DI Herculis Revisited: Starspots, Gravity Darkening, and 3-D Obliquities

Abstract

DI Herculis is an eclipsing binary famous for a longstanding disagreement between theory and observation of the apsidal precession rate, which was resolved when both stars were found to be severely misaligned with the orbit. We used data from the Transiting Exoplanet Survey Satellite (TESS) to refine our knowledge of the stellar obliquities and sharpen the comparison between the observed and theoretical precession rates. The TESS data show variations with a 1.07-day period, which we interpret as rotational modulation from starspots on the primary star. This interpretation is supported by the detection of photometric anomalies during primary eclipses consistent with starspot crossings. The secondary eclipse light curve shows a repeatable asymmetry which we interpret as an effect of gravity darkening. By combining the TESS data with previously obtained data, we determined the three-dimensional spin directions of both stars. Using this information, the updated value of the theoretical apsidal precession rate (including the effects of tides, rotation, and general relativity) is 1.35 +0.58 −0.50 arcsec/cycle. The updated value of the observed rate (after including new TESS eclipse times) is 1.41 +0.39 −0.28 arcsec/cycle. Given the agreement between the observed and theoretical values, we fitted all the relevant data simultaneously assuming the theory is correct. This allowed us to place tighter constraints on the stellar obliquities, which are 75 +3 −3 and 80 +3 −3 degrees for the primary and secondary stars, respectively.

Were the Obliquities in DI Herculis Excited by an Unseen Tertiary Companion?

“…We have also updated the theoretically-predicted apsidal precession rate (consisting of contributions from general relativity, and tidal and rotational distortion) in the absence of a tertiary companion, using recent photometric detections of the spin period by Liang et al (2022). This updated apsidal motion rate remains consistent with the observed value obtained by Claret et al (2010) (see Fig.…”

“…The former case requires a close tertiary companion to be present, while the latter case allows for a much more distant companion. We also take advantage of a recent determination of the stars' spin periods (≈1 day) based on quasiperiodic photometric variability (Liang et al 2022). Using the spin periods, we update the calculation of the expected apsidal motion rate in the absence of a tertiary companion, and demonstrate that the theoretical prediction remains consistent with the apsidal motion rate derived by Claret et al (2010).…”

“…We assume the spin period of each star is 1.0 ± 0.1 day. This assumption is based upon a recent analysis of DI Herculis using data from the Transiting Exoplanet Survey Satellite (Liang et al 2022).…”

Stellar obliquities in exoplanetary systems

“…Obliquities have been measured in a few double-star systems (see , for a listing) but none of them fulfill all these requirements. There are two systems, DI Her and CV Vel, in which both components have radiative envelopes and measured obliquities -but in those systems, the time variations in λ and v sin i are caused by the precession of the misaligned stars around the total angular momentum vector (Reisenberger & Guinan 1989;Albrecht et al 2009;Philippov & Rafikov 2013;Albrecht et al 2014;Liang et al 2022).…”