# Presentation: 844297

## Physics of GRB Jets

**Jonathan Granot**
- KIPAC @ Stanford

- “GRBs: the first 3 hours”, Sanrotini, August 31, 2005

## Outline of the Talk:

- Observational evidence for jets in GRBs
- The jet **dynamics**: degree of **lateral expansion**
- Semi-Analytic models
- Simplifying the dynamical Eqs.: 2D  1D
- Full hydrodynamic simulation
- The Jet** Structure: **how can we tell what it is
**Afterglow polarization**
**Statistics** of the prompt & afterglow emission
**Afterglow light curves**
- Off-axis viewing angles & X-ray flashes
**Conclusions**

## Observational Evidence for Jets in GRBs

- The energy output in -rays assuming isotropic emission approaches (or even exceeds) Mc2
-   difficult for a stellar mass progenitor
**True energy** is much smaller for a narrow** jet**
- Achromatic break
-    or steepening of the
-    afterglow light
-    curves (“jet break”)

- Optical light curve of
- GRB 990510

- (Harrison et al. 1999)

## Dynamics of GRB Jets: Lateral Expansion

- Simple (Semi-) Analytic Jet Models (Rhoads 97, 99; Sari, Piran & Halpern 99,...)Typical Simplifying Assumptions:

- A uniform jet with sharp edges (even at t > tjet)
- The shock front is a part of a sphere within θ < θjet
- The velocity is in the radial direction (even at t > tjet)
- Lateral expansion in a velocity of cs  c/3 or  c in the local rest frame
- The jet dynamics are obtained by solving simple 1D equations for conservation of energy and momentum
- Most works assume a uniform external medium (ISM)

## Simplified Jet Dynamics:

- (Rhoads 97, 99; Sari, Piran & Halpern 99,...)

- dR  jetdR + csdt’  (jet + cs/c)dR
- djet/dR  (dR/dR - jet)/R  cs/cR
-  jet **~** 0 + cs/c
- The jet edges become visible when  **~ **1**/** 0
- Sideways expansion becomes large  **~ **(cs/c) **/ **0

- R

- R

- jet

- jet  R / R

## Quick & Dirty:

- E/c2 **~** ext(R)R3(jet)22 + jet **~**** **cs/c** ** R ~ const

**More Careful Analysis **(Rhoads 99)**:**

- E/c2  M2  d(2)/dR = -(2/M)dM/dR
- dM/dR = 2(jetR)2ext(R)**,** 2/M  4c2/E
- jet  jet- 0  (cs/c)dr/r ~ (cs/cR)dr/
-  **~**** **(cs/c0)exp(-R/Rjet)**,** jet **~** 0(Rjet/R)exp(R/Rjet)
- t  dr/2c2 **~ **Rjet/4c2      t-1/2
- Rjet= [E/ext(cs)2]1/3  (comparable to the Sedov length for the true energy, if cs ~ c)

## Most models predict a jet break but differ in the details:
The time of the jet break tjet (by a factor of ~20)
Temporal slope F( > m,t > tjet)  t-,  ~ p (±15%)
The sharpness of the jet break (~1- 4 decades in time)
Kumar & Panaitescu (2000) predicted a significantly smoother jet break for a stellar wind environment (this was reproduced in other works, but not observed yet) 

- The time of the jet break tjet (by a factor of ~20)
- Temporal slope F( > m,t > tjet)  t-,  ~ p (±15%)
- The sharpness of the jet break (~1- 4 decades in time)
- Kumar & Panaitescu (2000) predicted a significantly smoother jet break for a stellar wind environment (this was reproduced in other works, but not observed yet)

**Implications: **Afterglow Light Curve

## Simplifying the Dynamics: 2D  1D

- Integrating the hydrodynamic equations over the radial direction significantly reduces the numerical difficulty
- This is a reasonable approximation as most of the shocked fluid is within a thin layer of width ~ R/102

- (Kumar & JG 2003)

##       Numerical Simulations:(JG et al. 2001; Cannizzo et al. 2004; Zhang & Macfayen 2006)

- The difficulties involved:

- The hydro-code should allow for both  **≫** 1 and   1
- Most of the shocked fluid lies within in a very thin shell behind the shock ( ~ R/102)  hard to resolve
- A relativistic code in **at least** **2D** is required
- A complementary code for calculating the radiation

- Very few attempts so far

## Movie of Simulation

## Proper Density:(logarithmic color scale)

- Bolometric Emissivity:(logarithmic color scale)

## The Velocity Field (on top of lab frame density)

- (JG, Miller, Piran, Suen & Hughes 2001)

## The Jet Dynamics: very modest lateral expansion

- There is slow material at the sides of the jet while most of the emission is from its front

## Main Results of Hydro-Simulations:

- The assumptions of simple models fail:
- The shock front is not spherical
- The velocity is not radial
- The shocked fluid is not homogeneous
- There is only very mild lateral expansion as long as the jet is relativistic
- Most of the emission occurs within θ < θ0
**Nevertheless**, despite the differences, there is a **sharp** achromatic **jet break** [for  > m(tjet)] at tjet close to the value predicted by simple models

## Comparison to (Semi-) Analytic Models:

- Similarities:
- An achromatic **jet break** at tjet for  > m(tjet)
- The value of tjet is similar
- Temporal slope, F ( > m, t > tjet)  t-, is close to the analytic value   p ( =1 .12p for p = 2.5 and is even closer to p for p < 2.5)
- Differences:
- The jet dynamics are very different
- For  < m(tjet) (radio)  changes more gradually and moderately at tjet and changes more sharply only at a later time when m decreases below obs
- Jet break is sharper than in most analytic models, and is somewhat sharper for θobs= 0 than for θobs  θ0

## Why do we see a Jet Break: 

- Aberration of light or
-  ‘relativistic beaming’

- Source frame

- Observer
- frame

- 1/

- The observer sees mostly emission from within an angle of 1/ around the line of sight

- 1/

- Direction to observer

**Relativistic Source:**

- The edges of the jet become visible when  drops below 1/jet , causing a jet break

- For v ~ c, jet ~ 1/ so there is not much “missing” emission from
-  > jet & the jet break is due to the decreasing dE/d + faster fall in (t)

## Lateral Expansion: Evolution of Image Size(Taylor et al. 04,05; Oren, Nakar & Piran 04; JG, Ramirez-Ruiz & Loeb 05)

- GRB 030329

- Image diameter

- (JG, Ramirez-Ruiz & Loeb 2005)

- Model 1: v ~ c
- Model 2: v ≪** **c
- while  ≳ 2

## The Structure of GRB Jets:

## How can we determine the jet structure?1. Afterglow polarization light curvesthe polarization is usually attributed to a jet geometry

- Log()

- 0

- Log()

- Log(dE/d)

- core

- -2

**Structured jet****††****:**

**Uniform jet****†****:**

**†**Rhoads 97,99; Sari et al. 99, ...

**††**Postnov et al. 01;
-      Rossi et al. 02;
-      Zhang & Meszaros 02

- Log(t)

- P

- tjet

- (Sari 99; Ghisellini & Lazzati 99)

- Log(t)

- P

- tjet

- (Rossi et al. 2003)

- Log(dE/d)

-  p = const

-  p flips by 90o at tjet

**Ordered **
**B-field*********:**

*****Granot & Königl 03

- Log(t)

- P

-  p = const

- t0

- while for jet  models
- P(t≪tj) ≪ P(t~tj)

## Linear polarization at the level of  P ~ 1%-3%              was detected in several optical afterglows
In some cases P varied, but usually p  const
Different from predictions of uniform or structured jet

- In some cases P varied, but usually ****p  const
- Different from predictions of uniform or structured jet

- (Covino et al. 1999)

- (Gorosabel et al. 1999)

**Afterglow Polarization: ****Observations**

- GRB 020813

**t****j**

## The Afterglow polarization is affected not only by the jet structure but also by factors such as
the B-field structure in the emitting region
Inhomogeneities in the ambient density or in the jet (JG & Königl 2003; Nakar & Oren 2004)
“refreshed shocks” - slower ejecta catching up with the afterglow shock from behind (Kumar & Piran 2000; JG, Nakar & Piran 03; JG & Königl 03) 
Therefore, afterglow polarization is not a very “clean” method to learn about the jet structure

- the **B-field structure** in the **emitting region**
**Inhomogeneities** in the ambient density or in the jet (JG & Königl 2003; Nakar & Oren 2004)
- “**refreshed shocks**” - slower ejecta catching up with the afterglow shock from behind (Kumar & Piran 2000; JG, Nakar & Piran 03; JG & Königl 03)
- Therefore, **afterglow polarization** is **not** a very “**clean**” method to learn about the jet structure

**Afterglow Pol. & Jet Structure:**** Summary**

## Both the UJ & USJ models provide an acceptable fit
Provides many constraints 
    but not a “clean” method 
    to study the jet structure

- Provides many constraints
-     but not a “clean” method
-     to study the jet structure

**Jet Structure from log N - log S distribution**(Guetta, Piran & Waxman 04; Guetta, JG & Begelman 05; Firmiani et al. 04)

- (Guetta, JG & Begelman 2005)

- Cumulative distribution

- differential distribution

-  different binning

## dN/d appears to favor the USJ model
dN/ddz disfavors the USJ model
It is still premature to draw strong conclusions due to the inhomogeneous sample & various selection effects
Not yet a “clean” method for extracting the jet structure

- dN/ddz disfavors the USJ model
- It is still premature to draw strong conclusions due to the inhomogeneous sample & various selection effects
- Not yet a “clean” method for extracting the jet structure

**Jet Structure from ****t****jet**** (****z****) distribution**

- (Perna, Sari & Frail 2003)

- (Nakar, JG & Guetta 2004)

## Uniform “top hat” jet - extensively studied  

**Afterglow  Light Curves: ****Uniform Jet**** **(Rhoads 97,99; Panaitescu & Meszaros 99; Sari, Piran & Halpern 99; Moderski, Sikora & Bulik 00; JG et al. 01,02)

- e=0.1, B=0.01, p=2.5, θ0=0.2, θobs=0, z=1, Eiso=1052 ergs, n=1 cm-3

- (JG et al. 2001)

## It is a “smooth edged” version of a “top hat” jet
Reproduces on-axis light curves nicely

- It is a “smooth edged” version of a “top hat” jet
- Reproduces on-axis light curves nicely

**Afterglow Light Curves: ****Gaussian**** ****Jet**** **(Zhang & Meszaros 02; Kumar & JG 03; Zhang et al. 04)

- (Kumar & JG 2003)

## Works reasonably well but has potential problems

**Afterglow LCs: ****Universal Structured**** ****Jet**(Lipunov, Postnov & Prohkorov 01; Rossi, Lazzati & Rees 02; Zhang & Meszaros 02)

- (Rossi et al. 2004)

## LCs Constrain the power law indexes ‘a’ & ‘b’: dE/d  -a, 0  -b
1.5 ≲ a ≲ 2.5, 0 ≲ b ≲ 1

- LCs Constrain the power law indexes ‘a’ & ‘b’: dE/d  -a, 0  -b
- 1.5 ≲ a ≲ 2.5, 0 ≲ b ≲ 1

**Afterglow LCs: ****Universal Structured**** ****Jet**

- (JG & Kumar 2003)

## Usual light curves + extra features: bumps, flatening

**Afterglow LCs: ****Two Component**** ****Jet**** **(Pedersen et al. 98; Frail et al. 00; Berger et al. 03; Huang et al. 04; Peng, Konigl & JG 05; Wu et al. 05)

- (Peng, Konigl & JG 2005)

- (Berger et al. 2003)

- (Huang et al. 2004)

- GRB 030329

## The bump at tdec,w for an on-axis observer is wide & smooth

- The bump at tdec,w for an on-axis observer is wide & smooth

**Two Component**** ****Jet: ****GRB 030329**

- (Lipkin et al. 2004)

- (JG 2005)

- Abrupt
- deceleration

- gradual
- deceleration

## The bump in the light curve when the narrow jet becomes visible is smooth & wide

- The **bump** in the light curve when the narrow jet becomes visible is **smooth & wide**

**Two Component**** ****Jet: ****XRF 030723**

- (Fynbo et al. 2004)

- (JG 2005)

- (Huang et al. 2005)

## Afterglow Light Curves: Off-Axis Viewing Angles 

- Afterglow Light Curves: **Off-Axis** Viewing Angles

- Granot et al. (2002)

- θ0=0.2, Ejet=3•1051 erg, n=1 cm-3, z=1, p=2.5, e=0.1,
- B=0.01

- Model 2: uniform jet + sharp edges & v = cs

## Prompt Emission: Off-Axis Viewing Angles 

- Epeak  -1, f  -a where   1+[(obs -0)]2 & a  2 for 0 < obs ≲ 20; a  3 for obs ≳ 20
- The prompt emission from **large **off-axis viewing angles,  ≫ 1 or obs ≳ 20, will not be detected (“orphan afterglows”)
-  The prompt emission from slightly off-axis viewing angles might still be detected, but peaks at lower Epeak & has a much smaller fluence f (X-ray flashes or X-ray rich GRBs)

## Suggest a roughly uniform jet with reasonably sharp edges, where GRBs, XRGRBs & XRFs are similar jets viewed from increasing viewing angles (Yamazaki, Ioka & Nakamura 02,03,04)

**Light Curves of X-ray Flashes & XRGRBs**

- (JG, Ramirez-Ruiz & Perna 2005)

- XRF 030723

- XRGRB 041006

## Afterglow L.C. for Different Jet Structures:

- Uniform conical jet with sharp ejdges: 
- Gaussian jet in both 0 & dE/d: might still work
- Constant 0 + Gaussian dE/d: not flat enough
- Core + dE/d  -3 wings: not flat enough

- obs / 0/c = 0, 0.5, 1, 1.5, 2 , 2.5, 3, 4, 5, 6 ** **(JG, Ramirez-Ruiz & Perna 2005)

## Conclusions:

- Numerical studies show **very little lateral expansion** while the jet is relativistic & produce a **sharp jet break** (similar to that seen in afterglow observations)
- There are some important differences in the resulting afterglow light curves, especially at  < m(tjet) (radio)
- The most promising way to **constrain** the **jet structure** is through the **afterglow light curves**
- XRF & XRGRB afterglow light curves favor a roughly uniform jet with rather sharp edges

## Afterglow Light Curves from Simulations