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
• 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
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)
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)
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
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.