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.     

Stability of outer planetary orbits around binary stars: A comparison of Hill's and Laplace's stability criteria

A comparison is made between the stability criteria of Hill and that of Laplace to determine the stability of outer planetary orbits encircling binary stars. The restricted, analytically determined results of Hill's method by Szebehely and co-workers and the general, numerically integrated results of Laplace's method by Graziani and Black are compared for varying values of the mass parameter =m ₂/(m ₁+m ₂). For 00.15, the closest orbit (lower limit of radius) an outer planet in a binary system can have and still remain stable is determined by Hill's stability criterion. For >0.15, the critical radius is determined by Laplace's stability criterion. It appears that the Graziani-Black stability criterion describes the critical orbit within a few percent for all values of .     

Stable planetary orbits around one component in nearby binary stars. II

Numerical simulations are made within the framework of the plane restricted three-body problem, in order to find out if stable orbits for planets around one of the two components in double stars can exist. For any given set of initial parameters (the mass ratio of the two stars and the eccentricity of their orbit around each other), the phase-space of initial positions and velocities is systematically explored.In previous works, systematic exploration of the circular model as well as studies of more realistic (elliptic) cases such as Sun-Jupiter and the nearby Centauri and Sirius systems, large stable planetary orbits were found to exist around both components of the binary, up to distances from each star of the order or more than half the binary's periastron separation.The first results presented here for the Coronae Borealis system confirm the previous studies.     

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.     

Stable planetary orbits around one component in nearby binary stars

Numerical simulations are made within the frame of the elliptic planar restricted three-body problem, in order to search if stable orbits exist for planets around one of the two components in double stars. The values taken for the binary's mass ratio and eccentricity correspond to real nearby double and multiple systems; the Centauri and Sirius systems are investigated here. Large stable planetary orbits, already known to exist through a systematic exploration of the circular model and for the Sun-Jupiter elliptic case, are found to exist around Centauri A and Centauri B, as well as around Sirius A, up to distances from each star of the order of more than half the binary's periastron separation. Similar studies for other nearby well-known double and multiple systems will follow.     

Polar orbits around binary stars

Oks proposes the existence of a new class of stable planetary orbits around binary stars, in the shape of a helix on a conical surface whose axis of symmetry coincides with the interstellar axis, and rotates with the same orbital frequency as the binary pair. We show that this claim relies on the inappropriate use of an effective potential that is only applicable when the stars are held motionless. We also present numerical evidence that the only planetary orbits whose planes are initially orthogonal to the interstellar axis that remain stable on the time scale of the stellar orbit are ordinary polar orbits around one of the stars, and that the perturbations due to the binary companion do not rotate the plane of the orbit to maintain a fixed relationship with the axis.     

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.     

Empirical evidence for tidal evolution in transiting planetary systems

Most transiting planets orbit very close to their parent star, causing strong tidal forces between the two bodies. Tidal interaction can modify the dynamics of the system through orbital alignment, circularization, synchronization and orbital decay by exchange of angular moment. Evidence for tidal circularization in close-in giant planet is well known. Here, we review the evidence for excess rotation of the parent stars due to the pull of tidal forces towards spin-orbit synchronization. We find suggestive empirical evidence for such a process in the present sample of transiting planetary systems. The corresponding angular momentum exchange would imply that some planets have spiralled towards their star by substantial amounts since the dissipation of the protoplanetary disc. We suggest that this could quantitatively account for the observed mass–period relation of close-in gas giants. We discuss how this scenario can be further tested and point out some consequences for theoretical studies of tidal interactions and for the detection and confirmation of transiting planets from radial velocity and photometric surveys.     

Masses of binary stars for general orbits

The masses of a pair of stars in the visual binary system have been estimated. The angle between the orbital plane of the stars and the plane of the sky has been taken into account. Inclination of the major axes of the orbits of the stars with the line of interaction between the orbital plane and the plane of the sky has also been considered. These two inclinations are also computed in terms of the observed quantities. Major and minor axes of actual orbits of the stars are determined.     

Properties of planetary nebulae with WR central stars

A review of the observational properties of the Wolf-Rayet central stars is given.     

The separation of the stars in the binary nucleus of the planetary nebula Abell 35

Using the Planetary Camera on board the Hubble Space Telescope , we have measured the projected separation of the binary components in the nucleus of the planetary nebula Abell 35 to be larger than 0.08 arcsec but less than 0.14 arcsec. The system has been imaged in three filters centred at 2950, 3350 and 5785 Å. The white dwarf primary star responsible for ionizing the nebula is half as bright as its companion in the 2950-Å filter, causing the source to be visibly elongated. The 3350-Å setting, on the other hand, shows no elongation as a result of the more extreme flux ratio. The F300W data allow the determinination of the projected separation of the binary. At the minimum distance of 160 pc to the system, our result corresponds to 18 ± 5 au. This outcome is consistent with the wind accretion induced rapid rotation hypothesis, but cannot be reconciled with the binary having emerged from a common-envelope phase.     

Symmetric and asymmetric 3:1 resonant periodic orbits with an application to the 55Cnc extra-solar system

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.     

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.     

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     

Higher order resonant orbits

I present higher-order resonances of the tube and boomerang orbits in the co-rotating meridional plane of an axisymmetric galactic potential. Additionally, mirror forms of these orbits are given.     

Parent stars of extrasolar planets – VIII. Chemical abundances for 18 elements in 31 stars

We present the results of detailed spectroscopic abundance analyses for 18 elements in 31 nearby stars with planets (SWPs). The resulting abundances are combined with other similar studies of nearby SWPs and compared to a sample of nearby stars without detected planets. We find some evidence for abundance differences between these two samples for Al, Si and Ti. Some of our results are in conflict with a recent study of SWPs in the SPOCS data base. We encourage continued study of the abundance patterns of SWPs to resolve these discrepancies.     

Metallicity, planetary formation and migration

Recent observations show a clear correlation between the probability of hosting a planet and the metallicity of the parent star. As radial velocity surveys are biased, however, towards detecting planets with short orbital periods, the probability–metallicity correlation could merely reflect a dependence of migration rates on metallicity. We investigated the possibility, but find no basis to suggest that the migration process is sensitive to the metallicity. The indication is, therefore, that a higher metallicity results in a higher probability for planet  formation .