visco-elastic model has the advantages of (i) not introducing any discontinuity for vanishing tidal frequencies, as is the case for synchronous rotation, and (ii) to allow for a complete calculation of the tidal effect [*without any assumption on the eccentricity*]{} for an ideal viscoelastic body.

Indeed, as shown by @Gre09, the frequency dependence of the phase lag of a perfect viscoelastic oscillator is given by $$\tan (\epsilon)=\frac{\sigma}{\tau (\omega_0^2-\sigma^2)}$$ where $\tau$ is a viscous damping timescale and $\omega_0$ the natural frequency of the oscillator. In an incompressible gaseous body, the restoring force acting against the tidal deformation is the self gravity of the body. Thus $\omega_0$ can be estimated through the free-fall time as $2\pi/\omega_0\approx\frac{1}{4}\sqrt{\frac{3\pi}{2\,G\bar{\rho}}}\approx 30$ minutes for Jupiter mean density. For tidal periods of several days $\omega_0\gg\sigma$, and for weakly viscous fluid, the phase lag reads $$\epsilon(\sigma)\approx\frac{\sigma}{\tau\, \omega_0^2}\equiv\,\sigma\, \Delta t$$ which is the frequency dependence corresponding to the constant time lag model.

On the contrary, constant-$Q$ models described by @GS66, @JGB08, @FRH08 were derived using perturbative developments of Kepler equations of motion both in eccentricity and inclination. Such Fourier decomposition is in fact necessary in an “lag and add” approach with a given frequency dependence of the phase lag ($\epsilon(\sigma)$). 