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Opening angles of collapsar jets θ j ~CxΓ 0 -1 (C~1/5). Akira MIZUTA Computational Astrophysics Lab. RIKEN Kunihito IOKA KEK. SN&GRB 2013 @ YITP Kyoto Univ. 14.11.2013. Reference: MA, Nagataki, Aoi, ApJ, 732 , 26 (2011) MA&Ioka ApJ, 777 , 162 (2013)(arXiv:13040163). - PowerPoint PPT Presentation
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Opening angles of collapsar jets θj~CxΓ0
-1 (C~1/5)
Akira MIZUTAComputational Astrophysics
Lab. RIKEN
Kunihito IOKAKEK
SN&GRB 2013 @ YITP Kyoto Univ. 14.11.2013
Reference: MA, Nagataki, Aoi, ApJ, 732, 26 (2011)MA&Ioka ApJ, 777, 162 (2013)(arXiv:13040163)
MIZUTA SN-GRB 2013 @YITP
GRB jet : relativistic collimated outflow 1010cm105cm 1012-13cm 1015-17cm
●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).
Numerical Hydro Jet propagation(jet structures, opening angle, effect of progenitor structure,photospheric emission etc., i.e.,Aloy+00, Zhang, MacFadyen+03,04Mizuta+06,09,11,13Morsony+07,10, Lazzati+09,13Nagakura+11,12, Suzuki+13 ….
MIZUTA SN-GRB 2013 @YITP
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) [email protected]
Progenitor: 14Msun ,radius=4x1010cm
MIZUTA SN-GRB 2013 @YITP
θ0=10degrees100s injection
Bulletfree expandion
θ0=10degrees 30s injection
Γ
Γ
Γ
Photospheric emission (dissipated photosphere)
OA10 [0:100s]
OA10[0:30s]
Only dissipated region
Γ
Γ
Γ
log10(ρ)
log10(ρ)
log10(ρ)
Γ
MA+ 2011 , inprep
MIZUTA SN-GRB 2013 @YITP
Γ final>=200
Γ final>=2
Morsony +07
“Opening angle” measured at a certain radius
θ0
●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.
Motivation
MIZUTA SN-GRB 2013 @YITP
Γ0 ~θ0 -1
What determines opening angel of jets ?
Before shock breakout, the jet is influenced by the interactionwith 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 ?
Γ0 ~θ0 -1
Naive expectation Our model
relativistic beamingeffect
stellar envelopes
stellar envelopesΓ0 ~θ0
-1
MIZUTA SN-GRB 2013 @YITP
Elongated:Aspect ratio is not correct
High resolution grid points are devotedat 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)
High resolution hydro. simulations(2D, axisymmetric)ρ
Γ pprogenitor surface
MIZUTA SN-GRB 2013 @YITP
Bromberg, Levinson (2009)
Γ0=20
Γ0=10
Γ0=5
Collimation shockthin layer
Why is high resolution necessary ?
θ0=1/Γ0
Cocoon gas
Initial jet size should be small
MIZUTA SN-GRB 2013 @YITP
baryon loading due to numerical diffusion
hΓ is conserved to the reverse shock
hΓ (=const along stream line, steady state :Bernoulli's principle)
Why is high resolution necessary ?
Along jet axis
Γ0=5
mass density Lorentz factor
So many internal oblique shocks have a potential for (Ito's talk on Tuesday day)
MIZUTA SN-GRB 2013 @YITP
After collimation shock, jet is almost cylindrical shape as Bromberg et al. suggested.Some oblique shocks can be seen inside the jet
jet injection
Bromberg + (2011)collimation shock +Cylindrical jet (Γ~Γ0)
Γ0=5 Γ0=2.5
See also Komissarov & Falle 1998
Cocoon confinement (Before jet breakout)
Cocoonpressure
Γ0
θ0~1/Γ0
MIZUTA SN-GRB 2013 @YITP
Light jet and Overpressured Cocoon
Cocoon
Begelman & Cioffi (1988)Bromberg +(2011)
Light jet : ρj hjΓ2< ρ ambient
~c
Momentum balance on rest frame of the head of the jet (Marti+97...)
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
Given: Lj , ρj , v j, ρa ==> jet dynamics ?
MIZUTA SN-GRB 2013 @YITP
Comparison with analytical model
Widthof cylindrical jet
Conversion position
The results of numerical simulation are good agreement with theoretical model (Bromberg + 2011, see also Komissarov +1997). Since the mass gradient near the stellar surface is so steep, the theoretical model can not be applied.
Shock breakout
Jet height
MIZUTA SN-GRB 2013 @YITP
Probe particles allowsus to follow theLagrangian motion offluid elements.
Every hydro time step probe particles movewith the local velocity which is derivedhydro calculation, i.e.,
xnew=xold+vr,zxΔt
Probe particles
Every 0.01s, 32 particles are injected with the jet
32 particles
Initial jet nozle
MIZUTA SN-GRB 2013 @YITP
Probe particles
ρ t=2.3s ρ t=4.6s
cocoon
Before jet breakout the particles which reach the head of the jet moveinto the cocoon (shocked jet).
MIZUTA SN-GRB 2013 @YITP
Particle trajectories injected at t=5 s
MIZUTA SN-GRB 2013 @YITP
Stellar surface
Particle trajectories injected at t=5 s
θj
Γ -1 Γ -1
Off-center explosion occurs. The location of off- center depends onparticle (when and where the particle is injected).
MIZUTA SN-GRB 2013 @YITP
shock breakout
~0.02=1/10X1/5
~0.04=1/5X1/5
~0.08=1/2.5X1/5
10s after shock breakθj~Γ0
-1 x1/5
Time evolution of jet opening angles
MIZUTA SN-GRB 2013 @YITP
θj~Γ0-1
x1/5
θj~Γ0-1
θj~1/5Γ0-1
MIZUTA SN-GRB 2013 @YITP
Collimation shock after jet breakout (Pc (z) ∝ z-λ )
λ<2 (to converge the collimation shock) (Komissarov 97)
where
Lorentz factor aftercollimation shock
where
MIZUTA SN-GRB 2013 @YITP
Pc~ const Γ~Γ0
Pc~ r-2 at break acceleration Γ~5xΓ0
Pc~ r-4 after reak
p
1D Pressure profile in the cocoon
Cocoon confinement
X
Off-center pressure profile
Slope is not enough steep (-1.8).Expansion with weak cocoon confinementalso works before the free expansion.
MIZUTA SN-GRB 2013 @YITP Fong et al. (2012)
(t_d>2s))
(t_d<2s))
Distribution of opening angle of GRB jets
θj=1/5Γ0<0.2 rad~10greesConstant luminosity and uniform jetcan not explain large opening angle jets==>Structured jet ?Or, long duration (more than a few tensof seconds) activity
MIZUTA SN-GRB 2013 @YITP
Summary●High resolution calculation of jet from collapsars to reducenumerical baryon loading●Just before and after jet breakout, we observe jet breakoutacceleration which leads Lorentz factor about 5xΓ0 . Numerical
results and analytical model suggests that both collimation shockand 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 longerinjection or structured jet is necessary.
Future works●3D simulations to see the multidimensional effect (see, Matsumoto's poster) and magnetized jet should be studied.
MIZUTA SN-GRB 2013 @YITP
MIZUTA SN-GRB 2013 @YITP
High resolution case (after shock breakout )ρ
Γ p
progenitor surface
So many oblique shocks
Radial mass density profile
Model 16TI (Woosley & Heger 2006)14M_sun, R*~4x1010cm
MIZUTA SN-GRB 2013 @YITP
Fluid rest frame isotropic emission
Observer frameLorentz transformation
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 isotopicadiabatic expansion.
Relativistic beaming effect
=1/Γ
c
GRBs are radiation from relativistic jets (Γ>100)).
Γ
(for Φ'=π/2)
Γ
MIZUTA SN-GRB 2013 @YITP
Γ0=2.5mass 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.
MIZUTA SN-GRB 2013 @YITP Fong et al. (2012)
Opening angle of GRB jets
Opening angles of the jet are estimatedfrom the light curve in the afterglow. θj~ several degrees. Opening angles are more than10 degrees for some GRBs