Black Hole Accretion Disk Models

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Black Hole Accretion Disk Models

• Lecture 5 Driving explosions from disks around
  black holes C. Fryer (UA/LANL)

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Energy Sources

• Two Main Energy Sources
• In Astrophysics
• Gravitational Potential
• Energy
• II) Nuclear Energy

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Energy Transport

• Radiation (photons, neutrinos)

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Energy Transport

• Radiation (photons, neutrinos)
• Magnetic Fields

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Variabilities of GRBs limits models to compact
objects (NS, BH)
Variability size scale/speed of
light Again, Neutron Stars and Black Holes
likely Candidates (either in an Accretion disk
or on the NS surface). 2 p 10km/cs .6 ms cs
1010cm/s
NS, BH
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Black Hole Accretion Disk Models Compact
Mergers And collapse of Stars.
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Neutron Star Models

• Magnetar Models ruled out because most
  supernovae do NOT produce GRBs.
• Supranova models ruled out for long-duration
  bursts by duration times

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Black Hole Accretion Disks

• Structure of a Relativistic Disk
• Neutrino Driven Explosions from a Black Hole
  Accretion Disk
• Magnetic Field Driven Explosions from a Black
  Hole Accretion Disk

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Black Hole Accretion Disk Models Material
accreting Onto black hole Through disk
Releases potential Energy. If this Energy can
be Harnessed to Drive a relativistic Jet, a GRB
is formed.
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Accretion Disks from Gamma Ray Burst Models
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Relativistic Disks

• Mass ( Particle)Conservation Continuity
  Equation
• Momentum Equation 1) Radial Velocity, 2)
  angular momentum
• Energy Equation

Gammie Popham 1998, Popham Gammie 1998 GRBs
Popham, Woosley, Fryer 1999 Di
Matteo, Perna, Narayan 2002
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Accretion Disks in General Relativity
Boyer-Lindquist Coordinates Kerr Metric
ds2 -1-2/(rm)dt2 - 4asin2q/(rm)dtdf
m/(1-2/ra2/r2) dr2 r2m dq2 r2
sin2q1a2/r22a2sin2q/(r3m)df2
Where m1a2cos2q/r2, with scalings set
GMc1 For the gravitational constant, black
hole mass, speed of light respectively.
aJc/GM2 and J is the angular momentum of the
black hole.
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Particle Number Conservation GRs version of
mass conservation
Particle-number conservation
(rum)m 0 where r is the rest-mass density
g-1/2(g1/2um),m
r-2(r2rur),r m-1sin-1q(msinquq),q 0
Where g is Det(gmn) r4sin2qm2
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Particle Number Conservation - Continued
Average over the disk scale height (vertical
averaging) and define Hq as the
characteristic angular scale of the flow about
the equator (assume this scale is the same for
all flow variables). Integral of dqdf g1/2f
4pHq f(qp/2) where f is the flow variable
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Particle Number Conservation - Continued
(rum)m r-2(r2rur),r 0
Using angle-average
(4pHqr2rur),r 0
Integrating once in radius
4pHqr2rur -dM/dt where dM/dt is the rest-mass
accretion rate
4pHqr2rV(1-2/ra2/r2)1/2/(1-V2)1/2 -dM/dt
Where V is the radial velocity.
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Pressure Scale Height
Hq2 p/(rhnz2) where nz2 l2-a2(E2-1)/r4 l
is the specific angular momentum, E-ut, the
energy at infinity and h (rpu)/r is the
relativistic enthalpy.
Assumes uq and uq,q are small Abramowicz, Lanza
Percival 1997
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Radial Momentum Conservation in General Relativity
hrm(Tmn)n 0 where hmngmnumun is the
projection tensor and Tmn is the stress energy
tensor.
V/(1-V2)dV/drfr-1/(rh) dp/dr where h is defined
by the sound speed cs2Gp/(hr), fr -r-2Agf2/D
(1-W/W)(1-W/W-)
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Radial Momentum Conservation Continued
fr -r-2Agf2/D (1-W/W)(1-W/W-)
gf2(1-bf2)-1/2 A1a2/r22a2/r3
D1-2/ra2/r2 Wuf/utwlD1/2/r2A3/2g, and
W,- ,-(r3/2,-a)-1
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Angular Momentum Conservation
dM/dt l h 4pHqrtfr dM/dt j where dM/dt j is
the inward flux of the angular momentum (j is an
eigenvalue of the problem and is solved
numerically) and tfr is the viscous stress
tensor (see 4.2 of Gammie Popham 1998 for
details).
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Energy Conservation in General Relativity
um(Tmn)n 0
(Ellis 1971)
urdu/dr-ur(up)/r dr/dr F - L
u(r,T) rTg(T)
VD/(1-V2)1/2(du/dT dT/dr p/rdr/dr)F-L
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Energy Conservation in General Relativity
VD/(1-V2)1/2(du/dT dT/dr p/rdr/dr)F-L
F is the dissipation function L is the cooling
function L5x1033(T/1011K)9 ergs cm-3 s-1
9.0x1033(r/1010g cm-3)(T/1011K)6 Xnuc ergs
cm-3 s-1 Xnuc is the fraction of nucleons
e/e- annihilation
electron capture
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Characteristics of Disk dM/dt 1Msun s-1, a0,
a0.1, MBH3Msun
Popham, Woosley, Fryer 1999
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dM/dt0.01,0.1,1,10Msuns-1 a0.,a0.1,MBH3Msun
a0.,0.5,0.95 dM/dt 0.1Msuns-1,a0.1,MBH3Msun
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a0.1,0.03,0.01 a0MBH3Msun
dM/dt0.01Msuns-1
dM/dt0.1Msuns-1
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Evolution of Black Hole Spin as a Function of
Total accreted Mass for a Thin disk.
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Models With Modified Potentials Seem to Get the
Same Rough Result! MacFadyen Woosley
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Neutrinos Not Optically thin!
Accretion rates 10,1 and 0.1 solar masses per
second. Thick lines (di Matteo, Perna Narayan
2002), Thin Lines (Popham et al. 1999)
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Di Matteo, Perna Narayan 2002
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Neutrino Driven Jets Neutrinos from accretion
disk deposit their energy above the disk. This
deposition can drive an explosion.
Densities above 1010-1011 g cm-3 Temperatures
above a few MeV
Disk Cools via Neutrino Emission
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Neutrino Driven Jets
e,e- pair plasma
Neutrino Annihilation
Scattering
Absorption
Densities above 1010-1011 g cm-3 Temperatures
above a few MeV
Disk Cools via Neutrino Emission
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Neutrino Driven Jets Energy Deposition

• ksc(5a21)/24 s0lte2ngt/(mec2)2 r/mu (YnYp),
• kab(3a21)/4 s0lte2ngt/(mec2)2 r/mu (Yn,Yp),
• where a-1.26, mu 1.66x10-24g is the atomic mass
  unit, mec20.511 MeV is the electron rest-mass
  energy, s01.76x10-44 cm2, en is the neutrino
  energy, r is the density above the rotation axis
  and Yn and Yp are the number fractions of free
  neutrons and protons respectively (0.5 each).
• ktotal 1.5x10-17 r (kBTne/4MeV)2 cm-1

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Neutrino Annihilation
e
n
e-
n
Lnn(nini) A1SDLkni/d2k SDLkni/d2k
ltegtniltegtni(1-cosq)2
A2SDLkni/d2k SDLkni/d2k
ltegtniltegtni/ltegtniltegtni(1-cosq)
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Neutrino Annihilation
dk
dk
q
Lni
Lni
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Energy Deposition from Neutrino Annihilation
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From energy deposition to Jet Neutrino
Acceleration

• aabs/scat (ktdr/mshell)Ln/c 1.5x10-17
  (kBoltzTn/4 MeV)2 Ln/(4pr2c), where mshell
  r4pr2dr is the mass of a shell of radius r and
  thickness dr, Ln is the neutrino luminosity and
  kt is the total absorptionscattering
  cross-section for neutrinos.
• aannihilation Lnn(r) dr/c 1/mshell
  Lnn(r)/(pcr2r), where Lnn is the energy deposited
  at a given radius r by neutrino annihilation.

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From energy deposition to Jet Jet is launched
when acceleration from neutrinos overcomes
gravitational acceleration.

• aabs/scat aannihilation gt -agrav

1.5x10-17 (kBoltzTn/4 MeV)2 Ln/(4pr2c)
Lnn(r)/(pcr2r) gt GMBH/r2
There exists a critical density in the evacuated
polar region, Below which an explosion is
launched
rcrit 4Lnn(r)/-1.5x10-17 (kBoltzTn/4 MeV)2Ln
(4pcGMBH)
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This critical density corresponds to a critical
infall rate along the rotation axis

• dMcrit/dt 0.536pr2rcritvff for a 30o cone where
  we assume the infalling material is moving at
  free-fall velocities vff(2GMBH/r)1/2.
• The free-fall accretion rate as a function of
  mass for stellar models. We can determine the
  mass and time after collapse that the jet is
  launched!

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Non-Rotating Stars
Rauscher et al. 2002
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Rotating vs. non-Rotating
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Neutrino Summary

• Critical Densities for most-likely accretion
  disks 104-108 g/cm3
• For Collapsars type I, this corresponds to black
  hole masses of 10-25 Msun and delays between
  collapse and jet of 30-300s. Does the
  neutrino-driven Collapsar type I model work?
• Alternatives magnetic fields, Collapsar type II
  (MacFadyen Woosley 1999)

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Magnetic Field Driven Jets

• And then the theorist raises his magic. I mean
  magnetic wand and viola, there are jets -
  Shri Kulkarni
• Lots of Mechanisms proposed, but most boil down
  to a reference to the still unsolved mechanism
  behind the jet mechanism for Active Galactic
  Nuclei (Generally the Blandford-Znajek
  Mechanism).
• We are extrapolating from a non-working model
  dangerous at best.

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Magnetic Field Mechanism Sources of Energy

• Source of Magnetic Field Dynamo in accretion
  disk.
• Source of Jet Energy -
  I) Accretion Disk
  II) Black Hole Spin

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Magnetic Dynamos

• Duncan Thompson (1993) High Rossby Number
  Dynamo (convection driven) Bsat(4prvconvective2
  )1/2
• Akiyama et al. (2003) Shear-driven Dynamo
  Bsat2(4prr2W2(dlnW/dlnr)2
• Popham et al. (1999) Disk Dynamo
  Bsat2h(4prvtot2)

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Schematic Cross-Section of a black hole and
magnetosphere
The poloidal field is shown in solid lines,
typical particle velocities are shown with
arrows. In the magnetosphere, spark gaps (SG)
form
that create electron/ positron pairs.
Blandford Znajek 1977
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Electromagnetic structure of force-free
magnetosphere with (a) radial and (b)
para- boloidal magnetic fields. For paraboloidal
fields, the Energy appears to be Focused alonge
the rotation Axis. The overall efficiency of
Electromagnetic energy Extraction from a disk
Around a black hole is Difficult to calculate
with Any precision
Blandford Znajek (1977)
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Magnetic Jet Power

• Blandford-Znajek L3x1052 a2 dM/dt erg/s with
  B2x1015(L/1051 erg/s)1/2 (MBHa)-1
• Popham et al. 1999 (Based on BZ)
  L1050a2(B/1015G)2 erg/s, Bhrv2 where h1
• Katz 1997 (Parker Instability)
  L1051(B/1013G)(W/104s-1)5(h/106cm) (r/1013g
  cm-3)-1/2(r/106cm)6 erg/s

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Magnetic Field Summary

• Magnetically driven jets could possibly produce
  much more energy than neutrino annihilation
  (easily enough for GRBs). If it works for AGN,
  it must work for GRBs.
• Most estimates extrapolate from an already faulty
  AGN jet model. No physics calculation or
  derivation has yet to be made.

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Summary
Black Hole Accretion Disk Structure can be
determined With simple post-newtonian Corrections.
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Summary
Black Hole Accretion Disk Structure can be
determined With simple post-newtonian Corrections.
Neutrinos can be solved. The energetics may be
low.
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Summary
Black Hole Accretion Disk Structure can be
determined With simple post-newtonian Corrections.
Neutrinos can be solved. The energetics may be
low. Magnetic Fields can get Any answer one
likes.
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Tomorrow Compact Binary Mergers

• Binary Terminology
• NS/NS mergers formation scenarios and
  simulations
• BH/NS mergers
• BH/WD mergers
• Rates and Distribution comparison to
  observations