Time-lags

When IC scattering is the process responsible for producing high energy photons, there will be a hard time-lag between the photons at high and low energies. The photon time-lags between two different energy bands essentially represents the difference in photon pathlength (or escape time) between bands: photons in the higher energy bands tend to have undergone more scattering events, and therefore have traveled further in order to escape the corona.

The time-lags between two energy bands can be found by Fourier transforming the light curves in the high and low energy bands (Miyamoto et al. 1988, van der Klis et al. 1987). The discrete Fourier transforms are then used to form the complex cross power spectrum by multiplying the complex Fourier amplitudes from the low-energy band with the complex-conjugate of the complex Fourier amplitudes from the high-energy band. The imaginary part of the complex cross power spectrum is then divided by its real part, and the arctangent of the result is taken to find the phase lags between bands. Dividing the phase lags by then gives the hard time-lag curve as a function of Fourier frequency.

Figure 3: Intensity in the two X-ray bands against photon escape time. In this graph the inner cloud radius is chosen to be 0.03 light seconds, while we assume that the physical radius of the outer cloud is 1.0 light second. The light curves are produced by Comptonization of a soft photon flare (delta function) at time zero. The cloud parameters are the best fit parameters (taken from MCS) for BATSE-COMPTEL data taken when Cyg X-1 was in the low state. They are, inner cloud:

, optical depth , outer cloud:

, optical depth .

Observationally this technique has been used to investigate the time variability of galactic black hole candidates (for instance, van der Hooft et al. 1999). On the theoretical side, Kazanas, Hua & Titarchuk (1997) have used this method to investigate the effect of density gradients in the Comptonizing plasma, and Böttcher and Liang (1998, 1999) have attempted to determine the location of the photon source with respect to the plasma. It has been found by these authors that the time-lag curve is sensitive to variations in electron density.

Figure 4: Comparison of the phase lag for two models: the solid curve is the MCS model where the ratio of the inner and outer radii is 0.001, while the dotted curve is for a homogeneous sphere. In both cases the outer radius is fixed at one light second. The difference between the two curves at low frequencies is due to the outer shell of plasma, and therefore tells us whether the outer shell exists.

Figure 5: Comparison of time-lags for various values of the ratio of the inner cloud radius to the outer cloud radius for the MCS model. Curve (a) is for the model where the ratio of the inner and outer radii is 0.1, while curve (c) is for a ratio of 0.05. Also shown are the lags due to the inner core only (curve b). The radius of the inner core in that case is taken to be 0.1 light seconds. The difference between curves (a) and (b) at low frequencies is due to the outer shell of plasma.

Figure 6: Comparison of time-lags for various small values of the ratio of the inner cloud radius to the outer cloud radius for the MCS model. Curve (a) is for a ratio of 0.03, curve (b) is for a ratio of 0.02, curve (c) is for a ratio of 0.01 and Curve (d) is for a ratio of 0.005.

Figure 7: Comparison of time-lags for various large values of the ratio of the inner cloud radius to the outer cloud radius for the MCS geometry. Curve (a) is for a ratio of 0.8, curve (b) is for a ratio of 0.5, and curve (c) is for a ratio of 0.3. Curve (d) are the lags due to a cloud of radius one light second with parameters equal to the outer shell plasma in the MCS model. The crosses indicate the lag for a different pair of energy bands (2-10, 10-400 keV), where the ratio of radii is 0.1. Because these energy bands do not extend to gamma-ray energies no photons from the outer-corona contribute to the lags and the curve is flat with no rise at low Fourier frequencies.

In this work we therefore use this method to determine the effect a density discontinuity of the type suggested by MCS has on the time-lag curve. Here we treat the radius of the inner cloud as a free parameter, while the outer radius is fixed at one light second. Changing the value of the outer radius (for any fixed ratio of the two radii) simply changes the location of the curve in frequency - time-lag parameter space without changing the shape of the curve. The electron temperature and optical depth of the clouds, as well as the temperature of the blackbody source photons, are the same as for section three.

We consider the lags between two different bands, 2-10 keV and 0.01-10 MeV (which we in this paper call bands one and two, respectively), where we have taken the central illumination of the cloud by the source photons to be an instantaneous delta function in time. Because so few photons scatter in the low optical depth outer cloud, the second band has been made large in order to make the two light curves of comparible intensity.

Figure 3 shows the intensity light curves as a function of time in the two bands, where the inner cloud radius is 0.03 light seconds. We see that there is a flattening of the light curve at around 0.1-0.2 seconds due to photons scattering in the outer cloud. Although the overall spectrum is largely insensitive to the ratio of the inner and outer cloud radii, variations in this ratio do result in different light curves as more or fewer photons scatter in the outer cloud, and therefore result in different hard time-lags when Fourier transformed.

Figure 4 shows the phase lag between bands one and two as a function of Fourier frequency for two different models. The solid curve is the MCS model with temperatures and optical depths as in section three. The dotted curve is the phase lag for a homogeneous cloud of plasma with temperature and optical depth equal to that of the inner core of the MCS model. In both cases the outer radius is set to one light second. We see that the solid curve develops a second peak at low frequencies, while the dotted curve is single-peaked. This rise in the solid curve at low frequencies is due to the change in temperature and optical depth in the outer shell of plasma in the MCS model. This therefore indicates the possibility of determining if the MCS geometry exists in BHCs.

Figures 5, 6 and 7 show the hard time-lag curves between bands one and two as a function of Fourier frequency for various values of the inner cloud radius. In figure 5 for comparison we also show the time-lags due to the inner cloud only (that is, with the outer plasma shell removed). With the outer shell added there is a larger lag at low frequencies than for the inner cloud alone. This is the key difference between simple spherical corona models and the multi-temperature model due to MCS.

Figure 6 shows that for small inner-radius values, the lags become almost constant at high frequencies as the inner radius is made small, but there is still a peak at low frequencies. For large inner-radius values (figure 7), we are seeing the inner cloud/core only, as fewer photons scatter in the relatively smaller outer cloud/shell. In figure 7 we also show the lags due to a spherical corona with electron temperature and optical depth equal to that of the outer cloud/shell in the MCS model. This is equivalent to making the inner cloud radius and optical depth approach zero.

The crosses in figure 7 show the lags for a different pair of energy bands (2-10, 10-400 keV, where the ratio of radii is 0.1). Because these bands do not extend up to gamma-ray energies, fewer high energy photons from the outer-corona have contributed to the time-lag curve. The lags are therefore almost constant with frequency as we are essentially seeing just the inner-corona. In this case the lags tell us little about the outer-corona.

The various time-lag curves can now in principle be used to determine the ratio of the inner and outer radii, as well as set the overall scale of the corona. Given the temperatures and optical depths (from spectral modeling) of the inner and outer-coronae, the shape of the hard time-lag curve determines the ratio of the inner and outer radii, while the position of the hard time-lag curve in the frequency - time-lag parameter space determines the magnitude of the outer radius. Thus if the optical depths and the values of the radii are known, then the density of the corona is known, and we can in principle know all the physical parameters of the corona (assuming the MCS core plus shell model is true).

It is also possible that the outer radius can be found from other criteria, perhaps by identifying the break in the power spectral density of a light curve with the value of the outer radius as suggested by Böttcher & Liang (1998).

© Copyright Astronomical Society of Australia 1997