Spin-orbit coupling in the solar system

Weakly Dissipative Systems in Celestial Mechanics

“…Using the relations cos δ =r ·r T , sin δ =r T ∧r, we obtain that MacDonald's torque takes the expression (see, e.g., [16])…”

“…Among the dissipative terms, the strongest contribution is due to the internal non-rigidity of the satellite. This tidal torque may assume different mathematical formulations; among the others we quote the classical MacDonald's ( [24]) and Darwin's ( [11]) torques which are reviewed in the following subsections (see [16], [26]). Let us summarize by saying that MacDonald expression assumes a phase lag depending linearly on the angular velocity, while Darwin's formulation Fourier decomposes the tidal potential, assigning to each component a constant amplitude.…”

“…Bearing in mind the discussions about the capture into resonance for the dissipative standard map, we proceed to illustrate some classical results (see [16], [27]) about resonance capture in the spin-orbit problem. We will see that such event strongly depends on the form of the dissipation.…”

“…We plot in Figure 8 the behavior of 1 2γ 2 versus γ as derived from (8) (see [16]). We denote by γ max the point at whichγ = 0.…”

“…In particular, we focus on the spin-orbit interaction for which we present some explicit expressions of dissipative forces known as MacDonald's and Darwin's torques. In this context we discuss the occurrence of capture into resonance, which depends on the specific form of the dissipation (see, e.g., [10], [12], [15], [16]). We also provide a short discussion of the restricted planar, circular, 3-body problem and related sources of dissipation (see, e.g., [1], [2], [25], [31]).…”