Magnetohydrodynamic Effects in (Propagating) Relativistic Ejecta

1. Magnetohydrodynamic Effects in (Propagating) Relativistic Ejecta Yosuke Mizuno Center for Space Plasma and Aeronomic Research University of Alabama in Huntsville Collaborators B. Zhang (UNLV), B. Giacomazzo (MPIG, AEI), K.-I. Nishikawa (NSSTC/UAH), P. E. Hardee (UA), S. Nagataki (YITP, Kyoto Univ.), D. H. Hartmann (Clemson Univ.) Mizuno et al. 2009, ApJ, 690, L47 High Energy Phenomena in Relativistic Outflows II, Oct. 26-30, Buenos Aires

2. Role of Magnetic Field in Propagating Relativistic Jet/Ejecta • The magnetic fields play an important role in relativistic jets/ejecta (e.g., jet formation) • The degree of magnetization (quantified by s; magnetic to kinetic energy flux ratio) is poorly constrained by observations. • GRB afterglow modeling indicates GRB ejecta are more magnetized than the ambient medium (e.g., Zhang et al. 2003, Gomboc et al. 2008) • possibly important dynamic role for magnetic fields in GRB jet/ejecta • Useful diagnostic for jet magnetization can be obtained from interaction between decelerating jet/ejecta and ambient medium • Addition of magnetic field in the jet changes the condition for formation and strength of a reverse shock (RS) (e.g., Kennel & Coroniti 1984)

3. Role of Magnetic Field in Propagating Relativistic Jet/Ejecta (cont.) • Analytical studies of the deceleration of a GRB flow with magnetic field (Zhang & Kobayashi 2005) suggest some new behavior that does not exist in pure hydrodynamic model (e.g., Sari & Piran 1995) • However, a consensus as to the conditions for the existence of the RS has not yet been achieved (e.g., Zhang & Kobayashi 2005; Giannios et al. 2008) GRB blast wave model

4. Purpose of This Study • We investigate the interaction between magnetized relativistic jet/ejecta and unmagnetized external medium • For GRB case, the interaction with external medium takes place after acceleration, collimation, and prompt emission phase are over • A Riemann Problem is solved both analytically and numerically over a broad range of magnetization (s=magnetic to kinetic energy flux ratio).

5. The Riemann Problem • Consider a Riemann problem consisting of two uniform initial states • Right (external medium): cold fluid with constant rest-mass density and essentially at rest. • Left (jet/ejecta): higher density, higher pressure, relativistic velocity normal to the discontinuity surface • To investigate the effect of magnetic fields, put toroidal (By) components of magnetic field in the jet region (left state) • Typical case • Density: rho_L=100.0, rho_R=1.0 • Pressure: p_L=1.0, p_R=0.01 • Velocity: Vx_L=0.995c (g=10), Vx_R=0.0c • Adiabatic index: 4/3 • Calculation box:0.8-1.2 (transition at 1.0) • To solve the Riemann problem, RMHD exact solution calculation code developed by Giacomazzo & Rezzolla (2006) is used

6. Ejecta-Medium Interaction s: magnetization parameter =Emag/Ekin In Riemann profile, S: shock, C: contact discontinuity R: rarefaction wave Solid: Gas pressure Dashed: Mag pressure • s=0.1 (black) • SCS profile, reverse shock (RS) propagate in the ejecta • s=1.0 (red) • SCS profile, RS becomes weaker and propagate faster than s=0.1 case • These features are expected from analytical work (Zhang & Kobayashi 05) • s=10.0 (green) • RCS profile, reverse rarefaction wave (RR) propagate in the ejecta • Density, pressure in the ejecta decrease • Flow velocity increases (accelerated) • s=20.0 (blue) • RCS profile, jet is more accelerated • s=2.7 (yellow) • Critical sigma value, neither reverse shock nor a rarefaction wave is established density Lorentz factor

7. Physical Conditions for Reverse Shock or Magnetic Acceleration Four regions: (1) unshocked medium, (2) shocked medium, (3) shocked ejecta, (4) unshocked ejecta Based on relativistic shock jump conditions with adiabatic indexG=4/3 and Thermal pressure generated in the forward shock region • Constant speed across the contact discontinuity, g2=g3 • Relation between gas pressure and internal energy, p2=u2/3 The condition for existence of the reverse shock/ reverse rarefaction wave pressure balance between forward shock (pgas, 2) and ejecta (pmag, 4) Reverse shock: pgas,2 > pmag, 4; Rarefaction wave: pgas, 2< pmag, 4 Critical sigma value

8. Dependence on magnetization parameter • Magnetization, s in flow increases, pressure ratio decreases with s and makes smooth transition from RS to RR regime • Transition of RS to RR • s~0.7, 2.7, 10.6 in gL=5, 10, 20 • The critical s increases with gL, so that a RS can exist in the high-s regime if flow Lorentz factor is sufficiently large • Another condition for a RS: shock propagation speed in the fluid frame is higher than the speed of the Alfven wave, g’RS,RR > g’A • pFS/pB and g’RS,RR/g’A reach unity at the same critical s • Two RS conditions have intrinsically the same physical origin (see Giannios et al. 2008) Gas pressure in FS to magnetic pressure in flow ratio Pressure of shocked flow RS regime RR regime Shocked Lorentz factor Lorentz factor of propagating RS or RR to Alfven Lorentz factor ratio Initial magnetization

9. Terminal Lorentz factor • The terminal Lorentz factor after magnetic acceleration can be estimated by pressure balance between the forward shock and shocked ejecta region • From the definition of magnetized parameter, s=B2/gr, • this condition becomes • This analytical estimation is good agreement with the exact solution of the Riemann problem

10. Magnetic Acceleration Efficiency Terminal Lorentz factor Acceleration Efficiency • A jet with a higher initial Lorentz factor reaches a higher terminal Lorentz factor • But a lower initial flow Lorentz factor experiences a higher acceleration efficiency RS regime RR regime Acceleration efficiency Initial flow Lorentz factor +: analytical estimation

11. Discussion • The observed paucity of bright optical flashes in GRBs (e.g., Roming et al. 2006) may be attributed to highly magnetized ejecta (if optical flashes are related the emission from RS) • The magnetic acceleration mechanism suggests that s and g are not independent parameters at the deceleration radius. • For high-s flow, ejecta would experience magnetic acceleration at small radii, before reaching the coasting regime; the coasting Lorentz factor (initial Lorentz factor for the afterglow) = terminal Lorentz factor (same mechanism also seen in Mimica et al. 2009) • Our results suggest the possibility of magnetic acceleration occurring where highly magnetized jet material overtakes more weakly magnetized jet material. It may be related to variable emission observed in some TeV blazars which suggests very high Lorentz factor in AGN jets (Aharonian et al. 2007)

12. Summery • We have investigated the interaction between magnetized relativistic jet/ejecta and unmagnetized static ambient medium • We confirm that the reverse shock propagating in the flow becomes weak when the jet is magnetized • We found the new acceleration mechanism by the rarefaction wave propagating in the jet/ejecta when the flow is strongly magnetized • Critical magnetization for new acceleration mechanism depends on the initial jet velocity; • For the magnetic acceleration the jet with higher initial Lorentz factor needs strong magnetization • Terminal Lorentz factor depends on the magnetization of jet/ejecta • Recently Mimica et al. (2009) have performed 1D RMHD simulations of radially expanding magnetized GRB ejecta and found same acceleration mechanism has occurred when GRB ejecta is highly-magnetized.