On the effect of tidal deformation on planetary phase curves

With the continuous improvement in the precision of exoplanet observations, it has become feasible to probe for subtle effects that can enable a more comprehensive characterization of exoplanets. A notable example is the tidal deformation of ultra-hot Jupiters by their host stars, whose detection can provide valuable insights into the planetary interior structure. In this work, we extend previous research on modeling deformation in transit light curves by proposing a simple approach to account for tidal deformation in phase curves. The planetary shape is modeled as a function of the second fluid Love number for radial deformation $h_{2f}$. We show that the effect of tidal deformation manifests across the full orbit of the planet as its projected area varies with phase, thereby allowing us to better probe the planet's shape in phase curves than in transits. Comparing the effects and detectability of deformation by different space-based instruments, we find that the effect of deformation is more prominent in infrared observations where the phase curve amplitude is the largest. A single JWST phase curve observation of a deformed planet, such as WASP-12b, can allow up to 17$\sigma$ measurement of $h_{2f}$ compared to 4$\sigma$ from transit-only observation. Such high precision $h_{2f}$ measurement can constrain the core mass of the planet to within 19\% of the total mass, thus providing unprecedented constraints on the interior structure. Due to the lower phase curve amplitudes in the optical, the other instruments provide $\leq4\sigma$ precision on $h_{2f}$ depending on the number of phase curves observed. We also find that detecting deformation from infrared phase curves is less affected by uncertainty in limb darkening, unlike detection in transits. Finally, the assumption of sphericity when analyzing the phase curve of deformed planets can lead to biases in several system parameters