Periodic Trojan Orbits for the Resonance 1/12

Deprit, André; Rabe, Eugene

doi:10.1086/110812

Cited by 34 publications

(41 citation statements)

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“…the fast detection of outer perturbing planets and the determination of their orbital and physical parameters from their perturbations on the transit timing of the inner planet, by the help of the analytical description of the perturbed transiting O−C curve. For the mathematical description, they use the explicit perturbation theory of Hori (1966) and Deprit (1969), based on canonical transformations and on the use of Lie-series. This theory does not require the hierarchical assumption; i.e., the α = a 1 /a 2 parameter, although less than unity, is not required to be small.…”

Section: Discussionmentioning

confidence: 99%

Transit timing variations in eccentric hierarchical triple exoplanetary systems

Borkovits

Csizmadia

Forgács‐Dajka

et al. 2011

A&A

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Aims. We study the long-term time scale (i.e. period comparable to the orbital period of the outer perturber object) transit timing variations (TTVs) in transiting exoplanetary systems that contain another more distant (a 2 a 1 ) planetary or stellar companion. Methods. We give an analytical form of the O−C diagram (which describes such TTVs) in a trigonometric series, which is valid for arbitrary mutual inclinations, up to the sixth order in the inner eccentricity. Results. We show that the dependence of the O−C on the orbital and physical parameters can be separated into three parts. Two of these are independent of the real physical parameters (i.e. masses, separations, periods) of a concrete system and only depend on dimensionless orbital elements, so can be analysed in general. We find that, for a specific transiting system, where eccentricity (e 1 ) and the observable argument of periastron (ω 1 ) are known, say, from spectroscopy, the main characteristics of that TTV, which is caused by a possible third-body can be mapped simply. Moreover, as the physical attributes of a given system only occur as scaling parameters, the real amplitude of the O−C can also be estimated for a given system, simply as a function of the m 3 /P 2 ratio. We analyse the above-mentioned dimensionless amplitudes for different arbitrary initial parameters, as well as for two particular systems, CoRoT-9b and HD 80606b. We find in general that, while the shape of the O−C strongly varies with the angular orbital elements, the net amplitude (departing from some specific configurations) depends only weakly on these elements, but strongly on the eccentricities. As an application, we illustrate how the formulae work for the weakly eccentric CoRoT-9b and the highly eccentric HD 80606b. We also consider the question of detection, as well as the correct identification of such perturbations. Finally, we illustrate the operation and effectiveness of Kozai cycles with tidal friction (KCTF) in the case of HD 80606b.

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Section: Discussionmentioning

confidence: 99%

Transit timing variations in eccentric hierarchical triple exoplanetary systems

Borkovits

Csizmadia

Forgács‐Dajka

et al. 2011

A&A

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“…First, the differences remain small although the chosen time of integration is very sizable. Indeed, the computed root mean square (rms) can be considered to be very small because the comparison is made without any preliminary fitting of the mean initial conditions (Deprit 1969;Henrard 1970;Valk et al 2009a). Moreover, the RMS on both the argument of perigee and the longitude of the ascending node are mostly influenced by the singular transformation when projecting the integrated non-singular state vector into Keplerian orbital elements.…”

Section: Valk Semi-analytical Studymentioning

confidence: 99%

NIMASTEP: a software to modelize, study, and analyze the dynamics of various small objects orbiting specific bodies

Delsate

Compère

2012

A&A

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NIMASTEP is a dedicated numerical software developed by us, which allows one to integrate the osculating motion (using Cartesian coordinates) in a Newtonian approach of an object considered as a point-mass orbiting a homogeneous central body that rotates with a constant rate around its axis of smallest inertia. The code can be applied to objects such as particles, artificial or natural satellites, or space debris. The central body can be either any terrestrial planet of the solar system, any dwarf-planet, or even an asteroid. In addition, very many perturbations can be taken into account, such as the combined third-body attraction of the Sun, the Moon, or the planets, the direct solar radiation pressure (with the central body shadow), the non-homogeneous gravitational field caused by the non-sphericity of the central body, and even some thrust forces. The simulations were performed using different integration algorithms. Two additional tools were integrated in the software package; the indicator of chaos MEGNO and the frequency analysis NAFF. NIMASTEP is designed in a flexible modular style and allows one to (de)select very many options without compromising the performance. It also allows one to easily add other possibilities of use. The code has been validated through several tests such as comparisons with numerical integrations made with other softwares or with semi-analytical and analytical studies. The various possibilities of NIMASTEP are described and explained and some tests of astrophysical interest are presented. At present, the code is proprietary but it will be released for use by the community in the near future. Information for contacting its authors and (in the near future) for obtaining the software are available on the web site http://www.fundp.ac.be/en/research/projects/ page_view/10278201/

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“…This transformation will be chosen such as to decouple the reaction coordinate from the bath modes. To this end, we will use time-dependent Lie transformations 22 in the formulation of Dragt and Finn. 27 First we extend the phase space from (q c , p c ) to…”

Section: Time-dependent Normal Form Theorymentioning

confidence: 99%

“…In this paper, we combine the concept of the TS trajectory with the method of normal form (NF) expansions based on Lie transformations 22 which has been shown to be an effective tool to calculate invariant manifolds in autonomous systems [1][2][3][4][5][6][7][8][9] (see also Refs. 23,24 for a quantum version of Lie transformations).…”

Section: Introductionmentioning

confidence: 99%

Transition state theory for laser-driven reactions

Kawai

Bandrauk

Jaffé

et al. 2007

The Journal of Chemical Physics

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Recent developments in Transition State Theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, we construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and non-reactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. We illustrate the power of our approach on a driven Hénon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of Transition State Theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses.

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Periodic Trojan Orbits for the Resonance 1/12

Cited by 34 publications

References 0 publications

Transit timing variations in eccentric hierarchical triple exoplanetary systems

Transit timing variations in eccentric hierarchical triple exoplanetary systems

NIMASTEP: a software to modelize, study, and analyze the dynamics of various small objects orbiting specific bodies

Transition state theory for laser-driven reactions

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