The angle $psi$ between a planets orbital axis and the spin axis of its parent star is an important diagnostic of planet formation, migration, and tidal evolution. We seek empirical constraints on $psi$ by measuring the stellar inclination $i_{rm s}$ via asteroseismology for an ensemble of 25 solar-type hosts observed with NASAs Kepler satellite. Our results for $i_{rm s}$ are consistent with alignment at the 2-$sigma$ level for all stars in the sample, meaning that the system surrounding the red-giant star Kepler-56 remains as the only unambiguous misaligned multiple-planet system detected to date. The availability of a measurement of the projected spin-orbit angle $lambda$ for two of the systems allows us to estimate $psi$. We find that the orbit of the hot-Jupiter HAT-P-7b is likely to be retrograde ($psi=116.4^{+30.2}_{-14.7}:{rm deg}$), whereas that of Kepler-25c seems to be well aligned with the stellar spin axis ($psi=12.6^{+6.7}_{-11.0}:{rm deg}$). While the latter result is in apparent contradiction with a statement made previously in the literature that the multi-transiting system Kepler-25 is misaligned, we show that the results are consistent, given the large associated uncertainties. Finally, we perform a hierarchical Bayesian analysis based on the asteroseismic sample in order to recover the underlying distribution of $psi$. The ensemble analysis suggests that the directions of the stellar spin and planetary orbital axes are correlated, as conveyed by a tendency of the host stars to display large inclination values.
تحميل البحث