On the Love Numbers of an Andrade Planet

Abstract

Abstract 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 Eα,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.

Keywords

• Andrade
• rheology
• transient
• Mittag‐Leffler
• Love numbers
• planets