Signatures of Core-envelope Rotational Misalignment in the Mixed-mode Asteroseismology of Kepler-56

Abstract Existing asteroseismic rotational measurements assume that stars rotate around a single axis. However, tidal torques from misaligned companions, or their possible engulfment, may bring the rotational axis of a star's envelope out of alignment with its core, breaking azimuthal symmetry. I derive perturbative expressions for asteroseismic signatures of such hitherto unexamined rotational configurations, under the “shellular approximation” of constant rotation rates on radially stratified mass shells. In the aligned case, the distribution of power between multiplet components is determined by the inclination of the rotational axis; radial differential misalignment causes this to vary from multiplet to multiplet. I examine in particular detail the phenomenology of gravitoacoustic mixed modes as seen in evolved sub- and red giants, where near-resonance avoided crossings may break geometrical degeneracies. Upon applying the revised asteroseismic observational methodology that results from this theoretical discussion to revisit Kepler-56—a red giant with a misaligned planetary system—I find that its core and envelope rotate around different rotational axes. While the rotational axis of its core is indeed misaligned from the orbit normal of its transiting planets (consistently with earlier studies), its envelope's rotational axis is close to lying in the sky plane, and may well be aligned with them. More detailed asteroseismic modeling, and spectroscopic follow-up, will be required to fully elucidate the full spin–orbit geometry of the Kepler-56 system, and potentially discriminate between hypotheses for how it formed.