Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets

By M. Lee, S. Peale. 2001

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The discovery by Marcy and coworkers of two planets in 2 : 1 orbital resonance about the star GJ 876 has been supplemented by a dynamical fit to the data by Laughlin & Chambers, which places the planets in coplanar orbits deep in three resonances at the 2 : 1 mean-motion commensurability. The selection of this almost singular state by the dynamical fit means that the resonances are almost certainly real, and with the small amplitudes of libration of the resonance variables, indefinitely stable. Several unusual properties of the 2 : 1 resonances are revealed by the GJ 876 system. The libration of both lowest order mean-motion resonance variables and the secular resonance variable, θ1 = λ1 - 2λ2 + ϖ1, θ2 = λ1 - 2λ2 + ϖ2, and θ3 = ϖ1 - ϖ2, about 0° (where λ1,2 are the mean longitudes of the inner and outer planet and ϖ1,2 are the longitudes of periapse) differs from the familiar geometry of the Io-Europa pair, where θ2 and θ3 librate about 180°. By considering the condition that 1 = 2 for stable simultaneous librations of θ1 and θ2, we show that the GJ 876 geometry results from the large orbital eccentricities ei, whereas the very small eccentricities in the Io-Europa system lead to the latter's geometry. Surprisingly, the GJ 876 configuration, with θ1, θ2, and θ3 all librating, remains stable for e1 up to 0.86 and for amplitude of libration of θ1 approaching 45° with the current eccentricities—further supporting the indefinite stability of the existing system. Any process that drives originally widely separated orbits toward each other could result in capture into the observed resonances at the 2 : 1 commensurability. We find that forced inward migration of the outer planet of the GJ 876 system results in certain capture into the observed resonances if initially e1 ≲ 0.06 and e2 ≲ 0.03 and the migration rate |2/a2| ≲ 3 × 10-2(a2/AU)-3/2 yr-1. Larger eccentricities lead to likely capture into higher order resonances before the 2 : 1 commensurability is reached. The planets are sufficiently massive to open gaps in the nebular disk surrounding the young GJ 876 and to clear the disk material between them, and the resulting planet-nebular interaction typically forces the outer planet to migrate inward on the disk viscous timescale, whose inverse is about 3 orders of magnitude less than the above upper bound on |2/a2| for certain capture. If there is no eccentricity damping, eccentricity growth is rapid with continued migration within the resonance, with ei exceeding the observed values after a further reduction in the semimajor axes ai of only 7%. With eccentricity damping i/ei = -K|i/ai|, the eccentricities reach equilibrium values that remain constant for arbitrarily long migration within the resonances. The equilibrium eccentricities are close to the observed eccentricities for K ≈ 100 if there is migration and damping of the outer planet only, but for K ≈ 10 if there is also migration and damping of the inner planet. This result is independent of the magnitude or functional form of the migration rate i as long as i/ei = -K|i/ai|. Although existing analytic estimates of the effects of planet-nebula interaction are consistent with this form of eccentricity damping for certain disk parameter values, it is as yet unclear that such interaction can produce the large value of K required to obtain the observed eccentricities. The alternative eccentricity damping by tidal dissipation within the star or the planets is completely negligible, so the observed dynamical properties of the GJ 876 system may require an unlikely fine-tuning of the time of resonance capture to be near the end of the nebula lifetime.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/8bc9b28ef14d1226cce6ca9f161eddedd7d3c5e4

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Citations in the corpus

THE LICK-CARNEGIE EXOPLANET SURVEY: A URANUS-MASS FOURTH PLANET FOR GJ 876 IN AN EXTRASOLAR LAPLACE CONFIGURATION (2010, background): “Murray et al. (2002), Lee & Peale (2002), Chiang et al. (2002), Thommes & Lissauer (2003), Beauge et al. (2003), Kley et al. (2004), Lee (2004), Kley et al. (2005), and Lee & Thommes (2009) examined scenarios of how the system may have come to be in its current dynamical configuration.”
Planet-Disk Interaction and Orbital Evolution (2012, background): “Continued migration in resonance drives the eccentricities upward, and they only remain small during migration if the planets also experience eccentricity damping (Lee & Peale, 2002).”
Dynamical Outcomes of Planet-Planet Scattering (2007, background): “Convergent migration in a dissipative disk can lead to resonant capture into a stable MMR, particularly the 2:1 MMR (Lee & Peale 2002).”
The minimum-mass extrasolar nebula: in situ formation of close-in super-Earths (2012, background): “Even for Gliese 876 b and c — a pair of giant planets trapped deeply in a 2:1 resonance — the amount by which these planets migrated differentially could have been as small as ∼7% (Lee & Peale 2002; see also Rivera et al. 2010).”
OVERSTABLE LIBRATIONS CAN ACCOUNT FOR THE PAUCITY OF MEAN MOTION RESONANCES AMONG EXOPLANET PAIRS (2013, background): “The fact that Lee & Peale (2002) did not find any overstable librations for the GJ 876 planets is consistent with our analytic findings, since the roughly Jupiter mass planets and the range of values of τe/τn considered in their study should result in damping of librations and permanent resonance…”
DISSIPATIVE DIVERGENCE OF RESONANT ORBITS (2012, background): “For example, planets embedded in protoplanetary disks experience dissipative forces exerted by the gaseous nebula (Lee & Peale 2002), while planets that reside on orbits that are in close proximity to their host stars are subject to tidal 5 friction (Bodenheimer et al. 2001) (in this work, we shall…”
STATISTICAL STUDY OF THE EARLY SOLAR SYSTEM'S INSTABILITY WITH FOUR, FIVE, AND SIX GIANT PLANETS (2012, background): “The exact nature of this coupling, however, depends on several disk parameters that are poorly constrained (e.g., Lee & Peale 2002).”
Three-dimensional simulations of multiple protoplanets embedded in a protostellar disc (2008, background): “…HD 82943, HD 7352), whose resonances are thought to have been established through differential type II migration (e.g. Snellgrove et al. 2001; Lee & Peale 2002; Kley et al. 2004), our simulations show that differential type I migration may lead to the formation of multiple MMRs between…”
THE RUNTS OF THE LITTER: WHY PLANETS FORMED THROUGH GRAVITATIONAL INSTABILITY CAN ONLY BE FAILED BINARY STARS (2009, background): “This mechanism is invoked by Lee & Peale (2002) to generate convergent migration and resonance capture in the planets orbiting GJ 876.”
Debris disks as signposts of terrestrial planet formation (2011, background): “…that expected if multiple giant planets interact with the gas disk such that they end up trapped in mean-motion resonances (Snellgrove et al. 2001; Lee & Peale 2002; Kley et al. 2004), provided that the majority of these resonances are broken before or shortly after the dispersal of the gas disk.”
The Core Accretion Model Predicts Few Jovian-Mass Planets Orbiting Red Dwarfs (2004, background): “The system displays clear signs of having undergone migration (Lee & Peale 2002), indicating that abundant gas was present in the system when the planets formed.”
The Occurrence and Architecture of Exoplanetary Systems (2014, background): “…for resonances, and because the theory of disk migration—by which wide-orbiting planets are conveyed to smaller orbits through gravitational interactions with the protoplanetary disk—predicted that planet pairs would often be caught into resonances (Goldreich & Tremaine 1980, Lee & Peale 2002).”
TIDALLY HEATED TERRESTRIAL EXOPLANETS: VISCOELASTIC RESPONSE MODELS (2009, background): “Mean Motion Resonances Several mean motion resonances (MMR) have been observed in extrasolar planet systems (Lee & Peale 2002; Gu et al. 2003; Kley et al. 2005).”
CAPTURE OF TROJANS BY JUMPING JUPITER (2013, result): “Using artificial forces we will also slowly damped planetary eccentricities and inclinations (Lee & Peale 2002), in a manner that is consistent with the original evolutions.”
Structure and Evolution of Nearby Stars with Planets. II. Physical Properties of ~1000 Cool Stars from the SPOCS Catalog (2006, background): “Systems with multiple giant planets of comparable masses can easily achieve high eccentricities through dynamical instabilities (Ford et al. 2005) or crossing orbital resonances (Lee & Peale 2002; Kley et al. 2004).”
Five New Multicomponent Planetary Systems (2005, background): “…the planets in multiplanet systems shape the final configuration of the system and shed light on the dissipative medium when the interactions occurred (Bryden et al. 2000; Rivera & Lissauer 2001; Laughlin & Chambers 2001; Lee & Peale 2002; Chiang & Murray 2002; Ford et al. 2003; Ida & Lin 2004).”
Quantifying the Uncertainty in the Orbits of Extrasolar Planets (2003, background): “In this case, the two precession rates are comparable, allowing for a secular resonance in which ∆ω remains small (Lee & Peale 2002).”
Secular Evolution of Hierarchical Planetary Systems (2002, background): “The outer two planets of the And system (Rivera & Lissauer 2000; Lissauer & Rivera 2001; Chiang et al. 2001) and the two planets about GJ 876 (Laughlin & Chambers 2001; Lee & Peale 2002) are also likely in the secular resonance involving $1 $2, but in both of these cases$1 $2 librates about 0 .”
Origins of Eccentric Extrasolar Planets: Testing the Planet-Planet Scattering Model (2007, background): “…Juric & Tremaine 2008), d) interactions of orbital migration with mean-motion resonances (Chiang & Murray 2002; Kley 2000; Kley et al. 2004, 2005; Lee & Peale 2002; Tsiganis et al. 2005), e) resonances between secular perturbations and precession induced by general relativity, stellar…”