The purpose of this undergraduate research project is to guide the student on topics of celestial mechanics and applications on the problem of rigid body rotation and resonance "spin-orbit". Emphasis will be given on the formulation of the problem and applications to some planets with physical parameters (e.g. radius and mass) consistent with formation of a terrestrial type (e.g. rock) which belong to multiple planet systems. As an example we study the planets CoRoT-7 b, Kepler-9 d, 55 Cnc e, Gliese 876 d, and the newly discovered Kepler-10 b, Kepler-11 b. The methodology is both analytic and numeric. The analytical part consists of: i) deduction of the equation of motion of the problem of rotation of a non-spherical body with symmetry, perturbed by a central point; ii) modeling the same problem through the inclusion of the third body in the star-planet system; iii) formulation of the concept of resonance "spin-orbit," in which the planet's orbital period is a multiple of its period of rotation. Topics on dynamic systems (e.g., equilibrium points, chaos, surfaces of section) will be initiated in this step. Tthe numerical simulations are performed with numerical models developed in the analytical part. The main efforts will be focused on: i) numerical integration of differential equations, ii) construction of surfaces of section (s.s.). In a first step we consider the planet's orbit not disturbed by a third body in the star-planet system. In this case the orbital eccentricity and semi-major axis of the planet is constant (consider the planar case). At this stage the technique of s.s., widely used in dynamic system will be applied to our problem. Then the elements of the planet will vary due to the action of a third body. Thus, in this work we are developing a more realistic model of planetary rotation (and original one), than those written in the field of two bodies. The results in both cases are compared. Since the technique of s.s. is no longer applicable in the disturbed case, we quantitatively analyze the evolution of the characteristic angles of rotation (e.g. physical libration) studying individual orbits in dynamically important regions of phase space obtained in the case undisturbed.
News published in Agência FAPESP Newsletter about the scholarship: