GRB Afterglow Spectra Daniel Perley Astro September* 2005 * International Talk Like a Pirate Day

Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects log P 1/2 -(p+1)/2 m t cool =  m e c / σ T cβ 2  e 2  2 (B 2 /8π)  c = σ T  B 2 t 4 3 6π mec6π mec t cool α 1 /  e  c α 1 / t 'critical' e - : t = t cool  e =  c

Remember, P pk is const, so constant amount of energy is emitted at all  i m < < e pk (  i ) α  i 2 log e e 1/2 mm ii m e i

Subject Daniel Perley19 September 2005 GRB Afterglow Spectra Title Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Background Daniel Perley19 September 2005 GRB Afterglow Spectra The GRB Standard Model ISM Shocked Gas Earth SHOCK Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Background Daniel Perley19 September 2005 GRB Afterglow Spectra Relativistic Shock SHOCK ISM Γ number density n o energy density E o = n o m p c 2 energy per particle E o /n o = m p c 2 From Brian’s lecture… n′ = 4  n o E′ = 4  2  n o m p c 2 E′/n′ =  m p c 2  = Γ √2 >Compression< by 4  Energy Increase by factor  Deceleration by factor √ 2 Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Energy Deposition Where does the energy go? energy per particle E o /n o = m p c 2 E′/n′ =  m p c 2 Energy Increase by factor  Protons Electrons Magnetic field Other particles? E p = ε p E′ E e = ε e E′ B = ε B E′ Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Energy Deposition Daniel Perley19 September 2005 GRB Afterglow Spectra Proton/Electron Energy SHOCK ISMShocked Gas Γ  ee Extreme (relativistic) ‘temperature’ of shocked gas described by  p,  e Bulk motion of shocked gas relative to observer Particle energy deposited in random motions. Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Proton Energy Not particularly interesting on its own. Protons necessarily drag electrons with them at the same bulk velocity. Share energy with electrons: electron  factors necessarily much higher. Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Electron Energy Faster-moving electrons will radiate more efficiently by all important processes. ee Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Electron Energy Distribution Q: How is electron energy distributed? A:………? Hypothesis: Power-law? (Seen in SNe, NR shocks) Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Log N Log  N α  -p N α [Complicated] Model as power-law: Energy Deposition Electron Energy Distribution Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Electron Energy Distribution Simplify: cut-off power law at minimum energy Log N  Log  N  α  -p N α [Complicated] mm Minimum energy Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Electron Energy Distribution Mimimum energy determined by total energy density: n = ∫ N  e d  e E e = m e c 2 ∫  e N  e d  e = C  m 1-p = m e c 2 C  m 2-p Infinite if p<2 N  e e  e  -p mm n eN eN e e mm  e  1-p E 1 1-p C = (1-p)  m p-1 n 1 2-p = m e c 2  m n 1-p 2-p  m  p-2 p-1  EeEe n me c2n me c2 p-2 p-1  mpmp meme  εeεe ≈ 610 ε e  Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Electron Energy Distribution N e e  -p mm Limits on p:  N e e mm n = ∫ N  e d  e = 1 1-p N α  1-p C  m 1-p E = m e c 2 ∫  N  e d  e = m e c 2 2-p = C = (1-p) n  m 1-p C  m 2-p Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Magnetic Energy Strong post-shock magnetic field expected from equipartition. Generation mechanism unknown/complicated – various plasma effects B2B2 8π8π = ε B E′ = ε B 4  2  n o m p c 2 = 32π ε B  2  n o m p c 2 B2B2 B = 32π ε B n o m p  c ≈ ( 0.4 gauss) ε B 1/2 ( ) 1/2  nono cm -3 B Energy Deposition Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Emission Mechanisms How does it cool? Bremsstrahlung P α  e 3/2 n 2 Inverse Compton P = σ T cβ 2  e 2 U ph Synchrotron P = σ T cβ 2  e 2 U B Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Relativistic Cyclotron Relativistic modification to cyclotron frequency: ω cyc = e B  m c Most emission is not at this frequency. Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Synchrotron Beaming Emission is highly pulsed – we see emission for only 1/  2 of total emission time. ω cyc = e B  m c E t 1/ω cyc 1/  2 ω cyc - One factor of  from beaming angle - Additional factor of   from "Doppler" boost 1/  Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

1e - Synchrotron Spectrum E t 1/ω cyc 1/  2 ω cyc = E t δ(t-n/ω cyc ) = Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

1e - Synchrotron Spectrum E t = E t δ(ω-nω cyc ) ^ ^ Fourier transformed: ω cyc  2 ω cyc Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra 1e - Synchrotron Spectrum log P log  1/3 e - /ω cyc  1/2 pk Total Power: P = σ T cβ 2  2 U B α  2 U B Peak Freq.: pk ~  2 ω cyc 4 3 More precisely… Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra 1e - Synchrotron Spectrum log P  1/3 e - /ω cyc  1/2 pk log More precisely… Shocked frame: α  e 2 α const Total Power: P = σ T cβ 2  e 2 U B Peak Freq.: pk ≈  e 2 ω cyc / 2π Peak Power: P pk ≈ P / pk 4 3 Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra 1e - Synchrotron Spectrum log P log  1/3 e - /ω cyc  1/2 pk Shocked frame: α  e 2 α const Total Power: P = σ T cβ 2  e 2 U B Peak Freq.: pk ≈  e 2 ω cyc / 2π Peak Power: P pk ≈ P / pk 4 3 Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra 1e - Synchrotron Spectrum log P log  1/3 e - /ω cyc  1/2 pk Total Power: P = σ T cβ 2  e 2  2 U B Peak Freq.: pk ≈  e 2  ω cyc / 2π Peak Power: P pk ≈ P / pk Observer frame: α  e 2  2 α  e 2  α  4 3 Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Uncooled Multi-e - Spectrum log P log 1/3 pk exp Material contains many electrons at different velocities (  e ) – true spectrum is a combination of individual spectra, according to electron energy distribution. log N  log  e -p mm Electron distribution Electron spectrum Uncooled Sychrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Uncooled Multi-e - Spectrum Can just do a weighted sum (convolution) – but need to convert x-axis from  e to pk. log N  log  e -p mm Electron distribution From before, pk α  e 2 log N log pk m e - distribution: N   α  e -p Solve: N = N  α  e -p  e -1 α -(p+1)/2 dd d d dd α ee -(p-1)/2 Sign error?? Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Uncooled Multi-e - Spectrum Electron distribution log N log pk m -(p-1)/2 log P log 1/3 pk exp Electron spectrum Total Spectrum Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Uncooled Multi-e - Spectrum log N log pk m -(p-1)/2 log P log 1/3 pk exp log P 1/3 Daniel Perley19 September 2005 GRB Afterglow Spectra -(p-1)/2 m Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Uncooled Multi-e - Spectrum Daniel Perley19 September 2005 GRB Afterglow Spectra "Broken" Power law: Below m, emission dominated by low-  e - Above m, emission from electrons with peak (  ) = m log P 1/3 -(p-1)/2 m log pk Uncooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooled Synchrotron Characteristic Cooling Time Daniel Perley19 September 2005 GRB Afterglow Spectra This analysis is too simplistic for GRBs. Calculate characteristic cooling time: log P 1/3 -(p-1)/2 m t cool = E / P =  m e c / σ T cβ 2  2 U B ≈ 4 × s ( ) -2  B gauss Potentially much shorter than time since GRB (shock passage) log pk Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooled Synchrotron Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling e - Spectrum If an electron's energy changes significantly over the time since the energy injection, use an "averaged" spectrum for that electron.  e = Initial electron energy (at injection)  c ≡ Final electron energy (after cooling) ≈ Energy of the highest-  e - that hasn't cooled Determined by observational timescale: t obs =  c m e c / σ T cβ 2  c 2 U B  c = 4 3 6π mec6π mec σ T B 2 t obs Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooled Synchrotron Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling e - Spectrum Electron radiates as it cools, with a simple synchrotron spectrum corresponding to the instantaneous energy  i.  e = Initial electron energy  c ≡ Final electron energy log P log 1/3 pk exp Instantaneous spectrum ii Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling e - Spectrum Peak power radiated at each  i is the same: log P log 1/3 pk (  i ) exp Instantaneous spectrum  e = Initial electron energy  c ≡ Final electron energy ii P(  i ) = const log P  e e cc Electron evolution log  i Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

log P log 1/3 pk exp log N  log  e -p mm Electron distribution Electron spectrum log pk e e 2 mm ii m e i e e 2 mm ii m e i

Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling e - Spectrum From before, pk α  e 2 Power distribution: P   = const Solve: P = P  =  -1 = -1/2 dd d d dd α ee log P e c Another convolution - need to transform  e to pk. log P  cc Electron evolution const -1/2 e e log  i Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling e - Spectrum log P e c log 1/3 pk exp Instantaneous spectrumElectron evolution -1/2 Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooling e - Spectrum log P e c log 1/3 pk exp Instantaneous spectrumElectron evolution log P 1/3-1/2 Daniel Perley19 September 2005 GRB Afterglow Spectra -1/2 Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooling e - Spectrum Daniel Perley19 September 2005 GRB Afterglow Spectra log P 1/3 -1/2 c e Broken power law: > e : Exponential cut-off (model as no emission) c < < e : Instantaneous emission when electron passed through appropriate  < c : Post-cooling emission log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooling e - Spectrum Daniel Perley19 September 2005 GRB Afterglow Spectra log P 1/3 -1/2 m e Higher initial energy simply extends the curve to higher frequencies. log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling e - Spectrum t cool α 1 /  e  c α 1 / t 'critical' e - : t = t cool  e =  c log N  log  e -p mm So for c < < e : P = -1/2 Cooled Synchrotron

Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling Regimes Two possibilities for multi-electron spectra: log N  log  e -p mm log N  log  e -p mm  c <  m cc cc  c >  m ALL electrons will cool on given timescale : Fast cooling SOME electrons will cool on given timescale : Slow cooling Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Fast Cooling log N  log  e -p mm  c <  m cc ALL electrons will cool on given timescale : Fast cooling Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Fast Cooling log N log e -(p-1)/2 m c log P 1/3 -1/2 Cooled synchrotron spectrum Electron distribution Sum for multi-e - using the new spectrum: -p/2 c e Fraction of N > log P 1/3-1/2 log Daniel Perley19 September 2005 GRB Afterglow Spectra c m -1/2 -(p-2)/2 ?? Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Fast Cooling log P 1/3 -1/2 c m -p/2 Broken power law: > m : Emission from electrons with  e > , during passage through appropriate  c < < m : Emission from all electrons, during passage through appropriate  < c : Emission from all electrons at all times log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Slow Cooling log N  log  e -p mm cc  c >  m SOME electrons will cool on given timescale : Slow cooling Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Slow Cooling Fast-cooling electrons have fast-cooling spectrum, but with effective  m →  c (no -1/2 segment) log N  log  e -p mm cc -p/2 1/3 c log P log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Slow Cooling Non-cooling electrons have an uncooled-population spectrum, but cut off at c. log N  log  e -p mm cc 1/3 -(p-1)/2 m c log log P Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Slow Cooling By their powers combined… log N  log  e -p mm cc 1/3 m c -(p-1)/2 -p/2 1/3 -(p-1)/2 -p/2 log P log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Slow Cooling log P 1/3-(p-1)/2 m c -p/2 Broken power law: > c : Emission from cooling electrons with  e >   during passage through appropriate  m < < c : Emission from slow electrons with initial (constant) energy  < m : Emission from slow electrons with min.  m log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Daniel Perley19 September 2005 GRB Afterglow Spectra Cooling Comparison log P 1/3-(p-1)/2 m c -p/2 log P 1/3 -1/2 c m -p/2 Fast cooling Slow cooling log Cooled Synchrotron Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Cooled Synchrotron Daniel Perley19 September 2005 GRB Afterglow Spectra Synchrotron Self-Absorption Photon can be re-absorbed to excite an electron in a magnetic field (inverse of synchrotron emission.) Synchrotron emission/absorption will be in equilibrium below a certain frequency a : below this point the shocked gas is optically thick and will radiate as a blackbody (P α 2 ) 1/3 log P log 2 a Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Synchrotron Summary Daniel Perley19 September 2005 GRB Afterglow Spectra Complete Comparison log P 1/3-(p-1)/2 m c -p/2 log P 1/3 -1/2 c m -p/2 Fast cooling Slow cooling 2 2 log a a Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Observing Daniel Perley19 September 2005 GRB Afterglow Spectra Theory vs. Observations GRB – Galama et al t burst = 12.1 days > Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Subject Daniel Perley19 September 2005 GRB Afterglow Spectra Observable Parameters An instantaneous spectrum gives several key pieces of information: a c m p F pk z ε e ε B n o E' Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Subject Daniel Perley19 September 2005 GRB Afterglow Spectra Intervening ISM Effects Cosmological redshift will not affect power-law - all radiation scaled down by (1+z) Will see deviation from power-law in some frequency ranges in some cases: Galactic extinction (can be calculated/removed) Host extinction (similar to Galactic, but at higher frequencies, and cannot be estimated independently of GRB) Hydrogen absorption features (associated with high-z) Background GRB Standard Model Relativistic Shock Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e - Spectrum Multi-e - Spectrum Cooled Synchrotron Cooling Time 1e - Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects