Celestial Mechanics

The lessons will cover the following topics: -Fundamental variables for rotational motion. -Andoyer variables. -Hamiltonian formulation. -Precession motion of the Earth and Moon. -Vectorial formulation. -Rotational motion of a perturbed planet.

Impact Monitoring is the procedure by which the possibility, for anyone of the known asteroids, of impacting the Earth within a finite time span is assessed. Lecture 1: Probabilistic interpetation of orbit determination. Error models for observations. Target planes, propagation of the probability density, Impact Probability (IP). Lacture 2: Line Of Variations (LOV) method for impact monitoring. Computation of the LOV, non-uniqueness. Trace of the LOV on target planes, showers and trails, finidng the minimum distance. Off-LOV check, computation of the IP. Lecture 3: Risk scales, extension of the time interval for impact monitoring. Dynamical uncertainty, non-gravitational perturbations, Yarkovsky effect. impact monitoring with non-gravitational perturbations, use of models and of a priori. Lecture 4: The problem of the imminent impactors. Historical cases. The Admissible region. The Manifold Of Variations (MOV). The target function on the MOV and propagation of the probability density. Data quality and IP. Imminent impactors and the announcement problem.

The ability to simulate the entry of space debris and asteroids is one of the critical aspects for the risk analysis and impact footprint determination processes. These lectures give a brief overview of the methods and tools currently used to predict the evolution of these objects during the passage through the atmosphere, from entry to impact or to destruction.

In this series of lectures we are going to present an overview of orbital dynamics of small solar-system bodies (in particular, asteroids) and artificial satellites. Emphasis will be given to the main dynamical mechanisms, affecting the long-term dynamical evolution of main-belt, Trojan and Near-Earth asteroids (NEAs). After presenting the fundamentals of the three-body problem and the structure of the disturbing function, we will discuss the construction of simplified models (averaged Hamiltonians and symplectic maps), using the theory of Lie transforms. We will aply these techniques to derive suitable models for studying secular dynamics as well as mean motion resonant dynamics of asteroids. With the help of these models we will review the main characteristics of the motion of main-belt asteroids, Trojans and NEAs and assess their long-term evolution. The effect of the non-conservative Yarkovsky force will also be examined, in the spirit of adiabatic theory. Finally, the same techniques will be applied to the problem of orbital motion of an artificial satellite, around a (non-spherical) primary with a non-trivial gravity field. Special techniques for finding long-term stable solutions will also be presented.

The analytical theory of close encounters allows us to establish relationships between the post-encounter orbital elements of the small body and the coordinates on the b-plane. We exploit these relationships in order to study various situations of interest in planetary science and in astrodynamics.

Since the beginning of space flight in 1957, humans have left a troubling legacy of garbage in space. Discarded rocket bodies, derelict satellites, and explosion debris now litter Earth orbit. These debris pose a risk to human space flight, robotic space flight, and even safety of humans on the ground. What do we know about this environment? What are we doing about it? This talk will answer these questions and discuss the path for the future.

We consider a natural extension of Newton's equations of the N-body problem to spaces of constant curvature. We first present some qualitative results regarding the motion of the bodies, focusing on relative equilibria and rotopulsators, which generalize the notion of homographic orbits from Euclidean to curved space. Then we write the equations in intrinsic coordinates and discuss the advantages and disadvantages of this approach. Finally we come up with a new and simple form of the equations that brings together the Euclidean case (of Gaussian curvature k=0), the hyperbolic case (of Gaussian curvature k.lt.0), and the elliptic case (of Gaussian curvature k.gt.0). Thus Newton's classical equations can be regarded in a broader context, namely that in which the motion of the bodies takes place in spaces of constant curvature. The equations of motion depend on the curvature k, and the Euclidean case is recovered when k=0. This conclusion could not be drawn from previously known forms of the equations of motion in curved space since taking k.to.0, for both k.gt.0 and k.lt.0, led to undetermined limits. This new form of the equations of motion allows the study of the classical case, k=0, in a larger framework and will help us better understand Newton's original approach.

The theory of adiabatic invariants has many applications in dynamical astronomy and plasma physics. It deals with the computation of approximate invariants of motion in oscillatory systems, when one of the frequencies can be considered as a small parameter. The classical theory, however, does not cover the case of resonances appearing in such systems. We will present a method to construct adiabatic invariants in cases of resonance by standard tools of canonical perturbation theory. The present method can be considered as a combination of two algorithmic techniques. These are: i) `book-keeping', i.e., a formal treatment of the splitting of the Hamiltonian in terms of different orders of smallness, ii) `detuning' (suggested by G. Pucacco and collaborators), i.e. the development of formal series in terms of a small parameter expressing the difference between near and exact resonance. As an example, we will construct resonant adiabatic invariants in a `magnetic bottle' Hamiltonian. Such Hamiltonians describe the `mirror' motions e.g. of charged particles in the magnetospheres of planets or in artificial magnetic traps. We will present a comparison of normal forms with numerical results. We will then discuss the asymptotic behavior, and finally the limits of applicability of the adiabatic normal form series. (Collaboration with G. Contopoulos, M. Harsoula)

Control theory is a branch of mathematics focusing on the problem of controlling, guiding systems on which one may act by means of a control, like a car, a space shuttle, or a chemical reaction or in more general any process that we aim at steering to a certain final desired target state. I will overview several potential applications of that theory and in particular I will focus on aerospace problems like the orbit transfer or interplanetary space mission design.

In this talk, I will present some recent results regarding the resonant structure as well as the stability and long term diffusion phenomena for hypothetical trojan companions to giant exoplanets. In particular, I will discuss a theoretical formalism appropriate for the study of secondary resonances in the space of proper elements (actions), defined for tadpole motions. I will also introduce some numerical results obtained within the framework of the elliptic restricted three body problem using i) FLI stability maps and ii) integrations of ensembles of orbits within the web of secondary resonances.

Since the beginning of space exploration a large number of space debris accumulated in the vicinity of the Earth, from near atmosphere to the geosynchronous region. The impact of operative spacecraft or satellites with large enough space debris could result in a dramatic situation. Understanding the overall orbital evolution of space debris is essential for maintenance and control strategies, as well as for space debris mitigation.

In this talk, we present a description of the dynamics of space debris in the 1:1 and 2:1 resonances by using the Hamiltonian formalism. We consider a model including the geopotential contribution and we compute the secular and the resonant parts of the Hamiltonian function for each resonance. Determining the leading terms of the expansions in specific resonant regions, we explain the main dynamical features of each resonance in a very effective way. Then, by computing the Fast Lyapunov Indicators, we provide a cartographic study of each resonance, yielding the regular and chaotic behavior of the 1:1 and 2:1 resonant regions. This study allows to determine easily the location of the equilibrium points, the amplitudes of the libration islands and the main dynamical stability features of each resonance. The results are validated by a comparison with a model developed in Cartesian coordinates, including the geopotential, the lunar and solar gravitational attractions and the solar radiation pressure. Joint work with Alessandra Celletti.