Spherical accretion: Bondi, Michel and rotating black holes
Alejandro Aguayo-Ortiz, Emilio Tejeda, Olivier Sarbach, Diego López-Cámara
MMon. Not. R. Astron. Soc. , 1–15 (0000) Printed 26th February 2021 (MN L A TEX style file v2.2)
Spherical accretion: Bondi, Michel and rotating black holes
Alejandro Aguayo-Ortiz , (cid:63) Emilio Tejeda, Olivier Sarbach, & Diego L ´opez-C´amara Universidad Nacional Aut´onoma de M´exico, Instituto de Astronom´ıa, AP 70-264, CDMX 04510, M´exico C´atedras CONACyT – Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Edificio C-3, Ciudad Universitaria, 58040 Morelia,Michoac´an, Mexico Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Edificio C-3, Ciudad Universitaria, 58040 Morelia, Michoac´an, Mexico C´atedras CONACyT – Universidad Nacional Aut´onoma de M´exico, Instituto de Astronom´ıa, AP 70-264, CDMX 04510, M´exico
ABSTRACT
In this work we revisit the steady state, spherically symmetric gas accretion problem fromthe non-relativistic regime to the ultra-relativistic one. We first perform a detailed compar-ison between the Bondi and Michel models, and show how the mass accretion rate in theMichel solution approaches a constant value as the fluid temperature increases, whereas thecorresponding Bondi value continually decreases, the difference between these two predictedvalues becoming arbitrarily large at ultra-relativistic temperatures. Additionally, we extend theMichel solution to the case of a fluid with an equation of state corresponding to a monoatomic,relativistic gas. Finally, using general relativistic hydrodynamic simulations, we study spher-ical accretion onto a rotating black hole, exploring the influence of the black hole spin onthe mass accretion rate, the flow morphology and characteristics, and the sonic surface. Theeffect of the black hole spin becomes more notorious as the gas temperature increases and asthe adiabatic index γ stiffens. For an ideal gas in the ultra-relativistic limit ( γ = 4 / ), we finda reduction of 10 per cent in the mass accretion rate for a maximally rotating black hole ascompared to a non-rotating one, while this reduction is of up to 50 per cent for a stiff fluid( γ = 2 ). Key words: accretion, accretion discs – gravitation – hydrodynamics – methods: numerical.
Gas accretion onto a compact gravitating object is one of the moststudied problems in astrophysics. In one of the pioneering worksof accretion theory, Bondi (1952) found an analytic solution for thespherically symmetric, steady-state accretion flow of an infinite gascloud onto a Newtonian point-mass potential. This model has beenwidely extended and applied in many different fields in astrophys-ics, from the study of star formation to cosmology. See Armitage(2020) for a recent historical review of this subject.One of the first studies of spherical accretion onto black holeswas the extension of the Bondi solution into the general relativisticregime performed by Michel (1972). In this study, Michel found ananalytic solution describing the spherical accretion of a polytropicgas onto a Schwarzschild black hole. A formal mathematical ana-lysis of the Michel solution for a generic equation of state (EoS) canbe found in Chaverra et al. (2016). Following Michel’s procedure,other authors have found semi-analytic generalizations for differ-ent types of spherically-symmetric (non-rotating) black hole solu-tions (e.g. Chaverra & Sarbach 2015; Miller & Baumgarte 2017;Yang et al. 2021; Abbas & Ditta 2021). In a recent work, Richards (cid:63)
E-mail: [email protected], [email protected],[email protected], [email protected] et al. (2021a) explore the non-relativistic and ultra-relativistic lim-its of Michel’s solution, mainly focusing on a gas with a stiff EoS(values of the adiabatic index larger than 5/3).In past decades, the spherical accretion Bondi model has beenrevisited and extended, by including different additional physicalingredients. For example, some authors have taken into account thefluid’s self gravity by solving the coupled Einstein-Euler systemin spherical symmetry, either with an analytical treatment (Malec1999) or by performing numerical simulations (Lora-Clavijo et al.2013). Some works have considered the extension of a Bondi-likesolution by introducing a low angular momentum fluid (Abramow-icz & Zurek 1981; Proga & Begelman 2003; Mach et al. 2018),finding a transition between a quasi-spherical accretion flow andthe formation of a thick torus in the equatorial plane. Similarly,there have been works studying spherical accretion in the presenceof magnetic fields, either assuming a central dipole (Toropin et al.1999), or by including a three-dimensional, large-scale weak mag-netic field (e.g. Igumenshchev & Narayan 2002). Together withmagnetic fields, some works have included the effects of a radiationfield, addressing the problem either with a simplified approach (Be-gelman 1978, where the author considered a radiation-dominatedfluid) or with a self-consistent, radiative-transfer treatment usingnumerical simulations (McKinney et al. 2014; Weih et al. 2020). Inthis regard, there have also been studies that extract the shadow of © 0000 RAS a r X i v : . [ a s t r o - ph . H E ] F e b A. Aguayo-Ortiz et al. the spherically accreted, optically thin cloud around a non-rotatingblack hole (Narayan et al. 2019). Other studies have consideredthe effects of thermal conduction on magnetized spherical accre-tion flows (Sharma et al. 2008), vorticity (Krumholz et al. 2005), orstudied the spherical accretion of a relativistic collisionless kinetic(i.e. a Vlasov) gas (Rioseco & Sarbach 2017a).Recent works have also studied deviations away from spher-ical symmetry by introducing large-scale, small-amplitude dens-ity anisotropies, finding that even a slight equator-to-poles densitycontrast can drastically modify Bondi’s solution, giving rise to aninflow-outflow configuration consisting of equatorial accretion anda bipolar outflow. The resulting steady-state configuration, dubbedchoked accretion, was studied in Aguayo-Ortiz et al. (2019) at theNewtonian level and, within a general relativistic framework, in Te-jeda et al. (2020) and Aguayo-Ortiz et al. (2021) for Schwarzschildand Kerr black holes, respectively. In these series of works, it wasfound that the total mass flux that reaches the central accretor is ofthe order of magnitude of the corresponding Bondi mass accretionrate, while all the excess flux is redirected by the density gradientas outflow. Under the conditions explored so far, the Bondi massaccretion rate acts as a threshold value delimiting whether a givenincoming flow becomes choked and prone to the ejection of a bi-polar outflow.Among the astrophysical applications of the spherical accre-tion model, we mention the study of gas accretion in an expandingUniverse (Colpi et al. 1996), the formation and growth of primor-dial black holes in the early stages of the Universe (Zel’dovich &Novikov 1967; Carr 1981; Karkowski & Malec 2013; Lora-Clavijoet al. 2013), and the accretion onto a mini black hole from the in-terior of a neutron star (Kouvaris & Tinyakov 2014; G´enolini et al.2020; Richards et al. 2021b). On the other hand, the Bondi solu-tion allows to estimate, by providing useful characteristic scaletools, the accretion and growth rate of the central supermassiveblack hole at the centre of galaxies (Maraschi et al. 1974; Mos-cibrodzka 2006; Ciotti & Pellegrini 2017; Moffat 2020) and activegalactic nuclei (Krolik & London 1983; Russell et al. 2013, 2015),where observations provide information only from regions far awayfrom the central accretor. Similarly, the Bondi prescription is oftenused in cosmological simulations as a sub-grid model to estimatethe accretion rate of gas onto supermassive black holes at galacticcentres (Dav´e et al. 2019).On the other hand, the analytic study of accretion flows ontorotating black holes has proven more challenging. Notably, Pet-rich, Shapiro & Teukolsky (1988) found a full analytic solutionthat describes the accretion of an irrotational, ultra-relativistic stifffluid onto a rotating Kerr black hole. However, a main caveat ofthis solution is that it requires a rather specific, unphysical EoS, inwhich the sound speed equals the speed of light. Assuming a moregeneral EoS, Beskin & Pidoprygora (1995) studied the problem ofspherical accretion onto a slowly rotating black hole by means of aperturbative analysis, and Pariev (1996) extended this work to thecase of a rapidly rotating black hole. Both Beskin & Pidoprygora(1995) and Pariev (1996) considered only small deviations awayfrom a Bondi background solution, in other words, these studieswhere limited to the case of non-relativistic values for the gas tem-perature at infinity. Even though this assumption might be reason-able in many astrophysical settings, the determination of the effect An ultra-relativistic stiff fluid corresponds to the relativistic generalisa-tion of an incompressible fluid in Newtonian hydrodynamics (Tejeda 2018). of the black hole spin on the accretion flow given an arbitrary gastemperature remains an open problem.The applications of Bondi’s model in most of the aforemen-tioned works consider the gas accretion in the non-relativistic re-gime, not to mention that they neglect the rotation of the black hole.The reason for this is that the Bondi scale factors are estimated andmeasured at distances far away from the central black hole, where itis safe to neglect relativistic effects. Nevertheless, in order to ana-lyse the exact differences between the Bondi solution and the re-lativistic extension performed by Michel, as well as to assess theeffect of the black hole spin, it is important to perform a quantit-ative study of the consequences of having relativistic gas temper-atures and strong gravity fields in the vicinity of a rotating blackhole.In this work we study the spherically symmetric gas accretionproblem from the non-relativistic regime to the ultra-relativisticone, considering both rotating and non-rotating black holes. Wefirst perform a detailed comparison between the Bondi (1952)and Michel (1972) models by studying the behaviour of the re-lativistic solution across a wide range of values of the gas tem-perature. In particular, we discuss in detail the isothermal, the non-relativistic, and the ultra-relativistic limits of the Michel solution.We then extend Michel’s solution to the case of a monoatomic gasobeying a relativistic EoS (J¨uttner 1911; Taub 1948; Synge 1957).We also revisit Petrich et al. (1988)’s analytic solution and applyit to the particular case of a spherically symmetric accretion flowonto a Kerr black hole. Finally, by means of two dimensional (2D)general relativistic hydrodynamic simulations, we perform a quant-itative study of the effect that the black hole spin has on the spher-ical accretion problem, focusing in particular on its effects on themass accretion rate and on the flow morphology for several EoS.As part of this study, we show how, under the appropriate limits,the obtained numerical results coincide with the analytic solutionsof Michel (1972) and Petrich et al. (1988).The paper is organised as follows. In Section 2 we discuss theanalytic solutions of Bondi (1952), Michel (1972) and Petrich et al.(1988). In Section 3 we present our numerical study of the spher-ical accretion of a perfect fluid onto a rotating black hole. Finally,in Section 4 we present a summary of the main results found inthis article and give our conclusions. Technical details regardingthe isothermal and non-relativistic limits of the Michel solution,the correct determination of the sonic surface for flows on rotatingblack holes, and orthonormal frames are discussed in appendices.
In this section we review three analytic solutions describing asteady-state, spherical accretion flow onto a central massive object.We start by revisiting the Bondi (1952) solution and perform a de-tailed comparison with the relativistic extension found by Michel(1972). Then, we extend the latter solution by considering the morerealistic equation of state for a monoatomic relativistic gas intro-duced by J¨uttner (1911). Finally, in order to give a descriptionof accretion onto a rotating black hole, we also discuss the ultra-relativistic, stiff solution found by Petrich et al. (1988) in the caseof spherical symmetry. By ‘spherically symmetric’ accretion problem in Kerr spacetime, werefer to the gas state being spherically symmetric asymptotically far awayfrom the central black hole. Clearly, a rotating black hole does not admit aspherically symmetric solution at finite radii.© 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes In the Bondi (1952) analytic solution, one considers an infinite,spherically symmetric gas cloud accreting onto a Newtonian cent-ral object of mass M . At large distances, the gas cloud is assumedto be at rest and characterised by a homogeneous density ρ ∞ andpressure P ∞ . Note that, using an ideal gas EoS, we can alternat-ively describe the state of the fluid in terms of the dimensionlessgas temperature Θ defined as: Θ = k B T ¯ m c = Pρ c , (2.1)where c is the speed of light, k B Boltzmann’s constant, and ¯ m theaverage rest mass of the gas particles. As reference values, Θ (cid:39) T / (10 K) for atomic hydrogen gas and Θ (cid:39) T / (10 K) for anelectron-positron plasma.Under the assumptions of steady-state and spherical sym-metry, the equations governing the Bondi accretion flow are thecontinuity equation and the radial Euler equation, i.e. r dd r (cid:0) r ρ v (cid:1) = 0 , (2.2a) v d v d r + 1 ρ d P d r + GMr = 0 , (2.2b)where v = | d r/ d t | is the radial velocity of the fluid.Considering that, in addition to the ideal gas EoS, the fluidobeys a polytropic relation P = Kρ γ , with K = const . and γ the adiabatic index (assumed to lie in the range (cid:54) γ (cid:54) ), equa-tions (2.2a) and (2.2b) can be integrated: π r ρ v = ˙ M = const ., (2.3a) v (cid:104) − GMr = (cid:104) ∞ = const ., (2.3b)where (cid:104) = (cid:18) γγ − (cid:19) Pρ = γ Θ c γ − C γ − (2.4)is the specific enthalpy and C := (cid:112) ∂P/∂ρ the adiabatic speed ofsound. Note that equation (2.4) is only valid for γ > . In the iso-thermal case, where γ = 1 and Θ ≡ Θ ∞ = C ∞ /c , equation (2.4)needs to be replaced by (cid:104) − (cid:104) ∞ = C ∞ ln (cid:18) ρρ ∞ (cid:19) . (2.5)In addition to the steady-state and spherical symmetry condi-tions, Bondi also assumed that the flow is transonic, i.e. that thereexists a radius r s at which the fluid’s radial velocity equals the localspeed of sound. From equations (2.2a) and (2.2b), it is simple tocalculate that the fluid at the sonic radius, r s , satisfies r s = GM v s , (2.6a) v s = C s = C ∞ (cid:18) − γ (cid:19) / . (2.6b)The transonic solution found by Bondi is unique and max-imises the accretion rate onto the central object, which, in turn, isgiven by ˙ M B = 4 π λ B ( GM ) ρ ∞ C ∞ , (2.7)where λ B is a numerical factor of order one that depends only on γ and is given by λ B = 14 (cid:18) − γ (cid:19) − γ γ − . (2.8) Figure 1.
Mach number as a function of radius for the case of a γ = 4 / polytrope and for three asymptotic temperatures. Note that the non-relativistic limit ( Θ ∞ (cid:28) ) corresponds to the Bondi solution. The greydashed line shows the point at which, for a given Θ ∞ , the correspond-ing solution crosses the black hole’s event horizon. In all cases the accre-tion flow has transitioned from subsonic to supersonic before crossing theevent horizon. The horizontal axis is scaled in units of the Bondi radius r B = GM/C ∞ . The accretion rate given in equation (2.7) is only valid for γ (cid:54) / . In order to discuss the γ > / case, one must necessarilyaccount for general relativistic effects as we shall see in Section 2.2.Particular values for λ B in equation (2.8) are λ B (5 /
3) = 1 / ,λ B (4 /
3) = 1 / √ (cid:39) . ,λ B (1) = e / / (cid:39) . . An interesting characteristic of the Bondi solution is that itcan be written in a scale-free form with respect to the mass of thecentral object M and the thermodynamic state of the fluid ( ρ ∞ , P ∞ ) by adopting r B = GM/C ∞ , C ∞ , and ρ ∞ as units of length,velocity, and density, respectively. In other words, a global solutionof the Bondi accretion problem is fully characterised once a givenvalue for the adiabatic index γ is provided. Once the value for themass accretion rate of Bondi’s solution for a given γ is known, onecan go back to equations (2.3a)–(2.4) and solve numerically thecorresponding algebraic system of non-linear equations to obtain ρ , P , and v as a function of radius. See Figure 1, for an example wherewe show the resulting Mach number ( M = v/ C ) as a function ofradius, for the solution with γ = 4 / (red line). As mentioned in the introduction, a general relativistic extension ofthe Bondi solution was presented by Michel (1972) who considereda Schwarzschild black hole as central accretor. In what follows wereview Michel’s solution and discuss its main differences with re-spect to the Bondi model. It is important to remark that the Michelsolution assumes an ideal gas EoS that follows a polytropic rela-tion ( P ∝ ρ γ ). Note however that this assumption is limited ingeneral. For example, for a monoatomic ideal gas, it is only validat non-relativistic temperatures (for which γ = 5 / ), or at ultra-relativistic temperatures (for which γ = 4 / ). In order to studythe whole temperature domain in a consistent way, the polytropic © 0000 RAS, MNRAS000
3) = 1 / √ (cid:39) . ,λ B (1) = e / / (cid:39) . . An interesting characteristic of the Bondi solution is that itcan be written in a scale-free form with respect to the mass of thecentral object M and the thermodynamic state of the fluid ( ρ ∞ , P ∞ ) by adopting r B = GM/C ∞ , C ∞ , and ρ ∞ as units of length,velocity, and density, respectively. In other words, a global solutionof the Bondi accretion problem is fully characterised once a givenvalue for the adiabatic index γ is provided. Once the value for themass accretion rate of Bondi’s solution for a given γ is known, onecan go back to equations (2.3a)–(2.4) and solve numerically thecorresponding algebraic system of non-linear equations to obtain ρ , P , and v as a function of radius. See Figure 1, for an example wherewe show the resulting Mach number ( M = v/ C ) as a function ofradius, for the solution with γ = 4 / (red line). As mentioned in the introduction, a general relativistic extension ofthe Bondi solution was presented by Michel (1972) who considereda Schwarzschild black hole as central accretor. In what follows wereview Michel’s solution and discuss its main differences with re-spect to the Bondi model. It is important to remark that the Michelsolution assumes an ideal gas EoS that follows a polytropic rela-tion ( P ∝ ρ γ ). Note however that this assumption is limited ingeneral. For example, for a monoatomic ideal gas, it is only validat non-relativistic temperatures (for which γ = 5 / ), or at ultra-relativistic temperatures (for which γ = 4 / ). In order to studythe whole temperature domain in a consistent way, the polytropic © 0000 RAS, MNRAS000 , 1–15 A. Aguayo-Ortiz et al. restriction must be dropped and a relativistic EoS (as derived, forexample, from relativistic kinetic theory, Synge 1957) must be ad-opted. We discuss the extension of the Michel solution to such arelativistic EoS in Section 2.3.As in the Newtonian case, the governing equations are the con-servation of mass and energy, i.e. the continuity equation and the re-quirement for the energy-momentum tensor to be divergence-free, ( ρ U µ ) ; µ = 0 , (2.9a) ( T µν ) ; µ = 0 , (2.9b)where the semicolon stands for covariant derivative, U µ is the fluidfour-velocity, T µν = ρ h U µ U ν + p g µν is the stress-energy tensorof a perfect fluid, and h = 1+ (cid:104) is the specific relativistic enthalpy.In order to ease the notation, we adopt geometrised units in which G = c = 1 .It is useful to recall at this point that, in the relativistic regime,the fluid’s sound speed is defined as C := ρh ∂h∂ρ = γh Pρ = γh Θ , (2.10)where, for the second equal sign, we have substituted the polytropicrelation for a perfect fluid. Also note that equation (2.10) can berecast to express h in terms of C or Θ as h = 11 − C / ( γ −
1) = 1 + γγ − . (2.11)Under the conditions of steady-state and spherical symmetry,equations (2.9a) and (2.9b) reduce to dd r (cid:0) r ρ U r (cid:1) = 0 , (2.12a) dd r (cid:0) r ρ h U t U r (cid:1) = 0 , (2.12b)which, upon integration, can be rewritten as π r ρ u = ˙ M = const ., (2.13a) h (cid:18) − Mr + u (cid:19) / = h ∞ = const ., (2.13b)where u = | U r | .As in the Newtonian case, there exists a unique, transonicsolution where the fluid is at rest asymptotically far away from thecentral object and that is regular across the black hole’s event ho-rizon (Chaverra & Sarbach 2015; Chaverra et al. 2016). In orderto find the defining conditions that are satisfied at the sonic point r s , it is useful to combine equations (2.12a) and (2.12b) into thefollowing differential equation (cid:20) − C u (cid:18) − Mr + u (cid:19)(cid:21) u d u d r = − Mr + 2 C r (cid:18) − Mr + u (cid:19) . (2.14)By requiring that both sides of this equation vanish simultaneouslyat r s , the following conditions arise r s = 12 Mu s , (2.15a) u s = C s C s . (2.15b)On the other hand, a relationship between the fluid state atinfinity and at the sonic point can be obtained by substituting equa-tions (2.15a) and (2.15b) into equation (2.13b). Doing this results in the following cubic equation for h s (see Tejeda et al. 2020, Ap-pendix A) h s − (3 γ − h ∞ h s + 3( γ − h ∞ = 0 , (2.16)as well as the corresponding equation for the sound speed C s = 13 (cid:18) h s h ∞ − (cid:19) . (2.17)The polynomial in equation (2.16) has three real roots but onlyone satisfies h s > and thus has physical meaning. This root isgiven by h s = 2 h ∞ (cid:114) γ −
23 sin (cid:16)
Ψ + π (cid:17) , (2.18)where Ψ = 13 arccos (cid:34) γ − h ∞ (cid:18) γ − (cid:19) − / (cid:35) . (2.19)Substituting these results back into equation (2.13a), the massaccretion rate can be expressed in terms of the asymptotic state ofthe fluid as ˙ M M = 4 π λ M M ρ ∞ C ∞ , (2.20)where now the numerical factor λ M depends not only on γ but alsoon the asymptotic state of the fluid and is given by λ M = 14 (cid:18) h s h ∞ (cid:19) γ − γ − (cid:18) C s C ∞ (cid:19) − γγ − . (2.21)In Figure 2 we show the dependence of λ M on Θ ∞ for severaldifferent values of the adiabatic index γ . From this figure, it is clearthat for γ (cid:54) / in the non-relativistic limit (Θ ∞ (cid:28) λ M → λ B , as expected. We stress that the mass accretion rate given inequation (2.20) is only a measure for the flux of rest-mass (particlenumber times the average rest-mass per particle) onto the centralblack hole. If interested in computing the actual growth rate of theblack hole’s mass, the total energy advected by each fluid particleshould be taken into account by computing the energy accretionrate (for further details see Aguayo-Ortiz et al. 2021). Since for thepresent problem the fluid is assumed to be at rest at infinity, oneonly needs to multiply ˙ M M in equation (2.20) by h ∞ to obtain thisrate, i.e. ˙ E M = 4 π λ M M ρ ∞ (cid:18) γ Θ ∞ + 1 γ − (cid:19) / γ Θ ∞ . (2.22)In contrast to the Bondi solution, where the asymptotic speedof sound C ∞ is the only characteristic velocity, the Michel solutionnaturally features the speed of light as an additional characteristicvelocity. Consequently, the Michel solution can only be renderedscale invariant with respect to M and ρ ∞ . Therefore, in additionto the adiabatic index γ , to completely describe a given solutionone must also specify C ∞ or, alternatively, Θ ∞ . In what follows weshall use Θ ∞ as the dynamically relevant parameter describing thestate of the fluid asymptotically far away from the central object.Examples of the resulting Mach number M for a As long as γ > and h ∞ > the cubic polynomial on the left-handside of equation (2.16) is positive for h s = 0 and negative for h s = 1 ,which implies that it has three real roots lying in the intervals ( −∞ , , (0 , and (1 , ∞ ) , respectively. See also Chaverra et al. (2016); Richardset al. (2021a) for alternative ways to characterise the sonic radius using C s .© 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes − − − . Θ ∞ λ M . . / . / . Figure 2.
Numerical factor λ M in the definition of the mass accretion rateof the Michel solution (equation 2.21) as a function of the dimensionlesstemperature Θ ∞ and for different values of γ as indicated by the labels ontop of each curve. In the non-relativistic limit Θ ∞ (cid:28) and for values of γ (cid:54) / , the curves asymptotically approach the values corresponding to λ B (dashed horizontal lines) of the Bondi solution in equation (2.7). In theultra-relativistic limit Θ ∞ (cid:29) , the curves asymptotically approach thevalue given in equation (2.26). γ = 4 / polytrope and various asymptotic temperatures: Θ ∞ = 10 − , . , are also shown in Figure 1. Note that,in the relativistic case we have defined M = V / C , where V is thenorm of the fluid three-velocity calculated as V = (cid:18) − Mr (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) U r U t (cid:12)(cid:12)(cid:12)(cid:12) . (2.23)As we can see from this figure, in all cases the accretion flowtransitions from being subsonic to supersonic before crossing theevent horizon (indicated by the grey dashed line). Also, the non-relativistic limit ( Θ ∞ (cid:28) ) coincides with the Bondi solution.In Figures 3 and 4 we show, respectively, the sonic radius r s and the mass accretion rate ˙ M M as functions of Θ ∞ and for severalrepresentative values of γ . By examining these figures, and analy-sing in detail the result obtained in equation (2.20), three interes-ting limits can be identified (isothermal, non-relativistic and ultra-relativistic). In what follows, we list the main conclusions that canbe drawn in each case, leaving detailed calculations to Appendix A. (i) Isothermal limit The isothermal limit corresponds to the condition when γ → . From Figure 3 we note that, within this limit and for alltemperatures Θ ∞ , the sonic radius recedes without limit from theevent horizon. As we prove in Appendix A, the Michel solution(with a suitable rescaling) converges to the Newtonian Bondi solu-tion with an EoS as given by equation (2.5). Thus, the isothermalcase can be entirely described within the context of Newtonianphysics, even for large temperatures that would ordinarily be as-sociated with an ultra-relativistic regime. (ii) Non-relativistic limit This limit is described by the condition Θ ∞ (cid:28) which im-plies h ∞ → . As expected, and as is already apparent from Fig-ures 1 and 2, in this limit the Michel solution converges to the Bondione and expressions like the mass accretion rate (equation 2.20) re- − − − . Θ ∞ − − . ( r s − M ) / M . . . / / . Figure 3.
Distance between the sonic radius r s and the black hole’s eventhorizon radius r + = 2 M in the Michel solution as a function of the dimen-sionless temperature Θ ∞ and for different values of γ as indicated by thelabels on top of each curve. − − − − . Θ ∞ ˙ M M / π M ρ ∞ γ = 2 γ = 5 / γ = 4 / γ = 1 . γ = 1 . γ = 1 . Figure 4.
Mass accretion rate of the Michel solution ˙ M M as a function of Θ ∞ and for several representative values of γ . The dashed lines representthe corresponding Bondi mass accretion rate ˙ M B in cases with γ (cid:54) / .The dotted lines show the corresponding energy accretion rate (see equa-tion 2.22). Note that once Θ ∞ (cid:38) − , the differences between Bondi’sand Michel’s solutions become of order one and that this difference actuallydiverges as Θ ∞ → ∞ . duce to their non-relativistic counterparts (equation 2.7). Neverthe-less, this is only true for γ (cid:54) / . When γ > / a qualitativechange takes place in Michel’s solution. From Figure 3 it is clearthat for γ < / the value of r s grows to infinity as Θ − ∞ (as inthe Bondi solution), while it converges to a finite distance from theevent horizon for γ > / . This is indicative that the cases with γ > / cannot be described with Newtonian physics, even in thelow temperature limit. As shown in Appendix A, when Θ ∞ (cid:28) and γ > / , one finds that r s ≈ M and h s > , while the result-ing mass accretion rate converges to ˙ M M → πh γ − γ − s C − γγ − s M ρ ∞ C − γ − ∞ . (2.24) (iii) Ultra-relativistic limit © 0000 RAS, MNRAS , 1–15 A. Aguayo-Ortiz et al.
Finally, we discuss the case where Θ ∞ (cid:29) . From Figure 3it is clear that r s converges to a finite value strictly larger than theevent horizon radius for all values of γ , with the exception of a stiffEoS γ = 2 , in which case r s → r + as Θ ∞ → ∞ . Moreover,within this limit equation (2.18) reduces to h s h ∞ → (cid:112) γ − , (2.25)from which one also obtains C s / C ∞ → and, hence, λ M →
14 (3 γ − γ − γ − . (2.26)This limit value grows monotonically from e / (cid:39) . to as γ increases from to (see Figure 2). From this result, and as is alsoclear from Figure 4, one sees that the mass accretion rate becomesindependent of Θ ∞ , rapidly approaching the constant value ˙ M M → πM ρ ∞ (3 γ − γ − γ − ( γ − − / . (2.27)In comparison, the Bondi mass accretion rate steadily decreasesas Θ − / ∞ as the temperature increases. Therefore, the differencebetween ˙ M B and ˙ M M becomes arbitrarily large when Θ ∞ (cid:29) .Also note that the energy accretion rate is not monotonic; it de-creases for small temperatures but increases for large ones, even-tually growing linearly in Θ ∞ (see equation 2.22 and Figure 4).A similar qualitative behaviour has been observed for the accre-tion of a Vlasov gas (Rioseco & Sarbach 2017b). This is a remark-able difference between the Bondi and Michel solutions that, to thebest of our knowledge, had not been discussed in the literature be-fore. We can track the reason behind this behaviour by examiningequations (2.7) and (2.20). Even though the factor C − ∞ appears inboth expressions, this factor behaves drastically differently when Θ ∞ (cid:38) . For a perfect fluid in Newtonian hydrodynamics, onesimply has C = γ Θ and, thus, as the temperature increases sodoes the speed of sound without limit. Consequently, and as can beseen in Figure 4, the Bondi mass accretion rate decreases to smallvalues as the asymptotic gas temperature increases. On the otherhand, in relativistic hydrodynamics one has C = γ ( γ − γ − γ Θ , that, in the limit Θ ∞ (cid:29) , implies that the speed of sound at-tains a maximum value given by C ∞ → √ γ − . Therefore, when Θ ∞ (cid:29) , ˙ M M becomes independent of Θ ∞ . In the previous subsection we revisited spherical accretion of a fluidthat follows an ideal gas EoS and that is restricted to obey a poly-tropic relation. As mentioned before, assuming a monoatomic gas,this restriction is only valid in the non-relativistic limit ( Θ ∞ (cid:28) )with γ = 5 / or in the ultra-relativistic one ( Θ ∞ (cid:29) ) with γ = 4 / . Nevertheless, as it was shown by Taub (1948), in therelativistic case ( Θ ∞ ∼ ) the polytropic restriction is not physicaland has to be dropped.In this subsection we extend the Michel solution to the caseof a gas obeying an appropriate EoS for the relativistic regime. In the comparison presented in Malec (1999) it is stated that, due to re-lativistic effects, the Michel mass accretion is enhanced by, at most, a factorof 10 as compared to the Bondi value, whereas in our case this factor isunbounded. Note, however, that the adopted EoS in that work is P = K(cid:15) γ ,with (cid:15) the energy density. As derived from relativistic kinetic theory, the EoS of an ideal,monoatomic gas can be written as (J¨uttner 1911; Synge 1957; Falle& Komissarov 1996) h = K (1 / Θ) K (1 / Θ) , (2.28)where, as before Θ =
P/ρ , and K n is the n th-order modifiedBessel function of the second kind. By additionally imposing the adiabatic condition (i.e. isen-tropic flow), one obtains the following relation between ρ and Θ (see e.g., Appendix B of Chavez Nambo & Sarbach 2020) ρρ ∞ = f (Θ) f (Θ ∞ ) , (2.29a) f (Θ) = Θ K (1 / Θ) exp (cid:20) K (1 / Θ) K (1 / Θ) (cid:21) . (2.29b)Meanwhile, the speed of sound in this case is given by C = ¯ γ Θ h , (2.30)where ¯ γ is the effective adiabatic index, defined as ¯ γ := ρP ∂P∂ρ = h Θ C . (2.31)In contrast to the polytropic gas treatment discussed before, ¯ γ is not a constant but rather a function of the temperature that can becalculated explicitly as ¯ γ = h (cid:48) h (cid:48) + Θ , (2.32)where the prime refers to derivatives with respect to the argumentof the modified Bessel functions, i.e. h (cid:48) = d[ K ( x ) /K ( x )] / d x .With this definition of ¯ γ it follows that, as expected, for non-relativistic temperatures, ¯ γ → / while, in the ultra-relativisticlimit, ¯ γ → / .In order to derive the appropriate governing equations in thiscase, we first notice that equation (2.16) should be replaced with h s = h ∞ (1 + 3 C s ) = h ∞ (cid:20) s h s (cid:18) h (cid:48) s h (cid:48) s + Θ s (cid:19)(cid:21) , (2.33)which, in contrast to equation (2.16), does not allow for an analyticsolution. Nevertheless, it can be easily solved numerically usingany standard root finding algorithm.The corresponding mass accretion rate is obtained by evaluat-ing equation (2.13a) at the sonic point, i.e. ˙ M = 4 π r s ρ s u s , (2.34)and, by applying the conditions given by equations (2.15a) and(2.15b) that, together with equation (2.29a), result in ˙ M = πM ρ ∞ (1 + 3 C s ) C s / f (Θ s ) f (Θ ∞ ) . (2.35)In practice, to calculate the resulting mass accretion rate for a givenasymptotic state ( ρ ∞ , Θ ∞ ) , we numerically solve equation (2.33)to obtain Θ s , from which we can compute h s and C s via equa-tions (2.28) and (2.30), respectively, and then substitute these val-ues into equation (2.35).In Figure 5 we show the resulting mass accretion rate as afunction of Θ ∞ for the relativistic EoS and compare it with the We adopt the definition of the modified Bessel function as presented in https://dlmf.nist.gov/10.25 .© 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes − − − . Θ ∞ ˙ M M / π M ρ ∞ γ = 5 / γ = 4 / . (2006) Figure 5.
Mass accretion rate in Michel’s model for a gas described by arelativistic EoS (Synge 1957). The resulting rate converges to a γ = 5 / polytrope when Θ ∞ (cid:28) while it behaves as a γ = 4 / polytrope for Θ ∞ (cid:29) . Also shown is the result of using the approximation to the re-lativistic EoS by Ryu et al. (2006). corresponding values for γ = 5 / , / polytropes. From this fig-ure we can see that the result obtained with the relativistic EoSprovides a smooth transition between the polytropic approxima-tions as the temperature transitions from non-relativistic values tothe ultra-relativistic regime. We also show the approximation to therelativistic EoS proposed by Ryu et al. (2006), where, h = 2 6Θ + 4Θ + 13Θ + 2 , (2.36)and which provides an accurate estimate to the mass accretion rateto within 2 % . This comparison is relevant for this work, given that,for some of the numerical simulations presented in Section 3, wehave adopted this proxy for the implementation of the relativisticEoS. The analytic solutions revisited so far consider a non-rotating blackhole as the central accretor. The inclusion of the black hole’s spinbreaks the spherical symmetry of the problem, resulting in a newscenario for which it is not clear whether it admits a closed, analyticsolution in general. As mentioned in the Introduction, a notableexception is the solution derived by Petrich, Shapiro & Teukolsky(1988) (PST henceforth), that we shall now briefly review. In thatwork, the authors found a full analytic solution for accretion onto aKerr black hole which is, however, restricted to the special case ofan ultra-relativistic stiff fluid. Within this approximation, the fluidrest-mass energy is neglected as compared to its internal energy,while the stiff condition means that a γ = 2 polytrope is beingconsidered. Under these conditions, the thermodynamic variablesof the fluid are simply related as P = Kρ , h = 2 K ρ. (2.37) Both Shapiro (1974) and Zanotti et al. (2005) have proposed a Michel-like solution for spherical accretion onto a rotating Kerr black hole that isbuilt on the assumption that the polar angular velocity vanishes everywhere.However, as we show in Section 3, this condition is not satisfied for a gen-eral perfect fluid.
Moreover, the spacetime metric is considered as fixed and corres-ponding to a Kerr black hole of mass M and spin parameter a , inother words, the accreting gas is assumed to be a test fluid with anegligible self-gravity contribution. With the further assumptionsof steady-state and irrotational flow, the fluid is described as thegradient of a scalar potential Φ such that h U µ = Φ ,µ , (2.38)and, by imposing the normalization condition of the four-velocity, h = (cid:112) − Φ ,µ Φ ,µ . (2.39)By substituting equation (2.38) into equation (2.9a), it followsthat Φ satisfies the linear wave equation Φ ; µ,µ = 1 √− g (cid:0) √− g g µν Φ ,µ (cid:1) ,ν = 0 , (2.40)where g µν and √− g are, respectively, the inverse and the determ-inant of the Kerr metric. In what follows we shall adopt Kerr-typecoordinates ( t, r, θ, φ ) in which the line element assumes the form d s = − (cid:18) − Mr(cid:37) (cid:19) d t + (cid:18) Mr(cid:37) (cid:19) d r + 4 Mr(cid:37) d t d r − aMr(cid:37) sin θ d t d φ − a (cid:18) Mr(cid:37) (cid:19) sin θ d r d φ + (cid:37) d θ + Σ sin θ(cid:37) d φ , (2.41)with the functions (cid:37) = r + a cos θ, (2.42a) Σ = (cid:0) r + a (cid:1) − a ∆ sin θ, (2.42b) ∆ = r − Mr + a . (2.42c)By requiring that the fluid is uniform and at rest asymptot-ically far away from the central object, the solution is given byAguayo-Ortiz et al. (2021) Φ = h ∞ (cid:20) − t + 2 M ln (cid:18) r − r − r + − r − (cid:19)(cid:21) , (2.43)where r ± = M ± √ M − a are the roots of the equation ∆ = 0 ,with r + corresponding to the event horizon and r − to the Cauchyhorizon of the Kerr black hole. It is clear that Φ is regular every-where outside the Cauchy horizon r > r − .Substituting the velocity potential in equation (2.43) intoequation (2.38), leads to hh ∞ U t = 1 + 2 Mr(cid:37) (cid:18) r + r + r − r − (cid:19) , (2.44a) hh ∞ U r = − Mr + (cid:37) , (2.44b) hh ∞ U θ = 0 , (2.44c) hh ∞ U ϕ = 2 aMr(cid:37) ( r − r − ) , (2.44d) We use the same notation as Aguayo-Ortiz et al. (2021) and warn thereader that the symbol (cid:37) refers to the metric coefficient defined in equa-tion (2.42a) which should be distinguished from the similar-looking symbol ρ which denotes the rest-mass density.© 0000 RAS, MNRAS , 1–15 A. Aguayo-Ortiz et al. while, by combining equations (2.37) and (2.39), one obtains ρρ ∞ = hh ∞ = (cid:115) M(cid:37) r ( r + r + ) + 2 Mr + r − r − . (2.45)Note that, although the fluid’s four-velocity has a non-vanishing azimuthal component when a (cid:54) = 0 , its angular mo-mentum is zero since U µ ξ µ ( φ ) = U φ = 0 , where ξ µ ( φ ) = δ µ isthe Killing vector field associated with the axisymmetry of Kerrspacetime. Also note that, both the four-velocity and the fluid dens-ity, are well-defined for all r > r − (including at the event horizon)but diverge as one approaches the Cauchy (inner) horizon r → r − .In the non-rotating case equation (2.45) reduces to ρ = ρ ∞ (cid:115) Mr + (cid:18) Mr (cid:19) + (cid:18) Mr (cid:19) , (2.46)which agrees with the findings in Section 4.2 of Chaverra & Sar-bach (2015), with a compression rate of ρ ( r + ) /ρ ∞ = 2 at the ho-rizon. In the rotating case, this compression rate can be consider-ably higher, with ρ ( r + ) /ρ ∞ → ∞ in the maximally rotating limit | a | → M .The resulting mass accretion rate for the potential flow de-scribed by equation (2.43) is given by ˙ M PST = 8 πMr + ρ ∞ = 4 π ( r + a ) ρ ∞ . (2.47)Interestingly, from equation (2.47) we see that, in this special caseof an ultra-relativistic stiff fluid, the resulting mass accretion rate isproportional to the event horizon area A = 4 π ( r + a ) (Car-roll 2003), and, consequently, for fixed M and ρ ∞ , ˙ M PST de-creases as | a | increases, having the finite limit ˙ M PST = 8 πM ρ ∞ when | a | → M . We also note that, for a non-rotating black hole, ˙ M PST = 16 πM ρ ∞ , which coincides exactly with the result givenin equation (2.27) when γ = 2 .One inconvenience of assuming an ultra-relativistic stiff EoS,is that the speed of sound equals the speed of light, leading to amodel with a limited applicability in astrophysics. Nevertheless, itrepresents a fully hydrodynamic exact solution that is very usefulas a benchmark test for the validation of general relativistic hy-drodynamic numerical codes in a fixed Kerr spacetime. In the nextsection we relax this restriction on the EoS. In the previous sections we reviewed, along with the Bondi andMichel models, the analytic PST solution. This is the only exactsolution that considers a rotating black hole as central accretor. Thissolution corresponds to an upper limit in both the temperature of thegas (Θ ∞ (cid:29) and in the adiabatic index ( γ = 2 ). Unfortunately,for a more general EoS, or even just a different value of γ , it isapparently not possible to find a closed analytic solution. Therefore,we explore the spherical accretion of a perfect fluid with a moregeneral EoS onto a rotating Kerr black hole by means of generalrelativistic hydrodynamic numerical simulations. Specifically, weshall focus on the dependence of the resulting accretion flow onthe spin parameter a , the asymptotic gas temperature Θ ∞ , and thefluid EoS. We also compare the results with the analytic solutionspresented in Section 2. We perform a total of 311 numerical simulations using the opensource code
AZTEKAS . This code solves the general relativistichydrodynamic equations, written in a conservative form using avariation of the “3+1 Valencia formulation” (Banyuls et al. 1997)for time independent, fixed metrics (Del Zanna et al. 2007). Thespatial integration is carried out using a grid-based, finite volumescheme coupled with a high resolution shock capturing methodfor the flux calculation, and a monotonically centred second or-der spatial reconstructor. The time integration is performed using asecond order total variation diminishing Runge-Kutta method (Shu& Osher 1988). The evolution of the equations is performed on aKerr background metric, using the same horizon penetrating Kerr-type coordinates as in Section 2.4.The set of primitive variables used in the code consists of therest-mass density ρ , pressure P , and the three-velocity vector v i asmeasured by Local Eulerian Observers associated with the chosencoordinate system. Both ρ and P are thermodynamic quantitiesmeasured at the co-moving reference frame, and the vector v i iscomputed as v i = γ ij v j where v i = U i αU t + β i α , i = r, θ (3.1)with α , β i and γ ij the lapse, shift vector and three-metric of the3+1 formalism (Alcubierre 2008), respectively. For all the simulations we adopt a two-dimensional (2D) axisym-metric numerical domain with spherical coordinates ( r, θ ) ∈ [ R in , R out ] × [0 , π/ , where R in and R out are the inner and outerradial boundaries, respectively. We use a uniform polar grid and anexponential radial grid (see Aguayo-Ortiz et al. 2019, for details)and fix the numerical resolution to 128 ×
64 grid cells, unless other-wise stated. Reflective boundaries are set at θ = 0 and θ = π/ .The inner radial boundary, at which we impose a free-outflow con-dition, is placed within the event horizon ( R in < r + ). On theother hand, the outer radial boundary is set with the correspond-ing Michel solution. With this external boundary condition, the do-main size must be sufficiently large as to avoid introducing numer-ical artefacts in the resulting steady-state solution. By performing aquantitative study varying R out , we find that we can be confidentof the independence on the domain size by taking R out = 10 r B in the non-relativistic regime (cid:0) Θ ∞ (cid:46) − (cid:1) , and R out = 40 r s inthe relativistic one (cid:0) Θ ∞ (cid:38) − (cid:1) , where r B and r s are the Bondiand sonic radii, respectively. In other words, for a given Θ ∞ , weset R out = 10 max( r B , r s ) . In what regards the initial condi-tions, we start our simulations with a static ( v i = 0) and uniform ( ρ = ρ ( R out ) , P = P ( R out )) gas distribution.The mass accretion rate evolves as a function of time withperiodic and exponentially damped oscillations (in agreement withthe results from Aguayo-Ortiz et al., 2019). The numerical simu-lations are left to run until the time variation of the resulting massaccretion rate drops below 1 part in , a criterion that we take as The code can be downloaded from https://github.com/aztekas-code/aztekas-main . See Aguayo-Ortiz et al. (2018); Te-jeda & Aguayo-Ortiz (2019); Aguayo-Ortiz et al. (2019); Tejeda et al.(2020), for further details regarding the characteristics, test suite and dis-cretization method of
AZTEKAS . © 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes l og (cid:12)(cid:12)(cid:12) − ˙ M / ˙ M M (cid:12)(cid:12)(cid:12) Θ ∞ Ideal γ = 2Ideal γ = 5 / γ = 4 / − − − − − − Figure 6.
Relative error in the mass accretion rate between the numericalresults ( ˙ M ) and the Michel analytic solution ( ˙ M M ) , as a function of theasymptotic temperature Θ ∞ . We plot the results for γ = 4 / , / , andthe relativistic EoS. signalling the onset of the steady-state condition. We compute themass accretion rate according to ˙ M = 4 π (cid:90) π/ ρ Γ (cid:18) v r − β r α (cid:19) √− g d θ, (3.2)where Γ = 1 / (cid:112) − γ ij v i v j is the Lorentz factor. In order to validate our numerical results, we exploit the knownanalytic solutions discussed in Section 2 and use them as bench-mark in our test runs.Considering first a non-rotating black hole, in Figure 6 weshow the relative error in the steady-state mass accretion ratebetween the Michel analytical solution ( ˙ M M ) and the numericalresults as a function of the asymptotic temperature Θ ∞ . We showthe results for γ = 4 / , / , as well as the fit to the relativisticEoS given by Ryu et al. (2006). For simplicity, in what follows weshall refer to this fit as the “relativistic EoS”. In all cases the numer-ical error is less than 5% in the non-relativistic regime ( Θ ∞ (cid:28) )and less than 1% in the relativistic regime ( Θ ∞ (cid:29) ), which isconsistent with the numerical resolution being used.We also perform additional numerical tests to validate the im-plementation of a non-zero spin parameter in our setup. In order toapproximate the ultra-relativistic stiff EoS and to compare with thePST analytic solution, we perform simulations using an adiabaticindex γ = 2 and an asymptotic temperature Θ ∞ = 10 , for differ-ent values of a . The result of this comparison is shown in Figures 7and 8, from where we find an excellent agreement between bothsolutions, with a relative error of less than 1%. We explain thesetwo figures in further detail in the next subsection. In order to quantify the spherical accretion flow onto a rotating Kerrblack hole and analyse its dependence on the black hole’s spin para-meter a , we perform a series of simulations varying both a and Θ ∞ ,both for a polytrope with γ = 4 / , / , as well as for the relativ-istic EoS. ˙ M / ˙ M M Θ ∞ Ideal γ = 2Ideal γ = 5 / γ = 4 / . . . .
91 0.1 1 1010 − − Ultra-relativistic stiff EoS (PST)
Figure 7.
Mass accretion rate as a function of Θ ∞ , for a rotating black holewith a/M = 0 . . The first three lines show the results for an ideal gasEoS with different values of γ , and the last line represents the relativisticEoS. The black dashed line represents the ultra-relativistic stiff EoS lowerlimit. The mass accretion rate is normalised using the corresponding valuein the non-rotating case ( ˙ M M ). For the spin parameter, we take a uniformly distributed setof values between a = 0 (non-rotating black hole) and a/M =0 . . On the other hand, for the temperatures we choose a list ofrepresentative values between Θ ∞ = 10 − and , in order tostudy the behaviour of the solution in the transition from the non-relativistic regime to the ultra-relativistic one. We explore the variation in the mass accretion rate for the rotatingblack hole case a > , as compared with the non-rotating case.We find that the larger difference is obtained for a maximally ro-tating black hole, as is to be expected considering the analytic PSTsolution (see equation 2.47).In Figure 7 we show the steady-state mass accretion rate as afunction of the asymptotic temperature (for γ = 4 / , / , andthe relativistic EoS) for the case of a rotating black hole with a spinparameter a/M = 0 . . The mass accretion rate is normalised bythe corresponding Michel value ( a = 0 ). As can be seen from thisfigure, all simulations are bounded between the non-rotating blackhole value ( ˙ M/ ˙ M M = 1) and the ultra-relativistic stiff EoS case ( ˙ M/ ˙ M M (cid:39) . . In the non-relativistic regime (Θ ∞ (cid:28) , themass accretion rate for all γ values converges to the correspond-ing Michel solution, although this convergence appears to be muchslower in the case γ = 2 . Thus, we conclude that in this regimethe effects of the spin on ˙ M are negligible for γ (cid:54) / . In theultra-relativistic regime (Θ ∞ (cid:29) , ˙ M decreases by a factor of ∼ , , and for the solutions with γ = 4 / , / , and ,respectively. Note how the solution for γ = 2 in the Θ ∞ (cid:29) limitmatches the ultra-relativistic stiff analytical value. In order to study the dependence of the spherical accretion solutionon the spin parameter, we perform a series of simulations varyingthe value of a . For these runs, we also consider three values ofthe asymptotic temperature corresponding to the non-relativistic, © 0000 RAS, MNRAS000
Mass accretion rate as a function of Θ ∞ , for a rotating black holewith a/M = 0 . . The first three lines show the results for an ideal gasEoS with different values of γ , and the last line represents the relativisticEoS. The black dashed line represents the ultra-relativistic stiff EoS lowerlimit. The mass accretion rate is normalised using the corresponding valuein the non-rotating case ( ˙ M M ). For the spin parameter, we take a uniformly distributed setof values between a = 0 (non-rotating black hole) and a/M =0 . . On the other hand, for the temperatures we choose a list ofrepresentative values between Θ ∞ = 10 − and , in order tostudy the behaviour of the solution in the transition from the non-relativistic regime to the ultra-relativistic one. We explore the variation in the mass accretion rate for the rotatingblack hole case a > , as compared with the non-rotating case.We find that the larger difference is obtained for a maximally ro-tating black hole, as is to be expected considering the analytic PSTsolution (see equation 2.47).In Figure 7 we show the steady-state mass accretion rate as afunction of the asymptotic temperature (for γ = 4 / , / , andthe relativistic EoS) for the case of a rotating black hole with a spinparameter a/M = 0 . . The mass accretion rate is normalised bythe corresponding Michel value ( a = 0 ). As can be seen from thisfigure, all simulations are bounded between the non-rotating blackhole value ( ˙ M/ ˙ M M = 1) and the ultra-relativistic stiff EoS case ( ˙ M/ ˙ M M (cid:39) . . In the non-relativistic regime (Θ ∞ (cid:28) , themass accretion rate for all γ values converges to the correspond-ing Michel solution, although this convergence appears to be muchslower in the case γ = 2 . Thus, we conclude that in this regimethe effects of the spin on ˙ M are negligible for γ (cid:54) / . In theultra-relativistic regime (Θ ∞ (cid:29) , ˙ M decreases by a factor of ∼ , , and for the solutions with γ = 4 / , / , and ,respectively. Note how the solution for γ = 2 in the Θ ∞ (cid:29) limitmatches the ultra-relativistic stiff analytical value. In order to study the dependence of the spherical accretion solutionon the spin parameter, we perform a series of simulations varyingthe value of a . For these runs, we also consider three values ofthe asymptotic temperature corresponding to the non-relativistic, © 0000 RAS, MNRAS000 , 1–15 A. Aguayo-Ortiz et al. γ = 2Relativistic EoS ˙ M / ˙ M M Θ ∞ = 10 − Θ ∞ = 10 − Θ ∞ = 10 Ultra-relativistic stiff EoS (PST)0 . . . . ˙ M / ˙ M M a/M .
91 0 0 . . . . Figure 8.
Mass accretion rate as a function of the spin parameter for the γ =2 (top panel) and the relativistic EoS (bottom panel), and different valuesof the asymptotic temperature Θ ∞ . The mass accretion rate is normalisedby its value in the non-rotating case ˙ M M . The black dashed line in the toppanel represents the solution obtained with the ultra-relativistic stiff EoS(PST) model. intermediate, and ultra-relativistic regimes. In Figure 8 we showour analysis of this dependence adopting two fluid models: the stifffluid ( γ = 2 , top panel) and the relativistic EoS (bottom panel).The former case allows us to study the behaviour of the simulationsfor an extreme adiabatic index (for which the spin effects are morenoticeable), while the latter constitutes a more realistic EoS. As inFigure 7, the γ = 2 and Θ ∞ = 10 case matches the analytic PSTsolution, providing yet another code validation, but now for a widerange of spin values.As can be seen in Figure 8, the mass accretion rate decreasesas the spin parameter a increases. Moreover, the dependence on a becomes more notorious as higher temperatures are considered. Inthe case of the stiff fluid (top panel), we find that the mass accretionrate is reduced by up to a factor of for a maximally rotatingblack hole as compared to a non-rotating one. On the other hand,this reduction is at most of ∼ in the case of the relativistic EoS(bottom panel). It is interesting to note that all the numerical res-ults follow a qualitatively similar dependence on a as the analyticPST solution: the accretion rate decreasing as the spin parameterincreases. In this regard, it is interesting to mention the recent work by Cie´slik& Mach (2020) who study the spherical accretion of a Vlasov gas onto a(charged) Reissner-Nordstr¨om black hole which is often considered as asimpler model for the Kerr spacetime since it shares many of its qualitativeproperties. In this model, the charge parameter plays the role of the spin
The dependence on the spin parameter has been studied so far byconsidering only its effect on the mass accretion rate. This is im-portant since one of the most relevant results of any accretion modelis the associated mass growth of the central object. Nevertheless, itis also of interest to study the overall morphology of the resultingaccretion flow in order to understand the global effect of the spin.To study the effect of the spin on the velocity field, as wellas on the rest-mass density profile, we perform a simulation of afluid obeying the relativistic EoS, with an asymptotic temperature Θ ∞ = 0 . , and a spin parameter a/M = 0 . .In Figure 9 we show the steady-state rest-mass density ρ/ρ ∞ at the equatorial plane and the spatial components of the velocityfield U ˆ r , U ˆ θ , U ˆ φ (measured in an orthonormal reference frame,see Appendix B). The solid black arrows show the fluid stream-lines and the solid white line represents the location of the sonicsurface (see Appendix C for its invariant determination). Note thatthe azimuthal flow shown in the rest-mass density and in the U ˆ φ field, is due exclusively to the frame dragging of the black hole.As can be seen from Figure 9, the polar component U ˆ θ exhib-its a quadrupolar-like morphology, which is in contrast to the non-rotating case where U ˆ θ = 0 . This is interesting since in the PSTsolution this component of the four-velocity is exactly zero, inde-pendently of the value of the spin parameter (see equation 2.44c).On the other hand, the isocontours for U ˆ r depart from sphericalsymmetry close to the event horizon, in particular inside the sonicsurface. However, apart from the inspiraling effect due to the framedragging, the fluid streamlines do not deviate significantly fromthose of the spherically symmetric inflow.In order to analyse the behaviour of the fluid velocity, we com-pute the latitudinal average at each radius, defined as, (cid:68) U ˆ i (cid:69) θ = (cid:90) π/ U ˆ i √− g d θ (cid:90) π/ √− g d θ . (3.3)In Figure 10 we show this average for U ˆ r (upper-panel) and U ˆ θ (lower-panel) as a function of r for a/M = 0 . and Θ ∞ = 0 . .We also use two different numerical resolutions in order to showthat our results are robust with respect to the grid size. The blackdotted line represents the non-rotating Michel solution in the radialvelocity case (top-panel), and the average numerical error that weobtain from our simulations in the polar velocity (bottom-panel).We find that the average of U ˆ r is larger for a rotating black holethan for a non-rotating one. Also, in the rotating case, the averageof U ˆ θ is comparable in size to U ˆ r at the horizon and decreasesapproximately as /r for r > r + . The fact that U ˆ θ is differentfrom zero is relevant in view of previous work (see Shapiro 1974;Zanotti et al. 2005), which discuss spherical accretion models inKerr spacetime based on the assumption U θ = 0 .In Figure 11 we show the shape of the sonic surface by plottingthe sonic radius as a function of the polar angle, for different valuesof the spin parameter. As shown in Appendix C, this surface con-sists of those points at which the magnitude of the three-velocity asmeasured by zero angular momentum observers (ZAMOs, Bardeen1970) transitions from sub- to supersonic values. As can be seen parameter, and similar to our findings, the authors of that study find that themass accretion rate decreases as the charge parameter increases.© 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes Figure 9.
Isocontour plots of the steady-state of a simulation of the spherical accretion problem onto a rotating black hole with a/M = 0 . , for a gas obeyingthe relativistic EoS and Θ ∞ = 0 . . The figures show the normalised rest-mass density ρ/ρ ∞ at the equatorial plane (top-left) and the spatial orthonormalcomponents of the four-velocity ( U ˆ r [top-right], U ˆ θ [bottom-left] and U ˆ φ [bottom right]) projected on the R − z plane, where R = √ r + a sin θ and z = r cos θ . The black solid arrows show the fluid streamlines, whereas the black dashed lines the isocontour levels. The white solid line shows the locationof the sonic surface, see Figure 11 for further details. The outer boundary in this simulation is R out ≈ M . from this figure, for the non-rotating case the sonic surface corres-ponds to the sphere r s = const . , as expected. As the spin para-meter increases, the sonic surface contracts unevenly giving rise toa slightly oblate shape in the R − z plane. This flattening at thepoles is more notorious as a/M → . For the maximum value ex-plored in this work ( a/M = 0 . ), the equator-to-poles differencein radii is of around . In this work we have studied the spherical accretion problem fromthe non-relativistic regime to the ultra-relativistic one, for both ro-tating and non-rotating black holes. We have focused on steady-state solutions for a perfect fluid obeying an ideal gas EoS andparametrised its thermodynamic state far away from the black holeusing the dimensionless temperature Θ ∞ = P ∞ /ρ ∞ . We have alsoassumed that the gravitational field is dominated by the black hole,such that the fluid’s self-gravity can be neglected. We first revis- ited the analytic solutions of Bondi (1952) and Michel (1972), andprovided a quantitative comparison between them. Next, we exten-ded Michel’s solution to the case of an ideal gas obeying a relativ-istic EoS (J¨uttner 1911; Synge 1957). Finally, we studied the spher-ical accretion problem in the case of a rotating black hole, first bywriting the exact ultra-relativistic, stiff solution (Petrich et al. 1988)in the spherically symmetric case and then by performing generalrelativistic hydrodynamic simulations of a general perfect fluid.Concerning the comparison between the Bondi and Michelsolutions, we have shown rigorously that Michel’s solution re-duces to the Bondi one when the non-relativistic limit is considered( Θ ∞ (cid:28) ) and when γ (cid:54) / , as expected. Importantly, when γ > / , the obtained global solution is intrinsically relativistic,even for non-relativistic asymptotic temperatures, in accordancewith Richards et al. (2021a). Additionally, we derived appropriateanalytic expressions for the mass accretion rate for the Michel solu-tion in the ultra-relativistic limit ( Θ ∞ (cid:29) . Moreover, within thislimit and for a stiff EoS ( γ = 2 ), we have shown that the result- © 0000 RAS, MNRAS , 1–15 A. Aguayo-Ortiz et al. /r l og (cid:10) U ˆ r (cid:11) θ × × a = 0 − − l og (cid:10) U ˆ θ (cid:11) θ r/M − − − − − − Figure 10.
Solutions of the angular averaged values of U ˆ r (top panel) and U ˆ θ (bottom panel) as a function of r , for a/M = 0 . and for two differ-ent resolutions. The parameters used in this plots are Θ ∞ = 0 . and therelativistic EoS. The grey dotted lines show the Michel’s a = 0 solution inthe top panel, and the average numerical error in the polar velocity in thebottom panel. Note that the difference between the two resolutions in thebottom panel for large radii is of the same order as the average numericalerror. The black dashed represents the approximate behaviour of (cid:104) U ˆ θ (cid:105) θ . r / M θ . . . . . π/ π/
20 0 .
25 0 . .
75 0 . Figure 11.
Resulting sonic surface as a function of the polar angle, for dif-ferent values of the spin parameter a/M (as indicated by the label on top ofeach curve). This figure shows that, for a rotating black hole, the sonic sur-face contracts to smaller radii and ceases to be characterised by a constantradius. ing mass accretion rate coincides exactly with the result obtainedby Petrich et al. (1988). Furthermore, we have shown that in theisothermal limit, in which γ → , the entire accretion flow can bedescribed in a Newtonian way, i.e. the Michel solution reduces tothe Bondi one for all asymptotic temperatures Θ ∞ .Regarding the relativistic regime, we have found that the dif-ference between the mass accretion rates as obtained in the Bondiand Michel solutions grows arbitrarily as the asymptotic temper-ature increases. The reason behind this relies in the fact that, atultra-relativistic temperatures ( Θ ∞ (cid:29) , the Michel mass accre-tion rate reaches a minimum constant value, whereas the Bondi onedecreases without limit (Figure 4). The discrepancy between thesetwo values is already noticeable (of order one) for Θ ∞ ∼ . .Moreover, we have extended the Michel solution by considering therelativistic EoS of an ideal, monoatomic gas (J¨uttner 1911; Synge1957), which is a more accurate description for a perfect fluid inthis regime (Figure 5).We have also extended, by means of numerical simulations,the Michel solution to the case of a rotating Kerr black hole. Themain purpose of this numerical exploration was to analyse theeffect of the black hole spin on the mass accretion rate and theflow morphology. We ran a series of 2D general relativistic hydro-dynamic simulations covering a space parameter comprising theasymptotic temperature Θ ∞ , the gas EoS, and the spin parameterof the black hole. We have validated our results by comparing themwith the known analytic solutions, as well as by performing a seriesof careful resolution and domain-size convergence tests.The numerical results show that the influence of the blackhole’s rotation is only larger than a few percent in the relativistic re-gime (Θ ∞ (cid:38) . or for γ > / (Figure 7). As the spin parameterincreases, the mass accretion rate decreases as compared with thenon-rotating case. This effect is stronger for larger values of Θ ∞ and γ . Nevertheless, even in the most extreme case ( Θ ∞ (cid:29) and γ = 2 ), the reduction in the accretion rate is no larger than (Figure 8). The simulations in this work allowed us to study themorphology of the fluid’s density profile and velocity field near theevent horizon, showing in the latter a behaviour considerably differ-ent from the non-rotating black hole case, even in for mildly relativ-istic temperatures. We have shown that the black hole rotation in-duces an azimuthal velocity component (entirely due to relativisticframe-dragging), a non-zero polar angular velocity component, aswell as a non-spherically symmetric radial component (Figures 9and 10). Furthermore, the sonic surface ceases to be characterisedby a constant radial coordinate (Figure 11).Our results imply that the relativistic features of a black holecan be safely neglected when considering the spherical accretion ofa fluid with a non-relativistic asymptotic temperature ( Θ ∞ (cid:28) )and γ (cid:54) / . However, this is not true for relativistic and ultra-relativistic values of the asymptotic temperature ( Θ ∞ (cid:38) . ). Inthis regime, a proper relativistic description must be used in orderto compute the mass accretion rate, as the Bondi and Michel solu-tions conduce to completely different values. On the other hand,the black hole’s rotation, even in the ultra-relativistic case and for aclose-to-maximally rotating black hole, does not change the result-ing mass accretion rate by more than 50% (for γ (cid:54) ) with respectto the non-rotating case. For a more realistic EoS ( γ = 4 / ) thischange is even smaller and lies below 10%. Thus, it is safe to neg-lect the black hole spin when considering an order of magnitudeestimation, but it should be taken into account when performing amore accurate calculation.The results presented in this work could be useful for study-ing spherical accretion onto rotating and non-rotating black holes © 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes in extreme environments where the ambient gas approaches relativ-istic temperatures ( Θ ∞ ∼ ), or that are well approximated bya stiff EoS ( γ > / ). Examples of such scenarios might rangefrom primordial black holes accreting during the radiation era in theearly universe evolution (especially between the quark and leptonepochs when K < T < K ) (Jedamzik 1997; Lora-Clavijo et al. 2013), to mini black holes accreting from within aneutron star (whose core can be modelled, as a first approximation,with a γ = 2 polytrope) (Capela et al. 2013; G´enolini et al. 2020). DATA AVAILABILITY
All of the simulations presented in this work can be reproduced us-ing the “Spherical accretion” setup of the
AZTEKAS code that canbe found on the Github repository ( https://github.com/aztekas-code/aztekas-main ). Any further data underly-ing this paper will be shared upon request to the corresponding au-thor.
ACKNOWLEDGEMENTS
This work was partially supported by CONACyT Ciencia deFrontera Project No. 376127 “Sombras, lentes y ondas gravitat-orias generadas por objetos compactos astrof´ısicos” and by a CICgrant to Universidad Michoacana. The authors acknowledge thesupport from the Miztli-UNAM supercomputer (project LANCAD-UNAM-DGTIC-406). AAO acknowledge support from CONACyTscholarship (No. 788898).
APPENDIX A: LIMITS OF THE MICHEL SOLUTION
In this appendix we make a few remarks regarding the followingtwo limits of the Michel solution: the isothermal limit for whichthe adiabatic index γ → and the non-relativistic limit for whichthe asymptotic temperature is Θ ∞ (cid:28) . (i) Isothermal limit In the limit when γ → , we show that the Michel solutionapproaches the Newtonian (Bondi) flow solution with an EoS givenby equation (2.5). To this end, we first use the cubic equation (2.16)and find, for small values of δ := γ − > , h s h ∞ = 1 + 32 δ − (cid:18)
98 + 32Θ ∞ (cid:19) δ + O ( δ ) , (A1a) C ∞ = δ (cid:20) − δ Θ ∞ + O ( δ ) (cid:21) , (A1b) C s C ∞ = 1 + O ( δ ) , (A1c)from which ˙ M M πM ρ ∞ C − ∞ → e / , (A2)which coincides with the Bondi solution in equation (2.7).Next, we introduce the dimensionless quantities x := rM C ∞ , z := ρρ ∞ , ν := uc , λ := ˙ M M πM C ∞ ρ ∞ . (A3) in terms of which eqs. (2.13a,2.13b) can be rewritten as x νz γ +12 = λ (cid:18) hh ∞ (cid:19) / , (A4a) − x + h ∞ h z γ − ν = 1 C ∞ (cid:34)(cid:18) h ∞ h (cid:19) − (cid:35) . (A4b)For small values of δ , one finds, using h = 1 + γρ δ /δ , hh ∞ = 1 + δ log( z ) + O ( δ ) . (A5)Introduced into equations (A4a),(A4b), using equation (A1b) andtaking the limit δ → yields (assuming that x , z and ν have finitevalues in this limit) x νz = λ, − x + 12 ν = − log z, (A6)which agrees precisely with the Newtonian equations (2.3a,2.3b)with the EoS (2.5), for x , z and λ defined as in Eq. (A3) and ν = v/ C ∞ (note that c/ C ∞ → in the limit δ → ). Taking into accountthe limit (A2) this yields the transonic flow solution discussed insubsection 2.1 which has been shown in Ref. Chaverra & Sarbach(2016) to be the correct γ → limit of the Bondi flow. (ii) Non-relativistic limit In the low-temperature limit Θ ∞ → one has h ∞ → , andin this limit equation (2.16) has two positive roots h s = 1 , h s = 12 (cid:16)(cid:112) γ − − (cid:17) , (A7)the third one being negative and hence unphysical. For γ < / the second positive root is smaller than one, and hence unphysicalas well and the correct limit is h s = 1 . Computing the first-ordercorrection in Θ ∞ one finds h s h ∞ = 1 + 3 γ − γ Θ ∞ + O (Θ ∞ ) , (A8)from which C s C ∞ = 25 − γ + O (Θ ∞ ) , (A9)and substituting into equation (2.20) it follows that ˙ M M → ˙ M B when Θ ∞ → and γ < / . When γ > / it turns out thecorrect root is the second one in equation (A7), see Chaverra et al.(2016), and the corresponding squared sound speed at the sonicpoint is C s = 13 ( h s − > . (A10)It follows from equation (2.20) that ˙ M M → πh γ − γ − s C − γγ − s M ρ ∞ C − γ − ∞ , (A11)and the mass accretion rate decays slower than C − ∞ . Note that h s > and C s > imply that the flow does not lie in the New-tonian regime close to the sonic point when γ > / . © 0000 RAS, MNRAS , 1–15 A. Aguayo-Ortiz et al.
APPENDIX B: ORTHONORMAL FRAME ADAPTED TOTHE KERR-TYPE COORDINATES
The orthonormal frame adapted to the constant time slices in theKerr-type coordinates ( t, φ, r, θ ) used in this article is given by e ˆ t = (cid:114) Mr(cid:37) (cid:18) ∂∂t − Mr(cid:37) + 2 Mr ∂∂r (cid:19) , (B1a) e ˆ r = 1 (cid:113) Mr(cid:37) ∂∂r , (B1b) e ˆ θ = 1 (cid:37) ∂∂θ , (B1c) e ˆ φ = 1 (cid:37) sin θ (cid:18) ∂∂φ + a sin θ ∂∂r (cid:19) , (B1d)(B1e)and it is well-defined for all r > . The corresponding componentsof the four-velocity vector field, such that U µ ∂∂x µ = U ˆ t e ˆ t + U ˆ r e ˆ r + U ˆ θ e ˆ θ + U ˆ φ e ˆ φ (B2)are given by U ˆ t = 1 (cid:113) Mr(cid:37) U t , (B3a) U ˆ r = (cid:114) Mr(cid:37) (cid:18) U r + 2 Mr(cid:37) + 2 Mr U t − a sin θU φ (cid:19) , (B3b) U ˆ θ = (cid:37)U θ , (B3c) U ˆ φ = (cid:37) sin θU φ . (B3d) APPENDIX C: INVARIANT DETERMINATION OF THESONIC SURFACE
In relativistic fluids, it is not immediately obvious how to determinethe sonic surfaces, that is, the boundary separating the events atwhich the flow is subsonic from those at which it is supersonic.Indeed, the fluid’s sound speed C is a scalar, while the velocity U µ of the fluid is a four-vector. One could consider instead of U µ themagnitude of the three-velocity V with respect to a specific familyof observers and define the sonic surface by those events for which V = C , but this definition would clearly be observer-dependent.A definition which does provide an invariant characterizationof the sonic surface is based on the sonic metric, G µν := ρh C (cid:2) g µν + (cid:0) − C (cid:1) U µ U ν (cid:3) , (C1)first introduced by Moncrief (1980), for the purpose of analysingthe propagation linearised, acoustic perturbations of an isentropic,vorticity-free flow on a background spacetime with metric g µν . Thesonic metric (C1) is a Lorentzian metric whose set of null vectorsat a given spacetime event e form a cone (the sound cone) thatcan be shown to lie inside the light cone at e provided C < .Another useful property of the sonic metric is that it inherits thesymmetries of the spacetime and the flow configuration: if X is aKilling vector field, such that the Lie derivative £ X of g µν , U µ , ρ and h vanish, then it follows that £ X G µν = 0 , that is, the sonicmetric is invariant with respect to X .For the solutions described in this article, where both thespacetime metric and the flow are steady-state and axisymmetric, it follows that Eq. (C1) describes a steady-state and axisymmetricgeometry which is asymptotically flat since the flow’s four-velocityis constant at infinity. A sonic surface corresponds to the “event ho-rizon” of this geometry, that is, the surface which separates thoseevents that can send an acoustic signal to infinity from those thatcannot. Due to the aforementioned symmetries of the sonic geo-metry, this surface must be a Killing horizon, i.e. a null surface ofthe form (see, e.g. Heusler 1996) H := { x : G µν ( x ) X µ X ν = 0 } , (C2)whose normal vector, X µ = δ µt + Ω H δ µφ , (C3)is a superposition of the Killing vector fields of the Kerr metric,where here the constant Ω H describes the angular velocity of thehorizon. The requirement of H being a null surface (with respect tothe sonic metric) with normal X µ implies the condition ∇ α [ G µν ( x ) X µ X ν ] = − κX α , (C4)the proportionality factor κ describing the “surface gravity” associ-ated with the horizon. Assuming a regular horizon, such that κ (cid:54) = 0 ,the four equations (C4) imply G tt + Ω H G tφ = 0 , (C5a) G tφ + Ω H G φφ = 0 , (C5b) G tr + Ω H G φr = − κ ∂N∂r , (C5c) G tθ + Ω H G φθ = − κ ∂N∂θ (C5d)with N := G µν X µ X ν = G tt + 2Ω H G tφ + Ω H G φφ . Note thatthe first two conditions (C5a,C5b) imply that X µ is null on H ,i.e. N = 0 , as required. They determine the location of the sonicsurface H through the requirement that the determinant of the × matrix ( G ab ) a,b = t,φ vanishes. In view of definition (C1) this yields det (cid:2) g ab + (1 − C ) U a U b (cid:3) = 0 . (C6)In turn, either Eq. (C5c) or Eq. (C5d) can be used to determine thesurface gravity κ , but this will not be needed here. In terms of the Kerr-type coordinates ( t, φ, r, θ ) used in thisarticle, the determinant condition (C6), together with the property U φ = 0 satisfied by the flow, leads to the condition g tt − g tφ g φφ + (1 − C ) U t = 0 , (C7)which yields an implicit relation between r and θ . This conditionacquires a much clearer interpretation when rewriting it in terms ofthe flow’s Lorentz factor Γ measured by a ZAMO, which gives (1 − C )Γ = 1 , (C8)i.e. the sonic surface is determined by those events for whichthe flow, as measured by ZAMOs, changes from sub- to super-sonic. From Eq. (C5b) and U φ = 0 it also follows that Ω H = − g tφ /g φφ = Ω ZAMO , i.e. the angular velocity of the sonic hori-zon is equal to the angular velocity of the ZAMO at H . Note that the condition N = 0 on H implies that Eq. (C4), when con-tracted with a tangent vector to H is automatically satisfied, such that onlyone of the two equations (C5c,C5d) needs to be considered.© 0000 RAS, MNRAS , 1–15 pherical accretion onto rotating black holes References
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