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Blandford-Znajek mechanism and GRB jets
Serguei Komissarov
University of Leeds, UK
Maxim Barkov
University of Leeds, UK,
Space Research Institute, Russia
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?):
short duration
GRBs
long duration
GRBs
Inferred high speed:
Variability
compactness
too high opacity to
unless the Lorentz factor > 100
Total energy emitted in gamma rays:
Assuming isotropic emission up to Eg = 1054erg (wide distribution).
With beaming correction Eg ~ 1051erg (“standard energy reservoir”)
Total energy in the jet:
From afterglow observations Ejet = few 1051erg ~ Eg
High-velocity supernovae or hypernovae Es ~ 1052erg.
Swift input
“Canonical” X-ray afterglow lightcurve (Swift)
Zhang (2007)
log10Fx (0.3 – 10keV)
0
1
5
2
3
4
(
1
- occasionally observed)
2
3
4
log10(t/sec)
5
0 - prompt emission;
1 - steep decay phase;
2 - shallow decay phase;
3 - normal decay phase;
4 - post “jet break” phase;
5 - X-ray flares.
New question marks:
Jet breaks (?)
standard energy (?)
Restarting of central engine, wide Lorentz factor distribution, multi-component
ejecta, and many other ideas.
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.
Collapsing stellar
envelope
1% of the energy is sufficient
to explain hypernovae and
GRB-jets
Mechanisms of tapping the disk energy
Neutrino heating
fireball
photons
Magnetic braking
MHD wind
B
Eichler et al.(1989), Aloy et al.(2000)
MacFadyen & Woosley (1999)
Nagataki et al.(2006)
?
B
Blandford & Payne (1982)
Proga et al. (2003)
Fujimoto 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),
Y - the magnetic flux of black hole.
Y =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!
B
(Komissarov 2004, Koide 2004,
McKinney & Gammie, 2004 )
Lorentz factor
wave front
B
Komissarov (2004)
• 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
;
The solution develops steady-state
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,
r = 2rg= 2GM/c2 :
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):
k = 1.2
k = 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 k
is indeed close to unity.
It depends on a but
weakly.
log10r
log10r
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 (!);
Solid body rotation.
Uniform magnetization
R=4500 km
Y= 4x1027-4x1028 G cm2
black hole
M=3M3
a=0.9
v
B
v
v
v
v
B
free fall
accretion
(Bethe 1990)
outer boundary,
R= 104 km
• 2D axisymmetric
GRMHD;
• Kerr-Schild metric;
• Starts at 1s from
collapse onset.
• Lasts for < 1s
Results
• No explosion in models with k < 0.3;
movie.
log10r
• Bipolar explosions in models with k > 0.3;
movie: log10 p/pm and v; small scale
movie: log10 p/pm; large scale
The critical value of k 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
“exhaust”
“relieved” magnetic lines
magnetic “cushion”
accretion disk
Critical value
log10 (B2/4prc2)
Unloading of black hole magnetosphere
accretion disk
magnetic “cushion”
black hole
“exhaust”
“relieved” magnetic lines
accretion shock
stagnation point
log10 (B2/4prc2)
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 a-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
log10 B
return
log10 Bf/Bp
Integral jet energy flux
event horizon
return
Weak dependence of k on a
return