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BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
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: vvh@astro.uni-jena.de
BSAC VII, 01.06.2010, V.Hambaryan
• Introduction
• Method
• Results & Outlook
Outline of talk
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Radio Pulsar Basics
• spin characterized by spin period rate of change of period
PP0
P
...)()()( 00
ttPtPtP
time
P
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Pulsar Basics cont...
322 4
21
PP
IIIdtd
E
spin-downluminosity
P
Pc
2 characteristic
age
GPPB2/1
19102.3
magneticfield
Assumes magnetic dipole braking in a vacuum
BSAC VII, 01.06.2010, V.Hambaryan
The pulsar HRD
binary
1800 pulsars 1800 pulsars knownknown
143 pulsars with 143 pulsars with period less than 10 period less than 10 msms
A whole zoo of new A whole zoo of new and interseting and interseting objects objects
--AXPs/SGRs--AXPs/SGRs
--CCOs--CCOs
--RRATs--RRATs
--INSs--INSs
BSAC VII, 01.06.2010, V.Hambaryan
Neutron star mass and radius linking with measurable
phenomena
Arzoumanian (2009)
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Gravitational redshift
EXO 0748 (Cottam et al. 2002)
No second observation of this kind
No second observation of this kind
BSAC VII, 01.06.2010, V.Hambaryan
Radius of RXJ1856 is R = 17 km (Trümper et al., 2004)M / R = 0.153 M_sun/km for EXO 0748?? (Cottam et al., 2002)M / R = 0.096 „ for X7 47 Tuc (Heinke et al. ,2006) „ „ for LMXRBs (Suleimanov & Poutanen, 2006)M / R = 0.089 „ for Cas A (Wyn & Heinke, 2009)M / R = 0.087 „ for RBS 1223 (Hambaryan & Suleimanov, 2010)
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)
0.087
0.096
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
• Bayesian methodology
• Bayesian Periodicity search
• Bayesian Variability detection
Method
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
What is a Bayesian approach?
• Three-fold task:
Why it?
What the method is?
How it works?
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
What the method is?
• Bayesian methodology
• Classical approach or Sampling Statistics
P(D|MI) or P(D|MI)
Given the data D, how probable is variation in the data, given model M,model parameters , and any other relevant prior information I ?
Inverse: How probable are models or model parameters given data?
P(DMI) or P(M|DI)
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
What the method is?
• Bayesian approch: details
P(D,M,I) = P() P(D|M,I) P(D|M,I)
Given the data D, how probable are model M, model parameters ?
P(D,M,I) = Posterior probability
P(D|M,I) = Direct probability
P(M,I) = Prior probability
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
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 distributionMarginalize over uninteresting parameters (some prefer to look at thepeak of the posterior without marginalizing)
• 4. Report the resultsFor comparing models: it may include, likelihood ratios, probabilitiesFor parameters: one might report the posterior mode, or mean and variance
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Bayesian variability testing
• High energy astronomy and modern equipments allow:Register arrival times of individal photonswith high accuracy
Time binnig technique give rise to certaindifficulties:
• many different binnings of the data have to be considered
• the bins must be large enough so that therewill 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 thesizes and locations of the bin
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Bayesian variability testing
• Observational interval T consisting of m discrete moments of time m = T/t (t spacecraft‘s „clock tick“) • Registered n photon arrival times D (ti,ti+1,...,ti+n-1)
Compare two hypothesis
• is any point from T dividing into two parts with length T1 & T2 at which the Poisson processswitches from count rate to
•Second hypothesis – two-rate Poisson processmodel M2: parameters and
•First hypothesis –constant rate Poisson process model M1: one parameter, i.e. count rate
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Change point detection mthodology deals withsets of sequentlly ordered observations (as intime) and undertakes to determine whether the fundamental mechanism generating theobservations has changed during the time the data have been gathered
Bayesian variability testing
• To detect so called „change points“
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Bayesian variability testing
BSAC VII, 01.06.2010, V.Hambaryan
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....
• Epoch folding
• Rayleigh test
i = ti/P – INT(ti/P)
Z12 = 2/N cos2sin2
Simple model of rotating NS
62
30
20
00
ttttttt
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
!!...!
!),(
),(
1
,),(
1,|
21
12
0
2
0
mm
m
m
m
nnn
N
andd
C
wheredC
MDp
W
W
Whi
lo
Normal
(epoch folding) GLTwo more parameters:Qpo start & Qpo end
(via MCMC)
PSR 0540-693
FFT failed (Gregory & Loredo,1996)
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Simulation photon arrival times
ti = -ln (RANDOMU / )
Bayesian variability and periodicity testing
Simple signal simulation
Period = 7.56sec.
Pulse duration, count rates and(pulsed fraction)were selected randomaly
Event start time,duration, count rates andwere selected randomaly
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Bayesian periodicity detection
Simple periodic signal simulation
BSAC VII, 01.06.2010, V.Hambaryan
SGR 1806-20 giant flare on 27 Dec 2004
XHBands
noIROPT
PL
keVBB
GxB
kpcd
ssxP
sP
index
T
surf
,
,
36.1
65.0
101.2~
15~
109.54
56.7
15
111
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
5822
At least two more frequenciesdetected by our method …
QPO frequencies as expected by Colaiuda, Beyer, Kokkotas (2009)
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
197
90.1687.16
Pr%68
88.16
34.
ratioOdds
Hz
rangeobability
Hzf
No
qpo
BSAC VII, 01.06.2010, V.Hambaryan
GL method application to the SGR flare: Rotational cycles # 24 & 32
30
38.2135.21
Pr%68
36.21
32.
ratioOdds
Hz
rangeobability
Hzf
No
qpo
197
88.3683.36
Pr%68
84.36
24.
ratioOdds
Hz
rangeobability
Hzf
No
qpo
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 flaresBurst 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 wellpiecewise constant (apodizing or tempering) flare decayQpo start & end times will be included as free parameters and derived
via MCMC approach
BSAC VII, 01.06.2010, V.Hambaryan21/04/23 13:37
Conclusions…
• To bin or not to bin ...• To be and not to bin
• There are three kinds of lies:
• lies• damned lies• and statistics Mark Twain
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