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Innermost Stable Circular Orbits and Epicyclic Frequencies Around a Magnetized Neutron Star

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Innermost Stable Circular Orbits and Epicyclic Frequencies Around a Magnetized Neutron Star 129129 Univ.Univ. Sci. Sci. 201 2013,4, V Vol. 1 (): 3 doi:doi: 10.11144/Javeriana.SC1 10.11144/Jav -. FreelyFreely available available on on line line ORIGINALORIGINAL ARTICLEPAPER Innermost stable circular orbits and epicyclic frequencies around a magnetized neutron star Andrés F. Gutiérrez-Ruiz1, Leonardo A. Pachón1, César A. Valenzuela-Toledo2 B Abstract A full-relativistic approach is used to compute the radius of the innermost stable circular orbit (ISCO), the Keplerian, frame-dragging, precession and oscillation frequencies of the radial and vertical motions of neutral test particles orbiting the equatorial plane of a magnetized neutron star. The space-time around the star is modelled by the six parametric solution derived by Pachón et al. (2012) It is shown that the inclusion of an intense magnetic field, such as the one of a neutron star, have non-negligible effects on the above physical quantities, and therefore, its inclusion is necessary in order to obtain a more accurate and realistic description of physical processes, such as the dynamics of accretion disks, occurring in the neighbourhood of this kind of objects. The results discussed here also suggest that the consideration of strong magnetic fields may introduce non-negligible corrections in, e.g., the relativistic precession model and therefore on the predictions made on the mass of neutron stars. Keywords: Relativistic precession frequencies; innermost stable circular orbits; neutron stars. Edited by Humberto Rafeiro & Alberto Acosta Introduction 1 Grupo de Física Atómica y Molecular, Instituto de Física, Fac- ultad de Ciencias Exactas y Naturales, Universidad de Antioquia Stellar models are generally based on the Newto-nian UdeA; Calle 70 No. 52-21, Medellín, Colombia. universal law of gravitation. However, consid-ering 2 Departamento de Física, Universidad del Valle, A.A. 25360, the size, mass or density, there are five classes of Santiago de Cali, Colombia. stellar configurations w here o ne c an recognize significant d eviations f rom t he N ewtonian theory, Received: 25–06–2013 Accepted: 13–09–2013 Published online: 23–12–2013 namely, white dwarfs, neutron stars, black holes, su- permassive stars and relativistic star clusters (Misner Citation: Gutiérrez-Ruiz AF, Valenzuela-Toledo CA, Pachón LA (2014) Innermost stable circular orbits and epicyclic et al. 1973). In the case of magnetized objects, such as frequencies around a magnetized neutron star. Universitas white dwarfs ( 105 T) or neutron stars ( 1010 T), Scientiarum 19(1): 63–73 doi: 10.11144/Javeriana.SC19-1.isco the Newtonian∼ theory not only fails in ∼describing Funding: Fundación para la Promoción de la Investigación y la the gravitational field g enerated b y t he m atter dis- Tecnología del Banco de la República; Vicerrectoría de Investi- tribution, but also in accounting for the corrections gaciones, Universidad del Valle; Comité para el Desarrollo de la from the energy stored in the electromagnetic fields. Investigación—CODI, Universidad de Antioquia. Despite this fact and due to the sheer complexity of Electronic supplementary material: N/A an in-all-detail calculation of the gravitational and SICI: 2027-1352(201401/03)19:1<0xx:ISCOAEFAAMNS>2.0.TS;2- electromagnetic fields i nduced b y t hese astrophysi- Universitas Scientiarum,, JournalJournal ofof thethe FacultyFaculty ofof Sciences,Sciences, PontificiaPontificia UniversidadUniversidad Javeriana,Javeriana, isis licensedlicensed underunder thethe CreativeCreative CommonsCommons 2.52.5 ofof Colombia:Colombia: AttributionAttribution -- NoncommercialNoncommercial -- NoNo DerivativeDerivative Works.Works. 64 Innermost stable sircular orbits cal objects, one usually appeals to approaches based magnetic field on these particular quantities. on post-Newtonian corrections (Aliev & Özdemir The paper is organized as follows. In section De- 2002, Preti 2004), which may or may not be enough scription of the space-time around the source, we briefly in order to provide a complete and accurate descrip- describe the physical properties of the PRS solution, tion of the space-time around magnetized astrophys- the general formulae to calculate the parameters that ical objects. In particular, these approaches consider, characterize the dynamics around the star are pre- e.g., that the electromagnetic field is weak compared sented in section Characterization of the dynamics to the gravitational one and therefore, the former around the source. Sections Influence of the dipolar does not affect the space-time geometry (Aliev & magnetic field in the ISCO radius and Keplerian and Özdemir 2002, Preti 2004, Mirza 2005, Bakala et al. epicyclic frequencies are devoted to the study of the 2010, 2012). That is, it is assumed that the electro- influence of the magnetic field on the ISCO radius magnetic field does not contribute to the space-time and on the Keplerian and epicyclic frequencies, re- curvature, but that the curvature itself may affect the spectively. The effect of the magnetic field on the en- electromagnetic field. Based on this approximation, ergy E and the angular momentum L are outlined in the space-time around a stellar source is obtained section Energy and angular momentum. Finally, the from a simple solution of the Einstein’s field equa- conclusions of this paper are given in the Concluding tions, such as Schwarzschild’s or Kerr’s solution, su- remarks. perimposed with a dipolar magnetic field (Aliev & Özdemir 2002, Preti 2004, Mirza 2005, Bakala et al. Description of the space-time around the 2010, 2012). Although, to some extend this model source may be reliable for weakly magnetized astrophysical sources, it is well known that in presence of strong According to Papapetrou (1953), the metric ele- magnetic fields, non-negligible contributions to the ment d s2 around a rotating object with stationary space-time curvature are expected, and consequently and axially symmetric fields can be cast as on the physical parameters that describe the physics 2 2 in the neighbourhood of these objects. d s = f (d t !dφ) − −1 2γ 2 2 2 2 In particular, one expects contributions to the ra- + f − [e (dρ + d z ) + ρ dφ ], dius of the innermost stable circular orbit (ISCO) (Sanabria-Gómez et al. 2010), the Keplerian fre- where f , γ and ! are functions of the quasi- quency, frame-dragging frequency, precession and os- clyndrical Weyl-Papapetrou coordinates (ρ, z). The cillation frequencies of the radial and vertical mo- non-zero components of metric tensor, which are re- tions of test particles (Stella & Vietri 1999, Bakala et lated to the metric functions f , ! and γ, are al. 2010, 2012), and perhaps to other physical prop- ρ2 erties such as the angular momentum of the emitted 2 gφφ = f (ρ, z)!(ρ, z) , radiation (Tamburini et al. 2011), which could reveal f (ρ, z) − (1) some properties of accretion disks and therefore of gt t = f (ρ, z), the compact object (Bocquet et al. 1995, Konno et − al. 1999, Broderick et al. 2000, Cardall et al. 2001). and In other words, to construct a more realistic theo- gt = f (ρ, z)!(ρ, z), retical description that includes purely relativistic ef- φ fects such as the modification of the gravitational in- e2γ(ρ,z) 1 1 (2) g g . teraction by electromagnetic fields, it is necessary to zz = ρρ = = zz = ρρ f (ρ, z) g g use a complete solution of the full Einstein-Maxwell field equations that takes into account all the possi- By using the Ernst procedure and the line ele- ble characteristics of the compact object. In this pa- ment in equation (1), it is possible to rewrite the per, we use the six parametric solution derived by Einstein- Maxwell equations in terms of two com- Pachón et al. (2006) (hereafter PRS solution), which plex potentials (ρ, z) and Φ(ρ, z) [see Ernst (1968) E provides an adequate and accurate description of the for details] exterior field of a rotating magnetized neutron star 2 2 (Pachón et al. 2006, 2012), to calculate the radius of (Re + Φ ) =( + 2Φ∗ Φ) , E j j r E rE r · rE (3) the ISCO, the Keplerian, frame-dragging, precession 2 2 (Re + Φ ) Φ =( + 2Φ∗ Φ) Φ, and oscillation frequencies of neutral test-particles E j j r rE r · r orbiting the equatorial plane of the star. The main where ∗ stands for complex conjugation. The above purpose of this paper is to show the influence of the system of equations can be solved by means of Universitas Scientiarum Vol. 19 (1): 63–73 www.javeriana.edu.co/scientiarum/ojs Gutiérrez-Ruiz et al. 65 the Sibgatullin integral method (Sibgatullin 1991, where the Mi s denote the moments related to the Manko & Sibgatullin 1993), according to which the mass distribution and Si s to the current induced by Ernst potentials can be expressed as the rotation. Besides, the Qi s are the multipoles re- lated to the electric charge distribution and the Bi s to 1 the magnetic properties. In the previous expressions, 1 Z e(ξ )µ(σ)dσ (z,ρ) = Æ , m corresponds to the total mass, a to the total angu- E π 1 2 lar moment per unit mass (a J M , being J the 1 σ = = 0 − − 1 total angular moment), q to the total electric charge. 1 Z f (ξ )µ(σ)dσ In our analysis, we set the electric charge parameter Φ(z,ρ) = , q to zero because, as it is case of neutron stars, most π Æ 2 1 σ of the astrophysical objects are electrically neutral. 1 − − The parameters k, s and µ are related to the mass where ξ = z + iρσ and e(z) = (z,ρ = 0) and quadrupole moment, the current octupole, and the E f (z) = Φ(z,ρ = 0) are the Ernst potentials on the magnetic dipole, respectively. symmetry axis. As shown below, these potentials Using Eqs. (4) and (5), the Ernst potentials ob- contain all information about the multipolar struc- tained by Pachón et al.
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