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GRB Jet Dynamics

Jonathan Granot Open University of Israel. GRB Jet Dynamics. Gamma-Ray Burst Symposium, Marbella, Spain, Oct. 8, 2012. Jet angular structure & evolution stages Magnetic acceleration: overview & recent results Jet dynamics during the afterglow: brief overview

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GRB Jet Dynamics

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  1. Jonathan Granot Open University of Israel GRB Jet Dynamics Gamma-Ray Burst Symposium, Marbella, Spain, Oct. 8, 2012

  2. Jet angularstructure & evolutionstages Magnetic acceleration: overview & recent results Jet dynamics during the afterglow: brief overview Analytic vs. numerical results: a discrepancy? Recent numerical & analytic results: finally agree Simulations of an afterglow jet propagating into a stratified external medium:ρextR−kfork = 0,1,2 Implications for GRBs: jet breaks, radio calorimetry Outline of the talk:

  3. The Angular Structure of GRB Jets: • Jet structure: unclear(uniform, structured, hollow cone,…) • Affects Eγ,isoEγ & observed GRB rate true rate • Viewing-angle effects (afterglow & prompt - XRF) • Can also affect late time radio calorimetry wide jet: 0 ~ 20-50 w n narrow jet: 0 > 100 • Here I consider mainly a uniform “top hat” jet (JG 2005)

  4. Stages in the Dynamics of GRB Jets: • Launching of the jet: magnetic (B-Z?) neutrino annihilation? • Acceleration: magnetic or thermal? • For long GRBs: propagation inside progenitor star • Collimation: stellar envelope, accretion disk wind, magnetic • Coasting phase that ends at the deceleration radius Rdec • At R >Rdecmost of the energy is in the shocked external medium: the composition & radial profile are forgotten, but the angular profile persists (locally: BM76 solution) • Once Γ <1/θ0 at R >Rjetjet lateralexpansion is possible • Eventually the flow becomes spherical approaches the self-similar Sedov-Taylor solution

  5. The σ-problem: for a “standard” steady ideal MHDaxisymmetric flow • Γ∞~σ01/3&σ∞~σ02/31forasphericalflow;σ0=B02/4πρ0c2 • However, PWN observations (e.g. the Crab nebula) implyσ1 after the wind termination shock – the σproblem!!! • A broadly similar problem persists in relativistic jet sources • Jet collimation helps, but not enough: Γ∞~σ01/3θjet-2/3, σ∞~(σ0θjet)2/3 & Γθjet≲σ1/2 (~1 for Γ∞~Γmax~σ0) • Stillσ∞≳1inefficientinternalshocks, Γ∞θjet1inGRBs • Suddendropinexternalpressurecangive Γ∞θjet1 butstillσ∞≳1(Tchekhovskoyetal.2009)inefficientinternalshocks

  6. Alternatives to the “standard” model • Axisymmetry: non-axisymmetric instabilities (e.g. the current-driven kink instability) can tangle-up the magnetic field (Heinz & Begelman 2000) • If then the magnetic field behaves as an ultra-relativistic gas: magnetic acceleration as efficient as thermal • Ideal MHD:a tangled magnetic field can reconnect (Drenkham & Spruit 2002; Lyubarsky 2010 - Kruskal-Schwarzschild instability (like R-T) in a “striped wind”) magneticenergy  heat (+radiation)  kineticenergy • Steady-state: effects of strong time dependence (JG, Komissarov & Spitkovsky 2011; JG 2012a, 2012b)

  7. v ImpulsiveMagneticAcceleration:ΓR1/3 our simulation vs. analytic results  Useful case study: B Initial value of magnetization parameter: t0 ≈ R0/c Δ tc ≈ Rc/c 1 2 3 (JG, Komissarov & Spitkovsky 2011) vacuum “wall” • 1. ΓE≈σ01/3by R0~Δ0 • 2. ΓER1/3between R0~Δ0 & Rc~σ02R0 and then ΓE≈σ0 • 3. AtR>Rcthesellspreadsas ΔR & σ~Rc/R rapidlydrops • Complete conversion of magnetic to kinetic energy! • This allows efficient dissipation by shocks at large radii

  8. Impulsive Magnetic Acceleration: single shell propagating in an external medium acceleration&decelerationaretightlycoupled(JG 2012) II. Magnetized “thick shell” “Thin shell”, low-σ: strong reverse shock, peaks atTGRB “Thick shell”, high-σ: weak or no reverse shock, Tdec~TGRB likeII, but the flow becomes independent of σ0 a Newtonian flow (if ρextisvery high, e.g. inside a star) II*.ifρextdrops very sharply I. Un-Magnetized “thin shell”

  9. Many sub-shells: acceleration, collisions (JG 2012b) steady impulsive    Flux freezing(ideal MHD): Φ ~ BrΔ = constant EEM ~ B2r2Δ 1/Δ        Δ      r Δgap Δ      r Δ r      shell width Δ grows constant shell width Δ • For a long lived variable source (e.g. AGN), each sub shell canexpandby1+Δgap/Δ0σ∞= (Etotal/EEM,∞−1)−1 ~ Δ0/Δgap • For a finite # of sub-shells the merged shell can still expand • Sub-shells can lead to a low-magnetization thick shell & enable the outflow to reach higher Lorentz factors

  10. Afterglow Jet Dynamics:2D hydro-simulationsvery modest lateral expansion Proper emissivity Proper density (JG et al. 2001) • Emissionmostlyfromfront, slowmaterialatthesides Bolometric Emissivity(logarithmic color scale) • Proper Density(logarithmic color scale)

  11. Analyticvs.Numericalresults:aproblem? • Analytic results (Rhoads1997,99; Sari,Piran&Halpern99): exponential lateral expansion at R > Rjet e.g.Γ~ (cs/c0)exp(-R/Rjet), jet~ 0(Rjet/R)exp(R/Rjet) • Supportedbyaself-similarsolution (Gruvinov 2007) • Hydro-simulations: verymildlateralexpansion while jet is relativistic (also for simplified 2D  1D) θ95%/θ0 same Eiso • (Wygoda et al. 2011) large lateral expansion for Γ ≤ 1/θ0 99% 95% 90% Modest θ0 small region of validity jet= 0exp(t/tjet) jet= 0exp(t/tjet) 50% • (Wygoda et al. 2011) (Zhang & MacFadyen 2009)

  12. Analyticvs.Numericalresults:aproblem? • van Eerten & MacFadyen 11’ • No exponential lateral expansion even for θ0 = 0.05 • Lateral expansion is instead onlylogarithmic • Affectsjetbreakshape + tj & late time radio calorimetry • Lyutikov 2011 • Lateral expansion becomes significantonlyforΓ≤θ0−1/2 • Based on thin shell approx. • (Kumar & JG 2003) • (van Eerten & MacFadyen 2011)

  13. Generalized Analytic model (JG & Piran 2012) • Lateral expansion: • 1. newrecipe: βθ/βr~1/(Γ2Δθ)~1/(Γ2θj)(based on ) • 2. old recipe: βθ= uθ/Γ = u’θ/Γ ~ βr/Γ(based on u’θ ~ 1) • Generalized recipe: • Newrecipe: lowerβθforΓ>1/θ0buthigherβθforΓ<1/θ0 • Does not assume Γ≫1 or θj≪1 (&variable: Γu=Γβ) • Sweeping-upexternalmedium:trumpetvs.conicalmodels

  14. Generalized Analytic model (JG & Piran 2012) • Main effect of relaxing the Γ≫1, θj≪ 1 approximation: quasi-logarithmic (exponential) lateral expansion for θ0≳ 0.05 • conical ≠ rel. for r≳rcwhile trumpet ≠ rel. for θj≳ 0.2 New recipe relativistic ρextR−k trumpet conical rc=[(3-k)/2]2/(3-k)

  15. Comparison to Simulations (JG & Piran 2012) • There is a reasonable overall agreement between the analytic generalized models and the hydro-simulations • Analytic models: over-simplified, but capture the essence • 2D hydro-simulation by F. De Colle et al. 2012, with θ0= 0.2,k = 0 trumpet conical simulation relativistic simulation

  16. Afterglow jet in stratified external media(De Colle, Ramirez-Ruiz, JG & Lopez-Camara 2012) • Previous simulations were all for k = 0 where ρextR−k • Larger(e.g. k =1, 2) are motivated by the stellar wind of a massive star progenitor for long GRBs k = 0 k = 1 k = 2 ΓBM=2 ΓBM=5 Logarithmic color map of ρ ΓBM=10 θ0 = 0.2, Eiso = 1053 erg, next(Rjet) ~ 1 cm−3

  17. Afterglow jet in stratified external media(De Colle, Ramirez-Ruiz, JG & Lopez-Camara 2012) • Previous simulations were all for k = 0 where ρextR−k • Larger(e.g. k =1, 2) are motivated by the stellar wind of a massive star progenitor for long GRBs • At the same Lorentz factor larger k show larger sideways expansion since they sweep up mass and decelerate more slowly (e.g. MR3−k, ΓR(3−k)/2 in the spherical case) and spend more time at lower Γ(and βθdecreases withΓ) k = 1 k = 0 k = 2 ΓBM=10, 5

  18. Afterglow jet in stratified external media(De Colle, Ramirez-Ruiz, JG & Lopez-Camara 2012) • Swept-up mass: a lot at the sides of the jet at large angles • Energy, emissivity: near the head • Spherical symmetry approached later for larger k tNR(Eiso)

  19. Afterglow jet in stratified external media(De Colle, Ramirez-Ruiz, JG & Lopez-Camara 2012) • For k = 0 the growth of Ris stalled at tNR(Eiso) while R continues to grow  helps approach spherical symmetry • Less pronounced for larger kas the slower accumulation of mass enables Rto grow more  become spherical more slowly R R t / tjet

  20. The shape of the jet break • Jet break becomes smoother with increasing k (as expected analytically; Kumar & Panaitescu 2000 – KP00) • However, the jet break is significantly sharper than found by KP00  better prospects for detection • Varying θobs < θ0 dominates over varying k≲ 2 Temporal index Lightcurves k = 0 k = 2

  21. Late time Radio emission & Calorimetry • The bump in the lightcurve from the counter jet is much lesspronouncedforlargerk (as the counter jet decelerates & becomes visible more slowly)  hard to detect • The error in the estimated energy assuming a spherical flow depends on the observation time tobs & on k Flux Ratio: 2D/1D(Ejet) Radio Lightcurves

  22. Magnetic acceleration: likely option worth further study Jet lateral expansion: analytic models & simulations agree For θ0≳ 0.05 the lateral expansion is quasi-logarithmic(exponential), due to small dynamic range 1/θ0 >Γ≫ 1 For θ0≪ 0.05 there is an exponential lateral expansion phase early on (but such narrow GRB jets appear rare) Jet becomes first sub-relativistic, then (slowly) spherical Jet in a stratified external medium: ρextR−kfork = 0,1,2 larger k jets sweep-up mass & slow down more slowly sideways expansion is faster at t <tj & slower at t >tjbecome spherical slower; harder to see counter jet Jet break is smoother for larger k but possibly detectable Jet break sharpness affected more by θobs < θ0 than k≲ 2 Radio calorimetry accuracy affected both by tobs & k Conclusions:

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