On the Arnold diffusion mechanism in Medium Earth Orbit

Abstract

Space debris mitigation guidelines represent the most effective method to preserve the circumterrestrial environment. Among them, end-of-life disposal solutions play a key role. A growing effort is devoted to exploit natural perturbations to lead the satellites towards an atmospheric reentry, reducing the disposal cost, also if departing from high-altitude regions. In the case of the Medium Earth Orbit region, home of the navigation satellites (like Galileo), the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude can get into the atmospheric drag domain and the satellite can eventually reenter. In this work, we show how an Arnold diffusion mechanism can trigger the eccentricity growth. Focusing on the case of Galileo, we consider a hierarchy of Hamiltonian models, assuming that the main perturbations on the motion of the spacecraft are the oblateness of the Earth and the gravitational attraction of the Moon. First, the Moon is assumed to lay on the ecliptic plane and periodic orbits and associated stable and unstable invariant manifolds are computed for various energy levels, in the neighborhood of a given resonance. Along each invariant manifold, the eccentricity increases naturally, achieving its maximum at the first intersection between them. This growth is, however, not sufficient to achieve reentry. By moving to a model where the inclination of the Moon is taken into account, the problem becomes non-autonomous and the satellite is able to move along different energy levels. Under the ansatz of transversality of the manifolds in the autonomous case, checked numerically, Poincaré-Melnikov techniques are applied to show how diffusion can be attained, by constructing a sequence of homoclinic orbits that connect invariant tori at different energy levels.

Figures (13)

• Figure 1: The normally hyperbolic invariant cylinder and its invariant manifolds. In the left figure $i_{\mathrm{M}}=0$ and therefore $J$ is a first integral. The cylinder is foliated by invariant tori whose invariant manifolds intersect transversally within $J=\text{constant}$ giving rise to a homoclinic channel. In the right figure, $i_{\mathrm{M}}>0$ (small enough). Then, $J$ is not a first integral anymore and one can construct heteroclinic connections between different tori in the cylinder (diamond intersections in the picture). See Corollary \ref{['corollary:NHIMiM0']} and Theorem \ref{['theorem:invariantObjectsPerturbed']}, respectively, for a detailed description of the notation.
• Figure 2: The pseudo-orbit (left) and the shadowing orbit (right): The blue orbit shadows (follows closely) the pseudo-orbit formed by heteroclinic orbits connecting different points in the cylinder and pieces of cylinder orbits.
• Figure 3: Examples of periodic orbits for $i_M=0$, non-dimensional units. The colorbar reports the value of ${\mathcal{H}}_{\mathrm{CP}}$. Here it is shown the behavior of a range of energies larger than the one we are interested in.
• Figure 4: The value of the eigenvalue greater than 1, $\lambda$, as a function of the ${\mathcal{H}}_{\mathrm{CP}}$ for the hyperbolic periodic orbits.
• Figure 5: The period (left) $\mathcal{T}$ of the hyperbolic periodic orbits as a function of ${\mathcal{H}}_{\mathrm{CP}}$. On the right, $n_{\mathrm{Saros}} \mathcal{T}$ (where $n_{\mathrm{Saros}}\equiv n_{\Omega_{\mathrm{M}}}$) as a function of ${\mathcal{H}}_{\mathrm{CP}}$.

Theorems & Definitions (30)

• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 4.1
• Theorem 4.2
• Definition 4.3
• Definition 4.4
• Remark 4.5
• Corollary 5.2
• Corollary 5.3