Trojan Planets Research Articles

Do tides destabilize Trojan exoplanets?

One outstanding problem in extrasolar planet studies is why no co-orbital exoplanets have been found, despite numerous searches among the many known planetary systems, many of them in other mean-motion resonances. Here we examine the hypothesis that dissipation of energy by tides in Trojan planets is preventing their survival.The Appendix of this paper generalizes the conventional theory of tides to include tidal forces independent of dissipation, as well as the effects of one body on tides raised by another. The main text applies this theory to a model system consisting of a primary of stellar mass, a secondary of sub-stellar mass in a circular orbit about the primary, and a much lighter Trojan planet librating with small amplitude about an equilateral point of the system.Next, we linearize the equations of motion about the Trojan points, including the tidal forces, and solve for the motion of the Trojan. The results indicate that tides damp out the Trojan’s motion perpendicular to the orbital plane of the primary and secondary, as well as its epicycles due to its eccentricity; but they pump up the amplitude of its tadpole librations exponentially. We then verify our analytic solutions by integrating the non-linearized equations of motion numerically for several sample cases. In each case, we find that the librations grow until the Trojan escapes its libration, which leads to a close encounter with either the primary or the secondary.

On the Detection of Habitable Trojan Planets in the Kepler Circumbinary Systems

Abstract We present the results of a study of the prospect of detecting habitable Trojan planets in the Kepler Habitable Zone circumbinary planetary systems (Kepler-16, -47, -453, -1647, and -1661). We integrated the orbits of 10,000 separate N-body systems (N = 4, 6), each with a one Earth-mass body in a randomly selected orbit near the L 4 and L 5 Lagrangian points of the host HZ circumbinary planet. We find that stable Trojan planets are restricted to a narrow range of semimajor axes in all five systems and limited to small eccentricities in Kepler-16, -47, and -1661. To assess the prospect of the detection of these habitable Trojan planets, we calculated the amplitudes of the variations they cause in the transit timing of their host bodies. Results show that the mean amplitudes of the transit timing variations (TTVs) correlate with the mass of the transiting planet and range from 70 minutes for Kepler-16b to 390 minutes for Kepler-47c. Our analysis indicates that the TTVs of the circumbinary planets caused by these Trojan bodies fall within the detectable range of timing precision obtained from the Kepler telescope’s long-cadence data. The latter points to Kepler data as a viable source to search for habitable Trojan planets.

Dust trapping around Lagrangian points in protoplanetary disks

Aims.Trojans are defined as objects that share the orbit of a planet at the stable Lagrangian pointsL4andL5. In the Solar System, these bodies show a broad size distribution ranging from micrometer (μm) to centimeter (cm) particles (Trojan dust) and up to kilometer (km) rocks (Trojan asteroids). It has also been theorized that earth-like Trojans may be formed in extra-solar systems. The Trojan formation mechanism is still under debate, especially theories involving the effects of dissipative forces from a viscous gaseous environment.Methods.We perform hydro-simulations to follow the evolution of a protoplanetary disk with an embedded 1–10 Jupiter-mass planet. On top of the gaseous disk, we set a distribution ofμm–cm dust particles interacting with the gas. This allows us to follow dust dynamics as solids get trapped around the Lagrangian points of the planet.Results.We show that large vortices generated at the Lagrangian points are responsible for dust accumulation, where the leading Lagrangian pointL4traps a larger amount of submillimeter (submm) particles than the trailingL5, which traps mostly mm–cm particles. However, the total bulk mass, with typical values of ~Mmoon, is more significant inL5than inL4, in contrast to what is observed in the current Solar System a few gigayears later. Furthermore, the migration of the planet does not seem to affect the reported asymmetry betweenL4andL5.Conclusions.The main initial mass reservoir for Trojan dust lies in the same co-orbital path of the planet, while dust migrating from the outer region (due to drag) contributes very little to its final mass, imposing strong mass constraints for the in situ formation scenario of Trojan planets.

The TROY project: Searching for co-orbital bodies to known planets

Context. 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. Aims. In this context, the main goal of the TROY project is to detect exotrojans for the first time and to measure their occurrence rate (η-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. Methods. We used archival radial velocity data of 46 close-in (P < 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. Results. We identify nine systems for which the archival data provide >1σ evidence for a mass imbalance between L4 and L5. Two of these systems provide >2σ 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.

CHARACTERIZATION OF KEPLER-91B AND THE INVESTIGATION OF A POTENTIAL TROJAN COMPANION USING EXONEST

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 $60^o$ 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.

Eclipse timing variations to detect possible Trojan planets in binary systems

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 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 $\sigma$ 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.

On the Possibility of Habitable Trojan Planets in Binary Star Systems.

Approximately 60% of all stars in the solar neighbourhood (up to 80% in our Milky Way) are members of binary or multiple star systems. This fact led to the speculations that many more planets may exist in binary systems than are currently known. To estimate the habitability of exoplanetary systems, we have to define the so-called habitable zone (HZ). The HZ is defined as a region around a star where a planet would receive enough radiation to maintain liquid water on its surface and to be able to build a stable atmosphere. We search for new dynamical configurations-where planets may stay in stable orbits-to increase the probability to find a planet like the Earth.

Detection of Earth-mass and super-Earth Trojan planets using transit timing variation method

We have carried out an extensive study of the possibility of the detection of Earth-mass and super-Earth Trojan planets using transit timing variation method with the Kepler space telescope. We have considered a system consisting of a transiting Jovian-type planet in a short period orbit, and determined the induced variations in its transit timing due to an Earth-mass/super-Earth Trojan planet. We mapped a large section of the phase space around the 1:1 mean-motion resonance and identified regions corresponding to several other mean-motion resonances where the orbit of the planet would be stable. We calculated TTVs for different values of the mass and orbital elements of the transiting and perturbing bodies as well as the mass of central star, and identified orbital configurations of these objects (ranges of their orbital elements and masses) for which the resulted TTVs would be within the range of the variations of the transit timing of Kepler's planetary candidates. Results of our study indicate that in general, the amplitudes of the TTVs fall within the detectable range of timing precision obtained from the Kepler's long-cadence data, and depending on the parameters of the system, their magnitudes may become as large as a few hours. The probability of detection is higher for super-Earth Trojans with slightly eccentric orbits around short-period Jovian-type planets with masses slightly smaller than Jupiter. We present the details of our study and discuss the implications of its results.

Effects of Trojan exoplanets on the reflex motions of their parent stars

This paper collects and corrects certain results on the radial velocity variations of stars in Trojan systems, where two exoplanets separated by 60° of longitude share the same orbit. Radial velocity measurements can seriously misjudge the masses and densities of planets in such systems.I also derive a simple new method for combining tidal perturbations from two or more sources, such as the tides raised on Earth by the Sun and Moon. This method then shows that the observable effects of tides raised on stars by Trojan planets are also misleading. However, the combination of both tidal and radial velocity measurements can determine the mass of each planet. I conclude with a discussion of possible scenarios for the formation of co-orbital exoplanets.

Trojans in Exosystems with Two Massive Planets

AbstractWe take as dynamical model for extrasolar planetary systems a central star like our Sun and two giant planets m1 and m2 like Jupiter and Saturn. We change the mass ratio μ=m2/m1 of the two large planets for a wide range of 1/16 < μ < 16. We also change the ratio between the initial semi-major axes (ν=a2/a1) in the range of 1.2 < ν < 3 to model the different architecture of extrasolar planetary systems hosting two giant planets. The results for possible Trojans (Trojan planets) in the equilateral equilibrium points of the inner planet m1 and the outer planet m2 were derived with the aid of numerical integration. It turned out that in many configurations – depending on the mass ratios μ and the semi-major axes ratio ν – giant planets may host Trojans.

On the stability of possible Trojan planets in the habitable zone: an application to the systems HD 147513 and HD 210277

In this paper, we study the dynamical stability of fictitious terrestrial planets in the 1:1 mean-motion resonance with a gas giant moving in the habitable zone (HZ). We investigate the stability of Trojan planets both in a general study and for two specific known extrasolar planetary systems (HD 147513 and HD 210277). In the general study, we determine the stability of the Lagrangian point L4. The numerical simulations have been carried out using the spatial elliptic-restricted three-body problem, where we placed the test particles (TPs) exactly at L4, which is located 60° ahead of the gas giant at the same distance to the star. In the stability study, we have concentrated on the dependences between the eccentricity and the inclination of the Trojan planet. Going a step further, we have investigated two specific systems where the known gas giant moves in the HZ. The two specific extrasolar systems are investigated by using the general study to define the region in eccentricity and inclination where, in principle, stable motion is possible. To find out whether the existence of a Trojan planet is possible, in addition we have calculated the stable area in the semimajor axis (aTP) and the argument of perihelion (ωTP) for an actual planetary mass (2 M⊕). Thus, we can conclude that the region around the Lagrangian points L4 and L5 allows stable motion in the system HD 147513, because the gas giant of this system has an eccentricity of 0.26, which lies well within the stable region. The known planet in the system HD 210277 has a much higher eccentricity and thus it cannot harbour any terrestrial planets in the Lagrangian points.

THE THREE TROJAN PROBLEM

This paper is devoted to the stability study of three Trojan planets around a central star of 1 M Sun in a distance of 1 AU. In an earlier study, the stability of this coorbital configuration was computed if the mutual angular separation is 47.4° [Salo & Yoder, 1988]. We investigated by using numerical integrations the largeness of the stable region around these stable equilibrium points, depending on the mass parameter and the distances of the three planets. It turned out that even initial deviations in semimajor axis in the order of 5% can lead to very stable orbits. Also differences in the initial angular distance from the equilibrium points up to 70° do not mean that the orbits will be unstable. But the extensions in semimajor axis and angles depend sensitively on the masses involved, which were given the same values for our investigations in a first step. Test computations for different masses of the three planets involved show that they may be quite large without destroying the stability.

Case studies of habitable Trojan planets in the system of HD 23079

AbstractWe investigate the possibility of habitable Trojan planets in the HD 23079 star–planet system. This system consists of a solar-type star and a Jupiter-type planet, which orbits the star near the outer edge of the stellar habitable zone in an orbit of low eccentricity. We find that in agreement with previous studies Earth-mass habitable Trojan planets are possible in this system, although the success of staying within the zone of habitability is significantly affected by the orbital parameters of the giant planet and by the initial condition of the theoretical Earth-mass planet. In one of our simulations, the Earth-mass planet is captured by the giant planet and thus becomes a habitable moon.

Orbital stability of systems of closely-spaced planets, II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 $${{\rm M}_{\oplus}}$$ planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 108, 5 × 105, and 2 × 105 years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t c /t 0) = b β + c, where t c is the crossing time, t 0 = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.

Dynamics of possible Trojan planets in binary systems

This paper is devoted to the dynamical stability of possible Trojan planets in binaries and in binary systems where one of the substellar companions is not larger than a brown dwarf. Using numerical integrations, we investigated how the size of the stable region around the Lagrangian point L 4 depends on the mass parameter and the eccentricity of the secondary star. An additional goal of this work was to create a catalogue of all possible candidates, which could be useful for future observations to detect such objects.

Stability of Trojan planets in multi-planetary systems

Today there are more than 340 extra-solar planets in about 270 extra-solar systems confirmed. Besides the observed planets there exists also the possibility of a Trojan planet moving in the same orbit as the Jupiter-like planet. In our investigation we take also into account the habitability of a Trojan planet and whether such a terrestrial planet stays in the habitable zone. Its stability was investigated for multi-planetary systems, where one of the detected giant planets moves partly or completely in the habitable zone. By using numerical computations, we studied the orbital behaviour up to 107 years and determined the size of the stable regions around the Lagrangian equilibrium points for different dynamical models for fictitious Trojans. We also examined the interaction of the Trojan planets with a second or third giant planet, by varying its semimajor axis and eccentricity. We have found two systems (HD 155358 and HD 69830) that can host habitable Trojan planets. Another aim of this work was to determine the size of the stable region around the Lagrangian equilibrium points in the restricted three body problem for small mass ratios μ of the primaries μ ≤ 0.001 (e.g. Neptune mass of the secondary and smaller masses). We established a simple relation for the size depending on μ and the eccentricity.

On the growth and stability of Trojan planets

Aims. We investigate the stability of those low-mass Trojan planets that form in a protoplanetary disc and subsequently accrete gas to become gas giants.Methods. We calculate their evolution before, during, and after gas disc dispersal. A two-dimensional hydrodynamics code combined with an N -body solver is used to evolve the system of disc and planets. Gas disc dispersal is simulated in a simple manner by assuming global exponential decay of the disc mass, leading to the stalling of migration after semi-major axes have approximately halved from their initial values. We consider Trojan pairs with different initial masses and gas accretion rates and gas disc models with different masses and viscosities. An N -body code, adapted to model disc forces, is used to examine large-scale migration and the formation of very short period Trojan planets.Results. For each combination of planetary pair and disc model that we consider in our hydrodynamic simulations, each Trojan system remains stable before, during, and after disc dispersal. The long-term stability of these systems in the absence of gas is tested using N -body simulations, and all systems remain stable for those evolution times equal to 109 years. Eccentricities remain low () in all cases. Increases in the amplitude of libration about the L 4 /L 5 Lagrange points accompany the inward migration, and during very large-scale migration Trojan systems may be disrupted prior to the onset of disc dispersal.Conclusions. The stability of Trojan pairs during rapid type I migration, during the transition to type II migration with the accompanying gap formation in the gas disc, and during gas loss when the disc disperses, indicates that isolated Trojan planet systems are very stable. If a common mechanism exists for their formation, we suggest they may be readily observed in nature.

Stability of inclined orbits of terrestrial planets in habitable zones

In long-term stability studies of terrestrial planets moving in the habitable zone (HZ) of a sun-like star, we distinguish four different configurations: (i) planets moving in binary star systems, (ii) the inner type (where the gas giant moves outside the HZ), (iii) the outer type (where the gas giant is closer to the star, than the HZ) and (iv) the Trojan type (where the gas giant moves in the HZ). Since earlier calculations indicated, that the stability of the motion in the HZ also depends on the inclination of the terrestrial planet orbits, we present a detailed numerical investigation to show correlations between the eccentricity, the mass and the distance of the giant planet for various inclinations of the terrestrial planets. The orbital stability of the HZ was examined for all four configurations stated above. While we could find hardly any stable orbits for the first three types for inclinations higher than 40 ∘ , the Trojan planets can be stable up to an inclination of 60 ∘ . Additionally, we could also find some stabilizing effects of the inclination for the first three types. As dynamical model we used the elliptic restricted three-body problem, which consists of two massive and one mass-less body. This allows an application to all detected and future extrasolar single planet systems.

Survey of the stability region of hypothetical habitable Trojan planets

Aims. In this work we study the dynamical possibility in extrasolar planetary systems that a terrestrial planet can exist in 1:1 mean motion resonance with a Jovian-like planet. We compiled a catalogue of hypothetical habitable Trojan planets, to be able to make a stability forecast for further extrasolar planetary systems discovered in the future. When speaking of habitability we also took the influence of the spectral type of the central star into account. Methods. We integrated some 10 6 orbits of fictitious Trojans around the Lagrangian points for up to 10 7 orbital periods of the primary bodies and checked the stability of the orbital elements and their chaoticity with the aid of the Lyapunov characteristic indicator and maximum eccentricity. The computations were carried out using the dynamical model of the elliptic, restricted three-body problem that consists of a central star, a gas giant moving in the habitable zone, and a hypothetical (massless) terrestrial planet. Results. Our investigations have shown that 7 exoplanetary systems can harbour habitable Trojan planets with stable orbits (HD 93083, HD 17051, HD 28185, HD 27442, HD 188015, HD 99109, and HD 221287, which is a recently discovered system). The comparison of the investigated systems with our catalogue showed matching results, so that we can use the catalogue in practice.

On the dynamics of Trojan planets in extrasolar planetary systems

AbstractIn this article we examine the motion of fictitious Trojan planets close to the equilateral Lagrangean equilibrium points in extrasolar planetary systems. Whether there exist stable motion in this area or not depends on the massratio of the primariy bodies in the restricted three body problem, namely the host star and the gasgiant. Taking into account also the eccentricity of the primaries we show via results of extensive numerical integrations that Trojan planets may survive only for e < 0.25. We also show first results of a mapping in the 1:1 resonance with a gas giant on an eccentric orbit which is applied to the extrasolar planetary systems HD 17051. We furthermore study the influence of an additional outer planet which perturbs the motion of the gasgiant as well as the Trojan cloud around its L4 Lagrangean point.