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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 $i_{\rm p}$ to the plane of the sky and of the sky-projected spin-orbit misalignment $λ$ at two epochs about six years apart allowed for the determination of the longitude of the ascending node $Ω$ and of the orbital inclination $I$ to the apparent equatorial plane at the same epochs. As a consequence, average rates of change $\dotΩ_{\rm exp},~\dot I_{\rm exp}$ of this two orbital elements, accurate to a $\approx 10^{-2}~{\rm deg}~{\rm yr}^{-1}$ level, were calculated as well. By comparing them to general theoretical expressions $\dotΩ_{J_2},~\dot I_{J_2}$ 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 $i^{\star} = 142^{+10}_{-11}~{\rm deg},~J_2^{\star} = (2.1^{+0.8}_{-0.5})\times 10^{-4}.$ As a by-product, the angle between $S^{\star}$ and the orbital angular momentum $L$ is as large as about $ψ\approx 100$ deg $(ψ^{2008} = 99^{+5}_{-4}~{\rm deg},~ψ^{2014} = 103^{+5}_{-4}~{\rm deg})$, and changes at a rate $\dotψ= 0.7^{+1.5}_{-1.6}~{\rm deg}~{\rm yr}^{-1}$. 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.