# Presentation: 259908

## The Why & How of X-Ray Timing

- Z. Arzoumanian
- (USRA/NASA-GSFC)
- with thanks to C. Markwardt

- Why should I be interested?
- What are the tools?
- What should I do?

## Typical Sources of X-Ray Variability

- Isolated pulsars            (ms–10 s)
- X-ray binary systems
- Accreting pulsars (ms–10s s)
- Eclipses         (10s min–days)
- Accretion disks  (~ms–years)
- Flaring stars & X-ray bursts

- 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, stellar-sized objects (& super-massive black holes?)

## Why should I be interested?

- Rotation of stellar bodies
- pulsation period
- stability of rotation

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

- Accretion phenomena
- broadband variability
- “quasiperiodic” oscillations (QPOs)
- bursts & “superbursts”

- credit for frame-dragging image: J. Bergeron, Sky & Telescope

- 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

## Rotational modulation: Pulsars

- Crab pulsar

## Accretion Science: QPOs from Neutron Star Binaries

- kHz QPO maximum frequency constrains NS equations of state

- Excluded

- Sco X-1

## Questions that timing analysis should address

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

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

## Basics

- Basic variability measure, variance:* *2 = <*x*2> – <*x*>2
*	** * Root Mean Square

- Sampling interval ∆t and frequency *f**samp** = *1/∆t
*Nyquist *frequency,
*	**f**Nyq** = *1/2 *f**samp*,
-     is the highest signal frequency that can be accurately recovered

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

## Slide 8

## Fourier Analysis

- Answers the question: how is the variability of a source distributed in frequency?

- 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

## What is a QPO?

- A *quasiperiodic oscillation* is a “sloppy” oscillation—can be due to:
-   intrinsic frequency   variations
-   finite lifetime
-   amplitude modulation
- Q-value = *f**o** /**f*

## Fourier Transform & FFT

- Given a light curve {*x*} with *N* samples, Fourier coefficients are:
- 	*a**j **= ∑**k** x**k* exp(2π*ijk*/*N*),	[ *j = *–*N*/2,...,0,*N*/2–1],
- 	usually computed with a Fast Fourier Transform (FFT) algorithm, e.g., with the powspec tool.

- Power density spectrum (PDS):
- 		*P**j** = *2⁄*N*ph |*a**j*|2
- 	[Leahy normalization]
- 	Use  *Pj**/*<CR>  (fractional RMS normalization) to plot (rms/mean)2 Hz–1, often displayed and rebinned in a log-log plot.

## Estimating Variability from observations

- Find “area” *A* under curve in power spectrum,
- 	*A* = ∫ *P *d ≈ ∑*j** P**j** *∆,
- 	where *P**j** *are the PDS values, and ∆ = 1/*T**  *is the Nyquist spacing.

- Fractional RMS is	*r* = √ *A*/<CR>
- For coherent pulsations,
- 		*f**p* = √ *2(P–2)*/<CR>is the *pulsed fraction*, i.e., (peak–mean)/mean

## Estimating Variability for Proposals

- To estimate amplitude of variations, or exposure time, for a desired significance level...

-  Broadband noise:
*r*2  =  2*n* √ ∆*/**I**√ **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:
*f**p  *= 4 *n**** */*I T*
-  Example:
- X-ray binary, 0–10 Hz, 3 detection, 5 ct/s source, 10 ks exposure
-  3.8% threshold RMS

## 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 normali-zation

- Good—Hypothesis testing used in, e.g., spectroscopy also works for a PDS
- Bad —mean value is 2, variance is 4! Typical noise measure-ment is 2±2
- Adding more lightcurve points won’t help: makes more finely spaced frequencies

## 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 2*M* 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 – *N*bins Prob(*MP**j*,2*M*),
- 	where *N*bins is the number of PDS bins (i.e., trials), and Prob(2,) is the hypothesis test.

## Tips

- Pulsar (coherent pulsation) searches are most sensitive when *no rebinning* is done
- QPO searches need to be done with *multiple rebinning* scales
- Beware of signals introduced by
- instrument, e.g., CCD read time
- dead time
- orbit of spacecraft
- rotation period of Earth (and harmonics)

## 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 can’t 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.

## SAX J1808: The First Accreting Millisecond Pulsar

- Lightcurve

- PDS

- Residuals

## Step 3. Pulsar or eclipses found?

*Barycenter* the data: corrects to arrival times at solar system’s center of mass (tools: fxbary/axbary)

- Refine timing analysis to boost signal-to-noise.

- 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.

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

## Step 3. Broadband feature(s) found?

- 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
- Compute RMS

- Refined analysis best done interactively (IDL? MatLab?).

## Rebinning to find a QPO: 4U1728–34

- To detect a weak QPO buried in a noisy spectrum, finding the right frequency resolution is essential!

## 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; *Z*2
- Vaughan et al. 1994, ApJ 435, 362
- noise statistics