C. Beaugé

Extrasolar Planets in Mean-Motion Resonance: Apses Alignment and Asymmetric Stationary Solutions

In recent years several pairs of extrasolar planets have been discovered in the vicinity of mean-motion commensurabilities. In some cases, such as the GJ 876 system, the planets seem to be trapped in a stationary solution, the system exhibiting a simultaneous libration of the resonant angle θ1 = 2λ2 - λ1 - 1 and of the relative position of the pericenters. In this paper we analyze the existence and location of these stable solutions, for the 2 : 1 and 3 : 1 resonances, as functions of the masses and orbital elements of both planets. This is undertaken via an analytical model for the resonant Hamiltonian function. The results are compared with those of numerical simulations of the exact equations. In the 2 : 1 commensurability, we show the existence of three principal families of stationary solutions: (1) aligned orbits, in which θ1 and 1 - 2 both librate around zero, (2) antialigned orbits, in which θ1 = 0 and the difference in pericenter is 180°, and (3) asymmetric stationary solutions, in which both the resonant angle and 1 - 2 are constants with values different from 0° or 180°. Each family exists in a different domain of values of the mass ratio and eccentricities of both planets. Similar results are also found in the 3 : 1 resonance. We discuss the application of these results to the extrasolar planetary systems and develop a chart of possible planetary orbits with apsidal corotation. We estimate, also, the maximum planetary masses in order for the stationary solutions to be dynamically stable.

Planetary migration and extrasolar planets in the 2/1 mean-motion resonance

In this paper, we present a new set of corotational solutions for the 2/1 commensurability, including previously known solutions and new results. Comparisons with observed exoplanets show that current orbital fits of three proposed resonant planetary systems are consistent with apsidal corotations. We also discuss the possible relationship between the current orbital elements fits of known exoplanets in the 2/1 mean-motion resonance and the expected orbital configuration due to migration. We find that, as long as the orbital decay was sufficiently slow to be approximated by an adiabatic process, all captured planets should be in apsidal corotations. In other words, they should show a simultaneous libration of both the resonant angle and the difference in longitudes of pericenter.

Evolution of Migrating Planet Pairs in Resonance

Numerical simulations of the evolution of planets or massive satellites captured in the 2/1 and 3/1 resonances, under the action of an anti-dissipative tidal force. The evolution of resonant trapped bodies show a richness of solutions: librations around stationary symmetric solutions with aligned periapses (Δϖ = 0) or anti-aligned periapses (Δϖ = π), librations around stationary asymmetric solutions in which the periapses configuration is fixed, but with Δϖ taking values in a wide range of angles. Many of these solutions exist for large values of the eccentricities and, during the semimajor axes drift, the solutions show turnabouts from one configuration to another. The presented results are valid for other non-conservative anti-dissipative forces leading to adiabatic convergent migration and capture into one of these resonances.

THE ORBITS OF THE EXTRASOLAR PLANETS HD 82943c AND b

The published orbits of the planets HD 82943b and HD 82943c correspond to a system bound to a catastrophic event in less than 100,000 yr. Alternative sets of elements and masses, which fit the available observational data and correspond to regular motions, are presented in this paper. The planets HD 82943c and b are in a 2 : 1 mean-motion resonance and are trapped in the neighborhood of a (0, 0) apsidal corotation.

Co-orbital terrestrial planets in exoplanetary systems: a formation scenario

Aims. We study the formation of a hypothetical terrestrial-type body in the equilateral Lagrange points of a giant extrasolar planet. Starting from a swarm of planetesimals in stable tadpole orbits, we simulate its dynamical and collisional evolution under a wide range of different initial conditions and masses for both the Trojan population and its planetary companion. We also analyze the effects of gas drag from the interaction of the planetesimals with the nebular disk. Methods. The formation process is simulated with an N -body code that considers full gravitational interactions between the planetesimals and the giant planet. Gas interaction is modeled with Stokes and Epstein drags, where the drag coefficients are chosen following the results of full hydrodynamic simulations performed with the 2D public hydro-code FARGO. Results. In both gas-free and gas-rich scenarios, we have been able to obtain a single final terrestrial-type body in a stable tadpole orbit around one of the triangular Lagrange points of the system. However, due to gravitational instabilities within the swarm, the accretional process is not very efficient and the mass of the final planet never seems to exceed ~0.6 Earth masses, even when the total mass of the swarm is five times this value. Finally, we also included an orbital decay of the giant planet due to a typexa0II migration. Although the accretional process shows evidence of a lower efficiency, a small terrestrial planet is still able to form, and follows the giant planet towards the habitable zone of the hosting star.

On the mass determination of super-Earths orbiting active stars: the CoRoT-7 system

Context. Due to the star activity, the masses of CoRoT-7b and CoRoT 7c are uncertain. Investigators of the CoRoT team have proposed several solutions, all but one of them larger than the initia l determinations of 4.8�0.8 MEarth for CoRoT-7b and 8.4�0.9 MEarth for CoRoT 7c. Aims. This investigation uses the excellent HARPS radial velocity measurements of CoRoT-7 to re-determine the planet masses and to explore techniques able to determine mass and elements of planets discovered around active stars when the relative variation of the radial velocity due to the star activity cannot be considere d as just noise and can exceed the variation due to the planets. Methods. The main technique used here is a self-consistent version of the high-pass filter used by Queloz et al. (2009) in the first ma ss determination of CoRoT-7b and CoRoT-7c. The results are compared to those given by two alternative techniques: (1) The approach proposed by Hatzes et al. (2010) using only those nights in which 2 or 3 observations were done; (2) A pure Fourier analysis. In all cases, the eccentricities are taken equal to zero as indicat ed by the study of the tidal evolution of the system; the periods are also kept fixed at the values given by Queloz et al. Only the observation s done in the time interval BJD 2,454,847 ‐ 873 are used because they include many nights with multiple observations; otherwise it is not possible to separate the effects of the rotation fourth harmonic (5.91 d=Prot/4) from the alias of the orbital period of CoRoT-7b (0.853585 d). Results. The results of the various approaches are combined to give for the planet masses the values 8.0�1.2 MEarth for CoRoT-7b and 13.6�1.4 MEarth for CoRoT 7c. An estimation of the variation of the radial velocity of the star due to its activity is also given. Conclusions. The results obtained with 3 different approaches agree to give masses larger than those in previous determinations. From the existing internal structure models they indicate that C oRoT-7b is a much denser super-Earth. The bulk density is 11�3.5 g.cm −3 . CoRoT-7b may be rocky with a large iron core.

Dynamics of two planets in co-orbital motion

We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities; we also assume that both planets share the same orbital plane. Initially, we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analysed in more detail using a semi-analytical model. n n n nApart from the well-known quasi-satellite orbits and the classical equilibrium Lagrangian points L4 and L5, we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at (Δλ, Δϖ) = (±60°, ∓120°), where Δλ is the difference in mean longitudes and Δϖ is the difference in longitudes of pericentre. The position of these anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities and are found for eccentricities as high as ∼0.7. n n n nFinally, we also applied a slow mass variation to one of the planets and analysed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.

FAST INVERSION METHOD FOR DETERMINATION OF PLANETARY PARAMETERS FROM TRANSIT TIMING VARIATIONS

The transit timing variation (TTV) method relies on monitoring changes in timing of transits of known exoplanets. Non-transiting planets in the system can be inferred from TTVs by their gravitational interaction with the transiting planet. The TTV method is sensitive to low-mass planets that cannot be detected by other means. Here we describe a fast algorithm that can be used to determine the mass and orbit of the non-transiting planets from the TTV data. We apply our code, ttvim.f, to a wide variety of planetary systems to test the uniqueness of the TTV inversion problem and its dependence on the precision of TTV observations. We find that planetary parameters, including the mass and mutual orbital inclination of planets, can be determined from the TTV data sets that should become available in near future. Unlike the radial velocity technique, the TTV method can therefore be used to characterize the inclination distribution of multi-planet systems.

Tidal decay and orbital circularization in close-in two-planet systems

The motion of two planets around a Sun-like star under the combined effects of mutual interaction and tidal dissipation is investigated. The secular behaviour of the system is analysed using two different approaches. First, we solve the exact equations of motion through the numerical simulation of the system evolution. In addition to the orbital decay and circularization, we show that the final configuration of the system is affected by the shrinking of the inner orbit. Our second approach consists of the analysis of the stationary solutions of mean equations of motion based on a Hamiltonian formalism. n n n nWe consider the case of a hot super-Earth planet with a more massive outer companion. As a real example, the CoRoT-7 system is analysed, solving the exact and mean equations of motion. The star–planet tidal interaction produces orbital decay and circularization of the orbit of CoRoT-7b. In addition, the long-term tidal evolution is such that the eccentricity of CoRoT-7c is also circularized and a pair of final circular orbits is obtained. A curve in the space of eccentricities can be constructed through the computation of stationary solutions of mean equations including dissipation. The application to the CoRoT-7 system shows that the stationary curve agrees with the result of numerical simulations of exact equations. n n n nA similar investigation performed in a super-Earth–Jupiter two-planet system shows that the doubly circular state is accelerated when there is a significant orbital migration of the inner planet, in comparison with previous results where migration is neglected.

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 planets 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 accordancemorexa0» to results found in real resonant extrasolar systems.«xa0less