Three-body Mean Motion Resonances Research Articles

Kepler-80 Revisited: Assessing the Participation of a Newly Discovered Planet in the Resonant Chain

In this paper, we consider the chain of resonances in the Kepler-80 system and evaluate the impact that the additional member of the resonant chain discovered by Shallue & Vanderburg has on the dynamics of the system and the physical parameters that can be recovered by a fit to the transit timing variations (TTVs). Ultimately, we calculate the mass of Kepler-80 g to be 0.8 ± 0.3M ⊕ when assuming all planets have zero eccentricity, and 1.0 ± 0.3 M ⊕ when relaxing that assumption. We show that the outer five planets are in successive three-body mean-motion resonances (MMRs). We assess the current state of two-body MMRs in the system and find that the planets do not appear to be in two-body MMRs. We find that while the existence of the additional member of the resonant chain does not significantly alter the character of the Kepler-80 three-body MMRs, it can alter the physical parameters derived from the TTVs, suggesting caution should be applied when drawing conclusions from TTVs for potentially incomplete systems. We also compare our results to those of MacDonald et al., who perform a similar analysis on the same system with a different method. Although the results of this work and MacDonald et al. show that different fit methodologies and underlying assumptions can result in different measured orbital parameters, the most secure conclusion is that which holds true across all lines of analysis: Kepler-80 contains a chain of planets in three-body MMRs but not in two-body MMRs.

Web of resonances and possible path of evolution of the small Uranian satellites

Satellite systems around giant planets are immersed in a region of complex resonant configurations. Understanding the role of satellite resonances contributes to comprehending the dynamical processes in planetary formation and posterior evolution. Our main goal is to analyse the resonant structure of small moons around Uranus and propose different scenarios able to describe the current configuration of these satellites. We focus our study on the external members of the regular satellites interior to Miranda, namely Rosalind, Cupid, Belinda, Perdita, Puck, and Mab, respectively. We use N-body integrations to perform dynamical maps to analyse their dynamics and proximity to two-body and three-body mean-motion resonances (MMR). We found a complicated web of low-order resonances amongst them. Employing analytical prescriptions, we analysed the evolution by gas drag and type-I migration in a circumplanetary disc (CPD) to explain different possible histories for these moons. We also model the tidal evolution of these satellites using some crude approximations and found possible paths that could lead to MMRs crossing between pairs of moons. Finally, our simulations show that each mechanism can generate significant satellite radial drift leading to possible resonant capture, depending on the distances and sizes.

The dynamics of the TRAPPIST-1 system in the context of its formation

ABSTRACT TRAPPIST-1 is an 0.09 M⊙ star, which harbours a system of seven Earth-sized planets. Two main features stand out: (i) all planets have similar radii, masses, and compositions; and (ii) all planets are in resonance. Previous works have outlined a pebble-driven formation scenario where planets of similar composition form sequentially at the H2O snowline (∼0.1 au for this low-mass star). It was hypothesized that the subsequent formation and migration led to the current resonant configuration. Here, we investigate whether the sequential planet formation model is indeed capable to produce the present-day resonant configuration, characterized by its two-body and three-body mean motion resonances structure. We carry out N-body simulations, accounting for type-I migration, stellar tidal damping, disc eccentricity-damping, and featuring a migration barrier located at the disc’s inner edge. Due to the sequential migration, planets naturally form a chain of first-order resonances. But to explain the period ratios of the b/c/d-system, which are presently in higher order resonances, we find that planets b and c must have marched across the migration barrier, into the gas-free cavity, before the disc has dispersed. We investigate both an early and late cavity infall scenario and find that the early infall model best matches the constraints, as well as being more probable. After the dispersal of the gaseous disc, stellar tidal torque also contributes towards a modest separation of the inner system. We outline how the insights obtained in this work can be applied to aid the understanding of other compact resonant planet systems.

The functional relation between three-body mean motion resonances and Yarkovsky drift speeds

ABSTRACT We examined the motion of asteroids across the three-body mean motion resonances (MMRs) with Jupiter and Saturn and with the Yarkovsky drift speed in the semimajor axis of the asteroids. The research was conducted using numerical integrations performed using the Orbit9 integrator with 84 000 test asteroids. We calculated time delays, dtr, caused by the seven three-body MMRs on the mobility of test asteroids with 10 positive and 10 negative Yarkovsky drift speeds, which are reliable for Main Belt asteroids. Our final results considered only test asteroids that successfully crossed over the MMRs without close approaches to the planets. We have devised two equations that approximately describe the functional relation between the average time 〈dtr〉 spent in the resonance, the strength of the resonance SR, and the semimajor axis drift speed da/dt (positive and negative) with the orbital eccentricities of asteroids in the range (0, 0.1). Comparing the values of 〈dtr〉 obtained from the numerical integrations and from the derived functional relations, we analysed average values of 〈dtr〉 in all three-body MMRs for every da/dt. The main conclusion is that the analytical and numerical estimates of the average time 〈dtr〉 are in very good agreement, for both positive and negative da/dt. Finally, this study shows that the functional relation we obtain for three-body MMRs is analogous to that previously obtained for two-body MMRs.

Survivability of moon systems around ejected gas giants

ABSTRACT We examine the effects that planetary encounters have on the moon systems of ejected gas giant planets. We conduct a suite of numerical simulations of planetary systems containing three Jupiter-mass planets (with the innermost planet at 3 au) up to the point where a planet is ejected from the system. The ejected planet has an initial system of 100 test-particle moons. We determine the survival probability of moons at different distances from their host planet, measure the final distribution of orbital elements, examine the stability of resonant configurations, and characterize the properties of moons that are stripped from the planets. We find that moons are likely to survive in orbits with semi-major axes out beyond 200 planetary radii (0.1 au in our case). The orbital inclinations and eccentricities of the surviving moons are broadly distributed and include nearly hyperbolic orbits and retrograde orbits. We find that a large fraction of moons in two-body and three-body mean-motion resonances also survive planetary ejection with the resonance intact. The moon–planet interactions, especially in the presence of mean-motion resonance, can keep the interior of the moons molten for billions of years via tidal flexing, as is seen in the moons of the gas giant planets in the solar system. Given the possibility that life may exist in the subsurface ocean of the Galilean satellite Europa, these results have implications for life on the moons of rogue planets – planets that drift through our Galaxy with no host star.

Element history of the Laplace resonance: a dynamical approach

Context. We consider the three-body mean motion resonance defined by the Jovian moons Io, Europa, and Ganymede, which is commonly known as the Laplace resonance. In terms of the moons’ mean longitudes λ1 (Io), λ2 (Europa), and λ3 (Ganymede), this resonance is described by the librating argument φL ≡ λ1 − 3λ2 + 2λ3 ≈ 180°, which is the sum of φ12 ≡ λ1 − 2λ2 + ϖ2 ≈ 180° and φ23 ≡ λ2 − 2λ3 + ϖ2 ≈ 0°, where ϖ2 denotes Europa’s longitude of perijove. Aims. In particular, we construct approximate models for the evolution of the librating argument φL over the period of 100 yr, focusing on its principal amplitude and frequency, and on the observed mean motion combinations n1 − 2n2 and n2 − 2n3 associated with the quasi-resonant interactions above. Methods. First, we numerically propagated the Cartesian equations of motion of the Jovian system for the period under examination, and by comparing the results with a suitable set of ephemerides, we derived the main dynamical effects on the target quantities. Using these effects, we built an alternative Hamiltonian formulation and used the normal forms theory to precisely locate the resonance and to semi-analytically compute its main amplitude and frequency. Results. From the Cartesian model we observe that on the timescale considered and with ephemerides as initial conditions, both φL and the diagnostics n1 − 2n2 and n2 − 2n3 are well approximated by considering the mutual gravitational interactions of Jupiter and the Galilean moons (including Callisto), and the effect of Jupiter’s J2 harmonic. Under the same initial conditions, the Hamiltonian formulation in which Callisto and J2 are reduced to their secular contributions achieves larger errors for the quantities above, particularly for φL. By introducing appropriate resonant variables, we show that these errors can be reduced by moving in a certain action-angle phase plane, which in turn implies the necessity of a tradeoff in the selection of the initial conditions. Conclusions. In addition to being a good starting point for a deeper understanding of the Laplace resonance, the models and methods described are easily generalizable to different types of multi-body mean motion resonances. Thus, they are also prime tools for studying the dynamics of extrasolar systems.

Asteroids in three-body mean motion resonances with planets

We have identified all asteroids in three-body mean-motion resonances in all possible planets configurations. The identification was done dynamically: the orbits of the asteroids were integrated for 100,000 yrs and the set of the resonant arguments was numerically analyzed. We have found that each possible planets configuration has a lot of the resonant asteroids. In total 65,972 resonant asteroids (≈14.1% of the total number of 467,303 objects from AstDyS database) have been identified.

Identification of asteroids trapped inside three-body mean motion resonances: a machine-learning approach

Identification of asteroids trapped inside three-body mean motion resonances: a machine-learning approach

Asteroids in three-body mean-motion resonances with Jupiter and Mars

We identify the asteroids in three-body mean-motion resonances with Jupiter and Mars on the set of all known on April 2016 numbered asteroids (467308 objects). The resonant objects are identified by the direct analysis of the behavior (libration/circulation) of the resonant arguments on 100000 yrs. All essential perturbations during the integration of the equations of the motion are taken into account. The number of the asteroids in different resonances has been calculated for all possible resonances with the order less or equal 6.

The Laplace resonance in the Kepler-60 planetary system

Abstract We investigate the dynamical stability of the Kepler-60 planetary system with three super-Earths. We determine their orbital elements and masses by transit timing variation (TTV) data spanning quarters Q1–Q16 of the Kepler mission. The system is dynamically active but the TTV data constrain masses to ∼4 M⊕ and orbits in safely wide stable zones. The observations prefer two types of solutions. The true three-body Laplace mean-motion resonance (MMR) exhibits the critical angle librating around ≃45° and aligned apsides of the inner and outer pair of planets. In the Laplace MMR formed through a chain of two-planet 5:4 and 4:3 MMRs, all critical angles librate with small amplitudes ∼30° and apsidal lines in planet's pairs are anti-aligned. The system is simultaneously locked in a three-body MMR with librations amplitude ≃10o. The true Laplace MMR can evolve towards a chain of two-body MMRs in the presence of planetary migration. Therefore, the three-body MMR formed in this way seems to be more likely state of the system. However, the true three-body MMR cannot be disregarded a priori and it remains a puzzling configuration that may challenge the planet formation theory.

SUBLIMATION-DRIVEN ACTIVITY IN MAIN-BELT COMET 313P/GIBBS

We present an observational and dynamical study of newly discovered main-belt comet 313P/Gibbs. We find that the object is clearly active both in observations obtained in 2014 and in precovery observations obtained in 2003 by the Sloan Digital Sky Survey, strongly suggesting that its activity is sublimation-driven. This conclusion is supported by a photometric analysis showing an increase in the total brightness of the comet over the 2014 observing period, and dust modeling results showing that the dust emission persists over at least three months during both active periods, where we find start dates for emission no later than 2003 July 24+/-10 for the 2003 active period and 2014 July 28+/-10 for the 2014 active period. From serendipitous observations by the Subaru Telescope in 2004 when the object was apparently inactive, we estimate that the nucleus has an absolute R-band magnitude of H_R=17.1+/-0.3, corresponding to an effective nucleus radius of r_e~1.00+/-0.15 km. The object's faintness at that time means we cannot rule out the presence of activity, and so this computed radius should be considered an upper limit. We find that 313P's orbit is intrinsically chaotic, having a Lyapunov time of T_l=12000 yr and being located near two 3-body mean-motion resonances with Jupiter and Saturn, 11J-1S-5A and 10J+12S-7A, yet appears stable over >50 Myr in an apparent example of stable chaos. We furthermore find that 313P is the second main-belt comet, after P/2012 T1 (PANSTARRS), to belong to the ~155 Myr old Lixiaohua asteroid family.

Massive identification of asteroids in three-body resonances

An essential role in the asteroidal dynamics is played by the mean motion resonances. Two-body planet–asteroid resonances are widely known, due to the Kirkwood gaps. Besides, so-called three-body mean motion resonances exist, in which an asteroid and two planets participate. Identification of asteroids in three-body (namely, Jupiter–Saturn–asteroid) resonances was initially accomplished by Nesvorný and Morbidelli (Nesvorný D., Morbidelli, A. [1998]. Astron. J. 116, 3029–3037), who, by means of visual analysis of the time behaviour of resonant arguments, found 255 asteroids to reside in such resonances. We develop specialized algorithms and software for massive automatic identification of asteroids in the three-body, as well as two-body, resonances of arbitrary order, by means of automatic analysis of the time behaviour of resonant arguments. In the computation of orbits, all essential perturbations are taken into account. We integrate the asteroidal orbits on the time interval of 100,000yr and identify main-belt asteroids in the three-body Jupiter–Saturn–asteroid resonances up to the 6th order inclusive, and in the two-body Jupiter–asteroid resonances up to the 9th order inclusive, in the set of ∼250,000 objects from the “Asteroids – Dynamic Site” (AstDyS) database. The percentages of resonant objects, including extrapolations for higher-order resonances, are determined. In particular, the observed fraction of pure-resonant asteroids (those exhibiting resonant libration on the whole interval of integration) in the three-body resonances up to the 6th order inclusive is ≈0.9% of the whole set; and, using a higher-order extrapolation, the actual total fraction of pure-resonant asteroids in the three-body resonances of all orders is estimated as ≈1.1% of the whole set.

OBSERVATIONAL AND DYNAMICAL CHARACTERIZATION OF MAIN-BELT COMET P/2010 R2 (La Sagra)

We present observations of comet-like main-belt object P/2010 R2 (La Sagra) obtained by Pan-STARRS 1 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 at La Silla, and the Isaac Newton Telescope on La Palma. An antisolar dust tail is observed from August 2010 through February 2011, while a dust trail aligned with the object's orbit plane is also observed from December 2010 through August 2011. Assuming typical phase darkening behavior, P/La Sagra is seen to increase in brightness by >1 mag between August 2010 and December 2010, 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 H_R=17.9+/-0.2 mag, corresponding to a nucleus radius of ~0.7 km, assuming an albedo of p=0.05. Using optical spectroscopy, we find no evidence of sublimation products (i.e., gas emission), finding an upper limit CN production rate of Q_CN<6x10^23 mol/s, from which we infer an H2O production rate of Q_H2O<10^26 mol/s. Numerical simulations indicate that P/La Sagra is dynamically stable for >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.

Numerical exploration of resonant dynamics in the system of Saturnian major satellites

We numerically investigate the long-term dynamics of the Saturnian system by analyzing the Fourier spectra of ensembles of orbits taken around the current orbits of Mimas, Enceladus, Tethys, Rhea and Hyperion. We construct dynamical maps around the current position of these satellites in their respective phase spaces. The maps are the result of a great deal of numerical simulations where we adopt dense sets of initial conditions and different satellite configurations. Several structures associated to the current two-body mean-motion resonances, unstable regions associated to close approaches between the satellites, and three-body mean-motion resonances in the system, are identified in the map.

Portrait of Theobalda as a young asteroid family

The (778) Theobalda asteroid family attracted little attention so far, but our study shows that it is important in several aspects. In this paper we investigate the origin and evolution of Theobalda family. Firstly, we identify the family as a statistically relevant group in the space of synthetic proper elements. Using the hierarchical clustering method and adopted cut-off velocity of d_{cutoff}=85 m/s we found that Theobalda family currently consists of 128 members. This family is located in the outer belt, near proper semi-major axis a_{p}~3.175 au. This region is crossed by several three-body mean motion resonances which give rise to significant chaotic zones. Consequently, the majority of family members reside on chaotic orbits. Using two independent methods, chaotic chronology and backward integration, we found Theobalda family to be only 6.9 +/- 2.3 Myr old. We have also estimated, that the family was likely produced by the cratering impact on a parent body of diameter D_{PB}~78 +/- 9 km.

Dynamical portrait of the Lixiaohua asteroid family

In this paper we present a comprehensive analysis of the dynamics in the region of the (3556) Lixiaohua asteroid family. The family lies in a particularly interesting region of the phase space, crossed by several two-body and three-body mean motion resonances. Also, members of this family can have close encounters with large asteroids, such as Ceres. We have identified the mean motion resonances which contribute to the long-term dynamical evolution of the family and our results confirm that the members of this family can be classified into a number of groups, exhibiting different dynamical behavior. We show for the first time that in the Lixiaohua region, apart from the chaotic diffusion in proper eccentricity and inclination (e p and I p ), there is at least one extended chaotic zone where several resonances overlap, thus giving rise to chaotic diffusion in proper semi-major axis (a p ) as well. Using a code of Monte Carlo type, we simulate the evolution of the family, according to the model which combines the chaotic diffusion (in a p , e p and I p ), Yarkovsky/YORP thermal effect and random walk in a p due to the close encounters with massive asteroids. These simulations show that all these effects should be taken into account in order to accurately explain the observed distribution of family members in the space of proper elements, although a “minimal” model that accounts for chaotic diffusion in (e p , I p ), Yarkovsky-induced drift in a p and random walk in a p due to the close encounters with the most massive asteroids is enough to grossly characterize the shape of the family.

Dynamic picture of the inner asteroid belt: implications for the density, size and taxonomic distributions of real objects

In this paper, we construct a dynamic portrait of the inner asteroidal belt. We use information about the distribution of test particles, which were initially placed on a perfectly rectangular grid of initial conditions, after 4.2 Myr of gravitational interactions with the Sun and five planets, from Mars to Neptune. Using the spectral analysis method introduced by Michtchenko et al., the asteroidal behaviour is illustrated in detail on the dynamical, averaged and frequency maps. On the averaged and frequency maps, we superpose information on the proper elements and proper frequencies of real objects, extracted from the data base, AstDyS, constructed by Milani and Kneževic. A comparison of the maps with the distribution of real objects allows us to detect possible dynamical mechanisms acting in the domain under study; these mechanisms are related to mean-motion and secular resonances. We note that the two- and three-body mean-motion resonances and the secular resonances (strong linear and weaker non-linear) have an important role in the diffusive transportation of the objects. Their long-lasting action, overlaid with the Yarkovsky effect, may explain many observed features of the density, size and taxonomic distributions of the asteroids.

Is the HR 8799 extrasolar system destined for planetary scattering?

Abstract The recent discovery of a three-planet extrasolar system of HR 8799 by Marois et al. is a breakthrough in the field of the direct imaging. This great achievement raises questions on the formation and dynamical stability of the system, because Keplerian fits to astrometric data disrupt during ∼0.2 Myr. We search for stable, self-consistent N-body orbits with the so-called GAMP (genetic algorithm with MEGNO penalty) method that incorporates stability constraints into the optimization algorithm. Our searches reveal only small regions of stable motions in the phase space of three-planet, coplanar configurations. Most likely, if the planetary masses are in 10 MJ range, they may be stable only if the planets are involved in two- or three-body mean motion resonances (MMRs). We found that 80 per cent systems found by GAMP that survived 30 Myr backwards integrations, eventually become unstable after 100 Myr. It could mean that the HR 8799 system undergo a phase of planet–planet scattering. We test a hypothesis that the less certain detection of the innermost object is due to a blending effect. In such a case, two-planet best-fitting systems are mostly stable, on quasi-circular orbits and close to the 5:2 MMR, resembling the Jupiter–Saturn pair.

On the Lyapunov exponents of the asteroidal motion subject to resonances and encounters

AbstractIn theoretical as well as practical issues of the asteroidal hazard problem, it is important to be able to assess the degree of predictability of the orbital motion of asteroids. Some asteroids move in a virtually predictable way, others do not. The characteristic time of predictability of any motion is nothing but the Lyapunov time (the reciprocal of the maximum Lyapunov exponent) of the motion. In this report, a method of analytical estimation of the maximum Lyapunov exponents of the orbital motion of asteroids is described in application for two settings of the problem. Namely, the following two types of the motion are considered: (1) the motion close to the ordinary or three-body mean motion resonances with planets, and (2) the motion in highly eccentric orbits subject to moderately close encounters with planets. Whatever different these settings may look, the analytical treatment is universal: it is performed within a single framework of the general separatrix map theory. (Recall that the separatrix maps describe the motion near the separatrices of a nonlinear resonance.) The analytical estimates of the Lyapunov times are compared to known numerical ones, i.e., to known estimates obtained by means of numerical integration of the orbits.

Long‐Term Stability and Dynamical Environment of the PSR 1257+12 Planetary System

We study the long-term dynamics of the PSR 1257+12 planetary system. Using the recently determined accurate initial condition by Konacki & Wolszczan, who derived the orbital inclinations and the absolute masses of planets B and C, we investigate the system stability by long-term, 1 Gyr direct integrations. No secular changes of the semimajor axes, eccentricities, and inclinations appear during such an interval. This stable behavior is confirmed with the fast indicator MEGNO. The analysis of the orbital stability in the neighborhood of the nominal initial condition reveals that the PSR 1257+12 system is localized in a wide stable region of the phase space but close to a few weak two- and three-body mean motion resonances. The long-term stability is additionally confirmed by a negligible exchange of the angular momentum deficit between the innermost planet A and the pair of outer planets B and C. An important feature of the system that helps sustain the stability is the secular apsidal resonance between planets B and C with the center of libration about 180°. We also find useful limits on the elements of the innermost planet A that are otherwise unconstrained by the observations. Specifically, we find that the line of nodes of planet A cannot be separated by more than about ±60° from the nodes of the bigger companions B and C. This limits the relative inclination of the orbit of planet A to the mean orbital plane of planets B and C to moderate values. We also perform a preliminary study of the short-term dynamics of massless particles in the system. We find that a relatively extended stable zone exists between planets A and B. Beyond planet C, the stable zone appears already at distances 0.5 AU from the parent star. For moderately low eccentricities beyond 1 AU, the motion of massless particles does not suffer from strong instabilities and this zone is basically stable, independent of the inclinations of the orbits of the test particles to the mean orbital plane of the system. This is an encouraging result supporting the search for a putative dust disk or a Kuiper Belt, especially with the Spitzer Space Telescope.