Relationship between the moment of inertia and the \(k_2\) Love number of fluid extra-solar planets

Context: Tidal and rotational deformation of fluid giant extra-solar planets may impact their transit light curves, making the \(k_2\) Love number observable in the upcoming years. Studying the sensitivity of \(k_2\) 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 \(k_2\) Love number, aiming at obtaining analytical relationships valid, at least, for some particular ranges of the model parameters. We also seek a general, approximate relationship useful to constrain \(N\) once observations of \(k_2\) will become available. Methods: For two-layer fluid extra-solar planets, we explore the relationship between \(N\) and \(k_2\) by analytical methods, for particular values of the model parameters. We also explore approximate relationships valid over all the possible range of two-layer models. More complex planetary structures are investigated by the semi-analytical propagator technique. Results: A unique relationship between \(N\) and \(k_2\) cannot be established. However, our numerical experiments show that a `rule of thumb' can be inferred, valid for complex, randomly layered stable planetary structures. The rule robustly defines the upper limit to the values of \(N\) for a given \(k_2\), and agrees with analytical results for a polytrope of index one and with a realistic non-rotating model of the tidal equilibrium of Jupiter.