Opacity Build-up in Impulsive Opacity Build-up in Impulsive Relativistic SourcesRelativistic Sources

Jonathan GranotJonathan GranotUniversity of HertfordshireUniversity of Hertfordshire

(Royal Society Wolfson Research Merit Award Holder)(Royal Society Wolfson Research Merit Award Holder)

High Energy Phenomena in Relativistic Outflows September 26, 2007, Dublin, Ireland

Collaborators: Collaborators: Johann Cohen-Tanugi,Johann Cohen-Tanugi, Eduardo do Couto e SilvaEduardo do Couto e Silva (KIPAC, Stanford Linear Accelerator Center)(KIPAC, Stanford Linear Accelerator Center)

Outline of the Talk:Outline of the Talk: MotivationMotivation (GRBs, flares in Blazars, GLAST) (GRBs, flares in Blazars, GLAST)

Description of the Description of the modelmodel

Outline of the Outline of the calculationcalculation

ResultsResults: opacity, light curves & spectra: opacity, light curves & spectra

Some Some intuitionintuition

ConclusionsConclusions

Motivation (1): We consider the We consider the opacityopacity to to pair productionpair production

( (γγγγ →→ e e++ee−−) ) within the source within the source (flaring region)(flaring region) Opacity effects are expected to be important in Opacity effects are expected to be important in

GLAST LAT energy range GLAST LAT energy range ((~ 20~ 20 MeV - 300MeV - 300 GeVGeV)) Above some photon energy Above some photon energy εε11, , ττγγγγ > 1 > 1 & the & the

spectrum is expected to cut off exponentiallyspectrum is expected to cut off exponentially Lack of such a cutoff up to an observed photon Lack of such a cutoff up to an observed photon

energy energy εεmaxmax ⇒⇒ ΓΓ 100[L 100[L0,520,52((εεmaxmax))αα-1-1// RR1313]]1/21/2αα

wherewhere εε = E = Ephph/m/meecc22 and and LLεε = L= L00εε1-1-αα

This was used to put a lower limit on assuming This was used to put a lower limit on assuming R ~ R ~ ΓΓ22cc∆∆t t wherewhere ∆∆t = t = observed variability timeobserved variability time

Motivation (2): Observing the high energy cutoff due to Observing the high energy cutoff due to ττγγγγ will will

determine determine ΓΓ22ααRR (instead of just a lower limit) (instead of just a lower limit) Together withTogether with anan independentindependent estimateestimate of of ΓΓ this this

can determine can determine RR and check if indeed and check if indeed R ~ R ~ ΓΓ22cc∆∆t t Some sources are highly variable, suggesting Some sources are highly variable, suggesting

impulsive emission (GRBs,impulsive emission (GRBs, flaresflares inin Blazars,…)Blazars,…) Initially there is no photon field & the opacity Initially there is no photon field & the opacity

builds-up with time builds-up with time ⇒⇒ eveneven εε > > εε11(steady state)(steady state) photons can initially escape, as long as photons can initially escape, as long as εε11(t) >(t) > εε

⇒⇒ a distinct temporal & spectral signaturea distinct temporal & spectral signature

Simple (yet rich) Semi-Analytic Model Ultra-relativistic (Ultra-relativistic (ΓΓ ≫≫11) spherical thin () spherical thin (∆∆ ≪≪ R/R/ΓΓ22) )

shell emits in a finite interval shell emits in a finite interval RR00 ≤ R ≤ R ≤ R ≤ R00++∆∆RR Isotropic emission in the shell co-moving frameIsotropic emission in the shell co-moving frame For simplicity For simplicity ΓΓ

22 ∝∝ RR-m-m,, L’ L’εε’’ ∝∝ ((εε’)’)1-1-ααRRb b is assumed is assumed while the formalism is more generalwhile the formalism is more general

observerat infinity

GRB

0R RR ∆+0

emissionregion

0RR ≤∆

0RR > >∆

impulsive

quasi-steadyγγ → γγ → ee++ee--

“turnson”

“turnsoff”

expanding shellexpanding shellCorresponds to a singleCorresponds to a singleflare/spike in light curveflare/spike in light curve

Calculation of the observed Flux: Flux calculation: integration over the equal Flux calculation: integration over the equal

arrival time surface of photons to the observerarrival time surface of photons to the observer The photon field is calculated at all spaceThe photon field is calculated at all space && timetime The pair-production optical depth is calculated The pair-production optical depth is calculated

byby integratingintegrating alongalong thethe trajectorytrajectory ofof eacheach photonphoton

observer

Equal arrival time surfaces

RL(T)

∆∆RRR0TT== photonphoton arrivalarrival timetime toto observerobserver

θθ = = emission angle from the l.o.s.emission angle from the l.o.s.t = t = emission time (in lab frame)emission time (in lab frame)

`

equal arrival time surfaceequal arrival time surface::

Expanding spherical ultra-relativistic shell observer

at infinity

θ t,0

t0

photon front

equal arrival times surface of photons

to the observer(EATS)

radius where the GRB source“turns on”

the shell emits a test photon

)(tR 1t

test photon is behind the shell

test photon trajectory

t1

test photon is ahead of shell

t3t2

test photon crosses shell

€

Ω i , n i = s o l i d a n g l e a n d n u m b e r d e n s i t y o f

t h e p o t e n t i a l l y i n t e r a c t i n g p h o t o n s

σ * = t h e p a i r p r o d u c t i o n c r o s s s e c t i o n

χ = c e n t e r o f m o m e n t u m e n e r g y

o f p h o t o n s i n u n i t s o f m e c 2

€

τ γγ ( R t , 0 , θ t , 0 , ε t ) = d s d ε i∫∫ d Ω i σ * χ ε t , ε i , µ t , i( )[ ]∫ 1 − µ t , i( ) d n i

d ε i d Ω i

R(t1)

R0

Rt,0

Calculating the γγ → e+e- Optical Depth

γt,0= γ(Rt,0)

€

s = p a t h l e n g t h a l o n g t e s t p h o t o n t r a j e c t o r y

ε t , ε i = e n e r g i e s o f t e s t p h o t o n a n d p o t e n t i a l l y

i n t e r a c t i n g p h o t o n i n u n i t s o f m e c 2

µ t , i = c o s i n e o f t h e a n g l e b e t w e e n t h e

d i r e c t i o n s o f t h e t w o p h o t o n s

Calculating the γγ → e+e− Optical Depth

At each point along the test photon trajectory the local At each point along the test photon trajectory the local photon field is calculated by integrating along the equal photon field is calculated by integrating along the equal arrival time surface to that space-time point:arrival time surface to that space-time point: EATS-IIEATS-II

Results: Light Curves & Instantaneous Spectra

Time integrated spectrum

Time of instantaneous

spectrum

one dynamical

time

T0 = time when first photon reaches the observer at infinity

1 GeV

Time Integrated Spectrum: Power law High Energy Tail

GBM LAT

1 GeV25 MeV

8 keV 300 GeV

1 MeV

GLAST:

Temporal signature:High energy photons, above the break in time integrated spectrum escape mainly near the onset of aflare or spike in the light curve

Theoretical Calculations

γγ → γγ → ee++ee--

high energy photons reach the observer

near the onset ofthe flare / spike

in light curve

sourceopaqueto γ-rays

γ-raysescapefreely

The opacity builds-up & saturates on a dynamical time scale

Validity of the Model Assumptions: Thin Shell: in internal shocks tcool ≪ tdynamic ⇒ thin

cooling layer behind the shock Spherical geometry: reasonably valid in GRBs;

should not qualitatively affect the main results Power law emission spectrum: only marginally

valid for GRBs ⇒ will be generalized Neglecting external opacity: valid for GRBs; not so

clear how valid in Blazar flares (but can be distinguished by lack of τγγ time dependence)

Single spike/flare: reasonably valid for spikes after quiescent period; vicinity to previous spike or flare would effect manly high energies ε ≫ ε1*

Conclusions: OpacityOpacity effectseffects cancan constrainconstrain thethe emissionemission radius & radius &

outflow Lorentz factor outflow Lorentz factor ⇒⇒ composition as well (composition as well (ee±± // pp // BB))

We developed a semi-analytic time dependent model for We developed a semi-analytic time dependent model for a single flare / spike in the light curvea single flare / spike in the light curve

Relevant for GRBs & perhaps also flares in BlazarsRelevant for GRBs & perhaps also flares in Blazars γγγγ →→ ee++ee−− opacity hasopacity has distinct observable signatures: distinct observable signatures:

Power law Power law high-energyhigh-energy tail tail in the in the time integratedtime integrated spectrumspectrum (instantaneous spec.(instantaneous spec. : exponential cutoff): exponential cutoff)

Photons above the spectral break would arrive Photons above the spectral break would arrive mainly near the onset of spikes in light curvemainly near the onset of spikes in light curve

We plan to improve our model - use a more realistic low We plan to improve our model - use a more realistic low energy spectrum & compare with GLAST dataenergy spectrum & compare with GLAST data

Why is there an exponential cutoff in the spectrum of a (quasi-) steady source?

RPhoton 1

Photon

2

If the emission and “absorption” are in the same region (e.g. by the same material), then photons can escape only from a thin layer of width ~R/τ at the edge of the emitting region: Lesc ~ Lemit/τ

For γγ → e+e− attenuation occurs also outside of the emitting region ⇒ τ2 ~ τ1 ~ σ nphR for steady sources ⇒ exponential cutoff

This assumes a ~ uniform nph

in emission region ⇒ requires reasonably localized emission Holds for a relativistic source