Non-resonant Orbits Research Articles

Particle Motion under the Conservative Piece of the Self-Force is Hamiltonian.

We consider the motion of a point particle in a stationary spacetime under the influence of a scalar, electromagnetic, or gravitational self-force. We show that the conservative piece of the first-order self-force gives rise to Hamiltonian dynamics, and we derive an explicit expression for the Hamiltonian on phase space. Specialized to the Kerr spacetime, our result generalizes the Hamiltonian function previously obtained by Fujita etal. [Classical Quantum Gravity 34, 134001 (2017).CQGRDG0264-938110.1088/1361-6382/aa7342], which is valid only for nonresonant orbits.

Chaotic diffusion of asteroids in the exterior 1:2 mean motion resonance with Mars

ABSTRACT The inner asteroid belt between 2.1 and 2.5 au is of particular dynamical significance because it is the dominant source of both chondritic meteorites and near-Earth asteroids. This inner belt is bounded by an eccentricity-type secular resonance and by the 1:3 mean motion resonance with Jupiter. Unless asteroid perihelia are low enough to allow scattering by Mars, escape requires transport to one of the bounding resonances. In addition Yarkovsky forces are generally ineffective in changing either the eccentricity and/or inclination for asteroids with diameter ≳30 km. Thus, large asteroids with pericentres far from Mars may only escape from the inner belt through large changes in their eccentricities. In this paper, we study chaotic diffusion of orbits near the 1:2 mean motion resonance with Mars in a systematic way. We show that, while chaotic orbital evolution in both resonant and non-resonant orbits increase the dispersion of the inclinations and eccentricities, it does not significantly change their mean values. We show further that, while the dispersive growth is greatest for resonant orbits, at high e the resonance acts to mitigate asteroid scattering by Mars - making the asteroid lifetime in the belt longer than it would have been for a non-resonant orbit. For asteroids of all sizes in both resonant and non-resonant orbits, the changes in eccentricity needed to account for the observations cannot be achieved by gravitational forces alone. The role of resonant trapping in protecting asteroids from encounters with Mars is also analysed.

Pluto’s Resonant Orbit Visualized in 4D

Abstract Orbital resonance phenomena are notoriously difficult to communicate in words due to the complex dynamics arising from the interplay of gravity and orbital angular momentum. A well known example is Pluto’s 3:2 mean motion resonance with Neptune. We have developed a python software tool to visualize the full three-dimensional aspects of Pluto’s resonant orbital dynamics over time. The visualizations include still images and animated movies. By contrasting Pluto’s resonant dynamics with the dynamics of a nearby non-resonant orbit, this tool enables better understanding and exploration of complex planetary dynamics phenomena.

A Self-consistent, Time-dependent Treatment of Dynamical Friction: New Insights Regarding Core Stalling and Dynamical Buoyancy

Abstract Dynamical friction is typically regarded as a secular process, in which the subject (“perturber”) evolves very slowly (secular approximation) and has been introduced to the host over a long time (adiabatic approximation). These assumptions imply that dynamical friction arises from the LBK torque with nonzero contribution only from purely resonant orbits. However, dynamical friction is only of astrophysical interest if its timescale is shorter than the age of the universe. In this paper we therefore relax the adiabatic and secular approximations. We first derive a generalized LBK torque, which reduces to the LBK torque in the adiabatic limit, and show that it gives rise to transient oscillations due to nonresonant orbits that slowly damp out, giving way to the LBK torque. This is analogous to how a forced, damped oscillator undergoes transients before settling to a steady state, except that here the damping is due to phase mixing rather than dissipation. Next, we present a self-consistent treatment that properly accounts for time dependence of the perturber potential and circular frequency (memory effect), which we use to examine orbital decay in a cored galaxy. We find that the memory effect results in a phase of accelerated, super-Chandrasekhar friction before the perturber stalls at a critical radius, R crit, in the core (core stalling). Inside R crit the torque flips sign, giving rise to dynamical buoyancy, which counteracts friction and causes the perturber to stall. This phenomenology is consistent with N-body simulations, but has thus far eluded proper explanation.

Quasi-Static Pull-in: an Instability in Electrostatic Actuators

We identify a new instability in electrostatic actuators dubbed quasi-static pull-in. We report experimental evidence of the instability and study its characteristics in two types of micro actuators operating in ambient air. We found that the underlying mechanism is a fast-slow dynamic interaction between slowly-varying electrostatic excitation and fast resonator response that instigate large non-resonant oscillatory orbits and eventually disappears in a global Shilnikov bifurcation. Based on these findings, we formulate and present a new taxonomy of pull-in instabilities in electrostatic actuators.

Ejecta cloud from the AIDA space project kinetic impact on the secondary of a binary asteroid: II. Fates and evolutionary dependencies

This paper presents a quantitative study of the evolution of the ejecta cloud released from a hypervelocity impact on a binary asteroid. The Asteroid Impact & Deflection Assessment (AIDA) mission project in collaboration between NASA and ESA aims to perform an asteroid deflection demonstration, using a half-ton projectile that will perform a hypervelocity impact on the surface of the secondary of the binary near-Earth asteroid (65803) Didymos, called hereafter Didymoon. We performed numerical simulations of the post-impact dynamics of the ejecta cloud in the framework of the current mission scenario of AIDA. Our analysis relies on a classification of the orbits as a function of the ejecta fates, e.g., a collision with one of the binary components or the escape from the region of influence of the system. A grid search of launching sites of ejecta was defined over the globe of Didymoon, and considering a wide range of possible ejection speeds, we determined the dependency of ejecta fate on launching sites (projectile impact sites) and speeds. This range enables us to track all the complex cases that include different types of dynamical fates. The results reveal the detailed proportions of the ejecta that are either orbiting, escaping or re-accreting on the primary/secondary at the end of the considered timescale, as a function of the ejection speed, which allows us to explore the global characteristics of the ejecta dynamical fates. Two major mechanisms are found to be working broadly during the post-ejection evolution of the ejecta cloud: 1) ejecta on mean motion resonance orbits with Didymoon produce long-term quasi-periodic showers onto Didymoon over at least a couple of weeks after the projectile impact, 2) ejecta on non-resonant orbits produce a rapid and high re-accretion flux. This rapid and high flux occurs just once because ejecta on such orbits leave the system unless they experience a collision during their first encounter. For both mechanisms, swing-bys of Didymoon are found to occur. These swing-bys are a source of chaotic motion because the outcome of the swing-by is extremely sensitive to the ejecta initial conditions. Moreover, for all ejecta speeds, a zone free of ejecta is noticed to emerge around the mid-latitude zone of the celestial sphere about two months after the projectile impact. Also, the extent of this zone depends on the ejecta speed. For the second part of this study, we performed full-scale simulations of the ejecta cloud released from 6 hypothetical impact sites. To define the initial conditions of the ejecta based on cratering scaling laws, we considered two kinds of material composing Didymoon’s subsurface and then combined a power-law size distribution of the ejecta with an ejection speed distribution. We find that the ejecta cloud evolution can be divided in two periods. It starts with a first violent period (<10 h) with fast re-accretion or ejection of the ejecta from the system. A second period is found to be more sensitive to the launching site than the first one. During this second period, ejecta will either re-accrete or being ejected from the system, depending both on their sizes and on their average survival time in close proximity of the binary components. There is thus a size-sorting effect dictated by the solar radiation pressure, which proves to be efficient to move out of the system the dust-size ejecta (<1 mm) for all considered launching sites and material types. On the other hand, the larger ejecta, being less or not affected by the solar radiation pressure, can survive longer in the system.

Dynamical rearrangement of super-Earths during disk dispersal

Context.The Kepler mission has provided a large sample to statistically analyze the orbital properties of the super-Earth planets. We hypothesize that these planets formed early and consider the problem of matching planet formation theory to the current observations. Two scenarios, disk migration and in-situ formation, have been proposed to explain their origin. In the migration scenario planets migrate inward due to planet-disk interaction, whereas in the in-situ scenario planets assemble locally. Therefore, planets formed by migration are expected to end up in resonances, whereas those formed in-situ are expected to stay in short period ratios and in non-resonant orbits. Both predictions are at odds with observations. Aims. We investigate whether a preferred formation scenario can be identified through a comparison between the magnetospheric rebound model and the Kepler data. Methods. We conduct N-body simulations of two-planet systems during the disk dispersal phase, and make a statistical comparison between the simulations and the Kepler observations. Results. Comparing the two scenarios, we find that magnetospheric rebound tends to erase the difference in the orbital configuration that was initially presented. After disk dispersal, not all planets are in resonance in the migration scenario, whereas planets do not remain in compact configurations in the in-situ scenario. In both scenarios, the orbits of planets increase with the cavity expansion, and their period ratios have a wider distribution. Conclusions. From a statistical perspective, the magnetospheric rebound model reproduces several observed properties of Kepler planets, such as the significant number of planets are not in resonances and planet pairs can end up at large period ratios. The disparity in orbital configuration between the two formation scenarios is substantially reduced after disk dispersal.

Long-term evolution and stability of Saturnian small satellites: Aegaeon, Methone, Anthe and Pallene

Aegaeon, Methone, Anthe, and Pallene are four Saturnian small moons, discovered by the Cassini spacecraft. Although their orbital characterization has been carried on by a number of authors, their long-term evolution has not been studied in detail so far. In this work, we numerically explore the long-term evolution, up to $10^5$ yr, of the small moons in a system formed by an oblate Saturn and the five largest moons close to the region: Janus, Epimetheus, Mimas, Enceladus, and Tethys. By using frequency analysis we determined the stability of the small moons and characterize, through diffusion maps, the dynamical behavior of a wide region of geometric phase space, $a$ vs $e$, surrounding them. Those maps could shed light on the possible initial number of small bodies close to Mimas, and help to better understand the dynamical origin of the small satellites. We found that the four small moons are long-term stable and no mark of chaos is found for them. Aegaeon, Methone, and Anthe could remain unaltered for at least $\sim0.5$Myr, given the current configuration of the system. They remain well-trapped in the corotation eccentricity resonances with Mimas in which they currently librate. However, perturbations from nearby resonances, such as Lindblad eccentricity resonances with Mimas, seem responsible for largest variations observed for Methone and Anthe. Pallene remains in a non-resonant orbit and it is the more stable, at least for 64 Myr. Nonetheless, it is affected by a quasi-resonance with Mimas, which induces long-term orbital oscillations of its eccentricity and inclination.

NEPTUNE’S ORBITAL MIGRATION WAS GRAINY, NOT SMOOTH

ABSTRACT The Kuiper Belt is a population of icy bodies beyond the orbit of Neptune. The complex orbital structure of the Kuiper Belt, including several categories of objects inside and outside of resonances with Neptune, emerged as a result of Neptune’s migration into an outer planetesimal disk. An outstanding problem with the existing migration models is that they invariably predict excessively large resonant populations, while observations show that the non-resonant orbits are in fact common (e.g., the main belt population is ≃2–4 times larger than Plutinos in the 3:2 resonance). Here we show that this problem can be resolved if it is assumed that Neptune’s migration was grainy, as expected from scattering encounters of Neptune with massive planetesimals. The grainy migration acts to destabilize resonant bodies with large libration amplitudes, a fraction of which ends up on stable non-resonant orbits. Thus, the non-resonant-to-resonant ratio obtained with the grainy migration is higher, up to ∼10 times higher for the range of parameters investigated here, than in a model with smooth migration. In addition, the grainy migration leads to a narrower distribution of the libration amplitudes in the 3:2 resonance. The best fit to observations is obtained when it is assumed that the outer planetesimal disk below 30 au contained 1000–4000 Plutos. We estimate that the combined mass of Pluto-class objects in the original disk represented 10%–40% of the estimated disk mass ( ). This constraint can be used to better understand the accretion processes in the outer solar system.

SOPHIE velocimetry ofKeplertransit candidates

As part of our follow-up campaign of Kepler planets, we observed Kepler-117 with the SOPHIE spectrograph at the Observatoire de Haute-Provence. This F8-type star hosts two transiting planets in non-resonant orbits. The planets, Kepler-117 b and c, have orbital periods $\simeq 18.8$ and $\simeq 50.8$ days, and show transit-timing variations (TTVs) of several minutes. We performed a combined Markov chain Monte Carlo (MCMC) fit on transits, radial velocities, and stellar parameters to constrain the characteristics of the system. We included the fit of the TTVs in the MCMC by modeling them with dynamical simulations. In this way, consistent posterior distributions were drawn for the system parameters. According to our analysis, planets b and c have notably different masses ($0.094 \pm 0.033$ and $1.84 \pm 0.18$ M$_{\rm J}$) and low orbital eccentricities ($0.0493 \pm 0.0062$ and $0.0323 \pm 0.0033$). The uncertainties on the derived parameters are strongly reduced if the fit of the TTVs is included in the combined MCMC. The TTVs allow measuring the mass of planet b, although its radial velocity amplitude is poorly constrained. Finally, we checked that the best solution is dynamically stable.

Secular orbital evolution of planetary systems and the dearth of close-in planets around fast rotators

Recent analyses of Kepler space telescope data reveal that transiting planets with orbital periods shorter than 2–3 d are generally observed around late-type stars with rotation periods longer than ∼5–10 d. We investigate different explanations for this phenomenon and favour an interpretation based on secular perturbations in multiplanet systems on non-resonant orbits. In those systems, the orbital eccentricity of the innermost planet can reach values close to unity through a process of chaotic diffusion of its orbital elements in the phase space. When the eccentricity of the innermost orbit becomes so high that the periastron gets closer than ∼0.05 au, tides shrink and circularize the orbit producing a close-in planet on a time-scale ≲50 Myr. The probability of high eccentricity excitation and subsequent circularization is estimated and is found to increase with the age of the system. Thus, we are able to explain the observed statistical correlation between stellar rotation and minimum orbital period of the innermost planet by using the stellar rotation period as a proxy of its age through gyrochronology. Moreover, our model is consistent with the entire observed distributions of the rotation and orbital periods Porb for 3 ≲ Porb ≲ 15 d.

Leveraging Resonant-Orbit Manifolds to Design Transfers Between Libration-Point Orbits

Resonant orbits have been widely employed in mission design for planetary flyby trajectories (Jupiter Europa Orbiter) and, more recently, as a source of long-term stability (Interstellar Boundary Explorer). Yet, resonant orbits have not been explored extensively as transfer mechanisms between nonresonant orbits in multibody systems. To highlight the benefit of employing resonant orbits for transfers and given the increased interest in employing libration-point orbits for a variety of purposes, planar and three-dimensional transfers from a low Earth orbit to the vicinity of the Earth–moon libration points via resonant arcs are constructed. Solutions are efficiently generated in the circular restricted three-body model, and transitioned to a higher-fidelity model for validation. Direct optimization techniques are applied to further reduce the propellant requirements, and a system translation process is defined to allow for the quick translation of Earth–moon transfers to other three-body systems.

Spin-up of low-mass classical bulges in barred galaxies

Secular evolution is one of the key routes through which galaxies evolve along the Hubble sequence. Not only the disk undergoes morphological and kinematic changes, but also a preexisting classical bulge may be dynamically changed by the secular processes driven primarily by the bar. We study the influence of a growing bar on the dynamical evolution of a low mass classical bulge such as might be present in galaxies like the Milky Way. Using self-consistent high resolution {\it N}-body simulations, we study how an initially isotropic non-rotating small classical bulge absorbs angular momentum emitted by the bar. The basic mechanism of this angular momentum exchange is through resonances and a considerable fraction of the angular momentum is channeled through Lagrange point (-1:1) and ILR (2:1) orbits. In the phase of rapid dynamical growth, also retrograde non-resonant orbits absorb significant angular momentum. As a result of this angular momentum gain, the initially non-rotating classical bulge transforms into a fast rotating, radially anisotropic and triaxial object, embedded in the similarly fast rotating boxy bulge formed from the disk. Towards the end of the evolution, the classical bulge develops cylindrical rotation. By that time, its inner regions host a "classical bulge-bar" whose distinct kinematics could serve as direct observational evidence for the secular evolution in the galaxy. Some implications of these results are discussed briefly.

On the detection of (habitable) super-Earths around low-mass stars using Kepler and transit timing variation method

We present the results of an extensive study of the detectability of Earth-sized planets and super-Earths in the habitable zones of cool and low-mass stars using transit timing variation method. We have considered a system consisting of a star, a transiting giant planet, and a terrestrial-class perturber, and calculated TTVs for different values of the parameters of the system. To identify ranges of the parameters for which these variations would be detectable by Kepler, we considered the analysis presented by Ford et al. (Transit timing observations from Kepler: I. Statistical analysis of the first four months. ArXiv:1102.0544, 2011) and assumed that a peak-to-peak variation of 20 s would be within the range of the photometric sensitivity of this telescope. We carried out simulations for resonant and non-resonant orbits, and identified ranges of the semimajor axes and eccentricities of the transiting and perturbing bodies for which an Earth-sized planet or a super-Earth in the habitable zone of a low-mass star would produce such TTVs. Results of our simulations indicate that in general, outer perturbers near first- and second-order resonances show a higher prospect for detection. Inner perturbers are potentially detectable only when near 1:2 and 1:3 mean-motion resonances. For a typical M star with a Jupiter-mass transiting planet, for instance, an Earth-mass perturber in the habitable zone can produce detectable TTVs when the orbit of the transiting planet is between 15 and 80 days. We present the details of our simulations and discuss the implication of the results for the detection of terrestrial planets around different low-mass stars.

Theoretical predictions of mass, semimajor axis and eccentricity distributions of super-Earths

AbstractWe discuss the effects of close scattering and merging between planets on distributions of mass, semimajor axis and orbital eccentricity, using population synthesis model of planet formation, focusing on the distributions of close-in super-Earths, which are being observed recently. We found that a group of compact embryos emerge interior to the ice line, grow, migrate, and congregate into closely-packed convoys which stall in the proximity of their host stars. After the disk-gas depletion, they undergo orbit crossing, close scattering, and giant impacts to form multiple rocky Earths or super-Earths in non-resonant orbits around ~ 0.1AU with moderate eccentricities of ~ 0.01–0.1. The formation of these planets does not depend on model parameters such as type I migration speed. The fraction of solar-type stars with these super-Earths is anti-correlated with the fraction of stars with gas giants. The newly predicted family of close-in super-Earths makes less clear “planet desert” at intermediate mass range than our previous prediction.

TOWARD A DETERMINISTIC MODEL OF PLANETARY FORMATION. VI. DYNAMICAL INTERACTION AND COAGULATION OF MULTIPLE ROCKY EMBRYOS AND SUPER-EARTH SYSTEMS AROUND SOLAR-TYPE STARS

Radial velocity and transit surveys indicate that solar-type stars bear super-Earths, with mass and period up to ~ 20 M_E and a few months, are more common than those with Jupiter-mass gas giants. In many cases, these super-Earths are members of multiple-planet systems in which their mutual dynamical interaction has influenced their formation and evolution. In this paper, we modify an existing numerical population synthesis scheme to take into account protoplanetary embryos' interaction with their evolving natal gaseous disk, as well as their close scatterings and resonant interaction with each other. We show that it is possible for a group of compact embryos to emerge interior to the ice line, grow, migrate, and congregate into closely-packed convoys which stall in the proximity of their host stars. After the disk-gas depletion, they undergo orbit crossing, close scattering, and giant impacts to form multiple rocky Earths or super-Earths in non-resonant orbits around ~ 0.1AU with moderate eccentricities of ~0.01-0.1. We suggest that most refractory super-Earths with period in the range of a few days to weeks may have formed through this process. These super-Earths differ from Neptune-like ice giants by their compact sizes and lack of a substantial gaseous envelope.

Formation and dynamical evolution of the Neptune Trojans - the influence of the initial Solar system architecture

In this work, we investigate the dynamical stability of pre-formed Neptune Trojans under the gravitational influence of the four giant planets in compact planetary architectures, over 10 Myr. In our modelling, the initial orbital locations of Uranus and Neptune (aN) were varied to produce systems in which those planets moved on non-resonant orbits, or in which they lay in their mutual 1:2, 2:3 and 3:4 mean-motion resonances (MMRs). In total, 420 simulations were carried out, examining 42 different architectures, with a total of 840000 particles across all runs. In the non-resonant cases, the Trojans suffered only moderate levels of dynamical erosion, with the most compact systems (those with aN less than or equal 18 AU) losing around 50% of their Trojans by the end of the integrations. In the 2:3 and 3:4 MMR scenarios, however, dynamical erosion was much higher with depletion rates typically greater than 66% and total depletion in the most compact systems. The 1:2 resonant scenarios featured disruption on levels intermediate between the non-resonant cases and other resonant scenarios, with depletion rates of the order of tens of percent. Overall, the great majority of plausible pre-migration planetary architectures resulted in severe levels of depletion of the Neptunian Trojan clouds. In particular, if Uranus and Neptune formed near their mutual 2:3 or 3:4 MMR and at heliocentric distances within 18 AU (as favoured by recent studies), we found that the great majority of pre-formed Trojans would have been lost prior to Neptune's migration. This strengthens the case for the great bulk of the current Neptunian Trojan population having been captured during that migration.

N-BODY SIMULATIONS OF PLANETARY ACCRETION AROUND M DWARF STARS

We have investigated planetary accretion from planetesimals in terrestrial planet regions inside the ice line around M dwarf stars through N-body simulations including tidal interactions with disk gas. Because of low luminosity of M dwarfs, habitable zones (HZs) are located in inner regions. In the close-in HZ, type-I migration and the orbital decay induced by eccentricity damping are efficient according to the high disk gas density in the small orbital radii. In the case of full efficiency of type-I migration predicted by the linear theory, we found that protoplanets that migrate to the vicinity of the host star undergo close scatterings and collisions, and 4 to 6 planets eventually remain in mutual mean motion resonances and their orbits have small eccentricities and they are stable both before and after disk gas decays. In the case of slow migration, the resonant capture is so efficient that densely-packed ~ 40 small protoplanets remain in mutual mean motion resonances. In this case, they start orbit crossing, after the disk gas decays and eccentricity damping due to tidal interaction with gas is no more effective. Through merging of the protoplanets, several planets in widely-separated non-resonant orbits with relatively large eccentricities are formed. Thus, the final orbital configurations of the terrestrial planets around M dwarfs sensitively depend on strength of type-I migration. We also found that large amount of water-ice is delivered by type-I migration from outer regions and final planets near the inner disk edge around M dwarfs are generally abundant in water-ice except for the innermost one that is shielded by the outer planets, unless type-I migration speed is reduced by a factor of more than 100 from that predicted by the linear theory.

New Observations and a Possible Detection of Parameter Variations in the Transits of Gliese 436b

We present ground-based observations of the transiting Neptune-mass planet Gl 436b obtained with the 3.5 m telescope at Apache Point Observatory and other supporting telescopes. Included in this is an observed transit in early 2005, over 2 years before the earliest reported transit detection. We have compiled all available transit data to date and perform a uniform modeling using the JKTEBOP code. We do not detect any transit timing variations of amplitude greater than ~1 minute over the ~3.3 year baseline. We do however find possible evidence for a self-consistent trend of increasing orbital inclination, transit width, and transit depth, which supports the supposition that Gl 436b is being perturbed by another planet of 12 M⊕ in a nonresonant orbit.

Discovery of a binary Centaur

We have identified a binary companion to (42355) 2002 CR 46 in our ongoing deep survey using the Hubble Space Telescope's High Resolution Camera. It is the first companion to be found around an object in a non-resonant orbit that crosses the orbits of giant planets. Objects in orbits of this kind, the Centaurs, have experienced repeated strong scattering with one or more giant planets and therefore the survival of binaries in this transient population has been in question. Monte Carlo simulations suggest, however, that binaries in (42355) 2002 CR 46-like heliocentric orbits have a high probability of survival for reasonable estimates of the binary's still-unknown system mass and separation. Because Centaurs are thought to be precursors to short period comets, the question of the existence of binary comets naturally arises; none has yet been definitively identified. The discovery of one binary in a sample of eight observed by HST suggests that binaries in this population may not be uncommon.