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.

Context. The mass and radius of a planet directly provide its bulk density, which can be interpreted in terms of its overall composition. Any measure of the radial mass distribution provides a first step in constraining the interior structure. The fluid Love number k 2 provides such a measure, and estimates of k 2 for extrasolar planets are expected to be available in the coming years thanks to improved observational facilities and the ever-extending temporal baseline of extrasolar planet observations. Aims. We derive a method for calculating the Love numbers k n of any object given its density profile, which is routinely calculated from interior structure codes. Methods. We used the matrix-propagator technique, a method frequently used in the geophysical community. Results. We detail the calculation and apply it to the case of GJ 436b, a classical example of the degeneracy of mass-radius relationships, to illustrate how measurements of k 2 can improve our understanding of the interior structure of extrasolar planets. We implemented the method in a code that is fast, freely available, and easy to combine with preexisting interior structure codes. While the linear approach presented here for the calculation of the Love numbers cannot treat the presence of nonlinear effects that may arise under certain dynamical conditions, it is applicable to close-in gaseous extrasolar planets like hot Jupiters, likely the first targets for which k 2 will be measured.

Context. Increasing our knowledge of the interior structure, composition and density distribution of exoplanets is crucial to make progress in the understanding of exoplanetary formation, migration and habitability. However, the directly measurable mass and radius values offer little constraint on interior structure, because the inverse problem is highly degenerate. Therefore there is a clear need for a third observable of exoplanet interiors. This third observable can be the k 2 fluid Love number which measures the central mass concentration of an exoplanet. Aims. The aims of this paper are (i) to develop a basic model to fit the long-term radial velocity and TTV variations caused by tidal interactions, (ii) to apply the model to the WASP-18Ab system, and (iii) to estimate the Love number of the planet. Methods. Archival radial velocity, transit and occultation timing data are collected and fitted via the model introduced here.Results. The best model fit to the archival radial velocity and timing data of WASP-18Ab was obtained with a Love number of the massive (∼ 10M Jup ) hot Jupiter WASP-18Ab: k 2,Love = 0.62 +0.55 −0.19 . This causes apsidal motion in the system, at a rate of ∼ 0.0087 ± 0.0033 • /days 31.3 ± 11.8 arcseconds/day. When checking possible causes of periastron precession, other than the relativistic term or the non-spherical shape of the components, we found a companion star to the WASP-18 system, named WASP-18B which is a probable M6.5V dwarf with ∼ 0.1 M at 3519 AU distance from the transit host star. We also find that small orbital eccentricities may be real, rather than an apparent effect caused by the non-spherical stellar shape.

Data from transit light curves, radial velocity and transit timing observations can be used to probe the interiors of exoplanets beyond the mean density, by measuring the Love numbers h 2 and k 2 . The first indirect estimate of k 2 for an exoplanet from radial velocity and transit timing variations observations has been performed by taking advantage of the years-spanning baseline. Not a single measurement of h 2 has been achieved from transit light curves, mostly because the photometric precision of current observing facilities is still too low. We show that the Imaging Spectrograph instrument on-board the Hubble Space Telescope could measure h 2 of the hot Jupiter WASP-121b if only few more observations were gathered. We show that a careful treatment of the noise and stellar limb darkening must be carried out to achieve a measurement of h 2 . In particular, we find that the impact of the noise modelling on the estimation of h 2 is stronger than the impact of the limb darkening modelling. In addition, we emphasize that the wavelet method for correlated noise analysis can mask limb brightening. Finally, using presently available data, we briefly discuss the tentative measurement of h 2 = 1.39 +0.71 −0.81 in terms of interior structure. Additional observations would further constrain the interior of WASP-121b and possibly provide insights on the physics of inflation. The possibility of using the approach presented here with the Hubble Space Telescope provides a bridge before the high-quality data to be returned by the James Webb Space Telescope and PLATO telescope in the coming decade.

In this chapter, the key technologies and the instrumentation required for the subsurface exploration of ocean worlds are discussed. The focus is laid on Jupiter's moon Europa and Saturn's moon Enceladus because they have the highest potential for such missions in the near future. The exploration of their oceans requires landing on the surface, penetrating the thick ice shell with an ice-penetrating probe, and probably diving with an underwater vehicle through dozens of kilometers of water to the ocean floor, to have the chance to find

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