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Relativistic MHD Jets in GRB Steady State Rarefaction. K. Sapountzis National 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 - PowerPoint PPT Presentation
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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 flux magnetization parameterinertial energy flux
labels each poloidal field line
, , 0z
US
U UF A G A A
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 conditionsOne 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 ModelWe 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 M cmM 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
21
iPM
i
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