A numerical investigation of coorbital stability and libration in three dimensions

Motivated by the dynamics of resonance capture, we study numerically the coorbital resonance for inclination180 >=I>=0 in the circular restricted three-body problem. We examine the similarities and differences between planar and three dimensional coorbital resonance capture and seek their origin in the stability of coorbital motion at arbitrary inclination. After we present stability maps of the planar prograde and retrograde coorbital resonances, we characterize the new coorbital modes in three dimensions. We see that retrograde mode I (R1) and mode II (R2) persist as we change the relative inclination, while retrograde mode III (R3) seems to exist only in the planar problem. A new coorbital mode (R4) appears in 3D which is a retrograde analogue to an horseshoe-orbit. The Kozai-Lidov resonance is active for retrograde orbits as well as prograde orbits and plays a key role in coorbital resonance capture. Stable coorbital modes exist at all inclinations, including retrograde and polar obits. This result confirms the robustness the coorbital resonance at large inclination and encourages the search for retrograde coorbital companions of the solar system's planets.Comment: accepted for publication in Celestial Mechanics and Dynamical Astronom

Robust Power and Subcarrier Allocation for OFDM-based Cognitive Radio Networks Considering Spectrum Sensing Uncertainties

‎In this paper‎, ‎we address power and subcarrier allocation for cooperative cognitive radio (CR) networks in the presence of spectrum sensing errors‎. ‎First‎, ‎we derive the mutual interference of primary and secondary networks affecting each other by taking into account spectrum sensing errors‎. ‎Then‎, ‎taking into account the interference constraint imposed by the cognitive network to the primary user and the power budget constraint of cognitive network‎, ‎we maximize the achievable data rates of secondary users‎. ‎Besides‎, ‎in a multi secondary user scenario‎, ‎we propose a suboptimal but low complexity power and subcarrier allocation algorithm to solve the formulated optimization problem‎. ‎Our numerical results indicate that the proposed power loading scheme increases the cognitive achievable data rates compared to classical power loading algorithms that do not consider spectrum sensing errors‎