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Laplace-like resonances with tidal effects

We generalize the Laplace resonance among three satellites, S1, S2 , and S3, by considering different ratios of the mean-longitude variations. These resonances, which we call Laplace-like, are classified as first order in the cases of the 2:1&2:1, 3:2&3:2, and 2:1&3:2 resonances, second order in the case of the 3:1&3:1 resonance, and mixed order in the case of the 2:1&3:1 resonance. We consider a model that includes the gravitational interaction with the central body together with the effect due to its oblateness, the mutual gravitational influence of the satellites S1, S2, and S3 and the secular gravitational effect of a fourth satellite S 4 , which plays the role of Callisto in the Galilean system. In addition, we consider the dissipative effect due to the tidal torque between the inner satellite and the central body. We investigate these Laplace-like resonances by studying different aspects: (i) we study the survival of the resonances when the dissipation is included, taking two different expressions for the dissipative effect in the case of a fast- or a slowly rotating central body, (ii) we investigate the behavior of the Laplace-like resonances when some parameters are varied, specifically, the oblateness coefficient, the semimajor axes, and the eccentricities of the satellites, (iii) we analyze the linear stability of first-order resonances for different values of the parameters, and (iv) we also include the full gravitational interaction with S 4 to analyze its possible capture into resonance. The results show a marked difference between first-, second-, and mixed-order resonances, which might find applications when the evolutionary history of the satellites in the Solar System are studied, and also in possible actual configurations of extrasolar planetary systems.