Hello everybody! Welcome to my personal page! Here begins a series of posts related to my previous and recently work.
I would like to start with a common, but complex problem: the orbital stability of the planar circular restricted three-body problem. Assume a massless particle orbiting a central body and perturbed by a massive body, for example can be an asteroid or moon, a star or the Sun, and a planet or Jupiter, respectively. Through a dynamic map I want to make a portrait of how stable are the orbits for different test particles depending on the initial eccentricity and semi-major axis.
The map generated is a dynamical map of Δe for 2048 x 576 initial conditions in the (a, e) plane of osculating elements with Jupiter as the perturber and the Sun as central body. All angles were taken to be equal to zero and the total integration time was set to T = 1000 orbits of the perturber, not sufficient to eject most of the unstable orbits but sufficient to slow-growing in the eccentricity. In the figure you can see the first mean motion resonance (MMR) 2:1 (left), 3:2 (middle) and 4:3 (right) in black, and the third mean motion resonance between 2:1 and 3:2, corresponding to 7:4 and 8:5. Also, you can observe a beautiful unstable region in the right. The zones of maximum variation of the eccentricity appear in yellow-orange (unstable orbits), while those associated with medium change are indicated in purple (both branches of the separatrix of 2:1 MMR for example), and finally, small change are indicated in black (stable orbits).
All initial conditions were integrated with a Bulirsch-Stoer integrator. This map has 1179648 independent initial conditions evolved in parallel.
This figure is a beautiful example to begin to better understand the stability of these systems and the beginning of resonance overlap and how this generates a large unstable region.