analysis of the neutrino-antineutrino annihilationgrb07/presentations/...salmonson & wilson...

Analysis of the Neutrino-Antineutrino Annihilation

near Accreting Stellar Black Holes

Reiner Birkl

29.3.2007

Max-Planck-Institut für Astrophysik

M.-A. Aloy, H.-Th. Janka, E. Müller [email protected]

Neutrino-Antineutrino Annihilation Reiner Birkl, Ringberg 1 / 20

Outline

1 Introduction

2 Theoretical fundamentals

3 Numerical implementation

4 Results

5 Conclusions

Outline

1 Introduction

4 Results

5 Conclusions

Outline

1 Introduction

4 Results

5 Conclusions

Outline

1 Introduction

4 Results

5 Conclusions

Outline

1 Introduction

4 Results

5 Conclusions

IntroductionTheoretical fundamentalsNumerical implementation

ResultsConclusions

MotivationPrevious approaches

Motivation

Focus:

Stellar size black hole surrounded by a hot accretion torus

Emission of neutrinos and antineutrinos

Annihilation of neutrinos and antineutrinos into e+e−γ-Plasma

Fireball expands along the symmetry axis

Motivation: A detailed parameter study!

Simplications:

Stationarity

Axisymmetry

(Anti)neutrinosphere

Isotropic black body for fermions (µ = 0)

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Motivation

Focus:

Simplications:

Stationarity

Axisymmetry

ResultsConclusions

Previous approaches

Idealized (anti)neutrinosphere

Salmonson & Wilson (1999)Asano & Fukuyama (2000, 2001)Miller et al. (2003)

(Anti)neutrinosphere based on accretion torus

Jaroszynski (1993, 1996)

Limitations:

Spheres and discs for idealized modelsOnly consideration of GR (≈ 2)and angular momentum (≈ 2)

ResultsConclusions

Previous approaches

Salmonson & Wilson (1999)

Asano & Fukuyama (2000, 2001)Miller et al. (2003)

Limitations:

ResultsConclusions

Previous approaches

Salmonson & Wilson (1999)Asano & Fukuyama (2000, 2001)

Miller et al. (2003)

Limitations:

ResultsConclusions

Previous approaches

Limitations:

ResultsConclusions

Previous approaches

Limitations:

ResultsConclusions

Previous approaches

Limitations:

ResultsConclusions

Previous approaches

Limitations:

ResultsConclusions

Previous approaches

Limitations:

Spheres and discs for idealized models

Only consideration of GR (≈ 2)and angular momentum (≈ 2)

ResultsConclusions

Previous approaches

Limitations:

ResultsConclusions

OverviewAnnihilation rate formula

Overview

Kerr metric

Emission point'

Isotropic black bodyfor fermions in local co-moving frame

f (~x , ~p) =1

1 + eE

kBT (~x)

Motion equation:

(raytracing)

dλ2+ Γα

dλ= 0

Boltzmann equation:

dλf (~x (λ) ,~p (λ)) = 0

Annihilation point'

Observer at restin global (r , θ, φ)coordinates

Annihilation ratein local observerframe (tetrad)

ResultsConclusions

Overview

Kerr metric

Emission point'

f (~x , ~p) =1

1 + eE

kBT (~x)

Motion equation:

(raytracing)

dλ2+ Γα

dλ= 0

Boltzmann equation:

dλf (~x (λ) ,~p (λ)) = 0

Annihilation point'

ResultsConclusions

Overview

Kerr metric

Emission point'

f (~x , ~p) =1

1 + eE

kBT (~x)

Motion equation:

(raytracing)

dλ2+ Γα

dλ= 0

Boltzmann equation:

dλf (~x (λ) ,~p (λ)) = 0

Annihilation point'

ResultsConclusions

Overview

Kerr metric

Emission point'

f (~x , ~p) =1

1 + eE

kBT (~x)

Motion equation:

(raytracing)

dλ2+ Γα

dλ= 0

Boltzmann equation:

dλf (~x (λ) ,~p (λ)) = 0

Annihilation point'

ResultsConclusions

Overview

Kerr metric

Emission point'

f (~x , ~p) =1

1 + eE

kBT (~x)

Motion equation:

(raytracing)

dλ2+ Γα

dλ= 0

→Boltzmann equation:

dλf (~x (λ) ,~p (λ)) = 0

Annihilation point'

ResultsConclusions

Overview

Kerr metric

Emission point'

f (~x , ~p) =1

1 + eE

kBT (~x)

Motion equation:

(raytracing)

dλ2+ Γα

dλ= 0

→Boltzmann equation:

dλf (~x (λ) ,~p (λ)) = 0

Annihilation point'

ResultsConclusions

Annihilation rate formula (Ruert et al. 1997)

Qαi ≡ Qα

i (~x) := deposition of 4-momentum per time and vol-ume at ~x , caused by the annihilation of neutrino-antineutrino pairs of avor i ∈ e, µ, τ

Qαi =

∫d3pd3p Aα

(~p, ~p

)fνi fνi'

6((C1 + C2)νi νi

I4πdΩ

I4πdΩ(1− cos∆θ)2

dE (pα + pα)E3E3fνi fνi

+C3,νi νim2

I4πdΩ

I4πdΩ (1− cos∆θ)

Weak interaction cross section σ0 = 1.76 · 10−48m2

(C1 + C2)νeνe≈ 2.34, (C1 + C2)νxνx

≈ 0.50

C3,νeνe ≈ 1.06, C3,νxνx ≈ −0.16

cos∆θ = ~n · ~n

ResultsConclusions

Qαi ≡ Qα

Qαi =

∫d3pd3p Aα

(~p, ~p

)fνi fνi

6((C1 + C2)νi νi

I4πdΩ

+C3,νi νim2

I4πdΩ

I4πdΩ (1− cos∆θ)

(C1 + C2)νeνe≈ 2.34, (C1 + C2)νxνx

≈ 0.50

C3,νeνe ≈ 1.06, C3,νxνx ≈ −0.16

cos∆θ = ~n · ~n

ResultsConclusions

Qαi ≡ Qα

Qαi =

∫d3pd3p Aα

(~p, ~p

)fνi fνi'

6((C1 + C2)νi νi

I4πdΩ

+C3,νi νim2

I4πdΩ

I4πdΩ(1− cos∆θ)

(C1 + C2)νeνe≈ 2.34, (C1 + C2)νxνx

≈ 0.50

C3,νeνe ≈ 1.06, C3,νxνx ≈ −0.16

cos∆θ = ~n · ~n

ResultsConclusions

Qαi ≡ Qα

Qαi =

∫d3pd3p Aα

(~p, ~p

)fνi fνi'

6((C1 + C2)νi νi

I4πdΩ

+C3,νi νim2

I4πdΩ

I4πdΩ(1− cos∆θ)

(C1 + C2)νeνe≈ 2.34, (C1 + C2)νxνx

≈ 0.50

C3,νeνe ≈ 1.06, C3,νxνx ≈ −0.16

cos∆θ = ~n · ~n

ResultsConclusions

MethodsNeutrino raytracing

Methods

Monte-Carlo direction integration

Adaptive stepsize fourth-order Runge-Kutta method for(anti)neutrino raytracing (constraint updating)

(Anti)neutrinosphere hitting algorithm:

General method: mesh renement algorithm

Thin disc method: calculation of equatorial plane intersection point

ResultsConclusions

Methods

ResultsConclusions

Methods

ResultsConclusions

Methods

ResultsConclusions

Methods

ResultsConclusions

Neutrino raytracing

ResultsConclusions

Preliminaries (••)Idealized models (F, GR + geometry, •, rotation)Equilibrium models (F•, rotation + torus mass, rotation inuence origin)

Annihilation rate in x-z-plane

Only annihilationof νe and νe isconsidered!

lg Q [erg s-1 cm-3]

24 25 26 27 28 29 30 31

ResultsConclusions

Annihilation rate in x-z-plane

Only annihilationof νe and νe isconsidered!

lg Q [erg s-1 cm-3]

24 25 26 27 28 29 30 31

ResultsConclusions

Energy deposition per time

Distant energy deposition rate:(Jaroszynski 1993)

Etot,∞νν :=

drdθdφ√−g Q

Eup,∞νν :=

drdθdφ√−g Q

Eciencies:

qtot,∞νν := E

tot,∞νν /L∞ν

qup,∞νν := E

up,∞νν /L∞ν

g = det (gαβ)

L∞ν := L

∞νe + L

∞νe

ResultsConclusions

Energy deposition per time

Distant energy deposition rate:(Jaroszynski 1993)

Etot,∞νν :=

drdθdφ√−g Q

Eup,∞νν :=

drdθdφ√−g Q

Eciencies:

qtot,∞νν := E

tot,∞νν /L∞ν

qup,∞νν := E

up,∞νν /L∞ν

g = det (gαβ)

L∞ν := L

∞νe + L

∞νe

ResultsConclusions

Idealized models - Parameter space

Black hole:

Mass M (0, 2M)

Dimensionless angularmomentum a (0, 1)

(Anti)neutrinosphere:

Geometry (disc, torus, sphere)

Size (radii, up to about 10M)

4-Velocity uα (Lagrangian angularmomentum l := u3

u0; 0, 5)

Temperature TC (5 · 1010K)

ResultsConclusions

Idealized models - Parameter space

Black hole:

Mass M (0, 2M)

Dimensionless angularmomentum a (0, 1)

Geometry (disc, torus, sphere)

Size (radii, up to about 10M)

4-Velocity uα (Lagrangian angularmomentum l := u3

u0; 0, 5)

Temperature TC (5 · 1010K)

ResultsConclusions

General relativistic and geometry eects - Plots

lg Q [erg s-1 cm-3]

24 25 26 27 28 29 30 31

ResultsConclusions

General relativistic and geometry eects - Plots

lg Q [erg s-1 cm-3]

24 25 26 27 28 29 30 31

ResultsConclusions

General relativistic and geometry eects - Table

Model M a Geometry Radii l TC L∞ν Etot,∞νν

Eup,∞νν

qtot,∞νν

qup,∞νν

name M M MGR 1010 K 1052 ergs

1049 ergs

10−3 10−3

D 2 0 disk 6 ↔ 7.7 0 5 0.33 0.36 0.23 1.1 0.70

DN 0 0 disk 6 ↔ 7.7 0 5 0.39 0.15 0.12 0.38 0.31

T 2 0 torus 6 ↔ 7.1; 1 0 5 0.30 0.44 0.11 1.5 0.37

TN 0 0 torus 6 ↔ 7.1; 1 0 5 0.39 0.16 0.095 0.41 0.24

S 2 0 sphere 3.4 0 5 0.16 0.083 0.083 0.52 0.52

SN 0 0 sphere 3.4 0 5 0.39 0.066 0.066 0.17 0.17

GR results:

Disc:≈ 2

Torus and sphere: ≈ 25%

Geometry results:

Energy deposition rate at innity:

Disc (highest)TorusSphere (lowest)

This result is true with and without

GR-eects

ResultsConclusions

Eup,∞νν

qtot,∞νν

qup,∞νν

1049 ergs

10−3 10−3

D 2 0 disk 6 ↔ 7.7 0 5 0.33 0.36 0.23 1.1 0.70

DN 0 0 disk 6 ↔ 7.7 0 5 0.39 0.15 0.12 0.38 0.31

T 2 0 torus 6 ↔ 7.1; 1 0 5 0.30 0.44 0.11 1.5 0.37

TN 0 0 torus 6 ↔ 7.1; 1 0 5 0.39 0.16 0.095 0.41 0.24

S 2 0 sphere 3.4 0 5 0.16 0.083 0.083 0.52 0.52

SN 0 0 sphere 3.4 0 5 0.39 0.066 0.066 0.17 0.17

GR results:

Disc:≈ 2

Geometry results:

GR-eects

ResultsConclusions

Eup,∞νν

qtot,∞νν

qup,∞νν

1049 ergs

10−3 10−3

D 2 0 disk 6 ↔ 7.7 0 5 0.33 0.36 0.23 1.1 0.70

DN 0 0 disk 6 ↔ 7.7 0 5 0.39 0.15 0.12 0.38 0.31

T 2 0 torus 6 ↔ 7.1; 1 0 5 0.30 0.44 0.11 1.5 0.37

TN 0 0 torus 6 ↔ 7.1; 1 0 5 0.39 0.16 0.095 0.41 0.24

S 2 0 sphere 3.4 0 5 0.16 0.083 0.083 0.52 0.52

SN 0 0 sphere 3.4 0 5 0.39 0.066 0.066 0.17 0.17

GR results:

Disc:≈ 2

Geometry results:

GR-eects

ResultsConclusions

Rotation

lg Q [erg s-1 cm-3]

24 25 26 27 28 29 30 31

Rotation results:

Black hole rotation (A) without inuence

Disc rotation (l) with nearly no inuence

ResultsConclusions

Rotation

lg Q [erg s-1 cm-3]

24 25 26 27 28 29 30 31

Rotation results:

Black hole rotation (A) without inuence

Disc rotation (l) with nearly no inuence

ResultsConclusions

Equilibrium models - Parameter space

The Equilibrium models are based on equilibrium accretion tori calculatedby Miguel A. Aloy (University of Valencia).

Black hole:

Mass M (3M)

Dimensionless angularmomentum a

(0.01 ≤ a ≤ 1)

Torus mass mtor

(0.05M ≤ mtor ≤ 0.5M)

Photon entropy sγ (1)

ResultsConclusions

Black hole:

Mass M (3M)

(0.01 ≤ a ≤ 1)

Torus mass mtor

(0.05M ≤ mtor ≤ 0.5M)

ResultsConclusions

Black hole:

Mass M (3M)

(0.01 ≤ a ≤ 1)

Torus mass mtor

(0.05M ≤ mtor ≤ 0.5M)

ResultsConclusions

Typical equilibrium model

lg Q [erg s-1 cm-3]

25 26 27 28 29 30 31 32 33 34

Antineutrinosphere inside ofneutrinosphere

Eup,∞νν = 3.7 · 1052 erg

-1.0 -0.5 0.0 0.5 1.0cos

E.01.05E.01.5E.8.05E.8.5

E[a][mtor/M]

Antineutrinosphere hotter thanneutrinosphere

Higher temperature near BH

ResultsConclusions

Typical equilibrium model

lg Q [erg s-1 cm-3]

25 26 27 28 29 30 31 32 33 34

Antineutrinosphere inside ofneutrinosphere

Eup,∞νν = 3.7 · 1052 erg

-1.0 -0.5 0.0 0.5 1.0cos

E.01.05E.01.5E.8.05E.8.5

E[a][mtor/M]

Antineutrinosphere hotter thanneutrinosphere

Higher temperature near BH

ResultsConclusions

Black hole rotation and torus massmtor = 0.1M

a = 0.8

Results:

Highest eciency is reached at a ≈ 0.6

Increasing mtor leads to a higher eciency

ResultsConclusions

Black hole rotation and torus massmtor = 0.1M a = 0.8

Results:

ResultsConclusions

Black hole rotation and torus massmtor = 0.1M a = 0.8

Results:

ResultsConclusions

Why does a larger a lead to a larger Q0?

mtor = 0.1M

IM1 parametersI M2 parameters xtot,∞νν xup,∞νν

aνν = 1, ator = 0.01 aνν = ator = 0.01 1.1 1.0

aνν = 0.01, ator = 1 aνν = ator = 0.01 1.9 1.6

Results:

Applying a = 0→ a = 1 leads to a ≈ 2 higher annihilation rate

This is not caused by modied ν-trajectories, but by the accretiontorus being nearer to the BH

ResultsConclusions

Why does a larger a lead to a larger Q0?

mtor = 0.1M

IM1 parametersI M2 parameters xtot,∞νν xup,∞νν

aνν = 1, ator = 0.01 aνν = ator = 0.01 1.1 1.0

aνν = 0.01, ator = 1 aνν = ator = 0.01 1.9 1.6

Results:

Applying a = 0→ a = 1 leads to a ≈ 2 higher annihilation rate

This is not caused by modied ν-trajectories, but by the accretiontorus being nearer to the BH

ResultsConclusions

Conclusions

Improvements in my approach:

Detailed parameter studyArbitrary (anti)neutrinosphere

Main results:

GR-eects of torus and sphere only 25%Energy deposition rate at innity: Disc (highest), torus, sphere(lowest)Factor ≈2 due to a = 0→ a = 1 comes exclusively from theaccretion torus being nearer to the BH

Future work:

Optically thin regionsLower boundaries for the energy available for GRBs (Jaroszynski)Time dependent numerical solutions

ResultsConclusions

Conclusions

Detailed parameter study

Arbitrary (anti)neutrinosphere

Main results:

Future work:

ResultsConclusions

Conclusions

Main results:

Future work:

ResultsConclusions

Conclusions

Main results:

Future work:

ResultsConclusions

Conclusions

Main results:

GR-eects of torus and sphere only 25%

Energy deposition rate at innity: Disc (highest), torus, sphere(lowest)Factor ≈2 due to a = 0→ a = 1 comes exclusively from theaccretion torus being nearer to the BH

Future work:

ResultsConclusions

Conclusions

Main results:

GR-eects of torus and sphere only 25%Energy deposition rate at innity: Disc (highest), torus, sphere(lowest)

Factor ≈2 due to a = 0→ a = 1 comes exclusively from theaccretion torus being nearer to the BH

Future work:

ResultsConclusions

Conclusions

Main results:

Future work:

ResultsConclusions

Conclusions

Main results:

Future work:

ResultsConclusions

Conclusions

Main results:

Future work:

Optically thin regions

Lower boundaries for the energy available for GRBs (Jaroszynski)Time dependent numerical solutions

ResultsConclusions

Conclusions

Main results:

Future work:

Optically thin regionsLower boundaries for the energy available for GRBs (Jaroszynski)

Time dependent numerical solutions

ResultsConclusions

Conclusions

Main results:

Future work:

References

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Miller, W. A., George, N. D., Kheyfets, A., & McGhee, J. M., 2003, O-Axis NeutrinoScattering in Gamma-Ray Burst Central Engines, ApJ 583, 833841

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analysis of the neutrino-antineutrino annihilationgrb07/presentations/...salmonson & wilson...

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