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Stellar structure, magnetism, and the variational principle

A. Čadež* and A. Mohorič

M. Calvani

  • *Contact author: andrej.cadez@fmf.uni-lj.si
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Phys. Rev. D 114, 023006 – Published 6 July, 2026

DOI: https://doi.org/10.1103/z77c-cy6d

Abstract

Matter interacts with two fundamental long-range forces: gravity and electromagnetism. While elementary building blocks of matter always contribute to the gravitational potential, the electromagnetic vector potential was expected to cancel out in large systems because of the symmetry between positive and negative charges. Yet, long range electromagnetic phenomena are present in astrophysical systems, so that the cancellation cannot be considered as perfect. The aim of this paper is to find a possible model for stationary aggregation of matter to make a “star” which consistently includes angular momentum and electromagnetic phenomena. We recast the standard polytropic stellar model as a variational problem and extend it to include the kinetic energy of (rigid) rotation and the electromagnetic energy of interaction between positively and negatively charged baryonic matter. We argue that the electromagnetic contribution to the action should be represented by the minimal energy needed to induce the stellar magnetic dipole moment. This energy consists of two parts; the pure electromagnetic energy, which can be expressed as a surface integral, and the free-energy difference between the magnetized and unmagnetized state of matter. This second part of the action is formulated by considering the magnetized quantum state of a Fermi gas of electrons in a sea of cold ions. Differential forms calculus is a useful tool to perform the mathematical analysis. Our mathematical model includes electromagnetism to stellar modeling in a way that is consistent with the linearized version of general relativity. The complete model presents a complex open boundary problem. The problem can be solved exactly under simplifying circumstances. The distribution of stellar objects in the phase diagram, based on the simplified model, exhibits a pattern which may prompt research to provide a deeper understanding of the natural balance between matter, gravitation, and electromagnetism.

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References (33)

  1. T. G. Cowling, Nature (London) 201, 1061 (1964).
  2. V. C. A. Ferraro, Astrophys. J. 119, 407 (1954).
  3. J. Fuller and S. Mathis, Mon. Not. R. Astron. Soc. 520, 5573 (2023).
  4. Y. Kojima, S. Kisaka, and K. Fujisawa, Mon. Not. R. Astron. Soc. 502, 2097 (2021).
  5. J. P. Ostriker and F. D. A. Hartwick, Astrophys. J. 153, 797 (1968).
  6. R. H. Cameron and M. Schüssler, Space Sci. Rev. 219, 60 (2023).
  7. J. J. Monaghan and I. W. Roxburgh, Mon. Not. R. Astron. Soc. 131, 13 (1965).
  8. J. J. Monaghan, Mon. Not. R. Astron. Soc. 131, 105 (1965).
  9. M. Harwit, Astrophysical Concepts (Springer, New York, 2006), 10.1007/978-0-387-33228-4.
  10. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).
  11. A. Čadež and M. Javornik, Astrophys. Space Sci. 77, 299 (1981).
  12. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).
  13. R. E. Rosensweig, Annu. Rev. Fluid Mech. 19, 437 (1987).
  14. A. Irbah, R. Mecheri, L. Damé, and Djelloul Djafer, Astrophys. J. Lett. 875, L26 (2019).
  15. G. F. Burgio and I. Vidaña, Universe 6, 119 (2020).
  16. M. Caleb, I. Heywood, K. Rajwade et al., Nat. Astron. 6, 828 (2022).
  17. D. N. Sob’yanin, Phys. Rev. D 107, L081301 (2023).
  18. M. Kilic, A. Kosakowski, A. G. Moss, P. Bergeron, and A. A. Conly, Astrophys. J. Lett. 923, L6 (2021).
  19. B. Vermette, C. Salcedo, K. Mori et al., Astrophys. J. 954, 138 (2023).
  20. L. L. Amorim, S. O. Kepler, B. Külebi , S. Jordan, and A. D. Romero, Astrophys. J. 944, 56 (2023).
  21. P. Dufour, S. Blouin, S. Coutu et al., Proceedings of the 20th European White Dwarf Workshop (2017), arXiv:1610.00986.
  22. F. Hardy, P. Dufour, and S. Jordan, Mon. Not. R. Astron. Soc. 520, 6135 (2023).
  23. G. Jewett, M. Kilic, P. Bergeron , A. Moss, S. Blouin, W. R. Brown, A. Kosakowski, S. Toonen, and M. A. Agüeros, Astrophys. J. 974, 12 (2024).
  24. M. S. Hernandez, M. R. Schreiber, J. D. Landstreet, S. Bagnulo, S. G. Parsons, M. Chavarria, O. Toloza, and K. J. Bell, Mon. Not. R. Astron. Soc. 528, 6056 (2024).
  25. T. Shenar, G. A. Wade, P. Marchant et al., Science 381, 761 (2023).
  26. J. P. Rozelot, A. G. Kosovichev, and A. Kilcik, arXiv:2501.10821.
  27. M. D. Fivian, H. S. Hudson, R. P. Lin, and H. Jabran Zahid, Science 322, 560 (2008).
  28. T. J. Okamoto and T. Sakurai, Astrophys. J. Lett. 852, L16 (2018).
  29. A. A. Abdo, M. Ackermann, W. B. Atwood et al., Astrophys. J. 696, 1084 (2009).
  30. A. A. Abdo, M. Ackermann, M. Ajello et al., Astrophys. J. 713, 154 (2010).
  31. P. S. Ray and P. M. S. Parkinson, High-Energy Emission from Pulsars and their Systems (Springer, Berlin, Heidelberg, 2011), p. 37, 10.1007/978-3-642-17251-9_3.
  32. M. P. Surnis, K. M. Rajwade, B. W. Stappers et al., Mon. Not. R. Astron. Soc. 526, L143 (2023).
  33. M. Zhou, V. Zhdankin, M. W. Kunz , Nuno F. Loureiro, and Dmitri A. Uzdensky, Proc. Natl. Acad. Sci. U.S.A. 119, e2119831119 (2022).

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Phys. Rev. D 114, 023006– Published 6 July, 2026
  • Received 17 October 2025
  • Revised 21 March 2026
  • Accepted 14 May 2026
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