Relativistic MHD Jets in GRB Steady State Rarefaction

K. SapountzisNational and Kapodistrian University of Athens

Kyoto 2013

Contents

Formulation of the problem and assumptions

Description of the model

Initial conditions and general results

Previous simpler works

Results on the Collapsar GRB model

Comparison and Conclusions

Motivation

Study of the super-fast magnetosonic jets applied to the Gamma Ray Bursts environment and especially in the Collapsar Model where rarefaction occurs

General application in astrophysical jets under a specific external pressure profile (in progress)

Geometry of the Collapsar outflowGeometry of Rarefaction Wave

In contrast with most sophisticated numerical simulation that solve the full time dependent problem we analyze the steady state case

Velocities are relativistic and the trivial assumption of an ideal conducting plasma was made.

Our model is axisymmetric

R MHDequations

Steady State Ideal

Cold Outflow Axisymmetric

Framework

Assumption of cold outflow (specific enthalpy unity) since any initial thermal content will have been converted to bulk kinetic long before the break up

0t

0

enthalpy per mass 1

Basic Annotation

1·

2 pA B dS

: physical parameter

total energy flux (inertial + poynting)

rest mass energy flux

Measure the length along a field line

poynting energy fluxmagnetization parameter

inertial energy flux

labels each poloidal field line

, , 0z

U

S

U UF A G AA

The Algorithm Schema I

Under analytical consideration we reach to a system of 5 first order pdes

The system has 5 characteristics

Coordinates of the poloidal plane

Line inclination

Quantities instead of,A z A

The Algorithm Schema II

Interpolation

Procedure

1) For a specific we find the intermediate points by linear interpolation on the characteristics inclination

2) We solve the system of the 5 characteristics and determine the new values of the quantities

The shape of the boundary line is obtained self consistently

d

In the inner (axis) and outer boundary one characteristic is missing. We fill the missing information by the physical requirements

3) Inner the magnetic field must be continuous and due to axis-symmetry

4) For the wall we set the pressure outflow to be equal with its environment . In rarefaction

0ax

int 0extP P

02

22 2

2

1 /

1 / 1 2 /

2 1/ 1 /

jpi

o

o oj ii

o o

BB

BB

The initial conditionsThe initial configuration settled in a plane cross section

and consists of a homogenous velocity flow with no azimuthal velocity.

Poloidal streamlines (and poloidal field lines) parallel to z-axis

510 0i iz const

Magnetic field has been chosen as to satisfy core like in the innermost structure the transfield component of momentum equation

Model parameter

Scale of the magnetic field

Using the above expressions we calculate and the other two

Core radius

100, 0u

0

0 ( )A ,S

The initial conditions

One more quantity or integral has to be given

or Determines the maximum attainable Lorentz factor max

Close to axis expressions alters in order to simulate the core and proper behavior

Results

Physical Shape of the flow: The flow accelerates without further collimation, actually its opening-angle increases slightly

0.0075PM

0.023RW

Reflection

Results

μ: maximum attainable Lorentz Factor

2/3r

The efficiency is high at the initial “Rarefaction” stage

The reflected wave ceases acceleration

Previous Work - ComparisonWe used the set of r-self similar solutions to solve the planar symmetric casegeneralization of hydro (eg Landau) and hydro relativistic (Granik 1982) K. Sapountzis, N. Vlahakis , MNRAS, 2013

Assumptions

• Relativistic, planar symmetric, ideal• Thermal content included• Poloidal magnetic field negligible ( ) generalization available, to be submitted

Obtained

• Semi-analytical system, 2 ordinary differential equations • Easier to handle analytically in specific limits (eg cold one)

FA r f

p yB B

Previous Works - Comparison

Conclusions on the self similar models

• Magnetic driven acceleration acts in shorter spatial scales comparing to hydrodynamic one

• The front is the envelope of the fast magnetosonic disturbances

• The efficiency is very high

• Appearance in cold limit of Rarefaction Wave at

• Extension of rarefaction region

• It follows the scaling

sin 0.023iRW

i

2/3r 2

0.00751

iPM

i

2

22 2

2

1 sin /

no curvature of the poloidal field lines

1 sin / 1 2 sin /

2 1sin / 1 sin /

jpi

o

o oj ii

o o

BB

r

r rBB

r r

GRB Collapsar Model

We used a more sophisticated set of initial conditions

~ 1j i

z

tan ii

iz

j

Initial surface of integration

Light cylinder radius defined by Ω at base of the flow (accretion disk)

3/2

5

7

light cylinder 4.24·10

24 light cylinder 10

d

j s

d s

Rc Mcm

M R

M M R R cm

Spatial scaling

7ˆ ~ 5 10 cm

GRB Collapsar Model

The integral of energy flux – magnetization parameter

Model* γi ωj σj ω0

g100 100 1000 5 434

g50 50 2000 11 217

g20 20 5000 29 86*Common ζ=0.6

Collapsar – Results

Higher results

• Further reflection distance

• Further shock wave creation

• Smoother deceleration

isin i

RWi

2

1i

PMi

Collapsar – Results

1.50

1.33 1.27

Collapsar – Results

Higher results

• Lower extension of the rarefied area

• Higher angles of the reflection wave to the boundary and thus stronger shock creation

i

Collapsar – Results

Conclusions

• Rarefaction is an effective mechanism converting magnetic energy (poynting) to bulk kinetic

• It arises naturally at the point where the outflow crosses the envelop of the star and continues to the interstellar medium

• Allows potential panchromatic breaks to appear due to the geometry it provides

• Reflection occurs and the reflecting wave ceases acceleration and causes a significant deceleration

• Our model determines the shape of the last field line self-consistently

• The curvature seems not to play important role in the first phase of acceleration and so the simpler planar symmetric models can be used

• The spatial distances are in accordance with the ones that fireball - internal shock require

This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund.

K. SapountzisNational and Kapodistrian University of Athens