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.

A plague of magnetic spots among the hot stars of globular clusters

Abstract

For more than six decades, the quest to understand the formation of hot (about 20,000−30,000 K) extreme horizontal branch (EHB) stars in Galactic globular clusters has remained one of the most elusive in stellar evolutionary theory. Here we report on two discoveries that challenge the idea of the stable luminosity of EHB stars. The first mode of EHB variability is periodic and cannot be ascribed to either binary evolution or pulsation. Instead, we attribute it here to the presence of magnetic spots: superficial chemical inhomogeneities whose projected rotation induces the variability. The second mode of EHB variability is aperiodic and manifests itself on timescales of years. In two cases, six-year-long light curves display superflare events that are several million times more energetic than solar analogues. We advocate a scenario in which the two EHB variability phenomena are different manifestations of diffuse, dynamo-generated, weak magnetic fields. Magnetism is therefore a key player driving the formation and evolution of EHB clusters stars and, likewise, operating in the Galactic field counterparts. Our conclusions bridge similar variability/magnetism phenomena in all radiative-enveloped hot-stars: young main-sequence stars, old EHBs and defunct white dwarfs.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Fig. 1: The phased near-ultraviolet light curves of the EHB variable stars in three globular clusters.
Fig. 2: The EHB variable stars in the near-ultraviolet colour–magnitude diagrams.
Fig. 3: Modelling of the stellar spot in the NGC2808 EHB variable star vEHB-12.
Fig. 4: The uSDSS light curve of the EHB Padua-1 showing a complete superflare event.

Data availability

All the raw data (and associated calibrations) used in this paper are available for download in the ESO Science archive under the respective programme ID (see Methods), at http://archive.eso.org. Processed data supporting the findings of this study are available from the corresponding author upon request.

Code availability

All the codes used in this study are available at: Phoebe, http://phoebe-project.org/; KSint, http://eduscisoft.com/KSINT/index.php; EXOFAST, http://astroutils.astronomy.ohio-state.edu/exofast/limbdark.shtml; ISIS, http://www2.iap.fr/users/alard/package.html; VARTOOL, https://www.astro.princeton.edu/~jhartman/vartools.html; SM, https://www.astro.princeton.edu/~rhl/sm/; ALAMBIC, https://esosoft.univie.ac.at/software/esomvm/; DAOPHOT, http://www.star.bris.ac.uk/~mbt/daophot/; IRAF, https://iraf-community.github.io/.

References

  1. Brown, T. M. et al. The Hubble Space Telescope UV legacy survey of galactic globular clusters. VII. Implications from the nearly universal nature of horizontal branch discontinuities. Astrophys. J. 822, 44 (2016).

    ADS  Google Scholar 

  2. Castellani, M. & Castellani, V. Mass loss in globular cluster red giants: an evolutionary investigation. Astrophys. J. 407, 649 (1993).

    ADS  Google Scholar 

  3. Heber, U. Hot subluminous stars. Publ. Astron. Soc. Pacif. 128, 082001 (2016).

    ADS  Google Scholar 

  4. Moni Bidin, C. et al. The lack of close binaries among hot horizontal branch stars in NGC 6752. Astron. Astrophys. 451, 499–513 (2006).

    ADS  Google Scholar 

  5. Moni Bidin, C., Villanova, S., Piotto, G. & Momany, Y. A lack of close binaries among hot horizontal branch stars in globular clusters. II. NGC 2808. Astron. Astrophys. 528, A127 (2011).

    ADS  Google Scholar 

  6. Moehler, S. et al. The hot horizontal-branch stars in ω Centauri. Astron. Astrophys. 526, A136 (2011).

    Google Scholar 

  7. Latour, M., Randall, S. K., Calamida, A., Geier, S. & Moehler, S. SHOTGLAS. I. The ultimate spectroscopic census of extreme horizontal branch stars in ω Centauri. Astron. Astrophys. 618, A15 (2018).

    ADS  Google Scholar 

  8. Moni Bidin, C. et al. A hot horizontal branch star with a close K-type main-sequence companion. Astrophys. J. Lett. 812, L31 (2015).

    ADS  Google Scholar 

  9. Lucatello, S. et al. The incidence of binaries in globular cluster stellar populations. Astron. Astrophys. 584, A52 (2015).

    Google Scholar 

  10. Catelan, M. Horizontal branch stars: the interplay between observations and theory, and insights into the formation of the Galaxy. Publ. Astron. Soc. Pacif. 320, 261–309 (2009).

    MATH  Google Scholar 

  11. Gratton, R. et al. What is a globular cluster? An observational perspective. Astron. Astrophys. Rev. 27, 8 (2019).

    ADS  Google Scholar 

  12. Momany, Y. et al. A new feature along the extended blue horizontal branch of NGC 6752. Astrophys. J. Lett. 576, L65–L68 (2002).

    ADS  Google Scholar 

  13. Kilkenny, D., Koen, C., O’Donoghue, D. & Stobie, R. S. A new class of rapidly pulsating star—I. EC 14026-2647, the class prototype. Mon. Not. R. Astron. Soc. 285, 640–644 (1997).

    ADS  Google Scholar 

  14. Brown, T. M., Landsman, W. B., Randall, S. K., Sweigart, A. V. & Lanz, T. The discovery of pulsating hot subdwarfs in NGC 2808. Astrophys. J. Lett. 777, L22 (2013).

    ADS  Google Scholar 

  15. Randall, S. K. et al. Pulsating hot O subdwarfs in ω Centauri: mapping a unique instability strip on the extreme horizontal branch. Astron. Astrophys. 589, A1 (2016).

    Google Scholar 

  16. Green, E. M. et al. Discovery of a new class of pulsating stars: gravity-mode pulsators among subdwarf B stars. Astrophys. J. Lett. 583, L31–L34 (2003).

    ADS  Google Scholar 

  17. Samus’, N. N., Kazarovets, E. V., Durlevich, O. V., Kireeva, N. N. & Pastukhova, E. N. General catalogue of variable stars: version GCVS 5.1. Astron. Rep. 61, 80–88 (2017).

    ADS  Google Scholar 

  18. Bernhard, K., Hümmerich, S., Otero, S. & Paunzen, E. A search for photometric variability in magnetic chemically peculiar stars using ASAS-3 data. Astron. Astrophys. 581, A138 (2015).

    ADS  Google Scholar 

  19. Mikulášek, Z. et al. An overview of the properties of a sample of newly-identified magnetic chemically peculiar stars in the Kepler field. In Physics of Magnetic Stars. ASP Conf. Ser. Vol. 518 (eds. Kudryavtsev, D. O. et al.) 117–124 (ASP, 2019).

  20. Bagnulo, S., Landi Degl’Innocenti, M., Landolfi, M. & Mathys, G. A statistical analysis of the magnetic structure of CP stars. Astron. Astrophys. 394, 1023–1037 (2002).

    ADS  Google Scholar 

  21. Brown, T. M. et al. A universal transition in atmospheric diffusion for hot subdwarfs near 18,000 K. Astrophys. J. 851, 118 (2017).

    ADS  Google Scholar 

  22. Paunzen, E. et al. Search for stellar spots in field blue horizontal-branch stars. Astron. Astrophys. 622, A77 (2019).

    Google Scholar 

  23. Krtička, J. et al. The nature of light variations in magnetic hot stars. Contrib. Astron. Observatory Skalnate Pleso 48, 170–174 (2018).

    ADS  Google Scholar 

  24. Shavrina, A. V. et al. Spots structure and stratification of helium and silicon in the atmosphere of He-weak star HD 21699. Mon. Not. R. Astron. Soc. 401, 1882–1888 (2010).

    ADS  Google Scholar 

  25. Glagolevskij, Y. V. & Chuntonov, G. A. Composite model for the magnetic field of HD 21699. Astrophysics 50, 362–371 (2007).

    ADS  Google Scholar 

  26. Cassisi, S. & Salaris, M. Old Stellar Populations: How to Study the Fossil Record of Galaxy Formation (Wiley-VCH, 2013).

  27. Cantiello, M. & Braithwaite, J. Envelope convection, surface magnetism, and spots in A and late B-type stars. Astrophys. J. 883, 106 (2019).

    ADS  Google Scholar 

  28. Cantiello, M. & Braithwaite, J. Magnetic spots on hot massive stars. Astron. Astrophys. 534, A140 (2011).

    ADS  Google Scholar 

  29. Moni Bidin, C. et al. Spectroscopy of horizontal branch stars in ω Centauri. Astron. Astrophys. 547, A109 (2012).

    Google Scholar 

  30. Schaefer, B. E. Astrophysics: startling superflares. Nature 485, 456–457 (2012).

    ADS  Google Scholar 

  31. Schaefer, B. E., King, J. R. & Deliyannis, C. P. Superflares on ordinary solar-type stars. Astrophys. J. 529, 1026–1030 (2000).

    ADS  Google Scholar 

  32. Schaefer, B. E. Flashes from normal stars. Astrophys. J. 337, 927 (1989).

    ADS  Google Scholar 

  33. Reed, M. D. et al. Analysis of the rich frequency spectrum of KIC 10670103 revealing the most slowly rotating subdwarf B star in the Kepler field. Mon. Not. R. Astron. Soc. 440, 3809–3824 (2014).

    ADS  Google Scholar 

  34. Balona, L. A. Flare stars across the H-R diagram. Mon. Not. R. Astron. Soc. 447, 2714–2725 (2015).

    ADS  Google Scholar 

  35. Fontaine, G., Brassard, P., Charpinet, S. & Chayer, P. The need for radiative levitation for understanding the properties of pulsating sdB stars. Mem. Soc. Astron. Ital. 77, 49 (2006).

    ADS  Google Scholar 

  36. Miller Bertolami, M. M., Battich, T., Còrsico, A. H., Christensen-Dalsgaard, J. & Althaus, L. G. Asteroseismic signatures of the helium core flash. Nat. Astron. 4, 67–71 (2020).

    ADS  Google Scholar 

  37. Landstreet, J. D., Bagnulo, S., Fossati, L., Jordan, S. & O’Toole, S. J. The magnetic fields of hot subdwarf stars. Astron. Astrophys. 541, A100 (2012).

    ADS  Google Scholar 

  38. Bagnulo, S., Fossati, L., Landstreet, J. D. & Izzo, C. The FORS1 catalogue of stellar magnetic field measurements. Astron. Astrophys. 583, A115 (2015).

    ADS  Google Scholar 

  39. Somers, G., Cao, L. & Pinsonneault, M. H. The SPOTS models: a grid of theoretical stellar evolution tracks and isochrones for testing the effects of starspots on structure and colors. Astrophys. J. 891, 29 (2020).

    ADS  Google Scholar 

  40. van Saders, J. L. et al. Weakened magnetic braking as the origin of anomalously rapid rotation in old field stars. Nature 529, 181–184 (2016).

    ADS  Google Scholar 

  41. Balona, L. A. Evidence for spots on hot stars suggests major revision of stellar physics. Mon. Not. R. Astron. Soc. 490, 2112–2116 (2019).

    ADS  Google Scholar 

  42. Balona, L. A. et al. Rotational modulation in TESS B stars. Mon. Not. R. Astron. Soc. 485, 3457–3469 (2019).

    ADS  Google Scholar 

  43. Dupuis, J., Chayer, P., Vennes, S., Christian, D. J. & Kruk, J. W. Adding more mysteries to the DA white dwarf GD 394. Astrophys. J. 537, 977–992 (2000).

    ADS  Google Scholar 

  44. Kilic, M. et al. A dark spot on a massive white dwarf. Astrophys. J. Lett 814, L31 (2015).

    ADS  Google Scholar 

  45. Koester, D. & Chanmugam, G. Review: physics of white dwarf stars. Rep. Prog. Phys. 53, 837–915 (1990).

    ADS  Google Scholar 

  46. Reding, J. S., Hermes, J. J. & Clemens, J. C. An exploration of spotted white dwarfs from K2. In Proc. 21st Eur. Workshop on White Dwarfs 1–6 (Univ. Texas Libraries, 2018).

  47. Mestel, L. The magnetic field of a contracting gas cloud. I. Strict flux-freezing. Mon. Not. R. Astron. Soc. 133, 265 (1966).

    ADS  Google Scholar 

  48. Ferrario, L., Wickramasinghe, D. T. & Kawka, A. Magnetic fields in isolated and interacting white dwarfs. Adv. Space Res. https://doi.org/10.1016/j.asr.2019.11.012 (2020).

  49. Le Fèvre, O. et al. Commissioning and performances of the VLT-VIMOS instrument. Proc. SPIE 4841, 1670–1681 (2003).

    ADS  Google Scholar 

  50. Kuijken, K. OmegaCAM: ESO’s newest imager. Messenger 146, 8–11 (2011).

    ADS  Google Scholar 

  51. Stetson, P. B. DAOPHOT: a computer program for crowded-field stellar photometry. Publ. Astron. Soc. Pacif. 99, 191 (1987).

    ADS  Google Scholar 

  52. Alard, C. & Lupton, R. H. A method for optimal image subtraction. Astrophys. J. 503, 325–331 (1998).

    ADS  Google Scholar 

  53. Montalto, M. et al. A new search for planet transits in NGC 6791. Astron. Astrophys. 470, 1137–1156 (2007).

    ADS  Google Scholar 

  54. Eastman, J., Siverd, R. & Gaudi, B. S. Achieving better than 1 minute accuracy in the heliocentric and barycentric Julian dates. Publ. Astron. Soc. Pacif. 122, 935 (2010).

    ADS  Google Scholar 

  55. Hartman, J. D. & Bakos, G. Á. VARTOOLS: a program for analyzing astronomical time-series data. Astron. Comput. 17, 1–72 (2016).

    ADS  Google Scholar 

  56. Schwarzenberg-Czerny, A. Fast and statistically optimal period search in uneven sampled observations. Astrophys. J. Lett. 460, L107 (1996).

    ADS  Google Scholar 

  57. Schwarzenberg-Czerny, A. & Beaulieu, J.-P. Efficient analysis in planet transit surveys. Mon. Not. R. Astron. Soc. 365, 165–170 (2006).

    ADS  Google Scholar 

  58. Kaluzny, J. et al. Cluster AgeS Experiment catalog of variable stars in the globular cluster ω Centauri. Astron. Astrophys. 424, 1101–1110 (2004).

    ADS  Google Scholar 

  59. Kaluzny, J. & Thompson, I. B. Variable stars in the globular cluster NGC 6752. Acta Astron. 59, 273–289 (2009).

    ADS  Google Scholar 

  60. Rozyczka, M. et al. The Cluster AgeS Experiment (CASE). Variable stars in the field of the globular cluster M22. Acta Astron. 67, 203–224 (2017).

    ADS  Google Scholar 

  61. Rozyczka, M. et al. The Cluster AgeS Experiment (CASE). Variable stars in the field of the globular cluster M10. Preprint at https://arxiv.org/abs/2001.01529 (2020).

  62. Milone, A. P. et al. The Hubble Space Telescope UV legacy survey of galactic globular clusters. III. A quintuple stellar population in NGC 2808. Astrophys. J. 808, 51 (2015).

    ADS  Google Scholar 

  63. Ferraro, F. R., Paltrinieri, B., Fusi Pecci, F., Rood, R. T. & Dorman, B. Multimodal distributions along the horizontal branch. Astrophys. J. 500, 311–319 (1998).

    ADS  Google Scholar 

  64. Bedin, L. R. et al. The anomalous Galactic globular cluster NGC 2808. Mosaic wide-field multi-band photometry. Astron. Astrophys. 363, 159–173 (2000).

    ADS  Google Scholar 

  65. Pietrinferni, A., Cassisi, S., Salaris, M. & Hidalgo, S. The BaSTI stellar evolution database: models for extremely metal-poor and super-metal-rich stellar populations. Astron. Astrophys. 558, A46 (2013).

    ADS  Google Scholar 

  66. Nardiello, D. et al. The Hubble Space Telescope UV legacy survey of galactic globular clusters—XVII. Public catalogue release. Mon. Not. R. Astron. Soc. 481, 3382–3393 (2018).

    ADS  Google Scholar 

  67. Cool, A. M. et al. HST/ACS imaging of omega Centauri: optical counterparts of Chandra X-ray sources. Astrophys. J. 763, 126 (2013).

    ADS  Google Scholar 

  68. Moni Bidin, C., Moehler, S., Piotto, G., Momany, Y. & Recio-Blanco, A. Spectroscopy of horizontal branch stars in NGC 6752. Anomalous results on atmospheric parameters and masses. Astron. Astrophys. 474, 505–514 (2007).

    ADS  Google Scholar 

  69. D’Orazi, V. et al. Lithium abundances in globular cluster giants: NGC 1904, NGC 2808, and NGC 362. Mon. Not. R. Astron. Soc. 449, 4038–4047 (2015).

    ADS  Google Scholar 

  70. Coelho, P. R. T. A new library of theoretical stellar spectra with scaled-solar and α-enhanced mixtures. Mon. Not. R. Astron. Soc. 440, 1027–1043 (2014).

    ADS  Google Scholar 

  71. Jones, D. & Boffin, H. M. J. Binary stars as the key to understanding planetary nebulae. Nat. Astron. 1, 0117 (2017).

    ADS  Google Scholar 

  72. Prša, A. et al. Physics of eclipsing binaries. II. Toward the increased model fidelity. Astrophys. J. Suppl. 227, 29 (2016).

    ADS  Google Scholar 

  73. Bertelli, G., Girardi, L., Marigo, P. & Nasi, E. Scaled solar tracks and isochrones in a large region of the Z-Y plane. I. From the ZAMS to the TP-AGB end for 0.15–2.5 M stars. Astron. Astrophys. 484, 815–830 (2008).

    ADS  Google Scholar 

  74. Hillwig, T. C. et al. Observational confirmation of a link between common envelope binary interaction and planetary nebula shaping. Astrophys. J. 832, 125 (2016).

    ADS  Google Scholar 

  75. Gilliland, R. L. et al. A lack of planets in 47 tucanae from a Hubble Space Telescope search. Astrophys. J. Lett. 545, L47–L51 (2000).

    ADS  Google Scholar 

  76. Nascimbeni, V., Bedin, L. R., Piotto, G., De Marchi, F. & Rich, R. M. An HST search for planets in the lower main sequence of the globular cluster NGC 6397. Astron. Astrophys. 541, A144 (2012).

    ADS  Google Scholar 

  77. Wallace, J. J., Hartman, J. D. & Bakos, G. Á. A search for transiting planets in the globular cluster M4 with K2: candidates and occurrence limits. Astron. J. 159, 106 (2020).

    ADS  Google Scholar 

  78. Santander-Garca, M. et al. The double-degenerate, super-Chandrasekhar nucleus of the planetary nebula Henize 2-428. Nature 519, 63–65 (2015).

    ADS  Google Scholar 

  79. Vos, J., Németh, P., Vučković, M., Østensen, R. & Parsons, S. Composite hot subdwarf binaries—I. The spectroscopically confirmed sdB sample. Mon. Not. R. Astron. Soc. 473, 693–709 (2018).

    ADS  Google Scholar 

  80. Grundahl, F., Catelan, M., Landsman, W. B., Stetson, P. B. & Andersen, M. I. Hot horizontal-branch stars: the ubiquitous nature of the “jump” in Strömgren u, low gravities, and the role of radiative levitation of metals. Astrophys. J. 524, 242–261 (1999).

    ADS  Google Scholar 

  81. Pietrukowicz, P. et al. Blue large-amplitude pulsators as a new class of variable stars. Nat. Astron. 1, 0166 (2017).

    ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge discussions with S. Bagnulo, A. Bressan, A. Bianchini, A. Renzini and P. Ochner, and we thank M. Dima for help in producing movies of the stellar spots. D.J. acknowledges support from the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation and Universities (MCIU) and the European Regional Development Fund (FEDER) under grant AYA2017-83383-P. D.J. also acknowledges support under grant P/308614 financed by funds transferred from the Spanish Ministry of Science, Innovation and Universities, charged to the General State Budgets and with funds transferred from the General Budgets of the Autonomous Community of the Canary Islands by the Ministry of Economy, Industry, Trade and Knowledge.

Author information

Authors and Affiliations

Authors

Contributions

Y.M. and S.Z. designed the study and coordinated the activity. Y.M., S.Z., M.M., H.M.J.B., D.J., M.G., I.S., L.M., C.M.B., V.D’O. and H.L. reduced and analysed the data. M.M. and S.Z. developed the spot modelling programme and related simulations. S.C., L.G. and D.J. provided theoretical modelling. G.P., A.P.M., P.B.S., Y.B. and E.M. contributed to the assembly of the photometric catalogues. Y.M. wrote the paper. S.Z., D.J., H.M.J.B., I.S., S.C., L.G. and H.L. contributed to the discussion and presentation of the paper. All authors contributed to the discussion of the results and commented on the manuscript.

Corresponding author

Correspondence to Y. Momany.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Extended data

Extended Data Fig. 1 Estimating the EHB variable frequency.

Upper-left panel displays our NGC2808 VIMOS diagram highlighting all identified variables, the box delimits the EHB sample used to normalize the EHB variable stars frequency, and a ZAHB model is used to confirm the EHB variable stars temperature range (~17,500–24,500 K). The two EHB variable stars with open orange symbols are confirmed EHB stars, as identified in the higher resolution HST catalogue (upper-right panel). Lower panels show the position of all EHB variable stars identified in HST diagrams.

Extended Data Fig. 2 No binarity signature detected in the NGC2808 vEHB-1 variable.

Upper panel displays the phased Hα radial velocity curve of a comparison RR Lyrae star proving a successful detection of velocity variations in the data-set. The lower panel displays the velocity curve of our photometric variable vEHB-1 present in the same data-set. The error bars display the 1-σ error (~3.5 km/s) estimated at the vEHB-1 luminosity. No substantial velocity variations for vEHB-1 are observed. For clarity, the NGC2808 average radial velocity has been subtracted.

Extended Data Fig. 3 No binarity signature detected in the NGC6752 vEHB-1/2 variables.

Upper panel displays the phased Hγ,δ radial velocity curve of a comparison SX Phoenicis star59 proving a successful detection of velocity variations in the data-set. The lower panels display the velocity curves of the 2 EHB photometric variables (and the candidate EHB photometric59 variable vEHB-4/V17) present in the same data-set. The error bars display the 1-σ error (~3.0 km/s estimated at the vEHB-1 luminosity. No substantial velocity variations for the vEHB-1/2 are observed. For clarity, the NGC6752 average radial velocity has been subtracted.

Extended Data Fig. 4 The long-term stable variability of vEHB-1 in NGC6752.

Bottom plot shows all the uSDSS OmegaCAM measurements of vEHB-1 collected over a six-year period. The upper plots show the phased light curves sub-divided over six years. A typical 1-σ photometric error bar is plotted. The solid light-blue line is the best fitting model (Period  19.5 days) calculated using the six years’ measurements.

Extended Data Fig. 5 The aperiodic long-term Padua variables in NGC6752.

Light blue squares display the six-year archival OmegaCAM@VST data, while black squares display those originating from our three-year monitoring. A typical 1-σ photometric error bar is plotted. The Padua-2 mini-burst is incomplete but discernible.

Extended Data Fig. 6 Rotational variability and superflare event in a Galactic field sdB star.

Upper panel displays the folded TESS light curve of a Galactic field sdB star showing α2CVn spot-induced variability. Filled squares are the 2.5-σ clipped median values every 300 data points, while the error bars reflect the 1-σ RMS of the clipped flux values. Lower panel proves the occurrence of an energetic (~1035 erg) superflare event in this field sdB star. Both phenomena necessitate the presence of magnetic fields.

Supplementary information

Supplementary Information

Supplementary Figures 1-7, Supplementary Discussion, and Supplementary Table 1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Momany, Y., Zaggia, S., Montalto, M. et al. A plague of magnetic spots among the hot stars of globular clusters. Nat Astron 4, 1092–1101 (2020). https://doi.org/10.1038/s41550-020-1113-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41550-020-1113-4

This article is cited by

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