Blandford-Znajek mechanism and relativistic jets
Serguei Komissarov University of Leeds, UK Plan of this Lecture
• Physics of the Blandford-Znajek (BZ) mechanism • Origin of relativistic astrophysical flows; • BZ versus other mechanisms; • BZ in GRBs; • Open issues; Penrose Mechanism (Penrose 1969) • Split the BH’s space-time into time t and space (t = const) with spatial coordinates {xi}. With appropriate selection of coordinates the metric is stationary and Minkowskian at infinity. • - integral of motion for free-falling particles (geodesic motion);
• So is , where m0 is the rest mass. • m is the called the red-shifted mass-energy (or the mass-energy at infinity ). • When the particle is far away from the BH the local observer at rest finds that • - the particle mass-energy in his frame. • When the particle is near the BH the local observer at rest finds that m is not equal to the mass-energy of the particle as measured in his frame. • When the BH swallows the particle its mass-energy changes by m. The total mass-energy of the system is conserved.
• Inside the ergosphere of rotating black hole m can be negative (while m0γ > 0). • The BH mass can decrease! Such particle cannot simply fall from infinity. Energy from Ergosphere Ergosphere
Ergoregion
The mass-energy of BH can be reduced by the amount
- event horizon radius
- spin parameter - gravitational radius no rotation:
maximal rotation: !!! Electromagnetic Mechanism
• Wald (1974) Vacuum solution for a Kerr black hole placed in a uniform magnetic field. Electric field is also generated.
Similarity with the vacuum solution for rotating neutron stars.
• Blandford & Znajek (1977) Plasma solution (Force-Free Electrodynamics or Magnetodynamics) for a Kerr BH with monopole magnetic field and slow rotation (a <<1). Outgoing Poynting flux – robust mechanism for extracting the rotational energy of astrophysical BHs! 3+1 Electrodynamics of Black Holes
Tamm (1928), Landau(1951), Komissarov (2004)
Space-time behaves as electromagnetically active medium !
In flat space-time E=D and B=H everywhere but in BH space time they differ.
The ergoregion is the location of the BZ’s “power engine”. Paired-wind structure of BZ-driven flows
Ergosphere
The outflow carries away positive electromagnetic The inflow carries into the BH redshifted energy negative electromagnetic energy at infinity
The flux of redsifted energy is outgoing everywhere. Membrane Paradigm (Thorne, Price & Macdonald, 1986)
Imagine that the BH horizon is a Ωh rotating conducting sphere with finite _ _ _ _ resistivity. In magnetic field its charges separate due to the Lorentz force + + and give you the electric field. + + + This gives qualitatively correct intuition + + most of the time, but not always. + Hides the real phenomenon. _ _ _ _ Places undue emphasis on the horizon and makes no use of the ergoregion.
Contributed to the lasting controversy surrounding the BZ mechanism Punsly & Coroniti (1990) Validity of BZ mechanism
The validity of BZ mechanism has been fully confirmed in time-dependent numerical simulations.
• Force-free limit (Magnetodynamics): Komissarov (2001,2004), McKinney (2006), Komissarov & McKinney(2007)
• Relativistic Magnetohydrodynamics:
McKinney & Gammie(2004), Komissarov (2004), Koide(2004), Nagataki(2009)
Now, the monopole problem, solved analytically by Blandford & Znajek (1977), serves as a test problem for numerical codes!
• Relativistic jets from BHs are produced in realistic simulations of astrophysical systems:
McKinney(2006), Barkov & Komissarov (2008,2009), McKinney & Blandford (2009) Resource and Power of the BZ mechanism
• Maximum available energy (a =1): B B black hole jet Sufficient to explain hypernovae Ψ associated with long GRBs! disk wind
• Jet power (split-monopole):
- angular velocity of the BH; Ψ – magnetic flux accumulated by the BH
- 3 seconds to drive a hypernova
Ψ = 1027 Gauss cm2 – the highest observed magnetic flux of massive stars) Competition: Disk or BH ?
Disk binding energy (for a =1):
• Neutrino heating (GRBs) • Magnetic braking
relativistic disk wind fireball B B
e.g. Eichler et al.(1989), e.g. Blandford (1976), Blandford & Payne (1982) Aloy et al.(2000), Li et al.(1982), Vlahakis & Konigl(2003,2004) MacFadyen & Woosley (1999) ,
Magnetic power of the disk against the BZ power.
Where is most of the magnetic flux? (Ghosh & Abramowicz, 1997; Livio et al.1999)
McKinney & Gammie (2004) Barkov & Komissarov (2008) Igumenshchev (2008,2009) Can disk magnetic winds be ultra-relativistic? (The issue of mass-loading)
In principle, Sun can drive relativistic wind, but
Wind speed:
corona The mass-loading of the wind is too heavy. The same is true for disks?
BH is not a star – matter is not lifted from wind the event horizon. The mass loading is completely different (e.g. pair production) and much weaker. Activation of the BZ mechanism (GRBs). (Komissarov & Barkov 2009)
Iron core of a rotating star collapses into a black hole. The equatorial zone of the stellar envelope forms an accretion disk. The polar zone falls straight onto the BH.
Accretion disk Accretion shock Woosley (1993), MacFadyen & Woosley (1999)
The neutrino mechanism can drive an outflow only after ~ 10s after the onset of the collapse. What is about the BZ mechanism? Collapsing stellar envelope Mechanisms in play MHD waves must be able to escape from the black hole ergoregion
The Alfven speed must be close to the speed of light
The energy density of magnetic field exceed that rest mass energy density of matter.
where Numerical Simulations Numerical simulations confirm this condition
κ = 1.2 κ = 1.6 • Kerr black hole, a = 0.9; • Polytropic EOS; • Initially cold plasma • Monopole magnetic field; • GRMHD, 2D, axisymmetry;
The critical value of κ is indeed close to unity.
It depends on a but weakly.
log10ρ log10ρ Strong Magnetic Fields During the first ten seconds of stellar collapse the accretion rate is too high and requires very strong magnetic field.
Maximum surface flux of magnetic stars -
BZ-mechanism cannot be activated straight away. Need to wait till the low density polar funnel is formed in the accretion flow.
Strong stellar magnetic fields (Heger et al. 2005)
magnetic braking of progenitors magnetic braking of stellar cores
no disk, no rapidly rotating BH delayed formation of accretion disk Binary progenitor. A way out?
• Magnetic star in a close synchronized binary;
Ω B Ω
e.g. Lee et al. (2002), Izzard et al. (2004), Barkov & Komissarov (2009) Compact Merger ?
• Merger of magnetic star with a compact companion (NS,BH);
Ω B B
(e.g. Fryer & Woosley 1998. Zhang & Fryer 2001 ). Observations Help • The plateau phase and flares in the light curves of Swift afterglows – the smoking gun of magnetic mechanism?
Continuous energy injection Zhang (2007) long-acting central engine (alternative explanations)
cannot be neutrino mechanism due to the drop of accretion rate
The BZ mechanism will continue to operate until the whole of the star is accreted, up to 104s in the close binary scenario (Barkov & Komissarov 2009). Super-collapsars of the early Universe
Formation of first stars in the early Universe Bromm et al.(2002), Gao et al.(2007), Ohkubo et al.(2009), Natarajan et al. (2009)
6 • Collapsed halos of dark matter of ~10 M3 at z ~20;
3 • The primordial metal-free gas fragments into clamps of ~10 M3
• Slow cooling means no further fragmentation;
• First stars (Population III) can be very massive , 100M3 < M <1000M3;
• Stars with M > 260M3 collapse into black holes with very little mass loss; ( Fryer et al. 2001 )
• Expected rate ~5000 per galaxy like Milky Way; seeds for SMBHs ( Madau & Rees 2001 )
Collapse of massive rotating Pop III stars (supercollapsars) Numerical simulations of Fryer et al. (2001);
Mstar = 300M3
• A black hole with mass ~78 M3 is formed;
• Later a thick accretion disk is formed;
• The disk is too large and cool; The neutrino annihilation mechanism is not effective.
• GRB jets can only be produced via the magnetic mechanism ! Expect soft, ~20keV, long lasting bursts, > 10,000s, visible to Swift-BAT at a rate of few per year (Komissarov & Barkov, 2009, arXiv).
Conclusions
• Blandford-Znajek process is a well established GR effect similar in nature to the Penrose process;
• Most likely source of the relativistic jets in AGN and XRB;
• Potentially important for GRB, particularly if long-lasting central engine is required (e.g. afterglow plateau). May cause long soft long lasting bursts from the massive Pop III stars in the early Universe (not detected yet?);
• Key open issues: 1) origin of magnetic field; 2) its dynamics in accretion disks; 3) mass loading mechanisms of BH/accretion disk outflows; Blandford-Znajek mechanism and GRB jets
Serguei Komissarov University of Leeds, UK
Maxim Barkov University of Leeds, UK, Space Research Institute, Russia
TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA Plan of this talk
• Gamma-Ray-Bursts – very brief review; • Collapsar model for long GRBs; • Activation of BZ- mechanism in collapsing stars; • GRMHD simulations of collapsars; • Discussion I. Gamma Ray Bursts
Bimodal distribution (two types of GRBs?):
long duration short duration GRBs GRBs Inferred high speed: Variability compactness too high opacity to unless the Lorentz factor > 100
Total energy emitted in gamma rays:
54 Assuming isotropic emission up to Eγ = 10 erg (wide distribution).
51 With beaming correction Eγ ~ 10 erg (“standard energy reservoir”)
Total energy in the jet:
51 From afterglow observations Ejet = few 10 erg ~ Eγ
52 High-velocity supernovae or hypernovae Es ~ 10 erg. ejecta, and many other ideas. Restarting ofcentral engine, wide Lorentz factor distribution, multi-component
log10Fx (0.3 – 10keV) “ Canonical” X 0 ( -occasionally observed) 1 1 2 - ray afterglow log 2 3 10 5 (t/sec)
lightcurve 4 3 Zhang (2007) ( 5 Swift 4
) Swift 5 -X-rayflares. 4 -post“jet break” phase; 3 -normal decay phase; 2 -shallow decay phase; 1 -steep decay phase; 0 -prompt emission; standard energy (?) Jet breaks (?) questionNew marks: input
II. Collapsar model of central engines of long GRBs
Iron core of a rotating star collapses into a black hole – “failed supernova”; Stellar envelope collapses into a hyper-accreting neutrino-cooled disk; (Woosley 1993, MacFadyen & Woosley 1999).
Accretion disk Accretion shock Disk binding energy (a =1):
Most of the dissipated energy is radiated in neutrinos which almost freely escape to infinity.
1% of the energy is sufficient Collapsing stellar to explain hypernovae and envelope GRB-jets Mechanisms of tapping the disk energy
Neutrino heating Magnetic braking
fireball MHD wind photons
B B
Blandford & Payne (1982) Eichler et al.(1989), Aloy et al.(2000) Proga et al. (2003) MacFadyen & Woosley (1999) Fujimoto et al.(2006) Nagataki et al.(2006) ? Mizuno et al.(2004) Tapping the rotational energy of black hole
Blandford & Znajek (1977), Meszaros & Rees (1997)
Black hole rotational energy (a =1):
Power of the Blandford-Znajek mechanism:
a - spin parameter of the black hole (0 < a < 1), Ψ - the magnetic flux of black hole. Ψ =1027G cm2 is the highest value observed in magnetic stars: Ap, white dwarfs, neutron stars (magnetars).
Ghosh & Abramowicz (1997), Livio et al.(1999): the electromagnetic power of the accretion disk may dominate the BZ power (?) Blandford-Znajek effect
Vacuum around black holes behaves as electromagnetically active medium:
Steady-state Faraday eq.:
Strong electric field is generated when BH is immersed into externally supported vacuum magnetic field!
In contrast to a unipolar inductor or a neutron star this field is not due to the electric charge separation on a conductor! Blandford-Znajek effect
When free charges are introduced this field can sustain electric currents along the field lines penetrating the black hole ergosphere.
For the perfectly conducting case with insignificant inertia of plasma the magnetosphere is described by Magnetodynamics (MD -- inertia-free relativistic MHD).
Blandford and Znajek(1977) found a perturbative stationary solution for monopole magnetospheres of slowly rotating black holes. It exhibited outflows of energy and angular momentum.
See for details: Komissarov (2004, mnras, 350, p427) Komissarov (2008arXiv0804.1912K )
Can we capture the BZ-effect with modern numerical RMHD schemes? Yes! (Komissarov 2004, Koide 2004, McKinney & Gammie, 2004 ) B Lorentz factor
wave front • GRMHD, 2D, axisymmetry; • Kerr-Schild coordinates, a=0.9; • Inner boundary is inside the event horizon; B • Outer free-flow boundary is far away; • Initially non-rotating monopole field and zero plasma speed in FIDOs frame; • Magnetically-dominated regime ;
The solution develops steady-state Komissarov (2004) paired-wind behind the expanding spherical wave front. Numerical solution versus analytical
- BZ-solution; - MHD at r = 50M; - MHD at r = 5M;
This magnetically-dominated MHD solution is very close to the steady-state MD solution; Blandford-Znajek (1977) for a << 1; Komissarov(2001) for a = 0.9. III. Activation of BZ mechanism
What is the condition for activation of the BZ-mechanism with finite inertia of plasma?
MHD waves must be able to escape from the black hole ergosphere !?
Alfven speed , , free fall speed
(Newtonian results)
Apply at the ergosphere, 2 r = 2rg= 2GM/c :
Thus, the energy density of magnetic field must exceed that of matter for the BZ-mechanism to be activated! In terms of , the integral mass accretion rate, and , the magnetic flux threading the black hole hemisphere, this condition reads
[ in fact, we anticipate ]
In the context of the collapsar model for Gamma Ray Bursts with and this requires
Note that the highest magnetic flux of magnetic stars measured so far is only
“Test-this-idea” simulations (in preparation):
κ = 1.2 κ = 1.6 • GRMHD, 2D, axisymmetry; • Kerr black hole, a = 0.9; • Polytropic EOS; • Free-fall accretion of initially cold plasma with zero angular momentum; • Monopole magnetic field;
The critical value of κ is indeed close to unity.
It depends on a but weakly.
log10ρ log10ρ IV. Collapsar GRMHD simulations
Based on Barkov & Komissarov (2008) and more recent results
Free fall model of collapsing star: Bethe (1990) + ad hoc rotation (MacFadyen & Woosley 1999) and magnetic field; Gravity: gravitational field of Kerr black hole only; no self-gravity; Microphysics: • neutrino cooling (Thompson et al.,2001); • realistic equation of state, (HELM, Timmes & Swesty, 2000); • dissociation of nuclei (Ardeljan et al., 2005); • no neutrino heating (!); black hole Solid body rotation. Uniform magnetization M=3M3 a=0.9 R=4500 km Ψ= 4x1027-4x1028 G cm2
v • 2D axisymmetric B GRMHD; v v • Kerr-Schild metric; v • Starts at 1s from collapse onset. v • Lasts for < 1s B
free fall outer boundary, accretion R= 104 km (Bethe 1990) Results
• No explosion in models with κ < 0.3;
movie. log10ρ
• Bipolar explosions in models with κ > 0.3;
movie: log10 p/pm and v; small scale
movie: log10 p/pm; large scale
The critical value of κ is smaller because of the angular momentum in the accreting matter. (see figure) Explosions are powered mainly by the black hole via the Blandford-Znajek mechanism
• No explosion if a = 0;
• ~70% of total magnetic flux is accumulated by the black hole ( see plot) This is in conflict with Olivie et al. (1999) but agrees with Newtonian simulations by Igumenshchev (2007);
• Energy flux in the jet ~ energy flux through the horizon; possible disk contribution < 20%; ( see plot )
• The observer jet power agrees very well with the theoretical BZ power:
( see plot ) Unloading of black hole magnetosphere
accretion shock
stagnation point black hole “relieved” magnetic lines “exhaust”
magnetic “cushion”
accretion disk
Critical value
2 2) log10 (B /4πρc Unloading of black hole magnetosphere
accretion disk
magnetic “cushion” black hole “exhaust” “relieved” magnetic lines
accretion shock
stagnation point
2 2) log10 (B /4πρc IV. Discussion
We have shown how BZ-mechanism could drive GRB explosions. However, this requires both fast rotation and strong magnetic field of stellar cores of GRB progenitors. This is problematic for solitary stars:
• Evolutionary models of solitary massive stars show that even much weaker magnetic fields (Taylor-Spruit dynamo) result in too slow rotation – no collapsar disk (Heger et al. 2005)
• Low metalicity may save the collapsar model with neutrino mechanism (Woosley & Heger 2006) but BZ mechanism needs much stronger magnetic field. Solitary magnetic stars (Ap and WD) are slow rotators (with solid body rotation). Disk dynamo. A possible way out?
- turbulent magnetic field (scale ~ H, disk height) - turbulent velocity of α-disk
Application to the neutrino-cooled disk (Popham et al. 1999):
The inverse-cascade in disk corona (Tout & Pringle 1996) may give larger scales. For the scale ~ R
This seems a bit small for activation of BZ-mechanism! However,
• The accretion rate through the polar region may strongly decline several seconds after the collapse (Woosley & MacFadyen 1999), reducing the magnetic flux required for explosion;
• Neutrino heating (excluded in the simulations) may also help to reduce the required magnetic flux. Two-stage GRB explosions!? Binary progenitor. Another possible way out?
The fast rotation of highly magnetized star may arise
1. In a very close synchronized binary; 2. After spiral-in of compact star (NS or BH) during the common envelope phase (e.g. Zhang & Fryer 2001 ).
In both cases the hydrogen envelope of progenitor is dispersed leaving, as required, a bare helium core. V. Conclusions
BHs of collapsars can drive powerful GRB explosions via BZ-mechanism provided (i) BHs accumulate very large magnetic flux , ~ 1027 - 3x1028 Gcm2; (ii) BHs rotate rapidly, a~1. The condition on magnetic field strength can be lowered if the rate of accretion directly onto the black hole is reduced; late explosions (?), neutrino assistance (?) .
The magnetic magnetic field is either (i) generated in the collapsar disk or (ii) relic field of the progenitor star. The latter implies close binary models in order to explain the rapid rotation of progenitor.
unit length=4km t=0.4s
log B log B /Bp 10 return 10 φ Integral jet energy flux
event horizon return Weak dependence of κ on a
return