6 High-Frequency Variability in Neutron-Star Low-Mass X-ray Binaries
291
the radius as a function of the Keplerian orbital frequency around a 2-M neutron
star. As expected, if the QPO frequency is equal to the orbital frequency at that
radius, the radius decreases as the QPO frequency increases. Notice, however,
that the match of the orbital frequency as a function of the radius with the QPO
frequency in that Figure would imply that, contrary to what most models propose
(see Sect. 6.4), the lower, not the upper, kHz QPO would reflect the Keplerian
frequency at the inner disc radius.
Figure 6.16 shows the spectral parameters of the X-ray corona as a function of
the kHz QPO frequencies in 4U 1636−53 ([139], see also [72]). The fits to the
energy spectra yield Γ and kT e , the power-law index and the electron temperature
of the Comptonised component, respectively, whereas the optical depth, τ , of the
corona is a function of the other two parameters [153]. In the left panel the red
and black points correspond to, respectively, the lower and the upper kHz QPO. The
right panel shows the same parameters but with the frequency of the lower kHz QPO
shifted up by 300 Hz. From this Figure it is apparent that there is a smooth relation of
the frequency of the QPOs and the parameters of the corona. Given that the corona
is driven by the soft photons in the disc, it is no surprise that both the inner disc
radius [11] and the corona parameters [72, 139] change with QPO frequency in a
systematic way. The dependence of the rms amplitude of the lower and upper kHz
QPOs upon the spectral parameters of the corona (see plots in [138]), however, do
not match in the same way; in other words, one cannot apply a shift to the relation
of the rms amplitude of one of the kHz QPOs vs. any of the spectral parameters
and make it match the same plot of the other kHz QPO [138]. As we discuss below,
the same applies to the quality factor and phase lags. The fact that, except for a
frequency shift, the relation of the parameters of the corona vs. the QPO frequency
is the same for both QPOs, whereas the relation of the rms amplitude is different,
indicates that the dynamical mechanism that drives the frequency of both kHz QPOs
can be the same, whereas the radiative mechanisms that modulate the QPO signals
must be different.
6.8 Beyond QPO Frequencies
6.8.1 The Fractional rms Amplitude of the kHz QPOs
For both kHz QPOs, the spectrum of the fractional rms amplitude of the variability
is hard. For instance, in 4U 1608−52 the fractional rms amplitude of the lower kHz
QPO increases from ∼5% at ∼3 keV up to ∼20% at 20–25 keV [24, 56, 110]. A
similar trend is seen for the lower kHz QPOs of 4U 1728−34 and Aql X-1 [110,
121], 4U 1636−53 [139], and the only kHz QPO in EXO 0748−676 (see [57], and
Fig. 6.17a). For the upper kHz QPO the trend is similar, although the increase of the
fractional rms amplitude with energy is less steep (Fig. 6.17a, b; but notice that, as
we show below, the total rms amplitude and the slope of the rms spectrum of both
291
the radius as a function of the Keplerian orbital frequency around a 2-M neutron
star. As expected, if the QPO frequency is equal to the orbital frequency at that
radius, the radius decreases as the QPO frequency increases. Notice, however,
that the match of the orbital frequency as a function of the radius with the QPO
frequency in that Figure would imply that, contrary to what most models propose
(see Sect. 6.4), the lower, not the upper, kHz QPO would reflect the Keplerian
frequency at the inner disc radius.
Figure 6.16 shows the spectral parameters of the X-ray corona as a function of
the kHz QPO frequencies in 4U 1636−53 ([139], see also [72]). The fits to the
energy spectra yield Γ and kT e , the power-law index and the electron temperature
of the Comptonised component, respectively, whereas the optical depth, τ , of the
corona is a function of the other two parameters [153]. In the left panel the red
and black points correspond to, respectively, the lower and the upper kHz QPO. The
right panel shows the same parameters but with the frequency of the lower kHz QPO
shifted up by 300 Hz. From this Figure it is apparent that there is a smooth relation of
the frequency of the QPOs and the parameters of the corona. Given that the corona
is driven by the soft photons in the disc, it is no surprise that both the inner disc
radius [11] and the corona parameters [72, 139] change with QPO frequency in a
systematic way. The dependence of the rms amplitude of the lower and upper kHz
QPOs upon the spectral parameters of the corona (see plots in [138]), however, do
not match in the same way; in other words, one cannot apply a shift to the relation
of the rms amplitude of one of the kHz QPOs vs. any of the spectral parameters
and make it match the same plot of the other kHz QPO [138]. As we discuss below,
the same applies to the quality factor and phase lags. The fact that, except for a
frequency shift, the relation of the parameters of the corona vs. the QPO frequency
is the same for both QPOs, whereas the relation of the rms amplitude is different,
indicates that the dynamical mechanism that drives the frequency of both kHz QPOs
can be the same, whereas the radiative mechanisms that modulate the QPO signals
must be different.
6.8 Beyond QPO Frequencies
6.8.1 The Fractional rms Amplitude of the kHz QPOs
For both kHz QPOs, the spectrum of the fractional rms amplitude of the variability
is hard. For instance, in 4U 1608−52 the fractional rms amplitude of the lower kHz
QPO increases from ∼5% at ∼3 keV up to ∼20% at 20–25 keV [24, 56, 110]. A
similar trend is seen for the lower kHz QPOs of 4U 1728−34 and Aql X-1 [110,
121], 4U 1636−53 [139], and the only kHz QPO in EXO 0748−676 (see [57], and
Fig. 6.17a). For the upper kHz QPO the trend is similar, although the increase of the
fractional rms amplitude with energy is less steep (Fig. 6.17a, b; but notice that, as
we show below, the total rms amplitude and the slope of the rms spectrum of both