1. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 1 Periodicity Search Methods for Gamma-Ray Pulsars Developed and applied to data of SAS-2, COS-B, and EGRET The gamma-ray sky (EGRET, >100 MeV) Crab Vela Geminga 1706-44 1509-58 1952+32 (l.e.) 1055-52

2. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 2 small source detection rates: typical psr flux ~ 10 -6  cm -2 s -1 effective area~ 10 2 – 10 3 cm 2 src detection rate 1  / 10 3 – 10 4 sec strong background: S/B ~ 0.1 - 1 long integration times of days – weeks no contemporaneous radio ephemeris available Characteristics of classical gamma-ray pulsar data:

3. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 3 High-Energy Lightcurves

4. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 4 Pulsar - dot Distribution 0. 2 Hz 1000 Hz Search region 0. 2 Hz

5. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 5 How many test  do we have to investigate? Step-size: the ‚independent Fourier interval‘.

6. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 6 For a complete search: Assume a stretch of data that is 1 week long: t obs = 6x10 5 sec f 1 =0.2 Hz to f 2 =1000 Hz : S f ~ 10 3 x 6x10 5 x m ~ 10 9 f 1,dot = 10 -9 s -2 to f 2,dot = 10 -17 s -2 : S fdot ~ 10 -9 x 4x10 11 x m ~ 10 3 Total number of searches: ~ 10 12

7. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 7 Steps to prepare data for a periodicity search: 1. extract photons from the map that belong (with a high probability) to the source : gtselect 2. apply barycentric time corrections : gtbary 3. derive periodicity indicators from the time series - folding and light curve assessment - Fourier transformation - any other method… 4. estimate significance and look for corroborating evidence

8. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 8 Step 1: extract photons from the map that belong (with a high probability) to the source: Simple: cookie cutter: gtselect Classical EGRET method based on PSF: Accept photons if  < 5.85° (E/100 MeV) -0.534 Refinement 1: apply a weight factor to photons dependent on angular distance and energy Refinement 2: accept photons if probability for origin from pulsar exceeds given threshold in view of the neighbouring sources and background

9. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 9 Geminga Crab DC2 Counts Map: Galactic Anticenter

10. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 10 DC2: Vela Region gtpsearch: 5° radius

11. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 11 Step 2: apply barycentric time corrections: gtbary Need: good source position tt SSC

12. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 12 Step 3: If needed: Preprocess time series to take into account period derivatives or binary motions (shrink or expand time scale): cancelpdot=yes derive periodicity indicators from the time series - folding and light curve assessment - Fourier transformation - any other method … estimate significance and look for corroborating evidence

13. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 13 Folding methods (1): Calculate phases from time series (Taylor expansion):  =  (t) =  0 + ft i + f dot t 2 /2 + f ddot t 3 /6 + … Derive lightcurve: histogram mod(  i,1) in n phase bins Inspect resulting lightcurve for deviations from uniformity: Chi-square test:  2 = (x i - ) 2  i=1 n x x 1

14. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 14 Folding methods (2): Fourier power over m harmonics (Buccheri et al., 1983): = Z m 2 2 n  i=1 n  n cos(2  k  i )] 2 sin(2  k  i )] 2 } + [ { [  k=1 m H-statistic test (De Jager et al., 1989): Z m 2 H  max ( - 4m + 4) 1  m  20

15. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 15 Some basic mathematics for Fourier analysis of time series Given an event rate of the form: Z(t) =   (t-t i ) where the t i are distributed uniformely in [0,T] Fourier Transform: X(f,T) =  Z(t) e -i2  ft dt = cos(2  ft i ) – i sin(2  ft i ) One sided Power Density: H(f)= |X(f,T)| 2 = { [ i=1 N  T  N  N 2 N 2 N  T  N  N cos(2  ft i )] 2 sin(2  ft i )] 2 } + [

16. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 16 Following Buccheri, Özel, and Sacco, 1987: For random arrival times H(f) has a  2 probability distribution with 2 d.o.f. A periodic signal of N p counts (in total of N counts) concentrated in a duty cycle of  leads to a PDF of H‘ = 2+2  N p (N p -1)/N ~ N p 2 / N and the significance is calculated from  2 2 : exp(-H(f)/2) If M trials were made S = M. exp(-H(f) max /2)

17. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 17 Significance limitations Mattox et al., 1996: The significance of detection depends exponentially On the ratio: N p 2 N T Source counts Total counts > 50 needed

18. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 18 Apply to the selected set of arrival times: Calculate PDF for test frequencies spaced by the ‚independent Fourier interval‘  f = 1/T (eventually use oversampling by a factor of ~2-3) Sum PDF for series of harmonics to increase signal (use FFT like Mattox et al., 1996; Chandler et al., 2001) Check for significant peaks and derive light-curve etc. Fourier Procedure:

19. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 19 FFT on a supercomputer Mattox et al., 1996

20. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 20 Evolutionary Search Brazier & Kanbach, 1996: -split T in shorter intervals - calculate full search in first interval - select significant frequencies - limit search in 2nd intl. to selected frequency regions - continue to rest - the signal survives…

21. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 21 Autocorrelation: Basic Idea (Marcus Ziegler et al.) t ake only differences with  t < max_diff typical max_diff = 10 000s ~ 3 hours typical EGRET viewingperiod ~ 1 000 000s Calculate the Fourier-Transform of the time differences of the photon arrival times  t n.

22. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 22 Dependence on max_diff The dependence of the signal width on max_diff Pulse width ~ 1/max_diff Simmulated Pulsar at 10Hz Power RMS off peak is called Noise S mall max_diff + Small number of differences (fast) + Coarse stepping in Frequency space (fast) -Large noise (Small S/N ratio) Large max_diff + Good S/N ratio - Large number of differences (very slow)

23. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 23 Blind Search for VELA VELA Viewing Period VP 7 max_diff 10 000s Scan region 1 Hz – 100 Hz Number of Photons 1 197 Number of differences 22 700 Number F trials 2 000 000 Calculations52 800 000 000 took 4h 30 min F0 catalog11.19888756 F0 from search 11.19882249 F1 catalog -0.1557 E-10 F1 from search 0.0850 E-10 F0 trials with S/N > 10 Refined search around good F0 candidates

24. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 24 Blind Search for GEMINGA GEMINGA Viewing Period VP 10 max_diff 10 000s Scan region 1 Hz – 100 Hz Number of Photons 1 200 Number of differences 12 300 Number F trials 2 000 000 Calculations 2 400 000 000 took 3h 30 min F0 catalog4.2177501 F0 from search 4.2176815 F1 catalog -0.00195 E-10 F1 from search -0.00935 E-10 F0 trials with S/N > 10 Refined search around good F0 candidates

25. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 25 The large Fdot=F1 of CRAB Simulate Pulsar F0 = 10Hz F1 = -3.0 E-10 F1 GEMINGA -0.00195 E-10 F1 VELA -0.15666 E-10 F1 CRAB -3.86228 E-10 Scan in F1 @ 10 Hz, max_diff 10 000s Scan in F1 CRAB max_diff 10 000s

26. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 26 F0 and F1 Scan for CRAB Scan in F CRAB max_diff 10 000s F1 steps @ 10Hz 0.05 E-10 F1 steps @ 30Hz 0.15 E-10 F0 catalog30.2254 F0 from search 29.9493 F1 catalog -3.8623 E-10 F1 from search -3.7719 E-10 Epoch CRAB 40000 Epoch Search48393 Calculations took 4d 16h

27. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 27 Autocorrelation: on a (visible) photon stream from the Crab using an APD detector(‚OPTIMA‘) and a commercial correlator unit* (D. Dravins et al., Lund University) * correlator.com, 15 Colmart Way, Bridgewater, NJ 08807

28. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 28 Summary: Folding methods are useful small P-P dot ranges to refine lightcurves or find periodicity inside an extrapolated ephemeris Fourier power on lightcurves (including harmonics) is an extension of epoch folding with well defined significance levels. Full scale Fourier transformations have been successful to find Geminga in EGRET data: FFT on supercomputer (Mattox et al., 1996) Evolutionary search (Brazier & Kanbach, 1996) Autocorrelation methods could be even more sensitive because phase coherence is less essential

29. GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006 29 Some References Buccheri, R., et al., A&A, 175, 353 (1987) Buccheri, R., et al., A&A, 128, 245 (1983) De Jager O.C. et al., A&A, 221, 180 (1989) Chandler, A.M. et al., ApJ, 556, 59, (2001) Mattox, J.R., et al., A&A Suppl., 120. 95, (1996) Brazier, K.T.S. & Kanbach, G., A&A Suppl., 120. 85, (1996)