Effects of perturbing forces on the orbital stability of planetary systems

We consider dynamical effects of additional perturbative forces due to the non-point mass nature of stars and planets: effects such as quadrupolar distortion and tidal friction in the systems of exo-planets. It is shown that these forces should not be neglected while modelling the dynamics of planetary systems, especially taking into account the undefined real masses of the planets due to unknown orbital inclinations and the unsatisfactory application of Keplerian fits to the radial velocity data in multiple planetary systems.Comment: 5 pages, 2 figures, accepted by ApJ Le

Global dynamics and stability limits for planetary systems around HD 12661, HD 38529, HD 37124 and HD 160691

In order to distinguish between regular and chaotic planetary orbits we apply a new technique called MEGNO in a wide neighbourhood of orbital parameters determined using standard two-body Keplerian fits for HD 12661, HD 38529, HD 37124 and HD 160691 planetary systems. We show that the currently announced orbital parameters place these systems in very different situations from the point of view of dynamical stability. While HD 38529 and HD 37124 are located within large stability zones in the phase space around their determined orbits, the preliminary orbits in HD 160691 are highly unstable. The orbital parameters of the HD 12661 planets are located in a border region between stable and unstable dynamical regimes, so while its currently determined orbital parameters produce stable regular orbits, a minor change within the margin of error of just one parameter may result in a chaotic dynamical system.Comment: 12 pages, 3 figures, accepted ApJ, revised version following the referee's repor

Towards Multiple-Star Population Synthesis

The multiplicities of stars, and some other properties, were collected recently by Eggleton & Tokovinin, for the set of 4559 stars with Hipparcos magnitude brighter than 6.0 (4558 excluding the Sun). In this paper I give a numerical recipe for constructing, by a Monte Carlo technique, a theoretical ensemble of multiple stars that resembles the observed sample. Only multiplicities up to 8 are allowed; the observed set contains only multiplicities up to 7. In addition, recipes are suggested for dealing with the selection effects and observational uncertainties that attend the determination of multiplicity. These recipes imply, for example, that to achieve the observed average multiplicity of 1.53, it would be necessary to suppose that the real population has an average multiplicity slightly over 2.0. This numerical model may be useful for (a) comparison with the results of star and star cluster formation theory, (b) population synthesis that does not ignore multiplicity above 2, and (c) initial conditions for dynamical cluster simulations

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In the present paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10^(-3) to 10^(3). We also present more numerical results for short term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.Comment: Accepted for publication in MNRA

Conditions of Dynamical Stability for the HD 160691 Planetary System

The orbits in the HD 160691 planetary system at first appeared highly unstable, but using the MEGNO and FLI techniques of global dynamics analysis in the orbital parameter space we have found a stabilizing mechanism that could be the key to its existence. In order to be dynamically stable, the HD 160691 planetary system has to satisfy the following conditions: (1) a 2:1 mean motion resonance, combined with (2) an apsidal secular resonance in (3) a configuration $P_{c}(ap) - S - P_{b}(ap)$ where the two apsidal lines are anti-aligned, and (4) specific conditions on the respective sizes of the eccentricities (high eccentricity for the outer orbit is in particular the most probable necessary condition). More generally, in this original orbital topology, where the resonance variables $\theta_{1}$ and $\theta_{3}$ librate about $180^{\circ}$ while $\theta_{2}$ librates about $0^{\circ}$, the HD 160691 system and its mechanism have revealed aspects of the 2:1 orbital resonances that have not been observed nor analyzed before. The present topology combined with the 2:1 resonance is indeed more wide-ranging than the particular case of the HD 160691 planetary system. It is a new theoretical possibility suitable for a stable regime despite relatively small semi-major axes with respect to the important masses in interactions.Comment: 21 pages, 8 figures, 1 table, accepted version to ApJ (31 Jul 2003

Planets in binary systems: is the present configuration indicative of the formation process?

The present dynamical configuration of planets in binary star systems may not reflect their formation process since the binary orbit may have changed in the past after the planet formation process was completed. An observed binary system may have been part of a former hierarchical triple that became unstable after the planets completed their growth around the primary star. Alternatively, in a dense stellar environment even a single stellar encounter between the star pair and a singleton may singificantly alter the binary orbit. In both cases the planets we observe at present would have formed when the dynamical environment was different from the presently observed one. We have numerically integrated the trajectories of the stars (binary plus singleton) and of test planets to investigate the abovementioned mechanisms. Our simulations show that the circumstellar environment during planetary formation around the primary was gravitationally less perturbed when the binary was part of a hierarchical triple because the binary was necessarely wider and, possibly, less eccentric. This circumstance has consequences for the planetary system in terms of orbital spacing, eccentricity, and mass of the individual planets. Even in the case of a single stellar encounter the present appearance of a planetary system in a binary may significantly differ from what it had while planet formation was ongoing. However, while in the case of instability of a triple the trend is always towards a tighter and more eccentric binary system, when a single stellar encounter affects the system the orbit of the binary can become wider and be circularized.Comment: 5 pages, 5 figures Accepted for publication on A&