Dynamic portrait of the planetary 2/1 mean-motion resonance - II. Systems with a more massive inner planet

This paper presents the second part in our study of the global structure of the planar phase space of the planetary three-body problem, when both planets lie in the vicinity of a 2/1 mean-motion resonance. While Paper I was devoted to cases where the outer planet is the more massive body, the present work is devoted to the cases where the more massive body is the inner planet. As before, outside the well-known Apsidal Corotation Resonances (ACR), the phase space shows a complex picture marked by the presence of several distinct regimes of resonant and non-resonant motion, crossed by families of periodic orbits and separated by chaotic zones. When the chosen values of the integrals of motion lead to symmetric ACR, the global dynamics are generally similar to the structure presented in Paper I. However, for asymmetric ACR the resonant phase space is strikingly different and shows a galore of distinct dynamical states. This structure is shown with the help of dynamical maps constructed on two different representative planes, one centred on the unstable symmetric ACR and the other on the stable asymmetric equilibrium solution. Although the study described in the work may be applied to any mass ratio, we present a detailed analysis for mass values similar to the Jupiter-Saturn case. Results give a global view of the different dynamical states available to resonant planets with these characteristics. Some of these dynamical paths could have marked the evolution of the giant planets of our Solar system, assuming they suffered a temporary capture in the 2/1 resonance during the latest stages of the formation of our Solar system.

First stereocontrolled acetylation of a hydroxypropargylpiperidone by lipase CALB

Hydroxypropargylpiperidones rac-1-3 were efficiently obtained by a one-pot three-component coupling reaction; enantioenriched propargylpiperidones were then obtained by a kinetic resolution process using the lipase from Candida antarctica. Lipase CALB has been shown to efficiently catalyse the stereocontrolled acetylation of hydroxypropargylpiperidones rac-3 by promoting stereodiscrimination at the carbinolic centre. The enzymatic catalytic processes allow the separation of the (S,R)- and (S,S)-3 diastereoisomers into the corresponding acetates produced as a (R,S)- and (R,R)-6 diastereoisomeric pair. The CALB was able to discriminate the stereogenic centre of the secondary (R)-enantiomer of rac-3 according to the Kaslauzkas rule. The remote stereogenic centre was not discriminated by the lipase. The functionalised enantioenriched diastereoisomers obtained are important building blocks in organic synthesis. (C) 2010 Elsevier Ltd. All rights reserved.