1. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 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

2. BSAC VII, 01.06.2010, V.Hambaryan Introduction Method Results & Outlook Outline of talk

3. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Radio Pulsar Basics spin characterized by spin period rate of change of period time P

4. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Pulsar Basics cont... spin-down luminosity characteristic age magnetic field Assumes magnetic dipole braking in a vacuum

5. BSAC VII, 01.06.2010, V.Hambaryan The pulsar HRD binary  1800 pulsars known  143 pulsars with period less than 10 ms  A whole zoo of new and interseting objects --AXPs/SGRs --AXPs/SGRs --CCOs --CCOs --RRATs --RRATs --INSs --INSs

6. BSAC VII, 01.06.2010, V.Hambaryan Neutron star mass and radius linking with measurable phenomena Arzoumanian (2009)

7. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Gravitational redshift EXO 0748 (Cottam et al. 2002) No second observation of this kind

8. 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

9. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Bayesian methodology Bayesian Periodicity search Bayesian Variability detection Method

10. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 What is a Bayesian approach? Three-fold task:  Why it?  What the method is?  How it works?

11. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 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)

12. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 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

13. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 How it works? 1. Specify the hypothesis  Carefully specifying the models M i 2. Assign direct probailities  Assign direct probabilites appropriate to data (Poisson, Bernulii,...)  Assign priors for parameters for each M i 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

14. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 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

15. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41    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 (t i,t i+1,...,t i+n-1 ) Compare two hypothesis   is any point from T dividing into two parts with length T 1 & T 2 at which the Poisson process switches from count rate   to  Second hypothesis – two-rate Poisson process model M 2 : parameters      and  First hypothesis –constant rate Poisson process model M 1 : one parameter, i.e. count rate 

16. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 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 Bayesian variability testing To detect so called „change points“   

17. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Bayesian variability testing   

18. 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 = t i /P – INT(t i /P) Z 1 2 = 2/N  cos2      sin2      Simple model of rotating NS

19. BSAC VII, 01.06.2010, V.Hambaryan Our method for periodicity search: Bayesian statistics (GL)

20. BSAC VII, 01.06.2010, V.Hambaryan GL method II Normal (epoch folding) GL Two more parameters: Qpo start & Qpo end (via MCMC) PSR 0540-693 FFT failed (Gregory & Loredo,1996 )

21. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Simulation photon arrival times  t i = -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  and   were selected randomaly

22. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 Bayesian periodicity detection Simple periodic signal simulation

23. BSAC VII, 01.06.2010, V.Hambaryan SGR 1806-20 giant flare on 27 Dec 2004

24. 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)

25. BSAC VII, 01.06.2010, V.Hambaryan GL method application to the SGR flare: preliminary results 58 22 At least two more frequencies detected by our method … QPO frequencies as expected by Colaiuda, Beyer, Kokkotas (2009)

26. BSAC VII, 01.06.2010, V.Hambaryan GL method application to the SGR flare: preliminary results

27. BSAC VII, 01.06.2010, V.Hambaryan GL method application to the SGR flare: Rotational cycles # 34

28. BSAC VII, 01.06.2010, V.Hambaryan GL method application to the SGR flare: Rotational cycles # 24 & 32

29. 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

30. BSAC VII, 01.06.2010, V.Hambaryan 20/12/2015 15:41 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