WASP-131 b with ESPRESSO I: A bloated sub-Saturn on a polar orbit around a differentially rotating solar-type star

In this paper, we present observations of two high-resolution transit datasets obtained with ESPRESSO of the bloated sub-Saturn planet WASP-131~b. We have simultaneous photometric observations with NGTS and EulerCam. In addition, we utilised photometric lightcurves from {\tess}, WASP, EulerCam and TRAPPIST of multiple transits to fit for the planetary parameters and update the ephemeris. We spatially resolve the stellar surface of WASP-131 utilising the Reloaded Rossiter McLaughlin technique to search for centre-to-limb convective variations, stellar differential rotation, and to determine the star-planet obliquity for the first time. We find WASP-131 is misaligned on a nearly retrograde orbit with a projected obliquity of $\lambda = 162.4\substack{+1.3 \\ -1.2}^{\circ}$. In addition, we determined a stellar differential rotation shear of $\alpha = 0.61 \pm 0.06$ and disentangled the stellar inclination ($i_* = 40.9\substack{+13.3 \\ -8.5}^{\circ}$) from the projected rotational velocity, resulting in an equatorial velocity of $v_{\rm{eq}} = 7.7\substack{+1.5 \\ -1.3}$~km s$^{-1}$. In turn, we determined the true 3D obliquity of $\psi = 123.7\substack{+12.8 \\ -8.0}^{\circ}$, meaning the planet is on a perpendicular/polar orbit. Therefore, we explored possible mechanisms for the planetary system's formation and evolution. Finally, we searched for centre-to-limb convective variations where there was a null detection, indicating that centre-to-limb convective variations are not prominent in this star or are hidden within red noise.