Relation between the moment of inertia and the k2 Love number of fluid extra-solar planets

Abstract

Context. Tidal and rotational deformation of fluid giant extra-solar planets may impact their transit light curves, making the k2 Love number observable in the near future. Studying the sensitivity of k2 to mass concentration at depth is thus expected to provide new constraints on the internal structure of gaseous extra-solar planets. Aims. We investigate the link between the mean polar moment of inertia N of a fluid, stably layered extra-solar planet and its k2 Love number. Our aim is to obtain analytical relations valid, at least, for some particular ranges of the model parameters. We also seek a general approximate relation useful for constraining N once observations of k2 become available. Methods. For two-layer fluid extra-solar planets we explore the relation between N and k2 via analytical methods, for particular values of the model parameters. We also explore approximate relations valid over the entire range of two-layer models. More complex planetary structures are investigated by the semi-analytical propagator technique. Results. A unique relation between N and k2 cannot be established. However, our numerical experiments show that a rule of thumb can be inferred that is valid for complex, randomly layered stable planetary structures. The rule robustly defines the upper limit to the values of N for a given k2, and agrees with analytical results for a polytrope of index one and with a realistic non-rotating model of the tidal equilibrium of Jupiter.