bsac vii, 01.06.2010, v.hambaryan 20/12/2015 15:41 bayesian probability theory in astronomy: timing...

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: [email protected]

BSAC VII, 01.06.2010, V.Hambaryan

• Introduction

• Method

• Results & Outlook

Outline of talk


Radio Pulsar Basics

• spin characterized by spin period rate of change of period

PP0

P

...)()()( 00

ttPtPtP

time

P


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


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


Neutron star mass and radius linking with measurable

phenomena

Arzoumanian (2009)


Gravitational redshift

EXO 0748 (Cottam et al. 2002)

No second observation of this kind

No second observation of this kind


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


• Bayesian methodology

• Bayesian Periodicity search

• Bayesian Variability detection

Method


What is a Bayesian approach?

• Three-fold task:

Why it?

What the method is?

How it works?


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)


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


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


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



• 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


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


• To detect so called „change points“


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


Our method for periodicity search:Bayesian statistics (GL)


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)


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


Bayesian periodicity detection

Simple periodic signal simulation


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


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)


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)


GL method application to the SGR flare: preliminary results


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


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


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


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

bsac vii, 01.06.2010, v.hambaryan 20/12/2015 15:41 bayesian probability theory in astronomy: timing...

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