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Bayesian Probability theory in astronomy:
Timing analysis of Neutron Stars
VII BSAC, Chepelare, Bulgaria, 01.06.2010
Valeri Hambaryan
Astrophysical Institute and University Observatory, Friedrich
Schiller University of Jena, Germany
E-Mail: [email protected]
05/05/2017 17:37
BSAC VII, 01.06.2010, V.Hambaryan
Outline of talk
• Introduction
• Method
• Results & Outlook
BSAC VII, 01.06.2010, V.Hambaryan
Radio Pulsar Basics
• spin characterized by
spin period
P
P
rate of change of period

P 0

P(t )  P(t0 )  P(t  t0 )  ...
P
time
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BSAC VII, 01.06.2010, V.Hambaryan
Pulsar Basics cont...

d 1 2 
P
2
E   I   I   4 I 3
dt  2
P



P
c 
spin-down
luminosity
characteristic
age

2P

1/ 2


B  3.2 10  P P  G


19
magnetic
field
Assumes magnetic dipole braking in a vacuum
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BSAC VII, 01.06.2010, V.Hambaryan
The pulsar HRD
 1800 pulsars
known
 143 pulsars with
period less than 10
ms
 A whole zoo of new
and interseting
objects
--AXPs/SGRs
--CCOs
--RRATs
--INSs
binary
BSAC VII, 01.06.2010, V.Hambaryan
Neutron star mass and radius
linking with measurable
phenomena
Arzoumanian (2009)
BSAC VII, 01.06.2010, V.Hambaryan
Gravitational redshift
EXO 0748 (Cottam et al. 2002)
No second
observation
of this kind
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BSAC VII, 01.06.2010, V.Hambaryan
M / R = 0.153 ??
Unable to identify
FeXV-FeXVI
(Rauch,Suleimanov &
Werner, 2008 )
XMM-Newton non
detection
(Cottam et al., 2008)
Spin frequency 552Hz
(Galloway et al., 2009)
Radius of RXJ1856 is R
M / R = 0.153 M_sun/km
M / R = 0.096
„
„
„
M / R = 0.089
„
M / R = 0.087
„
0.096
0.087
= 17 km (Trümper et al., 2004)
for EXO 0748?? (Cottam et al., 2002)
for X7 47 Tuc (Heinke et al. ,2006)
for LMXRBs (Suleimanov & Poutanen, 2006)
for Cas A (Wyn & Heinke, 2009)
BSAC VII, 01.06.2010,
for RBS 1223 (Hambaryan & Suleimanov,
2010) V.Hambaryan
Method
• Bayesian methodology
• Bayesian Variability detection
• Bayesian Periodicity search
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BSAC VII, 01.06.2010, V.Hambaryan
What is a Bayesian
approach?
• Three-fold task:
What the method is?
How it works?
 Why it?
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BSAC VII, 01.06.2010, V.Hambaryan
What the method is?
• Classical approach or Sampling Statistics
Given the data D, how probable is variation in the data, given model M,
model parameters q, and any other relevant prior information I ?
P(D|qMI) or P(D|MI)
• Bayesian methodology
Inverse: How probable are models or model parameters given data?
P(q|DMI) or P(M|DI)
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BSAC VII, 01.06.2010, V.Hambaryan
What the method is?
• Bayesian approch: details
Given the data D, how probable are
model M, model parameters q?
P(q|D,M,I) = P(q|M,I) P(D|q,M,I) | P(D|M,I)
P(q|D,M,I) = Posterior probability
P(D|q,M,I) = Direct probability
P(q|M,I) = Prior probability
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How it works?
• 1. Specify the hypothesis
 Carefully specifying the models Mi
• 2. Assign direct probailities
 Assign direct probabilites appropriate to data (Poisson, Bernulii,...)
 Assign priors for parameters for each Mi
• 3. „Turn the crank“
 Apply Bayes‘ Theorem to get posterior probability densty distribution
Marginalize over uninteresting parameters (some prefer to look at the
peak of the posterior without marginalizing)
• 4. Report the results
For comparing models: it may include, likelihood ratios, probabilities
For parameters: one might report the posterior mode, or mean and variance
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Bayesian variability testing
• High energy astronomy and modern equipments allow:
Register arrival times of individal photons
with high accuracy
Time binnig technique give rise to certain
difficulties:
• many different binnings of the data have
to be considered
• the bins must be large enough so that there
will be enough photons to provide a good
stastistical smaple
• larger bins will dilute short variations &
overllooks a considerable amount of info
• introduces a dependency of results on the
sizes and locations of the bin
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Bayesian variability testing
• Observational interval T consisting
Compare two hypothesis
of m discrete moments of time
m = T/dt
•First hypothesis –constant rate Poisson process
(dt  spacecraft‘s „clock tick“)
• Registered n photon arrival times model M1: one parameter, i.e. count rate  l
D (ti,ti+1,...,ti+n-1)
•Second hypothesis – two-rate Poisson process
model M2: parameters l1,l2 and 
•  is any point from T dividing
into two parts with length
T1 & T2 at which the Poisson process
switches from count rate l1 to l2
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
l1
l2
BSAC VII, 01.06.2010, V.Hambaryan
Bayesian variability testing
• To detect so called „change points“
Change point detection mthodology deals with
sets of sequentlly ordered observations (as in
time) and undertakes to determine whether
the fundamental mechanism generating the
observations has changed during the time
the data have been gathered

l1
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l2
BSAC VII, 01.06.2010, V.Hambaryan
Bayesian variability testing

l1
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l2
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Detection of periodicty and
QPOs
Different methods have been developed for periodicdy search:
Leahy et al., 1983, ApJ, 272,256; Scargle, 1989,ApJ,343,874;
Swanepoel & De Beer, 1990, ApJ,350,754; Gregory & Loredo (GL),
1992, ApJ,398,146; Bai, 1992, ApJ, 397,584; Cincotta et al.,
1995, ApJ, 449,231; Cicuttin et al., 1998, ApJ,498,666 , De Jager 2001....
Simple model of rotating NS


F(t )  F  (t  t )  (t  t )   (t  t )  F
2
0
0
3
0
0
2
6
• Epoch folding
Fi = ti/P – INT(ti/P)
• Rayleigh test
Z12 = 2/N [(Scos2fi) 2(Ssin2fi) 2]
BSAC VII, 01.06.2010, V.Hambaryan
Our method for periodicity search:
Bayesian statistics (GL)
BSAC VII, 01.06.2010, V.Hambaryan
GL method II
p ( | D, M m ) 
 d
C 
lo 
h i
C

2
 df
0
1
, where
W m ( , f )
1

1
0 W ( , f )  and
m

2
W m ( , f ) 
PSR 0540-693
N!
n1! n2 !...nm !
Normal
Two more parameters:
Qpo start & Qpo end
(via MCMC)
GL
(epoch folding)
FFT failed (Gregory & Loredo,1996)
BSAC VII, 01.06.2010, V.Hambaryan
Bayesian variability and periodicity testing
Simulation photon arrival times
Dti = -ln (RANDOMU / l )
Simple signal simulation
l1
l2
l1
Period = 7.56sec.
Pulse duration, count rates l1 and l2
(pulsed fraction) were selected randomaly
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Event start time,duration, count rates l1
and l2 were selected randomaly
BSAC VII, 01.06.2010, V.Hambaryan
Bayesian periodicity
detection
Simple periodic signal simulation
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BSAC VII, 01.06.2010, V.Hambaryan
SGR 1806-20 giant flare on
27 Dec 2004
P  7.56 s

P  54.9 x10 11 s s 1
d ~ 15 k pc
Bsurf ~ 2.1x1015 G
BBT  0.65 k eV
PLindex  1.36
OPT , IR   no
Bands   H , X
BSAC VII, 01.06.2010, V.Hambaryan
SGR 1806-20 giant flare on
27.12.2004
Application of DFT for short (3sec) time intervals & averaging
(Israel et al 2005, Watts et al. 2006,Strohmayer et al. 2006)
However, DFT transform will give optimal frequency estimates:
The number of data values N is large,
There is no constant component in the data,
There is no evidence of a low frequency,
The frequency must be stationary (i.e. amplitude and phase are constant),
The noise is white
(Bretthorst 1988,2001,2002, Gregory 2005)
BSAC VII, 01.06.2010, V.Hambaryan
GL method application to the SGR
flare: preliminary results
58
22
QPO frequencies as expected by
Colaiuda, Beyer, Kokkotas (2009)
At least two more frequencies
detected by our method …
BSAC VII, 01.06.2010, V.Hambaryan
GL method application to the SGR
flare: preliminary results
BSAC VII, 01.06.2010, V.Hambaryan
GL method application to the SGR
flare: Rotational cycles # 34
No. 34
f qpo 16.88 Hz
68% Pr obability range
16.87  16.90 Hz
Odds ratio  197
BSAC VII, 01.06.2010, V.Hambaryan
GL method application to the SGR
flare: Rotational cycles # 24 & 32
No. 32
f qpo  21.36 Hz
68% Pr obability range
21.35  21.38 Hz
Odds ratio  30
No. 24
f qpo  36.84 Hz
68% Pr obability range
36.83  36.88 Hz
Odds ratio  197
BSAC VII, 01.06.2010, V.Hambaryan
Problems & Plans…
Smaller flares, smaller vibrations?
Giant flares are rare and unpredictable events.
Could the more regular intermediate and normal flares also excite seismic
vibrations?
Analaysis should be performed:
Intermediate & normal SGR flares
Burst active and quiter periods
 Constrain and refine QPO models with frequency detections
 Prediction of QPOs also in neutron stars with lower magnetic fields
 search for smaller flares, activity phases on neutron stars with lower magnetic fields
(AXPs & M7)
More complex model is needed for data analysis:
modified GL method taking into account rotational light curve as well
piecewise constant (apodizing or tempering) flare decay
Qpo start & end times will be included as free parameters and derived
via MCMC approach
BSAC VII, 01.06.2010, V.Hambaryan
Conclusions…
• To
bin or not to bin ...
• To be and not to bin
• There are three kinds of lies:
• lies
• damned lies
• and statistics
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Mark Twain
BSAC VII, 01.06.2010, V.Hambaryan