advisor seminar tu-m nchen ws 2005/06 time variability studies · 2006-01-27 · – binary stars...

Advisor seminar TU-München WS 2005/06

Time variability studies

27.1.2006Martin Mühlegger

[email protected]

M. Mühlegger Time variability studies 1/23

mailto:[email protected]

Outline

1.Photon counting vs. Flux measurement

2.Time variability in Astrophysics

3.Data Corrections

4.Relevant timing systems

5.Analysis Techniques


1. Photon counting versus Flux measurement

At radio wavelengths: only Flux measurement possible (Dipole antennas!) E=hν of single photon too small to evoke a signal.

Binning fixed before observation

From optical wavelengths on: single Photon detection possible.(Silicon bandgap: E=1.14 eV ~> λ=1.09μm )Time stamp for every individual photon.

Binning is done afterwards


2. Time variability in Astrophysics

Long term variability: Timescales hours --- days --- months

examples:

– Binary stars orbiting each other and eclipsing

– Long term stellar oscillations: e.g. Cepheids

– Rotating white dwarfs with inhomogeneous B-Field

-> Data is represented as LIGHTCURVE.


Time variability in Astrophysics

Short term variability: Timescales hours --- seconds --- milliseconds

examples:

– GRBs

– Short term stellar oscillations

– Pulsars

-> For periodic objects data is represented as PULSE PROFILE.


Time variability in Astrophysics

Creating a Pulse Profile: “Folding” (not in math. sense)

... and the sections are added together.(coherent addition)

-> S/N improved


Lightcurve is divided into “sections” according to the object's period...

phase

3. Data Corrections

Barycentering

Knowledge required about

– Positions of Planets

– Position of Earth

– Position of Observatory with respect to the Earth

– Source position

Δt: up to 1000s


Solar system Ephemeris

Data Corrections

Shapiro delay

Initially a test of General Relativity, in Pulsar astronomy used to correct for gravitational fields in the solar system.

The closer the signal passes the sun, the more it is delayed by the longer pathlength due to space curvature.

Δt: up to 250μs


Data Corrections

Further corrections

– Pulsar signal travels into Solar System's gravitational field-> gravitational “blueshift”

– This blueshift is different at different seasons (elliptical orbit!)

– Dispersion in the ISM

--> All corrections are included in the TEMPO code.

http://pulsar.princeton.edu/tempo orhttp://www.atnf.csiro.au/research/pulsar/tempo/


http://pulsar.princeton.edu/tempo

4. Relevant timing systems

TOA: Time Of Arrival measured by the spacecraft clock (SCC)or the ground based telescope's clock (e.g. GPS)

UTC: Time of photon arrival in Universal Time Coordinated

UTC = TOA + S(t)


Contains Satellite's or Telescope's Ephemeris

Relevant timing systems

TAI: International Atomic Time = Average of some dedicated atomic clocks around the world

=> TAI = UTC + L.S.


TAI

UTC

31. December 1. January

54 55 56 57 58 59 60 0 1 2 3 4 5 6 7

54 55 56 57 58 59 60 1 2 3 4 5 6 7 8= 0

L.S.

1.1.1972: LS := 10now: LS = 33

Relevant timing systems

TT: Terrestrial (Dynamical) Time accounts for gravitational effects on the different atomic clocks contributing to TAI.

TT = TAI + 32.184s

TDB: Barycentric Dynamical Time

TDB = TT + F(TT) = ti


Contains Earth's Ephemeris and other effects

5. Analysis Techniques

Goal: Find the pulsar in the P , P diagram


Period [s]

Crab Pulsar

log P

Analysis Techniques

Fourier analysis

f t =∑i=1

N

t−t i F =∑i=1

N

cos 2 t i ∑i=1

N

sin 2 t i

Power spectrum: ∣ F ∣2=∑i=1

N

cos 2 t i 2 ∑

i=1

N

sin 2 t i 2

observe 1 month later

P=P2−P11month


FT

Crab FFT

P , P

P= 1

Analysis Techniques

Folding ( either from fourier or from radio observations)

t i = TDB for photon number i with i = 1,...,N N:Total number of photons

t ref = reference point in time

= pulsar frequency at reference time

, = 1st and 2nd derivative of frequency at reference time

i t = residual phase of photon #i ≈t i−t ref mod1


i t =Fractionof [⋅t i−t ref 12⋅⋅t i−t ref

216⋅⋅t i−t ref

3]

P , P

Analysis Techniques

Pulse profile assessment: χ2-Test

badly folded pulse profile well folded pulse profile

is an indicator for the goodness of the pulse profile.

n: number of bins x : number of photons in binb: bin index x : average number of photons


2= 1x∑b=1

n

xb−x 2

xx

Analysis Techniques

Pulse profile assessment: Zn2 - Test (Buccheri et al., 1983)

Z n2= 2N ∑k=1

n [∑i=1

N

cos 2 ki 2∑

i=1

N

sin 2ki 2 ]

-> calculate Z n2 for n=1,...,10 and look for maximum value

The respective n is the harmonical content of the Pulse profile.

Do this for various P , P and look where you get the highest Z n2 .

But: Z n2 are not comparable for different n!


Analysis Techniques

Pulse profile assessment: H – Test (De Jager et al., 1989)

Do a 3D-plot of H over P , P and look for the maximum.


H= max1m20

Z m2−4m4

limits the number of tests

and corrects for different values of m.

Analysis Techniques

Blind searches: evolutionary method (Brazier & Kanbach, 1996)

Idea: - Split observation into smaller time intervals - do P search with P=const.- do Z2

2 – Test on frequencies separated by

Advantage: quicker than scanning the whole P− P range

Disadvantage: if missing the right P in the first step -> fails.


=1/T

Analysis Techniques

Blind searches: Bayes method (Gregory & Loredo, 1992)

1 st step: check signal for periodicity

Two complementary hypotheses:

p(Mc|D,I): Probability of not having any periodicity in the Data (= constant countrate)

p(Mper|D,I): Probability of having a periodic signal in the Data D with prior Information I

Parameters: Period P Phase Φ Average countrate cm-1 parameters for the shape of the pulse profile

(m = number of phasebins)

Odds for periodic signal:


O per=p M per∣D , I p M c∣D , I

Analysis Techniques

Parameter space made up by prior Information (e.g. Pmin = 1.5 ms)

Marginalization: p M per∣D =∫ p D∣M per , I × p M per , I p D

dI for

means that 3 bins arethe best solution


Bayes method

I= , , c

Analysis Techniques

2 nd step: derive signal parameters

-> marginalize again for

with 68.3% of the prob.dist. lying in the range +0.8 and -0.2 μHz of the peak.


Bayes method

I= , c ,m1,m2,m3

=19.85287838Hz

PSR 0540-693PSR 0540-693

References

Brazier & Kanbach: “A new, fast pulsar search method for sparse data”, A&A suppl.Ser. 120:85-87 (1996)

Buccheri et al.: “Search for pulsed γ-ray emission from radio pulsars in the COS-B data” A&A 128:245-251 (1983)

Gregory & Loredo: “A new method for the detection of a periodic signal of unknown shape and period”, ApJ, 398:146-168 (1992)

Gregory & Loredo: ”Bayesian periodic signal detection: analysis of ROSAT observations of PSR 0540-693”, ApJ, 473:1059-1066 (1996)

De Jager et al.: “A powerful test for weak periodic signals with unknown lightcurve shape in sparse data”, A&A, 221:180-190 (1989)

Lyne & Graham Smith: “Pulsar Astronomy”, Cambridge University Press, Cambridge 1998

http://www.ptb.de/de/org/4/44/441/ssec.htm (about leap seconds)


http://www.ptb.de/de/org/4/44/441/ssec.htm

advisor seminar tu-m nchen ws 2005/06 time variability studies · 2006-01-27 · – binary stars...

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