Plasma injection and outflow formation in Kerr black holes
SSeptember 13, 2018 11:45 WSPC/INSTRUCTION FILE Hepro4
International Journal of Modern Physics: Conference Seriesc (cid:13)
World Scientific Publishing Company
Plasma injection and outflow formation in Kerr black holes
Noemie Globus and Amir Levinson
School of Physics & Astronomy, Tel Aviv University Tel Aviv 69978, Israel
Received Day Month YearRevised Day Month YearWe discuss the role plasma injection plays in the formation of outflows in Kerr spacetime.Using a model for the double flow established in the polar region of a rotating black hole,we study the interplay between the different processes that can power the outflow. Inparticular, we find two types of flows with distinct properties that depend on the rateat which energy is deposited in the magnetosphere. We discuss the implications of thisresult for gamma ray bursts outflows.
Keywords : Black holes; magnetohydrodynamics; relativistic plasma dynamics.PACS numbers: 04.70.-s, 47.75.+f, 95.30.Qd
1. Introduction
Accreting black holes are thought to power the relativistic jets that form in activegalactic nuclei (AGNs), microquasars, and gamma-ray bursts (GRBs). A plausibleproduction mechanism for those jets is magnetic extraction of the spin energy ofa Kerr black hole . A key feature of this process is a double trans-magnetosonicplasma flow which is launched from a stagnation radius located between the innerand outer light surfaces, and sustained by a plasma source in the magnetosphere. Ingeneral, there is a range of plasma injection rates within which the Blandford-Znajek(BZ) process can be activated; it has to be sufficiently high to provide the chargedensity required to establish an MHD flow, but low enough to avoid overloading ofmagnetic field lines that leads to a shutdown of the BZ process.The nature of the plasma source in most of the relativistic systems mentionedabove is not well understood yet. In AGNs the problem seems to be how to in-ject enough charged particles on open magnetic field lines. Direct feeding by thesurrounding accretion flow seems unlikely, as charged particles would have to crossmagnetic field lines on timescale shorter than the accretion time in order to reachthe polar outflow. Free neutrons that may be produced in a radiative inefficientaccretion flow (RIAF) can cross field lines, however, they will decay over a dis-tance typically much shorter than the horizon scale of a supermassive black hole.Thus, even if existent at sufficient quantity they will not reach the inner regions.Pair production on open filed lines via annihilation of MeV photons is a plausible a r X i v : . [ a s t r o - ph . H E ] N ov eptember 13, 2018 11:45 WSPC/INSTRUCTION FILE Hepro4 Noemie Globus & Amir Levinson plasma source, but requires sufficiently hot accretion flow, and may be relevantonly to faint sources like M87 and Sgr A ∗ . It has been shown that the density ofthe charges thereby injected depends sensitively on the accretion rate and the con-ditions in the RIAF , . Naive estimates , although highly uncertain, suggest thatin case of M87 this process cannot provide complete screening at accretion ratesthat correspond to the inferred jet power, and to a fit of the observed SED by anadvection-dominated accretion flow (ADAF) model. As a consequence, an intermit-tent gap may form at the base of the flow in which the injected density is amplifiedto the required level by copious pair cascades induced by the potential drop in thegap. This gap may be the source of the variable TeV emission observed in M87 and,conceivably, some other non-blazar AGNs , , .In powerful blazars the accretion flow is much cooler, hence direct charge injec-tion should not ensue. However, the disk luminosity is much higher and it could bethat once pair creation is initiated in the gap by a stray charge, it will be sustainedforever in a cyclic process, owing to inverse Compton scattering and pair creationon the dense target photon field, as demonstrated in the case of pulsars , .In GRBs the plasma injection rate is anticipated to be always well above thatrequired to establish a MHD flow. However, under certain conditions, it may leadto overloading and a consequent shutdown of the BZ process. In what follows, wedescribe some recent studies , wherein the role of plasma injection in the magne-tosphere of a Kerr black hole has been carefully examined.
2. A model for loaded MHD flows
We consider relativistic jets that are created near the polar region of a Kerr blackhole. As mentioned above, the magnetosphere consists of a double-flow structure,whereby two plasma streams are launched, along every magnetic flux tube, in oppo-site directions from a stagnation radius r st ( θ ) located between the two light surfaces,where θ is the inclination angle of the flux tube. To analyze this structure, we haveconstructed a semi-analytic model that incorporates plasma injection in a self-consistent manner. The injection process is modeled by prescribed source terms inthe MHD equations, that determine the rate of change in mass, energy, angularmomentum and entropy along streamlines. This generalize the equations of Ref. 10to the Kerr geometry.We identified two distinct types of solutions, that correspond to regimes wherethe BZ process is switch-on or switch-off (see Figure 1). These two types of flows arecharacterized by the sign of the specific energy on the horizon, henceforth denotedby E H . For negative energy solutions ( E H <
0) the energy flux is always directedoutwards, implying energy extraction of the black hole spin energy. At sufficientlylow injection rates the emitted power is shown to converge to the BZ result derivedin the force free limit (see below). The dynamics of the flow in this regime is governedentirely by the frame dragging potential induced by the black hole. In case of positiveenergy solutions ( E H >
0) the dynamics of the flow is dictated by the externaleptember 13, 2018 11:45 WSPC/INSTRUCTION FILE Hepro4
Plasma injection and outflow formation in Kerr black holes plasma source. If the plasma injected in the magnetosphere is relativistically hot,then a pressure driven, double-flow is launched from the stagnation radius, wherebya fraction of the injected energy is absorbed by the black hole and the rest emergesat infinity in the form of a relativistic jet. If the injected plasma is cold, an outflowmay not form at all.Which type of flow will form under given conditions depends merely on theplasma injection rate, as discussed in the next section. Fig. 1. Illustration of the double-flow structure: In a Kerr spacetime there exist two types offlows, depending on the plasma injection rate. At supercritical loads the outflow is powered by theexternal energy source (upper panel). At subcritical loads, the outflow is powered by the blackhole rotational energy (lower panel). eptember 13, 2018 11:45 WSPC/INSTRUCTION FILE Hepro4 Noemie Globus & Amir Levinson
3. A critical load
Takahashi et al. have shown that two conditions must be satisfied in order forenergy to be extracted from a Kerr black hole: (i) the frame dragging potentialmust exceed the angular velocity of the magnetic field lines near the horizon, and(ii) the Alfv´en point of the inflow must be located inside the ergosphere. When theseconditions are satisfied, the specific energy of a fluid element near the horizon, asmeasured at infinity, is negative, E H <
0, implying an outward energy flux on thehorizon.A question of interest is how the efficiency of the extraction process depends onthe load. This question was not addressed in Ref. 11. In order to analyze the effectof the load, we computed the structure of the ideal MHD inflow emanating from thestagnation radius, for different plasma injection rates. To simplify the analysis, weinvoked an infinitely thin injection zone, whereby the mass injection profile is givenby q n ∝ δ ( r − r st ), and likewise for the energy-momentum source terms. Then, thespecific energy E is conserved at r < r st , and its value is uniquely determined by theregularity condition at the fast magnetosonic point. The value of the enthalpy fluxinjected at the stagnation radius fixes the location of the slow point a . For furtherdetails see Ref. 9.To elucidate key features, we consider first the zero temperature limit. An ex-ample is shown in Figure 2 for an equatorial flow, where the extracted power P is plotted against the injected mass flow rate ˙ M . The numerical values were com-puted assuming magnetic flux of Ψ = 9 × G cm . As seen, the extracted powerconverges to the force-free result, P F F , derived in Ref. 1 (marked by the horizontaldashed line) at sufficiently small loads, but is strongly suppressed as the load ap-proaches the critical value ˙ M c (cid:39) P F F /c . From this, we concluded that a rotatingblack hole can transfer its rotational energy to the outflow only along field lines onwhich the accretion rate satisfies˙ M < − (cid:18) M BH M (cid:12) (cid:19) − (cid:18) Ψ G cm (cid:19) g ( a, θ ) M (cid:12) s − , (1)where g ( a, θ ) = a ( r H + a ) sin θ/ [ r H ( r H + a cos θ )].In Ref. 9 it is shown that similar results are obtained in the general case of ahot flow. The critical condition generalizes to P inj (cid:39) P F F /c , where P inj is totalpower injected in the magnetosphere which, in the zero temperature limit, reducesto P inj = ˙ M c .
4. Application to GRBs
The immediate consequence of the above results for long GRBs is that following thestellar collapse, the polar region must be devoid of matter in order for a jet to form. a in the cold case, the slow point coincides with the stagnation point eptember 13, 2018 11:45 WSPC/INSTRUCTION FILE Hepro4 Plasma injection and outflow formation in Kerr black holes θ = π/
2) vs. the injected mass flow rate in theregime where energy extraction is switched on, for different black hole angular momenta a . Eachpoint corresponds to a cold inflow solution. The horizontal dashed line gives the force-free resultderived in Ref. 1. The critical load ˙ M c can be readily obtained from the figure in each case. But even then, there is another plasma source in the magnetosphere, namely anni-hilation of MeV neutrinos that emanate from the hyper-accretion disk surroundingthe black hole. The plasma thereby deposited is relativistically hot, and so a polaroutflow will be driven either by the black hole or by the pressure of the injectedplasma, provided that the central region is baryon poor, as explained above.MeV neutrinos are emitted from the inner disk region, with a sensitive de-pendence of the neutrino luminosity on accretion rate. Recent calculations of theannihilation rate around a Kerr black hole , yield a net energy deposition rateof ˙ E ν ¯ ν (cid:39) ˙ m / ( M BH / M (cid:12) ) − / x − . erg s − , for accretion rates (henceforthmeasured in units of M (cid:12) s − ) in the range 0 . < ˙ m acc <
1, where x mso is theradius of the marginally stable orbit in units of m = GM BH /c . From our analysiswe estimate that for accretion rates˙ m acc < . (cid:18) M BH M (cid:12) (cid:19) − / (cid:18) Ψ G cm (cid:19) / f ( a, θ ) , (2)eptember 13, 2018 11:45 WSPC/INSTRUCTION FILE Hepro4 Noemie Globus & Amir Levinson the jet is powered by the black hole (the first flow type), whereas for higher ratesit is powered by the neutrino source. The function f ( a, θ ) satisfies f (0 , θ ) = 0, butotherwise depends weakly on a . For θ = π/ . . ≥ a ≥ . , we alreadyanalyzed the structure of a neutrino driven flow in a Schwarzschild geometry. Werestricted our analysis to unmagnetized and non rotating flows, since the hole itselfis non rotating ( a = 0) and the particles follow radial geodesics near the horizon.We adopted an energy deposition profile of the form − q t ∝ x − b with b (cid:39) . a = 0 .
95 and b (cid:39) . a = 0, where a is the normalized black hole spin .We derived the double transonic structure and computed the power of the outflowfor different energy deposition profiles . We requiered that both the inflow and theoutflow solutions pass smootly through their sonic points. We started the integrationat the inner sonic point and adjusted the stagnation pressure to cross the outer one.We found that for a given choice of the energy deposition profile, there exists aunique solution that passes through the inner and the outer sonic points. Near thehorizon, the inflow moves along radial geodesics, while the position of the outersonic point is determined by the pressure at the stagnation point.We concluded that the outflow production efficiency (cid:15) , defined as the fraction of˙ E ν ¯ ν that emerges at infinity, is typically large, with (cid:15) = 0 .
58, 0 .
73, 0 .
83 for b = 5, 4and 3 .
5, respectively. We also found that the specific entropy in the outflow is largerthan usually thought (see Figure 2 of Ref. 8), owing to the delayed acceleration ofthe flow, and pointed out the implications for prompt emission.
References
1. R. D. Blandford and R. L. Znajek,
Mon. Not. R. Astron. Soc. , 433 (1977).2. A. Levinson and F. Rieger,
Astrophys. J. , 123 (2011).3. M. Mo´scibrodzka, C. F. Gammie, J. C. Dolence and H. Shiokawa,
Astrophys. J. ,9 (2011).4. A. Levinson,
Phys. Rev. Lett. , , 912 (2000).5. A. Neronov and F. A., Aharonian, Astrophys. J. , 85 (2007).6. A. Timokhin,
Mon. Not. R. Astron. Soc. , 209 (2010).7. A. Timokhin and J. Arons,
Mon. Not. R. Astron. Soc. , 429 (2013).8. A. Levinson and N. Globus,
Astrophys. J. , 159 (2013).9. N. Globus and A. Levinson, Loaded MHD flows in Kerr spacetime, to appear in
Phys.Rev. D15 .10. A. Levinson,
Astrophys. J. , 510 (2006).11. M. Takahashi, S. Nitta, Y. Tatematsu and A. Tomimatsu,
Astrophys. J. , 206(1990).12. W.-X. Chen and A. Beloborodov,
Astrophys. J. , 383 (2007).13. I. Zalamea and A. Beloborodov,
Mon. Not. R. Astron. Soc.410