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Tidal shear energy dissipation and stellar structure

Published online by Cambridge University Press:  09 November 2026

L. Maranan,
S. J. ArthurOpen the ORCID record for S. J. Arthur[Opens in a new window] and
G. KoenigsbergerOpen the ORCID record for G. Koenigsberger[Opens in a new window]
L. Maranan*
Affiliation:
Instituto de Investigación en Ciencias Básicas y Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, Morelos, México
S. J. Arthur*
Affiliation:
Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Morelia, Michoacán, México
G. Koenigsberger*
Affiliation:
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, México
*
email: leomar.maranan@uaem.edu.mx
email: j.arthur@irya.unam.mx
email: gloria@icf.unam.mx
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Abstract

Asynchronous rotation in binary stars induces a localized differential rotation structure which, in the presence of viscous turbulence, can inject energy into the outer stellar layers. This paper explores the effect of such energy injection in a 162 $M_\odot$ star early in its evolutionary lifetime. We find that in response to the injected energy, the radius, luminosity and mass-loss rate increase, compared to a standard model. We also find that the computed turbulent viscosity values imply synchronization timescales for the inner layers that exceed the main sequence lifetime. Hence, binary stars on the main sequence may not be in true corotation even if their outer layers appear to be rotating synchronously, and they may appear more luminous and have larger mass-loss rates than their analogous single-star counterparts.

Keywords

Massive binariesBinary evolutionTidal interactionsAsynchronous rotation

Information

Type
Poster Paper
Information
Proceedings of the International Astronomical Union , Volume 21 , Symposium S402: Massive Stars Across Redshifts in the Era of JWST and Large Surveys , January 2025, pp. 290 - 292
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The International Astronomical Union

1. Shear energy dissipation and stellar structure

Binary stars whose rotation angular velocity differs from the instantaneous orbital angular velocity undergo oscillatory motions that induce differential rotation (Koenigsberger et al. Reference Koenigsberger, Moreno and Langer2021). In the presence of viscous turbulence, a fraction of the energy involved in the shearing motions can be converted into heat. This is an ingredient generally neglected in stellar structure calculations. In this paper we present results of an experiment to determine the effect on its structure of injecting heat into the outer layers of a massive star.

The Modules for Experiments in Stellar Astrophysics (MESA) spherically symmetric stellar evolution code (version 12778, Paxton et al. Reference Paxton2015) was used to evolve a binary system with ZAMS masses 186 plus $+$ 186 upper M Subscript circled dot $M_\odot$ , initial orbital period upper P 0 equals 2.8 $P_0=2.8$ d and SMC metallicity up to an age of 0.729 Myr, at which time an estimate of the internal tidal shear energy dissipation rate at several depths was injected. At this age, partway through the main sequence, the stellar mass is 162.5 upper M Subscript circled dot $M_\odot$ , the radius is 17.66 upper R Subscript circled dot $R_\odot$ , and the orbital period is 3.48 d.

The energy dissipation rate ModifyingAbove upper E With dot left parenthesis r comma theta comma phi right parenthesis $\dot{E}(r,\theta,\phi)$ was calculated in a 3D grid with the TIDESnvv codeFootnote † (Moreno et al. Reference Moreno, Koenigsberger and Harrington2011; Koenigsberger & Estrella-Trujillo Reference Koenigsberger and Estrella-Trujillo2024), assuming a polytrope with index 2.7, which approximates the internal structure of the MESA model at 0.729 Myr. The perturbed star was assumed to be initially rotating with a uniform, supersynchronous rotation rate omega Subscript normal r normal o normal t $\omega_\mathrm{rot}$ such that omega Subscript normal r normal o normal t Baseline divided by normal upper Omega Subscript normal o normal r normal b Baseline equals 1.4 $\omega_\mathrm{rot}/\Omega_\mathrm{orb}=1.4$ , where normal upper Omega Subscript normal o normal r normal b $\Omega_\mathrm{orb}$ is the orbital angular velocity at periastron. The TIDESnvv calculation was performed for 10 layers, with radii as listed in column 1 of Table 1. The total ModifyingAbove upper E With dot $\dot{E}$ value in each layer was averaged over the orbital cycle, interpolated to the MESA radial grid, and activated when the model reached an age of 0.729 Myr.

Table 1.

Results from the TIDESnvv calculation.

A table comparing various parameters across different layers of a star.
Table 1 Long description

A table with 10 rows and 6 columns comparing various parameters across different layers of a star. The columns are labeled as follows: R layer (solar radii), nu (viscosity in cm^2/s), t diss (dissipation time in years), tau sync (synchronization time in years), and E (energy dissipation rate in erg/s/layer). The table lists values for each parameter across different layers. For example, Row 1: R layer 7.59, nu 8.4 x 10^13, t diss 105.3, tau sync 1.2 x 10^9, E 1.9 x 10^35. Row 2: R layer 8.65, nu 1.5 x 10^14, t diss 79.3, tau sync 2.9 x 10^8, E 1.8 x 10^36.

Notes: r Subscript normal l normal a normal y normal e normal r $r_\mathrm{layer}$ is the midpoint of each layer modeled. left angle bracket nu right angle bracket $\left\lt\nu\right\gt$ is the turbulent viscosity average over the layer and over an orbital cycle, and is computed as nu equals lamda script l Subscript t Baseline normal upper Delta u Subscript l $\nu=\lambda \ell_t \Delta u_l$ , where lamda $\lambda$ is the fraction of local kinetic energy converted into heat, script l Subscript t $\ell_t$ is the characteristic length, here taken to be the width of the layers, and normal upper Delta u Subscript t $\Delta u_t$ is the relative velocity of volume elements with respect to their neighbors. left angle bracket ModifyingAbove upper E With dot right angle bracket $\left\lt\dot{E}\right\gt$ is the tidal shear energy dissipation rate in each layer, averaged over orbital cycle, assuming lamda equals 1 $\lambda=1$ (see Koenigsberger & Estrella-Trujillo Reference Koenigsberger and Estrella-Trujillo2024, for details). The synchronization timescale for each layer is StartFraction 1 Over tau Subscript normal s normal y normal n normal c Baseline EndFraction equals StartFraction 1 Over t Subscript normal d normal i normal s normal s Baseline EndFraction StartFraction left parenthesis normal upper Omega minus omega right parenthesis Over normal upper Omega EndFraction q squared StartFraction upper M r squared Over upper I EndFraction left parenthesis StartFraction r Over a EndFraction right parenthesis Superscript 6 $\frac{1}{\tau_\mathrm{sync}} = \frac{1}{t_\mathrm{diss}} \frac{(\Omega-\omega)}{\Omega}q^2 \frac{Mr^2}{I}\left(\frac{r}{a}\right)^6$ , where t Subscript normal d normal i normal s normal s $t_\mathrm{diss}$ = r squared divided by left angle bracket nu right angle bracket $r^2/\left\lt\nu\right\gt$ , I is the moment of inertial and a is the orbital separation (Zahn Reference Zahn2008).

Figure 1 shows the equatorial radius upper R 1 $R_1$ , total luminosity upper L 1 $L_1$ , mass-loss rate ModifyingAbove upper M With dot Subscript 1 $\dot{M}_1$ and total mass upper M 1 $M_1$ as a function of time of the heated model, compared to the standard model. A sharp jump in upper R 1 $R_1$ , upper L 1 $L_1$ , and ModifyingAbove upper M With dot Subscript 1 $\dot{M}_1$ is observed in the heated model as the stellar structure adjusts to the extra energy in its outer layers, and these parameters remain high during the next tilde 1 $\sim1$ Myr. The mass-loss rate increases because it depends directly on the luminosity (Vink et al. Reference Vink, de Koter and Lamers2001; Brott et al. Reference Brott2011). As a consequence of the larger mass-loss rate, the total mass of the star is reduced with respect to the standard model. Thus, we conclude that heat injection in a MESA massive-star binary model causes an increase in mass-loss rate, equatorial radius and luminosity with respect to a standard model.

Figure 1.

Evolution of radius, luminosity, mass-loss rate, and mass as function of age for the tidally-heated model (red) and the standard model.

Four line graphs depict the evolution of stellar properties over time.

2. Implications and conclusions

TIDESnvv computes a turbulent viscosity nu Subscript t Baseline equals lamda script l Subscript t Baseline normal upper Delta u Subscript l $\nu_t=\lambda \ell_t \Delta u_l$ at each timestep of the calculation using the instantaneous velocity of each volume element relative to its neighboring elements normal upper Delta u Subscript l $\Delta u_l$ and the element characteristic size script l $\ell$ . nu Subscript t $\nu_t$ is a crucial ingredient in the calculation of synchronization timescales tau Subscript normal s normal y normal n normal c $\tau_\mathrm{sync}$ .

The average nu Subscript t $\nu_t$ value in each layer of our model is listed in column 2 of Table 1 and the resulting values of tau Subscript normal s normal y normal n normal c $\tau_\mathrm{sync}$ , calculated using the expression given by Zahn (Reference Zahn2008), in column 4 of this table. Whereas the surface layers synchronize with the orbital motion on short timescales, the timescales for the inner layers exceed the main sequence lifetime. Hence, the tidal perturbations due to asynchronous rotation persist throughout most of the lifetime of the star.

Our values of tau Subscript normal s normal y normal n normal c $\tau_\mathrm{sync}$ are lower limits because in the calculation for this experiment we set lamda equals 1 $\lambda=1$ . This proportionality factor lamda element of left bracket 0 comma 1 right bracket $\lambda\in[0,1]$ represents the fraction of kinetic energy that is dissipated and converted into heat, and in general, lamda less than 1 $\lambda\lt1$ . A smaller nu Subscript t $\nu_t$ implies longer synchronization times. Thus, from an observational standpoint, the surface equatorial velocity may be in corotation but deeper layers may not be. As a consequence, the entire stellar structure may differ from that of a single-star model and several modes of stellar oscillations may be active.

There are several caveats to our results. Tidal perturbations are aspherical, with the largest amplitudes generally occurring around the equatorial latitudes. However, MESA calculates the 1D, spherically symmetric stellar structure, so we have adopted the average tidal shear energy dissipation over an entire shell for input into MESA. In reality, the actual values in the equator are significantly larger that in the poles, an effect that has unknown consequences on the stellar structure. Also, our turbulent viscosities are estimated using a simplistic formulation and a refinement is required. Finally, it is not clear how tidal perturbations interact with surface convective layers, a phenomenon that requires further investigation.

3. Acknowledgements

We gratefully acknowledge the work of Edmundo Moreno in writing the TIDES codes and the support from UNAM PAPIIT grant IN105723.

Footnotes

† TIDESnvv is available at https://zenodo.org/records/10799022

References

Brott, I., et al. 2011, A&A, 530, A115 Google Scholar
Estrella-Trujillo, D., Arthur, S. J. & Koenigsberger, G. 2023, A&A, 670, A44 Google Scholar
Koenigsberger, G. & Estrella-Trujillo, D. 2024, A&A, 685, A145 Google Scholar
Koenigsberger, G., Moreno, E. & Langer, N. 2021, A&A, 653, A127 Google Scholar
Moreno, E., Koenigsberger, G. & Harrington, D. M. 2011, A&A, 528, A48 Google Scholar
Paxton, B. et al. 2015, ApJS, 220, 15 CrossRefGoogle Scholar
Vink, J. S., de Koter, A. & Lamers, H. J. G. L. M., 2001, A&A, 369, 574 Google Scholar
Zahn, J.-P. 2008, EAS Publication Series, 29, 2008 CrossRefGoogle Scholar
Figure 0

Table 1.Results from the TIDESnvv calculation.Table 1 long description.

Figure 1

Figure 1.Evolution of radius, luminosity, mass-loss rate, and mass as function of age for the tidally-heated model (red) and the standard model.