The TROY project: Searching for co-orbital bodies to known planets. I. Project goals and first results from archival radial velocity

The TROY project: Searching for co-orbital bodies to known planets. I. Project goals and first results from archival radial velocity

Authors:

Lillo-Box et al

Abstract:

The detection of Earth-like planets, exocomets or Kuiper belts show that the different components found in the solar system should also be present in other planetary systems. Trojans are one of these components and can be considered fossils of the first stages in the life of planetary systems. Their detection in extrasolar systems would open a new scientific window to investigate formation and migration processes. In this context, the main goal of the TROY project is to detect exotrojans for the first time and to measure their occurrence rate (eta-Trojan). In this first paper, we describe the goals and methodology of the project. Additionally, we used archival radial velocity data of 46 planetary systems to place upper limits on the mass of possible trojans and investigate the presence of co-orbital planets down to several tens of Earth masses. We used archival radial velocity data of 46 close-in (P less than 5 days) transiting planets (without detected companions) with information from high-precision radial velocity instruments. We took advantage of the time of mid-transit and secondary eclipses (when available) to constrain the possible presence of additional objects co-orbiting the star along with the planet. This, together with a good phase coverage, breaks the degeneracy between a trojan planet signature and signals coming from additional planets or underestimated eccentricity. We identify nine systems for which the archival data provide 1-sigma evidence for a mass imbalance between L4 and L5. Two of these systems provide 2-sigma detection, but no significant detection is found among our sample. We also report upper limits to the masses at L4/L5 in all studied systems and discuss the results in the context of previous findings.

On the Rotation of Co-orbital Bodies in Eccentric Orbits

On the rotation of co-orbital bodies in eccentric orbits

Authors:

Leleu et al

Abstract:

We investigate the resonant rotation of co-orbital bodies in eccentric and planar orbits. We develop a simple analytical model to study the impact of the eccentricity and orbital perturbations on the spin dynamics. We show there are three different families of spin-orbit resonances, one depending on the eccentricity, one depending on the orbital libration frequency, and another depending on the motion of the pericenter. The final rotation can be captured in any of these resonances. In some regions of the phase space the resonant islands may overlap, giving rise to chaotic rotation.

Kepler-91b *MAY* Have a Trojan Companion

Characterization of Kepler-91b and the Investigation of a Potential Trojan Companion Using EXONEST

Authors:

Placek et al

Abstract:

Presented here is an independent re-analysis of the Kepler light curve of Kepler-91 (KIC 8219268). Using the EXONEST software package, which provides both Bayesian parameter estimation and Bayesian model testing, we were able to re-confirm the planetary nature of Kepler-91b. In addition to the primary and secondary eclipses of Kepler-91b, a third dimming event appears to occur approximately 60o away (in phase) from the secondary eclipse, leading to the hypothesis that a Trojan planet may be located at the L4 or L5 Lagrange points. Here, we present a comprehensive investigation of four possibilities to explain the observed dimming event using all available photometric data from the Kepler Space Telescope, recently obtained radial velocity measurements, and N-body simulations. We find that the photometric model describing Kepler-91b and a Trojan planet is highly favored over the model involving Kepler-91b alone. However, it predicts an unphysically high temperature for the Trojan companion, leading to the conclusion that the extra dimming event is likely a false-postive.

Finding Co-Orbital Exoplanets Through Radial Velocity & Astrometic Detection Methods

Detectability of quasi-circular co-orbital planets. Application to the radial velocity technique

Authors:

Leleu et al

Abstract:

Several celestial bodies in co-orbital configurations exist in the solar system. However, co-orbital exoplanets have not yet been discovered. This lack may result from a degeneracy between the signal induced by co-orbital planets and other orbital configurations. Here we determine a criterion for the detectability of quasi-circular co-orbital planets and develop a demodulation method to bring out their signature from the observational data. We show that the precision required to identify a pair of co-orbital planets depends only on the libration amplitude and on the planet's mass ratio. We apply our method to synthetic radial velocity data, and show that for tadpole orbits we are able to determine the inclination of the system to the line of sight. Our method is also valid for planets detected through the transit and astrometry techniques.

How to Detect Trojan Configured Exoplanets in Binary Star Systems

Eclipse timing variations to detect possible Trojan planets in binary systems

Authors:

Schwarz et al

Abstract:

This paper is devoted to study the circumstances favourable to detect Trojan planets in close binary star systems by the help of eclipse timing variations (ETVs). To determine the probability of the detection of such variations with ground-based telescopes and space telescopes (like former missions CoRoT and Kepler and future space missions like PLATO, TESS and CHEOPS), we investigated the dynamics of binary star systems with a planet in tadpole motion. We did numerical simulations by using the full three-body problem as a dynamical model. The stability and the ETVs are investigated by computing stability/ETV maps for different masses of the secondary star and the Trojan planet. In addition, we changed the eccentricity of the possible Trojan planet. By the help of the libration amplitude σ, we could show whether or not all stable objects are moving in tadpole orbits. We can conclude that many amplitudes of ETVs are large enough to detect Earth-like Trojan planets in binary star systems. As an application, we prepared a list of possible candidates.

A Study of the Motion of two Exoplanets in Co-orbital Resonance

Rigorous treatment of the averaging process for co-orbital motions in the planetary problem

Authors:

Robutel et al

Abstract:

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamiltonian, we estimate the size of the transformation that maps this Hamiltonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale.

Is CoRoT candidate SRc01 E2 1066 the First Detected Binary Exoplanet???

Extrasolar Binary Planets II: Detectability by Transit Observations

Authors:

Lewis et al

Abstract:

We discuss the detectability of gravitationally bounded pairs of gas-giant planets (which we call "binary planets") in extrasolar planetary systems that are formed through orbital instability followed by planet-planet dynamical tides during their close encounters, based on the results of N-body simulations by Ochiai, Nagasawa and Ida (Paper I). Paper I showed that the formation probability of a binary is as much as ∼10% for three giant planet systems that undergo orbital instability, and after post-capture long-term tidal evolution, the typical binary separation is 3--5 times the sum of physical radii of the planets. The binary planets are stable during main sequence lifetime of solar-type stars, if the stellarcentric semimajor axis of the binary is larger than 0.3 AU. We show that detecting modulations of transit light curves is the most promising observational method to detect binary planets. Since the likely binary separations are comparable to the stellar diameter, the shape of the transit light curve is different from transit to transit, depending on the phase of the binary's orbit. The transit durations and depth for binary planet transits are generally longer and deeper than those for the single planet case. We point out that binary planets could exist among the known inflated gas giant planets or objects classified as false positive detections at orbital radii greater than 0.3 AU, propose a binary planet explanation for the CoRoT candidate SRc01 E2 1066, and show that binary planets are likely to be present in, and could be detected using Kepler-quality data.

Co-orbital Companions Split Spin-orbit Resonances

Spin-orbit coupling and chaotic rotation for eccentric coorbital bodies

Authors:

Leleu et al

Abstract:

The presence of a co-orbital companion induces the splitting of the well known Keplerian spin-orbit resonances. It leads to chaotic rotation when those resonances overlap.

One to One Resonant Orbits in 3d Exoplanetary Systems

1/1 resonant periodic orbits in three dimensional planetary systems

Authors:

Antoniadou et al

Abstract:

We study the dynamics of a two-planet system, which evolves being in a 1/1 mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the planar to the spatial case. We find that such bifurcations exist only for planetary mass ratios ρ=m2m1 less than 0.0205. For ρ in the interval 0 less than ρ less than 0.0205, we compute the generated families of spatial periodic orbits and their linear stability. These spatial families form bridges, which start and end at the same planar family. Along them the mutual planetary inclination varies. We construct maps of dynamical stability and show the existence of regions of regular orbits in phase space.

Examining the Formation of Exoplanets in Trojan Points

Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems

Authors:

Páez et al

Abstract:

We investigate the dynamics of small trojan exoplanets in domains of secondary resonances within the tadpole domain of motion. We consider the limit of a massless trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of more planets (the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the trojan's dynamics. In the theoretical part of this paper, we develop a Hamiltonian formalism in action-angle variables, which allows to treat in a unified way resonant dynamics and secular effects on the trojan body in both the ERTBP or the RMPP. Our formalism leads to a decomposition of the Hamiltonian in two parts, H=Hb+Hsec. Hb, called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom. Hsec contains only terms depending on slow (secular) angles. Hb is formally identical in the ERTBP and the RMPP, apart from a re-definition of angular variables. An important physical consequence is that the slow chaotic diffusion proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism better approximated by the paradigm of `modulational diffusion'. In the numerical part, we focus on the ERTBP for making a numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we provide a survey of the resonant web for characteristic mass parameters of the primary, in which the secondary resonances from 1:5 to 1:12 (ratio of the short over the synodic period) and their resonant multiplets appear. We give numerical examples of diffusion of weakly chaotic orbits in the resonant web. We make a statistics of the escaping times in the resonant domain, and find power-law tails of the distribution of escaping times for slowly diffusing chaotic orbits.

Stable Co-orbital Islands

Spin-orbit resonances and rotation of coorbital bodies in quasi-circular orbits

Authors:

Robutel et al

Abstract:

The rotation of asymmetric bodies in eccentric Keplerian orbits can be chaotic when there is some overlap of spin-orbit resonances. Here we show that the rotation of two coorbital bodies (two planets orbiting a star or two satellites of a planet) can also be chaotic even for quasi-circular orbits around the central body. When dissipation is present, the rotation period of a body on a nearly circular orbit is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We further show that the rotation becomes chaotic when the natural rotational libration frequency, due to the axial asymmetry, is of the same order of magnitude as the orbital libration frequency.

Exoplanets With Large Companions Will Remain Habitable Longer

Having a companion in old age is good for people — and, it turns out, might extend the chance for life on certain Earth-sized planets in the cosmos as well.

Planets cool as they age. Over time their molten cores solidify and inner heat-generating activity dwindles, becoming less able to keep the world habitable by regulating carbon dioxide to prevent runaway heating or cooling.

But astronomers at the University of Washington and the University of Arizona have found that for certain planets about the size of our own, the gravitational pull of an outer companion planet could generate enough heat — through a process called tidal heating — to effectively prevent that internal cooling, and extend the inner world's chance at hosting life.

UW astronomer Rory Barnes is second author of a paper published in the July issue of the Monthly Notices of the Royal Astronomical Society. The lead authors are graduate student Christa Van Laerhoven and planetary scientist Richard Greenberg at the University of Arizona.

Tidal heating results from the gravitational push and pull of the outer companion planet on its closer-in neighbor, Barnes said. The effect happens locally, so to speak, on Jupiter's moons Io and Europa. The researchers showed that this phenomenon can take place on exoplanets — those outside the solar system — as well.

Using computer models, the researchers found the effect can occur on older Earth-sized planets in noncircular orbits in the habitable zone of low-mass stars, or those less than one-quarter the mass of the Sun. The habitable zone is that swath of space around a star just right to allow an orbiting rocky planet to sustain liquid water on its surface, thus giving life a chance.

"When the planet is closer to the star, the gravitational field is stronger and the planet is deformed into an American football shape. When farther from the star, the field is weaker and the planet relaxes into a more spherical shape," Barnes said. "This constant flexing causes layers inside the planet to rub against each other, producing frictional heating."

The outer planet is necessary, Barnes added, to keep the potentially habitable planet's orbit noncircular. When a planet's orbit is circular, the gravitational pull from its host star is constant, so its shape never changes, and there is no tidal heating.

And so, the researchers conclude, any discoveries of Earth-sized planets in the habitable zone of old, small stars should be followed by searches for outer companion planets that might improve the inner world's chance at hosting life.

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How Binary Exoplanets can Form Through Tidal Capture

Extrasolar Binary Planets I: Formation by tidal capture during planet-planet scattering

Authors:

Ochiai et al

Abstract:

We have investigated i) the formation of gravitationally bounded pairs of gas-giant planets (which we call "binary planets") from capturing each other through planet-planet dynamical tide during their close encounters and ii) the following long-term orbital evolution due to planet-planet and planet-star {\it quasi-static} tides. For the initial evolution in phase i), we carried out N-body simulations of the systems consisting of three jupiter-mass planets taking into account the dynamical tide. The formation rate of the binary planets is as much as 10% of the systems that undergo orbital crossing and this fraction is almost independent of the initial stellarcentric semi-major axes of the planets, while ejection and merging rates sensitively depend on the semi-major axes. As a result of circularization by the planet-planet dynamical tide, typical binary separations are a few times the sum of the physical radii of the planets. After the orbital circularization, the evolution of the binary system is governed by long-term quasi-static tide. We analytically calculated the quasi-static tidal evolution in later phase ii). The binary planets first enter the spin-orbit synchronous state by the planet-planet tide. The planet-star tide removes angular momentum of the binary motion, eventually resulting in a collision between the planets. However, we found that the binary planets survive the tidal decay for main-sequence life time of solar-type stars (~10Gyrs), if the binary planets are beyond ~0.3 AU from the central stars. These results suggest that the binary planets can be detected by transit observations at less than 0.3AU.

Orbital Configurations for Three and Four Body Co-orbital Exoplanet Systems

Periodic orbits for 3 and 4 co-orbital bodies

Authors:

Verrier et al

Abstract:

We investigate the natural families of periodic orbits associated with the equilibrium configurations of the the planar restricted 1+n body problem for the case 2≤n≤4 equal mass satellites. Such periodic orbits can be used to model both trojan exoplanetary systems and parking orbits for captured asteroids within the solar system. For n=2 there are two families of periodic orbits associated with the equilibria of the system: the well known horseshoe and tadpole orbits. For n=3 there are three families that emanate from the equilibrium configurations of the satellites, while for n=4 there are six such families as well as numerous additional connecting families. The families of periodic orbits are all of the horseshoe or tadpole type, and several have regions of neutral linear stability.

Co-Orbital Planets Between Supearth to Saturn Mass Predicted to be Absent

Disruption of co-orbital (1:1) planetary resonances during gas-driven orbital migration

Authors:

Pierens et al

Abstract:

Planets close to their stars are thought to form farther out and migrate inward due to angular momentum exchange with gaseous protoplanetary disks. This process can produce systems of planets in co-orbital (Trojan or 1:1) resonance, in which two planets share the same orbit, usually separated by 60 degrees. Co-orbital systems are detectable among the planetary systems found by the Kepler mission either directly or by transit timing variations. However, no co-orbital systems have been found within the thousands of Kepler planets and candidates. Here we study the orbital evolution of co-orbital planets embedded in a protoplanetary disk using a grid-based hydrodynamics code. We show that pairs of similar-mass planets in co-orbital resonance are disrupted during large-scale orbital migration. Destabilization occurs when one or both planets is near the critical mass needed to open a gap in the gaseous disk. A confined gap is opened that spans the 60 degree azimuthal separation between planets. This alters the torques imparted by the disk on each planet -- pushing the leading planet outward and the trailing planet inward -- and disrupts the resonance. The mechanism applies to systems in which the two planets' masses differ by a factor of two or less. In a simple flared disk model the critical mass for gap opening varies from a few Earth masses at the inner edge of the disk to 1 Saturn-mass at 5 AU. A pair of co-orbital planets with masses in this range that migrates will enter a region where the planets are at the gap-opening limit. At that point the resonance is disrupted. We therefore predict an absence of planets on co-orbital configurations with masses in the super-Earth to Saturn mass range with similar masses.

Ways Exoplanet Systems can "Hide" Co-Orbital Planets

Radial Velocities of Stars with Multiple Co-orbital Planets

Author:

Dobrovolskis

Abstract:

To date, well over a thousand planets have been discovered orbiting other stars, hundreds of them in multi-planet systems. Most of these exoplanets have been detected by either the transit method or the radial velocity method, rather than by other methods such as astrometry or direct imaging. Both the radial velocity and astrometric methods rely upon the reflex motion of the parent star induced by the gravitational attraction of its planets. However, this reflex motion is subject to misinterpretation when a star has two or more planets with the same orbital period. Such co-orbital planets may effectively "hide" from detection by current algorithms.

In principle, any number of planets can share the same orbit; the case where they all have the same mass has been studied most. Salo and Yoder (A & A 205, 309--327, 1988) have shown that more than 8 planets of equal mass sharing a circular orbit must be equally spaced for dynamical stability, while fewer than 7 equal-mass planets are stable only in a configuration where all of the planets remain on the same side of their parent star. For 7 or 8 equal-mass planets, both configurations are stable.

By symmetry, it is clear that the equally-spaced systems produce no reflex motion or radial velocity signal at all in their parent stars. This could lead to their being overlooked entirely, unless they happen to be detected by the transit method. It is equally clear that the lopsided systems produce a greater radial velocity signal than a single such planet would, but a smaller signal than if all of the planets were combined into one. This could seriously mislead estimates of exoplanet masses and densities. Transit data and ellipsoidal (tidal) brightness variations in such systems also are subject to misinterpretation. This behavior is also representative of more natural systems, with co-orbital planets of different masses.