On the Love Numbers of an Andrade Planet

The Andrade rheological model is often employed to describe the response of solar system or extra-solar planets to tidal perturbations, especially when their properties are still poorly constrained. While for uniform planets with steady-state Maxwell rheology the analytical form of the Love numbers was established long ago, for the transient Andrade rheology no closed-form solutions have been yet determined, and the planetary response is usually studied either semi-analitically in the frequency domain or numerically in the time domain. Closed-form expressions are potentially important since they could provide insight into the dependence of Love numbers upon the model parameters and the time-scales of the isostatic readjustment of the planet. First, we focus on the Andrade rheological law in 1-D and we obtain a previously unknown explicit form, in the time domain, for the relaxation modulus in terms of the higher Mittag-Leffler transcendental function E_α,β(z) that generalizes the exponential function. Second, we consider the general response of an incompressible planetary model — often referred to as the “Kelvin sphere” — studying the Laplace domain, the frequency domain and the time domain Love numbers by analytical methods. Through a numerical approach, we assess the effect of compressibility on the Love numbers in the Laplace and frequency domains. Furthermore, exploiting the results obtained in the 1-D case, we establish closed-form — although not elementary — expressions of the time domain Love numbers and we discuss the frequency domain response of the Kelvin sphere with Andrade rheology analytically.