Sweeping through a second order resonance

The adiabatic invariant theory and its extension to the case of separatrix crossing have been instrumental in setting up simple models describing the effects of the passage through a first order resonance forced by small non conservative forces. These models have been helpful in understanding the processes of capture into resonance and of formation of gaps at resonances. Second order resonances (3/1 or 5/3 ...) have not been so thoroughly investigated and present some special features. We shall show how to disencumber first the problem from its secular term and how to modelize the mixed resonance term involving the sum of the resonant angles. A modelization of the two single satellite resonant terms has already been proposed by Lemaître and Borderies and Goldreich. We shall then assemble those three simple models to show the effect of a passage through the resonance under the assumption that the natural precession rates are large enough for the three individual resonances to be sufficiently separated.