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1.
《New Astronomy》2007,12(3):169-181
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. 相似文献
2.
Action-minimizing solutions of the one-dimensional <Emphasis Type="Italic">N</Emphasis>-body problem
We supplement the following result of C. Marchal on the Newtonian N-body problem: A path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution when the dimension d of the physical space \({\mathbb {R}}^d\) satisfies \(d\ge 2\). The focus of this paper is on the fixed-ends problem for the one-dimensional Newtonian N-body problem. We prove that a path minimizing the action functional in the set of paths joining two given configurations and having all the time the same order is always a true (collision-free) solution. Considering the one-dimensional N-body problem with equal masses, we prove that (i) collision instants are isolated for a path minimizing the action functional between two given configurations, (ii) if the particles at two endpoints have the same order, then the path minimizing the action functional is always a true (collision-free) solution and (iii) when the particles at two endpoints have different order, although there must be collisions for any path, we can prove that there are at most \(N! - 1\) collisions for any action-minimizing path. 相似文献
3.
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. 相似文献
4.
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. 相似文献
5.
An iterative approach is used to construct spherically symmetric equilibrium models with an anisotropic velocity distribution. The potentialities of the method have been tested on models with known distribution functions, the Osipkov-Merritt models. It is shown that models that differ significantly from the Osipkov-Merritt models can be constructed. An N-body model of a dark halo with a density distribution that approximates the results of cosmological simulations (the Navarro-Frenk-White model) has been constructed. The anisotropy profile has been taken to be similar to that yielded by cosmological simulations. The constructed models can serve as direct input data for investigating the dynamics and stability of such systems in N-body simulations. 相似文献
6.
We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z 0 values, where z 0 lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses. 相似文献
7.
《New Astronomy》2021
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. 相似文献
8.
We present a new formulation of the viscosity in planetary rings, where particles interact through their gravitational forces and direct collisions. In the previous studies on the viscosity in self-gravitating rings, the viscosity consists of three components, which are defined separately in different ways. The complex definitions make it difficult to evaluate the viscosity in N-body simulation of rings. In our new formulation, the viscosity is expressed in terms of changes in orbital elements of particles due to particle interactions. This makes the expression of the viscosity simple. The new formulation gives a simple way to evaluate the viscosity in N-body simulation. We find that for practical evaluation of the viscosity of planetary rings, only energy dissipation at direct inelastic collisions is needed.For tenuous particle disks (i.e., optically thin disks), we further derive a formula of the viscosity. The formula requires only a numerical coefficient that can be obtained from three-body calculation. Since planetesimal disks are also tenuous, the viscosity in planetesimal disks can be also obtained from this formula. In a subsequent paper, we will evaluate this coefficient through three-body calculation and obtain the viscosity for a wide range of parameters such as the restitution coefficient and the radial location in rings. 相似文献
9.
V. N. Snytnikov V. A. Vshivkov E. A. Kuksheva E. V. Neupokoev S. A. Nikitin A. V. Snytnikov 《Astronomy Letters》2004,30(2):124-137
We developed a three-dimensional numerical model to investigate nonstationary processes in gravitating N-body systems with gas. We used efficient algorithms for solving the Vlasov and Poisson equations that included the evolutionary processes under consideration, which ensures rapid convergence at high accuracy. We give examples of the numerical solution of the problem on the growth of physical instability in the model of a flat rotating disk with a gaseous component and its three-dimensional dynamics under various initial conditions including a nonzero velocity dispersion along the rotation axis. 相似文献
10.
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. 相似文献
11.
Generalized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c 2 h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL 2 collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN≥4 bodies, and the main confinement condition, the Easton inequality, can also be extended but it no longer provides separate regions of trapped motion, whatever is the value ofz for the wholeN-body system,N≥4. However, thez criterion of hierarchical stability applies to every 3-body subsystem, whosez ‘integral’ will of course vary in time because of the perturbations from the other bodies. In theN=4 case we decompose the system into two 3-body subsystems whosec 2 h ‘integrals’,z 23 andz 34, att=0 are assumed to be smaller than the corresponding critical values \(\tilde z_{23} \) and \(\tilde z_{34} \) , so that both the subsystems are initially hierarchically stable. Then the hierarchical arrangement of the 4 bodies cannot be broken until eitherz 23 orz 34 is changed by an amount \(\tilde z_{ij} - z_{ij} \left( 0 \right)\) ; that is the whole system is hierarchically stable for a time spain not shorter than the minimum between \(\Delta t_{23} = {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} {\dot z_{23} }}} \right. \kern-0em} {\dot z_{23} }}\) and \(\Delta t_{34} = {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} {\dot z_{34} }}} \right. \kern-0em} {\dot z_{34} }}\) . To estimate how long is this stability time, two main steps are required. First the perturbing potentials have to be developed in series; the relevant small parameters are some combinations of mass ratios and length ratios, the? ij of Roy and Walker. When an appropriate perturbation theory is based on the? ij , the asymptotic expansions are much more rapidly decreasing than the usual expansions in powers of the mass ratios (as in the classical Lagrange perturbation theory) and can be extended also to cases such as lunar theory or double binaries. The second step is the computation of the time derivatives \(\dot z_{ij} \) (we limit ourselves to the planar case). To assess the long term behaviour of the system, we can neglect the short-periodic perturbations and discuss only the long-periodic and the secular perturbations. By using a Poisson bracket formalism, a generalization of Lagrange theorem for semimajor axes and a generalization of the classical first order theories for eccentricities and pericenters, we prove that thez ij do not undergo any secular perturbation, because of the interaction with the other subsystem, at the first order in the? ik . After the long-periodic perturbations have been accounted for, and apart from the small divisors problems that could arise both from ordinary and secular resonances, only the second order terms have to be considered in the computation of Δt 23, Δt 34. A full second order perturbative theory is beyond the scope of this paper; however an order-of-magnitude lower estimate of the Δt ij can be obtained with the very pessimistic assumption that essentially all the second order terms affect in a secular way thez ij . The same method could be applied also toN≥5 body systems. Since almost everyN-body system existing in nature is strongly hierarchical, the product of two? ij is very small for almost all the real astronomical problems. As an example, the hierarchical stability of the 4-body system Sun, Mercury, Venus, and Jupiter is investigated; this system turns out to be stable for at least 110 million years. Although this hierarchical stability time is ～10 times less than the real age of the Solar System, taking into account that many pessimistic assumptions have been done we can conclude that the stability of the Solar System is no more a forbidden problem for Celestial Mechanics. 相似文献
12.
The Ricci curvature criterion is used to investigate the relative instability of various configurations of N-body gravitational systems. Systems with double massive centers are shown to be less stable than homogeneous systems or systems with single massive centers. In general, this is a confirmation that the Ricci curvature criterion is efficient in studying N-body systems by means of relatively simple computations. 相似文献
13.
14.
Thierry Combot 《Celestial Mechanics and Dynamical Astronomy》2012,114(4):319-340
We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree ?1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H 0, C 0) with (H 0, C 0) ?? (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided. 相似文献
15.
16.
We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included. 相似文献
17.
Mercury's capture into the 3:2 spin–orbit resonance can be explained as a result of its chaotic orbital dynamics. One major objective of MESSENGER and BepiColombo spatial missions is to accurately measure Mercury's rotation and its obliquity in order to obtain constraints on internal structure of the planet. Analytical approaches at the first-order level using the Cassini state assumptions give the obliquity constant or quasi-constant. Which is the obliquity's dynamical behavior deriving from a complete spin–orbit motion of Mercury simultaneously integrated with planetary interactions? We have used our SONYR model (acronym of Spin–Orbit N-bodY Relativistic model) integrating the spin–orbit N-body problem applied to the Solar System (Sun and planets). For lack of current accurate observations or ephemerides of Mercury's rotation, and therefore for lack of valid initial conditions for a numerical integration, we have built an original method for finding the libration center of the spin–orbit system and, as a consequence, for avoiding arbitrary amplitudes in librations of the spin–orbit motion as well as in Mercury's obliquity. The method has been carried out in two cases: (1) the spin–orbit motion of Mercury in the 2-body problem case (Sun–Mercury) where an uniform precession of the Keplerian orbital plane is kinematically added at a fixed inclination (S2K case), (2) the spin–orbit motion of Mercury in the N-body problem case (Sun and planets) (Sn case). We find that the remaining amplitude of the oscillations in the Sn case is one order of magnitude larger than in the S2K case, namely 4 versus 0.4 arcseconds (peak-to-peak). The mean obliquity is also larger, namely 1.98 versus 1.80 arcminutes, for a difference of 10.8 arcseconds. These theoretical results are in a good agreement with recent radar observations but it is not excluded that it should be possible to push farther the convergence process by drawing nearer still more precisely to the libration center. We note that the dynamically driven spin precession, which occurs when the planetary interactions are included, is more complex than the purely kinematic case. Nevertheless, in such a N-body problem, we find that the 3:2 spin–orbit resonance is really combined to a synchronism where the spin and orbit poles on average precess at the same rate while the orbit inclination and the spin axis orientation on average decrease at the same rate. As a consequence and whether it would turn out that there exists an irreducible minimum of the oscillation amplitude, quasi-periodic oscillations found in Mercury's obliquity should be to geometrically understood as librations related to these synchronisms that both follow a Cassini state. Whatever the open question on the minimal amplitude in the obliquity's oscillations and in spite of the planetary interactions indirectly acting by the solar torque on Mercury's rotation, Mercury remains therefore in a stable equilibrium state that proceeds from a 2-body Cassini state. 相似文献
18.
We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies. 相似文献
19.
We analyze the relationship between the mass of a spherical component and the minimum possible thickness of stable stellar disks. This relationship for real galaxies allows the lower limit on the dark halo mass to be estimated (the thinner the stable stellar disk is, the more massive the dark halo must be). In our analysis, we use both theoretical relations and numerical N-body simulations of the dynamical evolution of thin disks in the presence of spherical components with different density profiles and different masses. We conclude that the theoretical relationship between the thickness of disk galaxies and the mass of their spherical components is a lower envelope for the model data points. We recommend using this theoretical relationship to estimate the lower limit for the dark halo mass in galaxies. The estimate obtained turns out to be weak. Even for the thinnest galaxies, the dark halo mass within four exponential disk scale lengths must be more than one stellar disk mass. 相似文献
20.
Juan F. Navarro 《Celestial Mechanics and Dynamical Astronomy》2018,130(2):16
The aim of this article is to present a method for the integration of the equations of motion of the N-body ring problem by means of recurrent power series. We prove that the solution is convergent for any set of initial conditions, excluding those corresponding to binary collisions. 相似文献