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1.
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
2.
Esther Barrabés Josep Maria Cors Conxita Pinyol Jaume Soler 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):55-66
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. 相似文献
3.
R. F. Arenstorf 《Celestial Mechanics and Dynamical Astronomy》1978,17(4):331-355
Letn2 mass points with arbitrary masses move circularly on a rotating straight-line central-configuration; i.e. on a particular solution of relative equilibrium of then-body problem. Replacing one of the mass points by a close pair of mass points (with mass conservation) we show that the resultingN-body problem (N=n+1) has solutions, which are periodic in a rotating coordinate system and describe precessing nearlyelliptic motion of the binary and nearlycircular collinear motion of its center of mass and the other bodies; assuming that also the mass ratio of the binary is small. 相似文献
4.
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. 相似文献
5.
The oldest open clusters in our Galaxy set the lower limit to the age of the Galactic Disk (9–10 Gyr). Although they appear
to be very rich now, it is clear that their primordial populations were much larger. Often considered as transitional objects,
these populous open clusters show structural differences with respect to globular clusters so their dynamics and characteristic
evolutionary time scales can also be different. On the other hand, their large membership lead to different dynamical evolution
as compared with average open clusters. In this paper, the differential features of the evolution of rich open clusters are
studied using N-body simulations, including several of the largest (104 stars) published direct collisional N-body calculations so far, which were performed on a CRAY YMP. The disruption rate of rich open clusters is analysed in detail
and the effect of the initial spatial distribution of the stars in the cluster on its dynamics is studied. The results show
that cluster life-time depends on this initial distribution, decreasing when it is more concentrated. The effect of stellar
evolution on the dynamical evolution of rich clusters is an important subject that also has been considered here. We demonstrate
that the cluster's life-expectancy against evaporation increases because of mass loss by evolving high-mass stars.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
Angelo B. Mingarelli Chiara M. F. Mingarelli 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):391-401
In an effort to understand the nature of almost periodic orbits in the n-body problem (for all time t) we look first to the more basic question of the oscillatory nature of solutions of this problem (on a half-line, usually
taken as R
+). Intimately related to this is the notion of a conjugate point(due to A. Wintner) of a solution. Specifically, by rewriting the mass unrestricted general problem of n-bodies in a symmetric form we prove that in the gravitational Newtonian n-body problem with collisionless motions there exists arbitrarily large conjugate points in the case of arbitrary (positive)
masses whenever the cube of the reciprocal of at least one of the mutual distances is not integrable at infinity. The implication
of this result is that there are possibly many Wintner oscillatorysolutions in these cases (some of which may or may not be almost periodic). As a consequence, we obtain sufficient conditions
for all continuable solutions (to infinity) to be either unbounded or to allow for near misses (at infinity). The results
also apply to potentials other than Newtonian ones. Our techniques are drawn from results in systems oscillation theory and
are applicable to more general situations.
Dedicated to the memory of Robert M. (Bob) Kauffman, formerly Professor of the
University of Alabama in Birmingham 相似文献
7.
We have used merger-trees realizations to study the formation of dark matter haloes. The construction of merger-trees is based
on three different pictures about the formation of structures in the Universe. These pictures include the spherical collapse
(SC), the ellipsoidal collapse (EC) and the non-radial collapse (NR). The reliability of merger-trees has been examined comparing
their predictions related to the distribution of the number of progenitors, as well as the distribution of formation times,
with the predictions of analytical relations. The comparison yields a very satisfactory agreement. Subsequently, the mass-growth
histories (MGH) of haloes have been studied and their formation scale factors have been derived. This derivation has been
based on two different definitions that are (a) the scale factor when the halo reaches half its present day mass and (b) the
scale factor when the mass-growth rate falls below some specific value. Formation scale factors follow approximately power
laws of mass. It has also been shown that MGHs are in good agreement with models proposed in the literature that are based
on the results of N-body simulations. The agreement is found to be excellent for small haloes but, at the early epochs of the formation of large
haloes, MGHs seem to be steeper than those predicted by the models based on N-body simulations. This rapid growth of mass of heavy haloes is likely to be related to a steeper central density profile
indicated by the results of some N-body simulations. 相似文献
8.
New doubly-symmetric families of comet-like periodic orbits in the spatial restricted (N + 1)-body problem 总被引：1，自引：0，他引：1
For any positive integer N ≥ 2 we prove the existence of a new family of periodic solutions for the spatial restricted (N +1)-body problem. In these solutions the infinitesimal particle is very far from the primaries. They have large inclinations
and some symmetries. In fact we extend results of Howison and Meyer (J. Diff. Equ. 163:174–197, 2000) from N = 2 to any positive integer N ≥ 2.
相似文献
9.
We prove the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic
invariant tori for (an “outer solar-system” model of) the planar (N + 1)-body problem.
相似文献
10.
Christopher McCord 《Celestial Mechanics and Dynamical Astronomy》2004,89(2):99-118
In the N-body problem, it is a simple observation that relative equilibria (planar solutions for which the mutual distances between
the particles remain constant) have constant moment of inertia. In 1970, Don Saari conjectured that the converse was true:
if a solution to the N-body problem has constant moment of inertia, then it must be a relative equilibrium. In this note, we confirm the conjecture
for the planar three-body problem with equal masses.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
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
12.
The aim of this paper is to study the global geometry of non-planar 3-body motions in the realms of equivariant Differential
Geometry and Geometric Mechanics. This work was intended as an attempt at bringing together these two areas, in which geometric
methods play the major role, in the study of the 3-body problem. It is shown that the Euler equations of a three-body system
with non-planar motion introduce non-holonomic constraints into the Lagrangian formulation of mechanics. Applying the method
of undetermined Lagrange multipliers to study the dynamics of three-body motions reduced to the level of moduli space [`(M)]{\bar{M}} subject to the non-holonomic constraints yields the generalized Euler-Lagrange equations of non-planar three-body motions
in [`(M)]{\bar{M}} . As an application of the derived dynamical equations in the level of [`(M)]{\bar{M}} , we completely settle the question posed by A. Wintner in his book [The analytical foundations of Celestial Mechanics, Sections
394–396, 435 and 436. Princeton University Press (1941)] on classifying the constant inclination solutions of the three-body problem. 相似文献
13.
How the Method of Minimization of Action Avoids Singularities 总被引：4，自引：0，他引：4
C. Marchal 《Celestial Mechanics and Dynamical Astronomy》2002,83(1-4):325-353
The method of minimization of action is a powerful technique of proving the existence of particular and interesting solutions of the n-body problem, but it suffers from the possible interference of singularities. The minimization of action is an optimization and, after a short presentation of a few optimization theories, our analysis of interference of singularities will show that:(A) An n-body solution minimizing the action between given boundary conditions has no discontinuity: all n-bodies have a continuous and bounded motion and thus all eventual singularities are collisions;(B) A beautiful extension of Lambert's theorem shows that, for these minimizing solutions, no double collision can occur at an intermediate time;(C) The proof can be extended to triple and to multiple collisions. Thus, the method of minimization of action leads to pure n-body motions without singularity at any intermediate time, even if one or several collisions are imposed at initial and/or final times.This method is suitable for non-infinitesimal masses only. Fortunately, a similar method, with the same general property with respect to the singularities, can be extended to n-body problems including infinitesimal masses. 相似文献
14.
Florin Nicolae Diacu 《Celestial Mechanics and Dynamical Astronomy》1989,46(2):119-128
Relations between the rectilinear, collinear and syzygy solutions of the N-body problem are first pointed out. It is shown that, along a solution, the set of the non-collinear syzygy configuration instants is formed by isolated points. Then we restrict the study to the planar 3-body problem and prove that for Dirichlet-stable solutions, a non-syzygy solution cannot be as close as possible to a syzygy one. It is also true that, in the case of a syzygy solution, the orbit of one particle crosses the line of the other two and can not be tangent to this line in the transition point. Finally we prove that the set of initial conditions leading to non-collinear syzygy solutions is non-empty and open. 相似文献
15.
The chain regularization method (Mikkola and Aarseth 1990) for high accuracy computation of particle motions in smallN-body systems has been reformulated. We discuss the transformation formulae, equations of motion and selection of a chain of interparticle vectors such that the critical interactions requiring regularization are included in the chain. The Kustaanheimo-Stiefel (KS) coordinate transformation and a time transformation is used to regularize the dominant terms of the equations of motion. The method has been implemented for an arbitrary number of bodies, with the option of external perturbations. This formulation has been succesfully tested in a generalN-body program for strongly interacting subsystems. An easy to use computer program, written inFortran, is available on request. 相似文献
16.
Sergei M. Poleshchikov 《Celestial Mechanics and Dynamical Astronomy》2003,85(4):341-393
The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position. 相似文献
17.
Joseph J. F. Liu Philip M. Fitzpatrick 《Celestial Mechanics and Dynamical Astronomy》1975,12(4):463-487
Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w
0)2, wheren is the satellite's orbital mean motion andw
0 its initial rotational angular speed. 相似文献
18.
LetN2 mass points (primaries) move on a collinear solution of relative equilibrium of theN-body problem; i.e. suitably fixed on a uniformly rotating straight line. Consider the motion of a massless particle in the gravitational field of these primaries with arbitrarily given masses. An existence proof for periodic solutions (i.e. closed trajectories in a rotating coordinate system) will be given, in which the particle performs nearly keplerian elliptic motions about (and close to) any one of the primaries. 相似文献
19.
G. Antonacopoulos 《Astrophysics and Space Science》1979,62(1):217-224
In this paper we give the Hamiltonian function for aN-body system up to the 2-P.N.A. Then as an example, from the LagrangianL
m
of a test particle we derive the equations of its motion up to the 2-P.N.A. in the field of a heavy bodym
2at rest. 相似文献
20.
Celestial coordinate reference systems in curved space-time 总被引：1，自引：1，他引：0
S. M. Kopejkin 《Celestial Mechanics and Dynamical Astronomy》1988,44(1-2):87-115