Jet formation in GRBs: A semi-analytic model of MHD flow in Kerr geometry with realistic plasma injection
aa r X i v : . [ a s t r o - ph . H E ] A ug Jet formation in GRBs: A semi-analytic model of MHD flow in Kerr geometrywith realistic plasma injection
Noemie Globus and Amir Levinson ABSTRACT
We construct a semi-analytic model for MHD flows in Kerr geometry, that incorporates energyloading via neutrino annihilation on magnetic field lines threading the horizon. We computethe double-flow structure for a wide range of energy injection rates, and identify the differentoperation regimes. At low injection rates the outflow is powered by the spinning black hole via theBlandford-Znajek mechanism, whereas at high injection rates it is driven by the pressure of theplasma deposited on magnetic field lines. In the intermediate regime both processes contributeto the outflow formation. The parameter that quantifies the load is the ratio of the net powerinjected below the stagnation radius and the maximum power that can be extracted magneticallyfrom the black hole.
1. Introduction
An issue of considerable interest in the theory of gamma-ray bursts (GRBs) is the nature of the gamma-ray emitting jet. The conventional wisdom has been that the jet is produced by a hyper-accreting black holethat results from a neutron star merger in case of short GRBs (Eichler et al. 1989), or the core-collapse of amassive star in case of long GRBs (MacFadyen, Woosley 1999). The black hole is likely to be immersed ina strong magnetic field ( B H ≃ G) seeded by the progenitor and advected inwards during the formationof the central engine. Feedback from a rapidly rotating black hole may dominate the torque experiencedby the surrounding torus, leading to a state of suspended accretion in long GRBs (van Putten & Ostriker,2001; van Putten & Levinson 2003), provided that mass loading of magnetic field lines anchored to the diskis, somehow, strongly suppressed (Komissarov & Barkov 2009; Globus & Levinson 2013; hereafter GL13).Rapid heating of the inner regions of the hyper-accretion disk, or the torus in the suspended accretion stateif established, leads to prodigious emission of MeV neutrinos (and anti-neutrinos), with luminosities in therange L ν = 10 − erg s − , depending on accretion rate and specific angular momentum a of the blackhole (e.g., Popham et al. 1999; Chen & Beloborodov 2007).In the context of the picture outlined above, two competing jet formation mechanisms have been widelydiscussed in the literature; magnetic extraction of the spin down power of a Kerr black hole, and outflowformation via neutrino annihilation in the polar region, above the horizon (Paczy´nski 1990, Levinson &Eichler 1993, Levinson 2006). These two processes are commonly treated under idealized conditions: modelsof Blandford-Znajek jets usually invoke the force-free limit and ignore loading of magnetic field lines (but c.f.,Komissarov & Barkov 2009), whereas models of jets driven by ν ¯ ν annihilation (MacFadyen & Woosley 1999,Fryer & M´esz´aros 2003) are usually constructed within the pure hydrodynamic limit. In general, however,both processes might be at work, and it is desirable to characterize the interplay between them. The approachundertaken in this paper is to treat ν ¯ ν annihilation in the magnetosphere as external plasma load. It hasbeen shown elsewhere (GL13) that injection of relativistically hot plasma on horizon threading field lines School of Physics & Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
2. Model
The double-flow structure established in the magnetosphere is illustrated in Figure 1: plasma inflow intothe black hole and outflow to infinity are ejected from a stagnation radius, r st , located between the innerand outer light surfaces. The plasma consists of relativistically hot e ± pairs created via annihilation of MeVneutrinos emitted from the surrounding accretion disk. The exact location of the stagnation surface depends,quite generally, on the energy injection profile, and is treated as an eigenvalue of the MHD equations. Thedouble flow possesses six critical surfaces, corresponding to the characteristic phase speeds of the three MHDwaves propagating in the medium: two slow magnetosonic, two Alfv´enic and two fast magnetosonic. Weconsider an infinitely conducting, stationary and axisymmetric flow. In general, the flow is characterized by astream function Ψ( r, θ ) that defines the geometry of magnetic flux surfaces, and by the following functionalsof Ψ: the angular velocity of magnetic field lines Ω F (Ψ), the ratio of mass and magnetic fluxes η (Ψ), and theenergy, angular momentum and entropy per baryon, denoted by E (Ψ), L (Ψ) and s (Ψ), respectively. Thesequantities are given explicitly in Equations (A7)-(A10).The ideal MHD condition implies that Ω F (Ψ) is conserved along magnetic flux tubes, as usual. Allother quantities change along streamlines, owing to plasma injection by the external source, according toEquations (A11)-(A14). We assume that in the acceleration zone the plasma is relativistically hot with anegligible baryonic content. We can therefore adopt the equation of state w = ρc ¯ h = 4 p . To simplify theanalysis we invoke a split monopole configuration for the magnetic field lines, described by a stream functionof the form Ψ( r, θ ) = Ψ (1 − cos θ ). The energy, angular momentum and entropy fluxes, Equation (A6),then have only a radial component: ǫ r = ρ E u r , l r = ρ L u r , s r = ρsu r . Note that E , L and s diverge in thelimit ρ → ǫ r , l r and s r , remain finite and are welldefined also in the baryon-free case. For the relativistic equation of state adopted above the entropy per unitvolume is given by S = ρs = 4 p/kT ∝ p / , whereby s r ∝ u r p / , hence the pressure p can be used as a freevariable instead of S . Since our analysis encompases the force-free limit, we find it convenient to use ǫ r , l r and p as our free variables. With the above simplifications, Equations (A11)-(A13) reduce to:1 √− g ∂ r ( √− gǫ r ) = − q t , (1) 3 –1 √− g ∂ r ( √− gl r ) = q ϕ , (2)34 ∂ r ln p = − ∂ r ln(Σ u r ) − u α q α pu r , (3)here √− g = Σ sin θ , and Σ = r + a cos θ . This set needs to be augmented by an equation of motion forthe velocity u r . Instead of u r we use the poloidal velocity u p = √ u r u r = √ g rr u r . Its rate of change alongstreamlines is derived in appendix A and can be written in the form, ∂ r ln u p = N ad + N q D , (4)where N ad , N q and D are functionals of ǫ r , l r , p, u p , Ω F , given explicitly by Equations (A23)-(A25). Equations(1)-(4) form a complete set that governs the structure of the double MHD flow. The solutions for the radialprofiles of the free variables ǫ r , l r , p , and u p , depend on the particular choice of the angle θ that characterizesmagnetic flux surfaces. The angular velocity Ω F ( θ ) is given as an input. The energy and angular momentumflow rates per solid angle (along a particular flux surface) are defined, respectively, as:˙ E ( r, θ ) ≡ Σ ǫ r , ˙ L ( r, θ ) ≡ Σ l r . (5) ν ¯ ν → e + e − The energy-momentum deposition rate by the reaction ν ¯ ν → e + e − was computed in a number of works,under different simplifying assumptions (Popham et al. 1999, Chen & Beloborodov 2007, ZB11). In whatfollows we use the recent analysis by ZB11 which includes general relativistic effects. Following ZB11 wedenote by Q αν ¯ ν the local energy-momentum deposition rate measured by a zero-angular-momentum observer(ZAMO). In general, those rates are functions of the Boyer-Lindquist coordinates r , θ and ϕ , as can beseen from Figures 2 and 3 in ZB11. In terms of the metric components defined in appendix A, g ϕϕ = ̟ , g tϕ = − ωg ϕϕ , g tt = − α + ω g ϕϕ and g rr , we have the following relations between the ZAMO rates Q αν ¯ ν andthe source terms q α measured by a distant observer: αq t = Q tν ¯ ν , ̟q ϕ = Q ϕν ¯ ν + ̟ωQ tν ¯ ν /α , √ g rr q r = Q rν ¯ ν .From that we obtain − q t = αQ tν ¯ ν + ̟ωQ ϕν ¯ ν , (6) q ϕ = ̟Q ϕν ¯ ν , (7) u α q α = − αu t Q tν ¯ ν + ̟Q ϕν ¯ ν ( u ϕ − ωu t ) + u p Q rν ¯ ν . (8)The total power deposited in the magnetosphere can be expressed as,˙ E totν ¯ ν = Z r ≥ r H (cid:0) αQ tν ¯ ν + ̟ωQ ϕν ¯ ν (cid:1) √− gdrdθdϕ . (9)A fit to the numerical results by ZB11 yields: ˙ E totν ¯ ν ≃ ˙ m / acc x − . mso erg s − for a black hole mass M BH =3 M ⊙ , and accretion rates (henceforth measured in units of M ⊙ s − ) in the range 0 . < ˙ m acc <
1, where x mso is the radius of the marginally stable orbit in units of m = GM BH /c .Unfortunately, ZB11 do not exhibit results for the azimuthal term Q ϕν ¯ ν . It is also difficult to fit theirresult for Q rν ¯ ν . We shall therefore set Q ϕν ¯ ν = Q rν ¯ ν = 0. This should not alter much q t and u α q α , as the firstterm on the right hand side of Equations (6) and (8) dominates anyhow. However, for this choice q ϕ = 0, 4 –implying that the angular momentum flow rate, ˙ L ( r, θ ) ≡ Σ l r , is conserved, as readily seen from Equation(2). For our radial flow model it is sufficient to use the angle-averaged energy deposition rate. We adopt theform Q tν ¯ ν ( r ) = ˙ Q f ( x ) , (10)where x = r/m is a fiducial radius, and f ( x ) is normalized such that f (1) = 1. From figures 2 and 3 in ZB11we obtain the approximate profile f ( x ) ≃ x − b , with b = 4 . a ≡ a/m = 0 . b = 3 . a = 0. The injected power per solid angle, from the horizon to a given radius x , is then givenby ˙ E ν ¯ ν ( x, θ ) = m ˙ Q Z xx H α Σ f ( x ′ ) dx ′ . (11)The cumulative power distribution in the upper hemisphere (0 ≤ θ ≤ π/
2) is˙ E ν ¯ ν ( x ) = 2 π Z π/ ˙ E ν ¯ ν ( x, θ ) sin θdθ. (12)It is related to the total power through ˙ E totν ¯ ν = ˙ E ν ¯ ν ( x = ∞ ). The amount absorbed by the black hole alonga particular field line equals the power per solid angle injected in the inflow section (between the horizonand the stagnation radius): ˙ E inν ¯ ν ( θ ) = ˙ E ν ¯ ν ( x st , θ ) = m ˙ Q Z x st x H α Σ x − b dx. (13)The rest, ˙ E outν ¯ ν ( θ ) = ˙ E totν ¯ ν ( θ ) − ˙ E inν ¯ ν ( θ ), where ˙ E totν ¯ ν ( θ ) ≡ ˙ E ν ¯ ν ( ∞ , θ ), emerges at infinity. The total powerintercepted by the black hole in one hemisphere is˙ E inν ¯ ν = 2 π Z π/ ˙ E inν ¯ ν ( θ ) sin θdθ = ˙ E ν ¯ ν ( x st ) . (14)A plot of ˙ E ν ¯ ν ( x ) (Equation (12)), is exhibited in Figure 2. For our computations we use b = 4 . a = 0 .
95 (the solid line in Figure 2). κ ( θ )Let us denote by ˙ E H ( θ ) = ˙ E ( r H , θ ), ˙ E st ( θ ) = ˙ E ( r st , θ ), and ˙ E ∞ ( θ ) = ˙ E ( ∞ , θ ) the angular distribution ofthe power at the horizon, stagnation radius and infinity, respectively, where ˙ E ( r, θ ) is defined in Equation(5). Integration of Equation (1) yields ˙ E H ( θ ) = ˙ E st ( θ ) − ˙ E inν ¯ ν ( θ ) , (15)˙ E ∞ ( θ ) = ˙ E st ( θ ) + ˙ E outν ¯ ν ( θ ) , (16)where Equations (6), (10) and (13) have been employed.Now, the specific energy of the injected plasma is positive, hence ˙ E inν ¯ ν ( θ ) ≥
0, ˙ E outν ¯ ν ( θ ) ≥
0, as can beinferred from Equation (11). In the situations envisaged here ˙ E ∞ ( θ ) >
0, but ˙ E H ( θ ) can be negative orpositive, depending on the load. In the force-free limit ˙ E totν ¯ ν ( θ ) →
0, whereby Equations (15) and (16) yield 5 –˙ E ∞ ( θ ) = ˙ E H ( θ ) >
0. The extracted power per solid angle is given, in this limit, by (Blandford & Znajek,1977): ˙ E H ( θ ) = P BZ ( θ ) ≡ c π α Ω (1 − α Ω ) (cid:18) ˜ am (cid:19) ( x H + ˜ a ) sin θx H ( x H + ˜ a cos θ ) Ψ , (17)in terms of the black hole spin ˜ a , magnetic flux Ψ , and the dimensionless parameter α Ω = Ω F /ω H . Ingeneral ˙ E H ( θ ) < P BZ ( θ ), as readily seen from Equation (15).Henceforth, we shall quantify the load on a specific streamline θ by the parameter κ ( θ ) ≡ ˙ E inν ¯ ν ( θ ) P BZ ( θ ) . (18)At κ ( θ ) << θ is nearly force-free. At κ ( θ ) >> E st ( θ ) ≃ E H ( θ ) ≃ − ˙ E inν ¯ ν ( θ ) <
0, namely the energy injected below the stagnationradius on the flux surface θ is completely absorbed by the black hole. As shown below, the transition betweenthe two regimes occurs, in general, at κ ( θ ) ≃
3. Integration method
In general, the double flow must pass smoothly through 6 critical surfaces. Solutions that satisfy thisrequirement can be obtained, in principle, only if the trans-field equation is solved simultaneously withEquations (A11)-(A13), as the exact location of the critical surfaces is contingent upon the actual shape ofthe magnetic surfaces. Such an analysis is beyond the scope of this paper. Fixing the geometry of magneticfield lines renders the system of MHD equations, Eqs. (1)-(4), over constrained. The reason is that there are4 regularity conditions (the regularity conditions at the inner and outer Alfv´en surfaces are automaticallysatisfied), but only 3 adjustable parameters; the location of the stagnation point, r st , at which u p = 0, andthe values of ǫ r and p at r st , henceforth denoted by ǫ rst , and p st , respectively. The value of the angularmomentum flux at r st is related to ǫ rst through the Bernoulli condition, Equation (A31): l rst = Ω − F ǫ rst .Since we are interested in determining the energy flux on the horizon, we seek solutions that are regularon all 3 inner surfaces, and on the outer slow magnetosonic surface, but not necessarily on the outer fastmagnetosonic surface. In practice we find that any solution that crosses the outer slow magnetosonic surface,is also regular on the outer Alfv´en surface.Our strategy is to start with some initial guess for the three adjustable parameters, r st , ǫ rst , and p st ,whereupon Equations (1)-(4) are integrated numerically from r st inwards to the horizon, and outwards tothe outer Alfv´en surface, along a given field line θ . The integration is repeated many times, where in eachrun the values r st , ǫ rst , p st are readjusted until a solution that crosses the desired critical points smoothlyis obtained. In all the examples presented below we neglected the change in linear and angular momentumowing to plasma injection, that is, we set Q ϕν ¯ ν = Q rν ¯ ν = 0 in Equations (6)-(9). Equation (2) then readilyyields ˙ L = Σ l r = const. The black hole spin ˜ a , angular velocity of the streamlines Ω F , and load parameter κ are given as input parameters. In practice, however, κ cannot be determined a priori, since r st is unknown.We therefore use instead of κ the dimensionless parameter ˜ p B ≡ Ψ c/ (32 π ˙ Q m ) as an indicator for theload. Once a solution is obtained and r st is determined, κ is computed by employing Equations (13), (17)and (18). 6 –
4. Results
The family of solutions can be divided into two classes that are distinguished by the sign of the energyflux on the horizon, ǫ rH . This devision is dictated by the load parameter κ ( θ ), as discussed further below(Figure 4). We find that in case of underloaded solutions, defined as those for which κ ( θ ) <<
1, the specificenergy is negative in the entire region encompassed by the plasma inflow (below the stagnation radius),including the horizon, whereby ǫ rH >
0. In case of overloaded solutions ( κ ( θ ) >>
1) we find ǫ rH < κ = 10 − ) and overloaded ( κ = 20) equatorial flows are exhibited, for a blackhole spin parameter ˜ a = 0 .
95, and an energy deposition profile f ( x ) = x − . . In the right panel we alsoexhibit solutions with κ = 0 .
4, 1, 6, and κ = ∞ (a purely hydrodynamic flow), that are not shown in theleft panel for clarity. As seen, the energy flux of the underloaded solutions is positive everywhere, whereasthat of overloaded solutions changes sign below the stagnation radius. The location r at which ˙ E ( r , θ ) = 0approaches r st as κ ( θ ) → ∞ . We think that this peculiar behavior stems from the fact that in the regimeΩ F < ω H the Poynting flux measured by a distant observer is always driven by the black hole (i.e., by framedragging). To elucidate this point we employ Equation (A16) to obtain the Poynting flux on the horizon: (cid:18) F rθ F θt π (cid:19) H = − ̟ H Ω F ( ω H − Ω F ) M H + ̟ H ( ω H − Ω F ) ( ǫ rH − ω H l rH ) . (19)Now, ǫ rH − ω H l rH = ρ H u rH ( E H − ω H L H ) is always negative, since the ZAMO energy is always positive, viz., E ZAMO = E − ω L >
0, and u rH <
0. Thus, for any value of the load parameter, the electromagnetic flux onthe horizon is positive if ω H > Ω F . Note also that in the force-free limit M H →
0, and (19) reduces to thefamiliar result, ǫ rH = ( F rθ F θt ) H / π , whereas in the pure hydrodynamic case M H → ∞ and the Poyntingflux vanishes, as expected. Figure 4 shows the electric current, I = α̟B ϕ , for two overloaded solutions,and it is seen that it never changes sign. We find that this is true in general in the regime 0 < Ω F < ω H ,implying that for any value of κ there is a continuous flow of Poynting energy from the horizon outwards,against the inflow of injected plasma. In particular, at the stagnation radius B ϕ ( r st ) <
0, and from Equation(A7) we obtain ǫ rst = − (cid:18) ̟ Ω F B ϕ B r π √ g rr (cid:19) st = (cid:18) F rθ F θt π (cid:19) st > . (20)Since for overloaded solutions ǫ rH <
0, it is evident that the energy flux must vanish at some radius r < r st ,as seen in the right panel of Figure 3. Our interpretation is that the outward flow of electromagnetic energydriven by the black hole is counteracted by an inward flow of kinetic energy injected on magnetic flux tubes.When the latter exceeds the former the net energy flux becomes negative. For κ ( θ ) << κ ( θ ) >> ǫ rH < ǫ rH >
0. For reference, powers are normalized to the equatorial BZ power, This point was not properly understood in GL13. The claim made there, that for overloaded solutions B ϕ must vanish at r st is incorrect. However, the conclusion regarding the activation of the BZ process remains valid, as confirmed in the presentanalysis. P BZ ( π/ E inν ¯ ν ( θ ) /P BZ ( π/ θ = π/
2) is just the load parameter κ ( θ ) defined in Equation (18). For other streamlines, the loadparameter is obtained by multiplying values on the horizontal axis by the factor P BZ ( π/ /P BZ ( θ ). Thedashed lines delineate the normalized power extracted from the black hole, ˙ E H ( θ ) /P BZ ( π/ E ∞ ( θ ) /P BZ ( π/ F = ω H / F = ω H / F , even though the structure of theflow does depend on this parameter (see Figure 6 below). This analysis confirms that the transition fromunderloaded to overloaded flows occurs at κ ( θ ) ≃
1. We also computed solutions for different black holespins, and found the same behavior (see, for example, Figure 4 in GL13). One caveat is the possibility of anonlinear feedback of the load on the magnetic flux in the vicinity of the horizon. Such a feedback may, inprinciple, change somewhat the activation condition, but not in a drastic way. It may be possible to test itusing numerical simulations.As explained above, in our model the angular velocity Ω F is given as an input. In reality it is determinedby global conditions. For nearly force-free flows numerical simulations indicate that Ω F ≃ ω H /
2. Wetherefore used this value for the underloaded solutions. However, when the inertia of the injected plasmabecomes important, it is likely to affect Ω F . To study how the properties of the flow depend on this parameter,we sought solutions with different values of Ω F ( θ ), but the same value of κ ( θ ). An example is presented inFigure 6, and it is seen that while the velocity profile depends on Ω F , the power profile is insensitive to thechoice of this parameter. In particular, it does not affect at all the activation condition. We find this trendis quite general, and therefore conclude that the result exhibited in Figure 5 is robust.
5. Conclusion
We constructed a semi-analytic model for the double-transonic flow established in the magnetosphereof a Kerr black hole under conditions anticipated in GRBs, incorporating plasma deposition on magneticfield lines via annihilation of MeV neutrinos emitted by the surrounding hyper-accretion flow. We examinedthe effect of energy loading on the properties of the flow, and identified the different operation regimes. Wefind that magnetic extraction of the black hole spin energy ensues, as long as the power deposited below thestagnation radius separating the inflow and outflow sections is smaller than the force-free BZ power. Thetransition from underloaded flows that are powered by the black hole spin energy, to overloaded flows thatare powered by the neutrino source is continuous, as seen in Figure 5.To relate the load parameter derived in Equation (18) to the accretion rate ˙ m acc (henceforth measuredin units of M ⊙ s − ), we employ the scaling relation derived in ZB11. As mentioned above, their analysis,that exploit an advanced disk model, yields a total energy deposition rate of˙ E totν ¯ ν ≃ ( M BH / M ⊙ ) − / ˙ m / acc x − . mso erg s − , (21)for accretion rates in the range 0 . < ˙ m acc <
1, where x mso is the radius of the marginally stable orbit inunits of m . Combining the latter result with the activation condition derived from Figure 5, and using theangle averaged energy deposition rate, yields a rough estimate for the accretion rate at which a transitionfrom underloaded to overloaded solutions occurs:˙ m c ≃ (cid:18) M BH M ⊙ (cid:19) − / (cid:18) Ψ G cm (cid:19) / f (˜ a ) . (22) 8 –Here Ψ is the magnetic flux accumulated in the vicinity of the horizon, and the function f (˜ a ) is displayedin Figure 7. According to this relation, when ˙ m acc < ˙ m c the outflow is powered by the BZ process, whereasfor ˙ m acc > ˙ m c it is driven by the neutrino source. In reality, the magnetic flux Ψ should also depend onthe accretion rate, however, the sensitivity of this relation to the assumptions underlying the specific diskmodel adopted for its calculation renders it highly uncertain. Furtheremore, the presence of sufficientlystrong magnetic field in the inner disk regions may affect the neutrino luminosity. For illustration, we usethe disk model of Chen & Beloborodov (2007) to estimate the magnetic flux. Unfortunately, it is difficult toderive scaling relations from the results presented in this paper, but from Figures 1 and 2 there we obtainedΨ ∼ × √ ξ B G cm for a black hole mass M BH = 3 M ⊙ , angular momentum ˜ a = 0 .
95, viscosityparameter α vis = 0 .
1, and accretion rate ˙ m acc = 0 .
2, assuming that the magnetic pressure in the innerregions of the disk is a fraction ξ B of the total pressure. For this choice we infer that with ξ B on the orderof a few percents, as naively expected, the transition from underloaded to overloaded flows may occur ataccretion rates ˙ m acc > . θ of magnetic surfaces (see Figure 5), and the approximateuniformity of the angular distribution of the energy deposition rate indicated in Figures 2 and 3 of ZB11,suggest that for accretion rates ˙ m acc < ∼ ˙ m c , and unless the magnetic flux near horizon is extremely high, theoutflow produced in the polar region may consist of an inner core inside which the power is dominated bythe thermal energy of the hot plasma, and outside which it is dominated by the Poynting flux driven byframe dragging.This research was supported by a grant from the Israel Science Foundation no. 1277/13 A. Derivation of the flow equations
The stress-energy tensor of a magnetized fluid takes the form, T αβ = ¯ hρc u α u β + pg αβ + 14 π (cid:18) F αγ F βγ − g αβ F (cid:19) , (A1)here u α is the four-velocity measured in units of c, ¯ h = ( ρc + e int + p ) /ρc the dimensionless specific enthalpy, ρ the baryonic rest-mass density, p the pressure, and g µν the coefficients of the metric tensor of the Kerrspacetime. In the following we use geometrical units ( c = G = 1), unless otherwise stated, and expressthe Kerr metric in the regular Boyer-Lindquist coordinates, ds ≡ g µν dx µ dx ν with the non-zero metriccoefficients given by: g rr = Σ / ∆, g θθ = Σ, g ϕϕ ≡ ̟ = A sin θ/ Σ, g tt = − α + ω g φφ , g tφ = − ωg φφ , interms of ∆ = r + a − mr , Σ = r + a cos θ , A = ( r + a ) − a ∆ sin θ , α = p Σ∆ /A , and ω = 2 mra/A .The parameters m and a are the mass and specific angular momentum per unit mass of the hole, α is thetime lapse and ω the frame dragging potential between a zero-angular-momentum observer (ZAMO) and anobserver at infinity. The angular velocity of the black hole is defined as the value of ω on the horizon, viz., ω H ≡ ω ( r = r H ) = a/ (2 mr H ), here r H = m + √ m − a is the radius of the horizon, obtained from thecondition ∆ H = 0.The dynamics of the flow is governed by the energy-momentum equations:1 √− g ( √− gT αβ ) ,α + Γ βµν T µν = q β , (A2) 9 –mass conservation: 1 √− g ∂ α ( √− gρu α ) = q n , (A3)and Maxwell’s equations: F βα ; α = 1 √− g ( √− gF βα ) ,α = 4 πj β , (A4) F αβ,γ + F βγ,α + F γα,β = 0 , (A5)subject to the ideal MHD condition F µν u ν = 0. Here, q β denotes the source terms associated with energy-momentum transfer by an external agent, q n is a particle source, and Γ βµν denotes the affine connection.The energy, angular momentum and entropy fluxes, can be expressed explicitly as ǫ a ≡ − T at = ρu a E , l a ≡ T aϕ = ρu a L , s a = ( ρ/m N ) u a s, (A6)in terms of the energy per baryon, E = − ¯ hu t − α̟ Ω F πη B ϕ , (A7)angular momentum per baryon, L = ¯ hu ϕ − α̟B ϕ πη , (A8)and the entropy per baryon s , where Ω F = v ϕ − v p B ϕ ̟B p (A9)is the angular velocity of magnetic field lines, η = ρu p B p (A10)is the ratio of mass and magnetic fluxes, and the index a runs over r and θ . In the above equations u p = ± ( u r u r + u θ u θ ) / is the poloidal velocity, where the plus sign applies to outflow lines and the minus signto inflow lines, v p = u p /γ , with γ = u t α being the Lorentz factor measured by a ZAMO, v ϕ = u ϕ /u t , B p =( B r + B θ ) / /α is the redshifted poloidal magnetic field, and B r = F θϕ / √ A sin θ , B θ = √ ∆ F ϕr / √ A sin θ and B ϕ = √ ∆ F rθ / Σ the magnetic field components measured by a ZAMO (see van Putten and Levinson2012 for details). Note that with our sign convention the value of η is defined to be positive on outflow linesand negative on inflow lines.For a stationary and axisymmetric ideal MHD flow, Equations (A2)-(A5) can be reduced to (GL13)1 √− g ∂ a ( √− gǫ a ) = − q t , (A11)1 √− g ∂ a ( √− gl a ) = q ϕ , (A12) kT √− g ∂ a ( √− gs a ) = − u a q α , (A13) u a ∂ a η = u p q n B p , (A14) u a ∂ a Ω(Ψ) = 0 . (A15) 10 –It can be readily seen that for q n = q µ = 0, the quantities Ω(Ψ), E (Ψ), L (Ψ), η (Ψ) and s (Ψ) are conservedon magnetic flux surfaces Ψ( r, θ ) =const, as is well known (e.g., Camenzind 1986).From (A7)-(A10) we obtain the expressions: B ϕ = − πη E α̟ α ˜ L − ̟ (Ω F − ω )(1 − ˜ Lω ) k − M (A16) u t = E ¯ h α (1 − Ω F ˜ L ) − M (1 − ω ˜ L ) α ( k − M ) , (A17) u ϕ = E ¯ h α Ω F (1 − Ω F ˜ L ) − M ω (1 − ω ˜ L ) − M ˜ Lα ̟ − α ( k − M ) , (A18)here ˜ L = L / E , M as the poloidal Alfv´enic Mach number, defined through M ≡ π ¯ hη c /ρ = u p /u A , and u A = B p / (4 π ¯ hρc ). Combining the latter relations with the normalization condition u α u α = − u p + 1 = (cid:18) E ¯ h (cid:19) k k − k M − k M ( k − M ) , (A19)where k = α − ̟ (Ω F − ω ) , (A20) k = (1 − ˜ L Ω) , (A21) k = ˜ L ̟ − (1 − ˜ Lω ) α . (A22)In terms of the free variables, ǫ r , l r , p , used in our integration we have: η E = √ Σ ǫ r / ( √ ∆ B p ), and E / ¯ h = ǫ r / (4 pu r ), ˜ L = l r /ǫ r .The equation of motion (4) is obtained upon differentiating Equation (A19) along a given streamline,using Equations (A11)-(A13) with the source terms q t = − α ˙ Q f ( x ), q φ = 0 and u α q α = u t q t , which arederived in section 2.1, and noting that in the split monopole geometry, the redshifted poloidal field reducesto B p = Ψ / (2 π √ Σ∆), the poloidal velocity is u p = p Σ / ∆ u r , and the convective derivative reduces to u α ∂ α = u r ∂ r = p ∆ / Σ u p ∂ r . This yields the following expressions for the functionals D , N ad and N q inEquation (4): D = (cid:0) k − M (cid:1) "(cid:0) u p − c s (cid:1) (cid:0) k − M (cid:1) + (cid:18) E ¯ h (cid:19) M ( k k + k )( k − M ) , (A23) N ad = " − (cid:0) u p (cid:1) (cid:0) k − M (cid:1) c s + (cid:18) E ¯ h (cid:19) M ( k k + k ) ∂ x ln B p (A24) − (cid:18) E ¯ h (cid:19) (cid:2) M (cid:0) k − M (cid:1) ∂ x k ad + (cid:0) k k − k M − k M (cid:1) ∂ x k (cid:3) ,N q = − q t ǫ r (cid:18) E ¯ h (cid:19) ( k − M ) (cid:20)(cid:0) k − M (cid:1) (1 − Ω F ˜ L ) + M α (1 − ω ˜ L ) (cid:21) − q t ǫ r (cid:18) E ¯ h (cid:19) (cid:2) − k M − k (cid:0) k − k M + 3 M (cid:1)(cid:3) α (1 − Ω F ˜ L ) − M (1 − ω ˜ L ) α ( k − M ) , (A25) 11 –where the derivatives are defined by ∂ x k = ∂ x ( α ) − (Ω F − ω ) ∂ x ( ̟ ) + 2 ̟ (Ω F − ω ) ∂ x ω, (A26) ∂ x k ad = 2 α (1 − ˜ Lω ) ∂ x ln α + 2 ˜ Lωα (1 − ˜ Lω ) ∂ x ln ω − L ̟ ∂ x ln ̟ . (A27) A.1. The stagnation point
At the stagnation point x = x st , where u p = 0, Equations (1)-(4) with q ϕ = 0 yield: ∂ x u p | x = xst = − (Σ q t ) st p st α st ( k st A st ) / , (A28) ∂ x (Σ ǫ r ) | x = xst = − (Σ q t ) st , (A29)34 ∂ x ln( p ) | x = xst = − x st − st − x st Σ st − Σ st f ( x st ) ǫ rst (1 − ω st / Ω F )8 p st α st k st ˜ p B , (A30)where − (Σ q t ) st = α st Σ st ˙ Q f ( x st ), and the parameter ˜ p B ≡ Ψ c/ (32 π ˙ Q m ) is an indicator for the load.The Bernoulli condition can be rewritten(1 − ˜ L st Ω F ) ǫ rst (4 pu r ) st = p k st , (A31)implying ˜ L st Ω F = 1, since as we have shown, there is always extraction of angular momentum from theblack hole so that ǫ rst > A.2. The Alfv´en surfaces
The location of the Alfv´en surfaces is defined by the roots of the denominator in Equations (A16)-(A18), M A = k A = α A − ̟ A (Ω F − ω A ) , here the subscript A denotes values on this surface. The latter equationhas two roots, corresponding to the inner and outer Alfv´en surfaces. The requirement that B ϕ , u t and u ϕ in Equations (A16)-(A18) are continuous there imposes a condition on the ratio of the ZAMO energy, E ZAMO = E − ω L , and the energy E of an observer at infinity: (cid:18) EE ZAMO (cid:19) A = 1 − α − A ̟ A ω A ( ω A − Ω F ) . (A32)In the regime where the BZ process is activated, ǫ rH = ρu r E >
0, and we must have E < u r <
0, and in particular at the inner Alfv´en point ( IA ). Since the ZAMOenergy is always positive, the latter condition, combined with Equation (A32), readily implies ω IA > Ω F ,and defines the range of Alfv´en radii that are permitted for energy extraction: α IA p ω IA ( ω IA − Ω F ) < ̟ IA . (A33)The shaded area in Figure 8 marks this range for solutions with Ω F = ω H /
2. In the outflow section allenergies are positive, yielding ω OA < Ω F , and α OA p ω OA ( ω OA − Ω F ) > ̟ OA , (A34)at the outer Alfv´en point ( OA ). 12 – B. Pure hydrodynamic flows
The equations governing a purely hydrodynamic flow can be obtained formally from the MHD equationsderived above upon taking the limit M → ∞ :( u p − c s ) ∂ x ln u p = N ad + N q , (B1) N ad = (1 + u p ) c s (cid:18) x Σ + x − (cid:19) − (cid:18) E ¯ h (cid:19) ∂ x k ad , (B2) N q = − q t α ǫ r (1 − ˜ Lω ) (cid:18) E ¯ h (cid:19) " (cid:18) E ¯ h (cid:19) k . (B3)34 ∂ x ln ˜ p = − ∂ x ln u p − x Σ − x − − q t (1 − ˜ Lω ) α ǫ r (cid:18) E ¯ h (cid:19) , (B4) ∂ x (Σ ǫ r ) = − Σ q t , (B5)where now ˜ L is a free parameter that describes the family of solutions, ˜ p = p/ ˙ Q t d is the normalized pressure,with t d = GM BH /c . The Bernoulli condition (A31) implies, in this limit, ǫ rst = 0, as expected in the absenceof magnetic fields. It can be readily shown that when M → ∞ the slow-magnetosonic and Alfven speedsapproach zero, whereas the fast-magnetosonic speed approaches the sound speed, c s = 1 / √
2. Consequently,the above system of equations has critical points at u p = ± c s , as can be directly verified.The regularity conditions at the sonic points, obtained from Equations (B2) and (B3), read:2 (cid:18) x c Σ c + x c − c (cid:19) + 3 ( ∂ x k ad ) c k ,c = − s − k ,c √ Σ c f ( x c )(1 − ˜ L c ω c )˜ p c √ ∆ c α c , (B6)2 (cid:18) x c Σ c + x c − c (cid:19) + 3 ( ∂ x k ad ) c k ,c = + s − k ,c √ Σ c f ( x c )(1 − ˜ L c ω c )˜ p c √ ∆ c α c , (B7)denoting the sonic point of the inflow (outflow) by x c ( x c ), and noting that u p = − / √ x c , and u p = 1 / √ x c . As seen, the existence of two sonic points, that is, x c = x c , is a consequence of energyinjection. When f ( x ) = 0 the solutions of (B6) and (B7) merge, and the system has only one critical point,for either an inflow or an outflow, depending on the boundary conditions.At the stagnation point, x = x st , the above equations yield ∂ x u p | x = xst = f ( x st ) √ Σ st (1 − ˜ L st ω st )4˜ p st √ ∆ st α st p − k ,st , (B8) ∂ x (Σ ǫ r ) | x = xst = − (Σ q t ) st , (B9) ∂ x ln(˜ p ) | x = xst = 2 k ,st ( ∂ x k ad ) st , (B10)where ˜ p st = ˜ p ( x st ) is the normalized stagnation pressure. Thus, for a given choice of f ( x ) the solution isfully determined once x st and ˜ p st are known, since ǫ rst = 0. 13 – REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
14 –Fig. 1.— Illustration of the double-transonic flow model.Fig. 2.— Net power deposited on magnetic field lines via neutrino annihilation as a function of radius x (Equation (12)), for two different injection profiles, f ( x ) = x − . and f ( x ) = x − . (˜ a = 0 . κ = 10 − ) and overloaded ( κ = 20) solutions, with f ( x ) = x − . , ˜ a = 0 .
95, and θ = 90 ◦ . The region above (below) the horizontal dotted line u r = 0, correspondsto the outflow (inflow) sections. The inner and outer slow magnetosonic points (SMP), Alfv´en points (AP),and fast magnetosonic points (FMP) are indicated. The vertical red line delineates the horizon. Right panel:profiles of the outflow power per solid angle for different values of the load parameter κ . The κ = 10 − and κ = 0 . I ( x ) = B ϕ ̟α , for equatorial flow solutions. 16 –Fig. 5.— Dependence of the outflow power on the load for two different streamlines, θ = 90 ◦ , and θ = 30 ◦ .For reference, powers are normalized by the equatorial BZ power, P BZ ( π/ E H ( θ ) /P BZ ( π/ E ∞ ( θ ) /P BZ ( π/ F = ω H /
2, and the triangles to solutions with Ω F = ω H / κ = 0 .
4, anddifferent angular velocities Ω F , as indicated. The left panel displays the velocity profiles and the right panelthe corresponding power profiles. 17 –Fig. 7.— A plot of the function f (˜ a ) defined in Equation (22).Fig. 8.— Effect of the loading efficiency on the position of the Alfv´en surfaces. The profile of ˜ L Ω F is shownfor 4 solutions corresponding to different values of κ , and ˜ a = 0 .
95, Ω F = ω H /