchy searches can be as expensive as O(N³), making analysis highly inefficient for multiple data sets with N?¹⁰³. A simple test case is provided to illustrate the method. Timing tests demonstrating O(NlogN) scaling with the new code on real data are presented. We apply our method to data from asteroid satellite simulations [Durda et al., 2004. Icarus 167, 382-396; Erratum: Icarus 170, 242; reprinted article: Icarus 170, 243-257] and note interesting multi-particle configurations. The code is available at http://www.astro.umd.edu/zoe/companion/ and is distributed under the terms and conditions of the GNU Public License.     

Velocity oscillations in solar plage regions

The properties of wave propagation in a perfectly electrically conducting, plane-stratified, inviscid, compressible atmosphere premeated by a horizontal magnetic field which varies with height are investigated. It is shown that a diagnostic diagram can be constructed through a generalization of the propagation equation to account for the presence of a magnetic field.The effect of the magnetic field on the oscillations in solar plages around the temperature minimum is studied and compared with the non-magnetic case based on the Bilderberg Continuum Atmosphere. Due to the joint action of ionization and the magnetic field, a potential well for internal gravity waves is formed. The properties of the trapped waves are in qualitative agreement with the observations.The Brunt-Väisälä frequencies in the presence (N) and absence (N ₀) of the magnetic field around the temperature minimum are shown to obey the relation N/N ₀ (1 + ²)^–1/2, where is the ratio between the Alfvén and the sonic speeds. The modified Brunt-Väisälä frequency (N) is decreased as the strength of the magnetic field increases. The magnetic effect makes the potential well shallower, thus shrinking the domain for the trapped gravity modes.     

GRMHD/RMHD simulations & stability of magnetized spine-sheath relativistic jets

A new general relativistic magnetohydrodynamics (GRMHD) code “RAISHIN” used to simulate jet generation by rotating and non-rotating black holes with a geometrically thin Keplarian accretion disk finds that the jet develops a spine-sheath structure in the rotating black hole case. Spine-sheath structure and strong magnetic fields significantly modify the Kelvin-Helmholtz (KH) velocity shear driven instability. The RAISHIN code has been used in its relativistic magnetohydrodynamic (RMHD) configuration to study the effects of strong magnetic fields and weakly relativistic sheath motion, c/2, on the KH instability associated with a relativistic, γ=2.5, jet spine-sheath interaction. In the simulations sound speeds up to and Alfvén wave speeds up to ∼0.56c are considered. Numerical simulation results are compared to theoretical predictions from a new normal mode analysis of the RMHD equations. Increased stability of a weakly magnetized system resulting from c/2 sheath speeds and stabilization of a strongly magnetized system resulting from c/2 sheath speeds is found.     

Resonance trapping of circumstellar dust particles by an alleged planet

The dynamical evolution of dust particles forming a circumstellar disk around Pictoris is followed by numerical simulations on a Connection Machine. The disk appears to be cleared inside a radius of about 20 AU. We integrate simultaneously the orbits of 8,000 dust particles subjected to Poynting-Robertson drag and perturbed by one alleged planet. The simulations show that a planet revolving about Pictoris at a mean distance of 20 AU with a mass of at least 2 * 10^–5 central stellar mass can confine the disk by outer resonance trapping. The azimuthal density distribution of particles which shows very strong variations. appears to be stationary in a frame rotating with the planet.     

N-Body Simulations of Planetesimal Evolution: Effect of Varying Impactor Mass Ratio

We present results from direct N-body simulations of collisions between gravitational aggregates of varying size as part of a study to parameterize planetesimal growth in the Solar System. We find that as the ratio of projectile to target mass departs from unity, the impact angle has less effect on the outcome. At the same time, the probability of planetesimal growth increases. Conversely, for a fixed impact energy, collisions between impactors with mass ratio near unity are more dispersive than those with impactor mass ratio far from unity. We derive an expression for the accretion probability as a function of mass ratio. For an average mass ratio of 1:5, we find an accretion probability of ∼60% over all impact parameters. We also compute the critical specific dispersal energy Q*_D as a function of projectile size. Extrapolating to a projectile size of 1 m with a 1-km target, we find Q*_D=10³−10⁴ J kg⁻¹, in agreement with several other collision models that use fundamentally different techniques. Our model assumes that the components of each gravitational aggregate are identical and indestructible over the range of sampled impact speeds. In future work we hope to incorporate a simple fracture model to extend the range of applicable speeds and we plan to implement our results in a large-scale planetesimal evolution code.     

Integration error over very long time spans

The main limit to the time span of a numerical integration of the planetary orbits is no longer set by the availability of computer resources, but rather by the accumulation of the integration error. By the latter we mean the difference between the computed orbit and the dynamical behaviour of the real physical system, whatever the causes. The analysis of these causes requires an interdisciplinary effort: there are physical model and parameters errors, algorithm and discretisation errors, rounding off errors and reliability problems in the computer hardware and system software, as well as instabilities in the dynamical system. We list all the sources of integration error we are aware of and discuss their relevance in determining the present limit to the time span of a meaningful integration of the orbit of the planets. At present this limit is of the order of 10⁸ years for the outer planets. We discuss in more detail the truncation error of multistep algorithms (when applied to eccentric orbits), the coefficient error, the method of Encke and the associated coordinate change error, the procedures used to test the numerical integration software and their limitations. Many problems remain open, including the one of a realistic statistical model of the rounding off error; at present, the latter can only be described by a semiempirical model based upon the simpleN ² formula (N=number of steps, =machine accuracy), with an unknown numerical coefficient which is determined only a posteriori.     

Application of an Improved Method of Image Segmentation and Some Considerations on Identification and Tracking of Umbral Dots

An improved method of image segmentation is introduced. The object-tracking algorithm, originally developed by Sobotka, Brandt, and Simon (Astron. Astrophys. 328, 682, 1997) is modified with special attentions on splitting and merging of umbral dots (UDs), definition of the umbral boundary, and the birth-frames and the death-frames of UDs. By applying the new method of image segmentation and the object-tracking algorithm on a 67-min series of white-light images of a large pore (Sobotka et al., Astrophys. J. 511, 436, 1999), the physical characteristics of 20 “resolved” UDs with umbral origin were recorded. The most probable lifetime of the UDs is between 7 and 10 min. Umbral dots show a typical size of about 230 km. Their mean speeds are smaller than 2 km s⁻¹ with a distribution around a value less than 1 km s⁻¹. However, their average velocities are less than 0.8 km s⁻¹. Brighter (fainter) UDs are formed in the brighter (dimmer) region of the pore. There is no correlation between time-averaged area or time-averaged speeds and lifetimes. Also, the time-averaged peak intensities of UDs do not show any well-defined dependence on the corresponding time-averaged areas. It seems that there is a relation between average velocities of UDs and their time-averaged peak intensities, with brighter UDs moving more slowly.     

A note on the algorithm of symplectic integrators

For a Hamiltonian that can be separated into N+1(N\geq 2) integrable parts, four algorithms can be built for a symplectic integrator. This research compares these algorithms for the first and second order integrators. We found that they have similar local truncation errors represented by error Hamiltonian but rather different numerical stability. When the computation of the main part of the Hamiltonian, H ₀, is not expensive, we recommend to use S ^* type algorithm, which cuts the calculation of the H ₀ system into several small time steps as Malhotra(1991) did. As to the order of the N+1 parts in one step calculation, we found that from the large to small would get a slower error accumulation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical study of the relaxation of one-dimensional gravitational systems

The effective relaxation of one-dimensional gravitational systems towards micro-canonical equilibrium is demonstrated in a series of computer experiments, the Kolmogorov-Smirnov goodness of fit test being used to give a measure of the evolution of the distribution function. The relaxation timescalet _R is shorter thanNt _C, and thus much shorter than the generally accepted estimatet _RN ²t_C,N being the number of particles in the system andt _C a characteristic dynamical time.     

Absolute Parameters of Early-Type Close Binaries in the LMC

A project is presented which aims at high-precision determination of absolute parameters of close early-type eclipsing binaries in the LMC. We will use multi-object spectrographs (MOS) to measure RV curves of a large number of program stars selected from the MACHO archive. Spectroscopic mass ratios will be used as input for our light curve analysis code MORO. Application of the Simplex-based algorithm FITSB2 will achieve spectrum disentangling and fitting of NLTE model (TLUSTY) atmospheres, yielding orbital and atmospheric parameters. The method was extensively tested by application to time series of synthetic binary spectra for the expected range of S/N and MOS instrumental resolution. log g and T _eff were reestablished with a precision of better than 5%, radial velocity amplitudes with errors of <3%. An important by-product of our project will be the improvement of the distance modulus of the LMC, a topic which is still being intensely discussed.     

Prediction of satellites in Earth's gravitational field with axial symmetry using Burdet's regularized theory

In this paper, economical and stable recurrence formulae for the Earth's zonal potential and its gradient for Burdet's regularized theory will be established for any number N of the zonal harmonic coefficients. A general recursive computational algorithm based on these formulae is also established for the initial value problem of Burdet oscillator for the prediction of artificial satellites in the Earth's gravitational field with axial symmetry. Applications of the algorithm for the problem of the final state prediction are illustrated by numerical examples of three test orbits each for two geopotential models corresponding to N = 2 and N = 36.A final state of any desired accuracy is obtained for each case study, a result which shows the flexibility of the algorithm.Dept. of Astronomy, KAU     

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.     

VSOP and ground-based VLBI observations of Markarian 421

We have produced 22 VLBI images of the TeV blazar Markarian 421 at 11 epochs, including a Space VLBI observation with the HALCA satellite. We measure the speeds of the three innermost jet components to be 0.64±0.33, 0.48±0.09, and 0.06±0.09c (H₀=65 km s⁻¹ Mpc⁻¹). Interpretation of these subluminal speeds in terms of the high Doppler factor demanded by the TeV observations is discussed.     

On the abundance of chlorine in the Sun

A low-noise photoelectric scan which includes the predicted position of the Cli transition 4s ⁴ P _5/2-4 _p ⁴ D ⁰ _7/2 provides inconclusive evidence for the presence of the line in the solar photospheric spectrum. An upper limit logN(Cl) 5.5 is derived. It is pointed out that the fundamental vibration rotation band of HC1 at 3.3 should be detectable in the sunspot spectrum unless logN(Cl) < 4.6. Sunspot spectra may also provide the isotopic abundance ratio N(Cl³⁵)/N(Cl³⁷).A new derivation of the chlorine abundance for the Orion nebula is presented: logN(Cl) 5.8. It is suggested that a cosmic abundance logN(Cl) = 5.5 to 5.8 be adopted.Operated by the Association of Universities for Research in Astronomy Inc., under contract with the National Science Foundation.