Tests of Curvature Effects in the
Temporal and Spectral Properties of
GRB Pulses
Ashwin Shenoy1
In collaboration with
Eda Sonbas2, Charles Dermer3,
Kalvir Dhuga1, Leonard Maximon1,
William Parke1 and Glen MacLachlan1
1. Physics Department, The George Washington University
2. NASA Goddard Space Flight Center
3. Naval Research Laboratories, Space Sciences Division
An Outline of this presentation

Motivation.

A relativistic colliding shell model.

Some previous work.

Selection of GRBs.

A test of curvature.

Results and Summary.

Proposed future work.
Motivation

Curvature dominated models with different cooling mechanisms
such as synchrotron emission
f Єpk(t) ~ Єpkλ


Tests of this form will allow us to distinguish between, or
validate/constrain such models.
These relations could lead to an estimate of the bulk lorentz
factor of the jet.
Predictions of the Model (Dermer 2004.)

Thin shells

Spherically symmetric and emit homogenously.

The spectrum follows a broken power law.

Curvature effects dominate at later times and f Єpk(t) ~ Єpk3
Some Previous Work

Soderberg and Fenimore (2001).
- Intensity vs. time, analogous to
f Єpk(t) ~ Єpk3 - (a+b)
- 2 GRBs, with a negative result.

Borgonovo and Ryde (2001).
- BATSE GRBs: f Єpk(t) ~ Єpkλ with 0.6 ≤ λ ≤ 3.
- Pulse properties within a GRB are similar.
Light curve selection and extraction


GRBs with high fluxes and count
rates. (GBM circulars, Nava et al.
2009).
Choice of 3/4 brightest NaI detectors
from GBM quicklook products.
In this case: Detectors 0,6,7 and 9.



- Background-subtracted light curves.
- 200 μs binning.
- Energy range: ~ 8 keV - 1MeV.
Light curves were re-binned until pulse
structures were visible.
GRBs with distinct peaks were
selected and analyzed.
Acknowledgement: Narayan Bhat.
GRB 081224887




Single peak GRB with
T90 ~ 50 secs
Pulse has a FRED
profile
The Peak is at ~2.6 secs.
FWHM for this pulse is
about 10 secs.
Temporal fit of the Pulse-Decay


Decay portion of the
pulse is fit by C*(t-tp)α
We chose the squareroot of the count rates
as their errors.
Pulse fit parameters of selected GRBs
GRB
Name
α
α
error
Best fit range
(secs)
Chi2
Range for
Spectral
analysis
081224887
-0.45
0.012
5.0 - 14.0
1.23
4 – 10
(1-second)
110625881
-0.93
0.034
11.5 - 13.5
1.63
11 – 13.5
(0.5-second)
081009140
-5.66
0.133
42.0 - 49.0
1.23
Poor
Spectral fit
Light Curve Segments for Spectral fits




3 - 4 brightest
detectors were chosen.
Pre- and post- burst
regions were selected
for background
subtraction.
Light curves were
background-subtracted
with a polynomial fit.
Divided into 1 second
segments (4.0 – 9.0
seconds)
Spectral Fitting



For each time
segment the spectrum
was fit with a Band
function.
The Epeak was
determined
The flux in the Epeak
region was extracted.
Results for Band function fits for GRB
081224887
Time
Epeak (keV)
Segment
(secs)
α
β
Flux
10-7 ergs/cm2. s
4 -5
207.0 +/- 24.6
-0.44 +/- 0 .09
-1.90 +/- 0.12
2.98 +/- 0.17
5-6
183.3 +/- 27.2
-0.49 +/- 0.12
-1.89 +/- 0.13
2.60 +/- 0.16
6-7
178.9 +/- 26.7
-0.53 +/- 0.13
-2.09 +/- 0.22
2.00 +/- 0.15
7-8
154.3 +/- 42.4
-0.63 +/- 0.20
-1.75 +/- 0.12
2.47 +/- 0.17
8 -9
159.1 +/- 44.2
-0.69 +/- 0.21
-1.98 +/- 0.27
1.88 +/- 0.18
Єpk3
Summary



We have selected bright GRBs with FRED like pulse shapes.
We analyzed the decay portion of these pulses by extracting
spectral parameters via a band function fit for time slices of
the pulse and extracted the Epeak and the fluxes under the Epeak
region.
We looked for curvature effects in GRB prompt emission
data via the relation
- f Єpk(t) ~ Єpk3

We tested this relation for a few candidate GRBs and we
found encouraging results.
Proposed Future Work

Identify other candidate GRBs including short GRBs for GBM

Extend the analysis to BATSE and Swift GRBs.

Test other relations such as (Kumar and Paneitescu, 2000)
- α~2-β
- Fv (t) ~ v β (t-tpk)α

Move from a relatively simple kinematic model to a more
sophisticated phenomenological model
http://iopscience.iop.org/0004-637X/614/1/284/pdf/60132.web.pdf

Test the rise profiles of GRB pulses.
The Talk Ends Here
Following are
extra/discarded
slides
The colliding shell model
We looked at the light curves of ~70 bright GRBs. On close inspection, only the
following 3 GRBs had distinct single peak-pulse profiles without secondary
overlapping peaks. We fit the temporal profile of these peaks with a simple power
law. As is customary, we chose the square root of the counts as the error for each
count. The table below provides the parameters of the fit.
GRB Name α % error Reduced Chi^2 Best fit range for α (in secs) β
GRB 090718762 4.59 +/- 0.382 8.34 1.45 23.5 - 26.5
GRB 081224887 0.45 +/- 0.012 2.68 1.23 5.0 - 14.0 ~ 1.30
GRB 081009140 5.66 +/- 0.133 2.36 1.23 42.0 - 49.0
GRB 110625881 0.81 +/1.63 11.5 13.5
As the table shows, only one GRB (GRB 081224887) satisfies the equation
α ~ 2 - β eqn.1
We then tested the following 2 equations for this GRB.
f ~ Єpk3 eqn.2, And
Fν ~ ν-β (t-tpk)-α eqn.3.
This is a single peak GRB with the peak at ~2.6 secs. The following 2
figures show the light curve and the zoomed in profile of the pulse
respectively. We fit the decaying portion of the pulse with a power law of
the form C*(t-tpk)-α
To obtain the νFν corresponding to this peak photon energy, E_peak, we chose an energy
range, E_peak ± 20 keV and determined the νFν values in this range. Following are the
slides detailing the spectral fitting procedure corresponding to one time-segment (3- 5 s).
To obtain the νFν corresponding to this peak photon energy, E_peak, we chose an energy
range, E_peak ± 20 keV and determined the νFν values in this range. Following are the
slides detailing the spectral fitting procedure corresponding to one time-segment (3- 5 s).
The figure shows the time-segment extraction and the background fit to the light curve.
The unfolded spectrum (convoluted with the detector response) corresponding to the light
segment.
These steps applied for N7 and N9 detectors as well.
Spectral Fitting
Rmfit supplies several test statistics. We
used C-Stat which is -2log(likelihood)
and performed a spectral fit for each of
the 3 detectors using both the Band
function and a simple power law.
The fluence and other parameters are extracted in the range E_peak +/- 20 keV.
To confirm our results obtained using RMFITS, the same fit was performed with
XSPEC. The results were found to be compatible for these two methods.
f(t) vs. Єpk -β (t-tpk)-α correlation for
each time segment.
To test this relation the spectral analysis was performed on a time resolved spectrum
(64 ms resolution). It was found that the spectral index for an energy range E_peak
+/- 20 keV obtained by a power law fit was not very different from that for the entire
spectral range. We therefore decided to fix the value of ν to νpk. This allowed us to
obtain an νF(ν = vpk, t) flux and a β(t) for every 64 ms bin over the 2 second length
of the segment.
Once we included errorbars in our plots, we found that the flux (F) errors became
large beyond 9 secs suggesting that the 64ms resolution was too fine beyond this
range to allow for sufficient photon counts.
Also, the y errorbars were obtained by treating the parameters α , β and vpk as
independent parameters. This leads to an overestimate of the error given that α
and β are correlated and so are vpk and β. We will address these issues in our next
run.
Following are the plots for the 3 relevant time segments.
νFν vs ν-β+1 (t-tp)-α for 3-5 second
ν-β+1 (t-tp)-α
νFν
vFν vs ν-β+1 (t-tp)-α for 5-7 seconds
ν-β+1 (t-tp)-α
νFν
vFν vs ν-β+1( t-tp)-α for 7-9 second
ν-β+1 (t-tp)-α
νFν
Proposed Future Work
•
We noted a hard to soft evolution of the E_peak across several time segments. We think
it worthwhile to determine the spectral lags for this GRB to try and connect the lag to the
curvature effect.
•
We will also search for new single peak GBM GRBs and will apply these tests according
to your suggestions and comments.
•
In place of a simple power law for the spectral fit, we will attempt to fit the spectrum with
a broken powerlaw or log-parabola.
•
We will also use a time resolution for the spectral analysis that reduces the flux errors
and use Monte-carlo simulations to improve our error estimates of derived quantities
while accounting for correlations between the free parameters.