Retrieval of the Fluid Love Number k₂ in Exoplanetary Transit Curves

Abstract

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 ≤ 65 ppm/ √ 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.

The CHEOPS mission

“…Planets close to their host star are tidally distorted into an ellipsoidal shape. Potentially, this effect is detectable in the transit light curve which would allow the measurement of the planet's Love number providing further insight into the planet's internal structure [1,18]. The main challenge is to separate ellipsoidal deformation from stellar limb darkening effects.…”

HST/STIS Capability for Love Number Measurement of WASP-121b

“…A radial deformation of 5%, which translates into roughly 3600 km, is clearly not negligible. Therefore, we adopt the shape model described by Hellard et al (2019), where the radius is given by:…”

Tidal evolution of exoplanetary systems hosting potentially habitable exoplanets. The cases of LHS-1140 b-c and K2-18 b-c

“…Such approximation is reasonable in the applications of this work since the planets are relatively far from their host star. However, such approximation may not be valid in the case of close-in planets subjected to large tidal distortions (see e.g the discussions presented in Hellard et al (2019)). The expression for the potential acting on the companion considering the resulting triaxial shape of the primary rotated of an angle ϕ B = ϕ + δ w.r.t the axis x can be approximated, to first order in the flattenings, by…”