Skip to main content
SpringerLink
Log in

Relation between gravitational mass and baryonic mass for non-rotating and rapidly rotating neutron stars

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

With a selected sample of neutron star (NS) equations of state (EOSs) that are consistent with the current observations and have a range of maximum masses, we investigate the relations between NS gravitational mass Mg and baryonic mass Mb, and the relations between the maximum NS mass supported through uniform rotation (Mmax) and that of nonrotating NSs (MTOV). We find that for an EOS-independent quadratic, universal transformation formula \(({M_b} = {M_g}\; + A\; \times \;M_g^2)\), the best-fit A value is 0.080 for non-rotating NSs, 0.064 for maximally rotating NSs, and 0.073 when NSs with arbitrary rotation are considered. The residual error of the transformation is ∼ 0.1M ⊙ for non-spin or maximum-spin, but is as large as ∼ 0.2M ⊙ for all spins. For different EOSs, we find that the parameter A for non-rotating NSs is proportional to \(R_{1.4}^{- 1}\) (where R 1.4 is NS radius for 1.4M ⊙ in units of km). For a particular EOS, if one adopts the best-fit parameters for different spin periods, the residual error of the transformation is smaller, which is of the order of 0.01M ⊙ for the quadratic form and less than 0.01M ⊙ for the cubic form (\(({M_b} = {M_g}\; + \,{A_1}\; \times \;M_g^2\; + \,{A_2}\; \times \;M_g^3)\)). We also find a very tight and general correlation between the normalized mass gain due to spin Δm = (M max − M TOV)/M TOV and the spin period normalized to the Keplerian period \(\mathcal{P}\), i.e., \({\log _{10}}{\rm{\Delta}}m = (- 2.74 \pm 0.05){\log _{10}}{\mathcal P} + {\log _{10}}(0.20 \pm 0.01)\), which is independent of EOS models. These empirical relations are helpful to study NS-NS mergers with a long-lived NS merger product using multi-messenger data. The application of our results to GW170817 is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. M. Lattimer, The nuclear equation of state and neutron star masses, Annu. Rev. Nucl. Part. Sci. 62(1), 485 (2012)

    Article  ADS  Google Scholar 

  2. A. K. Harding, The neutron star zoo, Front. Phys. 8(6), 679 (2013)

    Article  ADS  Google Scholar 

  3. J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, M. H. van Kerkwijk, M. Kramer, C. Bassa, V. S. Dhillon, T. Driebe, J. W. T. Hessels, V. M. Kaspi, V. I. Kondratiev, N. Langer, T. R. Marsh, M. A. McLaughlin, T. T. Pennucci, S. M. Ransom, I. H. Stairs, J. van Leeuwen, J. P. W. Verbiest, and D. G. Whelan, A massive pulsar in a compact relativistic binary, Science 340(6131), 1233232 (2013)

    Article  Google Scholar 

  4. S. Lawrence, J. G. Tervala, P. F. Bedaque, and M. C. Miller, An upper bound on neutron star masses from models of short gamma-ray bursts, Astrophys. J. 808(2), 186 (2015)

    Article  ADS  Google Scholar 

  5. C. L. Fryer, K. Belczynski, E. Ramirez-Ruiz, S. Rosswog, G. Shen, and A. W. Steiner, The fate of the compact remnant in neutron star mergers, Astrophys. J. 812(1), 24 (2015)

    Article  ADS  Google Scholar 

  6. B. Margalit and B. D. Metzger, Constraining the maximum mass of neutron stars from multi-messenger observations of GW170817, Astrophys. J. 850(2), L19 (2017)

    Article  ADS  Google Scholar 

  7. L. Rezzolla, E. R. Most, and L. R. Weih, Using gravitational-wave observations and quasi-universal relations to constrain the maximum mass of neutron stars, Astrophys. J. 852(2), L25 (2018)

    Article  ADS  Google Scholar 

  8. M. Ruiz, S. L. Shapiro, and A. Tsokaros, GW170817, general relativistic magnetohydrodynamic simulations, and the neutron star maximum mass, Phys. Rev. D 97(2), 021501 (2018)

    Article  ADS  Google Scholar 

  9. A. Rowlinson, P. T. O’Brien, N. R. Tanvir, B. Zhang, P. A. Evans, N. Lyons, A. J. Levan, R. Willingale, K. L. Page, O. Onal, D. N. Burrows, A. P. Beardmore, T. N. Ukwatta, E. Berger, J. Hjorth, A. S. Fruchter, R. L. Tunnicliffe, D. B. Fox, and A. Cucchiara, The unusual X-ray emission of the short Swift GRB 090515: Evidence for the formation of a magnetar? Mon. Not. R. Astron. Soc. 409(2), 531 (2010)

    Article  ADS  Google Scholar 

  10. A. Rowlinson, P. T. O’Brien, B. D. Metzger, N. R. Tanvir, and A. J. Levan, Signatures of magnetar central engines in short GRB light curves, Mon. Not. R. Astron. Soc. 430(2), 1061 (2013)

    Article  ADS  Google Scholar 

  11. H. J. Lü, B. Zhang, W. H. Lei, Y. Li, and P. Lasky, The millisecond magnetar central engine in short GRBs, Astrophys. J. 805(2), 89 (2015)

    Article  ADS  Google Scholar 

  12. P. D. Lasky, B. Haskell, V. Ravi, E. J. Howell, and D. M. Coward, Nuclear equation of state from observations of short gamma-ray burst remnants, Phys. Rev. D 89(4), 047302 (2014)

    Article  ADS  Google Scholar 

  13. V. Ravi and P. D. Lasky, The birth of black holes: neutron star collapse times, gamma-ray bursts and fast radio bursts, Mon. Not. R. Astron. Soc. 441(3), 2433 (2014)

    Article  ADS  Google Scholar 

  14. H. Gao, B. Zhang, and H. J. Lü, Constraints on binary neutron star merger product from short GRB observations, Phys. Rev. D 93(4), 044065 (2016)

    Article  ADS  Google Scholar 

  15. A. Li, B. Zhang, N. B. Zhang, H. Gao, B. Qi, and T. Liu, Internal X-ray plateau in short GRBs: Signature of supramassive fast-rotating quark stars? Phys. Rev. D 94(8), 083010 (2016)

    Article  ADS  Google Scholar 

  16. J. M. Lattimer and M. Prakash, Neutron star observations: Prognosis for equation of state constraints, Phys. Rep. 442(1–6), 109 (2007)

    Article  ADS  Google Scholar 

  17. F. X. Timmes, S. E. Woosley, and T. A. Weaver, The neutron star and black hole initial mass function, Astrophys. J. 457, 834 (1996)

    Article  ADS  Google Scholar 

  18. C. L. Fryer, K. Belczynski, G. Wiktorowicz, M. Dominik, V. Kalogera, and D. E. Holz, Compact remnant mass function: Dependence on the explosion mechanism and metallicity, Astrophys. J. 749(1), 91 (2012)

    Article  ADS  Google Scholar 

  19. O. Pejcha and T. A. Thompson, The landscape of the neutrino mechanism of core-collapse supernovae: Neutron star and black hole mass functions, explosion energies, and nickel yields, Astrophys. J. 801(2), 90 (2015)

    Article  ADS  Google Scholar 

  20. J. M. Lattimer and A. Yahil, Analysis of the neutrino events from supernova 1987A, Astrophys. J. 340, 426 (1989)

    Article  ADS  Google Scholar 

  21. A. Burrows, D. Klein, and R. Gandhi, The future of supernova neutrino detection, Phys. Rev. D 45(10), 3361 (1992)

    Article  ADS  Google Scholar 

  22. J. Gava, J. Kneller, C. Volpe, and G. C. McLaughlin, Dynamical collective calculation of supernova neutrino signals, Phys. Rev. Lett. 103(7), 071101 (2009)

    Article  ADS  Google Scholar 

  23. G. Camelio, A. Lovato, L. Gualtieri, O. Benhar, J. A. Pons, and V. Ferrari, Evolution of a proto-neutron star with a nuclear many-body equation of state: Neutrino luminosity and gravitational wave frequencies, Phys. Rev. D 96(4), 043015 (2017)

    Article  ADS  Google Scholar 

  24. S. Rosswog, T. Piran, and E. Nakar, The multimessenger picture of compact object encounters: Binary mergers versus dynamical collisions, Mon. Not. R. Astron. Soc. 430(4), 2585 (2013)

    Article  ADS  Google Scholar 

  25. A. Bauswein, S. Goriely, and H. T. Janka, Systematics of dynamical mass ejection, nucleosynthesis, and radioactively powered electromagnetic signals from neutron-star mergers, Astrophys. J. 773(1), 78 (2013)

    Article  ADS  Google Scholar 

  26. K. Hotokezaka, K. Kiuchi, K. Kyutoku, H. Okawa, Y. Sekiguchi, M. Shibata, and K. Taniguchi, Mass ejection from the merger of binary neutron stars, Phys. Rev. D 87(2), 024001 (2013)

    Article  ADS  Google Scholar 

  27. R. Fernandez, D. Kasen, B. D. Metzger, and E. Quataert, Outflows from accretion discs formed in neutron star mergers: effect of black hole spin, Mon. Not. R. Astron. Soc. 446(1), 750 (2015)

    Article  ADS  Google Scholar 

  28. C. Y. Song, T. Liu, and A. Li, Outflows from black hole hyperaccretion systems: Short and long-short gamma-ray bursts and “quasi-supernovae”, Mon. Not. R. Astron. Soc. 477(2), 2173 (2018)

    Article  ADS  Google Scholar 

  29. B. P. Abbott, et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017)

    Article  ADS  Google Scholar 

  30. D. Radice, A. Perego, S. Bernuzzi, and B. Zhang, Long-lived remnants from binary neutron star mergers, Mon. Not. R. Astron. Soc. 481(3), 3670 (2018)

    Article  ADS  Google Scholar 

  31. J. M. Lattimer and M. Prakash, Neutron star structure and the equation ofstate, Astrophys. J. 550(1), 426 (2001)

    Article  ADS  Google Scholar 

  32. M. Coughlin, T. Dietrich, K. Kawaguchi, S. Smartt, C. Stubbs, and M. Ujevic, Toward rapid transient identification and characterization of kilonovae, Astrophys. J. 849(1), 12 (2017)

    Article  ADS  Google Scholar 

  33. G. Bozzola, N. Stergioulas, and A. Bauswein, Universal relations for differentially rotating relativistic stars at the threshold to collapse, Mon. Not. R. Astron. Soc. 474(3), 3557 (2018)

    Article  ADS  Google Scholar 

  34. A. M. Studzińska, M. Kucaba, D. Gondek-Rosińska, L. Villain, and M. Ansorg, Effect of the equation of state on the maximum mass of differentially rotating neutron stars, Mon. Not. R. Astron. Soc. 463(3), 2667 (2016)

    Article  ADS  Google Scholar 

  35. D. Gondek-Rosińska, I. Kowalska, L. Villain, M. Ansorg, and M. Kucaba, A New View on the Maximum Mass of Differentially Rotating Neutron Stars, Astrophys. J. 837(1), 58 (2017)

    Article  ADS  Google Scholar 

  36. A. Bauswein and N. Stergioulas, Semi-analytic derivation of the threshold mass for prompt collapse in binary neutron-star mergers, Mon. Not. R. Astron. Soc. 471(4), 4956 (2017)

    Article  ADS  Google Scholar 

  37. L. R. Weih, E. R. Most, and L. Rezzolla, On the stability and maximum mass of differentially rotating relativistic stars, Mon. Not. R. Astron. Soc. 473(1), L126 (2018)

    Article  ADS  Google Scholar 

  38. E. M. Butterworth and J. R. Ipser, On the structure and stability of rapidly rotating fluid bodies in general relativity (I): The numerical method for computing structure and its application to uniformly rotating homogeneous bodies, Astrophys. J. 204, 200 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  39. N. Stergioulas and J. L. Friedman, Comparing models of rapidly rotating relativistic stars constructed by two numerical methods, Astrophys. J. 444, 306 (1995)

    Article  ADS  Google Scholar 

  40. F. Douchin and P. Haensel, A unified equation of state of dense matter and neutron star structure, Astronomy and Astrophysics 380, 151 (2001)

    Article  ADS  Google Scholar 

  41. R. B. Wiringa, V. Fiks, and A. Fabrocini, Equation of state for dense nucleon matter, Phys. Rev. C 38(2), 1010 (1988)

    Article  ADS  Google Scholar 

  42. A. Akmal and V. R. Pandharipande, Spin-isospin structure and pion condensation in nucleon matter, Phys. Rev. C 56(4), 2261 (1997)

    Article  ADS  Google Scholar 

  43. S. Goriely, N. Chamel, and J. M. Pearson, Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas (XII): Stiffness and stability of neutron-star matter, Phys. Rev. C 82(3), 035804 (2010)

    Article  ADS  Google Scholar 

  44. S. Typel, G. Röpke, T. Klähn, D. Blaschke, and H. H. Wolter, Composition and thermodynamics of nuclear matter with light clusters, Phys. Rev. C 81(1), 015803 (2010)

    Article  ADS  Google Scholar 

  45. H. Müther, M. Prakash, and T. L. Ainsworth, The nuclear symmetry energy in relativistic Brueckner-Hartree-Fock calculations, Phys. Lett. B 199(4), 469 (1987)

    Article  ADS  Google Scholar 

  46. H. Müller and B. D. Serot, Relativistic mean-field theory and the high-density nuclear equation of state, Nucl. Phys. A. 606(3–4), 508 (1996)

    Article  ADS  Google Scholar 

  47. G. B. Cook, S. L. Shapiro, and S. A. Teukolsky, Rapidly rotating neutron stars in general relativity: Realistic equations of state, Astrophys. J. 424, 823 (1994)

    Article  ADS  Google Scholar 

  48. J. P. Lasota, P. Haensel, and M. A. Abramowicz, Fast rotation of neutron stars, Astrophys. J. 456, 300 (1996)

    Article  ADS  Google Scholar 

  49. C. Breu and L. Rezzolla, Maximum mass, moment of inertia and compactness of relativistic stars, Mon. Not. R. Astron. Soc. 459(1), 646 (2016)

    Article  ADS  Google Scholar 

  50. Y. W. Yu, L. D. Liu, and Z. G. Dai, A long-lived remnant neutron star after GW170817 inferred from its associated kilonova, Astrophys. J. 861(2), 114 (2018)

    Article  ADS  Google Scholar 

  51. S. Z. Li, L. D. Liu, Y. W. Yu, and B. Zhang, What powered the optical transient AT2017gfo associated with GW170817? Astrophys. J. 861(2), L12 (2018)

    Article  ADS  Google Scholar 

  52. S. Ai, H. Gao, Z. G. Dai, X. F. Wu, A. Li, B. Zhang, and M.Z. Li, The allowed parameter space of a long-lived neutron star as the merger remnant of GW170817, Astrophys. J. 860(1), 57 (2018)

    Article  ADS  Google Scholar 

  53. L. Piro, E. Troja, B. Zhang, G. Ryan, H. van Eerten, R. Ricci, M. H. Wieringa, A. Tiengo, N. R. Butler, S. B. Cenko, O. D. Fox, H. G. Khandrika, G. Novara, A. Rossi, and T. Sakamoto, A long-lived neutron star merger remnant in GW170817: Constraints and clues from X-ray observations, Mon. Not. R. Astron. Soc. 483(2), 1912 (2019)

    Article  ADS  Google Scholar 

  54. B. D. Metzger, Welcome to the multi-messenger era! Lessons from a neutron star merger and the landscape ahead, arXiv: 1710.05931 (2017)

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11722324, 11603003, 11633001, 11690024, and 11873040, the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23040100 and the Fundamental Research Funds for the Central Universities. A. B. acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No. 759253 and by the Sonderforschungsbereich SFB 881 “The Milky WaySystem” (Subproject A10) of the German Research Foundation (DFG).

Author information

Authors and Affiliations

  1. Department of Astronomy, Beijing Normal University, Beijing, 100875, China

    He Gao, Shun-Ke Ai & Zhou-Jian Cao

  2. Department of Physics and Astronomy, University of Nevada, Las Vegas, NV, 89154, USA

    Shun-Ke Ai & Bing Zhang

  3. Department of Astronomy, Xiamen University, Xiamen, 361005, China

    Zhen-Yu Zhu & Ang Li

  4. Institute for Theoretical Physics, Max-von-Laue-Strasse 1, 604 38, Frankfurt, Germany

    Zhen-Yu Zhu

  5. Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai, 264209, China

    Nai-Bo Zhang

  6. GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291, Darmstadt, Germany

    Andreas Bauswein

Authors
  1. He Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shun-Ke Ai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhou-Jian Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhen-Yu Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Nai-Bo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Andreas Bauswein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to He Gao.

Additional information

arXiv: 1905.03784.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, H., Ai, SK., Cao, ZJ. et al. Relation between gravitational mass and baryonic mass for non-rotating and rapidly rotating neutron stars. Front. Phys. 15, 24603 (2020). https://doi.org/10.1007/s11467-019-0945-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-019-0945-9

Keywords

Search

Navigation