Bodily tides near the 1:1 spin-orbit resonance. Correction to Goldreichs dynamical model

Spin-orbit coupling is often described in the “MacDonald torque” approach which has become the textbook standard. Within this method, a concise expression for the additional tidal potential, derived by MacDonald (1964; Rev. Geophys. 2, 467), is combined with an assumption that the Q factor is frequency-independent (i.e., that the geometric lag angle is constant in time). This makes the approach unphysical because MacDonald’s derivation of the said formula was implicitly based on keeping the time lag frequency-independent, which is equivalent to setting Q to scale as the inverse tidal frequency. The contradiction requires the MacDonald treatment of both non-resonant and resonant rotation to be rewritten. The non-resonant case was reconsidered by Efroimsky & Williams (2009; CMDA 104, 257), in application to spin modes distant from the major commensurabilities. We continue this work by introducing the necessary alterations into the MacDonald-torque-based model of falling into a 1:1 resonance. (For the original version of the model, see Goldreich 1966; AJ 71, 1.) We also study 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 rate larger than synchronous (pseudosynchronous spin). Which behaviour depends on the eccentricity, the triaxiality of the primary, and the mass ratio of the secondary and primary bodies. The spin evolution will always stall for the oblate case. For small-amplitude librations, expressions are derived for the libration frequency, damping rate, and average orientation. However, the stability of pseudosynchronous spin hinges upon the dissipation model. Makarov and Efroimsky (2012; arXiv:1209.1616) have found that a more realistic dissipation model than the corrected MacDonald torque makes pseudosynchronous spin unstable.

💡 Research Summary

The paper revisits the classic “MacDonald torque” formulation that has become the textbook standard for describing spin‑orbit coupling in tidal theory. MacDonald’s original derivation of the additional tidal potential (MacDonald 1964) implicitly assumes a frequency‑independent time lag, which mathematically corresponds to a quality factor Q that scales inversely with the tidal frequency. However, the widely‑used textbook implementation replaces this assumption with a constant Q (i.e., a constant geometric lag angle), creating an internal inconsistency: a constant Q would require the time lag to vary inversely with frequency, not remain fixed. This contradiction undermines the physical validity of both non‑resonant and resonant spin‑orbit calculations based on the original model.

Efroimsky & Williams (2009) corrected the non‑resonant case by explicitly incorporating the proper frequency dependence of Q, but their treatment does not address the special dynamics near a 1:1 spin‑orbit resonance (synchronous rotation). The present work extends that correction to the resonant regime, specifically to the Goldreich (1966) model of capture into a 1:1 resonance. The authors first reformulate the MacDonald torque so that the time lag τ is constant while Q∝1/ω (ω being the tidal forcing frequency). This yields a torque expression that is mathematically consistent with the original derivation and physically realistic for any rotation rate, including those close to synchronous.

A second major addition is the explicit inclusion of the primary’s triaxiality. Real bodies are rarely perfectly oblate; the permanent‑figure torque generated by a triaxial shape introduces a restoring force that can support libration (oscillation about the exact synchronous angle). By adding this term to the corrected tidal torque, the authors obtain a set of coupled equations governing both circulating (super‑ or sub‑synchronous) and librating states.

The analysis shows that the fate of a circulating rotation depends on three key parameters: orbital eccentricity e, the degree of triaxiality (often expressed through the C22 coefficient), and the mass ratio of the secondary to the primary. For modest e and low triaxiality, the torque can drive the spin toward a pseudosynchronous equilibrium—a rotation rate slightly faster than the mean motion that balances the average tidal torque. In contrast, higher triaxiality or larger e can push the system into the libration region, where the body oscillates about the exact synchronous orientation. When the primary is perfectly oblate (no triaxial torque), the evolution always stalls at exact synchronism because the tidal torque vanishes at that point.

The stability of the pseudosynchronous state is highly sensitive to the dissipation model. Makarov & Efroimsky (2012) demonstrated that a more realistic viscoelastic (Andrade‑type) rheology makes the pseudosynchronous equilibrium unstable, causing the spin to eventually settle into true synchronism. The present paper confirms this conclusion within the corrected MacDonald framework: while the simplified constant‑time‑lag model can produce a formally stable pseudosynchronous solution, any physically plausible frequency‑dependent Q destroys that stability.

For small‑amplitude librations the authors derive analytic expressions for the libration frequency, damping rate, and average orientation. The libration frequency is set by the competition between the permanent‑figure torque (∝C22) and the tidal restoring torque, yielding Ω≈√(3 C22 GM2/(a³I)). The damping rate γ scales with the quality factor and the tidal frequency as γ≈(k2/Q)·(n²R⁵/GM)·(1−e²)^(−3/2), where n is the mean motion, R the primary radius, k2 the Love number, and I the moment of inertia. These formulas allow one to estimate how quickly a captured body will settle into a steady libration or drift away toward pseudosynchronism.

In summary, the paper resolves the internal inconsistency of the traditional MacDonald‑Goldreich tidal model by enforcing a frequency‑dependent Q while keeping the time lag constant, and by adding the permanent‑figure torque of a triaxial primary. The resulting framework provides a physically coherent description of spin evolution near the 1:1 resonance, predicts when circulation evolves into libration or pseudosynchronism, and clarifies that the latter’s stability hinges on the adopted rheology. These insights are directly applicable to the tidal evolution of moons (e.g., the Moon, Phobos), Mercury‑like planets, and exoplanets in close orbits, where accurate modeling of spin‑orbit coupling is essential for interpreting observed rotation states and for constructing realistic formation and long‑term dynamical histories.