[PPT] - Short Gamma Ray Bursts The MRI print out on the launched jet Time PowerPoint Presentation

SLIDE 1

Kostas Sapountzis

Center for Theoretical Physics – Polish Academy of Science

Short Gamma Ray Bursts

The MRI print out on the launched jet – Time Variability Warsaw 2018

SLIDE 2

Outline

Short GRBs properties – The framework
MRI Characteristics
Initial configuration

Useful quantities and simulation characteristics

Results - Conclusion

SLIDE 3

Kouveliotou et al 1993, ApJ, L, 413, L101 Fishman G., Meegan C., 1995, AnRevAstronAstroph, 33, 415

SLIDE 4

The general SGRBs Framework

Intensively transient phenomena, prompt emission in γ-rays, peak few 100 KeV Isotropic Energy up to 1054 erg (less if collimated) Highly relativistic (compactness problem), γ>100. Low baryon loading, 10-5 MO Lei et al, 2013, ApJ, 765, 125 Two distinct phenomena (duration, hardness, location) Intense variability of the prompt radiation G. A. MacLachlan et al. 2013, MNRAS, 432, 857 Two candidates of SGRB: BH-NS NS-NS (LIGO,VIRGO,FERMI)

Abbott, B. P., et al. "Gravitational Waves and Gamma Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A." 2017, Phys. Rev. Lett. 119, 161101 Berger E., Focus on the Electromagnetic Counterpart of the Neutron Star Binary Merger GW170817

SLIDE 5

The zoo of the sGRB progenitors

Compact objects binaries outcome NS-NS

Heavy NS
supramassive NS Rigid Body Rotation (τ~τsd)

ρ d ⃗ v d t =−⃗ ∇ P−ρ⃗ ∇ Φ

⋅⃗ r ,∫ d

3 x

1 2 d

2

dt

2 (∫ ρ x 2d 3 x)−2∫ ρv 2d 3 x

=1 2 d

2I

dt

2 −2T

∫ P ⃗

∇⋅⃗ x d

3x

3∫ Pd

3 x=3 Π

P∝{ ρ

5/3

ρ

4/3

→

1 2 d I

2

dt

2 =2T +W +3 Π

T=1 2 I Ω

2∝ M R 2Ω 2∝ J 2

M R

2

∬d

3 x d 3 x' G ρ(⃗

x)⃗ x ⃗ ∇ ρ( ⃗ x ') |⃗ x− ⃗ x '| =1 2∬d

3 x d 3 x' G ρ(⃗

x) ρ( ⃗ x') (⃗ x− ⃗ x')(⃗ x− ⃗ x ') |⃗ x− ⃗ x'|

3

=1 2∫ d

3 x ρ(⃗

x)Φ(⃗ x)=W 1 2 d2(x2) dt

2 −v 2

Π=Μ⟨ P ρ ⟩∝{ Μ

5/3/R 2

Μ

4/3/ R

W ∝G M 2 R

τ sd= Ω −dΩ/dt = 3c

3 M

5 R

4 B 2Ω 2=1.2⋅10 3(

10 Km R )

4

(

10

15 G

B )

2

(

M 1.4 M ⨀)( T 1ms)

2

s dE dt =−2| ¨ pM|2 3c

3

pm=pm, 0sin Ωt E=1/2I Ω

2

dΩ dt =−2Ω3 pm0

2

3c

3 I

Ι=(2/5)M R

2

⃗ B= pm0 4 π r

3 (2cosθ ^

r+sinθ ^ θ)

SLIDE 6

The zoo of the sGRB progenitors

HMNS Differential Rotation

(winding of poloidal field → toroidal + Alfven waves → angular momentum redistribution)

t A= R u A ∼10

2(

10

12G

B0 )( 20 Km R )

1/2

(

M 3 M ⨀)

1/2

s

^ A:Ω= Ωc 1+ ^ r

2sin 2/ ^

A

2

The result depends on Members Initial State (mass, self-rotation, EOS, B) Details of the merging process ~7% (mass ejection, gravitational waves, ν)

0=−α 3 GM 2 R +κ3 J M R

2+ β3

M 4/3 R 0=−α 3/2 GM 2 R +κ3/2 J M R

2+ β3/2

M 5/3 R

2

Baumgarte T., Shapiro S., Shibata M., 2000, 528, L29

BH-NS Individual Object Magnetar

Break up velocity

SLIDE 7

Abbott et al 2017, ApJ, 848, L13

Δt=(1.74±0.5) s E p=158.1−33

+180 KeV

Eiso=(4.58±0.19 )10

46erg

GW170817 – GRB 170817A

90% credible interval

Granot et al 2017, ApJ, 850, L24

high spin prior restriction( χ<0.89) m1=(1.81±0.45) M ⨀ m2=(1.11±0.25) M ⨀ M tot=2.82−0.09

+0.47 M ⨀

low spin prior restriction( χ<0.05) m1=(1.48±0.12)M ⨀ m2=(1.26±0.10) M ⨀ M tot=2.74−0.01

+0.04 M ⨀

SLIDE 8

The MagnetoRotational Instability (MRI)

A purely HD Keplerian disk is stable, Rayleigh stability criterion
The MHD instability due to Balbus & Hawley, 1991 (Velikhov

1959, Chandrasekhar 1960) review.

Mechanical analogy
Characteristics

∂ ∂ R

2 (R 2Ω)>0⇒ stable

Ω∝ R−3/2

ω

4−ω 2[κ 2+2(⃗

k⋅⃗ v A)

2]+(⃗

k⋅⃗ v A)

2[(⃗

k⋅⃗ v A)

2+ d Ω 2

dlnR ]=0 ωmax=1 2| dΩ dln R| λmax=2π (v A

z )

4 Ω

√(4 Ω

2+κ 2)

| dΩ

2

dln R|

−1/2

SLIDE 9

The MagnetoRotational Instability (MRI)

The GR in Kerr Gammie C., 2004, ApJ, 614, 309

ω

4−ω 2[κ 2+2(⃗

k⋅⃗ v A)

2]+(⃗

k⋅⃗ v A)

2[(⃗

k⋅⃗ v A)

2− 3

r

3

D C ]=0

B=1+ a r

3/2

C=1−3 r +2ar

3/2

D=1−2 r + α

2

r

2

ωmax, τ

2

=− 1 16 1 r

3(

D C)

2

ωmax ,t

2

=− 9 16 Ω

2 D 2

C λmax=2π(v A

z )

Ω f (r ,α)

d

2x μ

dτ

2 =−Γ νλ μ dx ν

dτ d x

λ

dτ

circular Perturbations x

μ→x μ+ξ μ

Assume a spring: 1

2 γ

2hμν ξ μξ ν Into the Lagrangian: L=1

2 gμν ˙ x

μ ˙

x

ν−1

2 γ

2hμ ν ξ μξ ν

d

2x μ

dτ

2 =−∂σ Γ νλ μ u νu λ ξ σ−2 Γ νλ μ u ν ξ λ−γ 2hν μξ ν

d

2x μ

dτ

2 =−∂σ Γ νλ μ u νu λ ξ σ−2 Γ νλ μ u ν ξ λ

u

μ=d x μ

dτ ={B/√C,0,0 ,1/(r

3/2√C)}=u t{1,0,0, Ω}

Novikov – Thorne (1973)

ξ∝e−iωt

ω

4 (ω 2−v 2) (ω 2−k 2)=0

v

2= 1

r

3

1−4 a/r

3/2+3a 2/r 2

C k

2= 1

r

3

1−6/r+8 a/r

3/2−3 a 2/r 2

C

ξ∝e

−iωt

ω

4−ω 2(κ 2+2γ 2)+γ 2(γ 2− 3

r

3

D C )=0

http://www.inp.demokritos.gr/~sbonano/RGTC/

SLIDE 10

The MagnetoRotational Instability (MRI)

Real world (simulations) problems

Nonlinear coupling of the instability modes:

What’s the proper resolution to resolve MRI properly? Sanot et al. 2004 Hawley J. et al 2011

In 3D the situation is even more complicate
Empirical tests with different resolutions

Q MRI

θ

=2π v A

θ

Ω dx

θ

Q MRI

θ

>6−8

Q MRI

φ

=2π v A

φ

Ω dx

φ

Q MRI

θ

QMRI

φ

>200

(dx

φ/dx r )mid≤4 Sanot et al. 2004, ApJ, 605, 321 Hawley J. et al, 2011, 738, 84

SLIDE 11

Simulation Set Up

arXiv:1802.02786

The disrupted NS as a FM torus (hydro dynamic, steady state)
The magnetic field circular wire

K,E the complete elliptic functions, R at Pmax of FM torus, A0 controls the initial plasma-β

The ISM density has the lowest value
System in total not in balance

See Fishbone L., Moncrief V., 1976, ApJ, 207, 962 Aφ= A0

√r

2+ R 2+2r Rsinθ

(2−k

2) K(k 2)−2 E(k 2)

k

2

k=√ 4 R sinθ r

2+R 2+2r R sinθ

SLIDE 12

Simulation Set Up

Model Torus radii (rg) ISM density A0 TMRI QMRI rin rmax Harm units Harm units tg HD-Therm 50 60 1.6 · 10-9 10 630 9 HD-Mag 50 60 8.6 · 10-8 200 630 151 MD-Therm 20 25 1.0 · 10-8 1.6 174 9 MD-Mag 20 25 3.9 · 10-7 32 174 173 LD-Therm 10 12 4.0 · 10-8 0.32 61 13 LD-Mag 10 12 2.5 · 10 -7 3.1 61 122

Medium magn Weakly magn

SLIDE 13

Quantities Connection with SR jet theory

Τ ν

μ=

ρξu

μuν

⏟

(T m)ν

μ

+b

κbku μuν+ 1

2 b

κbk δ ν μ−b μb ν

⏟

(Tem)ν

μ

ξ : specific enthalpy σ = (T em)t

r

(T m)t

r

: magnetization parameter μ = T t

r

ρu

r :

total plasma energy

σ = Poynting (Thermal+inertial) energy flux μ = Total energy flux mass flux μ = γ ξ(1+σ)

Vlahakis N. & Konigl A. MNRAS, 2003, 596, 1080

Magnetic acceleration beyond finite resolution region SR

SLIDE 14

Simulation MD-Mag

SLIDE 15

Simulation MD-Mag

Gammie C., McKinney J., Toth G, 2003, ApJ, 589, 444. Noble S., Gammie C. , McKinney, J. C., Del Zanna L., 2006, ApJ, 641, 626.

HARM:

α= 0.9 Res: 1020 x 512 hslope: 0.3 Γ = 4/3 Gudinov, HLL, Shock capturing, fixed Kerr space

SLIDE 16

Snapshots MD-Mag

(40,200) (10,200) Point of MRI Resolution BZ Activity

Yuang H., Zhang F., Lehner L., Phys. Rev. D, 91, 124055

Ω F = Ftθ Fθ φ

Ω H = α 2 (1+√1−a

2)

SLIDE 17

Analysis of the results MD

SLIDE 18

Analysis of the results LD

SLIDE 19

Analysis of the results HD

MRI from shorter radii?

SLIDE 20

Conclusions

A well formed jet, intense variable, highly magnetized and of

lower density outflow is launched

MRI is accurately reflected on the time variability of the

ejected outflow using the μ ~ γmax quantity

The MRI print out is more intense at the inner part of the flow
The Blandford-Znajek BH mechanism is functioning effectively
The precise characteristic of the initial torus, a new era?

SLIDE 21

Tia n k y

u

f

r

y

u

r a t t e n t i

n

Further Comments – Questions: Kostas Sapountzis ( kostas@cft.edu.pl ) Agnieszka Janiuk (agnes@cft.edu.pl)