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. Apart from the well-known quasi-satellite orbits and the classical equilibrium Lagrangian points L(4) and L(5), we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at (delta lambda, delta pi) = (+/- 60 degrees, -/+ 120 degrees), where delta lambda is the difference in mean longitudes and delta pi 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 similar to 0.7. Finally, 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.

Conformational preferences for some 2-substituted N-methoxy-N-methylacetamides through spectroscopic and theoretical studies

The analysis of the IR carbonyl band of the 2-substituted N-methoxy-N-methylacetamides Y-CH(2)C(O)-N(OMe)Me (Y = F1, OMe 2, OPh 3, Cl 4), supported by B3LYP/6-311++G(3df, 3pd) calculations along with the NBO analysis for 1-4, indicated the existence of cis-gauche conformers i.e. (c) and (g) for 1 and 3, (c(1), c(2)) and (g(1), g(2)) for 2, and (c) and (g(1), g(2)) for 4. In the gas phase, the g conformer population prevails over the c one, for 1 and 3, the (c(1) + c(2)) population prevails over the (g(1) + g(2)) one for 2, and the (g(1) + g(2)) conformer population is more abundant than (c) one for 4. In n-hexane solution, the cis conformer is more abundant for 1-3. The occurrence of Fermi resonance in the nu(CO) region, in n-hexane, precludes the estimative of relative populations of the (c, g(1), g(2)) conformers for 4. The SCI-PCM calculations agree with the solvent effect on the nu(CO) band component relative intensities for 1-3. NBO analysis showed that the n(N) -> pi.(CO), orbital interaction is the main factor which stabilizes the gauche (g, g(1), g(2)) conformers for 1-4 into a larger extent relative to the cis (c, c(1), c(2)) ones. The n(y) -> pi(.)(Co,) sigma(C-Y) -> pi.(CO,) pi(CO) -> sigma(C-Y) and 7co orbital interactions still contribute, but into a minor extent for the stabilization of the gauche conformers relative to the cis ones. The existence of some pyramidalization at the nitrogen atom of the Weinreb amides 1-4 is responsible for the occurrence of Y(delta)-(4)center dot center dot center dot O(delta)-(9) and Y(delta)-(4)center dot center dot center dot N(delta)-(7) short contacts in the gauche (g, g(1), g(2)) conformers, which originates strong repulsive Coulombic interactions, acting in opposition to the large orbital stabilization of the gauche conformer with respect to the cis one. Therefore, a delicate balance of the Coulombic and orbital interactions seems to be responsible for the observed stabilization of the gauche (g, g(1), g(2)) and cis (c, c(1), c(2)) conformers, both in the gas phase and in the solution for 1-4. However, the cis conformer predominance, in non polar solvents, for the 2-substituted N-methoxy-N-methyl acetamides 1-3, bearing in a first raw (fluorine and oxygen) atoms, is in the opposite direction to the gauche conformer preference for the corresponding 2-substituted N,N-dialkyl-acetamides. (C) 2010 Elsevier B.V. All rights reserved.

Magnetism of Co overlayers and nanostructures on W(001): A first-principles study

The magnetic properties of Co nanostructures and a Co monolayer on W(001) have been studied in the framework of density functional theory. Different geometries such as planar and three-dimensional clusters have been considered, with cluster sizes varying between 2 and 13 atoms. The calculations were performed using the real-space linear muffin-tin orbital method (RS-LMTO-ASA). With respect to the stability of the magnetic state, we predict an antiferromagnetic (AFM) structure for the ground state of the planar Co clusters and a ferromagnetic (FM) state for the three-dimensional clusters. For the three-dimensional clusters, one of the AFM arrangements leads to frustration due to the competing FM and AFM exchange interactions between different atoms in the cluster, and gives rise to a non-collinear state with energy close to that of the FM ground state. The relative role of the Co-Co and Co-W exchange interactions is also investigated. (C) 2007 Elsevier B.V. All rights reserved.