Matrix-propagator approach to compute fluid Love numbers and applicability to extrasolar planets

Abstract

Context. The mass and radius of a planet directly provide its bulk density, which can be interpreted in terms of its overall composition. Any measure of the radial mass distribution provides a first step in constraining the interior structure. The fluid Love number $k_2$ provides such a measure, and estimates of $k_2$ for extrasolar planets are expected to be available in the coming years thanks to improved observational facilities and the ever-extending temporal baseline of extrasolar planet observations. Aims. We derive a method for calculating the Love numbers $k_n$ of any object given its density profile, which is routinely calculated from interior structure codes. Methods. We used the matrix-propagator technique, a method frequently used in the geophysical community. Results. We detail the calculation and apply it to the case of GJ 436b, a classical example of the degeneracy of mass-radius relationships, to illustrate how measurements of $k_2$ can improve our understanding of the interior structure of extrasolar planets. We implemented the method in a code that is fast, freely available, and easy to combine with preexisting interior structure codes. While the linear approach presented here for the calculation of the Love numbers cannot treat the presence of nonlinear effects that may arise under certain dynamical conditions, it is applicable to close-in gaseous extrasolar planets like hot Jupiters, likely the first targets for which $k_2$ will be measured.