Advisor seminar TU-München WS 2005/06
Time variability studies
27.1.2006Martin Mühlegger
M. Mühlegger Time variability studies 1/23
Outline
1.Photon counting vs. Flux measurement
2.Time variability in Astrophysics
3.Data Corrections
4.Relevant timing systems
5.Analysis Techniques
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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
M. Mühlegger Time variability studies 3/23
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.
M. Mühlegger Time variability studies 4/23
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.
M. Mühlegger Time variability studies 5/23
Time variability in Astrophysics
Creating a Pulse Profile: “Folding” (not in math. sense)
... and the sections are added together.(coherent addition)
-> S/N improved
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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
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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
M. Mühlegger Time variability studies 8/23
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/
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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)
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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.
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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
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Contains Earth's Ephemeris and other effects
5. Analysis Techniques
Goal: Find the pulsar in the P , P diagram
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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
M. Mühlegger Time variability studies 14/23
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
M. Mühlegger Time variability studies 15/23
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
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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!
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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.
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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.
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=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:
M. Mühlegger Time variability studies 20/23
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
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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.
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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)
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