Coupled rotational motion of Mercury

¹ Depart. of Mathematics, Facultés Univ. N.D. de la Paix, 8 Rempart de la Vierge, 5000 Namur, Belgium e-mail: Nicolas.Rambaux@fundp.ac.be
² Royal Observatory of Belgium, 3 Avenue Circulaire, 1180 Brussels, Belgium

Received: 30 October 2006
Accepted: 2 March 2007

Abstract

We present a simple dynamical model of the rotation of Mercury in which the Hermean rotation is composed of two commensurabilities: (i) a 3:2 spin-orbit resonance between fast variables and (ii) a 1:1 synchronous precession of both orbital and rotational nodes. We investigate the coupling between these two degrees of freedom. First, we study the global phase space of Mercury and quantify the libration areas. Second, we concentrate on the present location of Mercury. The impact of the slow degree of freedom on the fast one can be modeled through the adiabatic invariant, whereas the impact of the fast degree of freedom on the slow one is clearly represented by Poincaré sections. In addition, the adiabatic invariant theory leads to a simple analytical model of the rotation of Mercury where the two coupled degrees of freedom are taken into account. This model can be used in different applications that require a first-order rotational motion such as the one describing the influence of the precession and rotation of the planet on the orbit of an artificial satellite around Mercury.