The Why

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The Why How of X-Ray Timing

• Tod Strohmayer
• (NASA-GSFC)
• with thanks to Z. Arzoumanian, C. Markwardt

• Why should I be interested?
• What are the methods and tools?
• What should I do?

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Typical Sources of X-Ray Variability

• Isolated pulsars (ms10 s)
• X-ray binary systems
• Accreting pulsars (ms10s s)
• Eclipses (10s mindays)
• Accretion disks (msyears)
• Transients (X-ray novae)
• Flaring stars X-ray bursters
• Magnetars

• Probably not supernova remnants, clusters, or the
  ISM
• But there could be variable serendipitous sources
  in the field, especially in Chandra and XMM
  observations

In short, compact objects ( super-massive black
holes?) are, in general, intrinsically variable.
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What can Timing Tell Us? (or, why should I be
interested?)
credit for magnetar image R. Mallozzi, UAH, MSFC
credit for herx1 image Stelzer et al. 1999
credit for scox1 image Van der Klis et al. 1997
credit for frame-dragging image J. Bergeron, Sky
Telescope

• Timing gt characteristic timescales PHYSICS
• Timing measurements can be extremely precise!!

• Binary orbits
• orbital period
• sizes of emission regions and occulting objects
• orbital evolution

• Accretion phenomena
• broadband variability
• quasiperiodic oscillations (QPOs)
• bursts superbursts
• Energy dependent delays (phase lags)

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What can Timing Tell Us? (cont)
credit for magnetar image R. Mallozzi, UAH, MSFC
credit for herx1 image Stelzer et al. 1999
credit for scox1 image Van der Klis et al. 1997
credit for frame-dragging image J. Bergeron, Sky
Telescope

• Rotation of stellar bodies
• pulsation periods
• stability of rotation
• torques acting on system

The X-ray sky is highly variable, on many
timescales! RXTE/PCA monitoring of the Galactic
center region. Thanks to Craig Markwardt.
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Example Accreting ms Pulsars, orbits, phase lags
credit for magnetar image R. Mallozzi, UAH, MSFC
credit for herx1 image Stelzer et al. 1999
credit for scox1 image Van der Klis et al. 1997
credit for frame-dragging image J. Bergeron, Sky
Telescope
XTE J1751-305 accreting ms pulsar.
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Example Burst Oscillations

• Expanding layer slows down relative to bulk of
  the star.
• Change in spin frequency crudely consistent with
  expected height increase, but perhaps not for
  most extreme variations.
• X-ray burst expands surface layers by 30 meters.

Frequency (Hz)
Oscillation frequency
4U 1702-429
Time
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Example ms QPOs from Neutron Star Binaries
Sco X-1
4U 1728-34
Excluded

• Sub-ms oscillations seen from gt 20 NS binaries.
• kHz QPO maximum frequency constrains NS equations
  of state

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Example Magnetar QPOs

• A sequence of frequencies was detected 28, 53.5,
  and 155 Hz!
• Amplitudes in the 7 11 range.
• 4 frequencies in SGR 190014, a sequence of
  toroidal modes?

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Rotational modulation Pulsars
Crab pulsar
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Questions that timing analysis should address

• Does the X-ray intensity vary with time?
• On what timescales?
• Periodic or aperiodic? What frequency?
• How coherent? (Q-value)
• Amplitude of variability
• (Fractional) RMS?
• Any variation with time of these parameters?
• Can the variability be modeled?

• Any correlated changes in spectral properties or
  emissions at other wavelengths?

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Basics
A light curve (for each source in the FOV) is a
good first step

• Sampling interval ?t and frequency fsamp 1/?t
• Nyquist frequency,
• fNyq 1/2 fsamp,
• is the highest signal frequency that can be
  accurately recovered

• Basic variability measure, variance ?2 ltx2gt
  ltxgt2
• ? ? Root Mean Square

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Fourier Power Spectral Analysis
Answers the question how is the variability of a
source distributed in frequency (on what
timescales is the source variability)?

• Long-timescale variations appear in low-frequency
  spectral bins, short-timescale variations in
  high-frequency bins
• If time-domain signal varies with non-constant
  frequency, spectral response is smeared over
  several bins

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Types of Variability, QPOs

• A quasiperiodic oscillation is a sloppy
  oscillationcan be due to
• intrinsic frequency variations
• finite lifetime
• amplitude modulation
• Q-value fo /?f

2
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Fourier Transform FFT

• Given a light curve x with N samples, Fourier
  coefficients are
• aj ?k xk exp(2pijk/N), j N/2,,0,N/21,
• usually computed with a Fast Fourier Transform
  (FFT) algorithm, e.g., with the powspec tool, or
  the IDL fft(x) function.

• Power density spectrum (PDS)
• Pj 2/Nph aj2
• Leahy normalization
• Use Pj/ltCRgt (fractional RMS normalization) to
  plot (rms/mean)2 Hz1, often displayed and
  rebinned in a log-log plot.

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Estimating Variability from observations

• Find area A under curve in power spectrum,
• A ? P d? ?j Pj ??,
• where Pj are the PDS values, and ?? 1/T is
  the Nyquist spacing.

• Fractional RMS isr ( A / ltCRgt)1/2
• For coherent pulsations,
• fp (2(P-2)/ltCRgt)1/2is the pulsed fraction,
  i.e., (peakmean)/mean

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Estimating Variability for Proposals
To estimate amplitude of variations, or exposure
time, for a desired significance level

• Broadband noise
• r2 2n? v ??/Iv T
• where r RMS fraction n?number of sigma
  of statistical significance demanded ??
  frequency bandwidth (e.g., width of QPO) I
  count rate T exposure time

• Coherent pulsations
• fp 4 n? /I T
• Example
• X-ray binary, 010 Hz, 3? detection, 5 ct/s
  source, 10 ks exposure
• ? 3.8 threshold RMS

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Power Spectrum Statistics

• Any form of noise will contribute to the PDS,
  including Poisson (counting) noise
• Distributed as ?2 with 2 degrees of freedom
  (d.o.f.) for the Leahy normalization

• GoodHypothesis testing used in, e.g.,
  spectroscopy also works for a PDS
• Bad mean value is 2, variance is 4!? Typical
  noise measure-ment is 22
• Adding more lightcurve points wont help makes
  more finely spaced frequencies

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Statistics Solutions

• Average adjacent frequency bins
• Divide up data into segments, make power spectra,
  average them (essentially the same thing)
• Averaging M bins together results in noise
  distributed as ?2/M with 2M d.o.f.? for
  hypothesis testing, still chi-squared, but with
  more d.o.f.

• However, in detecting a source, you examine many
  Fourier bins, perhaps all of them. Thus, the
  significance must be reduced by the number of
  trials. Confidence is
• C 1 Nbins? Prob(MPj,2M),
• where Nbins is the number of PDS bins (i.e.,
  trials), and Prob(?2,?) is the hypothesis test.

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Power Spectrum Statistics Averaging

• Any form of noise will contribute to the PDS,
  including Poisson (counting) noise. Averaging
  reduces the variance.
• Running average of 38 individual PDSs from
  independent X-ray bursts from EXO 0748-676.
• Distributed as ?2 with 2Nin degrees of freedom,
  where Nin is the number of independent frequency
  bins averaged.
• This is the expected distribution, the true noise
  distribution could be different

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Statistics Solutions

• However, in detecting a source, you examine many
  Fourier bins, perhaps all of them. Thus, the
  significance must be reduced by the number of
  trials. Confidence is
• C 1 Nbins? Prob(MPj,2M),
• where Nbins is the number of PDS bins (i.e.,
  trials), and Prob(?2,?) is the hypothesis test,
  based on the number of bins averaged in your PDS.

Example from EXO 0748-676, RXTE data. Noise
power distribution estimated by fitting to
observed histogram.
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Tips

• Pulsar (coherent pulsation) searches are most
  sensitive when no rebinning is done, ie., you
  want the maximum frequency resolution (in
  principle).
• QPO searches need to be done with multiple
  rebinning scales. In general, you are most
  sensitive to a signal when your frequency
  resolution matches (approximately) the frequency
  width of the signal.
• Beware of signals introduced by
• instrument, e.g., CCD read time
• dead time
• orbit of spacecraft
• rotation period of Earth (and harmonics)

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What To Do

• Step 1. Create light curves for each source in
  your field of view ? inspect for features, e.g.,
  eclipses. Usually, this is enough to know whether
  to proceed with more detailed analysis, but you
  cant always see variability by eye!
• Step 2. Power spectrum. Run powspec or equivalent
  and search for peaks. A good starting point,
  e.g., for RXTE, is to use FFT lengths of 500 s.

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SAX J1808 The First Accreting Millisecond Pulsar
Lightcurve
Residuals
PDS
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Step 3. Pulsar or eclipses found?
Refine timing analysis to boost signal-to-noise.

• Barycenter the data corrects to arrival times at
  solar systems center of mass (tools
  fxbary/axbary)

• Refined timing
• Epoch folding (efold)
• Rayleigh statistic (Z 2)
• Arrival time analysis (Princeton TEMPO?)

• Hint best to do timing analysis (e.g., epoch
  folding especially) on segments of data if they
  span a long time baseline, rather than all at
  once.

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Step 3. Broadband feature(s) found?
Refined analysis best done interactively (IDL?
MatLab?).

• Plot PDS
• Use ?2 hypothesis testing to derive significance
  of features
• Rebin PDS as necessary to optimize significance
• If detected with good significance, fit to
  simple-to-integrate model(s), e.g., gaussian or
  broken power-law, lorentzians.
• Compute RMS
• Is the variability time dependent, energy
  dependent?

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Rebinning to find a QPO 4U172834

• To detect a weak QPO buried in a noisy spectrum,
  finding the right frequency resolution is
  essential!
• It is important to have an idea of the kind of
  signals (strength, width) you are looking for.

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Suggested Reading

• van der Klis, M. 1989, Fourier Techniques in
  X-ray Timing, in Timing Neutron Stars, NATO ASI
  282, Ögelman van den Heuvel eds., Kluwer
• Press et al., Numerical Recipes
• power spectrum basics
• Lomb-Scargle periodogram

• Leahy et al. 1983, ApJ 266, 160
• FFTs power spectra statistics pulsars
• Leahy et al. 1983, ApJ 272, 256
• epoch folding Z2
• Vaughan et al. 1994, ApJ 435, 362
• noise statistics