308
M. Méndez and T. M. Belloni
factor of at least ∼2. Therefore, the lags of the upper kHz QPO cannot be due only
to a light travel-time delay between the disc and the neutron star either.
Since the lags of the lower kHz QPO are soft, whereas inverse Compton
scattering in a corona should produce hard lags (Sect. 6.4; but see below), a different
mechanism was required to model the lags of the lower kHz QPO. Reverberation
due to reflection of hard photons from the corona off the accretion disc had been
used to explain the soft lags of the broad-band noise component in the power
spectrum of active galactic nuclei [49, 189, 190] and galactic black-hole candidates
([39, 160], see also [74]). Reverberation was therefore a promising mechanism to
explain the lag spectrum of the lower kHz QPOs.
Figure 6.30a shows the lag spectrum of the lower kHz QPO in 4U 1608−52 [30]
with the best-fitting reverberation model. While the lags of the QPO decrease more
or less monotonically as the energy increases, the best-fitting model predicts that
above E ∼ 8 keV the lags should increase with energy, contrary to the observations.
This led to the conclusion [30] that the lags of the lower kHz QPO in 4U 1608−52
cannot be due only to reverberation. Subsequently, the same idea was tested on the
upper kHz QPO in 4U 1728−34 [36]; the result of the fits of a reverberation model
to the lags of this QPO is shown in Fig. 6.30b. From this Figure it is apparent that,
while the lags of the QPO first decrease with energy up to E ∼ 5–6 keV, and
increase again above that energy, the model predicts that the lags should follow the
opposite behaviour. This led to the conclusion that, as with the case of the lags of the
lower kHz QPO in 4U 1608−52, the lags of the upper kHz QPO in 4U 1728−34
cannot be solely due to reverberation. Notice that, with few exceptions [66, 101],
so far reverberation models have been used only to fit and explain the time/phase
lags, and not the amplitude, of the observed light curves of accreting supermassive
black holes in active galactic nuclei and stellar-mass black holes in galactic X-ray
binaries, whereas a consistent model of the radiative mechanism that fits the data
should be able to explain both the rms and lag spectrum of the variability in these
sources.
Figure 6.31 shows some of the results of the most extensive study yet of the lags
and rms spectra of the kHz QPOs in LMXBs [157]. That work presents the energydependent rms amplitude, time lags and intrinsic coherence function, plus the time
lags as a function of frequency of the lower kHz QPOs in 14 sources, and the same
information for the upper kHz QPO in six out of those 14 sources. In Fig. 6.31 we
show only the rms and lag spectra of the lower kHz QPO in those 14 sources (we
already showed the rms spectra of the upper kHz QPO in six of those 14 sources in
Fig. 6.17b). The results shown in [157] reinforce the importance of considering both
the rms and lag spectra to try and model the radiative properties of the kHz QPOs.
Models that involve inverse Compton scattering did not seem appropriate to
explain the lags of the lower kHz QPO, given that those lags are soft, whereas
Comptonisation would only produce hard lags (see Sect. 6.4). Contrary to those
expectations, inverse Compton scattering could work, and produce soft lags, if there
is feedback from the corona to the disc ([92], see also [91]). In this scenario, soft
photons from the disc are up-scattered in the corona; part of those up-scattered
photons go to the observer, and produce the power-law like component in the
M. Méndez and T. M. Belloni
factor of at least ∼2. Therefore, the lags of the upper kHz QPO cannot be due only
to a light travel-time delay between the disc and the neutron star either.
Since the lags of the lower kHz QPO are soft, whereas inverse Compton
scattering in a corona should produce hard lags (Sect. 6.4; but see below), a different
mechanism was required to model the lags of the lower kHz QPO. Reverberation
due to reflection of hard photons from the corona off the accretion disc had been
used to explain the soft lags of the broad-band noise component in the power
spectrum of active galactic nuclei [49, 189, 190] and galactic black-hole candidates
([39, 160], see also [74]). Reverberation was therefore a promising mechanism to
explain the lag spectrum of the lower kHz QPOs.
Figure 6.30a shows the lag spectrum of the lower kHz QPO in 4U 1608−52 [30]
with the best-fitting reverberation model. While the lags of the QPO decrease more
or less monotonically as the energy increases, the best-fitting model predicts that
above E ∼ 8 keV the lags should increase with energy, contrary to the observations.
This led to the conclusion [30] that the lags of the lower kHz QPO in 4U 1608−52
cannot be due only to reverberation. Subsequently, the same idea was tested on the
upper kHz QPO in 4U 1728−34 [36]; the result of the fits of a reverberation model
to the lags of this QPO is shown in Fig. 6.30b. From this Figure it is apparent that,
while the lags of the QPO first decrease with energy up to E ∼ 5–6 keV, and
increase again above that energy, the model predicts that the lags should follow the
opposite behaviour. This led to the conclusion that, as with the case of the lags of the
lower kHz QPO in 4U 1608−52, the lags of the upper kHz QPO in 4U 1728−34
cannot be solely due to reverberation. Notice that, with few exceptions [66, 101],
so far reverberation models have been used only to fit and explain the time/phase
lags, and not the amplitude, of the observed light curves of accreting supermassive
black holes in active galactic nuclei and stellar-mass black holes in galactic X-ray
binaries, whereas a consistent model of the radiative mechanism that fits the data
should be able to explain both the rms and lag spectrum of the variability in these
sources.
Figure 6.31 shows some of the results of the most extensive study yet of the lags
and rms spectra of the kHz QPOs in LMXBs [157]. That work presents the energydependent rms amplitude, time lags and intrinsic coherence function, plus the time
lags as a function of frequency of the lower kHz QPOs in 14 sources, and the same
information for the upper kHz QPO in six out of those 14 sources. In Fig. 6.31 we
show only the rms and lag spectra of the lower kHz QPO in those 14 sources (we
already showed the rms spectra of the upper kHz QPO in six of those 14 sources in
Fig. 6.17b). The results shown in [157] reinforce the importance of considering both
the rms and lag spectra to try and model the radiative properties of the kHz QPOs.
Models that involve inverse Compton scattering did not seem appropriate to
explain the lags of the lower kHz QPO, given that those lags are soft, whereas
Comptonisation would only produce hard lags (see Sect. 6.4). Contrary to those
expectations, inverse Compton scattering could work, and produce soft lags, if there
is feedback from the corona to the disc ([92], see also [91]). In this scenario, soft
photons from the disc are up-scattered in the corona; part of those up-scattered
photons go to the observer, and produce the power-law like component in the