1. Opening angles of collapsar jets
θj~CxΓ0-1 (C~1/5)
Akira MIZUTA
Computational Astrophysics Lab. RIKEN
Kunihito IOKA
KEK
SN&GRB YITP Kyoto Univ
Reference: MA, Nagataki, Aoi, ApJ, 732, 26 (2011)
MA&Ioka ApJ, 777, 162 (2013)(arXiv: ) 1

2. GRB jet : relativistic collimated outflow
105cm
1010cm cm cm
Numerical Hydro Jet propagation
(jet structures, opening angle,
effect of progenitor structure,
photospheric emission etc., i.e.,
Aloy+00, Zhang, MacFadyen+03,04
Mizuta+06,09,11,13
Morsony+07,10, Lazzati+09,13
Nagakura+11,12, Suzuki+13 ….
GRBs are emitted form highly relativistic outflows (Γ>=100).
The outflow is believed to be well confined, i.e., jet.
At least some long GRBs originate from SN (collapsar model Woosley 93, MacFadyen Woosley99).
MIZUTA SN-GRB 2

3. JET: Lj=5.e50 erg/s [0:100s] θ0=10 degrees
AM, Nagataki + (2011)
JET: Lj=5.e50 erg/s [0:100s] θ0=10 degrees
Γ0=5, ε0/c2=80(h0~106) Γ_max~hΓ(>500)
Progenitor: 14Msun ,radius=4x1010cm
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4. Γ Γ Γ Photospheric emission (dissipated photosphere) log10(ρ) log10(ρ)
θ0=10degrees
100s injection
θ0=10degrees
30s injection
log10(ρ)
log10(ρ) Γ Γ
Bullet
free expandion Γ
Only dissipated region Γ
OA10
[0:100s]
OA10
[0:30s] Γ
log10(ρ) Γ Γ
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MA , inprep 4

5. Motivation
Numerical simulations show the opening angles of the jet are smaller than the initial opening angle at least for first a few tens of seconds. Complex structure appears.
The structure causes variety of photospheric light curves for different viewing angle observers.
We examine what determines opening angles from collapsar jets by numerical simulation and analytical way.
“Opening angle” measured at a certain radius θ0
Γ final>=2
Γ final>=200
Morsony +07
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6. What determines opening angel of jets ?
Naive expectation
Our model
relativistic beaming
effect
stellar envelopes
Γ0 ~θ0 -1
Γ0 ~θ0 -1
Γ0 ~θ0 -1
stellar envelopes
Before shock breakout, the jet is influenced by the interaction
with stellar envelopes.
Before shock breakout , the jet is not well accelerated (Γ~O(10)).
Numerical simulations by Zhang et al.,Mizuta et al., Morsony et al.
Lazzati et al., Nagakura et al.
Analytical work by 　Bromberg et al. (2011)
How about the jet evolution after jet breakout ?
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7. ρ p Γ High resolution hydro. simulations(2D, axisymmetric)
High resolution grid points are devoted
at least in the jet and a part of cocoon.
Δzmin=Δr min=107cm or =5x106cm
Ej=5x1050erg/s, r=8x107cm
η=h0Γ0=533, Γ0=2.5, 5, 10
Progenitor : Woosley & Heger(2006)
M~14Msun, R*=4x1010cm
Code: MA+04,06,
Riemann solver (Marquina),
2nd order accuracy in
space(MUSCL & time(TVD Runge Kutta)
progenitor surface p Γ
MIZUTA SN-GRB
Elongated:Aspect ratio is not correct 7

8. Why is high resolution necessary ?
Collimation shock
thin layer
Γ0=20
Cocoon gas
Γ0=10
Γ0=5
θ0=1/Γ0
Bromberg, Levinson (2009)
Initial jet size should be small
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9. Why is high resolution necessary ?
hΓ (=const along stream line, steady state :Bernoulli's principle)
hΓ is conserved to the reverse shock
Along jet axis
baryon loading
due to numerical diffusion
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10. Γ0=5 mass density Lorentz factor
So many internal oblique shocks have a potential for (Ito's talk on Tuesday day)
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11. Cocoon confinement (Before jet breakout)
pressure Γ0
θ0~1/Γ0
Γ0=5
Γ0=2.5
jet injection
After collimation shock, jet is almost
cylindrical shape as Bromberg et al. suggested.
Some oblique shocks can be seen inside the jet
Bromberg + (2011)
collimation shock +
Cylindrical jet (Γ~Γ0)
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See also Komissarov & Falle 1998 11

12. Cocoon Light jet and Overpressured Cocoon
Begelman & Cioffi (1988)
Bromberg +(2011)
Given: Lj , ρj , v j, ρa ==> jet dynamics ?
Light jet : ρj hjΓ2< ρ ambient ~c
Cocoon
Energy in the cocoon
~ Lj x t
~ Pc x V c ~ Pc x (vh x t)x (vc x t)
Pc can be derived as a function of time
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Momentum balance on rest frame of the head of the jet (Marti+97...) 12

13. Comparison with analytical model
Shock breakout
Jet height
Conversion
position
Width
of cylindrical jet
The results of numerical simulation are good agreement with theoretical model (Bromberg , see also Komissarov +1997). Since the mass gradient near the stellar surface is so steep, the theoretical model can not be applied.
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14. Probe particles xnew=xold+vr,zxΔt
Every hydro time step probe particles move
with the local velocity which is derived
hydro calculation, i.e.,
xnew=xold+vr,zxΔt
Probe particles allows
us to follow the
Lagrangian motion of
fluid elements.
32 particles
Initial jet nozle
Every 0.01s, 32 particles are injected with the jet
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15. Probe particles cocoon ρ ρ
t=2.3s ρ
t=4.6s
cocoon
Before jet breakout the particles which reach the head of the jet move
into the cocoon (shocked jet).
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16. Particle trajectories injected at t=5 s
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17. Particle trajectories injected at t=5 s
Stellar surface
Γ -1
Γ -1
Off-center explosion occurs. The location of off- center depends on
particle (when and where the particle is injected).
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18. 10s after shock break θj~Γ0-1 x1/5
Time evolution of jet opening angles
shock breakout
10s after shock break
θj~Γ0-1 x1/5
~0.08=1/2.5X1/5
~0.04=1/5X1/5
~0.02=1/10X1/5
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19. θj~Γ0-1 x1/5
θj~Γ0-1
θj~1/5Γ0-1
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20. Collimation shock after jet breakout (Pc (z) ∝ z-λ )
Lorentz factor after
collimation shock
where
where
λ<2 (to converge the collimation shock) (Komissarov 97)
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21. p 1D Pressure profile in the cocoon Pc~ r-2 at break Pc~ const Γ~Γ0
acceleration Γ~5xΓ0
Pc~ const Γ~Γ0
Pc~ r-4 after reak
Cocoon
confinement
Off-center pressure profile X
Slope is not enough steep (-1.8).
Expansion with weak cocoon confinement
also works before the free expansion. p
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22. Distribution of opening angle of GRB jets
(t_d>2s))
(t_d<2s))
θj=1/5Γ0<0.2 rad~10grees
Constant luminosity and uniform jet
can not explain large opening angle jets
==>Structured jet ?
Or, long duration (more than a few tens
of seconds) activity
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Fong et al. (2012) 22

23. Summary
●High resolution calculation of jet from collapsars to reduce
numerical baryon loading
●Just before and after jet breakout, we observe jet breakout
acceleration which leads Lorentz factor about 5xΓ0 . Numerical
results and analytical model suggests that both collimation shock
and weak confinement work.
●The opening angle after the break for first several seconds
θj~CxΓ0-1 (C~1/5)
●Constant luminosity and non-structured jet with a few tens
injection can not explain large opening angle jets. Much longer
injection or structured jet is necessary.
Future works
●3D simulations to see the multidimensional effect (see, Matsumoto's poster) and magnetized jet should be studied.
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24. MIZUTA SN-GRB 24

25. ρ p Γ High resolution case (after shock breakout ) progenitor surface
So many oblique shocks
progenitor surface p Γ
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26. Radial mass density profile
Model 16TI (Woosley & Heger 2006)
14M_sun, R*~4x1010cm

27. Relativistic beaming effect
GRBs are radiation from relativistic jets (Γ>100)).
Fluid rest frame
isotropic emission
Observer frame
Lorentz transformation Γ c Γ
(for Φ'=π/2)
=1/Γ
Angle Φ' in fluid rest fame is transformed to angle Φ.
Radiation/ fluid motion concentrates in half opening angle 1/Γ.
Relativistic beaming effect can be applied for relativistic isotopic
adiabatic expansion.
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28. Γ0=2.5 mass density Lorentz factor
θ0~1/Γ0 is larger than that of Γ0=5.0.
As the radius of the jet increases, the momentum flux to push the stellar envelopes drops.
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29. Opening angle of GRB jets
Opening angles of the jet are estimated
from the light curve in the afterglow.
θj~ several degrees.
Opening angles are more than
10 degrees for some GRBs
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Fong et al. (2012) 29