Formulation for the Relativistic Blast Waves

Z. Lucas Uhm

Research Center of MEMS Space Telescope (RCMST) & Institute for the Early Universe (IEU),

Ewha Womans University, Seoul, South Korea

Friday, April 23rd 2010Deciphering the Ancient Universe with Gamma-Ray

Bursts,Kyoto, Japan

Formulation for the Relativistic Blast

Waves

Uhm, Z. Lucas 2010 submitted

(arXiv:1003.1115)

• A central engine ejects a relativistic outflow – ejecta• Forward shock (FS) & Reverse shock (RS) develop• FS sweeps up the ambient medium, and RS

propagates through the ejecta

(Meszaros & Rees 1997)

Relativistic Blast Waves

Schematic Diagram of a Relativistic Blast Wave

• Blast – a compressed hot gas between FS & RS• General class of explosions with arbitrary radial

stratification of ejecta and ambient medium• Non-relativistic RS & mildly-relativistic RS

• How to find a dynamical evolution of the blast wave for this general problem ?

Jump Conditions

3 jump conditions for 4 independent unknowns: the shock has 1 free parameter

Kappa varies in between 1/3 and 2/3, depending on the shock strength

Shock strength described by relative Lorentz factor

(Blandford & McKee 1976)

Relation between kappa and mean Lorentz factor

(Uhm 2010 submitted)

Jump conditions for a monoenergetic gas

Exact solutions for a monoenergetic gasApply to shocks of arbitrary strength, relativistic or non-relativistic

Conservation laws across FS and RS are applied

Radially stratified ejecta

Continuity equation for ejecta∇α (ρej uα) = 0

Lagrangian coordinate τ r(τ,t) = vej (τ) * (t - τ)

(Uhm 2010 submitted)

Trajectory of the RS through ejecta

Given by jump condition at RS

(Uhm 2010 submitted)

Two different methods are described for finding the evolution of the blast Lorentz factor

(1)Customary pressure balance pr = pf (2)Mechanical model (Beloborodov & Uhm 2006)

Customary pressure balance : pr =pf

Depends only on input parameters

“Mechanical model” for relativistic blast waves

(Beloborodov & Uhm 2006)

Need to solve coupled differential equations

Example model

• An example burst is specified by the luminosity Lej(τ) = L0 = 1052 erg/s and

the Lorentz factor Γej(τ) = 500 - 9τ

for 0 ≤ τ ≤ τb = 50 s

• Total isotropic energy ejected by the burst is Eb = L0 τb = 5 * 1053 ergs

• Ambient medium density is assumed to be n1 = 1 cm-3

• These define the problem completely

Dynamics found for the customary pressure balance pr = pf

(Uhm 2010 submitted)

• (a) τr-shell passing through the RS at radius rr

• (b) the ejecta density nej(RS) of the τr-shell

• (c) the Lorentz factor Γej(RS) of the τr-shell and Γ of the blast

• (d) the relative Lorentz factor γ43

• (e) pressure p = pr = pf across the blast

• This numerical solution does not satisfy the energy-conservation law for adiabatic blast wave

Energy of adiabatic blast

Lagrangian description

(Uhm 2010 submitted)

Total energy found for the customary pressure balance

• Customary pressure balance pr = pf violates the energy-conservation law significantly for the adiabatic blast wave

• Total energy Etot of the entire system (blast + unshocked ejecta)

• Etot = Eblast + E4

Dynamics found for the mechanical model

(Uhm 2010 submitted)

• Numerical solutions for the blast-wave driven by the same example burst

• Solid (blue) curves are calculated using the mechanical model

• For comparison, the solution of customary pressure balance is also shown in dotted (red) curves

Total energy found for the mechanical model

(Uhm 2010 submitted)

• Mechanical model becomes a successful remedy for the the energy-violation problem

We suggest that one should use the mechanical model to solve for the dynamics of a blast wave in order to correctly find the afterglow light-curves!!

Summary

• We present a detailed description of our blast-wave modeling technique for a very general class of GRB explosions with arbitrary radial stratification of ejecta and ambient medium. See arXiv:1003.1115 for details.

• We demonstrate that the customary pressure balance for the blast wave violates the energy-conservation law significantly for adiabatic blast wave.

• We show that the energy-violation problem is

successfully resolved by the mechanical model.