Restricted N-body Problem Research Articles

Dynamical friction and feedback on galactic bars in the general fast–slow regime

ABSTRACT Current theories of dynamical friction on galactic bars are based either on linear perturbation theory, which is valid only in the fast limit where the bar changes its pattern speed rapidly, or on adiabatic theory, which is applicable only in the slow limit where the bar’s pattern speed is near-constant. In this paper, we study dynamical friction on galactic bars spinning down at an arbitrary speed, seamlessly connecting the fast and slow limits. We treat the bar–halo interaction as a restricted N-body problem and solve the collisionless Boltzmann equation using the fast-angle-averaged Hamiltonian. The phase-space distribution and density wakes predicted by our averaged model are in excellent agreement with full 3D simulations. In the slow regime where resonant trapping occurs, we show that, in addition to the frictional torque, angular momentum is transferred directly due to the migration of the trapped phase-space: trapped orbits comoving with the resonance typically gain angular momentum, while untrapped orbits leaping over the trapped island lose angular momentum. Due to the negative gradient in the distribution function, gainers typically outnumber the losers, resulting in a net negative torque on the perturber. Part of this torque due to the untrapped orbits was already identified by Tremaine and Weinberg, who named the phenomenon dynamical feedback. Here, we derive the complete formula for dynamical feedback, accounting for both trapped and untrapped orbits. Using our revised formula, we show that dynamical feedback can account for up to 30 per cent of the total torque on the Milky Way’s bar.

Oscillatory motions in restricted N-body problems

We consider the planar restricted N-body problem where the N−1 primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale's horseshoe. This implies the occurrence of chaotic motions, namely the oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first non-null perturbative term and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, and for a specific value of a parameter related to the angular momentum vector, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. When the Melnikov function corresponding to the principal part of the perturbation is zero we compute the next non-zero Melnikov function proving that it has simple zeroes. The theory is illustrated for various cases of restricted N-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed.

On the advantages of exploiting the hierarchical structure of astrodynamical models

In this paper an algorithm is developed that combines the capabilities and advantages of several different astrodynamical models of increasing complexity. Splitting these models in a strict hierarchical order yields a clearer grasp on what is available. With the effort of developing a comprehensive model overhead, the equations for the spacecraft motion in simpler models can be readily obtained as particular cases. The proposed algorithm embeds the circular and elliptic restricted three-body problems, the four-body bicircular and concentric models, an averaged n-body model, and, at the top hierarchic ladder, the full ephemeris SPICE-based restricted n-body problem. The equations of motion are reduced to the assignment of 13 time-varying coefficients, which multiply the states and the gravitational potential to reproduce the proper vector field. This approach yields an efficient and quick way to check solutions for different dynamics and parameters. We show that in bottom-up applications, a gradual increase of model complexity benefits accuracy, the chances of success and the convergence rate of a continuation algorithm. Case studies are simple periodic orbits and low-energy transfers.

Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three equilibrium Zn-orbits from which families of Lyapunov orbits emerge. Numerical continuation of these families using a boundary value formulation is used to construct the bifurcation diagram for the case n=7, also including some secondary and tertiary bifurcating families. We observe symmetry-breaking bifurcations in this system, as well as certain period-doubling bifurcations.

Dynamic Characteristics of Periodic Motions in a Restricted N-Body Problem

AbstractWe investigate the existence and follow the evolution and dynamic stability of periodic motions in a restricted version of the N-body problem, where the positions of the mass bodies reside persistently on a planar polygon ring and its geometrical centre. The results show the existence of both (linearly) stable and unstable periodic orbits around the body placed on the ring centre (inside the ring), around all the periphery bodies (outside the ring) and around one or more of the periphery bodies. The stable periodic trajectories form regions of gravitational stability, whereas some classes of the unstable orbits exhibit bifurcations to non-symmetric periodic orbits of the same period, and less often period-doubling bifurcations.

Existence and Structural Stability for Certain Configurations of Rotating Barotropic Fluids

We consider rotating configurations of one or more gas balls. We show that under a non-degeneracy condition these structures are structurally stable as solutions of the equilibrium equations, while not making any claims about their dynamic stability. Under a different non-degeneracy condition we show that we can replace the mass points of solutions of the restricted n-body problem by gas balls, if the scale is chosen sufficiently large.

Global bifurcation of planar and spatial periodic solutions in the restricted n-body problem

The paper deals with the study of a satellite attracted by n primary bodies, which form a relative equilibrium. We use orthogonal degree to prove global bifurcation of planar and spatial periodic solutions from the equilibria of the satellite. In particular, we analyze the restricted three body problem and the problem of a satellite attracted by the Maxwell’s ring relative equilibrium.

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) = z0 values, where z0 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 Poincare 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.

The role of close passages of galaxies and the asymmetry of their dark haloes in the formation of their spiral patterns

The influence of close passages of galaxies on the shapes of disk galaxies and the distribution of stars in them is studied for several types of interactions in the framework of the restricted N-body problem. Depending on the conditions adopted, either two spiral density waves or ring structures are formed in the stellar disk of the galaxy. These structures can generate star formation fronts with the corresponding shape, as are observed in disk galaxies. Our calculations can also be applied to study the influence of the passage of a nearby star on a protoplanetary disk. The formation of ring structures there could specify the type of planet formation in the outer regions of the planetary system and the distribution of semimajor axes for the planetary orbits. We use the same model to study the generation and evolution of spiral density waves in the stellar disks of galaxies as a result of the recently found asymmetry of the gravitational potential in the massive dark haloes in disk galaxies. The dipole component of the gravitational field of the halo can continuously permanently generate the spiral structure in disk galaxies.

Coalescence Problems in the Theory of Expanding Wrinkled Premixed Flames

Abstract In the framework of a Michelson-Sivashinsky (MS) evolution equation for the front shape we study cylindrically-expanding premixed flames, focusing attention on the spontaneous dynamics of collections of finite-amplitude wrinkles To begin with, we consider sharp crests described by pole-decomposed solutions to the MS equation, for which the flame dynamics is reduced to a finite set of Complex ODE's. The latter are simplified in the limit of large flame radii to result in a restricted N-body problem (with long-range interactions) for the real angular locations of the wrinkle crests. The corresponding statistical problem of coalescence/expansion competition is treated approximately, by a mean-field method, and yields analytical predictions for the cell-size distributions vs. time. Comparisons with spectral integrations of the MS equation and with a simulation of the restricted N-body problem reveal fair agreements Next we consider initial conditions outside the previous class, yet again representing...

Numerical mean elements for asteroid orbits

In the context of the search of asteroid families (i.e. identification of minor planets as potential fragments of an old bigger body), the calculation of proper elements plays an important role. They are quasi-invariants of the motion, obtained by a double averaging process of the restricted N-body problem; firstly the osculating elements are averaged over the short periodic terms (namely the longitudes of the asteroid and of the perturbing planets) so to get the mean elements, and secondly, the mean elements are averaged over the long periodic terms (longitudes of the pericenters and of the nodes of the asteroid and of the perturbing planets) to obtain the proper elements.

The restricted P + 2 body problem

In this paper we study a special case of the restricted n-body problem, called by us the restricted P + 2 body problem. The equilibrium configuration which the P + 1 bodies with mass form consists of one central mass encircled by a ring of P equally spaced particles of equal mass, the ring rotating at a specific angular velocity. We briefly discuss the stability of this configuration. We consider the dynamics of an infinitesimal mass under the influence of such a configuration. First the equilibrium points will be discussed, then the zero-velocity curves. We show that there are 3 P, 4 P or 5 P equilibrium points, depending on the ratio of the ring particle mass to the central body mass. Next motion about the equilibrium points is considered. We show that if the ring particle mass is small enough there will be P stable equilibrium points. Also if the number of particles, P, is large enough and the ratio of the ring particle mass to the central body mass is large enough there will be P different stable equilibrium points. Finally an analysis of the dynamics of the infinitesimal mass will be performed under the restriction that the particle does not cross or come close to the ring and lies in the plane of the ring. Under this restriction an approximate potential can be found which can be made arbitrarily close to the real potential under some circumstances. The dynamics of the particle under the approximate potential are integrable. We find a periodic orbit in this case with the Poincaré-Lindstedt method using the mass of the ring as a small parameter. The predictions from this approximate solution of the problem compare well with numerical integrations of the actual system.

Periodic orbits near infinity in the restrictedN-body problem

This paper shows that there exist two families of periodic solutions of the restrictedN-body problem which are close to large circular orbits of the Kepler problem. These solutions are shown to be of general elliptic type and hence are stable. If the restricted problem admits a symmetry, then there are symmetric periodic solutions which are close to large elliptic orbits of the Kepler problem.

Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the ( N − 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N − 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N − 1 particles move approximately on a large circular orbit of the two body problems and the remaining N − 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.