Retrieval of the fluid Love number $k_2$ in exoplanetary transit curves

We are witness to a great and increasing interest in internal structure, composition and evolution of exoplanets. However, direct measurements of exoplanetary mass and radius cannot be uniquely interpreted in terms of interior structure, justifying the need for an additional observable. The second degree fluid Love number, $k_2$, is proportional to the concentration of mass towards the body's center, hence providing valuable additional information about the internal structure. When hydrostatic equilibrium is assumed for the planetary interior, $k_2$ is a direct function of the planetary shape. Previous attempts were made to link the observed tidally and rotationally induced planetary oblateness in photometric light curves to $k_2$ using ellipsoidal shape models. Here, we construct an analytical 3D shape model function of the true planetary mean radius, that properly accounts for tidal and rotational deformations. Measuring the true planetary mean radius is critical when one wishes to compare the measured $k_2$ to interior theoretical expectations. We illustrate the feasibility of our method and show, by applying a Differential Evolution Markov Chain to synthetic data of WASP-121b, that a precision $\leq$ 65 ppm/$\sqrt{2\,min}$ is required to reliably retrieve $k_2$ with present understanding of stellar limb darkening, therefore improving recent results based on ellipsoidal shape models. Any improvement on stellar limb darkening would increase such performance.