Tidal Torques. Critical Review of Some Techniques.

Details Tidal Torques. Critical Review of Some Techniques. Tidal Torques. Critical Review of Some Techniques.

May 2008

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008dda....39.0902e&link_type=abstract

American Astronomical Society, DDA meeting #39, #9.02

Mathematics

Logic

Scientific paper

We 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. One method 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. Though such models indeed make the conventional expressions for the torque diverge for 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).

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