1. Can we probe the Lorentz factor of gamma-ray bursts from GeV-TeV spectra integrated over internal shocks ?
Junichi Aoi　(YITP, Kyoto Univ.)
co-authors:
Kohta Murase
Keitaro Takahashi
Kunihito Ioka
Shigehiro Nagataki
the method for proving the Lorentz factor of GRB.
TeV particle astrophysics /July/09

2. Gamma-ray bursts: prompt emission
highly variable light curve　~0.1 sec → cδT ~ 3×109cm
High energy photons ε~ 1 MeV
emission from
compact region
Compactness problem
Large number density of high energy photons nγ~1026 cm-3
Optically thick against gg → e+e-
high energy photons can not escape from
emission region.
conflict with observations
~1MeV

3. Gamma-ray bursts: prompt emission
highly variable light curve　~0.1 sec → cδT ~ 3×109cm
High energy photons ε~ 1 MeV
emission from
compact region
Compactness problem
Large number density of high energy photons nγ~1026 cm-3
Optically thick against gg → e+e-
One solution:
blue shift with relativistic outflows
In this case, there may be also cutoff
due to gg → e+e-
There is no cutoff observation.
~ecut
~1MeV

4. Importance of cufoff energy observation gg → e+e- : electron-positron pair-production
optical depth against gg → e+e-
definition of cutoff energy tgg(ecut)=1 ‘
solve these equations for Γ
inside of emission region
low energy photon
high energy photon
cutoff energy is important to constrain the
Lorentz factor.

5. Internal shock model relativistic outflow Compact object
Jet
Rs~1013~1015cm
(1)
relativistic outflow from
a system including compact object E~1051erg
(2)
Inhomogeneous part
collides with each other.
(3)
shock dissipation
bulk kinetic energy
　 → internal energy
(4)
emission from accelerated
electron

6. our study
previous study
our study
emission from a single shell
time independent
calculating energy spectrum of emission from multiple shells
using the internal shock model
consider electron-positron pair production;　 γγ→e+e- .
examine a time-integrated spectrum
aim: probe the Lorentz factor of a GRB from the time-integrated spectrum

7. method A ・・・ fluctuation of initial released energy by collision
N shells (Γ,n,l,r), spherical symmetry, equal separation
Random initial Lorentz factor (log-normal distribution)
A ・・・ fluctuation of initial
　　Lorentz factor distribution
released energy by collision
energy, momentum conservation Γs Γm Γr
inelastic collision
Calculate Eint for every collisions
1. calculate a spectrum of emission from a single shell
2. sum up spectra from each shell　
→　compare this spectrum with sensitivity curve of Fermi

8. result ~energy spectrum~
energy spectra of a single pulse
z=1, luminosity=1052erg s-1
There is cutoff due to pair
production in each spectrum
gg → e+e-
Flux is lower than a sensitivity
curve when emission comes
from broad region.
We have to sum up (time integrate) spectra

9. result ~energy spectrum~
time-integrated 　energy spectrum
z=1, luminosity=1052erg s-1
No pair-production cutoff
(Rem：Cutoff around 1011eV
is due to Cosmic Infrared
Background.)
Slope becomes softer above
some energy.
pair-break energy
cutoff is smeared by summation
fluctuation of initial Lorentz factor distribution
features
large A: slope gradually becomes softer at large energy.
large A: pair-break energy is smaller than for small A.
shells collide at smaller radius for large A.

10. On estimate of Lorentz factors
Cutoff is hidden by emission from multiple shells
Cutoff is smeared.
Maximum energy photon does not give the lower limit of Lorentz factor.
We can use a pair-break energy instead of cutoff energy. ~
the minimum cutoff energy in two shell collision
Γtrue: true value
Γupper: derived from pair-break energy
Γco: derived from maximum energy
advantages
The pair-break energy is produced by the inner collision with a short pulse.
It is smaller than a (maximum) cutoff energy in general. It can be smaller than
the CIB attenuation energy.
This spiky pulse is easier to observe than a broad pulse.

11. Comparison with recent observation of Fermi
GRB080916C
There is not observation of a cutoff energy and a pair-break energy.
observed maximum energy is 3 GeV in the main pulse
light curve observed by Fermi
there is a single pulse at
time interval (b).
(abdo+ 09)
minimum Lorentz factor ~ 890
light curve observed by INTEGRAL
There are multi pulses at time interval (b). Assume
a pair-break energy is determined by one of them.
assume there is a pair-break energy
Lorentz factor
(Greiner+ 09)

12. conclusion
The conventional exponential cutoff should be modified to a steepened power-law in practical observations that integrate emissions from different internal shocks.
There may be a pair-break energy around ~1 GeV.
　　This break energy may be observed by Fermi
We can use the pair-break energy to probe the Lorentz factor of GRBs. The smearing effect generally reduce the previous estimates of the Lorentz factor.
GRB C:
The Lorentz factor can be ~ 600, which is below but consistent with the previous result of ~900.