Apsidal Corotation Research Articles

Secular evolution of resonant planets in the coplanar case

Aims. We study the secular evolution of two planets in mutual deep mean-motion resonance (MMR) in the planar elliptic three-body problem framework for different mass ratios. We do not consider any restriction in the eccentricity of the inner planet e1 or in the eccentricity of the outer planet e2. Methods. The method we used is based on a semi-analytical model that consists of calculating the averaged resonant disturbing function numerically. It is assumed for this that all the orbital elements (except for the mean longitudes) of both planets are constant on the resonant timescale. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We constructed two phase portraits, called the ℋ1 and ℋ2 surfaces, in the three-dimensional spaces (e1, Δϖ, σ) and (e2, Δϖ, σ), where Δϖ is the difference between the planetary longitude of perihelia and σ is the critical angle. These surfaces, which are related through the angular moment conservation, allow us to find the apsidal corotation resonances (ACRs) and to predict the secular evolution of e1, e2, Δϖ, and σ (libration center). Results. While studying the 1:1, 2:1, 3:1, and 3:2 MMR, we found that large excursions in eccentricity can exist in some particular cases. We compared the secular variations of e1, e2, Δϖ, and σ predicted by the model with a numerical integration of the exact equations of motion for different mass ratios. We obtained good matches. Finally, the model was applied to study the secular evolution of the resonant exoplanet systems HD 73526 and HD 31527. They both have a pair of planets and are very close to the deep MMR condition. In the first system, we found that the pair of planets that constitutes the system evolves in a symmetrical ACR, whereas in the second system, we found that planets c and d, which are in an unusual 16:3 MMR, are close to an ACR, but outside its dynamical region, where Δϖ circulates.

The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

We revisit the secular 3D planetary three-body problem, aiming to provide a unified formalism representing all basic phenomena in the phase space as the mutual inclination between the planetary orbits increases. We propose a ‘book-keeping’ technique allowing to decompose the Hamiltonian as mathcal {H}_textrm{sec}=mathcal {H}_textrm{planar}+mathcal {H}_textrm{space}, with mathcal {H}_textrm{space} collecting all terms depending on the planets mutual inclination i_textrm{mut}. We numerically compare several models obtained by multipole (Legendre) or Laplace–Lagrange expansions of mathcal {H}_textrm{sec}, aiming to define suitable truncation orders for these models. We explore the transition, as i_textrm{mut} increases, from a ‘planar-like’ to a ‘Lidov–Kozai’ regime. Using a numerical example far from hierarchical limits, we find that the structure of the phase portraits of the (integrable) planar case is reproduced to a large extent also in the 3D case. A semi-analytical criterion allows to estimate the level of i_textrm{mut} up to which the dynamics remains nearly integrable. We propose a normal form method to compute the basic periodic orbits (apsidal corotation orbits A and B) in this regime. We explore the sequence of saddle-node and pitchfork bifurcations by which the A and B families are connected to the highly inclined periodic orbits of the Lidov–Kozai regime. Finally, we perform a numerical study of phase portraits for different planetary mass and distance ratios and qualitatively describe the approach to the corresponding hierarchical limits.

Proximity of exoplanets to first-order mean-motion resonances

ABSTRACT Planetary formation theories and, more specifically, migration models predict that planets can be captured in mean-motion resonances (MMRs) during the disc phase. The distribution of period ratios between adjacent planets shows an accumulation in the vicinity of the resonance, which is not centred on the nominal resonance but instead presents an offset slightly exterior to it. Here, we extend on previous works by thoroughly exploring the effect of different disc and planet parameters on the resonance offset during the disc migration phase. The dynamical study is carried out for several first-order MMRs and for both low-mass Earth-like planets undergoing type-I migration and giant planets evolving under type-II migration. We find that the offset varies with time during the migration of the two-planet system along the apsidal corotation resonance family. The departure from the nominal resonance increases for higher planetary masses and stronger eccentricity damping. In the Earth to super-Earth regime, we find offset values in agreement with the observations when using a sophisticated modelling for the planet–disc interactions, where the damping time-scale depends on the eccentricity. This dependence causes a feedback that induces an increase of the resonance offsets. Regarding giant planets, the offsets of detected planet pairs are well reproduced with a classical K-factor prescription for the planet–disc interactions when the eccentricity damping rate remains low to moderate. In both regimes, eccentricities are in agreement with the observations too. As a result, planet–disc interactions provide a generic channel to generate the offsets found in the observations.

Secular evolution of resonant small bodies: semi-analytical approach for arbitrary eccentricities in the coplanar case

ABSTRACT We study the secular evolution of a particle in deep mean motion resonance (MMR) with a planet in the planar elliptic restricted three body problem. We do not consider any restriction neither in the planet’s eccentricity ep nor in the particle’s eccentricity e. The methodology used is based on a semi-analytical model that consists on calculating the averaged resonant disturbing function numerically, assuming for this that in the resonant scale of time all the orbital elements of the particle are constant. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We construct two-dimensional surfaces (called $\mathcal {H}$ surfaces) in the three-dimensional space (σ, e, ϖ) that allow us to predict the secular evolution of these three variables. The 2:1 MMR is used as example to show some results. We found four apsidal corotation resonance (ACR) families, two symmetric and two asymmetric. One of the symmetric families exists for almost any ep value. The other one for ep > 0.3 and the asymmetric ones for ep > 0.44. We corroborate the secular variations in e and ϖ predicted by the model through numerical integrations even when the initial conditions are displaced from those ACR. Some peculiar examples are presented for the 3:1 and 3:2 MMR showing large excursions in eccentricity. As an application, the Planet 9 is investigated as a possible responsible of high eccentric distant TNOs.

A semi-analytic model for the study of 1/1 resonant dynamics of the planar elliptic restricted co-orbital problem

Mean motion resonances (MMRs) are widespread in our Solar System. Moreover, resonant dynamics has always been an essential topic in planetary research. Recently, the research about exoplanets and the potential Planet Nine with large eccentricity has given rise to our interests in the secular dynamics inside MMRs in the elliptic model. In this paper, we study the fixed points of the averaged Hamiltonian and the long-term stable regions of the 1/1 resonance (or co-orbital motion) in the elliptic restricted three-body problem (ERTBP) systematically. Numerical integrations prove those test particles trapped in “apsidal co-rotation”, where both the resonant angle ϕ res and the secular angle ⧍ϖ (or apsidal longitude differences) librate simultaneously, always survive the long-term simulations. Furthermore, utilizing a semianalytical method based on the adiabatic approach, three families of long-term fixed points of the averaged Hamiltonian of the planar ERTBP inside the 1/1 resonance have been found. We call them QS-points, Hpoints, and T-points here, whose values of the (ϕ res, ⧍ϖ) are (0°, 180°), (180°, 0°), and (± 60°, ± 60°), respectively. All the fixed points of the averaged Hamiltonian of the co-orbital motion in the ERTBP are presented in the e-e′ plane (′ represents the elements of the planet in this paper). We find that QS-points and T-points always exist for the arbitrary eccentricity of a planet, while H-points only exist for the cases of low e′ and very high e. Furthermore, we measure the libration width in terms of eccentricity, ⧍e, around these stable equilibrium points in the e-⧍ϖ phase-space portraits. The “apsidal co-rotation” around all the stable equilibrium points is presented in the e-e′ plane. All these results are effectively confirmed by numerical experiments. The long-term stable zones around these periodic orbits in the e-e′ plane are significant for the research of the co-orbital motion in the ERTBP. Above all, these practical approaches that we proposed can also be used to study the secular dynamics of other MMRs.

Modeling Radial Velocity Data of Resonant Planets to Infer Migration Histories

Abstract A number of giant-planet pairs with period ratios ≲2 discovered by the radial velocity (RV) method may reside in mean motion resonances. Convergent orbital migration and resonant capture at the time of formation would naturally explain the present-day resonant orbital configurations of these systems. Planets that experience smooth migration and eccentricity-damping forces due to a protoplanetary disk should not only be captured into mean motion resonances but also end up in a specific dynamical configuration within the resonance, sometimes referred to as apsidal corotation resonance (ACR). Here we develop a method for testing the hypothesis that a planet pair resides in an ACR by directly fitting RV data. The ACR hypothesis strongly restricts the number of free parameters describing the RV signal, and we compare fits using this highly restricted model to fits using a more conventional two-planet RV model by using nested sampling simulations. We apply our method to HD 45364 and HD 33844, two systems hosting giant-planet pairs in 3:2 and 5:3 resonances, respectively. The observations of both systems are consistent with ACR configurations, which are formally preferred based on the Bayes factors computed from nested sampling simulations. We use the results of our ACR model fits to constrain the possible migration histories of these systems.

Secular dynamics in extrasolar systems with two planets in mutually inclined orbits

AbstractWe revisit the problem of the secular dynamics in two-planet systems in which the planetary orbits exhibit a high value of the mutual inclination. We propose a ‘basic hamiltonian model’ for secular dynamics, parameterized in terms of the system’s Angular Momentum Deficit (AMD). The secular Hamiltonian can be obtained in closed form, using multipole expansions in powers of the distance ratio between the planets, or in the usual Laplace-Lagrange form. The main features of the phase space (number and stability of periodic orbits, bifurcations from the main apsidal corotation resonances, Kozai resonance etc.) can all be recovered by choosing the corresponding terms in the ‘basic Hamiltonian’. Applications include the semi-analytical determination of the actual orbital state of the system using Hamiltonian normalization techniques. An example is discussed referring to the system of two outermost planets of the ν-Andromedae system.

Modeling the evection resonance for Trojan satellites: application to the Saturn system

Context. The stability of satellites in the solar system is affected by the so-called evection resonance. The moons of Saturn, in particular, exhibit a complex dynamical architecture in which co-orbital configurations occur, especially close to the planet where this resonance is present. Aims. We address the dynamics of the evection resonance, with particular focus on the Saturn system, and compare the known behavior of the resonance for a single moon with that of a pair of moons in co-orbital Trojan configuration. Methods. We developed an analytic expansion of the averaged Hamiltonian of a Trojan pair of bodies, including the perturbation from a distant massive body. The analysis of the corresponding equilibrium points was restricted to the asymmetric apsidal corotation solution of the co-orbital dynamics. We also performed numerical N-body simulations to construct dynamical maps of the stability of the evection resonance in the Saturn system, and to study the effects of this resonance under the migration of Trojan moons caused by tidal dissipation. Results. The structure of the phase space of the evection resonance for Trojan satellites is similar to that of a single satellite, differing in that the libration centers are displaced from their standard positions by an angle that depends on the periastron difference ϖ2 −ϖ1 and on the mass ratio m2∕m1 of the Trojan pair. In the Saturn system, the inner evection resonance, located at ~8 RS, may capture a pair of Trojan moons by migration; the stability of the captured system depends on the assumed values of the dissipation factor Q of the moons. On the other hand, the outer evection resonance, located at >0.4 RHill, cannot exist at all for Trojan moons, because Trojan configurations are strongly unstable at distances from Saturn longer than ~0.15 RHill. Conclusions. The interaction with the inner evection resonance may have been relevant during the early evolution of the Saturn moons Tethys, Dione, and Rhea. In particular, Rhea may have had Trojan companions in the past that were lost when it crossed the evection resonance, while Tethys and Dione may either have retained their Trojans or have never crossed the evection. This may help to constrain the dynamical processes that led to the migration of these satellites and to the evection itself.

Forming Gliese 876 through Smooth Disk Migration

We run a suite of dissipative N-body simulations to determine which regions of phase space for smooth disk migration are consistent with the GJ876 system, an M-dwarf hosting three planets orbiting in a chaotic 4:2:1 Laplace resonance. We adopt adaptive mesh refinement (AMR) methods which are commonly used in hydrodynamical simulations to efficiently explore the parameter space defined by the semi-major axis and eccentricity damping timescales. We find that there is a large region of phase space which produces systems in the chaotic Laplace resonance and a smaller region consistent with the observed eccentricities and libration amplitudes for the resonant angles. Under the assumptions of Type I migration for the outer planet, we translate these damping timescales into constraints on the protoplanetary disk surface density and thickness. When we strongly (weakly) damp the eccentricities of the inner two Laplace planets, these timescales correspond to disk surface densities around ten thousand (a few hundred) grams per square centimeter and disk aspect ratios between 1-10%. Additionally, smooth migration produces systems with a range of chaotic timescales, from decades and centuries to upwards of thousands of years. In agreement with previous studies, the less chaotic regions of phase space coincide with the system being in a low energy double apsidal corotation resonance. Our detailed modeling of multi-planetary systems coupled with our AMR exploration method enhances our ability to map out the parameter space of planet formation models, and is well suited to study other resonant chain systems such as Trappist-1, Kepler-60, and others.

Migration of planets in circumbinary discs

Aims. The discovery of planets in close orbits around binary stars raises questions about their formation. It is believed that these planets formed in the outer regions of the disc and then migrated through planet-disc interaction to their current location. Considering five different systems (Kepler-16, -34, -35, -38, and -413) we model planet migration through the disc, with special focus on the final orbital elements of the planets. We investigate how the final orbital parameters are influenced by the disc and planet masses. Methods. Using two-dimensional, locally isothermal, and viscous hydrodynamical simulations, we first model the disc dynamics for all five systems, followed by a study of the migration properties of embedded planets with different masses. To strengthen our results, we apply two grid-based hydrodynamical codes using different numerics (PLUTO and FARGO3D). Results. For all systems, we find that the discs become eccentric and precess slowly. We confirm the bifurcation feature in the precession period – gap-size diagram for different binary mass ratios. The Kepler-16, -35, -38, and -413 systems lie on the lower branch and Kepler-34 on the upper one. For systems with small binary eccentricity, we find a new non-monotonic, loop-like feature. In all systems, the planets migrate to the inner edge of the disc cavity. Depending on the planet-disc mass ratio, we observe one of two different regimes. Massive planets can significantly alter the disc structure by compressing and circularising the inner cavity and they remain on nearly circular orbits. Lower-mass planets are strongly influenced by the disc, their eccentricity is excited to high values, and their orbits are aligned with the inner disc in a state of apsidal corotation. In our simulations, the final locations of the planets are typically too large with respect to the observations because of the large inner gaps of the discs. The migrating planets in the most eccentric discs (around Kepler-34 and -413) show the largest final eccentricity in agreement with the observations.

New Constraints on Gliese 876—Exemplar of Mean-motion Resonance

Abstract Gliese 876 harbors one of the most dynamically rich and well-studied exoplanetary systems. The nearby M4V dwarf hosts four known planets, the outer three of which are trapped in a Laplace mean-motion resonance. A thorough characterization of the complex resonant perturbations exhibited by the orbiting planets, and the chaotic dynamics therein, is key to a complete picture of the system’s formation and evolutionary history. Here we present a reanalysis of the system using 6 yr of new radial velocity (RV) data from four instruments. These new data augment and more than double the size of the decades-long collection of existing velocity measurements. We provide updated estimates of the system parameters by employing a computationally efficient Wisdom–Holman N-body symplectic integrator, coupled with a Gaussian process (GP) regression model to account for correlated stellar noise. Experiments with synthetic RV data show that the dynamical characterization of the system can differ depending on whether a white-noise or correlated-noise model is adopted. Despite there being a region of stability for an additional planet in the resonant chain, we find no evidence for one. Our new parameter estimates place the system even deeper into resonance than previously thought and suggest that the system might be in a low-energy, quasi-regular double apsidal corotation resonance. This result and others will be used in a subsequent study on the primordial migration processes responsible for the formation of the resonant chain.

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

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, Beauge 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 1 sigma 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 chi(2) 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 is 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.

Orbital stability of coplanar two-planet exosystems with high eccentricities

The long-term stability of the evolution of two-planet systems is considered by using the general three body problem (GTBP). Our study is focused on the stability of systems with adjacent orbits when at least one of them is highly eccentric. In these cases, in order for close encounters, which destabilize the planetary systems, to be avoided, phase protection mechanisms should be considered. Additionally, since the GTBP is a non-integrable system, chaos may also cause the destabilization of the system after a long time interval. By computing dynamical maps, based on Fast Lyapunov Indicator, we reveal regions in phase space with stable orbits even for very high eccentricities (e>0.5). Such regions are present in mean motion resonances (MMR). We can determine the position of the exact MMR through the computation of families of periodic orbits in a rotating frame. Elliptic periodic orbits are associated with the presence of apsidal corotation resonances (ACR). When such solutions are stable, they are associated with neighbouring domains of initial conditions that provide long-term stability. We apply our methodology so that the evolution of planetary systems of highly eccentric orbits is assigned to the existence of such stable domains. Particularly, we study the orbital evolution of the extrasolar systems HD 82943, HD 3651, HD 7449, HD 89744 and HD 102272 and discuss the consistency between the orbital elements provided by the observations and the dynamical stability.

Dynamics of the 3/1 planetary mean-motion resonance: an application to the HD60532 b-c planetary system

In this paper, we use a semi-analytical approach to analyze the global structure of the phase space of the planar planetary 3/1 mean-motion resonance. The case where the outer planet is more massive than its inner companion is considered. We show that the resonant dynamics can be described using two fundamental parameters, the total angular momentum and the spacing parameter. The topology of the Hamiltonian function describing the resonant behaviour is investigated on a large domain of the phase space without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The families of the Apsidal Corotation Resonances (ACR) parameterized by the planetary mass ratio are obtained and their stability is analyzed. The main dynamical features in the domains around the ACR are also investigated in detail by means of spectral analysis techniques, which allow us to detect the regions of different regimes of motion of resonant systems. The construction of dynamical maps for various values of the total angular momentum shows the evolution of domains of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems.

Comparative orbital evolution of transient Uranian co-orbitals: exploring the role of ephemeral multibody mean motion resonances

Uranus has three known co-orbitals: 83982 Crantor (2002 GO9), 2010 EU65 and 2011 QF99. All of them were captured in their current resonant state relatively recently. Here, we perform a comparative analysis of the orbital evolution of these transient co-orbitals to understand better how they got captured in the first place and what makes them dynamically unstable. We also look for additional temporary Uranian co-orbital candidates among known objects. Our N-body simulations show that the long-term stability of 2011 QF99 is controlled by Jupiter and Neptune; it briefly enters the 1:7 mean motion resonance with Jupiter and the 2:1 with Neptune before becoming a Trojan and prior to leaving its tadpole orbit. During these ephemeral two-body mean motion resonance episodes, apsidal corotation resonances are also observed. For known co-orbitals, Saturn is the current source of the main destabilizing force but this is not enough to eject a minor body from the 1:1 commensurability with Uranus. These objects must enter mean motion resonances with Jupiter and Neptune in order to be captured or become passing Centaurs. Asteroid 2010 EU65, a probable visitor from the Oort cloud, may have been stable for several Myr due to its comparatively low eccentricity. Additionally, we propose 2002 VG131 as the first transient quasi-satellite candidate of Uranus. Asteroid 1999 HD12 may signal the edge of Uranus’ co-orbital region. Transient Uranian co-orbitals are often submitted to complex multibody ephemeral mean motion resonances that trigger the switching between resonant co-orbital states, making them dynamically unstable. In addition, we show that the orbital properties and discovery circumstances of known objects can be used to outline a practical strategy by which additional Uranus’ co-orbitals may be found.

Possible solution to the riddle of HD 82943 multiplanet system: the three-planet resonance 1:2:5?

We carry out a new analysis of the published radial velocity data for the planet-hosting star HD82943. We include the recent Keck/HIRES measurements as well as the aged but much more numerous CORALIE data. We find that the CORALIE radial velocity measurements are polluted by a systematic annual variation which affected the robustness of many previous results. We show that after purging this variation, the residuals still contain a clear signature of an additional $\sim 1100$ days periodicity. The latter variation leaves significant hints in all three independent radial velocity subsets that we analysed: the CORALIE data, the Keck data acquired prior to a hardware upgrade, and the Keck data taken after the upgrade. We mainly treat this variation as a signature of a third planet in the system, although we cannot rule out other interpretations, such as long-term stellar activity. We find it easy to naturally obtain a stable three-planet radial-velocity fit close to the three-planet mean-motion resonance 1:2:5, with the two main planets (those in the 1:2 resonance) in an aligned apsidal corotation. The dynamical status of the third planet is still uncertain: it may reside in as well as slightly out of the 5:2 resonance. We obtain the value of $\sim 1075$ days for its orbital period and of $\sim 0.3 M_{\rm Jup}$ for its minimum mass, while the eccentric parameters are uncertain.

Artificial satellites: resonance effects produced by terrestrial tide

Abstract: Effects due to resonances in the orbital motion of artificial satellites disturbed by the terrestrial tide are analyzed. The nodal co-rotation resonance, apsidal co-rotation resonance and the Lidov-Kozai’s mechanism are studied. The effects of the resonances are analyzed through the variations of the metric orbital elements. Libration and circulation motions for high orbits with high eccentricities are verified for the Lidov-Kozai’s mechanism.

The 2:1 librating-circulating planetary configuration produced by a hybrid scenario

AbstractDifferent migration scenario of two giant planets may play a major role in forming the diverse resonant planetary configurations. The studies on the HD 128311 and HD 73526 systems show that two gas giants are captured in a 2:1 resonance but not in apsidal corotation, because one of the resonant argument circulates over the dynamical evolution. Herein we explore potential scenarios to produce the 2:1 librating-circulating resonance configuration. In the simulations, we find that both colliding or scattering events at early stage of dynamical evolution can induce the configurations trapped into resonance. In this sense, the librating-circulating resonance configuration is more likely to form by a hybrid mechanism of scattering and collision.

A hybrid mechanism forming a2:1librating-circulating resonant configuration in the planetary system

A diversity of resonance configurations may be formed under different migration of two giant planets. And the researchers show that the HD 128311 and HD 73526 planetary systems are involved in a 2:1 mean motion resonance but not in apsidal corotation, because one of the resonance argument circulates over the dynamical evolution. In this paper, we investigate potential mechanisms to form the 2:1 librating-circulating resonance configuration. In the late stage of planetary formation, scattering or colliding among planetesimals and planetary embryos can frequently occur. Hence, in our model, we consider a planetary configuration of two giants together with few terrestrial planets. We find that both colliding or scattering events at very early stage of dynamical evolution can influence the configurations trapped into resonance. A planet-planet scattering of a moderate terrestrial planet, or multiple scattering of smaller planets in a crowded planetary system can change the resonant configuration. In addition, collision or merging can alter the masses and location of the giant planets, which also play an important role in shaping the resonant configuration during the dynamical evolution. In this sense, the librating-circulating resonance configuration is more likely to form by a hybrid mechanism of scattering and collision.

DETECTABILITY AND ERROR ESTIMATION IN ORBITAL FITS OF RESONANT EXTRASOLAR PLANETS

We estimate the conditions for detectability of two planets in a 2/1 mean-motion resonance from radial velocity data, as a function of their masses, number of observations and the signal-to-noise ratio. Even for a data set of the order of 100 observations and standard deviations of the order of a few meters per second, we find that Jovian-size resonant planets are difficult to detect if the masses of the planets differ by a factor larger than {approx}4. This is consistent with the present population of real exosystems in the 2/1 commensurability, most of which have resonant pairs with similar minimum masses, and could indicate that many other resonant systems exist, but are currently beyond the detectability limit. Furthermore, we analyze the error distribution in masses and orbital elements of orbital fits from synthetic data sets for resonant planets in the 2/1 commensurability. For various mass ratios and number of data points we find that the eccentricity of the outer planet is systematically overestimated, although the inner planet's eccentricity suffers a much smaller effect. If the initial conditions correspond to small-amplitude oscillations around stable apsidal corotation resonances, the amplitudes estimated from the orbital fits are biased toward larger amplitudes, in accordancemore » to results found in real resonant extrasolar systems.« less