On periodic orbits and resonance in extrasolar planetary systems

We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω ₂ ? ω ₁) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω ₂ ? ω ₁) librates or rotates are not different, but are closely connected to the exact periodic motion.     

2/1 resonant periodic orbits in three dimensional planetary systems

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50^?–60^?, which may be related with the existence of real planetary systems.     

Periodic orbits in three-dimensional planetary systems

Three-dimensional planetary systems are studied, using the model of the restricted three-body problem for Μ =.001. Families of three-dimensional periodic orbits of relatively low multiplicity are numerically computed at the resonances 3/1, 5/3, 3/5 and 1/3 and their stability is determined. The three-dimensional orbits are found by continuation to the third dimension of the vertical critical orbits of the corresponding planar problem     

Stability of planetary orbits in binary systems

The possible existence of stable orbits is investigated in binary systems using Hill's method. Analytical stability conditions are established for satellites, for inner planets and for outer planets, allowing arbitrary values for the mass-ratio of the binary.Presented at the Symposium Star Catalogues, Positional Astronomy and Celestial Mechanics, held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978.     

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60^° of mutual planetary inclination, but in most families, the stability does not exceed 20^°–30^°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.     

Stochasticity of planetary orbits in double star systems. II

Cosmogonical theories as well as recent observations allow us to expect the existence of numerous exo-planets, including in binaries. Then arises the dynamical problem of stability for planetary orbits in double star systems. Modern computations have shown that many such stable orbits do exist, among which we consider orbits around one component of the binary (called S-type orbits). Within the framework of the elliptic plane restricted three-body problem, the phase space of initial conditions for fictitious S-type planetary orbits is systematically explored, and limits for stability had been previously established for four nearby binaries which components are nearly of solar type. Among stable orbits, found up to distance of their sun of the order of half the binarys periastron distance, nearly-circular ones exist for the three binaries (among the four) having a not too high orbital eccentricity. In the first part of the present paper, we compare these previous results with orbits around a 16 Cyg B-like binarys component with varied eccentricities, and we confirm the existence of stable nearly-circular S-type planetary orbits but for very high binarys eccentricity. It is well-known that chaos may destroy this stability after a very long time (several millions years or more). In a first paper, we had shown that a stable planetary orbit, although chaotic, could keep its stability for more than a billion years (confined chaos). Then, in the second part of the present paper, we investigate the chaotic behaviour of two sets of planetary orbits among the stable ones found around 16 Cyg B-like components in the first part, one set of strongly stable orbits and the other near the limit of stability. Our results show that the stability of the first set is not destroyed when the binarys eccentricity increases even to very high values (0.95), but that the stability of the second set is destroyed as soon as the eccentricity reaches the value 0.8.     

On the 2/1 resonant planetary dynamics – periodic orbits and dynamical stability

The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general problem is studied in a systematic way. Starting from the Keplerian unperturbed system, we obtain the resonant families of the circular restricted problem. Then, we find all the families of the resonant elliptic restricted three-body problem, which bifurcate from the circular model. All these families are continued to the general three-body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio ρ. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values  [ρ∈ (0, ∞)]  and, therefore we include the passage from external to internal resonances. Thus, we can obtain all possible stable configurations in a systematic way. As an application, we consider the dynamics of four known planetary systems at the 2/1 resonance and we examine if they are associated with a stable periodic orbit.     

Radiation forces and the formation of planetary systems

We briefly support on some new results about the influence of the rotation and finite size of a stellar radiation source on dust particle orbits, emphasizing the possibility of stable orbits, in the equatorial plane, for dust sizes near the radiation pressure limit.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

The physicochemical mechanism of the formation of planetary systems

The planets and their satellites are formed in accordance with similar mechanisms as a result of spatially periodic condensation of gaseous matter during the formation of the central body.Using the diffusion theory one can calculate the age of the planets and explain the nature of the Titius-Bode law.     

Numerical experiments on planetary orbits in double stars

This is a numerical study of orbits in the elliptic restricted three-body problem concerning the dependence of the critical orbits on the eccentricity of the primaries. They are defined as being the separatrix between stable and unstable single periodic orbits. As our results are adapted to the existence of planetary orbits in double stars we concentrated first on the P-orbits (defined to surround both primaries). Due to the complexity of the elliptic problem there is no analytical approach possible. Using the results of some 300 integrated orbits for 10³ to 3. 10³ periods of the primaries we established lower and upper bounds for the critical orbits for different values of the eccentricity.     

Intrinsic stability of periodic orbits

Families of orbits of a conservative, two degree-of-freedom system are represented by an unsteady velocity field with componentsu(x, y, t) andv(x, y, t). Intrinsic stability properties depend on velocity field divergence and curl, whose dynamical evolution is determined by a matrix Riccati equation. Near equilibrium, divergence-free or irrotational fields are dynamically compatible with the conservative force field. It is shown that a necessary condition for stable periodic orbits is satisfied when the orbitaveraged divergence is zero, which results in bounded normal variations. A sufficient condition for stability is derived from the requirement that tangential variations do not exhibit secular growth.In a steady, divergence-free field, velocity component functionsu(x, y) andv(x, y) may be continuedanalytically from any initial condition, except when velocity is parallel to U or at equilibria. In an unsteady field, the orbit-averaged divergence is zero when the vorticity function is periodic. When such a field exists, initial conditions for stable periodic orbits (i.e., characteristic loci) may be determinedanalytically.     

Characteristics of resonant periodic orbits

We study the families of periodic orbits in a time-independent two-dimensional potential field symmetric with respect to both axes. By numerical calculations we find characteristic curves of several families of periodic orbits when the ratio of the unperturbed frequencies isA ^1/2/B ^1/2=2/1. There are two groups of characteristic curves: (a) The basic characteristic and the characteristics which bifurcate from it. (b) The characteristics which start from the boundary line and the axisx=0.     

Periodic orbits of the planetary type and their stability

A review is presented of periodic orbits of the planetary type in the general three-body problem and fourbody problem and the restricted circular and elliptic tnreebody problem. These correspond to planetary systems with one Sun and two or three planets (or a planet and its satellites), the motion of asteoids and also planetary systems with two Suns. The factors which affect the stability of the above configurations are studied in connection with resonance or additional perturbations. Finally, the correspondence of the periodic orbits in the restricted three-body problem with the fixed points obtained by the method of averaging or the method of surface of section is indicated.     

An approach to the radial distribution of planetary orbits

The problem of determination of the radial distribution of the planetary orbits is approached under the assumption that the average present radial sizes of the orbits were already determined when the protoplanetary cloud flattened by initial angular momentum aggregated into a set of concentric rings from which the planetary material was ultimately collected. The object of this argument is to derive a consistent stationary distribution of orbits so that the problem of the non-stationary formation of the orbital rings is not here considered. Under the flattening assumption the 3D Poisson equation is replaced by the 2D Helmholtz equation (inhomogeneous) which is solved by use of an averaging theorem generalization of the well-known averaging theorem for the homogeneous Helmholtz equation. Augmenting the ring potentials obtained by specializing the mass distribution in the disk by a solar potential term and a rotational potential, differentiation leads to a generalization of the Kepler 3D law suitable for the many-body problem of a solar system with circular orbits. In this way a system of transcendental equations involving Bessel functions of the first and second kind are obtained which must be satisfied by the orbital radii. Naturally the restriction to circular orbits represents only an approximation to the orbital determination problem, but considering that no arguments have previously been available for the determination even of circular orbits it would seem to represent an advance.     

Analytical method for the effects of the asteroid belt on planetary orbits

Analytic expressions are derived for the perturbation of planetary orbits due to a thick constant density asteroid belt. The derivations include extensions and adaptations of Plakhov's analytic expressions for the perturbations in five of the orbital elements for closed orbits around Saturn's rings. The equations of Plakhov are modified to include the effect of ring thickness and additional equations are derived for the perturbations in the sixth orbital element, the mean anomaly. The gravitational potential and orbital perturbations are derived for the asteroid belt with and without thickness, and for a hoop approximation to the belt. The procedures are also applicable to Saturn's rings and the newly discovered rings of Uranus.The effects of the asteroid belt thickness on the gravitational potential coefficients and the orbital motions are demonstrated. Comparisons between the Mars orbital perturbations obtained using the analytic expressions and those obtained using numerical integration are discussed. The effects of the asteroid belt on the Earth based ranging to Mars are also demonstrated.     

Stability of resonant planetary orbits in binary stars

This paper contains a numerical study of the stability of resonant orbits in a planetary system consisting of two planets, moving under the gravitational attraction of a binary star. Its results are expected to provide us with useful information about real planetary systems and, at the same time, about periodic motions in the general four-body problem (G4) because the above system is a special case of G4 where two bodies have much larger masses than the masses of the other two (planets). The numerical results show that the main mechanism which generates instability is the destruction of the Jacobi integrals of the massless planets when their masses become nonzero and that resonances in the motion of planets do not imply, in general, instability. Considerable intervals of stable resonant orbits have been found. The above quantitative results are in agreement with the existing qualitative predictions     

Analytic continuation of periodic orbits from equilibria of two-degree-of-freedom systems in rotating coordinates

A new theory is formulated for the analytic continuation of periodic (and aperiodic) orbits from equilibrium solutions of a two-degree-of-freedom dynamical system in rotating coordinates:% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acbiGab8xDayaacaGaa8xlaiaa-jdacaWFUbGaeqyXduNaa8xpaiaa% -zfadaWgaaWcbaGccaWF4baaleqaaOGaaiilaiqbew8a1zaacaGaey% 4kaSIaaGOmaiaad6gacaWG1bGaeyypa0Jaa8NvamaaBaaaleaakiaa% -LhaaSqabaGccaGGSaGabmiEayaacaGaeyypa0JaamyDaiaacYcace% WG5bGbaiaacqGH9aqpcqaHfpqDaaa!54CD!\[\dot u - 2n\upsilon = V_x ,\dot \upsilon + 2nu = V_y ,\dot x = u,\dot y = \upsilon \]Away from resonance, a family of nonlinear, normal-mode orbits defines an autonomous velocity field u(x, y), u(x, y) represented by convergent algebraic-series expansions in the two position variables. This approach is useful for determining the global structure of solution curves and nonlinear stability of normal modes using Liapunov's direct method. At resonance, the series coefficients are time dependent because stationary modes are incompatible with the equations of motion. By eliminating small divisors, explicit time dependence provides a natural transition from non-resonance to resonance cases within the same theory.     

Keplerian periodic orbits in the isosceles problem

The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision. The basic tool used is the study of the Poincaré maps associated with the periodic solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

The present status of periodic orbits

Recent results on periodic orbits are presented and it is shown that the periodic orbits can be used in the study of planetary systems and triple or multiple stellar systems. Triple stellar systems are stable even for close approaches of the three components. Also stable triple systems exist with nearly zero angular momentum. For the planetary systems a global view is obtained from which it is clear which configurations are stable or unstable and also what factors affect the stability. Also, the relation between resonance and instability is studied by making use of periodic orbits.