Role of electric charge in shaping equilibrium configurations of fluid tori encircling black holes
Jiří Kovář, Petr Slaný, Zdeněk Stuchlík, Vladimír Karas, Claudio Cremaschini, John C. Miller
RRole of electric charge in shaping equilibrium configurations of fluid tori encirclingblack holes
Jiˇr´ı Kov´aˇr, ∗ Petr Slan´y, and Zdenˇek Stuchl´ık
Institute of Physics, Faculty of Philosophy and Science, Silesian University in OpavaBezruˇcovo n´am. 13, CZ-74601 Opava, Czech Republic
Vladim´ır Karas
Astronomical Institute, Academy of Sciences, Boˇcn´ı II, CZ-14131 Prague, Czech Republic
Claudio Cremaschini
SISSA & INFN, Via Bonomea 265, I-34136 Trieste, Italy
John C. Miller
SISSA & INFN, Via Bonomea 265, I-34136 Trieste, Italy andDepartment of Physics (Astrophysics), University of OxfordKeble Road, Oxford OX1 3RH, U.K.
Astrophysical fluids may acquire non-zero electrical charge because of strong irradiation or chargeseparation in a magnetic field. In this case, electromagnetic and gravitational forces may act togetherand produce new equilibrium configurations, which are different from the uncharged ones. Followingour previous studies of charged test particles and uncharged perfect fluid tori encircling compact ob-jects, we introduce here a simple test model of a charged perfect fluid torus in strong gravitationaland electromagnetic fields. In contrast to ideal magnetohydrodynamic models, we consider herethe opposite limit of negligible conductivity, where the charges are tied completely to the movingmatter. This is an extreme limiting case which can provide a useful reference against which to com-pare subsequent more complicated astrophysically-motivated calculations. To clearly demonstratethe features of our model, we construct three-dimensional axisymmetric charged toroidal configura-tions around Reissner-Nordstr¨om black holes and compare them with equivalent configurations ofelectrically neutral tori.
I. INTRODUCTION
Equilibrium toroidal configurations of perfect fluidplay an important role in studies of geometrically thickaccretion discs around compact objects [1]. The isobaricsurfaces also have toroidal topology and in order for ac-cretion to occur there must be a critical, marginally-closed isobaric surface with a cusp through which mattercan outflow from the disc onto the compact object. Inthe following, we focus on black-hole systems and ignoreself-gravity of the disc material. Shapes and propertiesof the tori, such as pressure and density profiles, are thendetermined by the black-hole spacetime geometry, an ap-propriately chosen rotation law (giving the distributionof specific angular momentum), and the fluid parameters.Perfect fluid tori in Schwarzschild and Kerr back-grounds were extensively discussed in the original funda-mental papers establishing this line of work [2, 3]. Lateron, many studies appeared generalizing these modelsand including further details [4–10], describing tori alsoin Schwarzschild-de Sitter, Kerr-de Sitter and Reissner-Nordstr¨om-de Sitter spacetimes, where presence of theso-called static radius [11] implies also the existence oftori with an outer cusp. ∗ Electronic address:
The material in accretion discs contains charged par-ticles (which may or may not be quasi-neutral in bulk)and the central black hole might also be charged. Thecharged or quasi-neutral fluid creates its own electromag-netic field which would then couple with that of the blackhole, leading to a different and much more complicateddescription of the motion. Using the equations for thedynamics of the fluid and specifying its ‘internal’ proper-ties (conductivity, viscosity, equation of state, etc), onecan solve the system so as to obtain profiles for the four-velocity, pressure, matter density and charge density [12].However, the system of equations is rather complex, andin general requires the use of sophisticated numerical ap-proaches and codes, even if simplifying assumptions aremade such as taking infinite electrical conductivity (thelimit of ‘ideal magnetohydrodynamics’), no self-gravity,etc. On the other hand, some characteristic features ofthe motion of quasi-neutral or charged fluid, can also bestudied relatively simply in a semi-analytic way [13–20].The approximation of ideal magnetohydrodynamics(MHD) is reasonable in many astrophysically-relevantsituations involving fluids in motion [21]. However, thereare other physical circumstances in which it is impor-tant to include the effects of finite conductivity [22–24],and there the behavior becomes more complex, especiallywhen strong gravitational and external magnetic fieldsare also present. In order to address some of these effectsin their mutual interplay, it can be useful to look also a r X i v : . [ a s t r o - ph . H E ] O c t at the opposite limit to that of ideal MHD: the limit ofnegligibly small conductivity.Here, we examine the problem of the interaction be-tween charged moving matter and the gravitational andelectrostatic fields of the black hole, concentrating on anidealized situation which allows us to illustrate some oth-erwise very complicated effects. We consider a simple testmodel in which the matter is taken to be slightly chargedand electrically non-conductive (dielectric), with the aimof providing an extreme reference case against which tocompare subsequent more detailed calculations. We pro-ceed by first specifying a prescribed form for the angularmomentum distribution, and then solving the dynamicalequations to find the profiles of pressure, mass densityand charge density. This approach can be seen as gen-eralizing the studies of uncharged perfect-fluid tori men-tioned above by adding the charge. Note that, through-out, our tori are assumed to be composed of test fluids inwhich both the self-gravitational and self-electromagneticfields are neglected. This gives a useful simplification,which is acceptable for weakly-charged, low-mass torithat have hardly any effect on the spacetime geometry orthe ambient electromagnetic field. This approach helpsus in building a semi-analytic model. In principle, in-clusion of self-gravity of the tori (following, e.g. [25–27])and self-electromagnetic effects (see, e.g. [16]) would bepossible, but that would enormously complicate the sit-uation.The charged dielectric perfect-fluid tori can also beseen as generalizing studies of charged test particles or-biting around black holes [28–36]. Various aspects ofthe charged test particle motion were also discussed in[37, 38], concerning the possibility of collimated ejectionalong the axis of a rotating magnetized black hole, whileinvestigation of stable off-equatorial lobes of charged par-ticles was discussed in [39–41]. In general, the motion oftest particles is bounded within effective potential wellsand this represents a model for a very dilute toroidalstructure consisting of non-interacting particles; here,we add the non-electrical interaction between them, thepressure. Commonly, pressure leads to geometricallythick structures extending further out of the equatorialplane.In section II, we present the basic equations. In sec-tion III, we apply them to the case of a torus arounda charged, non-rotating black hole described by theReissner-Nordstr¨om metric. Nevertheless, the presentedapproach is suitable for a description of charged tori nearto any models of compact objects with well-defined ge-ometry and electromagnetic field. We chose the Reissner-Nordstr¨om black hole because of its extremely clear ex-ternal electric field and geometry, given in an analyticform. For illustrative purposes we set the charge of thecentral black hole considerably exceeding astrophysicallyrealistic values; our paper presents a toy model exhibitingthe physical mechanism of mutual interaction betweencharged fluid and a black hole. In section IV, we dis-cuss the form of the isobaric surfaces for a torus with constant specific angular momentum composed of an un-charged barotropic perfect fluid. This is then extendedto charged tori in section V, where we also present acomparison between equivalent charged and unchargedcases. This work involves making a number of simplify-ing assumptions, and we discuss the nature and impactof these (including the zero-conductivity limit) in sectionVI. Section VII is the conclusion. Throughout the paper,we use the geometrical system of units ( c = G = 1) andmetric signature +2. II. BASIC EQUATIONS
In general, the motion of charged perfect fluid is de-scribed by two sets of general relativistic MHD equations.These are the conservation laws and Maxwell’s equations ∇ β T αβ = 0 , (1) ∇ β F αβ = 4 πJ α , (2)where the 4-current density J α , which satisfies the con-tinuity equation ∇ α J α = 0 , (3)can be expressed in terms of the charge density q , con-ductivity σ and 4-velocity U α of the fluid by using Ohm’slaw J α = qU α + σF αβ U β , (4)with the electromagnetic tensor F αβ being given in termsof the vector potential by F αβ = ∇ α A β − ∇ β A α . Thiselectromagnetic tensor describes the vacuum externalelectromagnetic field of the compact object (which per-vades the fluid), and also the internal electromagneticfield of the fluid itself, i.e., F αβ = F αβ EXT + F αβ INT . (5)The stress-energy tensor T αβ consists of matter and elec-tromagnetic parts T αβ = T αβ MAT + T αβ EM , (6)where T αβ MAT = ( (cid:15) + p ) U α U β + pg αβ , (7) T αβ EM = 14 π (cid:18) F αγ F βγ − F γδ F γδ g αβ (cid:19) . (8)Besides the pressure p and energy density (cid:15) , the otherfluid variables are the rest-mass density ρ and the spe-cific internal energy ε = (cid:15)/ρ −
1. The thermodynamicaldescription of the fluid is specified by supplying an appro-priate equation of state p = p ( (cid:15), q ), which also involvesthe charge density of the fluid, describing the contribu-tion of the Coulomb interaction between the fluid parti-cles to the total pressure.We build our model by considering a non-conductive( σ = 0) charged test fluid (taken to be a perfect fluid)in an axially symmetric spacetime and use sphericalpolar coordinates ( t, r, θ, φ ). The fluid rotates in the φ -direction with 4-velocity U α = ( U t , U φ , , (cid:96) = − U φ /U t and angular velocity(related to distant observers) Ω = U φ /U t , related by theformulae Ω = − (cid:96)g tt + g tφ (cid:96)g tφ + g φφ , (9)( U t ) = g tφ − g tt g φφ (cid:96) g tt + 2 (cid:96)g tφ + g φφ (10)By writing out the covariant derivative of the electro-magnetic part of the stress-energy tensor (8) appearingthe left hand side of the conservation law (1), moving itto the right hand side, and using the Maxwell equations(2), with ∇ β F αβ EXT = 0 , ∇ β F αβ INT = 4 πJ α , (11)we obtain the equation ∇ β T αβ MAT = F αβ J β , with J β beingthe 4-current density due to the motion of the chargedfluid torus. We are not here including the effects of theelectromagnetic field generated by this 4-current: our toriare considered as being composed of ‘test matter’ fromthe electromagnetic point of view as well as from thegravitational one, i.e., F αβ INT (cid:28) F αβ EXT and we can write F αβ = F αβ EXT . Then we get the master equation ∇ β T αβ MAT = F αβ EXT J β . (12)Note that in this approach we do not need to solveMaxwell’s equations (2), since the electromagnetic fieldis prescribed. Also, because of the non-conductivity, wehave J α = qU α . (13)The equations of motion (12) give two non-linear par-tial differential equations for the pressure p , whose pro-files we are wanting to find: ∂ r p = − ( (cid:15) + p ) (cid:18) ∂ r ln ( U t ) − Ω ∂ r (cid:96) − Ω (cid:96) (cid:19) − q F rα U α ≡ R ,∂ θ p = − ( (cid:15) + p ) (cid:18) ∂ θ ln ( U t ) − Ω ∂ θ (cid:96) − Ω (cid:96) (cid:19) − q F θα U α ≡ T , (14)where R = R ( r, θ ) and T = T ( r, θ ). These equations arenot integrable unless the integrability condition ∂ θ R = ∂ r T (15)is satisfied, and this is therefore a requirement.The existence of a solution is guaranteed if q = 0,so that the last terms in equations (14) vanish and weget the Euler equation describing a rotating uncharged perfect fluid (see, e.g., papers [2, 3]). In this case, whenthe angular momentum distribution (cid:96) = (cid:96) ( r, θ ) is chosen,a solution of the Euler equation for any barotropic fluid(having p = p ( (cid:15) )) can be derived from Boyer’s condition[2, 3] (cid:90) p d pp + (cid:15) = W in − W, (16) W in − W = ln ( U t ) in − ln ( U t ) + (cid:90) (cid:96)(cid:96) in Ωd (cid:96) − Ω (cid:96) , (17)where the subscript ‘in’ refers to the inner edge of thetorus in the equatorial plane. This condition enables usto straightforwardly determine the isobaric surfaces inthe torus in terms of the equipotential surfaces of the‘gravito-centrifugal’ potential W ( r, θ ): for equilibriumtoroidal configurations composed of barotropic perfectfluid, the equipotential surfaces of W correspond to sur-faces of constant pressure (or energy density) in the fluid.When q (cid:54) = 0, the situation is more complicated andequations (14) are no longer integrable for arbitrary q ( r, θ ) = const and arbitrarily chosen (cid:96) = (cid:96) ( r, θ ). Itis then necessary either to specify (cid:96) = (cid:96) ( r, θ ) (witheven (cid:96) = const being possible) and find an appropriate q = q ( r, θ ) which is consistent with that or, vice versa,to specify q = q ( r, θ ) (with even q = const being possi-ble) and find an appropriate (cid:96) = (cid:96) ( r, θ ). However, thisis strictly ‘necessary’ only if the equation of state is pre-scribed. Otherwise, one could also absorb the constraintinto that. The charged tori must clearly have distribu-tions of charge and angular momentum which satisfy theintegrability condition. III. ISOBARIC SURFACES INREISSNER-NORDSTR ¨OM GEOMETRY
The Reissner-Nordstr¨om metric, representing thespace-time outside a charged, non-rotating black hole,provides a suitable mathematically-simple test examplefor experimenting with ideas before moving on to morecomplicated examples having direct astrophysical rele-vance. In the dimensionless ( M = 1) Schwarzschild co-ordinates, the only free parameter in the line element ofthe Reissner-Nordstr¨om geometryd s = − (cid:18) − r + Q r (cid:19) d t + (cid:18) − r + Q r (cid:19) − d r + r (d θ + sin θ d φ ) (18)is the dimensionless charge Q , which takes values | Q | ≤ | Q | > ≡ r − r + Q = 0. The ambient electric field is staticand spherically symmetric, like the space-time, and isdescribed by the vector potential A α = ( A t , , , A t = − Qr . (19)In this background, the pressure equations (14) reduceto the form ∂ r p = − ( p + (cid:15) ) (cid:18) ∂ r ln | U t | − Ω ∂ r (cid:96) − Ω (cid:96) (cid:19) + q U t ∂ r A t ,∂ θ p = − ( p + (cid:15) ) (cid:18) ∂ θ ln | U t | − Ω ∂ θ (cid:96) − Ω (cid:96) (cid:19) , (20)where U t = − r sin θ √ ∆ (cid:112) r sin θ − (cid:96) ∆ . (21)To proceed further, it is then necessary to choose anequation of state. For an uncharged perfect fluid, a suit-able choice is to have a polytropic relation between thepressure and the rest-mass density p = κρ Γ , (22)with κ and Γ being a polytropic constant and index.This widely-used relation is a convenient simple formwhich embodies conservation of entropy (as appropriatefor a perfect fluid). For our calculations, we have used κ = 10 and Γ = 2. This value of Γ is mathematicallyconvenient for making analytic integrations and, while itis rather high for physical applications, the conveniencemakes its use consistent within the spirit of the presentsimple model. We have chosen a high value of κ , be-cause electrostatic corrections to the equation of statethen become negligible, so that we can continue to usethis polytropic equation of state consistently even in thecharged case (we comment further on this in section VIC). Moreover, we use values of ρ which are sufficiently lowso that the medium is non-relativistic and the contribu-tion of the internal energy to the total energy densityis then negligible as well, i.e., (cid:15) ≈ ρ . This approxima-tion is consistent also with the assumption of negligibleself-electromagnetic-field.In order to find a solution for the pressure p , it is usefulto rewrite equations (20) as equivalent equations for thedensity ∂ r ρ = ( κρ Γ + ρ ) (cid:16) ∂ r ln | U t | − Ω ∂ r (cid:96) − Ω (cid:96) (cid:17) − q U t ∂ r A t − Γ κρ Γ − ,∂ θ ρ = ( κρ Γ + ρ ) (cid:16) ∂ θ ln | U t | − Ω ∂ θ (cid:96) − Ω (cid:96) (cid:17) − Γ κρ Γ − , (23)which can be solved more easily than the ones for thepressure. On the other hand, in the uncharged case, thepressure profiles are determined from relation (16) which, for a polytropic equation of state, gives the following rel-atively simple analytic formula p = (cid:32) e Γ − ( W in − W ) − κ (cid:33) ΓΓ − . (24)(Note that this formula is valid only in the region where W in ≥ W ).In the next sections, we consider the commonly-usedcondition of constant specific angular momentum, (cid:96) =const [2–8, 8–10]. Tori with (cid:96) = const are particularlysimple mathematically and are generally representativeof those with a more general angular momentum profile,although some care needs to be taken when consideringperturbations (these configurations are only marginallystable with respect to convective instability [42]). IV. UNCHARGED TORI, (cid:96) = const
For a barotropic fluid, the isobaric surfaces coincidewith the equipotential surfaces of the potential W ( r, θ ),which are given by the formula W ( r, θ ) = ln | U t | = ln r √ ∆ sin θ (cid:112) r sin θ − (cid:96) ∆ . (25)One of the reality conditions here, ∆ ≥
0, restricts theexistence of equipotential surfaces to the stationary re-gion of the spacetime. The other one, r sin θ − (cid:96) ∆ > θ = π/ (cid:96) < (cid:96) ( r ; Q ) ≡ r ∆ , (26)where the function (cid:96) ( r ; Q ) plays the role of the effec-tive potential governing photon geodesic motion in theequatorial plane (see [43] where the more general casewith a non-zero cosmological constant is discussed). Forthe purpose of classification, we only need to considerpositive values of (cid:96) ph ( r ; Q ), i.e., we define (cid:96) ph ( r ; Q ) ≡ r √ ∆ . The function (cid:96) ph ( r ; Q ) has one local extremum(a minimum) outside of the outer black-hole horizon, (cid:96) ph , c ( Q ), located at r ph , c = (3 + (cid:112) − Q ), corre-sponding to the circular photon orbit in the equatorialplane.The character of the equipotential surfaces is well rep-resented by the behavior of the potential W ( r, θ ) inthe equatorial plane, i.e., by the function W π/ ( r ) ≡ W ( r, θ = π/ W , i.e. those satisfying the conditions ∂ r W ( r, θ ) = ∂ θ W ( r, θ ) = 0, correspond to loci with zero pressure gra-dients, the fluid has to follow geodesic motion there. Thelocal extrema of W are located only in the equatorialplane. Evaluating the necessary condition ∂ r W π/ ( r ) = l c (cid:73) r ; Q (cid:77) l ph (cid:73) r ; Q (cid:77) l ph, c (cid:73) Q (cid:77) l mb (cid:73) Q (cid:77) l ms (cid:73) Q (cid:77) (cid:72) r (cid:76) (cid:123) FIG. 1: Behavior of the function (cid:96) c ( r ; Q ) governing ex-trema of the potential W ( r, θ ) in the equatorial plane of theReissner-Nordstr¨om spacetime with Q = 0 .
1. For a fixedvalue of the angular momentum (cid:96) , we can determine the po-sition of the potential maxima (smaller radius) and minima(larger radius). The potential W ( r, θ ) is not defined in theshaded region limited by the function (cid:96) ph ( r ; Q ). In black-hole spacetimes ( Q ≤ (cid:96) c ( r ; Q ) and (cid:96) ph ( r ; Q ), shownin this figure. The horizontal dashed lines denote the valuesof (cid:96) ms ( Q ) . = 3 . (cid:96) mb ( Q ) . = 3 .
995 and (cid:96) ph , c ( Q ) . = 5 . Q = 0 .
1, the value of Q being considered here, and thevertical solid line shows the position of the outer black-holehorizon. From the discussion of the behavior of W ( r, θ ) (seeFig. 2), it follows that equilibrium toroidal configurations ofa barotropic fluid with (cid:96) = const exist only for (cid:96) > (cid:96) ms ( Q ).
0, these extrema are given by the condition (cid:96) = (cid:96) ( r ; Q ) ≡ r ( r − Q )∆ . (27)Again, for the purpose of classification, we only need toconsider positive values of (cid:96) c ( r ; Q ). This function hasone local minimum (cid:96) ms ( Q ) corresponding to the spe-cific angular momentum of the marginally stable orbit.In addition to the limiting values (cid:96) ph , c ( Q ) and (cid:96) ms ( Q ),there is also another one, (cid:96) mb ( Q ), corresponding to thespecific angular momentum of a particle moving alongthe marginally bound equatorial circular geodesic, deter-mined by the conditions ∂ r W π/ ( r ) = 0 and W π/ ( r ) = 0.Independently of the charge parameter | Q | ≤ (cid:96) ph ( r ; Q ) and (cid:96) c ( r ; Q ). We show this in Fig. 1, drawn for Q = 0 . Q is rather high from an as-trophysical point of view [44], it still gives only verysmall deviations of the space-time away from that of theSchwarzschild metric; taking a value this high is useful forclearly demonstrating the effects which we are wantingto investigate.)Equilibrium toroidal configurations of a barotropicfluid with constant specific angular momentum exist only for (cid:96) > (cid:96) ms ( Q ), which takes values ranging from l ms (0) = 3 . l ms (1) = 3 . l < l mb ( Q ),which takes values ranging from l mb (0) = 4 to l mb (1) =3 . l mb (0) = 4 for the Schwarzschild casewas also found for self-gravitating tori [26]. The radiigiven by the condition (cid:96) = (cid:96) c ( r ; Q ) then correspondto motion of fluid elements along the unstable circulargeodesic (smaller radius) and the stable one (larger ra-dius). The stable circular geodesic represents the ‘center’of the torus (the potential W ( r, θ ) has a local minimumthere while the pressure is maximal there). The unstablecircular geodesic marks a critical point (cusp), where thepotential W ( r, θ ) has a local maximum; the correspond-ing equipotential surface is self-crossing and is referred toas the ‘critical surface’. As well as this critical equipoten-tial surface, there is also the characteristic null equipo-tential surface W ( r, θ ) = 0, which crosses the equatorialplane at infinity.The behavior of the potential W ( r, θ ) can be summa-rized in the following way:For (cid:96) ∈ (0 , (cid:96) ms ), there are no extrema of the po-tential W , and there are no closed equipotential surfacesand no critical equipotential surface. The null equipo-tential surface is open towards the black hole (Fig. 2A).For (cid:96) = (cid:96) ms , there is one inflexion point of the po-tential W in the equatorial plane at which the criticalsurface has its critical point, corresponding to a ring.The null equipotential surface is open towards the blackhole (Fig. 2B).For (cid:96) ∈ ( (cid:96) ms , (cid:96) mb ), there is a negative local maxi-mum and a negative local minimum of the potential W in the equatorial plane. In this case, closed equipotentialsurfaces exist which are bounded by the critical surfacethat self-crosses at the inner cusp. The null equipotentialsurfaces is open towards the black hole (Fig. 2C).For (cid:96) = (cid:96) mb , there is a zero local maximum and anegative local minimum of the potential W in theequatorial plane. The closed equipotential surfaces arebounded by the critical surface which coincides with thenull equipotential surface (Fig. 2D).For (cid:96) ∈ ( (cid:96) mb , (cid:96) ph , c ), there is a positive local maxi-mum and a negative local minimum of the potential W in the equatorial plane. The closed equipotentialsurfaces are bounded by the outer null equipotentialsurface. The critical surface is now open outwardsaway from the black hole, and self-crosses between theradii where the null surfaces cross the equatorial plane(Fig. 2E). (cid:45) (cid:45) (cid:45) (cid:137) sin (cid:72) Θ (cid:76) r (cid:137) c o s (cid:72) Θ (cid:76) A: (cid:123) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) W Π (cid:144) A: (cid:123) (cid:61) (cid:45) (cid:45) (cid:45) (cid:137) sin (cid:72) Θ (cid:76) r (cid:137) c o s (cid:72) Θ (cid:76) B: (cid:123) (cid:61) (cid:123) ms (cid:85) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) W Π (cid:144) B: (cid:123) (cid:61) (cid:123) ms (cid:85) (cid:45) (cid:45) (cid:45) (cid:137) sin (cid:72) Θ (cid:76) r (cid:137) c o s (cid:72) Θ (cid:76) C: (cid:123) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) W Π (cid:144) C: (cid:123) (cid:61) (cid:45) (cid:45) (cid:45) (cid:137) sin (cid:72) Θ (cid:76) r (cid:137) c o s (cid:72) Θ (cid:76) D: (cid:123) (cid:61) (cid:123) mb (cid:85) (cid:45) (cid:45) (cid:45) (cid:45) W Π (cid:144) D: (cid:123) (cid:61) (cid:123) mb (cid:85) (cid:45)
505 r (cid:137) sin (cid:72) Θ (cid:76) r (cid:137) c o s (cid:72) Θ (cid:76) E: (cid:123) (cid:61) (cid:45) W Π (cid:144) E: (cid:123) (cid:61) (cid:45) (cid:45) (cid:45) (cid:137) sin (cid:72) Θ (cid:76) r (cid:137) c o s (cid:72) Θ (cid:76) F: (cid:123) (cid:61) (cid:45) (cid:45) W Π (cid:144) F: (cid:123) (cid:61) FIG. 2: Typical behavior of the potential W ( r, θ ) shown in terms of poloidal sections through the equipotential surfaces andthe equatorial profile W π/ ( r ) in a Reissner-Nordstr¨om spacetime with parameter Q = 0 .
1. Taking progressively increasingvalues of the specific angular momentum (cid:96) , samples are shown of 4+2 qualitatively different types of behavior of the potential,differing in the properties of the critical (dashed) equipotential surface and the null (thick) equipotential surface.
For (cid:96) = (cid:96) ph , c , the potential W diverges at r ph , c and the local maximum no longer exists. The negativelocal minimum of the potential W is still present. Theclosed equipotential surfaces are bounded by the outernull equipotential surface. The critical surface is nolonger present.For (cid:96) > (cid:96) ph , c , the only extremum of the potential W is the negative minimum. The closed equipotentialsurfaces are bounded by the outer null equipotentialsurface. There is no longer any critical surface, but thereis a forbidden region for fluid elements with prescribedspecific angular momentum, delimited by the radiisatisfying the relation (cid:96) = (cid:96) ph ( r ; Q ) (Fig. 2F). The behavior of the potential W ( r, θ ) is qualita-tively the same as in the pure Schwarzschild case [3].The additional charge of the black hole Q here influencesthe values of (cid:96) ms ( Q ), (cid:96) mb ( Q ) and (cid:96) ph , c ( Q ).Profiles of the pressure and mass-density can now bedetermined from relations (24) and (22). For doing this,it is necessary to choose relevant values for the parame-ters of the polytropic equation of state (22), for the spe-cific angular momentum and for the location of the inneredge of the torus. Here we present two examples, for toriwith Γ = 2, κ = 10 and (cid:96) = 3 . FIG. 3: Profiles of the potential W ( r, θ ), pressure p ( r, θ ) and rest-mass density ρ ( r, θ ) of uncharged tori, shown in termsof poloidal sections through the equipotential, isobaric and iso-density surfaces, respectively, and their equatorial behavior W π/ ( r ), p π/ ( r ) and ρ π/ ( r ), for a Reissner-Nordstr¨om spacetime with Q = 0 .
1. Two examples are shown with different valuesfor the radius at the inner edge of the torus: r in = r cusp . = 4 .
544 in the upper figures and r in = 6 in the lower ones. The centers(pressure maxima) of both tori are located at r cent . = 8 . W π/ ( r ) indicate the physically relevantparts of the profiles, delimited by the inner and outer edges of the tori. The specific angular momentum (cid:96) = 3 . κ = 10 in each case. cusp, r in = r cusp . This is the most interesting case, sinceit can be used as a model for the inner parts of a thickaccretion disk from which matter can flow in a standardway onto the black hole. To obtain the position of thecusp, we solve equation (27), which yields two real rootsabove the event horizon; r I . = 4 .
544 and r II . = 8 . r in > r cusp . Here we simply chose the position ofthe inner edge to be at r in = 6. Note that the location ofthe inner edge must be chosen so as to be in between thecusp ( r I ) and the center of the torus ( r II ), the positionsof both of which are determined from the potential W independently of the choice of r in . V. CHARGED TORI, (cid:96) = const
In order to obtain the pressure or density profiles fromequations (20) or (23), it is necessary to determine the charge density distribution q ( r, θ ) in the torus. This mustsatisfy the integrability condition (15). By expressing q ( r, θ ) in the form q ( r, θ ) = q ρ ( r, θ ) k ( r, θ ) , (28)where q is a constant and k ( r, θ ) is a correction function,and using Γ = 2, we can rewrite equation (23) in the form ∂ r ρ = − κ (cid:18) ( κρ + 1) ∂ r ln | U t | − q k U t ∂ r A t (cid:19) ,∂ θ ρ = − κ (cid:18) ( κρ + 1) ∂ θ ln | U t | (cid:19) . (29)Note that, as we express by relation (28), it is feasibleto take the charge density distribution as being directlyproportional to the rest-mass density distribution and toa correction function which represents variations in thecharge per unit mass q k ( r, θ ) through the torus, as wediscuss in section VI A. From the mathematical point ofview, the correction function plays the role of an ‘inte-gration factor’, which must be chosen so that equations(29) are integrable.Now, due to the integrability condition (15), the cor-rection function k ( r, θ ) has to satisfy the relation2 sin θ ( (cid:96) ∆ − r sin θ ) ∂ θ k + 3 k(cid:96) ∆ cos θ = 0 , (30)which can be solved to give k ( r, θ ) = γ sin / θ (cid:18) (cid:96) ∆ − r sin θ(cid:96) ∆ − r (cid:19) / , (31)where γ ( r ) is a function representing a constant of in-tegration over θ for a given value of k in the equatorialplane, k ( r, θ = π/ ≡ γ ( r ). For the purposes of this pa-per, it is convenient to choose γ ( r ) = 1. The charge den-sity distribution function with the correction function inthe form (31) ensures the integrability of equations (29),and thus the existence of a unique solution for p ( r, θ ) and ρ ( r, θ ).Integrating the second of equations (29) over the lati-tude, we obtain the following expression for the rest-massdensity: ρ ( r, θ ) = 2 / κ C (cid:0) r sin θ − (cid:96) ∆ (cid:1) / − √ sin θκ √ sin θ . (32)The unknown function C ( r ), which depends only on theradial coordinate r , stands as a constant of this integra-tion. Its value can be determined by substituting thedensity formula (32) into the first density equation (29)and assuming the charge-density distribution accordingto relations (28) and (31). This leads to the ordinarydifferential equation2 r ∆ ∂ r C + (2 r − r + Q ) C = r √ ∆ Qq / κ ( r − (cid:96) ∆) / . (33)Unfortunately, there is no analytic solution for equation(33), and so C ( r ) must be determined numerically. Sincethe torus is delimited by the zero-pressure (and zero-density) surface, we can find the necessary initial con-dition from the fact that ρ ( r in ) = 0. From the relation(32) we obtain C ( r in ) = 12 / κ [ r − (cid:96) ( r − r in + Q )] / . (34)As mentioned earlier, the most interesting configura-tion is the one delimited by the self-crossing zero isobaric(and iso-density) surface, i.e., the marginally boundedtorus, which has its inner edge in the equatorial planecoincident with the cusp ( r in = r cusp ). We will be con-centrating on this type of torus from now on. The valuesof the specific angular momentum (cid:96) and the charges Q and q then completely determine the shape of the torusand the positions of its center and of its inner and outeredges. At the cusp, the pressure and density vanish andhave a saddle point (a minimum in the r -direction and a maximum in the θ -direction), in contrast with the cen-ter of the torus, where there is a local maximum. (Ingeneral, the density at the inner edge of the torus mustbe zero and when the inner edge is also a cusp, the den-sity profile has extrema there.) The location of the inneredge r in , which needs to be known in order to evaluatethe condition (34), can then be obtained from the first ofequations (29) by setting ∂ r ρ (cid:12)(cid:12) r = r in = 0, ρ ( r in ) = 0, and θ = π/
2. This gives the following implicit expression for r in : q Qr (cid:113) ∆( r − (cid:96) ∆) + ( Q − r in ) r + (cid:96) ∆ = 0 . (35)Note that when q = 0, we get the relation ( Q − r in ) r + (cid:96) ∆ = 0, in agreement with the formula (27) derived forthe case of the uncharged torus.To clearly illustrate how the charge on the torus affectsits equilibrium structure, we constructed marginally-bounded tori with the same matter parameters and spe-cific angular momentum as in the uncharged case, i.e.,we took κ = 10 and (cid:96) = 3 .
8, and we also consideredthe same charge of the central black hole Q = 0 .
1. Oursample tori are characterized by the parameters q = 0 . q = − . γ ( r ) = 1.For q = 0 .
4, equation (35) yields two real roots abovethe event horizon; r I . = 4 .
378 and r II . = 9 . .
4. Choosing r in = r I , the numerical integration gives a regular thickcharged torus. On the other hand, choosing r in = r II ,we get a degenerate torus (an infinitesimally thin ring)located just at r = r II .For q = − .
4, equation (35) again yields two real rootsabove the event horizon; r I . = 4 .
767 and r II . = 7 . − .
4. Again, the choice r in = r I leads to the regular thick charged torus, while r in = r II gives the degenerate torus.In principle, one could also construct charged tori with r I < r in < r II , as for the uncharged tori. However, thisintroduces further complications, and we focus here onlyon the more interesting critical (cusp) configurations.In Figs 4 and 6, we show the profiles of rest-mass den-sity and pressure. The related charge density distribu-tions q ( r, θ ) and correction functions k ( r, θ ) are shown inFigs 5 and 7. As can be seen from these, for the same Q and (cid:96) , the positively charged tori are more extended thanthe uncharged ones while the negatively charged tori areless extended. From the right-hand panels of Figs 4 and6, it can be seen that the densities in the more extendedtori are larger than those in the less extended ones andso the masses of the more extended tori are clearly alsolarger. We come back to this in more detail in section VIB. FIG. 4: Profiles of the pressure p ( r, θ ) and rest-mass density ρ ( r, θ ) for a positively charged torus ( q = 0 . (cid:96) = 3 .
8) in aReissner-Nordstr¨om spacetime with Q = 0 .
1, shown in terms of poloidal sections through the isobaric and iso-density surfaces(left), and in terms of their equatorial profiles p π/ ( r ) and ρ π/ ( r ) (right). The torus terminates at r in = r cusp . = 4 .
378 and r out . = 24 .
72, and its center is located at r cent . = 9 . k ( r, θ ) (left) and the charge density q ( r, θ ) (right)for the same positively charged torus as in Fig. 4. VI. DISCUSSIONA. Correction function and specific charge
The dimensionless correction function k ( r, θ ) has beenintroduced in equation (28) for convenience in calculating the torus configuration. It represents the variation withposition of the specific electric charge (charge per unitmass), ¯ q = q k ( r, θ ). As shown in the previous section,setting k = 1 so that the charge per unit mass is the sameeverywhere, does not give an equilibrium configuration: k must be allowed to vary so as to satisfy the integrability0 FIG. 6: Profiles of the pressure p ( r, θ ) and rest-mass density ρ ( r, θ ) for a negatively charged torus ( q = − . (cid:96) = 3 .
8) in aReissner-Nordstr¨om spacetime with Q = 0 .
1, shown in terms of poloidal sections through the isobaric and iso-density surfaces(left), and in terms of their equatorial profiles p π/ ( r ) and ρ π/ ( r ) (right). The torus terminates at r in = r cusp . = 4 .
767 and r out . = 11 .
62, and its center is located at r cent . = 7 . k ( r, θ ) (left) and the charge density q ( r, θ ) (right)for the same negatively charged torus as in Fig. 6. condition (15) and this requires the behavior (31). Inthe equatorial plane k ( r, θ = π/
2) = γ ( r ), and the choice γ ( r ) = 1 as a boundary condition is convenient for ourpresent simplified model. From Figs (5) and (7), it canbe seen that the required variations in k ( r, θ ) away fromthis are actually very small (with the maximum being on the equatorial plane).Our choice of γ ( r ) = const (with the constant nor-malised to 1) corresponds to a case where the maximumof the charge density q ( r, θ ) is located just at the centerof the torus, where there is the maximum of the density ρ ( r, θ ), as can be seen from relation (28). Other choices of1 γ ( r ) could describe more physically relevant situations,but with the maximum of q ( r, θ ) not necessarily being lo-cated at the center of the torus. For instance, by choosing γ ( r ) = 1 /r the specific net charge of the fluid in the torusgrows in the equatorial plane from the outer edge to theinner edge, where it is maximal. Moreover, the maxi-mum of q ( r, θ ) is shifted from the center of the torus. Ofcourse, at the inner edge, the net charge density q ( r, θ )goes to zero together with the matter density ρ ( r, θ ).The values q = ± . γ = 1, used for our rep-resentative cases, give a specific charge ¯ q in the torusaround 10 times smaller in magnitude than that for aproton, and so the medium can be thought of as havingone particle in 10 with a net charge while the rest areneutral. We note that if we decrease Q below our stan-dard value of 0 . q in such away that the product Q ¯ q remains unchanged, then we getessentially identical results. This is because the deviationof the space-time geometry away from Schwarzschild isextremely small for these values of Q , and so the rele-vant effect of the charge is almost entirely electromag-netic (depending on Q ¯ q ) rather than having a significantgravitational contribution (depending just on Q ). B. Total electric charge and mass of the torus
In our model, we neglect the effects of the electromag-netic field generated by the torus, which is acceptablewhen this self-field is much weaker than the external elec-tromagnetic field associated with the central compact ob-ject. The total charge of the torus is given by Q = (cid:90) V q √− g d r d θ d φ = 4 πq (cid:90) r out r in (cid:90) π θ ρ kρ √− g d θ d r , (36)where θ ρ = arcsin (cid:32) √ κ (cid:96)C √ κ r C − (cid:33) (37)is the function determining the upper boundary of thepoloidal projection of the zero isodensity surface, g = − r sin θ is the determinant of the metric tensor, and r out is the radial position of the outer edge of the torusin the equatorial plane. For the representative tori whichwe are considering, with Γ = 2, κ = 10 , Q = 0 . (cid:96) = 3 .
8, we obtain the total charge on the torus as being Q . = 2 . × − for q = 0 . Q . = − . × − for q = − . M = (cid:90) V ρ √− g d r d θ d φ = 4 π (cid:90) r out r in (cid:90) π θ ρ ρ √− g d θ d r . (38)For the three representative tori which we are consid-ering, we obtain the total rest-mass as being M . =6 . × − (positively charged), M . = 1 . × − (negatively charged) and M . = 9 . × − (uncharged).The ratio Q / M is then . = ± . q = q k ( r, θ ) = ± . k ( r, θ ) . = ± .
4, since k ( r, θ ) ≈ C. Parameters of the equation of state
We set Γ = 2 for illustration purposes, since this choicesimplifies the integration of the density equations (23).This value of Γ is inconveniently high for possible relatedastrophysical applications, but we stress that our modelis an extremely simplified one, purposely intended forinvestigating the behavior of a test case under extremeconditions (we are also taking a rather large value for theblack-hole charge Q , only one sign of charge for particlesin the torus and zero conductivity there). For such a testcase, it would not be appropriate to introduce additionalcomplications here in order to bring just one aspect ofthe model (the value of Γ) closer to astrophysical appli-cations.In general, electrostatic corrections should also be in-cluded in the equation of state, especially at higher mat-ter densities. However, this is not trivial to do (see, e.g.[45], for the electrostatic correction in the case of dustyplasmas). We have used a very high value of κ , whichenables us to neglect electrostatic corrections without in-consistency, because this enables us to demonstrate ourapproach more clearly. D. Distribution of specific angular momentum
We have considered perfect-fluid tori with a prescribedspecific angular momentum (cid:96) = − U φ /U t , which we set tobe constant through the torus. In an uncharged case, L = U φ and E = − U t would be constants of motion connectedwith the assumed axial symmetry and stationarity of thespacetime. For a charged torus, the constants of motionare the generalized quantities ˜ L = U φ + ¯ qA φ and ˜ E = − U t − ¯ qA t .Note that the condition (cid:96) = const is imposed forsimplicity of the calculations; it is not essential for themethod and can be relaxed. The Rayleigh criterion forlinear stability against radial convection requires (cid:96) to bea non-decreasing function of the distance from the axisof rotation, and so (cid:96) = const uncharged tori are just on2the stability limit. For charged tori, the stability con-dition needs to be formulated in terms of a generalizedquantity ˜ (cid:96) = ˜ L/ ˜ E [30, 46]. In the Reissner-Nordstr¨omelectric field, the only non-zero component of the vectorpotential is A t and one has˜ (cid:96) = ˜ L ˜ E = LE − q kA t , (39)which reduces to (cid:96) for q = 0. Stability depends heavilyon the specific charge distribution (i.e. on the behaviorof the correction function k ( r, θ )) and on the signs of thecharges of the torus and the black hole. E. Approximation of negligible conductivity
The assumption of high electrical conductivity of themedium is appropriate for many astrophysical plasmaswith a high degree of ionization, and the ideal MHDframework can then be employed. Under the conditionsof high conductivity and vanishing inertial effects of theplasma particles, the local electric field quickly becomesneutralized by rearranging the plasma flows (giving theconditions for the force-free approximation). A quasi-neutral medium then arises in which the volume densityof net electric charge is negligible.However, there is an ongoing debate about the con-ditions that may lead to the presence of non-vanishingnet charges, with an important role being played by elec-tric forces acting parallel to the magnetic field lines inthe local co-moving frame. For example, a large-scalemagnetic field may cause spatial separation of electriccharges of different signs and their gradual accumulationin different parts of the system. Pulsar magnetospheresprovide an example of such systems, with the charge sep-aration being caused by the dipole-type magnetic field ofthe neutron star [47, 48]. Black holes embedded in or-dered magnetic fields of external origin can also act ina similar way but then, for a low-density medium, thehydrodynamical description needs to be modified in or-der to describe the conditions of a collisionless plasma(since the particle mean-free paths are then comparablewith the characteristic length-scale of the system, givenby the gravitational radius of the central black hole).One can imagine also another relevant scenario: a neu-tral fluid containing a few free charges, such as the caseof dusty plasmas. In general, when charges feel an exter-nal electromagnetic field, they move generating a current,but if the fluid is dense enough and is highly collisional,the charges are less able to move and the conductivitybecomes almost zero (see [49] for a discussion). Such apicture is actually compatible with our model since, asmentioned earlier, only a very small fraction of particleswith net charge is required in order to give the parametervalues used in our representative examples.
F. Zero conductivity and consistency of the model
The various limiting situations which we have beenmentioning (hydrodynamical versus collisionless plasma;infinite conductivity versus zero conductivity; self-gravitating matter versus test particles and fluids), arerelevant under quite different circumstances and obvi-ously require different approaches. The approach whichwe have adopted in the present paper allows us to cap-ture the behavior of an idealized but non-trivial systemwhere the fluid motion is governed by the combined ac-tion of a global (large-scale) electromagnetic field, thegravitational field of the central body, and pressure gra-dients operating within the fluid together with a non-zeroelectric charge distribution.The basic assumptions of our model are: 1) the fluidis a single-component test fluid (we ignore its self-gravityand its own electromagnetic field); 2) the fluid flow isstationary, with the 4-velocity having only time and az-imuthal components. If the conductivity σ were non-zero, the second term on the right-hand side of the Ohm’slaw equation (4), which is proportional to σ , would giverise to a radial electric current unless there were a signif-icant self-field (contradicting the first basic assumption).Since our fluid is taken to be a single-species one, hav-ing a radial electric current would imply the existenceof a radial mass current (contradicting the second basicassumption). Having σ = 0 is therefore necessary forself-consistency. VII. CONCLUSIONS
In this paper we have presented a model for a simpletest case of an electrically-charged perfect-fluid torus ro-tating in strong gravitational and electromagnetic fieldsproduced by a central compact object. Distributionsof either specific angular momentum or charge densitythrough the torus first need to be specified (we chose tospecify the specific angular momentum distribution) andthen pressure and density profiles can be calculated. Anequation of state must be provided in order to close theset of equations. We have investigated the limiting caseopposite to that of ideal magnetohydrodynamics, con-sidering a non-conductive (dielectric) perfect fluid ratherthan infinite conductivity as in ideal magnetohydrody-namics. Our case can describe a fluid in which bulk hy-drodynamic motion predominates over electromagneticeffects and the fluid has almost infinite electric resistiv-ity. We are not here including the self-gravitational andself-electromagnetic fields of the matter in the torus, andso the treatment applies for low-mass, slightly-chargedtori.For illustrating the model, we constructed both pos-itively and negatively charged barotropic tori, with apolytropic equation of state and constant specific angularmomentum distribution, encircling a positively-chargedReissner-Nordstr¨om black hole. We compared the re-3sulting pressure and density profiles with an equivalentuncharged case, and also calculated the shapes of thetori. Taking the polytropic index Γ = 2, allows for the‘pressure’ equation to be integrated in a relatively simpleway so as to give semi-analytic results. The large valuefor the polytropic constant κ = 10 leads to tori wherethe Coulomb interaction between the charged particles ofthe fluid can be neglected in comparison with the stan-dard pressure due to the matter. The constructed tori areonly slightly electrically charged in comparison with thecharge of the black hole, thus generating a relatively weakelectromagnetic field which can safely be neglected. It isstriking that even with a very small value for the charge-to-mass ratio in the torus, significant effects are neverthe-less seen. However, it is necessary to stress out that theconstructed tori carry the specific charge Q / M . = ± . Q/M = 0 .
1; since for anyastrophysical body it would be practically impossible tomaintain the specific charge
Q/M > − [44], the re-sults should be considered as illustrating samples only.The aim of this paper has been to introduce a newmodel of a dielectric charged torus encircling a chargedcompact object. We have proceeded by using a num-ber of simplifying assumptions: the compact object is aReissner-Nordstr¨om black-hole; the torus is composed oftest matter; the matter has a polytropic equation of statewith prescribed Γ; the torus has constant specific angu- lar momentum, (cid:96) = const, and the specific charge is con-stant everywhere in the equatorial plane, γ = 1. Despitethe great simplifications coming from these assumptions,the scenario is still physically reasonable and non-trivial.Moreover, a large variety of degrees of freedom can becaptured and the free parameters can be convenientlychosen in order to describe an astrophysically relevantsituation. Of course, more complicated choices wouldthen require more complicated calculations. Such calcu-lations are now in progress, but are beyond the scope ofthe present paper. Acknowledgments
The Opava Institute of Physics and Prague Astro-nomical Institute have been operated under the projectsMSM 4781305903 and AV 0Z10030501, and further sup-ported by the Center for Theoretical AstrophysicsLC06014 in the Czech Republic. JK, VK and ZSthank the Czech Science Foundation (ref. P209/10/P190,205/07/0052, 202/09/0772). We also gratefully acknowl-edge support from CompStar, a Research NetworkingProgramme of the European Science Foundation andthank an anonymous referee for advice and critical com-ments which have led to improvement of our paper. [1] J. Frank, A. King, and D. Raine,