Shadow of a Schwarzschild black hole surrounded by a Bach–Weyl ring

We have studied the shadows of a Schwarzschild black hole surrounded by a Bach–Weyl ring through the backward ray-tracing method. The presence of Bach–Weyl ring leads to that the photon dynamical system is non-integrable and then chaos would appear in the photon motion, which affects sharply the black hole shadow. The size and shape the black hole shadow depend on the black hole parameter, the Bach–Weyl ring mass and the Weyl radius between black hole and ring. Some self-similar fractal structures also appear in the black hole shadow, which originates from the chaotic lensing. We also study the change of the image of Bach–Weyl ring with the ring mass and the Weyl radius. Finally, we analyze the invariant manifolds of Lyapunov orbits near the fixed points and discuss further the formation of the shadow of a Schwarzschild black hole with Bach–Weyl ring.

Invariant Manifolds and the Capture of Comet Shoemaker-Levy 9★

The impact of Comet Shoemaker-Levy 9 (SL9) with Jupiter in 1994 was the ultimate confirmation of Eugene Shoemaker’s theory that impacts are a common, fundamental process in the Solar System. On Earth, asteroid impacts have produced several near extinction level events. We tend to visualise these collisions as billiard balls hitting one another. But how, exactly, do they occur? The actual dynamics is much more complex and subtle because it is highly non-linear and involves chaos in the three-body problem. In this paper we investigate the effects of Lyapunov orbits, halo orbits, and their associated invariant manifolds, on the orbital motion of SL9. We demonstrate that periodic orbits act as gateways to Jupiter, and their invariant manifolds controlled the dynamics of SL9 during the initial capture phase.

Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors

The purpose of this paper is to apply Lagrangian Descriptors, a concept used to describe phase space structure, to autonomous Hamiltonian systems with two degrees of freedom in order to detect periodic solutions. We propose a method for Hamiltonian systems with saddle-center equilibrium and apply this approach to the classical Hénon–Heiles system. The method was successful in locating the unstable Lyapunov orbits in phase space.

Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with L2-minimization

In this work, we develop a new method to design energy minimum low-thrust missions (L2-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multiple shooting method in order to release the constraint. We finally optimize the terminal points on the periodic orbits. Moreover, we use continuation methods on position and on thrust, in order to gain robustness. A more general Halo to Halo mission, with different energies, is computed in the last section without heteroclinic orbits but using invariant manifolds to initialize shooting methods with a similar approach.

Connecting interacting galaxies with manifolds

It is well known that the interaction between two disk galaxies generates tidal spiral arms and a connection in the form of a bridge. Here we address the question of the formation of tidal arms and bridges from a dynamical point of view. We model the bridges and tails observed in interacting galaxies using the invariant manifolds associated to the Lyapunov orbits of the Lagrangian points of the galactic system, when the two galaxies are considered as two point masses in a circular orbit.