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Accretion disk in the Hartle–Thorne spacetime

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Abstract

We consider the circular motion of test particles in the gravitational field of a rotating deformed object described by the Hartle–Thorne metric. This metric represents an approximate solution to the vacuum Einstein field equations, accurate to second order in the angular momentum J and to first order in the mass quadrupole moment Q. We calculate the orbital parameters of neutral test particles on circular orbits (in accretion disks) such as angular velocity, \(\Omega\), total energy, E, angular momentum, L, and radius of the innermost stable circular orbit, \(R_{ISCO}\), as functions of the total mass, M, spin parameter, \(j=J/M^2\) and quadrupole parameter, \(q=Q/M^3\), of the source. We use the Novikov-Thorne-Page thin accretion disk model to investigate the characteristics of the disk. In particular, we analyze in detail the radiative flux, differential luminosity, and spectral luminosity of the accretion disk, which are the quantities that can be measured experimentally. We compare our results with those obtained in the literature for the Schwarzschild and Kerr metrics and the q-metric. It turns out that the Hartle–Thorne metric and the Kerr metric lead to similar results for the predicted flux and the differential and spectral luminosities, whereas the q-metric predicts different values. We compare the predicted values of M, j, and q with those of realistic neutron star models. Furthermore, we compare the values of \(R_{ISCO}\) with the static and rotating radii of neutron stars.

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Data Availability Statement

All data generated or analyzed during this study are included in this article.

Notes

  1. Nevertheless, the incoming observations can also shed some new light on possible deviations from general relativity in the incoming years. Consequently, the possible existence of exotic compact objects cannot be ignored, as most observations of black hole candidates do not allow one to study the geometry near such astrophysical sources yet.

  2. The luminosity of the accretion disk in the Kerr spacetime has been extensively investigated in the scientific literature.

  3. It was shown that the static Hartle–Thorne solution with \(j=0\) reduces to the approximate Erez-Rosen solution in the limiting case of a small deformation [27]. However, before finding the coordinate transformations to establish the relationship between the parameters of the solutions, it was necessary to generalize the Erez-Rosen metric by applying a Zipoy-Voorhees transformation, which introduces a new parameter that must be fixed in order to obtain the required transformations [27].

  4. Some details about the Zipoy-Voorhees transformations that are necessary to compare the Erez-Rosen and Hartle–Thorne solutions can be found in Ref. [38].

  5. This solution contains as a specific case the solution combining both Erez Rosen and Kerr solutions. Using the prescriptions provided in Refs. [27, 34], it was demonstrated that this particular Quevedo-Mashhoon solution in the limiting case of slow rotation and small deformation was equivalent to the Hartle–Thorne solution [33].

  6. Additionally, it was shown that the so-called Sedrakyan-Chubaryan solution is equivalent to the the Hartle-Thorne solution [48].

  7. Sometimes, this metric is known in the literature as the Zipoy-Voorhees metric, \(\delta\)-metric and \(\gamma\)-metric.

  8. The formulas for the angular velocity \(\Omega\), angular momentum L, and energy E in the Hartle–Thorne spacetime were initially derived in Ref. [52]. However, it is important to exercise caution when referring to [52] due to the presence of certain typographical errors.

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Acknowledgements

YeK acknowledges Grant No. AP19575366, TK acknowledges Grant No. AP19174979, KB and OL acknowledge Grant No. AP19680128, MM and AU acknowledge Grant No. BR21881941 from the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan. KB is grateful to the Departments of Physics and Mathematics at the University of Camerino (UniCam) for the academic mobility provided by Erasmus+ program “I CAMERIN01” (2022-1-IT02-KA171-HED-000073309), during the period in which this manuscript has been written. He is particularly grateful to prof. Carlo Lucheroni for his economical support at UniCam. OL is grateful to INAF, National Institute of Astrophysics, for the support and in particular to Roberto della Ceca, Gaetano Telesio and Filippo M. Zerbi for discussions. It is also a pleasure to acknowledge Carlo Cafaro and Roberto Giambò for fruitful discussions on the subject of this paper. The work of HQ was partially supported by UNAM-DGAPA-PAPIIT, Grant No. 114520, and CONACYT-Mexico, Grant No. A1-S-31269.

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Author notes

  1. Talgar Konysbayev, Yergali Kurmanov, Orlando Luongo, Marco Muccino, Hernando Quevedo, Ainur Urazalina have contributed equally to this work.

Authors and Affiliations

  1. National Nanotechnology Laboratory of Open Type, Al-Farabi av. 71, Almaty, 050040, Kazakhstan

    Kuantay Boshkayev, Talgar Konysbayev, Yergali Kurmanov & Ainur Urazalina

  2. Al-Farabi Kazakh National University, Al-Farabi av. 71, Almaty, 050040, Kazakhstan

    Kuantay Boshkayev, Talgar Konysbayev, Yergali Kurmanov, Orlando Luongo, Marco Muccino, Hernando Quevedo & Ainur Urazalina

  3. Institute of Nuclear Physics, Ibragimova, 1, Almaty, 050032, Kazakhstan

    Kuantay Boshkayev, Marco Muccino & Ainur Urazalina

  4. International Engineering Technological University, Al-Farabi av. 93 G/5, Almaty, 050060, Kazakhstan

    Yergali Kurmanov

  5. Scuola di Scienze e Tecnologie, Divisione di Fisica, Università di Camerino, Via Madonna delle Carceri 9, Camerino, 62032, Italy

    Orlando Luongo & Marco Muccino

  6. SUNY Polytechnic Institute, 13502 Utica, New York, 10001, USA

    Orlando Luongo

  7. Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via Alessandro Pascoli, 23c, Perugia, 06123, Italy

    Orlando Luongo

  8. INAF - Osservatorio Astronomico di Brera, Via Brera 28, Milano, 20121, Italy

    Orlando Luongo

  9. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Cto. Exterior S/N, C.U., Coyoacán, Mexico, 34000, Mexico

    Hernando Quevedo

  10. Dipartimento di Fisica and ICRA, Università di Roma “La Sapienza”, Città Universitaria di Roma - Sapienza, Piazzale Aldo Moro, 2, Roma, 00185, Italy

    Hernando Quevedo

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  1. Kuantay Boshkayev
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  2. Talgar Konysbayev
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Correspondence to Yergali Kurmanov.

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Boshkayev, K., Konysbayev, T., Kurmanov, Y. et al. Accretion disk in the Hartle–Thorne spacetime. Eur. Phys. J. Plus 139, 273 (2024). https://doi.org/10.1140/epjp/s13360-024-05072-8

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