Relative equilibria in the unrestricted problem of a sphere and symmetric rigid body

We consider the unrestricted problem of two mutually attracting rigid bodies, an uniform sphere (or a point mass) and an axially symmetric body. We present a global, geometric approach for finding all relative equilibria (stationary solutions) in this model, which was already studied by Kinoshita (1970). We extend and generalize his results, showing that the equilibria solutions may be found by solving at most two non-linear, algebraic equations, assuming that the potential function of the symmetric rigid body is known explicitly. We demonstrate that there are three classes of the relative equilibria, which we call "cylindrical", "inclined co-planar", and "conic" precessions, respectively. Moreover, we also show that in the case of conic precession, although the relative orbit is circular, the point-mass and the mass center of the body move in different parallel planes. This solution has been yet not known in the literature.Comment: The manuscript with 10 pages, 5 figures; accepted to the Monthly Notices of the Royal Astronomical Societ

Equilibria in the secular, non-coplanar two-planet problem

We investigate the secular dynamics of a planetary system composed of the parent star and two massive planets in mutually inclined orbits. The dynamics are investigated in wide ranges of semi-major axes ratios (0.1-0.667), and planetary masses ratios (0.25-2) as well as in the whole permitted ranges of the energy and total angular momentum. The secular model is constructed by semi-analytic averaging of the three-body system. We focus on equilibria of the secular Hamiltonian (periodic solutions of the full system), and we analyze their stability. We attempt to classify families of these solutions in terms of the angular momentum integral. We identified new equilibria, yet unknown in the literature. Our results are general and may be applied to a wide class of three-body systems, including configurations with a star and brown dwarfs and sub-stellar objects. We also describe some technical aspects of the semi-numerical averaging. The HD 12661 planetary system is investigated as an example configuration.Comment: 18 pages, 17 figures, accepted to Monthly Notices of the Royal Astronomical Societ

The orbital architecture and stability of the μ\mu Arae planetary system

We re-analyze the global orbital architecture and dynamical stability of the $\mu$ Arae planetary system. We have updated the best-fit elements and minimal masses of the planets based on literature radial velocity (RV) measurements, now spanning 15 years. This is twice the RVs interval used for the first characterization of the system in 2006. It consists of a Saturn- and two Jupiter-mass planets in low-eccentric orbits resembling the Earth-Mars-Jupiter configuration in the Solar system, as well as the close-in warm Neptune with a mass of ~14 Earth masses. Here, we constrain this early solution with the outermost period to be accurate to one month. The best-fit Newtonian model is characterized by moderate eccentricities of the most massive planets below 0.1 with small uncertainties ~0.02. It is close but meaningfully separated from the 2e:1b mean motion resonance of the Saturn-Jupiter-like pair, but may be close to weak three-body MMRs. The system appears rigorously stable over a wide region of parameter space covering uncertainties of several $\sigma$. The system stability is robust to a five-fold increase in the minimal masses, consistent with a wide range of inclinations, from 20 to 90 deg. This means that all planetary masses are safely below the brown dwarf mass limit. We found a weak statistical indication of the likely system inclination I~20-30 deg. Given the well constrained orbital solution, we also investigate the structure of hypothetical debris disks, which are analogs of the Main Belt and Kuiper Belt, and may naturally occur in this system.Comment: Several errors in the text have been removed and references corrected and expanded. This manuscript has 23 pages (20 pages+3 pages of supplementary on-line material), 2 tables and 14+2 multi-panel figures. Accepted to Monthly Notices of the RAS. Your comments are welcome

Modeling the RV and BVS of active stars

We present a method of modeling the radial velocity (RV) measurements which can be useful in searching for planets hosted by chromospherically active stars. We assume that the observed RV signal is induced by the reflex motion of a star as well as by distortions of spectral line profiles, measured by the Bisector Velocity Span (BVS). The RVs are fitted with a common planetary model including RV correction term depending linearly on the BVS, which accounts for the stellar activity. The coefficient of correlation is an additional free parameter of the RV model. That approach differs from correcting the RVs before or after fitting the "pure" planetary model. We test the method on simulated data derived for single-planet systems. The results are compared with the outcomes of algorithms found in the literature.Comment: 6 pages, 2 figures, proceedings of the conference "Extrasolar planets in multi-body systems: theory and observations" (August 2008, Torun, Poland

The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects

We describe numerical tools for the stability analysis of extrasolar planetary systems. In particular, we consider the relative Poincare variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator MEGNO (a measure of the maximal Lyapunov exponent) that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best fit solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best fit solutions are found in dynamically active region of the phase space. The long term stability of the system is determined by a net of low-order two-body and three-body mean motion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myrs of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behavior.Comment: 11 pages (total), 8 figures. Accepted for publication in MNRAS. The definitive version will be/is available at http://www.blackwellpublishing.com. The astro-ph version is prepared with low resolution figures. To obtain the manuscript with full-resolution figures, please visit http://www.astri.uni.torun.pl/~chris/mnrasIII.ps.g