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Black Hole Accretion Disk Models

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).

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Black Hole Accretion Disk Models

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  1. Black Hole Accretion Disk Models Lecture 5: Driving explosions from disks around black holes – C. Fryer (UA/LANL)

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

  3. Energy Transport • Radiation (photons, neutrinos)

  4. Energy Transport • Radiation (photons, neutrinos) • Magnetic Fields

  5. 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

  6. Black Hole Accretion Disk Models: Compact Mergers And collapse of Stars.

  7. 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

  8. 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

  9. 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.

  10. Accretion Disks from Gamma Ray Burst Models

  11. 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

  12. 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.

  13. 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

  14. 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

  15. 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.

  16. 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

  17. 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-)

  18. 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

  19. 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).

  20. 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

  21. 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

  22. Characteristics of Disk: dM/dt = 1Msun s-1, a=0, a=0.1, MBH=3Msun Popham, Woosley, & Fryer 1999

  23. dM/dt=0.01,0.1,1,10Msuns-1 a=0.,a=0.1,MBH=3Msun a=0.,0.5,0.95; dM/dt= 0.1Msuns-1,a=0.1,MBH=3Msun

  24. a=0.1,0.03,0.01; a=0;MBH=3Msun dM/dt=0.01Msuns-1 dM/dt=0.1Msuns-1

  25. Evolution of Black Hole Spin as a Function of Total accreted Mass for a Thin disk.

  26. Models With Modified Potentials Seem to Get the Same Rough Result! MacFadyen & Woosley

  27. 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)

  28. Di Matteo, Perna & Narayan 2002

  29. 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

  30. 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

  31. Neutrino Driven Jets – Energy Deposition ksc=(5a2+1)/24 s0<e2n>/(mec2)2r/mu (Yn+Yp), kab=(3a2+1)/4 s0<e2n>/(mec2)2r/mu (Yn,Yp), 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

  32. Neutrino Annihilation e+ n e- n L+nn(nini)= A1SDLkni/d2kSDLk’ni/d2k’ [<e>ni+<e>ni](1-cosq)2 +A2SDLkni/d2kSDLk’ni/d2k’ [<e>ni+<e>ni]/[<e>ni<e>ni](1-cosq)

  33. Neutrino Annihilation dk dk’ q Lni Lni

  34. Energy Deposition from Neutrino Annihilation

  35. 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.

  36. 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)]

  37. 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!

  38. Non-Rotating Stars Rauscher et al. 2002

  39. Rotating vs. non-Rotating

  40. 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)

  41. 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.

  42. 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

  43. 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)

  44. 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

  45. 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)

  46. 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

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