Electron Synchrotron Emission in GRB-SN Interaction: First Results 1,2, Remo Ruffini 1,2, Narek Sahakyan 3 1 Sapienza - Università di Roma, Rome, Italy 2 ICRANet, Pescara, Italy 3 ICRANet - Armenia, Yerevan, Armenia
Outline Introduction Kinetic equation: numerical solution Code testing Electron and synchrotron spectra under different initial injection Future plans
Introduction Long GRBs classified as BdHNe (E iso > 10 52 erg) show characteristic time power law behaviour in late afterglow phase (Giovanni s talk) Synchrotron cooling is proposed as the dominant cooling mechanism Sponge model fragmented ejecta with bubbles moving inside Kinetic energy losses come from relative drag of bubbles within the ejecta (first approximation matter collection from surrounding media).
Introduction
Introduction Bubble regime Medium regime Time power law index k constant constant 3 free expansion free expansion 1 Sedov expansion Sedov expansion 2.2 Sedov expansion constant size 4.6 constant size Sedov expansion 1.25 Sedov expansion free expansion 2.2 free expansion Sedov expansion 4.6 constant size free expansion 3 free expansion constant size 7 Data Giovanni s talk 1.5 Toy model calculation - more precise MHD approach is necessary but this gives us an general idea.
Introduction To reproduce the photon spectra and the light curves it is essential to know the evolution of electron spectra under synchrotron cooling it is necessary to solve kinetic equation.
Introduction Kinetic equation For a spatially homogeneous source n(γ, t) n(γ, t) = ( γ(γ, t) n(γ, t)) + q(γ, t) t γ t esc Energy losses Including adiabatic, synchrotron and inverse Compton losses γ = v R γ + 4σ T c m c 2 (u B + u 0 F KN ) γ 2...
Introduction Solving kinetic equation In case of time independent injection rate q(γ): n(γ, t t 0 ) = 1 γ 0 γ 1 q(γ dz ) exp dγ, (1) γ(γ) γ(z)τ(z) where γ 0 is defined through γ t t 0 = γ 0 γ γ dγ γ(γ ), (2) Injection rate and energy losses are expected to be time dependent!!! we need numerical approach.
Kinetic equation: numerical solution We use the fully implicit difference scheme proposed by Chang & Cooper (1970) and implemented by Chiaberge & Ghisellini (1999). Energy mesh ( ) j 1 γmax jmax 1 γ j = γ min, γj = γ j+1/2 γ j 1/2, γ min Including the time step t we define n i j = n(γ j, i t), q i j = q(γ j, i t), γ i j = γ(γ j, i t).
Kinetic equation: numerical solution Discretization of kinetic equation V 3 j n i+1 j+1 + V 2 jn i+1 j + V 1 j n i+1 j 1 = ni j + q i j t, V 1 j = 0, V 2 j = 1 + t + t γ j 1/2, V 3 j = t γ j+1/2 t esc γ j γ j System of equations forms a tridiagonal matrix which can be solved numerically (Press et al., 1989).
Code testing 10 2 Test of kinetic equation code for α inj. = 2.2 without cutt-off versus analytic solution 10 1 10 0 analytic mesh = 50 mesh = 100 mesh = 300 mesh = 500 10 1 γ α N 10 2 10 3 10 4 10 5 10 5 10 6 10 7 γ 10 8 10 9 10 10
Code testing 10 1 Cut-off dependence on maximum Lorentz factor γ max With α = 2, t esc = 10 20, j max = 200 and γ min = 0.1 10 0 10 1 γ α N 10 2 10 3 10 4 10 5 γ max = 10 8 γ max = 10 9 γ max = 10 10 γ max = 10 11 γ max = 10 12 10 0 10 3 10 6 10 9 10 12 γ
Electron and synchrotron spectra 10 9 Electron distribution for B = 1G and t esc. = 10 20 s (Δt = 1s) Q(γ, t)=10 6 γ -2 θ(500-t[s]) 10 8 γ 2 N 10 7 t = 300s t = 400s t = 500s t = 505s t = 525s t = 550s 10 6 10 5 10 6 10 7 10 8 10 9 γ Figure: Evolution of electron spectra for energy power law injection with sudden cut-off time q = q 0γ 2 θ(t 0 t)
Electron and synchrotron spectra 10 3 Synchrotron spectra for Q(γ, t) = 10-6 γ -2 Θ(500-t[s]) 10 4 ε 2 F(ε) (erg cm -2 s -1 ) 10 5 10 7 t = 500s t = 502s t = 504s t = 506s t = 508s t = 510s t = 520s t = 530s 10 0 10 3 10 6 10 9 10 12 10 15 10 18 ε (ev) Figure: Synchrotron spectra for energy power law injection with sudden cut-off time q = q 0γ 2 θ(t 0 t)
Figure: Electron and synchrotron spectra for injection with form q = q 0γ 2 exp( γ/γ 0)(t + t 0) α and magnetic field B = 1 G. 10 12 10 6 10 12 10 9 10 6 10 12 10 9 10 6 N(γ) 1 N(γ) 10 3 1 α = 1.5 α = 2.0 α = 1.0 10 9 10 9 1 10 100 10 4 10 5 10 6 10 7 1 10 100 10 4 10 5 10 6 10 7 1 10 100 10 4 10 5 10 6 10 7 γ γ γ 0,1 0,1 0,1 N(γ) 10 3 1 ε 2 F(ε) (erg cm -2 s -1 ) 0,01 10 3 10 4 10 5 ε 2 F(ε) (erg cm -2 s -1 ) ε 2 F(ε) (erg cm -2 s -1 ) 10 3 1 1000 10 6 10 9 10 3 1 1000 10 6 10 9 ε (ev) ε (ev) t = 100s t = 250s t = 500s t = 1000s t = 1000s t = 3000 10 3 1 1000 10 6 10 9 ε (ev)
Electron and synchrotron spectra 10 15 10 12 10 9 10 6 N(γ) 1000 1 10 3 100s 200s 300s 400s 500s 1000s 1500s 2000s 1000 10 4 10 5 γ Figure: Electron spectra for constant mono-energetic injection q = q 0 rect( γ γ 0 ) and magnetic field B = 10 G. γ
Electron and synchrotron spectra 10 9 ε 2 F(ε) (erg cm -2 s -1 ) 10 6 10 3 10 0 ε (ev) 100 s 200 s 300 s 400 s 500 s 1000 s 1500 s 2000 s 10 3 10 0 10 3 10 6 Figure: Synchrotron spectra for constant mono-energetic injection q = q 0 rect( γ γ 0 ) and magnetic field B = 10 G. γ
Electron and synchrotron spectra Monoenergetic like injection with turnoff time at 100 s and decaying magnetic field 10 6 10 3 50s 100s 150s 200s 250s 500s 1000s 2000s N(γe) 10 0 10 3 10 9 10 3 10 4 10 5 γe Figure: Electron spectra for mono-energetic injection with time cut-off q = q 0 rect( γ γ 0 γ ) θ(t0 t) and magnetic field decay B(t) = B0 (R(t)/R0) 1.5.
Electron and synchrotron spectra Monoenergetic like injection with turnoff time at 100s and decaying magnetic field ε 2 F(ε) (erg cm -2 s -1 ) 1000 1 50s 100s 150s 200s 250s 500s 1000s 2000s 10 3 10 3 1 1000 10 6 ε (ev) Figure: Synchrotron spectra for mono-energetic injection with time cut-off q = q 0 rect( γ γ 0 γ ) θ(t0 t) and magnetic field decay B(t) = B0 (R(t)/R0) 1.5.
Summary and future work For certainty code requires calculation in energy range at least one order of magnitude greater then typical electron energies. We already managed to explore some interesting scenarios like the turn off of injection and time power law injection with or without time dependence in energy losses This code has shown to be quite stable and fast and presents itself as a powerful tool for various astrophysical phenomena. Concise understanding of particle acceleration through analytical approach and/or numerical modeling presents itself as a future step. Turbulence, magnetic reconnection, diffusive shock acceleration...
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