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### Black Hole Accretion Disk Models

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

Energy Sources

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

Energy Transport

- Radiation (photons, neutrinos)

Energy Transport

- Radiation (photons, neutrinos)
- Magnetic Fields

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

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

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

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.

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

Accretion Disks in General Relativity

Boyer-Lindquist Coordinates – Kerr Metric:

ds2 = -[1-2/(rm)]dt2 - 4asin2q/(rm)dtdf +

m/(1-2/r+a2/r2) dr2 + r2m dq2 +

r2 sin2q[1+a2/r2+2a2sin2q/(r3m)]df2

Where m=1+a2cos2q/r2, with scalings set G=M=c=1

For the gravitational constant, black hole mass,

speed of light respectively. a=Jc/GM2 and J is

the angular momentum of the black hole.

Particle Number Conservation: GR’s 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

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(q=p/2) where

f is the flow variable

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/r+a2/r2)1/2/(1-V2)1/2 = -dM/dt

Where V is the radial velocity.

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 = (r+p+u)/r is the relativistic enthalpy.

Assumes uq and uq,q are small –

Abramowicz, Lanza & Percival 1997

Radial Momentum Conservation in General Relativity

hrm(Tmn);n =0 where hmn=gmn+umun is the

projection tensor and Tmn is the stress

energy tensor.

V/(1-V2)dV/dr=fr-1/(rh) dp/dr where h is defined

by the sound speed cs2=Gp/(hr),

fr = -r-2Agf2/D (1-W/W+)(1-W/W-)

Radial Momentum Conservation Continued

fr = -r-2Agf2/D (1-W/W+)(1-W/W-)

gf2=(1-bf2)-1/2

A=1+a2/r2+2a2/r3

D=1-2/r+a2/r2

W=uf/ut=w+lD1/2/r2A3/2g, and

W+,- = +,-(r3/2+,-a)-1

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).

Energy Conservation in General Relativity

um(Tmn);n =0

(Ellis 1971)

urdu/dr-ur(u+p)/r dr/dr = F - L

u(r,T) = rTg(T)

V[D/(1-V2)]1/2(du/dT dT/dr –p/rdr/dr)=F-L

Energy Conservation in General Relativity

V[D/(1-V2)]1/2(du/dT dT/dr –p/rdr/dr)=F-L

F is the dissipation function

L is the cooling function

L~5x1033(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

Characteristics of Disk: dM/dt = 1Msun s-1, a=0, a=0.1, MBH=3Msun

Popham, Woosley, & Fryer 1999

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)

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

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

Neutrino Driven Jets – Energy Deposition

ksc=(5a2+1)/24 s0

kab=(3a2+1)/4 s0

where a=-1.26, mu= 1.66x10-24g is the atomic mass unit, mec2=0.511 MeV is the electron rest-mass energy, s0=1.76x10-44 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 ~ 1.5x10-17r (kBTne/4MeV)2 cm-1

Neutrino Annihilation

e+

n

e-

n

L+nn(nini)=

A1SDLkni/d2kSDLk’ni/d2k’ [

+A2SDLkni/d2kSDLk’ni/d2k’

[

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 absorption+scattering 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.

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

aabs/scat + aannihilation > -agrav

1.5x10-17 (kBoltzTn/4 MeV)2 Ln/(4pr2c)+Lnn(r)/(pcr2r) > 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)]

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!

Non-Rotating Stars

Rauscher et al. 2002

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)

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.

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

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 – Bsat2~h(4prvtot2)

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

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)

Magnetic Jet Power

- Blandford-Znajek: L~3x1052 a2 dM/dt erg/s with B~2x1015(L/1051 erg/s)1/2 (MBHa)-1
- Popham et al. 1999 (Based on BZ): L~1050a2(B/1015G)2 erg/s, B~hrv2 where h~1%
- Katz 1997 (Parker Instability): L~1051(B/1013G)(W/104s-1)5(h/106cm) (r/1013g cm-3)-1/2(r/106cm)6 erg/s

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.

Summary

Black Hole Accretion Disk

Structure can be determined

With simple post-newtonian

Corrections.

Neutrinos can be solved.

The energetics may be low.

Summary

Black Hole Accretion Disk

Structure can be determined

With simple post-newtonian

Corrections.

Neutrinos can be solved.

The energetics may below.

Magnetic Fields can get

Any answer one likes.

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

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