Resonant Planets Research Articles

Planetary migration and extrasolar planets in the 2/1 mean-motion resonance

In this paper, we present a new set of corotational solutions for the 2/1 commensurability, including previously known solutions and new results. Comparisons with observed exoplanets show that current orbital fits of three proposed resonant planetary systems are consistent with apsidal corotations. We also discuss the possible relationship between the current orbital elements fits of known exoplanets in the 2/1 mean-motion resonance and the expected orbital configuration due to migration. We find that, as long as the orbital decay was sufficiently slow to be approximated by an adiabatic process, all captured planets should be in apsidal corotations. In other words, they should show a simultaneous libration of both the resonant angle and the difference in longitudes of pericenter.

Frequency map analysis of the 3/1 resonance between planets b and c in the 55 Cancri system

We investigate the dynamical stability of the 3/1 resonant planets b and c in the extrasolar planetary system around 55 Cancri by applying Laskar's frequency map analysis. We find that the region with low diffusion speed extends to high eccentricity for both planets and that the observed system is deeply embedded in this region. The dynamics while in resonance and the influence of planet e on the resonant couple are also analysed. Using a simple model for the capture of the planets in resonance during migration we show that evolutionary tracks from smaller to the presently observed eccentricities lie in the stable region.

The insignificance of P-R drag in detectable extrasolar planetesimal belts

This paper considers a simple model in which dust produced in a planetesimal belt migrates in toward the star due to P-R drag suffering destructive collisions with other dust grains on the way. Assuming the dust is all of the same size, the resulting surface density distribution can be derived analytically and depends only on the parameter η0 = 5000τeff (r0) √ M� /r0/β; this parameter can be determined observationally with the hypothesis that β = 0.5. For massive belts in which η0 � 1 dust is confined to the planetesimal belt, while the surface density of more tenuous belts, in which η0 � 1, is constant with distance from the star. The emission spectrum of dust from planetesimal belts at different distances from different mass stars shows that the dust belts which have been detected to date should have η0 � 1; dust belts with η0 � 1 are hard to detect as they are much fainter than the stellar photosphere. This is confirmed for a sample of 37 debris disk candidates for which η0 was determined to be >10. This means that these disks are so massive that mutual collisions prevent dust from reaching the inner regions of these systems and P-R drag can be ignored when studying their dynamics. Models for the formation of structure in debris disks by the trapping of particles into planetary resonances by P-R drag should be reconsidered. However, since collisions do not halt 100% of the dust, this means that in the absence of planetary companions debris disk systems should be populated by small quantities of hot dust which may be detectable in the mid-IR. Even in disks with η0 � 1 the temperature of dust emission is shown to be a reliable tracer of the planetesimal distribution meaning that inner holes in the dust distribution imply a lack of colliding planetesimals in the inner regions.

The (In)stability of Planetary Systems

We present results of numerical simulations which examine the dynamical stability of known planetary systems, a star with two or more planets. First we vary the initial conditions of each system based on observational data. We then determine regions of phase space which produce stable planetary configurations. For each system we perform 1000 ~1 million year integrations. We examine upsilon And, HD83443, GJ876, HD82943, 47UMa, HD168443, and the solar system (SS). We find that the resonant systems, 2 planets in a first order mean motion resonance, (HD82943 and GJ876) have very narrow zones of stability. The interacting systems, not in first order resonance, but able to perturb each other (upsilon And, 47UMa, and SS) have broad regions of stability. The separated systems, 2 planets beyond 10:1 resonance, (we only examine HD83443 and HD168443) are fully stable. Furthermore we find that the best fits to the interacting and resonant systems place them very close to unstable regions. The boundary in phase space between stability and instability depends strongly on the eccentricities, and (if applicable) the proximity of the system to perfect resonance. In addition to million year integrations, we also examined stability on ~100 million year timescales. For each system we ran ~10 long term simulations, and find that the Keplerian fits to these systems all contain configurations which may be regular on this timescale.

Resonances and stability of extra-solar planetary systems

This paper reviews recent results on the dynamics of multiple-planet extra-solar systems, including main sequence stars and the pulsar PSR B1257+12 and, comparatively, our own Solar System. Taking into account the degree of gravitational interaction of the planets, the known planetary systems may be separated into four main groups: (Ia) Planets in Mean-motion resonance (Ib) Low-eccentricity near-resonant pairs; (II) Non-resonant planets with a significant secular dynamics; and (III) Weakly interacting planet pairs. Different analytical and numerical tools can help to understand the structure of the phase space, to identify stability mechanisms and to categorize different types of motions in the cases of more significant dynamical interaction. The origin of resonant configurations is discussed in the light of the hypothesis of planetary migration. To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Formation of Stars and Planets

A01 Modeling the Resonant Planetary System GJ 876 A02 First Classification of Dusty Protoplanets A03 IDL-Models of Infrared Emission from Dust Objects (Applied to Dust in Comets) A04 Theoretical Predicitions and Observational Tests of the Migration Hypothesis A05 Towards Characterization of Exoplanetary Spectra with the VLT Interferometer A06 Strongly Variable Cosmic Ray Particle Environments at Planets during the History of the Solar System A07 Rotational Evolution of Very Low Mass Stars and Brown Dwarfs A08 Rotation and Excitation Properties of Jets from Young Stellar Objects A09 BS Indi: An Enigmatic Binary in the Horologium Association A10 Disc Encounters – The Low- and High-Mass Case

The stabilising mechanism of the HD 82943 planetary system

We carry out simulations to investigate the dynamics of the HD 82943 planetary system with two resonant Jupiter-like planets, and to reveal possible stabilizing mechanism for the system. By following different coplanar configurations in the neighborhood of the best-fit orbits, we find that all the stable cases are involved in the 2:1 mean motion resonance and that the alignment of the periastra of the two planets also plays important part in the secular orbital evolution, indicating that these two kinds of mechanisms could be responsible for the dynamics of the system under study.

A Critical Examination of Li Pollution and Giant‐Planet Consumption by a Host Star

We examine the overall likelihood that the consumption of a giant planet could pollute its host star. First, we show that the detection of 6Li in SWPs could be used to distinguish between giant planet formation theories, and can be used to detect the consumption of giant planets independent of planet mass. Second, we examine the probability that giant planets between 1 and 3 M_J could be destroyed in the outer convection zone of stars slightly more massive than the Sun, finding that heated giant planets would be efficiently destroyed near the surface of the star, while the cores of cold giant planets may be able to survive to the base of the star's convection zone. Third, we examine the tidal interaction between a planet and its host star, showing that this interaction can lead to tidal disruption of the planet and accretion of its material, or orbital decay followed by hydrodynamical interaction. Throughout, we consider the case of HD 82943, a star known to have two planets and having a preliminary detection of 6Li. Using stellar models including diffusion, we estimate the mass of the HD 82943 to be about 1.2 M_sun and its age to be about 0.5-1.5 Gyr. The observed 7Li abundance for HD 82943 is consistent with stars of similar T_eff and age in the open cluster NGC 752. We describe a possible dynamical history for a hypothetical planet in the presence of the two resonant planets currently known. (abridged)

On disc driven inward migration of resonantly coupled planets with application to the system around GJ876

We consider two protoplanets gravitationally interacting with each other and a protoplanetary disc. The two planets orbit interior to a tidally maintained disc cavity while the disc interaction induces inward migration. When the migration is slow enough, the more rapidly migrating outer protoplanet approaches and becomes locked in a 2:1 commensurability with the inner one. This is maintained in subsequent evolution. We study this evolution using a simple analytic model, full hydrodynamic 2D simulations of the disc planet system and longer time N -body integrations incorporating simple prescriptions for the effects of the disc on the planet orbits. The eccentricities of the protoplanets are found to be determined by the migration rate and circularization rate induced in the outer planet orbit by the external disc. We apply our results to the recently discovered resonant planets around GJ876. Simulation shows that a disc with parameters expected for protoplanetary discs causes trapping in the 2:1 commensurability when the planets orbit in an inner cavity and that eccentricities in the observed range may be obtained.

Asteroid orbits in mixed resonances: some numerical experiments

Dynamics of orbits of minor planets in mixed resonances “Jupiter–Saturn–Asteroid” of low order is investigated with the help of different numerical experiments. Several thousands of fictitious asteroids with different initial eccentricities and inclinations are integrated numerically with the simultaneous computation of Lyapunov–Exponent in models with two (Jupiter and Saturn) and one (Jupiter or Saturn) disturbing bodies in order to analyse, under what conditions the joint perturbations from Jupiter and Saturn can lead to the origin of chaos in motion of minor planets. It is shown, that in the middle part of the main asteroid belt the joint perturbations from Jupiter and Saturn lead to the phenomenon of “stable chaos” in asteroid motion. In the inner part of the asteroid belt the large variations in eccentricity appear in the model with two disturbing bodies for orbits with small inclinations. These variations can lead to close encounters of asteroids with Mars, that makes this region very unstable.

Eccentricity Evolution of Migrating Planets

We examine the eccentricity evolution of a system of two planets locked in a mean motion resonance, in which either the outer or both planets lose energy and angular momentum. The sink of energy and angular momentum could be a gas or planetesimal disk. We analytically calculate the eccentricity damping rate in the case of a single planet migrating through a planetesimal disk. When the planetesimal disk is cold (the average eccentricity is much less than 1), the circularization time is comparable to the inward migration time, as previous calculations have found for the case of a gas disk. If the planetesimal disk is hot, the migration time can be an order of magnitude shorter. We show that the eccentricity of both planetary bodies can grow to large values, particularly if the inner body does not directly exchange energy or angular momentum with the disk. We present the results of numerical integrations of two migrating resonant planets showing rapid growth of eccentricity. We also present integrations in which a Jupiter-mass planet is forced to migrate inward through a system of 5-10 roughly Earth-mass planets. The migrating planets can eject or accrete the smaller bodies; roughly 5% of the mass (averaged over all the integrations) accretes onto the central star. The results are discussed in the context of the currently known extrasolar planetary systems.

Dynamical Evolution of Main Belt Meteoroids: Numerical Simulations Incorporating Planetary Perturbations and Yarkovsky Thermal Forces

In the Yarkovsky effect, the recoil from asymmetric, reradiated thermal energy causes objects to undergo semimajor axis drift as a function of their spin, orbit, and material properties. We consider the role played by this mechanism in delivering meteoroids from parent bodies in the main belt to chaotic resonance zones where they can be transported to Earth-crossing orbits. Previous work has approximated the dynamical evolution of meteoroids via Yarkovsky forces, mostly through the use of the perturbations equation and simplified dynamics (e.g., Monte Carlo codes). In this paper, we calculate more precise solutions by formulating the seasonal and diurnal variants of this radiation force and incorporating them into an efficient N-body integrator capable of tracking test bodies for tens of millions of years with all relevant planetary perturbations included. Tests of our code against published benchmarks and the perturbation equations verify its accuracy. Results from long-term numerical integration of meter-sized bodies started from likely meteoroid parent bodies (e.g., 4 Vesta) indicate that dynamical evolution in the inner main belt can be complex. Chaotic effects produced by weaker planetary resonances allow many meteoroids to reach Mars-crossing orbits well before entering the 3:1 mean-motion resonance with Jupiter or the ν 6 secular resonance. Outward-evolving meteoroids sometimes become captured in these weaker resonances, increasing e and/or i while a stays constant. Conversely, inward-evolving meteoroids frequently jump across mean-motion resonances with Jupiter, bypassing potential “escape hatches” from the main belt. Despite these effects, our simulations indicate that most stony meteoroids reach Earth-crossing orbits via the 3:1 or ν 6 resonance after tens of Myr of evolution in the main belt. These time scales correspond well to the measured cosmic ray exposure ages of chondrites and achondrites. The source of these meteorites, however, is less clear, since Yarkovsky drift allows nearly any body in the main belt to add to the cumulate meteoroid flux. Our results suggest that small parent bodies dominate the meteoroid flux if the main belt size distribution at sub-km sizes is in collisional equilibrium, while big parent bodies dominate if observed population trends for km-sized bodies persist to smaller sizes.

Solar-planetary cycles in Jupiter's great red spot darkness

The change in the darkness of the Great Red Spot (GRS) of Jupiter (1894–1974) has been analysed with Fourier (FFT), Maximum Entropy and Power spectrum (Blackman-Tukey window) (PSA) methods of spectrum analysis. Significance, non-randomness and stationarity tests assigned high variance to periodicities of 33 ± 4, 13–15, about 11, 9 and 3 yrs. The highest correlation between solar activity and GRS darkness was found for the 14th and 16th solar cycle. The periodicities obtained are interpreted as the combined eftects of solar activity, planetary resonances and internal jovian mechanisms.

The Long-Term Dynamical Evolution of Comet Swift-Tuttle

Periodic comet Swift-Tuttle is the largest known object in the Solar System whose current orbit allows it to make repeated close approaches to the Earth. As such it poses a significant impact hazard. In addition, P/Swift-Tuttle possesses the second longest observing are of any comet, with recorded observations spanning two millennia. These data place extremely tight constraints on the current orbit. This paper analyzes the likely dynamical evolution of P/Swift-Tuttle for 40,000 years centered on the present. The evolution is uncertain prior to the earliest observations in 69 BC; however, given that nongravitational forces appear to be negligible, the future evolution is clear until an exceptionally close encounter with Earth in 4479 AD. Beyond this date, the current librations about the 1:11 mean-motion resonance with Jupiter will continue until at least 7000 AD, while the orbital descending node is likely to remain within 0.1 AU of Earth's orbit for the next 20,000 years. The resonant librations increase the orbit's Lyapunov time by a factor of two in the future and strongly influence the movement of the orbital nodes. The librations are maintained by indirect rather than direct planetary perturbations, which explains why a retrograde comet of such long orbital period is located in a resonance, and why Jupiter is the resonant planet. After 4479 the probability of impacting Earth is roughly 2 × 10 -8 per revolution until 12,000 AD, while the chance of a collision in 4479 is of order 1 in 10 6.

Planetary Distance Law and Resonance

AbstractIn this paper the relation between the planetary distance law and the resonant structures is shown, in that the resonance relation has been expressed in terms of Roche’s constant (Rawal 1984,1986,1989). This brings forth a coherent, elegant and unified picture of the Solar System and satellite systems.

Planetary distance law and resonance

The relation between the planetary distance law and the resonant structures in the solar system and in the satellite systems is shown, in that, the resonance relation has been expressed in terms of Roche’s (Bode’s) constant defined in the text. This brings forth a coherent, elegant and unified picture of the formation and structure of the solar system and the satellite systems. The Roche’s (Bode’s) constant is seen to play a central role in this unified picture, in that, it also appears to govern the resonance phenomenon in the systems

Resonant Planetary Waves in a Spherical Atmosphere

Abstract A global model of planetary wave propagation in a spherical atmosphere is used to examine the spectrum of free or resonant planetary waves of the solstitial stratosphere. These free modes are located by forcing the model with a weak periodic vertical velocity along the lower boundary and looking for a resonant response in wave amplitude. The modes correspond to the natural traveling oscillations in the earth's atmosphere, of which the 5-day wave is the best known example. The 15-day wave observed by Madden (1978) and others is found to be such a resonant mode. We find that the strong stratospheric winds cause the 15-day wave to become baroclinic by trapping the wave between the earth's surface and the strong winds at the stratopause. The strong winds effectively reduce the atmospheric damping which greatly reduces the amplitude of barotropic waves with periods >10 days. The computed meridional structure of the 15-day wave is in reasonable agreement with Madden's (1978) observations at extratropic...

Resonance capture in certain nearly Hamiltonian systems

The difFerentia1 equation fi = F(t, 8, e), in which 0 is an angular variable and F is small and periodic with period 2rr in t and 0, represents a material body spinning about a fixed axis subject to a small, periodic, not necessarily conservative torque. Mathematically the equation is a paradigm case of non. . linear oscillations. The equation 19 = 0, of which it is a perturbation, is formally linear but exhibits the characteristic behavior of a nonlinear integrable Hamiltonian system because Q is an angular variable, The solutions e=wt+e,, where w and 0, are initial conditions, are geometrically periodic with period 297/w, and dependence of the period upon the amplitude or initial conditions, as seen in the unperturbed Duffing equation 1+ cw: + /5’S = 0, is the chief property of nonlinear oscillations. When 0 == 0 is perturbed to a nonintegrable Hamiltonian system, the results of Poincare-Birkhoe and Kolmogorov-Arnol’d-Moser apply and a complicated global picture results, which however has the simple consequence that the angular velocity w = 8 is not subject to very much “drift” along any orbit. This is due to the presence of invariant curves surrounding the cylindrical phase space of the variables (0, w) near circles w = constant for which w is “badly irrational..” At the other extreme, when the perturbation contains substantial friction, all orbits may tend to a single limit cycle on the cylinder, with cu fluctuating around a single final value. The cases studied in this paper lie between these two extremes, when the perturbation consists of a small Hamiltonian part and a still smaller friction. The problem is then to determine the possible “final values” of w (in a sense to be made precise later), and show how their number increases as the friction goes to zero. The delicacy of the problem is seen in the fact that the result depends upon the higher harmonics of 0 present in F, and that no matter how rapidly the coefficients of these harmonics decay, their presence will be felt when the friction is small enough. The application motivating this work is to planetary spin resonance, in