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Analysis of Fermi GRB T90 distribution
Mariusz Tarnopolski
Astronomical Observatory
Jagiellonian University
23 July 2015
Cosmology School, Kielce
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 1 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBsχ2 �ttingMaximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBs
χ2 �ttingMaximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBsχ2 �tting
Maximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBsχ2 �ttingMaximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBsχ2 �ttingMaximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBsχ2 �ttingMaximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Presentation plan
1 Introduction and overview
2 T90 distributions of Fermi GRBsχ2 �ttingMaximum log-likelihood �tting
3 Hurst Exponents (HEs) & Machine Learning (ML)
4 On the limit between short and long GRBs
5 Conclusions
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 2 / 41
Introduction and overview
Satellites
CGRO/BATSE
Swift/BATÐ→RHESSI
BeppoSAX/GRBM
Fermi/GBM
HETE-2/FREGATE
INTEGRAL/SPI-ACS
SUZAKU
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 3 / 41
Introduction and overview
KONUS (Mazets et al. 1981)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 4 / 41
Introduction and overview
BATSE 1B
Kouveliotou et al. (1993) �tted a quadratic function between the twopeaks of 222 GRBs and determined its minimum to be at (1.2 ± 0.4) s,which rounded of to the next integer bin edge, is 2.0 s.Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 5 / 41
Introduction and overview
BATSE 3B (Horváth 1998)
-2 -1 0 1 2 3
0
10
20
30
40
50
logT90
Counts
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 6 / 41
Introduction and overview
BATSE 4B (current) (Horváth 2002)
-2 -1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
logT90
Horvath
chi2
MLE
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 7 / 41
Introduction and overview
Swift (Horváth et al. 2008)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 8 / 41
Introduction and overview
Swift (Huja et al. 2009)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 9 / 41
Introduction and overview
Swift (Huja & �ípa 2009)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 10 / 41
Introduction and overview
RHESSI (�ípa et al. 2009)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 11 / 41
Introduction and overview
BeppoSAX (Horváth 2009)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 12 / 41
T90 distributions of Fermi GRBs χ2 �tting
A mixture of Gaussians:
fk =k
∑i=1
AiNi(µi , σ2i )
fk =k
∑i=1
Ai√2πσi
exp(− (x−µi )2
2σ2i
)
is �tted to a histogram of log (T90).A signi�cance level of α = 0.05 isadopted; 25 binnings are applied,de�ned by the bin widths w from0.30 to 0.06 with a step of 0.01.The corresponding number of binsrange from 15 to 69.
w=0.27
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.26
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.25
-1 0 1 2 3
0
50
100
150
200
250
w=0.2
-1 0 1 2 3
0
50
100
150
200
w=0.13
-1 0 1 2 3
0
50
100
150
logT90
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 13 / 41
T90 distributions of Fermi GRBs χ2 �tting
A mixture of Gaussians:
fk =k
∑i=1
AiNi(µi , σ2i )
fk =k
∑i=1
Ai√2πσi
exp(− (x−µi )2
2σ2i
)
is �tted to a histogram of log (T90).A signi�cance level of α = 0.05 isadopted; 25 binnings are applied,de�ned by the bin widths w from0.30 to 0.06 with a step of 0.01.The corresponding number of binsrange from 15 to 69.
w=0.27
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.26
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.25
-1 0 1 2 3
0
50
100
150
200
250
w=0.2
-1 0 1 2 3
0
50
100
150
200
w=0.13
-1 0 1 2 3
0
50
100
150
logT90
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 13 / 41
T90 distributions of Fermi GRBs χ2 �tting
Table 1: Parameters of a two-Gaussian �t
w i µi σi Ai χ2 p-val
0.271 -0.042 0.595 100.2
6.467 0.6922 1.477 0.465 325.7
0.261 -0.063 0.569 92.30
12.23 0.2702 1.475 0.473 318.8
0.251 -0.125 0.510 79.55
22.00 0.0242 1.453 0.494 316.1
0.201 -0.049 0.611 73.34
22.09 0.0772 1.473 0.476 243.6
0.131 -0.030 0.607 48.56
33.74 0.1422 1.480 0.468 157.2
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 14 / 41
T90 distributions of Fermi GRBs χ2 �tting
w=0.27
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.26
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.25
-1 0 1 2 3
0
50
100
150
200
250
w=0.2
-1 0 1 2 3
0
50
100
150
200
w=0.13
-1 0 1 2 3
0
50
100
150
logT90
w=0.27
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.26
-1 0 1 2 3
0
50
100
150
200
250
300
w=0.25
-1 0 1 2 3
0
50
100
150
200
250
w=0.2
-1 0 1 2 3
0
50
100
150
200
w=0.13
-1 0 1 2 3
0
50
100
150
logT90
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 15 / 41
T90 distributions of Fermi GRBs χ2 �tting
Table 2: Parameters of a three-Gaussian �t
w i µi σi Ai χ2 p-val
1 -0.030 0.603 102.0
0.27 2 1.466 0.455 317.1 5.333 0.502
3 2.027 0.201 6.014
1 -0.210 0.461 71.48
0.26 2 1.119 0.450 128.6 6.819 0.448
3 1.598 0.421 208.4
1 -0.137 0.492 77.73
0.25 2 1.414 0.480 300.4 13.52 0.095
3 1.939 0.128 14.49
1 -0.204 0.493 57.30
0.20 2 1.221 0.488 144.0 17.71 0.087
3 1.665 0.396 113.1
1 -0.058 0.581 46.61
0.13 2 1.453 0.464 153.7 29.42 0.167
3 1.903 0.092 4.328
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 16 / 41
T90 distributions of Fermi GRBs χ2 �tting
∆χ2 = χ21 − χ22.= χ2(∆ν)
Table 3: Improvements of a three-Gaussian over a two-Gaussian �t
w ∆χ2 p-value
0.27 1.134 0.7670.26 5.411 0.1440.25 8.480 0.0370.20 4.380 0.2230.13 4.320 0.229
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 17 / 41
T90 distributions of Fermi GRBs χ2 �tting
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2.5BATSE
3B
BATSE4B
SwiftSwift
SwfitRHESSI
Beppo
SAX
Fermi
logT
90
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 18 / 41
T90 distributions of Fermi GRBs χ2 �tting
Results 1
T90 distribution of Fermi GRBs is bimodal � no evidence fora (phenomenological) third (intermediate) class
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 18 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
It feels like a waste of data to bin ∼ 1600 events into a few dozens of bins.
Having a distribution with a PDF given by f = f (x ; θ) (possibly a mixture),where θ = {θi}pi=1 is a set of parameters, the log-likelihood function isde�ned as
L =N
∑i=1
log f (xi ; θ),
where {xi}Ni=1 are the datapoints from the sample to which a distribution is�tted. The �tting is performed by searching a set of parameters θ forwhich the log-likelihood L is maximized.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 19 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
It feels like a waste of data to bin ∼ 1600 events into a few dozens of bins.
Having a distribution with a PDF given by f = f (x ; θ) (possibly a mixture),where θ = {θi}pi=1 is a set of parameters, the log-likelihood function isde�ned as
L =N
∑i=1
log f (xi ; θ),
where {xi}Ni=1 are the datapoints from the sample to which a distribution is�tted. The �tting is performed by searching a set of parameters θ forwhich the log-likelihood L is maximized.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 19 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
For nested as well as non-nested models, the Akaike information criterion(AIC ) may be applied. The AIC is de�ned as
AIC = 2p − 2Lp.
A preferred model is the one that minimizes AIC . The formulation of AICpenalizes the use of an overly excessive number of parameters, hencediscourages over�tting. Among candidate models with AICi , let AICmin
denote the smallest. Then,
Pri = exp(∆i
2) ,
where ∆i = AICmin −AICi , can be interpreted as the relative (compared toAICmin) probability that the i-th model minimizes the AIC .
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 20 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
For nested as well as non-nested models, the Akaike information criterion(AIC ) may be applied. The AIC is de�ned as
AIC = 2p − 2Lp.
A preferred model is the one that minimizes AIC . The formulation of AICpenalizes the use of an overly excessive number of parameters, hencediscourages over�tting. Among candidate models with AICi , let AICmin
denote the smallest.
Then,
Pri = exp(∆i
2) ,
where ∆i = AICmin −AICi , can be interpreted as the relative (compared toAICmin) probability that the i-th model minimizes the AIC .
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 20 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
For nested as well as non-nested models, the Akaike information criterion(AIC ) may be applied. The AIC is de�ned as
AIC = 2p − 2Lp.
A preferred model is the one that minimizes AIC . The formulation of AICpenalizes the use of an overly excessive number of parameters, hencediscourages over�tting. Among candidate models with AICi , let AICmin
denote the smallest. Then,
Pri = exp(∆i
2) ,
where ∆i = AICmin −AICi , can be interpreted as the relative (compared toAICmin) probability that the i-th model minimizes the AIC .
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 20 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
A mixture of k standard normal (Gaussian) N (µ,σ2) distributions:
f(N )
k(x) =
k
∑i=1
Aiϕ(x − µiσi
) =k
∑i=1
Ai√2πσi
exp(−(x − µi)2
2σ2i
)
A mixture of k skew normal (SN) distributions:
f(SN)
k(x) =
k
∑i=1
2Aiϕ(x − µiσi
)Φ(αix − µiσi
)
A mixture of k sinh-arcsinh (SAS) distributions:
f(SAS)
k(x) =
k
∑i=1
Aiσi
[1 + ( x−µiσi)2]− 12
βi cosh [βi sinh−1 ( x−µiσi) − δi]×
F(SAS)
k(x) =
k
∑i=1
× exp [−12sinh [βi sinh−1 ( x−µiσi
) − δi]2]
A mixture of k alpha-skew-normal (ASN) distributions:
f(ASN)
k(x) =
k
∑i=1
Ai
(1 − αi x−µiσi)2+ 1
2 + α2i
ϕ(x − µiσi
)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 21 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
A mixture of k standard normal (Gaussian) N (µ,σ2) distributions:
f(N )
k(x) =
k
∑i=1
Aiϕ(x − µiσi
) =k
∑i=1
Ai√2πσi
exp(−(x − µi)2
2σ2i
)
A mixture of k skew normal (SN) distributions:
f(SN)
k(x) =
k
∑i=1
2Aiϕ(x − µiσi
)Φ(αix − µiσi
)
A mixture of k sinh-arcsinh (SAS) distributions:
f(SAS)
k(x) =
k
∑i=1
Aiσi
[1 + ( x−µiσi)2]− 12
βi cosh [βi sinh−1 ( x−µiσi) − δi]×
F(SAS)
k(x) =
k
∑i=1
× exp [−12sinh [βi sinh−1 ( x−µiσi
) − δi]2]
A mixture of k alpha-skew-normal (ASN) distributions:
f(ASN)
k(x) =
k
∑i=1
Ai
(1 − αi x−µiσi)2+ 1
2 + α2i
ϕ(x − µiσi
)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 21 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
A mixture of k standard normal (Gaussian) N (µ,σ2) distributions:
f(N )
k(x) =
k
∑i=1
Aiϕ(x − µiσi
) =k
∑i=1
Ai√2πσi
exp(−(x − µi)2
2σ2i
)
A mixture of k skew normal (SN) distributions:
f(SN)
k(x) =
k
∑i=1
2Aiϕ(x − µiσi
)Φ(αix − µiσi
)
A mixture of k sinh-arcsinh (SAS) distributions:
f(SAS)
k(x) =
k
∑i=1
Aiσi
[1 + ( x−µiσi)2]− 12
βi cosh [βi sinh−1 ( x−µiσi) − δi]×
F(SAS)
k(x) =
k
∑i=1
× exp [−12sinh [βi sinh−1 ( x−µiσi
) − δi]2]
A mixture of k alpha-skew-normal (ASN) distributions:
f(ASN)
k(x) =
k
∑i=1
Ai
(1 − αi x−µiσi)2+ 1
2 + α2i
ϕ(x − µiσi
)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 21 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
A mixture of k standard normal (Gaussian) N (µ,σ2) distributions:
f(N )
k(x) =
k
∑i=1
Aiϕ(x − µiσi
) =k
∑i=1
Ai√2πσi
exp(−(x − µi)2
2σ2i
)
A mixture of k skew normal (SN) distributions:
f(SN)
k(x) =
k
∑i=1
2Aiϕ(x − µiσi
)Φ(αix − µiσi
)
A mixture of k sinh-arcsinh (SAS) distributions:
f(SAS)
k(x) =
k
∑i=1
Aiσi
[1 + ( x−µiσi)2]− 12
βi cosh [βi sinh−1 ( x−µiσi) − δi]×
F(SAS)
k(x) =
k
∑i=1
× exp [−12sinh [βi sinh−1 ( x−µiσi
) − δi]2]
A mixture of k alpha-skew-normal (ASN) distributions:
f(ASN)
k(x) =
k
∑i=1
Ai
(1 − αi x−µiσi)2+ 1
2 + α2i
ϕ(x − µiσi
)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 21 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
●
●
●
●
●
2 3 4 5 6
3432
3434
3436
3438
3440
3442
number of components
AIC
AIC vs. number of components in a mixture of standard normaldistributions. The minimal value corresponds to a three-Gaussian.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 22 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
PD
F
HaL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 HbL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
HcL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 HdL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
HeL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 HfL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
HgL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 HhL
-1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
logT90
Distributions �tted tologT90 data. Colordashed curves are thecomponents of the (blacksolid) mixturedistribution. The panelsshow a mixture of (a) twostandard Gaussians, (b)three standard Gaussians,(c) two skew-normal, (d)three skew-normal, (e)two sinh-arcsinh, (f) threesinh-arcsinh, (g) onealpha-skew-normal, and(h) twoalpha-skew-normaldistributions.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 23 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
æ
æ
æ
æ
æ
æ
æ
æ
HaL HbL HcL HdL HeL HfL HgL HhL
3430
3435
3440
3445
3450
3455
2-
G3-
G2-
SN
3-
SN
2-
SA
S
3-
SA
S
1-
ASN
2-
ASN
Distribution
AIC
æ AIC
ç
ç
ç
ç
ç
ç çç
0.0
0.2
0.4
0.6
0.8
1.0
Pr
ç Pr
AIC and relative probability (Pr) for the models examined.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 24 / 41
T90 distributions of Fermi GRBs Maximum log-likelihood �tting
Results 2
Log-likelihood method supported the non-existence of a third(intermediate) component in the T90 distribution of Fermi.
A two-component mixture of skewed distributions (2-SN and 2-SAS)describes the data better than a three-Gaussian.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 25 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Methods � HE � de�nition
HE is a measure of persistency/long-term memory/self-similarity of aprocess.
Two ways of de�ning:
1 a process Y (t) (non-stationary) is self-similar with self-similarityparameter H, if
Y (λt) .= λHY (t)2 a process X (t) (stationary) is self-similar if ∃α ∈ (0,2):
limτ→∞
ρ(t)∝ τ−α, α = 2 − 2H
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 26 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Methods � HE � properties
0 < H ≤ 1
H = 0.5 for a random walk (Brownion motion)
H < 0.5 for anti-persistent (anti-correlated, short memory) process
H > 0.5 for persistent (correlated, long memory) process
H = 1 for periodic time series
fractal dimension D = 2 −H
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 27 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
HE
Counts
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 28 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Methods � MVTS
Light curves are binned in to narrow time bins. Optimum bin-width atwhich the non-statistical variability in the light curve becomes signi�cant.Prompt duration emission and equal duration of background region.Ratio of the variances of the GRB and the background divided by thebin-width as a function of bin-width. For binnings beyond the minimum thesignal is indistinguishable from Poissonian �uctuations. Left fromminimum, signi�cant variability in the light curve may vanish (coarsebinning). Optimum bin-width is at the minimum.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 29 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Methods � SVM
Not probabilistic, but methods exist (probability calibration, e.g. distanceto the hyperplane) to make it probabilistic.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 30 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Methods � Monte Carlo & SVM
(2220) = 231 subsamples from short GRBs; for each, 435 subsamples of 42
(out of 46) long GRBs → training set. Remaining � validation set.
≈ 105 realisations.
Success ratio r : rshort and rlong; rtot =2rshort+4rlong
6.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 31 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 32 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 33 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 34 / 41
Hurst Exponents (HEs) & Machine Learning (ML)
Results 3
1 H and MVTS alone give unsatisfactory classi�cations
2 T90 works as expected
3 (H, logMVTS) � unsatisfactory
4 (H, logT90) � better than H and logT90 alone
5 (logMVTS, logT90) � �←� worse, �→� better6 complementing (logMVTS, logT90) with HEs � accuracy increased by
7%; comparable to T90 alone.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 35 / 41
On the limit between short and long GRBs
As in (Kouveliotou et al. 1993), the limitting T90 value is found as alocal minimum.
ML �t of a two-Gaussian instead of a parabola.
Datasets: BATSE 1B (for comparison; 226 events), BATSE current,Swift, BeppoSAX and Fermi (∼ 1000 − 2000 events).
Parameter errors: parametric bootstrap.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 36 / 41
On the limit between short and long GRBs
(a)
-2 -1 0 1 2 30.0
0.1
0.2
0.3
0.4
0.5 (b)
-2 -1 0 1 2 30.00.10.20.30.40.50.6
(c)
-2 -1 0 1 2 30.00.10.20.30.40.50.6 (d)
-2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
(e)
-2 -1 0 1 2 30.00.10.20.30.40.50.60.7
logT90
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 37 / 41
On the limit between short and long GRBs
Table 4: Parameters of the �ts. Errors are estimated using the bootstrap method.
Label Dataset N i µi δµi σi δσi Ai δAi min. δmin.
(a) BATSE 1B 2261 −0.393 0.099 0.465 0.069 0.272
0.040 2.158 0.0492 1.460 0.056 0.532 0.044 0.728
(b)BATSE
20411 −0.095 0.051 0.627 0.033 0.336
0.018 3.378 0.272current 2 1.544 0.018 0.429 0.013 0.664
(c) Swift 9141 −0.026 0.255 0.740 0.120 0.139
0.042 � �2 1.638 0.031 0.528 0.023 0.861
(d) BeppoSAX 10031 0.626 0.186 0.669 0.075 0.355
0.084 � �2 1.449 0.035 0.393 0.027 0.645
(e) Fermi 15961 −0.072 0.073 0.525 0.044 0.215
0.021 2.049 0.2482 1.451 0.021 0.463 0.014 0.785
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 38 / 41
On the limit between short and long GRBs
Results 4
Datasets from Swift and BeepoSAX are unimodal, hence no new limitmay be inferred.
BATSE 1B and Fermi are consistent with the conventional 2 s value.
A limit of 3.38 ± 0.27 s was obtained for BATSE current.
This leads to diminishing the fraction of long GRBs in the sampe by4%.
T90 is an ambiguous GRB type indicator.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 39 / 41
Conclusions
Conclusions
Both χ2 and ML �tting lead to a bimodal T90 distribution
This is a hint that there are only two GRB classes
Two types of two-component skewed distributions are a better �t thana three-Gaussian
It is unlikely that the third, intermediate-duration, GRB class is a realphysical phenomenon
It was suggested (Zitouni 2015) that an assymetric T90 distributionmay be due to an assymetric distribution of envelope masses of theprogenitors
HE might serve as a GRB class indicator � including it in the SVMscheme increased accuracy by 7%
A division between short and long GRBs at T90 of 3.38 s is moreappropriate for the BATSE current dataset than the the conventionalvalue of 2 s.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 40 / 41
Conclusions
Conclusions
Both χ2 and ML �tting lead to a bimodal T90 distribution
This is a hint that there are only two GRB classes
Two types of two-component skewed distributions are a better �t thana three-Gaussian
It is unlikely that the third, intermediate-duration, GRB class is a realphysical phenomenon
It was suggested (Zitouni 2015) that an assymetric T90 distributionmay be due to an assymetric distribution of envelope masses of theprogenitors
HE might serve as a GRB class indicator � including it in the SVMscheme increased accuracy by 7%
A division between short and long GRBs at T90 of 3.38 s is moreappropriate for the BATSE current dataset than the the conventionalvalue of 2 s.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 40 / 41
Conclusions
Conclusions
Both χ2 and ML �tting lead to a bimodal T90 distribution
This is a hint that there are only two GRB classes
Two types of two-component skewed distributions are a better �t thana three-Gaussian
It is unlikely that the third, intermediate-duration, GRB class is a realphysical phenomenon
It was suggested (Zitouni 2015) that an assymetric T90 distributionmay be due to an assymetric distribution of envelope masses of theprogenitors
HE might serve as a GRB class indicator � including it in the SVMscheme increased accuracy by 7%
A division between short and long GRBs at T90 of 3.38 s is moreappropriate for the BATSE current dataset than the the conventionalvalue of 2 s.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 40 / 41
References
[1] Tarnopolski M., 2015, A&A, in press (arXiv:1506.07324)
[2] Tarnopolski M., 2015 (arXiv:1506.07801)
[3] Tarnopolski M., 2015 (arXiv:1506.07862)
[4] Tarnopolski M., 2015 (arXiv:1507.04886)
[5] www.oa.uj.edu.pl/M.Tarnopolski
Thank you for your attention.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 41 / 41
References
[1] Tarnopolski M., 2015, A&A, in press (arXiv:1506.07324)
[2] Tarnopolski M., 2015 (arXiv:1506.07801)
[3] Tarnopolski M., 2015 (arXiv:1506.07862)
[4] Tarnopolski M., 2015 (arXiv:1507.04886)
[5] www.oa.uj.edu.pl/M.Tarnopolski
Thank you for your attention.
Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 41 / 41