A Motivating Exploration on Lunar Craters and Low-Energy Dynamics in the Earth -- Moon System

It is known that most of the craters on the surface of the Moon were created by the collision of minor bodies of the Solar System. Main Belt Asteroids, which can approach the terrestrial planets as a consequence of different types of resonance, are actually the main responsible for this phenomenon. Our aim is to investigate the impact distributions on the lunar surface that low-energy dynamics can provide. As a first approximation, we exploit the hyberbolic invariant manifolds associated with the central invariant manifold around the equilibrium point L_2 of the Earth - Moon system within the framework of the Circular Restricted Three - Body Problem. Taking transit trajectories at several energy levels, we look for orbits intersecting the surface of the Moon and we attempt to define a relationship between longitude and latitude of arrival and lunar craters density. Then, we add the gravitational effect of the Sun by considering the Bicircular Restricted Four - Body Problem. As further exploration, we assume an uniform density of impact on the lunar surface, looking for the regions in the Earth - Moon neighbourhood these colliding trajectories have to come from. It turns out that low-energy ejecta originated from high-energy impacts are also responsible of the phenomenon we are considering.Comment: The paper is being published in Celestial Mechanics and Dynamical Astronomy, vol. 107 (2010

Peaks and Jumps Reconstruction with B-Splines Scaling Functions

We consider a methodology based in B-splines scaling functions to numerically invert Fourier or Laplace transforms. The methodology is particularly well suited when the original function or its derivatives present peaks or jumps due to discontinuities in the domain

From manifolds to Lagrangian coherent structures in galactic bar models

We study the dynamics near the unstable Lagrangian points in galactic bar models using dynamical system tools in order to determine the global morphology of a barred galaxy. We aim at the case of non-autonomous models, in particular with secular evolution, by allowing the bar pattern speed to decrease with time. We have extended the concept of manifolds widely used in the autonomous problem to the Lagrangian coherent structures (LCS), widely used in fluid dynamics, which behave similar to the invariant manifolds driving the motion. After adapting the LCS computation code to the galactic dynamics problem, we apply it to both the autonomous and non-autonomous problems, relating the results with the manifolds and identifying the objects that best describe the motion in the non-autonomous case. We see that the strainlines coincide with the first intersection of the stable manifold when applied to the autonomous case, while, when the secular model is used, the strainlines still show the regions of maximal repulsion associated to both the corresponding stable manifolds and regions with a steep change of energy. The global morphology of the galaxy predicted by the autonomous problem remains unchanged

From manifolds to Lagrangian coherent structures in galactic bar models

We study the dynamics near the unstable Lagrangian points in galactic bar models using dynamical system tools in order to determine the global morphology of a barred galaxy. We aim at the case of non-autonomous models, in particular with secular evolution, by allowing the bar pattern speed to decrease with time. We have extended the concept of manifolds widely used in the autonomous problem to the Lagrangian coherent structures (LCS), widely used in fluid dynamics, which behave similar to the invariant manifolds driving the motion. After adapting the LCS computation code to the galactic dynamics problem, we apply it to both the autonomous and non-autonomous problems, relating the results with the manifolds and identifying the objects that best describe the motion in the non-autonomous case. We see that the strainlines coincide with the first intersection of the stable manifold when applied to the autonomous case, while, when the secular model is used, the strainlines still show the regions of maximal repulsion associated to both the corresponding stable manifolds and regions with a steep change of energy. The global morphology of the galaxy predicted by the autonomous problem remains unchanged

Warp evidence in precessing galactic bar models

Most galaxies have a warped shape when they are seen edge-on. The reason for this curious form is not completely known so far, so in this work we apply dynamical system tools to contribute to its explanation. Starting from a simple, but realistic model formed by a bar and a disc, we study the effect of a small misalignment between the angular momentum of the system and its angular velocity. To this end, a precession model was developed and considered, assuming that the bar behaves like a rigid body. After checking that the periodic orbits inside the bar continue to be the skeleton of the inner system even after inflicting a precession to the potential, we computed the invariant manifolds of the unstable periodic orbits departing from the equilibrium points at the ends of the bar to find evidence of their warped shapes. As is well known, the invariant manifolds associated with these periodic orbits drive the arms and rings of barred galaxies and constitute the skeleton of these building blocks. Looking at them from a side-on viewpoint, we find that these manifolds present warped shapes like those recognised in observations. Lastly, test particle simulations have been performed to determine how the stars are affected by the applied precession, this way confirming the theoretical results

Warp evidence in precessing galactic bar models

Most galaxies have a warped shape when they are seen edge-on. The reason for this curious form is not completely known so far, so in this work we apply dynamical system tools to contribute to its explanation. Starting from a simple, but realistic model formed by a bar and a disc, we study the effect of a small misalignment between the angular momentum of the system and its angular velocity. To this end, a precession model was developed and considered, assuming that the bar behaves like a rigid body. After checking that the periodic orbits inside the bar continue to be the skeleton of the inner system even after inflicting a precession to the potential, we computed the invariant manifolds of the unstable periodic orbits departing from the equilibrium points at the ends of the bar to find evidence of their warped shapes. As is well known, the invariant manifolds associated with these periodic orbits drive the arms and rings of barred galaxies and constitute the skeleton of these building blocks. Looking at them from a side-on viewpoint, we find that these manifolds present warped shapes like those recognised in observations. Lastly, test particle simulations have been performed to determine how the stars are affected by the applied precession, this way confirming the theoretical results

Application of high order expansions of two-point boundary value problems to astrodynamics

Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers