Tidal torques: a critical review of some techniques
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- Efroimsky, M. & Williams, J.G. Celest Mech Dyn Astr (2009) 104: 257. doi:10.1007/s10569-009-9204-7
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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.