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M. Méndez and T. M. Belloni
These results set up the stage for the idea [115] of a beat-frequency model of
the kHz QPOs. (Although published in 1998, the idea was first presented in detail
by the same authors in a preprint in 1996, arXiv:astro-ph/9609157.) In this model,
called the sonic-point beat-frequency model or, for short, the sonic-point model, the
upper kHz QPO is a beaming oscillation produced by a hot spot on the neutron-star
surface. This spot is the footprint of a stream of matter falling from material at the
sonic radius (see Sect. 6.4) onto the neutron star. When this stream hits the star it
heats a small area producing a footprint that rotates around the surface of the star at
the same frequency as that of the material in the disc, at the sonic radius, where
the stream originates. When the neutron star rotates, radiation from the pole(s)
illuminates periodically the part in the disc where the stream starts and, because of
radiation drag, increases momentarily the rate of mass that is injected into the stream
and falls onto the neutron star. When this extra amount of material hits the neutronstar surface at the footprint of the stream, the temperature of the spot increases. The
emission from the footprint is therefore modulated at a frequency that is equal to the
Keplerian frequency at the sonic radius minus the neutron-star spin frequency. In
this model, the luminosity modulation of the hot spot produces the lower kHz QPO
(Please read [115], for the full explanation of the model).
One obvious conclusion of this scenario is that the frequency difference between
the kHz QPOs, which is equal to the neutron-star spin frequency, 2 has to remain
constant when the QPO frequencies move (Sect. 6.4). This was the case for most
of the sources in which kHz QPO had been detected, except for Sco X-1 [165], in
which the frequency difference decreased systematically as the QPO frequencies
increased together. While this result posed a problem to the sonic-point model, the
situation could be explained if the clumps in the disc, where the stream originates,
spiralled in due to the strong radiation drag in this bright source [86]. In this case, the
frequency difference could be less than the neutron-star spin frequency and decrease
as the QPO frequencies increased. The situation became even more difficult for
the sonic-point model when this effect was observed in more sources, all much
weaker than Sco X-1 [105, 107, 108] and, especially, when the difference of the
QPO frequencies in some observations of 4U 1636−53 [70] turned out to be larger
than half the neutron-star spin frequency in this source, something that could not be
explained in the sonic model and its subsequent extensions.
Almost at the same time, a different model that could explain the dependence
of ν upp − ν low vs. the frequency of the QPO was proposed. As in the sonic-point
model, this relativistic-precession model [147, 148] considered that the frequency
of the upper kHz QPO is the Keplerian frequency at the inner edge of the disc;
but differently from the previous model, in this case the lower kHz QPO would be
the periastron precession frequency, equal to the difference between the Keplerian
and epicyclic radial frequencies at the inner edge of the disc. In this model the
frequency difference between the kHz QPOs is independent of the neutron-star spin
2 As explained in Sect. 6.4, in a modified version of the sonic-point model the frequency difference
between the kHz QPOs can also be equal to half the neutron-star spin frequency [87].
M. Méndez and T. M. Belloni
These results set up the stage for the idea [115] of a beat-frequency model of
the kHz QPOs. (Although published in 1998, the idea was first presented in detail
by the same authors in a preprint in 1996, arXiv:astro-ph/9609157.) In this model,
called the sonic-point beat-frequency model or, for short, the sonic-point model, the
upper kHz QPO is a beaming oscillation produced by a hot spot on the neutron-star
surface. This spot is the footprint of a stream of matter falling from material at the
sonic radius (see Sect. 6.4) onto the neutron star. When this stream hits the star it
heats a small area producing a footprint that rotates around the surface of the star at
the same frequency as that of the material in the disc, at the sonic radius, where
the stream originates. When the neutron star rotates, radiation from the pole(s)
illuminates periodically the part in the disc where the stream starts and, because of
radiation drag, increases momentarily the rate of mass that is injected into the stream
and falls onto the neutron star. When this extra amount of material hits the neutronstar surface at the footprint of the stream, the temperature of the spot increases. The
emission from the footprint is therefore modulated at a frequency that is equal to the
Keplerian frequency at the sonic radius minus the neutron-star spin frequency. In
this model, the luminosity modulation of the hot spot produces the lower kHz QPO
(Please read [115], for the full explanation of the model).
One obvious conclusion of this scenario is that the frequency difference between
the kHz QPOs, which is equal to the neutron-star spin frequency, 2 has to remain
constant when the QPO frequencies move (Sect. 6.4). This was the case for most
of the sources in which kHz QPO had been detected, except for Sco X-1 [165], in
which the frequency difference decreased systematically as the QPO frequencies
increased together. While this result posed a problem to the sonic-point model, the
situation could be explained if the clumps in the disc, where the stream originates,
spiralled in due to the strong radiation drag in this bright source [86]. In this case, the
frequency difference could be less than the neutron-star spin frequency and decrease
as the QPO frequencies increased. The situation became even more difficult for
the sonic-point model when this effect was observed in more sources, all much
weaker than Sco X-1 [105, 107, 108] and, especially, when the difference of the
QPO frequencies in some observations of 4U 1636−53 [70] turned out to be larger
than half the neutron-star spin frequency in this source, something that could not be
explained in the sonic model and its subsequent extensions.
Almost at the same time, a different model that could explain the dependence
of ν upp − ν low vs. the frequency of the QPO was proposed. As in the sonic-point
model, this relativistic-precession model [147, 148] considered that the frequency
of the upper kHz QPO is the Keplerian frequency at the inner edge of the disc;
but differently from the previous model, in this case the lower kHz QPO would be
the periastron precession frequency, equal to the difference between the Keplerian
and epicyclic radial frequencies at the inner edge of the disc. In this model the
frequency difference between the kHz QPOs is independent of the neutron-star spin
2 As explained in Sect. 6.4, in a modified version of the sonic-point model the frequency difference
between the kHz QPOs can also be equal to half the neutron-star spin frequency [87].