A note on Hansen's coefficients in satellite theory

Giacáglia, Giorgio Eugenio Oscare

doi:10.1007/bf01229062

Cited by 45 publications

(20 citation statements)

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“…We consider the expansion up to degree 2 in a/a M and we retain only the terms of the expansion which provide the average over the fast orbital phases (i.e., the mean anomalies of both the debris and the Moon). In (2.3) the following notation has been introduced: H 2 p 2p−2 (e) and G 210 (e M ) are, respectively, the Hansen coefficients X 2,2−2p 0 (e) and X −3,0 0 (e M ) = (1 − e M ) −3/2 (see [25]); [·] denotes the integer part; K 0 = 1, otherwise K m = 2 if m = 0; F 2mp (i) is the Kaula inclination function ( [32], see also [12]) that we define for the generic case F nmp (i) as…”

Section: 1mentioning

confidence: 99%

Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Celletti¹,

Galeş²,

Pucacco³

2016

SIAM J. Appl. Dyn. Syst.

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Using bifurcation theory, we study the secular resonances induced by the Sun and Moon on space debris orbits around the Earth. In particular, we concentrate on a special class of secular resonances, which depend only on the debris' orbital inclination. This class is typically subdivided into three distinct types of secular resonances: those occurring at the critical inclination, those corresponding to polar orbits, and a third type resulting from a linear combination of the rates of variation of the argument of perigee and the longitude of the ascending node.The model describing the dynamics of space debris includes the effects of the geopotential, as well as the Sun's and Moon's attractions, and it is defined in terms of suitable action-angle variables. We consider the system averaged over both the mean anomaly of the debris and those of the Sun and Moon. Such multiply-averaged Hamiltonian is used to study the lunisolar resonances which depend just on the inclination.Borrowing the technique from the theory of bifurcations of Hamiltonian normal forms, we study the birth of periodic orbits and we determine the energy thresholds at which the bifurcations of lunisolar secular resonances take place. This approach gives us physically relevant information on the existence and location of the equilibria, which help us to identify stable and unstable regions in the phase space. Besides their physical interest, the study of inclination dependent resonances offers interesting insights from the dynamical point of view, since it sheds light on different phenomena related to bifurcation theory.2010 Mathematics Subject Classification. 37N05,37L10,70F15.

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Section: 1mentioning

confidence: 99%

Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Celletti¹,

Galeş²,

Pucacco³

2016

SIAM J. Appl. Dyn. Syst.

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“…the functions H lpq (e) and G lhj (e S ) are the Hansen coefficients X l,l−2p l−2p+q (e), X −(l+1),l−2h l−2h+j (e S ) (see [32]), while the functions F lmp (i) and F lmh (i S ) are given in (3.4).…”

Section: 2mentioning

confidence: 99%

Dynamical models and the onset of chaos in space debris

Celletti

Efthymiopoulos

Gachet

et al. 2017

International Journal of Non-Linear Mechanics

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The increasing threat raised by space debris led to the development of different mathematical models and approaches to investigate the dynamics of small particles orbiting around the Earth. The choice of such models and methods strongly depend on the altitude of the objects above Earth's surface, since the strength of the different forces acting on an Earth orbiting object (geopotential, atmospheric drag, lunar and solar attractions, solar radiation pressure, etc.) varies with the altitude of the debris.\ud \ud In this review, our focus is on presenting different analytical and numerical approaches employed in modern studies of the space debris problem. We start by considering a model including the geopotential, solar and lunar gravitational forces and the solar radiation pressure. We summarize the equations of motion using different formalisms: Cartesian coordinates, Hamiltonian formulation using Delaunay and epicyclic variables, Milankovitch elements. Some of these methods lead in a straightforward way to the analysis of resonant motions. In particular, we review results found recently about the dynamics near tesseral, secular and semi-secular resonances.\ud \ud As an application of the above methods, we proceed to analyze a timely subject, namely the possible causes for the onset of chaos in space debris dynamics. Precisely, we discuss the phenomenon of overlapping of resonances, the effect of a large area-to-mass ratio, the influence of lunisolar secular resonances.\ud \ud We conclude with a short discussion about the effect of the dissipation due to the atmospheric drag and we provide a list of minor effects, which could influence the dynamics of space debris

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“…Next, we use the following expansion in terms of Hansen's coefficients X n,m r (e) (Cherniack, 1972;Giacaglia, 1976):…”

Section: )mentioning

confidence: 99%

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

Celletti

Galeş

Pucacco

et al. 2016

Celest Mech Dyn Astr

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We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula's expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane's expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time.We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.

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A note on Hansen's coefficients in satellite theory

Cited by 45 publications

References 1 publication

Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Dynamical models and the onset of chaos in space debris

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

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