Dynamics of resonances and equilibria of Low Earth Objects

The nearby space surrounding the Earth is densely populated by artificial satellites and instruments, whose orbits are distributed within the Low-Earth-Orbit region (LEO), ranging between 90 and 2 000 $km$ of altitude. As a consequence of collisions and fragmentations, many space debris of different sizes are left in the LEO region. Given the threat raised by the possible damages which a collision of debris can provoke with operational or manned satellites, the study of their dynamics is nowadays mandatory. This work is focused on the existence of equilibria and the dynamics of resonances in LEO. We base our results on a simplified model which includes the geopotential and the atmospheric drag. Using such model, we make a qualitative study of the resonances and the equilibrium positions, including their location and stability. The dissipative effect due to the atmosphere provokes a tidal decay, but we give examples of different behaviors, precisely a straightforward passage through the resonance or rather a temporary capture. We also investigate the effect of the solar cycle which is responsible of fluctuations of the atmospheric density and we analyze the influence of Sun and Moon on LEO objects.Comment: 39 pages, 10 figure

A study of the lunisolar secular resonance 2ω˙+Ω˙=02\dot{\omega}+\dot{\Omega}=0

The dynamics of small bodies around the Earth has gained a renewed interest, since the awareness of the problems that space debris can cause in the nearby future. A relevant role in space debris is played by lunisolar secular resonances, which might contribute to an increase of the orbital elements, typically of the eccentricity. We concentrate our attention on the lunisolar secular resonance described by the relation $2\dot{\omega}+\dot{\Omega}=0$, where $\omega$ and $\Omega$ denote the argument of perigee and the longitude of the ascending node of the space debris. We introduce three different models with increasing complexity. We show that the growth in eccentricity, as observed in space debris located in the MEO region at the inclination about equal to $56^\circ$, can be explained as a natural effect of the secular resonance $2\dot{\omega}+\dot{\Omega}=0$, while the chaotic variations of the orbital parameters are the result of interaction and overlapping of nearby resonances.Comment: 15 pages, 8 figure

Stability of nearly-integrable systems with dissipation

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, named the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and exponential stability times, which are established under smallness con- ditions on the parameters. We also provide some applications to concrete examples, which exhibit a linear or exponential stability behavior.Comment: 38 page

A study of the main resonances outside the geostationary ring

We investigate the dynamics of satellites and space debris in external resonances, namely in the region outside the geostationary ring. Precisely, we focus on the 1:2, 1:3, 2:3 resonances, which are located at about 66 931.4 km, 87 705.0 km, 55 250.7 km, respectively. Some of these resonances have been already exploited in space missions, like XMM-Newton and Integral. Our study is mainly based on a Hamiltonian approach, which allows us to get fast and reliable information on the dynamics in the resonant regions. Significative results are obtained even by considering just the effect of the geopotential in the Hamiltonian formulation. For objects (typically space debris) with high area-to-mass ratio the Hamiltonian includes also the effect of the solar radiation pressure. In addition, we perform a comparison with the numerical integration in Cartesian variables, including the geopotential, the gravitational attraction of Sun and Moon, and the solar radiation pressure. We implement some simple mathematical tools that allows us to get information on the terms which are dominant in the Fourier series expansion of the Hamiltonian around a given resonance, on the amplitude of the resonant islands and on the location of the equilibrium points. We also compute the Fast Lyapunov Indicators, which provide a cartography of the resonant regions, yielding the main dynamical features associated to the external resonances. We apply these techniques to analyze the 1:2, 1:3, 2:3 resonances; we consider also the case of objects with large area-to-mass ratio and we provide an application to the case studies given by XMM-Newton and Integral.Comment: 30 pages, 10 figure

The effect of Poynting-Robertson drag on the triangular Lagrangian points

We investigate the stability of motion close to the Lagrangian equilibrium points L4 and L5 in the framework of the spatial, elliptic, restricted three- body problem, subject to the radial component of Poynting-Robertson drag. For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to beta a1 n1 , where beta is the ratio of the solar radiation over the gravitational force, and a1, n1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray (1994)) on the asymmetry of the stability indices of L4 and L5 to a more realistic force model. Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.Comment: 47 pages, 8 figures

Halo orbits around the collinear points of the restricted three-body problem

We perform an analytical study of the bifurcation of the halo orbits around the collinear points $L_1$, $L_2$, $L_3$ for the circular, spatial, restricted three--body problem. Following a standard procedure, we reduce to the center manifold constructing a normal form adapted to the synchronous resonance. Introducing a detuning, which measures the displacement from the resonance and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio. In most cases, the analytical results thus obtained are in very good agreement with the numerical expectations, providing the bifurcation threshold with good accuracy. Care must be taken when dealing with $L_3$ for small values of the mass-ratio between the primaries; in that case, the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold.Comment: 35 pages, 3 figures, updated version accepted for publication on Physica

Some stability results on nearly–integrable systems (with dissipation)

The stability of nearly–integrable systems can be studied over different time scales and with different techniques. In this paper we review some classical methods, like the averaging technique, the classical perturbation theory, KAM theorem and Nekhoroshev’s stability for exponential times. We investigate also conformally symplectic systems, in particular nearly–integrable systems with dissipation, and we present some results about KAM and exponential stability in the dissipative context