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Adiabatic approximation in studies of mean motion resonances in celestial mechanics

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In the last decades of the XXth century, it became clear that when studying resonant effects in dynamics of celestial bodies, it is useful to pay attention to the behavior of approximate integrals of motion called adiabatic invariants (Henrard and Lemaitre 1983; Wisdom 1985). The standard scheme of the adiabatic approximation in the investigations of mean motion resonances (MMR) as a first step involves averaging over the fastest dynamic process, i.e. over the orbital motion of the objects in commensurability. In averaged equations of motion one should take a subsystem that describes the process of “intermediate” time scale – the variation of the resonant angle. This subsystem can be interpreted as a Hamiltonian system with one degree of freedom, depending on other variables as slowly varying parameters. Consequently, the value of the “action” variable for this subsystem will be an adiabatic invariant (AI). Studying then the properties of level surfaces of AI in the subspace of the slowest variables, we can draw conclusions about the secular evolution of the orbits of celestial bodies in MMR . More delicate situation arises in the case of nonuniqueness of the resonant modes allowed by the system (Sidorenko et al. 2014; Sidorenko 2018). In particular, in this case it is necessary to identify regions in the phase space, where resonant modes can coexist, to compare the probabilities of the capture into different modes, and to analyze the possibility of a transition between these modes. Fortunately, the theory of AI allows to do almost all of this. All phenomena under the discussion are illustrated by examples of their possible implementation in the dynamics of real celestial bodies.

This talk is part of the DAMTP Astro Mondays series.

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