Mikhail V. Medvedev (KU)

Transcription

1 KIAA GRB school Beijing, China, 9 June 2009 Mikhail V. Medvedev (KU) Students: Sarah Reynolds, Sriharsha Pothapragada Collaborators: Silva, Fonseca, Mori & others (Inst Superior Tecniko/GOLP, Portugal + UCLA) Nordlund, Hededal, Frederiksen, Haugboelle (Niels Bohr Inst. Denmark) Spitkovsky (Princeton) Preece, Nishikawa (UAH) Lazzati, Workman, Morsony (NCU/UC Boulder) Zakutnyaya (IKI, Russia)

2 Internal Shocks: Prompt Afterglows (external shock) - hours months Prompt g-rays - 2x10-3 2x10 2 seconds Idea: shocks Need many one per event (spike)

3 GRB Standard model Internal/External Shock Model fireball prompt emission afterglow G 2 G 1 g-rays ISM X-rays optical radio internal shock (collision of shells) external shock

4 Internal Shocks: Knots Jet from M87 knots HST image

5 External Shock Cassiopea A Chandra 2001 & VLA 1997 images forward shock (Gotthelf, et al. 2001)

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7 Collisional Shock Wave A shock forms when the wave crest starts to fall over, and the wave shoots out ahead. l l

8 Are GRBs collisional or collisionless? Coulomb cross-section s Coulomb ~πd 2, where d~e 2 /Γmc 2 the distance of closest approach Coulomb mean-free-path l Coulomb ~1/nσ Coulomb ~(10 31 cm)γ/n Nuc-nuc mean-free-path l nuclon ~1/nσ pp ~(10 25 cm)(γn) -1 Collisions are rare, prompt l ~ cm > R~ cm afterglow l~10 25 cm > R~10 18 cm Formally, a hydro shock cannot form! Collisionless shock - what is this? How magnetic fields appear? How particles are accelerated? What is the radiation mechanism?

9 Collisionless Regime Unlike a hydrodynamic shock, there is no single theory of a collisionless shock C. Kennel, R. Sagdeev, Ionized gas: Nonlinear waves Turbulence Wave particle interaction Particle kinetics Particles communicate via Electric and Magnetic fields Q.: Which model of a collisionless shock is correct? Sagdeev s A.: No, no! They all are correct! Different conditions.

10 Shock Zoo magnetized unmagnetized quasi-parallel quasi-perpendicular Laminar shocks (nonlinear waves) Alfvenic ion-acoustic magnetosonic Turbulent shocks (effective collisions) ion-acoustic Langmuir (E-static) fire hose (magnetic) two-stream [Weibel] (B-static) Mach

11 MHD regime Laminar shock nonlinear wave structure Non- Linear wave (soliton) + dissipation wave dispersion balances nonlinear steepening B

12 Collisionless Shock. MHD Magnetic field is needed to confine particles Dissipation is needed to match jump conditions N x L disp L disp ~ r L B

13 Kinetic regime Turbulent shock particles + fields Multiple stream region v dispersion cannot balance nonlinear steepening

14 Collisionless Shock. Kinetics Electric and/or magnetic fields are needed to randomize particles c/w p ω p ~ (4π e 2 n/m) 1/2 B E

15 A collisionless shock paradigm Electromagnetic fields can mediate interactions between plasma particles COLLISIONLESS SHOCK forms Magnetic field is generated at the shock front by the Weibel instability an instability which leads to formation of current filaments aligned with the shock propagation direction GRBs LSS & Cluster shocks Jets

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17 Zooming-in a Shock Electrons and protons form counter-propagating streams in front of the shock - unstable

18 Conditions at a shock shock ISM Anisotropic distribution of particles (counter-propagating streams) at the shock front e - p ISM Reflected Component

19 General remarks An anisotropic particle distribution is always unstable: counter-streaming or Weibel instability The instability generates magnetic fields, one always need to make sure that there is no a faster instability, which can isotropize particle distribution on a shorter time-scale!

20 General description Anisotropy: > characteristic energies (e.g., T) in the directions parallel and perpendicular to the shock propagation direction (or T) Introduce: plasma frequency anisotropy parameter w 2 (4.7, 3) 1/ 2 p( e, p) 4 e n / m( e, p) ~ 10 n A=( - )/ ~ (M-1)/(M+1) Growth rate Field scale Field strength g ~ (v/c) w p A 1/k ~ c/(w p A) B 2 /8 ~ ( - ) efficiency (from simulations)

21 Weibel instability /Weibel 59/ Vlasov equation Particle response Assume anisotropic PDF Dispersion relation

22 Weibel Instability /Weibel 59/ (cont.) For anisotropic Maxwellian PDF one obtains dispersion relation where assuming high anisotropy and then Dispersion relation Solution Growth rate

23 Weibel instability /relativistic/ (Medvedev & Loeb, 1999, ApJ) e v x B y J z J B current filamentation B - field produced x t ~ (g/n) 1/2 ms, l ~ 100 (gb/n) 1/2 km Produced magnetic field: * sub-equipartition * small-scale (<<Larmor)

24 Simulation Parameters e + e - cloud gv th = 0.1 c 3D Simulation 32 Million particles 200 x 200 x 100 cells (20 x 20 x 10 c 3 /w p 3 volume) (2 particles/species/cell) gv = 0.6 c e + e - cloud gv th = 0.1 c

25 Current Filamentation M = 6 Iso-surfaces: RED - positive J z BLUE - negative J z Contours are at n = 1.1 n 0 a) t = 10.1 w -1 p, b) t = 20.8 w -1 p, c) t = w -1 p, d) t = w -1 p

26 Magnetic Filaments M = 6 Iso-surfaces: From RED to GREEN : a) t = 10.1 w -1 p, b) t = 20.8 w -1 p, c) t = w -1 p, d) t = w -1 p

27 Instability saturation J z J inhibit current filamentation isotropize particle velocities l / r L ~ 1 B B ~ (g th +1) / [ 2 3/2 g th ] ~ 0.5 x y Magnetic fields scatter particles and provide effective collisions, l mfp ~ c / w p

28 Field evolution Linear instability, B ~ exp (w p t) Saturation, B ~ 0.1 Nonlinear stage, B ~ constant w p t Field is predominantly transverse, B-perp >> B-parallel

29 Particle distribution. Particles are randomized over pitch angle by the produced small-scale magnetic fields => Thermalization => Instability quenches

30 3D PIC simulations: -electron-positron pairs -relativistic particles ISM g cloud ejecta

31 More realistic shock simulation Relativistic shock, electron-proton plasma (gamma-ray burst) Edge-on view Proton density ISM-ejecta interface (co-moving frame) (Frederiksen, et al., 2003)

32 More realistic shock simulation Face-on view Shock plane, orthogonal to V shock Proton current density & B-field (Frederiksen, PhD, 2005) (Haugbolle, PhD, 2005

33 No proton isotropization Protons (blue) are still not izotropized after t ~ 70ω pp -1 Need longer simulations! (Frederiksen, et al., 2003)

34 Best ep-simulations (L. Silva & simul. Group, 2006) m p /m e = 100 Kin. Energy flux per particle

35 Electron density

36 Electron distribution

37 Electron-generated fields

38 Ion density

39 Ion-generated fields

40 Nonlinear stage. Early time z filament coalescence instability 2D gas of filaments B 0 = 2I 0 /cr 0, df = c -1 I 0 dl x B 0 size ~ separation ~ R 0 d 2 x/dt 2 ~ I 02 / (c 2 mnr 02 x) current I 0 t 0 ~ c (mn) 1/2 R 02 / I 0 x Coalescence: I N ~ I 0 2 N, R N ~ R 0 2 N/2 t 0 ~ t N y Magnetic field scale grows exponentially, t coal -1 ~ w p B 1/2

41 Nonlinear stage. Late time z filament coalescence instability 2D gas of filaments size ~ separation ~ R 0 current I 0 Coalescence: I N ~ I 0 2 N, R N ~ R 0 2 N/2 t 0 ~ t N x y After N ~ few, v=dx/dt ~ c Magnetic field scale grows linearly, l B ~ c t (Medvedev, et al, ApJ, 2005)

42 Field spectrum Scale grows exponentially for few w p -1 and then as a power-law ( B ~ constant) [w p /c] k ~ 1/t

43 Electron-positron simulations Evolution of B Electron-proton simulations m p /m e = 1860

44 The evolution of l B e - e + e - p medium medium Slope=0.8 Black non-relativistic Grey relativistic (R) l B ~ t As the spatial scale grows super-diffusively, the standard (diffusive) dissipation quenches (Medvedev, et al, ApJ, 2005)

45 Field dissipation Diffusive dissipation: 2 B B - ~ - B 2 2 t x l B Log B Weibel fields, ~0.8 Field scaling: l B t Solution: =0 diffusion <1/2 sub-diffusion >1/2 no dissipation =1/2 Log t B exp -2 [- lb ( t) dt] exp - A t, 1 exp t 1-2 (- t ), t 2-1 0, if 1 2 if 1 2 if 1 2 (Medvedev, et al, ApJ, 2005)

46 An important remark The growth of l B is NOT an MHD inverse cascade! It is a deeply KINETIC regime, r Larmor l B At (much) later times, when r Larmor << l B, the field turbulent evolution is described by the standard MHD INVERSE CASCADE.

47 The Weibel instability in brief Linear regime current filamentation B - field produced B(t) ~ B 0 exp(t/t) t 2 / w p ~ 10-3 s Kinetic energy is converted into magnetic field energy l 2 c / w p ~ 10 7 cm Saturation current filamentation inhibited isotropization of particle velocities l / r L ~ 1 B ~ (g th +1) / [ 2 3/2 g th ] ~ 0.5 Nonlinear regime filament coalescence instability 2D gas of filaments l (t) ~ c t Magnetic fields scatter particles and provide effective collisions, l mfp ~ c / w p Magnetic field scale grows linearly, t coal ~ c / l (Weibel, PRL, 1959; Medvedev & Loeb, ApJ, 1999, Medvedev, et al 2004)

48 Magnetized outflow: reconnection Small-scale field generation (Weibel instability) at a reconnection site Above --- Non-relativistic electron-positron pair plasma (Swisdak, Liu, J. Drake, ApJ, 2008) Not shown --- Relativistic electronpositron pair plasma (Zenitani & Hesse, PoP, 2008)

49 Laser-plasma interaction VULCAN laser syst. I > W/cm 2 ; Mylar target (Tatarakis, et al., PRL, 2003)

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51 (Credit: Spitkovky

52 (Credit: Spitkovky)

53 Major differences (Credit: Spitkovky)

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55 Rule of thumb Magnetized vs. Unmagnetized external medium Critical parameter: magnetization σ < 1% = unmagnetized Weibel > 1% = magnetized no Weibel + shock front ~ Larmor + field amplification is still possible via Alfvenic- or hose-type instability [Bell & Lucek; Malkov & Diamond ] (Spitkovski s simulations)

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57 2D ion shock: structure & PDF (Martins, Silva 2009)

58 2D ion shock: ion momentum (Martins, Silva 2009)

59 2D ion shock: ion energy (Martins, Silva 2009)

60 (Credit: Spitkovky) 2D ion+b shock: obliqueness

61 Electron heating density ions ε e ~30% electrons (Credit: Spitkovky) Electron heating= 50%(cross-shock electrostatic potential) + 50%(intrinsic heating by Weibel turbulence)?

62 Non-Fermi electron heating (Hededal, et al, 2005, PhD)

63 Electrostatic model Motion of electrons in electrostatic fields of ion currents local acceleration -- all electrons go trough filaments, but at different times lengthen cooling time by filling factor -- reversible on short time-scale -- irreversible in the long term: virialization of electrons in the potential + time-variable potential λ = l/(c/ω p ) E e - l current B (Medvedev, 2006)

64 Reconnection model Reconnection events during current coalescence can accelerate electrons -- permanent acceleration of electrons -- the efficiency depends on the filling factor of the filaments -- the characteristic energy is, again, ~ ebl (as in the electrostatic model) Typical size of the reconnection region ~ filament size ~ c/ω pp B ~< 2I 0 E ind dl= Φ/ t U electron ~ e E ind l ~ e (v/c)b λ(c/ω pp ) E ind and if the filling factor is not too small (not << 1), then, again: I 0 v ~ c B I 0 (Medvedev & Spitkovsky, 2007)

65 Electron distribution

66 Comparison to GRB afterglow Using ε e & ε B from Panaitescu (2005) we infer the value of λ for: best fit model (lowest χ 2 ) all good models (χ 2 /d.o.f. < 4) The Weibel shock theory works for afterglows! It is very surprising.

67 Field decays fast behind the shock (Chang et al 2008)

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69 Foreshock! P-parallel, protons P-parallel, electrons (Simulation by Spitkovsky)

70 downstream The model y B(γ(x)) x=0 x n CR n CR n CR x 1 x 2 >x 1 x 3 >x 2 γ 1 γ γ 2 > γ 1 γ γ 3 > γ 2 γ B B B λ(γ) λ(γ) λ(γ) (Medvedev & Zakutnyaya, ApJ, 2009)

71 Self-similar foreshock Consequences Assume steady state and neglect nonlinear effects: large effect region of pre-conditioning upstream is strongly of upstream magnetized on Weibel instability enough nonlinear to feedback explain efficient of B-fields acceleration (Li & Waxman, 2006) large-scale CR distribution fields function long life-time increase Shock radiative structure efficiency of afterglow shocks source CR of acceleration magnetic fields in galaxy clusters, at LSS formation shocks time evolution (Medvedev, of generated Silva, Kamionkovski, fields 2005) Valid at Typical field: B-field spectrum near a shock (Medvedev & Zakutnyaya, ApJ, 2009)

72 Simulations confirm (pair shock) (Keshet et al 2008)

73 Simulations confirm (pair shock) field strength increases spectrum flattens (Keshet et al 2008)

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75 Cooling & Weibel time-scales Inside the ejecta: Downstream an internal shock: from simulations Synchrotron cooling time Electron/proton dynamical time

76 Cooling & Weibel time-scales shock (Medvedev & Spitkovsky, in press) prompt prompt, large Γ-internal

77 Cooling & Weibel time-scales shock (Medvedev & Spitkovsky, in press) afterglow afterglow, strong explosion

78 Summary shock theory & simulations: magnetic field generation and saturation shock formation and evolution e + e - and e - p plasmas filament mergers nonlinear Weibel turbulence Fermi ion acceleration non-fermi electron heating foreshock is of TREMENDOUS importance internal shocks can be modeled numerically