Tidal torques: a critical review of some techniques
By M. Efroimsky, James G. Williams. 2008
163 citations recorded by the source corpus.
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.
Subfields: Astronomy and Astrophysics, Computational Mechanics, Oceanography
View Paper: https://www.semanticscholar.org/paper/e8cfa4c2a5f38ca7dbc34062d56bedb1074705df
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Citations in the corpus
- Tidal locking of habitable exoplanets (2017, background): “A substantial body of research is devoted to tidal theory, (e.g. Darwin 1880; Goldreich and Soter 1966; Hut 1981; Ferraz-Mello et al. 2008; Wisdom 2008; Efroimsky and Williams 2009; Leconte et al. 2010), and the reader is referred to these studies for a more complete description of the derivations a”
- Is tidal heating sufficient to explain bloated exoplanets? Consistent calculations accounting for finite initial eccentricity (2010, unknown): “Moreover, for an incompressible body, a reasonable assumption for giant planets, the tidal dissipation function is given by (Goldreich 1963; Efroimsky & Williams 2009): Q−1(σ) = −∆cycleE(σ) 2πEpeak(σ) = tan (σ) 1− (π2 − (σ)) tan (σ) , (17) where ∆cycleE(σ) is the energy dissipated by the body at…”
- Tidal obliquity evolution of potentially habitable planets (2011, unknown): “By analogy with the driven harmonic oscillator, the tidal dissipationfu ction Q has been introduced, where Q−1(χl) = tan ( ε(χl) ) 1− ( π 2 − ε(χl) ) tan ( ε(χl) ) ≈ tan ( ε(χl) ) (1) (Efroimsky & Williams 2009).”
- TIDALLY HEATED TERRESTRIAL EXOPLANETS: VISCOELASTIC RESPONSE MODELS (2009, background): “Corrections involving higher order terms and small angle approximations such as those discussed by Efroimsky & Williams (2009) for tides in a primary, are more likely to become non-negligible for extreme tidal exoplanets.”
- Tidal heating in icy satellite oceans (2014, background): “…are typically calculated assuming a viscoelastic rheology, such as Maxwell, Andrade or Burgers (e.g. Ross and Schubert (1989); Tobie et al. (2005); Efroimsky and Williams (2009); Castillo-Rogez et al. (2011); Nimmo et al. (2012)), though other processes (such as frictional heating, e.g. Nimmo and…”
- Bodily tides near spin–orbit resonances (2011, methodology): “Referring the Reader to Efroimsky & Williams (2009) for proofs and comments, we begin with an inventory of the key formulae describing the spin-orbit interaction.”
- The Physics of Bodily Tides in Terrestrial Planets, and the Appropriate Scales of Dynamical Evolution (2007, background): “The quandary is resolved through the observation that the bulge is lagging not only in its position but also in its height, for which reason the dissipation rate remains finite (Efroimsky 2007).”