Michael Efroimsky

Tidal torques: a critical review of some techniques

We review some techniques employed in the studies of torques due to bodily tides, and explain why the MacDonald formula for the tidal torque is valid only in the zeroth order of the eccentricity divided by the quality factor, while its time-average is valid in the first order. As a result, the formula cannot be used for analysis in higher orders of e/Q. This necessitates some corrections in the current theory of tidal despinning and libration damping (though the qualitative conclusions of that theory may largely remain correct). We demonstrate that in the case when the inclinations are small and the phase lags of the tidal harmonics are proportional to the frequency, the Darwin-Kaula expansion is equivalent to a corrected version of the MacDonald method. The latter method rests on the assumption of existence of one total double bulge. The necessary correction to MacDonald’s approach would be to assert (following Singer, Geophys. J. R. Astron. Soc., 15: 205–226, 1968) that the phase lag of this integral bulge is not constant, but is proportional to the instantaneous synodal frequency (which is twice the difference between the evolution rates of the true anomaly and the sidereal angle). This equivalence of two descriptions becomes violated by a nonlinear dependence of the phase lag upon the tidal frequency. It remains unclear whether it is violated at higher inclinations. Another goal of our paper is to compare two derivations of a popular formula for the tidal despinning rate, and emphasise that both are strongly limited to the case of a vanishing inclination and a certain (sadly, unrealistic) law of frequency-dependence of the quality factor Q—the law that follows from the phase lag being proportional to frequency. One of the said derivations is based on the MacDonald torque, the other on the Darwin torque. Fortunately, the second approach is general enough to accommodate both a finite inclination and the actual rheology. We also address the rheological models with the Q factor scaling as the tidal frequency to a positive fractional power, and disprove the popular belief that these models introduce discontinuities into the equations and thus are unrealistic at low frequencies. Although such models indeed make the conventional expressions for the torque diverge at vanishing frequencies, the emerging infinities reveal not the impossible nature of one or another rheology, but a subtle flaw in the underlying mathematical model of friction. Flawed is the common misassumption that damping merely provides phase lags to the terms of the Fourier series for the tidal potential. A careful hydrodynamical treatment by Sir George Darwin (1879), with viscosity explicitly included, had demonstrated that the magnitudes of the terms, too, get changed—a fine detail later neglected as “irrelevant”. Reinstating of this detail tames the fake infinities and rehabilitates the “impossible” scaling law (which happens to be the actual law the terrestrial planets obey at low frequencies). Finally, we explore the limitations of the popular formula interconnecting the quality factor and the phase lag. It turns out that, for low values of Q, the quality factor is no longer equal to the cotangent of the lag.

Bodily tides near spin–orbit resonances

Spin–orbit coupling can be described in two approaches. The first method, known as the “MacDonald torque”, is often combined with a convenient assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, because derivation of the expression for the MacDonald torque tacitly fixes the rheology of the mantle by making Q scale as the inverse tidal frequency. Spin–orbit coupling can be treated also in an approach called “the Darwin torque”. While this theory is general enough to accommodate an arbitrary frequency-dependence of Q, this advantage has not yet been fully exploited in the literature, where Q is often assumed constant or is set to scale as inverse tidal frequency, the latter assertion making the Darwin torque equivalent to a corrected version of the MacDonald torque. However neither a constant nor an inverse-frequency Q reflect the properties of realistic mantles and crusts, because the actual frequency-dependence is more complex. Hence it is necessary to enrich the theory of spin–orbit interaction with the right frequency-dependence. We accomplish this programme for the Darwin-torque-based model near resonances. We derive the frequency-dependence of the tidal torque from the first principles of solid-state mechanics, i.e., from the expression for the mantle’s compliance in the time domain. We also explain that the tidal torque includes not only the customary, secular part, but also an oscillating part. We demonstrate that the lmpq term of the Darwin–Kaula expansion for the tidal torque smoothly passes zero, when the secondary traverses the lmpq resonance (e.g., the principal tidal torque smoothly goes through nil as the secondary crosses the synchronous orbit). Thus, we prepare a foundation for modeling entrapment of a despinning primary into a resonance with its secondary. The roles of the primary and secondary may be played, e.g., by Mercury and the Sun, correspondingly, or by an icy moon and a Jovian planet. We also offer a possible explanation for the “improper” frequency-dependence of the tidal dissipation rate in the Moon, discovered by LLR.

The tidal history of Iapetus: Spin dynamics in the light of a refined dissipation model

[1]xa0We study the tidal history of an icy moon, basing our approach on a dissipation model, which combines viscoelasticity with anelasticity and takes into account the microphysics of attenuation. We apply this approach to Iapetus, the most remote large icy moon in the Saturnian system. Different authors provide very different estimates for Iapetuss despinning timescale, by several orders of magnitude. One reason for these differences is the choice of the dissipation model used for computing the spin evolution. As laboratory data on viscoelastic properties of planetary ices are sparse, many studies relied on dissipation models that turned out to be inconsistent with experiment. A pure water ice composition, generally assumed in the previous studies of the kind, yields despinning times of the order of 3.7 Gyr for most initial conditions. We demonstrate that through accounting for the complexity of the material (like second-phase impurities) one arrives at despinning times as short as 0.9 Gyr. A more exact estimate will remain unavailable until we learn more about the influence of impurities on ice dissipation. By including the triaxial-shape-caused torque, we encounter a chaotic behavior at the final stage of despinning, with the possibility of entrapments in the intermediate resonances. The duration of these entrapments turns out to be sensitive to the dissipation model. No long entrapments have been found for Iapetus described with our laboratory-based dissipation model.

The Serret Andoyer Formalism in Rigid-Body Dynamics: I. Symmetries and Perturbations

This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics, and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, that stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret-Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into action-angle ones, using the method of Sadov.

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

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.

Long-term evolution of orbits about a precessing oblate planet: 3. A semianalytical and a purely numerical approach

Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2004, 2005, 2006a, b). Here we continue this line of research by combining that analytical machinery with numerical tools. Our model includes three factors: the J2 of the planet, its nonuniform equinoctial precession described by the Colombo formalism, and the gravitational pull of the Sun. This semianalytical and seminumerical theory, based on the Lagrange planetary equations for the Keplerian elements, is then applied to Deimos on very long time scales (up to 1 billion years). In parallel with the said semianalytical theory for the Keplerian elements defined in the co-precessing equatorial frame, we have also carried out a completely independent, purely numerical, integration in a quasi-inertial Cartesian frame. The results agree to within fractions of a percent, thus demonstrating the applicability of our semianalytical model over long timescales. Another goal of this work was to make an independent check of whether the equinoctial-precession variations predicted for a rigid Mars by the Colombo model could have been sufficient to repel its moons away from the equator. An answer to this question, in combination with our knowledge of the current position of Phobos and Deimos, will help us to understand whether the Martian obliquity could have undergone the large changes ensuing from the said model (Ward 1973; Touma and Wisdom 1993, 1994; Laskar and Robutel 1993), or whether the changes ought to have been less intensive (Bills 2006; Paige etxa0al. 2007). It has turned out that, for low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations. This is an extension, to non-uniform equinoctial precession given by the Colombo model, of an old result obtained by Goldreich (1965) for the case of uniform precession and a low initial inclination. However, near-polar initial inclinations may exhibit considerable variations for up to ±10 deg in magnitude. This result is accentuated when the obliquity is large. Nevertheless, the analysis confirms that an oblate planet can, indeed, afford large variations of the equinoctial precession over hundreds of millions of years, without repelling its near-equatorial satellites away from the equator of date: the satellite inclination oscillates but does not show a secular increase. Nor does it show secular decrease, a fact that is relevant to the discussion of the possibility of high-inclination capture of Phobos and Deimos.

On the Theory of Bodily Tides

Different theories of bodily tides assume different forms of dependence of the angular lag δ upon the tidal frequency χ. In the old theory (Gerstenkorn 1955, MacDonald 1964, Kaula 1964) the geometric Iag angle is assumed constant (i.e., δ ∼ χ0), while the new theory (Singer 1968; Mignard 1979, 1980) postulates constancy of the time lag Δt (which is equivalent to saying that δ ∼ χ1).Each particular functional form of δ(χ) unambiguously determines the form of the frequency dependence of the tidal quality factor, Q(χ), and vice versa. Through the past 20 years, several teams of geophysicists have undertaken a large volume of experimental research of attenuation at low frequencies. This research, carried out both for mineral samples in the lab and for vast terrestrial basins, has led to a complete reconsideration of the shape of Q(χ). While in late 70s – early 80s it was universally accepted that at low frequencies the quality factor scales as inverse frequency, by now it is firmly established that Q ∼ χα, wh...