6 High-Frequency Variability in Neutron-Star Low-Mass X-ray Binaries
285
and, as can be readily seen from the identification of the lower kHz QPO, should be
equal to the epicyclic radial frequency at the inner radius of the disc. This epicyclic
frequency is 0 at the ISCO, first increases as the radial distance in the disc increases,
and then decreases again as the radial distance continues increasing. This implies
that the frequency difference between the kHz QPOs should decrease both at high
and low QPO frequencies, corresponding to small and large radial distances in the
disc. As indicated, this explained the observed decrease of ν upp − ν low with QPO
frequencies as the QPO frequencies increase in Sco X-1 [106, 165] and other sources
[70, 105, 107, 108], but also predicted a trend at low kHz QPO frequencies for
which there were no data at the time. A few years later, the neutron-star LMXB
Cir X-1 [28] showed exactly that (Fig. 6.6 in Sect. 6.4) and, since other predictions
of the model for low-frequency variability had already been validated (Fig. 6.7b
in Sect. 6.4), all this lent support to this model. Notice, however, that the model
relies on frequencies of test particles around the neutron star, and therefore does not
consider the hydrodynamical effects in the disc that may affect those frequencies.
We will come to this again in Sect. 6.8.4.
A third class of models considers wave patterns in the disc as the cause of the kHz
QPOs. These models also rely upon the three basic GR epicyclic frequencies
discussed in the previous models (and sometimes also upon the neutron-star spin),
but in this case those are not the frequencies of test particles orbiting the neutron star,
but characteristic frequencies in a hydrodynamical flow that determine how pressure
and gravity waves travel in the disc and, sometimes, lead to other frequencies that
are resonances of the basic ones (some examples of those ideas can be found in
[76–78, 80, 85, 124, 176], but the list is much longer). Among these models, one
that received some attention [1, 2] argued that a resonance in the disc appears when
the ratio of two of the epicyclic frequencies discussed above is the ratio of two
small integer numbers, e.g. 2:3. Such a preferred frequency ratio was reported for
the kHz QPOs in Sco X-1 [2], and the model gained popularity because, in two
cases in which two simultaneous high-frequency QPOs were observed in blackhole systems, those QPOs appear at frequencies that are in a 2:3 ratio (e.g., at 300
and 450 Hz in the black-hole LMXB GRO J1655-44; see [136, 149]). In essence,
this resonance model is equivalent to the example of a double pendulum discussed
in books of Mechanics (e.g. [89]) with, in this case, a mechanism that couples two
oscillating phenomena in the disc. The report of a 2:3 frequency ratio of the kHz
QPOs in Sco X-1 has been subsequently disputed [19, 21, 113], but the model is
still considered for high-frequency QPOs in black-hole LMXBs.
The models described in this section, and most of the models of the kHz
QPOs that appeared in the last 20 years, aim at explaining the frequencies of the
oscillations and, in that sense, are dynamical models of the QPO phenomenon.
Very few models have attempted to give an explanation of the other, radiative,
properties of the QPOs. We will discuss those radiative properties of the kHz QPOs
in Sect. 6.8, and we will also mention some of the latest attempts to try and explain
those properties.
285
and, as can be readily seen from the identification of the lower kHz QPO, should be
equal to the epicyclic radial frequency at the inner radius of the disc. This epicyclic
frequency is 0 at the ISCO, first increases as the radial distance in the disc increases,
and then decreases again as the radial distance continues increasing. This implies
that the frequency difference between the kHz QPOs should decrease both at high
and low QPO frequencies, corresponding to small and large radial distances in the
disc. As indicated, this explained the observed decrease of ν upp − ν low with QPO
frequencies as the QPO frequencies increase in Sco X-1 [106, 165] and other sources
[70, 105, 107, 108], but also predicted a trend at low kHz QPO frequencies for
which there were no data at the time. A few years later, the neutron-star LMXB
Cir X-1 [28] showed exactly that (Fig. 6.6 in Sect. 6.4) and, since other predictions
of the model for low-frequency variability had already been validated (Fig. 6.7b
in Sect. 6.4), all this lent support to this model. Notice, however, that the model
relies on frequencies of test particles around the neutron star, and therefore does not
consider the hydrodynamical effects in the disc that may affect those frequencies.
We will come to this again in Sect. 6.8.4.
A third class of models considers wave patterns in the disc as the cause of the kHz
QPOs. These models also rely upon the three basic GR epicyclic frequencies
discussed in the previous models (and sometimes also upon the neutron-star spin),
but in this case those are not the frequencies of test particles orbiting the neutron star,
but characteristic frequencies in a hydrodynamical flow that determine how pressure
and gravity waves travel in the disc and, sometimes, lead to other frequencies that
are resonances of the basic ones (some examples of those ideas can be found in
[76–78, 80, 85, 124, 176], but the list is much longer). Among these models, one
that received some attention [1, 2] argued that a resonance in the disc appears when
the ratio of two of the epicyclic frequencies discussed above is the ratio of two
small integer numbers, e.g. 2:3. Such a preferred frequency ratio was reported for
the kHz QPOs in Sco X-1 [2], and the model gained popularity because, in two
cases in which two simultaneous high-frequency QPOs were observed in blackhole systems, those QPOs appear at frequencies that are in a 2:3 ratio (e.g., at 300
and 450 Hz in the black-hole LMXB GRO J1655-44; see [136, 149]). In essence,
this resonance model is equivalent to the example of a double pendulum discussed
in books of Mechanics (e.g. [89]) with, in this case, a mechanism that couples two
oscillating phenomena in the disc. The report of a 2:3 frequency ratio of the kHz
QPOs in Sco X-1 has been subsequently disputed [19, 21, 113], but the model is
still considered for high-frequency QPOs in black-hole LMXBs.
The models described in this section, and most of the models of the kHz
QPOs that appeared in the last 20 years, aim at explaining the frequencies of the
oscillations and, in that sense, are dynamical models of the QPO phenomenon.
Very few models have attempted to give an explanation of the other, radiative,
properties of the QPOs. We will discuss those radiative properties of the kHz QPOs
in Sect. 6.8, and we will also mention some of the latest attempts to try and explain
those properties.