On the Dynamics of Space Debris: 1:1 and 2:1 Resonances

We study the dynamics of the space debris in the 1:1 and 2:1 resonances, where geosynchronous and GPS satellites are located. By using Hamiltonian formalism, we consider a model including the geopotential contribution for which we compute the secular and resonant expansions of the Hamiltonian. Within such model we are able to detect the equilibria and to study the main features of the resonances in a very effective way. In particular, we analyze the regular and chaotic behavior of the 1:1 and 2:1 resonant regions by analytical methods and by computing the Fast Lyapunov Indicators, which provide a cartography of the resonances. This approach allows us to detect easily the location of the equilibria, the amplitudes of the libration islands and the main dynamical stability features of the resonances, thus providing an overview of the 1:1 and 2:1 resonant domains under the effect of Earth’s oblateness. The results are validated by a comparison with a model developed in Cartesian coordinates, including the geopotential, the gravitational attraction of Sun and Moon and the solar radiation pressure.

[1]  Elena Lega,et al.  On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems , 2002 .

[2]  R Jastrow,et al.  Satellite Orbits. , 1961, Science.

[3]  Alessandro Rossi,et al.  Collision risk against space debris in Earth orbits , 2006 .

[4]  Alessandro Rossi,et al.  Resonant dynamics of Medium Earth Orbits: space debris issues , 2008 .

[5]  Anne Lemaitre,et al.  Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including Earth’s shadowing effects , 2008 .

[6]  H. Klinkrad.  Space Debris: Models and Risk Analysis , 2006 .

[7]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[8]  K. Howell,et al.  Dynamics of artificial satellite orbits with tesseral resonances including the effects of luni- solar perturbations , 1997 .

[9]  Elena Lega,et al.  On the Structure of Symplectic Mappings. The Fast Lyapunov Indicator: a Very Sensitive Tool , 2000 .

[10]  C. Galeş.  A Cartographic Study of the Phase Space of the Elliptic Restricted Three Body Problem: Application to the Sun–Jupiter–Asteroid System , 2012 .

[11]  Florent Deleflie,et al.  Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence , 2009 .

[12]  M. O. Terra,et al.  Artificial satellites orbits in 2:1 resonance: GPS constellation , 2012 .

[13]  C. Galé.  A cartographic study of the phase space of the restricted three body problem , 2010 .

[14]  W. M. Kaula.  Theory of satellite geodesy , 1966 .

[15]  C. Chao,et al.  Applied Orbit Perturbation and Maintenance , 2005 .

[16]  E. Lega,et al.  The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems. , 2013, Chaos.

[17]  A. Celletti.  Stability and chaos in celestial mechanics , 2010 .

[18]  B. Melendo,et al.  Long-term predictability of orbits around the geosynchronous altitude , 2005 .

[19]  G. Beutler.  Methods of celestial mechanics , 2005 .

[20]  A. Lemaitre,et al.  The impact of Earth’s shadow on the long-term evolution of space debris , 2013 .

[21]  N. K. Pavlis,et al.  Earth Gravitational Model 2008 , 2008 .

[22]  A web of secondary resonances for large A/m geostationary debris , 2009 .

[23]  Alessandro Rossi,et al.  Semi-analytical investigations of the long term evolution of the eccentricity of Galileo and GPS-like orbits , 2011 .

[24]  E. Lega,et al.  FAST LYAPUNOV INDICATORS. APPLICATION TO ASTEROIDAL MOTION , 1997 .

[25]  R. A. Gick,et al.  Long-term evolution of navigation satellite orbits: GPS/GLONASS/GALILEO , 2004 .

[26]  Luciano Anselmo,et al.  Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios , 2008 .

[27]  N. Delsate,et al.  Global dynamics of high area-to-mass ratios GEO space debris by means of the MEGNO indicator , 2008, 0810.1859.