# Tidal torques. A critical review of some techniques - Astrophysics

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Abstract: We point out that the MacDonald formula for body-tide torques is valid onlyin the zeroth order of e-Q, while its time-average is valid in the first order.So the formula cannot be used for analysis in higher orders of e-Q. Thisnecessitates corrections in the theory of tidal despinning and librationdamping.We prove that when the inclination is low and phase lags are linear infrequency, the Kaula series is equivalent to a corrected version of theMacDonald method. The correction to MacDonald-s approach would be to set thephase lag of the integral bulge proportional to the instantaneous frequency.The equivalence of descriptions gets violated by a nonlinearfrequency-dependence of the lag.We explain that both the MacDonald- and Darwin-torque-based derivations ofthe popular formula for the tidal despinning rate are limited to lowinclinations and to the phase lags being linear in frequency. TheDarwin-torque-based derivation, though, is general enough to accommodate both afinite inclination and the actual rheology.Although rheologies with Q scaling as the frequency to a positive power makethe torque diverge at a zero frequency, this reveals not the impossible natureof the rheology, but a flaw in mathematics, i.e., a common misassumption thatdamping merely provides lags to the terms of the Fourier series for the tidalpotential. A hydrodynamical treatment Darwin 1879 had demonstrated that themagnitudes of the terms, too, get changed. Reinstating of this detail tames theinfinities and rehabilitates the -impossible- scaling law which happens to bethe actual law the terrestrial planets obey at low frequencies.

Author: ** Michael Efroimsky, James G. Williams**

Source: https://arxiv.org/