opening angles of collapsar jets θ j ~cxΓ 0 -1 (c~1/5)

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 ….


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


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


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


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


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


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


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)


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


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 ?


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


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


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).


Particle trajectories injected at t=5 s


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).


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


θj~Γ0-1

x1/5

θj~Γ0-1

θj~1/5Γ0-1


Collimation shock after jet breakout (Pc (z) ∝ z-λ )

λ<2 (to converge the collimation shock) (Komissarov 97)

where

Lorentz factor aftercollimation shock

where


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


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.


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


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


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

opening angles of collapsar jets θ j ~cxΓ 0 -1 (c~1/5)

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