![]() ![]() ![]() We extend the 2D planar geometry to calculate the obliquity evolution. By analyzing the resonances individually and collectively, we determine the criteria for resonance overlap and the onset of chaos, as a function of orbital and geometric properties of the binary. After calculating the net gravitational torque on the satellite from the primary, as well as the associated equations of motion, we employ a Hamiltonian formalism that allows for a perturbative treatment of the spin–orbit and retrograde and prograde spin–spin coupling states. In this work, we adopt a simplified model of a gravitationally interacting primary and satellite pair, where each body's quadrupole moment is approximated by two diametrically opposed point masses. The ability of an object to sustain an aspheroidal shape largely determines whether or not it will exhibit nontrivial rotational behavior. Permanently deformed objects in binary systems can experience complex rotation evolution, arising from the extensively studied effect of spin–orbit coupling as well as more nuanced dynamics arising from spin–spin interactions. All our results are gathered in the CPM-ASTEROID database, which will be regularly updated by considering other asteroids. At last, we present an example of a successful operation of our orbit control with a total v of 0.495 m/s for 60 days. Finally, to solve the stabilization problem in the system, we apply a robust path following control law to control the orbital geometry of a spacecraft. However, no regions investigated could be a possible option for inserting a spacecraft into natural orbits around Apophis during the close approach with our planet. Moreover, from numerical analysis of orbits started on March 1, 2029, the less perturbed region is characterized by the variation of the semimajor axis of 40-day orbits, which do not exceed 2 km very close to the central body (a < 4 km, e < 0.4). The Earth is the one that most affects the investigated region making the vast majority of the orbits collide or escape from the system. The surfaces of section close to Apophis are build considering or not the gravitational perturbations of the Sun, the planets, and the SRP. That considerably reduces the computation processing time compared to previous methods to evaluate the gravitational potential. The physical properties from the polyhedral shape of the target are derived by assigning each tetrahedron to a point mass in its center. We aim at providing a preliminary approach on the dynamics of a spacecraft in orbit about the asteroid (99942) Apophis during its Earth close approach. We found good agreement between the two models. We considered the gravitational field of the asteroid, the gravitational attraction of the Sun, and the solar radiation pressure Our results are then compared to the results obtained using the Mascon gravitational model based on the polyhedral shape source. We investigated an initial inclination of 0 degree (direct orbits) and 180 degrees (retrograde orbits). Also, this model is used to carry out simulations to characterize regions with solutions that remain bounded or that escape from around each asteroid under analysis. The results show that the current triple-particle-linkage three-dimensional model gives better accuracy when compared to the axisymmetric triple-particle-linkage model available in the literature, and provides an advantage in terms of accuracy over the mass point model, while keeping computational time low. The model considered in this article is then applied to three real irregular asteroids 1620 Geographos, 433 Eros and 243, Ida. A non-linear optimization method is used to determine the parameters of our model, minimizing the errors of all the external equilibrium points with respect to the solutions calculated with a more realistic model, the polyhedron model, which are assumed to be the real values of the system. It is an extension of previous models that focused only on planar models. The proposed model consists of representing an elongated asteroid using a triple-particle-linkage system distributed in the three-dimensional space. The goal of the present paper is to develop a simplified model to describe the gravitational fields of elongated asteroids. ![]()
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