6 High-Frequency Variability in Neutron-Star Low-Mass X-ray Binaries
277
The initial observations of the kHz QPOs in Sco X-1 had already shown that
the frequency separation between the two QPO peaks was not always the same,
but decreased systematically, and significantly, by a few percent as the frequencies
of the two simultaneous kHz QPOs increased. The beat-frequency model could still
explain this behaviour if the material at the inner radius of the disc, where the beating
took place, suffered from radiation drag and the beating took place as that material
spiralled in towards the neutron star. Being a very luminous source, this effect could
be strong in the case of Sco X-1. But soon after several other less luminous sources,
starting with the atoll source 4U 1608−52, showed the same effect.
The situation got more complicated for this model when burst oscillations and
two simultaneous kHz QPOs were detected in 4U 1636−53, with the frequency
of the burst oscillations being twice the difference in frequency between the kHz
QPOs. The original beat-frequency model could not explain this. The model had
to be made more complex, by adding a possible excitation of vertical modes in
the accretion disc at a radial distance where the difference between the Keplerian
frequency at the inner edge of the disc (that causes the QPO at ν upp ) and the neutronstar spin frequency is equal to the vertical epicyclic frequency in the disc. Depending
on whether the material in the disc is smooth or clumped, the excited frequency,
which produces the lower kHz QPO, would be at ν upp − ν spin or ν upp − ν spin /2.
These complications for the beat-frequency model triggered other proposals to
explain the QPO frequencies. One of them, that could explain the phenomenology
rather naturally, was the idea of periastron precession of the innermost parts of the
accretion disc. Under this hypothesis, the upper kHz QPO was still identified as the
epicyclic azimuthal (Keplerian) frequency of a test particle at the inner edge of the
disc, under the influence of the general relativistic (GR) potential of the neutron
star. In this model, however, the lower kHz QPO would be the difference between
this azimuthal and the epicyclic radial frequency, the so-called periastron precession
frequency, at the same spot in the disc. The difference between the azimuthal and
the periastron precession frequency in the model changes generally in the same way
as in the observations [148], although the calculations do not fit the exact trend of
the observations.
One strong prediction of this model was that the difference between the
frequency of the two QPOs should not only decrease at high, but also at low QPO
frequencies, something that was later on observed in the system Cir X-1 (Fig. 6.6).
To be fair to history, the relativistic-precession model, as this model was called,
came about as an extension of the Lense-Thirring model [147] that was proposed
a year earlier to explain the correlation between the frequency of the upper kHz
QPO and a low-frequency QPO in neutron-star LMXBs. The Lense-Thirring and the
relativistic-precession models became one consistent model for both the low- and
the high-frequency variability. Notice, also, that in this model there is no relation
between the frequencies of the kHz QPOs and the spin of the neutron star, therefore
this model was also applicable to QPOs in black-hole systems.
If the kHz QPOs and the low-frequency QPOs are all GR frequencies in the disc
(but notice that the calculations assume test particles, so no disc hydrodynamics),
another prediction of this model is that the frequency of the low-frequency QPO
277
The initial observations of the kHz QPOs in Sco X-1 had already shown that
the frequency separation between the two QPO peaks was not always the same,
but decreased systematically, and significantly, by a few percent as the frequencies
of the two simultaneous kHz QPOs increased. The beat-frequency model could still
explain this behaviour if the material at the inner radius of the disc, where the beating
took place, suffered from radiation drag and the beating took place as that material
spiralled in towards the neutron star. Being a very luminous source, this effect could
be strong in the case of Sco X-1. But soon after several other less luminous sources,
starting with the atoll source 4U 1608−52, showed the same effect.
The situation got more complicated for this model when burst oscillations and
two simultaneous kHz QPOs were detected in 4U 1636−53, with the frequency
of the burst oscillations being twice the difference in frequency between the kHz
QPOs. The original beat-frequency model could not explain this. The model had
to be made more complex, by adding a possible excitation of vertical modes in
the accretion disc at a radial distance where the difference between the Keplerian
frequency at the inner edge of the disc (that causes the QPO at ν upp ) and the neutronstar spin frequency is equal to the vertical epicyclic frequency in the disc. Depending
on whether the material in the disc is smooth or clumped, the excited frequency,
which produces the lower kHz QPO, would be at ν upp − ν spin or ν upp − ν spin /2.
These complications for the beat-frequency model triggered other proposals to
explain the QPO frequencies. One of them, that could explain the phenomenology
rather naturally, was the idea of periastron precession of the innermost parts of the
accretion disc. Under this hypothesis, the upper kHz QPO was still identified as the
epicyclic azimuthal (Keplerian) frequency of a test particle at the inner edge of the
disc, under the influence of the general relativistic (GR) potential of the neutron
star. In this model, however, the lower kHz QPO would be the difference between
this azimuthal and the epicyclic radial frequency, the so-called periastron precession
frequency, at the same spot in the disc. The difference between the azimuthal and
the periastron precession frequency in the model changes generally in the same way
as in the observations [148], although the calculations do not fit the exact trend of
the observations.
One strong prediction of this model was that the difference between the
frequency of the two QPOs should not only decrease at high, but also at low QPO
frequencies, something that was later on observed in the system Cir X-1 (Fig. 6.6).
To be fair to history, the relativistic-precession model, as this model was called,
came about as an extension of the Lense-Thirring model [147] that was proposed
a year earlier to explain the correlation between the frequency of the upper kHz
QPO and a low-frequency QPO in neutron-star LMXBs. The Lense-Thirring and the
relativistic-precession models became one consistent model for both the low- and
the high-frequency variability. Notice, also, that in this model there is no relation
between the frequencies of the kHz QPOs and the spin of the neutron star, therefore
this model was also applicable to QPOs in black-hole systems.
If the kHz QPOs and the low-frequency QPOs are all GR frequencies in the disc
(but notice that the calculations assume test particles, so no disc hydrodynamics),
another prediction of this model is that the frequency of the low-frequency QPO