Some results on the global dynamics of the regularized restricted three-body problem with dissipation

Abstract

International audienceWe perform an analysis of the dynamics of the circular, restricted, planar three-body problem under the effect of different kinds of dissipation (linear, Stokes and Poynting-Robertson drags). Since the problem is singular, we implement a regularization technique in the style of Levi-Civita. The effect of the dissipation is often to decrease the semi-major axis; as a consequence the minor body collides with one of the primaries. In general, it is quite difficult to find non-collision orbits using random initial conditions. However, by means of the computation of the Fast Lyapunov Indicators (FLI), we obtain a global view of the dynamics. Precisely, we detect the regions of the phase space potentially belonging to basins of attraction. This investigation provides information on the different regions of the phase space, showing both collision and non-collision trajectories. Moreover, we find periodic orbit attractors for the case of linear and Stokes drags, while in the case of the Poynting-Robertson effect no other attractors are found beside the primaries, unless a fourth body is added to counterbalance the dissipative effect

Restricted Three Body Problem with Stokes Drag Effect

“…The unit of time is chosen so as to make the gravitational constant unity. The equations of motion of the infinitesimal mass 3 m in the synodic coordinate system (x, y) and dimensionless variables are …”

Periodic motions in a gravitational central field with a rotating external force

“…On the contrary, we will show that, when α = 0, the rotating term αe iωt /|x| β forces the particle to maintain an almost constant rotation speed, and to remain at an almost constant distance from the singularity. Detailed information about the role of dissipative effects on Celestial Mechanics and the most important mechanisms of dissipation can be found in [8,23], and the references therein.…”

“…If ε = 0, the linearized system at the equilibrium point E 1 is always non resonant, i.e., the eigenvalues of A 1 are never of the type ik, with k ∈ Z. Therefore, since, for j = 1, 2, the functions Γ j (s, ρ, η, ϕ, v; ε) are locally Lipschitz continuous in (ρ, η, ϕ, v), a classical perturbation theorem applies (see, e.g., [10, Chapter 14, Theorem 1.1]), providing the existence of a 2π-periodic solution of (8), for ε small enough. Correspondingly, by the changes of variables made above, we have a T -periodic solution of (1).…”

Transfers to distant periodic orbits around the Moon via their invariant manifolds

“…The controlled RTBP can be seen as a special Hamiltonian system with dissipation [2]. The equations refer to the planar, circular, restricted threebody problem, expressed in a synodic reference frame rotating with the angular velocity of the primaries.…”