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

We analyse the global structure of the phase space of the planar planetary 2/1 mean-motion resonance in cases where the outer planet is more massive than its inner companion. Inside the resonant domain, we show the existence of two families of periodic orbits, one associated to the librational motion of resonant angle (sigma-family) and the other related to the circulatory motion of the difference in longitudes of pericentre (Delta pi-family). The well-known apsidal corotation resonances (ACR) appear as intersections between both families. A complex web of secondary resonances is also detected for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system. The construction of dynamical maps for various values of the total angular momentum shows the evolution of the families of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems. For low-moderate eccentricities, several different stable modes exist outside the ACR. For larger eccentricities, however, all stable solutions are associated to oscillations around the stationary solutions. Finally, we present a possible link between these stable families and the process of resonance capture, identifying the most probable routes from the secular region to the resonant domain, and discussing how the final resonant configuration may be affected by the extension of the chaotic layer around the resonance region.

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.

PAIN IN FIBROMYALGIA AND DISCRIMINATIVE POWER OF THE INSTRUMENTS: VISUAL ANALOG SCALE, DOLORIMETRY AND THE MCGILL PAIN QUESTIONNAIRE

Objective: The aim of this study was to verify the discriminative power of the most widely used pain assessment instruments. Methods: The sample consisted of 279 subjects divided into Fibromyalgia Group (FM- 205 patients with fibromyalgia) and Control Group (CG-74 healthy subjects), mean age 49.29 +/- 10.76 years. Only 9 subjects were male, 6 in FM and 3 in CG. FM were outpatients from the Rheumatology Clinic of the University of Sao Paulo - Hospital das Clinicas (HCFMUSP); the CG included people accompanying patients and hospital staff with similar socio-demographic characteristics. Three instruments were used to assess pain: the McGill Pain Questionnaire (MPQ), the Visual Analog Scale (VAS), and the Dolorimetry, to measure pain threshold on tender points (generating the TP index). In order to assess the discriminative power of the instruments, the measurements obtained were submitted to descriptive analysis and inferential analysis using ROC Curve - sensibility (S), specificity (S I) and area under the curve (AUC) - and Contingence tables with Chi-square Test and odds ratio. Significance level was 0.05. Results: Higher sensibility, specificity and area under the curve was obtained by VAS (80%, 80% and 0.864, respectively), followed by Dolorimetry (S 77%, S177% and AUC 0.851), McGill Sensory (S 72%, S167% and AUC 0.765) and McGill Affective (S 69%, S1 67% and AUC 0.753). Conclusions: VAS presented the higher sensibility, specificity and AUC, showing the greatest discriminative power among the instruments. However, these values are considerably similar to those of Dolorimetry.