X-RAY VARIATIONS AT THE ORBITAL PERIOD FROM CYGNUS X-1 IN THE HIGH/SOFT STATE

The RXTE ASM Products Database provides light curves and intensities in the 1.5–3, 3–5, and 5–12 keV bands ("colors") for ≈350 sources. We use the Definitive Products version of the data that has been processed at MIT. In the resulting time series, there are often two measurements that occur at exactly the same time. This is because two of the SSCs have overlapping fields of view and can provide independent measurements. For our analysis, when this occurs we average the two measurements. For the ASM observations of Cyg X-1, 70% of the measurements carry a unique time stamp. The error bars on the light curves and colors are based on counting statistics convolved with a 3% systematic error, as estimated by the RXTE ASM team from Crab pulsar measurements.

We divide the ASM observations into periods of soft/high and consider the remainder of the period to be hard/low (Table 1 and Figure 1). There are short intervals that appear similar to the soft/high states that we do not single out, but treat as part of the remaining hard/low state. We choose the time interval of the Soft 3 period to correspond with the interval chosen by Lachowicz et al. (2006).

Zoom In Zoom Out Reset image size

To supplement our analysis of RXTE ASM data, we also use pointed RXTE PCA/HEXTE observations. We consider the data set of 202 observations from 1999 through 2004 examined in Wilms et al. (2006). These observations were scheduled biweekly for lengths of ∼5 ks. Wilms et al. fit the spectra between 3 and 200 keV to three models: a broken power law with exponential cutoff, the Comptonization model of Titarchuk (1994), and the Comptonization model of Coppi (1999). These fits were used to explore the hard and soft states, and led to the conclusion that there is a continuum between these states. However, the fits have not previously been used to investigate the 5.6 day orbital period.

We apply the Lomb Normalized Periodogram (LNP; Scargle 1982) and the Analysis of Variance (ANOVA; Davies 1991) methods to search for periodicity in the RXTE ASM observations of Cyg X-1.

The LNP is a generalization of the Fourier power spectrum to unevenly sampled data. In the case of evenly sampled time series, the LNP is identical to the power spectrum. When we apply this periodogram to the hardness ratio HR 1 (as defined in Figure 1) during the Soft 3 period, we find a peak power of 73 at a period of 5.61 days and a power of 47 in the neighboring bin at 5.59 days. We eliminated points with HR 1 below 0 or above 3, as negative hardness ratios are not physically meaningful and large hardness ratios result from division by count rates near 0 and carry large uncertainties. This only eliminated 2 of 6038 data points in the soft state. We show the LNP for each of the five soft states we have identified in Figure 2.

Zoom In Zoom Out Reset image size

For a power spectrum of uniformly sampled data, normalized so that the white noise level is 1, the probability of a signal level s would be exp(−s). An LNP power of 73 corresponds therefore to a false alarm probability of 2 × 10⁻³². However, the signal at 5.6 days does not appear against white noise, but instead against a continuum strongest at low frequencies. Examination of the distribution of powers shows that they obey a distribution closer to exp(−s/4), which would imply a false alarm probability of only ≈10⁻⁸, implying a much lower significance of the signal.

Application of ANOVA to the HR 1 ratio in the Soft 3 period gives a probability of 5 × 10⁻²³ that variation at 5.6 days is random. For this test, we used 16 phase bins. As with the power spectral methods, the ANOVA method may return a false positive in a test for periodicity if the power spectrum contains a significant continuum component, and the significance given by this method may be an overestimate.

We have found a clear orbital variation in the hardness ratio HR 1, but not in the individual energy bands in the same period ("Soft 3"). The count rates in the individual energy bands are more variable on all timescales than HR 1. For example, during Soft 3, the χ²_ν fit for a constant count rate in the 1.5–3 keV range is 62 with 6037 degrees of freedom. The χ²_ν fit for a constant HR 1 is 7.1. As a result of this greater nonorbital variability, it may be harder to detect any orbital variation in the actual count rates than in HR 1, which may increase near ϕ = 0 with the absorbing column, but which may be less affected by flares than the count rates.

If the Soft 3 orbital variation is nσ significant, then scaling from the duration of the observations, the Soft 1 and Soft 2 variability should be ≈0.4σ significant and the Soft 5 variability should be ≈0.7σ significant. However, the rms variability in HR 1 is 50% higher for Soft 5, so that we might expect only ≈0.5σ significance in the orbital variations. The rms variation in the three X-ray bands and two hardness ratios are within a factor of 2 for all the soft states, except for HR 2 in Soft 2, which appears anomalously high.

We also tested whether the HR 1 variability seen during the Soft 3 period would appear significant in shorter intervals of that same period. LNPs of the first or middle 200 days of Soft 3 show no significant variation at the orbital period, though the power spectrum of the last 200 days shows a peak as great as for the total Soft 3 period. There was no significant variation in HR 1 in the first 100 days of Soft 3, or in days 190 through 290, but within the last 100 days the power spectral peak reached 65. For the last 50 day interval in Soft 3, the power spectral peak is 47, but no significant peak was seen for the first or second 50 day periods, or another 50 day period near the end of Soft 3. Intervals of 200 days, 100 days, and 50 days are identical to the lengths of the Soft 1, Soft 4, and Soft 2 periods, respectively. In conclusion, it is possible that orbital variability as strong as that seen during the Soft 3 state could occur during all soft states but went undetected by RXTE/ASM during the shorter Soft 1, Soft 2, Soft 4, and Soft 5 states.

In Figures 3–7, we present light curves and hardness ratios during the Soft states 1–5, folded with the 5.6 days binary period.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We have demonstrated with both power spectra and epoch folding methods that the hardness ratio and count rate show the 5.6 day orbital periodicity during the Soft 3 state. However, previous studies of dips in the hard state have suggested that discrete dips may occur at any phase, but may simply be more likely to occur near ϕ = 0.

Cyg X-1 shows both dips in which the spectrum hardens (color dips) and also monochromatic dips (count dips) in which the count rate diminishes but the color does not change appreciably (Feng & Cui 2002). We thus have four classes of dips to compare: color and count dips in the soft and hard states. We choose criteria to define these dips.

3.2.1. Color Dips in the Soft State

Color dips in the hard state were already considered by Balucińska-Church et al. (2000). They selected as dips those observations with hardness ratios HR 1 >2.0 or HR 2 >2.5. We adjust these criteria to the changed spectrum during the soft state.

If we choose as dips those data points with $\mbox{HR\;1}/\overline{\mbox{HR\;1}}>1.96$, where $\overline{\mbox{HR\;1}}$ is the mean hardness ratio, we find only 14 observations to count as dips in the Soft 3 state. From Section 3.1 we know that HR 1 varies with the orbital period, so a less restrictive criterion should select individual observations (dips) with some preference for orbital phase. We thus choose some constant k < 1 and select dips based on $\mbox{HR\;1}/\overline{\mbox{HR\;1}}>1.96 k$. The results, shown in Figure 8, demonstrate that the strongest dips defined by HR 1 color cluster near ϕ = 0. Weaker dips depend less strongly on phase.

Zoom In Zoom Out Reset image size

For comparison, we show a histogram of color dips in the hard state in Figure 9. As there is more data for the hard state than for the Soft 3 state, this plot has less scatter. It shows more clearly the trend that stronger dips are more likely to occur near ϕ = 0.

Zoom In Zoom Out Reset image size

3.2.2. Count Dips in the Hard State

Dips selected by low count rate (count dips) are less frequent in the soft state and show no obvious orbital clustering. Count dips in the hard state cluster with phase similarly to color dips (Figure 10). To select dips based on count rate, we choose an interval for the constant b and select ASM observations with count rate $\mbox{CR}=b \overline{\mbox{CR}}$. We find that stronger count-selected dips are more likely to occur near ϕ = 0.

Zoom In Zoom Out Reset image size

While the 202 pointed RXTE PCA/HEXTE observations fit to models by Wilms et al. (2006) cover the Soft 3 period, they are not extensive enough to show the orbital variability in the soft state.

However, the pointed observations suffice to show the orbital variability in the hard state, already demonstrated with RXTE ASM observations. The broken power-law fits of Wilms et al. include the hydrogen column density N_H as a free parameter. In Figure 11, we show the fitted value of this parameter during the hard state folded over the 5.6 day period, where we have defined the hard state by F(5–10)/F(2–5)>0.3, where F(5–10) is the 5–10 keV flux according to the model and F(2–5) is the 2–5 keV flux. Eliminating observations that have N_H = 0 leaves 133 data points. Applying ANOVA, we find a false alarm probability 4 × 10⁻⁸ for a periodicity at 5.6 days. If instead we examine F(5–10)/F(2–5) during the hard state, we have 145 data points (Figure 12) which give a false alarm probability of 8 × 10⁻⁶.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Wilms et al. also fit the pointed RXTE observations to two Comptonization models, one based on Titarchuk (1994) and another based on Coppi (1999). The orbital variations in N_H and F(5–10)/F(2–5) are not as significant using fits with either of these models compared with the fits using the broken power-law model.

The RXTE pointed observations have much higher spectral resolution, timing resolution, and sensitivity compared with the ASM observations. Here we have shown the orbital variation in absorbing column using not just broadband colors, but detailed fits to spectra.

The ionization in the stellar wind depends on the X-ray spectrum. To compute the ionization in the wind, we use the XSTAR program (Bautista & Kallman, 2001). Under the assumption that the wind is optically thin to the X-rays from the vicinity of the black hole, the ionization and temperature equilibria can be computed based on the ionization parameter ξ = L/nr², where L is the X-ray luminosity, n is the hydrogen density, and r is the distance to the X-ray source.

XSTAR requires as input the luminosity L in the 13.6 eV to 13.6 keV range. Extending the power law to low energies, particularly during the soft state, would lead to an unrealistic spectrum. We model the spectrum using the Bulk Motion Comptonization (BMC) model (Shaposhnikov & Titarchuk 2007), which has been implemented in the XSPEC spectral fitting program. This model provides a physically motivated spectrum that is similar to a blackbody and a broken power law.

However, there may be an additional soft blackbody component with kT ∼ 0.1–0.2 keV, as observed by Ebisawa et al. (1996) with ASCA.

We therefore simulate the orbital variability in X-ray absorption using spectra given by the BMC fits of Shaposhnikov & Titarchuk (2007) with or without an added soft blackbody component of kT = 0.1 keV. We present here only the results without the soft blackbody, which has a greater effect on the UV P Cygni lines than on the X-ray absorption. For each model we identify both the 13.6 eV to 13.6 keV luminosity, which is what XSTAR requires, and the 2–200 keV X-ray luminosity, which describes the X-ray output in the range observed by RXTE.

We make a model similar to that of Wen et al. (1999), in which the orbital variation of X-rays results from the black hole passing behind larger column densities of partially ionized wind near ϕ = 0.

We fix the orbital inclination to 30 degrees, the radius of the O star to R_O = 1.387 × 10¹² cm, and the orbital separation to 2 R_O. We fix the radial velocity law in the wind to be v(R) = 1430(1 − R_O/R)^β with β = 0.75. The wind parameters are based on the fits to the UV P Cygni lines observed during the X-ray soft state with the HST STIS, as reported in Vrtilek et al. (2008).

We calculate, in phase intervals of Δϕ = 0.05, the column density of the sight line to the black hole at each ionization parameter between log ξ = 1 and log ξ = 5. This quantity, N(ϕ, log ξ), can be adjusted for various values of the wind mass-loss rate $\dot{M}$ and the X-ray luminosity L_x by

$$\begin{equation} N^\prime (\phi,\log \xi)=\frac{\dot{M}^\prime }{\dot{M}} N\left(\phi, \log \frac{L_x \dot{M}^\prime }{L_x^\prime \dot{M}} \xi \right), \end{equation} \tag{ 1 }$$

where $\dot{M}$ and L_x have been changed to take on values $\dot{M}^\prime$ and L'_x, respectively.

From XSTAR, we calculate the transmission of the source spectrum as it passes through each column at each value of log ξ, and thus the X-ray spectrum at each phase ϕ. We add the Compton scattering column density to the photoabsorption column predicted by XSTAR. This allows us to find those values of $\dot{M}$ and L_x that predict the orbital variation in the Cyg X-1 X-ray spectrum that results entirely from the black hole's motion through a smooth, partially ionized stellar wind.

The results of fitting the model to the orbital variation are shown in Table 2. The best-fit values provided for $\dot{M}$ and L_x are uncertain, as these parameters vary in tandem. The contours of reduced χ² provide a measure of which ranges of $\dot{M}$ and L_x are acceptable, although the models only provide fits with χ²_ν ≈ 1 for HR 1 during the Soft 3 state. In the hard state, the models strongly favor $L_{\rm x, 37}/\dot{M}_{-6}<1$, $\dot{M}_{-6}<4$, while in the soft state, they favor $L_{\rm x, 37}/\dot{M}_{-6}<1.5$ and $\dot{M}_{-6}<2$, where we use L_x,37 ≡ L_x × 10⁻³⁷ erg s⁻¹ and $\dot{M}_{-6}\equiv \dot{M}\times 10^{6}\; {M_{\odot }}$ yr⁻¹. For the soft state, the χ² contours given by fits to the HR 1 color and to the individual X-ray bands are complementary, limiting $L_x/\dot{M}$ and $\dot{M}$, respectively.

Table 2. Model Fits to the Cyg X-1 Variability in Three RXTE ASM Energy Bands or to the Hardness Ratio Between the Two Low-energy Bands

State	Data	Model	χ2ν	Lx/1037	$\dot{M}\times 10^{6}$	Fca
				(erg s−1)	(M☉ yr−1)
Soft	1.5–3, 3–5, 5–10 keV	Smooth wind	3.5	0.25	0.36	NA
Soft	1.5–3, 3–5, 5–10 keV	Partial covering	3.3	NA	NA	0.14
Soft	3–5 keV/1.5–3 keV	Smooth wind	1.10	0.25	0.72	NA
Soft	3–5 keV/1.5–3 keV	Partial covering	0.9	NA	NA	0.28
Hard	1.5–3, 3–5, 5–10 keV	Smooth wind	3.2	1.7	1.8	NA
Hard	1.5–3, 3–5, 5–10 keV	Partial covering	3.1	NA	NA	0.30
Hard	3–5 keV/1.5–3 keV	Smooth wind	2.8	1.5	2.2	NA
Hard	3–5 keV/1.5–3 keV	Partial covering	2.6	NA	NA	0.30

Notes. We give here the X-ray luminosity L_x and wind mass-loss rate $\dot{M}$ of the best fit. ^aCovering fraction.

Download table as:  ASCIITypeset image

The discrete X-ray dips seen in Cyg X-1, particularly near ϕ = 0 (Balucińska-Church et al. 2000), may be related to the dips seen in such X-ray binaries as Hercules X-1 (e.g., Shakura et al. 1999), or the class known as "dippers" (Frank et al. 1987). In these systems, dips may be associated with the gas stream feeding the accretion disk through Roche lobe overflow. The focused wind in Cyg X-1 could provide a similar absorbing structure. However, a full treatment of a focused wind would be beyond the scope of this paper, and would probably require hydrodynamic modeling.

Instead of dealing with complex asphericity, we pursue an alternative to the smooth, partially ionized wind model in which the wind is still assumed to be spherically symmetric and the orbital variation in the X-ray spectrum still results from the line of sight at ϕ = 0 passing through more of the dense wind regions. However, we assume that the absorption is caused by neutral clumps, and that the chance of such a clump passing through the line of sight is linearly proportional to the wind column density. We fix the clump column density to N_H = 4 × 10²² cm⁻², following the ASCA observations of X-ray dips (Ebisawa et al. 1996). We allow two free parameters: the covering fraction of the clump when the wind column density reaches its maximum and the mass-loss rate in the wind. The latter influences the X-rays only through Compton scattering, as the wind outside of the dense clumps is entirely ionized. The results do not constrain $\dot{M}$ very well, although at too high values, too great an orbital variation is seen. The covering fraction parameter serves as a time average of the actual covering fraction during dips with the zero covering fraction in periods when dips are not present.

4.3.1. Hard State

4.3.2. Soft State