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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-

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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 ωdφ) − −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 ω 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 , ω 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)ω(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)ω(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 2 ∗ , ( + Φ ) =( + Φ Φ) (3) the ISCO, the Keplerian, frame-dragging, precession E | |2 ∇2E ∇E ∇ · ∇E (Re + Φ ) Φ =( + 2Φ∗ Φ) Φ, and oscillation frequencies of neutral test-particles E | | ∇ ∇E ∇ · ∇ 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

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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. (2006) are ture of the astrophysical source [see also Pachón & A+ B C Sanabria-Gómez (2006) for a discussion on the sym- = , Φ = , (8) metries of these potentials]. The auxiliary unknown E A B A B − − function µ(σ) must satisfy the integral and normal- which leads to the following metric functions: ization conditions ¯ ¯ ¯ AA BB + C C 1 f − Z µ σ e ξ ˜e η 2f ξ f˜ η dσ = ¯ ¯ ( )[ ( ) + ( ) + ( ) ( )] (A B)(A B) Æ = 0, − − − 1 2 AA¯ BB¯ C C¯ (9) 1 (σ τ) σ 2γ + − − − (4) e = − 1 Y6 Z µ(σ)dσ KK¯ r π. n Æ = n=1 1 σ2 1 − − ¯ ¯ ¯ ¯ Im[(A+ B)H (A+ B)G C I ] Here η = z + iρτ and σ,τ [ 1,1], ˜e(η) = e∗(η∗), ω = − − . (10) ∈ − ¯ ¯ ¯ f˜ f . AA BB + C C (η) = ∗(η∗) − For the PRS solution (Pachón et al. 2006), the The analytic expressions for the functions A, B, C , Ernst potentials were chosen as: H, G, K, and I can be found in the original reference

3 2 (Pachón et al. 2006) or in Appendix of Pachón et al. z z (m + ia) kz + i s e z , (2012). A Mathematica 8.0 script with the numeri- ( ) = 3 − 2 − z + z (m ia) kz + i s cal implementation of the solution can be found at −2 − (5) http: gfam.udea.edu.co lpachon scripts nstars. q z + iµz // / / / f z . ∼ ( ) = 3 2 z + z (m ia) kz + i s Characterization of the dynamics around − − The electromagnetic and gravitational multipole mo- the source ments of the source were calculated by using the In the framework of general relativity, the dynam-ics Hoenselaers & Perjés method (Hoenselaers & Perjés of a particle may be analyzed via the Lagrangian 1990) and are given by (Pachón et al. 2006): formalism as follows. Let us consider a particle of 2 rest mass m0 = 1 moving in a space-time character- M0 = m, M2 = (k a )m,... ized by the metric tensor g , thus the Lagrangian of −3 (6) µν S1 = a, S3 = m(a 2ak + s),... the particle is given by − − 1 µ ν 2 = gµν x˙ x˙ , (11) Q0 = q, Q2 = a q aµ + kq,... L 2 − − 3 B1 = µ + aq,... B3 = aµ + µk a q (7) where the dot denotes differentiation with respect to − − µ + 2akq q s,... the proper time τ, x (τ) are the coordinates. Since −

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the fields are stationary and axisymmetric, there are which arising from the marginal stability condition. two constants of motion related to the time coordi- This condition together with the equations (16) and nate t and azimuthal coordinate φ [see Ryan (1995) (17) can be write in terms of the metric functions as for details], these are given by: [see Stute & Camenzind (2002)] ∂ ‚ d t Œ ‚ dφŒ E = L = gt t gtφ , (12) − ∂ t˙ − dτ − dτ 5 ω, ω, f ρ(2f f, ρ) ρ ρρ − ρ ‚ Œ ‚ Œ 2 4” 2 2— ∂ d t dφ + ω f 2f + (f,ρρ f f,ρ)ρ L g g , (13) ,ρ − = L = tφ + φφ q ∂ φ˙ dτ dτ 2 2 4 + ω f ω f + f, (2f f, ρ) ρ ,ρ ρ − ρ where E and L are the energy and the canonical an- ” 2 2— 2f f ρ 4f f ρ 2f gular momentum per unit mass, respectively. For ( ,ρ + ,ρρ ) + ,ρ × − • real massive particle, the four velocity is a time-like 2f f 3f f 2 4f 2 f f 3 2 µ ν + ρ( ,ρρ) ρ ,ρ ρ + ,ρρ vector with normalization gµν u u = 1. If the mo- − − − ˜ tion takes place in the equatorial plane of the source 2” q 2 4 — +f fρρρ ω,ρρ f ω f + f,ρ(2f f,ρρ) z = 0, this normalization condition leads to − ,ρ − = 0. (18) ‚ dρŒ2 ‚ d t Œ2 ‚ dφŒ2 gρρ = 1+ gt t + gφφ dτ − dτ dτ As is usual in the literature, the physical ISCO ‚ d t Œ‚ dφŒ radius reported here corresponds to evaluation of + 2gtφ . (14) pg at the root of equation (18). This equation is dτ dτ φφ solved for fixed total mass of the star M, the dimen- 2 From equation (14), one can identify an effective sionless spin parameter j = J /M (being J the angu- potential that governs the geodesic motion in the lar momentum), the quadrupole moment M2 and the equatorial plane [see e.g. Bardeen et al. (1972)] magnetic dipolar moment µ [see Table VI of Pappas 2 2 & Apostolatos (2012)]. E gφφ + 2ELgtφ + L gt t V ρ 1 . (15) eff( ) = 2 Keplerian, oscillation and precession frequencies − g g gt t tφ − φφ The Keplerian frequency ΩK at the ISCO can be ob- For circular orbits, the energy and the angular mo- tained from the equation of motion of the radial co- mentum per unit mass, are determined by the condi- ordinate ρ. This equation is easily obtained by using tions Veff(ρ) = 0 and dVeff/dρ = 0. Based on these the Lagrangian (11), conditions, one can obtain expressions for the en- ergy and the angular momentum of a particle that 1• moves in a circular orbit around the star see e.g. 2 ˙2 [ gρρρ¨ gρρ,ρρ˙ + gφφ,ρφ Stute & Camenzind (2002) , namely, − 2 − ] ˜ p g t˙2 g t˙ ˙ 0. (19) f + t t,ρ + tφ,ρ φ = E = Æ , L = E(ω + χ ), (16) 1 f 2χ 2/ρ2 − By imposing the conditions of circular orbit or con- 2 2 stant orbital radius, dρ/dτ = 0 and d ρ/dτ = 0 § q ª 2 2 4 and taking into account that dφ/dτ = ΩKd t/dτ, ρ[ ω,ρ f ω f + f,ρρ(2f f,ρρ) − − ,ρ − one gets [see e.g. Ryan (1995)] χ = ” — , f (2f f,ρρ) − q 2 (17) dφ gtφ,ρ (gtφ,ρ) gφφ,ρ gt t,ρ − ± − ΩK = = , (20) where the colon stands for a partial derivative respect d t gφφ,ρ the lower index. The radius of innermost stable cir- cular orbit (ISCO’s radius) is determined by solving numerically for ρ the equation where “+” and “ ” denotes the Keplerian frequency for corotating and− counter-rotating orbits, respec- 2 2 d Veff/dρ = 0, tively.

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The epicyclic frequencies are related to the oscilla- come via equivalence between matter and energy, 2 tions frequencies of the periastron and orbital plane E = mc . For the particular case of a magnetar ( of a circular orbit when we apply a slightly radial 1010 T), the electromagnetic energy density is around∼ 25 3 2 4 and vertical perturbations to it. According to Ryan 4 10 J/m , with an E/c mass density 10 times (1995), the radial and vertical epicyclic frequencies larger× than that of lead. Hence, the relevant physi- are given by the expression cal quantities should depend upon the magnetic field, and in particular on the magnetic dipole moment µ. ¨ αα – ‚ Œ 1 g gφφ ν g g 2 α = ( t t + tφΩk ) 2 Table 1. Realistic numerical solutions for rotat- 2π − 2 ρ ,αα ‚ Œ ing neutron stars derived by Pappas & Apostolatos gt φ (2012). Here, M0 is the total mass of the star, j is the 2(gt t + gt Ωk )(gt + g Ωk ) 2 φ φ φφ 2 dimensionless spin parameter: j J /M (being J − ρ ,αα = 0 ‚ g Œ ™« the angular momentum), M2 is the quadrupole mo- g g 2 t t , ment and S is the current octupole moment see + ( tφ + φφΩk ) 2 3 [ ρ ,αα Table VI of Pappas & Apostolatos (2012)]. p 3 4 where α = ρ, z . The periastron ν and the nodal Model M0 [km] j M2 [km ] S3 [km ] { } ρ p M17 4.120 0.588 -51.8 -210.0 νz frequencies are defined by: M18 4.139 0.635 -62.6 -279.0 p Ωk M19 4.160 0.682 -74.9 -365.0 ν = να, (21) α 2π − M20 4.167 0.700 -79.8 -401.0 which are the ones with an observational interest (Stella & Vietri 1999). The radial oscillation fre- 19.30 Table 1: M20 quency vanishes at the ISCO radius and therefore, M19 the radial precession frequency equals to the Keple- 19.25 M18 rian frequency. M17

Finally, the frame dragging precession frequency [km] 19.20 or Lense–Thirring frequency is given by see e.g. νLT [ ISCO R Ryan (1995)] 19.15

1 gtφ νLT = . (22) 19.10 2 g 0 1 2 3 4 5 6 − π φφ µ [km2 ]

νLT is related to purely relativistic effects only (Mis- ner et al. 1973). In this phenomenon, the astro- Fig. 1. ISCO radius as a function of magnetic physical source drags the test particle into the di- dipole parameter µ. The physical parameters for rection of its rotation angular velocity. In absence the star correspond to the models M17-M20 listed of electromagnetic contributions, as in the case of in Table 1. An increase of µ leads to a decrease of the Kerr solution, the frame dragging comes from the ISCO radius. the non-vanishing angular momentum of the source. Figure 1 shows the ISCO radius as a function of In the presence of electromagnetic contributions in the parameter µ for four particular realistic numer- non-rotating sources, it was shown by Herrera et al. ical solutions for rotating neutron stars models de- (2006) that the non-zero circulation of the Poynting rived by Pappas & Apostolatos (2012). The models vector is able to induced frame dragging, in which used coincide with the models 17,18,19 and 20 of Ta- case it is electromagnetically induced. In the cases dis- ble VI in that reference and correspond to results cussed below, due to the presence of fast rotations for the Equation of State L (see Table 1). The low- and magnetic fields, the frame dragging will be in- est multipole moments of the PRS solution, namely, duced by a combination of these two processes. mass, angular moment and mass quadrupole were fixed t o t he n umerical o nes o btained by Pappas & Influence of the dipolar magnetic field in Apostolatos (2012) (see Table 1). Since the main ob- the ISCO radius jective here is to analyze the influence o f t he mag- netic dipole, the current octupole parameter s was As discussed in the Introduction, contributions set to zero. Note that this does not mean that the from the energy stored in the electromagnetic fields current octupole moment vanishes (see in Eq. (6).

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The parameter µ was varied between 0 and 6.4 km2, 1148 which corresponds to magnetic fields dipoles around M20 31 2 M19 0 and 6.3 10 Am , respectively. In all these cases, M18 the ISCO× radius decreases for increasing µ, this can 1146 M17 be understood as a result of the dragging of iner- [Hz] tial frames induced by the presence of the magnetic k dipole, we elaborate more on this below. Ω 1142

11400 1 2 3 4 5 6 Keplerian and epicyclic frequencies µ [km2 ] Some of the predictions from General Relativ-ity (a) (RG) such as the dragging of inertial systems 1091 M20 (frame-dragging or Lense-Thirring effect) (Everitt M19 1088 M18 et al. 2011), the geodesic precession (geodesic effect M17 or de Sitter precession) (Everitt et al. 2011) or the 1082 analysis of the periastron precession of the orbits [Hz] 1085 z (Lucchesi & Peron 2010) have been experimental ν verified. However, since they have observed in the 1077 vicinity of the Earth, these observations represent 1074 experimental support to RG only in the weak field 0 1 2 3 4 5 6 limit. Thus, it is fair to say that the RG has not been µ [km2 ] checked in strong field limit [see e.g. Psaltis (2008)]. (b) In this sense, the study of compact objects such as 175 black holes, neutron stars, magnetars, etc., is cer- M20 170 M19 tainly a topic of great interest, mainly, because these M18 objects could be used as remote laboratories to test M17 fundamental physics, in particular, the validity of [Hz] general relativity in the strong field limit yet (Stella LT 156 ν & Vietri 1999, Pachón et al. 2012). The relevance of studying the Keplerian, epicyclic and Lense-Thirring frequencies relies on the fact that 1420 1 2 3 4 5 6 2 they are usually used to explain the quasiperiodic µ [km ] oscilation phenomena present in some Low Mass (c) X-ray Binaries (LMXRBs). In this kind of systems a compact object accretes from another one (which Fig. 2. Keplerian frequency (a), nodal precession is usually a normal star) and the X-ray emissions frequency (b) and, Lense-Thirring frequency (c) as could be related to the relativistic motion of the a function of the µ parameter. All the frequencies accreted matter e.g. rotation, oscillation and pre- increase for increasing µ. cession. Stella & Vietri (1999) showed that in the slow rotation regime, the periastron precession and In the framework of the RPM, it is assumed that the Keplerian frequencies could be related to the the motion of the accretion disk is determined by phenomena of the kHz quasiperiodic oscillations the gravitational field alone and thereby, it is nor- (QPOs) observed in many accreting neutron stars mal to assume that the exterior gravitational field in LMXRBs (Stella & Vietri 1999). This model is of the neutron star is well described by the Kerr known as the Relativistic Precession Model (RPM) metric. However, most of the neutron stars have (Stella & Vietri 1999) and identifies the lower and (i) quadrupole deformations that significantly differ higher QPOs frequencies with the periastron pre- from Kerr’s quadrupole deformation (Laarakkers & cession and the Keplerian frequencies respectively. Poisson 1999) and (ii) a strong magnetic field. The in- The RPM model has been used to predict the values fluence of the non-Kerr deformation was discussed, of the mass and angular momentum of the neutron e.g., by Johannsen & Psaltis (2010) and Pachón et al. stars in this kind of systems [see e.g. Stella & Vietri (2012). Based on observational data, it was found (1999) and Stella et al. (1999), for details]. that non-Kerr deformations dramatically affect, e.g.,

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matplotlib Gutiérrez-Ruiz et al. 69

predictions on the mass of the observed sources. Be- whereas the Keplerian frequency increases [see Fig. low, we consider the influence of the magnetic field 2(a)]. However, this same opposite trend is already and find that it introduces non-negligible corrections present in the dynamics around a Kerr source when in the precession frequencies and therefore, on the the angular momentum of the source is increased predictions made by the Kerr-based RPM model. [cf. equations (2.13) for the angular momentum and In the previous section, it was shown that the (2.16) for the Keplerian frequency in Bardeen et al. magnetic field affects the value of the ISCO radius (1972)]. In Kerr’s case, the co-rotating test parti- of a neutral test particle that moves around the exte- cles are dragged toward the source thus inducing a rior of a magnetized neutron star. Below, it is shown shorter ISCO radius [cf. equation (2.21) in Bardeen that other physicals quantities such as the Keplerian et al. (1972)], and since the leading order in the Ke- and epicyclic frequencies, which are used to describe plerian frequency goes as 1/ρ2, a larger frequency the physics of accretion disk, are also affected. The is expected. By contrast, the∼ contra-rotating test par- influence on the Lense-Thirring frequency is also dis- ticles are “repelled” by the same effect thus resulting cussed. in an increase of the ISCO radius and in a decrease Figure 2 shows the influence of the magnetic field of the Keplerian frequency [see also Fig. 2 in Pachón in the Keplerian [Fig. 2(a)], nodal precession [Fig. et al. (2012)]. 2(b) and Lense-Thirring Fig. 2(c) frequencies. ] [ ] 0.922 In all cases, the frequencies are plotted as a M20 0.921 M19 function of the parameter µ while fixing the others M18 free param-eters (mass, angular moment and mass 0.921 M17 quadrupole) according to the realistic numerical solutions in Table 1. As it is expected for [km] 0.920 ISCO shorter ISCO radius (see previous section), all E 0.920 frequencies increase with increasing . The changes µ 0.919 in the Lense-Thirring fre-quency come from the electromagnetic contribution discussed above. 0.9190 1 2 3 4 5 6b) µ [km2 ] Energy and angular momentum (a) 12.12 M20 In order to understand the results presented 12.10 M19 above, we consider that it is illustrative to consider M18 12.08

] M17

first the effect that the magnetic field has on the en- 2 ergy E and the angular momentum L needed to de- 12.06 [km scribed marginally stable circular orbits [see equa- 12.04 ISCO tions (16) and (17)]. In doing so, we fix t he total L 12.02 mass M0, the spin parameter j and the quadrupole 12.00 moment M , according to the values in Table 1. Fi- 2 11.98 ures 3(a) and 3(b) show E and L at the ISCO as a 0 1 2 3 4 5 6 g µ [km2 ] function of the dipolar moment µ. By contrast to the case of the Keplerian frequency, an increase of (b) µ decreases the value of the energy and the angular Fig. 3. Energy (a) and angularmomentum (b) of momentum needed to find an ISCO.Complementa- a test particle at the ISCO radius versus magnetic ily, in figures 4(a) and 4(b), E and L are depicted as a r dipole parameter µ given by the PRS solution. We function of ρ for various values of the dipolar mo- can see that reduction of the energy and angular ment. Since the energy and the angular momentum momentum of the particle while the the magnetic associated to the ISCO correspond to the minima of dipole of the star is increased. the curves E(ρ) and L(ρ) (indicated by triangles), figure shows that an increase of the magnetic dipole Having in mind the situation in Kerr’s case and moment induces a decrease of the ISCO radius (see by noting that the frame dragging can be induced by Fig. 1 above) and simultaneously a decrease of the current multipoles of any order (Herrera et al. 2006), energy and the angular momentum. the goal now is to compare the characteristics of At a first sight, for an increasing magnetic field, it the contributions from the angular momentum and may seem conspicuous that the angular momentum, the dipole moment to the Keplerian frequency based of co-rotating test particles, decreases [see Fig. 4(b)] on the approximate expansions derived by Sanabria-

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Gómez et al. (2010) and appeal then the general the- Hence, the role of the dipole moment of the ory of multipole moments to track the contribu- source in the dynamics of neutral test particles is tion of dipole moment to the current multipole mo- qualitatively analogous to the role of the angular mo- ments. If the expression for the Keplerian frequency mentum, albeit it induces corrections of higher or- [Eq. (21)] and for the angular momentum [Eq. (23)] ders in the multipole expansion of the effective po- in Sanabria-Gómez et al. (2010) are analyzed in de- tential. This is a subject that deserves a detailed dis- tail, one finds that the signs of the contributions of cussion and will be explained somewhere else. the angular momentum and the magnetic dipole of the source coincide, this being said, one could argue Conclusion that the contribution of the dipole moment is related to an enhancement of the current multipole moment The influence of weak electromagnetic fields on the of the source. This remark is confirmed by the gen- dynamics of test particles around astrophysical eral multipole expansion discussed by Sotiriou & objects has mainly been studied for the case of or- Apostolatos (2004). In particular, the influence of biting charged particles. Based on a analytic solu- the dipole moment in the higher current-multipole tion of the Einstein-Maxwell field equations, here it moments is clear from Eq. (23)–(25) of this reference. is shown that a strong magnetic field, via the energy mass relation, modifies the dynamics of neutral test particles. In particular, it is shown that an intense 0.91965 µ [km2 ] =1 magnetic field induces corrections in the Keplerian, µ [km2 ] =3 precession and oscillation frequencies of the radial 2 0.91960 µ [km ] =6 and vertical motions of the test particles as well as in the dragging of inertial frames. 0.91955 In particular, it was shown that if the angular mo- mentum and the dipole moment are parallel (note E[km] 0.91950 that j and µ have the same sign), then the ISCO radius of co-rotating orbits decreases for increasing dipole moment, this leads to an increase of the Kep- lerian, and precession and oscillation frequencies of the radial and vertical motions frequency. The angu- 0.91945 14.1 14.3 14.5 14.7 14.9 lar momentum and the energy of the ISCO decrease ρ[km] for increasing dipole moment as a consequence of (a) the dragging of inertial frames. The kind of geodetical analysis performed here

µ [km2 ] =1 is widely used for instance, in the original RPM µ [km2 ] =3 11.996 model (Stella & Vietri 1999, Stella et al. 1999), and µ [km2 ] =6 int its subsequent reformulations, of the HF QPOs observed in LMXBs. However, Lin et al. (2011) and Török et al. (2012) have concluded that although ] 2 11.991 these models of HF QPOs, which neglect in the in-

L[km fluence of strong magnetic fields, are qualitatively sat- 11.990 11.989 isfactory, they do not provide satisfactory fits to the observational data. Hence, in order to improve (i) the level of physical description of these models and (ii) the fit to observational data, a more detailed anal- 111.985 14.1 14.3 14.5 14.7 14.9 ysis on the role of the extremely strong magnetic ρ[km] field in the structure of the spacetime is necessary (b) and will be performed in the future. Fig. 4. Energy (a) and angular momentum (b) of Acknowledgements a test particle at the ISCO radius versus magnetic dipole parameter µ given by the PRS solution. We This work was partially supported by Fundación can see that reduction of the energy and angular para la Promoción de la Investigación y la Tec- momentum of the particle while the the magnetic nología del Banco la República grant number 2879. dipole of the star is increased. C.A.V.-T. is supported by Vicerrectoría de Inves-

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tigaciones (UniValle) grant number 7859. L.A.P. Herrera L, González GA, Pachón LA, Rueda JA ac-knowledges the financial support by the Comité (2006) Frame dragging, vorticity and electro- para el Desarrollo de la Investigación –CODI– of magnetic fields in axially symmetric stationary Universi-dad de Antioquia. spacetimes. Class Quant Grav 23(7):2395–2408 doi:10.1088/0264-9381/23/7/011 Conflict of interest Hoenselaers C, Perjés Z (1990) Multipole mo- The authors declare no conflict of interest. ments of axisymmetric electrovacuum space- times. Class. Quantum Grav. 7(10):1819–1825 References doi:10.1088/0264-9381/7/10/012

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