Maxim BarkovUniversity of Leeds, UK,

Space Research Institute, Russia

Serguei KomissarovUniversity of Leeds, UK

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 durationGRBsshort 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 E = 1054erg (wide distribution).

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

Total energy in the jet:

From afterglow observations Ejet = few 1051erg ~ E

High-velocity supernovae or hypernovae Es ~ 1052erg.

log

10F

x(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 shockAccretion 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

BB

Neutrino heating Magnetic braking

fireballMHD 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-Znajekmechanism:

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

a - spin parameter of the black hole (0 < a < 1), - the magnetic flux of black hole. =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 thesteady-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 finiteinertia 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;

= 1.2 = 1.6

The critical value of is indeed close to unity.

log10 log10

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 holeM=3M3

a=0.9

Solid body rotation.Uniform magnetization

R=4500 km= 4x1027-4x1028 G cm2

outer boundary, R= 104 km

free fallaccretion

(Bethe 1990)

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

• No explosion in models with < 0.3;

• Bipolar explosions in models with > 0.3;

movie. log10

movie: log10 p/pmand v; small scale

movie: log10 p/pm; large scale

Results

The critical value of 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 (B2/4c2)

accretion shock

Unloading of black hole magnetosphere

accretion disk

magnetic “cushion”

stagnation point

black hole “exhaust”

“relieved” magnetic lines

log10 (B2/4c2)

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 -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 B/Bplog10 B

unit length=4km t=0.4s

return

event horizon

Integral jet energy flux

return

return

Weak dependence of on a