The Librating Companions in HD 37124, HD 12661, HD 82943, 47 Uma and GJ 876: Alignment or Antialignment?

We investigated the apsidal motion for the multi-planet systems. In the simulations, we found that the two planets of HD 37124, HD 12661, 47 Uma and HD 82943 separately undergo apsidal alignment or antialignment. But the companions of GJ 876 and $\upsilon$ And are only in apsidal lock about $0^{\circ}$. Moreover, we obtained the criteria with Laplace-Lagrange secular theory to discern whether a pair of planets for a certain system are in libration or circulation.Comment: 13 Pages, 3 figures, 2 tables, Published by ApJ Letters, 591, July 1, 2003 (Figures now included to match the publication

The Dynamical Simulations of the Planets Orbiting GJ 876

We have performed simulations to investigate the dynamics of the M dwarf star GJ 876 in an attempt to reveal any stabilizing mechanism for sustaining the system.We simulated different coplanar and noncoplanar configurations of two-planet systems and other cases.From the simulations,we found that the 2 :1 mean-motion resonance between two planets can act as an effective mechanism for maintaining the stability of the system.This result is explained by a proposed analytical model.Using this model,we studied the region of motion of the inner planet by varying the parameters of the system,and we detected that the analytical results are well consistent with the numerical simulations.Comment: 17 pages, 8 figures available through authors, to be published in ApJ, June 20,2002 (V572, see figures

The Secular Evolution and Dynamical Architecture of the Neptunian Triplet Planetary System HD 69830

We perform numerical simulations to study the secular orbital evolution and dynamical structure in the HD 69830 system with the best-fit orbital solutions by Lovis and coworkers (2006). In the simulations, we show that the triplet Neptunian system can be stable at least for 2 Gyr and the stability would not be greatly influenced even if we vary the planetary masses. In addition, we employ the Laplace-Lagrange secular theory to investigate the long-term behaviors of the system, and the outcomes demonstrate that this theory can well describe the secular orbital evolution for all planets, where the secular periods and amplitudes in the eccentricities well agrees with those of the direct numerical integrations. We first reveal that the secular periods of the eccentricity $e_{1}$ and $e_{2}$ are identical about 8,300 yr. Moreover, we explore the planetary configuration of three Neptune-mass companions with one massive terrestrial planet in 0.07 AU $\leq a \leq 1.20$ AU, to examine the asteroid structure in this system. We underline that there are stable zones at least $10^{5}$ yr for low-mass terrestrial planets locating between 0.3 and 0.5 AU, and 0.8 and 1.2 AU with final low eccentricities. Still, we also find that the secular resonance $\nu_{1}$ and $\nu_{2}$ of two inner planets can excite the eccentricities of the terrestrial bodies, and the accumulation or depletion of the asteroid belt are also shaped by orbital resonances of the outer planets, i.e., 5:2 and 1:2 MMRs with Planet D... (abridged)Comment: 19 Pages, 3 figures, accepted for publication in Ap