316
M. Méndez and T. M. Belloni
Most of the models that have been proposed to explain the frequencies of the kHz
QPOs assume that one of the QPOs reflects the Keplerian orbital motion at some
preferred radius in the accretion disc. For instance, in the sonic-point beat-frequency
model [115], the upper kHz QPO is produced at the radius where the radial flow
velocity in the disc turns from subsonic to supersonic (the sonic radius), and the
lower kHz QPO is a beat between the upper kHz QPO and the spin frequency of
the neutron star. Besides the main peaks at ν low and ν upp , this model predicts some
other (weaker) harmonics and sidebands of these QPOs at specific frequencies; for
instance, in this model there should be a relatively strong harmonic of the lower
QPO at 2ν low (see table 3 of [115], for a list of other harmonic and sideband peaks
predicted by the model).
In the relativistic-precession model [148], the upper kHz QPO is also assumed
to be Keplerian, and the lower kHz QPO is the periastron precession frequency of a
slightly non-circular inner accretion disc. Oscillating disc models (e.g., [133]) in a
GR potential yield the same frequencies as those for test particles in the relativisticprecession model, but also predict other frequencies that are linear combinations of
the three basic relativistic frequencies in the disc. For instance, in an oscillating disc
in which the Keplerian and the periastron precession frequencies are excited, there
should be a peak at a frequency equal to 2ν upp − ν low .
Finding one of these other peaks would, on one hand, favour one model over the
other, opening up a path to understand the dynamics of the disc and the production
of the QPOs and, on the other hand, would confirm that the upper kHz QPO is
due to Keplerian orbital motion in the disc. Unfortunately, the only attempt to find
any of these peaks using data of Sco X-1 [106] gave negative results. The 95%
confidence upper limits of a peak at 2ν low and 2ν upp − ν low are, respectively, 0.12
and 0.26 times the amplitude of the upper kHz QPO in this source (the amplitude
of the upper kHz QPO in Sco X-1 is between 0.6% and 2.5%; see table 1 in [106],
for the upper limits at other frequencies). The signal of these other peaks could be
attenuated in the corona [29, 84, 112, 115] depending on the frequency of the peak,
and the radius and optical depth of the corona. Considering this effect, the upper
limits of an unattenuated signal at those two frequencies would be a factor between
0.15 and 0.30 of the unattenuated amplitude of the upper kHz QPO in Sco X-1
(see table 2 in [106], for the unattenuated upper limits at other frequencies). In
conclusion, in Sco X-1 none of the secondary QPO peaks predicted by the two main
classes of models of the kHz QPOs are detected, with upper limits that imply that
these secondary peaks are one to 2 orders of magnitude weaker in power than the
upper kHz QPO.
If the amplitude of the signal of the QPOs is modulated, one would expect to see
sidebands to the main oscillation, at frequencies that are equal to the frequency of
the QPOs plus or minus the frequency of the mechanism that modulates the QPO
amplitude. This could happen if, for instance, on very short scales the amplitude of
the kHz QPOs depends upon mass accretion rate. Although there is no guarantee that
this is the case, given that on long time scales the amplitude of the QPOs depends
upon QPO frequency (e.g., Fig. 6.5a in Sect. 6.4) and, in turn, QPO frequency
M. Méndez and T. M. Belloni
Most of the models that have been proposed to explain the frequencies of the kHz
QPOs assume that one of the QPOs reflects the Keplerian orbital motion at some
preferred radius in the accretion disc. For instance, in the sonic-point beat-frequency
model [115], the upper kHz QPO is produced at the radius where the radial flow
velocity in the disc turns from subsonic to supersonic (the sonic radius), and the
lower kHz QPO is a beat between the upper kHz QPO and the spin frequency of
the neutron star. Besides the main peaks at ν low and ν upp , this model predicts some
other (weaker) harmonics and sidebands of these QPOs at specific frequencies; for
instance, in this model there should be a relatively strong harmonic of the lower
QPO at 2ν low (see table 3 of [115], for a list of other harmonic and sideband peaks
predicted by the model).
In the relativistic-precession model [148], the upper kHz QPO is also assumed
to be Keplerian, and the lower kHz QPO is the periastron precession frequency of a
slightly non-circular inner accretion disc. Oscillating disc models (e.g., [133]) in a
GR potential yield the same frequencies as those for test particles in the relativisticprecession model, but also predict other frequencies that are linear combinations of
the three basic relativistic frequencies in the disc. For instance, in an oscillating disc
in which the Keplerian and the periastron precession frequencies are excited, there
should be a peak at a frequency equal to 2ν upp − ν low .
Finding one of these other peaks would, on one hand, favour one model over the
other, opening up a path to understand the dynamics of the disc and the production
of the QPOs and, on the other hand, would confirm that the upper kHz QPO is
due to Keplerian orbital motion in the disc. Unfortunately, the only attempt to find
any of these peaks using data of Sco X-1 [106] gave negative results. The 95%
confidence upper limits of a peak at 2ν low and 2ν upp − ν low are, respectively, 0.12
and 0.26 times the amplitude of the upper kHz QPO in this source (the amplitude
of the upper kHz QPO in Sco X-1 is between 0.6% and 2.5%; see table 1 in [106],
for the upper limits at other frequencies). The signal of these other peaks could be
attenuated in the corona [29, 84, 112, 115] depending on the frequency of the peak,
and the radius and optical depth of the corona. Considering this effect, the upper
limits of an unattenuated signal at those two frequencies would be a factor between
0.15 and 0.30 of the unattenuated amplitude of the upper kHz QPO in Sco X-1
(see table 2 in [106], for the unattenuated upper limits at other frequencies). In
conclusion, in Sco X-1 none of the secondary QPO peaks predicted by the two main
classes of models of the kHz QPOs are detected, with upper limits that imply that
these secondary peaks are one to 2 orders of magnitude weaker in power than the
upper kHz QPO.
If the amplitude of the signal of the QPOs is modulated, one would expect to see
sidebands to the main oscillation, at frequencies that are equal to the frequency of
the QPOs plus or minus the frequency of the mechanism that modulates the QPO
amplitude. This could happen if, for instance, on very short scales the amplitude of
the kHz QPOs depends upon mass accretion rate. Although there is no guarantee that
this is the case, given that on long time scales the amplitude of the QPOs depends
upon QPO frequency (e.g., Fig. 6.5a in Sect. 6.4) and, in turn, QPO frequency