Dynamic portrait of the planetary 2/1 mean-motion resonance - I. Systems with a more massive outer planet

We analyse the global structure of the phase space of the planar planetary 2/1 mean-motion resonance in cases where the outer planet is more massive than its inner companion. Inside the resonant domain, we show the existence of two families of periodic orbits, one associated to the librational motion of resonant angle (sigma-family) and the other related to the circulatory motion of the difference in longitudes of pericentre (Delta pi-family). The well-known apsidal corotation resonances (ACR) appear as intersections between both families. A complex web of secondary resonances is also detected for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system. The construction of dynamical maps for various values of the total angular momentum shows the evolution of the families of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems. For low-moderate eccentricities, several different stable modes exist outside the ACR. For larger eccentricities, however, all stable solutions are associated to oscillations around the stationary solutions. Finally, we present a possible link between these stable families and the process of resonance capture, identifying the most probable routes from the secular region to the resonant domain, and discussing how the final resonant configuration may be affected by the extension of the chaotic layer around the resonance region.

Dynamic portrait of the planetary 2/1 mean-motion resonance - II. Systems with a more massive inner planet

This paper presents the second part in our study of the global structure of the planar phase space of the planetary three-body problem, when both planets lie in the vicinity of a 2/1 mean-motion resonance. While Paper I was devoted to cases where the outer planet is the more massive body, the present work is devoted to the cases where the more massive body is the inner planet. As before, outside the well-known Apsidal Corotation Resonances (ACR), the phase space shows a complex picture marked by the presence of several distinct regimes of resonant and non-resonant motion, crossed by families of periodic orbits and separated by chaotic zones. When the chosen values of the integrals of motion lead to symmetric ACR, the global dynamics are generally similar to the structure presented in Paper I. However, for asymmetric ACR the resonant phase space is strikingly different and shows a galore of distinct dynamical states. This structure is shown with the help of dynamical maps constructed on two different representative planes, one centred on the unstable symmetric ACR and the other on the stable asymmetric equilibrium solution. Although the study described in the work may be applied to any mass ratio, we present a detailed analysis for mass values similar to the Jupiter-Saturn case. Results give a global view of the different dynamical states available to resonant planets with these characteristics. Some of these dynamical paths could have marked the evolution of the giant planets of our Solar system, assuming they suffered a temporary capture in the 2/1 resonance during the latest stages of the formation of our Solar system.

Dynamics of two satellites in the 2/1 Mean-Motion resonance: application to the case of Enceladus and Dione

The dynamics of a pair of satellites similar to Enceladus-Dione is investigated with a two-degrees-of-freedom model written in the domain of the planar general three-body problem. Using surfaces of section and spectral analysis methods, we study the phase space of the system in terms of several parameters, including the most recent data. A detailed study of the main possible regimes of motion is presented, and in particular we show that, besides the two separated resonances, the phase space is replete of secondary resonances.

Dynamics of two planets in the 3/2 mean-motion resonance: Application to the planetary system of the pulsar PSR B1257+12

This paper considers the dynamics of two planets, as the planets B and C of the pulsar PSR B1257+12, near a 3/2 mean-motion resonance. A two-degrees-of-freedom model, in the framework of the general three-body planar problem, is used and the solutions are analyzed through surfaces of section and Fourier techniques in the full phase space of the system.

Dynamics of Enceladus and Dione inside the 2:1 mean-motion resonance under tidal dissipation

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Mean Motion in Stochastic Plasmas With a Space-Dependent Diffusion Coefficient

Radial transport in the tokamap, which has been proposed as a simple model for the motion in a stochastic plasma, is investigated. A theory for previous numerical findings is presented. The new results are stimulated by the fact that the radial diffusion coefficients is space-dependent. The space-dependence of the transport coefficient has several interesting effects which have not been elucidated so far. Among the new findings are the analytical predictions for the scaling of the mean radial displacement with time and the relation between the Fokker-Planck diffusion coefficient and the diffusion coefficient from the mean square displacement. The applicability to other systems is also discussed. (c) 2009 WILEY-VCH GmbH & Co. KGaA, Weinheim

Mean motion resonances and the stability of a circumbinary disk in a triple stellar system

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Resonant and Secular Orbital Interations

In stable solar systems, planets remain in nearly elliptical orbits around their stars. Over longer timescales, however, their orbital shapes and sizes change due to mutual gravitational perturbations. Orbits of satellites around a planet vary for the same reason. Because of their interactions, the orbits of planets and satellites today are different from what they were earlier. In order to determine their original orbits, which are critical constraints on formation theories, it is crucial to understand how orbits evolve over the age of the Solar System. Depending on their timescale, we classify orbital interactions as either short-term (orbital resonances) or long-term (secular evolution). My work involves examples of both interaction types. Resonant history of the small Neptunian satellites In satellite systems, tidal migration brings satellite orbits in and out of resonances. During a resonance passage, satellite orbits change dramatically in a very short period of time. We investigate the resonant history of the six small Neptunian moons. In this unique system, the exotic orbit of the large captured Triton (with a circular, retrograde, and highly tilted orbit) influences the resonances among the small satellites very strongly. We derive an analytical framework which can be applied to Neptune's satellites and to similar systems. Our numerical simulations explain the current orbital tilts of the small satellites as well as constrain key physical parameters of both Neptune and its moons. Secular orbital interactions during eccentricity damping Long-term periodic changes of orbital shape and orientation occur when two or more planets orbit the same star. The variations of orbital elements are superpositions of the same number of fundamental modes as the number of planets in the system. We investigate how this effect interacts with other perturbations imposed by external disturbances, such as the tides and relativistic effects. Through analytical studies of a system consisting of two planets, we find that an external perturbation exerted on one planet affects the other indirectly. We formulate a general theory for how both orbits evolve in response to an arbitrary externally-imposed slow change in eccentricity.

Observational and Dynamical Characterization of Main-belt Comet P/2010 R2 (La Sagra)

We present observations of the recently discovered comet-like main-belt object P/2010 R2 (La Sagra) obtained by Pan-STARRS1 and the Faulkes Telescope-North on Haleakala in Hawaii, the University of Hawaii 2.2 m, Gemini-North, and Keck I telescopes on Mauna Kea, the Danish 1.54 m telescope (operated by the MiNDSTEp consortium) at La Silla, and the Isaac Newton Telescope on La Palma. An antisolar dust tail is observed to be present from 2010 August through 2011 February, while a dust trail aligned with the object's orbit plane is also observed from 2010 December through 2011 August. Assuming typical phase darkening behavior, P/La Sagra is seen to increase in brightness by >1 mag between 2010 August and December, suggesting that dust production is ongoing over this period. These results strongly suggest that the observed activity is cometary in nature (i.e., driven by the sublimation of volatile material), and that P/La Sagra is therefore the most recent main-belt comet to be discovered. We find an approximate absolute magnitude for the nucleus of HR = 17.9 ± 0.2 mag, corresponding to a nucleus radius of ~0.7 km, assuming an albedo of p = 0.05. Comparing the observed scattering surface areas of the dust coma to that of the nucleus when P/La Sagra was active, we find dust-to-nucleus area ratios of Ad /AN = 30-60, comparable to those computed for fellow main-belt comets 238P/Read and P/2008 R1 (Garradd), and one to two orders of magnitude larger than for two other main-belt comets (133P/Elst-Pizarro and 176P/LINEAR). Using optical spectroscopy to search for CN emission, we do not detect any conclusive evidence of sublimation products (i.e., gas emission), finding an upper limit CN production rate of Q CN 100 Myr, suggesting that it is likely native to its current location and that its composition is likely representative of other objects in the same region of the main belt, though the relatively close proximity of the 13:6 mean-motion resonance with Jupiter and the (3,-2,-1) three-body mean-motion resonance with Jupiter and Saturn mean that dynamical instability on larger timescales cannot be ruled out.

The scattered trans-Neptunian object 1998 XY95

On 1998 December 12 a new trans-Neptunian object, 1998 XY95, was discovered as part of a deep search. Recent observations of this object have placed it amongst the class of objects known as the scattered trans-Neptunian objects (TNOs). A total of 39 CCD images of 1998 XY95 were taken over two nights, and these were used to search for a light curve, but no significant periodicity was found. An examination of the possible orbital evolution gives no indication of how it may have arrived on its present orbit. The current best-fitting orbit is unstable, but remains within a band of semi-major axis approximately 2au wide. The bottom of this band is due to 3:1 mean motion resonance with Neptune, while the reason for the top of the band remains unclear.

Transit timing variations in WASP-10b induced by stellar activity

The hot-JupiterWASP-10bwas reported by Maciejewski et al. to showtransit timing variations (TTVs) with an amplitude of ~3.5 min. These authors proposed that the observed TTVs were caused by a 0.1M_Jup perturbing companion with an orbital period of ~5.23 d, and hence, close to the outer 5:3 mean-motion resonance with WASP-10b. To test this scenario, we present eight new transit light curves of WASP-10b obtained with the Faulkes Telescope North and the Liverpool Telescope. The new light curves, together with 22 previously published ones, were modelled with a Markov Chain Monte Carlo transit fitting code. Transit depth differences reported forWASP-10b are thought to be due to starspot-induced brightness modulation of the host star. Assuming the star is brighter at the activity minimum, we favour a small planetary radius. We find Rp = 1.039^+0.043 _-0.049R_Jup in agreement with Johnson et al. and Maciejewski et al. Recent studies find no evidence for a significant eccentricity in this system. We present consistent system parameters for a circular orbit and refine the orbital ephemeris ofWASP-10b. Our homogeneously derived transit times do not support the previous claimed TTV signal, which was strongly dependent on two previously published transits that have been incorrectly normalized. Nevertheless, a linear ephemeris is not a statistically good fit to the transit times of WASP-10b. We show that the observed transit time variations are due to spot occultation features or systematics. We discuss and exemplify the effects of occultation spot features in the measured transit times and show that despite spot occultation during egress and ingress being difficult to distinguish in the transit light curves, they have a significant effect in the measured transit times. We conclude that if we account for spot features, the transit times of WASP-10b are consistent with a linear ephemeris with the exception of one transit (epoch 143) which is a partial transit. Therefore, there is currently no evidence for the existence of a companion to WASP-10b. Our results support the lack of TTVs of hot-Jupiters reported for the Kepler sample. 

Reliability of orbital fits for resonant extrasolar planetary systems: the case of HD82943

We perform a statistical study of the process of orbital determination of the HD82943 extrasolar planetary system, using the current observational data set of N = 165 radial velocity (RV) measurements. Our aim is to analyse the dispersion of possible orbital fits leading to residuals compatible with the best solution, and to discuss the sensitivity of the results with respect to both the data set and the error distribution around the best fit. Although some orbital parameters (e.g. semimajor axis) appear well constrained, we show that the best fits for the HD82943 system are not robust, and at present it is not possible to estimate reliable solutions for these bodies. Finally, we discuss the possibility of a third planet, with a mass of 0.35M(Jup) and an orbital period of 900 d. Stability analysis and simulations of planetary migration indicate that such a hypothetical three-planet system could be locked in a double 2/1 mean-motion resonance, similar to the so-called Laplace resonance of the three inner Galilean satellites of Jupiter.

Modelling the secular evolution of migrating planet pairs

The subject of this paper is the secular behaviour of a pair of planets evolving under dissipative forces. In particular, we investigate the case when dissipative forces affect the planetary semimajor axes and the planets move inwards/outwards the central star, in a process known as planet migration. To perform this investigation, we introduce fundamental concepts of conservative and dissipative dynamics of the three-body problem. Based on these concepts, we develop a qualitative model of the secular evolution of the migrating planetary pair. Our approach is based on the analysis of the energy and the orbital angular momentum exchange between the two-planet system and an external medium; thus no specific kind of dissipative forces is invoked. We show that, under the assumption that dissipation is weak and slow, the evolutionary routes of the migrating planets are traced by the Mode I and Mode II stationary solutions of the conservative secular problem. The ultimate convergence and the evolution of the system along one of these secular modes of motion are determined uniquely by the condition that the dissipation rate is sufficiently smaller than the proper secular frequency of the system. We show that it is possible to reassemble the starting configurations and the migration history of the systems on the basis of their final states and consequently to constrain the parameters of the physical processes involved.

DETECTABILITY AND ERROR ESTIMATION IN ORBITAL FITS OF RESONANT EXTRASOLAR PLANETS

We estimate the conditions for detectability of two planets in a 2/1 mean-motion resonance from radial velocity data, as a function of their masses, number of observations and the signal-to-noise ratio. Even for a data set of the order of 100 observations and standard deviations of the order of a few meters per second, we find that Jovian-size resonant planets are difficult to detect if the masses of the planets differ by a factor larger than similar to 4. This is consistent with the present population of real exosystems in the 2/1 commensurability, most of which have resonant pairs with similar minimum masses, and could indicate that many other resonant systems exist, but are currently beyond the detectability limit. Furthermore, we analyze the error distribution in masses and orbital elements of orbital fits from synthetic data sets for resonant planets in the 2/1 commensurability. For various mass ratios and number of data points we find that the eccentricity of the outer planet is systematically overestimated, although the inner planet`s eccentricity suffers a much smaller effect. If the initial conditions correspond to small-amplitude oscillations around stable apsidal corotation resonances, the amplitudes estimated from the orbital fits are biased toward larger amplitudes, in accordance to results found in real resonant extrasolar systems.