Bill Atwood, SCIPP/UCSC, Jan., 2006 GLAST 1 A Time Differencing Technique for doing Blind Pulsar...

Bill Atwood, SCIPP/UCSC, Jan., 2006 GLASTGLAST1

A Time Differencing Technique for doing Blind Pulsar Searches

Bill Atwood, Brian Baughman, Marcus Ziegler, and Robert Johnson


Pulsar Basics at High Energies

1) The data is sparse. Crab: 1 photon every ~ 1000 turns

2) Faint sources could require months-to-years of exposure to find

3) Presents of Period Derivative compromises direct use of Fourier Transforms

4) However the Period Derivatives are very small and to lowest order cause a phase slip rather then shift the frequency significantly

Differencing Concept

If a time series has a periodicity – the time differences will exhibit the same.

Time differences cancel out long term phase slips and glitches

Differencing starts the "clock" over (and over, and over...)


Phase Slippage Caused by the Period Derivative

PeriodPeriod elongation caused by Period Derivative:

Time

PPΔP After N Periods PPΔP N

Phase Slip (in sec) after N Periods is just the sum of the P's2

2N

0 P

TPP

2

1PNP

2

1PP)(i

Or more conveniently the relative Phase Slip is

2

P

TP2

1

P

Geminga Light Curve

A reasonable value for canbe estimated from a "typical" light curve...

1.η


T(sec)

Crab

Geminga

Tmax(Crab) = 23k sec

Tmax(Geminga) = 106 sec

Estimation of the maximumlength of the time differencingwindow

For Geminga – an entire EGRETViewing period can be used, whilefor the Crab, due to its short periodand large period-derivative, onlyabout ¼ of a day is usable.

Additional Phase shift due to Frequency Drift

As time progresses the period (frequency) slowly change (as well as shifting in phase).Time differences from a differencing time window started at a time t after the start ofdata collection will be further limited:

22Diff2 tt)(T

P

P

2

1η(t)

Solving for TDiff:tP

Pηt

P

P2ηt(t)T

2max

2max2

Diff


TD

iff(m

ax)

(sec

)

t (sec)

Crab

Geminga

Crab – Approx. Soln.

Max. Allowed Time Differencingwindow allowing a Phase Slip< .1

Again – Crab is hard to find dueto its short period and large periodderivative

An estimation of the min. flux required can be arrived at by note that fortime differencing we need at least 2 photons within the diff. window.

Assuming the window is opened by a source photon, then in TDiff(max) weneed on average 1 photon.


Flux Limit for Time Differencing Blind Searches

),,,(

11),,,(

tPPTAtPPF

DiffEffMin

This must be compared with the overall flux limit for the mission

tFtFLim

1)( 0 where F0 is mission specific

Flu

x(cm

-2se

c-1)

t (sec)

Where the 2 Flux estimates cross is the limit.

Solving for t and re-inserting it forFLim one finds:

32

20

1),,(

P

P

AFPPF

EffLim


The Plane from Dave Thompson's Talk

LAT - 10-8LAT - 10-7EGRET - 10-6

The "one number" resultis that Time Differencing requires ~ 10X higher flux thenthe mission's limit

PP,

Finally we can sweep out contours in the plane for various values of FLim

20

23lim F

APFP Eff

PP,


Closing Remarks

1) Real Data with noise will corrupt things – see Marcus' Talk

2) Improvements possiblea) Selection of Start Photonb) More efficient freq. search alg.c) ...

3) Applications to GRBs and AGN Flares if there is a common underlying frequency – differencing allows compiling statistics directly.

Bill Atwood, SCIPP/UCSC, Jan., 2006 GLAST 1 A Time Differencing Technique for doing Blind Pulsar...

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