Dynamical investigation of minor resonances for space debris

We study the dynamics of the space debris in regions corresponding to minor resonances; precisely, we consider the resonances 3:1, 3:2, 4:1, 4:3, 5:1, 5:2, 5:3, 5:4, where a j:ℓ resonance (with j, ℓ∈Z) means that the periods of revolution of the debris and of rotation of the Earth are in the ratio j/ℓ. We consider a Hamiltonian function describing the effect of the geopotential and we use suitable finite expansions of the Hamiltonian for the description of the different resonances. In particular, we determine the leading terms which dominate in a specific orbital region, thus limiting our computation to very few harmonics. Taking advantage from the pendulum-like structure associated to each term of the expansion, we are able to determine the amplitude of the islands corresponding to the different harmonics. By means of simple mathematical formulae, we can predict the occurrence of splitting or overlapping of the resonant islands for different values of the parameters. We also find several cases which exhibit a transcritical bifurcation as the inclination is varied. These results, which are based on a careful mathematical analysis of the Hamiltonian expansion, are confirmed by a numerical study of the dynamical behavior obtained by computing the so-called fast Laypunov indicators. Since the Hamiltonian approach includes just the effect of the geopotential, we validate our results by performing a numerical integration in Cartesian variables of a more complete model including the gravitational attraction of Sun and Moon, as well as the solar radiation pressure.