Accurate characterization of the stellar and orbital parameters of the exoplanetary system WASP-33 b from orbital dynamics

By using the most recently published Doppler tomography measurements and accurate theoretical modeling of the oblateness-driven orbital precessions, we tightly constrain some of the physical and orbital parameters of the planetary system hosted by the fast rotating star WASP-33. In particular, the measurements of the orbital inclination $i_{\mathrm{p}}$ to the plane of the sky and of the sky-projected spin-orbit misalignment $\lambda$ at two epochs about six years apart allowed for the determination of the longitude of the ascending node $\mathrm{\Omega }$ and of the orbital inclination $I$ to the apparent equatorial plane at the same epochs. As a consequence, average rates of change ${\dot{\mathrm{\Omega }}}_{\mathrm{e}\mathrm{x}\mathrm{p}},{\dot{I}}_{\mathrm{e}\mathrm{x}\mathrm{p}}$ of this two orbital elements, accurate to a $\approx {10}^{-2}\mathrm{d}\mathrm{e}\mathrm{g}{\mathrm{y}\mathrm{r}}^{-1}$ level, were calculated as well. By comparing them to general theoretical expressions $\dot\Omega_{J_2},~\dot I_{J_2}$ for their precessions induced by an oblate star whose symmetry axis is arbitrarily oriented, we were able to determine the angle $i^{\star }$ between the line of sight the star’s spin $S^{\star }$ and its first even zonal harmonic $J_2^{\star }$ obtaining $i^{\star} = 142^{+10}{-11}~{\rm deg},~J_2^{\star} = (2.1^{+0.8}{-0.5})\times 10^{-4}.$Asaby-product,theanglebetween$S^{\star}$andtheorbitalangularmomentum$L$isaslargeasabout$\psi \approx 100$deg$(\psi^{2008} = 99^{+5}{-4}~{\rm deg},~\psi^{2014} = 103^{+5}{-4}~{\rm deg})$,andchangesatarate$\dot\psi = 0.7^{+1.5}_{-1.6}~{\rm deg}~{\rm yr}^{-1}$. The predicted general relativistic Lense-Thirring precessions, or the order of $\approx 10^{-3}~{\rm deg}~{\rm yr}^{-1}$, are, at present, about one order of magnitude below the measurability threshold.