Massive astrophysical compact halo object (MACHO) 80.7443.1718 is a high mass, eccentric binary system exhibiting the largest-known-amplitude tidally excited oscillations. The system’s ±20% photometric amplitude, along with the high mass of the primary star, ~35 M⊙, make this the most extreme of the class of periodically perturbed ‘heartbeat stars.’ Here, we use a hydrodynamic simulation to demonstrate that with each periapse passage, an unseen companion star raises tidal waves so large that they break, shock-heating and dissipating energy and angular momentum on the surface of the star. The shock-heated material forms a rapidly rotating circumstellar atmosphere, which is stripped and reassembled with each periapse passage. The dissipation of nonlinear tides through surface wave breaking explains the super-synchronous rotation of the primary star, the evolution of spectral emission features and the observed decay of the binary orbital period. Connecting these features demonstrates that MACHO 80.7443.1718 is a natural product of massive binary star evolution, and that it provides an ideal laboratory for the direct study of nonlinear tidal dissipation.
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The data needed to reproduce the results and figures of this article are available online via the Harvard Dataverse81.
The software used to run the simulation models described in this article and perform the analysis that reproduces the figures of this article is available online at https://github.com/morganemacleod/HBStarWaveBreaking and via Zenodo82.
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We acknowledge helpful conversations with C. Alcock, A. Dupree, V. Kalogera and J. Stone. We are particularly grateful to M. Vick for many discussions and collaboration on tidal wave breaking. This work was supported by the National Science Foundation under Grant No. 1909203 and by a Clay Postdoctoral Fellowship at the Smithsonian Astrophysical Observatory. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI1548562. In particular, use of XSEDE resource Stampede2 at TACC through allocation TG-AST200014 enabled this work. A.L. was supported in part by Harvard’s Black Hole Initiative, which is funded by GBMF and JTF grants.
The authors declare no competing interests.
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Extended Data Fig. 1 Comparison of the radiative envelope of a MESA stellar evolution model to a polytropic approximation with Γ = 1.307.
The polytrope model joins to an isothermal atmosphere outside of the stellar profile. The MESA model has evolved just past the ignition of Helium in the core when it reaches the conditions matching those observed in MACHO 80.7443.1718. The right-hand panel compares the propagation diagram of the MESA and polytropic models, showing the dimensionless Lamb frequency (Sl, evaluated for l = 2) and Brunt-Väisälä frequency (N). Acoustic, p-modes propagate above the envelope set by these two frequencies, while buoyancy, g-modes propagate below.
Extended Data Fig. 2 Lomb-Scargle periodogram of tidally-excited oscillations between periapse passages.
The periodogram of the simulation differs from that of the data, showing that different modes are excited most strongly. In the simulation case, we see most of the power near the rotational frequency, while in the data, the 25th harmonic of the orbital frequency is most-strongly excited.
This mode leads to oscillatory motions in the ϕ direction, which modulate the envelope angular frequency. In addition to oscillations in Ω/Ωenv, dissipation spins the envelope to higher average frequencies near the surface.
A vortex sheet marks the shear layer in the l = 1 g-mode. This shear interface becomes corrugated and breaks when the mach number in the rotating frame is on the order of unity. The associated dissipation drives the spin up of the outer stellar layers.
Extended Data Fig. 5 Slice through the midplane of model MACHO 80.7443.1718 primary star near orbital apoapse.
Through shocks and dissipation, the atmosphere layers acquire rotation similar to the Keplerian rate outside the stellar surface (the profile in the simulation outside 30R⊙ is similar to 2/3vkep), indicating that these high angular momentum layers will be rotationally supported even if they cool.
Supplementary text and Figs. 1–4.
Animation of main text Fig. 1. A slice of density through the orbital plane shows interaction between the envelope of the primary star and the companion. Each passage, waves are raised that crash to the stellar surface, creating a rotating, shock-heated atmosphere.
Animation of main text Fig. 1, but zoomed in on the primary-star envelope and moving in the orbiting reference frame. This provides a closer perspective on the tidal waves and shock-laced atmosphere that develop.
Three-dimensional rendering of the photospheric surface, as in main text Fig. 2. Colours represent surface flux, and the orientation is identical to the observer orientation relative to MACHO 80.7443.1718. Following each passage, the corrugated surface of breaking waves are visible near the stellar equator.
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MacLeod, M., Loeb, A. Breaking waves on the surface of the heartbeat star MACHO 80.7443.1718. Nat Astron (2023). https://doi.org/10.1038/s41550-023-02036-3