Analytical model of multi-planetary resonant chains and constraints on migration scenarios

Analytical model of multi-planetary resonant chains and constraints on migration scenarios


Author:

Delisle

Abstract:

Resonant chains are groups of planets for which each pair is in resonance, with an orbital period ratio locked at a rational value (2/1, 3/2, etc.). Such chains naturally form as a result of convergent migration of the planets in the proto-planetary disk. In this article, I present an analytical model of resonant chains of any number of planets. Using this model, I show that a system captured in a resonant chain can librate around several possible equilibrium configurations. The probability of capture around each equilibrium depends on how the chain formed, and especially on the order in which the planets have been captured in the chain. Therefore, for an observed resonant chain, knowing around which equilibrium the chain is librating allows for constraints to be put on the formation and migration scenario of the system. I apply this reasoning to the four planets orbiting Kepler-223 in a 3:4:6:8 resonant chain. I show that the system is observed around one of the six equilibria predicted by the analytical model. Using N-body integrations, I show that the most favorable scenario to reproduce the observed configuration is to first capture the two intermediate planets, then the outermost, and finally the innermost.

Mean Motion Resonances at High Eccentricities: The 2:1 and the 3:2 Interior Resonances

Mean Motion Resonances at High Eccentricities: The 2:1 and the 3:2 Interior Resonances

Authors:

Wang et al

Abstract:
Mean motion resonances (MMRs) play an important role in the formation and evolution of planetary systems and have significantly influenced the orbital properties and distribution of planets and minor planets in the solar system and in exoplanetary systems. Most previous theoretical analyses have focused on the low- to moderate-eccentricity regime, but with new discoveries of high-eccentricity resonant minor planets and even exoplanets, there is increasing motivation to examine MMRs in the high-eccentricity regime. Here we report on a study of the high-eccentricity regime of MMRs in the circular planar restricted three-body problem. Numerical analyses of the 2:1 and the 3:2 interior resonances are carried out for a wide range of planet-to-star mass ratio μ, and for a wide range of eccentricity of the test particle. The surface-of-section technique is used to study the phase space structure near resonances. We find that new stable libration zones appear at higher eccentricity at libration centers that are shifted from those at low eccentricities. We provide physically intuitive explanations for these transitions in phase space, and we present novel results on the mass and eccentricity dependence of the resonance widths. Our results show that MMRs have sizable libration zones at high eccentricities, comparable to those at lower eccentricities.

Exoplanet Predictions Based on Harmonic Orbit Resonances

Exoplanet Predictions Based on Harmonic Orbit Resonances

Authors:

Aschenwanden et al

Abstract:
The current exoplanet database includes 5454 confirmed planets and candidate planets observed with the KEPLER mission. We find 932 planet pairs from which we extract distance and orbital period ratios. While earlier studies used the Titius-Bode law or a generalized version with logarithmic spacing, which both lack a physical model, we employ here the theory of harmonic orbit resonances, which contains quantized ratios instead, to explain the observed planet distance ratios and to predict undetected exoplanets. We find that the most prevailing harmonic ratios are (2:1), (3:2), and (5:3), in 73\% of the cases, while alternative harmonic ratios of (5:4), (4:3), (5:2), (3:1) occur in 27\% of the other cases. Our orbital predictions includes 171 exoplanets, 2 Jupiter moons, one Saturn moon, 3 Uranus moons, and 4 Neptune moons. The accuracy of the predicted planet distances amounts to a few percent, which fits the data significantly better than the Titius-Bode law or a logarithmic spacing. This information may be useful for targeted exoplanet searches with Kepler data and to estimate the number of live-carrying planets in habitable zones.

Three planets around HD 27894. A close-in pair with a 2:1 period ratio and an eccentric Jovian planet at 5.4 AU

Three planets around HD 27894. A close-in pair with a 2:1 period ratio and an eccentric Jovian planet at 5.4 AU 

Authors:

Trifonov et al

Abstract:

Aims.

Our new program with HARPS aims to detect mean motion resonant planetary systems around stars which were previously reported to have a single bona fide planet, often based only on sparse radial velocity data.

Methods.

Archival and new HARPS radial velocities for the K2V star HD 27894 were combined and fitted with a three-planet self-consistent dynamical model. The best-fit orbit was tested for long-term stability.

Results.

We find clear evidence that HD 27894 is hosting at least three massive planets. In addition to the already known Jovian planet with a period Pb ≈ 18 days we discover a Saturn-mass planet with Pc ≈ 36 days, likely in a 2:1 mean motion resonance with the first planet, and a cold massive planet (≈ 5.3 MJup) with a period Pd ≈ 5170 days on a moderately eccentric orbit (ed = 0.39).

Conclusions.

HD 27894 is hosting a massive, eccentric giant planet orbiting around a tightly packed inner pair of massive planets likely involved in an asymmetric 2:1 mean motion resonance. HD 27894 may be an important milestone for probing planetary formation and evolution scenarios.

Resonant structure, formation and stability of the planetary system HD 155358

Resonant structure, formation and stability of the planetary system HD155358

Authors:

Silburt et al

Abstract:

Two Jovian-sized planets are orbiting the star HD155358 near exact mean motion resonance (MMR) commensurability. In this work we re-analyze the radial velocity (RV) data previously collected by Robertson et al. (2012). Using a Bayesian framework we construct two models - one that includes and one that excludes gravitational planet-planet interactions (PPI). We find that the orbital parameters from our PPI and noPPI models differ by up to 2σ, with our noPPI model being statistically consistent with previous results. In addition, our new PPI model strongly favours the planets being in MMR while our noPPI model strongly disfavours MMR. We conduct a stability analysis by drawing samples from our PPI model's posterior distribution and simulating them for 109 years, finding that our best-fit values land firmly in a stable region of parameter space.

We explore a series of formation models that migrate the planets into their observed MMR. We then use these models to directly fit to the observed RV data, where each model is uniquely parameterized by only three constants describing its migration history. Using a Bayesian framework we find that a number of migration models fit the RV data surprisingly well, with some migration parameters being ruled out.

Our analysis shows that planet-planet interactions are important to take into account when modelling observations of multi-planetary systems. The additional information that one can gain from interacting models can help constrain planet migration parameters.

The Influence of Orbital Resonances on the Water Transport to Objects in the Circumprimary Habitable Zone of Binary Star Systems

The Influence of Orbital Resonances on the Water Transport to Objects in the Circumprimary Habitable Zone of Binary Star Systems

Authors:

Bancelin et al

Abstract:
We investigate the role of secular and mean motion resonances on the water transport from a belt of icy asteroids onto planets or embryos orbiting inside the circumprimary habitable zone (HZ) of a binary star system. In addition, the host-star has an accompanying gas giant planet. For a comparison, we perform two case studies where a secular resonance (SR) is located either inside the HZ close to 1.0 au (causing eccentric motion of a planet or embryos therein) or in the asteroid belt, beyond the snow line. In the latter case, a higher flux of icy objects moving toward the HZ is expected. Collisions between asteroids and objects in the HZ are treated analytically. Our purely dynamical study shows that the SR in the HZ boosts the water transport; however, collisions can occur at very high impact speeds. In this paper, we treat for the first time, realistic collisions using a GPU 3D-SPH code to assess the water loss in the projectile. Including the water loss into the dynamical results, we get more realistic values for the water mass fraction of the asteroid during an impact. We highlight that collisions occurring at high velocities greatly reduce the water content of the projectile and thus the amount of water transported to planets or embryos orbiting inside the HZ. Moreover, we discuss other effects that could modify our results, namely the asteroid's surface rate recession due to ice sublimation and the atmospheric drag contribution on the asteroids' mass loss.

On the Spin States of Habitable Zone Exoplanets Around M Dwarfs: The Effect of a Near-Resonant Companion

On the Spin States of Habitable Zone Exoplanets Around M Dwarfs: The Effect of a Near-Resonant Companion


Authors:

Vinson et al

Abstract:
One longstanding problem for the potential habitability of planets within M dwarf systems is their likelihood to be tidally locked in a synchronously rotating spin state. This problem thus far has largely been addressed only by considering two objects: the star and the planet itself. However, many systems have been found to harbor multiple planets, with some in or very near to mean-motion resonances. The presence of a planetary companion near a mean-motion resonance can induce oscillatory variations in the mean-motion of the planet, which we demonstrate can have significant effects on the spin-state of an otherwise synchronously rotating planet. In particular, we find that a planetary companion near a mean-motion resonance can excite the spin states of planets in the habitable zone of small, cool stars, pushing otherwise synchronously rotating planets into higher amplitude librations of the spin state, or even complete circulation resulting in effective stellar days with full surface coverage on the order of years or decades. This increase in illuminated area can have potentially dramatic influences on climate, and thus on habitability. We also find that the resultant spin state can be very sensitive to initial conditions due to the chaotic nature of the spin state at early times within certain regimes. We apply our model to two hypothetical planetary systems inspired by the K00255 and TRAPPIST-1 systems, which both have Earth-sized planets in mean-motion resonances orbiting cool stars.

HD 155358's Interacting Gas Giants

The orbital configuration of the two interacting Jupiters in HD 155358 system
Authors:

Ma et al

Abstract:
Recent observation reveals two interacting Jupiters possibly trapped in a 2:1 mean motion resonance (MMR) around the star HD 155358. For the 2:1 MMR, Beaugé et al. found that as long as the orbital decay was sufficiently slow, the trapped planets should also be in apsidal co-rotation. So it is very interesting to explore whether HD 155358 did undergo such an evolution and presents an apsidal co-rotation. Based on the existing results of spectroscopic orbital determination, the global dynamics of the system shows that the two planets are in an apsidal co-rotation if the eccentricity of the outer planet (ec) takes values very close to the lower limit of its 1σ confidence interval. This makes us conjecture that the globally minimizing solution could be missed in the previous orbital determination. Using an efficient global optimization method, we do find a better solution, reducing χ2 from 1.4 to 1.2. This new solution is significantly different from the previous one, and in particular, with smaller ec. However, the increased possibility for the system to be trapped in a 2:1 MMR with apsidal co-rotation is still not high. A set of simulations of the adiabatic convergent migration process are then performed. The results consistently indicate that the 2:1 MMR forms before apsidal co-rotation. Finally, the long-term stability of the formed system and of its resonant structure are extensively explored, and the resulting statistics are given. A conclusion is that the 2:1 MMR with apsidal co-rotation is a very stable structure.

Simulations Fail to Explain SuperEarth Resonant Chains

Breaking the Chains: Hot Super-Earth systems from migration and disruption of compact resonant chains
Authors:

Izidoro et al

Abstract:
“Hot super-Earths” (or “Mini-Neptunes”) between 1 and 4 times Earth’s size with period shorter than 100 days orbit 30-50% of Sun-like type stars. Their orbital configuration – measured as the period ratio distribution of adjacent planets in multi-planet systems – is a strong constraint for formation models. Here we use N-body simulations with synthetic forces from an underlying evolving gaseous disk to model the formation and long-term dynamical evolution of super-Earth systems. While the gas disk is present, planetary embryos grow and migrate inward to form a resonant chain anchored at the inner edge of the disk. These resonant chains are far more compact than the observed super-Earth systems. Once the gas dissipates resonant chains may become dynamically unstable. They undergo a phase of giant impacts that spreads the systems out. Disk turbulence has no measurable effect on the outcome. Our simulations match observations if a small fraction of resonant chains remain stable, while most super-Earths undergo a late dynamical instability. Our statistical analysis restricts the contribution of stable systems to less than 25%. Our results also suggest that the large fraction of observed single planet systems does not necessarily imply any dichotomy in the architecture of planetary systems. Finally, we use the low abundance of resonances in Kepler data to argue that, in reality, the survival of resonant chains happens likely only in ∼5% of the cases. This leads to a mystery: in our simulations only 50-60% of resonant chains became unstable whereas at least 75% (and probably 90-95%) must be unstable to match observations.

The Origin and Orbital Resonances of the Kepler-29 System

The origin and 9:7 MMR dynamics of the Kepler-29 system

Authors:

Migaszewski et al

Abstract:

We analyse the Transit Timing Variation (TTV) measurements of a~system of two super-Earths detected as Kepler-29, in order to constrain the planets' masses and orbital parameters. A dynamical analysis of the best-fitting configurations constrains the masses to be ∼5 and ∼6 Earth masses for the inner and the outer planets, respectively. The analysis also reveals that the system is likely locked in the 9:7~mean motion resonance. However, a variety of orbital architectures regarding eccentricities and the relative orientation of orbits is permitted by the observations as well as by stability constraints. We attempt to find configurations preferred by the planet formation scenarios as an additional, physical constraint. We show that configurations with low eccentricities and anti-aligned apsidal lines of the orbits are a natural and most likely outcome of the convergent migration. However, we show that librations of the critical angles are not necessary for the Kepler-29 system to be dynamically resonant, and such configurations may be formed on the way of migration as well. We argue, on the other hand, that aligned configurations with e≳0.03 may be not consistent with the migration scenario.

Asymmetric Orbital Distribution near Mean Motion Resonance

Asymmetric Orbital Distribution near Mean Motion Resonance: Application to Planets Observed by Kepler and Radial Velocities

Author:

Xie

Abstract:

Many multiple-planet systems have been found by the Kepler transit survey and various radial velocity (RV) surveys. Kepler planets show an asymmetric feature, namely, there are small but significant deficits/excesses of planet pairs with orbital period spacing slightly narrow/wide of the exact resonance, particularly near the first order mean motion resonance (MMR), such as 2:1 and 3:2 MMR. Similarly, if not exactly the same, an asymmetric feature (pileup wide of 2:1 MMR) is also seen in RV planets, but only for massive ones. We analytically and numerically study planets' orbital evolutions near and in the MMR. We find that their orbital period ratios could be asymmetrically distributed around the MMR center regardless of dissipation. In the case of no dissipation, Kepler planets' asymmetric orbital distribution could be partly reproduced for 3:2 MMR but not for 2:1 MMR, implying that dissipation might be more important to the latter. The pileup of massive RV planets just wide of 2:1 MMR is found to be consistent with the scenario that planets formed separately then migrated toward the MMR. The location of the pileup infers a K value of 1-100 on the order of magnitude for massive planets, where K is the damping rate ratio between orbital eccentricity and semimajor axis during planet migration.

Understanding the Dynamics of TIghtly Packed Systems

The study of dynamics, the movement of objects in space under the influence of each other's gravity, gets more complicated the more objects you throw into your system. At this year’s Division for Planetary Sciences/European Planetary Science Congress meeting, the Exoplanet Dynamics session was packed full of talks on tightly-packed multi-planet systems and their instabilities.

link.

Mutual Inclinations of Multi-Planet Systems with External Companions

Hiding Planets Behind a Big Friend: Mutual Inclinations of Multi-Planet Systems with External Companions

Authors:

Lai et al

Abstract:

The {\it Kepler} mission has detected thousands of planetary systems with 1-7 transiting planets packed within 0.7~au from their host stars. There is an apparent excess of single-transit planet systems that cannot be explained by transit geometries alone, when a single planetary mutual inclination dispersion is assumed. This suggests that the observed compact planetary systems have at least two different architectures. We present a scenario where the "Kepler dichotomy" can be explained by the action of an external giant planet (or stellar) companion misaligned with the inner multi-planet system. The external companion excites mutual inclinations of the inner planets, causing such systems to appear as "Kepler singles" in transit surveys. We derive approximate analytic expressions (in various limiting regimes), calibrated with numerical calculations, for the mutual inclination excitations for various planetary systems and perturber properties (mass mp, semi-major axis ap and inclination θp). In general, the excited mutual inclination increases with mp/a3p and θp, although secular resonances may lead to large mutual inclinations even for small θp.

On the orbital evolution of a pair of giant planets in mean motion resonance

On the orbital evolution of a pair of giant planets in mean motion resonance

Authors:

André et al

Abstract:

Pairs of extrasolar giant planets in a mean motion commensurability are common with 2:1 resonance occurring most frequently. Disc-planet interaction provides a mechanism for their origin. However, the time scale on which this could operate in particular cases is unclear. We perform 2D and 3D numerical simulations of pairs of giant planets in a protoplanetary disc as they form and maintain a mean motion commensurability. We consider systems with current parameters similar to those of HD 155358, 24 Sextantis and HD 60532, and disc models of varying mass, decreasing mass corresponding to increasing age. For the lowest mass discs, systems with planets in the Jovian mass range migrate inwards maintaining a 2:1 commensurability. Systems with the inner planet currently at around 1 au from the central star could have originated at a few au and migrated inwards on a time scale comparable to protoplanetary disc lifetimes. Systems of larger mass planets such as HD 60532 attain 3:1 resonance as observed. For a given mass accretion rate, results are insensitive to the disc model for the range of viscosity prescriptions adopted, there being good agreement between 2D and 3D simulations. However, in a higher mass disc a pair of Jovian mass planets passes through 2:1 resonance before attaining a temporary phase lasting a few thousand orbits in an unstable 5:3 resonance prior to undergoing a scattering. Thus finding systems in this commensurability is unlikely.

DETECTING PLANET PAIRS IN MEAN MOTION RESONANCES VIA THE ASTROMETRY METHOD

DETECTING PLANET PAIRS IN MEAN MOTION RESONANCES VIA THE ASTROMETRY METHOD

Authors:

Wu et al

Abstract:

Gaia is leading us into a new era with a high astrometry precision of ~10 μas. Under such precision, astrometry can play an important role in detecting and characterizing exoplanets. In particular, we can identify planet pairs in mean motion resonances (MMRs), which constrain the formation and evolution of planetary systems. In accordance with observations, we consider two-Jupiter or two-super-Earth systems in 1:2, 2:3, and 3:4 MMRs. Our simulations show that the false alarm probabilities (FAPs) of a third planet are extremely small, while the two real planets can be fitted well with a signal-to-noise ratio (S/N)$\;\gt \;3$. The probability of reconstructing a resonant system is related to the eccentricities and the resonance intensity. Generally, when the S/N $\geqslant \;10$, if the eccentricities of both planets are larger than 0.01 and the resonance is quite strong, the probability of reconstructing the planet pair in MMRs is $\geqslant \;80 \% $. Jupiter pairs in MMRs are reconstructed more easily than super-Earth pairs with similar S/N when we consider dynamical stability. FAPs are also calculated when we detect planet pairs in or near MMRs. The FAPs for 1:2 MMRs are the largest, i.e., FAP $\gt 15 \% $ when S/N $\leqslant \;10$. Extrapolating from the Kepler planet pairs near MMRs and assuming a S/N ~ 3, we discover and reconstruct a few tens of Jupiter pairs and hundreds of super-Earth pairs in 2:3 and 1:2 MMRs within 30 pc. We also compare the differences between even and uneven data cadence and find that planets are better measured with more uniform phase coverage.

HIP 41378: an F Class Star Hosting 5 Transiting Exoplanets

Five Planets Transiting a Ninth Magnitude Star

Authors:

Vanderburg et al

Abstract:

The Kepler mission has revealed a great diversity of planetary systems and architectures, but most of the planets discovered by Kepler orbit faint stars. Using new data from the K2 mission, we present the discovery of a five planet system transiting a bright (V = 8.9, K = 7.7) star called HIP 41378. HIP 41378 is a slightly metal-poor late F-type star with moderate rotation (v sin(i) = 7 km/s) and lies at a distance of 116 +/- 18 from Earth. We find that HIP 41378 hosts two sub-Neptune sized planets orbiting 3.5% outside a 2:1 period commensurability in 15.6 and 31.7 day orbits. In addition, we detect three planets which each transit once during the 75 days spanned by K2 observations. One planet is Neptune sized in a likely ~160 day orbit, one is sub-Saturn sized likely in a ~130 day orbit, and one is a Jupiter sized planet in a likely ~1 year orbit. We show that these estimates for the orbital periods can be made more precise by taking into account dynamical stability considerations. We also calculate the distribution of stellar reflex velocities expected for this system, and show that it provides a good target for future radial velocity observations. If a precise orbital period can be determined for the outer Jovian planet through future observations, it will be an excellent candidate for follow-up transit observations to study its atmosphere and measure its oblateness.

The Impact of Secular Resonances on Habitable Zones in Circumbinary Exoplanetary Systems

The impact of secular resonances on habitable zones in circumstellar planetary systems of known binary stars

Authors:

Baszó et al

Abstract:

We present a survey on binary star systems with stellar separations less than 100 astronomical units. For a selection of 11 binaries with a detected (giant) planet in circumstellar motion we determine the conditions that would allow additional planets to be present inside or nearby the habitable zone (HZ) of the host star.

First we calculate the three-body HZ for these systems, in order to investigate the dynamics of bodies in those regions. After adding the giant planet's influence the final HZ is considerably modified in particular by mean motion and secular resonances. We apply a semi-analytical method to determine the locations of linear secular resonances, which is based on finding the apsidal precession frequencies of the massive bodies.

For very close-in giant planets we also take the general relativistic precession of the pericenter into account.

Our results demonstrate that there is a qualitative difference in the dynamics whether the giant planet is located exterior or interior to the HZ. An exterior giant planet is more likely to cause a secular resonance than an interior planet. Exterior planets play an important role for the HZ by enforcing highly eccentric orbits on potentially habitable planets. For interior planets generally we do not find a secular resonance in the Newtonian framework, but the general relativistic precession introduces another source for a secular resonance.

In 7 out of 11 systems secular resonances play a role. We compare our semi-analytical method to various purely analytical models, and show that it is more accurate in determining the location of secular resonances.

DYNAMICS AND TRANSIT VARIATIONS OF RESONANT EXOPLANETS

DYNAMICS AND TRANSIT VARIATIONS OF RESONANT EXOPLANETS

Authors:

Nesvorný et al

Abstract:

Transit timing variations (TTVs) are deviations of the measured midtransit times from the exact periodicity. One of the most interesting causes of TTVs is the gravitational interaction between planets. Here we consider a case of two planets in a mean motion resonance (orbital periods in a ratio of small integers). This case is important because the resonant interaction can amplify the TTV effect and allow planets to be detected more easily. We develop an analytic model of the resonant dynamics valid for small orbital eccentricities and use it to derive the principal TTV terms. We find that a resonant system should show TTV terms with two basic periods (and their harmonics). The resonant TTV period is proportional (m/M *)−2/3, where m and M * are the planetary and stellar masses. For m = 10−4 M *, for example, the TTV period exceeds the orbital period by about two orders of magnitude. The amplitude of the resonant TTV terms scales linearly with the libration amplitude. The ratio of the TTV amplitudes of two resonant planets is inversely proportional to the ratio of their masses. These and other relationships discussed in the main text can be used to aid the interpretation of TTV observations.

Astrometric Monitoring of the HR 8799 Planets

Astrometric Monitoring of the HR 8799 Planets: Orbit Constraints from Self-Consistent Measurements

Authors:

Konopacky et al

Abstract:

We present new astrometric measurements from our ongoing monitoring campaign of the HR 8799 directly imaged planetary system. These new data points were obtained with NIRC2 on the W.M. Keck II 10 meter telescope between 2009 and 2014. In addition, we present updated astrometry from previously published observations in 2007 and 2008. All data were reduced using the SOSIE algorithm, which accounts for systematic biases present in previously published observations. This allows us to construct a self-consistent data set derived entirely from NIRC2 data alone. From this dataset, we detect acceleration for two of the planets (HR 8799b and e) at greater than 3σ. We also assess possible orbital parameters for each of the four planets independently. We find no statistically significant difference in the allowed inclinations of the planets. Fitting the astrometry while forcing coplanarity also returns χ2 consistent to within 1σ of the best fit values, suggesting that if inclination offsets of ≲20o are present, they are not detectable with current data. Our orbital fits also favor low eccentricities, consistent with predictions from dynamical modeling. We also find period distributions consistent to within 1σ with a 1:2:4:8 resonance between all planets. This analysis demonstrates the importance of minimizing astrometric systematics when fitting for solutions to highly undersampled orbits.

Secular models and Kozai resonance for planets in coorbital non-coplanar motion

Secular models and Kozai resonance for planets in coorbital non-coplanar motion

Authors:

Guipponne et al

Abstract:

In this work, we construct and test an analytical and a semianalytical secular models for two planets locked in a coorbital non-coplanar motion, comparing some results with the case of restricted three body problem.

The analytical average model replicates the numerical N-body integrations, even for moderate eccentricities (≲ 0.3) and inclinations (≲10∘), except for the regions corresponding to quasi-satellite and Lidov-Kozai configurations. Furthermore, this model is also useful in the restricted three body problem, assuming very low mass ratio between the planets. We also describe a four-degree-of-freedom semianalytical model valid for any type of coorbital configuration in a wide range of eccentricities and inclinations.

{Using a N-body integrator, we have found that the phase space of the General Three Body Problem is different to the restricted case for inclined systems, and establish the location of the Lidov-Kozai equilibrium configurations depending on mass ratio. We study the stability of periodic orbits in the inclined systems, and find that apart from the robust configurations L4, AL4, and QS is possible to harbour two Earth-like planets in orbits previously identified as unstable U and also in Euler L3 configurations, with bounded chaos.