Effective resonant stability of Mercury

Abstract

Mercury is the unique known planet that is situated in a 3:2 spin-orbit resonance nowadays. Observations and models converge to the same conclusion: the planet is presently deeply trapped in the resonance and situated at the Cassini state 1, or very close to it. We investigate the complete non-linear stability of this equilibrium, with respect to several physical parameters, in the framework of Birkhoff normal form and Nekhoroshev stability theory. We use the same approach adopted for the 1:1 spin-orbit case, published in Sansottera et al. (2014), with a peculiar attention to the role of Mercury's non negligible eccentricity. The selected parameters are the polar moment of inertia, the Mercury's inclination and eccentricity and the precession rates of the perihelion and node. Our study produces a bound to both the latitudinal and longitudinal librations (of 0.1 radians) for a long but finite time (greatly exceeding the age of the solar system). This is the so-called effective stability time. Our conclusion is that Mercury, placed inside the 3:2 spin-orbit resonance, occupies a very stable position in the space of these physical parameters, but not the most stable possible one.

Kolmogorov variation: KAM with knobs (à la Kolmogorov)

“…where K and Γ are two positive constants. Furthermore, the classical approach is the only way to directly implement KAM theory in practical applications and it proved advantageous in different contexts, e.g., the construction of lower dimensional elliptic tori in planetary systems in [43,44], the study of the long term dynamics of exoplanets in [24,42,45], the investigation of the effective stability in the spin-orbit problem in [40,41], the design of an a priori control for symplectic maps related to betatronic motion in [39] and the continuation of periodic orbits on resonant tori in [32,33,38]. In the present paper too, we adopt the classical approach, which turns out to be better suited in order to devise a normal form algorithm that introduces a detuning of the initial frequencies that will be determined, step-by-step, along the normalization procedure.…”

Exponential Stability in the Perturbed Central Force Problem

“…Here we will proceed by an explicit construction using a symbolic manipulator. We recall that explicit computations of Birkhoff normal form have already been implemented numerically in many situations (see, e.g., [SLG13], [GLS14], [SLL14], [SLL15] and [GLS17]).…”

Rigorous estimates for the relegation algorithm

“…Thus, using computer algebra in order to perform high-order perturbation expansions, one can produce estimates on the long-time stability for realistic models. For instance, see, e.g., [20] and [21] for the study of the effective resonant stability in the spin-orbit problem and [8], [22], [23] for the problem of the long-time stability of some of the giant planets of the Solar system. The paper is organised as follows.…”