Serguei Komissarov University of Leeds, UK

Blandford-Znajek mechanism and relativistic jets

•  Physics of the Blandford-Znajek (BZ) mechanism •  Origin of relativistic astrophysical flows; •  BZ versus other mechanisms; •  BZ in GRBs; •  Open issues;

Plan of this Lecture

•  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.

(Penrose 1969)

Penrose Mechanism

The mass-energy of BH can be reduced by the amount

no rotation:

maximal rotation:

- event horizon radius

!!!

- spin parameter

- gravitational radius

Ergosphere

Ergoregion

Energy from Ergosphere

Vacuum solution for a Kerr black hole placed in a uniform magnetic field.

•  Wald (1974)

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!

Electromagnetic Mechanism

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”.

Ergosphere

The outflow carries away positive electromagnetic redshifted energy The inflow carries into the BH

negative electromagnetic energy at infinity

The flux of redsifted energy is outgoing everywhere.

Paired-wind structure of BZ-driven flows

(Thorne, Price & Macdonald, 1986)

Ωh

Imagine that the BH horizon is a 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)

Membrane Paradigm

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)

Validity of BZ mechanism

•  Maximum available energy (a =1):

•  Jet power (split-monopole):

B B

black hole jet

disk wind

Sufficient to explain hypernovae associated with long GRBs!

- 3 seconds to drive a hypernova

Ψ

- angular velocity of the BH; Ψ – magnetic flux accumulated by the BH

Ψ = 1027 Gauss cm2 – the highest observed magnetic flux of massive stars)

Resource and Power of the BZ mechanism

B B

•  Magnetic braking

disk wind

e.g. Eichler et al.(1989), Aloy et al.(2000), MacFadyen & Woosley (1999) ,

e.g. Blandford (1976), Blandford & Payne (1982) Li et al.(1982), Vlahakis & Konigl(2003,2004)

Disk binding energy (for a =1):

relativistic fireball

•  Neutrino heating (GRBs)

Competition: Disk or BH ?

Where is most of the magnetic flux? (Ghosh & Abramowicz, 1997; Livio et al.1999)

McKinney & Gammie (2004) Barkov & Komissarov (2008) Igumenshchev (2008,2009)

Magnetic power of the disk against the BZ power.

wind

corona

Wind speed:

In principle, Sun can drive relativistic wind, but

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 the event horizon. The mass loading is completely different (e.g. pair production) and much weaker.

Can disk magnetic winds be ultra-relativistic? (The issue of mass-loading)

Collapsing stellar envelope

Accretion shock Accretion disk

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.

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?

(Komissarov & Barkov 2009) Activation of the BZ mechanism (GRBs).

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

Mechanisms in play

Numerical simulations confirm this condition

•  Kerr black hole, a = 0.9; •  Polytropic EOS; •  Initially cold plasma •  Monopole magnetic field; •  GRMHD, 2D, axisymmetry;

κ = 1.2 κ = 1.6

The critical value of κ is indeed close to unity.

log10ρ log10ρ

It depends on a but weakly.

Numerical Simulations

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

magnetic braking of stellar cores magnetic braking of progenitors

delayed formation of accretion disk no disk, no rapidly rotating BH

(Heger et al. 2005)

Strong Magnetic Fields

•  Magnetic star in a close synchronized binary;

Ω

Ω

e.g. Lee et al. (2002), Izzard et al. (2004), Barkov & Komissarov (2009)

B

Binary progenitor. A way out?

•  Merger of magnetic star with a compact companion (NS,BH);

(e.g. Fryer & Woosley 1998. Zhang & Fryer 2001 ).

Ω

B B

Compact Merger ?

•  The plateau phase and flares in the light curves of Swift afterglows – the smoking gun of magnetic mechanism?

Zhang (2007)

cannot be neutrino mechanism due to the drop of accretion rate

Continuous energy injection

long-acting central engine (alternative explanations)

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).

Observations Help

•  Collapsed halos of dark matter of ~106 M3 at z ~20; •  The primordial metal-free gas fragments into clamps of ~103 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 )

Formation of first stars in the early Universe Bromm et al.(2002), Gao et al.(2007), Ohkubo et al.(2009), Natarajan et al. (2009)

Super-collapsars of the early Universe

•  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).

Numerical simulations of Fryer et al. (2001); Mstar = 300M3

Collapse of massive rotating Pop III stars (supercollapsars)

•  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;

Conclusions

Maxim Barkov University of Leeds, UK,

Space Research Institute, Russia

Serguei Komissarov University of Leeds, UK

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Blandford-Znajek mechanism and GRB jets

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

Bimodal distribution (two types of GRBs?):

long duration GRBs short duration

GRBs

I. Gamma Ray Bursts

Total energy emitted in gamma rays:

Inferred high speed: Variability compactness too high opacity to unless the Lorentz factor > 100

Assuming isotropic emission up to Eγ = 1054erg (wide distribution).

With beaming correction Eγ ~ 1051erg (“standard energy reservoir”)

Total energy in the jet: From afterglow observations Ejet = few 1051erg ~ Eγ

High-velocity supernovae or hypernovae Es ~ 1052erg.

log 1

0Fx

(0.3

–10

keV)

log10(t/sec)2 3 4 51

0

1

2 3

4

“Canonical” X-ray afterglow lightcurve (Swift)

Zhang (2007)

5

( - occasionally observed)

0 - prompt emission; 1 - steep decay phase; 2 - shallow decay phase; 3 - normal decay phase; 4 - post “jet break” phase; 5 - X-ray flares.

Restarting of central engine, wide Lorentz factor distribution, multi-component ejecta, and many other ideas.

New question marks: Jet breaks (?) standard energy (?)

Swift input

Collapsing stellar envelope

Accretion shock Accretion disk 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 to explain hypernovae and GRB-jets

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).

Mechanisms of tapping the disk energy

B B

Neutrino heating Magnetic braking

fireball MHD wind

Eichler et al.(1989), Aloy et al.(2000) MacFadyen & Woosley (1999) Nagataki et al.(2006) ?

Blandford & Payne (1982) Proga et al. (2003) Fujimoto et al.(2006) Mizuno et al.(2004)

photons

Black hole rotational energy (a =1):

Power of the Blandford-Znajek mechanism:

Blandford & Znajek (1977), Meszaros & Rees (1997)

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).

Tapping the rotational energy of black hole

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

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 )

When free charges are introduced this field can sustain electric currents along the field lines penetrating the black hole ergosphere.

Can we capture the BZ-effect with modern numerical RMHD schemes? Yes!

•  GRMHD, 2D, axisymmetry; •  Kerr-Schild coordinates, a=0.9; •  Inner boundary is inside the event horizon; •  Outer free-flow boundary is far away; •  Initially non-rotating monopole field and zero plasma speed in FIDOs frame; •  Magnetically-dominated regime ;

wave front

Lorentz factor

B

B

(Komissarov 2004, Koide 2004, McKinney & Gammie, 2004 )

Komissarov (2004) The solution develops steady-state paired-wind behind the expanding spherical wave front.

- 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.

Numerical solution versus analytical

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

Apply at the ergosphere, r = 2rg= 2GM/c2 :

Thus, the energy density of magnetic field must exceed that of matter for the BZ-mechanism to be activated!

III. Activation of BZ mechanism

(Newtonian results)

In terms of , the integral mass accretion rate, and , the magnetic flux threading the black hole hemisphere, this condition reads

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

[ in fact, we anticipate ]

“Test-this-idea” simulations (in preparation):

•  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;

κ = 1.2 κ = 1.6

The critical value of κ is indeed close to unity.

log10ρ log10ρ

It depends on a but weakly.

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:

IV. Collapsar GRMHD simulations Based on Barkov & Komissarov (2008) and more recent results

•  neutrino cooling (Thompson et al.,2001); •  realistic equation of state, (HELM, Timmes & Swesty, 2000); •  dissociation of nuclei (Ardeljan et al., 2005); •  no neutrino heating (!);

v

B

v

B

v

v

v

black hole M=3M3 a=0.9

Solid body rotation. Uniform magnetization

R=4500 km Ψ= 4x1027-4x1028 G cm2

outer boundary, R= 104 km

free fall accretion

(Bethe 1990)

•  2D axisymmetric GRMHD; •  Kerr-Schild metric; •  Starts at 1s from collapse onset. •  Lasts for < 1s

•  No explosion in models with κ < 0.3;

•  Bipolar explosions in models with κ > 0.3;

movie. log10ρ

movie: log10 p/pm and v; small scale

movie: log10 p/pm; large scale

Results

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 )

Critical value

Unloading of black hole magnetosphere

accretion disk

magnetic “cushion”

stagnation point

black hole “exhaust”

“relieved” magnetic lines

log10 (B2/4πρc2)

accretion shock

Unloading of black hole magnetosphere

accretion disk

magnetic “cushion”

stagnation point

black hole “exhaust”

“relieved” magnetic lines

log10 (B2/4πρc2)

accretion shock

IV. Discussion

•  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).

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:

- 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!

Disk dynamo. A possible way out?

•  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!?

However,

Binary progenitor. Another possible way out?

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.

The fast rotation of highly magnetized star may arise

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.

log10 Bφ/Bp log10 B

unit length=4km t=0.4s

return

event horizon

Integral jet energy flux

return

return

Weak dependence of κ on a