Periodicity Search Methods for Gamma-Ray Pulsars

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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, gt100 MeV)
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Characteristics of classical gamma-ray pulsar
data

• small source detection rates typical psr flux
  10-6 g cm-2 s-1 effective area 102 103
  cm2 src detection rate 1 g / 103 104 sec
• strong background S/B 0.1 - 1
• long integration times of days weeks
• no contemporaneous radio ephemeris available

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High-EnergyLightcurves
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Pulsar n-ndot Distribution
0.2 Hz
0.2 Hz
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How many test n, n 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 f10.2 Hz to f21000 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 q lt 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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DC2 Counts Map Galactic Anticenter
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DC2 Vela Region
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Step 2

• apply barycentric time corrections gtbary
• Need good source position

dt
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) cancelpdotyes der
ive 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)
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Folding methods (2)
Fourier power over m harmonics (Buccheri et
al., 1983)
H-statistic test (De Jager et al., 1989)
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Some basic mathematics for Fourier analysis of
time series
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Following Buccheri, Özel, and Sacco, 1987
For random arrival times H(f) has a c2
probability distribution with 2 d.o.f. A periodic
signal of Np counts (in total of N counts)
concentrated in a duty cycle of a leads to a PDF
of H 22aNp(Np-1)/N Np2 / N and the
significance is calculated from c22
exp(-H(f)/2) If M trials were made S M .
exp(-H(f)max/2)
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Significance limitations
Mattox et al., 1996 The significance of
detection depends exponentially On the ratio
Source counts Total counts
gt 50 needed
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Fourier Procedure
Apply to the selected set of arrival
times Calculate PDF for test frequencies spaced
by the independent Fourier interval df 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.
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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.)
Calculate the Fourier-Transform of the time
differences of the photon arrival times Dtn.
take only differences with Dt lt max_diff typical
max_diff 10 000s 3 hours typical EGRET
viewingperiod 1 000 000s
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Dependence on max_diff
The dependence of the signal width on max_diff
Simmulated Pulsar at 10Hz
Pulse width 1/max_diff
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 VELA
VELA Viewing Period VP 7 max_diff 10 000s Scan
region 1 Hz 100 Hz
F0 trials with S/N gt 10
Number of Photons 1 197 Number of
differences 22 700 Number F trials
2 000 000 Calculations 52 800 000
000 took 4h 30 min
Refined search around good F0 candidates
F0 catalog 11.19888756 F0 from search
11.19882249 F1 catalog -0.1557 E-10 F1
from search 0.0850 E-10
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Blind Search for GEMINGA
GEMINGA Viewing Period VP 10 max_diff 10
000s Scan region 1 Hz 100 Hz
F0 trials with S/N gt 10
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
Refined search around good F0 candidates
F0 catalog 4.2177501 F0 from search
4.2176815 F1 catalog -0.00195 E-10 F1
from search -0.00935 E-10
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The large FdotF1 of CRAB
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F0 and F1 Scan for CRAB
Scan in F CRAB max_diff 10 000s F1 steps _at_ 10Hz
0.05 E-10 F1 steps _at_ 30Hz 0.15 E-10
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., AA, 175, 353
(1987) Buccheri, R., et al., AA, 128, 245
(1983) De Jager O.C. et al., AA, 221, 180
(1989) Chandler, A.M. et al., ApJ, 556, 59,
(2001) Mattox, J.R., et al., AA Suppl., 120. 95,
(1996) Brazier, K.T.S. Kanbach, G., AA Suppl.,
120. 85, (1996)