Love numbers for an Andrade planet

The Andrade rheological law ε(t)=ε₀+βt^α has been introduced by Andrade in 1910 for the description of elongation in metal wires. Since then, this model has gained increasing popularity in geophysics and planetary sciences, being extremely effective in the description of numerous materials, including polycrystalline ices, amorphous solids and silicate rocks. Recently, many works in the field of planetology have adopted this model for the description of the response of solar system or extra-solar planets to tidal perturbations, especially for bodies whose properties are still poorly constrained. This is because the Andrade rheology can describe transient deformation using a low number of parameters, a highly valued characteristic for the study of planetary bodies for which few observational constraints are available, such as exoplanets. For the Moon, the Andrade rheology provides an accurate description of the viscoelastic tidal deformation, satisfying the observed frequency dependence of the quality factor.

While for uniform bodies described by a steady-state Maxwell rheology the analytical form of the time-dependent Love numbers (LNs) was established long ago, in the case of the transient Andrade model no closed-form solutions have been determined so far. This is mainly due to the fact that the planetary response is normally studied in the Fourier-transformed frequency domain or by numerical methods in the time domain. Closed-form expressions could be important since they have the potential of providing insight into the dependence of LNs upon the model parameters and the viscoelastic relaxation time-scales of the planet.

In this work, we focus on the Andrade rheological law in 1-D and we obtain a previously unknown explicit expression, in the time domain, for the relaxation modulus in terms of the Mittag-Leffler function E_α,β(z), a higher transcendental function that generalises the exponential function. Second, we consider the general response of a uniform, incompressible planetary model - the “Kelvin sphere” - studying the Laplace-transformed, the frequency domain and the time-domain LNs by analytical methods. By exploiting the results obtained in the 1-D case, we establish closed-form expressions of the time domain LNs and we discuss the frequency-domain response of the Kelvin sphere with Andrade rheology analytically.

Our findings exhibit a complex relation between the planet parameters and the resulting deformation. From the analysis of the frequency-dependent LNs we show that dissipation in Earth-like planets is strongly dependent upon the choice of the planet density, rigidity and viscosity, while the variation of the Andrade creep parameter α has an effect that is limited to short-period tidal forcing. Concurrently, the study of the time dependent LNs shows that α regulates the duration of the transient phase, while the remaining parameters set the value of elastic limit, and the rate at which  the fluid limit is reached. Finally, some examples concerning the tidal deformations of the Moon are presented to point out the relevance that the Andrade rheology assumes in this particular case.