The global solution of the N-body problem

The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum.A new blowing up transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum.The main result in this paper has appeared in Chinese in Acta Astro. Sinica. 26 (4), 313–322. In this version some mistakes have been rectified and the problems we solved are now expressed in a much clearer fashion.     

Hip-hop solutions of the 2N-body problem

Hip-hop solutions of the 2N-body problem with equal masses are shown to exist using an analytic continuation argument. These solutions are close to planar regular 2N-gon relative equilibria with small vertical oscillations. For fixed N, an infinity of these solutions are three-dimensional choreographies, with all the bodies moving along the same closed curve in the inertial frame.     

Implementation of an N-body code in a HP 1000 computer

We describe the implementation of Aarseth's NBODY2 code on a HP 1000 computer. We use the Extended Memory Array (EMA) feature with this code in order to investigate problems that include several hundreds of bodies, but the use of EMA requires some care in order to avoid large increases in computing time. The Vector Instruction Set (VIS) feature, a group of arithmetic subroutines that operates on arrays of floating point numbers and significantly reduces the computing time, turned out to be of little value for this application. We present the computing times demanded by two different problems for a variety of programs, including EMA and VIS. Finally, we present mass loss and mass accretion results for several simulations of galaxy-galaxy encounters performed with our implementation of the NBODY2 code.     

On the global solution of the N-body problem

In connection with the publication (Wang Qiu-Dong, 1991) the Poincaré type methods of obtaining the maximal solution of differential equations are discussed. In particular, it is shown that the Wang Qiu-Dong'sglobal solution of the N-body problem has been obtained in Babadzanjanz (1979). First the more general results on differential equations have been published in Babadzanjanz (1978).     

The homographic solutions in the N-body problem with a general attraction

The existence of homographic solutions of the N-body problem with a geneva attraction is verified, and the way which leads to obtaining certain types of homographic solutions is indicated. Basic properties of the solutions, such as the relations between the dynamical quantities and the initial conditions are presented. Furthermore, we proved that, for k is not equal to 3, if a homographic solution is not planar, it must be homothetic. And in this case, another important conclusion is that the configurations corresponding to any homographic solution are central configurations. Finally, we showed that along each homographic solution, motion of any individual mass point observes the same rules as the ones observed by mass points of a certain two-body system.     

A class of periodic solutions of the N-body problem

We prove existence and multiplicity of T-periodic solutions (for any given T) for the N-body problem in ^m (any m 2) where one of the bodies has mass equal to 1 and the others have masses ₂,..., _N , small. We find solutions such that the body of mass 1 moves close to x = 0 while the body of mass _i moves close to one of the circular solutions of the two body problem of period T/k _i, where ki is any odd number. No relation has to be satisfied by k ₂,...,k _N.     

Using the transition to action variables of the newtonian problem in the numerical solution of the N-body problem

We propose an approach for overcoming the problem of close encounters in collisional systems, globular and open star clusters. As is well known, the numerical integration step in such systems, for example, during the formation of close binary stars, begins to fragment and the rate of calculations goes down to a complete stop. We show that using the perturbation theory in the proposed approach, one can isolate the singularity and to increase considerably the integration step without losing the physical effects that affect significantly the evolution of star clusters.     

On the number of central configurations in the N-body problem

Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m ₁,..., m _n we denote by N(m ₁, ..., m _n, k) the number of central configurations' of the n-body problem in ^k modulus dilatations and rotations. If m _{n 1},..., m _n, k) is finite, then we give a bound of N(m ₁,..., m _n, k) which only depends of n and k.     

Boltzmann equation approach to the N-body problem with masses varying according to the Eddington-Jeans law

We study special aspects of the N-body problem with masses varying according to the Eddington-Jeans law with powers n=2 and 3. Our main result is that a particular set of variables can be found that allows one to write the pertinent Boltzmann and Poisson equations in a fashion similar to that corresponding to the usual fixed mass situation.     

The validity of the continuum limit in the gravitational N-body problem

This paper attempts to give quantitative as well as qualitative answers to the question of the analogy between smooth potentials and N-body systems. A number of simulations were performed in both integrable and nonintegrable smooth environments and their frozen N-body analogues, and comparisons were made using a number of different tools. The comparisons took place on both statistical and pointwise levels. The results of this study suggest that microscopic chaos associated with discreteness effects is always present in N-body configurations. This chaos is different from the macroscopic chaos which is associated with the bulk potential and persists even for very large N. Although the Lyapunov exponents of orbits evolving in N-body environments do not decrease as N increases, comparisons associated with the statistical properties, as well as with the power spectra of the orbits, affirm the existence of the continuum limit.     

Solving linearized equations of the N-body problem using the Lie-integration method

Several integration schemes exist to solve the equations of motion of the N -body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s, this method was applied to solve the equations of motion of the N -body problem by giving the recurrence formulae for the calculation of the Lie-terms. The aim of this work is to present the recurrence formulae for the linearized equations of motion of N -body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step-size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30–40 per cent faster than other integration methods.     

NBSymple,a double parallel,symplectic N-body code running on graphic processing units

We present and discuss the characteristics and performance, both in term of computational speed and precision, of a numerical code which integrates the equation of motions of N ‘particles’ interacting via Newtonian gravitation and move in an external galactic smooth field. The force evaluation on every particle is done by mean of direct summation of the contribution of all the other system’s particles, avoiding truncation error. The time integration is done with second-order and sixth-order symplectic schemes. The code, NBSymple, has been parallelized twice, by mean of the Compute Unified Device Architecture (CUDA) to make the all-pair force evaluation as fast as possible on high-performance Graphic Processing Units NVIDIA TESLA C1060, while the O(N) computations are distributed on various CPUs by mean of OpenMP Application Program. The code works both in single-precision floating point arithmetics or in double precision. The use of single-precision allows the use of the GPU performance at best but, of course, limits the precision of simulation in some critical situations. We find a good compromise in using a software reconstruction of double-precision for those variables that are most critical for the overall precision of the code. The code is available on the web site astrowww.phys.uniroma1.it/dolcetta/nbsymple.html.     

The stability of vertical motion in the N-body circular Sitnikov problem

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.     

Minimum energy configurations in the N-body problem and the celestial mechanics of granular systems

Minimum energy configurations in Celestial Mechanics are investigated. It is shown that this is not a well defined problem for point-mass celestial mechanics but well-posed for finite density distributions. This naturally leads to a granular mechanics extension of usual Celestial Mechanics questions such as relative equilibria and stability. This paper specifically studies and finds all relative equilibria and minimum energy configurations for N?=?1, 2, 3 and develops hypotheses on the relative equilibria and minimum energy configurations for N ? 1 bodies.     

GalevNB: a conversion from N-body simulations to observations

We present GalevNB(Galev for N-body simulations), a utility that converts fundamental stellar properties of N-body simulations into observational properties using the GALEV(GAlaxy EVolutionary synthesis models) package, and allowing direct comparisons between observations and N-body simulations.It works by converting fundamental stellar properties, such as stellar mass, temperature, luminosity and metallicity into observational magnitudes for a variety of filters used by mainstream instruments/telescopes,such as HST, ESO, SDSS, 2MASS, etc., and into spectra that span the range from far-UV(90 ) to near-IR(160 μm). As an application, we use GalevNB to investigate the secular evolution of the spectral energy distribution(SED) and color magnitude diagram(CMD) of a simulated star cluster over a few hundred million years. With the results given by GalevNB we discover a UV-excess in the SED of the cluster over the whole simulation time. We also identify four candidates that contribute to the FUV peak: core helium burning stars, second asymptotic giant branch(AGB) stars, white dwarfs and naked helium stars.     

Strong Chaos in N-body problem and Microcanonical Thermodynamics of Collisionless Self Gravitating Systems

An effective Microcanonical Thermodynamics of self gravitating systems(SGS) is proposed, analyzing the well known obstacles thought to prevent the formulation of a rigorous Statistical Mechanics (SM), as those due to the formal unboundedness of available phase space and to the unscreened, long range, nature of the interaction. The latter feature entails the well known inequivalence of statistical ensembles, puts clearly into question the meaning, for these systems, of the Thermodynamic Limit, and rules out the use of canonical and grand-canonical ensembles. As to the first obstacle, we argue nevertheless that a hierarchy of timescales exist such that, at any finite time, the volume of the effectively available region of phase space is indeed finite, and that the dynamics satisfies a strong chaos criterion, leading to a fast, increasingly uniform, spreading of orbits over an effectively invariant subset of the constant (N,V,E) surface; thus leading to the definition of a secularly evolving, generalized microcanonical ensemble, which allows to define an (almost extensive) effective entropy and to derive self-consistent definitions for other thermodynamic variables, giving thus an orthode for SGS. Moreover, a Second Law-like criterion allows to single out the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of SGS. This revised version was published online in July 2006 with corrections to the Cover Date.     

Application of the restricted hyperbolic three-body problem to a star-sun-comet system

The equations of motion for a third body of small mass are developed in the problem where the two primary bodies are in hyperbolic orbits about each other. The equations are applied to a hypothetical star-sun-comet system to determine the effect of the stellar encounter on the orbit of the comet.This paper is part of a doctoral thesis completed at the University of Illinois at Urbana-Champaign.     

N-body simulations of the Magellanic stream

A suite of high-resolution N -body simulations of the Magellanic Clouds–Milky Way system are presented and compared directly with newly available data from the H  i Parkes All-Sky Survey (HIPASS). We show that the interaction between Small Magellanic Clouds (SMC) and Large Magellanic Clouds results in both a spatial and kinematical bifurcation of both the stream and the leading arm. The spatial bifurcation of the stream is readily apparent in the HIPASS data, and the kinematical bifurcation is also tentatively identified. This bifurcation provides strong support for the tidal disruption origin for the Magellanic stream. A fiducial model for the Magellanic Clouds (MCs) is presented upon completion of an extensive parameter survey of the potential orbital configurations of the MCs and the viable initial boundary conditions for the disc of the SMC. The impact of the choice of these critical parameters upon the final configurations of the stream and leading arm is detailed.