Tatiana A. Michtchenko

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.

Secular dynamics of the three-body problem: application to the υ Andromedae planetary system

Abstract The discovery of extra-solar planetary systems with multiple planets in highly eccentric orbits (∼0.1–0.6), in contrast with our own Solar System, makes classical secular perturbation analysis very limited. In this paper, we use a semi-numerical approach to study the secular behavior of a system composed of a central star and two massive planets in co-planar orbits. We show that the secular dynamics of this system can be described using only two parameters, the ratios of the semi-major axes and the planetary masses. The main dynamical features of the system are presented in geometrical pictures that allows us to investigate a large domain of the phase space of this three-body problem without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The topology of the phase space is also investigated in detail by means of spectral map techniques, which allow us to detect the separatrix of a non-linear secular apsidal resonance. Finally, the qualitative study is supplemented by direct numerical integrations. Three different regimes of secular motion with respect to the secular angle Δ ϖ are possible: they are circulation, oscillation (around 0° and 180°), and high eccentricity libration in a non-linear secular resonance. The first two regimes are a continuous extension of the classical linear secular perturbation theory; the last is a new feature, hitherto unknown, in the secular dynamics of the three-body problem. We apply the analysis to the case of the two outer planets in the υ Andromedae system, and obtain its periodic and ordinary orbits, the general structure of its secular phase space, and the boundaries of its secular stability; we find that this system is secularly stable over a large domain of eccentricities. Applying this analysis to a wide range of planetary mass and semi-major axis ratios (centered about the υ Andromedae parameters), we find that apsidal oscillation dominates the secular phase space of the three-body coplanar system, and that the non-linear secular resonance is also a common feature.

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.

Resonant Structure of the Outer Solar System in the Neighborhood of the Planets

The stability of the outer solar system is studied using a new numerical method for detecting the chaoticity of planetary motion. We explore a large portion of the phase space where the outer solar system evolves and construct dynamical maps of the regions around the Jovian planets. We show that these regions are densely filled by two- and three-planet mean motion resonances that generate instabilities in planetary motion. We also show how close the actual positions of the planets are to these instabilities.

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.

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.

Modeling the 3-D secular planetary three-body problem: Discussion on the outer υ Andromedae planetary system

Abstract The three-dimensional secular behavior of a system composed of a central star and two massive planets is modeled semi-analytically in the frame of the general three-body problem. The main dynamical features of the system are presented in geometrical pictures allowing us to investigate a large domain of the phase space of this problem without time-expensive numerical integrations of the equations of motion and without any restriction on the magnitude of the planetary eccentricities, inclinations and mutual distance. Several regimes of motion of the system are observed. With respect to the secular angle Δϖ, possible motions are circulations, oscillations (around 0° and 180°), and high-eccentricity/inclination librations in secular resonances. With respect to the arguments of pericenter, ω 1 and ω 2 , possible motions are direct circulation and high-inclination libration around ±90° in the Lidov–Kozai resonance. The regions of transition between domains of different regimes of motion are characterized by chaotic behavior. We apply the analysis to the case of the two outer planets of the υ Andromedae system, observed edge-on. The topology of the 3-D phase space of this system is investigated in detail by means of surfaces of section, periodic orbits and dynamical spectra, mapping techniques and numerical simulations. We obtain the general structure of the phase space, and the boundaries of the spatial secular stability. We find that this system is secularly stable in a large domain of eccentricities and inclinations.

A frequency approach to identifying asteroid families

Context. Secular resonances usually have a complicated three-dimensional structure in a - e - i (or a - e - sin(i)) space, which is not easily represented in a two-dimensional projection. As a consequence, the classic approach to identifying asteroid families fails in some cases to identify objects that have migrated in such resonances because of the Yarkovsky effect. Aims. We propose an alternative approach by identifying asteroid families in the proper frequency domain (n, g, g+ s) rather than in the proper element domain (a, e, sin(i)). Methods. We applied the HCM method in the proper frequency domain (n, g, g + s) to identify four of the largest asteroid families: Vesta, Eunomia, Eos, and Koronis. We compared our results with those obtained with the classical method. In addition, we applied the extended metrics in the domain of asteroid colors in both proper element and frequency domains. Results. The frequency approach to determining families is an excellent tool for (i) more easily identifying the effect of nonlinear secular resonances on families and (ii) connecting to the family of origin objects that have migrated via the interplay of the Yarkovsky effect and nonlinear secular resonances.

The inner region of the asteroid Main Belt: a spectroscopic and dynamic analysis

Aims. To better understand the dynamical and collisional evolution of the inner Main Belt, we perform a visible spectroscopic survey and construct dynamical maps of the region. Methods. The survey was performed between March 2002 and August 2005, at diverse observatories. The dynamical analysis was performed by the integration of 3131 massless particles homogeneously distributed in the region. Results. We obtained new taxonomic classification for 88 asteroids representing a 13% increase in the sample of asteroids with know classification in the inner region of the Main Belt. The increase in the number of classified objects further confirms the notion that the inner region can be divided into three compositional zones: the innermost, where asteroids of the S-group concentrate, the outermost, where the C-group peaks, and in-between, with the highest concentration of V-type. The dynamical analysis shows that the region is covered by a dense web of mean-motion and secular resonances which may play an important role in the dynamical diffusion of the asteroids.