# Bodily tides near the 1:1 spin-orbit resonance: correction to Goldreich’s dynamical model

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DOI: 10.1007/s10569-012-9446-7

- Cite this article as:
- Williams, J.G. & Efroimsky, M. Celest Mech Dyn Astr (2012) 114: 387. doi:10.1007/s10569-012-9446-7

## Abstract

Spin-orbit coupling is often described in an approach known as “*the MacDonald torque*”, which has long become the textbook standard due to its apparent simplicity. Within this method, a concise expression for the additional tidal potential, derived by MacDonald (Rev Geophys 2:467–541, 1994), is combined with a convenient assumption that the quality factor *Q* is frequency-independent (or, equivalently, that the geometric lag angle is constant in time). This makes the treatment unphysical because MacDonald’s derivation of the said formula was, very implicitly, based on keeping the *time* lag frequency-independent, which is equivalent to setting *Q* scale as the inverse tidal frequency. This contradiction requires the entire MacDonald treatment of both non-resonant and resonant rotation to be rewritten. The non-resonant case was reconsidered by Efroimsky and Williams (Cel Mech Dyn Astron 104:257–289, 2009), in application to spin modes distant from the major commensurabilities. In the current paper, we continue this work by introducing the necessary alterations into the MacDonald-torque-based model of falling into a 1-to-1 resonance. (The original version of this model was offered by Goldreich (Astron J 71:1–7, 1996). Although the MacDonald torque, both in its original formulation and in its corrected version, is incompatible with realistic rheologies of minerals and mantles, it remains a useful toy model, which enables one to obtain, in some situations, qualitatively meaningful results without resorting to the more rigorous (and complicated) theory of Darwin and Kaula. We first address this simplified model in application to an oblate primary body, with tides raised on it by an orbiting zero-inclination secondary. (Here the role of the tidally-perturbed primary can be played by a satellite, the perturbing secondary being its host planet. A planet may as well be the perturbed primary, its host star acting as the tide-raising secondary). We then extend the model to a triaxial primary body experiencing both a tidal and a permanent-figure torque exerted by an orbiting secondary. We consider the effect of the triaxiality on both circulating and librating rotation near the synchronous state. Circulating rotation may evolve toward the libration region or toward a spin faster than synchronous (the so-called pseudosynchronous spin). Which behaviour depends on the orbit eccentricity, the triaxial figure of the primary, and the mass ratio of the secondary and primary bodies. The spin evolution will always stall for the oblate case. For libration with a small amplitude, expressions are derived for the libration frequency, damping rate, and average orientation. Importantly, the stability of pseudosynchronous spin hinges upon the dissipation model. Makarove and Efroimsky (Astrophys J, 2012) have found that a more realistic tidal dissipation model than the corrected MacDonald torque makes pseudosynchronous spin unstable. Besides, for a sufficiently large triaxiality, pseudosynchronism is impossible, no matter what dissipation model is used.