Families Of Asymmetric Periodic Orbits Research Articles

Inclined asymmetric librations in exterior resonances

Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $\pi$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincare surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L1, L2 and L3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.

Chaos, Order, and Periodic Orbits in 3 : 1 Resonant Planetary Dynamics

This study addresses the long-term evolution of possible two-planet extrasolar systems that are initially trapped in 3:1 mean motion resonance. A planar general three-body problem model is used, and its resonant dynamics are examined by computing periodic orbits in a rotating frame, Poincare maps, and maps of dynamical stability. We computed the families of symmetric resonant periodic orbits that obey bifurcations, giving rise to families of asymmetric periodic orbits. The linear stability of such orbits has also been computed, and their relation to the long-term stability of their nearby phase-space domain has been studied. The maps of dynamical stability reveal a complicated structure in the phase space, where chaos and order coexist and alternate as the initial eccentricities or the phases of the planets change. The regular orbits are classified into various types according to the librating or rotating evolution of the resonant angles. Apsidal symmetric librations are common in the domain of resonant motion, but asymmetric ones are associated exclusively with the existence of asymmetric periodic orbits. Such a stable asymmetric configuration seems to correspond to the companions b and c of the 55 Cnc extrasolar system, which are trapped in the 3:1 mean motion resonance according to the study of McArthur et al. However, a recent study by Fischer et al. shows the existence of a new planet (the companion f) in the system, and that the planets b and c are not in mean motion resonance.

Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m 1>m 2, the system HD82943 for the new masses, for the case m 1=m 2 and the same system HD82943 for the values of the masses m 1<m 2 given in previous work. In the case m 1≥ m 2 there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m 1<m 2 all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by $$\varpi_2-\varpi_1$$ and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.

Evolution with the mass parameter of families of asymmetric periodic solutions of the restricted three body problem

The two asymmetric bifurcations associated with the exterior commensurabilities of the formq+1: 1 are found to exist forq=1, 2, 3, 4 throughout the entire range of the mass parameter μ. The asymmetric families emanating from these bifurcation orbits for values of μ=0.05, 0.15, 0.25, 0.35, and 0.45 are completely determined. For μ=0.05 and μ=0.15 for each of the four pairs of critical orbits the family of asymmetric periodic orbits evolves continuously from one bifurcation orbit to the other. The families evolving from each pair of bifurcations for μ=0.25, 0.35, and 0.45 terminate on an asymptotic orbit to one of the two equilateral equilibrium points. The evolution of these four families of asymmetric periodic orbits therefore changes significantly for some value of μ between 0.15 and 0.25. An indication is given on how the evolution with μ of familyl and its branches can be studied.

Numerical investigation of the manifold I qp in the Sun-Jupiter case of the restricted three-body problem

Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.