Can the tidal quality factors of terrestrial planets and moons scale as positive powers of the tidal frequency?

TL;DR

This paper clarifies that the common belief about the unphysical nature of tidal quality factor scaling at low frequencies is based on a mathematical misconception, and demonstrates that such scaling laws are physically valid for terrestrial planets and moons.

Contribution

It identifies and corrects a mathematical flaw in the modeling of tidal torques, supporting the validity of frequency-dependent tidal quality factors.

Findings

Tidal quality factors scale as a positive power of frequency in planetary mantles.

The divergence at low frequencies is due to a flawed assumption about the relation between torque and quality factor.

Correcting the mathematical model removes the divergence, validating the scaling law.

Abstract

In geophysics and seismology, it is a common knowledge that the quality factors Q of the mantle and crust materials scale as the tidal frequency to a positive fractional power (Karato 2007, Efroimsky and Lainey 2007). In astronomy, there exists an equally common belief that such rheological models introduce discontinuities into the equations and thus are unrealistic at low frequencies. We demonstrate that, while such models indeed make the conventional expressions for the tidal 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 the tidal force and torque are inversely proportional to the quality factor. In reality, they are proportional to the sine of the tidal phase lag, while the inverse quality factor is commonly identified with the tangent of the lag. The sine and tangent of the lag are close everywhere {\it{except in the vicinity of the zero frequency}}. Reinstating of this detail tames the fake infinities and rehabilitates the "impossible" scaling law (which happens to be the actual law the mantles obey). This preprint is a pilot paper. A more comprehensive treatise on tidal torques is to be published (Efroimsky and Williams 2009).