Elsevier

Physics Reports

Volume 325, Issue 3, February 2000, Pages 83-114
Physics Reports

The Blandford–Znajek process as a central engine for a gamma-ray burst

https://doi.org/10.1016/S0370-1573(99)00084-8Get rights and content

Abstract

We investigate the possibility that gamma-ray bursts are powered by a central engine consisting of a black hole with an external magnetic field anchored in a surrounding disk or torus. The energy source is then the rotation of the black hole, and it is extracted electromagnetically via a Poynting flux, a mechanism first proposed by Blandford and Znajek (Mon. Nat. R. Astron. Soc. 179 (1997) 433) for AGN. Our reanalysis of the strength of the Blandford–Znajek power shows that the energy extraction rate of the black hole has been underestimated by a factor 10 in previous works. Accounting both for the maximum rotation energy of the hole and for the efficiency of electromagnetic extraction, we find that a maximum of 9% of the rest mass of the hole can be converted to a Poynting flow, i.e. the energy available to produce a gamma-ray burst is 1.6×1053(M/M) erg for a black hole of mass M. We show that the black holes formed in a variety of gamma-ray burst scenarios probably contain the required high angular momentum. To extract the energy from a black hole in the required time of ≲1000 s a field of 1015 G near the black hole is needed. We give an example of a disk-plus-field structure that both delivers the required field and makes the Poynting flux from the hole dominate that of the disk. Thereby we demonstrate that the Poynting energy extracted need not be dominated by the disk, nor is limited to the binding energy of the disk. This means that the Blandford–Znajek mechanism remains a very good candidate for powering gamma-ray bursts.

Introduction

Gamma-ray bursts presently provide great excitement in astronomy and astrophysics as optical observations by way of many instruments give considerable detail of the history of each burst. We are concerned here with the prodigious energy in each burst, the estimate for GRB971214 being ≳3×1053 erg [1], although this could be diminished if considerable beaming is involved in the central engine, as we will discuss.

Amazingly, 2×1054 erg is just the rest mass energy of our sun, so it seems immediately clear that the central engine for the GRB must be able to extract a substantial fraction of the rest mass energy of a compact object, neutron star or black hole, and convert it into energy of GRB.

The second criterion for the central engine is that it must be able to deliver power over a long interval up to ∼1000 s, since some GRBs last that long although other GRBs last only a fraction of a second. It must also be able to account for the vast diversity in pulses, etc., or, alternatively, one must have a number of diverse mechanisms.

We believe the need to deliver power over the long-time found in some bursters to be the most difficult requirement to fulfill, since the final merger time of the compact objects is only a fraction of a second and it is difficult to produce a high-energy source of, e.g., νν̄-collisions that goes on for more than 2 or 3 s.

For many years mergers of binary neutron stars were considered to be likely sources for the GRBs. The estimated merger rate in our Galaxy of a few GEM1 is of the right order for the occurrence of GRBs. The possible problem with binary mergers might be the ejected materials during the merging processes. Not more than ∼10−5M of baryons can be involved in the GRB, since it would not be possible to accelerate a higher mass of nucleons up to the Lorentz factors Γ∼100 needed with the energies available.

We find the merger of a neutron star with a black hole to be a particularly attractive mechanism. The baryon number “pollution” problem can be solved by the main part of the baryons going over the event horizon. In the Blandford–Znajek mechanism [2] we wish to invoke, a substantial proportion of the rotational energy of the black hole, which will be sent into rapid rotation by swallowing up the neutron star matter, can be extracted through the Poynting vector. The rate of extraction is proportional to the square of magnetic field strength, B2, as we shall discuss, so that power can be furnished over varying times, depending upon the value of B. With substantial beaming, we estimate that B∼1015 G would be sufficient to power the most energetic GRBs with ∼1053 erg.

Recently at least three magnetars, neutron stars with fields B∼1014–1015 G, have been observed. Their visible lifetime is only a few thousand years, because neutron stars slow down by emission of magnetic dipole radiation and join the “graveyard” where they no longer emit pulses. The time of observability is proportional to B−1, so the number of these high magnetic field stars may not be an order of magnitude less than the garden variety 1012 G neutron stars. It is also possible that existing magnetic fields can be increased by the dynamo effect.

Failed supernovae were suggested by Woosley [3] as a source of GRB. In this case the black hole would be formed in the center of a massive star, and surrounding baryonic matter would accrete into it, spinning it up. This mechanism is often discussed under the title of hypernovae [4].

More recently Bethe and Brown [5] found that in binary neutron star evolutions, an order of magnitude more low-mass black-hole, neutron-star binaries were formed than binary neutron stars. The low-mass black-hole mass of ∼2.4M looks favorable for the Blandford–Znajek mechanism.

In some calculations which begin with a neutron star binary, one of the neutron stars evolves into a black hole in the process of accretion, and the resulting binary might also be a good candidate for GRBs. In any case, there are various possibilities furnished by black-hole, neutron-star binaries.

In this paper we will discuss the Blandford–Znajek mechanism in quantitative detail. In Section 2, an overview of the proposed central engine for gamma-ray bursts using Blandford–Znajek process is given. The Blandford–Znajek process and the evolution of the black hole will be discussed in detail in 3 Blandford–Znajek process, 4 Evolution of a black hole via the Blandford–Znajek process, respectively. The structure of ambient magnetic field surrounding black hole and accretion disk is discussed in Section 5. In Section 6, we will give a rough estimation of the possible angular momentum of the black hole which might emerge as a final compact object during the merging or collapsing processes. The possible constraint from the surrounding accretion disk is discussed in Section 7 and the results are summarized and discussed in Section 8.

Section snippets

Overview of the proposed central engine for GRB

Two decades ago Blandford and Znajek [2], [6] proposed a process (BZ) in which rotational energy of black hole can be efficiently extracted. If there are sufficient charge distributions around the black hole to provide the force-free condition, then the magnetic field lines exert no force and corotate rigidly with the rotating black hole. The induced current loops which pass along the black hole's stretched horizon feel the forces by the magnetic field supported by the environment. Hence, these

Blandford–Znajek process

Consider a half hemisphere (radius R) rotating with angular velocity Ω and a circle on the surface at fixed θ (in the spherical polar coordinate system) across which a surface current I flows down from the pole. When the external magnetic field B is imposed to thread the surface outward normally, the surface current feels a force and the torque due to the Lorentz force exerted by the annular ring of width Rdθ isΔT=−Rsinθ×IRΔBdθ=−(I/2π)BΔAann.=−(I/2π)ΔΨ,where ΔΨ is the magnetic flux through

Evolution of a black hole via the Blandford–Znajek process

While the black hole is slowed down and a part of the rotational energy is carried out as Poynting outflow, the rest of the rotational energy increases the entropy of the black hole or its irreducible mass. The increasing rate of the irreducible mass is given by using , dMirrdt=Prot−PL=∫[(ΩH−ΩF)2/4πc]ω̃2BHΔΨ.The irreducible mass eventually becomes the mass of Schwarzschild black hole when it stops rotating. The difference between the initial Kerr black-hole mass, M0, and the final Schwarzschild

Magnetic field and force-free plasma

The rotating black hole immersed in the magnetic field induces an electric field around the black hole [6]. The electromagnetic field in the vicinity of rotating black hole immersed in the uniform magnetic field in free space has been obtained [6], [18], [19] as a solution of the source-free Maxwell equationFμν=0.From the analytic expression (see Appendix B) which gives an asymptotically uniform magnetic field at infinity (r→∞)B=Bẑ,r→∞,one can see that the radial component of the magnetic

Rotation of the black hole

It has been suggested that merging compact binary systems [8] or hypernovae [4] may result in a disk with rapidly rotating black hole at the center. In this section we will demonstrate how this is possible using semiquantitative arguments. The basic idea is that a substantial part of the orbital angular momentum of the binary system or the spin angular momentum of the progenitor of hypernova can be imparted onto the black hole.

Consider first the BH–NS merger. The typical distance for the

Magnetized accretion disks

A black hole by itself cannot keep magnetic fields on it for a long time. Magnetic fields diffuse away in a short time ∼RH/c [6], [10]. The most plausible environments which can support a magnetic field threading the black hole are accretion disks surrounding the black hole. There are two issues about accretion disks that we need to consider in order to decide whether the Blandford–Znajek process is a viable power source for gamma-ray bursts: The life time should be long enough to extract the

Conclusion

We have evaluated the power and energy that can be extracted from a rotating black hole immersed in a magnetic field, the Blandford–Znajek effect. We improve on earlier calculations to find that the power from a black hole of given mass immersed an external field is 10 times greater than previously thought. The amount of energy that can be extracted from a black hole in this way is limited by the fact that only 29% of the rest mass of a black hole can be in rotational energy, and that the

Acknowledgements

We would like to thank Roger Blandford, Sterl Phinney and Kip Thorne for guidance and useful discussions. HKL is very grateful to the Nuclear Theory Group, SUNY at Stony Brook for the hospitality during his one year stay, where the most of collaborated works on this paper has been done. This work is supported by the U.S. Department of Energy under Grant No. DE-FG02-88ER40388. HKL is also supported in part by KOSEF-985-0200-001-2 and BSRI-98-2441.

References (49)

  • S.R. Kulkarni

    Nature

    (1998)
  • R.D. Blandford et al.

    Mon. Not. R. Astron. Soc.

    (1977)
  • S.E. Wooseley

    Astrophys. J.

    (1993)
  • B. Paczyński

    Astrophys. J.

    (1998)
  • H. Bethe et al.

    Astrophys. J.

    (1998)
  • K.S. Thorne, R.H. Price, D.A. MacDonald, Black Holes; The Membrane Paradigm, Yale University Press, New Haven and...
  • S.R. Kulkarni et al., astro-ph/9902272...
  • P. Mészáros, M.J. Rees, R.A.M.J. Wijers, New Ustron. 4 (1999)...
  • R. Popham et al., Astrophys. J. 518 (1999)...
  • M. Livio, G.I. Ogilvie, J.E. Pringle, Astrophys. J. 512 (1999)...
  • P. Ghosh et al.

    Mon. Not. R. Astron. Soc.

    (1997)
  • C. Kouveliotou

    Nature

    (1998)
  • H. Janka, M. Ruffert, T. Eberl,...
  • C. Fendt

    Astron. Astrophys.

    (1997)
  • D. Macdonald et al.

    Mon. Not. R. Astron. Soc.

    (1982)
  • S. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, Wiley, New York,...
  • I. Okamoto

    Mon. Not. R. Astron. Soc.

    (1992)
  • R.M. Wald

    Phys. Rev. D

    (1974)
  • A.R. King et al.

    Phys. Rev. D

    (1975)
  • J. Bicak et al.

    Mon. Not. R. Astron. Soc.

    (1985)
  • See, for example K. Hirotani, I. Okamoto, Astrophys J. 497 (1998)...
  • S.E. Woosley et al.

    Astrophys J.

    (1992)
  • H. Janka, M. Ruffert, H.-Th. Janka,...
  • D.A. MacDonald

    Mon. Not. R. Astron. Soc.

    (1984)
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