Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Internal mixing of rotating stars inferred from dipole gravity modes

Nature Astronomy volume 5, pages 715–722 (2021)Cite this article

Subjects

Abstract

During most of their life, stars fuse hydrogen into helium in their cores. The mixing of chemical elements in the radiative envelope of stars with a convective core is able to replenish the core with extra fuel. If effective, such deep mixing allows stars to live longer and change their evolutionary path. Yet localized observations to constrain internal mixing are absent so far. Gravity modes probe the deep stellar interior near the convective core and allow us to calibrate internal mixing processes. Here we provide core-to-surface mixing profiles inferred from observed dipole gravity modes in 26 rotating stars with masses between 3 and 10 solar masses. We find a wide range of internal mixing levels across the sample. Stellar models with stratified mixing profiles in the envelope reveal the best asteroseismic performance. Our results provide observational guidance for three-dimensional hydrodynamical simulations of transport processes in the deep interiors of stars.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Light curves overplotted with amplitude spectra of six SPB stars.
Fig. 2: Gravity-mode period spacing patterns of six SPB stars.
Fig. 3: Schematic representation of the considered mixing profiles.
Fig. 4: Population of the eight model grids in terms of model capacity.
Fig. 5: Inferred internal mixing profiles for 26 SPB stars.
Fig. 6: Correlations among estimated parameters and inferred quantities for the sample.

Similar content being viewed by others

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The iterative prewhitening code is freely available and documented at https://github.com/IvS-KULeuven/IvSPythonRepository. The stellar evolution code, MESA, is freely available and documented at http://mesa.sourceforge.net/. The stellar pulsation code, GYRE, is freely available and documented at https://bitbucket.org/rhdtownsend/gyre/wiki/Home.

References

  1. Nomoto, K., Kobayashi, C. & Tominaga, N. Nucleosynthesis in stars and the chemical enrichment of galaxies. Annu. Rev. Astron. Astrophys. 51, 457–509 (2013).

    Article  ADS  Google Scholar 

  2. Maeder, A. & Meynet, G. The evolution of rotating stars. Annu. Rev. Astron. Astrophys. 38, 143–190 (2000).

    Article  ADS  Google Scholar 

  3. Salaris, M. & Cassisi, S. Chemical element transport in stellar evolution models. R. Soc. Open Sci. 4, 170192 (2017).

    Article  ADS  Google Scholar 

  4. Georgy, C. et al. Populations of rotating stars. I. Models from 1.7 to 15 M⊙ at Z = 0.014, 0.006, and 0.002 with Ω/Ωcrit between 0 and 1. Astron. Astrophys. 553, A24 (2013).

    Article  Google Scholar 

  5. Zahn, J. P. Circulation and turbulence in rotating stars. Astron. Astrophys. 265, 115–132 (1992).

    ADS  Google Scholar 

  6. Chaboyer, B., Demarque, P. & Pinsonneault, M. H. Stellar models with microscopic diffusion and rotational mixing. I. Application to the Sun. Astrophys. J. 441, 865 (1995).

    Article  ADS  Google Scholar 

  7. Mathis, S. & Zahn, J. P. Transport and mixing in the radiation zones of rotating stars. II. Axisymmetric magnetic field. Astron. Astrophys. 440, 653–666 (2005).

    Article  ADS  Google Scholar 

  8. Rogers, T. M., Lin, D. N. C., McElwaine, J. N. & Lau, H. H. B. Internal gravity waves in massive stars: angular momentum transport. Astrophys. J. 772, 21 (2013).

    Article  ADS  Google Scholar 

  9. Brott, I. et al. Rotating massive main-sequence stars. I. Grids of evolutionary models and isochrones. Astron. Astrophys. 530, A115 (2011).

    Article  Google Scholar 

  10. Deupree, R. G. Stellar evolution with arbitrary rotation laws. III. Convective core overshoot and angular momentum distribution. Astrophys. J. 499, 340–347 (1998).

    Article  ADS  Google Scholar 

  11. Chaboyer, B. & Zahn, J. P. Effect of horizontal turbulent diffusion on transport by meridional circulation. Astron. Astrophys. 253, 173–177 (1992).

    ADS  MATH  Google Scholar 

  12. Zahn, J. P. Convective penetration in stellar interiors. Astron. Astrophys. 252, 179–188 (1991).

    ADS  Google Scholar 

  13. Freytag, B., Ludwig, H.-G. & Steffen, M. Hydrodynamical models of stellar convection. The role of overshoot in DA white dwarfs, A-type stars, and the Sun. Astron. Astrophys. 313, 497–516 (1996).

    ADS  Google Scholar 

  14. Pinsonneault, M. Mixing in stars. Annu. Rev. Astron. Astrophys. 35, 557–605 (1997).

    Article  ADS  Google Scholar 

  15. Charbonnel, C. & Lagarde, N. Thermohaline instability and rotation-induced mixing. I. Low- and intermediate-mass solar metallicity stars up to the end of the AGB. Astron. Astrophys. 522, A10 (2010).

    Article  ADS  Google Scholar 

  16. Dotter, A. et al. The Dartmouth stellar evolution database. Astrophys. J. Suppl. Ser. 178, 89–101 (2008).

    Article  ADS  Google Scholar 

  17. Morel, T., Hubrig, S. & Briquet, M. Nitrogen enrichment, boron depletion and magnetic fields in slowly-rotating B-type dwarfs. Astron. Astrophys. 481, 453–463 (2008).

    Article  ADS  Google Scholar 

  18. Martins, F. et al. Observational effects of magnetism in O stars: surface nitrogen abundances. Astron. Astrophys. 538, A29 (2012).

    Article  Google Scholar 

  19. Tkachenko, A. et al. The mass discrepancy in intermediate- and high-mass eclipsing binaries: the need for higher convective core masses. Astron. Astrophys. 637, A60 (2020).

    Article  Google Scholar 

  20. Aerts, C., Christensen-Dalsgaard, J. & Kurtz, D. W. Asteroseismology (Springer, 2010).

  21. Miglio, A., Montalbán, J., Noels, A. & Eggenberger, P. Probing the properties of convective cores through g modes: high-order g modes in SPB and γ Doradus stars. Mon. Not. R. Astron. Soc. 386, 1487–1502 (2008).

    Article  ADS  Google Scholar 

  22. Bossini, D. et al. Uncertainties on near-core mixing in red-clump stars: effects on the period spacing and on the luminosity of the AGB bump. Mon. Not. R. Astron. Soc. 453, 2290–2301 (2015).

    Article  ADS  Google Scholar 

  23. Koch, D. G. et al. Kepler mission design, realized photometric performance, and early science. Astrophys. J. Lett. 713, L79–L86 (2010).

    Article  ADS  Google Scholar 

  24. Van Reeth, T., Tkachenko, A. & Aerts, C. Interior rotation of a sample of γ Doradus stars from ensemble modelling of their gravity-mode period spacings. Astron. Astrophys. 593, A120 (2016).

    Article  ADS  Google Scholar 

  25. Bouabid, M. P. et al. Effects of the Coriolis force on high-order g modes in γ Doradus stars. Mon. Not. R. Astron. Soc 429, 2500–2514 (2013).

    Article  ADS  Google Scholar 

  26. Aerts, C. et al. Forward asteroseismic modeling of stars with a convective core from gravity-mode oscillations: parameter estimation and stellar model selection. Astrophys. J. Suppl. Ser. 237, 15 (2018).

    Article  ADS  Google Scholar 

  27. Herwig, F. The evolution of AGB stars with convective overshoot. Astron. Astrophys. 360, 952–968 (2000).

    ADS  Google Scholar 

  28. Rogers, T. M. & McElwaine, J. N. On the chemical mixing induced by internal gravity waves. Astrophys. J. Lett. 848, L1 (2017).

    Article  ADS  Google Scholar 

  29. Mathis, S., Palacios, A. & Zahn, J. P. On shear-induced turbulence in rotating stars. Astron. Astrophys. 425, 243–247 (2004).

    Article  ADS  Google Scholar 

  30. Moravveji, E., Townsend, R. H. D., Aerts, C. & Mathis, S. Sub-inertial gravity modes in the B8V star KIC 7760680 reveal moderate core overshooting and low vertical diffusive mixing. Astrophys. J. 823, 130 (2016).

    Article  ADS  Google Scholar 

  31. Szewczuk, W. & Daszyńska-Daszkiewicz, J. K. I. C. 3240411-thehottestknownS. P. Bstarwiththeasymptoticg-modeperiodspacing Mon. Not. R. Astron. Soc. 478, 2243–2256 (2018).

    Article  ADS  Google Scholar 

  32. De Cat, P. & Aerts, C. A study of bright southern slowly pulsating B stars. II. The intrinsic frequencies. Astron. Astrophys. 393, 965–981 (2002).

    Article  ADS  Google Scholar 

  33. Szewczuk, W. & Daszyńska-Daszkiewicz, J. Domains of pulsational instability of low-frequency modes in rotating upper main sequence stars. Mon. Not. R. Astron. Soc. 469, 13–46 (2017).

    Article  ADS  Google Scholar 

  34. Anders, F. et al. Photo-astrometric distances, extinctions, and astrophysical parameters for Gaia DR2 stars brighter than G = 18. Astron. Astrophys. 628, A94 (2019).

    Article  Google Scholar 

  35. Pedersen, M. G., Escorza, A., Pápics, P. I. & Aerts, C. Recipes for bolometric corrections and Gaia luminosities of B-type stars: application to an asteroseismic sample. Mon. Not. R. Astron. Soc. 495, 2738–2753 (2020).

    Article  ADS  Google Scholar 

  36. Kippenhahn, R., Weigert, A. & Weiss, A. Stellar Structure and Evolution (Springer, 2012).

  37. Li, C. et al. Extended main-sequence turnoffs in the double cluster h and χ Persei: the complex role of stellar rotation. Astrophys. J. 876, 65 (2019).

    Article  ADS  Google Scholar 

  38. Johnston, C., Aerts, C., Pedersen, M. G. & Bastian, N. Isochrone-cloud fitting of the extended main-sequence turn-off of young clusters. Astron. Astrophys. 632, A74 (2019).

    Article  ADS  Google Scholar 

  39. Wu, T. et al. Asteroseismic analyses of slowly pulsating B star KIC 8324482: ultraweak element mixing beyond the central convective core. Astrophys. J. 899, 38 (2020).

    Article  ADS  Google Scholar 

  40. Wu, T. & Li, Y. High-precision asteroseismology in a slowly pulsating B star: HD 50230. Astrophys. J. 881, 86 (2019).

    Article  ADS  Google Scholar 

  41. Balona, L. A., Baran, A. S., Daszyńska-Daszkiewicz, J. & De Cat, P. Analysis of Kepler B stars: rotational modulation and Maia variables. Mon. Not. R. Astron. Soc. 451, 1445–1459 (2015).

    Article  ADS  Google Scholar 

  42. Pápics, P. I. et al. Signatures of internal rotation discovered in the Kepler data of five slowly pulsating B stars. Astron. Astrophys. 598, A74 (2017).

    Article  Google Scholar 

  43. Moravveji, E., Aerts, C., Pápics, P. I., Triana, S. A. & Vandoren, B. Tight asteroseismic constraints on core overshooting and diffusive mixing in the slowly rotating pulsating B8.3V star KIC 10526294. Astron. Astrophys. 580, A27 (2015).

    Article  ADS  Google Scholar 

  44. Triana, S. A. et al. The internal rotation profile of the B-type star KIC 10526294 from frequency inversion of its dipole gravity modes. Astrophys. J. 810, 16 (2015).

    Article  ADS  Google Scholar 

  45. Aerts, C. Probing the interior physics of stars through asteroseismology. Rev. Mod. Phys. 93, 015001 (2021).

    Article  ADS  Google Scholar 

  46. Raskin, G. et al. HERMES: a high-resolution fibre-fed spectrograph for the Mercator telescope. Astron. Astrophys. 526, A69 (2011).

    Article  Google Scholar 

  47. Pápics, P. I. et al. Two new SB2 binaries with main sequence B-type pulsators in the Kepler field. Astron. Astrophys. 553, A127 (2013).

    Article  Google Scholar 

  48. Tkachenko, A. Grid search in stellar parameters: a software for spectrum analysis of single stars and binary systems. Astron. Astrophys. 581, A129 (2015).

    Article  ADS  Google Scholar 

  49. Aerts, C., Molenberghs, G., Kenward, M. G. & Neiner, C. The surface nitrogen abundance of a massive star in relation to its oscillations, rotation, and magnetic field. Astrophys. J. 781, 88 (2014).

    Article  ADS  Google Scholar 

  50. Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (MESA): pulsating variable stars, rotation, convective boundaries, and energy conservation. Astrophys. J. Suppl. Ser. 243, 10 (2019).

    Article  ADS  Google Scholar 

  51. Aerts, C., Mathis, S. & Rogers, T. M. Angular momentum transport in stellar interiors. Annu. Rev. Astron. Astrophys. 57, 35–78 (2019).

    Article  ADS  Google Scholar 

  52. Seaton, M. J. Opacity Project data on CD for mean opacities and radiative accelerations. Mon. Not. R. Astron. Soc. 362, L1–L3 (2005).

    Article  ADS  Google Scholar 

  53. Przybilla, N., Nieva, M. F., Irrgang, A. & Butler, K. in New Advances in Stellar Physics: From Microscopic to Macroscopic Processes EAS Publications Series 63 (eds Alecian, G. et al.) 13–23 (EDP Sciences, 2013).

  54. Vink, J. S., de Koter, A. & Lamers, H. J. G. L. M. Mass-loss predictions for O and B stars as a function of metallicity. Astron. Astrophys. 369, 574–588 (2001).

    Article  ADS  Google Scholar 

  55. Björklund, R., Sundqvist, J. O., Puls, J. & Najarro, F. New predictions for radiation-driven, steady-state mass-loss and wind-momentum from hot, massive stars II. A grid of O-type stars in the Galaxy and the Magellanic Clouds. Astron. Astrophys. 648, A36 (2021).

    Article  ADS  Google Scholar 

  56. Böhm-Vitense, E. Über die wasserstoffkonvektionszone in sternen verschiedener effektivtemperaturen und leuchtkräfte. Mit 5 textabbildungen. Z. Astrophys 46, 108–143 (1958).

    ADS  Google Scholar 

  57. Ouazzani, R.-M. et al. A new asteroseismic diagnostic for internal rotation in γ Doradus stars. Mon. Not. R. Astron. Soc 465, 2294–2309 (2017).

    Article  ADS  Google Scholar 

  58. Townsend, R. H. D., Goldstein, J. & Zweibel, E. G. Angular momentum transport by heat-driven g-modes in slowly pulsating B stars. Mon. Not. R. Astron. Soc. 475, 879–893 (2018).

    Article  ADS  Google Scholar 

  59. Bellinger, E. P. et al. Fundamental parameters of main-sequence stars in an instant with machine learning. Astrophys. J. 830, 31 (2016).

    Article  ADS  Google Scholar 

  60. Claeskens, G. & Hjort, N. L. Model Selection and Model Averaging Cambridge Series in Statistical and Probabilistic Mathematics Vol. 27 (Cambridge Univ. Press, 2008).

Download references

Acknowledgements

We thank the MESA and GYRE code developers for their efforts, public dissemination and training initiatives to make their software so accessible to the worldwide astrophysics community. We thank S. Ekström of the Geneva Observatory for providing mixing profiles from Georgy et al.4 in electronic format. We acknowledge the work of the teams behind the NASA Kepler and ESA Gaia space missions. This work is based on observations with the HERMES spectrograph at the Mercator Telescope, which is operated at La Palma, Spain, by the Flemish Community. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and NASA’s Astrophysics Data System. The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 670519: MAMSIE), from the National Science Foundation (grant number NSF PHY-1748958), from the KU Leuven Research Council (grant number C16/18/005: PARADISE) and from the Research Foundation Flanders (FWO) by means of PhD Fellowships to M.M. and S. Gebruers under contract numbers 11F7120N and 11E5620N and a senior post-doctoral fellowship to D.M.B. under grant agreement number 1286521N. Funding for the Kepler Mission was provided by NASA’s Science Mission Directorate. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC); funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA).

Author information

Authors and Affiliations

  1. Institute of Astronomy, KU Leuven, Leuven, Belgium

    May G. Pedersen, Conny Aerts, Péter I. Pápics, Mathias Michielsen, Sarah Gebruers, Siemen Burssens, Stefano Garcia & Dominic M. Bowman

  2. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, USA

    May G. Pedersen

  3. Department of Astrophysics, IMAPP, Radboud University Nijmegen, Nijmegen, the Netherlands

    Conny Aerts

  4. Max Planck Institute for Astronomy, Heidelberg, Germany

    Conny Aerts & Sarah Gebruers

  5. Department of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, UK

    Tamara M. Rogers

  6. Planetary Science Institute, Tucson, AZ, USA

    Tamara M. Rogers

  7. I-BioStat, Universiteit Hasselt, Hasselt, Belgium

    Geert Molenberghs

  8. I-BioStat, KU Leuven, Leuven, Belgium

    Geert Molenberghs

Authors
  1. May G. Pedersen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Conny Aerts
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Péter I. Pápics
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mathias Michielsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sarah Gebruers
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Tamara M. Rogers
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Geert Molenberghs
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Siemen Burssens
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Stefano Garcia
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Dominic M. Bowman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

M.G.P. performed frequency analysis and mode identification, wrote code to include mixing profiles in MESA, computed asteroseismic observables, implemented and applied the modelling procedures, interpreted the results and wrote part of the text. C.A. defined the research, developed the modelling procedure, interpreted the results and wrote part of the text. P.I.P. constructed light curves from the raw Kepler data and discovered the targets to be new SPB stars. M.M. wrote code to include mixing profiles in MESA and assessed the capacity of observables used for the modelling. S. Gebruers determined abundances from spectroscopy. T.M.R. computed and provided envelope mixing profiles due to internal gravity waves. G.M. provided advice on the parameter estimation and statistical model selection and performed the cluster analysis. S.B., S. Garcia and D.M.B. contributed to the frequency analysis and interpretation. All authors contributed to the discussions and have read and iterated upon the text of the final manuscript.

Corresponding author

Correspondence to May G. Pedersen.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Astronomy thanks Joyce Guzik, Ernst Paunzen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Discussion, Figs. 1–34 and Tables 1–8.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pedersen, M.G., Aerts, C., Pápics, P.I. et al. Internal mixing of rotating stars inferred from dipole gravity modes. Nat Astron 5, 715–722 (2021). https://doi.org/10.1038/s41550-021-01351-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41550-021-01351-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing