glast dc-ii kick-off, g. kanbach, mar 2, 2006 1 periodicity search methods for gamma-ray pulsars...

GLAST DC-II kick-off, G. Kanbach, Mar 2, 2006

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Periodicity Search Methodsfor Gamma-Ray Pulsars

Developed and applied to data of SAS-2, COS-B, and EGRET

The gamma-ray sky (EGRET, >100 MeV)

CrabVela

Geminga

1706-44

1509-58

1952+32 (l.e.)

1055-52


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• small source detection rates: typical psr flux ~ 10-6 cm-2 s-1

effective area ~ 102 – 103 cm2

src detection rate 1 / 103 – 104 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:


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High-EnergyLightcurves


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Pulsar -dot Distribution

0.2 Hz0.2 Hz

1000 Hz

Search region

0.2 Hz


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How many test do we have to investigate?

Step-size: the ‚independent Fourier interval‘

.


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For a complete search:

Assume a stretch of data that is 1 week long: tobs = 6x105 sec

f1=0.2 Hz to f2=1000 Hz : Sf ~ 103 x 6x105 x m ~ 109

f1,dot= 10-9 s-2 to f2,dot= 10-17 s-2 : Sfdot ~ 10-9 x 4x1011 x m ~ 103

Total number of searches: ~ 1012


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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


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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


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Geminga

Crab

DC2 Counts Map: Galactic Anticenter


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DC2: Vela Regiongtpsearch:5° radius


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Step 2:

apply barycentric time corrections: gtbary

Need: good source position

t

SSC


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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


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Folding methods (1):

Calculate phases from time series (Taylor expansion):

= (t) = 0 + fti + fdott2/2 + fddott3/6 + …

Derive lightcurve: histogram mod(i,1) in n phase bins

Inspect resulting lightcurve for deviations from uniformity:

Chi-square test: 2 = (xi - )2i=1

nx

x1


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Folding methods (2):

Fourier power over m harmonics (Buccheri et al., 1983):

= Zm2 2

n i=1

n

i=1

ncos(2ki)]2 sin(2ki)]2

}+ [{ [

k=1

m

H-statistic test (De Jager et al., 1989):

Zm2H max ( - 4m + 4)

1 m 20


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Some basic mathematics for Fourier analysis of time series

Given an event rate of the form:

Z(t) = (t-ti) where the ti are distributed uniformely in [0,T]

Fourier Transform:

X(f,T) = Z(t) e-i2ftdt = cos(2fti) – i sin(2fti)

One sided Power Density:

H(f)= |X(f,T)|2 = { [

i=1

N

T

i=1

N

i=1

N

2N

2N

T

i=1

N

i=1

Ncos(2fti)]2 sin(2fti)]2 }+ [


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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 Np counts (in total of N counts) concentrated in a duty cycle of leads to a PDF of

H‘ = 2+2Np(Np-1)/N ~ Np2 / N

and the significance is calculated from 22 : exp(-H(f)/2)

If M trials were made S = M . exp(-H(f)max/2)


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Significance limitationsMattox et al., 1996:The significance of detection depends exponentiallyOn the ratio:

Np2

NT

Source counts

Total counts

> 50 needed


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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:


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FFT on a supercomputer

Mattox et al., 1996


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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…


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Autocorrelation: Basic Idea(Marcus Ziegler et al.)

take only differences with t < max_diff

typical max_diff = 10 000s ~ 3 hourstypical EGRET viewingperiod ~ 1 000 000s

Calculate the Fourier-Transform of the time differences of the photon arrival times tn.


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Dependence on max_diffThe dependence of the signal width on max_diff

-0 .001 -0.0005 0 0.0005 0.0010

0.2

0.4

0.6

0.8

1

FT+differnces

F + f in Hz0

Po

we

r

max_diff = 2000 s

-0. 001 -0.0005 0 0.0005 0.0010

0.2

0.4

0.6

0.8

1

FT+differnces

F + f in Hz0

Pow

er

max_diff = 10000 s

-0. 001 -0.0005 0 0.0005 0.0010

0.2

0.4

0.6

0.8

1

FT+differnces

F + f in Hz0

Pow

er

max_diff = 100000 s

Pulse width ~ 1/max_diff

Simmulated Pulsar at 10Hz

Power RMS off peak is called Noise

Small 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)


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Blind Search for VELAVELA Viewing Period VP 7max_diff 10 000sScan region 1 Hz – 100 Hz

S/N

F in Hz

VELA

S/N

F in Hz

VELA

Number of Photons 1 197Number of differences 22 700Number F trials 2 000 000Calculations 52 800 000 000took 4h 30 min

F0 catalog 11.19888756F0 from search 11.19882249

F1 catalog -0.1557 E-10F1 from search 0.0850 E-10

F0 trials with S/N > 10

Refined search around good F0 candidates


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Blind Search for GEMINGA

GEMINGA Viewing Period VP 10max_diff 10 000sScan region 1 Hz – 100 Hz

Number of Photons 1 200Number of differences 12 300Number F trials 2 000 000Calculations 2 400 000 000took 3h 30 min


F1 catalog -0.00195 E-10F1 from search -0.00935 E-10

0 20 40 60 80 100

10

20

30

40

50

60

70S/N

F in Hz

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700S/N

F in Hz

F0 trials with S/N > 10

Refined search around good F0 candidates


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The large Fdot=F1 of CRAB

Simulate Pulsar F0 = 10HzF1 = -3.0 E-10

F1 GEMINGA -0.00195 E-10F1 VELA -0.15666 E-10F1 CRAB -3.86228 E-10

Scan in F1 @ 10 Hz, max_diff 10 000s

-F1

Po

wer

-F1

Po

wer

Scan in F1 CRAB max_diff 10 000s


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F0 and F1 Scan for CRAB

Scan in F CRAB max_diff 10 000sF1 steps @ 10Hz 0.05 E-10F1 steps @ 30Hz 0.15 E-10


F1 catalog -3.8623 E-10F1 from search -3.7719 E-10

Epoch CRAB 40000Epoch Search 48393

Calculations took 4d 16h


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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


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Summary:Folding methods are useful small P-Pdot 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


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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)

glast dc-ii kick-off, g. kanbach, mar 2, 2006 1 periodicity search methods for gamma-ray pulsars...

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