6 High-Frequency Variability in Neutron-Star Low-Mass X-ray Binaries
305
6.8.3 The Energy-Dependent Lags and Coherence of the kHz
QPOs
As we described in Sect. 6.4, one can study the phase/time lags of the QPOs
using light curves in two different energy bands. Besides the lags, one can also
study the coherence function between the two light curves [65, 125, 173]. The
coherence function [23], γ 2 (ν), measures the degree of linear correlation between
two noiseless signals (light curves) as a function of the Fourier frequency. More
precisely, γ 2 (ν) should be called γ 2 (ν; E 1 , E 2 ) to indicate that it is the coherence
function between two light curves at energies E 1 and E 2 ; when the lags and the
coherence function are given as a function of energy, one means that those are the
quantities measured at E = E 2 with respect to the reference band, in this case
E 1 . However, since the observed light curves are not noiseless, but are affected by
Poisson counting noise (see Sect. 6.4), to study the degree of correlation between
two X-ray light curves one uses the intrinsic coherence function, usually denoted as
γ 2
I (ν), that corrects for this (see [173], for an explanation). For simplicity, here we
will use the term coherence function, and will write γ 2 (ν), to refer to the intrinsic
coherence function.
Figure 6.27 shows the time lags (top panel) and coherence function (bottom
panel) around the frequency of the lower kHz QPO in 4U 1608−53, using light
curves with energies around 3 and 8 keV [37]. This Figure shows that the two light
curves used to compute the lags and coherence function are perfectly correlated
(coherence function equal to 1 with small errors) in the frequency range at which
the QPO signal dominates, and uncorrelated (coherence function consistent with 0
Fig. 6.27 Time lags (top
panel) and coherence function
(bottom panel) as a function
of Fourier frequency, at
frequencies around that of the
lower kHz QPO in
4U 1608−52. The horizontal
line in the top panel is at the
zero time lag, while the
horizontal line in the bottom
panel shows the perfect
coherence of 1 [37]
-0.0008
-0.0004
0
0.0004
0.0008
Time delay [msec]
-10
0
10
680 685 690 695 700 705 710 715 720
Intrinsic coherence
ν [Hz]
305
6.8.3 The Energy-Dependent Lags and Coherence of the kHz
QPOs
As we described in Sect. 6.4, one can study the phase/time lags of the QPOs
using light curves in two different energy bands. Besides the lags, one can also
study the coherence function between the two light curves [65, 125, 173]. The
coherence function [23], γ 2 (ν), measures the degree of linear correlation between
two noiseless signals (light curves) as a function of the Fourier frequency. More
precisely, γ 2 (ν) should be called γ 2 (ν; E 1 , E 2 ) to indicate that it is the coherence
function between two light curves at energies E 1 and E 2 ; when the lags and the
coherence function are given as a function of energy, one means that those are the
quantities measured at E = E 2 with respect to the reference band, in this case
E 1 . However, since the observed light curves are not noiseless, but are affected by
Poisson counting noise (see Sect. 6.4), to study the degree of correlation between
two X-ray light curves one uses the intrinsic coherence function, usually denoted as
γ 2
I (ν), that corrects for this (see [173], for an explanation). For simplicity, here we
will use the term coherence function, and will write γ 2 (ν), to refer to the intrinsic
coherence function.
Figure 6.27 shows the time lags (top panel) and coherence function (bottom
panel) around the frequency of the lower kHz QPO in 4U 1608−53, using light
curves with energies around 3 and 8 keV [37]. This Figure shows that the two light
curves used to compute the lags and coherence function are perfectly correlated
(coherence function equal to 1 with small errors) in the frequency range at which
the QPO signal dominates, and uncorrelated (coherence function consistent with 0
Fig. 6.27 Time lags (top
panel) and coherence function
(bottom panel) as a function
of Fourier frequency, at
frequencies around that of the
lower kHz QPO in
4U 1608−52. The horizontal
line in the top panel is at the
zero time lag, while the
horizontal line in the bottom
panel shows the perfect
coherence of 1 [37]
-0.0008
-0.0004
0
0.0004
0.0008
Time delay [msec]
-10
0
10
680 685 690 695 700 705 710 715 720
Intrinsic coherence
ν [Hz]