The moon’s gravity causes a tidal bulge in Earth’s oceans, so that the water facing the moon is raised several metres. Similarly, close-orbiting exoplanets will have a tidally distorted shape, with a tidal bulge facing the host star. The amount of distortion can be quantified by the “Love number” h (named after the mathematician Augustus Love)
Specifically, h2 tells us the relative height of the tidal bulge, and would be zero for a perfectly rigid body that did not distort at all, and would be 2.5 for a perfectly fluid body that adapted fully to the tidal potential. Gas-giant planets have large envelopes of gaseous fluid, so would be expected to have fairly high values of h2. However, they also might have rocky or metallic cores, and so would have values less than 2.5. For example Jupiter has h2 = 1.6 while Saturn has h2 = 1.4.
A new paper by Hugo Hellard et al discusses whether h2 for a hot-Jupiter exoplanet can be measured from the shape of the transit lightcurve, given good-enough photometry such as that from the Hubble Space Telescope.
The main problem is that the transit profile is heavily affected by variations in the brightness of the stellar disc, in particular the limb darkening (a star’s limbs appear a bit dimmer, because a tangential line-of-sight into a gas cloud skims only the cooler, upper layers). Thus the Hellard et al paper discusses at length different ways to model the limb darkening.
The end-result, however, is a claim to have measured h2 for WASP-121b, with a value of h2 = 1.4 ± 0.8. This is not (yet) a strong constraint, but points to the potential in the future, and also flags up the need to understand and properly parametrise limb darkening.