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«J. F. Dolan Department of Astronomy San Diego State University San Diego, CA 92182-1221 tejfd ABSTRACT X-ray pulses with ...»

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J. F. Dolan

Department of Astronomy

San Diego State University

San Diego, CA 92182-1221



X-ray pulses with millisecond-long FWHM gave been detected in RXTE (Rossi X-Ray

Timing Explorer) observations of Cyg X-1. Their identity as short-timescale variations in the X-

ray luminosity of the source, and not stochastic variability in the X-ray flux, is established by their simultaneous occurrence and similar pulse structure in two independent energy bandpasses.

The light-time distance corresponding to the timescale of their FWHM indicates that they originate in the inner region of the accretion disk around the system’s black hole component. The fluence in the pulses can equal or exceed the fluence of the system’s average continuous flux over the duration of the pulses’ FWHM in several different bandpasses between 1 and 73 keV.

Millisecond pulses are detected during both high and low luminosity states of Cyg X-1, and during transitions between luminosity states.

Keywords: X-rays: Binaries – X-rays: Individual: Cyg X-1

1. INTRODUCTION Sunyaev (1973) pointed out that turbulent convection or magnetic reconnection should lead to hot spots – self-luminous clumps of emitting material – in a quiescent accretion disk surrounding a black hole. A quiescent disk, radiating solely by thermal processes, occurs when the mechanisms that transfer angular momentum outward in the disk have low efficiency, parameterized as α 1, where α = [vt /vs] + [H2/4πρvs2], and vs is the speed of sound, vt is the turbulent velocity, H is the magnetic field strength, and ρ is the density in the disk. The luminosity of theses flare patches may equal or exceed the average luminosity of the entire disk in some energy bandpasses.

If the lifetime of a flare patch exceeds its orbital period around the black hole, then Doppler effects and occultation by the black hole will modulate the flux from the flare patch received at the Earth with a timescale characteristic of this orbital period. A typical maximum to minimum luminosity ratio for the flux is calculated to be 16 (Sunyaev 1973; Cunningham & Bardeen 1972), so the emitted flux detected at Earth will appear to be a pulse of radiation.

Shakura & Sunyaev (1973) estimate the characteristic timescale of these fluctuations as ∆τ = (10-4 /α) m (r/3)3/2 s [1] where m is the mass of the black hole in solar masses and r is the distance of the flare patch from the black hole in units of the Schwarzschild radius RS = 2GM/c2. The innermost stable orbit around a Schwarzschild (non-rotating) black hole has r = 3, so for m = 10 a characteristic timescale 1 ms is expected for the pulses. Stable orbits do exist for r 3 for a Kerr black hole with angular momentum (in geometric units) a 0. For a approaching 1, its maximum value in geometric units, the characteristic timescale is 7 – 10 times shorter than that given in Eq. [1]. i.e., ∆τ ~ 1 ms (Sunyaev 1973; Novikov & Thorne 1973).

Rothschild et al. (1974, 1977) detected variable X-ray luminosity from Cyg X-1 that they described as bursts of radiation with a millisecond timescale. Detections occurred in a 2–35 keV bandpass during two different rocket flights with 320 µs time resolution in the first flight and 160 µs time resolution in the second. Both observations were made during the 2-10 keV low flux state of Cyg X-1. (The 2-10 keV low flux state is the hard spectral state of the source. It is also its low luminosity state [Dolan et al. 1979].) The average intensity profile of the bursts with this time resolution was consistent with a rectangular (“top-hat”) pulse.

The identification of the variability detected by Rothschild et al. in low count-rate data as ms-timescale pulses was questioned by Press & Schachter (1974), who pointed out that stochastic variability in the flux from the system could mimic short-timescale flares if it occurred on top of small, slow variations in the luminosity of the source. Weiskopf & Sutherland (1978) also questioned the identification of the variability as ms pulses because of the presence of shotnoise variability in the X-ray flux of Cyg X-1. Priedhorsky et al (1979) detected no ms bursts in an 84 s long observations of Cyg X-1. These authors also found shot noise to be an acceptable model of its variability.

To further complicate the situation, Giles (1981) reported ms-timescale bursts during a

6.4 s rocket observation of Cyg X-1, but cautioned that the evidence for ms-timescale bursts was not definitive because of the statistical limitations imposed by the small number of counts in each burst. Giles also found his data to be consistent with a shot-noise model of variability.

Meekins et al. (1984) also reported variability on a timescale of ~ 3 ms during a 9 minute observation of Cyg X-1 with HEAO-1, modeling it as the superposition of uncorrelated 3 ms and 300 µs shots. But Chaput et al. (2000) found no ms bursts from Cyg X-1 in 1.3 hours of 1-25 keV HEAO-1 and RXTE observations. The latter authors concluded that the Meekins et al.

results were spurious, and attributed them to instrumental effects. Uttley & McHardy (2001) then showed that the standard shot-noise models do not apply to the variable flux from Cyg X-1 because time series data from the source is not stationary on short timescales. (The rms variability scales linearly with the flux.) More recently, Gierlinski & Zdziarski (2003) reported 13 strong ms-timescale flares in over 600 hours of 2-60 keV observations of Cyg X-1 with RXTE. The flares occurred in both the hard and soft spectral states of the source. We report her that ms-timescale pulses are a common occurrence in the 1-15 keV flux from Cyg X-1 observed with RXTE. The pulses occur during both high and low luminosity states, and during the transitions between these states.


We analyzed 7.95 hr of archival RXTE observations of Cyg X-1 for millisecondtimescale pulses. The observations were obtained during 1996 February, May, June, August, and December; 1997 September; and 2000 December. During 3.69 hr of this time, Cyg X-1 was in the 2-6 keV high state, which is the soft spectral state / high luminosity state of the source. 1.77 hr of these high-state observations were obtained 2 d before a transition from the high state to the low state, which is the hard spectral state / low luminosity state of the source. 2.69 hr of observations were obtained during failed transitions between the low state and the high state, including 1.02 hr during a drop back from the high state to the low state.

All data were obtained using the Proportional Counter Assembly (PCA) detectors with a time resolution of either 122 µs or 244 µs. The data were binned in two separate energy bandpasses, each of which contained an approximately equal number of photons. The two bandpasses were independent, in the sense that each detected photon appeared in only one of the bandpasses.

We define a pulse as a statistically significant intensity variation having a full width at half-maximum (FWHM) duration shorter than a pre-selected value. The confidence level of the identification of a positive-going variation in counting rate as a pulse is then determined by a statistical estimate of the frequency of occurrence of stochastic variations in the detector’s counting rate as large as the pulse or larger. If the number of candidate pulses detected significantly exceeds the expectation value for the number of variations that large or larger above the mean flux given by Poisson statistics, then intensity variations caused by pulses are present in the data. (Note that a frequency of occurrence analysis can never positively identify any individual event as a pulse because, given a long enough observing time, stochastic variability can mimic any characteristic of a photon pulse. A frequency of occurrence analysis can show how likely or unlikely it is that an observed variation is a stochastic variation.) Because the X-ray flux from Cyg X-1 exhibits stochastic variability over timescales shorter than 1 s, the mean flux to which the significance level of a variation must be referred is the local mean flux of the source in a time interval that includes the variation. The two intervals from which the mean counting rate was estimated included the underlying counts in the wings of any pulse, but excluded the peak count rate channels. If the variation is stochastic and not a pulse, then excluding the peak count rate region from the calculation of the local mean biases the mean to lower values. We therefore adopted a mean flux level 10% above that calculated to attempt to account for this bias. In any case, because of the small probabilities involved, this procedure should discriminate between a stochastic variation and a pulse.

The maximum S/N in a detector’s counting rate produced by a pulse with a given fluence will occur when the sample interval of the data equals the FWHM of the pulse and the center of a sample interval is coincident with the maximum of the pulse (Dolan 2001). If the sample interval is longer than the FWHM of the pulse, the number of counts from the continuous flux will increase in proportion to the sample interval while the number of counts in the pulse will remain constant, lowering the S/N of the pulse. If the sample interval is shorter than the FWHM of the pulse, or if the center of the pulse is not the center of the sample interval, then the counts from the pulse will be divided between two or more sample intervals, each of which has, on average, the same number of counts from the continuous flux, also lowering the maximum S/N attained by the pulse in any one sample interval. Searching for ms-timescale pulses with only one sample time introduces a selection effect into the detection process that discriminates against pulses with a FWHM duration different from the time resolution of the observations.

We searched for pulses with FWHM between ~ 0.2 and ~ 10 ms. We therefore rebinned the original data into additional data sets modulo 2, 3, 5, 10, 20, and 40 times the original time resolution (0.122 or 0.244 ms) to improve the detectability of ms-timescale pulses. When we sum the counts in n successive sample intervals of the original data set to produce the counts in one rebinned sample interval, there are n different data sets we can produce by starting the furst sum with sample number 0, 1, 2,... n-1 of the original data set. We call the sample number with which we start the first sum the phase of the rebinned data set. Estimates of the statistical significance of any detected pulses must than take into account a factor representing the number of ways the original data set was rebinned.

Additional criteria can now be used to discriminate between pulses and stochastic


- a pulse should exist at the same time in two independent energy bandpasses; stochastic variability should be random in independent energy bandpasses;

- a pulse should exist at multiple phases of a rebinned data set with a given sample time.

Stochastic variability displays random structure at different phases of a rebinned data set;

- a pulse should exhibit a characteristic rise to and fall from maximum when examined with a sample time smaller than the pulse’s FWHM. Stochastic variability exhibits fluctuations between the counts in adjacent sample intervals that are random in direction.

The requirement that any variability meet these additional criteria in order to be identified as a pulse increases the confidence level of the identification of ms-timescale pulses beyond that resulting from the statistical significance criterion alone.

The FWHM duration of the event, in sample intervals, was estimated as the time at the center of the sample interval on the descending slope of the pulse containing half the counts of the peak sample interval (after subtracting the counts per sample in the continuous flux from each sample) minus the time at the center of the corresponding sample interval on the ascending slope. The uncertainty of the time of half-maximum is one-half a sample interval, so the uncertainty of the FWHM is √2/2 times the width of a sample interval. FWHM are calculated using the time resolution of each binning and phase at which a pulse is detected. We use the FWHM with the lowest associated uncertainty as the FWHM of the pulse. The FWHM used in calculating the probability of a stochastic variation producing the fluence of the event is the FWHM determined with the time resolution and phase of the data set in which the probability is calculated.

3. RESULTS Millisecond-timescale X-ray pulses exist in every 100 s duration data set we examined.

We list in Table 1 the properties of several such pulses, selected to illustrate the range of parameters they possess.

–  –  –

——————————————————————————————————————— Event Luminosity State Bandpass FWHM Spectral Hardness Ratio (keV) (ms) Pulse Bkg ———————————————————————————————————————

–  –  –

3.1. Event 960212 Fig. 1. Event 960212. The PCA counting rate at 4.88 ms sample time. The time is given as the number of sample intervals after the start of the data set. The event peaks in sample interval

4192. above: 1-73 keV bandpass; middle: 1-9.5 keV; below: 9.5-73 keV, all phase 1.

The event occurred on 1996 February 12 at 09:39:20 UT, when Cyg X-1 was in the low state. The counting rate in a 1-73 keV bandpass is shown in Fig. 1, binned at 4.88 ms sample time. The bin numbers on the x –axis are the number of 4.88 ms time intervals since the start of the data set. The event we identify as a pulse peaks at 105 counts in sample interval 4192. We assign a local continuous flux level of 40 counts per sample interval at the time of the event. The Poisson probability of observing a sample interval with 105 counts or more in a data set with a mean of 40 counts per sample is 10-23. There are 20,480 sample intervals in the 100 s long data set; the expectation value for the number of samples with a count of 105 (or larger) is 1.5 x 10-19.

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